cours ct part1 - u-bordeaux.frtheo.ism.u-bordeaux.fr/~castet/doc4/cours_ct_part1.pdf · hˆ=! 1 2...
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Theore&cal Chemistry
Frédéric Castet
frederic.castet@u-‐bordeaux.fr
Summary • The Hartree-‐Fock-‐Roothaan method • Pople and Dunning basis sets • Semiempirical models • Configura&on interac&on • Möller-‐Plesset perturba&on theory • Density func&onal theory • Time-‐dependent DFT
Download material (lectures & prac&cals) at the address hHp://blake.ism.u-‐bordeaux1.fr/~castet/doc4.html
Douglas Hartree (1897-‐1958)
Vladimir Fock (1898-‐1974)
THE HARTREE-‐FOCK-‐ROOTHAAN METHOD
Clemens Roothaan (1918-‐)
H = ! 12"µ
µ=1
2N
# !ZArµA=1
P
#µ=1
2N
# +1rµ$$=1
2N
#µ<$
2N
# = hµ
µ=1
2N
# +1rµ$$=1
2N
#µ<$
2N
#
Electronic Hamiltonian for a molecule with 2N electrons and P nuclei
! =1(2N)!
"1(r1) "2 (r1) ... "2N (r1)"1(r2 ) "2 (r2 ) ... "2N (r2 )... ... ... ...
"1(r2N ) "2 (r2N ) ... "2N (r2N )
THE HARTREE-‐FOCK-‐ROOTHAAN METHOD
Single determinant wavefuncNon
!µ (rµ ) = "µ (rµ )# $µ (%µ )Spin-‐orbitals
Spin func&on α or β
Molecular orbital
HF: search for the best varia%onal wavefuncNon
x
y
z θ
ϕ
O
–e
r
r = r,!,"{ }dr = dV = r2 sin!drd!d"
VariaNonal principle
E = !H!dr " Eexact#1. Introduc&on of varia&onal parameters !"! #1…#M( )
2. Op&miza&on of the parameters
! E!"i
= 0 # i
Linear CombinaNon of Atomic Orbitals (Roothaan)
Basis set of known atomic func&ons
Expansion coefficients = varia%onal parameters
!i r( ) = Cpi"p r( )p=1
M
#
Minimiza%on of the total energy
HF: search for the best orbitals
THE HARTREE-‐FOCK-‐ROOTHAAN METHOD
Total energy
E = 2Iii=1
N
! + 2Jij "Kij( )j=1
N
!i=1
N
!
E = !H!dr"! =
1(2N)!
"1(r1) "2 (r1) ... "2N (r1)"1(r2 ) "2 (r2 ) ... "2N (r2 )... ... ... ...
"1(r2N ) "2 (r2N ) ... "2N (r2N )
Jij =!i*(rµ )!i (rµ )! j
*(r" )! j (r" )rµ"
# drµdr"
Ii = !i*(rµ )h!i (rµ )" drµ
HF integrals
Kij =!i*(rµ )!i (r" )! j
*(r" )! j (rµ )rµ"
# drµdr"
1-‐electron integrals
Coulomb integrals
Exchange integrals
Sum over N doubly occupied molecular orbitals
THE HARTREE-‐FOCK-‐ROOTHAAN METHOD
Roothaan expansion
Ii = !i*(rµ )h!i (rµ )" drµ = CpiCqihpq
q=1
M
#p=1
M
#
M2 integrals
Jij =!i*(rµ )!i (rµ )! j
*(r" )! j(r" )rµ"
# drµdr" = CpiCqiCrjCsj pq( rs)s=1
M
$r=1
M
$q=1
M
$p=1
M
$
pq( rs) =!p (rµ )!q (rµ )!r (r" )!s (r" )
rµ"drµ dr"#
Integrals in the AO basis
hpq = !p (rµ )h!q (rµ )" drµ
M4 integrals
pq rs( ) = qp rs( ) = pq sr( ) = qp sr( ) = rs pq( ) = sr pq( ) = rs qp( ) = sr qp( )
THE HARTREE-‐FOCK-‐ROOTHAAN METHOD
!i r( ) = Cpi "p r( )!p=1
M
#Atomic basis func%ons
Energy minimizaNon
E = 2Iii=1
N
! + 2Jij "Kij( )j=1
N
!i=1
N
! F(r)!i (r) = "i!i (r)Minimisa6on
N Fock equa%ons
F(rµ ) = h(rµ )+ 2Ji (rµ )! Ki (rµ )"#
$%
i=1
N
&
J i (rµ )! j (rµ ) =!i*(r" )!i (r" )rµ"
# dV"
$
%&&
'
())! j(rµ )
Fock operator
Ki (rµ )! j (rµ ) =!i*(r" )! j (r" )rµ"
# dV"
$
%&&
'
())!i (rµ )
Coulomb operator
Exchange operator
1-‐electron Fock operator
The Fock operator depends on its own solu%ons φ(r)
Itera%ve process un%l self-‐consistence
Describes electron µ in the mean electrosta6c field of the other electrons
THE HARTREE-‐FOCK-‐ROOTHAAN METHOD
Matrix form of the Fock equaNons
F(r)!i (r) = "i!i (r)
with
Fock matrix F
!
