ab initio investigation of potential-energy surfaces...

Ab inifio investigation of potential-energy surfaces involved in the photophysics of benzene and pyrazine

A. L. Sobolewski ’ Institute of Physics, Polish Academy of Sciences, PL-02-668 Warsaw, Poland

C. Woywod and W. Domcke Institute of Physical and Theoretical Chemistry, Technical University of Munich, D-8046 Garching, Germany

(Received 17 November 1992; accepted 17 December 1992)

Potential-energy surfaces of the lowest singlet and triplet excited states of benzene and pyrazine have been calculated using complete-active-space self-consistent-field and multireference configuration interaction (MRCI) techniques. We have focused our attention on the saddle points and surface intersections associated with the reaction path to a biradical form called prefulvene. The barrier heights separating the prefulvenic minimum from the minimum of the planar aromatic form on the Z-T* excited singlet surface and on the ground-state surface have been estimated by large-scale MRCI calculations. The conical intersection of the lowest rfl excited singlet surface with the So surface has been mapped out in two dimensions, the reaction coordinate to prefulvene and the coordinate of maximum coupling perpendicular to it. The re1evanc.e of these ab initio potential-energy data for the interpretation of photophysical relaxation pathways in benzene and pyrazine (“channel- three” effect) is discussed.

I. INTRODUCTION

Benzene and its aza-derivatives (azines) form the basic structures of some of the most important organic compounds in nature. Benzene can be considered as a model compound for a wide class of aromatic hydrocarbons. Azines, on the other hand, form the skeleton of biologically important molecules such as nucleic acid bases or amino acids. As such, they deserve particularly careful examina- tion. One of the most important characteristics of these compounds is 7~ delocalization, which leads to remarkably reduced reactivity in the ground state.

Electronic excitation to the lowest valence states corresponds to the promotion of one of the electrons into the antibonding r* orbital. The exploration of the photophys- its and photochemistry associated with the valence excited states of benzene and the simple azines has been a subject of continued interest over decades.‘?’ While the spectros- copy and the photophysical dynamics of low vibronic levels of the first singlet state (Si) are well characterized, in particular for benzene and pyrazine (see, for example, Refs. 3-12), very little is still known about the very rapid nonradiative processes occurring after excitation of aromatic molecules with some excess energy above the S1 origin. In the S1(r7r*) state of benzene, the opening of a rapid nonradiative relaxation process about 3000 cm-’ above the origin is known as the “channel-three” phenomenon.13-18 In the azines, a similar effect is observed at somewhat higher excess energies above the S1 (nrr*) origin. l9 The quantum yield of fluorescence after excitation of benzene and the azines to higher singlet states (S,,S,,...) is extremely low, indicating the existence of a very efficient internal-conversion (IC) pathway to the electronic ground state. ‘1’

In addition to the intriguing photophysical dynamics,

which also includes the intersystem-crossing (ISC) process to the triplet manifold, benzene and the azines undergo remarkable photochemical transformations upon UV irra- diation.2@26 Benzvalene, fulvene, and Dewar benzene have been detected as photolysis products of benzene, although with very low quantum yields. The isomer production in- dicates that the rigidity of the aromatic (or aza-aromatic) ring is significantly reduced in excited electronic states.

It has since long been speculated that the complex photophysical relaxation processes in aromatic molecules and the photoisomerization dynamics are intimately connected phenomena. W9,2%27,28 As first pointed out by Bryce-Smith and Longuet-Higgins,2g a biradicalic structure called “prefulvene” is expected to play a central role as an intermediate in the photoisomerization of benzene. Oikawa et al. 3o have quantitatively characterized prefulvene as a local minimum on the lowest triplet potential-energy (PE) surface of benzene with the semiempirical MIND0/3 method as well as with ab initio unrestricted Hartree-Fock (UHF) calculations. It has also been shown by ab initio calculations that prefulvene exists as a local minimum on the lowest Z-T* excited singlet PE surface3’ and that completely analogous biradicalic forms exist for pyridine and pyrazine.32 It has been shown, moreover, that the PE surface of the lowest ?rr* excited singlet configuration intersects the So PE surface along the reaction coordinate connecting the planar forms of benzene, pyridine, and pyrazine with the prefulvenic minimum.31’32 In pyridine and pyrazine the low-lying nrr* excited singlet states have been found to be stable against out-of-plane deforma- tions;32p33 the S, (nr*> PE surface is intersected, however, at low energies by the S2(r7r*) PE surface.34 This multidimensional adiabatic connection of the Si surface with the

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5628 Sobolewski, Woywod, and Domcke: Photophysics of benzene and pyrazine

So surface via the S, surface is expected to open a fast IC channel in the Sl(mr*) states of the azines.32

While the close connection of photophysical relaxation dynamics and photochemistry is convincingly established by the above-mentioned quantum-chemical calculations, the available theoretical data are not yet sufficiently accurate to draw definitive conclusions. The PE surface computations have been performed with the UHF ansatz or rather limited multiconfiguration self-consistent-field (MCSCF) schemes employing basis sets of moderate quality. In the present work we have attempted to provide a more sophisticated quantum-chemical description of pho- tophysically and photochemically relevant PE surfaces of aromatic molecules, taking benzene and pyrazine as repre- sentative examples. Our main concern is the reaction path to the prefulvenic form and the associated intersection of the TIT* excited singlet surface with the So surface. Basis sets of double-6 (DZ) and double-c-polarization (DZP) quality are employed.35 The PE surfaces have been obtained with state-averaged complete-active-space (CAS) SCF calculations, with careful selection of the active space. To obtain more accurate results on the relative energy of relevant stationary points (local minima and saddle points), large-scale multireference configuration interaction (MRCI) calculations based on the CAS reference have also been performed.

the ultrafast IC dynamics in these molecules. As a by- product of this investigation, we have also located the saddle point which separates the prefulvenic minimum from the planar minimum on the So surface in Cl symmetry. This saddle point represents, most probably, the lowest barrier for isomerization of benzene or pyrazine on the electronic ground-state surface.

II. METHODS AND RESULTS OF CALCULATIONS

A commonly encountered bottleneck in such investi- gations of multidimensional global PE surfaces with ab initio methods is the determination of “unconventional” stationary points, that is, hitherto unknown local minima and saddle points. The unambiguous identification of minima and saddle points requires the determination of the complete matrix of second derivatives of the energy (the Hessian matrix). Since the Hessian of the CASSCF surface has to be obtained by numerical differentiation of the analytic energy gradients, the determination and characterization of stationary points at the CASSCF level with extended basis sets can become exceedingly time consuming for larger polyatomic systems with low symmetry (C, or C,). We found it necessary to accept a compromise between accuracy and computational cost and to perform the geometry optimizations with the split-valence 3-21G basis set.36 We have also determined the stationary points with the much cheaper STO-3G minimal basis37 and have used this basis set in the evaluation of the final force matrix and the vibrational frequencies in order to estimate the zero- point energy. The minimal basis geometries have been also used as the starting guesses for geometry optimization with the 3-21G basis set.

The complete-active-space self-consistent-field (CASSCF) method is nowadays a well-established approach for electronic-structure calculations, having the ad- vantage of including the most important electron correlation effects together with a full optimization of all orbitals.38 The method is especially well suited for situa- tions where the electronic structure varies strongly, as, for example, around a transition-state geometry along the reaction coordinate. In this method a configuration interaction (CI) calculation is performed involving all configurations, of given spin and spatial symmetry, which can be formed by distributing a number of active electrons among a set -of active orbitals (a full CI in limited space). In addition, a number of inactive orbitals are kept fully occupied in all configurations. The orbitals and CI coefficients are simultaneously optimized until the energy is stationary. The calculation is fully specified, apart from the choice of state spin and symmetry label, by the number of active orbitals of each symmetry type.

