www.elsevier.com/locate/cplett

Chemical Physics Letters 401 (2005) 276–281

Semiempirical molecular dynamics investigation ofthe excited state lifetime of ethylene

M. Barbatti a,*, G. Granucci b, M. Persico b, H. Lischka a

a Institute for Theoretical Chemistry and Structural Biology, University of Vienna, Waeringer Strasse 17, A-1090 Vienna, Austriab Dipartimento di Chimica e Chimica Industriale, Universita di Pisa, v.Risorgimento 35, 56126 Pisa, Italy

Received 20 October 2004; in final form 11 November 2004

Available online 8 December 2004

Abstract

Semiempirical molecular dynamics with surface hopping was employed to investigate the lifetime of excited states of ethylene.

Based on previous ab initio multireference configuration interaction results, a complete reparametrization of the AM1 semiempirical

parameters was performed. Depending on the initial vertical excitation energy, lifetimes from 105 to 139 fs were found for the V-

state decay. Comparison to the pump–probe experiments was performed in order to explain the large differences between the the-

oretically and experimentally obtained lifetimes. The results show that probe energies of at least 7.4 eV should be employed to ionize

the system for geometries close to the conical intersections.

� 2004 Elsevier B.V. All rights reserved.

1. Introduction

The ethylene molecule plays a fundamental role for

the understanding of photoisomerization processes and

particularly for the ultrafast energy conversion through

nonadiabatic transitions [1–4]. Although this molecule

has been intensively studied for decades, several ques-

tions about its photodynamics still remain open. In

particular, the lifetime of the excited state after photoex-

citation is still subject of discussion. While pump–probeexperimental values are around 20–40 fs [5–8], the direct

and quantum wavepacket dynamics predictions point to

values around 50 fs [9], 180 fs [10] or even more [2,4].

Ethylene is a sufficiently small molecule for which

dynamics with ab initio energy surfaces and wavepacket

dynamics can be performed [2,4,10], but already here

limitations concerning the size of the statistical samples

0009-2614/$ - see front matter � 2004 Elsevier B.V. All rights reserved.

doi:10.1016/j.cplett.2004.11.069

* Corresponding author. Fax: +43 1 4277 52793.

E-mail addresses: mario.barbatti@univie.ac.at (M. Barbatti),

hans.lischka@univie.ac.at (H. Lischka).

[2,10] or the number of degrees of freedom [4] appear.

Our present choice, a semiempirical, on-the-fly dynamicsapproach with surface hopping [9], has the advantage of

good statistics samples and the use of a complete set of

degrees of freedom, provided that a reliable parametri-

zation of the semiempirical method has been achieved.

Moreover, investigations on smaller systems such as eth-

ylene allow to gain confidence for applications on larger

systems for which ab initio dynamics becomes impracti-

cable [11].The computational efficiency of the semiempirical

methods in comparison with ab initio methods makes

the former the obvious choice for the thousands of en-

ergy and gradient calculations necessary to compute just

one trajectory. However, the central point of the molec-

ular dynamics is the quality of the potential energy sur-

faces, and semiempirical methods using standard

parametrizations cannot be expected to work well fordistorted geometries (especially in excited states) far

from the structures used during the parametrization.

Even though a re-parametrization has been performed

for the ethylene photochemistry previously [9], some

M. Barbatti et al. / Chemical Physics Letters 401 (2005) 276–281 277

artifacts appeared, which made it desirable to investigate

this matter again.

The aim of this letter is to report on the new achieve-

ments, which were applied especially to comparative

studies on the lifetime of ethylene in the V state with

the purpose to find explanations for the already men-tioned discrepancies between experimental and theoreti-

cal results.

2. Computational details

Quasi-classical molecular dynamics simulations were

performed using an on-the-fly strategy for computingenergies, gradients and nonadiabatic couplings [9].

Transition probabilities between electronic states were

taken into account by means of the surface hopping

method developed by Tully [12]. For all simulations, a

time step of 0.1 fs was used, which gave a good level

of energy conservation, corresponding to a maximum

deviation of 0.08% in 500 fs in the average total energy

(<0.01 eV). Initial conditions were sampled so as to re-produce the normal mode quantum harmonic oscillator

(QHO) in its vibrational ground state.

