Experiment and first principles calculation of spectra

Resonance Raman spectroscopy of the S 1 and S 2 states of pyrazine: Experiment and first principles calculation of spectra Gerhard Stock, Clemens Woywod, Wolfgang Domcke, Tim Swinney, and Bruce S. Hudson Citation: The Journal of Chemical Physics 103, 6851 (1995); doi: 10.1063/1.470689 View online: http://dx.doi.org/10.1063/1.470689 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/103/16?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Effect of electron correlation and shape resonance on photoionization from the S1 and S2 states of pyrazine J. Chem. Phys. 137, 194314 (2012); 10.1063/1.4765374 First‐principles Calculation of Excited State Spectra in QCD AIP Conf. Proc. 1354, 7 (2011); 10.1063/1.3587578 Magnetic properties of Gd2C: Experiment and first principles calculations J. Appl. Phys. 109, 07A924 (2011); 10.1063/1.3554257 First observation of the B ̃ A 1 1 state of SiH 2 and SiD 2 radicals by optical-optical double resonance spectroscopy J. Chem. Phys. 122, 154302 (2005); 10.1063/1.1881172 A b i n i t i o characterization of the S 1–S 2 conical intersection in pyrazine and calculation of spectra J. Chem. Phys. 96, 5298 (1992); 10.1063/1.462715

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Resonance Raman spectroscopy of the S 1 and S 2 states of pyrazine: Experiment and first principles calculation of spectra Gerhard Stock, Clemens Woywod, and Wolfgang Domcke Institute of Physical and Theoretical Chemistry, Technical University of Munich, D-85748 Garching, Germany

Tim Swinneya) and Bruce S. Hudson Department of Chemistry and Chemical Physics Institute, University of Oregon, Eugene, Oregon 97403

~Received 23 June 1995; accepted 20 July 1995! New experimental and theoretical data on the resonance Raman ~RR! spectroscopy of the S 1 and S 2 states of pyrazine are presented. Based on recent ab initio CASSCF ~completeactive-space-self-consistent-field! and MRCI ~multireference configuration interaction! calculations of Woywod et al. @J. Chem. Phys. 100, 1400 ~1994!#, we construct a vibronic coupling model of the conically intersecting S 1 and S 2 states of pyrazine, which includes the seven most relevant vibrational degrees of freedom of the molecule. Employing a time-dependent approach that treats the intramolecular couplings in a nonperturbative manner, we calculate RR cross sections for this model, taking explicitly into account the nonseparability of all vibrational modes. The combination of high-level ab initio calculations and multimode propagation techniques makes it possible, for the first time, to make first-principles predictions of RR spectra for vibronically coupled electronic states of an aromatic molecule. The theoretical data are compared to experimental gas-phase RR spectra which have been obtained for five different excitation wavelengths. The comparison reveals that the ab initio predictions match the experimental results in almost every detail. © 1995 American Institute of Physics.

I. INTRODUCTION

The photophysics and photochemistry of excited singlet states of simple aromatic molecules have attracted considerable interest over the past few decades.1,2 While the spectroscopy of low vibronic levels of the first singlet state ~S 1! is well established, in particular for representative simple systems such as benzene and pyrazine ~see, for example, Refs. 3–9!, very little is still known about the photophysical dynamics of the S 2 state and higher excited singlet states. The spectroscopic investigation of these states is often hampered by large line broadening of the electronic spectra, rendering the appearance of the absorption and fluorescence bands typically diffuse and rather structureless. Besides time-resolved techniques, which are about to become available due to recent development of uv femtosecond laser sources,10 resonance Raman ~RR! spectroscopy has proven to be very useful to obtain specific information on the vibrational and relaxational dynamics of short-lived higher singlet states.11–14 Even in the case of structureless absorption bands, RR spectra can yield valuable knowledge of the symmetries and geometries of excited states, the location of optically forbidden transitions, and the shapes of excitedstate potential-energy surfaces ~PESs! along displaced modes.11–14 Theoretical investigations of RR processes have employed either the time-independent15 or the time-dependent16 formulation of RR spectroscopy. The usual approach to simulate RR spectra is to assume a separable harmonic model for both ground and excited electronic states, whereby the vibrational frequencies and geometry a!