The flexibility of the method in choosing the active orbital space permits, in general, the construction of molecular wave functions which describe all states involved in a balanced way, including near degeneracies occurring along the reaction path of interest. On the other hand, the CASSCF approximation describes only a fraction of the electron correlation energy, i.e., it accounts for near- degeneracy effects that occur in the system, but does not include the so-called dynamic electron correlation.38 In the present work, the effect of dynamic electron correlation is taken into account by means of the internally contracted MRCI (CMRCI) method.39 In the CMRCI approximation all single and double excitations from the CASSCF reference into the external orbitals (apart from the frozen core) are included. Configurations with two electrons in the external orbital space are internally contracted by ap- plying pair excitation operators to the reference function as a whole.3g940 It has been shown that for most cases, while the internal contraction greatly reduces the number of vari- ational parameters, essentially no loss in accuracy occurs in comparison to uncontracted MRCI calculations.3g’41

For the understanding of the dynamics of photophysical and photochemical processes it is not sufficient, in general, to know just the stationary points of the PE surface(s) or one-dimensional energy profiles. As a first step towards a more complete characterization of the photo- physically relevant singlet surfaces in benzene and pyrazine, we have explicitly mapped out the conical intersection of the lowest rr* excited singlet state with the So state in two dimensions. It is clear that this conical intersection is of central importance for the microscopic description of

The basis sets used are 3-21G (Ref. 36) for geometry optimization and DZ, DZP (Ref. 35) otherwise. The STO-3G minimal basis also has been employed to obtain starting guesses for stationary points and to compute the Hessian. The numbers of contracted basis functions for benzene (pyrazine) are 36(34) for STO-3G, 66(62) for 3-21G, 66(62) for DZ, and 120( 110) for DZP.

All calculations presented in this paper have been performed using the GAMESS~~ and the MOLPRO~~ program packages.

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Sobolewski, Woywod, and Domcke: Photophysics of benzene and pyrazine 5629

TABLE I. CASSCF optimized geometries of benzene in D6h symmetry.

State SoM,) S,(Bz,,)

Basis set STO-3G 3-21G DZ Expt.= STO-3G 3-21G Expt.b

cc (A) CH Lb Energy (a.u.) Zero-point energy (cm-‘)

1.405 1.395 1.406 1.395 1.449 1.435 1.432 1.082 1.072 i.oi9 1.082 1.081 1.070 1.084

-227.997 48 ’ -229.494 19 -230.718 28 . . . -227.817 64 -229.315 45 ... 24 920.0 . . . . . . 21 77o.oc 23 370.0 . . . . . .

“Reference 60. bReference 61. ‘Reference 62.

A. Geometry optimization of stable forms

Geometry optimization of the stable planar and prefulvenic forms is performed by the use of the CASSCF analytic energy-gradient method implemented within the GAMESS package. A minimal active space which can be constructed for each case should include at least the molecular orbitals of all excited configurations of interest. Usually this includes the set of highest occupied SCF orbitals together with the lowest virtuals. In benzene, for example, a natural choice is to distribute six electrons over the six valence r orbitals of a2", elg, e,,, and b, symmetry (in the D6/1 point group) in order to describe the orbital properties of the lowest rr# excited valence states. In pyra- zinc, in addition to the TV* excitations analogous to those of benzene, there are n# transitions involving the non- bonding (or lone pair) orbitals on the nitrogens. One of these is n+ (a,), the symmetric combination of the lone pair orbitals nl and n2 localized on the two nitrogen atoms, and the other the asymmetric combination n- (b,,), in the D2h point group. Thus the minimal active space of pyrazine consists of 10 electrons distributed over eight molecular orbitals.

All calculations for the planar forms were performed within the D2,, point group with the x axis perpendicular to the molecular plane. Thus the minimal CASSCF calculations for benzene and pyrazine are characterized by ~~,~,~,~,~,~,~,~~~,~,~,~,~,~,~, 1) and (5,0,4,0,4,0,3/ 1,2,0,1,1,2,0,1), respectively, where the first set of eight numbers is the number of inactive orbitals with symmetry labels ag, &, blur b,, b,,, b2, b,, and a,,, respectively, while the last eight denote the number of the active orbitals

TABLE II. CASSCF optimized geometries of pyrazine in Dzh symmetry.

in a similar way. The resulting calculation comprises 52 and 176 configuration state functions (CSFs) of A, symmetry for benzene and pyrazine, respectively. Similar calculations performed for the B2J rr*) states, which correlate adiabatically to the prefulvenic forms, involve 60 and 200 CSFs of B2u symmetry, respectively. The calculated bond lengths and bond angles together with total and zero- point energies for the ground ( ?qg) and the n-r* excited ( ’ B2u) states of both forms are given in Tables I and II. In the calculation of the Hessian matrix the symmetry point group is reduced to C,; this increases the numbers of CSFs~ to 175 and 1176 for benzene and pyrazine, respectively.

The above discussion concerns the systems at the reference (highest symmetry) geometry of nuclei. The prefulvenic forms have C, symmetry, with broken aromatic plane; therefore, the classification of molecular orbitals in terms of V, IZ, or (T is no longer possible. In order to define the minimal active space for the CASSCF geometry optimization of these forms, we have performed larger CASSCF calculations correlating 10 valence electrons in 10 orbitals, taking as initial guess the geometries of the prefulvenic forms optimized within the UHF scheme.32 It appeared that a reasonable description of the wave function and energy can be obtained in each case by correlating six electrons in six orbitals. This result has been further confirmed by extended MRCI calculations showing that in all cases of interest the coefficient of the CASSCF reference configuration is larger than 0.930. Thus the minimal CASSCF calculations for both molecules at the C, symme; try are characterized by ( 11,7/4,2), where the first and the second set of two numbers denote the number of inactive

State

Basis set

cc (K) CN Lb CH (A, CNC (deg) CCH (deg) Energy (a.u.) Zero-point energy (cm- ‘)

STO-3G

1.406 1.378 1.085

113.1 120.1

-259.506 76 18 460.0

3-21G DZ Expt.’ STO-3G 3-21G

1.389 1.405 1.39-1.40 1.451 1.428 1.346 1.351 1.32-1.34 1.4.25 1.392 1.069 1.068 1.08 1.085 1.067

117.3 117.2 115-118 110.6 113.9 121.3 121.5 “117-120 119.3 120.3

-261.287 16 -262.678 95 . . . -259.337 83 -261.107 36 . . . . . . 1.. 17 680.0 . . .

“Compiled from Ref. 63.

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Sobolewski, Woywod, and Domcke: Photophysics of benzene and pyrazine

FIG. 1. Geometries of the prefulvenic forms of benzene and pyrazine in their lowest singlet state.

and active orbitals (with symmetry a’ and a”), respectively.

As mentioned in the Introduction, the geometry optimizations have been performed with the 3-21G basis set, using the corresponding geometries optimized with the STO-3G minimal basis set as starting guesses. The planar St, reference geometries of benzene and pyrazine of D6,, and D2h symmetry, respectively, have also been optimized with the DZ basis set. The complete force field and the vibrational frequencies have been evaluated with the STO-3G basis only, since such calculations would be extremely time consuming with larger basis sets at the CASSCF level. The zero-point energies of &,(A,,) and the lowest s-7I.Y’ excited singlet state obtained with the STO-3G basis are included in Tables I and II. It should be stressed that our aim is not the determination of accurate vibrational frequencies (for recent work on the electronic ground state of benzene, see Ref. 44). In the present context the force field is required for the unambiguous identification of the character of sta-

tionary points as well as for the qualitative prediction of differences in the zero-point energies.

The optimized geometry of the prefulvenic forms of benzene and pyrazine is shown in Fig. 1. The CASSCF/ STO-3G normal mode analysis performed at the optimized geometries gives all vibrational frequencies real, proving that both prefulvenic structures correspond to stable points on the potential-energy surface at this level of approximation. The most important geometrical data, i.e., the length of the newly formed C2-C6 bond (R26), as well as the angles formed by the atoms 1 (a) and 2 (fi) with respect to the plane spanned by the C2-C3+,-C!6 atoms are given in Table III. We decided to give the Cartesian coordinates of the symmetry nonequivalent atoms instead of describing the geometries of the low-symmetry systems in terms of internal coordinates. The optimized geometries are listed in the Appendix and may serve as useful starting values for future more sophisticated treatments.

