Page 1

A Multireference Configuration Interaction Investigation of the Excited-State Energy

Surfaces of Fluoroethylene (C2H3F)

Mario Barbatti,* Ade ´lia J. A. Aquino, and Hans Lischka†

Institute for Theoretical Chemistry, UniVersity of Vienna, Wa ¨hringerstrasse 17, A-1090 Vienna, Austria

ReceiVed: February 17, 2005; In Final Form: April 22, 2005

Multireference configuration interaction with singles and doubles (MR-CISD) calculations has been performed

for the optimization of conical intersections and stationary points on the fluoroethylene excited-state energy

surfaces. For the planar ground state geometry, the vertical spectrum including 3s and 3p Rydberg states was

calculated. From this geometry, a rigid torsion around the CC bond strongly reduces the energy gap between

S0 and S1 states. Furthermore, a search for the minimum of the crossing seam shows that there exists a

conical intersection close to the twisted structure and two additional ones for cis and trans pyramidalized

structures. These three intersections are connected by the same seam. We have shown that the Hula-Twist

process is an alternative way to the direct CC twisting in order to reach this part of the seam. Other conical

intersections were also located in the CH3CF and CH2FCH, H-migration, and C3Vstructures. The photodynamics

of the system is discussed based on topological features of these intersections.

1. Introduction

It is well-known that conical intersections act as funnels

allowing fast, radiationless transfer to another electronic state.

Actually, these intersections span a (N-2)-dimensional space,

the intersection seam.1,2The properties of the conical intersec-

tions make the crossing seam the very heart of the ultrafast

photodynamical processes and a great effort has been devoted

to its characterization, mainly of its minima. Despite the

significant progress achieved in the last years in the development

of methods2-5and in the study of the conical intersections for

even relatively large molecules,6-9several questions still arise,

such as those concerning the extension of the seam and the role

that each part is playing during the dynamics. Attempts to

answer this kind of question have been recently performed by

using ethylene as a prototype molecule.10,11The break of

symmetry introduced by substituted ethylenes, such as H2Cd

XH2or C2HnX4-n, can furnish valuable information about the

electronic structures involved in the crossing seam. Recently,

silaethylene (CSiH4)12and the methyleneimmonium cation

(CNH4+)13have been studied in our group. The haloethylenes

C2HnX4-n(X ) F, Cl, Br), and in particular fluorethylene (X

) F, n ) 3)sthe subject of the present workscomprise

examples of a different kind since the polarity of the π bond is

achieved by a link to the CdC bond and not by substitution of

a carbon atom.

An important topic in the research of haloethylenes is the

determination of the elementary processes resulting in the

molecular fragmentation, such as the hydrogen halide and

molecular hydrogen eliminations.14-17These eliminations can

involve the H and the halide atoms attached to the same C atom

(three-center RR-elimination) or atoms on different C atoms

(four-center R?-elimination). Since the work of Berry and

Pimentel,14which presented the first experimental results for

the HF elimination after a photoexcitation, numerous theoretical

and experimental investigations have been dedicated to this

subject,14,23especially in treating the ground-state structures and

processes. Kato and Morokuma15showed that in the case of

fluoroethylene the simple determination of the reaction path was

not sufficient to decide between the three- and four-center HF

elimination. On the basis of the HF translational energy release

after 157 nm (ππ*) photoexcitation, Sato and co-workers23

concluded that the four-center R?-elimination should be the

dominant process. However, the HX (X ) F, Cl, Br) rovibra-

tional distribution obtained from the photolysis of CH3X at 193

nm16,17has shown that the three-center process is the dominant

one.