Current address: Stanford Research Systems, 1290 D Reamwood Ave., Sunnyvale, CA 94089.

shifts are inferred from experimental data.11,12 Only few workers have performed ab initio calculations at the Hartree–Fock level, which allow to roughly estimate the geometry changes in the excited states.17,18 The S 1 [ 1 B 3u (n p * )] and S 2 [ 1 B 2u ( pp * )] excited states of pyrazine represent a classic example of vibronic coupling in aromatic systems. For symmetry reasons, the out-of-plane deformation mode n 10a (B 1g ) is the single normal mode of pyrazine which can couple the 1B 3u and 1B 2u states in first order. Manifestations of the vibronic activity of n10a have been observed in the absorption spectrum, in single-vibroniclevel fluorescence spectra, and in RR spectra of the S 1 state.5–9,19,20 The S 1 2S 2 vibronic coupling furthermore causes an ultrafast S 2 →S 1 internal conversion process as has been discussed in detail in Ref. 21. A considerable amount of theoretical investigations has been concerned with the S 1 2S 2 vibronic-coupling problem in pyrazine; see Ref. 9 for a comprehensive list of citations up to 1988. More recently, calculations of the RR and fluorescence spectra22,23 and the simulation of femtosecond pump–probe experiments, considering transient absorption24 as well as two-photon ionization signals25 have been reported. In particular, a series of electronic structure calculations of increasing accuracy ~with respect to the treatment of electron correlation! and completeness ~with respect to the vibrational modes considered! have been performed,23,26,27 which have identified a conical intersection28 of the S 1 and S 2 adiabatic PESs within the Franck–Condon region. Most recently, the conically intersecting S 1 and S 2 PESs have been characterized by CASSCF ~complete-active-space-selfconsistent field!/MRCI ~multireference configuration interaction! calculations.26,27

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Stock et al.: Raman spectroscopy of pyrazine

In this work we present new experimental and theoretical data on the RR spectroscopy of the S 1 and the S 2 states of pyrazine. While Raman spectra resonant to the S 1 (n p * ) state of pyrazine have been reported for excitations at several wavelengths,19,20 only a single S 2 RR spectrum ~at 266 nm! is available in the literature.20 Here, we report gas-phase RR spectra of pyrazine for five excitation wavelengths within the S 2 absorption band. These experimental data are compared with ab initio calculations of RR spectra. To obtain these spectra, we have extended the calculations of Ref. 27 to obtain the complete set of parameters of the Taylor expansion ~gradients and Hessian matrices! of the S 1 and S 2 surfaces at the S 0 equilibrium geometry. Based on the ab initio data we have constructed a vibronic-coupling model including seven nonseparable vibrational modes. For this model, RR cross sections have been obtained in a time-dependent approach, treating the intramolecular coupling in a nonperturbative manner. Apart from the electronic S 1 2S 2 gap, which had to be adjusted by less than 2%, and a phenomenological linewidth, which was inferred from the experimental absorption spectrum, we have used exclusively ab initio data for the calculation of the RR spectra. The comparison of the theoretical predictions with the new experimental gas-phase RR spectra obtained at five excitation wavelengths provides a stringent test of the theoretical model. II. AB INITIO CALCULATIONS A. Methods

Our aim is the characterization of the multidimensional PESs of the S 1 (n p * ) and S 2~pp*! of pyrazine within the range of nuclear coordinates which is relevant for the calculation of absorption and RR spectra. In previous works, the S 1 and S 2 PESs have been characterized within the subspace of the five totally symmetric normal modes n1 , n2 , n6a , n8a , n9a and the S 1 2S 2 coupling mode n10a of B 1g symmetry by the calculation of one-dimensional cuts along normal coordinates and two-dimensional surface calculations in selected subspaces.23,27 We have now extended these calculations to obtain the complete set of parameters of the Taylor expansion ~gradients and Hessian matrices! of the S 1 and S 2 surfaces at the S 0 equilibrium geometry. The S 0 equilibrium geometry has been optimized at the MP2 level ~second-order Mo” ller–Plesset perturbation theory! using the Cambridge Analytic Derivatives package ~CADPAC!.29 A double-zeta basis set with polarization functions on all atoms ~DZP!30 has been employed. For the S 0 surface in the vicinity of the equilibrium geometry, the MP2 treatment is adequate and has been shown to yield rather accurate geometry parameters and vibrational frequencies for pyrazine.23,31 The molecule is taken to lie in the y – z plane, with the z axis going through the nitrogen atoms. The S 0 harmonic force field in Cartesian coordinates has been obtained at the MP2 level and diagonalized to yield the normal coordinates and harmonic vibrational frequencies of pyrazine. We have employed the CASSCF method32 for the calculation of the S 1 and S 2 PESs of pyrazine. The CASSCF approach properly accounts for quasidegeneracy effects, which

is essential for the correct description of multidimensional surface crossings.32 A detailed description of these calculations for pyrazine has been given previously.27 The active space consists of six valence p orbitals ~three of which are occupied in the ground-state SCF configuration! as well as the two nonbonding orbitals on the nitrogens. The CASSCF calculation thus distributes ten electrons over eight orbitals. To obtain a balanced description of the S 0 , S 1 , and S 2 surfaces, all three states have been included in the CASSCF functional with equal weight. The state-averaged CASSCF calculation defines a single set of orbitals for the three lowest singlet states, which simplifies the calculation of transitiondipole-moment functions. The CASSCF calculations have been performed with the program system MOLPRO33 employing a DZP basis set.34 Electron-vibrational coupling constants have been calculated as derivatives of electronic excitation energies with respect to ground-state normal coordinates. The coupling constants are defined as35

k ~i k ! 5 g ~i kj ! 5

S D S D ]Ek ]Qi

~1!

