• 沒有找到結果。

The Role of the n pi* (1)A(u) State in the Photoabsorption and Relaxation of Pyrazine

N/A
N/A
Protected

Academic year: 2021

Share "The Role of the n pi* (1)A(u) State in the Photoabsorption and Relaxation of Pyrazine"

Copied!
9
0
0

加載中.... (立即查看全文)

全文

(1)

DOI: 10.1002/asia.201100472

The Role of the np*

1

A

u

State in the Photoabsorption and Relaxation of

Pyrazine

Chih-Kai Lin,*

[a, b]

Yingli Niu,

[a]

Chaoyuan Zhu,

[a]

Zhigang Shuai,

[c, d]

and

Sheng Hsien Lin

[a, b]

Introduction

Pyrazine is a benchmark system in photochemistry that has been widely studied for more than half a century. Being a diazine molecule, this aromatic compound differs from ben-zene by substituting a pair of nitrogen atoms for carbon ones in the para position. The lone-pair electrons reside in the nonbonding orbitals of the nitrogen atoms and thus gen-erate n–p* transitions in addition to p–p* ones upon

excita-tion in the valence shell. In terms of symmetry, the ground state is designated as S01Agof the D2hpoint group, and the lowest singlet np* and pp* excited states have conventional-ly been assigned as S11B3uand S21B2u, respectively.

The details in the ultraviolet absorption spectra of pyra-zine were first resolved in solution in 1950s,[1] and the data from vapors were obtained in the following decades.[2–7] People found that the 325 nm absorption band, that is, the np* 1B

3ustate, has a lot of sharp peaks that have been as-signed as n6a (ag), n9a (ag), n10a(b1g) fundamentals and their overtones as well as combination bands.[3, 4, 6, 7] The 265 nm band, that is, the pp*1B

2ustate, presents a distinctly differ-ent feature, which is quite intense and broad with few struc-tures, though the crests could be assigned with the aid of the resonant Raman spectrum.[3]The significant difference in in-tensities between the two absorption bands indicates that the 1B

2u ! 1Agtransition is strongly dipole-allowed, whereas the 1B

3u ! 1A

g one is just weakly allowed, and the appear-ance of odd-vibrational-quanta peaks of the non-totally-symmetric mode (n10ab1g) in the

1B

3u band reveals the “in-tensity borrowing” absorption through vibronic coupling. Moreover, the broad feature implies a quite short lifetime of the 1B

2ustate. The lifetimes of 1B

2uand 1B

3u states have been recorded as approximately 20 fs and approximately 22 ps, respectively, by using photoelectron spectra.[8–10] Suc-ceeding researchers proposed an ultrafast 1B

2u!1B3u relaxa-tion mechanism through conical intersecrelaxa-tion. Domcke and co-workers firstly carried out ab initio calculations combined with mathematical models to describe this internal conver-Abstract: The geometric, energetic,

and spectroscopic properties of the ground state and the lowest four singlet excited states of pyrazine have been studied by using DFT/TD-DFT, CASSCF, CASPT2, and related quan-tum chemical calculations. The second singlet np* state,1A

u, which is conven-tionally regarded dark due to the

dipole-forbidden 1A u !

1A

g transition, has been investigated in detail. Our new simulation has shown that the state could be visible in the absorption

spectrum by intensity borrowing from neighboring np* 1B

3u and pp* 1B2u states through vibronic coupling. The scans on potential-energy surfaces fur-ther indicated that the 1A

u state inter-sects with the1B

2ustates near the equi-librium of the latter, thus implying its participation in the ultrafast relaxation process.

Keywords: absorption · pyrazines · quantum chemistry · excited states · vibronic coupling

[a] Dr. C.-K. Lin, Dr. Y. Niu, Prof. C. Zhu, Prof. S. H. Lin Department of Applied Chemistry

National Chiao Tung University Hsinchu 30010 (Taiwan) Fax: (+ 886) 3-5723764

E-mail: ethene@gate.sinica.edu.tw [b] Dr. C.-K. Lin, Prof. S. H. Lin

Institute of Atomic and Molecular Sciences Academia Sinica

Taipei 10617 (Taiwan) [c] Prof. Z. Shuai

Key Laboratory of Organic Optoelectronics and Molecular Engineering

Department of Chemistry Tsinghua University Beijing 100084 (P.R. China) [d] Prof. Z. Shuai

Key Laboratory of Organic Solids

Beijing National Laboratory for Molecular Science Institute of Chemistry

Chinese Academy of Sciences Beijing 100190 (P.R. China)

(2)

sion process and simulated the absorption spectra.[11–16] More recently, Werner and Suzuki et al. carried out dynami-cal simulation on related states, thus providing insights into the relaxation of the1B

2ustate.

[9, 10, 17, 18] Islampour calculated the strength of vibronic coupling and the relaxation-rate constant using vibronic Hamiltonian matrix.[19]Lin and co-workers simulated the absorption spectra based on Fermis golden rule with Franck–Condon factors;[20, 21]they neglected the explicit contribution of conical intersection but still ob-tained good results consistent with experiments.

