Anharmonic Franck-Condon simulation of the absorption and fluorescence spectra for the low-lying S-1 and S-2 excited states of pyrimidine

(1)

Anharmonic Franck–Condon simulation of the absorption and ﬂuorescence

spectra for the low-lying S

1

and S

2

excited states of pyrimidine

Ling Yang

a,b,c

_{, Chaoyuan Zhu}

a,⇑

_{, Jianguo Yu}

c,⇑

_{, Sheng Hsien Lin}

a a

Department of Applied Chemistry, Institute of Molecular Science and Center for Interdisciplinary Molecular Science, National Chiao-Tung University, Hsinchu 30050, Taiwan

b

Institute of Theoretical and Simulation Chemistry, Academy of Fundamental and Interdisciplinary Science, Harbin Institute of Technology, Harbin 150080, PR China

c

Department of Chemistry, Beijing Normal University, Beijing 100875, PR China

a r t i c l e

i n f o

Article history: Received 9 August 2011 In ﬁnal form 14 March 2012 Available online 3 April 2012 Keywords:

Harmonic and anharmonic simulation Franck–Condon factors

Reorganization energy

Absorption and ﬂuorescence spectra Vibronic spectra

a b s t r a c t

Intensities and profiles of vibronic spectra of the low-lying singlet excited states were investigated with anharmonic and harmonic Franck–Condon simulations for pyrimidine. The first-order anharmonic cor-rection shows dynamic shift of spectra that is exactly same as difference of reorganization energy between ground and excited states. The first-order correction show intensity enhancement of absorption and intensity weakening of fluorescence for S1state, and dynamic shift is also significant. On the other hand, the first-order correction is negligible for S2state. The main spectral progressions are well described by totally symmetry modesm6a,m1andm12. One mode from non-total symmetrym16a contrib-utes to the weak band at 16a2_{transition for S}

1state. Four ab initio methods were employed in simulation; CASSCF, CASPT2, DFT and TD-DFT, and coupled-cluster singles-doubles (CCSD) and the equation-of-motion (EOM-CCSD) methods. They all work well, but CASSCF method show the best agreement with experiment for the weak-band intensities.

1. Introduction

Ab initio quantum chemistry methods provide a powerful tool to simulate molecular spectroscopy and dynamics. Two major steps are generally made for simulation. The ﬁrst part is statics in which molecular structures, vibrational frequencies and transi-tion energies are calculated within Born–Oppenheimer approxima-tion for electronically ground and excited states. The second part is dynamics in which wavefucntion overlap between ground and ex-cited states is computed within Franck–Condon (FC) approxima-tion[1–3]. These theoretical simulations can be very helpful for interpreting experimental observations such as electronic spectra like VUV absorption, ﬂuorescence, and the other nonradiative pro-cesses like electron and energy transfer. Exact simulation for mul-tidimensional FC overlap integrals is not practical for many-atom systems. Harmonic approximation with normal mode analysis is commonly utilized for simulation and anharmonic correction may be included for improvement. Further approximations are usually introduced for practical simulation like displaced oscillator approximation, distorted oscillator approximation, and normal

mode-mixing with including Duschinsky effect[4]. Various analyt-ical and numeranalyt-ical methods have been developed to compute FC overlap integrals with various applications[5–35].

Pyrimidine (1,3-diazine), C4H4N2, belongs to the group of

dia-zine ring molecules, whose skeletons serve as building blocks in nature, that may be used as a chemical or molecular model for the single nucleosides (thymine, cytosine and uracil) in nucleic acids [36–39]. The electronic spectroscopy of the azabenzenes (pyridine, pyrazine and pyrimidine) has been very interesting sub-ject due to its rich excited-state dynamical and photochemical properties[36,40,41], as well as its importance for biologically rel-evant spectroscopic processes[42,43]. A variety of computational simulations have been carried out for the S1and S2absorption and

the ﬂuorescence spectra of pyridine[41,44,45]and pyrazine[46,47]. Pyrimidine molecule has been studied by both experimental measurements and theoretical simulations for its low-lying excited states with applications of photophysics and photochemistry. VUV photoabsorption spectrum has been experimentally studied [36,38,39] and ab initio calculation within the level of multi-reference conﬁguration interaction method has been reported as well [48]. More recently, high resolution VUV photoabsorption spectrum has been obtained by Ferreira da Silva[49]. These studies suggested that the low-lying excited states (S1and S2) have C2v

group symmetry with transition types of 1_B

1 (n ?

p

⁄) and 1B2

(

p

?

p

⁄_{), respectively. In addition, Knight et al.}_[50] _{carried out}

extensive analysis of the emission spectrum by exciting a number

⇑Corresponding authors.

E-mail addresses: cyzhu@mail.nctu.edu.tw (C. Zhu), jianguo_yu@bnu.edu.cn (J. Yu).

Contents lists available atSciVerse ScienceDirect

Chemical Physics

(2)

of selectively vibronic levels in the lowest excited (n

p

⁄_{) singlet}

state. Computational studies have been carried out for equilibrium geometries, vibrational frequencies, and ground and low-lying excited state spectra of pyrimidine[51–59]. Malmqvist et al.[57] performed ab initio quantum chemistry studies for number of ver-tically excited singlet states of pyrimidine. Billes et al.[58] mea-sured and calculated vibrational frequencies at the Møller– Plesset perturbation and density functional theory levels. Later on, symmetry adapted cluster-configuration interaction (SAC-CI) [51] and equation of motion coupled-cluster (EOM-CC) [52–56] methods were adopted to produce fairly high accuracy for excita-tion energies. Fischer et al.[56]presented not only new experi-mental data for the lowest excited singlet and triplet states but also high-level calculations ((EOM-CCSD, CASPT2, and CIS) and density functional (B3LYP and TD-B3LYP)) for the first eight singlet and triplet valence excited states of pyrimidine. Åsbrink et al.[60] reported the photoelectron spectra of pyrimidine and other aza-benzenes up to 25 eV excitation energy, where the lowest ioniza-tion energy was attributed to the removal of an electron from the nitrogen lone pairs. Calculations based on response theory were utilized for calculating two-photon absorption spectra of the lowest electronic states by Luo et al.[59]. A main propose in the present study is to simulate intensities and profiles of vibronic manifold spectra of the low-lying singlet excited states, and to treat absorption and fluorescence in an equal footing by monic Franck–Condon overlapping integrals. The present anhar-monic correction can show how intensities and profiles of spectra changes simultaneously for both absorption and fluorescence.

In the present study, we ﬁrst calculate the equilibrium geome-tries, vibrational frequencies, vertical and adiabatic excitation energies of the ground state S0(1A1) and the low-lying S1(1B1)

and S2(1B2) excited states using CASSCF, CASPT2, CCSD/EOM-CCSD,

and DFT/TD-DFT methods. It should be noted that the variational principle in ab initio quantum chemistry methods can insure bet-ter accuracy for excitation energies with higher level method but it does not guarantee better accuracy for equilibrium geometries. Intensities and proﬁles of vibronic spectra in terms of the Franck–Condon overlapping integrals are most sensitive to geom-etry difference between excited and ground states. Therefore, the CASSCF method can provide accurate vibronic spectra since it treats electronic ground and excited states in an equal footing and it is especially good for geometry optimization of aromatic molecules in which resonance structures are essential. Then, we simulate the absorption and ﬂuorescence spectra for S1(1B1) state

and the absorption spectrum for S2(1B2) state by employing the

displaced harmonic oscillator approximation including the ﬁrst-or-der anharmonic effect. Furthermore, we analyze the distorted ef-fect that takes into account contribution from non-totally symmetry normal modes and it can be considered as diagonal-part correction of Duschinsky mode-mixing matrix. With analytical

formulation of absorption and fluorescence coefficients[35], we can explicitly demonstrate how anharmonic effect influences the shifts of spectral peaks, relative intensities and profiles of spectra with respect to harmonic Franck–Condon simulation.