,

0

] 2E k 1 2 ] Q i] Q j

,

~2!

0

where E k denotes the electronic excitation energy and the Q i (k) are dimensionless normal coordinates. The k(k) i and gi j have been evaluated by numerical differentiation at the CASSCF are nonzero only for the five totally symmetlevel. The k(k) i ric modes of pyrazine, while the g(k) i j are nonzero for all vibrational modes. The off-diagonal coupling constants g(k) ij , which are responsible for the Dushinsky rotation36 of normal coordinates, are nonzero only for modes belonging to the same symmetry species in the D 2h point group. The mode n10a of B 1g symmetry is responsible for S 1 2S 2 vibronic coupling and requires a special treatment. In Ref. 27 a diabatic electronic representation37 has explicitly been constructed. The vibronic-coupling constants l is then defined as the first derivative of the off-diagonal element of the electronic Hamiltonian, and g(k) 10a,10a as the second derivative of the diagonal elements with respect to Q 10a . We refer to Ref. 27 for details. B. Results

The harmonic vibrational frequencies of pyrazine obtained at the MP2 level with the DZP basis set are listed in Table I. They may be compared with experimental fundamental frequencies, which have been compiled in Ref. 9. With the exception of C–H stretch modes, for which anharmonic corrections are significant, the agreement is very good. This gives us confidence that the ground-state normal coordinates are reliably described at the MP2 level. MP2 harmonic frequencies of pyrazine have also been obtained by Zhu and Johnson, employing a slightly smaller basis set.31 The present results are in better overall agreement with the experimental data. for the five totally The linear coupling constants k(k) i symmetric modes n1 ,n2 ,n 6a , n 8a , n 9a have been calculated in

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Stock et al.: Raman spectroscopy of pyrazine TABLE I. Harmonic vibrational frequencies ~in cm21! of the 1A g ground state of pyrazine. Theoretical frequencies are given both at the MP2 and CASSCF levels of theory and are compared with experimental fundamental frequencies ~Ref. 9!. Symmetry

Mode

Experiment

MP2

CASSCF

Ag

n6a n1 n9a n8a n2

596 1015 1230 1582 3055

597 1027 1264 1633 3280

672 1072 1347 1797 3329

B 1g

n10a

919

914

970

B 2g

n4 n5

756 983

761 913

774 1039

B 3g

n6b n3 n8b n7b

704 1346 1525 3040

711 1384 1592 3254

774 1509 1649 3307

Au

n16a n17a

341 960

343 900

434 1069

B 1u

n12 n18a n19a n13

1021 1136 1416 3012

1032 1166 1456 3253

1251 1248 1680 3312

B 2u

n18b n14 n19b n20b

1063 1149 1416 3063

1092 1369 1456 3277

1177 937 1522 3320

B 3u

n16b n11

420 785

426 781

468 850

TABLE II. Gradients of the excitation energies of the S 1 and S 2 states of pyrazine with respect to dimensionless totally symmetric normal coordinates obtained at the CASSCF and MRCI levels of theory. Units are eV.

MRCI

the S 1 state and the modes n14 , n17a , n4 , and n5 in the S 2 state. Significant off-diagonal quadratic coupling constants, leading to Dushinsky rotation of the normal modes, are found in B 1u symmetry ~n12 ,n 18a , n 19a ,n13!, B 3g symmetry ~n3 ,n8b ! and A u symmetry ( n 16a , n 17a ) in the S 1 state; in the S 2 state, significant off-diagonal quadratic couplings are found in B 2u symmetry ~n14 ,n19b ! and B 1u symmetry ~n12 ,n18a !. By determining the gi j at the CASSCF level, we have neglected dynamic electron correlation effects; by determining vi at the MP2 level, on the other hand, dynamic correlation effects have been accounted for. It is thus not fully consistent to combine the CASSCF quadratic coupling constants with the MP2 ground-state frequencies in the construction of the model Hamiltonian. To correct for the uneven treatment of ground and excited states, we define scaled quadratic coupling constants g 8i j according to

g 8i j 5X i g i j X j ,

~3!

where

Ref. 27 at the CASSCF and MRCI levels. These data are reproduced in Table II for completeness. It can be seen from Table II that dynamical correlation effects are not of decisive importance for the energy gradients. In Ref. 27 the linear vibronic coupling constant l as well as the quadratic coupling constants g(k) 10a,10a have been obtained at the CASSCF and MRCI levels. The results are ~MRCI values in parentheses!: l50.1676 eV ~0.1825 eV! (2) and g (1) 10a,10a 5 g 10a,10a 520.0180 eV ~20.0180 eV!. The relatively large negative value of g10a,10a implies a significant reduction of the vibrational frequency of g10a in both excited states. The quadratic coupling constants for all other modes of pyrazine obtained at the CASSCF level are collected in Table III. Significant quadratic couplings ~i.e., large diagonal elements gii ! are found for the modes n12 ,n 8b , n 17a , n 18a ,n14 in

CASSCF

6853

Q1

Q2

Q 6a

Q 8a

Q 9a

k~1! k~2!