It should be noted that there are two nitrogen atoms with nonbonding orbitals, so the excitation of an n electron to the lowest p* orbital should be described by the linear com-bination of two equivalent configurations. It results in a pair of singlet np* states, 1B

3u and 1B2g. The former is more stable than the latter evaluated by the overlap integral of corresponding orbitals from a valence-bond scheme.[22] In a similar fashion, the excitation from n to the second-lowest p* generates1A

uand 1B

1gstates. All these np* states except 1B

3uare regarded as “dark” since the transitions are dipole-forbidden by symmetry. Nevertheless, they might not be completely invisible from the absorption spectrum because of possible vibronic couplings with the neighboring intensely dipole-allowed pp*1B

2ustate. Furthermore, all of these np* states would probably participate in the relaxation mecha-nism of the pp* 1B

2ustate as indicated by a recent dynami-cal simulation.[17]

The relative positions of these low-lying singlet excited states, in terms of energy, is therefore an important topic and has been investigated by various quantum chemical cal-culations including density functional theories (DFT) and their time-dependent (TD-DFT) components, configuration interaction singles (CIS), complete active-space self-consis-tent field (CASSCF) and that with the second-order Møller–Plesset perturbation (CASPT2), equation-of-motion coupled-cluster singles and doubles (EOM-CCSD), and so on.[23–25]It was found that the np*1B

3ustate is unambiguous-ly the lowest one, followed by the pp* 1B

2u state. Among the other np* states,1A

ulies close to, whereas1B2g and1B1g locate somewhat higher than, the1B

2ustate. Compared with spectroscopic observations,[26–28] TD-DFT, CASPT2, and EOM-CCSD yielded generally small errors in excitation en-ergies, whereas CIS and CASSCF showed apparent overesti-mations for some of these states.

The np*1A

ustate, noted as the S3state in some literature reports, is of special interest because it was predicted by CASPT2 and TD-DFT to be lying between conventional S1 1B

3uand S2 1B

2ustates.

[25]To avoid confusion among state se-quences, we skip the Snnotation hereafter. As the1Austate might no longer be dark through vibronic coupling, it should contribute to the absorption band located between1B

3uand 1B

2u peaks, most likely the weak broad feature around 290 to 310 nm.[3, 6]In this work, we have constructed the mathe-matical framework of vibronic couplings between these low-lying singlet excited states, carried out quantum chemical calculations on geometric optimizations, vibrational frequen-cies, potential-energy surfaces (PES) scans as well as

vibron-ic coupling constants, and simulated the absorption spec-trum that covers the range of all three states. In addition to the harmonic oscillator approximation, we have adopted the double-well potential model to describe certain vibrational modes.[13, 29] The combination of computed adiabatic poten-tials and mathematical treatments on vibronic couplings could readily reveal the role of the1A

ustate in the photoab-sorption and relaxation processes.

Theories

In general, the one-photon absorption cross section of a single molecule can be expressed by Fermis golden rule [Eq. (1)]:[30–33]

in which a and b indicate the initial and final electronic states, {vj} andfv0jg, are the corresponding vibrational quan-tum numbers, ˆm is the dipole moment operator, and Pafvjg is the Boltzmann distribution function for the initial vibration-al states. The vvibration-alues wbfv0

jg;afvjgand w refer to the energy gap between the two states and the excited energy (in the form of angular frequency), respectively. The value d(Dw) repre-sents the Dirac delta function, which is in practice substitut-ed by the Lorentzian function or the Gaussian function under homogeneous or inhomogeneous dephasing.

According to the Born–Oppenheimer adiabatic approxi-mation, the molecular wavefunction can be approximately defined by the direct product of electronic and vibrational parts, that is, in which r denotes the electronic coordinates, and Q is the dimensionless normal-mode coordinate matrix that can be transformed from the Cartesian displacement coordinates x [Eq. (2)]:

Here w1=2 and M1=2 are diagonal matrices, the elements of

which are square roots of vibrational angular frequencies and atomic masses, respectively. The columns in L represent the eigenstates of the Hessian matrix.

The electronic wavefunction is dependent on Q parametrically, and it can be expanded around a reference geometry Q0based on the Herzberg–Teller expansion [e.g., Eq. (3)]:

in which F0and E0refer to the eigenfunction and the eigen-state energy, respectively, of the electronic Schrçdinger equation at the reference geometry Q0, and H’ denotes the

(3)

vibronic coupling matrix elements with the following defini-tion [Eq. (4)]:

In the above equation, V represents the electro-static potential and lca

i is the vibronic coupling constant between states a and c with respect to the normal coordinate Qi. The transition matrix element in Equation (1) can be rewritten as Equation (5):

in which is the coordinate-dependent

transition dipole moment vector. With expansion up to the second order [Eq. (6)]:

in which m0

bais the zeroth-order transition dipole moment at the reference geometry Q0, whereas the first and second de-rivatives are [Eqs. (7) and (8)]:

Supposing the Duschinsky rotation effect is insignificant and the second-order term is negligible, Equation (5) becomes Equation (9):

For clarity, Equation (1) at zero temperature is divided into three parts [Eq. (10)]:

in which [Eqs. (11) and (12)]:

and [Eq. (13)]:

The vibronic coupling constant lican be estimated based on the method proposed by Domcke et al.[13, 15] Briefly, the po-tentials of two interacting electronic states b and c are de-scribed by Equation (14):

in which DE is the vertical excitation energy. The lbc i value as well as the gb;c

i parameters can be fitted following a PES scan along the coupling coordinate Qi.

Now consider the transitions from 0)1A

g ground state to 1)1B

3u, 2)1B2u, and 3)1Au excited states of pyrazine in the D2h point group. The

1A u !

1A

g transition is dipole-forbid-den, which means that m0

30 is zero at the ground-state equi-librium and hence both sFC(w) and sFC/HT(w) are zero. How-ever, the first derivative mðiÞ30 could be nonzero by intensity borrowing from neighboring1B

3u ! 1Agand1B2u ! 1Ag transi-tions, in which m0

10 and m 0

20 are nonzero, through vibronic coupling of mode i, the l31

i or l 32

i values of which are non-zero. As a result, the most contribution comes from the di-agonal terms of sHT(w) [Eq. (15)]:

The 1B 3u !

