1

The Character of the π-π* Valence State in Ethylene and Butadiene I

Long-standing controversy about the character of the V(11B1u) state in ethylene.SCF and π-CASSCF calculations give a V state, which is much too diffuse (SCF: <x2> ≈ 42-44 a0

2) as compared to the ground state (SCF: <x2> ≈ 11.7 a0

2)It is practically impossible to obtain convergence <x2> values with a CI approach. Excitation energies are systematically too high.McMurchie and Davidson (1977), and Davidson (1996) emphasize “complete spaces”, especially with respect to the π* orbital. Only differential electron correlation effects (π-π and σ-π correlation) are taken into account. This approach is good to determine the orbitals, but is not sufficient to compute excitation energies.

2

The Character of the π-π* Valence State in Ethylene and Butadiene II

For butadiene (and higher polyenes) a similar problem occurs. The character of the π* orbital and also of the 11Bustate depends strongly on the method of calculation.The π-π* state in ethylene and butadiene is dominated by a single configuration.Additionally, in butadiene we have a second valence state (2 1Ag), energetically close to the V state. This state has a strong multireference character. The excitation from the ground state is dipole-forbidden (dark state), but is considered very important for the spectroscopy and dynamics of the V state. Long controversy concerning the relative ordering of the two states.Simultaneous calculation of the valence states together with Rydberg states.

3

Ethylene Calculations

Two-step procedure (Müller 1999)Calculation of MOs: MR-CISD(σ-1ex) – the reference space is a CAS of two electrons in n π and m π* orbitals; all singles and doubles into all virtual orbitals with the restriction of only single excitation from the doubly occupied σ orbitals. This gives a full CI for the π electrons and simultaneous relaxation in the field of the σ orbitals. Different orbital sets (from CASSCF calculations for the N, V and T states) are used for the CI.Conventional MR-CISD, MR-CISD+Q and MR-AQCC calculations with increasing size of the reference space based on the natural orbitals (NOs) of the wave function of step 1.

4

Computational Details

cc-pVQZaug-cc-pVQZ + two diffuse p functions,g functions deleted

QZ

cc-pVTZaug-cc-pVTZ + two diffuse p functions

TZ

cc-pVDZaug-cc-pVDZ + two diffuse p functions

DZ

HydrogenCarbon

Basis sets

Complete basis set (CBS) limit: Cnn BeEE −

∞ +=

Reference spaces: Ref (2,2) – CAS(2,2); Ref (6,6) – CAS(6,6);Ref (12,12,1p): single excitations in CAS(12,12)

5

MR-CISD(σ-1ex) results for the V state

11.711.741.344

CASSCF(2,8)

16.116.115.81715

16.216.316.08114.514.525.011

T state MOs <x2>

N state MOs <x2>

V state MOs<x2>m π*n π

M M

TZ basis

6

MR-CI+Q and MR-AQCC results

-7.697.567.887.817.918.068.058.27CBS

-7.697.577.877.877.918.068.068.27QZ

7.697.717.597.877.987.918.078.078.27TZ

7.777.807.677.918.017.948.078.078.25DZ

AQCCCI+QCIAQCCCI+QCIAQCCCI+QCI

Ref (12,12,1p)Ref (6,6)Ref (2,2)Basis

Excitation energies (eV) for the 1 1B1u state

7

<x2> Values

TZ

DZ

17.5 (17.6)a17.2 (18.2)aRef(6,6)

18.2 (17.9)a17.1 (18.0)aRef(12,12,1p)

17.4 (17.9)a17.2 (19.5)aRef(2,2)

17.816.1Ref(12,12,1p)

16.516.1Ref(6,6)

16.516.2Ref(2,2)

MR-AQCCMR-CISD

a Energy derivative calculation

8

Ethylene Summary

Best estimate of the N−V excitation is 7.7 eV, close to the experimental band maximum of 7.66 eV (vibrationallycorrected 7.8 eV).It is not necessary too refer to nonadiabatic effects in order to get agreement with experiment.For the V state <x2> ≈ 16.5−17.0 a0

2, as compared to ≈ 90 a0

2 for the 2 1B1u and 11.7 a02 for the ground state.