Besides HF elimination, Kato and Morokuma15investigated

several reaction channels on the ground and first triplet energy

surfaces of fluoroethylene including H-migration and cis-trans

isomerization. Smith, Coffey, and Radom24systematically

studied the ground-state equilibrium geometry at several ab initio

levels. Martı ´nez-Nu ´n ˜ez and Va ´squez25investigated the frag-

mentation reactions in the ground state. Recently, they have

furnished theoretical results about the rovibrational distribution

of the HF fragments in the photodissociation process.22Also

recently, Ljubic ´ and Sabljic ´26and Lie and co-workers27have

investigated the reaction mechanisms of fluoroethylene and

ozone. Kunsa ´gi-Ma ´te ´ et al.21have employed AM1 semiempirical

molecular dynamics to investigate the influence of the environ-

ment on the HF elimination. To the best of our knowledge,

Bacskay28was the only one to perform ab initio calculations

on the first singlet-excited state of fluoroethylene investigating

the isomerization and fragmentation processes.

In the present work, our main goal is to characterize the

lowest excited-state energy surfaces of fluoroethylene with

particular emphasis on the crossing seam between the S0and

S1states, and to discuss the role that it has in the photochemistry

of this system. Although our research is not primarily concerned

with the fragmentation topic, we believe that the characterization

of the first singlet excited state and of the S0/S1crossing seam

can furnish new and valuable information about this process.

* Address correspondence to this author. E-mail: mario.barbatti@

univie.ac.at.

†E-mail: hans.lischka@univie.ac.at.

5168

J. Phys. Chem. A 2005, 109, 5168-5175

10.1021/jp050834+ CCC: $30.25 © 2005 American Chemical Society

Published on Web 05/20/2005

Page 2

We also present ab initio results for the vertical spectrum of

fluoroethylene, including in total six valence and Rydberg

singlet-excited states.

2. Computational Details

State-averaged multiconfiguration self-consistent field (SA-

MCSCF) and multireference configuration interaction with

singles and doubles (MR-CISD) calculations were carried out.

In the SA-MCSCF calculations equal weights were used for all

states. Two different active spaces were chosen in the MCSCF

calculations. For the description of valence states, the wave

function was a CAS(2,2) in the π and π* orbitals. State

averaging was performed over the states N (π2), V (ππ*), and

Z (π*2). The same CAS(2,2) was used as reference space in

the subsequent MR-CISD calculations. The final expansion

space for the MR-CISD calculations in terms of configuration

state functions (CSFs) consisted of the reference configurations

and of all single and double excitations thereof into all internal

and external orbitals. The interacting space restriction29was

applied. The three core orbitals were always kept frozen in the

post-MCSCF calculations. Analogous principles for the con-

struction of the CI wave function were applied for the larger

reference space used in the calculation of the Rydberg states

described below. The MR-CISD/CAS(2,2) calculations were

used for the determination of the minima on the crossing seam

(MXS), for the calculation of stationary points, and for the cuts

of the potential energy surface. In general, all optimizations were

performed without symmetry restrictions. The aug-cc-pVDZ

basis30,31was selected in these calculations. All stationary points

and MXSs were reoptimized with an aug′-cc-VTZ basis set,

which was derived from the original aug-cc-pVTZ set by

omitting the augmented f functions on the carbon atoms and

all d functions on the hydrogen atoms. Size-extensivity correc-

tions were taken into account by means of the extended

Davidson correction32,33and will be denoted by +Q.

For the joint calculation of valence and Rydberg states, the

ππ*-CAS(2,2) was augmented by four auxiliary (AUX) orbitals

representing the 3s and 3p Rydberg orbitals. Only single

excitations were allowed from the CAS to the AUX space. In

addition to the three valence states N, V, and Z, four Rydberg

states (π-3s, π-3px, π-3py, and π-3pz) were considered

leading to a state-averaging over seven states at the MCSCF

level. The same configuration space was used as reference space

in the MR-CISD calculations and is denoted as MR-CISD/SA-

7-CAS(2,2)+AUX(4). A d′-aug-cc-pVDZ basis30,31,34,35was

chosen to take into account the diffuseness of the Rydberg

orbitals. On this basis, d′ stands for the original d-augmented

set of diffuse basis functions but omitting the doubly augmented

d functions on the carbon atoms and the doubly augmented p

function on the hydrogen atoms.