1A

g transition is weakly dipole-allowed, and therefore sFC(w) is nonzero yet not large. The value of sHT(w) is also nonzero through vibron-ic coupling, but the cross-term sFC/HT(w) is zero due to the orthogonality of dipole moment components within this four-state framework. The1B

2u ! 1A

gtransition, on the other hand, is strongly dipole-allowed, and consequently sFC(w) dominates.

(4)

Computational Methods

To investigate the properties of relevant electronic states of pyrazine, we have carried out both DFT and ab initio calculations. In the DFT ap-proach, B3LYP and B971 functionals were applied for the geometric op-timization and vibrational modes of the1A

gground state, and their

time-dependent components (TD-DFT) were used to describe the excitation energies, transition dipole moments, and geometric properties of the low-lying singlet excited states. Results from the two functionals were found to be quite similar. In the ab initio category, CASSCF (or CAS for short) and CASPT2 in the state-average manner were applied. Several differ-ent-sized active spaces were tested, and it turned out that the (10,8) one—that is, ten electrons in six valence p orbitals plus two nonbonding orbitals—was adequate for computing structures and transition energies under the native D2hpoint group of pyrazine.[25]CIS with the

perturba-tive doubles [CIS(D)] was also applied to compare vertical excitation en-ergies. In addition to equilibrium geometries, conical intersection points were searched by CASSCF to deduce vibronic couplings between neigh-boring electronic states.

The Pople-type 6-311++GACHTUNGTRENNUNG(d,p) and the Dunning-type aug-cc-pVTZ basis sets were used in this work. The two basis sets turned out to make minor differences, and hence only results from the former are shown. DFT and TD-DFT calculations were carried out with Gaussian 09,[34]

whereas CASSCF and CASPT2 ones were performed with MOLPRO 2006.[35]

Results and Discussion

Equilibrium Geometries and Excitation Energies The optimized geometries of the ground state (1A

g), the lowest three singlet np* excited states (1B

3u, 1Au, and 1B2g), and the lowest singlet pp* (1B

2u) excited state of pyrazine computed by DFT/TD-DFT with the B3LYP functional, CASSCFACHTUNGTRENNUNG(10,8), and CASPT2ACHTUNGTRENNUNG(10,8) are listed in Table 1. To the best of our knowledge, this is the first report that con-cerns the equilibrium geometries of the optically dark 1A

u and 1B

2g states. Focusing on the 1Au state, the data shows that the CC bond length becomes noticeably longer and

the CNC angle is apparently larger than other states, whereas the CN bond length somewhat contracts. The overall effect is a significant expansion of the aromatic ring. According to the molecular orbital maps and excitation co-efficients calculated by TD-DFT as shown in Figure 1 and Table 2, the 1A

u state has a major configuration of the HOMO!LUMO+1 transition in which the nonbonding electron enters the CN-bonding/CC-antibonding orbital and results in the ring deformation.

The vertical (Franck–Condon) and the adiabatic (0–0 transition) energies from the ground state to these low-lying singlet states are listed in Table 3. At the Franck–Condon point, it is clear that1B

3uis the lowest and1B2gis the highest among the four states studied both experimentally and com-putationally. The results of TD-DFT and CASPT2 in this

Table 1. Optimized geometries of pyrazine in its ground state and the lowest four singlet excited states. The bond lengths (r) are in  and the angles (q) in degrees. All structures belong to the D2hpoint group.

State Method rCN rCC rCH qCNC qNCC qNCH ground1A g B3LYP 1.335 1.395 1.086 116.0 122.0 117.2 CASSCF 1.329 1.395 1.075 116.1 122.0 117.2 CASPT2 1.343 1.402 1.086 115.0 122.5 116.9 calcd[a] 1.346 1.402 1.085 115.6 122.3 117.5 exp[b] 1.338 1.397 1.083 115.7 122.2 117.9 np*1B 3u TD-B3LYP 1.343 1.396 1.084 120.0 120.0 120.5 CASSCF 1.353 1.383 1.072 119.6 120.2 119.5 CASPT2 1.348 1.409 1.085 119.5 120.3 120.7 calcd[c] 1.357 1.387 1.073 119.3 120.3 119.4 pp*1B 2u TD-B3LYP 1.367 1.428 1.086 110.2 124.9 116.6 CASSCF 1.368 1.432 1.073 112.2 123.9 116.8 CASPT2 1.376 1.439 1.086 110.4 124.8 116.6 calcd[d] 1.392 1.428 1.067 113.9 123.1 116.7 np*1A u TD-B3LYP 1.299 1.505 1.081 127.3 116.4 122.7 CASSCF 1.302 1.489 1.070 126.6 116.7 122.0 CASPT2 1.309 1.510 1.083 126.7 116.6 122.9 np*1B 2g TD-B3LYP 1.381 1.349 1.082 125.8 117.1 116.9 CASSCF 1.387 1.347 1.071 123.4 118.3 117.0 CASPT2 1.387 1.361 1.083 124.2 117.9 116.9

[a] MP2/DZP.[15][b] From electron diffraction and liquid-crystal NMR spectroscopy.[37][c] CASACHTUNGTRENNUNG(10,8)/DZP.[38][d] CASACHTUNGTRENNUNG(10,8)/3-21G.[39]

Figure 1. Selected molecular orbital maps of pyrazine calculated by DFT/ B3LYP. HOMO and HOMO2 show the nonbonding character, HOMO1 and HOMO3 are p bonding, and LUMO and LUMO+1 are p* antibonding.