9

Trans-Butadiene

States investigated: 11Ag, 21Ag, 11Bu(V), 11Bg(3s), 11Au(3pσ), 21Au(3pσ), 21Bu (3pπ) (Dallos 2004a)Two-step procedure as for ethylene

Frozen Core DOCC:All and (1a ) σ π u

Virtual Orbitals Frozen Virtual(Rydberg)

CAS;(1b ), n *(a )π πg u

S

D

D

10

6.1

6.2

6.3

6.4

6.5

6.6

6.7

6.8

6.9

7.0

7.1

CI(1Bu) CI+Q (1Bu) AQCC(1Bu) CI(2Ag) CI+Q (2Ag) AQCC(2Ag)

CAS(4,4)-R

CAS(4,4)-F

CAS(4,5)-R

CAS(4,5)-F

RCA1

Butadiene Results

aug’-cc-pVTZ basis

11

Butadiene Summary

Best values (this work): 6.18 eV (11Bu) and 6.55 eV (21Ag)EOM-CCSD( ) (Watts 1996) 6.13 eV (11Bu) and 6.76 eV(21Ag)CASPT2 (Serrano-Andrés 1993, Ostojić 2001)6.23, 6.06 eV (11Bu) and 6.27 eV (21Ag)

11Bu excitation (this work) is in very good agreement with EOM-CCSD( ), much better than for other CI calculations21Ag: the EOM result is too large (multireference character of this state)The CASPT2 excitation energy is definitely too low

T~

T~

12

Excited States of Formaldehyde

Four valence states are usually discussed: ground state, n-π*, σ- π* and π- π*. The n-π* state is the lowest excited state. The other ones are embedded in Rydberg states and have not been identified spectroscopically so far.How influence geometry changes (e.g. CO stretch) the energies?Energy minima for excited states

13

Potential energy curves for the four lowest 1A1 states

M. R. J. Hachey, P. J. Bruna and F. GreinJ. Phys. Chem. 99, 8050 (1995)

The π-π* is vertically highly excited

With CO stretching it is strongly stabilized showing several avoided crossings

The diabatic curve is repulsive

The Rydberg states are destabilized

14

GeometriesThe n-π* is pyramidalized, spectroscopically well investigatedThe σ- π* state is nonplanarThe π- π* state is planar (Merchán 1995 and references therein)

Full geometry optimizations (Dallos 2001):Reference spaces: MINVAL σ(5a1), π(1b1), n(2b2), π*(2b1)and σ*(6a1)This is the minimum number of orbitals to describe the11A1(N), 11A2(n-π*), 11B1(σ-π*) and 21A1(π-π*) statesStructures at sufficiently large CO distances are investigatedso that Rydberg states are higher in energyCASVAL : CAS(12,10) 3-7a1, 1-2b1 and 1-3b2

15

Basis sets: cc-pVDZ – cc-pVQZMCSCF state-averaging including four statesFull MR-CISD and MR-AQCC optimizations at given symmetriesHarmonic vibrationsConfirmation of all previous results except for the π-π* state

Investigation of the interaction of the 11B1(σ-π*) and21A1(π-π*) states (at bent structures 2A’ and 3A’)

16

21A1

11B1

Potential energy curves

17

Out-of-plane potential energy curves for the 21A’ and 31A’ statesCrossing of the planar 11B1 and

21A1 statesDependence on CO and HCH

18

Conical intersection for the 21A’ and 31A’ states Contour plot of the 21A’ state

19

Summary Formaldehyde

At the bent minimum of the 21A’ state the character of the wave function is mainly (62%) σ-π* characterFor the same geometry, the upper 31A’ state is mixture (35% π- π*)For the 31A’ there is no minimum (as a stationary point). The lowest energy point corresponds to the conical intersection and is a point of rapid transition to the lower stateThe π- π* state escapes a standard description as a local minimum

20

For molecules containing 2 – 4 heavy atoms very accurate MR-CI calculations are possible (program packages MOLPRO, DIESEL, GAMESS, COLUMBUS, …)In these cases geometry optimizations are not so criticalHow is the situation for larger molecules?