20.0808 20.2234

0.0281 0.0340

20.0938 0.1029

0.0193 0.0248

0.1511 0.0509

k~1! k~2!

20.0470 20.2012

0.0368 0.0211

20.0964 0.1193

20.0623 0.0348

0.1594 0.0484

X i 5 @ v i ~ MP2 ! / v i ~ CASSCF!# 1/2 .

~4!

The ground-state vibrational frequencies obtained at the CASSCF level have been included in Table I; the scaling factors X i can thus be taken from this table. The results collected in Tables I–III provide the quantitative basis for the construction of model Hamiltonians which systematically include all vibrational modes which couple significantly to the S 0→S 1 and S 0→S 2 electronic transitions in pyrazine.

III. CALCULATION OF RESONANCE-RAMAN SPECTRA A. Model Hamiltonian

For the calculation of optical spectra of vibronically coupled electronic states it is advantageous to formulate the model Hamiltonian in a diabatic electronic representation,37 which has explicitly been constructed in Ref. 27. Let uw0& denote the electronic ground state and uw1& and uw2& denote the S 1 (n p * ) and the S 2~pp*! state, respectively. Similar to previous work, we treat the strong vibronic S 1 2S 2 coupling via the mode n10a exactly to all orders, whereas the intramolecular coupling of the excited electronic states uw1&, uw2& with the well-separated electronic ground state uw0& is neglected. As has been found in experimental investigations,38 the S 1,2 →S 0 internal conversion process takes place on a picosecond time scale, and can therefore be ignored, as well as intersystem crossing, on the femtosecond timescale of RR spectroscopy. Neglecting furthermore vibronic couplings and possible intersections of S 1 and S 2 with higher singlet states, the model Hamiltonian reads H5

(

k50,1,2

u w k & h k ^ w k u 1 $ u w 1 & V 12 ^ w 2 u 1H.c.% .

~5!

The vibrational dynamics in the electronic ground state is described by the harmonic-oscillator Hamiltonian h 0

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TABLE III. The quadratic coupling constants g ~1! and g ~2! ~in eV! of the S 1 and S 2 PE functions of pyrazine obtained at the CASSCF level. A g modes

(D 2h →

Symmetry

D 2h )

Mode

Q1

Q 6a

Q 8a

Q 9a

(B 3u )S 1

Q1 Q2 Q 6a Q 8a Q 9a

20.000 46

0.000 25 0.000 14

20.001 59 20.000 29 0.001 92

20.000 92 0.003 70 0.002 63 20.011 94

0.000 12 20.002 75 0.001 17 0.003 60 20.003 73

(B 2u )S 2

Q1 Q2 Q 6a Q 8a Q 9a

0.002 76

0.000 93 20.000 49

20.004 37 0.000 86 20.006 73

20.001 03 20.000 24 0.000 30 0.008 85

0.001 00 20.000 51 0.002 07 0.001 73 0.002 67

Q2

A u modes

(D 2h →

Symmetry

Mode

(B 1 )S 1

Q 16a Q 17a Q 16a Q 17a

20.018 56

B 1g mode

(D 2h →

C 2h )

Symmetry

Mode

Q 10a

(B u )S 1 (B u )S 2

Q 10a Q 10a

20.018 00 20.018 00

B 2g modes

(D 2h →

C 2h )

Symmetry

Mode

Q4

Q5

Q4 Q5 Q4 Q5

20.023 06

20.003 57 20.012 60 0.007 00 20.036 10

B 3g modes

(D 2h →

C 2h )

Symmetry

Mode

Q 6b

Q3

Q 8b

Q 7b

Q 6b Q3 Q 8b Q 7b Q 6b Q3 Q 8b Q 7b

20.005 99

0.000 24 20.008 01

0.005 60 20.012 50 20.051 57

20.010 79

20.000 11 20.008 16

20.000 47 0.000 68 0.006 61

20.001 25 0.004 00 0.006 65 20.000 13 0.000 45 20.000 26 20.000 70 20.009 52

(B 2 )S 2

(B g )S 1 (A u )S 2

(A u )S 1

(B u )S 2

D 2) Q 16a 0.014 57

20.035 28

Q 17a 20.012 15 20.047 08 0.000 07 20.064 54

B 1u modes

(D 2h →

C 2v)