B. Reaction path calculation

In order to calculate PE functions along the reaction coordinate connecting the planar form with the prefulvenic form we define, as in Ref. 32, a vector of internal displacements which connects the reactant (planar form) with the product (prefulvenic form) in the space spanned by the totally symmetric coordinates (A’ in the C, point group) of a given molecule. Next, we generate several nuclear configurations along the reaction path, transform them to the Cartesian coordinates of the nuclei, and calculate the energy of the ground and the lowest excited valence states. The DZ basis set has been employed. The converged CASSCF orbitals were used as the initial guess at the neighboring nuclear geometry. In order to include as much as possible of the dynamic correlation energy at the CASSCF level of approximation, we have increased the active space in these calculations, correlating 10 electrons in 10 -orbitals. The extended active space has been constructed with the help of MRCI calculations performed with the above-discussed minimal CASSCF reference at crucial points along the reaction path. Within the C, point group the extended CASSCF calculations are labeled as ( 10,6/6,4). This corresponds to a total number of CSFs of

TABLE III. Some of the structural parameters of benzene and pyrazine at the CASSCF/3-21G optimized stationary points along the reaction path to the prefulvenic form (PF). SP, and SP, are the saddle points on the a-@ excited singlet surface and the So surface, respectively (see text). SP, has been optimized with the STO-3G basis set. All zero-point energies and imaginary frequencies have been evaluated with the STO-3G basis.

Molecule Benzene Pyrazine

Stationary point PF(‘A”) -1 spa PF( ‘A”) SPI SP”

r(GCd CA) a (deg) P (de& Energy (a.u.) Zero-point energy (cm-‘) Imaginary frequencies (cm-‘)

1.600 2.110 1.807 1.492 2.039 1.794 68.4 52.4 . . . 75.2 49.2 . . .

0.3 7.2 . . . 0.8 6.8 . . . -229.299 58 -229.283 50 -227.801 27 -261.098 44 -261.078 91 -259.340 54

24 050.0 23 690.0 23 780.0 . 17 980.0 17 350.0 17 540.0

. . . 580.0 960.0 . . . 590.0 1180.0



Sobolewski, Woywod, and Domcke: Photophysics of benzene and pyrazine

REACTION COORDINATE

FIG. 2. Potential-energy curves of the lowest singlet (solid lines) and triplet (dashed lines) states of benzene along the reaction path to the prefulvenic form, obtained at the CASSCF/DZ level.

9772 (9884) of symmetry A’ (A”). Because of conver- gence problems in the CASSCF calculations with some of the individual states, the orbitals have been optimized for the average energy of all states of interest of given multiplicity rather than for each individual state. This is another reason for using an extended active space for these calculations. The PE curves of the lowest singlet and triplet states resulting from the calculations described above are presented in Figs. 2 and 3 for benzene and pyrazine, respectively.

C. Optimization of saddle points

Inspection of the results presented in Figs. 2 and 3 shows that in both benzene and pyrazine the vertically excited ‘BZu states are separated by low barriers from the

0.0 0.2 0.4 0.6 0.8 1.0 REACTION COORDINATE

FIG. 3. Potential-energy curves of the lowest singlet (solid lines) and triplet (dashed lines) states of pyraxine along the reaction path to the prefulvenic form, obtained at the CASSCF/DZ level.

..A.

.

5631

. . : ,.

.

FIG. 4. Schematic representation of relevant minima, saddle points and barriers on the Se and lowest rr?r* excited singlet surface of benzene and pyrazine.

prefulvenic forms on the PE surfaces. Thus one can expect the existence of a saddle point over which the nuclear mo- tion must pass in order to convert to the prefulvenic form. The energetical difference between the system optimized in the ‘B,, electronic state in D6h/Dlh symmetry and the saddle point (SPi) on the ‘A” PE surface in C, symmetry represents the barrier for photoisomerization. One also can expect a saddle point on the ground-state PE surface (SP,), representing the transient point for rearomatization of the prefulvenic form. It results from the fact that the intersection of the ‘A’ and ‘A” PE curves shown in Figs. 2 and 3 exists only when the C, symmetry of nuclear frame is conserved. Any vibrational displacement of A” symmetry mixes the two electronic states and as an effect the ‘A’ and ‘A” PE curves will avoid crossing. In other words, the PE surfaces of both electronic configurations form a conical intersection in the space spanned by the internal displacements of the nuclei. The location of these saddle points and the relevant energy differences are shown schematically in Fig. 4.

Both saddle points (SPe and SP,) for each system were optimized by the methods described in Sec. II A. For the determination of the saddle point on the ‘A” PE surface (C, point group) we have used the same CASSCF as defined in Sec. II A for optimization of the prefulvenic forms, i.e., the CASSCF labeled as ( 11,7/4,2). At the saddle point on the ground-state PE surface, the nuclear frame has C, symmetry; there is thus only one (totally symmetric) representation. Therefore the CASSCF which correlates six electrons in six orbitals is labeled ( 18/6), i.e., it includes three occupied orbitals and three unoccupied orbitals of the SCF wave function. The saddle points were first optimized with the STO-3G basis set. The normal mode analysis performed-at the optimized geometries shows that in each case there exists only one mode of imaginary fre- quency, thus confirming that the stationary points are re- ally the saddle points on the PE surfaces at this level of


theory. The SP, saddle point has also been optimized for both systems with the 3-21G basis set. We were unable, however, to locate the SP, saddle point with this basis set and using the CASSCF/STO-3G Hessian. The calculation of the Hessian at Ci symmetry within the CASSCF/3-21G approach has appeared to be a prohibitively long task. In the following discussion concerning the SPe saddle point we use the CASSCFBTO-3G optimized geometry. It may be noted that the geometries optimized in C, symmetry with the two basis sets are not significantly different from each other for both benzene and pyrazine. The most relevant parameters characterizing the optimized saddle-point structures are given in Table III together with their total and zero-point energies as well as with the imaginary frequencies determined at the CASSCF/STO-3G geometry. The Cartesian coordinates are given in the Appendix.

D. Characterization of the conical intersection between the mu* excited singlet configuration and SO

Having determined the saddle point on the ground- state PE surface, we are able to characterize the singlet PE surfaces near their conical intersection. To do this, we have defined two vectors of internal displacements of A’ symmetry (reaction’ coordinate) and of A” symmetry (coupling coordinate). The first describes the reaction coordinate from the planar ( QA,=O) to the prefulvenic form (&I= I), and the second displaces the system from the reaction path (QA,,=O) into the direction of the saddle point on the ground-state PE surface ( QA” = 1). Along QA” maximum repulsion of the Se configuration and the gr* excited configuration takes place. The PE surfaces presented in Figs. 5 and 6 have been constructed on a grid of points calculated for displacements of nuclear geometries along the two vectors of internal displacements. The calculations were performed at the CASSCF level using the DZ basis set. The CASSCF waves function correlates 10 electrons in 10 orbitals, i.e., the CASSCF is labeled as (16/10). This corresponds to a total number of 19 404 CSFs in the C, point group.

FIG. 5. Conical intersection of the St (7Mir) and the Se potential-energy surfaces of benzene, shown in the two-dimensional space spanned by the reaction coordinate to prefulvene and the coordinate of maximum coupling. The surfaces have been obtained at the CASSCF level with the DZ basis set. The upper and lower panels show different perspective views of the intersection.

E. MRCI calculations

The results discussed so far indicate that (i) there exists a barrier which separates the planar minimum on the ‘B,, surface from the prefulvenic minimum in C, symmetry (SP,), and (ii) that the prefulvenic minimum is separated by a barrier from the So minimum in Ci symmetry (SP,). In order to determine quantitatively the height of these barriers and the relative energies of the local minima on the PE surfaces, we have performed additional bench- mark calculations with the DZ and DZP basis sets, taking account of dynamic correlation effects by the CMRCI method.

The CMRCI calculations have been performed with respect to the CASSCF reference function, with the same active space as defined in the preceding sections for the geometry optimizations. This means that, apart from pyrazine at planar geometry where the CASSCF reference correlates 10 electrons in 8 7c and n orbitals, all other CASSCF

references correlate 6 electrons in 6 orbitals. The orbitals in the CASSCF reference were optimized independently for each state. It has been found to be impossible to treat systems of this size at the MRCI level with a frozen core limited to the 1s orbitals of the heavy atoms only. In the present calculations the frozen core is defined as (4,0,2,0,3,0,2,0), (7,4), and (11) for the Dzh, C, and C, point groups, respectively. This MRCI produces a total number of configurations which varies between 1.2X lo6 (contracted to 6.0X 104) for benzene in Dzh symmetry in the DZ basis to 1.2~ 10’ (2.2~ 106) in C1 symmetry in the DZP basis. The corresponding numbers of configurations for pyrazine vary between 3.2X106 (1.7~10~) and 9.8 x lo7 (1.9X 106). On a CONVEX C 3220, the CPU time for the evaluation of one energy varies from 1.5 h for benzene at the planar geometry with the DZ basis to 7-9 h for benzene or pyrazine in C’, symmetry with the DZP basis. The evaluation of the energy in no-symmetry cases (SP, saddle points) took up to 100 min of CPU on a Cray Y-MP. The results .of the CMRCI calculations together with Davidson’s corrections for the estimation of the qua- druples contribution are given in Tables IV and V for benzene and pyrazine, respectively.