21

Pericyclic reactions

Cope rearrangement of 1,5-hexadiene

cyclohexane-1,4-diyl

aromatic TS

bis-allyl

(i) A concerted and synchronous mechanism – Woodward-Hoffmann allowed, ‘aromatic transition state’(ii) A non-synchronous mechanism involving a biradicaloidcyclohexane-1,4-diyl as stable intermediateBoth structures have C2h symmetry

Ventura do Monte 2003

22

Cope rearrangement

V.N. Staroverov and E.R. Davidson, JACS 122, 7377 (2000)

Previous results:CASSCF: prediction of a diyl and an aromatic TSCASPT2a and MRMP2b: only one stationary point (saddle point), however,only R is optimized, remaining geometry from CASSCF calculationDFT potential energy curves are controversial

a Hrovat, D. A.; Morokuma, K.; Borden, W. T. J. Am. Chem. Soc. 1994, 116, 1072b Kozlowski, P. M.; Dupuis, M.; Davidson, E. R. J. Am. Chem. Soc. 1995, 117, 774

23

Cope rearrangement

GOALS:Complete geometry optimizations at post-CASSCF levelPotential energy curve in R (C2h restriction), how many stationary points?Transition state or local minimum

METHODS:MR-CISD/MR-AQCC for geometry optimization and single-point calculations

24

Cope rearrangement – computational methods

MR-CISD/MR-AQCC based on CASSCF(6,6):7ag, 7bu, 5au, 5bg, 8ag, 8au

Reference spaces CAS(4,4) (7bu, 5au, 5bg, 8ag) and CAS(6,6)

Basis sets: 6-31G*, 6-31G**, 6-311G**, 6-311G(2d,1p)Largest CI dimension: 368 millionNew parallel CI program (Thomas Müller): one Davidson iteration~1 hour on 32 nodes of Schroedinger I (University of Vienna), AthlonXP 1700+

25

Size consistency

0.654.8MR-AQCC

20.231.3MR-CISD+Q

37.355.5MR-CISD

CAS(6,6)CAS(4,4)

Reference space

Energy difference E(bis-allyl)-2xE(allyl)

Energies in kcal/mol, 6-31G* basisfor allyl the CAS (3,3) reference space has been used always

26

1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.3 2.4-29

-28

-27

-26

-25

-24

-23

-22

-21

-20

∆E [k

cal/m

ol]

RCC[Å]

6-31G*/CAS(6,6) s.p. 6-31G*/CAS(6,6) opt. 6-311G(2d,1p)/CAS(6,6) opt. 6-311G(2d,1p)/CAS(6,6) s.p.

-30.135.0‘exp.’-32.933.1CASPT2

-29.733.46-311G (2d,1p)all-ref.sym.

-27.836.86-311G (2d,1p)1-ref.sym.

Stab.bAct.a

a activation energy relative to 1,5-hexadieneb stabilization energy relative to bisallyl

MR-AQCC potential energy curves

MR-AQCC activation and stabilization energies

27

Tetramethylene

Textbook Woodward-Hoffmann case:

≠

+

non-concertedconcerted

transition state (forbidden in the ground state)

diradical

+

..

28

Tetramethylene

C. Doubleday Jr., J. Am. Chem. Soc. 115, 11968 (1993)C. Doubleday Jr., J. Phys. Chem. 100, 15083, 1996N. W. Moriarty, R. Lindh and G. Karlström, Chem. Phys. Lett. 289, 442 (1998)S. Pedersen, J. L. Herek and A. H. Zewail, Science 266, 1359 (1994)

Previous results:CASSCF: gauche (GM) and trans (TM) minima, cyclization(G1, G2, CT) and gauche-trans (GT) saddle points.Higher-level calculations – MRCI, CASPT2: broad plateau onthe energy surfaceExperimentally, a barrier height of approx. 4 kcal/mol isobserved for decomposition of the biradical

Do local minima for the diradical structure exist and what are the barrier heights?