FC = "SC
Fpq = !p*" (r# )F(r# )!q (r# )dr# = hpq + Drs pq rs( )$ ps rq( )%& '(
s)
r)
Spq = !p*" (r# )!q (r# )dr#
Overlap matrix S
Dimension M x M
Elements
Dimension M x M
Elements
!i r( ) = Cpi"p r( )p=1
M
#
Dpq = niCpiCqii!
First-‐order density matrix D
Dimension M x M
Elements
Method of Linear Varia%ons
W. Ritz, J. Reine Angew. Math. 135, 1 (1909). See A. Szabo and N. S. Ostlund, Modern Quantum Chemistry: Introduc6on to Advanced Electronic Structure Theory, McGraw-‐Hill, New York, 1989.
THE HARTREE-‐FOCK-‐ROOTHAAN METHOD
General soluNon of the HF matrix equaNons for non orthogonal basis sets
By mul6plying by S–1/2 on each side
FC = SC!
!F
F!S"1/2 !S1/2 !C = S1/2 !S1/2C#
C = S!1/2 "C
S!1/2 "F"S!1/2! "## $## "S1/2 "C!"$ = S1/2C!#
!F !C !C
One seeks to obtain an eigenvalue equaNon of the form
Löwdin orthogonalizaNon P. O. Löwdin J. Chem. Phys. 1950, 18, 365
!! "F "C = "C #!C = eigenvectors of
!F !C = !C "
THE HARTREE-‐FOCK-‐ROOTHAAN METHOD
M basis func%ons ⇒ M MOs
ε MO
energy
Koopmans theorem: IP ≈ – ε
SoluNon of the HF equaNons
HOMO
LUMO
Virtual MOs
Occupied MOs
THE HARTREE-‐FOCK-‐ROOTHAAN METHOD
!(r) = ni" i
*(r)i=1
M
# "i(r)
Total electron density
!(r)space" dr = ni
i=1
M
# $i
*(r)$i(r)
space" dr = ni
i=1
M
# = N
Mulliken AO populaNons
!(r) = ni Cpiq=1
M
"p=1
M
" Cqi# p(r)
i=1
M
" #q(r) = Dpq# p
(r)#q(r)
q=1
M
"p=1
M
" Dpq = niCpiCqii=1
M
!
!(r)space" dr =N = DpqSpq
q=1
M
#p=1
M
# = Dpp + DpqSpqq$p
M
#%
&''
(
)**
p=1
M
# = Qp( )p=1
M
#
with
Qp = Dpp + DpqSpqq!p
M
" Electron popula%on in χp
nA = Qpp!A"
Mulliken atomic charges
!A = ZA " nA Net charge on atom A
ELECTRON DENSITY AND RELATED PROPERTIES
!µ = !
!µ!" dr = !
!µelec +
!µnuc( )!" dr
Dipole moment
!µelec = !eri
i" !!!!!!!!!!!!! !µnuc = eZA
!RA
A" !
!µelec = !2e "
i
*(r)# ri=1
occ
$ "i
*(r)dr
with and
!µelec = !2e CpiCqi "p (r)# r "q (r)dr
q=1
M
$p=1
M
$i=1
occ
$
= !e Dpq "p (r)# r "q (r)dr" #$$$ %$$$q=1
M
$p=1
M
$
ELECTRON DENSITY AND RELATED PROPERTIES
Dipole integrals
!µnuc = !
i
*(r)" eZA
!RA
A# !
i
*(r)dr = eZA
!RA
A#
VARIOUS TYPES OF BASIS SETS
!",n,l,mSTO r,#,$( ) =NYl,m #,$( )rn%1e%"r
Slater-‐type orbitals (STO)
STO are not efficient for evalua%ng the 3-‐ and 4-‐center integrals
!",n,l,mGTO r,#,$( ) =NYl,m #,$( )r2n%2%le%"r
2
The use of GTO ensures analy%cal solu%ons for all integrals appearing in the HF method
!i r( ) = Cpi "p r( )!p=1
M
#
Gaussian-‐type orbitals (GTO)
Atomic basis func%ons
PRACTICAL INTEREST OF GAUSSIAN FUNCTIONS
exp !"(r !A)2( )# exp !$(r !B)2( ) =Wexp !("+$)(r !P)2( )
Product of 2 Gaussian funcNons
The product of two Gaussian func6ons centered on A and B respec6vely is a Gaussian centered on the barycenter P of A and B
P = !A+"B!+"
W = exp !"#("+#)
A!B( )2$
%&
'
()with and
0
0,2
0,4
0,6
0,8
1
1,2
-‐20 -‐15 -‐10 -‐5 0 5 10 15 20
A=-‐4 α=1/50
B=+4 β=1/50
Product P=0, W=0.527
GAUSSIAN VS. SLATER FUNCTIONS: EXAMPLE OF HELIUM
He atom using a single STO
What is the best (varia%onal) STO to describe the ground state of He?
