AAPPÉÉNNDDIICCEESS

APÉNDICE A 331

Programa EAHFsto3g Código Fortran de la utilidad que calcula el espacio activo de un hidrocarburo insaturado a partir de la función de onda HF/sto-3g.

APÉNDICE B 339

Programa SCANXYZ Código Fortran del programa que genera estructuras moleculares por interpolación lineal a partir de dos estructuras moleculares

APÉNDICE C 347

Programa ISC Código Fortran del programa que junto con el cálculo ab initio de gradientes analíticos determina puntos de cruce de sistemas mínimos en energía.

APÉNDICE D 355

Programa RELAXATION Código Fortran del programa que recoge diversas utilidades en el estudio del proceso de relajación para una reacción fotoquímica diabática a través de un intersección cónica.

APÉNDICE E 369

Porgrama RUMER Código Fortran del programa que evalúa los productos escalares entre funciones de espín representadas en las funciones de base de Rumer.

APÉNDICE F 375

Integral de Solape (J) Deducción del término exponencial de la ecuación (6.40) como una integral de solape.

APÉNDICE G 377

Porgrama TET-ACC Código Fortran del programa que codifica el algoritmo TET-ACC

APÉNDICE H 389

Separatas de las Publicaciones Separatas de publicaciones referidas a la Tesis

AAPPÉÉNNDDIICCEE AA

Código Fortran de la utilidad (EAHFsto3g) que calcula el espacio activo de un hidrocarburo insaturado a partir de la función de onda HF/sto-3g.

Breve descripción del programa

EAHFsto3g 1. Lee el nombre del fichero de Gaussian98 que contiene el cálculo de la función de onda rhf/sto-3g en el cual hay que añadir las keywords: “pop=full” “GFInput” y “Nosymm”. (rutina principal) 2. Se lee la información básica sobre el problema concreto: número de átomos de la molécula, números atómicos, número de orbitales (ocupados y virtuales) entre otros. (subrutina: leeorbitales(geom,n,norb,norbvi,norboc,nombre,zatom)). 3. Genera la matriz de conectividad y determina los carbonos insaturados. (subrutina: matrizconect(geom,zatom,n,conect,typec)). 4. Se generan los vectores de base, que son aquellos que definen el espacio (π) sobre el que se va a proyectar la función de onda. (subrutina: vectorbase(geom,n,typec,zatom,conect,vbase)). 5. El usuario puede añadir vectores σ sobre los que construir el espacio activo (útil por ejemplo en anillos tensionados). (subrutina: sigmabase(geom,vbase)). 6. Proyecta (evaluando la parte radial de la función de onda sobre cada átomo) la función de onda sobre el conjunto de vectores antes determinado. (rutina principal). 7. Crea un fichero “coeflist” que contiene los valores de las proyecciones de cada OM sobre el espacio definido por los vectores de base (vectores π). (rutina principal). 8. Fin de la rutina principal.

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program EAHFsto3g C------------------------------------------------------------- C------------------------------------------------------------- C Programa que ofrece el espacio de simetría pi para una -- C hidrocarburo insaturado que contiene carbonos sp2 y sp3 -- C a partir de una funció de onda HF/sto-3g calculada con -- C Gaussian98 -- C -- C Luisma, Siena 2003 -- C------------------------------------------------------------- C------------------------------------------------------------- real coef1,coef2,coef3,coef4,coef5,geom,vbase,coefom, $scoef1,scoef2,scoef3,scoef4,scoef5 integer i,norb,norbvi,norboc,zatom,conect,n,typec, $nnn,mmm,contad character nombre*50,line*72 parameter (omax=1000,nmax=200) dimension coef1(omax),coef2(omax),coef3(omax), $coef4(omax),coef5(omax),geom(3,nmax),vbase(3,nmax), $zatom(nmax),conect(nmax,nmax),typec(nmax),coefom(omax) print *,'nombre del fichero' read *,nombre C Llamada a la subrutina que lee goem,zatom,norb... call leeorbitales(geom,n,norb,norbvi,norboc, $nombre,zatom) C Llamada a la subrutina que calcula la conectividad C y el tipo de carbonos call matrizconect(geom,zatom,n,conect,typec) C Llamada a la subrutina que obtiene el vector de base call vectorbase(geom,n,typec,zatom,conect,vbase) C Llamada a la subrutina que permite cambiar el vector C de base para introducir enlaces sigma call sigmabase(geom,vbase) open (unit=1,file=nombre) i=1 nnn=1 contad=1 mmm=-5 9 mmm=mmm+5 100 Read(1,20,End=1000) Line If (line(6:17).ne.'EIGENVALUES' ) then Goto 100 endif 20 format(A) 10 read (1,200,ERR=500)coef1(i),coef2(i), $coef3(i),coef4(i),coef5(i) scoef1=scoef1+coef1(i)**2 scoef2=scoef2+coef2(i)**2 scoef3=scoef3+coef3(i)**2 scoef4=scoef4+coef4(i)**2 scoef5=scoef5+coef5(i)**2 i=i+1 goto 10 500 i=1 if(zatom(contad).eq.1) then nnn=nnn+1

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elseif (zatom(contad).eq.17) then =nnn+9 nnn if (contad.eq.n) then nnn=1 contad=1 goto 9 endif contad=contad+1 goto 500 endif coefom(mmm+1)=coefom(mmm+1)+((coef1(nnn+2)*vbase(1,contad))**2 $/scoef1) coefom(mmm+1)=coefom(mmm+1)+((coef1(nnn+3)*vbase(2,contad))**2 $/scoef1) coefom(mmm+1)=coefom(mmm+1)+((coef1(nnn+4)*vbase(3,contad))**2 $/scoef1) coefom(mmm+2)=coefom(mmm+2)+((coef2(nnn+2)*vbase(1,contad))**2 $/scoef2) coefom(mmm+2)=coefom(mmm+2)+((coef2(nnn+3)*vbase(2,contad))**2 $/scoef2) coefom(mmm+2)=coefom(mmm+2)+((coef2(nnn+4)*vbase(3,contad))**2 $/scoef2) coefom(mmm+3)=coefom(mmm+3)+((coef3(nnn+2)*vbase(1,contad))**2 $/scoef3) coefom(mmm+3)=coefom(mmm+3)+((coef3(nnn+3)*vbase(2,contad))**2 $/scoef3) coefom(mmm+3)=coefom(mmm+3)+((coef3(nnn+4)*vbase(3,contad))**2 $/scoef3) coefom(mmm+4)=coefom(mmm+4)+((coef4(nnn+2)*vbase(1,contad))**2 $/scoef4) coefom(mmm+4)=coefom(mmm+4)+((coef4(nnn+3)*vbase(2,contad))**2 $/scoef4) coefom(mmm+4)=coefom(mmm+4)+((coef4(nnn+4)*vbase(3,contad))**2 $/scoef4) coefom(mmm+5)=coefom(mmm+5)+((coef5(nnn+2)*vbase(1,contad))**2 $/scoef5) coefom(mmm+5)=coefom(mmm+5)+((coef5(nnn+3)*vbase(2,contad))**2 $/scoef5) coefom(mmm+5)=coefom(mmm+5)+((coef5(nnn+4)*vbase(3,contad))**2 $/scoef5) if((mmm+5).GE.(norb-4)) then o 1000 go t endif =nnn+5 nnn if (contad.eq.n) then nnn=1 contad=1 goto 9 endif contad=contad+1 goto 500 200 format(23x,f8.5,2x,f8.5,2x,f8.5,2x,f8.5,2x,f8.5) 210 format(i3,1x,f8.6) 1000 do 1010 i=1,norb print 210,i,coefom(i) 1010 continue open(unit=2,file='coeflist') 1020 do 1030 i=1,norb write 210),i,coefom(i) (2,

continue 1030 print *,'nr. de orb.=',norb print *,'nr. de orb. occ.=',norboc close(1) end

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subroutine leeorbitales(geom,n,norb,norbvi,norboc, $nombre,zatom) C--------------------------------------------------------------- C C preparado como subrutina que lee el numero de atomos C (n), el de orbitales(norb) el numero de ellos ocupados C (norboc) y el numero de ellos vacíos (norbvi) C Ademas lee la geometria (geom(3,nmax)) C real geom integer n,zatom,i,n2,norb,norboc,norbvi parameter (nmax=200) character nombre*20,null*10,line*68 dimension zatom(nmax),geom(3,nmax) open (unit=1,file=nombre) 100 Read(1,20,End=1000) line If ( Line(1:29).ne.' Number Number Type ')then Goto 100 endif 20 format(A) read (1,*)null i=1 10 n=n2 read (1,200,ERR=300)n2,zatom(i),geom(1,i),geom(2,i),geom(3,i) i=i+1 goto 10 200 format(2x,i3,8x,i3,20x,f10.6,2x,f10.6,2x,f10.6) 300 print *,'numero de atomos=',n do 30 i=1,n print 200,n,zatom(i),geom(1,i), $geom(2,i),geom(3,i) 30 continue 400 Read(1,20,End=1000)line If ( Line(8:22).ne.'basis functions') then 00 Goto 4 endif read(1,410),norboc do 0 i=1,4 42 read )null (1,*20 continue 4

read (1,430,err=431),norb 430 format (11x,i3) goto 432 431 print *,'numero de orbitales?=' read *,norb 432 norbvi=norb-norboc print *,'numero de orbitales=',norb print *,'numero de orbitales ocupados=',norboc print *,'numero de orbitales virtuales=',norbvi 410 format (3x,i3) 1000 print *,'fin del fichero' close (1) return

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END subroutine matrizconect(geom,zatom,n,conect,typec) C-------------------------------------------------------------------- C C preparado como subrutina que genera la matriz de conectividad C a partir de una gemetría dada, asi como el tipo de atomo del C que se trata. C Entrada geom,zatom C la primera es la matriz conect(nmax,nmax) con valor 0 para los C atomos no enlazados y 1 para los enlazados C y la siguiente es un vector llamado typec(nmax) con C valores 0 para los hidrógenos valores 1 para C sp, C valor 2 para Csp2 y valor 3 para C sp3 real m geo integer n,zatom,i,j,n2,conect,typec,sum parameter (nmax=200) character nomb*20,null*10,line*68 dimension zatom(nmax),geom(3,nmax),conect(nmax,nmax), $typec(nmax) do 320 i=1,n do 310 j=1,n if(zatom(i).eq.1.and.zatom(j).eq.1) then t(i,j)=0 conec elseif(i.eq.j) then ect(i,j)=0 con

else dist=sqrt((geom(1,i)-geom(1,j))**2+(geom(2,i)-geom(2,j))**2+ $ (geom(3,i)-geom(3,j))**2) (z nd.za if(dist.LT.1.6) then

if atom(i).eq.6.a tom(j).eq.6) then

conect(i,j)=1 else conect(i,j)=0 endif else if(dist.LT.1.2) then ect(i,j)=1 con else ct(i,j)=0 cone endif endif endif 310 continue 3 20 continue

C Llamada a la subrutina que permite cambiar la conectividad call cambiaconect(conect) sum=0 do 340 j=1,n if(zatom(j).eq.6) then do 330 i=1,n sum=sum+conect(j,i)

continue 330 if(sum.eq.2) then typec(j)=1 elseif(sum.eq.3) then typec(j)=2 else

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(j)=3 typecendif

endif sum=0 340 continue do 350 i=1,n print *,zatom(i),typec(i) 350 continue return END subroutine vectorbase(geom,n,typec,zatom,conect,vbase) C-------------------------------------------------------------- C C programa preparado como subrutina para generar C el vector base para calcular el espacio activo C Entradas: C geom(3,nmax): geometría de la molecula C n: numero de atomos C typec(nmax): tipo de carbono C zatom(nmax): numero atomico C conect: matriz de conectividad C Salidas: C vbase(3,nmax):con vectores unitarios o nulos C para cada atomo real geom,vbase,norma,vect,x,vect12,vect23,vect31 integer n,zatom,i,j,k,conect,typec parameter (nmax=200) character nomb*20,null*10,line*68 dimension zatom(nmax),geom(3,nmax),conect(nmax,nmax), $typec(nmax),vbase(3,nmax),vect(3,3),x(3),vect12(3), $vect23(3),vect31(3) do 1 i=1,n do 2 j=1,3 vbase(j,i)=0.0 2 continue 1 continue k=1 do 10 i=1,n if(typec(i).eq.2) then do 20 j=1,n norma=sqrt((geom(1,i)-geom(1,j))**2+(geom(2,i)-

if (conect(i,j).eq.1)then

$ geom(2,j))**2+(geom(3,i)-geom(3,j))**2) vect(1,k)=(geom(1,j)-geom(1,i))/norma vect(2,k)=(geom(2,j)-geom(2,i))/norma vect(3,k)=(geom(3,j)-geom(3,i))/norma k=k+1 endif 20 continue vect12(1)=vect(2,1)*vect(3,2)-vect(3,1)*vect(2,2) vect12(2)=vect(3,1)*vect(1,2)-vect(1,1)*vect(3,2) vect12(3)=vect(1,1)*vect(2,2)-vect(2,1)*vect(1,2) vect23(1)=vect(2,2)*vect(3,3)-vect(3,2)*vect(2,3) vect23(2)=vect(3,2)*vect(1,3)-vect(1,2)*vect(3,3) vect23(3)=vect(1,2)*vect(2,3)-vect(2,2)*vect(1,3) vect31(1)=vect(2,3)*vect(3,1)-vect(3,3)*vect(2,1) vect31(2)=vect(3,3)*vect(1,1)-vect(1,3)*vect(3,1) vect31(3)=vect(1,3)*vect(2,1)-vect(2,3)*vect(1,1) vbase(1,i)=vect12(1)+vect23(1)+vect31(1) vbase(2,i)=vect12(2)+vect23(2)+vect31(2)

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vbase(3,i)=vect12(3)+vect23(3)+vect31(3) norma=sqrt(vbase(1,i)**2+vbase(2,i)**2+vbase(3,i)**2) vbase(1,i)=vbase(1,i)/norma vbase(2,i)=vbase(2,i)/norma vbase(3,i)=vbase(3,i)/norma endif k=1 10 continue do 30 i=1,n print *,vbase(1,i),vbase(2,i),vbase(3,i) 3 0 continue

open(unit=2,file='vector.xyz') do i=1,n 40 write(2,*),zatom(i),geom(1,i),geom(2,i),geom(3,i) 40 continue do 50 i=1,n write(2,*),'0',(geom(1,i)+vbase(1,i)),(geom(2,i)+vbase(2,i)), $(geom(3,i)+vbase(3,i)) 50 continue return END subroutine cambiaconect(conect) C-------------------------------------------------- C C Subrutina que permite cambiar la conectividad C integer conect,i,j parameter (nmax=200) dimension conect(nmax,nmax) character resp*1 goto 10 9 print *,'Debe responder "S" (si) o "N" (no)' 10 print *,'Quiere hacer cero algun elemento de la matriz $de conectividad?(S/N)' read *,resp if(resp.eq.'S') then print *,'primer atomo?=' read *,i print *,'segundo atomo?=' read *,j conect(i,j)=0 conect(j,i)=0 goto 10 elseif (resp.eq.'N') then 20 goto else 9 goto endif

goto 20 19 print *,'Debe responder "S" (si) o "N" (no)' 20 print *,'Quiere hacer uno algun elemento de la matriz $de conectividad?(S/N)' read *,resp if(resp.eq.'S') then

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print *,'primer atomo?=' read ,i * print *,'segundo atomo?=' read *,j conect(i,j)=1 goto 20 elseif (resp.eq.'N') then goto 30 else goto 19 endif 30 return END subroutine sigmabase(geom,vbase) C--------------------------------------------------- C Subrutina que genera vectores sigma para el vbase C real geom,vbase,norma integer i,j,k parameter (nmax=200) dimension geom(3,nmax),vbase(3,nmax) character resp*1 goto 10 9 print *,'Debe responder "S" (si) o "N" (no)' 10 print *,'Desea generar algun vector sigma?(S/N)' read *,resp (resp. print *,'Atomo que desea cambiar?='

if eq.'S') then

read ,i * print *,'Atomo con el que desea generar el enlace?=' read *,j norma=sqrt((geom(1,j)-geom(1,i))**2+(geom(2,j)-geom(2,i))**2 $+(geom(3,j)-geom(3,i))**2) do 20 k=1,3 vbase(k,i)=(geom(k,j)-geom(k,i))/norma 20 continue elseif (resp.eq.'N') then goto 30 else 9 goto endif goto 10 3 0 return

END

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AAPPÉÉNNDDIICCEE BB

Código Fortran del programa SCANXYZ que genera estructuras moleculares por interpolación lineal a partir de dos estructuras moleculares

Breve descripción del programa SCANXYZ

1. Recoge información de la geometría inicial y final en los ficheros: - nommax - nommin (los formatos son coordenadas cartesianas o PDB) (rutina principal). 2. Se chequea que dichas entradas sean coherentes (mismo número y orden de átomos) y se crean las masas atómicas. (rutina principal). 3. Se elige el centro de masas de ambas moléculas como el origen de coordenadas al que se van a referir todos los cálculos (subrutina centrodemasas(max,matom,n)). 4. S orientan las moléculas en base a sus tres ejes de inercia principales. Subrutinas: orientaz(max,min,matom,n) orientax(max,min,matom,n) orientay(max,min,matom,n) 5. Una vez orientadas las moléculas, se desplaza la geometría inicial en la dirección de la final en una cantidad dada por el usuario, y creando tantas estructuras como se desee. Asimismo se crea un fichero de salida de Gaussian98 con dicha estructura. (rutina principal). 6. Fin del programa.

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Program SCANXYZ C------------------------------------------------------------- C------------------------------------------------------------- C Programa que genera estructuras moleculares -- C a partir de una estructura "máxima" en la dirección -- C del vector dado por una estructura "mínima" -- C -- C Es útil para hacer: -- C scaners en la dirección de un vector -- C caminos de relajación a partir de una CoIn -- C búsqueda de TS tanto en estado fundamental -- C como excitado -- C caminos de reaccion en moléculas con alta simetría -- C -- C Luisma, Siena 2000 -- C -- C------------------------------------------------------------- C------------------------------------------------------------- real max,min,matom,tpc,maxn,ts,paso,pass integer n,i,j,k,atomax,atomin,zatom,num,mem,carga,multip parameter (nmax=200) dimension max(3,nmax),min(3,nmax),atomax(nmax),atomin(nmax), &zatom(nmax),matom(nmax),maxn(3,nmax),ts(3,nmax) character nommax*50,typef*3,nommin*50,out*20,ruta1*80, &ruta2*80,ruta3*80 print *,'Numero de atomos de la molecula? ' read ,n * print *,'Nombre del fichero con la estructura del max?' read *,nommax open unit=1,file=nommax) (1 print *,'Fichero pdb o coordenadas cartesianas?(PDB/CC) ' read *,typef if (typef.eq.'PDB') then do 10 i=1,n read (1,100),atomax(i),max(1,i),max(2,i),max(3,i) 10 continue else if(typef.eq.'CC ') then do 20 i=1,n read (1,101),atomax(i),max(1,i),max(2,i),max(3,i) 20 continue else print *,'solo ficheros PDB o CC !!!' goto 1 endif

print *,'Nombre del fichero con la estructura del min?' read *,nommin open (unit=2,file=nommin) 30 print *,'Fichero pdb o coordenadas cartesianas?(PDB/CC) ' read *,typef if (typef.eq.'PDB') then c lectura de las coordenadas cartesianas del fichero .ent c de la estructura con menor energía c do 40 i=1,n read (2,100),atomin(i),min(1,i),min(2,i),min(3,i) 40 continue else if(typef.eq.'CC ') then do 50 i=1,n read ,101),atomin(i),min(1,i),min(2,i),min(3,i) (2

continue 50 else print *,'solo ficheros PDB o CC !!!' goto 30 endif

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100 format(12x,i2,18x,f6.3,1x,f6.3,3x,f6.3) 101 format(1x,i2,4x,f9.6,2x,f9.6,2x,f9.6) C subprograma de chequeo de el orden de los atomos do 60 i=1,n if(atomax(i).NE.atomin(i)) then print *,'¡¡¡FICHERO CON ESTRUCTURAS NO VALIDO!!!' endif 60 continue do 70 i=1,n zatom(i)=atomax(i) 70 continue C Subprograma para la generacion de masas atomicas do i=1,n 80 if (zatom(i).EQ.1) then matom(i)=1.00783 elseif (zatom(i).EQ.6) then matom(i)=12.0000 endif C continuar aquí para otros átomos si fuese necesario 80 continue C fin del suprograma genreación de masas atómicas close(1) close(2) call centrodemasas(max,matom,n) call centrodemasas(min,matom,n) open (unit=1,file='anguloz') open (unit=2,file='angulox') open (unit=3,file='anguloy') do 500 i=1,100 call orientaz(max,min,matom,n) call orientax(max,min,matom,n) call orientay(max,min,matom,n) 500 continue close(1) close(2) close(3) c c entrada del numero de estructuras y el % que se desea que c se aleje de la estructura máxima c print *,'Numero de estructuras que desea generar?' read *,num print *,'tanto por ciento de avance hacia el minimo?' read *,tpc paso=(tpc/num)/100 c c entrada de los datos para el calculo en el gausian c nombre del fichero, memoria, ruta... c print *,'Como quiere que se llame el fichero de salida?' read ,out * print *,'Cuanta memoria en Mb desea usar?' read ,mem * print *,'ruta del calculo? (incluya #)y en cualquier' print *,'caso 3 lineas con sus respectivos #' read *,ruta1 read *,ruta2

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read ,ruta3 * print *,'Introduzca la carga' read ,carga * print *,'Introduzca la multiplicidad' read *,multip open(unit=3,file=out) C--------------------------------------------------------- do 90 i=1,3 do 110 j=1,n ts(i,j)=min(i,j)-max(i,j) 110 continue 9 0 continue

do 200 i=0,num pass=paso*i*100 do 202 k=1,3 do 201 j=1,n maxn(k,j)=max(k,j)+(ts(k,j)*(paso*i)) 201 continue 202 continue c c creacion del fichero de salida c write (3,*)'%chk=',out write (3,*)'%mem=',mem,'Mb' write (3,*) ruta1 write (3,*) ruta2 write (3,*) ruta3 write (3,*),'' write (3,*)'estructura numero',i,'. Supone un:',pass,'%' write (3,*),'' write (3,*),carga,' ',multip do 300 k=1,n write (3,105),zatom(k),maxn(1,k),maxn(2,k),maxn(3,k) 300 continue 105 format(1x,i2,4x,f10.6,2x,f10.6,2x,f10.6) write (3,*) write (3,214)'0.5 0.5' write (3,*)'' write (3,213)'--Link1--' 200 continue 213 format (0x,a9) 214 format (0x,a14) END

subroutine centrodemasas(geom,matom,n) C----------------------------------------------------------------------- C Orientacion con respecto al centro de masas de la molécula real geom,matom,sumass,rcx,rcy,rcz integer n parameter (nmax=200) dimension matom(nmax),geom(3,nmax) do 10 i=1,n sumass=sumass+matom(i) rcx=rcx+geom(1,i)*matom(i) rcy=rcy+geom(2,i)*matom(i) rcz=rcz+geom(3,i)*matom(i) 10 continue C print *,'suma de la masas atomicas',sumass C print *,'rcx=',rcx,' rcy=',rcy,' rcz=',rcz C read *,k

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do 30 i=1,n geom(1,i)=geom(1,i)-(rcx/sumass) geom(2,i)=geom(2,i)-(rcy/sumass) geom(3,i)=geom(3,i)-(rcz/sumass) 30 continue return END subroutine orientaz(max,min,matom,n) C---------------------------------------------------------------------- real max,min,matom,cseno,summas,vectoz,alfa,a,norma,normag, $normap,min2,nn integer n parameter (nmax=200) dimension max(3,nmax),min(3,nmax),matom(nmax),cseno(nmax), $ a(3,nmax),normag(nmax),normap(nmax),norma(nmax),min2(3,nmax) nn=real(n) summas=0.0 vectoz=0.0 do 10 i=1,n summas=summas+matom(i) cseno(i)=(max(1,i)*min(1,i)+max(2,i)*min(2,i))/( $(sqrt(max(1,i)**2+max(2,i)**2))* $(sqrt(min(1,i)**2+min(2,i)**2))) normag(i)=sqrt(max(1,i)**2+max(2,i)**2) normap(i)=sqrt(min(1,i)**2+min(2,i)**2) norma(i)=normag(i)*normap(i) print coseno=',cseno(i) *,'

continue 10 print *,'norma(1)=',norma(1) do 20 i=1,n a(1,i)=0.0 a(2,i)=0.0 a(3,i)=((matom(i)**2)*acos(cseno(i))*(max(1,i)*min(2,i)-max(2,i)* $min(1,i))/norma(i)) print *,a(3,i) 20 continue do 40 i=1,n vectoz=vectoz+a(3,i) min2(1,i)=min(1,i) min2(2,i)=min(2,i) min2(3,i)=min(3,i) 40 continue print *,'vector(coseno de alfa)=',vectoz print *,'suma de las masas atomicas al cuadrado=',summas C vector es el vector coseno de alfa alfa=vectoz/(nn*(summas**2)) write (1,*) alfa,(vectoz/((summas**2)*nn)) print *,alfa if(vectoz.LE.0.0) then do 51 i=1,n min(1,i)=min2(1,i)*cos(alfa)-min2(2,i)*sin(alfa) min(2,i)=min2(1,i)*sin(alfa)+min2(2,i)*cos(alfa) 5 1 continue

else do 61 i=1,n min(1,i)=min2(1,i)*cos(alfa)+min2(2,i)*sin(alfa) min(2,i)=-min2(1,i)*sin(alfa)+min2(2,i)*cos(alfa) 61 continue

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endif return END subroutine orientax(max,min,matom,n) C---------------------------------------------------------------------- real max,min,matom,cseno,summas,vectox,alfa,a,norma,normag, $normap,min2,nn integer n parameter (nmax=200) dimension max(3,nmax),min(3,nmax),matom(nmax),cseno(nmax), $a(3,nmax),normag(nmax),normap(nmax),norma(nmax),min2(3,nmax) nn=real(n) summas=0.0 vectox=0.0 do 10 i=1,n summas=summas+matom(i) cseno(i)=(max(2,i)*min(2,i)+max(3,i)*min(3,i))/( $(sqrt(max(2,i)**2+max(3,i)**2))* $(sqrt(min(2,i)**2+min(3,i)**2))) normag(i)=sqrt(max(2,i)**2+max(3,i)**2) normap(i)=sqrt(min(2,i)**2+min(3,i)**2) norma(i)=normag(i)*normap(i) 1 0 continue

do 20 i=1,n a(1,i)=0.0 a(2,i)=((matom(i)**2)*acos(cseno(i))*(max(2,i)*min(3,i)-max(3,i) $*min(2,i))/norma(i)) a(3,i)=0.0 20 continue do 40 i=1,n vectox=vectox+a(2,i) min2(1,i)=min(1,i) min2(2,i)=min(2,i) min2(3,i)=min(3,i) 4 0 continue

C vector es el vector coseno de alfa alfa=vectox/(nn*(summas**2)) write (2,*) alfa,(vectox/((summas**2)*nn)) if(vectox.LE.0.0) then do 51 i=1,n min(2,i)=min2(2,i)*cos(alfa)-min2(3,i)*sin(alfa) min(3,i)=min2(2,i)*sin(alfa)+min2(3,i)*cos(alfa) 51 continue else do 61 i=1,n min(2,i)=min2(2,i)*cos(alfa)+min2(3,i)*sin(alfa) min(3,i)=-min2(2,i)*sin(alfa)+min2(3,i)*cos(alfa) 6 1 continue

endif return END

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subroutine orientay(max,min,matom,n) C---------------------------------------------------------------------- real max,min,matom,cseno,summas,vectoy,alfa,a,norma,normag, $normap,min2,nn integer n parameter (nmax=200) dimension max(3,nmax),min(3,nmax),matom(nmax),cseno(nmax), $a(3,nmax),normag(nmax),normap(nmax),norma(nmax),min2(3,nmax) nn=real(n) summas=0.0 vectoy=0.0 do 10 i=1,n summas=summas+matom(i) cseno(i)=(max(1,i)*min(1,i)+max(3,i)*min(3,i))/( $(sqrt(max(1,i)**2+max(3,i)**2))* $(sqrt(min(1,i)**2+min(3,i)**2))) normag(i)=sqrt(max(1,i)**2+max(3,i)**2) normap(i)=sqrt(min(1,i)**2+min(3,i)**2) norma(i)=normag(i)*normap(i) 10 continue do 20 i=1,n a(1,i)=((matom(i)**2)*acos(cseno(i))*(max(3,i)*min(1,i)-max(1,i) $*min(3,i))/norma(i)) a(2,i)=0.0 a(3,i)=0.0 2 0 continue

do 40 i=1,n vectoy=vectoy+a(1,i) min2(1,i)=min(1,i) min2(2,i)=min(2,i) min2(3,i)=min(3,i) 40 continue C vector es el vector coseno de alfa alfa=vectoy/((summas**2)*nn) write (3,*) alfa,(vectoy/((summas**2)*nn)) (v do 51 i=1,n

if ectoy.LE.0.0) then

min(1,i)=min2(1,i)*cos(alfa)-min2(3,i)*sin(alfa) min(3,i)=min2(1,i)*sin(alfa)+min2(3,i)*cos(alfa) 51 continue else do 61 i=1,n min(1,i)=min2(1,i)*cos(alfa)+min2(3,i)*sin(alfa) min(3,i)=-min2(1,i)*sin(alfa)+min2(3,i)*cos(alfa) 61 continue endif return END

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AAPPÉÉNNDDIICCEE CC

Código Fortran del programa ISC que junto con el cálculo ab initio de gradientes analíticos determina puntos de cruce de sistemas mínimos en energía

Breve descripción del programa ISC

1. Lee la energía de los dos cálculos Gaussian para los estados singlete y triplete (subrutina: readener(enerS0,enerT1)). 2. Lee los vectores fuerza de ambos estados electrónicos y calcula el vector diferencia de gradientes. (subrutina: makegrad(grad,gradS0,gradT1,natom)). 3. Lee la geometría en coordenadas catesianas y los correspondientes números atómicos de cada átomo de la molécula (subrutina: readgeom(zatom,geomini,natom)). 4. Calcula el vector diferencia de gradientes, y el vector fuerzas en el estado de interés proyectado sobre el primero de los vectores. Compone el vector desplazamiento en la dirección de la suma de ambos. Crea la nueva geometría en la dirección del vector. (subrutina: makeNG(geomini,grad,gradS0,natom,enerS0,enerT1,zatom,gradT1)). 5. Chequea la convergencia de los dos vectores. Esta subrutina permite comprobar si el punto optimizado cumple ser mínimo en el subespacio de intersección. (subrutina: converg(grad,gradS0,natom,conver)). 6. Crea el fichero con la nueva geometría para realizar el cálculo del gradiente en el primer estado. (subrutina: makeIS0(geomini,natom,zatom)). 7. Crea el fichero con la nueva geometría para realizar el cálculo del gradiente en el segundo estado. (subrutina: makeIT1(geomini,natom,zatom)). 8. Añade la información de un nuevo punto al fichero de salida que contiene la geometría del nuevo punto, la diferencia de energía entre los estados, la norma del vector diferencia de gradientes y la norma del vector fuerzas proyectado en el estado optimizado. (subrutina: makeout(geomini,grad,gradS0,natom,enerS0,enerT1)) 9. Fin de la rutina principal. Nota: El proceso representa un solo punto de la optimización, este proceso se repite iterativamente con el uso de una shell en linux.

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program ISC C------------------------------------------------------------- C------------------------------------------------------------- C Este programa encuentra minimos de la función de energía -- C potencial E_0+E_1 sujeta a la condición E_1-E_0=0 -- C Donde E_0=E(S_0) y E_1=E(T_1) o -- C E_0=E(T_1) y E_1=E(S_0), es decir, puntos de ISC -- C -- C Luisma, Alcalá 2003 -- C------------------------------------------------------------- C------------------------------------------------------------- real geomini,grad,gradS0,gradT1, $enerS0,enerT1 integer tom,zatom na parameter (atomax=100) dimension geomini(3,atomax),zatom(atomax), $grad(3,atomax),gradS0(3,atomax),gradT1(3,atomax) character conver*1 C Número de átomos: natom=16 call leeener(enerS0,enerT1) C------------------------------------------------------- C Subrutina que lee las energías de los ficheros .log C de Gaussian98 (de los dos estados call makegrad(grad,gradS0,gradT1,natom) C------------------------------------------------------- C Subrutina que lee el gradiente en S_0 y T_1 y genera C el vector diferencia de gradientes call leegeom(zatom,geomini,natom) C------------------------------------------------------- C Subrutina que lee la geometría de un fichero de entrada C llamado Geometry.input en coordenadas cartesianas y con C el formato (15x,i1,20x,f10.6,2x,f10.6,2x,f10.6) call makeNG(geomini,grad,gradS0,natom,enerS0 $,enerT1,zatom,gradT1) C------------------------------------------------------- C Subrutina que crea el desplazamiento en el proceso de C optimización a partir de los gradientes calculados call converg(grad,gradS0,natom,conver) C------------------------------------------------------- C Subrutina que comprueba la convergencia del proceso C de optimización if (conver.eq.'Y') then 1000 goto endif call creaIS0(geomini,natom,zatom) C------------------------------------------------------- C Subrutina que crea la entrada del cálculo de Gaussian98 C para el estado S_0 si el cálculo no ha convergido

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call creaIT1(geomini,natom,zatom) C------------------------------------------------------- C Subrutina que crea la entrada del cálculo de Gaussian98 C para el estado T_1 si el cálculo no ha convergido 1000 print *,'Optimization complete!' call makeout(geomini,grad,gradS0,natom, $enerS0,enerT1) C------------------------------------------------------- C Subrutina que crea el fichero de salida con toda la C información relevante END subroutine leeener(enerS0,enerT1) C---------------------------------------------------- real enerS0,enerT1 integer i character line*60,null*1 100 format (a) 101 format (33x,e15.12) open(unit=1,file='S0.log') open(unit=2,file='T1.log') 10 read (1,100) line if (line(6:26).ne.'DO AN EXTRA-ITERATION') then 10 go to

else read (1,101) enerS0 endif close (1) 20 read (2,100) line if (line(6:26).ne.'DO AN EXTRA-ITERATION') then 20 go to else read 2,101) enerT1 (

endif close (2) return end subroutine makegrad(grad,gradS0,gradT1,natom) C--------------------------------------------------- real grad,gradS0,gradT1 integer i,natom parameter (atomax=100) dimension grad(3,atomax),gradS0(3,atomax),gradT1(3,atomax) character line*70,null*1 20 format(a) 100 format(26x,f12.9,3x,f12.9,3x,f12.9) open (unit=1,file='S0.log')

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open (unit=2,file='T1.log') 10 Read(1,20,End=1000) line If (line(38:43).ne.'Forces')then Goto 10 endif read (1,20)null read (1,20)null print *,'S0 Forces:' do 40 i=1,natom read (1,100,ERR=1000) gradS0(1,i),gradS0(2,i) $,gradS0(3,i) print 100,gradS0(1,i),gradS0(2,i) $,gradS0(3,i) 40

continue

101 Read(2,20,End=1000) line If (line(38:43).ne.'Forces')then Goto 101 endif read (2,20)null read (2,20)null print *,'T1 Forces:' do 400 i=1,natom read (2,100,ERR=1000) gradT1(1,i),gradT1(2,i) $,gradT1(3,i) print 100,gradT1(1,i),gradT1(2,i) $,gradT1(3,i) 40

0 continue

do 402 i=1,natom grad(1,i)=gradT1(1,i)-gradS0(1,i) grad(2,i)=gradT1(2,i)-gradS0(2,i) grad(3,i)=gradT1(3,i)-gradS0(3,i) 402 continue 1000 print *,'Gradient Difference Vector Generated' close (1) close (2) END

return

subroutine leegeom(zatom,geomini,natom) C--------------------------------------------------- real geomini integer natom,zatom parameter (atomax=100) dimension geomini(3,atomax),zatom(atomax) character line*60 open (unit=1,file='Geometry.input') do i=1,n m 1 ato10 Read(1,20,End=1000) zatom(i),geomini(1,i),geomini(2,i), $geomini(3,i) C natom=i 1 continue

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20 format(15x,i1,20x,f10.6,2x,f10.6,2x,f10.6) 1000 print *,'Geomtry readed' close (1) return END subroutine makeNG(geomini,grad,gradS0,natom,enerS0 $,enerT1,zatom,gradT1) C------------------------------------------------------- real grad,gradS0,gradT1,geomini,prodesc,f,NGeom $,ff,enerS0,enerT1,norm,gradST integer zatom,natom parameter (atomax=100) dimension grad(3,atomax),gradS0(3,atomax),gradT1(3,atomax), $geomini(3,atomax),NGeom(3,atomax),zatom(atomax), $gradST(3,atomax) C la relacion ff/f mantenida constante, a medida que aumenta C aumenta el peso de la componente en la optimización en T1 C frente al peso de la optimización de la diferencia de energía C valores originales: f=2*(enerT1-enerS0)*30 ff=0.05 f=2*(enerT1-enerS0)*30 ff=0.50 prodesc=0.0 norm=0.0 do 6 i=1,natom grad(1,i)=grad(1,i)*f grad(2,i)=grad(2,i)*f grad(3,i)=grad(3,i)*f 6 continue do 1 i=1,natom norm=norm+grad(1,i)*grad(1,i)+ $ grad(2,i)*grad(2,i)+grad(3,i)*grad(3,i) 1 continue C--------------------------- C Si se quiere optimizar solo E_0 se comenta el siguiente C Bucle y donde aparece en esta subrutina gradST se sustituye C por gradS0 C--------------------------- C do 7 i=1,natom C gradST(1,i)=(gradT1(1,i)+gradS0(1,i)) C gradST(2,i)=(gradT1(2,i)+gradS0(2,i)) C gradST(3,i)=(gradT1(3,i)+gradS0(3,i)) C7 continue do 2 i=1,natom prodesc=prodesc+grad(1,i)*gradT1(1,i)+ $ grad(2,i)*gradT1(2,i)+grad(3,i)*gradT1(3,i) 2 continue do 3 i=1,natom NGeom(1,i)=geomini(1,i)+ff*(gradT1(1,i)+grad(1,i)* $ (1-(prodesc/norm))) NGeom(2,i)=geomini(2,i)+ff*(gradT1(2,i)+grad(2,i)* $ (1-(prodesc/norm))) NGeom(3,i)=geomini(3,i)+ff*(gradT1(3,i)+grad(3,i)*

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$ (1-(prodesc/norm))) 3 continue

do 4 i=1,natom geomini(1,i)=NGeom(1,i) geomini(2,i)=NGeom(2,i) geomini(3,i)=NGeom(3,i) 4 continue open (unit=1,file='Geometry.input') do 5 i=1,natom 10 write(1,20) zatom(i),geomini(1,i),geomini(2,i), $geomini(3,i) 5 continue

20 format(15x,i1,20x,f10.6,2x,f10.6,2x,f10.6) return END subroutine converg(grad,gradS0,natom,conver) C------------------------------------------------------- real d,gradS0,norm,normS0 gra integer tom na parameter (atomax=100) dimension grad(3,atomax),gradS0(3,atomax) character conver*1 norm=0.0 do 1 i=1,natom norm=norm+grad(1,i)*grad(1,i)+grad(2,i)*grad(2,i)+ $grad(3,i)*grad(3,i) 1 continue mS0=0.0 nor do 2 i=1,natom normS0=normS0+gradS0(1,i)*gradS0(1,i)+gradS0(2,i)*gradS0(2,i)+ $gradS0(3,i)*gradS0(3,i) 2 continue if (norm.lt.0.0001) then if (normS0.lt.0.0001) then conver='Y' go to 10 endif endif go to 20 10 print *,'-------------------------' print *,'--Convergence Reached!!--' print *,'-------------------------' 20 print *,'END OF CHECKING CONVERGENCE' return

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END subroutine creaIS0(geomini,natom,zatom) C------------------------------------------------------- real geomini integer natom,zatom parameter (atomax=100) dimension geomini(3,atomax),zatom(atomax) 10 format(a) 11 format(15x,i1,20x,f10.6,2x,f10.6,2x,f10.6) open (unit=1,file='S0.com') write(1,10) '%chk=S0' write(1,10) '%mem=100Mb' write(1,10) '#p CASSCF(8,8,NRoot=1,NoCPMCSCF)/sto-3g* Nosymm ' write(1,10) '# guess=read SCF=(conver=8,MAxCycle=300)' write(1,10) '# opt=(MaxCycle=1)' write(1,10) ' ' write(1,10) 'S1' write(1,10) ' ' write(1,10) '0 1' do i=1,natom 1 write(1,11) zatom(i),geomini(1,i),geomini(2,i), $geomini(3,i) 1 write(1,10) ' '

continue

close (1) return END subroutine creaIT1(geomini,natom,zatom) C------------------------------------------------------- real geomini integer tom,zatom na parameter (atomax=100) dimension geomini(3,atomax),zatom(atomax) 10 format(a) 11 format(15x,i1,20x,f10.6,2x,f10.6,2x,f10.6) open (unit=1,file='T1.com') write(1,10) '%chk=T1' write(1,10) '%mem=100Mb' write(1,10) '#p CASSCF(8,8,NRoot=1,NoCPMCSCF)/sto-3g Nosymm ' write(1,10) '# guess=read SCF=(conver=8,MAxCycle=300)' write(1,10) '# opt=(MaxCycle=1)' write(1,10) ' ' write(1,10) 'T1' write(1,10) ' ' write(1,10) '0 3' do i=1,natom 1 write(1,11) zatom(i),geomini(1,i),geomini(2,i), $geomini(3,i) 1 continue write(1,10) ' '

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close (1) END

return

subroutine makeout(geomini,grad,gradS0,natom, $enerS0,enerT1) C------------------------------------------------------- real geomini,grad,gradS0,enerS0,enerT1,ppp,normS0,norm integer natom parameter (atomax=100) dimension geomini(3,atomax),grad(3,atomax), $gradS0(3,atomax) 100 format(a) 101 format(f9.6,3x,f9.6,3x,f9.6) C102 format (*) 103 format(a28,1x,f12.6) 104 format(a36,1x,f12.6) 105 format(a24,1x,f12.6) norm=0.0 do 1 i=1,natom norm=norm+grad(1,i)*grad(1,i)+grad(2,i)*grad(2,i)+ $grad(3,i)*grad(3,i) 1 continue mS0=0.0 nor do 2 i=1,natom normS0=normS0+gradS0(1,i)*gradS0(1,i)+gradS0(2,i)*gradS0(2,i)+ $gradS0(3,i)*gradS0(3,i) 2 continue open (unit=1,file='T1S0_IC.log') write(1,100) 'Geomtery' do 10 i=1,natom write(1,101) geomini(1,i),geomini(2,i),geomini(3,i) 10 continue write(1,100) ' ' write(1,100) '--------------------------------------------' write(1,100) 'Energy:' write(1,*) 'Energy S0=',enerS0 write(1,*) 'Energy T1=',enerT1 ppp=(enerT1-enerS0)*627.5 write(1,103) 'Energy Difference(Kcal/mol)=',ppp write(1,104) 'Norm of the energy difference vector',norm write(1,105) 'Norm of the force vector',normS0 close(1) return end

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AAPPÉÉNNDDIICCEE DD

Código Fortran del programa RELAXATION que recoge diversas utilidades en el estudio del proceso de relajación para una reacción fotoquímica diabática a través de un intersección cónica.