D s 4,”

z.oL-j&:: 0 s ~~C-rK~ <I*xII>,Nn.l.l. , (, ?Q ,$g”


‘-. II_ __




_ ‘.‘

FIG. 6. Conical intersection of the &(rrti) and the Sa potential-energy surfaces of pyrazine in the two-dimensional space spanned by the reaction coordinate and the coordinate of maximum coupling. The surfaces have been obtained at the CASSCF/DZ level. The upper and lower panels show different perspective views.

‘Ill. DISCUSSION.

A. *!ptimized geometries and barriers 3,. : L’ ‘Pi ,I _,. ‘?‘;,i. . ‘The CASSCF optimized parameters of ‘the ground-

state geometry of benzene and pyrazine obtained with three different basis sets are given in Tables I and II, respectively. The corresponding experimental data are also included. The basis sets STO-3G, 3-21G, and DZ are of increasing quality, but one can see that at this level of theory there is no obvious correlation between the quality of the basis set and the accuracy of the calculated geometry. Good results were obtained with the relatively simple split-valence 3-21G basis set. In the following we conse- quently refer to the CASSCF/3-2 1G optimized geometries, when possible. The zero-point energies and imaginary vibrational frequencies at saddle points were estimated at the level of CASSCFBTO-3G. As expected, the zero-point energies are lo%-20% too large at this level of theory.

The optimized geometry of the prefulvenic form of benzene (see Table III) differs only slightly from Kato’s result?31 but differs significantly from the UHF/3-21G geometry of Ref. 30. The differences mostly concern the geometry of the three-membered ring. The length of the newly formed. bond C&!6 predicted by CASSCF/3-2 1G (1.600 A) is close to the MIND0/3 result of Ref. 30 ( 1.592 A). The other geometrical parameters are similar in all of these treatments. This also concerns the optimized geometry at the C, saddle point on the ‘B,, PE surface (SP,). The bond length R(C,-C6) =2.110 A is slightly shorter than found by Kato (2.14 A). Our energetical bar-

TABLE IV. CASSCF, MRCI, and Davidson-corrected MRCI energies (relative to -232.0 a.u.) of the ground and the lowest excited states of benzene calculated with the DZ and DZP basis sets. Numbers in parentheses represent the energy relative to ground state (in eV). PF denotes the prefulvenic minimum, SP, and SPe saddle points on the St and S,, surfaces, respectively. C(0) is the coefficient of the CASSCF reference in the MRCI wave function.

State CASSCF C(O) MRCI MRCI-DC Expt.8

1 ‘A,, 1 ‘&u 1 ‘4, 1 'Bz, b

1 ‘A” (SP,) 1 ‘A” (PF) 1 ‘A’ (PF) 1 ‘A (SP,)

-0.117 59 -0.532 85 (5.02) -0.362 73 (9.65) -0.545 56 (4.68) -0.508 87 (5.68) -0.521 48 (5.33) -0.403 76 (8.54) -0.519 34 (5.39)

DZ basis

0.950 -0.945 69 0.950 -0.754 29 (5.21) 0.936 -0.633 43 (8.49) 0.949 -0.768 92 (4.81) 0.944 -0.742 30 (5.53) 0.948 -0.751 20 (5.29) 0.946 -0.632 77 (8.51) 0.945 -0.756 71 (5.14)

-0.969 47 -0.777 15 (5.23) 4.9 -0.671 22 (8.11) 7.0 -0.792 46 (4.81) -0.769 13 (5.45) -0.776 33 (5.25) -0.658 56 (8.46) -0.783 97 (5.05)

DZP basis

1 ‘4, -0.816 32 0.942 -1.17972 - 1.224 98 1 ‘B,, -0.633 46 (4.97) 0.942 -0.989 72 (5.17) -1.033 49 (5.21) 4.9 1 ‘5, -0.470 37 (9.41) 0.93 1 -0.873 64 (8.32) -0.934 59 (7.90) 7.0 1 ‘BZub -0.643 40 (4.70) 0.942 -0.998 71 (4.92) - 1.042 88 (4.95) 1 ‘A” (SP,) -0.614 74 (5.48) 0.938 -0.978 05 (5.48) - 1.025 74 (5.42) 1 ‘A” (PF) -0.633 58 (4.97) 0.941 -0.994 93 (5.03) - 1.041 09 (5.00) 1 ‘A’ (PF) -0.517 30 (8.13) 0.940 -0.877 17 (8.23) -0.923 54 (8.20) 1 ‘A (SP,) -0.632 72 (4.99) 0.938 -0.999 05 (4.91) - 1.047 33 (4.83) 1 ‘A” (PF) -0.637 71 (4.86) 0.940 -l.OCO 13 (4.88) - 1.046 80 (4.85) 1 3A’ (PF) -0.519 35 (8.08) 0.940 -0.879 59 (8.16) -0.926 (12 (8.13)

Y’ertical excitation energy estimated from Ref. 64. bAt the ’ Bzu optimized geometry.



TABLE V. CASSCF, MRCI, and Davidson-corrected MRCI energies (relative to -262.0 a.u.) of the ground and the lowest excited states of pyrazine calculated with the DZ and DZP basis sets. Numbers in parentheses represent the energy relative to ground state (in eV). PF denotes the prefulvenic minimum, SP, and SPe saddle points on the S, and Sc surfaces, respectively. C(0) is the coefficient of the CASSCF reference in the MRCI wave function.

_ :- State CASSCF C(O) MRCI MRCI-DC Expt.a

..~

1 ‘A, 1 ‘4, 1 ‘4” 2 ‘4. 1 ‘&u b

1 ‘A” (SP,) 1 ‘A” (PF) 1 ‘A’ (PFj 1 ‘A (SP,)

1 ‘A, 1 ‘4, 1 ‘&u 2 ‘&u 1 ‘Bzpb 1 ‘A” (SP,) 1 ‘A” (PF) 1 ‘A’ (PF) 1 ‘A (SPc) 1 ‘A” (PF) 1 3A’ (PF)

-0.678 40 -0.514 44 (4.46) -0.49098 (5.10) -0.301 27 (10.3) -0.505 15 (4.71) -0.476 68 (5.49) -0.503 54 (4.76) -0.419 69 (7.04) -0.490 37 (5.11)

-0.814 65 -0.640 35 (4.74) -0.629 65 (5.03) -0.446 41 (10.0) -0.637 09 (4.83) -0.617 16 (5.37) -0.648 56 (4.52) -0.572 36 (6.59) -0.635 87 (4.86) -0.641 92 (4.70) -0.515 15 (8.15)

DZ basis

0.951 -0.910 25 0.943 -0.757 27 (4.16) 0.950 -0.718 15 (5.22) 0.934 -0.574 92 (9.12) 0.948 -0.736 38 (4.73) 0.939 -0.726 89 (4.99) 0.944 -0.751 45 (4.32) 0.936 -0.697 87 (5.78) 0.939 -0.746 73 (4.45)

DZP basis

0.942 - 1.185 18 0.937 - 1.02598 (4.33) 0.942 -0.996 48 (5.13) 0.93 1 -0.855 17 (8.98) 0.941 - 1.005 40 (4.89) 0.935 - 1.003 77 (4.93) 0.938 - 1.Q32 62 (4.15) 0.934 -0.972 64 (5.78) 0.935 - 1.026 20 (4.32) 0.936 L- 1.035 87 (4.06) 0.934 -0.905 67 (7.60)

-0.933 86 -0.785 52 (4.03) 3.9 -0.741 38 (5.23) 4.9 -0.612 90 (8.73) 7.6 -0.760 92 (4.70) -0.757 84 (4.79) -0.781 19 (4.15) -0.736 88 (5.36) -0.779 11 (4.21)

- 1.230 52 - 1.077 74 (4.15) 3.9 -1.041 34 (5.14) 4.9 -0.915 99 (8.55) 7.6 -~1.051 55 (4.87) - 1.057 01 (4.72) - 1.084 50 (3.97) - 1.030 83 (5.43) - 1.080 78 (4.07) -1.091 13 (3.79) -0.96142 (7.32)

Yertical excitation energy estimated from Ref. 64. bAt the ’ B,, optimized geometry.

rier with respect to the ‘Blu optimized geometry (7000 cm-t) is lower (by 1000 cm-‘) than the result of Kato.31

Apart from minor differences in the lengths of some of the bonds, the CASSCF/3-21G prefulvenic form of pyrazine is very similar to prefulvene. The same is true for the C, saddle point on the ‘BZU PE surface (SP, ). This con- firms our earlier conclusion32 that along the reaction path to the prefulvenic form the r-electronic configurations are mostly involved: The nti excitations seem to be of minor importance in this respect. The barrier height on the ’ B2, PE surface of pyrazine (6200 cm- ’ ) is lower than the corresponding barrier in benzene. Characterization of the molecular geometry of SP, in terms of internal vibrational displacements from the planar reference geometry indi- cates that the SPr configuration lies very close to the concerted reaction path from the planar to the prefulvenic form, and is located roughly in the middle of this path.