In Section2, we briefly introduce ab initio methods for calcula-tion of electronic structures and normal-mode frequencies of the ground and two low-lying singlet excited states of pyrimidine. The displaced harmonic approximation with including the first-order anharmonic correction is also discussed for Franck–Condon factors. In Section3, the simulated results of electronic structures, absorption and fluorescence spectra are reported along with com-parison to experimental observations as well as the some other theoretical simulations. Concluding remarks are given in Section4.

2. Ab initio methods and anharmonic Franck–Condon factor 2.1. Ab initio methods

Gaussian03[61]and Molcas 7.5[62] program packages were employed for calculating the electronic structures of the ground state (S0(1A1)) and the ﬁrst two low-lying singlet excited states

(S1(1B1) and S2(1B2)). Numbering of atoms is given inFig. 1. The

equilibrium geometries for these three electronic states were cal-culated at CASSCF(6,6) level with consideration that pyrimidine consists of 6 active electrons and 6 active orbitals including 3 dou-bly occupied orbitals and 3 virtual orbitals. We have done the other combinations of active electrons and orbitals to test structure cal-culations and the results verify consistency of the present choice. As the dynamic correlation effects are not included in CASSCF method, we have done CASPT2 and MP2 corrections for both exci-tation energies and vibrational normal mode frequencies at CASSCF optimized geometries. On the other hand, we have employed the CASPT2, B3LYP/TD-B3LYP, and CCSD/EOM-CCSD methods to calcu-late the equilibrium geometries of three electronic states, and cor-responding vertical excitation energies as well as normal model frequencies. Basis sets of 6-311++g⁄⁄

and aug-cc-pVDZ[63]were adopted in cooperation with the methods mentioned above. The TD-B3LYP and EOM-CCSD methods [64] have been performed using G09[65]program. All 24 normal-mode frequencies for three electronic states were computed to conﬁrm the optimized geome-tries as true minima corresponding to their potential energy sur-faces. The anharmonic parameters were computed for the ground state with using MP2 and B3LYP methods in both G03 and G09 programs, respectively.

2.2. Anharmonic Franck–Condon factor

We start with perturbation expansion of the jth vibrational nor-mal-mode potential energy as[35]

VjðQÞ ¼ aj2Q2j þ kaj3Q3j þ k 2_a

j4Q4j þ ð1Þ

in which k is chosen as a perturbation parameter and Qjis

mass-weighted normal-mode coordinate. The ﬁrst-order correction in perturbation is zero for energy, but is nonzero for wave function with which absorption coefﬁcient is analytically derived as[35],

a

ð

x

Þ ¼2

px

3h j~

lba

j 2Z 1 1 dteitðxbaþX0xÞcbajtj exp P j Sjð1 þ 3

g

jÞf2

tj

þ 1 ð

tj

þ 1Þe itxj

tj

_eitxj_g " # ð2Þ

for excitation from electronic ground state a to excited state b that means

x

ba> 0 in Eq.(2)for adiabatic energy gap between b and a.

Fluorescence coefﬁcient is analytically derived as well[35]

(3)

Ið

x

Þ ¼2

px

3h j~

lba

j 2Z 1 1 dteitðjxbajþX0xÞcbajtj exp P j Sjð1 3

g

jÞf2

tj

þ 1 ð

tj

þ 1Þeitxj

tj

eitxjg " # ð3Þ

for excitation from electronic excited state a to ground state b that means

x

ba< 0 in Eq.(3)for adiabatic energy gap between b and a.

Other quantities are same for both Eqs. (2) and (3), where

m

j¼ ðehxj=kBT 1Þ1 is the average phonon distribution,

c

ba

represents the dephasing constant (with relation to the lifetime

s

ba= 1/

c

ba) between two electronic states, and ~

l

bais the electronic

transition dipole moment. The most important quantitiesX0and

g

j

stand for the ﬁrst-order anharmonic correction given by

X

0¼ 2P j

g

jSjxj ð4Þ and

gj

¼aj3dj aj2 ¼ aj3dj 0:5

x

2 j ð5Þ

where

x

jis harmonic vabrational normal-mode frequency, and the

Huang–Rhys factor Sj, the displacement dj, the second coefﬁcient aj2

and the third coefﬁcient aj3of potential energy in Eq. (1) are deﬁned

as Sj¼ 1 2h

xj

d 2 j ð6Þ dj¼ Q0j Qj¼P n Ljnðq0n qnÞ ð7Þ aj2¼ 1 2 @2V @Qj ¼1 2

x

2 j ð8Þ and aj3¼ 1 3! @3V @Q3j ¼1 3Kj3 ð9Þ

Inserting Eq. (9) into Eq. (5) leads to

g

j¼ 2Kj3dj 3

x

2 j ¼ Kj3d 3 j 3Sjh

xj

ð10Þ The q0

nand qnin Eq. (7) are the mass-weighted Cartesian

coor-dinates at the equilibrium geometries of the electronic excited and ground states, respectively. Transformation matrix L in Eq. (7) can be computed with frequency calculation in G03 and G09 programs. If the dimensionless ﬁrst-order anharmonic parameter

g

jis equal to zero, the absorption and ﬂuorescence coefﬁcients in

Eqs. (2) and (3) are exactly same as those in displaced harmonic oscillator approximation. An anharmonic parameter

g

jin Eq. (10)

is expressed in terms of the diagonal element Kj3of cubic force

con-stant estimated from G03 and G09 programs. The displacement dj

in Eq. (7) that is sensitive to geometry differences between two electronic states is essentially parameter to determine intensities and proﬁle of vibronic spectra.

The effective Huang–Rhys factors in Franck–Condon factors are no longer the same from the ﬁrst-order anharmonic correction; S0

j¼ ð1 3

g

jÞSj (+ for absorption in Eq.(2)and – for ﬂuorescence

emission in Eq.(3)). Immediate consequence is that mirror image between absorption and ﬂuorescence spectra is broken down accompanying with intensity enhancement of absorption against weakening of ﬂuorescence (or vice versa). At the same time, the harmonic 0–0 excitation energy is shifted byX0and this can be

interpreted as a dynamic correction to spectral position. However,

the ﬁrst-order correction inﬂuences little change of band shape in vibronic spectra and for detailed band-shape change the second-order correction is necessary along with the effect like Duschinsky mode-mixing.