Breve descripción del programa RELAXATION

1. Lee los nombres del fichero de Gaussian98 con el cálculo opt=conical (.log) y del fichero de salida. (rutina principal) 2. Lee la geometría del fichero (.log) que se desea analizar. (subrutina: readgeom(geom,n,name,zatom,nofg)). 3. Lee los vectores DC y GD correspondientes a dicha geometría. (subrutina: readvec(n,name,DC,UGD,nofg)). 4. Crea el fichero a proyectar en el subespacio formado por DC y GD, ya sea como combinación lineal de formación de enlaces, como combinación lineal de los propios vectores DC y GD o leído directamente de un fichero con el formato adecuado. (subrutina: readproy(vproy,geom,n,UGD,DC)). 5. Calcula diversas propiedades del vector a proyectar (angulo con respecto a los vectores de la base, ángulo que forman los vectores de la base…) Además escribe esta información en el fichero de salida. (rutina principal) 6. Obtiene el símbolo atómico de cada átomo de la molécula a partir de una base de datos. (subrutina: calczzatom(n,zatom,zzatom)). 7. Crea el fichero R.ent de HyperChem con la geometría de la Intersección Cónica. (subrutina: ent(namef,geom,n,zzatom)). 8. Crea los ficheros script de HyperChem para los vectores DC, GD y el vector propuesto (R). (subrutina: scr(namef,R,n)). (subrutina: scr(namef,DC,n)). (subrutina: scr(namef,UGD,n)). 9. Crea la salida con el scan entorno a la cónica a una distancia (radio) dada y en un número de geometrías dado. (subrutina: scan_DC_UGD (n,zatom,geom,DC,UGD)). 10. Fin de la rutina principal.

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program RELAXATION C------------------------------------------------------------- C------------------------------------------------------------- C Progarma que lee los dos vectores que definen el -- C subespacio donde desaparece la degeneración energética -- C en una CoIn y proyecta un vector dado en dicho subespacio,-- C además crea ficheros .ent y .scr de HyperChem con -- C información sobre todos los vectores, permite escasear -- C una CoIn así como preparar entradas de Gaussian98 para -- C realizar caminos de relajación desde la CoIn en distintas -- C direcciones a lo largo de todo el espacio de ramificación -- C -- C Luisma, Alcalá 2001 -- C -- C------------------------------------------------------------- C------------------------------------------------------------- real geom,DC,UGD,DCNOR,UGDNOR,rnor,rnorr,r, $vproy,temp,angDC,angUGD,DCUGD,angDU,vpRDC,vpRUGD, $angRDC,angRUGD integer n,zatom,nofg parameter(nmax=200) character name*50,nameout*50,ans*1,namef*30, $zzatom*2 dimension geom(nmax,3),DC(nmax,3),UGD(nmax,3), $R(nmax,3),vproy(nmax,3),zatom(nmax), $zzatom(nmax) print *,'Name of the file to read?' read ,name * print *,'Name of the output file?' read *,nameout open(unit=2,file=nameout) vpDC=0. vpUGD=0. RNORR=0. print *,'Reading Geometry...' C------------------------------------------------------ call leegeom(geom,n,name,zatom,nofg) C Llamada a la subrutina que lee goem,n,nombre,zatom print *,'Reding vectors...' C------------------------------------------------------ call readvec(n,name,DC,UGD,nofg) C Llamada a la subrutina que lee los vectores C Derivative Coupling y Unscaled Gradient Difference 10 print *,'Do you want to create an output-file' print *,'with the vectors DC and UGD?' read *,ans if (ans.eq.'Y'.or.ans.eq.'y') then goto 20 elseif (ans.eq.'N'.or.ans.eq.'n') then goto 30 else 10 go to endif C Calculo de la norma de los vectores DC,UGD 20 do 22 i=1,n DCNOR=DCNOR+DC(i,1)**2+DC(i,2)**2+DC(i,3)**2 UGDNOR=UGDNOR+UGD(i,1)**2+UGD(i,2)**2+ $UGD(i,3)**2 22 continue DCNOR=sqrt(DCNOR)

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UGDNOR=sqrt(UGDNOR) open(unit=3,file='DC.xyz') open(unit=4,file='UGD.xyz') do 21 i=1,n write (3,23) (geom(i,1)+(DC(i,1))/DCNOR), $(geom(i,2)+(DC(i,2))/DCNOR), $(geom(i,3)+(DC(i,3))/DCNOR) write (4,23) (geom(i,1)+(UGD(i,1))/UGDNOR), $(geom(i,2)+(UGD(i,2))/UGDNOR), $(geom(i,3)+(UGD(i,3))/UGDNOR) 21 continue do 24 i=1,n temp=temp+DC(i,1)**2+DC(i,2)**2+DC(i,3)**2 temp2=temp2+sqrt(temp) temp3=temp3+UGD(i,1)**2+UGD(i,2)**2+UGD(i,3)**2 temp4=temp4+sqrt(temp3) 24 print *,'Norma total DC=',sqrt(temp)

continue

print *,'Norma Parcial DC=',temp2 print *,'Norma total UGD =',sqrt(temp3) print *,'Norma Parcial UGD =',temp4 close(3) close(4) 23 format(1x,f10.5,3x,f10.5,3x,f10.5) C C Normalizacion de los vectore DC y UGD C do 00 i=1,n 1 do 110 ii=1,3 DC(i,ii)=(DC(i,ii)/DCNOR) UGD(i,ii)=(UGD(i,ii)/UGDNOR) 110 continue 100 continue 30 print *,'Reading vector to proyect...' C------------------------------------------------------ call readproy(vproy,geom,n,UGD,DC) C Llamada a la subrutina que lee el vector que se proyecta C en la base formada por los vectores DC y UGD C C Proyeccion de vproy sobre DC y UGD C vpDC=0. vpUGD=0. DCUGD=0. do 200 i=1,n do 210 ii=1,3 vpDC=vpDC+DC(i,ii)*vproy(i,ii) vpUGD=vpUGD+UGD(i,ii)*vproy(i,ii) DCUGD=DCUGD+UGD(i,ii)*DC(i,ii) 210 continue 200 continue angDC=(acos(vpDC)/3.141592654)*180. angUGD=(acos(vpUGD)/3.141592654)*180. angDU=(acos(DCUGD)/3.141592654)*180. print *,'Angle of the vector with DC =',angDC,' deg' print *,'Angle of the vector with UGD =',angUGD,' deg' print *,'Angle of the DC with UGD =',angDU,' deg'

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print *,'Proyection of R into DC=',vpDC print *,'Proyection of R into UGD=',vpUGD do 220 i=1,n do 230 ii=1,3 R(i,ii)=DC(i,ii)*vpDC+UGD(i,ii)*vpUGD 230 continue 220 continue print *,'Norm of the displacement?' read (*,500) RNOR 500 format (f10.5) 501 format (f10.5,2x,f10.5,2x,f10.5) 502 format (i3,4x,f10.5,2x,f10.5,2x,f10.5) RNORR=0. do 320 i=1,n RNORR=RNORR+R(i,1)**2+R(i,2)**2+R(i,3)**2 320 continue RNORR=sqrt(RNORR) vpRDC=0. vpRUGD=0. do 221 i=1,n do 222 ii=1,3 vpRDC=vpRDC+DC(i,ii)*(R(i,ii)/RNORR) vpRUGD=vpRUGD+UGD(i,ii)*(R(i,ii)/RNORR) 222 continue 2 21 continue

angRDC=(acos(vpRDC)/3.141592654)*180. angRUGD=(acos(vpRUGD)/3.141592654)*180. print *,' ' print *,'Angle of the vector with DC =',angRDC,' deg' print *,'Angle of the vector with UGD =',angRUGD,' deg' do 420 i=1,n do 430 ii=1,3 R(i,ii)=(R(i,ii)*RNOR)/RNORR write (2,500) R(i,ii),',' 430 continue 420 continue open(unit=3,file='R.xyz') do =1,n 450 i write (2,501) R(i,1),R(i,2),R(i,3) write (3,502) zatom(i),(R(i,1)+geom(i,1)), $(R(i,2)+geom(i,2)),(R(i,3)+geom(i,3)) 450 continue close (1) close (2) close (3) C------------------------------------------------------- C Generacion de zzatom(nmax) C------------------------------------------------------- call calculazzatom(n,zatom,zzatom) C-------------------------------------------------------- C C Comienzo de la generación de los ficheros HyperChem C C--------------------------------------------------------

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601 print *,'Create HyperChem files with DC,UDC and R vectors?' read *,ans if (ans.eq.'Y'.or.ans.eq.'y') then goto 600 elseif (ans.eq.'N'.or.ans.eq.'n') then goto 900 else goto 601 endif 600 namef='R.ent' print *,'Generating ENT file...' C------------------------------------------------------ call ent(namef,geom,n,zzatom) C Llamada a la subrutina que genera el fichero ENT namef='R.scr' print *,'Generating Script Files...' C------------------------------------------------------ call scr(namef,R,n) C Llamada a la subrutina que genera el fichero ENT namef='DC.scr' print *,'Generating Script Files...' C------------------------------------------------------ call scr(namef,DC,n) C Llamada a la subrutina que genera el fichero ENT namef='UGD.scr' print *,'Generating Script Files...' C------------------------------------------------------ call scr(namef,UGD,n) C Llamada a la subrutina que genera el fichero ENT 800 print *,'Do you want to create an input' print *,'scan of DC and UGD vectors?' read *,ans if (ans.eq.'y'.or.ans.eq.'Y') then C------------------------------------------------------ call scan_DC_UGD (n,zatom,geom,DC,UGD) else if (ans.eq.'n'.or.ans.eq.'N') then 900 goto else goto 800 end if 900 print *,'Job Complete' END subroutine leegeom(geom,n,nombre,zatom,count) C--------------------------------------------------------------- C C preparado como subrutina que lee el numero de atomos C (n) y la geometria (geom(3,nmax)) C

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real m geo integer n,i,n2,norb,norboc,norbvi, $nmax,omax,oamax,lmax,zatom,count,countt,nofg parameter (nmax=200) character nombre*50,null*10,line*68,ans*1 dimension zatom(nmax),geom(nmax,3) open (unit=1,file=nombre) count=0 countt=0 101 Read(1,21,End=50) line If ( Line(1:29).ne.' Number Number Type ')then 01 Goto 1 endif count=count+1 go to 101 21 format(A) 50 close (1) open (unit=1,file=nombre) 51 print *,'Read last geometry?' read *,ans if (ans.eq.'Y'.or.ans.eq.'y') then print *,'Number of Geometry=',count goto 102 elseif (ans.eq.'N'.or.ans.eq.'n') then 103 goto

else 1 goto 5 endif 103 print *,'Number of Geometry to read?' read *,nofg count=nofg 102 Read(1,21,End=1000) line if ( Line(1:29).ne.' Number Number Type ')then Goto 102 endif countt=countt+1 if (countt.eq.count) then 100 goto else 102 goto endif

C100 Read(1,20,End=1000) line C If ( Line(1:29).ne.' Number Number Type ')then C Goto 100 C endif 20 format(A) 100 read (1,*)null i=1 10 n=n2 read (1,200,ERR=300)n2,zatom(i),geom(i,1),geom(i,2),geom(i,3) i=i+1 goto 10 200 format(2x,i3,8x,i3,20x,f10.6,2x,f10.6,2x,f10.6) 300 print *,'numero de atomos=',n write (2,*)'numero de atomos=',n do =1,n 30 i print 200,i,zatom(i),geom(i,1), $geom(i,2),geom(i,3) 30 continue

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1000 print *,'Lectura de geometria terminada' close (1) return END subroutine readvec(n,name,DC,UGD,nofg) C--------------------------------------------------------------- C C Subrutina que lee los vectores C Derivative Coupling y Unscaled Gradient Difference real DC,UGD integer n,count,number,countt parameter (nmax=200) character name*50,line*70,ans*1,kk*1 dimension DC(nmax,3),UGD(nmax,3) count=0 countt=0 open (unit=1,file=name) 100 Read(1,20,End=120) line if (line(1:20).ne.' Derivative Coupling')then 00 Goto 1

endif count=count+1 go to 100 20 format(A) 120 close (1) print *,'Read the same number of vector vector?' read *,ans if (ans.eq.'Y'.or.ans.eq.'y') then open (unit=1,file=name) 251 read(1,20,End=1000) line If (line(1:20).ne.' Derivative Coupling')then 251 go to endif untt=countt +1 co if(countt.eq.nofg) then 170 goto else 251 go to endif elseif (ans.eq.'N'.or.ans.eq.'n') then go to 250 else go to 120 endif 250 print *,'Read last vector?' read *,ans if (ans.eq.'Y'.or.ans.eq.'y') then 161 go to elseif (ans.eq.'N'.or.ans.eq.'n') then o 150 go t else go to 120 endif

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150 print *,'Number of vector to read?' read *,number open (unit=1,file=name) 151 read(1,20,End=1000) line If (line(1:20).ne.' Derivative Coupling')then go to 151 endif countt=countt +1 if(countt.eq.number) then goto 170 else 151 go to endif 161 open (unit=1,file=name) 160 read(1,20,End=1000) line If (line(1:20).ne.' Derivative Coupling')then 160 go to endif countt=countt+1 if(countt.eq.count) then goto 170 else goto 160 endif print *,'Derivative Copuling Vector:' 170 do 171 i=1,n read (1,200) DC(i,1),DC(i,2),DC(i,3) C print *,'DC' print 0,DC(i,1),DC(i,2),DC(i,3) 20

continue 171 read 1,20) line ( print *,'----------------------------------------------' print *,'Unscaled Gradient Difference Vector:' 172 do 173 i=1,n read (1,200) UGD(i,1),UGD(i,2),UGD(i,3) C print *,'UGD' print 200,UGD(i,1),UGD(i,2),UGD(i,3) 173 continue 200 format (8x,f13.10,7x,f13.10,7x,f13.10) goto 1010 1000 print *,'End of file!! ERROR reading vector!!' 1010 close (1) END

return

subroutine readproy(vproy,geom,n,UGD,DC) C----------------------------------------------------------- C C Subrutina que lee el vector que se desea proyectar C sobre el espacio engendrado por los vectores CD y UGD real vproy,geom,norm,aa,ab,ac,coefUGD,coefDC $,UGD,DC,coefsum,normm character name*50,ans*1 parameter (nmax=200) integer ini,atend at dimension vproy(nmax,3),geom(nmax,3), $DC(nmax,3),UGD(nmax,3) norm=0. 200 do 201 i=1,n do 202 ii=1,3

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vproy(i,ii)=0. 202 continue 01 continue 2

5 print *,'Do you want to generate a bond vector only?' read *,ans if (ans.eq.'Y'.or.ans.eq.'y') then go to 204 elseif (ans.eq.'N'.or.ans.eq.'n') then go to 11 else 5 go to endif 11 print *,'Do you want to generate a LC of UGD and DC vectors?' read *,ans if (ans.eq.'Y'.or.ans.eq.'y') then 0 go to 2 elseif (ans.eq.'N'.or.ans.eq.'n') then 9 go to else go to 5 endif 20 print *,'Coefficient of the UGD vector?' read *,coefUGD print *,'Coefficient of the DC vector?' read *,coefDC C print *,'Norm of the displacement?(in A)' C read *,norm coefsum=sqrt(coefUGD**2+coefDC**2) coefUGD=coefUGD/coefsum coefDC=coefDC/coefsum do 25 i=1,n vproy(i,1)=coefUGD*UGD(i,1)+coefDC*DC(i,1) vproy(i,2)=coefUGD*UGD(i,2)+coefDC*DC(i,2) vproy(i,3)=coefUGD*UGD(i,3)+coefDC*DC(i,3) normm=normm+(coefUGD*UGD(i,1)+coefDC*DC(i,1))**2+ $(coefUGD*UGD(i,2)+coefDC*DC(i,2))**2+ $(coefUGD*UGD(i,3)+coefDC*DC(i,3))**2 25 continue normm=sqrt(normm) do 26 i=1,n do 27 j=1,3 vproy(i,j)=vproy(i,j)/normm 27 continue 26 continue goto 1000 9 print *,'Name of the file with the vector to project?' read *,name open (unit=3,file=name) do 10 i=1,n read (3,100) aa,ab,ac vproy(i,1)=aa-geom(i,1) vproy(i,2)=ab-geom(i,2) vproy(i,3)=ac-geom(i,3) 10 continue close (3) goto 231 204 do 205 i=1,n vproy(i,1)=0. vproy(i,2)=0. vproy(i,3)=0. 2 05 continue

207 print *,'Origin atom?' read *,atini print *,'Last atom?' read *,atend

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vproy(atini,1)=geom(atend,1)-geom(atini,1) vproy(atini,2)=geom(atend,2)-geom(atini,2) vproy(atini,3)=geom(atend,3)-geom(atini,3) norm=0. 231 do 210 i=1,n norm=norm+vproy(i,1)**2+vproy(i,2)**2+vproy(i,3)**2 210 continue norm=sqrt(norm) do 220 i=1,n do 230 ii=1,3 vproy(i,ii)=vproy(i,ii)/norm 230 continue 220 continue 206 print *,'Have you finished?' read *,ans if (ans.eq.'Y'.or.ans.eq.'y') then 000 goto 1 elseif (ans.eq.'N'.or.ans.eq.'n') then goto 207 else goto 206 endif 100 format(f8.5,2x,f8.5,2x,f8.5) 1000 close (3) print *,'Do you want the antibonding vector?' read *,ans if (ans.eq.'Y'.or.ans.eq.'y') then do 1001 i=1,n vproy(i,1)=-vproy(i,1) vproy(i,2)=-vproy(i,2) vproy(i,3)=-vproy(i,3) 1001 else

continue

print *,'End of reading vector' endif return END subroutine calculazzatom(n,zatom,zzatom) C------------------------------------------------------- C integer tom za character zzatom*2 parameter (nmax=200) dimension zatom(nmax),zzatom(nmax) do 10 i=1,n if (zatom(i).eq.1) then zzatom(i)=' H' elseif(zatom(i).eq.2) then zzatom(i)='He' elseif(zatom(i).eq.3) then m(i)='Li' zzato elseif(zatom(i).eq.4) then m(i)='Be' zzato

elseif(zatom(i).eq.5) then m(i)=' B' zzato elseif(zatom(i).eq.6) then m(i)=' C' zzato elseif(zatom(i).eq.7) then zzatom(i)=' N' elseif(zatom(i).eq.8) then zzatom(i)=' O' elseif(zatom(i).eq.9) then

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m(i)=' F' zzato elseif(zatom(i).eq.10) then m(i)='Ne' zzato elseif(zatom(i).eq.11) then zzatom(i)='Na' elseif(zatom(i).eq.12) then zzatom(i)='Mg' elseif(zatom(i).eq.13) then zzatom(i)='Al' elseif(zatom(i).eq.14) then m(i)='Si' zzato elseif(zatom(i).eq.15) then m(i)=' P' zzato elseif(zatom(i).eq.16) then m(i)=' S' zzato elseif(zatom(i).eq.17) then m(i)='Cl' zzato elseif(zatom(i).eq.18) then m(i)='Ar' zzato elseif(zatom(i).eq.19) then zzatom(i)=' K' elseif(zatom(i).eq.20) then zzatom(i)='Ca' elseif(zatom(i).eq.21) then zzatom(i)='Sc' elseif(zatom(i).eq.22) then m(i)='Ti' zzato elseif(zatom(i).eq.23) then m(i)=' V' zzato elseif(zatom(i).eq.24) then m(i)='Cr' zzato

elseif(zatom(i).eq.25) then m(i)='Mn' zzato elseif(zatom(i).eq.26) then m(i)='Fe' zzato elseif(zatom(i).eq.27) then zzatom(i)='Co' elseif(zatom(i).eq.28) then zzatom(i)='Ni' elseif(zatom(i).eq.29) then zzatom(i)='Cu' elseif(zatom(i).eq.30) then m(i)='Zn' zzato elseif(zatom(i).eq.31) then m(i)='Ga' zzato elseif(zatom(i).eq.32) then m(i)='Ge' zzato elseif(zatom(i).eq.33) then m(i)='As' zzato elseif(zatom(i).eq.34) then m(i)='Se' zzato elseif(zatom(i).eq.35) then zzatom(i)='Br' elseif(zatom(i).eq.36) then zzatom(i)='Kr' else print *,'Error reading z-number!!' print *,'We havent got the sympol of the',i,'atom' endif 10 continue return END subroutine ent(namef,geom,n,zzatom) C----------------------------------------------------------- C C Subrutina que genera el fichero ENT para Hyperchem

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real m geo integer n character namef*30,zzatom*2 parameter (nmax=200) dimension geom(nmax,3),zzatom(nmax) open (unit=1,file=namef) do 10 i=1,n write(1,200) 'HETATM',i,zzatom(i),i,geom(i,1), $geom(i,2),geom(i,3) 0 continue 1

200 format(a6,2x,i3,1x,a2,9x,i3,5x,f7.3,1x,f7.3,1x,f7.3) close(1) return END subroutine scr(namef,vector,n) C----------------------------------------------------------- C C Subrutina que genera el fichero .SCR para Hyperchem real tor vec integer n character namef*30,string,temp*10,temp2*1 parameter (nmax=200) dimension vector(nmax,3) string='' temp2=',' temp='' open(unit=1,file=namef) write (1,*) 'file-format pdb' write (1,*) 'open-file R.ent' write (1,*) 'ir-band-count = 1' write (1,*) 'ir-frequency(1) = 0.0' write (1,*) 'ir-intensity(1) = 0.0' C write (1,*) 'ir-normal-mode(1) = ' C do 30 i=1,n C do 40 ii=1,3 write (1,200) 'ir-normal-mode(1) = ', $(vector(i,1),',',vector(i,2),',',vector(i,3),',',i=1,n) C40 continue C30 continue write(1,*) 'menu-compute-vibrational-spectrum' write(1,*) 'exit-script' 200 format (a20,200(f10.5,a1)) 201 format (f10.5) 202 format (a11) 203 format (a20,a2000) 204 format (f10.5,a1) close(1) return END

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subroutine scan_DC_UGD (n,zatom,geom,DC,UGD) C----------------------------------------------------------- C C Subrutina que genera la entrada de Gaussian para hacer C el scan de la combinación lineal de los vectores DC y UGD real geom,DC,UGD,norm,alpha,UGDT,NORMM,proyUD integer n,nofg,count,mem,zatom,charge,mult,i,ii parameter (nmax=200) character namef*50,route*70,ans*1,namec*50 dimension geom(nmax,3),DC(nmax,3),UGD(nmax,3), $route(10),zatom(nmax),UGDT(nmax,3) C Creacion de UGDT C------------------------------ proyUD=0. NORMM=0. do 2 i=1,n do 3 ii=1,3 proyUD=proyUD+DC(i,ii)*UGD(i,ii) 3 continue 2 continue C print *,proyUD C read *,kkk do 60 i=1,n do 61 ii=1,3 UGDT(i,ii)=UGD(i,ii)-proyUD*DC(i,ii) 61 continue 0 continue 6

do 62 i=1,n NORMM=NORMM+UGDT(i,1)**2+UGDT(i,2)**2+UGDT(i,3)**2 62 continue C print *,NORMM C read *,kkk do 63 i=1,n do 64 ii=1,3 UGDT(i,ii)=UGDT(i,ii)/sqrt(NORMM) 64 continue 63 continue C------------------------------ 1 count= print *,'Name of the Output File?' read *,namef print *,'Name of the CheckPoint File?(include ".chk")' read *,namec print *,'Memory?(in Mb)' read *,mem print *,'Number of Geometries to do?' read *,nofg print *,'Charge of Molecule?' read *,charge print *,'Multiplicity of Molecule?' read *,mult print *,'Norm of the displacement?' read *,norm 10 print *,'Route section?(include #, 70 characters max)' read *,route(count) 15 print *,'Do you want to create another line of R.S.?' read *,ans if(ans.eq.'y'.or.ans.eq.'Y') then count=count +1 goto 10

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else if (ans.eq.'n'.or.ans.eq.'N') then 20 goto else goto 15 end if C Creación del fichero 20 open (unit=4,name=namef) alpha=(2*3.141592654)/nofg do 50 ii=1,nofg write (4,100) '%chk=',namec write (4,101) 'mem=',mem,'Mb' do i=1,count 30 write 102) route(i) (4,

continue 30 write (4,*) ' ' write (4,*) 'scan DC*cos(',alpha*ii,')', $'+ UGDT*sin(',alpha*ii,')' write (4,*) ' ' write (4,*) charge,' ',mult do 40 j=1,n write (4,103) zatom(j),(geom(j,1)+cos(alpha*ii)*norm* $DC(j,1)+sin(alpha*ii)*norm*UGDT(j,1)), $(geom(j,2)+cos(alpha*ii)*norm* $DC(j,2)+sin(alpha*ii)*norm*UGDT(j,2)), $(geom(j,3)+cos(alpha*ii)*norm* $DC(j,3)+sin(alpha*ii)*norm*UGDT(j,3)) 40 write (4,*) ' '

continue

write (4,*) '0.5 0.5' write (4,*) ' ' write 105) '--Link1--' (4,0 continue 5

100 format (a5,a50) 101 format (a4,i3,a2) 102 format (a70) 103 format (i2,5x,f10.5,2x,f10.5,2x,f10.5) 104 format(f8.5,2x,f8.5,2x,f8.5) 105 format(a9) close(4) 800 print *,'Do you want to create a relaxation vector?' read *,ans if (ans.eq.'y'.or.ans.eq.'Y') then C------------------------------------------------------ open unit=4,file='Relaxa.xyz') ( print *,'number of geometry?' read *,ii do 400 j=1,n write (4,104) (geom(j,1)+cos(alpha*ii)*norm* $DC(j,1)+sin(alpha*ii)*norm*UGDT(j,1)), $(geom(j,2)+cos(alpha*ii)*norm* $DC(j,2)+sin(alpha*ii)*norm*UGDT(j,2)), $(geom(j,3)+cos(alpha*ii)*norm* $DC(j,3)+sin(alpha*ii)*norm*UGDT(j,3)) 400 continue else if (ans.eq.'n'.or.ans.eq.'N') then 900 goto else 00 goto 8 end if

900 close (4) return END

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AAPPÉÉNNDDIICCEE EE

Código Fortran del programa que evalúa los productos escalares entre funciones de espín representadas en las funciones de base de Rumer.

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program rumer C------------------------------------------------------------- C------------------------------------------------------------- C Este programa calcula el producto escalar -- C de una función de espín de Rumer por cada -- C función de una base canónica dada -- C -- C Luisma, York 2002 -- C------------------------------------------------------------- C------------------------------------------------------------- real coeff,weight,Gweight,prodesc integer maxbas,spinb,cmtrxb,cmtrxv,chang,count,vector, $i,ii,iii,iiii,sig,mtrxtmp parameter (maxbas=100) dimension spinb(maxbas,4,2),coeff(maxbas),weight(maxbas), $Gweight(maxbas),cmtrxb(maxbas,4,2),cmtrxv(maxbas,4,2), $prodesc(maxbas),vector(4,2),sig(maxbas),mtrxtmp(maxbas,4,2) character spinf*30,p*5 101 format(41x,a5) 102 format(12x,i1,1x,i1,1x,i1,1x,i1,1x,i1,1x,i1,1x,i1,1x, $i1,6x,f10.6,1x,f10.6,1x,f10.6) print *,'Name of the SPINS output file?' read *,spinf open (unit=1,file=spinf) C-------------------------------------------------------------------- C El fichero de entrada tiene el siguiente formato: C k Spin-Coupling pattern Coefficient Weight GWeight C --------------------------------------------------------------- C 1: (1-2,3-4,5-6,7-8) 0.770677 0.641087 0.358555 C ........ C ........ C-------------------------------------------------------------------- nbasis=14 10 read (1,101)p if(p.ne.'Rumer') then 10 go to endif read (1,101)p read (1,101)p read (1,101)p do 0 i=1,nbasis 2 read(1,102) spinb(i,1,1),spinb(i,1,2),spinb(i,2,1), $spinb(i,2,2),spinb(i,3,1),spinb(i,3,2),spinb(i,4,1), $spinb(i,4,2),coeff(i),weight(i),Gweight(i) print 102, spinb(i,1,1),spinb(i,1,2),spinb(i,2,1), $spinb(i,2,2),spinb(i,3,1),spinb(i,3,2),spinb(i,4,1), $spinb(i,4,2),coeff(i),weight(i),Gweight(i) 20 continue close(1) C Vector del que se van a calcular sus componentes C Ordenarlo de tal manera que el primer número de cada C pareja sea el menor posible! ej (1-3,2-4,5-6,7-8) vector(1,1)=1 vector(1,2)=3 vector(2,1)=2 vector(2,2)=4 vector(3,1)=5 vector(3,2)=6 vector(4,1)=7

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vector(4,2)=8 C Fin de la entrada del vector del que se van a calcular las componentes do 30 i=1,nbasis do 31 ii=1,4 do 32 iii=1,2 cmtrxb(i,ii,iii)=0 cmtrxv(i,ii,iii)=0 32 continue 31 continue 30 continue C Evaluacion del valor absoluto del producto escalar do 33 i=1,nbasis cmtrxb(i,1,1)=1 cmtrxb(i,1,2)=1 33 continue count=1 chang=0 do 90 i=1,nbasis 60 do 40 ii=1,4 C print *,cmtrxb(i,ii,1) if (cmtrxb(i,ii,1).eq.count) then do 41 iii=1,4 do 42 iiii=1,2 if (vector(iii,iiii).eq.spinb(i,ii,1)) then if(cmtrxv(i,iii,1).eq.0) then cmtrxv(i,iii,1)=cmtrxb(i,ii,1) cmtrxv(i,iii,2)=cmtrxb(i,ii,1) chang=chang+1 endif

endif

if tor(iii,iiii).eq.spinb( 2)) then (vec i,ii, if(cmtrxv(i,iii,1).eq.0) then cmtrxv(i,iii,1)=cmtrxb(i,ii,1) cmtrxv(i,iii,2)=cmtrxb(i,ii,1) chang=chang+1 endif endif 42 continue 41 continue endif 4 0 continue

C print *,cmtrxv(i,1,1),cmtrxv(i,1,2),cmtrxv(i,2,1), C $cmtrxv(i,2,2),cmtrxv(i,3,1),cmtrxv(i,3,2), C $cmtrxv(i,4,1),cmtrxv(i,4,2) do 50 ii=1,4 if (cmtrxv(i,ii,1).eq.count) then do 51 iii=1,4 do 52 iiii=1,2 if (spinb(i,iii,iiii).eq.vector(ii,1)) then if(cmtrxb(i,iii,1).eq.0) then cmtrxb(i,iii,1)=cmtrxv(i,ii,1) cmtrxb(i,iii,2)=cmtrxv(i,ii,1) chang=chang+1 endif endif if nb(i,iii,iiii).eq.vecto 2)) then (spi r(ii, if(cmtrxb(i,iii,1).eq.0) then cmtrxb(i,iii,1)=cmtrxv(i,ii,1) cmtrxb(i,iii,2)=cmtrxv(i,ii,1) chang=chang+1 endif endif 52 continue

-371-

51 endif

continue

5 0 continue

C print *,cmtrxb(i,1,1),cmtrxb(i,1,2),cmtrxb(i,2,1), C $cmtrxb(i,2,2),cmtrxb(i,3,1),cmtrxb(i,3,2), C $cmtrxb(i,4,1),cmtrxb(i,4,2) CC read *,kk if (chang.eq.0) then do 53 ii=1,4 if(cmtrxb(i,ii,1).eq.0) then count=count+1 cmtrxb(i,ii,1)=count cmtrxb(i,ii,2)=count go to 60 endif

continue 53 prodesc(i)=(2**real(count))/16 count=1 chang=0 go to 91 else chang=0 go to 60 endif 91 print *,'Terminada la base n§',i 90 continue C Obtenci¢n del signo del producto escalar C****************** do 200 i=1,nbasis sig(i)=1 C Hacemos una copia del vector do 204 ii=1,4 do 205 iii=1,2 mtrxtmp(i,ii,iii)=cmtrxv(i,ii,iii) 205 continue 204 continue C Fin de la copia del vector do 201 j=1,4 do 02 jj=1,4 2 do 03 jjj=1,4 2 do 206 jjjj=1,4 mtrxtmp(i,1,1)=cmtrxv(i,j,1) mtrxtmp(i,1,2)=cmtrxv(i,j,2) if (jj.eq.j) then goto 202 endif mtrxtmp(i,2,1)=cmtrxv(i,jj,1) mtrxtmp(i,2,2)=cmtrxv(i,jj,2) if (jjj.eq.j) then goto 203 endif if (jjj.eq.jj) then 203 goto endif mtrxtmp(i,3,1)=cmtrxv(i,jjj,1) mtrxtmp(i,3,2)=cmtrxv(i,jjj,2) if (jjjj.eq.j) then 206 goto

endif if (jjjj.eq.jj) then goto 206 endif if (jjjj.eq.jjj) then

-372-

206 goto endif mtrxtmp(i,4,1)=cmtrxv(i,jjjj,1) mtrxtmp(i,4,2)=cmtrxv(i,jjjj,2) if(mtrxtmp(i,1,1).eq.cmtrxv(i,1,1).or. $mtrxtmp(i,1,1).eq.cmtrxv(i,1,2).or. $mtrxtmp(i,1,2).eq.cmtrxv(i,1,1).or. $mtrxtmp(i,1,2).eq.cmtrxv(i,1,2)) then if(mtrxtmp(i,2,1).eq.cmtrxv(i,2,1).or. $mtrxtmp(i,2,1).eq.cmtrxv(i,2,2).or. $mtrxtmp(i,2,2).eq.cmtrxv(i,2,1).or. $mtrxtmp(i,2,2).eq.cmtrxv(i,2,2)) then if(mtrxtmp(i,3,1).eq.cmtrxv(i,3,1).or. $mtrxtmp(i,3,1).eq.cmtrxv(i,3,2).or. $mtrxtmp(i,3,2).eq.cmtrxv(i,3,1).or.

,3,2).eq.cmtrxv(i,3,2)) then $mtrxtmp(i if(mtrxtmp(i,4,1).eq.cmtrxv(i,4,1).or. $mtrxtmp(i,4,1).eq.cmtrxv(i,4,2).or. $mtrxtmp(i,4,2).eq.cmtrxv(i,4,1).or. $mtrxtmp(i,4,2).eq.cmtrxv(i,4,2)) then do 207 ij=1,4 if(mtrxtmp(i,ij,1).eq.cmtrxv(i,ij,1).or. $mtrxtmp(i,ij,2).eq.cmtrxv(i,ij,2)) then sig(i)=sig(i) elseif(mtrxtmp(i,ij,1).eq.cmtrxv(i,ij,2).or. $mtrxtmp(i,ij,1).eq.cmtrxv(i,ij,2)) then sig(i)=-sig(i) else print *,'ERROR AL CALCULAR EL SIGNO DEL PRODESC' endif

continue 207 endif endif endif endif 206 continue 203 continue 202 continue 201 continue 200 continue do 1000 i=1,nbasis print *,'coeficiente (',i,' ) = ',prodesc(i)*real(sig(i)) 1 000 continue

end

-373-

APÉNDICE F Deducción del término exponencial de la ecuación (6.40) como una integral de solape. En el modelo discutido aquí, basado en la teoría del estado de transición, la energía térmica del aceptor está distribuida clásicamente entre los estados rovibrónicos de la molécula en el estado fundamental. Esta distribución continua es función de una única coordenada 1

Aξ y es el estado de partida para la excitación al nivel de energía (E). De

acuerdo con esto, la probabilidad de la transición debe ser proporcional a la probabilidad de la energía vibracional calculada a partir de la distribución de Boltzmann, la cual está unívocamente relacionada con la coordenada 1

Aξ , que satisface

la ecuación (6.38), esto es:

( )21

21

2( )AT

BE E

k T gP E eκ

− −

∝ G1

Esta expresión puede ser usada para representar el espectro de absorción T1←S0

normalizado del aceptor introduciendo una constante de normalización N( )Eε A:

( )21

21

2( )AT

BE E

k T gAE N e

κ

ε− −

= G2

La ecuación G2 muestra como el espectro de absorción T1←S0 del aceptor se representa

por una función gaussiana de anchura 12A

Bk Tgκσ = .