(5)

work and EOM-CCSD(T) from a previous report[24] were generally consistent with experimental findings, whereas CASSCF and CIS(D) showed some overestimates. The rela-tive positions of 1B

2u and 1Au, however, are not definite. CASSCF and EOM-CCSD(T) indicated that the latter lo-cates higher than the former, whereas TD-DFT, CASPT2,

and CIS(D) showed a reversed sequence. Moreover, TD-DFT and CASPT2 predicted that 1A

ubecomes nearly isoe-nergetic with1B

3u after relaxation to their equilibrium geo-metries (differed by only around 0.05 eV). Supposing this prediction to be correct, 1A

umight play an important inter-mediate role in the1B

2u!1B3urelaxation process by vibronic coupling with the two states. With a similar coupling mecha-nism, this dipole-forbidden state upon optical excitation should be no longer dark and probably reveals its contribu-tion between1B

3uand1B2uabsorption bands. These points of view are explored and verified in the following PES scans and spectral simulations.

Vibrational Modes and PES Scan

The vibrational frequencies of all relevant states have been calculated by different methods under the harmonic approx-imation as shown in Table 4. The Huang–Rhys factors of to-tally symmetric modes (ag), defined as S

HR

i  ðDQiÞ 2=2 in which DQi is the dimensionless displacement coordinate of mode i, calculated on the basis of DFT and CASSCF data, are listed in Table 5. The n6amode, which is the totally sym-metric mode with the lowest frequency, raises particular in-terest since it has a relatively large Huang–Rhys factor upon excitation to each state, thus implying a major component of spectral progression. It is also proposed as the major tuning mode in the ultrafast 1B

2u!1B3u relaxation mechanism be-cause the crossing point between the two surfaces along the Q6a coordinate was estimated to be quite close to the

1B 2u equilibrium.[13, 15]For this reason, we carried out a PES scan Table 2. Major excitation configurations to the lowest four singlet excited

states at the ground-state equilibrium geometry calculated by TD-B3LYP.

State Major configuration Coefficient

np*1B 3u #21 HOMO!#22 LUMO 0.707 pp*1B 2u #20 HOMO1!#22 LUMO #18 HOMO3!#23 LUMO+1 0.664 0.239 np*1A u #21 HOMO!#23 LUMO+1 0.706 np*1B 2g #19 HOMO2!#22 LUMO 0.704

Table 3. Vertical (DEFC) and adiabatic (DEad) excitation energies [eV]

from the ground state to the low-lying singlet excited states.

State np*1B 3u pp*1B2u np*1Au np*1B2g Excitation energy DEFC DEad DEFC DEad DEFC DEad DEFC DEad TD-B3LYP 3.94 3.83 5.40 5.18 4.61 3.87 5.57 4.98 TD-B971 3.95 3.84 5.41 5.18 4.63 3.87 5.63 4.96 CASSCFACHTUNGTRENNUNG(10,8) 4.87 4.72 5.11 4.84 5.90 5.22 5.93 5.29 CASPT2ACHTUNGTRENNUNG(10,8) 3.78 3.62 4.71 4.34 4.30 3.56 5.33 4.73 CIS(D) 4.47 – 5.49 – 4.86 - 6.38 – EOM-CCSD(T)[a] 3.95 4.64 4.81 5.56 exptl[b] 3.94 3.83 4.81 4.69 5.50 5.46

[a] With the 6-31+GACHTUNGTRENNUNG(d,p) basis set.[24][b] From UV absorption and

near-threshold electron energy-loss spectroscopy.[6, 7, 25, 28]

Table 4. Vibrational frequencies [cm1] of all 24 modes of pyrazine. 1A

g np*1B3u pp*1B2u np*1Au np*1B2g

Sym Mode[a] Exp[b] DFT[c] CAS[d] CASPT2[d] Exp[e] TDDFT CAS TDDFT CAS TDDFT CAS TDDFT CAS

ag n6a 596 612 643 601 585 620 550 546 567 650 – 517 545 n1 1015 1038 1074 1025 970 1024 1061 969 988 942 838 975 987 n9a 1230 1253 1324 1261 1104 1195 1313 1237 1303 1195 1057 1208 1275 n8a 1582 1614 1731 1622 1377 1534 1771 1532 1674 1417 1347 1665 1733 n2 3055 3172 3351 3234 – 3196 3377 3175 3367 3225 3404 3214 3396 b1g n10a 919 940 953 886 383 468 618 1158 681 338 342 821 789 b2g n4 756 772 776 662 518 511 461 627 491 561 369 601 529 n5 983 980 990 882 552 810 717 1034 749 585 528 874 886 b3g n6b 704 720 758 710 662 688 701 659 702 659 679 659 699 n3 1346 1373 1467 1377 – 1296 1440 1242 1428 1280 1403 931 1217 n8b 1525 1578 1664 1564 – ACHTUNGTRENNUNG(588i) 1664 1394 1590 ACHTUNGTRENNUNG(640i) – 1353 1457 n7b 3040 3151 3328 3215 – 3160 3353 3159 3348 3187 3337 3185 3371 au n16a 341 347 403 297 400 432 193 210 291 300 272 316 294 n17a 960 989 995 832 743 824 772 911 807 787 646 870 866 b1u n12 1021 1036 1114 1031 – 623 1002 970 992 761 – 923 950 n18a 1136 1165 1225 1163 – 1012 1114 1096 1129 1141 1161 1371 1513 n19a 1484 1511 1626 1515 – 1389 1553 1463 1551 1437 1508 1537 2439 n13 3012 3152 3330 3216 – 3170 3355 3159 3349 3208 3375 3199 3424 b2u n15 1063 1089 1041 1088 – 1079 1104 984 999 1157 1272 1112 1161 n14 1149 1218 1150 1330 – 1267 1469 1280 1448 1459 1573 1249 1332 n19b 1416 1440 1531 1445 – 1356 2003 1395 1876 820 799 1518 1572 n20b 3063 3166 3346 3230 – 3193 3371 3172 3363 3214 3392 3207 3391 b3u n16b 420 433 456 398 236 220 173 ACHTUNGTRENNUNG(181i) 250 445 385 ACHTUNGTRENNUNG(74i) 170 n11 785 801 831 763 898 723 638 700 673 572 585 765 763