The ﬁrst-order anharmonic correction can affect the reorganiza-tion energy of ground state ðk0¼ E00 E0Þ and excited stateðkex¼

E0

ex EexÞ as shown in Fig. 2. In comparison with ground-state

potential energy deﬁned in Eq.(1), excited-state potential energy is deﬁned as the left-handed shift with respect to ground state

VexjðQ0jÞ ¼ aj2ðQjþ djÞ2þ kaj3ðQjþ djÞ3þ k2aj4ðQjþ djÞ4þ

ð11Þ

for derivation of absorption and fluorescence coefficient in Eqs. (2) and (3). By adding each-mode contribution to reorganization energy together, we derive expression of reorganization energy within the first-order anharmonic correction as

k0¼P j Sjh

xj

ð1

g

jÞ ð12Þ and kex¼P j Sjh

xj

ð1 þ

g

jÞ ð13Þ

from which we can immediately ﬁnd that the reorganization energy differences between the excited state and the ground state is ex-actly equal to the dynamic shiftX0¼ k0 kex in Eq.(4), and this

is zero within harmonic case. 3. Results and discussion

3.1. Equilibrium geometries of S0, S1, and S2states

We optimized the equilibrium geometries of three electronic states (the ground state (S0(1A1)) and the ﬁrst two low-lying singlet

excited states (S1(1B1) and S2(1B2)) using CAS(6,6)/6-311++g⁄⁄,

CASPT2/aug-cc-pVDZ, CCSD/EOM-CCSD/cc-pVDZ and B3LYP/TD-B3LYP/6-311++g⁄⁄ _{methods. The measurements from gas-phase}

electron diffraction, rotational spectroscopy, and liquid–crystal NMR[66]predicted that the ground state of pyrimidine molecule has C2vgroup symmetry with a planar geometry. The present

cal-culations with B3LYP/TD-B3LYP, CCSD /EOM-CCSD, CASPT2, and CASSCF agree well with the experiment observation and the other

Fig. 2. The ground and excited states reorganization energy from the potential energy surfaces, k0¼ E00 E0and kex¼ E0ex Eex.

(4)

theoretical simulations[54,56,66,67], except that the CASSCF re-sults in which the C1–N9 bond (1.328 ÅA

0

) is the same to the exper-imental results [66] but ca. 0.01 ÅA

0

shorter than the other calculation results[54–56], while N9–C2 bond (1.329 ÅA0) is close to Fischer et al.’s CASSCF result (1.332 ÅA

0

)[56], however, both the present and the literature calculations are about 0.01 ÅA

0

shorter than the experimental data[36,67]. Optimized equilibrium geom-etry parameters (bond distances and bond angles) of the three electronic states S0(1A1), S1(1B1), and S2(1B2) are shown inTable 1.

By studying the REMPI and MRTI spectroscopy plus theoretical simulation, Riese and Grotemeyer[68]suggested that the equilib-rium geometry of the ﬁrst excited state S1belongs to the C2vgroup

symmetry with a planar geometry, which is conﬁrmed by Fischer et al.’s experimental and calculation results[56]. The present cal-culation did give the same results as the previous theoretical calcu-lations for bond distances and bond angles within C2vsymmetry

[54–56,68] at both CASSCF and EOM-CCSD levels, respectively. The present CASSCF calculations indicated that the ﬁrst excited state is a1_B

1symmetry with an excitation from molecular orbital

B2to orbital A2. This corresponds to an n ?

p

⁄transition and its

molecular orbitals are shown inFig. 3a as the pure excitation from HOMO ? LUMO. This transition is interpreted as electronic transition from the lone-pair orbitals at the N atoms to the

anti-p

orbitals. As a consequence of the electronic transition, the ring –N– angle (C–N–C) rises by 8°, the angles N–C–N and N–C–C decrease by 10° and 4° respectively, the C2–N9 bond distance is elongated to 1.396 Å, and the C1–N9 bond decreases to 1.300 Å in the S1(1B1) state in comparison with ground state S0(1A1).

The second excited state S2(1B2) is also shown to be C2vplanar

geometry. The present CASSCF and EOM-CCSD calculations proved that the second excited state is1_B

2symmetry with an excitation

from molecular orbital B1to orbital A2, which is in agreement with Table 1

The equilibrium geometries of ground state (S0(1A1)), the ﬁrst excited state (S1(1B1)) and the second excited state (S2(1B2)) of pyrimidine optimized by the present and other

(5)

Fischer’s results[56]. This corresponds to

p

?

p

⁄

transition and its molecular orbitals are shown inFig. 3b as the mixing excitation of the HOMO ? LUMO + 1 and the HOMO 1 ? LUMO. This is in consistent with the literature[36,38,39]. In comparison to S0(1A1)

geometry, the transition from

p

orbital to the anti-

p

orbital causes that all the bond distances in the ring and the C–H bonds elongate

by 0.04 Å and reduce by 0.002 Å, respectively, while the bond an-gles N–C–N and C–C–C increase by 1°, and the angle C–N–C and N–C–C decreases by 1° at CASSCF level. In summary, all methods above are almost equally good for geometry optimizations and more critical comparison with experiment should be shown in spectral simulation.

3.2. Harmonic frequencies and anharmonic parameters

24-normal-mode harmonic frequencies for three electronic states S0(1A1), S1(1B1), and S2(1B2) are calculated using CAS(6,6)/

6-311++g⁄⁄_, _{CASPT2/aug-cc-pVDZ,} _{CCSD/EOM-CCSD/cc-pVDZ,}

B3LYP/TD-B3LYP/6-311++g⁄⁄_{and methods, and only results from}

CAS and MP2 (for ground state) methods are shown inTable 2. In comparison with the Lord’s[69]experimental results measured by the infra-red and Raman spectroscopy, both MP2 and CASSCF methods show good agreement with experiment (the MP2 method performs slightly better than the CASSCF method).

As analyzed in Section2.1, the molecule structure changes due to the electronic transition from ground state to excited state, but what about change of corresponding frequencies. For nine total symmetry vibrational modes, the frequency differences between the excited states (S1(1B1) or S2(1B2)) and the ground state (S0(1A1)) are

consid-erably small (see inTable 2). Therefore, the displaced oscillator approximation can be considered as a good approximation for sim-ulating electronic spectroscopy. The Huang–Rhys factor in Eq. (6) and the displacement (dj) in Eq. (7) were estimated for the ﬁrst

and second excited states respectively as shown inTable 2. Five modes

m

6a,

m

1,

m

12,

m

13, and

m

2are shown signiﬁcant ﬁgures for the

ﬁrst excited state and four modes

m

6a,

m

1,

m

12, and

m

2for the second

excited state. This agrees well with explanations that the great changes in the inner angles of the ring account for the highest inten-sity of the bending mode 6a (seeFig. 4) in the vibronic spectrum, and the variations of bond lengths in the ring are responsible for valence bending modes 1 and 12 (seeFig. 4) in the vibronic spectrum[55]. Both CASSCF and MP2 methods produce satisfactory predictions for the most contributing modes in the present spectrum simulation.

Table 2

Experimental (exp) and calculated (MP2, CAS/s0) vibrational frequencies (cm1_{) for ground state, and the calculated frequencies differences between S}

1(1B1) and S0(1A1) states

(s1–s0), and between S2(1B2) and S0(1A1) states (s2–s0). Calculated Huang–Rhys factors (S(s1) and S(s2)), displacements (d(s1) and d(s2), in atomic units) and anharmonic

parameters (an(s1) and an(s2)) for S1(1B1) and S2(1B2).