El espectro de emisión T1→S0 normalizado del dador, representado por ( )f E se puede

aproximar también por una función gaussiana, de anchura Dσ (Serpa et al., 2003):

( )( )22

12

DT

DE E

Df E N e σ− −

= G3

La integral de solape entre los dos espectros normalizados: ( )Eε y ( )f E viene dada

por:

( ) ( ) ( ) ( )2 221

2 221 11

2 2 2 2

0 0

D AAT TT

B D AE E E E E EE E

k T gA D A DJ N N e e dE N N e e dE

κσ σ

∞ ∞− − − − − −− −

= ≡∫ ∫2

21 D

TDσ

2

G4

Es de esperar que la anchura de los dos espectros (emisión del dador y excitación del aceptor) sea muy diferente, ya que el estado fundamental del aceptor está caracterizado por una gran flexibilidad conformacional, en contraste con la estructura rígida del dador, y por lo tanto 2

A Dσ σ . En este caso, puede ser demostrado tras simples aunque

tediosas manipulaciones algebraicas, que la integral de solape en la ecuación G4 viene dada por:

( ) ( 211 2

1, , exp2

A DT T

B

J N T g E Ek T g

κκ⎛ ⎞

= −⎜ ⎟⎝ ⎠

)− G5

donde el término exponencial es el mismo que el de la ecuación (6.40).

-375-

AAPPÉÉNNDDIICCEE GG

Código Fortran del programa que codifica el algoritmo TET-ACC.

-377-

program TETACC C------------------------------------------------------------- C------------------------------------------------------------- C Este programa busca puntos a lo largo de la curva TET-ACC-- C curva que cumplen que g(S_0) es paralelo a g(T_1) -- C -- C Luisma, Alcalá 2003 -- C------------------------------------------------------------- C------------------------------------------------------------- recision R0,R1,grad,gradS0,gradT1,enerS0,enerT1 double p integer tom,zatom,atomax na parameter (atomax=100) dimension R0(3,atomax),R1(3,atomax),zatom(atomax) $,grad(3,atomax),gradS0(3,atomax),gradT1(3,atomax) call leeener(enerS0,enerT1) C-------------------------------------------------------------- C subrutina que lee la energía de un fichero .log de Gaussian98 C existen varias (distinto formato dependiendo del tipo de C cálculo. Todas están comentadas menos una C-------------------------------------------------------------- call makegrad(natom,gradS0,gradT1,grad) C--------------------------------------------------------------- C Subrutina que lee el gradiente en T_1 y S_0 desde los C ficheros .fchk de gaussian, también genera el vector C diferencia de gradientes: vector fuerza en T_1 – fuerza en S_0 C--------------------------------------------------------------- call makeNG(R0,R1,grad,gradS0,gradT1, $enerS0,enerT1,natom,zatom) C-------------------------------------------------------------- C Subrutina que crea el desplazamiento de optimización C del punto de la curva TET-ACC sobre la hiperesfera de C radio ||R1||. Además crea el fichero de salida con la C información (OUTPUT.log) más relevante. C-------------------------------------------------------------- call creaIS0(R0,R1,natom,zatom) C------------------------------------------------------- C Subrutina que crea la nueva entrada de Gaussian98 para C el nuevo cálculo del gradiente en S_0 C------------------------------------------------------- call creaIT1(R0,R1,natom,zatom) C------------------------------------------------------- C Subrutina que crea la nueva entrada de Gaussian98 para C el nuevo cálculo del gradiente en T_1 C------------------------------------------------------- END C subroutine leeener(enerS0,enerT1) CC------------------------------------------------------ C C double precision enerS0,enerT1 C character temp*9 C C101 format (a9,2x,f15.12) C open(unit=1,file='S0.log')

-378-

C open(unit=2,file='T1.log') C C11 read(1,101,err=11) temp,enerS0 C if (temp.ne.' Energy= ') then C10 go to 11 C endif C close (1) C C C20 read (2,101,err=20) temp,enerT1 C if (temp.ne.' Energy= ') then C21 go to 20 C endif C close (2) C C return C end subroutine leeener(enerS0,enerT1) C-------------------------------------------------- double precision enerS0,enerT1 character temp*9 101 format (a9,19x,f13.8) open(unit=1,file='S0.log') open(unit=2,file='T1.log') 11 read(1,101,err=11) temp,enerS0 11

if (temp.ne.' SCF Done') then go to

endif (1) close20 read (2,101,err=20) temp,enerT1 if (temp.ne.' SCF Done') then go to 20 endif close (2) return end C subroutine leeener(enerS0,enerT1) CC----------------------------------------------------- C C double precision enerS0,enerT1 C character temp*10 C C101 format(a) C102 format(33x,d15.8) C open(unit=1,file='S0.log') C open(unit=2,file='T1.log') C C11 read(1,101,err=11) temp C if (temp(6:10).ne.'DO AN') then C go to 11 C endif C read(1,102,err=11) enerS0 C close (1) C C C20 read (2,101,err=20) temp C if (temp(6:10).ne.'DO AN') then C go to 20 C endif C read(2,102,err=11) enerT1 C close (2) C C return C end

-379-

subroutine makegrad(natom,gradS0,gradT1,grad) C-------------------------------------------------- double precision cargS0,cargT1,gradS0,gradT1, $grad integer i,natom,item1,item2,atomax character null*30 parameter (atomax=100) dimension cargS0(3*atomax),cargT1(3*atomax), $grad(3,atomax),gradS0(3,atomax),gradT1(3,atomax) open(unit=1,file='S0.fchk') open(unit=2,file='T1.fchk') open(unit=3,file='chequeo') 100 format(a15,42x,i4) 101 format(a) 102 format(d16.9,d16.9,d16.9,d16.9,d16.9) 103 format(d16.9) 104 format(2d16.9) 105 format(3d16.9) 106 format(4d16.9) 107 format(3d16.9) 10 read(1,100,err=10)null,natom if(null(1:15).ne.'Number of atoms') then 10 go to

endif C20 read(1,101) null C if(null(1:29).ne.'Current cartesian coordinates') then C go to 20 C endif 40 read(1,101) null if(null(1:18).ne.'Cartesian Gradient') then 40 go to endif item1=int(3*natom/5) item2=(3*natom)-5*item1 do 0 i=1,item1 5 read(1,102,err=60) cargS0(5*(i-1)+1),cargS0(5*(i-1)+2), $cargS0(5*(i-1)+3),cargS0(5*(i-1)+4),cargS0(5*(i-1)+5) 50 continue if (item2.eq.1) then goto 61 elseif(item2.eq.2) then 62 goto elseif(item2.eq.3) then 63 goto elseif(item2.eq.4) then 64 goto endif go to 60 61 read(1,103,err=60) cargS0(item1*5+1) go to 60

-380-

62 read(1,104,err=60)cargS0(item1*5+1),cargS0(item1*5+2) go to 60 63 read(1,105,err=60)cargS0(item1*5+1),cargS0(item1*5+2), $cargS0(item1*5+3) go to 60 64 read(1,106,err=60) cargS0(item1*5+1),cargS0(item1*5+2), $cargS0(item1*5+3),cargS0(item1*5+4) go to 60 60 close(1) 70 read(2,101) null if(null(1:18).ne.'Cartesian Gradient') then 70 go to endif do 80 i=1,item1 read(2,102,err=90) cargT1(5*(i-1)+1),cargT1(5*(i-1)+2), $cargT1(5*(i-1)+3),cargT1(5*(i-1)+4),cargT1(5*(i-1)+5) 80 continue if (item2.eq.1) then 81 goto elseif(item2.eq.2) then 82 goto

elseif(item2.eq.3) then 83 goto elseif(item2.eq.4) then 84 goto endif go to 90 81 read(2,103,err=90)cargT1(item1*5+1) go to 90 82 read(2,104,err=90)cargT1(item1*5+1),cargT1(item1*5+2) go to 90 83 read(2,105,err=90)cargT1(item1*5+1),cargT1(item1*5+2), $cargT1(item1*5+3) go to 90 84 read(2,106,err=90)cargT1(item1*5+1),cargT1(item1*5+2), $cargT1(item1*5+3),cargT1(item1*5+4) go to 90 90 close(2) 110 format(d16.9,d16.9,d16.9) C gradS0 son las fuerzas en coordenadas cartesianas (cargS0) do 11 i=1,natom gradS0(1,i)=-cargS0(3*(i-1)+1) gradS0(2,i)=-cargS0(3*(i-1)+2) gradS0(3,i)=-cargS0(3*(i-1)+3) write 10) gradS0(1,i),gradS0(2,i),gradS0(3,i) (3,111 continue C gradT1 son las fuerzas en coordenadas cartesianas (cargT1)

-381-

do 12 i=1,natom gradT1(1,i)=-cargT1(3*(i-1)+1) gradT1(2,i)=-cargT1(3*(i-1)+2) gradT1(3,i)=-cargT1(3*(i-1)+3) write(3,110) gradT1(1,i),gradT1(2,i),gradT1(3,i) 12 continue C grad es la diferencia de fuerzas (T1 - S0) write(3,*) 'Gradient Difference Vector' do 13 i=1,natom grad(1,i)=gradT1(1,i)-gradS0(1,i) grad(2,i)=gradT1(2,i)-gradS0(2,i) grad(3,i)=gradT1(3,i)-gradS0(3,i) write(3,107) grad(1,i),grad(2,i),grad(3,i) 1 3 continue

write(3,*) 'S0 Gradient Vector' do i=1,natom 91 write(3,107) gradS0(1,i),gradS0(2,i),gradS0(3,i) 91 continue write(3,*) 'T1 Gradient Vector' do 92 i=1,natom write(3,107) gradT1(1,i),gradT1(2,i),gradT1(3,i) 92 continue close(3) return END subroutine makeNG(R0,R1,grad,gradS0,gradT1, $enerS0,enerT1,natom,zatom) C------------------------------------------------------- double precision normR1,R0,R1,normf,grad,normS0,normT1,gradT1, $gradS0,cosS0g,gradS0p,normS0p,cosT1g,gradT1p, $normT1p,normR,IncrR,ntemp,enerS0,enerT1,pppp,ppp,normInR, $cosSTR1 integer i,natom,zatom,atomax parameter (atomax=100) character line*80 dimension R0(3,atomax),R1(3,atomax), $zatom(atomax),grad(3,atomax),gradS0(3,atomax), $gradT1(3,atomax),gradS0p(3,atomax),gradT1p(3,atomax), $IncrR(3,atomax) open (unit=2,file='OUTPUT.log') !Fichero de salida open (unit=4,file='temp.log') !Fichero temporal de salida 30 read(2,112)line !Copia OUTPUT.log a temp.log if (line(1:3).ne.'END') then write(4,112)line go to 30 endif close(2) close(4) open (unit=2,file='OUTPUT.log') open (unit=4,file='temp.log') 31 read(4,112,end=32)line !Copia temp.log a OUTPUT.log y write(2,112)line !deja abierto este último go to 31 101 format(i2,4x,d16.9,d16.9,d16.9) 102 format(1x,a19,23x,f11.6,1x,a9) 103 format(1x,a39,3x,f11.6,1x,a18) 104 format(1x,a38) 105 format(1x,i2,6x,f11.7,2x,f11.7,2x,f11.7) C 105 format(1x,1i,6x,f11.7,2x,f11.7,2x,f11.7)

-382-

106 format 1x,a31,f11.7,a18) (107 format (1x,a35,2x,f11.5,1x,a7) 108 format(a24) 109 format(1x,a38,2x,f11.7,1x,a18) 110 format(a40) 111 format(a3) 112 format(a80) 113 format (a) 114 format (a31,1x,f11.5,1x,a8) 115 format(a30,2x,f11.5,1x,a7) 116 format(1x,a19,2x,f11.5) 117 format(d16.9,d16.9,d16.9) C118 format(d16.9) 32 close(4) open nit=1,file='R.geo') (u write (2,*) '....................................................' write (2,*) '......................R0 Vector.....................' do 1 i=1,natom read(1,101) zatom(i),R0(1,i),R0(2,i),R0(3,i) !Lee el vector R0(n)

write(2,101) zatom(i),R0(1,i),R0(2,i),R0(3,i) !Escribe en OUTPUT.log R0(n)

1 continue write (2,*) '....................................................' write (2,*) '......................R1 Vector.....................' do 2 i=1,natom read(1,101) zatom(i),R1(1,i),R1(2,i),R1(3,i) !Lee el vector R1 write(2,101) zatom(i),R1(1,i),R1(2,i),R1(3,i) !Escribe en OUTPUT.log R12 continue 1) close( write (2,*) '....................................................' do 00 i=1,natom !Escalamos los vectores 2 if (zatom(i).eq.1) then !fuerza dividiendo por la masa !ya que el desplazamiento de los goto 200

endif !núcleos es proporcional a F/m do 201 j=1,3 grad(j,i)=grad(j,i)/(zatom(i)*2) gradS0(j,i)=gradS0(j,i)/(zatom(i)*2) gradT1(j,i)=gradT1(j,i)/(zatom(i)*2) 201 continue 200 continue normR1=0 !Cálculo de la norma de R1 do 3 i=1,natom normR1=normR1+R1(1,i)**2+R1(2,i)**2+R1(3,i)**2 3 1=sqrt(normR1)

continue normR

write(2,102) 'Norm of R1 vector =',normR1, $'Angstroms' normf=0 !Cálculo norma de g = gT1- gS0 do 4 i=1,natom normf=normf+grad(1,i)**2+grad(2,i)**2+grad(3,i)**2 4 continue normf=sqrt(normf) write(2,103) 'Norm of (-)Gradient Difference Vector =', $normf,'Hartrees/Angstroms' write (2,*) 'Old Gradient Difference Vector:' do 5 i=1,natom write(2,117) grad(1,i),grad(2,i),grad(3,i) !Normalización del vector g grad(1,i)=grad(1,i)/normf grad(2,i)=grad(2,i)/normf grad(3,i)=grad(3,i)/normf 5 continue

write(2,*) ' ' !Display del vector g/||g|| write(2,104) 'Normalized Gradient Difference Vector:' !en OUTPUT.log do 6 i=1,natom

-383-

write 05) zatom(i),grad(1,i),grad(2,i),grad(3,i) (2,1continue 6

mS0=0 !Cálculo norma de gnor do 7 i=1,natom

S0

normS0=normS0+gradS0(1,i)**2+gradS0(2,i)**2+gradS0(3,i)**2 7 continue normS0=(sqrt(normS0)) write(2,106) 'Norm of (-)Gradient S0 Vector =', $normS0,'Hartrees/Angstroms' normT1=0 !Cálculo norma de gT1 do 8 i=1,natom normT1=normT1+gradT1(1,i)**2+gradT1(2,i)**2+gradT1(3,i)**2 8 continue 1=(sqrt(normT1)) normT write(2,106) 'Norm of (-)Gradient T1 Vector =', $normT1,'Hartrees/Angstroms' cosS0g=0 !Cálculo del coseno entre los do 9 i=1,natom !vectores gS0 y g cosS0g=cosS0g+grad(1,i)*gradS0(1,i)+ $grad(2,i)*gradS0(2,i)+grad(3,i)*gradS0(3,i) 9 continue cosS0g=cosS0g/normS0 print *,'cosS0g= ',cosS0g write(2,107) 'Angle Betwen -g and -g_s0 vectors =', $(acos(cosS0g)*(180.0000/3.141593)),'degrees' do 10 i=1,natom !Cálculo del vector

0

: S⊥g

gradS0p(1,i)=gradS0(1,i)-grad(1,i)*normS0*cosS0g gradS0p(2,i)=gradS0(2,i)-grad(2,i)*normS0*cosS0g gradS0p(3,i)=gradS0(3,i)-grad(3,i)*normS0*cosS0g 10 continue write(2,*) ' ' write(2,108) 'Proyected -g(S0) vector:' do 11 i=1,natom write(2,105) zatom(i),gradS0p(1,i),gradS0p(2,i), $gradS0p(3,i) 1 1 continue

normS0p=0 !Cálculo de la norma de 0S⊥g

g0Sg

0S⊥g

1q

2q

do 12 i=1,natom normS0p=normS0p+gradS0p(1,i)**2+gradS0p(2,i)**2+ $gradS0p(3,i)**2 12 continue 0p=sqrt(normS0p) normS write(2,109) 'Norm of the Proyected -g(S0) vector =', !Display de la norma de

0S⊥g

$normS0p,'Hartrees/Angstroms' cosT1g=0 !Cálculo del coseno entre los do 13 i=1,natom !vectores gT1 y g cosT1g=cosT1g+grad(1,i)*gradT1(1,i)+ $grad(2,i)*gradT1(2,i)+grad(3,i)*gradT1(3,i) 13 continue =cosT1g/normT1 cosT1g print *,'cosT1g= ',cosT1g write(2,107) 'Angle Betwen -g and -g_T1 vectors =', $(acos(cosT1g)*(180.0000/3.141593)),'degrees' do 14 i=1,natom !Cálculo del vector

1

: T⊥g

g

1q

2q

1Tg

1T⊥g

gradT1p(1,i)=gradT1(1,i)-grad(1,i)*normT1*cosT1g gradT1p(2,i)=gradT1(2,i)-grad(2,i)*normT1*cosT1g gradT1p(3,i)=gradT1(3,i)-grad(3,i)*normT1*cosT1g 14 continue write(2,*) ' ' write(2,108) 'Proyected -g(T1) Vector:' do 15 i=1,natom

-384-

write(2,105) zatom(i),gradT1p(1,i),gradT1p(2,i), $gradT1p(3,i) 1 5 continue

normT1p=0 do 16 i=1,natom !Cálculo de la norma de

1T⊥g

normT1p=normT1p+gradT1p(1,i)**2+gradT1p(2,i)**2+ $gradT1p(3,i)**2 16 continue 1p=sqrt(normT1p) normT write(2,109) 'Norm of the Proyected -g(T1) vector =', $normT1p,'Hartrees/Angstroms' C Escritura en OUTPUT.log de la diferencia de energías write(2,113) ' ' write(2,113) '--------------------------------------------' write(2,113) 'Energy:' write(2,*) 'Energy S0=',enerS0 write(2,*) 'Energy T1=',enerT1 ppp=(enerT1-enerS0)*627.5000 write(2,114) 'Energy Difference(Kcal/mol)= ',ppp $,'Kcal/mol' pppp=0 !Cálculo del coseno que forman do 24 i=1,natom ! gS0 y gT1 pppp=pppp+gradS0(1,i)*gradT1(1,i)+ $gradS0(2,i)*gradT1(2,i)+ $gradS0(3,i)*gradT1(3,i) 24 continue pppp=pppp/(normS0*normT1) C****************************************** C Chequeando Convergencia de la norma del C gradiente en E0 proyectado normR=0.020000000 C Este paso normR=0.03 funciona muy bien con el COT C en unidades atomicas C normR=0.01000000 se aumenta la cantidad de puntos if(normS0p.lt.0.001000) then do 33 i=1,natom R1(1,i)=R1(1,i)+R1(1,i)*(normR/normR1) R1(2,i)=R1(2,i)+R1(2,i)*(normR/normR1) R1(3,i)=R1(3,i)+R1(3,i)*(normR/normR1) 33 continue write(2,110) 'OPTIMIZED GEOMETRY ALONG THE TET-ACC FOUND!' write(2,110) ' GENERATING A NEW GEOMETRY ' go to 60 endif if (normS0p.lt.0.01) then if (abs(pppp).gt.0.995) then do 17 i=1,natom R1(1,i)=R1(1,i)+R1(1,i)*(normR/normR1) R1(2,i)=R1(2,i)+R1(2,i)*(normR/normR1) R1(3,i)=R1(3,i)+R1(3,i)*(normR/normR1) 17 continue write(2,110) 'OPTIMIZED GEOMETRY ALONG THE TET-ACC FOUND!' write(2,110) ' GENERATING A NEW GEOMETRY ' go to 60 endif endif C Fin del chequeo de la convergencia de de los vectores C proyectados gradiente en E0 y E1 C****************************************** C Formula de ajuste para alpha C alpha=1/(20*((1-abs(pppp))**2))

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C alpha=1.0000000 C Con la siguiente formula se obtiene un valor de C alpha=0.4 para cos(pppp)=0.98 y un valor de C alpha=1.0 para cos(pppp)=0.99 y un valor de C alpha=2.5 para cos(pppp)=0.995 if(abs(pppp).lt.0.98) then !Cálculo del factor de alpha=0.4 !desplazamiento: alpha else alpha=4*(0.002288/((1-abs(pppp))**(1.32))) endif C el factor de 4 esta para incrementar la rapidez write(2,116) 'Alpha coefficient=',alpha do 18 i=1,natom IncrR(1,i)=alpha*(gradS0p(1,i)+gradT1p(1,i)) IncrR(2,i)=alpha*(gradS0p(2,i)+gradT1p(2,i)) IncrR(3,i)=alpha*(gradS0p(3,i)+gradT1p(3,i)) 18 continue

( )0nR 1R

( )0 1S Tα ⊥ ⊥+g g1′=IncrR R

normInR=0 do 26 i=1,natom normInR=normInR+IncrR(1,i)**2+IncrR(2,i)**2+ $IncrR(3,i)**2 26 continue normInR=sqrt(normInR) cosSTR1=0 do 25 i=1,natom cosSTR1=cosSTR1+IncrR(1,i)*R1(1,i)+ $ IncrR(2,i)*R1(2,i)+IncrR(3,i)*R1(3,i) 25 continue cosSTR1=cosSTR1/(normR1*normInR) do 27 i=1,natom IncrR(1,i)=IncrR(1,i)-R1(1,i)*cosSTR1*normInR/normR1 IncrR(2,i)=IncrR(2,i)-R1(2,i)*cosSTR1*normInR/normR1 IncrR(3,i)=IncrR(3,i)-R1(3,i)*cosSTR1*normInR/normR1 27 continue write(2,*) ' ' write(2,*) ' Geometry Desplacement:' do i=1,natom 19 write(2,105) zatom(i),IncrR(1,i)-R1(1,i),IncrR(2,i)-R1(2,i), $IncrR(3,i)-R1(3,i) 1 9 continue

do 20 i=1,natom R1(1,i)=R1(1,i)+IncrR(1,i) R1(2,i)=R1(2,i)+IncrR(2,i) R1(3,i)=R1(3,i)+IncrR(3,i) 20 continue mp=0 nte do 21 i=1,natom ntemp=ntemp+R1(1,i)**2+R1(2,i)**2+R1(3,i)**2 21 continue ntemp=sqrt(ntemp) do 22 i=1,natom R1(1,i)=R1(1,i)*(normR1/ntemp) R1(2,i)=R1(2,i)*(normR1/ntemp) R1(3,i)=R1(3,i)*(normR1/ntemp) 22 continue

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write (2,*) ' New Geometry:' do i=1,natom 23 write(2,105) zatom(i),(R0(1,i)+R1(1,i)) $,(R0(2,i)+R1(2,i)),(R0(3,i)+R1(3,i)) 23 continue C Creando la nueva geometria 60 open(unit=1,file='R.geo') do 61 i=1,natom write(1,101) zatom(i),R0(1,i),R0(2,i),R0(3,i) 61 continue do 62 i=1,natom write(1,101) zatom(i),R1(1,i),R1(2,i),R1(3,i) 62 continue close(1) write(2,115) 'Angle Betwen g(E1) and g(E0) =', $(acos(pppp)*(180.000/3.14159)),'degrees' write(2,113)'----------------------------------------------------- $-----------------------------------' write(2,113)'--TETACC--TETACC--TETACC--TETACC--TETACC--TETACC--TET $ACC--TETACC--TETACC--TETACC--TETACC' write(2,113)'--TETACC--TETACC--TETACC--TETACC--TETACC--TETACC--TET $ACC--TETACC--TETACC--TETACC--TETACC' write(2,113)'----------------------------------------------------- $-----------------------------------' write(2,111)'END' write(2,*)' ' close(2) C Fin de la cracion de la nueva geometria return END subroutine creaIS0(R0,R1,natom,zatom) C------------------------------------------------------- recision R0,R1,geom double p integer natom,zatom,atomax parameter (atomax=100) dimension R0(3,atomax),R1(3,atomax),zatom(atomax), $geom(3,atomax) do 2 i=1,natom geom(1,i)=R0(1,i)+R1(1,i) geom(2,i)=R0(2,i)+R1(2,i) geom(3,i)=R0(3,i)+R1(3,i) 2 continue 10 format(a) 11 format(15x,i1,20x,d16.9,2x,d16.9,2x,d16.9) open (unit=1,file='S0.com') write(1,10) '%chk=S0' write(1,10) '%mem=120Mb' C write(1,10) '#p CASSCF(8,8,NoCPMCSCF,NRoot=1)/6-31g* Nosymm ' C write(1,10) '#p CASSCF(8,8,NoCPMCSCF,NRoot=1)/sto-3g Nosymm ' write(1,10) '#p B3Lyp/sto-3g Nosymm ' C write(1,10) '#p PM3 Nosymm ' write(1,10) '# guess=read SCF=(conver=8,MAxCycle=300)' write(1,10) '# force units=AU'

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write(1,10) ' ' write(1,10) 'S0' write(1,10) ' ' write(1,10) '0 1' do 1 i=1,natom write(1,11) zatom(i),geom(1,i),geom(2,i),geom(3,i) 1 continue write(1,10) ' ' close (1) return END subroutine creaIT1(R0,R1,natom,zatom) C------------------------------------------------------- recision R0,R1,geom double p integer tom,zatom,atomax na parameter (atomax=100) dimension R0(3,atomax),R1(3,atomax),zatom(atomax), $geom(3,atomax) do 2 i=1,natom geom(1,i)=R0(1,i)+R1(1,i) geom(2,i)=R0(2,i)+R1(2,i) geom(3,i)=R0(3,i)+R1(3,i) 2 continue 10 format(a) 11 format(15x,i1,20x,d16.9,2x,d16.9,2x,d16.9) open (unit=1,file='T1.com') write(1,10) '%chk=T1' write(1,10) '%mem=120Mb' C write(1,10) '#p CASSCF(8,8,NoCPMCSCF,NRoot=1)/6-31g* Nosymm ' C write(1,10) '#p CASSCF(8,8,NoCPMCSCF,NRoot=1)/sto-3g Nosymm ' C write(1,10) '#p PM3 Nosymm ' write(1,10) '#p B3Lyp/sto-3g Nosymm ' write(1,10) '# guess=read SCF=(conver=8,MAxCycle=300)' write(1,10) '# force units=AU' write(1,10) ' ' write(1,10) 'T1' write(1,10) ' ' write(1,10) '0 3' do i=1,natom 1 write(1,11) zatom(i),geom(1,i),geom(2,i), $geom(3,i) 1 continue write(1,10) ' ' close (1) return END

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AAPPÉÉNNDDIICCEE HH

Separatas de publicaciones referidas a la Tesis

H1. Org. Lett. 6, 1229 (2004) 391 H2. Angew. Chem., Int. Ed. in Eng. 39, 2095 (2000). 395 H3. Int. J. Quant. Chem., 86, 422 (2002). 399 H4. J. Am. Chem. Soc. 124, 13770 (2002) 403 H5. J. Comp. Chem. 23, 732 (2002). 423 H6. J. Phys. Chem. A. 107, 5472 (2003) 429 H7. J. Chem. Phys. 120, 1208 (2004) 437

Triplet versus Singlet PhotoreactionMechanism in the BarreleneDi-π-methane RearrangementLuis M. Frutos,* Unai Sancho, and Obis Castano

Departamento de Quı´mica Fısica, UniVersidad de Alcala´ , 28871 Alcala´ de Henares,Madrid, Spain

luisma.frutos@uah.es

Received January 2, 2004

ABSTRACT

A detailed study of the potential energy surfaces involved in the di-π-methane rearrangement (singlet and triplet states) reveals the factorsthat modulate the mechanisms (pathways) and reactivity in strained di-π-methane systems such as bicyclo[2.2.2]octa-2,5,7-triene (barrelene).

Since 1966, when Zimmerman and Grunewald reported thisreaction in bicyclo[2.2.2]octa-2,5,7-triene (barrelene),1 thedi-π-methane rearrangement (DPMR) had become an im-portant topic in organic photochemistry, which today stillcontinues to be an active field of research.2 Typically, DPMRtakes place when a molecule with two vinyl groups orequivalentπ-moieties (i.e., aryl groups) linked to the samesp3 carbon atom is irradiated to yield a vinylcyclopropane(Figure 1). Two possible mechanisms have been describedfor the di-π-methane rearrangement (Figure 1). In the presentcommunication, we advance the results of a theoretical studyon the photoreactivity of barrelene by direct as well as triplet-sensitized irradiation. Different reaction mechanisms as wellas their relative probabilities, and some of the factorscontrolling them, are discussed.

The irradiation of barrelene in the presence of a tripletsensitizer leads to the formation of the di-π-methane rear-rangement product with a high yield,3a while the directphotolysis of the molecule follows a different path (seebelow).

The mechanism proposed by Zimmerman,1,3 in which abiradical species (BR2) is an intermediate of the reaction, is

(1) Zimmerman, H. E.; Grunewald, G. L.J. Am. Chem. Soc.1966, 88,183-184.

(2) For a recent review see: Zimmerman, H. E.; Armesto, D.Chem.ReV. 1996, 96, 3065-3112.

(3) (a) Zimmerman, H. E.; Binklet, R. W.; Givens, R. S.; Sherwin, M.A. J. Am. Chem. Soc.1967, 98, 393. (b) Zimmerman, H. E.; Mariano, P.S. J. Am. Chem. Soc.1969, 91, 1718. (c) Zimmerman, H. E. InRearrangements in Ground and Excited States; de Mayo, P., Ed.; AcademicPress: London, 1980; Vol. 3, p 131.

Figure 1. (i) Zimmerman mechanism1,3 and (ii) Bernardi and Robbmechanism4,5 for the di-π-methane rearrangement in 1,3-pentadiene.The 1,4- (BR2) and 1,3-biradicals (BR3) are reaction intermediatespecies in the Zimmerman mechanism. BR3 is the biradical on S0formed in the nonadiabatic reaction involving a tetraradicaloid (S1/S0) conical intersection in the Bernardi and Robb mechanism.

ORGANICLETTERS

2004Vol. 6, No. 81229-1231

10.1021/ol049977p CCC: $27.50 © 2004 American Chemical SocietyPublished on Web 03/12/2004

strongly supported by theoretical results presented here, basedon the detailed study of the SEPs of electronic states in-volved in the reaction6 at the CASSCF(8,8)/6-31g*7 andCASPT2(8,8)/6-31g*8 levels of theory. The reaction takesplace in four steps (Figure 2).

In the first step, after the triplet-sensitized excitation, thebarrelene on T1 yields the 1,2-biradical (BR1). This BR1(minimum on T1) does not bear the orthogonalp orbitalspresent in analogous compounds without the structuralrestrictions of barrelene. An intersystem crossing (ISC) pointT1/S0

9a corresponding to this situation is located 39 kcal/mol above the BR1 in the PES.9b Hence, because of itsinstability; this ISC is not an effective channel of deactivationto the ground state, which, on the other hand, will recoverbarrelene.

After the formation of BR1, and since any biradical specieswould tend to adopt a structure that minimizes electron

repulsion, the molecule evolves in a first step, to the 1,3-bi-radical (BR2), and afterward to an 1-allyl,3-biradical (BR3),the most stable of all of them. These two biradicals werealso identified as minima in the PES of naphtho-barrelenes.10

As the system enters the 1,3-biradical region, the energy ofT1 and S0 becomes very close, giving rise to a quasidegen-erated situation (Figure 3).

The two-half-filled orbitals containing the unpaired elec-trons in BR3 are nearly perpendicular, while those in BR2are parallel. This could result in a much higher ISC rate forBR3 than for BR2, due to the angular dependence of thespin-orbit coupling (SOC) energy⟨T1|HSO|S0⟩, as illu-strated by the case of the trimethylene biradical,11 and morerecently for barrelene derivatives in a DPMR.10 Therefore,the surface crossing to the ground state would be clearlymuch more efficient at the BR3 region, not only becausethis is the most stable species, but also because of theenhanced SOC. Since the biradical BR3 is not a stableintermediate on the S0 surface (Figure 3), the fast pairing ofthe two electrons finally yields semibullvalene (SBV) as theonly reaction product.

On the other hand, direct irradiation of barrelene into thefirst absorption band at 239 nm12 populates the 11E′′electronic state, which presents a multiconfigurational wavefunction corresponding essentially to the HOMOf LUMOmonoexcitation, as calculated at the CASPT2(8,8)/6-31-g*level of theory,8 being the calculated transition energy (11A′1f 11E′′) 242 nm.

(4) Reguero, M.; Bernardi, F.; Jones, H.; Olivucci, M.; Ragazos, I. N.;Robb, M. A.J. Am. Chem. Soc.1993, 115, 2073-2074.

(5) Zimmerman has also noted that BR2 and BR3 are not necessarilyminima on the PES; see for example: Zimmerman, H. E.; Sulzbach, H.M.; Tollefson, M. B.J. Am. Chem. Soc.1993, 115, 6548.

(6) Analytical frecuencies, minimum energy paths, and stationary pointoptimizations were carried out with Gaussian 98, revision A.7 (see ref 7).Single-point calculations were carried out with MOLCAS 5.0 (see ref 8)

(7) Frisch, M. J.; Trucks, G. W.; Schlegel, H. B.; Scuseria, G. E.; Robb,M. A.; Cheeseman, J. R.; Zakrzewski, V. G.; Montgomery, J. A., Jr.;Stratmann, R. E.; Burant, J. C.; Dapprich, S.; Millam, J. M.; Daniels, A.D.; Kudin, K. N.; Strain, M. C.; Farkas, O.; Tomasi, J.; Barone, V.; Cossi,M.; Cammi, R.; Mennucci, B.; Pomelli, C.; Adamo, C.; Clifford, S.;Ochterski, J.; Petersson, G. A.; Ayala, P. Y.; Cui, Q.; Morokuma, K.; Malick,D. K.; Rabuck, A. D.; Raghavachari, K.; Foresman, J. B.; Cioslowski, J.;Ortiz, J. V.; Stefanov, B. B.; Liu, G.; Liashenko, A.; Piskorz, P.; Komaromi,I.; Gomperts, R.; Martin, R. L.; Fox, D. J.; Keith, T.; Al-Laham, M. A.;Peng, C. Y.; Nanayakkara, A.; Gonzalez, C.; Challacombe, M.; Gill, P. M.W.; Johnson, B. G.; Chen, W.; Wong, M. W.; Andres, J. L.; Head-Gordon,M.; Replogle, E. S.; Pople, J. A.Gaussian 98, revision A.7; Gaussian,Inc.: Pittsburgh, PA, 1998.

(8) CASPT2 calculations were made with:MOLCAS, version 5; Ander-sson, K.; Blomberg, M. R. A.; Fu¨lscher, M. P.; Karlstro¨m, G.; Lindh, R.;Malmqvist, P.-A° .; Neogrady, P.; Olsen, J.; Roos, B. O.; Sadlej, A. J.; Schu¨tz,M.; Seijo, L.; Serrano-Andre´s, L.; Siegbahn, P. E. M.; Widmark, P.-O.;Lund University: Lund, Sweden, 1999.

(9) (a) This ISC topology is responsible for the cis-trans isomerizationin acyclic compounds. (b) An algorithm developed by our laboratory forlocating minima in the intersystem crossing region working in conjunctionwith energy gradient calculated with Gaussian 98 (see ref 7) was used tolocate minima on the T1/S0 crossing subspace.

(10) Zimmerman, H. E.; Kutateladze, A. G.; Maekawa, Y.; Mangette,E. J. Am. Chem. Soc.1994, 116, 9795.

(11) Carlacci, L.; Doubleday: C.; Furlany, T. R.; King, H. F.; McIver,J. W. J. Am. Chem. Soc.1987, 109, 5323.

(12) Gedanken, A.; Meijere, A. deJ. Chem. Phys.1988, 88, 4153-4157.

Figure 2. Photoreaction of barrelene by triplet-sensitized irradia-tion, leading to the di-π-methane rearrangement (species in bracketsare not minima in the PES).

Figure 3. Schematic potential energy surface regions of T1 andS0 involved in the sensitized di-π-methane reaction. The most stablebiradical (BR3) and the intersystem crossing (ISC) region are alsoindicated, as well as the semibullvalene (SBV) photoproduct.

1230 Org. Lett., Vol. 6, No. 8, 2004

After the Franck-Condon (FC) excitation, the systemrelaxes vibrationally, reaching a conical intersection region(CoInR), rather than a conical intersection point (CoIn),corresponding to the [2+ 2] cycloaddition CoIn topology.13

No intermediate is reached in the relaxation on S1, and theS1fS0 radiationless transition occurs via a CoIn in the timescale of a few molecular vibrations, supporting the ultrafastcharacter of this reaction. Passing the CoInR, the systemevolves on the ground state through several intermediates(Figure 4) to yield 1,3,5,7-cyclooctatetraene (COT).

In the case of molecular systems free of the barrelenestructural restrictions (usually acyclic compounds), it hasbeen proposed from theoretical arguments that the DPMRmust involve decay to ground state via a CoIn.4 This CoIn,the most stable one, corresponds to an ethylenic rearrange-ment that yields the DPMR product.4 The same type of CoInis found in the barrelene potential energy surface (PES), andin fact, it is the most stable CoIn (in 19 kcal/mol). This CoInwould yield the di-π-methane rearrangement product, in aconcerted process (SBV). This CoIn, however, cannot bereached from the FC region on S1, despite its stability, andthe [2 + 2] cycloaddition CoInR would be the preferred

crossing (S1/S0) space, followed by the relaxation of thesystem to the ground state.

Comparison of the results described with those of ref 4points out the similar relative stability of the CoIns in systemsas different as 1,4-pentadiene and barrelene, due to the largestructural and electronic similarity between the CoIns in bothcompounds. Therefore, despite the structural differencesexisting between the two (di-π-methane) molecular systems,the photochemical reactivity by direct irradiation cannot beexplained on the basis of the differences in the CoIn stability.

On the contrary, the reactivity will be controlled by theCoIn region that can be accessed after the FC transition. Inthis way, if the molecule in the ground state presents twoneighbor ethylene groups, after the FC transition the systemwould most probably reach the [2+ 2] cycloaddition CoInR.On the other hand, if the system presents conformationalflexibility in such a way that the molecule in the ground-state preserves the separation of the two ethylene bonds, thedi-π-methane or the cis-trans isomerization CoInR couldbe accessible. Hence, the photochemical reactivity by directirradiation must be controlled mainly by the properties ofS0, and not by those of the excited state (S1), because theconformational stability on the ground state would determinethe CoInR reached after the FC transition, and consequently,the photoproducts obtained. Therefore, it should be possibleto control the photoproducts by designing, for example, thesubstituents of a di-π-methane system in order to optimize,in the S0 state, the desired conformation.

In conclusion, our theoretical study supports the Zimmer-man mechanism for the sensitized reaction of barrelene. Thismechanism must be common to many di-π-methane rear-rangements in molecules with structural restrictions such asbarrelene, in which the cis-trans isomerism is hindered and,therefore, cannot be an effective mechanism for relaxationto the ground state. The photoproducts are controlled by thestability of the biradical species with the higher SOC, theBR3, and it is not expected to depend on the FC transitionregion. In contrast, for the direct photoreaction, the photo-products must be determined by the FC transition region,and the stability of the CoIns should not play any importantrole.

Acknowledgment. We thank Professors A. Ulises Acun˜a,Julio AÄ lvarez-Builla, and Diego Armesto for helpful sug-gestions. Work was financed by Project BQU2000/0646 fromthe Spanish Ministry of Science and Technology (MCyT).L.M.F. is grateful to the Ministry of Education, Culture andSport (Spain) and University of Alcala´ for a DoctoralFellowship.

OL049977P

(13) (a) Gerhartz, W.; Poshusta, R. D.; Michl, J.J. Am. Chem. Soc.1977,99, 4263. (b) Olivucci, M.; Ragazos, I. N.; Bernardi, F.; Robb, M. A.J.Am. Chem. Soc.1993, 115, 3710. (c) Olivucci, M.; Bernardi, F.; Ottani, S.;Robb, M. A.J. Am. Chem. Soc.1994, 116, 2034.

