fluorobenzene and its rate of the internal conversion
Rongxing He, Ling Yang, Chaoyuan Zhu, Masahiro Yamaki, Yuan-Pern Lee, and Sheng Hsien Lin
Citation: The Journal of Chemical Physics 134, 094313 (2011); doi: 10.1063/1.3559454
View online: http://dx.doi.org/10.1063/1.3559454
View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/134/9?ver=pdfcov Published by the AIP Publishing
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Franck–Condon simulation of the
A
1B
2→
X
1A
1dispersed fluorescence
spectrum of fluorobenzene and its rate of the internal conversion
Rongxing He,1,2Ling Yang,1,3Chaoyuan Zhu,1,a)Masahiro Yamaki,4Yuan-Pern Lee,1
and Sheng Hsien Lin1,4
1Department of Applied Chemistry, Institute of Molecular Science and Center for Interdisciplinary Molecular
Science, National Chiao-Tung University, Hsinchu 30050, Taiwan
2College of Chemistry and Chemical Engineering, Southwest University, Chongqing 400715, China 3Department of Chemistry, Beijing Normal University, Beijing 100875, People’s Republic of China 4Institute of Atomic and Molecular Sciences, Academia Sinica, Taipei 106, Taiwan
(Received 16 August 2010; accepted 7 February 2011; published online 7 March 2011)
By using three different hybrid exchange-correlation functionals containing 20%, 35%, and 50% of exact Hartree–Fock (HF) exchange of the density functional theory and its time-dependent extension plus the Hartree–Fock and the configuration interaction of single excitation meth-ods, equilibrium geometries, and their 30 vibrational–normal-mode frequencies of the ground
S0(1A
1) and the first excited S1(1B2) states of fluorobenzene (FB) were calculated. The
dis-persed fluorescence spectrum and internal conversion (IC) rate of the A 1B
2 → X 1A1
transi-tion were simulated by Franck–Condon (FC) calculatransi-tions within the displaced harmonic oscil-lator approximation plus anharmonic and distorted corrections. The simulated spectral profile is
primarily described by the Franck–Condon progression from the ring-breathing modes v9 and
v10 which belong to totally symmetry modes. Anharmonic corrections simultaneously improve
the intensity order of 90
1 and 1001 bands and diminish 101 transition that is fairly strong in
har-monic simulations. It is concluded that the amount of Hartree–Fock exchange does impact the geometries and vibrational frequencies of FB molecule, but not the relative intensities of the transitions. It is anharmonic corrections that make the relative intensities of the transitions in good agreement with experimental results. Distorted corrections could assign most of the domi-nant overtones of out-of-plane nontotally symmetry modes, and the results agree well with the ex-perimental assignments. Furthermore, it was found that the internal conversion rate is dominated
by three promoting modes that are computed with lowing symmetry to C1. By choosing
dephas-ing width as 10 cm−1 that is consistent with spectral simulation, we obtained the lifetimes of the
A1B
2 → X1A1 de-excitation as 11 and 19 ns, respectively, from TD(B3LYP) and HF/CIS
calcu-lations in comparison with the experimental value 14.75 ns. © 2011 American Institute of Physics.
[doi:10.1063/1.3559454]
I. INTRODUCTION
Fluorobenzene (FB) molecule is the simplest prototypi-cal benzene derivative and the study of its spectroscopic and photophysical properties has a long history with focusing on its electronic ground [S0(1A1)] and first singlet excited
[S1(1B2)] states.1–5The presence of fluorine atom on aromatic
ring reduces the D6hpoint group symmetry of benzene to the
C2vsymmetry of FB and this makes the electronic structures
and spectroscopic characteristics of FB very different from
those of benzene.6–8 Therefore, it has been an interesting
subject for both experimental and theoretical investigation of vibrational assignments from its absorption and fluorescence
spectra.1–10 As early as 1925, Henri had studied the near
ultraviolet absorption spectrum of FB, and later in 1945 Wol-man remeasured this spectrum with more details and found that strong perturbation of aromatic ring by fluorine atom enhances the transitions originally forbidden in benzene but permitted in FB.11In order to assign vibrational and electronic
a)Electronic mail: cyzhu@mail.nctu.edu.tw.
transitions in terms of vibronic levels, a few experiments with the high resolution of infrared vibrational and electronic absorption spectra of FB have been performed.2–5,8–10,12Lipp and Seliskar obtained the most complete assignment of the
S1(1B2)← S0(1A1) excitation spectrum experimentally using
the hot and cool band spectra.2,3Padma and Jug13calculated the equilibrium geometries of electronic ground and first excited states, and adiabatic excitation energy between the two electronic states using configuration interaction (CI) and
semiempirical methods. Fogarasi and Császár14 simulated
the vibrational spectrum of the ground state using Hartree–
Fock method. More recently, Miura et al.15 computed the
vertical excitation energy and oscillator strength of the 1B
2
←1A
1transition of FB (corresponding to1E1u←1A1g
tran-sition in benzene) by the time-dependent density functional theory (TDDFT). Using more sophisticated experimental measurements of laser-induced fluorescence (LIF) and
dispersed fluorescence (DF) spectroscopy, Butler et al.4
provided almost complete assignment and interpretation of
this cold band1B2 ←1A1transitions of FB, and furthermore
they analyzed over 40 single vibronic levels of DF spectra
resulted in the assignment of 16 fundamental frequencies in the excited electronic state. Using the gas phase resonance en-hanced multiphoton ionization (REMPI) technique and zero electron kinetic energy spectroscopy, Pugliesi et al.5could re-fine the assignments of the lower wavenumber region (below
∼1000 cm−1) of the band 1B
2 ← 1A1 transitions of FB.
The Duschinsky effect (mode mixing) that was analyzed by high level ab initio calculations leaded to assignments of the considerable weak transitions observed in experiment
spectrum4,5 and relative intensities of the two-color (1 +
1) REMPI spectrum of FB were simulated by performing
numerical multidimensional Franck–Condon (FC) calcula-tions including standard linear Duschinsky normal coordinate
transformation procedure.5
A main purpose of the present work focuses on the
ab initio simulations of electronic spectra and internal
con-version (IC) of Fluorbenzene. From theoretical point of view, these two processes are partly governed by the same Franck–Condon factor. In order to demonstrate how accu-rate simulation we can do for internal conversion process, we must show that the same ab initio level of calcula-tion should perform well for electronic spectra. Time de-pendent density function method is used to calculate excited state in which the impact of Hartree–Fock exchange percent-age in the hybrid functional is extensively investigated and discussed.