Vibration normal modes[36,69] S0(1A1) S1(1B1) S2(1B2)

Sym N0 Mode Exp.[36] CAS(MP2) s1–s0 S(s1) d(s1) an(s1) s2–s0 S(s2) d(s2) an(s2)

A1 6a Cp 677 734(687) 38 1.566 0.265 0.004 72 0.044 0.044 0.001 1 CC 991 1068(1010) 80 0.299 0.089 0.035 85 1.402 0.193 0.077 12 Cp 1065 1135(1076) 41 0.454 0.160 0.043 130 0.153 0.093 0.025 9a Hp 1147 1218(1158) 31 0.048 0.059 0.007 93 0.020 0.038 0.005 19a CC 1398 1527(1437) 17 0.012 0.030 0.001 98 0.002 0.012 0.000 8a CC 1570 1715(1609) 79 0.086 0.053 0.019 81 0.000 0.002 0.001 20a CH 3038 3326(3200) 12 0.000 0.002 0.003 32 0.090 0.077 0.097 13 CH 3052 3352(3218) 34 0.319 0.145 0.300 23 0.014 0.031 0.064 2 CH 3074 3361(3243) 41 0.188 0.111 0.206 22 0.120 0.089 0.165 A2 16a Co 399 444(386) 193 0.000 0.000 0.000 170 0.000 0.000 0.000 17a Ho 927 1012(919) 546 0.000 0.000 0.000 406 0.000 0.000 0.000 B1 16b Co 344 408(299) 48 0.000 0.000 0.000 124 0.000 0.000 0.000 4 Co 721 752(674) 282 0.000 0.000 0.000 264 0.000 0.000 0.000 10b Ho 811 840(776) 212 0.000 0.000 0.000 250 0.000 0.000 0.000 17b Ho 955 990(880) 59 0.000 0.000 0.000 299 0.000 0.000 0.000 5 Ho 980 1028(928) 21 0.000 0.000 0.000 284 0.000 0.000 0.000 B2 6b Cp 623 670(625) 28 0.000 0.000 0.000 94 0.000 0.000 0.000 18b Hp 1071 1081(1092) 115 0.000 0.000 0.000 95 0.000 0.000 0.000 15 Hp 1159 1164(1243) 112 0.000 0.000 0.000 91 0.000 0.000 0.000 14 CC 1225 1318(1321) 50 0.000 0.000 0.000 141 0.000 0.000 0.000 3 Hp 1370 1494(1398) 66 0.000 0.000 0.000 25 0.000 0.000 0.000 19b CC 1466 1602(1492) 115 0.000 0.000 0.000 33 0.000 0.000 0.000 8b CC 1568 1718(1616) 182 0.000 0.000 0.000 168 0.000 0.000 0.016 7b CH 3086 3332(3205) 61 0.000 0.000 0.000 38 0.000 0.000 0.000

Fig. 3. Frontier molecular orbitals involved in (a) S0(1A1) ? S1(1B1) and (b)

(6)

Following the conventional scheme for the assignment of the vibra-tional normal modes[69], we found that there are the following typical vibrational modes as CC carbon–carbon stretching, CH car-bon–hydrogen stretching, Cp in-plane ring bend, Co out of plane ring bend, Hp in-plane CH bending, and Ho out of plane CH bending. The corresponding modes are given inTable 2. For example, the A1

symmetry mode with Huang–Rhys factor 1.566 for S1(1B1) excited

state corresponds to the vibration mode of 734 cm1_{in the CASSCF}

method, which can be assigned as

m

6amode, namely a in plane-ring

bend mode (Cp).

The cubic force constants (Kj3 in Eq.(9)) were calculated by

MP2/6-311++g⁄⁄ _{method and then they were converted into} Table 3

Calculated and observed vertical excitation energies (in eV) and its discrepancies D between theory and experiment with corresponding oscillator strengths f for1

B1and1B2 transitions. Method 1_B 1 D(1) f(1) 1B2 D(2) f(2) Exp. (max)[38,39,56] 4.20 0.00736 _5.20 _0.05_[36]_/0.028_[53] Exp. (max-new)[49] 4.183 5.22

Exp. (ZPE corrected)[52] 4.3 5.3

CASPT2[56,70] 3.81 0.39 4.93 0.27 TD-B3LYP[56] 4.31 0.11 5.87 0.67 TD-B3LYP 4.26 0.06 5.77 0.57 EOM-CCSD 4.62 0.42 0.0063 5.58 0.38 0.0303 EOM-CCSD[56] 4.74 0.54 5.52 0.32 EOM-CCSD(T)[52] 4.24 0.04 5.01 0.19 STEOM-CCSD[53,56] 4.40 0.20 5.04 0.16 CCSD R(3)[54] 4.55 0.35 5.44 0.24 SAC-CI SD-R[51] 4.32 0.12 0.0063 5.29 0.09 0.0327

Fig. 4. Five main vibrational modes calculated at CASSCF(6,6)/6-311++G⁄⁄

level.

Table 4

Calculated and observed adiabatic excitation energies (Ead, in eV) and its discrepancies D between theory and experiment for S1and S2states.

Method Ead(S1) D(1) Ead(S2) D(2)

Exp. (0–0-old assn.) 3.85[36,38,39,50] 5.00[36,38,39]

Exp. (0–0-new assn.)[49] 3.854 5.01(6)

CASPT2//CAS(6,6) 3.51 0.34 5.01 0.01 EOM-CCSD/aug-cc-pVDZ 3.56 0.29 5.17 0.17 TD-B3LYP 3.84 0.01 5.59 0.59 CASPT2//CASSCF(18,12)/6-31 + G⁄_[76] _3.75 0.10 EOM-CCSD/6-31 + G⁄ [76] 4.28 0.43 EOM-CCSD[56] 4.17 0.32 5.29 0.29 TD-B3LYP/6-31 + G⁄ [76] 3.88 0.03 CCS/cc-pVDZ[54] 5.44 1.59 6.62 1.62 CCSD/DZPR[54] 4.26 0.41 5.26 0.26 CIS[56] 5.32 1.47 6.48 1.48

(7)

anharmonic parameters

g

jby Eq. (10) as shown inTable 2. Except

for modes

m

13 and

m

2, anharmonic parameters

g

j are generally

small.

3.3. Vertical, adiabatic excitation energies

We calculated the vertical excitation energies as shown in Table 3 and adiabatic excitation energies as shown in Table 4. Combined the present calculations and the literature results [51–54,56,70,76], most of the methods overestimate the excitation energies, except that the present TD-B3LYP calculation shows the best agreement with the experimental observations for both verti-cal and adiabatic excitation energies of the ﬁrst excited S1(1B1)

state, and the SAC-CISD-R[51]and the present CASPT2//CAS(6,6) calculations show the best agreement with the experimental observations for vertical and adiabatic excitation energies of the second excited S2(1B2) state, respectively (see inTables 3 and 4).

Conventionally, computational studies of pyrimidine [51– 53,56,57,70–75]focused mainly on the evaluation of vertical exci-tation energies and oscillator strength with the aims of spectral assignment. For the vertical excitation from ground state to the ﬁrst excited state, experimental spectrum predicted the peak of the band maximum is located at 4.20 eV [38,39,56] with the zero-point energy correction up to 4.30 eV [52]. For the vertical excitation from ground state to the second excited state, the exper-iment predicated the peak of the band maximum is located at 5.20 eV [38,39,56] with the zero-point energy correction up to 5.30 eV[52]. Regarding to oscillator strengths of the excitation as shown inTable 3, both EOM-CCSD and SAC-CI[51]methods show good agreement with experiment. The experiment shows that the oscillator strength of n ?

p

⁄_{transition is about a quarter of that of}

p

?

p

⁄_{transition. Thus it leads to that the}1_B

1band intensity is

much smaller than1_B 2.