At the C1 saddle point on the ground-state PE surface (SPs) of benzene and pyrazine the C,, C3, Cs, and C6 atoms do not lie in one plane, and the pairs of bonds equiv- alent at the C, symmetry change their lengths in opposite directions. Thus for benzene at the CASSCF/STO-3G level we have R12/R16= 1.434 AD.516 A (instead of 1.493 A at the prefulvenic equilibrium), R,,/R,, = 1.522 A/1.429 A (1.514 A), and R,dR,,=1.356 A/1.470 A ( 1.404 A). The corresponding numbers for pyrazine are R12/R,6=1.427 ,A/1.500 %, (1.505 A), R2,/Rs6= 1.520 A/1.422 A (1.514 A), and R,JR,,=1.328 A/1.451 A

( 1.387 A), respectively. Both prefulvenic forms are protected by energetical barriers ( A2 = 3500 cm- ’ for benzene, and A,=4000 cm- ’ for pyrazine) against the rearOmati- zation at the CASSCF/STO-3G level of theory.

We have also optimized at the level of CASSCF/3-21G the triplet ground state (3A N ) of the prefulvenic forms and found both systems more stable than the singlet ‘A” state by about 1000 cm-’ for benzene, and 800 cm-’ for pyrazine, thus confirming the biradical character of these forms. The geometrical parameters of the prefulvenic forms optimized in the triplet state are almost the same as those found for the singlet state.

Generally, one can say that the main geometrical parameters of the stationary points of the TV* excited PE surfaces are similar for both molecular systems, and are virtually independent of the multiplicity of the electronic state.

B. Reaction path for interconversion to the prefulvenic form

As one can see from Figs. 2 and 3, the rrrr* excited singlet states of B,, symmetry (St of benzene, S2 of pyrazine) are adiabatically correlated to the ‘A” state of their prefulvenic forms along the reaction path retaining C, symmetry. In both cases the ground state Se of the planar form correlates to the first excited singlet state at the prefulvenic configuration. The PE curves of the lowest singlet config-




urations in the vicinity of the prefulvenic minimum obtained with the CASSCF wave function are very similar to those obtained in our previous work32 with the UHF wave function. It results simply from the fact that the CASSCF wave function of these species is dominated by a single electronic configuration, which can be labeled ( 14a’) ‘(8a”) i within the C, point group. The coefficient of this configuration in the CASSCF wave function is greater than 0.9 at the prefulvenic equilibrium of both benzene and pyrazine. The electronic structure of these forms is thus well approximated by a one-determinant wave function of the UHF type. This is no longer true for the molecular systems near their planar configuration and in the transition region. The lowest valence excited states of benzene result from the distribution of the electron-hole pair between two degenerate orbitals ( elg and e,,, in the Deb point group). As a result, the electronic wave functions of the lowest excited (singlet and triplet) states of B1,, B2u, and Et, symmetry are dominated by equal contributions from pairs of degenerate configurations. The wave functions of these states cannot be described, even in the zeroth-order approximation, by single determinants. This explains the qualitative difference between the PE curves calculated along the reaction path in this work and those of Ref. 32.

The previous result,32 although qualitative in nature, appears to be very useful in the interpretation of the PE curves obtained in the present more sophisticated study. It is convenient to perform the following discussion in terms of the Dzh point group. In our convention the irreducible representations of the Deb and &, symmetry groups are correlated as follows: al,+a,, aZr-*b3x, bIx-+blx, bzx+bzx, elx-+blx+bh, and ezx-+ax+b3x, where index x stands ei- ther for g or u. According to Ref. 32, the ground state of the prefulvenic form of benzene is adiabatically correlated to the ( 1 b,,) i ( 2b3J i electronic configuration at the planar geometry. This configuration is degenerate with the ( lb,) ’ ( la,) ’ configuration. Both electronic configurations give the leading contribution to the CASSCF wave function of the ‘B,, state (plus combination) and to one component of the ‘Elu state (minus combination), of benzene. The splitting of the ‘Blu and ‘Elu states reflects the strength of coupling between the above-defined configurations. The PE curves of the two electronic configurations show a very different behavior along the reaction coordinate to the prefulvenic form, i.e., the first of them stabilizes to become the prefulvenic ground state, while the second rises in energy. Near the prefulvenic minimum these two configurations are widely separated in energy and their interaction has a small effect on the stabilization of the ‘B2, state. We can thus identify the barrier on the ‘Bzu(?r@) PE surface as resulting from the interaction between the two above-mentioned configurations. This also explains the result that the 3A” state of prefulvene correlates to the 3El,, state rather than to the 3B2,, state as in the singlet manifold. This simply results from the different energetical ordering of these states at the planar geometry C3% < 3B2u, ‘B2u < ‘E,,). While both states ( B,, and E,,) contain an admixture of the ( Ibis) ’ ( 2b3,) ’ configuration, only the lower of the two states of given multiplicity cor-

relates adiabatically to the lowest state of the prefulvenic form. The first excited triplet state of the prefulvenic form has A’ symmetry and is almost degenerate with the ‘A’ state. It correlates adiabatically to the 3Blu state of the planar form (see Fig. 2).

All statements of the above discussion are valid also for pyrazine, with the only difference that the “active” component of the El, state is now replaced by the second singlet state of B,, symmetry. In addition, the VZ-* electronic states are now interloped by the nr* excitations. Cne can see that the latter states do not provide a reaction path to the prefulvenic form (Fig. 3). The Sl ( B,,) state of pyrazine intersects, however, the PE energy surface of the S2( Bzu) state near its bottom. The coupling of the nrr* and rr* states via the vibrational mode Y~,,~(B,,) produces a conical intersection between two PE surfaces.34

C. Characterization of the conical intersection of the m+ excited state with the electronic ground state

As already discussed in the preceding section, the ‘A” TG-* excited PE surface crosses with the IA’ ground-state surface along the reaction coordinate to the prefulvenic form which retains C, symmetry. Any distortion of the nuclear geometry along A” internal coordinates is expected to result in an avoided crossing of the two surfaces. There are 13 internal vibrational coordinates of A” symmetry for benzene, and 11 for pyrazine. Not all of them are important for the vibronic coupling near the crossing point, but the multimode character of the problem makes the visual- ization of the complete surface impossible. To catch the most essential aspects, we have characterized the conical intersection along two vectors of internal displacements, one of A’ symmetry (reaction coordinate) and a second of A” symmetry (coupling coordinate). The vectors have been defined in Sec. II D, and represent, roughly speaking, the steepest-descent vectors of internal displacements leading from the intersection point to the prefulvenic form, and to the ground-state saddle point (SP,).

The PE surfaces in the two-dimensional space spanned by the reaction coordinate and the coupling coordinate are shown in Figs. 5 and 6 for benzene and pyrazine, respectively. The upper and lower panels show different views of this intersection. In the upper panel the reaction coordinate QAt varies from left to right, the prefulvenic minimum lying at the right-hand rim of the grid. The coupling coordinate QA,f varies from back to front, the saddle point SP,, being located at QA”= 1. In the lower panels the coupling coordinate QA” varies from left to right and the reaction coordinate from front to back. The PE surfaces are symmetric with respect to QA,, =0 and are displayed, therefore, only for Q,tf>O.