Figure 4. Photochemistry of barrelene after direct irradiation. (a)Mechanism allowed by [2+ 2] cycloaddition and (b) forbiddendi-π-methane rearrangement mechanism (species in brackets arenot minima on the PES). (BOT) bicyclo[4.2.0]octa-2,4,7-triene,TCO ) tetracyclo[4.2.0.0.2,805,7]octene.)

Org. Lett., Vol. 6, No. 8, 2004 1231

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Note on the Theory of Bifurcation ofChemical Reactions

RAUL PALMEIRO, LUIS MANUEL FRUTOS, OBIS CASTAÑODepartamento de Química Física, Universidad de Alcalá, 28871 Alcalá de Henares (Madrid), Spain

Received 23 April 2001; revised 28 May 2001; accepted 28 June 2001

ABSTRACT: In the present communication two existing definitions of valley ridgeinflection, (VRI), point, one attributed to Valtazanos and Ruedenberg and the other one toBasilevsky are discussed. From our study we conclude that both definitions are notequivalent, and we demonstrate in which cases it occurs. However, they are used withoutany distinction in the bibliography. © 2002 John Wiley & Sons, Inc. Int J Quantum Chem86: 422–425, 2002

Key words: theory of bifurcation; valley ridge inflection point; second-order Jahn–Tellereffect; intrinsic reaction coordinate; potential energy surface

Introduction

A ccording to the Encyclopedia of ComputationalChemistry [1], valley ridge inflection (VRI)

points are defined as those where a projected fre-quency is equal to zero. This definition of a VRIpoint is attributed to Valtazanos and Ruedenberg[2]. However, it was really Basilevsky who, in anearlier work [3], introduced the correct definition ofVRI point. In that work, Basilevsky added precisionto this novel concept, using the results of the rig-orous mathematical formulation given by Hofacker[4] and Marcus [5] for the natural reaction coordi-nate(s) (NRC). These natural reaction coordinate(s){qi}i = 1,...,n are more suitable for a geometrical in-

Correspondence to: O. Castaño; e-mail: ocastano@iqfr.csic.es.Contract grant sponsor: D.G.E.S.Contract grant number: BQU2000-0646.

terpretation of the potential energy surface (PES)across the reaction path because of the rather sim-ple form of the NRC Hamiltonian and because theseare just the coordinates in which the Hamiltonianhas a clear physical interpretation. In other work,Basilevsky [6] extended the NRC construction tochemical reactions that involved polyatomic mole-cules.

Let us consider the curvilinear coordinates of thereal space Rn formed by the s parameter of the IRCcurve {xi = γi(s)} passing through each point of thePES and by the parameters of a surface orthogo-nal to that curve which defines the subspace Rn−1

that will be named by S. The first partial deriva-tives of the energy with respect to the curvilinearcoordinates of S are equal to zero while the eigen-values of the Hessian of the energy restricted tothe coordinates of S are invariant with respect toa coordinate change in this subspace S. The newcoordinates {qi}i = 2,...,n of S obey ∂2E/∂qi∂qj = 0, if

International Journal of Quantum Chemistry, Vol. 86, 422–425 (2002)© 2002 John Wiley & Sons, Inc.DOI 10.1002/qua.1109

BIFURCATION OF CHEMICAL REACTIONS

i �= j, with q1 defined as the s parameter form of theNRC. Thus, the eigenvalues of this Hessian, calledhenceforth the Hessian projected onto S, character-ize the nonstationary points of the reaction path asif they were stationary points. For the case of a VRIpoint, where there is at least one zero eigenvalue,Basilevsky emphasized the importance of studyingthis Hessian in the subspace S instead of studying itin the full space.

In their work Valtazanos and Ruedenberg [2]defined a VRI point as a nonstationary pointwhere the Hessian has an eigenvector orthogonalto the gradient with a zero eigenvalue. They re-stricted themselves to the particular case of the ringopening of the cyclopropylidene to allene whena breaking of symmetry occurs. Of course, theVRI point definition of Valtazanos and Ruedenbergsatisfies the more general conditions imposed byBasilevsky.

We will show here that, when a symmetry break-ing take place at VRI points, both definitions of VRIpoint coincide, i.e., a zero eigenvalue exists in bothHessians, in the Hessian of the full space and inthat of the subspace S. However, in the other caseswithout any symmetry breaking, VRI points obey-ing the Basilevsky’s definition may exist while thereis no zero eigenvalue of the Hessian in the full space.Moreover, it is possible to find points correspondingto a zero eigenvalue of the full Hessian, which, how-ever are not VRI points when Basilevsky’s criterionis applied.

As we shall show below, only when the eigenvec-tor of the projected Hessian with a zero eigenvalueis orthogonal to the IRC, the VRI point will fulfillnecessarily the criterion proposed by Valtazanosand Ruedenberg. It is, therefore, important tounderstand that Basilevsky’s criterion is the mostgeneral one when studying the theory of bifurcationof chemical reactions.

Method

One method for obtaining the directions of thecurves whose parameters are the NRC {qi}i = 2,...,n

belonging to the plane orthogonal to the IRC is todiagonalize the matrix Hp = (1 − Pg)H(1 − Pg) [7],where Pg is the projector onto the gradient. One ofthe eigenvectors of Hp is the gradient itself, the triv-ial eigenvector with a zero eigenvalue, while the re-maining ones have the directions of the parametriccurves. Up to date, a demonstration of the neces-sary and sufficient conditions for the coincidence of

both definitions of the VRI point, the Basilevsky’sand Valtazanos–Ruedenbeg’s does not exist. Herewe shall develop such a demonstration.

We know, by definition, that

d2E(x0)(v, w) =∑

i,j

(∂2E

∂xi∂xj

)0viwj. (1)

Let {wi}ni = 1 be an orthonormal basis set of

the n dimensional vectorial space, such thatd2E(x0)(wi, wj) = 0 if i �= j. This implies that{wi}n

i = 1 are the eigenvectors of the Hessian onto thefull space and d2E(x0)(wi, wi) are, respectively, theireigenvalues. On the other hand, let {vi}n

i = 1 be an-other orthonormal basis set of the same space suchthat v1 = g/|g| and d2E(x0)(vi, vj) = 0 if i �= j fori, j > 1. That is, {vi}i = 2,...,n are the nontrivial eigen-vectors of the Hessian Hp and {d2E(x0)(vi, vi)}n

i = 2are, respectively, their eigenvalues. The transformedHessian in the basis set {vi}n

i = 1 will be definedas Hnc. In this way, we can express Hnc as a func-tion of the eigenvalues of H in the VRI point x0 asfollows:

d2E(x0)(vi, vj)

= d2E(x0)

(n∑

k = 1

〈wk|vi〉wk,n∑

h = 1

〈wh|vj〉wh

)

=n∑

k = 1

n∑h = 1

〈wk|vi〉〈wh|vj〉 d2E(x0)(wk, wh)

=n∑

k = 1

〈wk|vi〉〈wk|vj〉 d2E(x0)(wk, wk). (2)

Considering the orthonormality of the basis sets, thefollowing holds:

n∑i = 1

〈wk|vi〉〈wh|vi〉 = δkh.

Hence, multiplying (2) by 〈wh|vi〉, and summingover i, the following expression is obtained:

n∑i = 1

〈wh|vi〉 d2E(x0)(vi, vj)

=n∑

k = 1

d2E(x0)(wk, wk)〈wk|vj〉(

n∑i = 1

〈wk|vi〉〈wh|vi〉)

= d2E(x0)(wh, wh)〈wh|vj〉. (3)

By definition of the basis set {vi}ni = 1 for i, j > 1

the expression d2E(x0)(vi, vj) is equal to zero wheni �= j, while it is equal to the eigenvalue of theeigenvector vi of Hp for i = j > 1. For i = 1 orj = 1, the expression d2E(x0)(vi, vj) is not neces-

INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY 423

PALMEIRO, FRUTOS, AND CASTAÑO

sarily equal to zero. According to this expression,(3) reduces to the following forms depending on thevalue of j:

n∑i = 1

〈wh|vi〉d2E(x0)(vi, v1)

= d2E(x0)(wh, wh)〈wh|v1〉〈wh|v1〉d2E(x0)(v1, vj) + 〈wh|vj〉d2E(x0)(vj, vj)

= 〈wh|vj〉 d2E(x0)(wh, wh)j > 1, h = 1, . . . , n. (4)

Since a nondegenerated VRI point is concerned,there exists one and only one eigenvector v2 of theHessian Hp with an eigenvalue equal to zero, i.e.,d2E(x0)(v2, v2) = 0.

With these constraint Eqs. (4) for j = 2 and h =1, . . . , n simplifies to

〈wh|v1〉 d2E(x0)(v1, v2) = d2E(x0)(wh, wh)〈wh|v2〉,h = 1, . . . , n. (5)

Using these Eq. (5), it is possible to know the nor-mal modes wi composing v2. These normal modescan be divided in two disjoint sets, respectively,orthogonal and nonorthogonal to v1. The first setcan or cannot break symmetry. The second onescompose v1, and since v1 pertains to the totally sym-metric representation, these normal modes mustpertain also to the totally symmetric representations(normal modes that do not break symmetry cannotbe obtained by adding vectors of representationsnontotally symmetric because they are linearly in-dependent).

Let us now suppose that there exists a vector wi

composing v2 (〈wi|v2〉 �= 0) orthogonal to v1, i.e.,〈wi|v1〉 = 0. According to (5), for all these vectorswi we have d2E(x0)(wi, wi) = 0, i.e., H posses a zeroeigenvalue also. Relating to Eqs. (4) for these vectorswi and for every j > 2, it is obtained that

〈wi|vj〉 = −〈wi|v1〉 d2E(x0)(v1, vj)d2E(x0)(vj, vj)

= 0, (6)

because d2E(x0)(vj, vj) �= 0 by hypotesis.Thus, wi is the same v2, or in other words, in the

case that v2 has any component wi orthogonal to v1,it should be the only component and its eigenvalueshould be equal to zero. In this case, v2 coincideswith a normal mode that can or cannot break sym-metry and both definitions of the VRI coincide.

The other remaining option is to take v2 as a lin-ear combination of several wi where none of themis orthogonal to v1 (〈wi|v1〉 �= 0), i.e., v2 would bea linear combination of several normal modes of to-

tally symmetric representations, that do not breakthe symmetry of the molecule. In this case, H doesnot necessarirly have a zero eigenvalue.

In this last case when v2 is a linear combinationof some normal modes, the necessary and sufficientcondition in order for the Hessian H to have at leastone zero eigenvalue is that d2E(x0)(v1, v2) = 0.

To verify that this condition is necessary, let ussee what happens if d2E(x0)(v1, v2) = 0. In this case,following (5) d2E(x0)(wh, wh)〈wh|v2〉 = 0, ∀h ≥ 1.Therefore, for all the eigenvectors wi of v2, it isfound that d2E(x0)(wi, wi) = 0, that is, its eigenval-ues are equal to zero. This way, if v2 is formed byonly one eigenvector wi(wi = v2), then H will nec-essarily have, at least one zero eigenvalue while ifit is a linear combination of some wi nonorthogo-nal to v1 the matrix H will be degenerated for thezero eigenvalue. It can be proven that the maximumdegeneration degree is 2 as it follows from the nextlemma.

LEMMA

Let m be the number of nontrivial eigenvectorsof Hp = (1 − Pg)H(1 − Pg) with zero eigenvalue(the eigenvector in the direction along the gradientis excluded). Then, the maximum degeneration ofthe zero eigenvalue of H will be equal m + 1.

In the basis set {vi}ni = 1 every wi can be expanded

as wi = ∑nj = 1〈wi|vj〉vj. Let us consider a point of the

PES where wi obeys the equation d2E(x0)(wi, wi) = 0and let be M = {j > 1 ∈ N | d2E(x0)(vj, vj) = 0} beingm = Card(M). Substituting in (4) for every j /∈ Mand j > 1 the following expression is obtained:

〈wi|vj〉 = −〈wi|v1〉 d2E(x0)(v1, vj)d2E(x0)(vj, vj)

. (7)

Thus, wi can be expressed as

wi =n∑

j = 1

〈wi|vj〉vj = 〈wi|v1〉v1 +∑j∈M

〈wi|vj〉vj

−∑j/∈M

〈wi|v1〉 d2E(x0)(v1, vj)d2E(x0)(vj, vj)

vj

= 〈wi|v1〉(

v1 −n∑

j/∈M

vjd2E(x0)(v1, vj)d2E(x0)(vj, vj)

)

+∑j∈M

〈wi|vj〉vj. (8)

This equation shows that the vectors wi with zeroeigenvalues in a point with m nontrivial eigenvec-tors vi with zero eigenvalues are linear combina-tions of m + 1 orthogonal vectors. Hence, we can

424 VOL. 86, NO. 5

BIFURCATION OF CHEMICAL REACTIONS

obtain a maximum of m + 1 orthogonal vectors wi

giving rise to a maximum degeneration of the zeroeigenvalue of m + 1 order.

In the VRI point x0, Hp has only one nontrivialzero eigenvalue. According to the lemma demon-strated above, the possible maximum degenerationof a zero eigenvalue of H is equal to 2. Thus, ifd2E(x0)(v1, v2) = 0 and v2 is composed of severalnormal modes, then it can only be a linear combina-tion of two of them. On the contrary, each of themwould have a zero eigenvalue, leading to a contra-diction with the previous lemma.

In the case where v2 is a linear combination oftwo normal modes with zero eigenvalues, it is easyto prove that d2E(x0)(v1, v2) = 0. Using Eq. (5) forevery wi with zero eigenvalue composing v2, it fol-lows that d2E(x0)(v1, v2) = 0.

It is thus shown that when d2E(x0)(v2, v2) = 0 andv2 is a linear combination of two normal modes,i.e., the symmetry is not broken, the conditiond2E(x0)(v1, v2) = 0 is a necessary and sufficient con-dition for both above-considered definitions of theVRI point to be equivalent, and H has a twofold de-generated zero eigenvalue.

In the study of Ramquet et al. [8] a PES thatpresents a VRI point without any zero Hessianeigenvalue has been reported (see Fig. 9, p. 4931).At this VRI point a symmetry breaking does notoccur because all the points on this PES own thesame symmetry. This example supports our above-demonstrated mathematical statement about themeaningless of the Ruedenberg definition of VRIpoint when the symmetry has remained in theVRI point along the NRC q2 with zero curvature[d2E(x0)(v2, v2) = 0].

Conclusions

The two definitions of the VRI point involved inour discussion were the following:

1. A nonstationary point where the Hessian ofthe energy has an eigenvector perpendicularto the gradient with zero eigenvalue.

2. A nonstationary point where the Hessian ofthe energy projected onto the subspace S or-thogonal to the gradient has a zero eigen-value.

From our study we can conclude that these twodefinitions are not equivalent, in spite of the usein the bibliography without any distinction. Theycan coincide in some cases, but this is not a gen-eral rule. They always coincide when the systembreaks symmetry in the VRI point. However, whenthe system does not break symmetry they coincideif ∂2E/∂q1∂q2 = 0, where q2 is the NRC that satis-fies the condition ∂2E/∂q2

2 = 0, and in such case, thezero eigenvalue of the Hessian of the full space istwofold degenerate.

ACKNOWLEDGMENTS

This work was supported by Grant BQU2000-0646 of the Spanish D.G.E.S.

References

1. Encyclopedia of Computational Chemistry; Schleyer, P. v. R.,Editor-in-chief; Wiley: New York, 1998; p. 2436.

2. Valtazanos, P.; Ruedenberg, K. Theor Chim Acta 1986, 69,281.

3. Basilevsky, M. V. Chem Phys 1977, 24, 81.

4. Hofacker, G. L. Z Naturforsch 1963, A18, 607.

5. Marcus, R. A. J Chem Phys 1968, 49, 2610, 2617.

6. Basilevsky, M. V. Theor Chim Acta 1987, 72, 63.

7. Schlegel, H. B. J Chem Soc Faraday Trans 1994, 90, 1569.8. Ramquet, M.-N.; Dive, G.; Dehareng, D. J Chem Phys 2000,

112, 4923.

INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY 425

Cyclooctatetraene Computational Photo- and ThermalChemistry: A Reactivity Model for Conjugated Hydrocarbons

Marco Garavelli,*,† Fernando Bernardi,*,† Alessandro Cembran,† Obis Castano,‡

Luis Manuel Frutos,‡ Manuela Merchan,§ and Massimo Olivucci*,|

Contribution from the Dipartimento di Chimica “G. Ciamician”, UniVersita di Bologna,Via Selmi 2, 40126 Bologna, Italy, Departamento de Quı´mica Fısica, UniVersidad de Alcala´ ,

28871 Alcala´ de Henares (Madrid), Spain, Departamento de Quı´mica Fısica,Instituto de Ciencia Molecular, Dr. Moliner, 50, 46100 Burjassot, Valencia, Spain,

and Istituto di Chimica Organica, UniVersita di Siena, Via Aldo Moro, I-53100 Siena, Italy eCentro per lo Studio dei Sistemi Complessi, Via Tommaso Pendola 37, Siena I-53100 Italy

Received May 24, 2002

Abstract: We use ab initio CASSCF and CASPT2 computations to construct the composite multistaterelaxation path relevant to cycloocta-1,3,5,7-tetraene singlet photochemistry. The results show that anefficient population of the dark excited state (S1) takes place after ultrafast decay from the spectroscopicexcited state (S2). A planar D8h-symmetric minimum represents the collecting point on S1. Nonadiabatictransitions to S0 appear to be controlled by two different tetraradical-type conical intersections, which aredirectly accessible from the S1 minimum following specific excited-state reaction paths. The higher-energyconical intersection belongs to the same type of intersections previously documented in linear and cyclicconjugated hydrocarbons and features a triangular -(CH)3- kink. This point mediates both cis f transphotoisomerization and cyclopropanation reactions. The lowest energy conical intersection has a boat-shaped structure. This intersection accounts for production of semibullvalene or for double-bond shifting.The mapping of both photochemical and thermal reaction paths (including also Cope rearrangements,valence isomerizations, ring inversions, and double-bond shifting) has allowed us to draw a comprehensivereactivity scheme for cyclooctatetraene, which rationalizes the experimental observations and documentsthe complex network of photochemical and thermal reaction path interconnections. The factors controllingthe selection and accessibility of a number of conjugated hydrocarbon prototype conical intersections andground-state relaxation channels are discussed.

1. Introduction

Direct UV-irradiation of cyclooctatetraene (COT, 1) may beof preparative value in the synthesis of semibullvalene (SBV,2).1 While in solution this method affords benzene (andacetylene) as an appreciable byproduct,1 in the vapor phase(∼70°C) UV irradiation (∼300 nm) has been found to minimizeconcomitant benzene formation,2 the method being one ofsimplicity and essentially quantitative yield (eq a).

On the basis of experimental observations, Zimmerman etal. presented an extended discussion of the possible mechanisticpathways for the1 f 2 photoisomerization.1 Low-temperatureirradiation was monitored by low-temperature infrared analy-sis: evidence was obtained for the direct formation of semi-bullvalene, or at least against the production of an S0 interme-diate. In particular, both the bicyclic triene bicyclo-[4,2,0]-octa-2,4,7-triene (BIC , 3) and the tricyclic diene tricyclo-[3,3,0,0]-octa-3,7-diene (4) were ruled out as possible intermediates,1 asshown in eqs b and c. A two-photon process for1 f 2photoconversion was thus proposed,1 where the first photon isabsorbed by1 (i.e., the stablecis,cis,cis,cis isomer of cyclooc-tatetraene), leading to production of the strainedtrans,cis,cis,cisisomer (COTtr , 5). In a subsequent step,5 was supposed toabsorb a second photon to produce semibullvalene rapidly,because no evidence for accumulation of such an intermediatecould be found in the low-temperature experiments (eq d).

Double-bond shifting is another process occurring in anti-aromatic [4n] systems such as cyclooctatetraene (n ) 2). It iswell established that the reversible interconversion betweendouble-bond shift isomers (aπ-skeletal rearrangement) may be

* To whom correspondence should be addressed. E-mail: (M.G.)mgara@ciam.unibo.it; (M.O.) olivucci@unisi.it; (F.B.) nando@ciam.unibo.it.

† Universitadi Bologna.‡ Universidad de Alcala´.§ Instituto de Ciencia Molecular.| Universitadi Siena.

(1) Zimmerman, H. E.; Iwamura, H.J. Am. Chem. Soc.1970, 92, 2015.(2) Turro, N. J.; Liu, J.-M.; Zimmerman, H. E.; Factor, R. E.J. Org. Chem.

1980, 45, 3511.

Published on Web 10/17/2002

13770 9 J. AM. CHEM. SOC. 2002 , 124, 13770-13789 10.1021/ja020741v CCC: $22.00 © 2002 American Chemical Society

induced either thermally or photochemically.3-7 The processoccurs when theπ-electrons “migrate” within the octagonalperimeter resulting in a site exchange between singly and doublybonded carbon atoms (1 f 1DBS), as shown in eq e.

While double-bond shifting is of no practical interest forparent cyclooctatetraene (yielding the degenerate product1DBS),it may be used to turn on or off conjugation betweenπ-sub-stituents placed in vicinal position (1,2-disubstituted-cyclooc-tatetraene). Thus, the double-bond shifting reaction may beexploited to design thermally or photochemically driven mo-lecular switches.7 This principle is described in eq f, where adouble-bond shifting is used to interconvert the system betweenan “on-state”, which possesses “through-conjugation” betweenthe π-donor (D) andπ-acceptor (A) substituents, and an “off-state”, where this through-conjugation is suppressed.

Thermal double-bond shifting (1 f 1DBS) and ring inversionbetween theD2d-symmetric equivalent tub forms of cyclooc-tatetraene (1 f 1′) have been investigated both experimentallyand computationally.3-7 NMR and computational studies haveshown the barrier for thermal cyclooctatetraene double-bondshifting to be about 14 kcal/mol,8,9 while the one for ringinversion is only 3-4 kcal/mol lower.9-11 In this respect, Borden

and co-workers3-5 showed that the thermal double-bond shiftingtransition state (TSDBS) has a planarD8h-symmetric structure(which violates Hund’s rule), while cyclooctatetraene ringinversion proceeds through a planarD4h transition state (TSRI);see eq g.

Cyclooctatetraene is also involved in two thermal valenceisomerization reactions (eq h). One (at∼100 °C) produces thebicyclic isomer3 with an energy barrier of 28.1 kcal/mol.12 Theother (observed at∼300°C)13 leads to the two semibullvaleneisomers2 and 2′, with an observed energy barrier of 42.2(0.2 kcal/mol.14 Semibullvalene appears to have a very lowenergy barrier (5.5 kcal/mol)15 for 2 f 2′ Cope rearrangement(eq h). All of these thermal processes (i.e., valence isomerizationand Cope rearrangement) have been the subject of extensiveand recent high level computational investigations.16-21

We have recently shown22 that photochemical semibullvaleneformation and double-bond shifting are two intrinsically com-petitive processes. In particular, we have provided computationalevidence that both photoinduced polycyclization to semi-bullvalene and double-bond shifting occur via an efficientradiationless decay of photoexcited cyclooctatetraene throughthe sameS1/S0 conical intersection funnel.

(3) Hrovat, D. A.; Borden, W. T.J. Am. Chem. Soc.1992, 114, 5879.(4) Wenthold, P. G.; Hrovat, D. A.; Borden, W. T.; Lineberger, W. C.Science

1996, 272, 1456.(5) Stevenson, C. D.; Brown, E. C.; Hrovat, D. A.; Borden, W. T.J. Am. Chem.

Soc.1998, 120, 8864.(6) Paquette, L. A.Acc. Chem. Res.1993, 26, 57.(7) Briquet, A. A. S.; Uebelhart, P.; Hansen, H. J.HelV. Chim. Acta1996, 79,

2282.(8) Anet, F. A. L.J. Am. Chem. Soc.1962, 84, 671.(9) Oth, J. F. M.Pure Appl. Chem.1971, 25, 573.

(10) Anet, F. A. L.; Bourn, A. J.; Lin, Y. S.J. Am. Chem. Soc.1964, 86, 3576.(11) Paquette, L. A.Pure Appl. Chem.1982, 54, 987.(12) Huisgen, R.; Mietzsch, F.Angew. Chem., Int. Ed. Engl.1964, 3, 83.(13) Martin, H.-D.; Urbanek, T.; Walsh, R.J. Am. Chem. Soc.1985, 107, 5532.(14) Hassenruck, K.; Martin, H.-D.; Walsh, R.Chem. ReV. 1989, 89, 1125 and

references therein.(15) (a) Cheng, A. K.; Anet, F. A. L.; Mioduski, J.; Meinwald, J.J. Am. Chem.

Soc.1974, 96, 2887. (b) Moskau, D.; Aydin, R.; Leber, W.; Gunther, H.;Quast, H.; Martin, H.-D.; Hassenruck, K.; Miller, L. S.; Grohmann, K.Chem. Ber.1989, 122, 925.

(16) Castano, O.; Frutos, L.-M.; Palmeiro, R.; Notario, R.; Andres, J.-L.;Gomperts, R.; Blancafort, L.; Robb, M. A.Angew. Chem., Int. Ed.2000,39, 2095.

(17) Castano, O.; Notario, R.; Gomperts, R.; Abboud, J.-L. M.; Palmeiro, R.;Andres, J.-L.J. Phys. Chem. A1998, 102, 4949.

(18) Andres, J.-L.; Castano, O.; Morreale, A.; Palmeiro, R.; Gomperts, R.J.Chem. Phys.1998, 108, 203.

(19) (a) Palmeiro, R.; Frutos, L.-M.; Castano, O.Int. J. Quantum Chem.2002,86, 422. (b) Castano, O.; Palmeiro, R.; Frutos, L.-M.; Andres, J.-L.J.Comput. Chem.2002, 23, 732.

(20) Jiao, H. J.; Schleyer, P. V.Angew. Chem., Int. Ed. Engl.1993, 32, 1760.(21) Jiao, H. J.; Nagelkerke, R.; Kurtz, H. A.; Williams, R. V.; Borden, W. T.;

Schleyer, P. V.J. Am. Chem. Soc.1997, 119, 5921.(22) Garavelli, M.; Bernardi, F.; Moliner, V.; Olivucci, M.Angew. Chem., Int.

Ed. 2001, 40, 1466.

Cyclooctatetraene Computational Photochemistry A R T I C L E S

J. AM. CHEM. SOC. 9 VOL. 124, NO. 46, 2002 13771

In the present paper, we significantly extend the investigationof photochemically relevant potential energy surfaces of cy-clooctatetraene. The results support immediate radiationlessdecay of the bright (S2) state into the dark lowest excited state(S1). Subsequent relaxation along the S1 path prompts populationof an excited (S1) planar minimum withD8h symmetry (COT* ,6). This is the collecting point for photoexcited molecules.Ground-state decay of6 is prompted by evolution along twobarrier-controlled S1/S0 conical intersections. One conicalintersection is similar to that documented for other unsaturatedhydrocarbons23-29 and displays a typical out-of-plane triangular-(CH)3- kink of triradical nature. The second conical intersec-tion corresponds to the one recently reported in a previousstudy.22 To draw a full comprehensive reactivity schemecomprising all of the thermal and photochemical reactions (andtheir interconnections), we have studied the following aspects:(i) the structure of the singlet manifold (S0, S1, S2, S3) in theFranck-Condon (FC) region of cyclooctatetraene, (ii) the initialrelaxation path (prompting population ofCOT* ) leading fromthe spectroscopic state S2 to the dark state S1, (iii) the alternativeroutes for the S1 evolution and S1 f S0 decay ofCOT* , and(iv) the thermal reaction paths connecting cyclooctatetraene andits photoproducts along the S0 energy surface (reinvestigatingalso previously studied processes such as valence isomerization,Cope rearrangement, ring inversion, and double-bond shift).

In the discussion section, the data i-iv will be used todetermine the substituent, steric, and electronic factors, playinga role in cyclooctatetraene photochemistry and associatedphotoproducts selection. The conical intersection structuresresponsible for the photochemistry ofcyclic conjugated hydro-carbons(i.e., photoproducts formation and distribution) will bediscussed, and a qualitative-predictive model will be proposedwith cyclooctatetraene taken as a paradigm. Speculations aboutapplicability and limits of cyclooctatetraene double-bond shift-based molecular switches will be finally considered.

2. Computational Methods

All CASSCF minimum energy path (MEP) computations on S1 andS0 have been carried out using the split-valence+ polarization 6-31G*

basis set. The active space in our computations is unambiguouslydefined: it comprises all of theπ electrons and valenceπ-orbitals ofthe polyene. Thus, a full active space of eight electrons in eightπ-orbitals has always been used for potential energy surface mapping,such as critical points and conical intersections optimizations, analyticalfrequency calculations, and minimum energy path computations, usingthe Gaussian 98 suite of quantum-chemical programs.30

IRD computations have been used for locating all of the possible S0

relaxation channels departing from the lower tip of a conical intersectionpoint.31,32 This is accomplished first via locating an initial direction ofrelaxation (IRD) on S0 (as close as possible to the conical intersec-tion),31,32and second via standard minimum energy path computationsfollowing that IRD. Briefly, an IRD corresponds to a localsteepestdescent direction, in mass-weighted Cartesians, from a given startingpoint. The IRD is calculated by locating the energy minimum on ahyperspherical (i.e.,n-1-dimensional) cross section of then dimensionalpotential energy surface (n is the number of vibrational degrees offreedom of the molecule) centered on the starting point (the S1/S0

conical intersection in this case). The radius of this hypersphere isusually chosen to be small (typically 0.25-0.5 au inmass-weightedCartesians) to locate the steepest direction in the vicinity of the startingpoint (i.e., the hypersphere center). The IRD is then defined as thevector joining the starting point to the energy minimum (ahypermini-mum). Once one or morehyperminimahave been determined, theassociated minimum energy path (emerging from these points) iscomputed asthe steepest descent line in mass-weighted Cartesians33

using the IRD vector to define the initial direction to follow.

These relaxation channels describe thestatic (i.e., nondynamical)evolution of the system after decay to the ground state and provideinsight into the mechanism of photoproduct formation in conditions ofvibrationally “cold” molecules (i.e., with infinitesimal velocity).Although molecules have more than infinitesimal kinetic energy (andin fact classical trajectories deviate from this static paths), neverthelessminimum energy paths may represent a convenient measure of theprogress of a molecule in a photochemical reaction, provided that itoccurs in a cold environment where slow excited-state motion or/andthermal equilibration is possible and the excited-state reactant has asmall/controlled amount of vibrationally excess energy. Under theseconditions, semiclassical dynamics yield the same mechanistic informa-tion as from topological investigation of the potential energy surface,because its structure is expected to play the dominant role in determiningthe initial molecular motion in the decay region. This strategy has beensuccessfully applied in previous computational investigations involvingconjugated hydrocarbons photochemistry23-27,32and has been validatedby semiclassical dynamics computations.32,34,35Thus, we are confidentthis approach may help to understand and rationalize the photochemicalbehavior of cyclooctatetraene.

To improve the computed energetics, the effect of dynamic electroncorrelation has also been included. The S1 and S0 energies along thefully optimized minimum energy paths have been recomputed by usingmulticonfigurational second-order perturbation theory via the CASPT2approach36 included in the MOLCAS-5 package.37 A reference CASSCF/6-31G* wave function with the standard (eight electrons in eightπ-orbitals) active space has always been used for all CASPT2 (PT2hereafter) computations. In addition, CASSCF/6-31G* zero-pointenergy (ZPE) corrections have been included when stated.

It is worth noting how PT2 corrections shift the position of theCASSCF optimized S1/S0 conical intersection points toward the relaxed(23) Celani, P.; Garavelli, M.; Ottani, S.; Bernardi, F.; Robb, M. A.; Olivucci,

M. J. Am. Chem. Soc.1995, 117, 11584.(24) Garavelli, M.; Celani, P.; Yamamoto, N.; Bernardi, F.; Robb, M. A.;

Olivucci, M. J. Am. Chem. Soc.1996, 118, 11656.(25) Garavelli, M.; Celani, P.; Bernardi, F.; Robb, M. A.; Olivucci, M.J. Am.

Chem. Soc.1997, 119, 11487.(26) Palmer, I. J.; Ragazos, I. N.; Bernardi, F.; Olivucci, M.; Robb, M. A.J.

Am. Chem. Soc.1993, 115, 673.(27) Garavelli, M.; Frabboni, B.; Fato, M.; Celani, P.; Bernardi, F.; Robb, M.

A.; Olivucci, M. J. Am. Chem. Soc.1999, 121, 1537.(28) Robb, M. A.; Garavelli, M.; Olivucci, M.; Bernardi, F.ReViews in

Computational Chemistry; 2000; Vol. 15, pp 87-146.(29) Bernardi, F.; Olivucci, M.; Robb, M. A.Chem. Soc. ReV. 1996, 25, 321.(30) Frisch, M. J.; Trucks, G. W.; Schlegel, H. B.; Scuseria, G. E.; Robb, M.

A.; Cheeseman, J. R.; Zakrzewski, V. G.; Montgomery, J. A., Jr.; Stratmann,R. E.; Burant, J. C.; Dapprich, S.; Millam, J. M.; Daniels, A. D.; Kudin,K. N.; Strain, M. C.; Farkas, O.; Tomasi, J.; Barone, V.; Cossi, M.; Cammi,R.; Menucci, B.; Pomelli, C.; Adamo, C.; Clifford, S.; Ochterski, J.;Petersson, G. A.; Ayala, P. Y.; Cui, Q.; Morokuma, K.; Malick, D. K.;Rabuck, A. D.; Raghavachari, K.; Foresman, J. B.; Cioslowski, J.; Ortiz,J. V.; Stefanov, B. B.; Liu, G.; Liashenko, A.; Piskorz, P.; Komaromi, I.;Gomperts, R.; Martin, R. L.; Fox, D. J.; Keith, T.; Al-Laham, M. A.; Peng,

C. Y.; Nanayakkara, A.; Gonzalez, C.; Challacombe, M.; Gill, P. M. W.;Johnson, B. G.; Chen, W.; Wong, M. W.; Andres, J.-L.; Gonzalez, C.;Head-Gordon, M.; Replogle, E. S.; Pople, J. A.Gaussian 98, revision A.6;Gaussian, Inc.: Pittsburgh, PA, 1998.

(31) Celani, P.; Robb, M. A.; Garavelli, M.; Bernardi, F.; Olivucci, M.Chem.Phys. Lett.1995, 243, 1.

(32) Garavelli, M.; Celani, P.; Fato, M.; Bearpark, M. J.; Smith, B. R.; Olivucci,M.; Robb, M. A. J. Phys. Chem. A1997, 101, 2023.

(33) Gonzalez, C.; Schlegel, H. B.J. Phys. Chem.1990, 94, 5523.(34) Garavelli, M.; Bernardi, F.; Olivucci, M.; Bearpark, M. J.; Klein, S.; Robb,

M. A. J. Phys. Chem. A2001, 105, 11496.(35) Vreven, T.; Bernardi, F.; Garavelli, M.; Olivucci, M.; Robb, M. A.; Schlegel,

H. B. J. Am. Chem. Soc.1997, 119, 12687.(36) Andersson, K.; Malmqvist, P.-Å.; Roos, B. O.J. Chem. Phys.1992, 96,

1218.(37) Andersson, K.; Blomberg, M. R. A.; Fu¨lscher, M. P.; Karlstro¨m, G.; Lindh,

R.; Malmqvist, P.-Å.; Neogra´dy, P.; Olsen, J.; Roos, B. O.; Sadlej, A. J.;Schutz, M.; Seijo, L.; Serrano-Andre´s, L.; Siegbahn, P. E. M.; Widmark,P.-O. Molcas 5.0; Lund University: Lund, 1999.

A R T I C L E S Garavelli et al.

13772 J. AM. CHEM. SOC. 9 VOL. 124, NO. 46, 2002

S1-excited cyclooctatetraene structure (6; see Scheme 1). Anyway, toevaluate the effect of dynamic electron correlation on the geometryoptimization is not trivial,38 and because all surface mapping has beenperformed using CASSCF, the CASSCF optimized S1/S0 conicalintersections have been taken as starting points for S0 relaxation pathsmapping.

The energies for the singlet manifold (S0, S1, S2, S3) reported inTable 1 and Figure 1 have been computed with the PT2 method andhave been used, together with the RASSI approach,39 to compute theoscillator strengths (f) for vertical electronic transitions (see Table 1).39

Again, a reference CASSCF wave function has been used with a 6-31G*

basis set and the standard (eight electrons in eightπ-orbitals) activespace. State-average CASSCF computations have been performedbetween states with the same symmetry, which have been averagedwith identical weights. These results have also been validated and

reproduced using a generally contracted basis set of atomic naturalorbitals (ANO-L) obtained from the C(14s9p4d)/H(8s4p) primitive setswith the C[4s3p2d]/H[2s1p] contraction scheme40,41(results not shown).In perturbation theories (like CASPT2), a very common problem isthe appearance of intruder states. For a given excited state, it is usuallyreflected in the relatively low weight (ω), with respect to that obtainedfor the ground state, of the corresponding CASSCF reference in thefirst-order wave function. In those cases, a number of calibrationcalculations have been carried out using a level-shift technique, andthe influence of weakly interacting intruder states on the computedexcitation energies has been checked. In particular, the so-calledimaginary level shift42 implemented in the MOLCAS-5 suite ofprograms37 has been employed, because it is capable of removingsingularities completely. For this purpose, excitation energies have beencomputed with different values of the level shift parameter (IMAG):0.0 (standard CASPT2 theory), 0.1, 0.2, 0.3, and 0.4 au. It is found

(38) Olivucci, M.; Page, C. S., in preparation.(39) Malmqvist, P.-Å.; Roos, B. O.Chem. Phys. Lett.1989, 155, 189. RASSI

has been used to compute the transition dipole moments from the CASSCFreference wave functions. The oscillator strength (f) is defined asf )2/3|Mif|2∆Efi, whereMif is the transition dipole moment (in au), and∆Efi isthe CASPT2 excitation energy (in au).

(40) Widmark, P.-O.; Malmqvist, P.-Å.; Roos, B. O.Theor. Chim. Acta1990,77, 291.

(41) Widmark, P.-O.; Persson, B. J.; Roos, B. O.Theor. Chim. Acta1991, 79,419.

(42) Forsberg, N.; Malmqvist, P.-Å.Chem. Phys. Lett.1997, 274, 196.

Scheme 1

Cyclooctatetraene Computational Photochemistry A R T I C L E S

J. AM. CHEM. SOC. 9 VOL. 124, NO. 46, 2002 13773

that the computed excitation energies are invariant within 0.05 eV withrespect to the IMAG parameter. Therefore, even if theω value at IMAG) 0.0 au is somewhat low in certain cases for the highest excited statehere considered (S3 state), the computed excitation energy canconfidently be trusted. All of these calculations were carried out withthe tools available in the MOLCAS-5 package.37

All of the computed energies are reported and summarized in Tables1 and 2. A nonrelaxed linear scan from theD2d FC structure (1) to theD8h excited (S1) minimum (6) is reported in Figure 1 and has beenperformed in mass-weighted internal coordinates. The direction of thisrigid scan appears to be parallel to the direction of the initialD2d-symmetric force vector computed at the FC point of the lowest dipole-allowed state (S2). Therefore, the reported scan follows the naturalrelaxation direction out of the FC region (i.e., the force direction) andpreserves theD2d symmetry all along the path (see section 3.1).