In the present work, we first simulated the dispersed fluorescence electronic spectrum of FB following LIF ex-citation of the 1B
2 ← 1A1 by employing simple products
of one-dimensional FC factors that corresponds to displaced harmonic oscillator approximation plus anharmonic and dis-torted effects. We believe that the DF spectral profile should be primarily described by the Franck–Condon progression in terms of the totally symmetry normal modes. Anharmonic correction to ground and excited states is assumed to be the same and thus their corrections are only to totally symmetry normal modes. The distorted effects that affect only nonto-tally symmetry normal modes should be small as it can be considered as diagonal-part correction of Duschinsky mode-mixing matrix. We calculated the internal conversion rate of
FB molecule following excitation of the1B
2←1A1transition
under the isolated molecular condition. Actually, the lifetimes and the radiationless transition rates from the single vibronic
levels of S1state were measured under the collision-free
con-dition by Abramson and co-workers.16,17 Reported lifetimes
and quantum yields of fluorescence come from 59 vibronic
states, and the lifetime of 0–0 transition is about 14.8 ns.16
Radiationless transition involving two electronic states is partly governed by the nonadiabatic coupling that is deter-mined by the off-diagonal matrix elements of the nuclear ki-netic operators. This nonadiabatic coupling can be expressed in terms of series of normal-mode vibronic couplings which can be calculated by ab initio quantum mechanic methods and rate of internal conversion is usually sensitive to accuracy of those vibronic couplings.
The present paper is organized as follows. SectionII
de-scribes computational details for applying ab initio methods to electronic structure, and for applying displaced oscilla-tor approximation of FC facoscilla-tors to DF spectrum and IC rate
FIG. 1. The molecular structure and the atom numbering of fluorobenzene.
constant. SectionIIIpresents the results and discussions and
Sec.IVprovides concluding remarks.
II. COMPUTATIONAL METHODS
A. Ab initio methods for electronic properties
It is important to notice that for molecule belonging to
the C2vpoint group symmetry the choice of B1 and B2
irre-ducible representations of electronic states depends on choice
of Cartesian axes. Typically, the molecular C2 axis is
cho-sen as the z-axis and then the molecule is placed in the yz-plane with x-axis to be perpendicular to the molecular yz-plane as shown in Fig.1. In this way, we can identify the first excited
state as B2symmetry.
The equilibrium geometries and its 30 vibrational fre-quencies of FB molecule were calculated in the framework of the density functional theory (DFT) for electronic ground
state S0 and time-dependent density functional theory for the
first excited state S1. We have reported calculations based
on three kinds of different exchange-correlation function-als. The Becke’s three-parameter hybrid functional with the
Lee–Yang–Parr correlation functional (B3LYP)18–21was
cho-sen as preliminary functional. Guthmuller and Champagne22
pointed out that the percentage of Hartree–Fock (HF) ex-change in the hybrid functionals has great impact. There-fore, we followed their choice with adding the other two hybrid functionals, BHandHLYP and B3LYP-35 (see
Ref. 22 for definition) containing 50% and 35% HF
ex-change in comparison with 20% in B3LYP functional. We have also done the Hartree–Fock calculation in comparison
with DFT for the ground state S0 and the configuration
in-teraction of single (CIS) excitation calculation in comparison
with TDDFT for the excited state S1. All the DFT, TDDFT,
HF, and CIS calculations were carried out using GAUSSIAN
09 program23 with basis set 6-311++G**. In the Franck–
Condon simulation of the DF spectrum, the geometrical pa-rameters and frequencies should be used consistently at the same level of ab initio methods, that is, DFT versus TDDFT and HF versus CIS, for the ground and the excited states, respectively.
For computing the IC rate constant of the S1(1B2)
CASSCF/6-311++G** level to evaluate the electronic nonadia-batic coupling matrix (vibronic couplings) between the
S0 and S1 states. This was carried out using MOLPRO
program.24
B. Franck–Condon simulation for DF spectra
The relative intensity of molecular fluorescence is theo-retically proportional to the third power of frequency spec-trum ω, but when it is converted into units used in
experi-mental measurement, the intensity must be divided by ω2.25
Therefore, within the Born–Oppenheimer approximation, flu-orescence coefficient I(ω) of the electronic transition from the initial state a to the final state b with vibrational quantaν and
ν’, respectively, can be expressed as26–28
I (ω) = 4π 2ω 3¯c | μba| 2 × ν ν Paν|bν | aν| 2 D(ωbν,aν+ ω), (1)
in which|bν| aν|2 is the Franck–Condon factor,
μbadenotes the electronic transition dipole moment, Paν the
Boltzmann factor, and c the speed of light. D(ωbν,aν+ ω) in
Eq.(1)is the Lorentzian line-shape function. If we use the dis-placed harmonic oscillator approximation plus the first-order
anharmonic correction, Eq.(1)can be reduced as29
I(ω) = 2πω 3¯ | μba| 2 ∞ −∞dt e −it(|ωba|+0−ω)−γba|t| ×exp − j Sj(1− 3ηj){2¯νj+ 1 − (¯νj+ 1)ei tωj − ¯νje−itωj} , (2)
where ωba= (Eb− Ea)/¯ denotes the electronic adiabatic
energy gap and Eb(Ea) stands for the electronic energy at
the equilibrium geometry of the state b(a),γbarepresents the
dephasing (or damping) width, ¯vi = (e¯ωi/kT − 1)−1 is the
phonon distribution, and Si denotes the Huang–Rhys factor
Si=
ωi
2¯d
2
i, (3)
in which ωi is the harmonic frequencies of the ith normal
mode. In this case, the displacement di in Eq.(3)can be
sim-ply calculated by
di = Qi− Qi =
j
Li j(qj− qj), (4)
where qj(qj) are the mass-weighted Cartesian coordinates at
the equilibrium geometry of the electronic state b (a), and the transformation matrix L in Eq. (4), along with qj(qj), were
calculated usingGAUSSIAN09 program.
The most important quantities0andηjin Eq.(2)stand
for the first-order anharmonic correction to Fracnk–Condon
factors and given by29,30
0= −2 j ηjSjωj, (5) ηj = 2Kj 3dj 3ω2 j , (6)
where Kj 3 is the third derivative of the ground-state
poten-tial energy surface with respect to normal mode Qjthat can
be calculated byGAUSSIAN09 program as well. (1− 3ηj)Sj
in Eq.(2)represents the certain effective Huang–Rhys factor
that influences profile of fluorescence spectra.