Although the calculations at the MR-CISD/SA-7-CAS-

(2,2)+AUX(4)/d′-aug-cc-pVDZ level produce satisfactory re-

sults in torsional angles larger than 20°, for the planar and

quasiplanar geometries the treatment at this level is not

sufficient. The π-3pxstate is located below the ππ* state, in

contradiction to the experimental results (see Subsection 3a).

This effect occurs due to a strong mixing between the π and

3pxmolecular orbitals, which also appears in other systems such

as ethylene36and butadiene.37

To obtain a more adequate description of the vertical

excitations, the aforementioned CAS(2,2)+AUX(4) reference

space was augmented by a restricted direct product (RDP)

space38constructed for all σ orbitals. The RDP space is

composed of 14 orbitals grouped in seven subspaces, one for

each σ bond and the lone pair, i.e., three [σ-σ*]CH, one

[σ-σ*]CC, one [σ-σ*]CF, and two [σ-σ*]n(F)pairs. Each σ-σ*

subspace is restricted to singlet pairing. The MCSCF calculation

based on the RDP wave function resulted in localized orbitals

very similar to those obtained in generalized valence bond

(GVB) calculations.39The molecular orbitals obtained in this

RDP(σ)+CAS(2,2)+AUX(4) space were used for the subse-

quent MR-CISD calculation. The reference space was composed

of all previously used CAS(2,2)+AUX(4) configurations plus

all single excitations from the σCCand σCForbitals to the CAS,

AUX, σCC*, and σCF* orbitals. The remaining σ orbitals were

transferred to the reference doubly occupied space and the

corresponding σ* orbitals to the virtual space. We denote this

level as MR-CISD/SA-7-[RDP+CAS(2,2)+AUX(4)].

Although the π* and 3pxorbitals resulting from the MCSCF

calculation in this improved RDP space are still strongly mixed,

the introduction of the σ-π correlation at the reference level

and of concomitant higher excitations at the MR-CISD have a

significant effect on the final result by stabilizing the V state

and reducing the mixing with the π-3pxstate.

The ground-state geometry and the vertical spectrum were

calculated also by means of the resolution of the identity

approximate coupled cluster singles and doubles (RI-CC2)

method.40-43For the geometry optimization of the ground state

the aug′-cc-pVTZ basis set was used. The vertical excitation

spectrum was obtained with the d′-aug-cc-pVDZ basis set.

For the geometry optimizations, analytic MR-CISD energy

gradients were computed by using the procedures described in

refs 44-47. Determinations of the minima on the crossing seam

(MXS) were performed by using the analytic MR-CI nonadia-

batic coupling vectors4and the direct inversion in the interactive

subspace (GDIIS) procedure developed in ref 48. Standard

GDIIS optimization49was used for the determination of station-

ary points. Natural internal coordinates were constructed ac-

cording to the directions given in ref 50.

Optimized geometries, energies, and conical intersections

were obtained with the COLUMBUS program system.51-54The

atomic orbital (AO) integrals and AO gradient integrals have

been computed with program modules taken from DALTON.55

The RI-CC2 calculations were performed with the TURBO-

MOLE program package.56

3. Results and Discussions

3.a. Vertical Excitations, CC Torsion, and Rydberg States.

The ground-state equilibrium geometry of the C2H3F system is

planar and belongs to the Cs point group (Figure 1). The

computed geometries are in good agreement with the experi-

mental ones24(see Table 1). The effect of improving the basis

set from the double to the triple-? level is to get an overall

reduction of the CC and CH bond distances by about 0.01 Å

(0.02 Å in the case of the CF bond distance). The same pattern

of bond-distance shortening is also observed in the optimization

of all other stationary structures and conical intersections that

will be discussed in Subsections 3.c and 3.d.