3. Computational Results

3.1. Spectroscopy and Relaxation from the FC Region:The S2 f S1 Deactivation Path. The electronic ground state(S0) for the D2d tube form (1) of cyclooctatetraene (i.e., thestarting reactant and ground-state minimum) has a1A1 sym-

metry. Electronic transitions from S0 to the excited S1 and S3

states (possessing a1A2 and 1A1 symmetry, respectively) areforbidden (see Table 1). In fact, for a dipole-allowed electronictransition, the product A× B × C (where A, C, and B representthe irreducible representation of the wave function in the initialstate, the final state, and the component (x,y,z) of the dipolemoment, respectively) has to belong to the totally symmetricrepresentation. Because the products A1 × B2 × A2 or A1 × E× A2 do not span the totally symmetric representation (A1 forD2d symmetry group), the S0fS1 electronic transition, relatedmainly to the HOMOf LUMO singly excited configuration,is forbidden. The same occurs for the S0 f S3 transition,involving the (HOMOf LUMO)2 two-electron promotion. Onthe other hand, transition to the electronic state S2 (a doubledegenerate state owning a1E symmetry and described mainlyby two coupled singly excited configurations: HOMO,HOMO-1 f LUMO,LUMO+1) is dipole-allowed (the oscil-lator strength for this transition being∼7 × 10-3). Hence, theS0(1A1) f S2(1E) electronic transition can be assigned as thelowest dipole-allowed absorption feature and, accordingly, theFC S2 state has been taken as the starting point for the excited-state relaxation of the system.

The optical absorption spectrum and electron-impact energy-loss spectrum of cyclooctatetraene can be described as a broadmaximum of low intensity over the region 310-260 nm (4.00-4.77 eV) and a strong feature with an intensity maximum at6.42 eV.43a As can be seen in Table 1, the computed verticalexcitation energies for1 are predicted with an exceedingly weakintensity. As stated above, two of them are actually dipole-

(43) (a) Frueholz, R. P.; Kuppermann, A.J. Chem. Phys.1978, 69, 3614. (b)Traetteberg, M.Acta Chem. Scand.1966, 1724. (c) Palmer, M. H.J. Mol.Struct.1988, 178, 79. (d) Van-Catledge, F. A.J. Am. Chem. Soc.1971,93, 4365. (e) Roos, B. O.; Mercha´n, M.; McDiarmid, R.; Xing, X.J. Am.Chem. Soc.1994, 116, 5927. (f) Serrano-Andre´s, L.; Lindh, R.; Roos, B.O.; Merchan, M. J. Phys. Chem.1993, 97, 9360. (g) Thediabaticcomponents of theadiabaticelectronic eigenfunctions describe the energyof a particular spin-coupling, while theadiabatic functions represent thesurfaces of the real states (the singlet ground and excited states, in thiscase): Olivucci, M.Superfici Adiabatiche e Diabatiche nel Trattamentodella ReattiVita Chimica, Ph.D. Thesis, Dipartimento di Chimica ‘G.Ciamician’; Universita’ di Bologna: Bologna, 1988.

Table 1. Total (E) and Relative (∆E) CASSCF and CASPT2 (6-31G*) Energies of the Singlet Manifold (S0, S1, S2, and S3) forCyclooctatetraene Relaxation out of the FC Region (See Discussion in the Text and Also the Energy Profiles in Figure 1): Values in au(Hartree) and kcal mol-1, Respectivelya

E + 307 aub ∆E (kcal mol-1) f ‚c

structuresd statee CASSCF CASPT2 (ω) CASSCF CASPT2 CASPT2

COT (D2d) S0(11A1) -0.618368 -1.537640(0.76) 0.0 0.0(1) S1(11A2) -0.388459 -1.390580(0.73) 144.3 92.3 forbidden

S2 (11E) -0.360835 -1.326304(0.75) 161.6 132.6 0.0068S3(21A1) -0.378099 -1.317077(0.63)f 150.8 138.4 forbidden

TSRI (D4h) S0(11A1g) -0.612008 -1.524331(0.76) 4.0 8.4S1(11A2g) -0.469375 -1.451949(0.74) 93.5 53.8 forbiddenS2(21A1g) -0.506827 -1.435574(0.76) 70.0 64.0 forbiddenS3(11Eu) -0.401875 -1.347476(0.48)f 135.8 119.3 0.0051

S2/S1 CIopt (D2d) S0(11A1g) -0.609744 -1.523380(0.76) 5.4 8.9S1(11A2g) -0.486312 -1.467654(0.76) 82.9 43.9 forbiddenS2(21A1g) -0.541434 -1.471382(0.74) 48.3 41.6 forbiddenS3(11Eu) -0.418990 -1.362325(0.73) 125.1 110.0 0.0056

COT* (D8h) S0(11B2g) -0.606038 -1.521029(0.75) 7.7 10.4(6) S1(11A1g) -0.555983 -1.489691(0.73) 39.2 30.1 forbidden

S2(11B1g) -0.492204 -1.472948(0.71) 79.2 40.6 forbiddenS3(11E1u) -0.424390 -1. 367131(0.51)f 121.7 107.0 forbidden

a The weight (ω) of the CASSCF reference wavefunction (i.e., the zeroth-order function) in the first-order (CASPT2) function is given within parentheses.Values for the oscillator strength (f) are also included.b State-average CASSCF calculations have been performed between states of the same symmetry.c See ref 39.d As defined in the text (see also Figure 1).e Ordering of states from the CASPT2 results.f Despite the relatively low weight, the excitationenergy is stable with respect to the variation of the imaginary level shift parameter42 (see section 2 for further details).

Figure 1. Nonrelaxed linear scan from1 (FC) to6 (COT* ) following theD2d-symmetric force vector computed on the FC spectroscopic state S2.Symmetry labels refer to theD2d symmetry. Distances (r) are expressed inmass-weightedatomic units (au): amu1/2 bohr.

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forbidden. Therefore, comparison of the current results with theexperimental data has to be limited to the low-energy region ofthe spectrum (e6.0 eV).

The lowest singlet excited-state S1 is of A2 symmetry inagreement with previous theoretical results [see, e.g., ref43a,c,d]. That the lowest singletf singlet transition is opticallyforbidden was clearly established by Frueholz and Kuppermannin 1978.43a The computed vertical excitation energy (4.00 eV,CASPT2 result in Table 1) is somewhat too low as comparedto the maximum of the low-intensity band (4.4 eV). It is worthrecalling that there is a crucial factor for deviation betweencomputed and recorded transition energies that affects bothcomputed and measured data: the equilibrium geometry for theexcited state differs substantially (see below) from that of theground state. The resulting broad Franck-Condon envelopeintroduces relatively large uncertainties into experimental bandmaximum determinations. In addition, because of the steepcurvature of the excited-state potential energy surface at theground-state equilibrium geometry (see Figure 1), small errorsin the geometry used for the ground state may lead to asignificant error in the computed vertical transition energy.Nevertheless, the computed ground-state geometry is very closeto the experimental data derived from electron diffraction.43b

Therefore, in the case that a higher accuracy would be required(to fully describe on theoretical grounds the observed low-intensity/low-energy band), vibronic interactions have to betaken into account. The band maximum and the verticaltransition seem to differ around half an eV, consistent with theexpected long Franck-Condon progression. A full theoreticaldescription of the electronic spectrum of cyclooctatetraene iswell beyond the scope of this work and is certainly a challengingtask: the implied theoretical difficulties are more closely relatedto a strained, nonplanar diene like norbornadiene (bicyclo-[2.2.1]-hepta-2,5-diene)43e than to even polyenes.43f However,work in progress considering also valenceσ/π, π/σ states andthe lowest Rydberg transitions enables us to confirm that thestrongest recorded feature in the electronic spectrum of cy-clooctatetraene can be related to a higher1E state (notdocumented here), as it was already suggested from pioneeringINDO calculations.43d

Analysis of the gradient at the FC point reveals that it is verysimilar for all of the studied states and possesses aD2d

symmetry. Full relaxation upon these states leads to similarplanar minima, owning the sameD8h symmetry. Furthermore,the topology of S1, S2, and S3 around the FC region is similar.From theD2d reactant1 to theD8h excited (S1) minimum 6, a

Table 2. Total (E) and Relative (∆E) CASSCF and CASPT2 (6-31G*) Energies for All of the S1 and S0 Structures Discussed in the Text:Values in au (Hartree) and kcal mol-1, Respectivelya

E + 307 aub ∆E (kcal mol-1) ZPEb

structuresc stated CASSCF CASPT2 (ω) CASSCF CASPT2 CASSCF

COT* (6)e S1 -0.55714 -1.48067 (0.76) 0.0 0.0 90.2TSDBS (6)e S0 0.60904 -1.51274 (0.76) -32.6 -20.1 85.2

TSb S1f -0.51403g -1.45236 (0.75)g 27.1 18.0

S1f -0.51662h 25.4 87.4

S0f -0.53032g -1.45058 (0.76)g 16.8 19.2

TSst S1 -0.47953 -1.40131 (0.73) 48.7 49.8 84.4S0 -0.53026 -1.44727 (0.75) 16.9 21.0

CIb S1f -0.51695g -1.46159 (0.75)g 25.2 12.0

S0f -0.52427g -1.44985 (0.75)g 20.6 19.3

S0f -0.52915h -1.44961 (0.76)h 17.6 19.5

CIst S1f -0.51102g -1.45208 (0.75)g 28.9 17.9

S0f -0.51457g -1.44156 (0.75)g 26.7 24.5

S0f -0.51930h -1.44069 (0.76)h 23.7 25.1

COT (1) S0 0.63567 -1.53381 (0.77) -49.2 -33.3 87.4

DIR S0 0.58801 -1.52402 -19.4 -27.2 86.9

TSCR S0 0.58754 -1.52500 (0.75) -19.1 -27.8 87.0

SBV (2) S0 0.60384 -1.53861 (0.76) -29.3 -36.4 89.2

DIR ′ (7) S0 0.53394 -1.46296 (0.75) 14.6 11.1

COTtr (5) S0 0.59518 -1.50068 (0.76) -23.9 -12.6

TSDIR ′ S0 0.52714 -1.46042 (0.75) 18.8 12.7

TS′ S0 0.51607 -1.43954 (0.76) 25.8 25.8

DIR ′′ S0 0.51870 -1.43649 (0.76) 24.1 27.7

TSSBV S0 0.50895 -1.43376 (0.76) 30.3 29.4

TS1VI S0 0.56892 -1.48922 (0.76) -7.4 -5.4 86.3

TS2VI S0 0.54163 -1.47279 (0.75) 9.7 4.9 85.5

BIC (3) S0 0.61099 -1.52346 (0.76) -33.8 -26.9

TSRI S0 0.61915 -1.51397 (0.77) -38.9 -20.9 87.9

a When computed, CASSCF/6-31G* zero-point energies (ZPE) are also reported (in kcal mol-1). The weight (ω) of the CASSCF reference wavefunction(i.e., the zeroth-order function) in the first-order (CASPT2) function is given within parentheses.b Single-state energy computations (unless differentlystated).c As defined in the text (see also Scheme 11).d S0 and S1 as from CASSCF energy computations; this corresponds (unless differently stated) to theorder established by CASPT2 computations.e COT* andTSDBS correspond to structure6. f CASSCF and CASPT2 energy order (S0 and S1) are inverted.g State-average computations between the lowest two singlet states (S0 and S1) with weights of 0.5 each.h When state averaging is used, a single-statecomputation is also performed for comparison.

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linear nonrelaxed scan of the potential energy surface for thesinglet states (S0, S1, S2, and S3) has been computed. It wasinspired analyzing the initialD2d-symmetric force vectorcalculated at the FC point of the lowest dipole-allowed state S2

(see Computational Methods). The corresponding PT2 energyprofiles have been reported in Figure 1. The results show thatalong this D2d-symmetric relaxation coordinate, a suddencrossing between the S2 and S3 states takes place (S3/S2 CI inFigure 1). Therefore, the S2 potential energy surface rapidlychanges its symmetry (from1E to 1A1) and electronic nature(from HOMO,HOMO-1 f LUMO, LUMO+1 to (HOMOfLUMO)2) as a result of a crossing with the steeper S3 (1A1)surface occurring along the initial ultrafast dynamics out of theS2 (1E) FC region. Following relaxation along the S2 (1A1)potential energy surface, the system is driven into a secondcrossing with the singlet S1 (1A2) state. This crossing (S2/S1

CI ) funnels S2 radiationless decay and prompts S1 population.Because the final S1 point 6 is diabatically connected (via S2/S1 CI ) with S2,43g it preserves both the electronic and thesymmetric (1A1) character of S2.

It is worth noting that, due to the sloped topology28,44,45ofthe conical intersection S2/S1 CI (see Figure 1), it is also possibleto populate the relaxedD8h-symmetric minimum on the S2 (1A2)state as a result of the initial dynamics through this crossing.44

Still, the energy gap separation between S2/S1 CI (or theoptimized- S2/S1 CIopt - conical intersection point) and therelaxed S2 minimum is very small (e4 kcal/mol; see Table 1and Figure 1). Therefore, due to excess vibrational energyfollowing relaxation from the FC region, an appreciablepopulation of the S2 state should not be expected because anefficient decay to S1 may be easily prompted overcoming thesmall activation barrier toward S2/S1 CI .

In conclusion, after the lowest S0(1A1) f S2(1E) dipole-allowed transition, the photoexcited system quickly reaches theS1 state and populates the relaxedD8h excited (S1) minimum6,while aD2d symmetry preserving planarization motion may beinvolved. Further validation for the computed relaxation comesfrom the gradient difference and derivative coupling vectorscomputed at the conical intersection point S2/S1 CI (see Figure2) and defining the branching space (i.e., the space which mustcontain all of the initial paths out of the crossing region).29 Thisanalysis reveals that any relaxation path starting from thisconical intersection must have two initial components: the firstone leads to the flattening of the system (Figure 2a), while thesecond one involves only aπ bond-length reorganization leadingto double bonds stretching and concomitant single bondscompressing (Figure 2b). Therefore, we have found that thetendency of the system (following decay from S2/S1 CI ) is fora full π-bonds delocalization leading to a flattening of themolecule. This motion is exactly the one which drives the systemtoward COT* , that is, the planarD8h structure6 on S1 (seeFigure 3a). Hence, we conclude that the most likely and favoredrelaxation path from the S2 FC point is that leading to theD8h

S1 minimum 6.46

3.2. The S1 Reaction Paths.In the previous section, we haveprovided evidence that photoexcited cyclooctatetraene relaxes

toward the planarD8h-symmetric minimum6 residing on thedark (HOMOf LUMO)2 doubly excited-state S1. Its structure(see Figure 3a) appears to be identical to the thermal double-bond shift transition state (TSDBS) already reported by Bordenet al. computationally,3-5 and also reinvestigated here (seesection 3.4). This is the typicalπ-delocalized structure expectedto characterize the relaxed S1 state for a Hu¨ckel antiaromatic[4n] system.3-5

As the S1/S0 energy gap is still quite large here (∼20 kcal/mol), we have looked for deformations along accessible reactionchannels leading to real crossing points. Two very different S1/S0 conical intersections (CIst andCIb) have been located on S1

(see Scheme 1). Both have a “quasi-tetraradical” electronicnature, but differ clearly in energy, geometry, and symmetry.CI st (see Figure 3b), which is higher in energy (about 18 kcal/mol above6; see Table 2), has aCs-symmetric structure andappears to be of the same type as the previously documentedcrossing points involved in the photochemistry of linear andcyclic conjugated hydrocarbons.23-27 This type of conical

(44) Klein, S.; Bearpark, M. J.; Smith, B. R.; Robb, M. A.; Olivucci, M.;Bernardi, F.Chem. Phys. Lett.1998, 292, 259.

(45) Atchity, G. J.; Xantheas, S. S.; Ruedenberg, K.J. Chem. Phys.1991, 95,1862.

(46) Garavelli, M.; Page, C. S.; Celani, P.; Olivucci, M.; Schmid, W. E.; Trushin,S. A.; Fuss, W.J. Phys. Chem. A2001, 105, 4458. Here we have reportedhow the dynamics through a conical intersection, if occurring along thelower-energy surface (as it is the case for motion through the first S3/S2CI ), must be accompanied by a change in the direction of motion whichgenerally follows the derivative coupling vector. In this case, the computedderivative coupling vector (which possesses aC2V symmetry) leads toD2d-symmetry breaking. Therefore, it is likely that, after travelling through thefirst S3/S2 CI , the real motion of the system occurs along aD2d-symmetrybreaking path, which deviates from theD2d-symmetric linear scan reportedin Figure 1. Still, we suppose that this change is minor and insignificantfor the discussion reported.

Figure 2. Gradient difference (a) and derivative coupling (b) vectorscomputed for the S2/S1 CI (see Figure 1) ofD2d symmetry.

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intersection has been shown23-27 to prompt a common radia-tionless deactivation channel for photoexcited polyenes by pro-viding the “funnel” for a fully efficient S1 f S0 internal con-version (i.c.) and subsequent transformations (e.g., cisf transphotoisomerization and ring-closing reactions).CIst is character-ized by an unusual out-of-plane distortion: a triangular “kink”of a-CH-CH-CH- segment of the polyene chain (see Figure3b). This-(CH)3- kink (whose central C2 atom is pyramidal-ized) displays three weekly coupledπ-electrons (see structurein Scheme 1) which are localized on three adjacent-CH- units,and a fourth electron which is delocalized along the remainingpart of the cyclic conjugated system (a pentadienyl moiety).

The excited-state minimum energy path describing the6 fCIst reaction process goes through a high energyCs-symmetrictransition state (TSst; see Figure 3c), which lies about 50 kcal/mol above6 (see Table 2 and Scheme 1). The transition vectorshown in Figure 3c describes the motion of the system on theway toCIst. It is worth noting that, along this path, C1 and C3

approach each other (i.e., the C1-C3 distance shortens goingfrom 2.60 Å in6 to 2.16 Å inCIst, and the C1-C2-C3 bendingangle reduces to 98°) with two unpaired (almost-localized)facing electrons. This will strongly affect the correspondingground-state relaxation channels departing fromCIst (see section3.3).

The 6 f CIst path describes an energetically demandingprocess pushing the system toward a “standard” kinked conicalintersection. However, as previously documented,22 a differentdeformation drives the system toward aC2V-symmetric boat form(see Scheme 1 and Figure 3d), decreasing the gap and makingthe crossing between the two potential energy surfaces possible.In this case, the system reaches an S1/S0 real crossing point

(CIb) at lower energy, which in fact represents the optimizedlowest energyconical intersection for the system (see Table 2for the energy differences). The computed minimum energy pathdriving the system towardsCIb appears to be easily accessibledue to the moderate 15.2 kcal/mol energy barrier (PT2- andZPE-corrected; see Table 2). This must be particularly true inthe gas phase. Here, in fact, vibrational cooling and energydissipation are much more inhibited than in the condensed phase,thus leaving the system with excess undissipated vibrationalenergy, which helps the molecule to easily overcome the barrierto CIb (still leaving the path toCIst less favored).

Analysis of the geometrical and electronic properties ofCIb

indicates that it corresponds to a “quasi-tetraradical” system,with two unpaired opposite facing electrons placed in 1,5relation on partially pyramidalized C centers (and two delocal-ized allyl systems in the remaining backbone). S1 motion towardCIb prompts 1,5 centers approaching (their distance shortensfrom 3.68 Å inCOT* to 2.89 Å inCIb); Figure 3e shows thegeometrical structure and the transition vector associated withthe optimizedC2V transition state (TSb) along the6 f CIb path.(Because of root-flipping problems,TSb optimization has beenperformed using state averaged orbitals (see Table 2).)

Scheme 1 reports the structures and the relative (toCOT* )PT2-corrected energies for all of the points discussed in thissection.47

3.3. Photochemistry: The S1 f S0 Deactivation Paths.Once the system reaches one of the two “tetraradicaloid” conicalintersections, efficient S1 f S0 internal conversion and branch-

(47) All PT2 computations give a well-balanced weight (0.76) for the CASSCFreference function in the first-order function. The reference PT2 absoluteenergy computed forCOT* is -308.48067 au.

Figure 3. CASSCF/6-31G* optimized structures forCOT* (6) (a), CIst (b), TSst (c), CIb (d), andTSb (e).

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ing of the reaction channel occur.28,29 In this section, we willanalyze and discuss all of the possible relaxation paths departingfrom the tip of the two conical intersection points, the drivingforce for these processes being the electron recoupling of thehighly unstable tetraradicaloid conical intersection structure.These channels have been computed via locating an initialdirection of relaxation (IRD) on S0, using an owner developedcode (see section 2 and refs 31,32 for details).

Relaxation from CIst. Two different IRDs have been locatedclose to the higher-energyCIst point, the first starting at 0.75au (mass-weightedatomic units: amu1/2 bohr) and the secondat 1.5 au. Each IRD defines a minimum energy path leadingtowards a specific photoproduct: a strained cyclooctatetraeneisomer (COTtr , 5) of C2 symmetry and a highly unstable bicyclicdiradical intermediate (DIR ′, 7) of Cs symmetry, respectively(see Scheme 2 and Table 2 for the energetics).

Formation of photoproduct7 occurs along a quite flatminimum energy path (MEP: see Scheme 2) which is the directprolongation on S0 of the motion gained by the molecule on S1

along the 6 f CIst reaction path (see the reactive mode

displayed in Figure 3c). Although this path is not favored (i.e.,it starts more distant and higher in energy; see Scheme 2),molecules may be channeled toward7 if the momentum gainedon the excited-state branch of the reaction coordinate is con-tinued on the ground state. As documented in the previoussection, this motion brings 1,3 C centers at a close proximity,leading to an increased overlap between the two facing localizedπ-orbitals (hosting two of the four unpaired electrons of theconical intersection point). Thus, if continued on S0, a C1-C3

σ-bond formation process occurs to give intermediate7 (seeFigure 4a). Still, two unpaired electrons remain, thus making7a very unstable system. The first electron is localized on thepyramidalized apex of the cyclopropane ring (C2), while thesecond is still delocalized on the pentadienyl moiety (see struc-tures in Scheme 2). In section 3.4, we will document all of thethermal processes involving this very reactive intermediate.

Formation of photoproduct5 (see Figure 4b) occurs via acis f trans one-step isomerization process along a barrierlessminimum energy path laying on the steeper side of the potentialaround the cone (see Scheme 2). Product5 represents the

Scheme 2

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strainedtrans,cis,cis,cis-cyclooctatetraene isomer (COTtr ) sup-posed by Zimmerman et al.1 to be involved as an intermediatein the 1 f 2 photoconversion process (see eq d). Althoughenergetically favored (the associated minimum energy pathstarting closer toCIst and at a lower energy), its relaxationcoordinate reveals that this channel requires a change in thedirection of the excited-state motion (6 f CIst). Thus, we canconclude that7 may be formed (provided the system reachesCIst) for dynamic reasons (S1 motion is continued on S0 alongthe flat minimum energy path to7), even if5 is energeticallymore favored.

Other two less favored IRDs have been located from theCIst

point. The first (at 3.5 au) is degenerate and leads back to thereactant (1 and 1DBS), and the second (at 8.5 au) drives theformation of the bicyclic isomer3. Although it has beenshown23-26,32 that disrotatory cyclobutene ring formation iseasily prompted by decay via a-(CH)3- kinked conicalintersection (see also section 4.2), still this process is very muchunfavored in this case (and, in fact, the existence of this channelis doubtful, due to the big distance). This is due to the inverteddirection of the pyramidalization found at carbon C2 (seeScheme 2), which favors trans double-bond coupling, on onehand, and puts geometric constraints to four-membered ringformation, on the other.

Relaxation from CIb. As previously documented,22 thelowest energy conical intersection (CIb) corresponds to a “quasi-tetraradical” system, with two unpaired opposite facing electronsplaced in 1,5 relation on partially pyramidalized carbon atoms(C1 and C5) and two allyl radicals in the remaining backbone.Again, recoupling between these four unpaired electronscharacterizes ground-state relaxation. We have seen22 that theS1 reactive process6 f CIb promotes 1,5 centers approachingand prompts formation of a diradical intermediate with a C1-C5 transannular bond (see the reactive mode displayed in Figure3e). The S0 relaxation path corresponding to this process hasindeed been located at a distance of 5 au from the conicalintersection tip (see Scheme 3). The resultingC2V bicyclo-[3.3.0]-octadienyl diradical (DIR ; see Figure 5a) corresponds to thestructure suggested by Iwamura et al.48 and by Martin et al.13

as the intermediate in the thermal valence isomerization ofcyclooctatetraene to semibullvalene.DIR is a flat S0 energyminimum at the CASSCF level, and, in fact, two very closed

transition states ofCs symmetry (TSCR; see Figure 5b, and itssymmetric- enantiomerically related- system) have beenlocated at the CASSCF level, connecting the diradical to thetwo semibullvalene isomers2 and 2′. Yet, the inclusion ofdynamic correlation energy (CASPT2 level; see Table 2) showsthat this structure is unstable along the symmetry breakingmotion which leads toTSCR and to theCs-symmetric semi-bullvalene (see Scheme 3 and Figure 5c). Thus, this structureturns out to correspond to the transition structure for the Coperearrangement of semibullvalene, in agreement with previousworks by Jiao et al.20,21and, more recently, by Castan˜o et al.16

Therefore, there must exist aC2V-symmetric bifurcation point19

splitting the S0 relaxation channel, which ultimately leads tothe final photoproducts2 and 2′ (i.e., the two possible semi-bullvalene isomers of the Cope rearrangement; see Scheme 3).These theoretical results are supported by studies of thermalisomerizations of substituted semibullvalenes and cycloocta-tetraenes,49-52 which reveal that the rates of Cope rearrange-ments and the2 f 1 rearrangement are somehow correlated(see also section 3.4).

Besides representing an easily accessible channel for a fastinternal conversion process,CIb also promotes the branchingof the reaction path. Other two degenerate and symmetric IRDshave been located in its proximity (1.5 au distance), whoseminimum energy paths lead back to theD2d reactant (1; seeFigure 5d) or to its double-bond-shifted isomer (1DBS) (seeScheme 3). Although starting closer to the conical intersectionand at a steeper side of the S0 potential energy surface, analysisof the relaxation coordinate and comparison with the excited-state motion (6 f CIb; see Figure 3e) reveal a drastic changein the direction of motion to populate these channels. Thus, wemay conclude that semibullvalene formation is likely to occurfor dynamic reasons (S1 motion is continued on S0 along thebarrierless minimum energy path to semibullvalene), whilecyclooctatetraene back recovery (and double-bond shifting) isfavored for energetic reasons, so that both processes may be

(48) Iwamura, H.; Morio, K.Bull. Chem. Soc. Jpn.1972, 45, 3599.(49) Jackman, L. M.; Fernandes, E.; Heubes, M.; Quast, H.Eur. J. Org. Chem.

1998, 2209, 9.(50) Quast, H.; Seefelder, M.Angew. Chem., Int. Ed.1999, 38, 1064.(51) Seefelder, M.; Quast, H.Angew. Chem., Int. Ed.1999, 38, 1068.(52) Quast, H.; Heubes, M.; Dietz, T.; Witzel, A.; Boenke, M.; Roth, W. R.

Eur. J. Org. Chem.1999, 813, 3.

Figure 4. CASSCF/6-31G* optimized structures forDIR ′ (7) (a) andCOTtr (5) (b).

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easily photoinduced, via suitably tuning reaction conditions.Remarkably, semibullvalene photoinduced formation has beenobserved to be more efficient in the gas phase1 (i.e., dynamiccontrol; see also section 4.1).

Finally, a fourth (unfavored) relaxation path has been locatedquite far fromCIb (8 au), which funnels a C2-C7 transannularσ-bond formation process leading to bicyclo-[4.2.0]-octa-2,4,6-triene (3). It is worth noting that, due to the big distance, doubtsexist about a direct connection withCIb, and population of thisdeactivation channel (prompted byCIb deactivation) seems tobe quite difficult. Moreover, cyclobutene ring formation suggestsa kinked conical intersection as a more suitable origin23-26,32

(although with an inverted direction in the pyramidalization withrespect to the one found at the carbon atom C2 inCIst). On theother hand,CIb represents the only deactivation funnel easilyaccessible on S1 by photoexcited cyclooctatetraene. In conclu-sion, although unfavored, still this path seems to be the onlyfeasible route for isomer3 photogeneration. This will allow us

to explain concomitant benzene and acetylene formation fol-lowing cyclooctatetraene irradiation (see section 4.4).1 Finally,isomer 3 photogeneration may also be seen as involvingcyclohexadiene ring formation. This is an alternative way tosee the process. This photoinduced reaction has been widelyinvestigated32 and involves a decay from a conical intersectionwhich is more similar toCIb (due to the presence of an allylradical) and which in fact resembles the relaxation path foundfrom CIb to 3.

Because of the small6 f CIb activation barrier, internalconversion is very likely to occur viaCIb. If this is the case,CIb provides the rationale for cyclooctatetraene photochemistry(i.e., photoinduced semibullvalene formation and double-bondshifting) as we have previously shown.22 Schemes 2 and 3 reportstructures and relative (to semibullvalene) PT2-corrected ener-gies.47

3.4. Thermal Chemistry: S0 Reaction Paths.Novel ground-state paths will be documented in the first subsection below,

Scheme 3

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which connectCIst and CIb photoproducts. In addition, alsopreviously studied thermal processes (i.e., valence isomerization,Cope rearrangement, ring inversion, and double-bond shift) havebeen reinvestigated here to have both an improved energetic(via PT2 computations) and a homogeneous full-comprehensivereactivity scheme embracing the full network of excited-stateand ground-state reaction channels. Finally, these results willbe used for comparison with the available experimental data.The good agreement shown below provides a validation for thecomputational method employed.

Thermal Paths Interconnecting CIst and CIb Photoprod-ucts. As shown in section 3.3, intermediate7 is very unstableand highly reactive. Provided it is formed, the cyclopropanering easily opens (i.e., breaking of the previously formed 1,3C-C σ bond), promoting a 2,3π-orbitals coupling (whichrestores the octagonalπ-system). This process involves anasymmetric low-energy barrier transition state (TSDIR ′; seeFigure 6a and Table 2). Thus,TSDIR ′ connects7 to 5 (seeScheme 4); isomer5 is formed due to the pyramidalization ofC2, which in fact favors a trans-double-bond coupling. Therefore,a minimum energy path exists which connects the two possiblephotoproducts (7 and 5) directly generated via deactivationthrough CIst (see section 3.3 ). Anyway, another interestingtransformation (all along aCs-symmetric minimum energy path)may involve7; via a coupling of the two left unpaired electrons,a 7 f 2 conversion occurs. At the CASSCF level, this processis a two-step reaction (see Table 2 for the energetics). First,7is converted to its isomerDIR ′′ (Figure 6c) via the transitionstateTS′, (Figure 6b) which corresponds to aC2-pyramidal-ization inversion process. Second, C2-C6 σ-bond formationoccurs to give semibullvalene via the transition stateTSSBV

(Figure 6d). Anyway, PT2 corrections show that this processcorresponds to a single-step reaction (as shown in Scheme 4)

with a unique transition state (TSSBV), the structures7, TS′,DIR ′′, andTSSBV being of increasing energy, respectively (seeTable 2). Scheme 4 reports also the relative (to semibullvalene)PT2-corrected energies.47

Cope Rearrangement.The bicyclo-[3.3.0]-octadienyl diradi-cal (DIR ) has already been recognized as the transition statefor Cope rearrangement (see section 3.3) between the twosemibullvalene isomers2 and2′, as previously pointed out bythe work of Jiao et al.20,21 and Castan˜o et al.16 The PT2- andZPE-corrected barrier for Cope rearrangement is 6.9 kcal/mol,very close to the experimental value of 5.5 kcal/mol15 and therecently computed value of 6.0 kcal/mol.16

Valence Isomerization. Castan˜o et al.16 have recentlyinvestigated the reversible semibullvalene to cyclooctatetraenevalence isomerization process (see eq h), calculating a barrierof 47.4 kcal/mol for2 f 1 interconversion and locating abifurcation point16,19along the minimum energy path downhillfrom theC2 transition state (TS2VI ; see Figure 7a) toward thetwo semibullvalene isomers2 and2′; see Scheme 5. Here, wehave computed (PT2 plus ZPE corrections) a significantlysmaller barrier for this process (37.6 kcal/mol), in much betteragreement with the reported 39.8 kcal/mol experimental value.14

Moreover, we have found that, although very close in energy,2 is slightly more stable than1 (only a 1.3 kcal/mol separation;see Table 2). Anyway, energy separation is below PT2 errors(around 1-2 kcal/mol), so that no conclusive statement maybe inferred. We have also reinvestigated the second (energeti-cally more favored) valence isomerization process involving1f 3 interconversion (for a previous computational study, seeref 18). The computed (PT2- and ZPE-corrected) energy barrierfor the correspondingCs-symmetric transition state (TS1VI ; seeFigure 7b) is 26.9 kcal/mol, very close to the experimental value

Figure 5. CASSCF/6-31G* optimized structures forDIR (a), TSCR (b), SBV (2) (c), COT (1) (d), andTSRI (e).

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of 28.1 kcal/mol.12 Geometry of3 is shown in Figure 7c. Scheme5 reports also the relative (to semibullvalene) PT2-correctedenergies.47

The mechanistic picture displayed in Scheme 5 is supportedby recent experimental observations for the thermal isomeriza-tions of substituted semibullvalenes and cyclooctatetraenes.49-52

Interestingly, Quast et al. have shown that the rates for the Copeand the2 f 1 rearrangements are somehow correlated. Thesefindings nicely fit with the theoretical results documented hereand in ref 16 and can be easily interpreted in terms of thepotential energy surface reported in Scheme 5. Moreover,formation of two noninterconvertible double-bond shiftingisomers of a highly substituted cyclooctatetraene (compounds

14b and 15b in ref 52) has been described. These compoundsvery likely arise from two isomeric transition structures of theTS2VI -type. Obviously, in the thermal rearrangements of semi-bullvalenes, one bifurcation point exists for each singleTS2VI -type transition state.

Double-Bond Shifting and Ring Inversion.As stated in theIntroduction, thermal cyclooctatetraene double-bond shifting (1f 1DBS) and ring inversion (1 f 1′) have been widelyinvestigated both experimentally and computationally.3-7 Thesestudies have lead to energy barriers of about 14 and 10 kcal/mol8-11 for the associated transition states,TSDBS and TSRI,respectively (see Figures 3a and 5e). As previously shown byBorden,3-5 CASSCF may properly describe these processes. In

Figure 6. CASSCF/6-31G* optimized structures forTSDIR ′ (a), TS′ (b), DIR ′′ (c), andTSSBV (d).

Scheme 4

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fact, via CASSCF/6-31G* optimizations (plus ZPE corrections),we calculate energies which are in very good agreement bothwith previous computations3-5 and with experiments:8-11 14.5and 10.9 kcal/mol, respectively (see Table 2). Anyway, whencorrecting for dynamic correlation (PT2 values in Table 2), thetwo transition states come very close in energy (about 13 kcal/mol both). They even invert their order,TSDBS being lower (11.0kcal/mol) thanTSRI (12.9 kcal/mol), when we include ZPEcorrections. Yet, the energy difference is within the expected

error bars of the CASPT2 method. Analysis of the optimizeddouble-bond shifting transition structure (TSDBS) reveals that ithas exactly the same geometry as the relaxed planarD8h-symmetric minimum found on S1 (6, COT* ) and shown inFigure 3a. Provided we consider this point (TSDBS) higher inenergy (as the CASSCF results3-5 and the experimental data8-11

show; see also Table 2), aD4h-symmetric bifurcation point iseasily predicted along the path connectingTSDBS to thecyclooctatetraene product wells. This point splits the minimum

Figure 7. CASSCF/6-31G* optimized structures forTS2VI (a), TS1VI (b), and the bicyclo-[4,2,0]-octa-2,4,7-trieneBIC (3) (c).

Scheme 5

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energy path into two branches, ultimately leading toward thetwo possible ring-inverted cyclooctatetraene isomers (1 and1′),thus avoiding the lower lying transition state for ring inversion(TSRI). Scheme 6 depicts this situation, reporting also thestructures and relative (to cyclooctatetraene) PT2-correctedenergies. This picture has been recently fully confirmed by thework of Castan˜o.19b

4. Discussion

4.1. Dynamics versus Energetic Control: Vibrationally“Hot” and Vibrationally “Cold” Photochemistry. In section3.3, we have shown that the semibullvalene relaxation pathdeparting fromCIb is, substantially, the continuation (throughCIb and DIR ) of the S1 path. Anyway, the two additionaldegenerate relaxation paths also originating in the proximity ofCIb and leading back to the reactant (1) and to its double-bondshifting isomer (1DBS) are energetically more favored becausethey begin closer (1.5 au distance) to theCIb point and developalong steeper regions of the S0 energy surface (see Scheme 3).Furthermore, their relaxation coordinate indicates that, withrespect to the S1 transition vector (see Figure 1e) and in contrastto the semibullvalene path, a drastic change in the direction of“motion” is needed to populate these paths.

The above results do not provide evidence for the formationof the previously proposed1 trans,cis,cis,cis-cyclooctatetraene(5) as an intermediate for the two-photon generation ofsemibullvalene. Although such a structure has been shown tobe formed via deactivation throughCIst (see section 3.3), stillthis path has been located much higher in energy to provide acompetitive deactivation funnel for photoexcited cyclooctatet-raene.

Interestingly, the documented S0 relaxation paths provide acomputationally based model for predicting the productsobserved in either vibrationally hot or vibrationally coldcyclooctatetraene photochemistry.1,2 In the case of a low amountof S1 vibrational excess energy,COT* would reach theCIb

region with a small momentum along the reactive mode shownin Figure 3. In these conditions, decay to S0 would result in thepopulation of the closer and steeper relaxation paths, yielding

ground-state cyclooctatetraene together with its double-bondshift tautomer only. This interpretation has been validated inour previous work on cyclohexadiene/hexatriene photoinducedinterconversion.32 On the other hand, in the presence of avibrationally excited reactant, at least part of the S1 populationwill be channeled along the transition vector with enoughmomentum to be displaced toward the semibullvalene relaxationpath. As a consequence, the photolysis of cyclooctatetraenewould yield a mixture of semibullvalene and double-bond-shifted cyclooctatetraene photoproducts. This model seems toprovide an explanation for the fact that semibullvalene can beefficiently produced in the gas phase (vibrationally hot) pho-tolysis of cyclooctatetraene.2 This model would also explainthe low photochemical reactivity displayed by polysubstitutedcyclooctatetraenes such as 1,3,5,7-tetramethyl-cyclooctatetraene,which does not lead to a detectable production of 1,3,5,7-tetramethyl-semibullvalene.1 This would be the result of a masseffect due to the 1,3,5,7-substituents. In fact, because asignificant displacement of these substituents is needed to boostpopulation along the semibullvalene relaxation channel (i.e.,COT* f CIb f SBV path; see Scheme 7), increasing the massof the substituents (i.e., replacing four hydrogen atoms incyclooctatetraene) will clearly slow this motion. Thus, inertiaand steric factors would favor reactant back formation anddouble-bond shifting. Moreover, for a polysubstituted cyclooc-tatetraene (such as 1,3,5,7-tetramethyl-cyclooctatetraene), thepaths to the lowest energy conical intersection (theCIb

equivalent) could even be higher in energy, restraining thischannel and, consequently, semibullvalene photogeneration.

Scheme 6

Scheme 7

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Finally, it is worth noting that double-bond shifting (andreactant back formation) may also arise via radiative/nonradia-tive vertical decay from structure6 (the S1 D8h-symmetricCOT* ). We have seen that6 is identical to the transition statefor double-bond shifting (TSDBS), which is about 20 kcal mol-1

lower in energy. Therefore, cyclooctatetraene deactivation wouldresult in a6 f TSDBS process, which populates the thermaldouble-bond shifting channel and favors double-bond shiftversus semibullvalene formation even more. This decay routeis favored in conditions (such as in a cold environment) wherethe system does not have enough energy to be channeled towardthe low-energy conical intersection (CIb) and/or the correspond-ing energy barrier is too high (as it may occur for somepolysubstituted cyclooctatetraenes).