C. Franck–Condon simulation for internal conversion
The internal conversion is due to the breakdown of
the Born–Oppenheimer approximation.31,32According to the
Fermi’s golden rule, IC rate constant from initial single
vibra-tional level iv to final f state at temperature T= 0 K can be
estimated by33 ki v→ f = 2π ¯ v |ψf v| ˆHB O |ψi v|D(Ef v− Ei v), (7)
where ˆHB O denotes the nonadiabatic coupling operator. The
coupling matrix element is estimated by ψf v| ˆHB O |ψi v = −¯2 l ii v| ∂f ∂ Ql ∂f v ∂ Ql , (8) wherei vandf vare vibrational wave functions for nuclear
motion, i andf represent wave functions for electronic
motion, and Ql is the mass-weighted vibrational normal
co-ordinate of the promoting mode. Since we are interesting the IC rate constant from the single vibronic level i v produced by the pumping laser, so that we can use the Condon and the displaced harmonic oscillator approximations under the collision-free condition to simplify Eq.(7)as34
ki v→ f = 1 ¯2 |Rl( f i )| 2 ∞ −∞dt exp i t (ωf i+ ωl)− γi f|t| − j j(1− ei tωj) k gvk(t), (9) where gvk(t)= vk nk=0 vk! (vk− nk)!(nk!)2 [Sk(ei tωk/2− e−itωk/2)2]nk. (10)
In the present work, we only consider vk= 0, and thus
gvk(t)= 1. Rl( f i ) in Eq.(9) denotes the vibronic coupling for single prompting mode l between the initial and final elec-tronic states and it is given by
Rl( f i )= −¯2 ωl 2¯ f∂/∂Ql|i , (11)
TABLE I. The equilibrium geometries of the ground state [S0(1A1)] and the first excited state [S1(1B2)] optimized by different levels of method for the
fluorobenzene. Bond lengths are given in angstroms and bond angles in degrees. All calculations are based on basis set 6-311++G**. The atom numbering is
shown in Fig.1. Method C1F7 C1C2 C2C3 C3C4 C2H8 C3H9 C4H10 F7C1C2 C6C1C2 C1C2C3 C2C3C4 C3C4C5 TD(B3LYP) S0 1.357 1.386 1.394 1.394 1.083 1.084 1.083 118.7 122.6 118.3 120.4 119.8 S1 1.342 1.426 1.424 1.424 1.080 1.081 1.083 117.6 124.8 117.2 119.4 122.0 TD(B3LYP-35) S0 1.346 1.381 1.389 1.389 1.079 1.080 1.079 118.7 122.5 118.4 120.4 119.8 S1 1.331 1.410 1.417 1.419 1.076 1.077 1.079 117.6 124.7 117.3 119.4 122.0 TD(BHandHLYP) S0 1.338 1.376 1.385 1.385 1.075 1.076 1.076 118.8 122.5 118.4 120.4 119.8 S1 1.322 1.406 1.412 1.415 1.073 1.073 1.076 117.7 124.7 117.3 119.3 122.0 HF/CIS S0 1.328 1.377 1.386 1.386 1.074 1.075 1.075 118.8 122.4 118.4 120.5 119.7 S1 1.312 1.406 1.410 1.415 1.072 1.072 1.074 117.9 124.2 117.6 119.5 121.5 RICC2a S0 1.349 1.386 1.393 1.393 1.078 1.079 1.079 118.8 122.4 118.5 120.5 119.8 S1 1.339 1.419 1.428 1.428 1.076 1.077 1.079 117.6 124.8 117.3 119.3 122.0 Expt.b S0 1.355 1.382 1.395 1.395 1.077 1.078 1.077 118.5 123.1 118.1 120.5 119.8 aSee Ref.5. bSee Ref.9.
where the electronic nonadiabatic coupling matrix elements
f|∂/∂ Ql|i were computed by the MOLPROprogram.24
For simplicity, the term ¯12|Rl( f i )|2 in Eq. (9) is called as
the electronic part of the IC rate constant. The integration
part in Eq.(9)is the Franck–Condon factor basically same as
the corresponding integration part in Eq.(2)for fluorescence
emission spectra without including anharmonic correction. Therefore, accuracy of spectral calculation can directly reflect accuracy of internal conversion calculation. In the present work, we consider the calculation of IC rate constant from the single vibronic levels of the first singlet excited state S1to
the ground state S0.
III. RESULTS AND DISCUSSIONS
A. Equilibrium geometries and vibrational frequencies
The geometrical optimizations were carried out using DFT and HF methods for electronic ground state, and TDDFT and CIS methods for the first excited state. In fact, the structural optimizations of FB have been performed using a set of ab initio methods and the results have been analyzed
in more detail.4,5 We presented our calculation results and
made a comparison with other theoretical calculation5 as
well as available experimental data.9 Optimized geometrical
parameters are given in Table I with the atom numbering
scheme adopted in Fig.1. TableIshows that for the
geome-tries of both S0 and S1 states all the geometrical parameters
calculated from the three functionals (B3LYP, B3LYP-35, and BHandHLYP) agree well each other and also agree with
Ref. 5’s calculations, and all of them agree with experiment
observation in reasonable accuracy. For example, for the ground state S0, the C1F7bond length is 1.357 (from B3LYP)
and 1.346 (from B3LYP-35) in comparison with experiment
value 1.355, C6C1C2 bond angle is 122.6 (from B3LYP)
and 122.5 (from B3LYP-35) in comparison with experiment value 123.1. The other geometry parameters calculated from B3LYP and B3LYP-35 show even smaller discrepancies in comparison with the corresponding experiment values. Overall, the geometry parameters calculated from the B3LYP
show closer to the experimental values than those of the B3LYP and BHandHLYP calculations. Moreover, we did HF/CIS calculation for the geometrical optimizations, and
its results shown in Table I are very similar to the results
calculated from BHandHLYP functional.
Calculated vibrational frequencies of S0 and S1 states
are displayed in Table II in comparison with other
theoret-ical calculations and available experimental data. It should be noted that frequency calculations are used not only for simulating fluorescence spectra but also for verifying opti-mized equilibrium geometries as true minima for the two
electronic states involved in the transition. Table II shows
that vibrational frequencies simulated with the three function-als (B3LYP, B3LYP-35, and BHandHLYP) are in very good agreement with the experimental observations2,4,6as well as the other theoretical calculations4,5for both electronic ground
S0 and the first excited S1 states, and this indicates that the
calculated frequencies can be used directly to simulate the fluorescence spectra without scaling. For example, the fre-quencies calculated from B3LYP method are just scaled by a factor of 0.98 to reproduce the experimental values very
well (see Table II). However, vibrational frequencies
simu-lated from HF and CIS methods are much larger than the corresponding experimental values. It should be pointed out that in the present calculation the normal mode v10is assigned
as the closest analogy to the ring-breathing mode in benzene
molecule.36 This is different from Ref.4 where it is normal
mode v8. In the present work we chose the Mulliken notation
to label the vibrations of FB molecule and the corresponding Wilson notation only quoted in the tables.