The absorption spectra of several haloethylenes were mea-

sured by Be ´langer and Sadorfy.57The spectrum of the C2H3F

species shows a broad peak for the V-state (ππ*) centered at

7.44 eV, superimposed by several sharp peaks assigned to

Rydberg transitions. Different from ethylene, the fluoroethylene

absorption spectrum does not present a long low-energy

progression. The π-3s transition was assigned to the 6.89 eV

peak. The 8.09 eV peak was assigned as the π-3pxtransition.

Except for the π-3pxand π-π* transitions, the RI-CC2 and

MR-CISD methods present quite similar transition energies (see

Excited-State Energy Surfaces of Fluoroethylene

J. Phys. Chem. A, Vol. 109, No. 23, 2005 5169

Page 3

Table 2) with good agreement for the experimental π-3s

transition. The addition of the Davidson corrections (+Q) to

the MR-CISD results produces a systematic increase of the

Rydberg excitation energies by about 0.3 eV. Our best MR-

CISD+Q result for the vertical π-π* transition is still 0.4 eV

above the experimental result. This shows that similar difficulties

with the description of this state occur as in ethylene (see ref

36 and references therein). Also the RI-CC2 calculations give

only a slightly better value. To resolve this discrepancy

significantly more extended calculations would have to be

performed and possible effects of the vibrational structure and

nonadiabatic couplings similar to ethylene58need to be con-

sidered. Moreover, we note that the experimentally observed

band is very broad and the determination of the band maximum

seems to be somewhat arbitrary. Since our aim is the investiga-

tion of energy surfaces and conical intersections far away from

the region of the vertical excitation, we did not pursue this

question further.

The values for the oscillator strength show that, as expected,

the optical absorption is dominated by the V state, but with

some contributions from the π-3pxand π-3s states. The high-

energy π*2state (Z) is shown in Table 2 just because it can be

obtained also within the ππ*-CAS(2,2) space. However, we

should bear in mind that between the V and the Z states a

multitude of other states will be located, including those

involving the σ*(C-F) orbital.

As is shown in Figure 2, the Rydberg states are destabilized

by the torsional coordinate, while the V and Z states decrease

in energy and ultimately become the lowest excited states. A

similar behavior was found in the case of ethylene.58,59This

means that the cis-trans isomerization can occur without barrier

in the V or Z states. The V state crosses all Rydberg states for

torsional angles between 0° and 30°, and after that it becomes

the S1state. Due to its high energy in the planar geometry, the

Z state crosses the Rydberg states in the region from 60° to

75°. Finally, it becomes the S2state.

For ethylene, the Z state crosses the V state at a torsional

angle of 86°10,60and these two states are almost degenerate at

90°. In the present case, the stabilization of the Z state is not

strong enough to allow this crossing and this state lies close to

the Rydberg 3s state for the 90° twisted geometry, about 3.4

eV above the V state. For a rigid torsion one finds only a

relatively small gap of 0.86 eV between the S0and S1states at

90°. If we optimize the twisted structure of the V state, this

gap is reduced to 0.62 eV (see Table 3). As we will show below,

these rather small gaps are a good indication that there is a

crossing between the ground and V states near the twisted

structure in contrast to the situation found for ethylene.

The qualitative difference in the torsional potential energy

curves for ethylene and fluoroethylene can be rationalized by

the 3 × 3 CI analytical model for biradicaloids developed by

Bonac ˇic ´-Koutecky ´ et al.61,62This model predicts that for the

type of nonsymmetric biradicaloids as is the case for fluoro-

ethylene the S1 state can become degenerate with S0 by

increasing the electronegativity difference of the CC bond. The

just-described results for ethylene and fluoroethylene fit very

well into this model.