In the upper panels the saddle point separating the prefulvenic minimum from the deep well of the aromatic form is clearly visible. It is seen that this saddle point is better developed in pyrazine than in benzene. This can be traced back to the fact that the prefulvenic minimum is more stable relative to the So reference energy in pyrazine than in benzene. ”

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The lower panels show that the repulsion of the surfaces along the coupling coordinate is more pronounced in benzene than in pyrazine. The comparatively strong repulsion of the surfaces also contributes to the lowering of the SPO energy in benzene.

TABLE VI. Barrier height A, (in wave numbers) on the n?r* excited singlet surface in C, symmetry, obtained with CASSCF/MRCI calculations using the DZ and DZP basis sets. See Fig. 4 for the definition of A,.

The photophysical implications of the surfaces in Figs. 5 and 6 will be discussed in Sec. III E below.

D. MRCI results

Molecule

Benzene

Pyrazine

Basis

DZ DZP

DZ DZP

CASSCF MRCI MRCI-DC

8ooo 5800 5100 6300 4500 3750

6200 2100 700 4400 400 -1200

The principal results of the MRCI studies are presented in Tables IV and V for benzene and pyrazine, respectively. One can see that in all cases the weight of the CASSCF reference in the total MRCI wave function is greater than 0.93. It should be recalled that the CASSCF data in Tables IV and V refer to the minimal active space (see Sec. II A), while the CASSCF PE curves in Figs. 2 and 3 have been obtained with an extended active space (see Sec. II B).

The only experimental data and theoretical results of a similar level of sophistication, which we can refer to, concern the vertical excitation energies at the ground-state geometry. As is well known from number of ab initio studies (for a review see Ref. 45 and references therein), the lowest ‘BzU(~~*) state of benzene and pyrazine has a covalent character and its excitation energy can be well reproduced even at a relatively crude level of theory like the CASSCF approximation. The dynamic electron correlation included in the MRCI wave function is almost the same for this state and for the ground state and thus does not effect significantly the excitation energy. The same holds for the ‘BJu( ns-*) state of pyrazine. The experimental vertical excitation energies of these states are satisfac- torily reproduced by our calculations. The remaining differences of the order of 0.2-0.3 eV can be attributed to the use of CASSCF/3-21G geometries in our studies. Calcula- tions of vertical excitation energies of benzene and pyrazine with extended basis sets have recently been reported by Roos and collaborators.46p47 Starting from CASSCF reference spaces, dynamic correlation effects have been included by MRCI as well as a multireference perturbation treatment. The present results are in good agreement with the data reported in Refs. 46 and 47. MRCI results for vertical excitation energies have also been reported recently by Palmer and Walker4s and by Buma, van der Waals, and van Hemert.4g

basis-set quality and on the level of theoretical treatment as typical for covalent and ionic structures, respectively, we can say that the other structures calculated by us show an intermediate behavior, that is, their energy depends mod- erately on the quality of the theoretical description. The relative energy of prefulvene as well as the saddle point SP, is slightly decreased by the inclusion of polarization functions, but is almost unchanged by the inclusion of dynamic electron correlation. The energy of the prefulvenic form of pyrazine appears to be more sensitive with respect to both factors. Our best estimate for the relative energy of its prefulvenic form (3.97 eV) shows that it is more stable by about 1 eV than prefulvene (5.0 eV>. This quantitative difference between the two systems is reflected in the qualitative topology of the PE surfaces as we have seen in the foregoing discussion.

In order to discuss the energetical parameters of the PE surfaces that are relevant for the intramolecular dynamics of the systems after electronic excitation, we refer to Fig. 4. The barriers A, and A2 have been referred to previously. In addition, we shall consider the vertical excitation energy of the prefulvenic form to its lowest singlet state (A,). The MRCI determinations for these parameters are given in Tables VI to VIII. The best estimates of the present work are the DZP/MRCI results with Davidson correction (MRCI-DC in Tables VI to VIII).

The ‘Elu state of benzene and the second singlet state of Bzu symmetry of pyrazine are dominated by ionic structures and are therefore more difficult to describe. Their accurate description requires more diffuse basis functions and their calculated energies are more sensitive to the level of electron-correlation treatment as well as to the presence of polarization functions in the theoretical description than in the case of covalent states.45 Our best estimate for the energies of these states is still higher by almost 1 eV than the experimental values.

In principle, the zero-point energies have to be included in the estimation of the barrier heights. The zero- point energies of minima and saddle points estimated at the CASSCF/STO-3G level have been given in Tables I-III. It is seen from these data that the zero-point energy of the saddle point SP, is only marginally larger than the zero- point energy at the ‘B2,, minimum for benzene. For pyrazine, on the other hand, the zero-point energy of SP, is smaller by about 300 cm- ’ than the zero-point energy at the ‘BZu minimum. The SP, zero-point energies are marginally larger than the SPl zero-point energies for both benzene and pyrazine. These rather moderate differences of the zero-point energies are not decisive and are neglected in the following discussion.

The MRCI energies of the prefulvenic forms as well as the transient species cannot be directly compared to experimental data or to other accurate theoretical results. Tak- ing the dependence of the ‘Bzu and the ‘Elu energies on the

The results of our MRCI estimations for the barrier Al are presented in Table VI. One can see that the height of the barrier in both molecules depends on the quality of the basis set and on the level of the theoretical treatment. Our best estimate for benzene (A, = 3750 cm-‘) is very close to the observed threshold of the “channel three.“i3*t4 In pyrazine, however, we find that A1 becomes negative as the level of the theoretical sophistication increases. This is an



TABLE VII. Barrier height AZ (in wave numbers) on the ground-state PE surface in C, symmetry, obtained with CASSCF/MRCI calculations using the DZ and DZP basis set. See Fig. 4 for the definition of A,.

Molecule

Benzene

Pyrazine

Basis

DZ DZP

DZ DZP

CASSCF MRCI MRCI-DC

500 -1200 -1700 200 -900 - 1400

2900 loo0 500 2800 1400 800

interesting result which means that the PE surface of the S2(~r*) state of pyrazine allows for barrierless photoisomerization.

The theoretically postulated prefulvenic forms of benzene and pyrazine may not be chemically stable in their singlet A” state, as is revealed by the MRCI results presented in Table VII. In case of benzene the barrier for rearomatization of prefulvene disappears at the MRCI level (A,= - 1400 cm-‘). The singlet state of the prefulvenic form of pyrazine, on the other hand, is protected by a small barrier against rearomatization at this level of theory (A, = 800 cm- ’ ) . It is obvious that these qualitative differences between the PE surfaces of benzene and pyrazine are just the reflection of the larger stability of the prefulvenic form of the latter system. One may expect that the observed trends will continue with increasing number of nitrogens in the aromatic ring (triazine, tetrazine, etc.).

The calculated barrier heights Al and AZ show a sys- tematic decrease with increasing level of sophistication of the theoretical treatment. It is noteworthy that this correlates with the trend observed for the ionic states ( ‘Elu of benzene and second ’ Bzu of pyrazine). This is not an accidental correlation, since the structure of the adiabatic ‘Bzu PE surface along the reaction coordinate of interest can be interpreted in terms of the interaction between two diabatic configurations, one resulting from the covalent ‘Bzu state and the second from a ionic state. Thus a theoretical treatment which is able to reproduce accurately the energy of the ionic configurations is expected to provide a good description of the reaction path. The best result of the present work still overestimates the vertical excitation energies of the ionic states by about 1 eV. It is then reasonable to assume that the barrier heights estimated in this work provide upper values for these quantities. The barrier height A1 of 4000 cm- ’ estimated by Kato for benzene at a much cruder level of theory31 seems to be, in the light of the present studies, an accidental result due to a fortuitous compensation of errors.

While the prefulvenic forms of benzene and pyrazine seem to represent only transient structures, they may be of some importance in time-resolved pump-probe experiments. In Table VIII we present estimates for the energy of the first allowed absorption band in the singlet and triplet manifolds. In the prefulvenic forms of both benzene and pyrazine the absorption is shifted to the red as compared to their parent planar forms. According to Table VIII, one expects absorption around 3.2 and 1.5 eV in the prefulvenic singlet forms of benzene and pyrazine, respectively.