In order to confirm that the displaced harmonic oscillator approximation is good approximation to be used for simulat-ing the DF electronic spectrum as well as IC rate constant, it
requires that difference between vibrational frequencies of S0
and S1states must be small for each of the eleven totally
sym-metry vibrational normal modes (a1-type). TableIIshows that
for example, the frequencies of the mode v10for the S0and S1
states are 819 and 793 cm−1 (from B3LYP calculations),
re-spectively, and its difference is just 26 cm−1. This difference
is just 8 cm−1 from B3LYP-35 method and 24 cm−1 from
TABLE II. Calculated and experimental vibrational frequencies (cm−1) of fluorobenzene for the S0(1A1) and S1(1B2) states.
S0 S1
Sym Modea B3b B35c BHd HF calc.e Expt.f Expt.g B3b B35c BHd CIS calc.e Expt.f Expt.g
a1 1(20a) 3203(3139) 3218 3298 3362 3242 3094 . . . 3230 3282 3325 3390 3262 . . . . 2(2) 3191(3128) 3207 3287 3350 3232 3080 . . . 3214 3266 3310 3374 3246 . . . . 3(13) 3170(3107) 3186 3265 3325 3210 3061 . . . 3187 3240 3284 3351 3221 . . . . 4(8a) 1634(1601) 1624 1715 1785 1632 1605 . . . 1560 1608 1648 1720 1547 . . . . 5(19a) 1522(1491) 1501 1587 1652 1520 1500 . . . 1448 1483 1513 1581 1431 . . . . 6(7a) 1234(1210) 1246 1301 1357 1252 1238 1239 1236 1275 1307 1363 1244 1220 1230 7(9a) 1175(1151) 1144 1216 1262 1177 1156 1156 1150 1173 1192 1240 1144 922 . . . 8(18a) 1038(1018) 1025 1076 1109 1036 1023 1023 988 1009 1028 1067 936 916 917 9(1) 1018(998) 1008 1054 1083 1014 1009 1009 959 983 1003 1037 978 968 969 10(12) 819(802) 820 853 879 818 809 810 793 812 829 855 780 765 765 11(6a) 524(514) 515 543 561 517 517 517 472 480 488 505 459 460 460 a2 12(17a) 965(946) 888 1022 1088 923 957 957 617 650 678 759 597 . . . 643 13(10a) 828(812) 747 873 927 839 818 818 484 517 544 631 487 . . . 509 14(16a) 421(413) 374 440 460 426 414 413 162 168 166 142 209 206 206 b1 15(5) 972(952) 895 1029 1095 955 978 978 785 814 840 891 738 . . . 755 16(17b) 905(887) 831 955 1010 902 895 895 658 691 719 785 611 . . . 661 17(10b) 765(749) 692 803 846 770 754 754 588 613 634 679 571 555 555 18(4) 677(664) 627 711 747 666 687 687 472 490 505 512 458 . . . 451 19(16b) 505(495) 444 530 558 512 498e 498 310 321 330 341 334 . . . 331 20(11) 237(232) 182 249 263 238 249e 233 174 181 186 196 174 182 167 b2 21(20b) 3201(3137) 3216 3296 3360 3239 . . . 3226 3278 3321 3386 3259 . . . . 22(7b) 3179(3116) 3195 3275 3336 3218 3069 . . . 3207 3259 3302 3366 3240 . . . . 23(8b) 1643(1610) 1635 1719 1784 1639 1605 . . . 1523 1554 1615 1824 1531 . . . . 24(19b) 1485(1456) 1460 1544 1603 1480 1460 . . . 1460 1534 1567 1573 1399 . . . . 25(14) 1344(1317) 1388 1386 1442 1430 1301 . . . 1421 1455 1485 1537 1691 1589 1589 26(3) 1325(1298) 1290 1347 1322 1322 . . . 1250 1301 1328 1351 1397 1284 . . . . 27(9b) 1180(1156) 1146 1211 1200 1178 1128 1128 1170 1191 1209 1245 1162 . . . . 28(15) 1090(1068) 1065 1127 1155 1088 1066 1066 1012 1040 1063 1107 987 976 955 29(6b) 627(615) 613 649 671 614 614 614 526 534 540 566 514 519 518 30(18b) 404(396) 381 421 440 402 400 404 392 401 408 427 384 387 388
aMulliken notation and, in parentheses, the Wilson notation are given.
bThe frequencies in parentheses have been scaled by a factor of 0.98. B3 denotes B3LYP functional. cB35 denotes B3LYP-35 functional.
dBH denotes BHandHLYP functional. eReference5(RICC2/def2-TZVPP). fReference6.
gReference4.
B. Electronic structures and excitation energies
We have done calculations of vertical excitation energies and its corresponding oscillator strengths for the first two singlet-excited states estimated at equilibrium geometry of the
ground state. Table IIIshows the vertical excitation energies
to S1 state 5.32, 5.51, and 5.67 eV, respectively, calculated
from B3LYP, B3LYP-35, and BHandHLYP functionals and
they all overestimate experimental value (4.69 eV),12but are
better than the previous value (5.81 eV) reported by Padma
et al.13which is just the same as the present CIS calculation. The present calculations showed the more percentage of HF exchange in the functional, the more overestimation
of vertical excitation energy to S1 state. This tendency
is consistent with the HF method that does not include the correlation energy. Besides, the present calculation based on the three functionals in text also shows lower
adiabatic energy gaps between S0 and S1 states than
that of Ref. 13. The experimental studies12,35–39 showed
that the substitution of fluorine in benzene does not
change the symmetry order of S1 and S2 states, and this
indicates that the 1B
2 corresponds to the first excited state
and1A
1corresponds to the second excited state. The present
calculations confirm all the experimental conclusion. On the other hand, the present calculations showed that the first
excited state S1 has aππ* transition feature, and thus the
significant changes in geometry come from the CC bond
lengths when the1B2 ← 1A1 transition occurs. The present
natural orbital calculation predicted that the S1 state results
mainly from the mixing excitation of the HOMO→ LUMO
(0.71 electron excited) and the HOMO − 1 → LUMO +
1 (0.29 electron excited), and its frontier molecular orbitals
are depicted in Fig. 2 and this agrees with calculations
reported by Pugliesi et al.5In comparison with experimental
vertical excitation energy to S2state 6.21 eV, B3LYP-35 and
BHandHLYP methods show the best agreement as shown in
Table III, while B3LYP method shows the underestimation
TABLE III. Calculated vertical transition energy ( E, unit is eV), adiabatic transition energy gap and oscillator strengths (f) from S0(1A1) to S1(1B2) together
with the experimental data and the previous calculations.