Although the calculations of vertical excitation energies and the oscillator strengths can qualitatively describe the peaks of the band maximum and the relative intensity for the spectra, this is not an entirely correct assumption (it may include errors of some tenths of an eV). For example, they cannot consider the origin bands for transitions in which the electronic transition causes a signiﬁcant

change in molecular geometry. A way to unequivocally compare equivalent quantities between theory and experiment is to com-pare 0–0 transition energies[51].

In the present study, simulations of vibronic spectra were based on the adiabatic excitation energies appeared in Eqs. (2) and (3) in which

x

ab+X0that includes X0 as a dynamic effect of vibronic

spectra should correspond to observed 0–0 excitation energy in experiment. The present work shows that this dynamic effect comes from the ﬁrst-order anharmonic correction. When we sim-ulate vibronic spectra with one method, we choose the corre-sponding adiabatic excitation energy of this method inTable 4. It should be noted that in the single vibronic manifold spectra the adiabatic excitation energies do not inﬂuence the band shape of spectra.

We utilized CSSCF/CSPT2 method to calculate conical intersec-tion (CI) between S1and S2states, and we found that the geometry

of CI is close to S1state and the energy of CI is higher than the

min-imum energy of S2 state. The present analytical formulation of

anharmonic Franck–Condon factor in Eqs. (2) and (3) can be valid for vibronic spectra with low vibrational number, so that this CI is not needed to be included in the present simulation. However, this CI conﬁrms that anharmonic effect for S1state can be large

as its geometry close to S1state.

3.4. Analysis of distorted effects and calculation of reorganization energies

We have analyzed the total symmetric vibrational modes with Huang–Rhys factors, and they play dominant roles in the progres-sions of the absorption[49,56]and ﬂuorescence[50]spectra of the S1state. There are few overtones of out-of-plane vibrations were

described and interpreted as contribution from the non-totally symmetric vibrational modes in the experimental spectra[49,50]. For example, the transition 16a2 _{with a}

2 symmetry has relative

strong/weak intensity in the absorption/ﬂuorescence spectrum and this is not included in the simulation with displaced harmonic oscillator approximation. We implement the distorted harmonic oscillator approximation [77] to analyze non-totally symmetry modes in comparison with total symmetric vibrational modes. As

Table 5

The calculated Franck–Condon factors in the displaced harmonic approximation (FCdisp) and the distorted harmonic approximation (FCdist).

C2vspecies Vibration normal modes cas/s0 cas/s1 FCdisp FCdist

A1 6a Cp 734 696 0.3271 0.0000 1 CC 1068 988 0.2220 0.0000 12 Cp 1135 1094 0.2882 0.0000 9a Hp 1218 1187 0.0457 0.0000 19a CC 1527 1544 0.0124 0.0000 8a CC 1715 1636 0.0791 0.0000 20a CH 3326 3338 0.0001 0.0000 13 CH 3352 3386 0.2321 0.0000 2 CH 3361 3402 0.1557 0.0000 A2 16a Co 444 251 0.0000 0.0371 17a Ho 1012 466 0.0000 0.0635 B1 16b Co 408 360 0.0000 0.0020 4 Co 752 470 0.0000 0.0258 10b Ho 840 628 0.0000 0.0103 17b Ho 990 931 0.0000 0.0005 5 Ho 1028 1049 0.0000 0.0000 B2 6b Cp 670 642 0.0000 0.0002 18b Hp 1081 966 0.0000 0.0016 15 Hp 1164 1052 0.0000 0.0013 14 CC 1318 1368 0.0000 0.0002 3 Hp 1494 1428 0.0000 0.0002 19b CC 1602 1487 0.0000 0.0007 8b CC 1719 1537 0.0000 0.0016 7b CH 3332 3393 0.0000 0.0000

(8)

shown inTable 2, some of the out-of-plane vibrations have a large change in frequencies. For example, for the mode

m

16awith a2

sym-metry, the frequency difference between S0 and S1 states is

193 cm1_{and in this case the distorted effect might be important.}

Within displaced approximation, Franck–Condon factor can be derived as[24,77] Fm0 i¼

H

bm 0 ij

H

a0i D E 2¼S m0 i i

m

0 i! eSi _ð14Þ where

m

0

idenotes the vibrational quantum number of the ith normal

mode and Siis the Huang–Rhys factor. Similarly, the Franck–Condon

factor in the distorted harmonic approximation is given by[24,77]

Fm0 i¼

H

bm 0 ij

H

a0i D E 2 ¼ ffiffiffiffiffiffiffiffiffiffiffi

xix

0 i p

xi

þ

x

0 i

x

0 i

xi

x

0 iþ

xi

m0 i

m

0 i! 2m0i1½ð

m

0 i=2Þ! 2 ð15Þ where

m

0

i can be only taken as an even integer number,

x

iand

x

0i

correspond to vibrational frequencies for different electronic states. We can see that, unless

x

0

i

x

i (or

x

0i

x

i), Fm0

i is much smaller

than unity. By applying Eqs. (14) and (15), we have computed the displaced Franck–Condon factors of 0–1 transitions and the dis-torted ones of 0–2 bands for all 24 vibrational modes and their val-ues are listed inTable 5. Obviously, the intensities of 0–2 transitions (overtones) of the out-of-plane modes are very small in comparison with 0–1 transitions (for example, the largest Franck–Condon factor arises from transition of 16a2_{is only 0.0371 based on the CASSCF} Table 6

Calculated reorganization energy (eV) for ground state (k0(S1), k0(S2)), excited S1(1B1) state (kex(S1)), and excited S2(1B2) state (kex(S2)).

Vibration normal modes[36,69] S0(1A1) S1(1B1) S2(1B2)

Sym N0 Mode CAS(MP2) k0(S1) kex(S1) k0(S2) kex(S2)

A1 6a Cp 734(687) 0.13 0.13 0.00 0.00 1 CC 1068(1010) 0.04 0.04 0.19 0.16 12 Cp 1135(1076) 0.06 0.06 0.02 0.02 9a Hp 1218(1158) 0.01 0.01 0.00 0.00 19a CC 1527(1437) 0.00 0.00 0.00 0.00 8a CC 1715(1609) 0.02 0.02 0.00 0.00 20a CH 3326(3200) 0.00 0.00 0.04 0.03 13 CH 3352(3218) 0.16 0.09 0.01 0.01 2 CH 3361(3243) 0.09 0.06 0.06 0.04

Total k with anharmonic correction 0.51 0.41 0.32 0.27

Total k from the PES 0.68 0.59 0.26 0.26

Fig. 5. S1(1B1) S0(1A1) absorption spectrum and S1(1B1) ? S0(1A1) ﬂuorescence spectrum of pyrimidine. Respectively, experimental data from (a) Ref.[49],[56]and (a0) Ref.