[a] Following assignments by McDonald and Rice[5]except that n

8aand n9aare interchanged according to Innes et al.[7]and Domcke et al.[13, 15][b] From

IR and Raman spectroscopy.[7][c] With the B3LYP functional. [d] With the (10,8) active space. [e] From UV-absorption and threshold-ionization

photo-electron spectroscopy.[5, 40]

(6)

along this coordinate around the ground-state equilibrium, similar to what was done by Domcke et al.,[13]yet we includ-ed more excitinclud-ed states in our TD-DFT and CASSCF/ CASPT2 calculations.

The TD-DFT results are demonstrated in Figure 2 togeth-er with a comparison to previous HF/MP2 calculations.[13] The results of 1A

g, 1B3u, and 1B2u states from both ap-proaches are rather consistent. However, the previous

report did not consider more than three states. As expected from our geometric optimizations for excited states, the1A

u state locates between 1B

3u and 1B2u at the Franck–Condon point (Q6a=0). It is surprising that the intersection point be-tween 1A

u and 1B2u surfaces along this coordinate is even closer to the1B

2ubottom. In addition to1Au, the1B2g poten-tial also intersects 1B

2uin the vicinity. It implies more than one relaxation channel after excitation to the1B

2ustate; all the 1B 2u! 1B 3u, 1A u, and 1B

2g processes should be competi-tive. This phenomenon was first predicted by Werner et al. in their dynamical simulation using TD-DFT,[17]but further consideration is still in demand.

The results from CASSCF and CASPT2 are illustrated in Figure 3. As mentioned above, CASSCF predicted higher energies for 1A

u and 1B2g states, and in consequence their crossing points with1B

2ulocate in an upper region. It could be a possible reason that Domcke and co-workers did not take these states into account. The CASPT2 data, on the

other hand, possess a feature similar to TD-DFT. It points out again that 1A

u and 1B2g states should be considered as well in the relaxation mechanism of the 1B

2u state, which will be an important topic of our future work.

It was found that some vibrational modes showed imagi-nary frequencies, thereby suggesting the failure of the har-monic approximation for these modes. Taking the n8b(b3g) mode of the1A

ustate as an example shown in Figure 4, the Table 5. Huang–Rhys factors of totally symmetric modes upon excitations

from the1A

gground state.

np*1B

3u pp*1B2u np*1Au np*1B2g

Mode TDDFT CAS TDDFT CAS TDDFT CAS TDDFT CAS

n6a 0.616 0.698 1.866 0.925 1.667 1.828 6.350 4.305

n1 0.030 0.149 0.753 1.338 0.409 0.397 0.416 0.851

n9a 0.339 0.494 0.033 0.050 0.208 0.139 0.639 1.043

n8a 0.050 0.008 0.009 0.008 3.412 2.632 0.418 0.661

n2 0.004 0.003 0.001 0.004 0.027 0.023 0.018 0.013

Figure 3. PES scan along the Q6a coordinate around the ground-state

equilibrium by a) CASSCF and b) CASPT2.

Figure 2. PES scan along the Q6a coordinate around the ground-state

equilibrium by TD-DFT with the B3LYP functional (solid lines) in this work and by HF/MP2 (dotted lines) from a previous report.[13]

Figure 4. PES scan along the n8bdouble-well mode of the1Austate by

TD-DFT (solid lines) and fitting to coupling states (dotted lines) accord-ing to Equation (14).

(7)

TD-DFT PES scan revealed the mode to have a very shal-low double-well potential in-stead of a harmonic one. It is the vibronic coupling between 1B

3uand1Austates through this b3g mode that distorts the shape of the two adiabatic po-tentials, thus making the lower one a double minimum.[29] Re-ferring to Equation (14), the l8b and g8b values were fitted as 1930 and100 cm1

, respectively. On the other hand, the double-well potential could be fitted in another way, say, the combinational of a quadratic potential and a Gaussian func-tion [Eq. (16)]:

VðQÞ ¼1

2w0Q2þ AeaQ

2

þ d ð16Þ

This is a purely mathematical approach that is helpful in evaluating vibrational levels and Franck–Condon factors of the double well.[36]The fitting parameters and corresponding levels for the n8bmode of the1Austate are listed in Table 6, which are utilized to simulate the absorption spectrum in the next section.