(TD)B3LYP (TD)BHandHLYP (TD)B3LYP-35 HF/CIS calc. Expt.a
State Sym Eb f Eb f Eb f Eb f Eb E f
S1 B2 5.32 0.013 5.67 0.016 5.51 0.015 6.16 0.019 5.81c 4.69 0.007
(5.18) (5.53) (5.37) (6.02) (5.78)
S2 A1 6.05 0.000 6.23 0.000 6.17 0.000 6.37 0.002 6.55c 6.21 0.081
aReference12.
bThe adiabatic energies are given in parentheses. cReference13, the used method was SINDO1.
dBasis set 6-311++G** is used for all the present calculations.
Let us now turn to discuss oscillator strengths of exci-tation transitions vertically from the ground state to excited
states. Relative strengths measured from experiment12 are
0.007 and 0.081 for excitation to S1and S2states, respectively,
as shown in TableIII. The present calculations do not
repro-duce this order of excitation strengths.
C. Franck–Condon simulation of the DF spectrum
The dispersed fluorescence spectrum of the A 1B2
→ X1A
1 transition was measured and the detailed analyses
were reported in the recent experiment studies,4where the DF
spectral profile is primarily described by the Franck–Condon progression in terms of the totally symmetry normal modes
(a1-type). Among these modes, v9 and v10 are the main
progression forming modes in the 1B
2 → 1A1 fluorescence
spectrum. The present calculations confirmed that modes
v10 and v9 have the largest Huang–Rhys factors (S= 0.420
and 0.408 for instance from B3LYP calculation as shown
in Table IV) and are assigned as the closest analogy to the
FIG. 2. The selected frontier molecular orbitals of fluorobenzene.
ring-breathing modes in benzene as shown in Fig. 3. This
differs from Ref.4in which v4and v9were interpreted as the
ring-breathing modes. The modes v2, v3, v4, and v5, which
are relative to the CH stretching and bending vibrations, have little contribution to FC factor as their Huang–Rhys factors are negligible small. Furthermore, the previous theoretical analysis4,5also indicated that there exist very weak bands in the DF spectrum, most of them are related to the nontotally
symmetric normal modes, e.g., v19 and v16 (b1 symmetry),
that can be interpreted in terms of Duschinsky mixing. In the present spectrum simulation, we utilized the un-scaled vibrational frequencies calculated at the same levels of ab initio method as was performed for optimizing the
ge-ometries of the electronic states in Sec. III A. The band
ori-gin (0–0 transition) is set up to be zero (cm−1) in the DF
spectrum as it was adopted in experimental study.4 In
or-der to simulate the experimental resolution of the DF
spec-trum, Lorentzian broadening width in Eq.(2)is tested asγab
= 10 cm−1 approximately. We know that this value contains
not only the truly dephasing but also the contribution from the
FIG. 3. The displacement vector of the three vibrational modes v10, v9,
TABLE IV. Calculated vibrational frequencies (Freq., in cm−1), cubic force constant (Kj3, in hartree * amu−3/2* bohr−3) of the diagonal elements of S0(1A1)
state, Huang–Rhys factors [S(s1)], Franck–Condon factors in the displaced oscillator approximation (FCdisp) and in the distorted oscillator approximation
(FCdist) (Value smaller than 0.0002 was neglected), and anharmonic parameters (ηj) for S− 1(1B2).
S0(1A1) S1(1B2)
B3LYP HF TD(B3LYP) CIS
Modea Freq. K
j3 Freq. Kj3 S(s1) FCdispb FCdistc ηj/2 S(s1) FCdispb FCdistc ηj/2
1(20a) 3203 − 0.605 3362 − 0.625 0.223 0.1784 0.0000 − 0.140 0.1905 0.1574 0.0000 − 0.120 2(2) 3191 0.433 3350 0.376 0.001 0.0009 0.0000 − 0.006 0.0006 0.0006 0.0000 − 0.004 3(13) 3170 − 0.117 3325 − 0.081 0.027 0.0263 0.0000 − 0.009 0.0216 0.0212 0.0000 − 0.005 4(8a) 1634 − 0.030 1785 − 0.031 0.002 0.0020 0.0000 − 0.004 0.0054 0.0054 0.0000 − 0.005 5(19a) 1522 0.003 1652 0.004 0.001 0.0009 0.0000 − 0.000 0.0016 0.0016 0.0000 − 0.000 6(7a) 1234 0.021 1357 0.031 0.085 0.0780 0.0000 0.032 0.0584 0.0551 0.0000 0.0301 7(9a) 1175 − 0.004 1262 − 0.004 0.006 0.0056 0.0000 − 0.002 0.0047 0.0047 0.0000 − 0.002 8(18a) 1038 − 0.005 1109 − 0.004 0.017 0.0168 0.0000 − 0.005 0.0120 0.0119 0.0000 0.0025 9(1) 1018 0.005 1083 0.006 0.408 0.2712 0.0000 − 0.025 0.3568 0.2497 0.0000 − 0.026 10(12) 819 − 0.005 879 − 0.006 0.420 0.2758 0.0000 − 0.050 0.3631 0.2525 0.0000 − 0.041 11(6a) 524 − 0.001 561 − 0.001 0.154 0.1323 0.0000 − 0.010 0.1293 0.1136 0.0000 − 0.008 12(17a) 965 0.000 1088 0.000 0.000 0.0000 0.0236 0.000 0.0000 0.0000 0.0156 0.0000 13(10a) 828 0.000 927 0.000 0.000 0.0000 0.0333 0.000 0.0000 0.0000 0.0177 0.0000 14(16a) 421 0.000 460 0.000 0.000 0.0000 0.0883 0.000 0.0000 0.0000 0.1183 0.0000 15(5) 972 0.000 1095 0.000 0.000 0.0000 0.0056 0.000 0.0000 0.0000 0.0053 0.0000 16(17b) 905 0.000 1010 0.000 0.000 0.0000 0.0123 0.000 0.0000 0.0000 0.0078 0.0000 17(10b) 765 0.000 846 0.000 0.000 0.0000 0.0085 0.000 0.0000 0.0000 0.0060 0.0000 18(4) 677 0.000 747 0.000 0.000 0.0000 0.0157 0.000 0.0000 0.0000 0.0171 0.0000 19(16b) 505 0.000 558 0.000 0.000 0.0000 0.0277 0.000 0.0000 0.0000 0.0283 0.0000 20(11) 237 0.000 263 0.000 0.000 0.0000 0.0117 0.000 0.0000 0.0000 0.0105 0.0000 28(15) 1090 0.000 1155 0.000 0.000 0.0000 0.0007 0.000 0.0000 0.0000 0.0002 0.0000 29(6b) 627 0.000 671 0.000 0.000 0.0000 0.0038 0.000 0.0000 0.0000 0.0036 0.0000
aMulliken notation, the Wilson notation are given in parentheses.
bFor the displaced approximation, only the Franck–Condon factors of 0–1 transitions are given. cThe data from 0–2 transitions in the distorted oscillator approximation.
instrumental broadening. The DF spectrum simulated from B3LYP, B3LYP-35, BHandHLYP, and CIS methods all show that the 0–0 transition is the strongest transition in the allowed
1B
2 → 1A1 electronic transition as shown in Figs.4 and5,
and this agrees with experiment observation. All simulated
DF spectra in Figs.4and5were performed in the framework
of displaced harmonic and anharmonic oscillator approxima-tion, respectively, in which the most prominent peaks have been assigned based on the present calculations in compari-son with experimental data. It can be seen that all methods including CIS reproduce qualitatively the essential character of the observed spectrum.