[50]; Simulated results with the present anharmonic correction, (b) and (b0_{) CASSCF, (c) and (c}0_{) EOM-CCSD, (d) and (d}0_{) TD-B3LYP; Simulated results with the present}

(9)

calculation). Therefore, we conclude that distorted effect is small in vibronic spectra of pyrimidine.

We calculated the reorganization energy using Eqs. (12) and (13) for S0, S1and S2states within the ﬁrst-order anharmonic

cor-rection and results are summarized inTable 6in which the reorga-nization energy contributed from each mode is also listed. A reorganization energy difference between the ground and the first excited state calculated from the first-order correction agrees with exact value 0.11 eV. However, this difference between the ground and the second excited state is about 0.05 eV (exact value is zero inTable 6). If we include the distorted contribution, the difference becomes 0.03 eV for the second excited state. Then, we conclude that the first-order correction to reorganization energy is signifi-cant to S1and negligible to S2 state. This agrees with conclusion

for spectral simulation in next subsection.

3.5. Anharmonic and harmonic Franck–Condon simulations

The present MP2 frequencies inTable 2are actually utilized in the simulation of both absorption and ﬂuorescence spectra for the two excited states within displaced harmonic and anharmonic oscillator approximations for CASSCF and EOM-CCSD methods, while the B3LYP frequencies is adopted for the TD-B3LYP method. The dephasing constants

c

ba in Eqs. (2) and (3) are chosen as

10 cm1 _{and 700 cm}1 _{for the excited states S}

1(1B1) and S2(1B2),

respectively. This choice includes resolution broadening in the experiment and it is similar to that of Refs.[41,44,45]. Temperature is taken as 298 K in the simulation as the experimental spectra were measured at room temperature. The intensity of simulated absorption and ﬂuorescence spectra is in the same unit for S1state

as shown inFig. 5.

Fig. 5shows that the main progressions of vibronic bands for the S1absorption and ﬂuorescence spectra that are well described

by mode

m

6aaccompanied with modes

m

1and

m

12. In fact, this can

be easily understood from that the Huang–Rhys factors for the modes

m

6a,

m

1, and

m

12are 1.566, 0.299, and 0.454 (seeTable 2),

respectively. The overall agreement between experimental (shown inFig. 5a[49,56] andFig. 5a0 _[50]_{) and the presently simulated}

spectra is generally good. However, the highest peak is assigned as the 0–0 vibronic transition for the absorption and the 6a1

0

tran-sition for ﬂuorescence spectra from experimental, while the 6a1 0is

the strongest transition for both the absorption and ﬂuorescence spectra from the present harmonic calculation. When the anhar-monic quantity

g

6a= 0.004 (see Table 2) is included and this

makes effective the Huang–Rhys factor for absorption as S0

6a= S6a(1 + 3

g

6a) = 1.547 and for ﬂuorescence as S06a=

S6a(1–3

g

6a) = 1.585, the

m

6atransition proﬁles and relative

inten-sity changes in the right direction as illustrated inFig. 5b and b0_.

Fig. 5e and e0 _{shows that the peak position of the 0–0 excitation}

from harmonic oscillator approximation has a big discrepancy with experiment observation for both absorption and ﬂuorescence spec-tra although the best static excitation energy |

x

ab| = 3.75 eV (see

Table 4) is chosen for simulation of the S1state. When anharmonic

corrections are included in the simulation,Fig. 5b and b0 _shows

that the peak position of the 0–0 excitation has blue shift

X0= 827 cm1with respect to harmonic oscillator approximation

and this leads to a very good agreement with experimental obser-vation. Actually, this dynamic shift was obtained with scaling fac-tor 0.8745 to all anharmonic constants

g

jin Table 2. This kind

scaling is widely employed for vibrational frequencies and we think that it is in the same reason to scaling anharmonic constants. We conclude that the ﬁrst-order anharmonic correction makes spectral peak positions shift and intensities change in the right direction simultaneously for absorption and ﬂuorescence spectra of S1 state; enhanced intensity of the absorption spectra and

weakened intensity of the ﬂuorescence. Finally, we have included the distorted effect for the non-total symmetric mode 16a2_{as is}

shown inFig. 5b, e, b0_{and e}0_{for a very weak band, which agrees}

very well with Knight’s experimental results. Furthermore, the present EOM-CCSD and TD-B3LYP spectral simulations with anhar-monic corrections are similar to the CASSCF simulation, except the highest peak for both absorption and ﬂuorescence of S1state now

is 0–0 vibronic transition, but the intensity of the weak band is much weaker than that of CASSCF and experiment. We can see that the simulated ﬂuorescence spectrum of S1state show better

agree-ment with the experiagree-mental results than the simulated absorption spectrum. The discrepancy may be because that the strong exper-imental absorption spectrum of S2 state affects the absorption

spectrum of S1state, while the weak ﬂuorescence spectrum of S2

state has little effects to the strong ﬂuorescence spectrum of S1

state during the experiment.

The largest Huang–Rhys factor for the S2(1B2) state listed in

Table 2is 1.402 for vibrational mode

m

1accompanied by the other

two; 0.044 for the mode

m

6aand 0.153 for mode

m

12.Fig. 6b shows

simulated absorption spectrum of the S2(1B2) state with the

dis-placed harmonic oscillator approximation at CASSCF level, and 11 0

is shown to be the strongest vibronic transition which agrees well with the experimental spectra[49].Fig. 6b shows that the both spectral peak position and the proﬁle is in very good agreement with experimental result. Anharmonic correction to the absorption spectrum of the S2(1B2) is negligible in the present simulation. In

addition,Fig. 6c and d show the EOM-CCSD and TD-B3LYP simu-lated spectral proﬁles which are similar to Fig. 6b. Compare to the strongest vibronic transition 11₀, the 0–0 band intensity from EOM-CCSD simulation is stronger than that of the CASSCF simula-tion and the experiment. However, the highest peak is assigned as the 0–0 vibronic transition for the TD-B3LYP simulation.

4. Concluding remarks

We have simulated absorption and ﬂuorescence spectra for S1(1B1) state and the absorption spectrum for S2(1B2) state using

harmonic and anharmonic oscillator approximation for pyrimidine molecule. We found that the ﬁrst-order anharmonic correction makes a signiﬁcant contribution to band shift of spectra for S1state

but it has no meaningful contribution to S2state. Franck–Condon Fig. 6. S2(1B2) S0(1A1) absorption spectrum of pyrimidine simulated results with

the present harmonic oscillator approximation. (a) Experimental data from Ref. [38], (b) CASSCF, (c) EOM-CCSD and (d) TD-B3LYP.

(10)

simulations with including the ﬁrst-order anharmonic correction show intensity enhancement of the absorption and intensity weak-ening of ﬂuorescence for the adiabatic S1(1B1) state and this agrees

well with experimental observation. Franck–Condon simulation of the absorption spectrum for the adiabatic S2(1B2) state shows good

agreement with experimental observation without the anharmonic correction.

We have optimized the equilibrium geometries of the electronic ground and the two lowest singlet excited states and then com-puted their 24-normal-mode frequencies that are all positive. All three electronic states have C2vgroup symmetry. We conﬁrmed

that our calculation results are basically the same as those from high-level ab initio calculation in the literatures [54–56]. This means that the equilibrium geometries from the present calcula-tion are accurate enough to be used for spectrum simulacalcula-tion.