Symmetry

Mode

Q 12

Q 18a

Q 19a

Q 13

(B 1 )S 1

Q 12 Q 18a Q 19a Q 13 Q 12 Q 18a Q 19a Q 13

20.076 72

20.037 15 20.032 68

0.021 60 0.016 75 20.014 10

20.009 97

0.010 10 20.013 02

20.005 55 0.006 30 20.013 52

20.010 20 20.009 30 0.002 15 20.001 28 0.002 87 20.000 14 0.000 58 20.009 39

Q 14

Q 19b

Q 20b 20.001 45 20.000 53 20.004 50 0.000 74 0.001 44 0.000 68 20.000 95 20.009 52

(B 2 )S 2

B 2u modes

(D 2h →

C 2v)

Symmetry

Mode

Q 18b

(B 1 )S 1

Q 18b Q 14 Q 19b Q 20b Q 18b Q 14 Q 19b Q 20b

20.003 07

0.000 15 0.031 99

0.001 89 20.000 92 20.006 20

20.013 05

20.008 40 0.107 78

0.003 16 0.037 50 0.008 41

B 3u modes

(D 2h →

C 2v)

Symmetry

Mode

Q 16b

Q 11

(A 1 )S 1

Q 16b Q 11 Q 16b Q 11

20.014 74

0.006 00 20.010 54 0.006 00 20.016 92

(A 1 )S 2

(B 2 )S 2

20.015 07

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Stock et al.: Raman spectroscopy of pyrazine

h 05

(i

S

D

vi ]2 2 1Q 2i , 2 ] Q 2i

~6!

where the sum runs over all seven vibrational modes included in the model ~see below!. The diagonal diabatic matrix elements h 1 and h 2 as well as the coupling element V 12 are expanded in terms of ground-state normal coordinates Q i up to second order, yielding h k 5h 0 1E k 1 V 12 5lQ 10a .

g ~i,kj! Q i Q j , (i k ~i k !Q i 1 ( i, j

~7! ~8!

Here, E k denotes the vertical excitation energy of state uwk &, and k(k) i represents the gradient of the excitation energy with respect to the totally symmetric coordinate Q i . The g(k) i, j account for changes in vibrational frequencies and rotations of the modes in the excited states ~the so-called Dushinsky effect36!. On the basis of the ab initio data presented above, we are able to specify the vibrational modes Q i which we need to include into the model for a successful description of optical spectra. The vibronic activity of the linearly coupled modes is easily estimated by their Poisson parameter ~1/2!k2/v2. It is clear from previous work that besides the coupling mode n10a at least three totally symmetric modes ~n1 , n6a , n9a ! are indispensable. According to Table II, one should also include n8a ~because of its considerable gradient k(1) 8a in the S 1 state!, whereas the n2 couplings are seen to be negligible. The spectroscopic effects of the quadratically coupling modes can be roughly characterized by their relative frequency shifts Dv/v. Table III reveals that the coupling mode n10a undergoes a considerable frequency shift in the excited states, whereas the quadratic couplings of the totally symmetric modes can be ignored. Table III furthermore shows that from the remaining 18 modes, which couple only in second order, the modes n4 , n5 , n12 , n14 , n16a , n16b , n17a exhibit a considerable relative frequency shift in either S 1 or S 2 . As there is no spectroscopic evidence of n12 and n17a , we have ignored these modes in the present model. From the remaining modes which show up at least weakly in the experimental absorption and emission spectra,5–9,19,20 we have included the modes n4 and n14 , which are the most important quadratically coupled modes in the S 1 and the S 2 state, respectively. Although the remaining 17 vibrational modes of pyrazine do hardly show vibronic activity, they still contribute to the overall decay of the electronic polarization ~the so-called electronic dephasing!, which is reflected, e.g., in the linewidth of the optical absorption spectrum.39 This pure electronic dephasing due to the remaining modes is included into the model in terms of a phenomenological dephasing time T 2 , which is adjusted to reproduce the homogeneous linewidth of the absorption spectrum of the S 2 state ~T 2535 fs!. For the S 1 state we introduce T 25400 fs as a purely technical parameter.40 To summarize, our model of pyrazine incorporates explicitly seven nonseparable vibrational modes, namely, the coupling mode n10a , four linearly coupled Condon active

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modes ~n1 , n6a , n8a , n9a !, and two quadratically coupled modes ~n4 and n14!. The spectroscopic effects of the remaining 17 weakly coupled modes are described in a phenomenological manner by the dephasing rate 1/T 2 . With the exception of the vertical S 1 2S 2 energy gap D, which was slightly adjusted to reproduce the experimental positions of the two absorption bands ~from Dab initio50.83 eV27 to Dexp50.846 eV!, all vibrational and vibronic coupling constants as well as vibrational frequencies are taken without any further modification from the ab initio data collected in Tables I–III.