29

Tetramethylene

Reference configuration sets:CAS(4,4)CAS(8,8)RAS(2,4)CAS(4,4)AUX(2,0)single (double)-excitations from RAS into CAS/AUXup to single (double) occupancy in AUXCAS(8,8)-1ex, CAS(8,8)-2ex

Basis sets6-31G* − 6-31G(2d,1p)

Ventura 2003

30

CT

G2

GM

GF

GT

TM TF

0.05

0.00

1.35

0.55

-1.37

-68.82 : CASSCF/6-31G*

-0.95

-3.05

-50.24

1.19

0.00

0.20

**

-2.29

-64.11 : MRAQCC/6-31G*

-0.24

-2.34

-43.02cbutane

1.21

0.00

*

*

-62.07 : MRAQCC/6-311G(2d,1p)

*

*

-44.45

*

1.33

0.00

-0.18

*

*

-63.37 : MRAQCC/6-311G**

-0.31

*

-42.78

(*) no stationary pointethylenes

+

(**) no convergence

31

Gauche Minimum Potential Energy CurvesR fixed, remaining geometry optimized

CASSCF(4,4) MR-AQCC

6-31G*

6-311G**

6-311G(2d,1p)

1.5 1.6 1.7 1.8 1.9-156.574-156.572-156.570-156.568-156.566-156.564-156.562-156.560-156.558

E[a

u]

RC2C3[Å]

CAS(4,4)/opt. CAS(8,8)/s.p.

1.5 1.6 1.7 1.8-156.722-156.720-156.718-156.716-156.714-156.712-156.710-156.708

E [a

u]

RC2C3[Å]

CAS(4,4)/opt. CAS(8,8)-1ex/s.p.

1.4 1.5 1.6 1.7 1.8 1.9-156.042-156.040-156.038-156.036-156.034-156.032-156.030-156.028-156.026-156.024-156.022

E [a

u]

RC2C3 [Å]

1.4 1.5 1.6 1.7 1.8 1.9-156.084-156.082-156.080-156.078-156.076-156.074-156.072-156.070-156.068

E[a

.u]

RC2C3[Å]

1.4 1.5 1.6 1.7 1.8 1.9

-156.090

-156.085

-156.080

-156.075

-156.070

E[a

u]

RC2C3 [Å]

1.4 1.5 1.6 1.7 1.8 1.9-156.690-156.688-156.686-156.684-156.682-156.680-156.678-156.676

E[a

u]

RC2C3 [Å]

CAS(4,4)/opt.

32

Cyclodimerization of ethyleneWoodward-Hoffmann Rules

MO Correlation Diagram State Correlation Diagram

33

Rectangular “face-to-face” approach

J. Michl and V. Bonačić-Koutecký, Electronic Aspects of Organic Photochemistry

Characterization of states:G: (π1+ π2)2 (π1- π2)2

S: (π1+ π2)2 (π1- π2)1(π1*+ π2*)1

At infinite separation: (G+V)states of ethylene

D: (π1+ π2)2 (π1*+ π2*)2

At infinite separation: 1(T+T)states of ethylene

34

Rectangular structure D2h Parallelogram structure C2h

Minimum in the D state? Conical intersectionBernardi et al., J. Chem. Soc. Faraday Trans. 90, 1617 (1994): Intersection between the G and D states – CASSCF calculations

35

Computational Methods

State-averaged CASSCF(4,4)MR-CISD, MR-CISD+Q – CAS(4,4) referencecc-pVDZ, aug-cc-pVDZ and cc-pVTZ basis sets

36

Face-to-face approach (D2h)RCC fixed, remaining geometry optimized for the S-state

MR-CISD+Q

0

2

4

6

8

10

0 2 4 6 8 10

R(CC)