The AM1 semiempirical method [13] was employed

to calculate the electronic energies. Molecular orbitals

were obtained from a closed shell calculation using the

floating occupation molecular orbitals (FOMO) method

[14], with the orbital energy width set as x = 0.2. Within

this orbital set a multi-electron configuration interaction(MECI) calculation [15] was performed with a complete

active space of two active electrons and two active orbi-

tals. Besides the ground state (N), two electronically ex-

cited states were computed, V and Z [16], for which the

main contribution in a vertical excitation comes from

the (p)1(p*)1 and (p*)2 configurations, respectively. In

order to reproduce correctly the main features of the

ethylene potential energy surface at highly distortedgeometries, the AM1 parameters set for carbon was

re-optimized by using the simplex method in order to

reproduce the multireference configuration interaction

with singles and doubles (MR-CISD) results obtained

recently [17]. This point will be discussed in more detail

in the following section.

From the initial geometries generated by the QHO

model vertical excitations were performed by acceptingonly those geometries that had the S1 state energy in the

interval Ev ± DEv. In order to simulate the excitation to

two different regions of the absorption band, we set

Ebv ¼ 6:2� 0:3 eV, which is the experimental excitation

energy used in [5,7] and Ecv ¼ 7:35� 0:15 eV, the com-

puted vertical excitation from the ground state minimum

at theAM1 level. For the former, 43 500 initial geometries

were constructed, from which just 680 had a V state lyingin the ±DEv range. For the other vertical energy, 2000 ini-

tial geometries were constructed and 746 were accepted.

The direct trajectories with surface hopping (DTSH)

approach as implemented in a development version of

the MOPACOPAC package [18] was used and is described in de-

tail in [9]. Ab initio calculations were performed in order

to obtain the ground state energy of the C2Hþ4 ion.

State-averaged CASSCF(2,2) calculations were per-formed for the three ethylene valence states N, V and

Z and the C2Hþ4 ground state. The same CAS(2,2) was

used as reference space in the subsequent multi-reference

configuration interaction calculations with singles and

doubles (MR-CISD). For each of the structures investi-

gated, the geometry of the neutral species was optimized

at the MR-CISD level and for this geometry the energy

of the ionic species was computed also. Details of the abinitio calculations can be found in [17].

3. The quality of the AM1 parameter fitting

As is well known (see, e.g., [1,4]), after vertical excita-

tion from the planar ground state to the first excited va-

lence state (V), the V state is stabilized by a torsionaround the CC axis. The ionic state Z is strongly stabi-

lized by this torsion as well, and the V and Z states

become almost degenerate around 90�. The V-excited

twisted-orthogonal structure can be further stabilized

by pyramidalization of one CH2 group or by H-migra-

tion, in the latter case resulting finally in the ethylidene

isomer. An extended S0/S1 crossing seam exists along

these two paths [17], with minima located in thetwisted-pyramidalized and the ethylidene intersection

seam. The most interesting structures are characterized

in Fig. 1.

As can be seen from Fig. 2, the AM1 default param-

eters [13] do not reproduce the conical intersection at the

twisted-pyramidalized structure (Fig. 1), which most

probably is the main funnel to the nonadiabatic transi-

tion to the ground state. A previous reparametrization[9] provided a good description of the twisted-pyrami-

dalized conical intersection. However, this previous set

of parameters also produced an undesired mixing of rand p molecular orbitals in the active space, which re-

sulted in a wrong energetic preference of a cis-double

pyramidalization of planar ethylene in the V state. The

main consequence was a fast double pyramidalization

occurring at the beginning of the dynamics. This processproduced a shortcut to the twisted-pyramidalized coni-

cal intersection and the lifetime for the S1 excited state

was predicted to be only 50 fs, in contrast to other

calculations [2,4,10].

In the present work, a new parameters set was devel-

oped in order: (i) to keep the good description for the

twisted-pyramidalized conical intersection of the first

re-parametrization, (ii) to add new and better ab initiotargets to the fitting procedure, and (iii) to eliminate

the spurious mixing in the active space.

Fig. 1. Selected geometrical parameters computed at the AM1 level for

the main structures studied in this work. Values in brackets are taken

from the MR-CISD results of [17]. For the twisted-pyramidalized

MXS, the pyramidalization angle is 95.2� [104.4�]. Distances are given

in A and angles in degrees. c.i. stands for conical intersection.