TABLE VIII. Vertical 1 ‘A” - 1 ‘A’ excitation energy A3 (in eV) for the prefulvenic forms of benzene and pyrazine, obtained with CASSCF/ MRCI calculations using the DZ and DZP basis sets. The corresponding data for the triplets ( 1 ‘A” -t 1 3,4’) are given in parentheses.

Molecule Basis CASSCF . r MRCI MRCI-DC

Benzene DZ 3.21 3.22 3.21 DZP 3.16 (3.22) 3.20 (3.28) 3.20 (3.28)

Pyrazine DZ 2.28 1.46 1.21 DZP 2.07 (3.45) 1.63 (3.54) 1.46 (3.53)

For the triplet species the absorption bands are expected at 3.3 and 3.5 eV, respectively. !

E. Photophysical implications

The extensive set of ab initio electronic-structure calculations reported in this work has been performed with the intention to develop a better understanding of some aspects of the complex photophysics of aromatic molecules. In this section we discuss the photophysical rele- vance of the PE surface data given in Tables IV-VIII and Figs. 2-6.

Let us first consider benzene. As mentioned in the In- troduction, it has repeatedly been speculated that the channel-three phenomenon in benzene is related to a fast isomerization process on the S, (Z-T*) PE surface. The work of Kato3’ has provided the first semiquantitative con- firmation of this ‘qualitative picture. As shown by Kato and confirmed by the present calculations, the biradical called prefulvene2’ exists as a low-lying stationary or quasista- tionary point on‘the lowest singlet PE surface of benzene. The St (QT~*) excited state of planar benzene is adiabatically connected to the singlet ground state (‘A”) of prefulvene along the concerted reaction path (Fig. 2). The electronic ground state of planar benzene correlates .adiabatically with the first excited singlet state (‘A’) of prefulvene (Fig. 2). Upon deformation out of C, symmetry (which is retained along the benzene-prefulvene reaction path) the ‘A’ and ‘A” states repel each other, resulting in a multidimensional conical intersection. A two-dimensional picture of this intersection has been given in Fig. 5.

From the CASSCF results shown in Figs. 2 and 5 and the MRCI results given in Tables VI and VII, the following qualitative picture of the photophysical dynamics in the S, (7rti) state of benzene emerges.’ The planar S1(~~*) minimum is separated by a barrier Al of about 3000 cm-’ (the best calculated result for A, is 3750 cm-’ and is considered as an upper limit) from the prefulvenic minimum. As soon as the excess energy in S, is sufficient to overcome (or to tunnel through) the barrier, the St-S, conical intersection becomes accessible to the nuclear mo- tion. The singular non-Born-Oppenheimer coupling associated with this intersection will cause ultrafast IC to the So surface.

From the analysis of rotationally resolved sub-Doppler spectra of the S, state of benzene, a qualitative picture for the relaxation pathways responsible for the channel-three effect has been developed.‘* The optically accessible vi-




bronic levels of St are thought to be coupled by anhar- manic and Coriolis interactions to dark backgrounds levels of the S, surface. These dark states, being excited in out- of-plane modes, are assumed to be short-lived owing to good Franck-Condon factors for IC to S’,. The onset of channel-three thus involves both an abrupt increase of the intrastate vibrational relaxation (IVR) rate as well as the IC rate (of the dark background states). This picture has been quantified by recent model calculations.5ti52

The picture derived from spectroscopic data is not in contradiction with the present ab initio PE data. The conical intersection of S1 with So, being located just “behind” the barrier towards isomerization, provides the microscopic mechanism (“funnel”) for ultrafast IC. The lowest vibronic levels of S, are well protected from the abyss of the conical intersection by the barrier A,. With increasing excess energy the optically bright vibronic levels feel the anharmonic distortion of the S, surface associated with the saddle point SPt, and efficient mixing with out-of-plane excited vibronic levels sets in. In this picture, illustrated by Figs. 2 and 5, the simultaneous abrupt onset of anharmonic mixing as well as IC at the channel-three threshold appears not as an accidental coincidence,53 but rather as a necessary consequence of the topology of the multidimensional adiabatic PE surfaces. It may also be noted that the present interpretation of the channel-three phenomenon is consistent with the absence of photochemical hydrogen scram- bling, as observed by Callomon and Somers.54

As discussed in Sec. III D, the isomerization barrier A, may alternatively be viewed as arising from the avoided crossing of two diabatic configurations. This alternative picture is useful for a qualitative understanding of the driv- ing forces behind the photoinduced isomerization dynamics of aromatic molecules.32 It has been shown in Ref. 28 that a simple model based on this diabatic picture can well reproduce the nonradiative decay kinetics at the channel- three threshold of benzene. It is now evident that the

I avoided crossing arises from the intersection of the S,( B,,) configuration with a higher excited valence configuration, rather than the S,( B,,) configuration, as has been tentatively assumed in Ref. 28. This reinterpretation does not affect the main conclusions of that work.

It should be clear from this discussion that the old- standing dispute whether the channel-three dynamics involves the St ( Z-T*) and So surfaces only or an additional electronic stateI is a semantic one. In the adiabatic picture, only the two lowest singlet surfaces of C6H6 are involved (Fig. 5). In the diabatic picture, the prefulvenic minimum is associated with a higher rrr* excited state which has been stabilized by out-of-plane deformation. The existence of a low barrier towards isomerization on the St surface and a conical intersection with the So surface may thus legitimately be ascribed to the participation of an additional electronic state.

The question whether prefulvene exists as a local minimum on the lowest adiabatic singlet surface of C6H6 cannot definitively be answered by the present study. We ob- serve that the barrier separating the aromatic and prefulvenic wells on the lowest singlet surface in C, sym-

metry decreases with increasing sophistication of the electronic-structure theory. We conclude from the present data that prefulvene exists as a clearly developed plateau on the ‘A surface, from which barrierless rearomatization to benzene is possible. Singlet prefulvene is thus not a chemically identifiable species, but might be detectable by transient absorption experiments. The present calculations predict an absorption maximum near 3.2 eV (390 nm) for the lowest singlet-singlet transition with a very small oscillator strength (f= 3.56 X 10e5). For the lowest triplet- triplet transition of prefulvene the predicted excitation energy is 3.28 eV with an oscillator strength f=4.02~ 10w4. The predicted dipole moment of prefulvene is rather small (D= 1.05 D). One would thus not expect significant stabilization of the prefulvenic form in condensed media.

Considering the topology of the PE surfaces of Fig. 5 and the fact that the IC process via a conical intersection is generally extremely rapid,55’56 we expect the formation as well as the decay of singlet prefulvene to occur on subpicosecond time scales. Solvation effects are not expected to modify significantly the intramolecular dynamics on this ultrafast time scale. The transient absorption experiments alluded to may therefore be performed in the condensed phase, e.g., liquid benzene. In the condensed phase the subpicosecond transient absorption of singlet prefulvene might be detectable despite the unfavorable oscillator strength of the ‘A’ c ‘A” transition. No interference from triplet states is expected at these time scales. The detection of subpicosecond transient absorption signals could even- tually discriminate the IC mechanism proposed here from alternative IC mechanisms proposed in the literature which do not involve a transient intermediate.50*52

The lack of a PE barrier between the prefulvenic and aromatic forms of benzene provides a consistent explana- tion for the observed low yield of isomerization products in the photolysis of benzene. Although the IC process is ini- tially triggered by the tendency of the St (VT*) state towards isomerization, the barrierless rearomatization to benzene will prevail after the IC process. The prefulvenic form of benzene is separated by PE barriers from the stable isomers of benzene. The location of these saddle points and the determination of precise barrier heights requires additional work.

We have focused in this work on the singlet surfaces, as these are exclusively involved in ultrafast photophysical processes after optical excitation from So. As a consequence of its biradical character, prefulvene possesses a triplet ground state (3A”). The triplet state is more stable (by 1200 cm-‘) than the singlet at the MRCI-DC level. The barrier which protects prefulvene from rearomatization on the triplet surface results from avoided crossing between the 3Elu and the 3Blu diabatic configurations. These configurations intersect each other at higher energies and at a smaller distance from the planar geometry than the corresponding singlet states ( ‘Bzu and ‘At,). Triplet prefulvene is thus more stable and might possibly be detected as a paramagnetic species. The question of the stability of triplet prefulvene will be considered in more detail elsewhere.