4.2. Prototype Conical Intersections in Conjugated Hy-drocarbons.We have provided evidence that the photochemicalbehavior of cyclooctatetraene is dominated by the presence ofa boat-shaped conical intersection funnel (CIb), featuring a newtype of molecular and electronic structure, as previouslydocumented in our work.22 The analysis of this point showsthatCIb has a tetraradical character with two unpaired electronslocated on two opposite centers (atoms C1 and C5) and the othertwo electrons residing on the allyl moieties C2-C3-C4 and C6-C7-C8 (see Scheme 8a). In the past,53 the same pattern andconical intersection have been documented for the case of theH4 clusters. For these elementary systems, other two geometricand electronic arrangements have been documented for conicalintersections, according to different tetraradical-type interactionpatterns such as rhomboid (Scheme 8b) and triangular (Scheme

8c), respectively. WhileCIb represents the first example founduntil now in conjugated hydrocarbons for a tetrahedral interac-tion scheme in low-energy conical intersections, the other twopatterns have the equivalent in previously optimized structures.For example, the four unpaired electrons of the low-energyconical intersections found in 2,3-di-tert-butyl-buta-1,3-diene27

and in the ethylene-ethylene cycloaddition31 interact accordingto a rhomboid pattern (see Scheme 8b), while benzene,cyclohexadiene, and linear conjugated chains in general havetriangular-type low-energy conical intersections (see Scheme8c).23-26,32 As is clearly shown in Scheme 8, each prototypeconical intersection has its specific recoupling scheme on theground state. Thus, a rhomboid pattern may prompt thesimultaneous formation of two newσ-bonds (leading to bicy-clobutane via a one-step concerted mechanism).27 On the otherhand,-(CH)3- kinked intersections may prompt both cisftrans photoisomerizations and 1,4/1,3σ-bond formations (lead-ing, for example, to cyclopropanation reactions), as seen inbenzene, cyclohexadiene, linear polyenes,23-26,32 and cyclooc-tatetraene (photoinduced formation of intermediate7 from CIst).Finally, tetrahedral interaction at the boat conformer ofCIb

promotes C1-C5 transannular bond formation, leading to theunstable diradical intermediateDIR .22

A major task in our study is to understand and rationalizethe factors playing a role in the selection and stability of low-energy conical intersections. As discussed above, each structurecontrols the outcome of a photochemical process. The knowl-edge of a simple predictive model could provide valuableguidelines for designing selective photochemical processesleading to complex molecular architectures (e.g., polycycliccarbon backbones) through irradiation of simple conjugated(53) Gerhartz, W.; Poshusta, R. D.; Michl, J.J. Am. Chem. Soc.1977, 99, 4263.

Scheme 8

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hydrocarbons.54 Examples of these processes are the productionof bicyclobutanes from butadienes,55 tetrahedranes from cy-clobutadienes,56 benzvalenes from benzenes,57 and semi-bullvalenes from cyclooctatetraenes.1,2,58 An example comesfrom our previous work on (substituted or unsubstituted)butadiene photochemistry.27 Here, we have shown how stericeffects (as the ones caused by strategically placed bulkytert-butyl substituents) may control the outcome of the photochemi-cal process, selecting the prototype (triangular or rhomboid)conical intersection funnel involved in the reaction. Cyclooc-tatetraene allows us to extend this investigation also to electronicfactors. A simple question may help us to catch the issue: whycyclic systems such as benzene (an aromatic hydrocarbon) andcyclohexadiene (and in general linear conjugated hydrocarbons)decay through a low-energy-(CH)3- kink conical intersection,while in cyclooctatetraene this point (i.e.,CIst) is very high inenergy and the favored reactive funnel corresponds to a differentprototype (tetrahedral) crossing (CIb)?

An answer may be found for conjugated cyclic systems viathe examination of all of the possiblelow-energy covalentvalence-bond (VB) configurations, arranging the four unpairedelectrons of a conical intersection. While for benzene (as wellas cyclohexadiene) the lower-energy arrangement may onlyinvolveonedelocalized allyl radical andthreeadjacent unpairedelectrons (i.e., the triangular kink, see Schemes 8c and 9a), incyclooctatetraene, due to its larger size, a new and more stabletetraradical-type configuration is possible. This configuration

corresponds to the electronic structure found inCIb andaccommodates a tetrahedral interaction pattern (see Scheme 8a).Here, we have onlytwo unpaired single-centered (C1, C5)electrons, plustwo delocalized allyl radicals (see Scheme 9b).Obviously, this configuration is electronically more favored thanthe kinked (triangular) one (CIst), where three unpaired electronsare localized on adjacent carbon centers (C1, C2, C3), and onlyone delocalized radical exists. Thus, in conclusion, it appearsthat the existence of different covalent configurations dependson the extension of the system and, in turn, on the energy ofthe orbitals which accommodate the electrons.

By further extending the length of the ring (e.g., to cyclo-decapentaene), we could end up with an even more stabletetraradical-type configuration for a conical intersection. Thisinvolves, according to our model, a four-centered interactionpattern withthreesingly occupied allyl orbitals and onlyonesingly occupied carbon-centeredπ-orbital (see Scheme 9c). Asecond less stable configuration involves onlytwo delocalized(an allyl and a pentadienyl) radicals and may generate a higher-energy conical intersection structure. Finally, a third unstableconfiguration (with onlyonedelocalized heptatrienyl radical)matches a triangular pattern and therefore corresponds to themost unstable-(CH)3- kink crossing point. Provided theseelectronic effects are the only interactions involved (e.g., stericinteractions are ca. equivalent for the different conical intersec-tions), we should end up with the energetic order shown inScheme 9c. On the other hand, unsubstituted linear conjugatedsystems (e.g., all-trans polyenes) can reach (from their side ofthe potential) only triangular-type conical intersections (i.e., the-(CH)3- kinked crossing). For the other interaction patternsto occur, major conformational changes are necessary.

Although qualitative, this picture seems to fit experimentaland computational evidence and could be of predictive value

(54) Gilbert, A. InCRC Handbook of Organic Photochemistry and Photobiology;Horspool, W. M., Song, P.-S., Eds.; CRC Press: Boca Raton, FL, 1995;pp 229-236.

(55) Hopf, H.; Lipka, H.; Traetteberg, M.Angew. Chem., Int. Ed. Engl.1994,33, 204.

(56) Maier, G.Angew. Chem., Int. Ed. Engl.1988, 27, 309.(57) Bryce-Smith, D.; Gilbert, A.Tetrahedron1976, 32, 1309.(58) Zimmerman, H. E.; Grunewald, G. L.J. Am. Chem. Soc.1966, 88, 183.

Scheme 9

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in the photochemistry of conjugated hydrocarbons. Moreover,this model may help us both to design and to control ad hocphotoinduced processes; via tuning the conditions which affectthe stability of VB configurations (and therefore the relatedconical intersection structures), it could be possible, in principle,to select among different photoproducts patterns.

Accurate investigation of larger terms (such as cyclodecap-entaene and longer polyene chains), complemented by explora-tion and characterization of the S1/S0 intersection space (i.e.,then-2-dimensional degenerate space), may shed new light intothis fascinating area.

Finally, it is worth noting that in cyclooctatetraene, the energybarriers toCIb andCIst (15.2 and 44.0 kcal mol-1, respectively)are significantly bigger than the ones computed for low-energyconical intersections in other conjugated systems such asbenzene (∼9 kcal mol-1), cyclohexadiene (<1 kcal mol-1), andlinear chains. In fact, relaxed S1-excited cyclooctatetraene(COT* ) has an extended delocalized structure (i.e., identical1.40 Å π-bonds and planarity, recalling an aromatic system).Therefore, we can think aboutCOT* as being stabilized by akind of aromatic effect which plays against the out-of-planedeformation needed to reach both the boat (CIb) and the kink(CIst) structures (because this motion breaks delocalization, andstabilization effects are lost). This feature makes the paths toout-of-plane crossing points more demanding in energy thanfor the other conjugated systems, which do not display such astabilization effect on S1. While this is obvious for linearpolyenes, also relaxed S1-excited benzene (which correspondsto an anti-Kekule minimum with long- 1.43 Å - π-bonds)26

has partially lost its aromatic stabilization. Therefore, in general,radiationless decay routes via conical intersections are lowerin energy for these systems.

4.3. Cyclooctatetraene as a Double-Bond Shifting Devicefor Photoswitches.Section 4.1 shows that photoinduced SBVformation and double-bond shifting are intrinsically competitiveprocesses, as these are prompted by decay at thesameconicalintersection funnel (CIb). This conclusion seems to imposesevere limitations to the practical use of cyclooctatetraene forthe construction of molecular switches or other types ofphotoswitchable devices. Anyway, we have seen how tuningreaction conditions (e.g., condensed/gas phase, reaction tem-perature, etc.) and/or substitutions (e.g., polysubstituted cy-clooctatetraene) may heavily affect relaxation paths and finalphotoproducts. For instance, in solution or in a cold matrix,where the excess vibrational energy is rapidly removed (i.e., incondition of low vibrational excess energy), the double-bondshifting process should dominate. In fact, if the system reachesCIb with a small momentum (e.g., a vibrationally cold system),or if it does not have enough energy to be channeled along thepath (so that it decays radiatively/nonradiatively fromCOT* ),and/or if the energy barrier is too high (as it may occur in somepolysubstituted cyclooctatetraene), decay to S0 would yieldreactant back formation and double-bond shift isomer only.Therefore, under these conditions, it should be possible, inprinciple, to favor double-bond shifting versus competitivesemibullvalene photogeneration.

A general cyclooctatetraene-based molecular switch has atleast two vicinal substituents (see eq f). As seen in section 4.1,substitution may lock semibullvalene formation so that (evenin a very simple switch) photoinduced double-bond shifting is

favored. This effect should be enhanced as the number ofsubstituents increases. Provided the basic photochemical proper-ties of cyclooctatetraene-based systems remain the same (i.e.,the qualitative shape of the potential energy surfaces involveddoes not change), we may conclude that photochemically con-trolled double-bond shifting can be used to design cyclooctatet-raene-based molecular switches. In particular, if the “on-state”(see eq f) absorbs at a different wavelength than the “off-state”(which is likely to occur for π-substituents), it should bepossible, in principle, to specifically control the conversion ofone isomer to the other, and vice versa.

4.4. Benzene and Acetylene Byproducts Photogeneration.In section 3.3, we have seen how a competitive (but less favoredthanCOT and double-bond shifting) bicyclo-[4.2.0]-octa-2,4,6-triene (3) photogeneration may occur along an additionalrelaxation path fromCIb, leading to C2-C7 transannular bondformation. Product3 may then absorb a second photon to givebenzene and acetylene1 (eq b), which in fact have been observedas concomitant byproducts in cyclooctatetraene irradiationexperiments (see eq a).1 This two-photon mechanistic explana-tion for concomitant benzene and acetylene formation seemsto be more realistic than the one involving a photochemicallyinduced two-step process for the generation of intermediate3.This process corresponds to hot-cyclooctatetraene recovery(from COT* internal conversion) followed by thermal1 f 3valence isomerization. Anyway, benzene formation is mainlyobserved in condensed-phase experiments at low temperatures,1

where generation of hot-reactant and intermediate3 (via thermal1 f 3 valence isomerization) should be very unlikely to occur.Under these conditions, a process (even if it is less favored)directly generating3 from CIb deactivation is certainly preferred.

5. Conclusions

Computational evidence shows that evolution of photoexcitedcyclooctatetraene out of the FC region prompts an efficientradiationless decay of the bright S2 (1E) state into the dark lowestS1 (1A1) state, leading to population of a planarD8h excitedminimum COT* . Nonadiabatic transitions to S0 appear to becontrolled by two different tetraradical-type conical intersections(CIst and CIb), directly connected toCOT* by excited-statereaction paths. Electronic factors playing a role in the stabilityand selection of low-energy conical intersection funnels havebeen analyzed, and a crude qualitative model has been elabo-rated, which allows prediction for the electronic structure oflow-energy crossings in unsaturated cyclic systems.CIst belongsto the family of the-(CH)3- kinked crossings found in smallercyclic terms and linear polyenes; it prompts ground-staterelaxation channels leading both to three- (or four-) memberedring formation (7, 3) and to cisf trans isomerization (5) (i.e.,the typical photoproducts observed from-(CH)3- kinkedintersections). Anyway, very remarkably, this channel is toohigh in energy to provide an efficient way for radiationlessdecay. In fact,CIb has only two unpaired electrons centered onsingle carbon atoms, plus two resonance stabilized allyl radicals.Consequently, different ground-state relaxation channels departfrom this point and are favored. In particular, we have shownhow photoinduced formation of semibullvalene (the primaryphotoproduct) and double-bond shifting are boosted byCOT*deactivation through this novel conical intersection point, whichin fact represents the locus for the ground-state branching ofthe photochemical process.

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We argue that this result provides not only a conclusivemechanistic explanation for the experimentally observed2 cy-clooctatetraenef semibullvalene photoisomerization in the gasphase (eq a), thus correcting previously proposed mechanistichypothesis,1 but also rationalizes photochemical double-bondshifting. In fact, both processes emerge from the same conicalintersection point (CIb), which thus provides the rationale forcyclooctatetraene photochemistry, interconnecting excited-statechannels versus ground-state relaxation paths. The main “chemi-

cal” result reported above is that production of semibullvaleneand double-bond shifting are intrinsically bounded processes.While for the condition of low vibrational excess energy (forinstance, in solution or in a cold matrix where the excessvibrational energy is rapidly removed), or for polysubstitutedcyclooctatetraenes, the double-bond shifting process shoulddominate, we expect semibullvalene formation to be competitivein the presence of a vibrationally excited reactant. This featureshould be properly considered when designing double-bond

Scheme 10

Scheme 11

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shift-based molecular switches or other types of double-bondshift-based molecular devices.

Thermal processes (i.e., thermal conversions between cy-clooctatetraene photoproducts, and reactions such as valenceisomerization, Cope rearrangement, double-bond shifting, ringinversion) have also been investigated to draw a comprehensivereactivity scheme, which rationalizes observations and embracesthe complex network of photochemical (S1) versus thermal (S0)reaction channels and their interconnections. A summary of thefull reaction network is shown in Scheme 10, while a cartoon-like drawing is shown in Scheme 11, which also reports therelative (toCOT* ) PT2-corrected energies for all of the keystructures discussed in the paper.

Acknowledgment. This work was supported by GrantsBQU2000-0646 and BQU2001-2926 of the Spanish D.G.E.S.Funds have been partly provided by the Universita` di Siena(Progetto di Ateneo A.A. 00/02), HFSP (RG 0229/2000-M),and the Universita` di Bologna (Funds for selected researchtopics). CINECA (Italy) is gratefully acknowledged (Grantk1ibozz1).

Supporting Information Available: Experimental data (PDF).This material is available free of charge via the Internet athttp://pubs.acs.org.

JA020741V

Cyclooctatetraene Computational Photochemistry A R T I C L E S

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Role of Bifurcation in the Bond Shiftingof Cyclooctatetraene

OBIS CASTANO,1 RAUL PALMEIRO,1 LUIS MANUEL FRUTOS,1 JOSE LUISANDRES2

1Departamento de Quımica Fısica, Universidad de Alcala, 28871 Alcala de Henares,Madrid, Spain

2Facultad de Ciencias, Universidad Nacional de Educacion a Distancia (UNED),C/ Colon, 114, 08222 Tarrasa, Spain

Received 23 March 2001; Accepted 27 July 2001Published online 0 Month 0000 in Wcom). DOI 10.1002/jcc.10013

Abstract: The present study of the cyclooctatetraene potential energy surface shows the presence of a bifurcation(valley ridge inflection point) in the intrinsic reaction coordinate path between the two transition states of D8h and D4h

symmetries. This result is of capital importance for the correct understanding of the bond shifting and ring inversionprocesses in this compound.

© 2002 Wiley Periodicals, Inc. J Comput Chem 23: 732–736, 2002

Key words: cyclooctatetraene; theory of bifurcation; valley ridge inflexion point; second-order Jahn-Teller effect;multiconfigurational theory; transition state structure

Introduction

Since its initial synthesis by Willstatter,1 1,3,5,7-cyclooctatetraene(COT), a member of the (CH)8 manifold, has been the subject ofnumerous experimental and theoretical studies. The simple Huckelrule2 for this 4n annulene (n � 2) predicts an unfavorable �-elec-tron delocalization energy. Widespread interest in COT startedafter Reppe’s development, which lead to large-scale commercialproduction of this hydrocarbon.3 COT is well known to havealternate both single and double bonds in a tube-shaped nonaro-matic structure of D2d symmetry and to be able to suffer threefundamental changes: ring inversion (RI), bond shifting (BS), andvalence isomerization (VI) (see refs. 4 and 5 and references there-in). Anet and coworkers6 and Oth7 have addressed some experi-mental data towards the understanding of the thermodynamicalproperties of COT and its derivatives. Anet et al.6 have examinedthe RI of COT itself by low temperature NMR studies and havefound for the Gibbs free energy of activation, �G‡, a value of 13.7kcal/mol. Oth7 has investigated the bond shifting and has estimatedvalues of 13.3 and 13.9 kcal/mol for �G‡ and �H‡, respectively.

However, the problem of setting up a quantitative theory forthese three processes taking place in COT has already been at-tempted by different authors using molecular mechanics8 and highlevel quantum chemical methods.9–11 The most relevant quantumchemical results have been presented by Hrovat and Borden.9 Inthis study, these authors have given a satisfactory quantitativeexplanation for the RI process. Their calculations, using multicon-figurational techniques and the complete active space self-consis-

tent field theory (CASSCF), presented geometries of the COTstructures involved in RI and BS processes that have been opti-mized using the 3-21G basis set. Energetic data have been recal-culated with single point energy calculations at the top of thesegeometries. These theoretical results support the existing NMRexperimental data of Anet and coworkers6 that the planar D4h

structure with alternate single and double bonds is the transitionstate (TS) of the ring inversion process.

More controversial are the results for the bond shifting. Hrovat andBorden9 have clearly shown that the D8h symmetric planar delocal-ized COT is the TS involved in this process, but they did not studywhether or not a molecule crossing the D8h transition state should passalso through a D4h planar localized COT. Elucidation of this aspect ofthe theory is of crucial importance for the correct understanding of theinterconversion between the different COT isomers.

In the present study, we have applied the bifurcation theory12,13

in order to get a deeper insight into this important aspect of theinterconversion of COT. We present solid theoretical arguments,and data that will show the BS process in COT avoiding the D4h

localized structure. In agreement with a theorem shown else-where,14 “between two TSs linked by the same intrinsic reactioncoordinate (IRC) path a bifurcation or valley ridge inflection (VRI)point always exists.” In our particular case, someplace between theD8h and D4h transition structures there must be a VRI point, fortwo reasons: these two structures are transition states of BS and RI

Correspondence to: O. Castano; e-mail: Obisd.castano@uah.es

Contract/grant sponsor: DGES; contract/grant number: BQU2000-0646

© 2002 Wiley Periodicals, Inc.

processes, and these two structures are linked by a IRC path, as thepresent study shows. At least, this important result implies, in astraightforward form, the small probability of the BS processpassing through a TS of D4h symmetry.

Results and Discussion

All structures presented in this study and reaction path calculationswere obtained at the CASSCF(8,8)/6-31G* level of theory. Theactive space includes eight �-electrons in the eight �-molecularorbitals. Geometries were fully optimized and stationary pointswere characterized by analytical harmonic frequency calculationsat the same level of theory. All the presented calculations wereperformed with the Gaussian98 suite of quantum chemical pro-grams.15

The higher energy of the D8h COT and its b2g transition vector(2363i cm�1) breaking symmetry in direction to the D4h COTstructure lead us to assume the existence of an IRC path connect-ing these two TSs. Moreover, as a consequence of the results of thepreviously mentioned theorem,14 a VRI point should exist betweenthese two stationary points of D8h and D4h symmetries. In order toverify these statements we started the IRC calculation on the D8h COTtransition state and we used a very small step (0.02 amu�1/2.Bohr)using the Gonzalez and Schlegel algorithm implemented in G98.After the determination of 18 points of the IRC path the systemreaches the TS pertaining to the D4h symmetry. To localize theVRI point we used our own program in conjunction with the G98.This program codifies the method proposed by Miller et al.16 andBaker and Gill.17 From geometries, gradients, and the Hessianmatrix of each IRC point, this code diagonalizes the Hessianmatrix and obtains the projected eigenvalues and eigenvectors forthe subspace orthogonal to translations, rotations, and gradient. AVRI point is localized between a point with new-negative pro-jected eigenvalue and the previously calculated point of the IRCpath. In our case, that took place between points 4 and 5 of thecalculated IRC path, and presents D4h symmetry. The single anddouble carbon-carbon bond lengths of the VRI point are in therange of 1.390–1.387 Å and 1.427–1.430 Å, respectively. InFigure 1, the force constant of the out-of-plane vibration modeagainst the energy descent along the IRC is represented.

At first glance, this result could be considered erroneous, be-cause the calculated Hessian at every point of the IRC is just anupdated Hessian, except for the D8h transition structure. To re-move this doubt, we have carried out a projected analytical fre-quency calculation on the fifth point of the IRC path. Meaninglessdifferences have been observed between updated and analyticalprojected frequencies. It was also necessary to verify the directionof the projected eigenvector corresponding to the new-negativeprojected eigenvalue. This projected eigenvector, which leads thesystem to the loss of planarity, lowers the D4h symmetry of theIRC path to the D2d one, and belongs to the irreducible represen-tation (IR) of symmetry b1u (reverse IRC) or b2u (forward IRC).On the other hand, because this projected eigenvector breaks thesymmetry of the IRC path, it must coincide with one of the normalmodes.18 To verify this point, we have expanded the b2u projectedeigenvector in terms of the normal modes of vibration and haveproved that it coincides with one of the normal modes.

Thus, we show a VRI point exists in the IRC path connectingD8h and D4h transition structures, in full agreement with thepreviously mentioned statement.14 This fundamental result clearlyshows the small probability the system has of going through theD4h transition structure when the bond shifting process occurs.Molecular dynamic calculations could confirm the results of thispaper. Our theoretical results are given in the schematic Figure 2.

To compare these theoretical results with the experimentalvalues of Anet,6 we calculated �H‡ and �G‡ at normal conditions.We also estimated the variation of �G‡ at the temperature ofAnet’s experiment (�10°C), being of the order of 0.1 kcal/mol,that is, practically the same result as that obtained at 298.15 K.These results are given in Table 1.

Here, it seems appropriate to remark on the excellent agreementbetween the measured differences of 3–4 kcal/mol for �G‡ andthe energies required for BS and RI in monosubstituted derivativesof COT.6

In order to elucidate the bifurcation mechanism in terms of thesecond-order Jahn-Teller (SOJT) effect, we also studied whetheror not, at the VRI point, the system breaks symmetry because anexcited electronic state �i (i � 0) of appropriate symmetry ap-proaches close enough energetically to the ground state �0.19a,b

Appropriate symmetry means the excited state �i must belong tothe same irreducible representation (b2u or b1u) of the normal mode

Figure 1. Variation of the force constant of the out-of-plane vibration mode along the IRC between D8h

and D4h COT. Energy descent is the energy relative to D8h COT.

Role of Bifurcation in the Bond Shifting of COT 733

Q with zero curvature K at the VRI point. In this way, Taketsuguet al.19a,b justify the presence of this VRI point recurring to theSOJT effect. To verify this statement we carried out single pointenergy calculations of the 1B1u and 1B2u excited states along theIRC path between D8h and D4h COT transition structures.20 In

Figure 3, the energy difference between 1B1u and 1B2u, and theground state along the IRC are represented.

Two different aspects deserve special attention in Figure 3. Theenergy difference between the excited and ground states alongthe IRC path seems way too large (over 175 kcal/mol) to explainthe breaking of the symmetry of the IRC path vibronic coupling asa perturbative effect; and, the energy difference between theground state and the first 1B2u excited states along the forward IRCand between the first 1B1u excited state and the ground state alongthe reverse IRC monotonally increases from 176 to 184 kcal/molat D8h and D4h transition structures, respectively.

Taketsugu et al.19c have presented IRC calculations that con-tain a VRI point, as in the case studied here, and in which a largeenergy gap appears between the ground state �0 and the firstexcited state with the appropriate symmetry �1 along the wholeIRC. These authors pointed out that the decrease of the energy

Figure 2. Potential energy hypersurface at 0 K for the RI and BS processes and the VRI point. Energiesrelative to COT of D2d symmetry in kcal/mol.

Table 1. Enthalpies and Free Energies, at 298 K, in Hartrees.

Isomer H298 G298 �H‡ �G‡

D2d �307.488875 �307.525450 0.0 0.0D4h �307.472271 �307.506986 10.4 11.6D8h �307.465879 �307.501235 14.4 15.2

Activation barriers �H‡ and �G‡ in kcal/mol.

734 Castano et al. • Vol. 23, No. 7 • Journal of Computational Chemistry

difference when approaching the VRI point leads necessarily to aSOJT effect, which would reverse the sign of K, and originates theVRI point. Firstly, the SOJT effect does not exist, by definition,when the energy difference is so high. Secondly, in the casediscussed here, the energy gap of the IRC shows a trend oppositeto Taketsugu’s statement, that is, it increases along the path.

At the VRI point the change of the sign can be studied con-sidering the following expression for the curvature:

K � K1 � K2 (1)

where K1 and K2 are given by

K1 � ��0��2H

�Q2��0� (2)

K2 � 2 �i�1

��0��H

�Q��i� 2

E0 � Ei(3)

and H is the adiabatic electronic Hamiltonian, which includesCoulomb interactions between electrons and nuclei. As Taketsuguet al.19a,b established, K2 can be approximated by the term thatcontains the excited state closest energetically that belongs to thesame symmetry representation as the nuclear displacement Q.When the energy difference between the ground state and thisexcited state is too high, this term has a small absolute value andpresents a meaningless change even if the energy difference (E0 �E1) between these two states around the VRI point increases ordecreases. Therefore, the change of the sign of the curvature K willdepend not only on the variation of (E0 � E1), but also on the valueK1, because small variations of the latter could be sufficient for asign change in K, and then a VRI point will appear.

Concluding Remarks

In the present study, the RI and BS processes in COT have beenreinvestigated. The principal goal of this paper was to show thatbetween the two COT transition structures of symmetry D8h andD4h connected by the same IRC, a VRI point exists. This result isa direct consequence of our early theorem,14 and clearly shows RIand BS processes to be independent between themselves, exclud-ing the BS process passing through the transition state of symme-try D4h. At the same time, the presence of a second-order Jahn-Teller effect at the VRI point, where the system breaks symmetry,is also excluded.

Acknowledgements

We are most grateful to “Servicios Informaticos de la Universidadde Alcala” for computer facilities and to the Spanish DGES(project BQU2000-0646) for financial support.

References

1. (a) Willstatter, R.; Waser, E. Dtsch Chem Gest 1911, 44, 3423; (b)Willstatter, R.; Heidelber, M. Dtsch Chem Gest 1913, 46, 517.

2. Huckel, E. Z Electrochem 1937, 43, 752.3. Reppe, W.; Schlichting, O.; Klager, K.; Toepel, T. Liebigs Ann 1948,

560, 1.4. Paquette, L. A. Pure Appl Chem 1982, 54, 987.5. Paquette, L. A. Acc Chem Res 1993, 26, 57.6. (a) Anet, F. A. L. J Am Chem Soc 1962, 84, 671; (b) Anet, F. A. L.;

Bourn, A. J.; Lin, Y. S. J Am Chem Soc 1964, 86, 3576.7. Oth, J. F. M. Pure Appl Chem 1971, 25, 573.8. Allinger, N. L.; Sprague, J. T.; Finder, C. J. Tetrahedron 1973, 29,

2519.9. Hrovat, D. A.; Borden, W. T. J Am Chem Soc 1992, 114, 5879.

10. Wenthold, P. G.; Hrovat, D. A.; Borden, W. T.; Linderber, W. C.Science 1996, 272, 1456.

Figure 3. Energy difference between the excited states and the ground state along the IRC path (forward and reverse).

Role of Bifurcation in the Bond Shifting of COT 735

11. Andres, J.-L.; Castano, O.; Morreale, A.; Palmeiro, R.; Gomperts, R.J Chem Phys 1998, 108, 203.

12. Basilevsky, M. V. Chem Phys 1977, 24, 81.13. Valtazanos, P.; Ruedenberg, K. Theor Chim Acta 1986, 69, 281.14. Castano, O.; Frutos, L. M.; Palmeiro, R.; Notario, R.; Andres, J.-L.;

Gomperts, R.; Blancafort, L.; Robb, M. A. Angew Chem Int Ed 2000,39, 2095.

15. Frisch, M. J.; Trucks, G. W.; Schlegel, H. B.; Scuseria, G. E.; Robb,M. A.; Cheeseman, J. R.; Zakrzewski, V. G.; Montgomery, J. A. Jr.;Stratmann, R. E.; Burant, J. C.; Dapprich, S.; Millam, J. M.; Daniels,A. D.; Kudin, K. N.; Strain, M. C.; Farkas, O.; Tomasi, J.; Barone, V.;Cossi, M.; Cammi, R.; Mennucci, B.; Pomelli, C.; Adamo, C.; Clif-ford, S.; Ochterski, J.; Petersson, G. A.; Ayala, P. Y.; Cui, Q.; Moro-kuma, K.; Malick, D. K.; Rabuck, A. D.; Raghavachari, K.; Foresman,J. B.; Cioslowski, J.; Ortiz, J.V.; Stefanov, B. B.; Liu, G.; Liashenko,A.; Piskorz, P.; Komaromi, I.; Gomperts, R.; Martin, R. L.; Fox, D. J.;Keith, T.; Al-Laham, M. A.; Peng, C. Y.; Nanayakkara, A.; Gonzalez,

C.; Challacombe, M.; Gill, P. M. W.; Johnson, B.; Chen, W.; Wong,M. W.; Andres, J. L.; Gonzalez, C.; Head-Gordon, M.; Replogle, E. S.;Pople, J. A. Gaussian 98, Revision A.7; Gaussian, Inc.: Pittsburgh, PA,1998.

16. Miller, W. H.; Handy, N. C.; Adams, J. E. J Chem Phys 1980, 72, 99.17. Baker, J.; Gill, P. M. W. J Comput Chem 1988, 9, 465.18. Palmeiro, R.; Frutos, L.-M.; Castano, O. Int J Quantum Chem, in

press.19. (a) Taketsugu, T.; Hirano, T. J Chem Phys 1993, 99, 9806; (b)

Taketsugu, T.; Hirano, T. J Mol Struct (Theochem) 1994, 130, 169;(c) Yanai, T.; Taketsugu, T.; Hirao, K. J Chem Phys 1997, 107,1137.

20. To obtain these excited states, the active space of CASSCF calcula-tions must be increased. Then, the a1g �*(COH) molecular orbital isincluded in the active space. Thus, ground state and 1B2u and 1B1u

excited states have been obtained on top of several points of this IRCpath.

736 Castano et al. • Vol. 23, No. 7 • Journal of Computational Chemistry

Theoretical Determination of the Singletf Singlet and Singletf Triplet ElectronicSpectra, Lowest Ionization Potentials, and Electron Affinity of Cyclooctatetraene

Luis-Manuel Frutos† and Obis Castano*,‡

Departamento de Quı´mica Fısica, UniVersidad de Alcala´ , Alcala de Henares, ES-28871 Madrid, Spain

Manuela Merchan* ,§

Departamento de Quı´mica Fısica, Instituto de Ciencia Molecular, UniVersitat de Vale`ncia, Dr. Moliner 50,Burjassot ES-46100 Valencia, Spain

ReceiVed: February 11, 2003; In Final Form: May 14, 2003

The singletf singlet and singletf triplet electronic spectra of cycloocta-1,3,5,7-tetraene are studied usingmulticonfigurational second-order perturbation theory (CASPT2) and extended atomic natural orbitals (ANOs)basis sets. The observed dipole-allowed features at 4.43, 6.02, and 6.42 eV and the spin-forbidden singletftriplet bands with maxima at 3.05, 4.05, and 4.84 eV (Frueholz, R. P.; Kuppermann, A.J. Chem. Phys.1978,69, 3614) are assigned as the transitions 11A1 f 11A2, 11A1 f 21B2(3pz), 11A1 f 31E, and 11A1 f 13A2,11A1 f 13E, 11A1 f 13B1, respectively. The lowest (3s) Rydberg singlet and triplet states are placed at 5.58(21A1) and 5.54 (13A1) eV. New assignments are tentatively suggested for the recorded higher-energy peaksin the singlet manifold. The three lowest ionization potentials are also characterized. At the highest level oftheory, the computed adiabatic electron affinity is 0.56 eV, in agreement with experimental determinations.As regards the role of cyclooctatetraene as an efficient triplet quencher of laser dye solutions, the computationsupports it could act as acceptor with donors having a triplet excited-state energy equal to or higher than 0.8eV.

1. IntroductionConsiderable effort has been devoted to understand the

thermal and photochemical reactivity of cyclooctatetraene (COThereafter) from both experimental and theoretical standpoints(for a recent review see Garavelliet al.1). The electronic spectraof COT have comparatively received less attention, and only afew experimental2-4 and theoretical1,5-8 studies are available.The ground state in the gas phase determined by electrondiffraction (ED) techniques has a tub-shaped structure withalternating bond lengths.9 Theoretical calculations1,8 have alsorepetitively predicted a ground-state structure belonging toD2d

symmetry consistent with the ED data. The COT molecule hasbeen considered along the years a classic prototype of a cyclicnonaromatic hydrocarbon. Because of its nonplanar structure,through-bond and through-space interactions between the doublebonds have to be taken into account in the theoretical descriptionof the excited states. The implied difficulties in the ab initiocharacterization of the electronic spectra of COT are moreclosely related to a strained, nonplanar diene like norbornadiene(bicyclo[2.2.1]hepta-2,5-diene)10 than to even polyenes such asall-trans-1,3,5,7-octatetraene.11 Apart from the intrinsic interest,characterization on theoretical grounds of the singletf singletand singletf triplet spectra can give further insight to theunderstanding of the behavior of COT as a triplet quencher.The use of COT as an efficient triplet scavenger for severallaser dye solutions is well-known,12,13 and the detailed mech-anism for the energy-transfer process has been recently ana-lyzed.14 In this context, a high-level ab initio description of theelectronic spectra of COT seems to be appropriate at present.

We report here a comprehensive theoretical research study ofthe excited states of COT, of both valence and Rydberg nature.

The gas-phase optical absorption of COT can be describedas a broad band of low intensity over the region 310-260 nm(4.00-4.77 eV) with a maximum at 4.39 eV and an intenseband with a shoulder at 6.05 eV and a maximum at 6.42 eV(limit of the experimental study).5,8 These singletf singlettransitions were also observed by electron energy-loss spec-troscopy (EELS)3 at 4.43, 6.02, and 6.42 eV. In addition,Frueholz and Kuppermann3 identified three singletf tripletfeatures with an intensity maximum at 3.05, 4.05, and 4.84 eVand several singletf singlet transitions at higher energies upto 14 eV. No clear experimental identification of the lowestRydberg states was, however, possible, and the assignmentsperformed by these authors were strongly biased by previoussemiempirical results where only valence excited states couldbe calculated.3 As discussed below, the lowest Rydberg stateof COT is firmly predicted from our results around 5.6 eV, andthe transition from the ground to a 3p-Rydberg state computedat 6.17 eV is proposed as the most plausible candidate for therecorded shoulder at about 6 eV.3,5,8 The latter was previouslyassociated with a valence feature.3

The excited states of COT are characterized by usingmulticonfigurational second-order perturbation theory throughthe CASPT2 method.15,16 The successful performance of theCASPT2 approach in computing spectroscopic properties is wellestablished.17-19 We believe that the present results provide amore complete picture for a better understanding of thespectroscopic behavior of the system and allow us to makeconfident assignments. The present research includes analysisof the vertical transitions related to the singletf singlet andsingletf triplet spectra,3 assignments for the lowest ionizationpotentials (IPs)20-22 of the COT molecule, computation of the

† E-mail: luisma.frutos@uah.es.‡ E-mail: obisd.castano@uah.es.§ E-mail: manuela.merchan@uv.es.* Corresponding author.

5472 J. Phys. Chem. A2003,107,5472-5478

10.1021/jp030193n CCC: $25.00 © 2003 American Chemical SocietyPublished on Web 06/21/2003

main features of the photoelectron (PE) spectrum of thecyclooctatetraene radical anion (COTRA),23,24determination ofthe adiabatic electron affinity (EA) for the neutral system,25 andan estimate for the band origin of the lowest singletf triplettransition. The latter issue is particularly relevant to support thenonvertical triplet energy transfer mechanism to COT, behavingas acceptor, in a quenching process.12 We shall find that anumber of previous assignments are clearly confirmed, but thesituation is shown to be even more complex in certain casesand new interpretations are then offered. The current resultsserve both to complement earlier studies and to bring a firmfoundation for a theoretical elucidation of some aspects involvedin the rich photochemistry of COT.

The paper is organized as follows. Computational details aredescribed in the next section. Results and analysis on thecomputed singletf singlet spectrum, IPs, low-lying tripletstates, EA, and PE spectrum of COTRA are subsequentlyconsidered, together with comparison to previous findings. Inlight of the present results, the role of COT in laser dyequenching is finally discussed. Our conclusions are summarizedin the last section.

2. Theoretical Methods and Computational Details

Generally contracted basis states of atomic natural orbital(ANO)-type obtained from the C(14s9p4d)/H(8s4p) primitivesets with the C[4s3p1d]/H[2s1p] contraction scheme26 were usedfor the geometry optimization of the ground state at the CASSCFlevel.27 Actual calculations were performed inC2V symmetry,imposing symmetry constraints of theD2d symmetry both inthe variation of the geometric parameters and in the CASSCFwave function. The orientation of the molecule is depicted inFigure 1.

The closed-shell restricted Hartree-Fock wave function has28 occupied molecular orbitals (MOs) denoted by (5;7;4;3;2),where the entries indicate the number of occupied MOsbelonging to the irreducible representations a1, e, b2, b1, and a2of the D2d point group. In the present case, with the mappingD2d/C2V a1 + b2/a1, e/b1 + b2, b1 + a2/a2, the SCF wave functioncan be represented by (9;7;7;5), corresponding to the numberof occupied MOs of the a1, b1, b2, and a2 symmetries of theC2Vpoint group. For the ground-state geometry optimization, theπ-valence space was employed. It comprises eight activeelectrons distributed among the eightπ valence MOs, (2;2;2;2)in C2V symmetry, keeping the remaining MOs inactive (7;6;6;5)(hereafterπ-CASSCF). The canonicalπ-valence MOs computedat the ground-state optimized geometry are depicted in Figure2, where the close-lying 3b1(D2d) MO of σ character is also

included. As illustrated in many previous applications,17-19

analysis of the nature and spacing of the canonical MOs isusually helpful and shall be employed later to rationalize themost important spectroscopic features obtained from morecomplex CASSCF wave functions.

The 3b1, 4b2, 7e, 5a1, 3a2, 9e, and 5b1 canonical MOs (D2d)are computed inC2V symmetry as the 5a2, 8a1, 7b1(7b2), 9a1,6a2, 9b1(9b2), and 9a2 MOs, respectively. The equilibrium geo-metry for the ground-state COTRA (12B1u of the D4h pointgroup) and the lowest triplet state (13A2g of theD8h symmetry)were also determined at the same level of calculation (π-CASSCF wave function employing the valence basis setC[4s3p1d]/H[2s1p]). Using the mappingD2d/D4h/D8h, theπ-va-lence MOs in increasing ordering of orbital energy transformas follows: b2/a2u/a2u, e/eg/e1g, a1 + a2/b2u + b1u/e2u, e/eg/e3g,and b1/a1u/b1u. Actual calculations of planar COT (D4h andD8h

point groups) were carried out inC1 symmetry. To calculateenergy differences (vertical and nonvertical excitation energies,IPs, and EA), as well as for the computation of transitionproperties, the basis set was supplemented with two s-, two p-,and two d-type diffuse functions (see exponents elsewhere28)placed at the origin of the molecule. The extended basis setconsists of 202 basis functions (666 primitives). The total SCFenergy for ground-state COT, employing the C[4s3p1d]/H[2s1p]+ 2s2p2d basis set, was calculated to be-307.619 000 au atthe D2d optimized geometry (see below).