B. Calculation of spectra

The interaction of the molecular system with the radiation field is given in the dipole approximation by H 1 ~ t ! 52

(

k51,2

mˆ k0 E ~ t ! 1H.c.,

~9!

where we have defined

mˆ k0 5 u w k & e–mk0 ~ Q! ^ w 0 u ,

~10!

and eE(t) represents the external electric field. The diabatic transition-dipole-moment functions mk0 ~Q! have been shown to depend only weakly on vibrational coordinates,27 which allows us to employ the Condon approximation. In the time-dependent formalism41– 43 the absorption cross section is given by

s A~ v ! 5

2 a p 2 v 2 Re 3 e

( E0 dt e i~ v 1 e !t F kk0 ~ t ! , k51,2 `

~11!

0

where a is the fine structure constant and e0 denotes the energy of the initially populated vibrational ground state. The time-dependent correlation functions 2t/T 2 8 F kk ^ vu ^ w 0 u mˆ 0k e 2iHt mˆ k 8 0 u w 0 & u 0& v ~ t ! 5e

~12!

describe the intramolecular excited-state dynamics of the system and are obtained by solving the field-free Schro¨dinger equation for the model Hamiltonian ~see below!. As discussed above, the dephasing rate 1/T 2 causes an additional phenomenological decay of F0(t), resulting in an additional homogeneous broadening of the absorption spectrum. In the case of a single optically accessible excited state ~say, uw2&!, the RR cross section for orientationally averaged scatterers, integrated over the scattered bandwidth and all scattered directions, can be written as16,22

s v←0~ v 1 ! 5

S D UE

8 a p 2 9 e c

2

v 3s v 1

`

0

U

2

dt e i ~ v 1 1 e 0 ! t F 22 v ~t! , ~13!

where uv& represents the final vibrational state, and v1 and vs denote the frequencies of the laser excitation and the spontaneous emission, respectively. In the case that both excited electronic states carry oscillator strength, interference effects of the radiation emitted from the two electronic states may occur, which have to be taken into account when averaging over the molecular orientations ~see Ref. 44 for a detailed discussion!.

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Stock et al.: Raman spectroscopy of pyrazine

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As described in detail in previous work,21,24 the intramolecular dynamics is treated numerically exactly by solving the time-dependent Schro¨dinger equation for the model Hamiltonian ~5! i

] u C ~ t ! & 5H u C ~ t ! & . ]t

~14!

The time-dependent state vector is expanded in a directproduct basis built from diabatic electronic states and harmonic-oscillator states for each of the seven vibrational modes. This way Eq. ~4! is converted into a system of firstorder differential equations for the expansion coefficients, which is solved employing a standard Runge–Kutta method. The harmonic-oscillator basis sets for the seven modes have max been truncated at occupation numbers N max 1 59, N 4 54, max max max max max N 6a 5 18, N 8a 5 4, N 9a 5 7, N 10a 5 15, N 14 5 5, resulting in a dimension of 1 360 800 of the Hamiltonian matrix for B 2u and B 3u vibronic symmetries, respectively. The state vector was propagated up to 2 ps, which took about 12 CPU h on a Cray YMP for each symmetry. As a check of convergence and to generate stick spectra for the absorption bands, we have also employed the Lanczos algorithm,45,46 which allowed us to increase the basis set to include up to 2.5 million basis states.

IV. EXPERIMENTAL METHODS

The experimental methods used in this work are basically those described in Ref. 13 except that a CCD multichannel detector ~Princeton Instruments! was used. Briefly, the excitation wavelengths are either harmonics of the Nd:YAG laser fundamental at 1064 nm or are obtained by stimulated Raman scattering from one of these harmonics in hydrogen gas. The vibrational shift in hydrogen gas is 4155 cm21. Anti-Stokes stimulated Raman shifting results in increases in the frequency of the radiation by multiples of this value. Specifically, 274 nm is 1064/3 plus 234155, 266 nm is 1064/4, 253 nm is 1064/2 plus 534155, 246 nm is 1064/3 plus 334155, and 240 nm is 266 plus 4155. The spectra shown are for the gas phase. Spectra of pyrazine were also obtained in acetonitrile and methanol solutions. The significant but small differences between the spectra obtained in solution and in the gas phase will be discussed in another publication.

V. RESULTS AND DISCUSSION

Although the present experimental RR spectra correspond to excitation of the S 2 state, we briefly discuss the spectroscopy of the S 1 state from the theoretical point of view. This is particularly of interest in order to exhibit the qualitative differences of the effects of the vibronic coupling on the S 1 and S 2 RR spectra, respectively. Unlike the diffuse absorption band of the S 2 state ~see below!, the S 1 absorption spectrum of pyrazine exhibits well resolved peaks ~mostly due to vibrational activity in the modes n6a , n9a , n10a !, which give rise to strong single-level fluorescence emission. Singlelevel fluorescence spectra have been calculated for the

FIG. 1. ~a! Experimental ~Ref. 20! and ~b! calculated preresonance Raman spectra of the S 1 (n p * ) state of pyrazine obtained for the excitation wavelength l5337 nm.