3.b. Description of Isomers. Besides the planar global

minimum CH2CHF, we have characterized two other isomers

of fluoroethylene on the S0surface, one with the structure CH3-

CF and another with the structure CH2FCH. The structure and

selected geometrical parameters of these isomers are shown in

Figure 1. The energy of each structure is given in Table 3. The

theoretical characterization of these two isomers at the SCF/4-

31G level was carried out many years ago by Kato and

Morokuma15and recently by Bacskay28for CH3CF at the DFT

and MR-CISD levels.

The isomer with the nonsubstituted CH3group is more stable

than that with the CH2F group, lying 2.32 eV (Table 3, L2Q

level) above the ground-state global minimum. The isomer with

the CH2F group, on its turn, has its ground state 3.30 eV above

the global minimum. Although we observe these strong differ-

ences in the ground state energies, the first-excited-state energies

of the two isomers are practically the same, as we can see from

Table 3.

Our result for the vertical S0-S1excitation energy of the CH3-

CF isomer is 2.91 eV (Table 3, L2Q level), which is in good

agreement with the 2.99 eV obtained by Bacskay at the MR-

CISD+Q/cc-pVTZ level, but with geometry optimization at the

B3LYP/cc-pVTZ level.

3.c. Minima on the Crossing Seam. As we discussed above,

the small S0/S1gap for the 90° twisted structure is an indication

that there should be a crossing near it. Indeed, we succeeded in

locating an intersection in a twisted structure with a slight

character of hydrogen migration. Its geometrical parameters and

energy are shown in Figure 3 and Table 4, respectively. Figure

4a shows the gradient difference vector g01and the nonadiabatic

Figure 1. Structures and selected parameters of the stationary structures

optimized at the L1 (L2) level defined in Table 3. Distances in Å and

angles in deg.

TABLE 1: Selected Geometric Parametersafor the Planar

Ground State Minimum

L1b

1.338

1.357

1.085

1.085

1.085

121.6

126.3

118.8

121.5

L2c

1.325

1.338

1.073

1.071

1.073

121.5

126.2

119.3

121.2

CC2d

1.327

1.358

1.090

1.088

1.089

122.0

126.4

118.9

121.9

exptle

1.329

1.346

1.077

1.081

1.081

121.5

125.4

118.6

120.9

C1C2

C1F

C1H

C2HC

C2HT

C2C1F

C2C1H

C1C2HC

C1C2HT

aDistances in Å and angles in degrees.bL1 ) MR-CISD/SA-3-

CAS(2,2)/aug-cc-pVDZ.cL2 ) MR-CISD/SA-3-CAS(2,2)/aug′-cc-

pVTZ.dRI-CC2/aug′-cc-pVTZ.eReference 24.

5170 J. Phys. Chem. A, Vol. 109, No. 23, 2005

Barbatti et al.

Page 4

coupling vector h01. The g-h space is composed mainly of CC

stretching, H-migration, and CC torsion. This situation is similar

to that for the CSiH412and CNH4+13molecules, for which the

simple torsion also ends in an intersection.

For the twisted-orthogonal structure of ethylene there is a

large S0/S1gap of 2.35 eV.59To reach the conical intersection

requires strong pyramidalization and partial hydrogen migration.

For fluoroethylene the twisted structure is already an intersection

and, furthermore, the seam continues along the pyramidalization

of the CH2group and reaches one of the two MXS at angles of

122.4° (cis pyramidalized) and 122.9° (trans pyramidalized).