According to the high resolution of experimental results
performed by Butler et al.,4 the DF spectrum is mostly
assigned as the totally symmetric normal mode progressions;
especially the mode v9 displays very strong intensity in the
spectral profile and in the present B3LYP calculation the
intensity of the vibronic line assigned as 90
1 fundamental
[see Fig. 3(b)] is slightly underestimated; the intensity of
90
1 transition in experiment is about 70% of that of the
0–0 line, but that in the present calculation is about 50%.
Harmonic Franck–Condon simulations in Fig. 4 indicate
that the strongest and second strongest vibronic transitions are the 100
1 and 901 (after the 0–0 transition) which seems
to be reversed in comparison with the experiment result. When anhramonic corrections are added, simulations in
Fig.5 from all methods except B3LYP-35 show that the 901
band is larger than 100
1 band in good agreement with
exper-imental observation. Moreover, Harmonic Franck–Condon
simulations in Fig. 4 indicate that there is strong peak in
high energy region of DF spectra and this corresponds to 101
transition. When anharmonic corrections are added, simu-lations in Fig. 5from all methods show that 101 transition is diminished. This is because that Huang–Rhys factor 0.22 for
mode v1is significantly reduced with anharmonic correction
as shown in Table IV. Within harmonic approximation
vibrational displacement vector for mode v1(CH stretching)
is very large as shown in Fig.3and even if CH bond lengths
change very small from S0 to S1 state, Huang–Rhys factor
is still as big as 0.22. However, anharmonic correction is
also large as shown in Table IV and it effectively cancels
out 10
1 transition. We can conclude that the amount of HF
exchange (from 20% in B3LYP, 35% in B3LYP-35, 50% in BHandHLYP, and 100% in HF) does impact the geome-tries and vibrational frequencies of FB molecule, but not the relative intensities of the transitions. It is anharmonic corrections that influence the relative intensities of the transitions.
The experimental spectra in Figs. 4(a) and 5(a) show
that the 1001 transition strongly overlaps with 1402 transition
with a just split 15 cm−1. According to the calculations
based on the nontotally symmetric vibrational transitions (see TableIV), it is found that the 140
2should be about one-third
of the 100
FIG. 4. The DF spectra of fluorobenzene from S1to S0transition calculated
by harmonic FC simulation (the relative energy of the 0–0 transition is set up
to be zero). (a) Experimental result from Ref.4. (b) (TD) B3LYP, (c) (TD)
B3LYP-35, (d) (TD)BHandHLYP, and (e) HF/CIS calculations.
in the present simulation, the 100
2transition and the
combina-tion band 100
11402 are nearly degenerate vibronic level pairs;
the vibrational origins of 100
2(2v10= 1637 cm−1) and 10011402
(v10 + 2v14 = 1613 cm−1) are separated by 24 cm−1 from
B3LYP calculation. This suggests that there should be strong coupling between these two vibronic transitions (v1and v14).
On the other hand, from B3LYP calculation as shown in
Figs. 4(b) and 5(b) we could assign the 1002 (1637 cm−1,
that is, 2v10), 1003 (2456 cm−1), 902 (2036 cm−1), and 903
(3054 cm−1) vibronic transitions as four fundamentals
lo-cated at 1613, 2434, 2016, and 3014 cm−1 in experiment,4,
respectively. The detailed assignments based on the present analysis are shown in very good agreement with the
exper-imental data as displayed in Table V. In the low energy
re-gion of the DF spectrum, the experimental observation and the present B3LYP simulation agree well for the
signifi-cant intensity assigned from normal mode v11, and this
cor-responds to Huang–Rhys factor S = 0.154 (from B3LYP)
as shown in Table IV. Intensity of 60
1 band in the
har-monic Franck–Condon simulation is lower than correspond-ing experiment intensity, but it is improved with anharmonic
correction as shown in Fig. 5. This fundamental v6
corre-sponds to the CF bond stretching in the present analysis. Fur-thermore, the experimental DF spectrum reported by Butler
et al.4showed that there are many moderate intensity peaks with the characteristics of combination between fundamen-tal modes. Most of these transitions can be assigned based
the present FC simulations. Figures4(a)and4(b)[Figs.5(a)
and5(b)] show very good agreement between experimental
result and the present B3LYP simulation for the DF spectrum in terms of the ordering and positioning of these
combina-FIG. 5. The same as Fig.4but including anharmonic corrections.
tion peaks and its intensity strengths, especially for extremely weak peak of the combination band of 902601 in the high fre-quency region.
Another issue is related with how nontotally symmetry modes contribute to the DF spectrum. As we know that a large number of overtones of out-of-plane vibrations were described and interpreted as contribution from the nonto-tally symmetric vibrational modes in the experimental DF
spectrum.4 For example, the transitions involving the a
2 and
b1vibrations,140
2, 1902, and 1802, have relative moderate
inten-sities which cannot included in the simulation with displaced oscillator approximation. In order to include contributions from the nontotally symmetric vibrational transitions, we
must use the distorted harmonic oscillator approximation.28
The distorted effect from the change of frequency between the ground and excited states is generally small, but it can be
observed experimentally when the ratio between |ωg− ωe|
and |ωg+ ωe| (ωg and ωe are the vibrational frequencies
of the ground state and the excited state) is large for the mode involved in the electronic excitation. As shown in
Table II, based on the present calculations some of the
out-of-plane vibrations have a large change in frequencies.
For example, for the mode v12 with a2 symmetry, the
fre-quency difference between S0 and S1 states is 348 cm−1
(from B3LYP) and in this case the distorted effect should be considered.