The electronic structure calculations conﬁrmed that the S1(1B1)

and S2(1B2) states have n

p

⁄and

pp

⁄conﬁgurations, respectively.

Both vertical and adiabatic excitation energies of the S1(1B1) and

S2(1B2) states are calculated and analyzed by comparing with

var-ious theoretical calculations and experimental results. Basically, even the best calculations for static adiabatic excitation energies of the S1(1B1) state differ from the experimental one, and the best

calculation for static adiabatic excitation energy of S2(1B2) state

agree exactly with the experimental one. This reﬂects that dynamic shift of excitation energies from anharmonic correction is signiﬁ-cant for the S1(1B1) state, but not for S2(1B2) state. This is same as

reorganization energy calculations that conﬁrm 0.1 eV discrepancy between k0(S1) and kex(S1) but little discrepancy between k0(S2)

and kex(S2).

The present studies indicate that the frequency for each of the nine total symmetric normal modes only slightly differs from one another for the three electronic states S0(1A1), S1(1B1) and S2(1B2).

Furthermore, the transformation matrices that transfer geometric structure conﬁguration from Jacobi to normal-mode coordinates for the three electronic states are also quite same for each of the nine total symmetric modes. Thus, displaced harmonic oscillator approximation is proved to be good approximation. In fact, Huang–Rhys factors directly indicate that the modes

m

6a,

m

1, and

m

12 contribute S1 absorption and ﬂuorescence spectra, and S2

absorption spectrum mostly, among which the main progression of S1bands comes from mode

m

6aand S2bands comes from mode

m

1. This agrees with experimental measurement[49]. Although all

ab initio CASSCF, CSPT2, CCSD/EOM-CCD and B3LYP/TD-B3LYP methods basically show good agreement with experimental results for vibronic spectra of pyrimidine molecule, the CASSCF shows the best agreement with experiment for weak-band intensities of vib-ronic spectra. This is because that CASSCF method provides equal footing calculation for electronic and excited states, and thus it produces the most accurate results for geometry differences be-tween the ground and excited states. It should be emphasized that vibronic spectra is the most sensitive to the geometry difference, not absolute geometry for particular electronic state.

The non-total symmetric vibrational mode 16a is taken into ac-count in the present spectrum simulation within the distorted har-monic approximation and it devotes a weak band 16a2. The other non-total symmetric modes are negligible in spectrum simulation. The present ﬁrst-order anharmonic corrections can only take into account diagonal part of anharmonicity so that it is not enough to correct detailed band shape in vibronic spectra. This is because that mode mixings due to off-diagonal part of anharmonicity are completely neglected as they belong to the second-order anhar-monic corrections along with Duschinsky mode mixings. The con-ventional Herzberg–Teller effect of intensity borrowing from the other nontotally symmetric vibrational modes and possible non-adiabatic coupling due to conical intersection are not considered

in the present studies as they both have little effect to totally sym-metric vibrational modes.

Acknowledgments

L.Y. thanks support from visiting graduate program in National Chiao-Tung University. This work is supported by National Science Council of the Republic of China under Grant Nos. 97-2113-M-009-010-MY3 and 100-2113-M-009-005-MY3. J.Y. would like to thank support from National Natural Science Foundation of People’s Republic of China under Grant Nos. 20733002, 20873008, and 21073014. C.Z. thanks the MOE-ATU project of the National Chiao Tung University for support.

References

[1] J. Franck, Trans. Faraday Soc. 21 (1925) 536. [2] E.U. Condon, Phys. Rev. 28 (1926) 1182. [3] E.U. Condon, Phys. Rev. 32 (1928) 872.

[4] F. Duschinsky, Acta Physicochim. URSS 7 (1937) 551. [5] T.E. Sharp, H.M. Rosenstock, J. Chem. Phys. 41 (1964) 3453. [6] S.H. Lin, J. Chem. Phys. 44 (1966) 3759.

[7] S.H. Lin, R. Bersohn, J. Chem. Phys. 44 (1966) 3768. [8] S.H. Lin, J. Chem. Phys. 58 (1973) 5760.

[9] L.S. Cederbaum, W. Domcke, J. Chem. Phys. 64 (1976) 603.

[10] E.V. Doktorov, I.A. Malkin, V.I. Manko, J. Mol. Spectrosc. 64 (1977) 302. [11] M.J.S. Dewar, E.G. Zoebisch, E.F. Healy, J.J.P. Stewart, J. Am. Chem. Soc. 107

(1985) 3902.

[12] M. Roche, Chem. Phys. Lett. 168 (1990) 556. [13] S.Y. Lee, J. Phys. Chem. – US 9 (4) (1990) 4420.

[14] D.C. Todd, G.R. Fleming, J.M. Jean, J. Chem. Phys. 97 (1992) 8915. [15] F. Zerbetto, J. Phys. Chem. – US 9 (8) (1994) 13157.

[16] P.T. Ruhoff, Chem. Phys. 186 (1994) 355.

[17] P.A. Malmqvist, N. Forsberg, Chem. Phys. 228 (1998) 227.

[18] T. Muller, P.H. Vaccaro, F. Perez-Bernal, F. Iachello, J. Chem. Phys. 111 (1999) 5038.

[19] D.K.W. Mok, E.P.F. Lee, F.T. Chau, D.C. Wang, J.M. Dyke, J. Chem. Phys. 113 (2000) 5791.

[20] P.T. Ruhoff, M.A. Ratner, Int. J. Quant. Chem. 77 (2000) 383.

[21] S. Schumm, M. Gerhards, K. Kleinermanns, J. Phys. Chem. A 104 (2000) 10648. [22] J.R. Reimers, J. Chem. Phys. 115 (2001) 9103.

[23] V. Zazubovich, I. Tibe, G.J. Small, J. Phys. Chem. B 105 (2001) 12410. [24] S.H. Lin, C.H. Chang, K.K. Liang, R. Chang, Y.J. Shiu, J.M. Zhang, T.S. Yang, M.

Hayashi, F.C. Hsu, Adv. Chem. Phys. 121 (2002) 1.

[25] H. Kikuchi, M. Kubo, N. Watanabe, H. Suzuki, J. Chem. Phys. 119 (2003) 729. [26] H. Hwang, P.J. Rossky, J. Phys. Chem. A 108 (2004) 2607.

[27] J.M. Luis, D.M. Bishop, B. Kirtman, J. Chem. Phys. 120 (2004) 813. [28] A. Hazra, H.H. Chang, M. Nooijen, J. Chem. Phys. 121 (2004) 2125. [29] M. Dierksen, S. Grimme, J. Phys. Chem. A 108 (2004) 10225. [30] M. Dierksen, S. Grimme, J. Chem. Phys. 122 (2005) 244101. [31] S. Yeganeh, M.A. Ratner, J. Chem. Phys. 124 (2006) 044108. [32] I. Pugliesi, K. Muller-Dethlefs, J. Phys. Chem. A 110 (2006) 4657. [33] I. Pugliesi, K. Muller-Dethlefs, J. Phys. Chem. A 110 (2006) 13045.

[34] T. Petrenko, O. Krylova, F. Neese, M. Sokolowski, New J. Phys. 11 (2009) 015001.