The two MXS are characterized in Figure 3 and Table 4. In

Figure 5 the path connecting the twisted intersection with the

cis- and trans-pyramidalized MXS is shown in terms of the

pyramidalization angle ?. The geometries used to calculate this

path were obtained by a simple geometrical linear interpolation

between the twisted conical intersection and each of the

pyramidalized MXSs. Very flat curves are obtained. Figure 5

shows that already this procedure leads to a path very close to

the seam. Therefore, we did not consider it necessary to optimize

this path completely. An analogous path on the seam exists in

TABLE 2: C2H3F Vertical Excitations from the Planar Ground State Structure

energy (eV)

MR-CISD+Qa

0.00f

7.42

8.06

8.26

7.86

8.37

13.05

state

N

π-3s

π-3pyz

π-3pyz

V

π-3px

Z

MCSCF

0.00d

5.89

6.49

6.72

8.33

6.82

14.09

MR-CISD

0.00e

7.15

7.78

7.98

7.86

8.17

13.51

CC2b

0.00g

7.12

7.74

7.99

7.72

8.51

exptlc

oscillator strengtha

11A′

11A′′

21A′

31A′

21A′′

31A′′

n1A′

6.98 0.06

0.00

0.00

0.29

0.10

0.00

7.44h

8.09

aMR-CISD+Q/SA-7-[RDP+CAS(2,2)+AUX(4)]/d′-aug-cc-pVDZ.bRI-CC2/d′-aug-cc-pVDZ.cReference 57.dE ) -176.988334 au.eE )

-177.365055 au.fE ) -177.422771 au.gE ) -177.550517 au.hMaximum of the absorption band.

Figure 2. Potential energy curves for the rigid torsion. The curves

are plotted in a diabatic way following the character of the wave

function.

TABLE 3: Ground- and Excited-State Energies of the

Stationary Structures Optimized for the State Sk(k ) 0, 1)a

energy (eV)

L1Q

0.00c

7.94

2.98

4.36

3.41

4.03

2.25

5.18

3.18

5.32

structure

planar

k

0

state

S0

S1

S0

S1

S0

S1

S0

S1

S0

S1

L1

0.00b

8.24

3.11

4.68

3.51

4.27

2.26

5.19

3.14

5.27

L2

0.00d

8.29

3.24

4.72

3.63

4.31

2.31

5.23

3.25

5.35

L2Q

0.00e

7.99

3.13

4.43

3.55

4.10

2.32

5.23

3.30

5.40

twisted0

twisted1

CH3CF0

CH2FCH0

aThe reference energy level is the planar ground state.bE )

-177.362514 au. L1 ) MR-CISD/SA-3-CAS(2,2)/aug-cc-pVDZ.cE

) -177.413880 au. L1Q ) MR-CISD+Q/SA-3-CAS(2,2)/aug-cc-

pVDZ.dE ) -177.490127 au. L2 ) MR-CISD/SA-3-CAS(2,2)/aug′-

cc-pVTZ.eE ) -177.549453 au. L2Q ) MR-CISD+Q/SA-3-CAS(2,2)/

aug′-cc-pVTZ.

Figure 3. Geometrical structures and selected parameters for the

conical intersections optimized at the L1 (L2) level defined in Table

3. The pyramidalization angles of the cis and trans pyramidalized MXSs

are 119.2° (122.4°) and 120.4° (122.9°), respectively. Distances in Å

and angles in deg.

TABLE 4: Energies of the S0/S1Conical Intersectionsa

energy (eV)

L1Q

3.92

3.96

4.28

4.92

5.50

5.88

6.11

structureb

cis pyramidal

trans pyramidal

twisted ci

CH2FCH MXS

CH3CF MXS

CH3CF C3Vc

H-migration

L1

4.10

4.12

4.39

4.83

5.47

5.82

6.38

L2

4.15

4.19

4.42

4.91

5.48

5.82

6.48

L2Q

3.91

3.94

4.21

5.00

5.50

5.86

6.24

aThe zero energy level is the planar ground state (see Table 3). The

computational levels (L) are defined in Table 3.bci ) conical

intersection. MXS ) minima on the crossing seam.cGeometry

optimized at the S1state restricted to the C3Vpoint group.