According to Eq. (1), the fluorescence coefficient I(ω)
is proportional to the Franck–Condon factor. Within the dis-placed approximation, the Franck–Condon factor can be de-rived as27,28 Fν i =bνia0i 2= Sν i i ν i! e−Si, (12)
where νi denotes the vibrational quantum number of the ith
normal mode and Siis the Huang–Rhys factor. Similarly, the
Franck–Condon factor in the distorted harmonic approxima-tion is given by27,28 Fvi = |bνi| a0i| 2 = ωiωi ωi+ ωi ω i− ωi ω i+ ωi vi vi! 2vi−1[(v i/2)!]2 , (13)
where vi can be only taken as an even integer
num-ber, ωiandωi correspond to vibrational frequencies for
dif-ferent electronic states. We can see that, unless ωi
ωi (or ωi ωi), Fνi is much smaller than unity. By
applying Eqs.(12)and(13), we have computed the displaced
Franck–Condon factors of 0–1 transitions and the distorted ones of 0–2 transitions for all 30 vibrational modes and their values are listed in TableIV. Generally, the intensities of 0– 2 transitions (overtones) of the out-of-plane modes are very small. For example, the largest distorted Franck–Condon fac-tor arises from transition of 1402 is only 0.0883 based on the B3LYP calculation. Using the displaced–distorted
approxi-mation with adding distorted FC factors of Eq. (13) into
Eq.(1), we simulated the DF spectrum of the1B2→1A1
elec-tronic transition again. Figure6(b)displays the spectrum
sim-ulated from B3LYP method in comparison with experiment
result given in Fig. 6(a). It can be seen that most of the
overtones are reproduced correctly. For example, the line at
474 cm−1 is assigned as the 200
2 transition, which agrees
well with the transition at 466 cm−1assigned experimentally.
Furthermore, the predicted band positions of overtones 140
2,
180
2, and 1202 show an excellent agreement with the
experi-ment. In the present calculations, the transitions of 190
2 and
90
1, 2902and 601appear at almost the same positions (that is, 1902
= 1010 cm−1≈90
1 = 1018 cm−1 and 2902 = 1254 cm−1≈601
= 1234 cm−1). It is should be noted that the intensities of
these out-of-plane vibrations are enlarged due to their notably distorted Huang–Rhys factors. However, the 0–1 transition of mode v29with b2-type symmetry is still not reproduced by the
present displaced–distorted calculation. This kind of the vi-brational transition is induced by the Herzberg–Teller effect, which is not included in the present FC simulation.
D. Internal conversion and the lifetime ofS1state In order to compute internal conversion constant, we have to compute the nonadiabatic coupling matrix elements
or vibronic couplings f|∂/∂ Ql|i (between S0 and S1
states) vertically at equilibrium geometry of S1 state. At the
equilibrium geometry of S1 state calculated by two methods;
(TD)B3LYP and HF/CIS, we employed CASSCF method for calculating nonadiabatic coupling matrix elements. We should mention that the equilibrium geometry optimized by (TD)B3LYP and HF/CIS methods may or may not corre-spond to true equilibrium optimized by CASSCF method, but on the other hand, vibronic couplings vary very slowly against change of geometry. We first computed vibronic cou-plings in Cartesian coordinate spaces and then transformed them into normal mode coordinates. However, we have to
lowering group symmetry to C1in order to perform vibronic
couplings by MOLPRO. We obtained 30 vibronic couplings
TABLE V. Theoretical frequencies and assignments of transitions in the DF spectrum using the displaced oscil-lator approximation. The relative energies of the origin bands in both experimental and theoretical spectra are set
up to be zero. The experimental data is taken from Ref.4.
Assignmenta Expt.d B3LYPb B35 BH HF/CIS
202(112)c 466 111(6a1) 519 524(514) 515 543 561 291(6b1) 611 101(121) 810 819(802) 820 853 879 142(16a2) 826 91(11) 1008 1018(998) 1008 1054 1083 61(7a1) 1239 1234(1210) 1246 1301 1357 111101(6a1121) 1326 1343(1316) 1335 1396 1440 182(42) 1369 11191(6a111) 1523 1542(1512) 1523 1597 1644 102, or 101142(122,or 12116a2) 1613 1637(1604) 1640 1706 1758 10191(12111) 1917 1837(1800) 1828 1907 1962 122(17a2) 1912 92(12) 2016 2036(1996) 2016 2108 2166 10161, or 82(1217a1, or 18a2) 2045 2053(2012) 2066 2154 2236 6191(7a111) 2243 2252(2208) 2254 2355 2440 103(123) 2434 2456(2407) 2460 2559 2637 93(13) 3014 3054(2996) 3024 3162 3249 11(20a1) 3203(3139) 3218 3298 3362 9261(127a1) 3251 3270(3206) 3262 3409 3523
aMulliken notation, the Wilson notation are given in parentheses. bThe scaled frequencies by a factor of 0.98 are listed in parentheses.
cThe subscript n (n= 1, 2, 3) denotes the transition level in the DF spectrum. For example, 20
2indicates the 0→2 transition
involving in the 20 mode.
FIG. 6. The same as in Fig.5but including distorted effect. (a) Experimental result from Ref.4. (b) Simulated results with B3LYP calculation in which the intensity of overtone 200
2is enlarged by a factor of 10 and the other overtones are enlarged by a factor of 4.
among which there are only three modes at the same or-der of magnitude and the rest of them are negligibly small. These three modes do not have clear correspondence to
modes obtained with C2v group symmetry. Therefore, we
can only label them according to order of frequency
mag-nitude in C1 symmetry, and they are 7th, 8th, and 15th
vi-brational normal modes computed by CASSCF. Then, three vibronic couplings are converted to the electronic part of the IC rate ¯12|Rl( f i )|2 by Eq.(11)and the results are given in
Table VI. Two methods produce almost same the coupling
matrix elements and its electronic part of the IC rate (see the coupling elements of the 7th normal mode are about 0.1482 and 0.1460 a.u., respectively, from (TD) B3LYP and HF/CIS
methods given in TableVI). This is because the optimized
ge-ometries of the excited state S1 performed by two methods
show small discrepancies (see TableI), besides vibronic
cou-plings vary slowly against change of geometry. .
Now we turn to compute the second part of IC rate of the transition1B2 →1A1, and that is the integral part in Eq.(9).
The dephasing widthγi f in Eq.(9)is chosen as four values;
5 cm−1, 10 cm−1, 15 cm−1and 20 cm−1, so that calculated IC rate ki v→ f [or lifetimeτl= 1/ki v→ f(l)] for single promoting
mode l is function of dephasing width. Then, we can estimate total lifetime as τT = 1 lki v→ f(l) = 1 l(1/τl) , (14)
where summation is over three promoting modes (7th, 8th,
and 15th in C1 group symmetry). All results are given in
TableVII. TableVIIshows that calculated IC rate constants
(the lifetimes) increase (decrease) with the increase of the de-phasing width for each of the three promoting modes. For ex-ample, the TD(B3LYP) calculation indicates that the IC rate
constant of mode 8th increases from 2.33 × 107 to 4.23 ×
107 s−1when dephasing width increases from 5 to 10 cm−1.