The molecular orbitals of the excited states have beenobtained from state average CASSCF calculations, where theaveraging included the states of interest for a given symmetry.

Figure 1. Orientation of the cycloocta-1,3,5,7-tetraene molecule (D2d

symmetry).

Figure 2. Highest five occupied andπ-valence unoccupied canonicalMOs computed with the ANO-type C,N,O[4s3p1d]/H[2s1p] basis setat the ground-stateπ-CASSCF equilibrium geometry of cycloocta-1,3,5,7-tetraene (see text). Orbital energies (within parentheses) are alsoincluded.

Electronic Properties of Cyclooctatetraene J. Phys. Chem. A, Vol. 107, No. 28, 20035473

The respective CASSCF wave functions were subsequently usedas reference function in a second-order perturbation treatmentby means of the CASPT2 method.15,16In addition, the effect ofweakly interacting intruder states was minimized by using theso-called imaginary level-shift technique.29 After a carefulanalysis on the influence of the shift value on the spectroscopicresults, the shift of 0.2 au was selected to collect the resultscompiled in the next section. A detailed account of the activespaces employed to compute the valence and Rydberg excitedstates can be found in Table 1.

In summary, theπ-valence active space was extended toinclude the Rydberg orbitals of the different symmetries, asappropriate. Initially, we wanted to study the singletf singletelectronic spectra up to 7 eV, since within this energy the mostimportant features of the spectrum can be expected. Neverthe-less, because of the large differential correlation energy betweensome valence excited states with respect to the ground state,the number of CASSCF roots had to be increased considerably.For instance, the valence state1B2(3b1 f 3a2) (D2d labeling)corresponds to the thirteen root of the first set of calculations(1A1(C2V)) listed in Table 1. As we shall see next, this state isplaced at 10.09 eV at the CASSCF level but at 6.14 eV oncedynamic correlation contributions are taken into account withinthe framework of the CASPT2 method. As a byproduct of therequired computation to consider all the excited states below 7eV, a number of additional valence and Rydberg excited stateswere obtained. In the second set of calculations described inTable 1, the E(ππ*) state responsible of the most intensetransition appears as the seventh root of the CASSCF computa-tion, above Rydberg states converging to the second and thirdIPs. To avoid their influence as intruder states in the treatmentof the valence1E states, those Rydberg states had to be explicitlyconsidered. Therefore, theπ-valence active space was conse-quently enlarged to include the 3s and 3dxy MOs belonging tothe a1 and a2 irreducible representations of theC2V point group.The vertical energy of each excited stated was referred to theground-state energy computed with the same active space. Sincevertical singletf singlet and singletf triplet electronic spectrawere computed inC2V symmetry and dynamic correlation effectshave a tremendous influence to determine accurately severalimportant transitions, the computation turned out to be particu-larly challenging from a technical standpoint.

The CASSCF state interaction (CASSI) method30,31was usedto calculate the transition dipole moments. In the formula forthe oscillator strength the CASSCF transition moment and theenergy difference obtained in the CASPT2 computation wereused.

All calculations were performed with the MOLCAS-5quantum chemistry software.32

3. Results and Discussion

3.1. Vertical Singletf Singlet Electronic Transitions.Thevertical singletf singlet electronic transitions were computedat theπ-CASSCF equilibrium geometry of COT. The optimizedparameters,r(CdC) ) 1.341 Å, r(CsC) ) 1.477 Å, r(CsH)) 1.078 Å, ∠(CdCsC) ) 127.1°, ∠(CdCsH) ) 117.6°,∠(CdCsCdC) ) 55.1°, are close to those reported in the EDstudy (r(CdC) ) 1.340 Å,r(CsC) ) 1.476 Å,∠(CdCsC) )126.1°, ∠(CdCsH) ) 117.6°).9 The results obtained by meansof the CASSCF/CASPT2 procedure are collected in Table 2.The first column identifies the excited states of the COTmolecule (labeling according to theD2d point group). The secondand third columns report the vertical excitation energy computedat the CASSCF and CASPT2 levels, respectively. The differencebetween those two energies is a measure of the contribution ofthe differential dynamic correlation effects to the vertical

TABLE 1: CASSCF Wave Functions Employed To Compute the Vertical Electronic Excited States of COT at theGround-State Optimized Geometry (D2d Symmetry)a

wave functionb Nec statesd Nconfigurations

e Nstatesf

Computed States in1A1(C2V) Symmetry (Inactive: 5;4;4;2)CASSCF(6;2;2;3) 10 5a1 f 3s, 3pz; 3dx2-y2, 3dz2; ππ*; ππ*; ππ*; 4b2 f 3s, 3pz; 3dx2-y2, 3dz2; 3b1 f 3a2; ππ* 108 001 14

Computed States in1B1(C2V) Symmetry (Inactive: 5;4;4;3)CASSCF(3;4;2;3) 8 5a1 f 3px,y, 3dxz,yz; ππ*; 7e f 3s, 3dxy; 4b2 f 3px,y; ππ*; 4b2 f 3dxz,yz 17 640 9

Computed States in1A2(C2V) Symmetry (Inactive: 5;4;4;3)CASSCF(2;2;2;3) 8 5a1 f 3a2; 5a1 f 3dxy; 4b2 f 3a2; 4b2 f 3dxy 1 308 4

a Actual calculations were carried out inC2V symmetry.b The ANO-type basis set C[4s3p1d]/H[2s1p]+ 2s2p2d Rydberg functions placed at theorigin were used (see text). Within parentheses are the number of orbitals of the symmetries a1, b1, b2, and a2, respectively, of the point groupC2V.Number of frozen orbitals: (2;2;2;2). Number of occupied SCF MOs: (9;7;7;5). The 3b1, 4b2, 7e, 5a1, and 3a2 MOs of D2d symmetry correspondto the 5a2, 8a1, 7b1(7b2), 9a1, and 6a2 MOs of theC2V symmetry, respectively. The character and ordering of the states are those obtained with theCASSCF method.c Number of active electrons.d Nature of the computed states according to theD2d point group (see Figure 2).e Number ofconfigurations in the CASSCF wave function.f States included in the average CASSCF wave function.

TABLE 2: Computed CASSCF and CASPT2 ExcitationEnergies (eV) and Related Oscillator Strengths (f) for theVertical Singlet-Singlet Electronic Transitions of COT (D2dSymmetry) Employing the C,N,O[4s3p1d]/H[2s1p]+ 2s2p2d(Rydberg Functions) ANO-Type Basis Seta

stateb CASSCF CASPT2 f experimental

Singlet Excited States up to 7 eV11A2(5a1f3a2) 6.70 3.79 forbidden 4.39,c 4.43,d 4.39e

11E(ππ*) 7.88 5.56 0.007521A1(5a1 f 3s) 5.81 5.58 forbidden21E(5a1f3px,y) 6.69 5.93 0.000431A1(ππ*) 6.84 6.14 forbidden11B2(3b1 f 3a2) 10.09 6.14 0.011121B2(5a1 f 3pz) 6.21 6.17 0.0532 6.05,c 6.02d

11B1(4b2 f 3a2) 8.02 6.36 forbidden31E(ππ*) 10.28 6.40 1.1096 6.42,c,d 6.16e

41A1(5a1 f 3dz2) 6.76 6.57 forbidden31B2(5a1 f 3dxy) 7.22 6.71 (-)f

41E(5a1 f 3dxz,yz) 7.35 6.78 0.001421B1(5a1 f 3dx2-y2) 6.75 6.80 forbidden

Additional Valence States1A1(ππ*) 8.72 7.41 forbidden1B2(ππ*) 8.38 7.68 0.00011A1(ππ*) 10.57 7.98 forbidden

Additional Rydberg States1E(7ef 3s) 8.12 7.04 0.0409 6.99d

1E(7ef 3dxy) 8.86 8.24 0.00021B2(4b2 f 3s) 8.93 8.39 0.0250 8.41d

1E(4b2 f 3px,y) 9.90 8.80 0.03901A1(4b2 f 3pz) 9.32 8.95 forbidden1E(4b2 f 3dxz,yz) 10.50 9.26 0.1346 9.05d

1A1(4b2 f 3dxy) 10.35 9.49 forbidden1A2(4b2 f 3dx2-y2) 9.86 9.58 forbidden1B2(4b2 f 3dz2) 9.88 9.62 0.0187

a Experimental data are also included.b The character and orderingof the states are those obtained with the CASPT2 method.c From theoptical spectrum.5,8 d EELS data.3 e From the absorption spectrum inhexane.4 f The state was computed in A2 (C2V) symmetry, andf couldnot be computed because the transition is dipole forbidden.

5474 J. Phys. Chem. A, Vol. 107, No. 28, 2003 Frutos et al.

excitation energy of the state. The computed oscillator strengthis listed in the fourth column. Finally, the available experimentaldata are compiled on the right-hand side of Table 2.

The lowest singlet excited state of COT hasA2 symmetry. Interms of the corresponding natural orbitals (NOs), the CASSCFwave function is dominated by a singly excited configuration.The NOs involved are homologous to the highest occupied MO(HOMO) and the lowest unoccupied MO (LUMO): the one-electron promotion 5a1 f 3a2 (cf. Figure 2). That the lowestsinglet f singlet transition is electric-dipole forbidden wasclearly established by Frueholz and Kuppermann3 in 1978. Thecomputed vertical excitation energy at about 3.8 eV (CASPT2result) is somewhat too low compared to the maximum of thelow-intensity band observed in the optical5,8 (4.39 eV) andEELS3 (4.43 eV) studies. However, the previous ab initio MRCIresult reported by Palmer7 predicted the lowest-energy band topeak at 4.37 eV. The origin of the discrepancy between theMRCI result and the present finding can be attributed to thelimited basis set (split-valence quality) employed for the MRCIcalculations.7 In fact, employing the current ANO-type basisset with the contraction C[3s2p]/H[2s], the CASPT2 methodyields 4.45 eV, consistent with the earlier MRCI result. Addingpolarization functions on the carbon atoms, C[3s2p1d]/H[2s]basis set, the excitation energy drops to 3.92 eV, consistent withour previous CASPT2/6-31G* result (4.00 eV).1 With the largestcontraction tested from the same primitive sets,26 C[5s4p2d1f]/H[3s2p1d], the vertical transition is computed to be 3.80 eV,nearly the same as that compiled in Table 2. Therefore, 3.8-4.0 eV is so far the best theoretical estimate for the lowestvertical feature of COT. This implies that the observed bandmaximum and the computed vertical transition have an energydifference of about half an electronvolt. Possible sources of sucha deviation have been discussed elsewhere.1

As can be inferred from Table 2, states denoted byππ*, withno explicit mention of the configurations involved, haveCASSCF wave functions with a character remarkably multi-configurational. That is, two or more configurations have largeweights in the corresponding CASSCF wave function. On thecontrary, when the CASSCF wave function has only oneconfiguration with a pronounced weight (larger than 70%), anexplicit assignment is given. In particular, the 11E(ππ*) and31E(ππ*) states are described mainly by a linear combinationof two singly excited configurations related to the HOMO- 1f LUMO and HOMOf LUMO + 1 one-electron promotions.According to the orbital energies given in Figure 2, the energydifference between the orbital energies of LUMO/HOMO- 1and LUMO+ 1/HOMO is 12.52 and 12.39 eV. Therefore, thosetwo one-electron promotions are nearly degenerate and canfurther interact, leading to minus and plus linear combinations,which rationalizes the CASSCF findings. Energetically, as aresult of the interaction, the minus and plus states are pusheddown and up, respectively. The electronic transition related tothe former is predicted with low intensity (the respectivetransition moments are subtracted), and the latter should carrymost of the intensity. This is indeed the case. The 11E(ππ*)and 31E(ππ*) states are computed vertically at 5.56 and 6.40eV, respectively. The transition from the ground to the 31E(ππ*)state is predicted to be the most intense feature of the spectrum,with a calculated oscillator strength of about 1.1. The oscillatorstrength for the transition 11A1 f 11E(ππ*) is, however,relatively small (0.008). The weights of the (HOMOf LUMO+ 1, HOMO - 1 f LUMO) one-electron promotions in therespective CASSCF wave function are (in percentage) (15, 29)(11E) and (22, 40) (31E). The apparent good agreement between

the computed excitation energy for the transition 11A1 f 31E-(ππ*) (6.40 eV) with respect to the observed maximum (6.42eV)3,5,8is somewhat fortuitous. The computed vertical excitationenergy for the 31E(ππ*) state is estimated to have an accuracyof (0.2 eV, somewhat larger than the usual error bars of themethod in similar applications (molecules of relatively smallmolecular size with extended ANO-type basis sets).17-19 Asstated above, the 31E(ππ*) state has large dynamic correlationcontributions, appearing as a high root in the average CASSCFprocedure, and many weakly interacting states behave asintruders. Most of them are removed with the employed activespace beyond theπ-valence active space, together with theimaginary level shift technique, but still some influence remains.The excitation energies for Rydberg states described in the samecalculation as the valence1E states (see Table 1) are, however,stable and expected to have the usual accuracy of the method((0.1 eV). Despite these technical problems, the most intensefeature of the spectrum can be unambiguously assigned to theelectronic transition 11A1 f 31E(ππ*), confirming the pioneer-ing theoretical assignment performed by Van-Catledge in 1971on the basis of INDO results.6 Apart from our preliminary resultsreported recently,1 only this INDO study6 characterizes thecomputed electronic transitions by both excitation energies andoscillator strengths. It is clear that both properties are equallyrelevant to perform spectroscopic assignments.

Between the 11E(ππ*) and 31E(ππ*) states, three valencestates are found. At the CASPT2 level, the 31A1(ππ*) and 11B2-(3b1 f 3a2) states are computed to be degenerated at 6.14 eV.The 11B1(4b2 f 3a2) state lies at 6.36 eV above the groundstate, close to the 31E(ππ*) state. The 31A1(ππ*) state has adoubly excited character, with a prominent weight (33.2%) ofthe doubly excited configuration (HOMOf LUMO)2. The11B2(3b1 f 3a2) state is described mainly by the one-electronpromotion from the highestσ orbital (3b1) to the LUMO. Theoscillator strength for the transition 11A1 f 11B2 is 0.011, andtherefore, it can contribute to some extent to the shoulderrecorded in the spectrum at about 6 eV.3,5,8 Incidentally, thetransition 11A1 f 11E(ππ*) was previously related to theobserved shoulder, but this is not confirmed by the presentresults. Instead, we propose the transition from the ground stateto the Rydberg state 21B2(5a1 f 3pz) computed at 6.17 eV, withoscillator strength 0.05, as the most plausible candidate respon-sible for the recorded shoulder in the gas phase at 6.02 eV.3

Our assignment of Rydberg character for the shoulder isconsistent with the fact that it is not observed in the absorptionspectrum of COT in hexane (band maximum at 4.39 and 6.16eV).4 As it is well-known Rydberg states as usually perturbedin condensed phases and collapse in solution.33

To design the number of Rydberg series to consider in thestudy, the lowest ionization potentials were calculated. Theresults are listed in Table 3, together with the selectedexperimental data taken from the PE spectrum.20

TABLE 3: Computed CASSCF and CASPT2 VerticalIonization Potentials (IPs) for COT Employing theC,N,O[4s3p1d]/H[2s1p]+ 2s2p2d (Rydberg Functions)ANO-Type Basis Set

IPa state CASSCF CASPT2 experimentalb

IP1 12A1(π-hole) 8.26 8.08 8.42IP2 12E(π-hole) 9.58 9.66 9.78IP3 12B2(π-hole) 11.32 10.80 11.15

12B1(σ-hole) 12.46 10.9512A2(5a1f3a2) 11.60 11.10

a The character and ordering of the states are those obtained withthe CASPT2 method. Results in eV.b Data taken from the PE spectrumof COT.20

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As previously suggested by different authors,3,7 according tothe Koopmans’ theorem, the lowest four IPs, determinedexperimentally20-22 at 8.42, 9.78, 11.15, and 11.55 eV, couldbe related to the 12A1, 12E, 12B2, and 12B1 states of the COTradical cation (cf. Figure 2). From the CASPT2 results (seeTable 3) one can easily conclude that the assignments basedon the Koopmans’ theorem are correct for the first (IP1) andsecond (IP2) ionization potentials. Three states are, however,found at about 11 eV (IP3): 12B2(π-hole), 12B1(σ-hole), and12A2(5a1 f 3a2). The latter is a non-Koopmans state and,therefore, is not visible in the PE spectrum. As pointed out byPalmer,7 the most intense feature in this energy region of thePE spectrum can probably be related to the 12B2(π-hole) state.Nevertheless, the results rule out the Koopmans-based assign-ment for the fourth IP (IP4) because the 12B1 state is placedaround 11 eV, that is, 0.7 eV below the observed IP4 at 11.55eV. Therefore, IP4 cannot be assigned from the current results.The computed gaps between the second-first (1.58 eV) and third-second PE bands (1.14 eV) are consistent with previoustheoretical predictions at the CASPT2/6-31G* level, 1.42 and1.08 eV, respectively.34 In the recorded PE spectrum20 of COT,the spacing between the first three bands is, however, equal(1.37 eV). Deviations between the theoretical and experimentalvertical ionization potentials (especially for IP1 and IP3) seemto point out that the observed peaks might not correspond tothe vertical ejection of the electron or/and the cation does notretain the COT structure (see discussion in ref 34).

The first and second vertical ionization potentials of COTlie within 1.58 eV of each other (CASPT2 results). As aconsequence, corresponding members of the Rydberg seriesconverging to these two ionization potentials are expected tolie within 1.5-1.6 eV of each other. Consistently, the energydifference between the1E(7ef 3s) and 21A1(5a1 f 3s) statesis computed to be 1.46 eV. The1B2(4b2 f 3s) state appears at8.39 eV, that is, 1.35 eV above the1E(7e f 3s) state. Thatenergy difference can be related to the energy difference IP3-IP2 (1.14 eV). As 1-1.5 eV approximates the usual 3s and 3dRydberg-state separation, the early series members leading tothe first ionization limit are not expected to overlap in energywith the remaining members. Therefore, the main target of thestudy was to characterize the 3s, 3p, and 3d members of theRydberg series converging to the first IP. As stated in theprevious section, to gather the valence states below 7 eV, anumber of additional Rydberg states were also obtained becauseof the intrinsic difficulties of the COT system and availablesoftware.32 (MOLCAS-5 does not offer the possibility to treatthe COT system withinD2d symmetry, and actual calculationswere performed within theC2V point group).

The lowest Rydberg state, the 21A1(5a1 f 3s) state, ispredicted at 5.58 eV. Frueholz and Kuppermann3 saw noevidence of the 5a1 f 3s member in their electron impactspectra. They remarked, however, that the 3s feature should benear 5.2 eV.3 Transition to the 21E(5a1 f 3px,y) state is computedsomewhat above, at 5.93 eV, with an exceedingly smalloscillator strength. The 21B2(5a1 f 3pz) state can be clearlyrelated to the observed shoulder at 6.02 eV.3 The 5a1 f 3dmembers are computed within the energy range 6.57-6.80 eV(average value 6.73 eV).

The average of the excitation energies corresponding to the5a1 f 3p is 6.01 eV. Adding to it the energy difference IP2-IP1 (1.58 eV) gives 7.59 eV, as an estimate for the 7ef 3paverage excitation energies, which were not directly computed.Adding 1.14 eV (IP3-IP2) to 7.59 eV is equal to 8.73 eV, theexpected average excitation energies of the members 4b2 f 3p,

which is consistent with the average value obtained for thecorresponding CASPT2 results (8.85 eV) (cf. Table 2). Employ-ing a similar reasoning, the 7ef 3d members are predictedaround 8.2-8.3 eV.

Despite the fact that we have a number of missing states inthe computed spectrum between 7 and 8 eV, the additionaltransitions observed by EELS3 at 6.99, 8.41, and 9.05 eV maybe tentatively assigned. The former can be related to the 11A1

f 1E(7ef 3s) transition, computed at 7.04 eV with oscillatorstrength 0.04. The two fairly intense transitions observed at 8.41and 9.05 eV are consistent with the results for the verticaltransition from the ground to the1B2(4b2 f 3s) and1E(4b2 f3dxz,yz) states with the excitation energies (oscillator strengths)8.39 eV (0.03) and 9.26 eV (0.13), respectively.

3.2. Low-Lying Triplet States. The CASSCF and CASPT2results computed for the low-lying triplet states are compiledin Table 4. Three valence singletf triplet vertical transitionsare found below the 13A1(5a1 f 3s) state, the lowest excitedstate of Rydberg character, placed at 5.54 eV. Those valencefeatures lie at 2.82, 3.84, and 4.69 eV above the ground stateand can be related to the three transitions reported by Frueholzand Kuppermann3 at 3.05, 4.05, and 4.84 eV, which wereidentified by these authors as singletf triplet transitions.Therefore, according to the current computation, those observedsingletf triplet transitions can be assigned as the 11A1 f 13A2-(5a1 f 3a2), 11A1 f 13E(ππ*), and 11A1 f 13B1(ππ*)transitions. Our assignments for the two lowest transitionsconfirm those performed by Palmer7 on the basis of limited CIcalculations and small basis sets, although the 13E(ππ*) statewas placed higher by more than 1 eV. The CASPT2 resultssuggest, however, that the 13B1(ππ*) state is responsible forthe third observed band. In the CI calculations of Palmer the23A2 state is found as the third-energy triplet state, although a1B1 state was also found to be energetically close. The CASPT2calculation places the 23A2 state (not shown in Table 4) about2.5 eV above the 13B1 state, that is about 0.7 eV above thelowest Rydberg triplet state.

A systematic deviation of the CASPT2 vertical singletftriplet transitions with respect to the EELS data is noted (cf.Table 4). The CASPT2 method seems to underestimate thoseby about 0.2 eV. It can be partially related to the employedzeroth-order Hamiltonian (Møller-Plesset type) that favorsopen-shell systems and leads to triplet states somewhat too low.It is a well-documented effect, and an interested reader canobtain further information about it elsewhere (see, for instance,ref 18, where a pedagogic status of the issue is developed, aswell as the original references therein). Since here there is nocontroversy in the assignments, additional checks employingmore sophisticated zeroth-order Hamiltonians were not per-formed. Moreover, from the available triplet-triplet energytransfer experiments contradictory results were suggested for

TABLE 4: Computed CASSCF and CASPT2 ExcitationEnergies (eV) for the Low-Lying Triplet Excited States ofCOT Employing the C,N,O[4s3p1d]/H[2s1p]+ 2s2p2d(Rydberg Functions) ANO-Type Basis Seta

state transition CASSCF CASPT2 experimental

13A2(5a1f3a2) S-T vertical 3.66 2.82 3.05,b 2.89c

S-T 0-0 1.42 0.78emission max. 0.66 0.22

13E(ππ*) S-T vertical 4.31 3.84 4.05b13B1(ππ*) S-T vertical 5.18 4.69 4.84b13A1(5a1f3s) S-T vertical 6.04 5.54

a Experimental data are also included.b EELS data.3 c From a kineticanalysis14 employing the experimental donor triplet energies reportedelsewhere.12,13

5476 J. Phys. Chem. A, Vol. 107, No. 28, 2003 Frutos et al.

the vertical singlet-triplet excitation energy.12,13 However,applying our recently proposed theoretical model14 to the sameexperimental data,12,13we predict that the vertical singlet-triplettransition energy is 2.89 eV, close to the computed value of2.82 eV.

To a good approximation, the 13A2(5a1 f 3a2) state can bedescribed by the one-electron promotion HOMOf LUMO.The π-CASSCF wave function of the 13E(ππ*) state iscomposed of a linear combination of two singly excitedconfigurations: HOMO- 1 f LUMO (49.6%) and HOMOf LUMO + 1 (37.4%). The 13B1(ππ*) state really has amulticonfigurational character with weights spread over manydifferent configurations equally important.

The 0-0 excitation energy and the phosphorescence maxi-mum were also computed for the lowest singletf triplettransition. We were motivated by the possible significance ofthe lowest-lying triplet state of COT in the triplet quenchingprocess of laser dye solutions (for instance, rhodamine bG). Theequilibrium geometry of the lowest triplet excited state isoctogonal (D8h symmetry).35 Theπ-CASSCF optimized param-eters, employing the ANO-type C[4s3p1d]/H[2s1p] basis set,arer(C-C) ) 1.404 Å andr(C-H) ) 1.077 Å. The adiabaticexcitation or 0-0 band origin is computed as the energydifference between the triplet state (13A2g(D8h)) and the groundstate (11A1(D2d)) at their respective equilibrium geometries. Nozero point energy (ZPE) corrections were included. In thismanner, the 0-0 excitation energy is computed to be 0.78 eVat the CASPT2 level. ZPE could contribute at most 0.1 eV,which is within the intrinsic error bars of the CASPT2 method.Within this level of approximation, the energy differencebetween the vertical and adiabatic excitation energy accountsfor the geometry relaxation of the triplet excited state. Therefore,simultaneously to the electronic excitation of COT, a progressivestructural reorganization toward planarity takes place. Becausethe energy difference between the vertical and 0-0 excitationenergies of the S0 f T1 transition is large, about 2 eV, COThas a pronounced nonvertical behavior and covers a wide rangeof triplet donors.12-14 Moreover, the origin of the S0 f T1

transition, about 0.8 eV, can be considered as an estimate ofthe lower limit for the triplet energy of a donor that the acceptorCOT could still react with.

The phosphorescence maximum is predicted by computingthe ground (11B2g(D8h)) and triplet (13A2g(D8h)) states at theequilibrium geometry of the triplet state. The computedphosphorescence maximum, 0.22 eV, lies in the infrared range.The current estimate is consistent with previous ab initio resultsfor the singlet-triplet splitting ofD8h COT reported by Hrovatand Borden.35 They found that the triplet state has a higher ZPEthan the singlet by 2.1 kcal/mol at the planar geometry. Thatthe ground state at theD8h symmetry is a singlet state (11B2g)and therefore the molecule violates the Hund’s rule is by nowwell established (see the elegant 1996 paper by Wenthold etal.,23 which is a landmark contribution on this issue). Analternative and interesting view is, of course, that in antiaromaticmolecules the Hund’s rule is not expected to hold, as was clearlysuggested two years later by Zilberg and Haas.36

Additional information about the lowest triplet state wasobtained from the PE spectrum of planar COTRA,23,24 wherephotodetachment to two distinct electronic states of neutral COTwas observed. Wentholdet al.23 identified these electronic statesas the 11A1g(D4h) state, corresponding to the transition state ofCOT ring inversion along the S0 hypersurface, and the 13A2g-(D8h) state, recorded at 1.10 and 1.62 eV, respectively.Incidentally, the ground-state potential energy hypersurface of

COT has been, and still is, a subject of great theoreticalattention.1,23,37-40 It is known that the ground state of neutralCOT has four equivalentD2d local minima connected by twoindependent reaction paths: ring inversion (D4h transition state)and bond shifting (D8h transition state). The vertical PE spectrumof planar COTRA and the EA of COT obtained at theπ-CASSCF and CASPT2 levels of theory are compiled, togetherwith available experimental data, in Table 5.

The ground-state (12B1u) equilibrium geometry of the COTRAsystem belongs to theD4h point group. Itsπ-CASSCF optimizedparameters arer(CdC) ) 1.375 Å,r(CsC) ) 1.437 Å,r(CsH) ) 1.081 Å, and ∠(CdCsC) 135.0°. Employing thisstructure, the ground state of the neutral system is computed tobe vertically at 1.11 eV and the lowest triplet state to be at1.47 eV, in agreement with the experimental data.23 As detailedin Table 5, those are results obtained at the CASPT2 level. Thenondynamical correlation effects treated by the CASSCF methodare certainly not enough to predict accurately (within 0.2 eV)the energy differences between the ground state of COTRA andthe considered states of the neutral system. It is amazing thatthe tremendous differential dynamical correlation between thetwo states can be handled just at a second-order level in abalanced way. In this type of difficult case, involving largedifferential dynamic correlation effects, the CASPT2 methodcertainly plays an outstanding role. In other words, a minus signin the CASSCF energy difference means that COTRA is notbound at that level of theory and the CASPT2 approach iscapable of recovering (qualitatively and quantitatively) the rightrelative position between the respective states of the anionicand neutral systems.

The vertical electron affinity is negative at both CASSCFand CASPT2 levels of theory. Therefore, we can confidentlyconclude that COTRA is not bound at the ground-state equi-librium geometry of COT, 11A1(D2d). In addition, the computedCASPT2 adiabatic electron affinity, 0.56 eV, is in agreementwith the most recent experimental determinations.25 The agree-ment is entirely due to the inclusion of dynamic correlationbecause the ground-state radical anion is above the ground stateof the neutral system by>1 eV at theπ-CASSCF level (-1.43eV in Table 5). A question arises here: Does the geometryoptimization of ground-state COTRA at the CASSCF level makeany sense? Previous experience dealing with true temporaryanion states (for instance, in biphenyl41 and p-semibenzo-quinone42 radical anions) gives a clue to answer this question.Yes, the CASSCF wave function represents in such a case alocalized solution, which can be regarded as a discrete repre-sentation of the temporary anion states (see the discussion inref 41 and the different resonance types in ref 42).

Assuming that the geometry of ground-state COTRA issimilar to theD4h structure of the neutral system correspondingto the transition state of COT for the ring inversion path, thebarrier height of that process can be easily estimated. From the

TABLE 5: Computed PE Spectrum of the PlanarCyclooctatetraene Radical Anion (COTRA) and the ElectronAffinity (EA) of COT Employing the C,N,O[4s3p1d]/H[2s1p]+ 2s2p2d (Rydberg Functions) ANO-Type Basis Seta

state CASSCF CASPT2 experimental

Ground State of COTRA (12B1u) (D4h Symmetry)11A1g -0.86 1.11 1.10b13A2g 0.16 1.47 1.62bvertical EA -2.79 -0.46adiabatic EA -1.43 0.56 0.58c

a Energy differences in eV. Experimental data are also included.b Taken from the PE spectrum of COTRA.23 c Determined by thekinetic method. See ref 25 and also references therein.

Electronic Properties of Cyclooctatetraene J. Phys. Chem. A, Vol. 107, No. 28, 20035477

computed PE spectrum one can obtain directly that the originof the triplet is 0.36 eV above the lowest energy planarD4h

singlet state. Combining this value with the computed 0-0excitation energy for the lowest singletf triplet transition (0.78eV), we estimate that the transition state for ring inversion ofCOT lies about 0.42 eV above the ground state, consistent withprevious experimental and theoretical determinations.1,23

4. Summary and Conclusions

The excited states of COT were studied using ab initioquantum-chemical methods based on multiconfigurational wavefunctions. Employing the optimized geometry of the ground stateat the CASSCF level, the vertical excitation energies werecomputed with the CASPT2 method. The calculation comprisedboth valence and Rydberg excited states in the singlet manifold.In particular, the 3s, 3p, and 3d members of the Rydberg seriesconverging to the first ionization potential were explicitly takeninto account. As a byproduct of the required computation toinclude all the excited states up to 7 eV, additional valence andRydberg states were also characterized. The computed excitedstates 11A2 (3.79 eV), 21B2(3pz) (6.17 eV), and 31E (6.40 eV)are related to the main singletf singlet features observedexperimentally, that is, the low-energy, low-intensity band(maximum at 4.43 eV), a shoulder at 6.02 eV, and an intenseband with a maximum at 6.42 eV, respectively. On the otherhand, the observed spin-forbidden singletf triplet bands withmaxima at 3.05, 4.05, and 4.84 eV are assigned as the transitions11A1 f 13A2, 11A1 f 13E, and 11A1 f 13B1, respectively. Thelowest (3s) Rydberg singlet and triplet states are placed at 5.58(21A1) and 5.54 (13A1) eV. In addition, new assignments aretentatively suggested for the recorded higher-energy peaks inthe singlet manifold at 6.99, 8.41, and 9.05 eV. The three lowestionization potentials are also characterized. The computedadiabatic electron affinity is 0.56 eV, in agreement withexperimental determinations. Nevertheless, the COT radicalanion is not bound at the equilibrium geometry of the neutralmolecule.

As regards the role of COT as an efficient triplet quencherof laser dye solutions, the computation supports the nonverticalprocess of triplet energy transfer. Within a nonvertical energy-transfer scheme, it is predicted that COT could act as acceptorwith donors having a triplet excited-state energy equal to orhigher than 0.78 eV, which corresponds to the computed S0 fT1 adiabatic excitation energy of the system. The experimentalconfirmation of this prediction would certainly be of great value.

Acknowledgment. Financial support through ProjectsBQU2000/0646, BQU2001-2926, and BQU2001-5037-ES fromthe Spanish Ministerio de Educacio´n, Cultura y Deporte(MECD) and by the Generalitat Valenciana is gratefullyacknowledged. L.M.F. thanks the MECD for a DoctoralFellowship.

References and Notes

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(2) Langseth, A.; Brodersen, S.Acta Chem. Scand.1949, 3, 778.(3) Frueholz, R. P.; Kuppermann, A.J. Chem. Phys.1978, 69, 3614.(4) Perkampus, H.-H. InUV-Vis Atlas of Organic Compounds;

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J. C.J. Am. Chem. Soc.1965, 87, 3430.

(6) Van-Catledge, F. A.J. Am. Chem. Soc.1971, 93, 4365.(7) Palmer, M. H.J. Mol. Struct.1988, 178, 79.(8) Hassenru¨ck, K.; Martin, H.-D.; Walsh, R.Chem. ReV. 1989, 89,

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(10) Roos, B. O.; Mercha´n, M.; McDiarmid, R.; Xing, X.J. Am. Chem.Soc.1994, 116, 5927.

(11) Serrano-Andre´s, L.; Lindh, R.; Roos, B. O.; Mercha´n, M. J. Phys.Chem.1993, 97, 9360.

(12) Forward, P. J.; Gorman, A. A.; Hamblett, I.J. Chem. Soc., Chem.Commun.1993, 250.

(13) Das, T. N.; Priyadarsini, K. I.J. Chem. Soc., Faraday Trans.1994,90, 963.

(14) Frutos, L. M.; Castan˜o, O.; Andres, J. L.; Mercha´n, M.; Acuna, A.U. Triplet energy transfer to cycloocta-1,3,5,7-tetraene: A theoreticaldescription of the anti-ArrheniusnonVertical process in terms of accuratepotential energy surfaces. Submitted for publication.

(15) Andersson, K.; Malmqvist, P.-Å.; Roos, B. O.; Sadlej, A. J.;Wolinski, K. J. Phys. Chem.1990, 94, 5483.

(16) Andersson, K.; Malmqvist, P.-Å.; Roos, B. O.J. Chem. Phys.1992,96, 1218.

(17) Roos, B. O.; Fu¨lscher, M. P.; Malmqvist, P.-Å.; Mercha´n, M.;Serrano-Andre´s, L. InTheoretical Studies of Electronic Spectra of OrganicMolecules; Langhoff, S. R., Ed.; Kluwer Academic Publishers: Dordrecht,The Netherlands, 1995; p 357.

(18) Roos, B. O.; Andersson, K.; Fu¨lscher, M. P.; Malmqvist, P.-Å.;Serrano-Andre´s, L.; Pierloot, K.; Mercha´n, M. In MulticonfigurationalPerturbation Theory: Applications in Electronic Spectroscopy; Prigogine,I., Rice, S. A., Eds.; J. Wiley & Sons: New York, 1996; p 219;AdV. Chem.Phys.1996, 93, 219.

(19) Merchan, M.; Serrano-Andre´s, L.; Fulscher, M. P.; Roos, B. O. InMulticonfigurational Perturbation Theory Applied to Excited States ofOrganic Compounds; Hirao, K., Ed.; World Scientific Publishing Com-pany: Amsterdam, 1999; Vol. 4, p 161.

(20) Batich, C.; Bischof, P.; Heilbronner, E.J. Electron Spectrosc. Relat.Phenom.1972/73, 1, 333.

(21) Bieri, G.; Burger, F.; Heilbronner, E.; Maier, J. P.HelV. Chim.Acta 1977, 60, 2213.

(22) Kobayashi, T.Phys. Lett.1979, A70, 292.(23) Wenthold, P. G.; Hrovat, D. A.; Borden, W. T.; Lineberger, W. C.

Science1996, 272, 1456.(24) Wenthold, P. G.; Lineberger, W. C.Acc. Chem. Res.1999, 32,

597.(25) Denault, J. W.; Chen, G. D.; Cooks, R. G.J. Am. Chem. Soc. Mass

Spectrom.1998, 9, 1141 and references therein.(26) Widmark, P.-O.; Malmqvist, P.-Å.; Roos, B. O.Theor. Chim. Acta

1990, 77, 291.(27) For reviews of the CASSCF methods, see different contributions

in: Ab Initio Methods in Quantum Chem.-II; Lawley, K. P., Ed.; J. Wiley& Sons Ltd.: New York, 1987.

(28) Rubio, M.; Mercha´n, M.; Ortı, E.; Roos, B. O.Chem. Phys.1994,179, 395.

(29) Forsberg, N.; Malmqvist, P.-Å.Chem. Phys. Lett.1997, 274, 196.(30) Malmqvist, P.-Å.Int. J. Quantum Chem.1986, 30, 479.(31) Malmqvist, P.-Å.; Roos, B. O.Chem. Phys. Lett.1989, 155, 189.(32) Andersson, K.; Baryz, M.; Bernhardsson, A.; Blomberg, M. R. A.;

Boussard, P.; Cooper, D. L.; Fleig, T.; Fu¨lscher, M. P.; Hess, B.; Karlstro¨m,G.; Lindh, R.; Malmqvist, P.-Å.; Neogra´dy, P.; Olsen, J.; Roos, B. O.; Sadlej,A. J.; Schimmelpfennig, B.; Schu¨tz, M.; Seijo, L.; Serrano-Andre´s, L.;Siegbahn, P. E. M.; Stålring, J.; Thorsteinsson, T.; Veryazov, V.; Wahlgren,U.; Widmark, P.-O.MOLCAS,Version 5.0; Department of TheoreticalChemistry, Chemical Centre, University of Lund: P. O. B. 124, S-221 00Lund, Sweden, 2000.

(33) Robin, M. B. Higher Excited States of Polyatomic Molecules;Academic Press: New York, 1975; Vols. I, II, and III.

(34) Bally, T.; Truttmann, L.; Dai, S.; Williams, F.J. Am. Chem. Soc.1995, 117, 7916.

(35) Hrovat, D. A.; Borden, W. T.THEOCHEM1997, 398-399, 211.(36) Zilberg, A.; Haas, Y.J. Phys. Chem. A1998, 102, 10851.(37) Andres, J. L.; Castan˜o, O.; Morreale, A.; Palmeiro, R.; Gomperts,

R. J. Chem. Phys.1998, 108, 203.(38) Wiberg, K. B.Chem. ReV. 2001, 101, 1317.(39) Bearpark, M. J.; Blancafort, L.; Robb, M. A.Mol. Phys.2002, 100,

1735.(40) Castan˜o, O.; Palmeiro, R.; Frutos, L. M.; Andre´s, J. L.J. Comput.

Chem.2002, 23, 732.(41) Rubio, M.; Mercha´n, M.; Ortı, E.; Roos, B. O.J. Phys. Chem.1995,

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Forsberg, N.J. Am. Chem. Soc.2000, 122, 6067.