Absorption Spectra The spectral shapes of the 1B

3u and 1B

2u absorption bands are quite different. The 1B

3u band, beginning from around 325 nm and extending to around 300 nm, has a number of sharp peaks that are clearly assigned to n6a(ag), n9a (ag), n10a (b1g), and so on, and their combinations.[3, 4, 6, 7] The 1B2u band, by contrast, is rather broad and covers around 270 nm to around 230 nm with few fine structures.[3]Efforts on as-signing peaks in this band have been made[3, 7, 21]and indicate the major components to be n1 (ag) and n6a (ag), although most peaks are hidden in the broad feature, which is regard-ed an evidence of the ultrafast relaxation of the 1B

2u state.[11–16]It is noteworthy that, in addition to the two dis-tinct bands, there also exists a weaker broad character around 310 to 280 nm.[3, 6]This does not seem to be just the tail of the 1B

3u band, since the progressions in this band decay rather fast toward the higher-energy region.[6, 15, 20] Based on the knowledge of excited states calculated by

TD-DFT and CASPT2 in this work as well as the prediction on relaxation dynamics by Werner et al.,[17] we have taken the 1A

ustate into account and made the first theoretical simula-tion on such an absorpsimula-tion feature. This dark np* state may become visible through intensity borrowing from dipole-al-lowed transitions to neighboring states, thereby providing a suitable explanation for the weak broad band in the absorp-tion spectrum.

As listed in Table 7, the zero-order transition dipole moment, m0

ca, was calculated by using TD-DFT. The vibronic coupling constant, lbc

i , and the first derivative of transition moment, mðiÞba, between related states upon coupling coordi-nate Qiwere calculated according to Equations (14) and (7), respectively. The absorption spectrum that covers the range from 330 to 230 nm was then simulated at zero temperature as shown in Figure 5. To compare this simulation with the experimental one,[3] the measured peak positions were Table 6. Calculated potential parameters and vibrational levels of the n8b

double-well mode of the1A

ustate according to Equation (16).

Potential parameters w0[2p cm1] A [cm1] a d [cm1] Barrier [cm1] ZPE [cm1] 1605 8303 0.6847 16 406 622 472 Vibrational levels v’ 0 1 2 3 4 Ev[cm1] 0 1002 2102 3272 4494

Table 7. Vibronic coupling constants and derivatives of transition dipole moments between 0)1A g, 1)1B3u,

2)1B

2u, and 3)1Austates.

Transition b ! a Coupling state c m0 ca [au] E0 b E0c [cm1] Coupling mode i lbc i [cm 1] mðiÞ ba [au] 1 (1B 3u) ! 0 (1Ag) 2 (1B2u) 0.856 7660 n10a(b1g) 1290[a] 0.135 3 (1A u) ! 0 (1A g) 1 (1B3u) 0.242 5440 n6b(b3g) 710 0.032 n3(b3g) 840 0.037 n8b(b3g) 1930 0.086 n7b(b3g) 350 0.016 3 (1A u) ! 0 (1Ag) 2 (1B2u) 0.856 2245 n4(b2g) 790 0.301 n5(b2g) 165 0.063

[a] Literature values: 1352 (CASSCF), 1472 (MRCI).[15]

Figure 5. Absorption spectra of pyrazine covering 1B

3u, 1B2u, and 1Au

bands. a) Experimental UV spectroscopy.[3] b) Simulated spectrum with

peaks broadened by homogeneous dephasing. c), d), and e) Calculated peak intensities and assignments in these bands.

(8)

adopted if applicable, whereas the peak intensities were esti-mated by transition dipole moments, vibronic couplings, and Franck–Condon factors obtained from first-principle calcula-tions.

In the1B

3uabsorption band, the major progressions com-posed of n6a(ag) and n9a(ag) are clearly seen. Both decrease monotonically as predicted by their Franck–Condon factors, the values of which are smaller than one (0.616 and 0.339, respectively, by TD-DFT). To fit experimental bandwidths, the homogeneous dephasing was set as 30 cm1 for the 0–0 transition and raised gradually as the vibrational energy in-creased. The appearance of the odd quantum peak of n10a (b1g) represents the intensity-borrowing feature from the neighboring 1B

2u state through vibronic coupling. This fea-ture has been successfully reproduced by the simple Herz-berg–Teller expansion described by Equation (15).

The dipole moment of the 1B

2u ! 1Ag transition is about 3.5 times larger than that of the 1B

3u ! 1Ag transition (m020= 0.856 au and m0

10=0.242 au by TD-DFT), thus giving the total intensity of the1B

2uband to be one order higher than the1B

3uband. The major components of progressions in the 1B

2uband are n6a(ag) and n1 (ag). The bandwidth was set as 240 cm1for the 0–0 transition and increased up to 400 cm1 for overtones and combination bands such as 13

0and 1 2 06a

2 0to reproduce the whole broad feature. In principle, 1B

2u can also borrow intensity from1B

3uby vibronic coupling through n10a, but the strength turned out to be too small relative to the direct transition to1B

2u, thereby resulting in a negligible contribution overwhelmed in the broad band.

It is noticeable that some weak hot bands took place as the experimental spectra were not recorded under low-tem-perature conditions.[3, 4, 6] In the 1B

3u band, there is a small absorption peak to the red of the 0–0 transition (Figure 5 a) which has been assigned as 16b1

1.

[4]In the1B

2uband, howev-er, possible hot-band peaks should be embedded in the broad feature and thus indistinguishable. The overall contri-bution of hot bands was not significant and the temperature effect was therefore not considered further in the present si-mulated spectrum (Figure 5 b).