2

4

6

8

10

12 Semiempirical (new parameters)

2

4

6

8

10

12Semiempirical (AM1 default parameters)

Ene

rgy

(eV

)

0 20 40 60 80 100 120

2

4

6

8

10

12

Pyramidalization angle (Degrees)

Ab Initio

Fig. 2. Potential energy curves for the rigid symmetrical pyramidal-

ization of one CH2 group calculated at the AM1 level with the new

parameters set (top) and with the AM1 default parameters set

(middle). At the bottom, the ab initio result according to [17] are

given. Zero degree corresponds to the twisted-orthogonal structure

(see Fig. 1). The planar ground-state ethylene energy represents the

energy zero.

278 M. Barbatti et al. / Chemical Physics Letters 401 (2005) 276–281

The optimization process of the AM1 parameter setwas performed as described in [11]. Table 1 shows the

structures, energies and weights used for the optimiza-

tion. Table 2 collects the new parameter set. In addition

to the targets used in the previous optimization [9], we

included the cis-doubly pyramidalized structure (re-

stricted to a pyramidalization angle of 70�) and the eth-

ylidene conical intersection. The weight w for each state

of each structure was chosen according to the estimatedrelevance for the photodynamical process. For instance,

the ground state energies for the planar structure (opti-

mized for the V state) and for the double-pyramidalized

structure received the lowest weights, since their occur-

rence in our dynamics runs should be rather unlikely.

On the other hand, the V state energy for the planar

structure (optimized for the ground state) and for the

twisted-orthogonal structure received the largestweights, given their importance for the first stages of

the dynamics. No drastic changes in the new parameter

set in comparison with the previous sets [9,13] were

observed.

Fig. 1 shows the geometrical parameters for a set ofstationary structures and conical intersections. The gen-

eral agreement between the AM1 and the MR-CISD re-

sults is quite good. A detailed comparison between the

two methods shows that the CC distance is systemati-

cally smaller at the AM1 level than at the MR-CISD

level. On the contrary, the CH distances are larger at

the former level. Concerning the conical intersections,

we note that the H-migration angle is 10.7� smaller usingthe AM1 method. As already discussed previously [17],

the twisted-pyramidalized minimum of the crossing

seam (MXS) has some degree of H-migration character.

This can be seen from the asymmetry of the CH distance

of the pyramidalized CH2 group. The present AM1 po-

tential energy surface (PES) overemphasizes this H-

migration character (Fig. 1). The most critical difference,

however, is the energy difference between the twisted-orthogonal geometry and the twisted-pyramidalized

MXS in the S1 state. While this value is 0.94 eV at the

MR-CISD level, it is only 0.24 eV at the AM1 level.

The overemphasis of the H-migration character and

Table 1

Ab initio energies used as targets in the AM1 parameter optimization

and corresponding semiempirical results obtained with the optimized

parameters

State Energy (eV) w

Semiempirical Ab initioa

Planar ground state

S0 0.00 0.00 –

S1 7.36 7.80 2.0

Planar V state

S0 0.17 0.45 0.5

S1 7.18 7.70 1.0

Twisted orthogonal V state (rigid torsion)

S0 2.08 3.00 1.0

S2 5.00 5.46 2.0

Twisted-pyramidalized MXS

S0 4.76 4.52 2.0

DE01 0.004 0.001 1.0

Doubly-pyramidalized (70�)S0 1.86 1.57 0.5

DE01 0.36 0.50 1.0

Ethylidene MXS

S0 3.90 4.57 2.0

DE01 0.003 0.001 1.0

DE01 is the energy difference between S0 and S1 states, w is the weight

of each state in the optimization function.a Ab initio results calculated at the MR-CISD + Q/SA-3-RDP/aug-

cc-pVDZ level. See [17] for details.

Table 2

Optimized AM1 parameters for carbon

Parameter Value

Uss �52.785103

Upp �39.092898

bs �15.681806

bp �7.746831

ns 1.822883

np 1.691083

Gss 12.190950

Gsp 11.527107

Gpp 12.0138080

Gp2 9.996804

Hsp 2.385988

0.4

0.6

0.8

1.00 50 100 150 200

0.0

0.2

0.4

0.6

0.8

1.0

S1

S

Ev

b = 6.2 ± 0.3 eV

τb = 139 ± 1 fsS

1+S

2

S1+S

2

S2

S1

S0

Ave

rage

occ

upat

ion

Ev

c = 7.35 ± 0.15 eV

τc = 105 ± 1 fs

M. Barbatti et al. / Chemical Physics Letters 401 (2005) 276–281 279

the small pyramidalization gradient have opposite ef-

fects on the dynamics, since the former favors the path

to conical intersections, while the second feature slows

down the motion toward the conical intersection. There-

fore, we expect some partial compensation in the com-

puted lifetimes.