The qualitative similarity of the PE surfaces obtained for pyrazine (Figs. 3 and 6) with those of benzene (Figs. 2 and 5) lends strong support to the idea that the tendency towards isomerization to a biradicalic form is a universal effect in rn=* excited states of aromatic molecules.32 The quantitative differences found in the present work between pyrazine and benzene with respect to reaction enthalpies and barriers correlate nicely with experimental observa- tions. The S2(’ Bzu(mr*>) absorption band of pyrazine is, in contrast to the S,( ‘B,,) band of benzene, completely diffuse. Yoshihara and co-workers1g found from measure- ments of the fluorescence lifetime that an efficient nonradiative decay channel opens already below the presumed origin of the Sz(r~-*) absorption band. This is in full ac- cord with our result that there exists no barrier towards isomerization to the prefulvenic form on the S2(rr@) surface of pyrazine. The conical intersection connecting the Sz( rr*) and So surfaces (Fig. 6) is thus directly accessible after S2 excitation, leading to ultrafast IC. The S,( Z-T*) surface of pyrazine is intersected, in addition, by the S,(m*) surface near the planar reference geometry (see Fig. 3). The spectroscopic and dynamic consequences of this S2-St conical intersection have been discussed elsewhere 34,51-59

The prefulvenic minimum on the lowest adiabatic ‘A PE surface of pyrazine lies lower in energy and is more clearly developed than in benzene (cf. Figs. 5 and 6), We expect that this local minimum remains existent in calculations with more extended basis sets and more complete inclusion of electron-correlation effects. The repulsion of the singlet adiabatic surfaces along the coupling coordinate is weaker in pyrazine than in benzene (cf. Figs. 5 and 6), indicating a smaller IC rate.

The prefulvenic form of pyrazine possess a significant dipole moment in its lowest singlet state (D=2.5 D), and will thus be additionally stabilized in polar solvents. The vertical excitation energy of the first excited singlet state of the prefulvenic form of pyrazine is predicted to be strongly shifted to the red (800 nm) (Table VI). The predicted oscillator strength of this transition is f = 1.06 X 10m3. The 3A” state is 1400 cm-’ more stable than the ‘A” state at the MRCI-DC level. The excitation energy of the lowest triplet-triplet transition is 3.53 eV with an oscillator strength f =2.23 x 10w3.

It can be inferred from these data that the singlet prefulvenic form of pyrazine might be more easily detectable as a transient species than prefulvene itself. The absorption at the unusually long wavelength of 800 nm should facili- tate the discrimination of this species from other possible intermediates and isomeric products. Considering that the prefulvenic form of pyrazine is protected by a barrier from rearomatization, we expect that the quantum yield of photochemical isomers should be larger than in benzene, especially in the condensed phase. In view of the significant permanent dipole moment of the prefulvenic form of pyrazine, the photochemical yield should also depend on the polarity of the solvent.

We expect that the qualitative trends emerging from the comparison of benzene and pyrazine can be extrapo-


lated towards triazine and tetrazine. The prefulvenic forms I of these systems should become increasingly stable and

more long-lived.

IV. CONCLUSIONS

We have performed ab initio calculations of the PE surfaces of the lowest singlet and triplet states of benzene and pyrazine which are more complete and more accurate than previous results. The CASSCF method has been adopted to obtain a qualitatively reliable global description of the surfaces. We have put particular emphasis on the determination of barrier heights and reaction enthalpies for the isomerization reaction of the lowest ~7r* excited state to the biradical form called prefulvene2g-31 using large- scale MRCI calculations. We have characterized the conical intersection of the rr# excited singlet surface with the So surface in a two-dimensional space spanned by the reaction coordinate and the coordinate of maximal coupling.

Considering that the computed barrier towards isomerization in benzene agrees very well with the observed channel-three threshold, we can be rather sure that we have indeed found, by following the reaction path to prefulvene, the lowest isomerization barrier and the minimum of the seam of intersection of the Sr(rr*) surface with-the So surface. Since the saddle point SP, separating the prefulvenic minimum from the planar minimum is a direct consequence of the conical intersection, we can be rather sure that we have also located the lowest barrier for isomerization on the singlet ground-state surface of benzene and pyrazine. The estimated barrier heights (SP, relative to the So minimum) of 4.83 and 4.07 eV for benzene and pyrazine, respectively, reflect the chemical stability of aromatic systems. The result that the lowest rr~-* excited surface of pyrazine lacks, in contrast to the Si (Z-Z-*) surface of benzene, any barrier towards isomerization has important implications for the understanding of the photophysics of this molecule.

The logical extension of the present work consists in the ab initio characterization of the conically intersecting singlet surfaces in more dimensions. Such calculations should provide the basis for a truly microscopic time- dependent wave-packet description of the IC and isomerization dynamics in aromatic molecules.

ACKNOWLEDGMENTS

This work has been supported through a grant of the Deutsche Forschungsgemeinschaft and the Committee for



Scientific Research of Poland. Part of the computations Prefulvenic form (‘A”) have been performed on a Cray Y-MP at the Leibniz Re- chenzentrum of the Bayerische Akademie der Wissen- * N1 1.2919 1.0369 0.0000 schaften. The authors would like to thank Dr. E. Riedle, Professor H. J. Neusser and Professor E. W. Schlag for stimulating discussions, and Professor H.-J. Werner for making the MOLPRO package available.

c2 0.0000 0.6956 0.7460 c3 0.0000 -0.7614 1.1025 N4 0.0120 - 1.5436 0.0000 H7 - 0.4244 1.4496 1.3704 H8 0.0043 -1.1702 2.0849

APPENDIX A

Cartesian coordinates (in A) of the stationary points on the S, and So PE surfaces of benzene are shown below. The geometries have been optimized at the CASSCF level with the 3-21G basis set, with the exception of the S’P, saddle point (C, symmetry), which has been determined with the STO-3% basis set.

Prefulvene (‘A”)

Cl 1.1740 1.2470 0.0000 c2 0.0000 0.7812 -0.8002 c3 o.oooo -0.6793 1.1540 c4 0.0049 - 1.4629 0.0000 H7 -0.5757 1.4779 1.3759 H8 0.0015 - 1.0427 2.1586 Hll 1.6026 2.2275 0.0000 H12 0.0394 -2.5319 0.0000

Saddle point SP , ( ‘A”)

Cl 0.8115 c2 0.0000 c3 0.0000 c4 0.0950 H7 -0.6049 H8 -0.1264 Hll 0.9352 H12 0.1630

Saddle point SPo( ‘A)

1.3349 0.0000 0.7103 1.0548

-0.7455 -~-- 1.1782 - 1.4917 0.0000

1.3083 r.7093 - 1.2020 2.1392

2.4014 0.0000 -2.5597 0.0000

Cl 0.9673 1.2978 0.0000 c2 0.0219 0.6634 1.0017 c3 -0.0949 - 0.7407 1.2424 c4 0.05 19 - 1.5121 o.oooo c5 0.0211 --d.7131 - 1.0957 C6 -0.1034 0.7747 -0.7977 H7 - 0.4629 1.3660 1.6690 H8 -0.6380 - 1.1224 2.0974 H9 0.1582 - 1.0442 -2.1168 HlO 0.7517 1.3793 -1.4302 Hll 1.1489 2.3685 0.0454 HI2 0.1609 -2.5873 -0.0239

APPENDIX 6

Cartesian coordinates (in A) of the stationary points on the St and So PE surfaces of pyrazine are shown below. The geometries have been optimized at the CASSCF level with the 3-21G basis set, with the exception of the SPo saddle point (Cl symmetry), which has been determined with the STO-3G basis set.

Saddle point SP, (‘A”)

Nl 0.7780 c2 0.0000 c3 0.0000 N4 0.0894 H7 - 0.4424 H8 -0.1166

Saddle point SPo( ‘A)

1.3486 0.0000 0.678 1 1.0194

* -0.7703 1.1175 - 1.5235 0.0000

1.2973 1.7707 - 1.2555 2.0620

Nl 0.9804 1.3264 o.oooo c2 p.055 1 0.6690 0.9813 c3 -0.0525 -0.7340 1.1886 N4 0.1122 - 1.5500 0.0000 c5 0.0566 -0.7301 - 1.0405 C6 -0.0586 0.7713 -0.8059 H7 -0.3628 1.3739 1.6926 H8 -0.6168 - 1.1517 2.0147 H9 0.2016 - 1.1020 -2.0509 HlO -0.5493 1.3880 - 1.5607

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