The 1A

u band is dipole-forbidden and thus must borrow intensity from either1B

3uor 1B

2uto be visible. The vibronic coupling between 1A

u and 1B2u occurs through b2g modes, among which n4 is the major contributor. The coupling be-tween1A

uand1B3u, on the other hand, requires b3gmodes in which n8bplays the leading role (Table 7). Due to the large value of m0

20, the contribution from 1A

u -1B

2u coupling domi-nates this band. The progressions in this band are mainly composed of n6a(ag) and n8a (ag). The n8a series is relatively important and extends to the1B

2uregion because of a large Huang–Rhys factor (3.412 by TD-DFT and 2.632 by CASSCF). As a result, the1A

uband presents a low-intensity but wide-range character, adequately filling the spectral window between the1B

3uand 1B

2ubands. It shows, however, that our calculated 1A

u band overestimates around 270 to 280 nm yet underestimates around 290 to 300 nm in critical comparison to the experimental spectrum. This may be at-tributed to possible inaccuracy of present computational

re-sults and should be examined with higher-level calculations in future works.

Conclusion

In the present studies, the equilibrium geometries, vibration-al frequencies, excitation energies, and vibronic couplings of the ground state (1A

g) and the lowest four singlet excited states (1B

3u,1B2u,1Au, and1B2g) of pyrazine have been calcu-lated by DFT/TD-DFT, CASSCF, and CASPT2 methods. The np* 1A

u state was of particular concern because of its predicted location between np*1B

3uand pp*1B2ustates. Al-though the 1A

u ! 1A

g transition is dipole-forbidden, it can borrow the intensity from the1B

2u ! 1Agand1B3u ! 1Ag tran-sitions through vibronic coupling. The 1A

uabsorption band therefore becomes slightly visible and covers the range be-tween 1B

3uand 1B2ubands, in agreement with experimental spectra. Furthermore, the existence of1A

uand 1B

2gstates in the vicinity of the1B

2ustate and the locations of their coni-cal intersections searched by PES scan reveal that these two optically dark states, in addition to the known bright 1B

3u state, should be important in the ultrafast relaxation mecha-nism of the 1B

2u state. The dynamic properties of these states will be the focus in our future work.

[1] F. Halverson, R. C. Hirt, J. Chem. Phys. 1951, 19, 711.

[2] M. Ito, R. Shimada, T. Kuraishi, W. Mizushima, J. Chem. Phys. 1957, 26, 1508.

[3] I. Suzuka, Y. Udagawa, M. Ito, Chem. Phys. Lett. 1979, 64, 333. [4] Y. Udagawa, M. Ito, I. Suzuka, Chem. Phys. 1980, 46, 237. [5] D. B. McDonald, S. A. Rice, J. Chem. Phys. 1981, 74, 4893. [6] I. Yamazaki, T. Murao, T. Yamanaka, K. Yoshihara, Faraday

Dis-cuss. Chem. Soc. 1983, 75, 395.

[7] K. K. Innes, I. G. Ross, W. R. Moomaw, J. Mol. Spectrosc. 1988, 132, 492.

[8] V. Stert, P. Famanara, W. Radloff, J. Chem. Phys. 2000, 112, 4460. [9] T. Horio, T. Fuji, Y.-I. Suzuki, T. Suzuki, J. Am. Chem. Soc. 2009,

131, 10392.

[10] Y.-I. Suzuki, T. Fuji, T. Horio, T. Suzuki, J. Chem. Phys. 2010, 132, 174302.

[11] R. Schneider, W. Domcke, Chem. Phys. Lett. 1988, 150, 235. [12] M. Seel, W. Domcke, J. Chem. Phys. 1991, 95, 7806.

[13] L. Seidner, G. Stock, A. L. Sobolewski, W. Domcke, J. Chem. Phys. 1992, 96, 5298.

[14] G. Stock, W. Domcke, J. Phys. Chem. 1993, 97, 12466.

[15] C. Woywod, W. Domcke, A. L. Sobolewski, H.-J. Werner, J. Chem. Phys. 1994, 100, 1400.

[16] G. Stock, C. Woywod, W. Domcke, T. Swinney, B. S. Hudson, J. Chem. Phys. 1995, 103, 6851.

[17] U. Werner, R. Mitric´, T. Suzuki, V. Bonacˇic´-Koutecky´, Chem. Phys. 2008, 349, 319.

[18] U. Werner, R. Mitric´, V. Bonacˇic´-Koutecky´, J. Chem. Phys. 2010, 132, 174301.

[19] R. Islampour, M. Miralinaghi, J. Phys. Chem. A 2009, 113, 2340. [20] R. X. He, C. Y. Zhu, C.-H. Chin, S. H. Lin, Sci. China Ser. B 2008,

51, 1166.

[21] R. X. He, C. Y. Zhu, C.-H. Chin, S. H. Lin, Chem. Phys. Lett. 2009, 476, 19.

[22] W. R. Wadt, W. A. Goddard, III, J. Am. Chem. Soc. 1975, 97, 2034. [23] M. P. Flscher, K. Andersson, B. O. Roos, J. Phys. Chem. 1992, 96,

(9)

[24] J. E. Del Bene, J. D. Watts, R. J. Bartlett, J. Chem. Phys. 1997, 106, 6051.

[25] P. Weber, J. R. Reimers, J. Phys. Chem. A 1999, 103, 9821.

[26] A. Bolovinos, P. Tsekeris, J. Philis, E. Pantos, G. Andritsopoulos, J. Mol. Spectrosc. 1984, 103, 240.

[27] Y. Okuzawa, M. Fujii, M. Ito, Chem. Phys. Lett. 1990, 171, 341. [28] I. C. Walker, M. H. Palmer, Chem. Phys. 1991, 153, 169. [29] P. Weber, J. R. Reimers, J. Phys. Chem. A 1999, 103, 9830.