0 50 100 150 2000.0

0.2S

2 0

Time (fs)

Fig. 3. S0, S1, S2 and S1 + S2 occupations as a function of time. Top:

Ecv ¼ 7:35� 0:15 eV, bottom: Eb

v ¼ 6:2� 0:3 eV.

4. Results and discussion

Using the new parametrization, we performed simu-

lations with the two ranges of vertical excitations,

Ebv ¼ 6:2� 0:3 eV and Ec

v ¼ 7:35� 0:15 eV, correspond-

ing to the beginning (b) and to the center (c) of the

absorption band, respectively. In both cases, after exci-

tation to the excited state, ethylene starts a torsional mo-

tion after just 8 fs. Transitions between the V and Z

states are common, mainly for geometries close to the

twisted-orthogonal structure, for which these states arealmost degenerate. These transitions start already after

10 fs (Fig. 3), and the number of trajectories in the S2(Z) state increases quickly, reaching up to 20% around

20 fs, after which the occupation starts to decrease. Dur-

ing the initial phase of the dynamics one can observe a

pattern of oscillations in the occupations of the S1 and

S2 states, corresponding to a transfer between these

two states, with an average period of 18 ± 2 fs(1852 ± 200 cm�1). The same occupation of the S2 state,

with oscillatory patterns between S1 and S2, was de-

scribed previously by Viel et al. [4]. These authors also

emphasized the importance of the asymmetrical scissor-

ing motion for the coupling between V and Z during the

first 50 fs of the dynamics [19].

After 18 fs in case of Ecv and 24 fs for Eb

v transitions to

the ground state appear (Fig. 3). After 50 fs, the occupa-tion of S0, n(S0) is 0.24 in the first case and just 0.09 for

the latter, but both evolve faster than in the AIMS [10]

and in the MCDTH [4] dynamics.

The lifetimes calculated for the two initial conditions

are shown in Table 3. The decay in n(S1) was fitted

with the function n = exp(�t/s), where s is the lifetime.

For Ecv the S1 lifetime sc1 ¼ 105 fs, for Eb

v sb1 ¼ 139 fs.

Table 4

280 M. Barbatti et al. / Chemical Physics Letters 401 (2005) 276–281

The average occupation n(S1) and the respective lifetime

includes all transitions from S1 to S0 as well as to S2. In

order to compare the simulated lifetimes with the exper-

imental results of [5] and [7], which used just one expo-

nential function to fit the ion signal, we should look at

the total excited state decay n(S1) + n(S2) (see Fig. 3).By using a single exponential function to fit

n(S1) + n(S2), we get sc1þ2 ¼ 137 fs and sb1þ2 ¼ 188 fs.

Mestagh et al. [7] pointed out that the ultrafast decay

of the ethylene molecule observed experimentally could

be partly a consequence of the experimental excitation

energy at 200 nm (6.2 eV). Since the center of the

absorption band is more than 1 eV higher, only very dis-

torted geometries presumably closer to the conical inter-section point would be excited. In the present

simulations, however, we found (Table 3) for the excita-

tion into the beginning of the progression band ðEbvÞ the

lifetime is actually larger than for the Ecv case. To explain

this point, we have to bear in mind that not only geo-

metrical criteria such as the distance to the conical inter-

section must be taken into account, but also energetic

criteria. In case of excitation into the center of the band,the system has more vibrational-excess energy. Thus, it

can move faster to the conical intersection than with

the excitation into the beginning of the band. The actual

ratio between these two factors will depend on the de-

tails of the excited-state energy surface and on the distri-

bution of the excess energy into individual vibrational

modes. Analysis of the dynamics data showed that con-

sidering the torsion mode only the ethylene moleculearrives slightly earlier at the twisted-orthogonal struc-

ture when excited into the beginning of the progression

band. This fact is in agreement with the geometrical

argument. However, in order to reach the conical inter-

section more complicated motions including pyramidal-

ization and hydrogen migration have to be performed

for which the higher vibrational energy of the excitation

into the band center seems to be the decisive factor.The lifetimes obtained in the present simulation are

smaller than the AIMS result but still of the same order

Table 3

Lifetimes of the excited states

Method Lifetime (fs)

DTSHc S1 ! S2, S0 139 ± 1a/105 ± 1b

DTSHc S1, S2 ! S0 188 ± 2a/137 ± 2b

DTSH [9] S1 ! S2, S0 50

AIMS [2] >250

AIMS [10] 180

MCDTH [4] >>100

Experimental [5] 30 ± 15

Experimental [6] 40 ± 20

Experimental [7] 20 ± 10

Experimental (s1 + s2) [8] 25 ± 5

a Ebv ¼ 6:2� 0:3 eV.

b Ecv ¼ 7:35� 0:15 eV.

c Present work.