1s(r) = !3/2
"1/2exp(#! r)!!!$!!!!?
! r1,r2( ) = 121s(r1)1s(r2 ) "(#1)$(#2 )%$(#1)"(#2 )[ ]
E = 2I+ J = 2 1s(r)h1s(r)! dr! "## $##
+ 1s1s 1s1s( )! "# $#
Total energy
!2 2" 2! 5! 8
d Ed!
= 0"! =2716
=1.6875 0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
0 0.5 1 1.5 2 2.5 3
1s o
rbita
l
Electron-nucleus distance r
E = !2 "278!
E = !2.84766!a.u.
He atom using a single GTO
What is the best (varia%onal) GTO to describe the ground state of He?
1s(r) = 2!"
#
$%
&
'(3/4
exp()!r2 )!!!*!!!!?
! r1,r2( ) = 121s(r1)1s(r2 ) "(#1)$(#2 )%$(#1)"(#2 )[ ]
E = 2I+ J = 2 1s(r)h1s(r)! dr+ 1s1s 1s1s( )
d Ed!
= 0"! = 0.766996
E = 3!" 8" 2#$
%&2!'
E = !2.300987!a.u.
Total energy
EGTO
>> ESTO
GAUSSIAN VS. SLATER FUNCTIONS: EXAMPLE OF HELIUM
GTO much less efficient than STO!
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
0 0.5 1 1.5 2 2.5 3
1s o
rbita
l
Electron-nucleus distance r
Slater-‐type orbital (STO)
Gaussian-‐type orbital (GTO)
CUSP
GAUSSIAN VS. SLATER FUNCTIONS: EXAMPLE OF HELIUM
CONTRACTED GAUSSIAN BASIS SETS
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
0 0.5 1 1.5 2 2.5 3
1s o
rbita
l
Electron-nucleus distance r
Slater-‐type orbital (STO)
Use of contracted Gaussian funcNons
1s(r) = aii=1
X
! " iG ! i
G =2"i#
$
%&
'
()3/4
exp(*"ir2 )
STO-‐2G
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
0 0.5 1 1.5 2 2.5 3
1s o
rbita
l
Electron-nucleus distance r
STO-‐3G Slater exponents α Contrac%on coefficients 0.6362421394D+01!0.1543289673D+00!0.1158922999D+01!0.5353281423D+00!0.3136497915D+00!0.4446345422D+00!
Slater exponents α Contrac%on coefficients 0.2432879285D+01!0.4301284983D+00!0.4330512863D+00!0.6789135305D+00!
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
0 0.5 1 1.5 2 2.5 3
1s o
rbita
l
Electron-nucleus distance r
Use of contracted Gaussian funcNons
-2.900
-2.850
-2.800
-2.750
-2.700
2 3 4 5 6
Tota
l SCF
-LCA
O e
nerg
y (a
.u.)
X in STO-XG
using a single STO function
Influence of the number of Gaussian contrac%ons on the total energy
CONTRACTED GAUSSIAN BASIS SETS
SCF-‐LCAO CALCULATION OF THE GROUND STATE OF THE HELIUM ATOM
PRACTICAL EXERCISE
Connect to the worksta6on bacon with your login and password ssh -Y [email protected]!Create a new directory named TP1 mkdir TP1!Go to the directory TP1 cd TP1!Copy the input file (data) used by the helium code cp /home/tp/Helium/Test/data .!Modify the data file as you need and save!kwrite data!Run the helium calcula6on!/home/tp/Helium/Src/helium < data > results!Open the result file (results) and note the relevant informa6on!kwrite results!
LOGIN AND PASSWORDS
tp00 -‐-‐> E9gLwpeP tp01 -‐-‐> wclzHhR7 tp02 -‐-‐> Ookl9DMz tp03 -‐-‐> ypCp7Faf tp04 -‐-‐> HVg09F7X tp05 -‐-‐> NH8l0c6q tp06 -‐-‐> FgMy9xkx tp07 -‐-‐> YP5IeeTc tp08 -‐-‐> iYEWXZB4 tp09 -‐-‐> I7DGJSg5 tp10 -‐-‐> z4PiqfQy
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