[30] S. H. Lin, Y. Fujimura, H. J. Neusser, E. W. Schlag, Multiphoton Spectroscopy of Molecules, Academic Press, New York, 1984, chap-ter 2.

[31] A. M. Mebel, M. Hayashi, K. K. Liang, S. H. Lin, J. Phys. Chem. A 1999, 103, 10674.

[32] K. K. Liang, R. Chang, M. Hayashi, S. H. Lin, Principle of Molecu-lar Spectroscopy and Photochemistry, National Chung Hsing Univer-sity Press, Taichung, 2001, chapters 2 and 5.

[33] C.-K. Lin, M.-C. Li, M. Yamaki, M. Hayashi, S. H. Lin, Phys. Chem. Chem. Phys. 2010, 12, 11432.

[34] Gaussian 09, Revision A.02, M. J. Frisch, G. W. Trucks, H. B. Schle-gel, G. E. Scuseria, M. A. Robb, J. R. Cheeseman, G. Scalmani, V. Barone, B. Mennucci, G. A. Petersson, H. Nakatsuji, M. Caricato, X. Li, H. P. Hratchian, A. F. Izmaylov, J. Bloino, G. Zheng, J. L. Son-nenberg, M. Hada, M. Ehara, K. Toyota, R. Fukuda, J. Hasegawa, M. Ishida, T. Nakajima, Y. Honda, O. Kitao, H. Nakai, T. Vreven, J. A. Montgomery, Jr., J. E. Peralta, F. Ogliaro, M. Bearpark, J. J. Heyd, E. Brothers, K. N. Kudin, V. N. Staroverov, R. Kobayashi, J.

Normand, K. Raghavachari, A. Rendell, J. C. Burant, S. S. Iyengar, J. Tomasi, M. Cossi, N. Rega, J. M. Millam, M. Klene, J. E. Knox, J. B. Cross, V. Bakken, C. Adamo, J. Jaramillo, R. Gomperts, R. E. Stratmann, O. Yazyev, A. J. Austin, R. Cammi, C. Pomelli, J. W. Ochterski, R. L. Martin, K. Morokuma, V. G. Zakrzewski, G. A. Voth, P. Salvador, J. J. Dannenberg, S. Dapprich, A. D. Daniels, . Farkas, J. B. Foresman, J. V. Ortiz, J. Cioslowski, D. J. Fox, Gaussian, Inc., Wallingford CT, 2009.

[35] MOLPRO is a package of ab initio programs written by H.-J. Werner, P. J. Knowles, R. Lindh, F. R. Manby, P. Celani, T. Korona, G. Rauhut, R. D. Amos, A. Bernhardsson, A. Berning, D. L. Cooper, A. J. Dobbyn, F. Eckert, C. Hampel, G. Hetzer, A. W. Lloyd, S. J. McNicholas, W. Meyer, M. E. Mura, A. Nicklaß, P. Pal-mieri, R. Pitzer, U. Schumann, H. Stoll, R. Tarroni, T. Thorsteinsson, 2006.

[36] C.-K. Lin, H.-C. Chang, S. H. Lin, J. Phys. Chem. A 2007, 111, 9347. [37] S. Cradock, P. B. Liescheski, D. W. H. Rankin, H. E. Robertson, J.

Am. Chem. Soc. 1988, 110, 2758.

[38] R. Berger, C. Fischer, M. Klessinger, J. Phys. Chem. A 1998, 102, 7157.

[39] A. L. Sobolewski, C. Woywod, W. Domcke, J. Chem. Phys. 1993, 98, 5627.

[40] L. Zhu, P. Johnson, J. Chem. Phys. 1993, 99, 2322.

Received: May 20, 2011 Published online: September 16, 2011

數據

Figure 1. Selected molecular orbital maps of pyrazine calculated by DFT/ B3LYP. HOMO and HOMO2 show the nonbonding character, HOMO1 and HOMO3 are p bonding, and LUMO and LUMO+1 are p* antibonding.
Table 3. Vertical (DE FC ) and adiabatic (DE ad ) excitation energies [eV]
Figure 3. PES scan along the Q 6a coordinate around the ground-state
Table 7. Vibronic coupling constants and derivatives of transition dipole moments between 0) 1 A g , 1) 1 B 3u ,

參考文獻

相關文件

Al atoms are larger than N atoms because as you trace the path between N and Al on the periodic table, you move down a column (atomic size increases) and then to the left across

substance) is matter that has distinct properties and a composition that does not vary from sample

Wang, Solving pseudomonotone variational inequalities and pseudocon- vex optimization problems using the projection neural network, IEEE Transactions on Neural Networks 17

Then, we tested the influence of θ for the rate of convergence of Algorithm 4.1, by using this algorithm with α = 15 and four different θ to solve a test ex- ample generated as

Particularly, combining the numerical results of the two papers, we may obtain such a conclusion that the merit function method based on ϕ p has a better a global convergence and

Then, it is easy to see that there are 9 problems for which the iterative numbers of the algorithm using ψ α,θ,p in the case of θ = 1 and p = 3 are less than the one of the

volume suppressed mass: (TeV) 2 /M P ∼ 10 −4 eV → mm range can be experimentally tested for any number of extra dimensions - Light U(1) gauge bosons: no derivative couplings. =>

Define instead the imaginary.. potential, magnetic field, lattice…) Dirac-BdG Hamiltonian:. with small, and matrix