(see Table 3). Concerning the MCDTH calculations [4],

the difference can be attributed to the restriction to the

6-dimensional PES employed in that simulation. The

freezing of the rocking and the CH-stretching modes

has as consequence a poor description of the twisted-

pyramidalized MXS and a complete lack of H-migrationcharacter. Both factors contribute to slowing down the

process of reaching the pyramidalized MXS.

All lifetimes obtained by dynamics simulations,

including the present ones, are very long compared to

the experimental ones (Table 3). A possible explanation

has already been suggested in [4,7,10]. The probe lasers

used for ionization of the excited state is operating at an

energy of 4.64 eV (267 nm) [5,7]. This energy is sufficientto ionize the V state ethylene only in the neighborhood

of the vertical excitation assuming the occurrence of

single-photon processes only.

Table 4 shows the vertical excitation energy of the V

state and the corresponding vertical energy of the ethyl-

ene cation for a few important structures. Both AM1

and MR-CISD results are given for comparison. For in-

stance, the ionization energy of the planar V state is 2.63eV, a value sufficiently small to be covered by the exper-

imentally available 4.6 eV for the ionization process.

However, for the twisted-pyramidalized MXS, for

example, we see in Table 4 that the required energy

for ionization is 7.2 eV, significantly above the available

4.6 eV. Similar observations can be made for the other

two conical intersections given in Table 4.

Fig. 4 shows the necessary energy to ionize ethylenefor each time step of the dynamics simulation. Averag-

ing over all trajectories was performed and the energies

of the ion and of the neutral molecule were computed at

the same geometry. The graph shows that a probe en-

ergy of 4.6 eV is not able to ionize the system after about

30 fs (indicated by snd, for non-detectable). This is a

much shorter time than the theoretically predicted life-

times and close to the experimental results. A similar

AM1 energies for the S1 state of ethylene and for the ground state of

the ethylene ion

Structure Energy (eV)a DE (eV)

S1(C2H4) S0 ðC2Hþ4 Þ

b

Planar 7.36 [7.55]c 9.99 [10.02]d 2.63 [2.47]

Twisted-orthogonal 4.90 [5.24]e 10.66 [11.35] 5.76 [6.11]

Twisted-pyramidalized MXS 4.76 [4.27] 11.97 [11.70] 7.21 [7.43]

Ethylidene MXS 3.90 [4.26] 10.81 [11.61] 6.91 [7.35]

DE is the energy difference between these two states.

Values in brackets are ab initio results.

Zero energy corresponds to the energy of the ground state equilibrium

geometry of ethylene.a Ab initio results calculated at the MR-CISD + Q/SA-4-CAS(2,2)/

aug-cc-pVDZ level. See [17] for details.b The S0 ðC2H

þ4 Þ energy is the IP without ZPE correction.

c Eexp = 7.66 eV.d Eexp = 10.5 eV.e Optimized for the V state.

0 20 40 60 80 100 120 140 160 180 2002

4

6

8

τb

τc

172 nm (7.21 eV)

τnd

267 nm (4.6 eV)Enoi

E-top

)Ve(

Time (fs)

Ev = 6.20 eV

Ev = 7.35 eV

Fig. 4. Average energy difference between the ground state of the

C2Hþ4 ion and the potential energy of ethylene at each time step

calculated at the AM1 level with vertical excitation energies of 6.2 and

7.35 eV.

M. Barbatti et al. / Chemical Physics Letters 401 (2005) 276–281 281

reduced lifetime was also obtained by Ben-Nun et al.

[10] assuming a 3.5 eV ionization threshold in AIMS

dynamics calculation. This simple analysis, which cer-tainly does not model completely the experimental situ-

ation, demonstrates nicely the problems with the

interpretation of the experimentally observed lifetimes.

We believe that, unless probe energies around 7.4 eV

(168 nm) are used, the measured lifetimes will corre-

spond just to the time the molecule needs to leave the

detection window while still being in the S1 state.