Excited-State Energy Surfaces of Fluoroethylene

J. Phys. Chem. A, Vol. 109, No. 23, 2005 5171

Page 5

ethylene connecting the pyramidalized MXS to the H-migration

conical intersection.59,63In fluoroethylene we are observing the

same feature, only displaced to small H-migration angles. An

important distinction between the pyramidalized MXSs in

ethylene and fluoroethylene is that in the former there is a strong

degree of migration in one of the H atoms of the pyramidalized

group, which results in a highly asymmetric structure. In

fluoroethylene, this asymmetry is not observed in any of the

pyramidalized MXSs and, therefore, the structures belong to

the Cspoint group.

In ethylene, Ben-Nun and Martinez10have shown that an

ethylidene (CH3CH) conical intersection exists. Later on,

Toniolo et al.64located the symmetry-required C3V-ethylidene

conical intersection. Recently, we have generalized these

results59by showing that part of the S0/S1 crossing seam of

ethylene lies also in the ethylidene region and that these two

conical intersections belong to this seam. In other previous

work,10we have shown that in the photochemical process, this

region of the seam is responsible from 10% to 30% of the S1

f S0conversions. In analogy to ethylene, we have located three

points of intersection in the CH3CF and CH2FCH regions of

the configuration space of fluoroethylene. The structures of the

CH3CF and CH2FCH MXSs are shown in Figure 3 and the

energies of both and also of the C3Vconical intersection (linear

CCF axis) are given in Table 4.

The C3V conical intersection was studied previously by

Bacskay.28In this work it was pointed out that this conical

intersection could be responsible for the internal conversion

during the CH3CF f CH2CHF isomerization process. A barrier

of 0.89 eV has been computed between the system initially

prepared in the S1state (vertical excitation) of the CH3CF isomer

and the C3V conical intersection. We note, however, that the

true MXS, as mentioned above, is distorted from the C3V

intersection. Our best result for this barrier is 0.27 eV,

significantly reduced in comparison to the 0.89 eV presented

by Bacskay. Therefore this MXS should be considered a better

candidate for the main S1-S0 funnel in this region of the

configuration space than the C3Vconical intersection.

For ethylene, the same seam connects the H-migration conical

intersection and the intersection in the ethylidene region of the

configuration space.59We expect that the same will occur for

fluoroethylene even though we did not search for this path. One

indication that this connection should exist in the present case

also is the H-migration conical intersection that we have found

for large H-migration angles (Figure 3 and Table 4). Probably,

this conical intersection does not correspond to a minimum but

to a saddle point on the crossing seam in analogy to the ethylene

case.63We did not follow this question in more detail here since

the energy of this intersection is relatively high.

It is well-known (see, e.g., Atchity et al.3and Jasper and

Truhlar65) that the topography of the region around a conical

intersection has an influence on the dynamics of the system.

For instance, depending on the inclination or on the symmetry

of the double cone, there will be a different probability of

returning to the upper state. Following Yarkony,2,66these

topographic features can be described in terms of a set of four

parameters:

where s01is the gradient sum vector and (x ˆ, y ˆ) are unit vectors

based on Schmidt-orthogonalized vectors g (energy gradient

difference) and h (nonadiabatic coupling vector):

Figure 4. (a) Difference gradient vector g01and nonadiabatic coupling

vector h01for the twisted conical intersection optimized at the MR-

CISD/SA-3-CAS(2,2)/aug′-cc-pVTZ level. (b) Linearized adiabatic

energies for the twisted conical intersection in the g-h (x-y) space.

Energy in eV and x and y in Å.

Figure 5. S1and S2potential energy curves for the pyramidalization

of the CH2group.

σx)s01‚x ˆ

dgh

(1a)

σy)s01‚y ˆ

dgh

(1b)

∆gh)(g2- h2)

dgh

2

(1c)

dgh) (g2+ h2)1/2

(1d)

x ˆ ) g01/g,

g ) ||g01||

h ) ||h01||

(2)

y ˆ ) h01/h, (3)

5172 J. Phys. Chem. A, Vol. 109, No. 23, 2005

Barbatti et al.