present model47 and are in good agreement with experiment.5,6 Since we wish to focus here on RR emission, these spectra will not be considered. As a representative example of S 1 Raman emission, Fig. 1 presents a preresonant Raman spectrum of pyrazine obtained with excitation at 337 nm. Panel ~a! shows the experimental data of Suzuka et al.,20 panel ~b! shows the theoretical spectrum obtained with the present seven-mode ab initio model. For better comparison with experimental data, the computed intensities are plotted at the energetical position determined by the measured ground state frequency of the corresponding vibrational quantum. To simulate the experimental resolution, the Raman lines of all reported spectra have been convoluted with a Lorentzian of 10 cm21 FWHM. The S 1 RR spectrum is dominated by the strong excitation of the fundamental of n10a , which derives its intensity through vibronic coupling from the S 2~pp*! state. The calculation reproduces nicely the strong excitation of n10a as well a the relative intensities of the excitations in the n1 , n6a , n8a , n9a vibrational modes, except that the fundamental of n1 is somewhat exaggerated. The peak at 1500 cm21 represents ~to about equal parts! the excitation of 10a16a and 234. Furthermore the calculation predicts significant intensity of 2314, which lies outside the frequency range covered by the experiment. Finally, it is noteworthy that all seven vibrational modes incorporated in the ab initio model do show up in the spectrum. Figure 2~a! shows the room temperature gas-phase absorption profile of the S 2 band of pyrazine.38 By analogy with the S 2 state of benzene,48 it can be assumed that the

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FIG. 3. Experimental ~a! and calculated ~b! resonance Raman spectra of the S 2~pp*! state of pyrazine obtained for the excitation wavelength l5274 nm.

FIG. 2. ~a! Experimental ~Ref. 38! and ~b! calculated absorption of the S 2~pp*! state of pyrazine. The stick spectrum in ~b! represents the vibronic energy levels and absorption intensities of the seven-mode model; the envelope is obtained by including a phenomenological electronic dephasing ~T 2535 fs!.

spectrum would not change under supersonic-jet conditions, i.e., it is truly homogeneously broadened. Figure 2~b! shows the calculated absorption spectrum, which reproduces the experiment in Fig. 2~a! in almost every detail. The envelope in Fig. 2~a! has been obtained by including a phenomenological electronic dephasing ~T 2535 fs!. The stick spectrum represents the vibronic energy levels and absorption intensities of the seven-mode model Hamiltonian ~5!, and has been obtained with the Lanczos algorithm.45,46 The very high density of vibronic levels reflects the dissolution of the S 2 Born– Oppenheimer levels into the dense manifold of levels of the lower-lying S 1 surface. It is clear that the irregularly spaced and diffuse structures of the absorption profile are not assignable in term of harmonic vibrational modes in the S 2 state. A comparison with previous four-mode calculations23,27 reveals that the stick spectrum shows significant differences as a consequence of the inclusion of three additional modes,

whereas the envelopes of the two calculations are rather similar. The remainder of this section is devoted to the discussion of S 2 RR spectra of pyrazine, in particular to the comparison of theoretical and experimental data. Figures 3–7 show the experimental @panels ~a!# and theoretical @panels ~b!# S 2 RR spectra of pyrazine obtained for excitations at 273.9, 252.6, 245.9, and 239.5, nm, respectively. For consistency, all spectra are plotted up to a Stokes shift of 3000 cm21, although for excitation near the center of the S 2 absorption band ~i.e., for l5253 nm and l5246 nm! further overtones and combinations of the modes n1 and n6a appear with measurable intensities at Stokes shifts beyond 3000 cm21. As in Fig. 1, experimental values of the vibrational frequencies of pyrazine in the electronic ground state have been used to determine the line positions of the theoretical spectra.

FIG. 4. Same as Fig. 3 except l5266 nm.

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FIG. 5. Same as Fig. 3 except l5253 nm.

FIG. 7. Same as Fig. 3 except l5239 nm.

For all excitation wavelengths, the fundamentals, overtones and combination levels of n1 and n6a dominate the observed spectra. The fundamental of n1 at '1000 cm21 corresponds to the strongest peak in all spectra. Excitation at 274 nm, near the red edge of thee S 2 absorption band, yields mainly excitation of the n1 and n6a fundamentals, whereas overtones and combination levels are weak @Fig. 3~a!#. Excitation near the center of the S 2 absorption band ~266 and 253 nm! leads to significantly enhanced intensity of n1 and n6a overtones and combination levels @Figs. 4~a!, 5~a!#. This enhancement of higher levels gradually disappears when the excitation frequency approaches the blue edge of the S 2 absorption band @Fig. 7~a!#. The theoretical RR spectra @panel ~b!# are seen to reproduce the enhancement of overtone intensity for excitation near the center of the S 2 absorption band. The relative intensities of the peaks corresponding to 1, 6a, 116a, 231, 231 16a, etc., are generally accurately reproduced. The modes 9a, 8a, 4 and 14, which are included in the calculation, lead