5478 J. Phys. Chem. A, Vol. 107, No. 28, 2003 Frutos et al.

A theory of nonvertical triplet energy transfer in terms of accuratepotential energy surfaces: The transfer reaction from p,p* triplet donorsto 1,3,5,7-cyclooctatetraene

Luis Manuel Frutosa) and Obis CastanoDepartamento de Quı´mica Fısica, Universidad de Alcala´, 28871 Alcala´ de Henares, Madrid, Spain

Jose Luis AndresFacultad de Ciencias, Universidad Nacional de Educacio´n a Distancia (UNED), C/Colo´n, 114,08222 Tarrasa, Spain

Manuela MerchanFısica, Instituto de Ciencia Molecular, Universitat de Vale`ncia, Dr. Moliner 50, Burjassot,E-46100 Valencia, Spain

A. Ulises AcunaInstituto de Quı´mica Fısica ‘‘Rocasolano,’’ C.S.I.C., Serrano 119, 28006 Madrid, Spain

~Received 22 July 2003; accepted 15 October 2003!

Triplet energy transfer~TET! from aromatic donors to 1,3,5,7-cyclooctatetraene~COT! is anextreme case of ‘‘nonvertical’’ behavior, where the transfer rate for low-energy donors isconsiderably faster than that predicted for a thermally activated~Arrhenius! process. To explain theanomalous TET of COT and other molecules, a new theoretical model based on transition statetheory for nonadiabatic processes is proposed here, which makes use of the adiabatic potentialenergy surfaces~PES! of reactants and products, as computed from high-level quantum mechanicalmethods, and a nonadiabatic transfer rate constant. It is shown that the rate of transfer depends ona geometrical distortion parameterg5(2g2/k1)1/2 in which g stands for the norm of the energygradient in the PES of the acceptor triplet state andk1 is a combination of vibrational forceconstants of the ground-state acceptor in the gradient direction. The application of the model toexisting experimental data for the triplet energy transfer reaction to COT from a series ofp,p*triplet donors, provides a detailed interpretation of the parameters that determine the transfer rateconstant. In addition, the model shows that the observed decrease of the acceptor electronicexcitation energy is due to thermal activation of CvC bond stretchings and C–C bond torsions,which collectively change the ground-state COT bent conformation (D2d) toward a planar tripletstate (D8h). © 2004 American Institute of Physics.@DOI: 10.1063/1.1631418#

I. INTRODUCTION

Triplet photosensitization, first characterized in con-densed phases by Terenin and Ermolaev,1 is a commonmechanism in photoinitiated bimolecular reactions2,3

~Scheme 1!, in which the electronic energy of an initiallyexcited triplet donor (3D) is transferred to an acceptor sin-glet state (1A) with an observed rate constantkexp

en :

Scheme 1:

3D11A ——→kexp

en

1D13A

The actual transfer of triplet energy~TET! takes place inthe range of picoseconds4,5 by an electronic exchangemechanism2,6 and, therefore, close approach of the two reac-tants is required~,1 nm!. When the balance between theenergies of the donor and acceptor triplet states,DET5ET

D2ETA , is positive ~‘‘exothermic’’ TET! the ob-

served rate in fluid solution is nearly diffusioncontrolled.2,5,7,8In the endothermic case (DET,0) the trans-fer rate can still prevail over other triplet quenching channelsif the energy deficit is not too large (uDET

u'3 – 4 kcal/mol, 1 kcal54.183 kJ), as in a conventionalthermally activated Arrhenius process. This type of kineticswas demonstrated by Sandros in a series of pioneering stud-ies in solution,7 where the well-known expression for the rateconstant of the activated transfer was introduced:kexp

en

5k8 exp(2DET /RT).On the other hand, it was realized at the outset9 that for

some acceptor compounds, such ascis-stilbene, the TET rateof very endothermic processes~in the above-noted sense!was considerably faster than that predicted by the Sandrosequation, an anomaly associated with acceptor moleculeswith significant conformational flexibility. In that seminalwork the termnonverticalTET was also introduced, arguingthat donor–acceptor interactions result in the breakdown ofFranck–Condon restrictions in such a way that the acceptorgeometry changed concurrently with its excitation. This de-manding interpretation has been replaced nowadays by morea!Electronic mail: luisma.frutos@uah.es

JOURNAL OF CHEMICAL PHYSICS VOLUME 120, NUMBER 3 15 JANUARY 2004

12080021-9606/2004/120(3)/1208/9/$22.00 © 2004 American Institute of Physics

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conventional ‘‘hot-band’’ concepts,2,10–12in which ‘‘vertical’’transitions from acceptor molecules, thermally activatedalong specific torsional modes, account for an observed ex-citation energy much lower than the ‘‘spectroscopic’’T1←S0 value. Recent developments of this idea included theconsideration of more than a single potential energycurve13,14 and, most remarkably, the definition of a rate con-stant distribution function associated with the geometricaldeformation.15 An important difficulty in the hot-band quali-tative analysis is, of course, the identification of the crucialdeforming vibrations that are most effective in lowering theexcitation energy.

Balzani and co-workers,16~a!,16~b! noting the similarity be-tween electron and triplet energy transfer reactions, devel-oped an expression for the molecular TET rate constant (ke)as a function of thedriving forceDG0 ([DET) of the pro-cess, and provided a comprehensive analysis of both normaland nonvertical processes. In addition, Orlandiet al.16~c! pre-sented an alternative quantum mechanical approach, basedon the treatment given by Ulstrup and Jortner for electrontransfer processes,17 in which ke is given by the product ofan electronic and a nuclear term:

ke52p

\U2J, ~1!

whereU stands for the electronic exchange interaction andJ~the nuclear term! is the Franck–Condon weighted density ofstates, which normally contains the dependence on energy,temperature, and solvent properties.

This last approach has been successfully applied to alarge variety of TET processes, as e.g., those giving rise to aninvertedregion18 and to triplet quenching through the wallsof hemicarcerands.19

Very recently, Serpaet al.20 discussed in great detail dif-ferent formulations of the vibrational factorJ for TET, basedon the golden rule. On the other hand, there is no simple wayof computing theU2 value from first principles for the mo-lecular systems of interest.21 Therefore, when a common ac-ceptor is excited by a series of donors with ‘‘similar’’ struc-ture, it is frequently assumed that theU2 value would remainrelatively constant.16~a!,16~b! According to that, the observedvariation of the rate constant has to be correlated withchanges in the overlap integralJ. Unfortunately, since thespectral data required to compute theJ integral are availableonly in a few cases, its value had to be estimated from dif-ferent approximations to the true~unknown! spectra.20 Innonvertical TET processes this estimation is even more dif-ficult, because the conformation of the~flexible! acceptor isassumed to change considerably on excitation to the tripletstate and the correspondingT1←S0 absorption would have avery low transition probability. There are a few examples12

where the expected correlation betweenke and the vibra-tional factor was not observed, indicating that either theU2

term cannot be taken as a constant in these experiments orthat the approximation used to computeJ was too unrealistic.

Here we present an alternative analysis of the nonverti-cal TET processes, developed from the same ‘‘hot band’’concept. The new approximation is based on the computationof accurate adiabatic potential energy surfaces~PESs! for all

the states~singlet and triplet! involved in the transfer step,combined with the use of transition state theory~TST!22 ofnonadiabatic processes23 to derive the rate constant for thecrossing between reactant and product surfaces. It is ex-pected that, in this way, the relationship between the energydependency of the transfer rate and the acceptor conforma-tional change may be better understood, and the crucial de-forming molecular motions clearly identified. In addition, theutility of this approximation is illustrated by its application tothe triplet energy transfer reaction of 1,3,5,7-cyclooctatetraene~COT!, for which accurate PESs are nowavailable.24,25This cyclic, very flexible nonaromatic polyene~COT! presents the first weak absorption band at 282 nm inhexane solution,26 and the lowestT1←S0 transition in therange of 65–70 kcal/mol, located by electron impact spec-troscopy as a broad, weakly structured band.27 A more recentestimate28 of the energy of the elusive COT triplet state, fromTET experiments in solution, yielded a value of 59 kcal/mol.In contrast, a much lower value~41 kcal/mol! was derivedfrom kinetic experiments that included a reversible transferstep.29 In the same work a long lifetime of 100ms was as-signed to3COT in cyclohexane solution, which seems diffi-cult to conciliate with the low energy of the relaxed tripletstate~;22 kcal/mol above theD2d ground state30! and thenoted molecular flexibility of the polyene.

Nowadays, COT is used as a sort of universal tripletscavenger in organic dye lasers, because it can improve theoperation of lasing dyes with triplet energies as low as 40kcal/mol.29 Not unexpectedly, when COT was used as accep-tor for a series of triplet donors, a strong deviation from theprediction of the Sandros equation was discovered,28 extend-ing over a large span of the energy deficit~.20 kcal/mol!,that reflects the large nonvertical behavior of the cyclic poly-ene. This exceptional deviation was explained as due to tor-sions around the annulene single bonds, which might largelydecrease the energy required to populate the triplet state.28

The interpretation of the experimental TET data of COTrequires an expression for the transfer rate constant as a func-tion of the triplet energy of a series of donor molecules. Thisexpression is derived here from the following sequence. In afirst step we introduce the adiabatic PES of the reactant andproduct collision complex; a gradient difference vector isdefined, that yields the difference between the energy gradi-ent in each of the two PESs. The interaction leading to sur-face crossing and triplet energy transfer is usually explainedas due to electron exchange~Dexter mechanism6! and de-pends on the proximity between donor and acceptor mol-ecules. Consequently, we next define an electron couplingvector, which yields the extent of this interaction. These twovectors are assigned to well-defined nuclear coordinates ofthe encounter complex. In a second step, the rate constant forthe crossing reaction is obtained, based on the nonadiabaticformulation of transition-state theory. Finally, the activationenergy of the transfer process is derived as a function of thedonor triplet energy for the case of nonvertical behavior.

1209J. Chem. Phys., Vol. 120, No. 3, 15 January 2004 Triplet energy transfer processes

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II. RESULTS AND DISCUSSION

A. The potential energy surfacesof the triplet energy transfer reaction

The transfer reaction~Scheme 1! can be formallyexpanded16 to separate the diffusive (kd , k2d ,) and energytransfer steps (ke , k2e) as:

Scheme 2:

3D11A�k2d

kd

@3D¯

1A#�k2e

ke

@1D¯

3A#�kd

k2d1D13A

The transfer process involves only the PES of reactant@3D¯

1A# and product @1D¯

3A# encounter complexes,which can be considered as a supermolecule in two differentelectronic states@Fig. 1~a!# ~in the following the more con-venient@* D¯A# @D¯* A# symbols will be used!. The en-ergy transfer process consists on the passage of the systemfrom the first to the second state. It is assumed here thatlong-lived exciplexes are not formed in the TET reaction insolution,11 due to the weak donor–acceptor interaction. Ifthis interaction is negligible, the relative energy of each com-plex would be given byE@* D¯A#5E(* D)1E(A) andE@D¯* A#5E(D)1E(* A). However, when the two PESsare near isoenergetic some sort of interaction must be presentfor the crossing to occur, and a coupling correction to theenergy should be introduced.

The PES of each collision complex is a space function ofinternal coordinates only, withF degrees of freedom, givenby F53(ND1NA)26, whereND(A) is the number of atomsof the donor~acceptor!, each with its associated coordinate.The total number of coordinates,Q, is separated here in twogroups,Q5$Qr ,t ,Qn%, whereQr ,t defines the relative posi-

tion of donor and acceptor~three coordinates for translationand three for rotation! and Qn corresponds to all molecularvibrations @3(ND1NA)212#. In the transfer step, the spinand spatial symmetry of the total wave function is preserved,resulting in an avoided crossing in the regions approachingenergy degeneracy31 @Fig. 1~b!#. The two special dimensionswhere the energy degeneracy disappears are defined by theelectronic coupling (vEC) and gradient difference (vGD) vec-tors. The first one~EC! indicates the maximum couplingstrength, while the second~GD! gives the difference betweenthe energy gradient vectors of the PES of two complexes at agiven nuclear configuration:@* D¯A# and @D¯* A# ~videinfra!. In the following paragraphs each vector is assigned toa well-defined subspace of coordinates.

The electron exchange between donor and acceptoris expressed by the coupling matrix termHi f 5^C@* D¯A#uH8uC@D¯A* #& ( i 5 initial, f 5final) inwhich

H85(i , j

1

r i j

is the electronic coupling Hamiltonian, where the indicesi,jdenote the donor and acceptor electrons, respectively. In afirst-order approximation, the wave functions can be taken asC@* D¯A#5`c* DcA , and C@D¯* A#5`cDc* A, where cstands for the electronic wave function of the isolated mol-ecules, and is a permutation operator which acts antisym-metrizing the wave function product under the permutationbetweeni and j electrons.6~c! The molecular wave functions,in a first approximation, can be taken as those of the highestoccupied molecular orbital~HOMO! and lowest unoccupiedmolecular orbital~LUMO! of both the singlet and the tripletstates of the reacting molecules.6~b! According to that,Hi f isa function only of the relative position of the donor andacceptor, given by theQr ,t coordinates, because the changeon the electronic orbital overlap due to internal vibrations isexpected to be comparatively much less important. There-fore, the electronic coupling vector (vEC) would depend onlyon the relative position of donor and acceptor, that is, on theQr ,t coordinates. Thus, for a given nuclear configurationQthe electronic coupling vector is given by

vEC5 (m51

6

umS ]Hi f ~Qr ,t!

]qmD

Q

, qmPQr ,t , ~2!

whereum is the corresponding unit vector andHi f the cou-pling energy between initial and final states.

It could also be possible to include an additional vi-bronic ~derivative! coupling term into the Hamiltonian, be-cause of the eventual near-degeneration of the PESs of thetwo electronic states. However, the contribution of the vi-bronic coupling to the electronic energy would be only asmall fraction of the electronic one. This can be shown in aqualitative way as follows:

In terms of perturbation theory, the vibronic couplingterm can be formulated as

FIG. 1. The triplet energy transfer reaction as pictured by the crossingbetween the potential energy surfaces~PES! of the donor–acceptor@D¯A#collision complex.~a! The potential energy surface as a function of themolecular vibrational coordinates (Qn). ET

D andETA are the~optical! excita-

tion energies of the donor and acceptor triplet states, respectively;E0† is the

energy at the crossing point of the adiabatic PES;Hi f represents the ampli-tude ~exaggerated! of the electron exchange interaction andDG the freeenergy of the process.~b! Another perspective of the PES, now as a functionof the donor–acceptor relative position, given by theQr ,t coordinates. Thearrows labeled as EC and GD show the direction of electronic coupling andenergy gradient difference vectors, respectively. The avoided crossing due toelectron exchange coupling perturbation (Hi f ) results in the lowering of theenergy barrier between reactants@* D¯A# and products@D¯* A#, com-pared with that in the adiabatic PES.

1210 J. Chem. Phys., Vol. 120, No. 3, 15 January 2004 Frutos et al.

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Wi f 5^C@* D¯A#u (i 5r ,t,n

]

]QiuC@D¯A* #&

'U^C@* D¯A#u (i 5r ,t,n

]~HD0 1HA

01H8!

]Qiu

3C@D¯A* #&U, ~3!

where HA,D0 is the electronic Hamiltonian of the isolated

molecule andH8 is the electronic coupling defined earlier. Ifthe internal conversion rate in each isolated molecule is neg-ligible ~otherwise the triplet energy transfer process wouldnot occur!, the first two terms in the sum of Eq.~3! vanish,that is

^C@* D¯A#u (i 5r ,t,n

]~HD,A0 !

]QiuC@D¯A* #&50,

and the resultant expression would be

Wi f 'U^C@* D¯A#u(i , j

]r i j21

]Qr ,tuC@D¯A* #&U. ~4!

Taking r i j 5R as a constant~the average distance be-tween electrons! Eq. ~4! can be further simplified to give

Wi f 'U^C@* D¯A#u]R21

]Qr ,tuC@D¯A* #&U

5uR22^C@* D¯A#uC@D¯A* #&u

5R22Si f ~Qr ,t!, ~5!

whereSi f represents the electron orbital overlap as a functionof the donor–acceptor relative position coordinates,Qr ,t .Since in the present approximation the dependence of theelectronic coupling on electron separation~R! is given byHi f 'R21Si f , the contribution of the vibronic termWi f tothe total electronic energy would be very small for the samevalue of the electron separation. In summary, the couplinginteraction between reactants and products PES arise essen-tially from electron exchange, with a small contribution fromvibronic coupling. The components of the vector associatedwith this interaction (vEC) extend only over rotational andtranslational coordinates (Qr ,t) describing the proximity ofdonor/acceptor within the collision complex.

On the other hand, the energy gradient vector for the twoencounter complexes and, therefore, the energy gradient dif-ference vector depends only on internal~vibrational! coordi-nates, because the donor–acceptor interaction in the transferprocess is so weak that no new bonds are formed~that mightstore energy!. Accordingly, the energy gradient differencewould not depend significantly on the donor–acceptor rela-tive position:

S ]~E@* D¯A#2E@D¯A* #!

]qmD

Q

'0, qmP$Qr ,t% ,

for any configurationQ.

Hence, the energy gradient difference vector for a configu-ration Q can be defined as

vGD5 (m51

N

umS ]~E@* D¯A#2E@D¯A* #!

]qmD

Q

, qmPQn . ~6!

In conclusion, thevEC vector is associated with theQr ,t sub-set of coordinates, whilevGD is associated with the vibra-tional subsetQn .

B. The energy transfer rate constant

The rate constant for the crossing between reactants andproducts energy surfaces can be expressed from transitionstate theory22 as

ke5xf Þ

f 1f 2¯

kBT

hexpS 2

V0

kBTD , ~7!

where f Þ is the partition function of the activated complexand f i the partition functions of reactants and products,V0 isthe activated complex energy,x the transmission coefficient,and the other symbols have the usual meaning.

The crossing step can be described as a nonadiabatictransition in a weakly coupled system.32 In that case, thevalue ofx is well described by23

x.4pHi f

2

\vusf2si u, ~8!

wherev is the relative velocity with which the system passesthe point of closest approach andusf2si u is the absolutemagnitude of the difference between the energy gradient ofthe two crossing surfaces. The weakly allowed nonadiabaticcrossing results in a lower energy barrier compared with thatcorresponding to the intersection of adiabatic surfaces (E0

†),that is: V05E0

†2Hi f @Fig. 1~b!#. Accordingly, the rate con-stant for nonadiabatic crossing becomes

ke54pHi f

2

\vusf2si uf Þ

f 1f 2¯

kT

hexpS Hi f

kBTDexpS 2E0

†

kBTD , ~9!

where E0† is the activated complex energy that would be

obtained fromadiabaticPES. As noted earlier, the value ofE0

† does not depend on the donor–acceptor separation withinthe collision complex, while that ofHi f changes very rapidly~exponentially! with that distance. Moreover, sinceHi f ,kBT, the exponential termeHi f /kBT'1 and the rateconstant given by Eq.~9! shows the exponential dependencewith the donor–acceptor distance observed in simplesystems.6~b!,33

C. The energy of the activated complex

In this section the value of the energy of the activatedcomplex would be derived, as a function of the triplet energyof the donor. The coupling termHi f @Eq. ~9!# depends onlyon the orbital overlap and, therefore, it is independent of thedonor energy. According to that, one only needs to determinethe dependence ofE0

† ~the adiabaticcomponent of the acti-vation energy! on the triplet donor energy. For moderate tem-peratures, the potential energy of the complex in the initialstate,@* D¯A#, defined asE0(Q)[E@* D¯A# (Qn), is well

1211J. Chem. Phys., Vol. 120, No. 3, 15 January 2004 Triplet energy transfer processes

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represented classically as a function of electronic energies

EelecA,* D , vibrational coordinates (qm,n

A,D), and harmonic force

constants (kmnA,* D) by

E0~Q!5E0~q1A ,q2

A ,...,q3NA26A ,q1

D ,q2D ,...,q3ND26

D !

5EelecA 1E

elec

* D1

1

2 (m

3NA26

(n

3NA26

kmnA qm

AqnA

11

2 (m

3ND26

(n

3ND26

kmn

* Dqm

DqnD1¯ , ~10!

where

kmnA,* D5S ]2E0

]qmD,A]qn

D,ADe

.

The e subscript is used here to identify the equilibrium con-figuration of the@* D¯A# complex. As noted earlier, due tothe very weak coupling there is no accumulation of energy inthe other subset of degrees of freedomQr ,t .

Similarly, the potential energy surface of the complexin the final state, @D¯A* #, defined as E1(Q)[E@D¯A* #(Qn), can be expressed by

E1~Q!5E1~q1A ,q2

A ,...,q3NA26A ,q1

D ,q2D ,...,q3ND26

D !

5Eelec

* A1Eelec

D 1 (m51

3NA26

am

* Aqm

A1 (m51

3ND26

amDqm

D

11

2 (m

3NA26

(n

3NA26

kmn

* Aqm

AqnA

11

2 (m

3ND26

(n

3ND26

kmnD qm

DqnD1¯ , ~11!

where

am

* A5S ]E1

]qmAD

e

, amD5S ]E1

]qmDD

e

,

kmn

* A,D5S ]2E1

]qmA,D]qn

A,DDe

.

Equation~11! corresponds to expansion of the@D¯A* #PES up to the quadratic terms, using the same origin and setof coordinates as those of Eq.~10!.

In the crossing spaceE0(Q)5E1(Q) and, therefore,from Eqs.~7! and ~8! we have

ETD2ET

A5 (m51

3NA26

am

* Aqm

A1 (m51

3ND26

amDqm

D

11

2 (m51

3NA26

(n51

3NA26

~kmn

* A2kmn

A !qmAqn

A

11

2 (m51

3ND26

(n51

3ND26

~kmnD 2k

mn

* D!qm

DqnD1¯ , ~12!

where

ETA[~E

elec

* A2Eelec

A ! and ETD[~E

elec

* D2Eelec

D !.

In those cases where the first, linear term provides the largestcontribution to the energy difference, Eq.~12! can be greatlysimplified to yield

ETD2ET

A' (m51

3NA26

am

* Aqm

A1 (m51

3ND26

amDqm

D . ~13!

This expression shows that the energy difference betweendonor and acceptor triplet states at the crossing region of thePES is matched by thermal activation of the internal degreesof freedom of the two reactants. It is clear that Eq.~13!would be more accurate the larger the value of theam,n* A,D

coefficients. In fact, these coefficients are the components ofthe energy gradient vector in the complex final state after avertical excitation; a large gradient value~i.e., largea coef-ficients! indicates an important change in the conformationof D, A or both in the transfer process. This is precisely oneof the characteristics of nonvertical processes, as discussedearlier, in which the deviation from the ‘‘normal’’ TET wasassociated with a flexible acceptor structure.34 On the otherhand, it should be noted that a large conformational changedoes not imply necessarily a large energy gradient.

Another distinctive property of the acceptor singlet statein nonvertical TET reactions is that small changes in vibra-tional coordinates (qi) result in large changes on the energydeficit ET

A2ETD , which also contributes to the accuracy of

the approximation in Eq.~13!.The triplet energy transfer processes of interest here are

those involving flexible acceptor molecules~as, e.g.,cis-stilbene and COT!. Therefore, Eq.~13!, can be further sim-plified for those cases in which the contribution of the donordeformation to the change in the excitation energy is negli-gible, to yield

ETD2ET

A5 (m51

3NA26

am*Aqm

A . ~14!

Finally, Eq. ~14! can be recast in a new orthonormalbasis set, in which the gradient vector for the vertical exci-tation of * A is one of the basis vectors, to give:

ETA2ET

D5uguj1A , ~15!

where

ugu5u¹W @E1~Q!#u5H (m51

3NA26

~am* A!2J 1/2

andj1A is the coordinate along the gradient vector.

In the above expressionugu ([g) is the norm of thegradient difference vector (vGD) between the gradient of theT1 andS0 surfaces, i.e.,g5g(T1)2g(S0). Since the excita-tion of the acceptorS0 state takes place from a configurationclose to the equilibrium position where the gradient is neg-ligible, theg value corresponds essentially to the gradient ofthe triplet state surface, only@g.g(T1)[g1#. According to

1212 J. Chem. Phys., Vol. 120, No. 3, 15 January 2004 Frutos et al.

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this, the acceptor excitation energy in the new basis set de-pends only on a single coordinatej1

A , which is that of thetriplet acceptor gradient vector:

ETA2ET

D5g1j1A . ~16!

This can be considered as the generalization of theabove-discussed ‘‘hot-band’’ concept, because, as shown byEq. ~16!, the energy deficit can be lessened by geometricaldistortions of the ground state acceptor along vibrational co-ordinates (j1

A) that correspond to a large change in the down-ward slope of the triplet potential energy surface. Distortionsin other directions are orthogonal vectors to the gradientgand, therefore, do not contribute, in this formalism, to theenergy expansion in Eq.~15!. In addition, it becomes clearthat the whole set of components of the gradient vector~bond stretchings, torsions, and bendings! is the importantfactor in determining the acceptor excitation energy.

The vibrational energy (E0,vib) of the initial complex@* D¯A# can be obtained from Eq.~10! by subtracting fromthe total energy the electronic energy, to give

E0,vib51

2 (m

3NA26

(n

3NA26

kmnA qm

AqnA

11

2 (m

3ND26

(n

3ND26

kmn* DqmDqn

D1¯ . ~17!

This expression can also be redefined in a new basis set,in which the coordinates arej1

A and those, for the acceptoronly, diagonalizing the Hessian matrix of the energy in thesubspace of coordinates orthogonal toj1

A , that is, j iA with

i 52,...,3NA26. If the donor coordinatesqiD are preserved,

we have

E0,vib51

2 (i

3NA26

(j

3NA26

k i jAj i

Aj jA

11

2 (m

3ND26

(n

3ND26

kmn* DqmDqn

D

51

2k11

A ~j1A!21 (

i 52

3NA26

k i1A j i

Aj1A

11

2 (i 52

3NA26

k i iA~j i

A!211

2 (m

3ND26

(n

3ND26

kmn* DqmDqn

D ,

~18!

where

k i jA5S ]2E0

]j iA]j j

AD0

.

The acceptor vibrational energy,E0,vib† , corresponding to the

configuration of the minimum energy in the crossing sub-space~i.e., the activation energy! and consistent with Eq.~16!, can be found by defining the Lagrangian function~L! as

L5E0,vib1l~ETA2ET

D2gj1A!. ~19!

Differentiating relative toj1A andjn

A gives

]L

]j1A 5k11

A j1A1 (

i 52

3NA26

k i1A j i

A1lg50,

~20!]L

]jnA 5k1n

A j1A1knn

A jnA50⇒jn

A52k1n

A

knnA j1

A .

Finally, by replacing in Eq.~18! the value given by Eqs.~16!and ~20!, the following expression results:

E0,vib† 5

1

2

k11A 2 (

i 52

3NA26~k i1

A !2

k i iA

g2 ~ETA2ET

D!2

[1

2

k1

g2 ~ETA2ET

D!2. ~21!

It should be recalled that the minimum energy at the crossingpoint is independent of the donor coordinates, because theenergy change~decrease! of the system depends on the ac-ceptor coordinatesj1

A @Eq. ~16!#.This important relationship@Eq. ~21!# gives the value of

the energy barrier for the direct energy transfer reaction as afunction of the force constants of ground-state vibrations(k1) for the acceptor, the energy gradient on the triplet-statesurface of the acceptor molecule~g!, and the difference be-tween the optical excitation energies of the acceptor and do-nor triplet states. Since the PESs as defined here do not in-clude any contribution from solvent effects, the energybarrier corresponds to a solvent-free environment or to onein which the solvent is unperturbed. This is, of course, anoversimplification but it is not expected to result in largeerrors in the computed activation energy, since TET does notinvolve a major redistribution of charges. On the other hand,the value of the preexponential factor in Eq.~9! may besensitive to solvent properties, due to the correspondingchanges in the velocity termv.

By substituting Eq.~21! into the expression for the trans-fer rate constant we have

ke54pHi f

2

\vusf2si uf Þ

f 1f 2¯

kBT

hexpF2S E0,vib

†

kBT

Hi f

kBTD G5

4pHi f2

\vusf2si uf Þ

f 1f 2¯

kBT

hexpS Hi f

kBTD3expF2

1

2kBT

k1

g2 ~ETA2ET

D!2G . ~22!

This expression applies for a triplet–triplet transfer reactionin which the acceptor singlet-state conformation changessubstantially on excitation, compared with that of the donormolecule. The preexponential factor in Eq.~22! can now beanalyzed in detail based on the properties of the above-discussed PES. Thus, the relative velocityv with which thecomplex crosses the coupling space would be determined bythe vibrational frequency of the modes contained ink1 .Since in the model discussed here the GD vector dependsonly on the properties of the acceptor PES~or, in otherwords, the decrease in excitation energy is due only to theacceptor deformation!, the value ofv would depend on the

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acceptor vibrational frequencies, as well as on the effect ofthe solvent vibrational modes coupled to that of the super-molecule and temperature.

On the other hand, the parameterusf2si u is just the mod-ule of the GD vector at the crossing space. In a series ofexperiments where the same acceptor molecule is used, thisterm remains constant because, as noted earlier, the gradientof the excited state is much larger than that of theS0 . Hence,usf2si u'usf u5constant.

The electronic coupling termHi f is a two-center, two-electron integral that depends mainly on the structure of boththe HOMO and LUMO of the donor–acceptor pair. How-ever, in a TET experiment involving a common acceptor anda series of donors of the same electronic type (p,p* , forexample! in which the spatial distribution of the MOs can betaken as relatively unchanged, theHi f value would not beexpected to depart substantially from an average value char-acteristic of the whole donor series. A similar reasoning canbe applied to the partition functions in Eq.~22!, by notingthat these functions can be expressed by two factors, whichdepend on the degrees of freedom of the donor and acceptor,respectively. Thus, if the conformation of the selected donormolecules does not change substantially in theT1→S0 tran-sition, the expression for the transfer rate constant to a com-mon acceptor may be approximated by

ke5Hi f2 k~T!expS Hi f

kBTDexpS 1

2kBT

k1

g2 ~ETA2ET

D!2D.k0

e~T!expS 21

2kBT

k1

g2 ~ETA2ET

D!2D . ~23!

In summary, the expression for the rate constant, as de-rived above, for a triplet–triplet transfer reaction between aseries of rigid triplet donors and a flexible acceptor contains~i! a preexponential factork0

e associated with the donor–acceptor orbital overlap throughHi f

2 eHi f /kBT and other fac-tors, which are not expected to depend on the triplet energydifference, as noted before,12,16,35 and ~ii ! an exponentialterm with a quadratic dependence on the energy differencebetween the donor and the acceptor triplet states.

In Eq. ~23! we may define ageometrical distortion pa-rameterg5(2g2/k1)1/2, which would eventually determinethe extent on nonvertical behavior. For acceptor moleculeswith large values of the energy gradient on the triplet surface,the easy activation of ground-state vibrational modes~lowforce constantsk1) along the gradient direction would bevery effective in reducing the triplet excitation energy~highg value!, resulting in a large rate constant even for endother-mic reactions. However, the occurrence of significant confor-mational changes can also be completely ineffective in thisregard, unless they give rise to a large change in the triplet-state energy gradient. Moreover, since the values ofg andET

A

are fixed for a specific acceptor molecule, Eq.~23! predictsan ‘‘inverted’’ region, where an increase in the donor energyavailable for excitation would eventually result in a lowertransfer rate,18,19~a!,20 in common with the Marcus theory ofelectron transfer.36 However, this effect would be difficult toobserve in TET reactions, due to the presence of higher en-ergy triplets and alternative reactive channels.37,38

The expression for the TET rate constant@Eq. ~23!# isamenable to experimental test, because theg values of manymolecules of interest can be computed with high accuracy bycurrent quantum mechanical methods, as shown in the fol-lowing for COT. It is important to remark that a computationof the complete PES for the singlet and triplet states is notnecessary in this approximation, as only that part of the3Asurface in the vicinity of the equilibrium configuration of1Ais required~Fig. 2!.

Finally, it may be instructive to compare the TST rateconstant expression given by Eq.~23! with that derived fromthe Fermi golden rule@Eq. ~1!#. In both cases, the preexpo-nential factor depends on the square of the electronic cou-pling (Hi f

2 ) between the intervening states. In addition, it canbe shown by lengthy algebraic manipulation that the expo-nential term in Eq.~23! is, in fact, the overlap integralJbetween the acceptorT1←S0 absorption and the donorT1→S0 emission if both spectra are approximated by singleGaussian functions with very different widths, i.e.,sA

2@sD2 .

D. Nonvertical triplet energy transfer in COT

The excitation energy dependence ofke given by Eq.~23! can be compared with that obtained from COT experi-mental energy transfer rate constants28,29 by means of theexpanded kinetic representation shown in Scheme 2. If allCOT triplet-state product molecules decay before being in-tercepted by the singlet donor~irreversible reaction!, the ex-perimental rate constant of the TET process,kexp

en , can beexpressed16 by

FIG. 2. The regions of the potential energy surfaces of cyclooctatetraene~COT! singlet and triplet states which are relevant for the energy transferprocess, computed as described in the main text. The PES are show here asa function of the dihedral angle,a, of the four consecutive carbon atoms~i.e., 1-2-3-4!, which gives the degree of the planarity of the molecule, andthe change in bond distancesDp5dC–C2dCvC , wheredC–C anddCvC arethe sum over all simple and double bond distances~i.e., 2-3, and 1-2!,respectively. Note that the extent ofp-charge delocalization is proportionalto the inverse ofDp. The arrow indicates the direction of the energy gradi-ent vector (vGD). This vector, which defines the relaxation of the initiallyexcited triplet, has large component in both thep-delocalization and flatten-ing coordinates of the molecule. The 0–0 energy of3COT(D8h) was takenfrom Ref. 30.

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kexpen 5

kd

11k2d

ke1

k2e

ke

. ~24!

The probability of reverse transfer (k2e) from triplet COT tothe donor within the collision complex is also likely to benegligible ~geminate irreversibility39!, because of the ex-pected very fast relaxation of the triplet energy~see above!.Therefore, if the ratiok2e /ke!1, Eq. ~24! reduces to

kexpen 5

kd

11k2d

ke

5hekd , ~25!

where he5ke /(ke1k2d) is the efficiency of the transferprocess in the encounter complex.5 Thus, when the energytransfer rate constant is larger than that of complex dissocia-tion (k2d), the reaction becomes controlled by diffusion.5

An expression directly comparable to the experimental loga-rithmic plot of the COT transfer rate constant as a function ofthe donor energy,28,29 can be obtained by replacingke in Eq.~25! by its value from Eq.~23!, to give ~in mol units!

logkexpen 5 logkd2 logF11

k2d

ke0 expF 1

RT S ETA2ET

D

g D 2G G . ~26!

The computed values~see Sec. III! of the COT geometricaldistortion parameter,g513.5 (kcal/mol)1/2, and triplet verti-cal excitation energy,ET

A565.0 kcal/mol, can be introducedin Eq. ~26! to give an expression for logkexp

en as a function ofthe excitation energyET

D of the triplet donor:

logkexpen 5 logkd2 logF11

k2d

ke0 expF 1

RT S 65.02ETD

13.5 D 2G G . ~27!

Figure 3 shows the satisfactory fit of Eq.~27! to the COTenergy transfer experimental data28,29 for the selected set ofdonor molecules, for a value of the ratiok2d /k0

e50.25. Asmentioned earlier, this series of donor molecules have incommon the same electronic orbital parentage (p,p* ) and arigid molecular frame.

The analysis of the components of the gradient vectoryields a detailed description of the geometrical distortionsthat facilitate the excitation of the triplet COT. In this case~Fig. 4!, the largest contribution is from CvC stretchingmodes, which decrease the bond order of the double bondsand increase thep-electron density at the single C–C bonds.The second important contribution is from torsional modesaround the four single C–C bonds of the molecule. In con-trast, the contribution to the gradient vector from torsionalmodes around double bonds is negligible.

As a result of these concerted motions, the boat (D2d)ground-state conformation of COT is deformed, favoring amore planar geometry of the molecule. Interestingly, high-level computations24 of COT and an experimental study ofthe transition-state spectroscopy of this compound30 bothpredict a flat octagonalD8h geometry for the COT tripletstate.

The rms error of the fit in Fig. 3 presents a single mini-mum ~not shown! for a value of 0.25 of thek2d /k0

e ratio, theonly effective fitting parameter, because that ofkd is rela-tively well defined by the plateau in the experimental data.Since this ratio is unknown ‘‘a priori,’’ it was taken hereidentical for all donor compounds used in the experiment~vide supra! and, therefore, the consistency of the fitting de-pends on the extent to which the computedk2d /k0

e ratio isclose to the correct value. Estimated values for the rate con-stant of the dissociation of the encounter complex (k2d) influid solvents, both from random walk8 and more elaboratediffusive arguments,16,39 fall in the range of 1 – 431010 s21

~a value of 1011 s21 was derived from macroscopic entropicconsiderations5!. Therefore, from the fitting value ofk2d /k0

e

one can estimate a COT preexponential factork0

e'1011 s21, which is in the same range as those deter-mined directly,4 from picosecond-resolved measurements ofke , for several aromatic donor/acceptor pairs not structurallyrelated to COT.

FIG. 3. Plot of the energy transfer rate constant to cyclooctatetraene from aseries ofp,p* triplet donors, as a function of the donor energyET

D . Theexperimental data, in benzene solution, were taken from Refs. 28~d! and 29~m!. The solid line shows the fitting of Eq.~27! in the text to the experi-mental data, using the theoretical values 13.5~kcal/mol!1/2 and 65.0 kcal/mol for the distortion parameterg and the COT triplet energyET

A ~300 K!,respectively. 1: 9-10, dichloroanthracene, 2: anthracene, 3: acridine, 4:pyrene, 5: fluorenone, 6: chrysene. 7: p-terphenyl, 8: 2-acetophenone, 9:naphtalene, 10: fluorene.

FIG. 4. Pictorial view of the relaxed singlet-ground state of cyclooctatet-raene (D2d symmetry! illustrating the direction~arrows! of the concertedCvC stretching and C–C torsional vibrations that decrease the excitationenergy to the triplet state.

1215J. Chem. Phys., Vol. 120, No. 3, 15 January 2004 Triplet energy transfer processes

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III. METHODS

The distortion parameterg was determined by calculat-ing the numeric gradient of theT1 PES and theS0 numericvibrational force constants~see Fig. 2! using the multicon-figurational ‘‘complete active space’’~CAS! Møller–Plessetsecond-order method CASMP2(8,8)/6-31g* , as imple-mented inGAUSSIAN 98;40 the active space in all CAS calcu-lation was taken to be that ofp-symmetry orbitals~thehigher four occupied orbitals and the lowest four unoccupiedorbitals!. The equilibrium geometry of theS0 (D2d) statewas optimized using the analytical gradient at theCASSCF(8,8)/6-31g* level of theory. TheET

A value was de-termined theoretically,24 in view of the limited experimentaldata available, using a complete active space second-order perturbation theory, with a CASSCF~8,8!/ANOC@4s3p2d#/H@2s1p# reference wave function as imple-mented inMOLCAS 5.0.41,42

IV. CONCLUSIONS

The theoretical model of the triplet energy transfer reac-tion derived here allows the identification of the physicalfactors and molecular changes involved in the anomalousnonvertical triplet excitation of flexible acceptors, and pro-vides an interpretation with atomic detail of the molecularmotions involved in the process. The present approach mayalso be of utility in predicting whether the energy transferrate constant of a specific donor–acceptor pair would show anonvertical dependence of the rate constant on the donortriplet energy. The application of this theory to existing ex-perimental TET data of COT in solution provides furtherinsight into the origin of the large deviation of the transferrate constant from Sandros equation. In addition, the ground-state conformational changes responsible for the lowering ofthe triplet excitation energy could be successfully identified.

ACKNOWLEDGMENTS

We thank Professor K. Razi Naqvi for crucial commentsand Dr. R. E. Dale for helpful suggestions. Work financed byProjects BQU2000/1500, BQU2000/0646, BQU2001/2926from the Spanish Ministry of Science and Technology~MCyT!. L.M.F. is grateful to the Ministry of Education,Culture and Sport~Spain! for a Doctoral Fellowship.

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