5. Conclusions

The present simulations of the photodynamics of eth-

ylene give for the lifetime of the S1 excited state values

between 105 and 139 fs, depending on the initial excita-

tion energy. The lifetime of the S1 and S2 states together,

which should be the quantity of interest in comparisonwith experimental data ranges from 137 to 188 fs. Our

simulations also show that probe energies of at least

7.4 eV should be employed in order to ionize the system

for geometries close to the conical intersections.

Our analysis follows the usual hypothesis that the va-

lence states dominate the ethylene photodynamics and

that the V-state decay occurs mainly through a conical

intersection with the ground state following torsionaland pyramidalization processes. However, based on

their new pump–probe experimental results using

multi-photon ionization techniques, Stert et al. [8] re-

cently proposed a quite distinct model, in which the

V-state decay would occur to a second excited state after

just 10 fs. From there, a non-detectable state would be

reached in additional 15 fs.

These ultrafast lifetimes are not compatible with therather long lifetimes connected to the just-mentioned tor-

sional mode followed by pyramidalization and/or hydro-

gen migration processes. Presently, there are two

independent and alternative models to explain such fast

experimental lifetimes. One possibility is the limitation

in the detection window as explained above. The other

is the influence of a second excited state as proposed byStert et al. [8]. At this moment, in our opinion there is

not enough theoretical and experimental evidence avail-

able in order to decide in favor of one of them, and both

options deserve to be investigated in more detail. The

former possibility can be tested in the laboratory by

using larger probe energies than has been used so far

and the latter one rises the need to include in future the-

oretical investigations other excited states than the threevalence states treated in the dynamics calculations.

Acknowledgements

The authors thank Prof. W. Fuß and Prof. M. Oliv-

ucci for valuable discussions. The authors acknowledge

support by the Austrian Science Fund within the frame-work of the Special Research Program F16 and Project

P14817-N03 and by COST, Project No. D26-0006-02.

Mario Barbatti thanks for the financial support from

the Brazilian funding agency CNPq.

References

[1] I. Ohmine, J. Chem. Phys. 83 (1985) 2348.

[2] M. Ben-Nun, T.J. Martınez, Chem. Phys. Lett. 298 (1998) 57.

[3] L. Freund, M. Klessinger, Int. J. Quantum Chem. 70 (1998) 1023.

[4] A. Viel, R.P. Krawczyk, U. Manthe, W. Domcke, J. Chem. Phys.

120 (2004) 11000.

[5] P. Farmanara, V. Stert, W. Radloff, Chem. Phys. Lett. 288 (1998)

518.

[6] P. Farmanara, O. Steinkellner, M.T. Wick, M. Wittmann, G.

Korn, V. Stert, W. Radloff, J. Chem. Phys. 111 (1999) 6264.

[7] J.M. Mestagh, J.P. Visticot, M. Elhanine, B. Soep, J. Chem.

Phys. 113 (2000) 237.

[8] V. Stert, H. Lippert, H.-H. Ritze, W. Radloff, Chem. Phys. Lett.

388 (2004) 144.

[9] G. Granucci, M. Persico, A. Toniolo, J. Chem. Phys. 114 (2001)

10608.

[10] M. Ben-Nun, J. Quenneville, T.J. Martınez, J. Phys. Chem. A 104

(2000) 5161.

[11] C. Ciminelli, G. Granucci, M. Persico, Chem. Eur. J. 10 (2004)

2327.

[12] J.C. Tully, J. Chem. Phys. 93 (1990) 1061.

[13] M.J.S. Dewar, E.G. Zoebisch, E.F. Healy, J.J.P. Stewart, J. Am.

Chem. Soc. 107 (1985) 3902.

[14] G. Granucci, A. Toniolo, Chem. Phys. Lett. 325 (2000) 79.

[15] D.R. Armstrong, R. Fortune, P.G. Perkins, J. Chem. Soc.,

Faraday II 68 (1972) 1839.

[16] A.J. Merer, R.S. Mulliken, Chem. Rev. 69 (1969) 639.

[17] M. Barbatti, J. Paier, H. Lischka, J. Chem. Phys. 121 (2004)

11614.

[18] J.J.P. Stewart, MOPACOPAC 2000, Fujitsu Limited, Tokio, Japan, 1999.

[19] A. Viel, R.P. Krawczyk, U. Manthe, W. Domcke, Angew. Chem.

Int. Ed. 42 (2003) 3434.