E(eV

) GDS

CASSCF

0

2

4

6

8

10

12

0 2 4 6 8 10 12

R(CC)E(

eV) G

DS

aug-cc-pVDZ basis

37

-156.70

-156.60

-156.50

-156.40

-156.30

-156.20

-156.10

-156.00

1.5 1.7 1.9 2.1 2.3 2.5 2.7 2.9

R(C-C) [A]E

[a.u

.] E(CI1)E(CI2)E(CI3)

G

SDMXS1

MXS2

MXS2: g-vector MXS2: h-vector

38

-156,60-156,55

-156,50-156,45

-156,40-156,35

-156,30-156,25

-156,20-156,15

90 95 100 105 110 115 120 125

<CCC [deg.]

ECI [

a.u.

] E(CI1)E(CI2)E(CI3)

C2h distortion

G

S

D

MXS3: g-vector MXS3: h-vector

39

MXS2

MXS3

MXS1

Starting from two separated ethylenes in the S state a direct reaction path to a conical intersection with the ground state is

found allowing rapid radiationless transition from S to G

40

Bond-Stretch Isomerism in the Benzo[1,2:4,5]dicyclobutadiene System

I

1

23

4

56

7

8

9

10

1

23

4

56

7

8

9

10

IIStructure I: …(1b1g)2 (1b2g)2 (2b3u)2 (2b1g)2 (1au)0

Structure II: …(1b1g)2 (1b2g)2 (2b3u)2 (1au)2(2b1g)0

Exist local minima for I and II and what is the barrier height?

Antol 2004

41

SA-CASSCF(10,10): 1b3u, 1b1g, 1b2g, 2b3u, 2b1g, 1au, 2b2g, 3b3u, 2au and 3b2g

The CAS(10,10) reference space is too big for MR-CI calculationsSelection of orbitals according to NO occupation

nocc= 1.956, 1.919, 1.898, 1.877, 1.730, 0.267, 0.128, 0.104,0.080 and 0.0401b3u moved to the reference-doubly occupied space3b2g moved to the virtual space1b1g, 1b2g moved to the RAS2b3u, 2b1g, 1au, 2b2g moved to the CAS3b3u, 2au moved to the AUX (auxiliary space)3b2g moved to the virtual space

42

Full MR-AQCC geometry optimizations in D2h symmetrySaddle-point determinationBasis sets: 6-31G* for geometry optimizations and 6-31G(2d,1p) for single point calculations

7.6

kcal

mol

−1

7.2

kcal

mol

−1

43

Cyclobutadiene

M. Eckert-Maksić, M. Vazdar, M. Barbatti, H. Lischka and Z. B. Maksić

Ref. space: π-CAS(4,4); RDP (restricted direct product): 4 σCCσCC* and

4 σCHσCH* orbitals, RAS(4 σCC )/π-CAS(4,4)/AUX(4 σCC*), singleexcitations RAS/CAS and CAS/AUXBasis sets: cc-pVDZ, aug-cc-pVDZ, cc-pVTZ and aug'-cc-pVTZMR-CISD and MR-AQCC calculations

rectangle rectanglesquare

D2h D4h D2h

44

0.0

10.0

20.0

30.0

40.0

50.0

60.0

70.0

80.0

90.0

100.0

0 0.5 1

λ

E /

kcal

mol

-1

1B2g

1A1g

3A2g

1B1g

1Ag

1B3g

3B3g

1Ag

0 1 0

45

MR-AQCC-CAS(4,4) Barrier

cc-pVDZ 7.3

aug-cc-pVDZ 7.4

cc-pVTZ 8.4

aug'-cc-pVTZ 8.3

MR-AQCC

Barrier

RDP(σCC)/aug-cc-pVDZ 7.5

RDP(σCC)/aug-cc-pVTZ 8.6

RDP(σCC, σCH)/aug-cc-pVDZ 7.3

Energy barriers in kcal/mol

46

NN

N N

Tetracyano-cyclobutadiene

Cyclobutadiene-oligomers