It should be emphasized that the electronic part of IC rate
1
¯2|Rl( f i )|2is independent to dephasing width, and thus it is
nuclear part of IC rate that is depending on dephasing width. How to determine dephasing width seems becoming a prob-lem. If we use consistent choice of dephasing width for both calculations of the DF spectrum and IC rate, we should choose
the dephasing width as 10 cm−1 that was used for the DF
spectrum simulation in Sec.III C. Atγi f = 10 cm−1, the
cal-culated total lifetimes of the decay1B2→1A1are 11 and 19
ns, respectively, from TD(B3LYP) and HF/CIS calculations
in comparison with the experimental value 14.75± 0.34 ns
(Ref. 16) [the lifetime of 0–0 de-exciting transition is
con-sidered so thatgvk(t)= 1 in Eq. (9)]. The present
calcula-tions show very good agreement with experiment for IC rate constant (or decay lifetime). Taking both the approximations introduced in the present calculations and the experimental uncertainties into consideration, we conclude that the differ-ence between the calculated and the experimental lifetimes (or the IC rates) is quite reasonable. We added anharmonic
TABLE VI. The coupling matrix elements and electronic part of the IC rate of three dominant promoting modes. See detail in context. The Mulliken notation is used to denote the vibrational mode.
TD(B3LYP) HF/CIS
f|∂/∂ Ql|i ¯12|Rl( f i )|2 f|∂/∂ Ql|i ¯12|Rl( f i )|2
Mode Sym. (a.u.) (×1015cm−1/s) (a.u.) (×1015cm−1/s)
7th a 0.1482 1.1996 0.1460 0.9352
8th a 0.1324 3.4187 0.1284 2.6908
TABLE VII. The evaluated IC rate (ki v) and lifetime (τl) for each of three promoting modes as well as the
total lifetimes (τT) for the1B2→1A1transition against different dephasing widths.
TD(B3LYP)a HF/CISa Expt.b
γ (cm−1) Mode ki v(107s−1) τl(ns) τT(ns) ki v(107s−1) τl(ns) τT(ns) τT(ns) 5 7th 0.65 154 21 0.50 200 44 14.75± 0.34 8th 2.33 43 0.85 118 15th 1.69 59 0.94 106 10 7th 1.48 68 11 0.84 119 19 8th 4.23 24 2.42 41 15th 3.54 28 2.10 48 15 7th 2.21 45 7.3 1.26 79 12 8th 6.34 16 3.64 27 15th 5.31 19 3.15 32 20 7th 2.95 34 5.4 1.68 60 9.3 8th 8.45 12 4.85 21 15th 7.08 14 4.20 24
aThe vertical transition energies in TableIIIare used for the calculations of IC rates. bReference16.
corrections to the second part of IC rate, and its results are the same as harmonic approximation.
IV. CONCLUDING REMARKS
In the present studies, we have simulated the DF spec-trum and IC rate constant of fluorobenzne following excita-tion of the 1B
2(S1) ← 1A1(S0) transition from the S1 state
to S0 state by using displaced harmonic oscillator
approx-imation with and without including anharmonic and dis-torted correction. Starting from optimization of equilibrium geometries and the corresponding normal mode frequencies
of S0 and S1 states, we obtained geometry parameters and
frequencies generally in good agreement with experimental results2,4,6 and the previous theoretical calculations.5 Three kinds of functionals (B3LYP, B3LYP-35, and BHandHLYP) with (TD)DFT method plus HF/CIS method were adopted for
ab initio calculations for ground and the first excited states.
Based on accurate geometry parameters and frequencies cal-culated by these methods, we could compute displacement
di between equilibrium geometries of S0 and S1 states and
Huang–Rhys factor Si accurately. Therefore, we are pretty
confident about the present simulations on the DF spectrum and IC rate constant. Displaced harmonic oscillator approxi-mation presented very good examinations of the spectral pro-file and the assignments of the active fundamental normal modes in the DF spectrum of fluorobenzene, and its results basically agree well with the previous assignments studied in the literatures.1–10 Actually, the present calculations proved that totally symmetry vibrational modes dominate spectral profile and the assignments, especially described by the
Franck–Condon progression from the v9 and v10 modes and
then we could also assign the dominant combination bands
with moderate intensities, such as the bands 11011001 and
100
11402, and so on. This is in good agreement with the
ex-periment. Anharmonic corrections (which correct only totally symmetry modes) improved harmonic simulations simultane-ously for the intensity order of 90
1and 1001bands and
diminish-ing 10
1transition. Anharmonic corrections also improved the
other small bands from the rest of totally symmetry modes. We conclude that the amount of HF exchange (from 20% in B3LYP, 35% in B3LYP-35, 50% in BHandHLYP, and 100% in HF) does impact the geometries and vibrational frequencies of FB molecule, but not the relative intensities of the transi-tions. It is anharmonic corrections that influence the relative intensities of the transitions.
We considered using the distorted corrections (which only correct nontotally symmetry modes) to explain the over-tones of out-of-plane vibrations which arise from the nonto-tally symmetric vibrational modes. The present calculations indicated that some of small peaks in the DF spectrum are due to contributions from the distorted effect. However, its con-tributions to spectra are basically small in comparison with totally symmetry modes.
By using the CASSCF method, we calculated the elec-tronic matrix elements of nonadiabatic coupling between the
S0 and S1 states and then we computed the single vibronic
level internal conversion rate of the1B
2(S1)→1A1(S0)
tran-sition within the collision-free condition. It was found that
the IC rate is sensitive to the dephasing width. Around γi f
= 10 cm−1that is adopted for spectrum simulation, the
calcu-lated total lifetimes of this decay are in good agreement with the experimental observation.
ACKNOWLEDGMENTS
R. He would like to thank Postdoctoral Fellowship supported by National Chaio Tung University. L. Yang would like to thank 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 no. 97–2113-M-009–010-MY3. C. Zhu would like to thank the MOE-ATU project of the National Chiao Tung University for support. R. He would like to thank the National Natural Science Foundation of China (no. 20803059) for sup-port. The first two authors contributed equally to the present paper.
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![TABLE I. The equilibrium geometries of the ground state [S 0 ( 1 A 1 )] and the first excited state [S 1 ( 1 B 2 )] optimized by different levels of method for the](https://thumb-ap.123doks.com/thumbv2/9libinfo/7631109.135103/5.918.82.838.128.356/table-equilibrium-geometries-ground-excited-optimized-different-levels.webp)