to weak but discernible peaks in the spectra. To exemplify that also weaker excitations are reproduced with high accuracy, consider the neighboring peaks 236a and 9a. In nice agreement of experiment and theory the relative intensity ratio R of the two peaks changes from R,1 ~Fig. 3! to R.1 ~Figs. 4 and 5! to R'1 ~Fig. 6! to R,1 ~Fig. 7!. These details indicate that the dynamics in the S 2 state of pyrazine, as far as it is reflected in the RR spectrum, is rather accurately described by the present ab initio vibronic-coupling model. It is particularly interesting to note that the nontotally symmetric coupling mode n10a , which dominates the S 1 RR spectrum ~Fig. 1!, is hardly observed under resonant excitation of the S 2 state of pyrazine. In agreement with experiment, the calculation predicts that the peaks 10a, 2310a and 6a110a appear with very low intensity for excitation wavelengths shorter than 274 nm. Considering the crucial role of the mode n10a for the S 2 →S 1 internal-conversion process as the promoting mode as well as an important accepting mode,21 the inactivity of this mode in the S 2 RR spectrum is remarkable. This observation underlines that the RR spectrum carries only limited information on the photophysical dynamics in the excited states. The agreement between experiment and calculation deteriorates for shorter excitation wavelengths. This is particularly conspicuous for excitation at 239 nm ~Fig. 7!. In this case the current level structure is reproduced by the calculation, but the calculated intensity of overtones and combination levels is too high relative to the intensity of the fundamentals. It is to be expected that for excitation at the blue side of the S 2 absorption band of pyrazine higher excited electronic states have to be considered in the calculation,49 i.e., the electronic three-state model ~5! may break down. Specifically, one expects off-resonance contributions to the fundamental transitions from higher-lying states. This presumes that n6a and n1 are achieved in some nearby higher state~s!, which could be tested with experiments at shorter wavelengths. The more immediate concern that such activity makes a significant contribution to the spectrum obtained at

FIG. 6. Same as Fig. 3 except l5246 nm.

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FIG. 8. Calculated resonance Raman spectrum of the S 2~pp*! state of pyrazine as obtained by the ab initio three-state seven-mode model. The S 2 Raman emission is plotted as a function of both excitation wavelength and Raman shift. We have added the absorption spectrum ~arbitrary scale! and indicated the excitation wavelengths used in the experiment.

239 nm requires a relative intensity determination so that spectra obtained at different wavelengths can be put on the same relative intensity scale. Overall, however, one can conclude that the present three-state seven-mode model is well justified and provides a rather comprehensive description of the Raman spectra in resonance with the S 1 and S 2 states of pyrazine. Figure 8, finally, presents a survey of the calculated S 2 RR spectra, showing the Raman emission as a function of both excitation wavelength and Raman shift, that is, excitation functions of the most prominent Raman lines. It is conspicuous that the shapes of the excitation functions of the 6a and 1 fundamentals and of overtones and combination levels are clearly different. The excitation functions of overtones of nontotally symmetric modes such as 2314 are again different, showing enhancement at considerably higher excitation energies than the fundamentals of totally symmetric modes. Figure 8 illustrates the wealth of information on excited-state dynamics which can in principle be obtained by RR spectroscopy. VI. CONCLUSIONS

We have presented new experimental and theoretical data on the RR spectroscopy of the S 1 and S 2 states of pyrazine. On the experimental side, we have obtained gas-phase RR spectra at five excitation wavelengths within the S 2 ab-

sorption band. On the theoretical side, we have extended recent CASSCF/MRCI calculations27 to obtain the complete set of parameters of the Taylor expansion ~gradients as well as Hessian matrices! of the S 1 and S 2 surface, at the S 0 equilibrium geometry. Based on these ab initio data, we have constructed a vibronic-coupling model which explicitly includes seven ~out of 24! vibrational modes. Employing a time-dependent approach, we have calculated RR cross sections for this three-state seven-mode model, treating all intramolecular couplings ~numerically! exactly. The calculated spectra have been found to be in excellent agreement with the experimental data. This demonstrates that the combination of modern ab initio electronic-structure theory and multimode time-dependent wave packet propagation techniques opens the possibility of first-principles predictions of RR spectra for polyatomic molecules. The degree of agreement between theory and experiment obtained in the present application demonstrates the adequacy of the modeling of the multidimensional diabatic PESs in terms of a Taylor expansion with respect to normal-mode displacements as well as the accuracy of the ab initio electronic-structure calculations. It is encouraging that predictive calculations of RR spectra are nowadays possible for molecules of this size, even when complex non-Born–Oppenheimer dynamics associated with surface crossings is involved.

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ACKNOWLEDGMENTS

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