[35] C.Y. Zhu, K.K. Liang, M. Hayashi, S.H. Lin, Chem. Phys. 358 (2009) 137. [36] (a) K.K. Innes, J.P. Byrne, I.G. Ross, J. Mol. Spectrosc. 22 (1967) 125;

(b) K.K. Innes, I.G. Ross, W.R. Moomaw, J. Mol. Spectrosc. 132 (1988) 492. [37] A.D. Boese, J.M.L. Martin, J. Phys. Chem. A 108 (2004) 3085.

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

[39] I. Yamazaki, T. Murao, T. Yamanaka, K. Yoshihara, Faraday Discuss. Chem. Soc. 75 (1983) 395.

[40] M. Chachisvilis, A.H. Zewail, J. Phys. Chem. A 103 (1999) 7408. [41] Z.-L. Cai, J.R. Reimers, J. Phys. Chem. A 104 (2000) 8389. [42] P. Wormell, J.E. Gready, Chem. Phys. 179 (1994) 55. [43] P.L. Muiño, P.R. Callis, J. Chem. Phys. 100 (1994) 4093.

[44] H. Wang, C. Zhu, J.-G. Yu, S.H. Lin, J. Phys. Chem. A 113 (2009) 14407. [45] D.-Y. Wu, M. Hayashi, Y.J. Shiu, K.K. Liang, C.H. Chang, S.H. Lin, J. Chin. Chem.

Soc. 50 (2003) 735.

[46] R. He, C. Zhu, C.-H. Chin, S.H. Lin, Sci. China Ser. B – Chem. 51 (2008) 1166. [47] R. He, C. Zhu, C.-H. Chin, S.H. Lin, Chem. Phys. Lett. 476 (2009) 19. [48] M.H. Palmer, I.C. Walker, M.F. Guest, A. Hopkirk, Chem. Phys. 147 (1990) 19. [49] F. Ferreira da Silva, D. Almeida, G. Martins, A.R. Milosavljevic´, B.P. Marinkovic´,

S.V. Hoffmann, N.J. Mason, Y. Nunes, G. Garcia, P. Limão-Vieira, Phys. Chem. Chem. Phys. 12 (2010) 6717.

[50] A.E.W. Knight, C.M. Lawburgh, C.S. Parmenter, J. Chem. Phys. 63 (1975) 4336. [51] Y. Li, J. Wan, X. Xu, J. Comput. Chem. 28 (2007) 1658.

[52] J.E. Del Bene, J.D. Watts, R.J. Bartlett, J. Chem. Phys. 106 (1997) 6051. [53] M. Nooijen, Spectrochim. Acta Part A 55 (1999) 539.

(11)

[54] A. Öhrn, O. Christiansen, Phys. Chem. Chem. Phys. 3 (2001) 730. [55] K.V. Berezin, L.M. Babkov, M.A. Kovner, J. Struct. Chem. 38 (1997) 281. [56] G. Fischer, Z.-L. Cai, J.R. Reimers, P. Wormell, J. Phys. Chem. A 107 (2003) 3093. [57] P.-A. Malmqvist, B.O. Roos, M.P. Fülscher, A.P. Rendell, Chem. Phys. 162 (1992)

359.

[58] F. Billes, H. Mikosch, S. Holly, J. Mol. Struct. (THEOCHEM) 423 (1998) 225. [59] Y. Luo, H. Ågren, S. Knuts, P. Jørgensen, Chem. Phys. Lett. 213 (1993) 356. [60] L. Åsbrink, C. Fridh, B.Ö. Jonsson, E. Lindholm, Int. J. Mass Spectrom. Ion Phys. 8

(1972) 215.

[61] M.J. Frisch, G.W. Trucks, H.B. Schlegel, G.E. Scuseria, M.A. Robb, J.R. Cheeseman, J.A. Montgomery Jr., T. Vreven, K.N. Kudin, J.C. Burant, J.M. Millam, S.S. Iyengar, J. Tomasi, V. Barone, B. Mennucci, M. Cossi, G. Scalmani, N. Rega, G.A. Petersson, H. Nakatsuji, M. Hada, M. Ehara, K. Toyota, R. Fukuda, J. Hasegawa, M. Ishida, T. Nakajima, Y. Honda, O. Kitao, H. Nakai, M. Klene, X. Li, J.E. Knox, H.P. Hratchian, 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, P.Y. Ayala, K. Morokuma, G.A. Voth, P. Salvador, J.J. Dannenberg, V.G. Zakrzewski, S. Dapprich, A.D. Daniels, M.C. Strain, O. Farkas, D.K. Malick, A.D. Rabuck, K. Raghavachari, J.B. Foresman, J.V. Ortiz, Q. Cui, A.G. Baboul, S. Clifford, J. Cioslowski, B.B. Stefanov, G. Liu, A. Liashenko, P. Piskorz, I. Komaromi, R.L. Martin, D.J. Fox, T. Keith, M.A. Al-Laham, C.Y. Peng, A. Nanayakkara, M. Challacombe, P.M.W. Gill, B. Johnson, W. Chen, M.W. Wong, C. Gonzalez, J.A. Pople, Gaussian 03, Revision E.01, Gaussian, Inc., Wallingford, CT, 2004. [62] G. Karlström, R. Lindh, P.-Å. Malmqvist, B.O. Roos, U. Ryde, V. Veryazov, P.-O.

Widmark, M. Cossi, B. Schimmelpfennig, P. Neogrady, L. Seijo, Comput. Mater. Sci. 28 (2003) 222.

[63] T.H. Dunning Jr., R.J. Harrison, J. Chem. Phys. 96 (1992) 6796. [64] J.F. Stanton, R.J. Bartlett, J. Chem. Phys. 98 (1993) 7029.

[65] M.J. Frisch, G.W. Trucks, H.B. Schlegel, 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. Sonnenberg, 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, O. Farkas, J.B. Foresman, J.V. Ortiz, J. Cioslowski, D.J. Fox, Gaussian 09, Revision A.02, Gaussian, Inc., Wallingford, CT, 2009.

[66] S. Cradock, P.B. Liescheski, D.W.H. Rankin, H.E. Robertson, J. Am. Chem. Soc. 110 (1988) 2758.

[67] L. Fernholt, C. Rømming, Acta Chem. Scand. Ser. A 32 (1978) 271. [68] M. Riese, J. Grotemeyer, Anal. Bioanal. Chem. 386 (2006) 59. [69] R.C. Lord, A.L. Marston, F.A. Miller, Spectrochim. Acta 9 (1957) 113. [70] M.P. Fülscher, K. Andersson, B.O. Roos, J. Phys. Chem. 96 (1992) 9204. [71] W.J. Buma, M.C.J.M. Donckers, E.J.J. Groenen, J. Am. Chem. Soc. 114 (1992)

9544.

[72] R. Bauernschmitt, R. Ahlrichs, Chem. Phys. Lett. 256 (1996) 454.

[73] D.J. Tozer, R.D. Amos, N.C. Handy, B.O. Roos, L. Serrano-Andrés, Mol. Phys. 97 (1999) 859.

[74] R.L. Ellis, G. Kuehnlenz, H.H. Jaffé, Theor. Chim. Acta (Berl) 26 (1972) 131. [75] D. Hegarty, M.A. Robb, Mol. Phys. 38 (1979) 1795.

[76] Z.-L. Cai, J.R. Reimers, J. Phys. Chem. A 109 (2005) 1576.