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Journal of Molecular Structure: THEOCHEM 814 (2007) 99–103

Franck–Condon simulation of photoelectron spectroscopyof O3

�: Including Duschinsky effects

Jun Liang *, Haiyan Zheng, Xiaowei Zhang, Renzhong Li, Zhifeng Cui

College of Physics and Electronic Information, Anhui Normal University, Wuhu 241000, PR China

Received 8 February 2007; received in revised form 27 February 2007; accepted 2 March 2007Available online 12 March 2007

Abstract

Geometry optimization and harmonic vibrational frequency calculations have been performed on the ~X1A1 state of O3 and ~X2B1 stateof O3

�. Franck–Condon analyses and spectral simulation were carried out on the first photoelectron band of O�3 . The theoretical spec-trum obtained by employing CCSD(T)/6-311+G(2d, p) values are in excellent agreement with the observed one. In addition, the equi-librium geometry parameters, re(OO) = 0.1355 ± 0.0005 nm and he(O–O–O) = 113.5 ± 0.5�, of the ~X2B1 state of O3

�, are derived byemploying an iterative Franck–Condon analysis procedure in the spectral simulation.� 2007 Elsevier B.V. All rights reserved.

Keywords: Ab initio calculations; Franck–Condon analysis; Spectral simulation; Anions

1. Introduction

Recently, we have determined the geometries of theanions HNO� and DNO� by applying Franck–Condon(FC) analyses to their photoelectron spectra [1]. In thestudy, the ab initio force constants were used in FC analy-sis via the reduced-mass-weighted atom displacementmatrix as obtained from an ab initio frequency calculation.With quantum chemical computing programs being readilyavailable, geometries and normal modes of small to med-ium size molecules in different electronic states can nowbe calculated routinely. Based on these methods, numerousapplications of FC calculations have been presented in theliterature [1–13], and most of these studies just focus on theinterpretation of experimentally known spectra. Becausethe geometry difference between two electronic states is amajor factor that influences FC intensities, the simulationof vibronic spectra of polyatomic molecules can beregarded as a valuable test with respect to the quality of

0166-1280/$ - see front matter � 2007 Elsevier B.V. All rights reserved.

doi:10.1016/j.theochem.2007.03.002

* Corresponding author. Tel.: +86 0553 5850091; fax: +86 0553 3869748.

E-mail address: jliang@mail.ahnu.edu.cn (J. Liang).

calculated geometries and as a starting point to obtainimproved structures. In addition, spectral simulations ofvibrational structure based on computed Franck–Condonfactors (FCF) could provide fingerprint type identificationof an observed spectrum, in terms of both the carrier andthe electronic states in the transition (see, for examples,[8,9], and references therein). Also, it has been demon-strated that spectral simulations can be very useful in estab-lishing vibrational assignments in an electronic spectrumobserved with complex vibrational structure [1–4,6,9,10].Moreover, even if the electronic spectrum is not rotation-ally resolved, if the geometrical parameters of one of theelectronic states is well established by, for example, micro-wave spectroscopic measurements, the geometrical param-eters of the other state can be determined by estimating thegeometry change between the states by the iterativeFranck–Condon analysis (IFCA) method [1–4,6–13].

In addition, the ozone molecule has received consider-able attention in recent year because of its pivotal role inatmospheric chemistry. While a hazardous pollutant nearthe earth surface, ozone protects lives on the earth byabsorbing ultraviolet solar rays in the stratosphere. Therehave been many theoretical and experimental studies on

100 J. Liang et al. / Journal of Molecular Structure: THEOCHEM 814 (2007) 99–103

O3 and O3� [14–26]. Research of the spectrum and dynam-

ics in the ground electronic state of O3 and its anion O3� is,

however, comparatively scarce. In this paper, geometryoptimization and harmonic vibrational frequency calcula-tions were performed on the ~X1A1 state of O3 and ~X2B1

state of O3�. Franck–Condon analyses and spectral simula-

tion were carried out on the first photoelectron band ofO3� (i.e., ~X1A1 � ~X2B1 transitions) [26]. Furthermore,

the equilibrium geometries of the ~X2B1 state of O3� are

derived, by employing the iterative Franck–Condon analy-sis procedure in the spectral simulation.

2. Theoretical methods and computational details

Geometry optimization and harmonic vibrational fre-quency calculations were carried out on the ~X1A1 statesof the neutral molecule O3, and the ~X2B1 state of the neg-ative ion O3

� at the QCISD, CCSD, QCISD(T) andCCSD(T) levels with the 6-311+G(2d,p) basis sets. TheQCISD, CCSD, QCISD(T) and CCSD(T) calculationswere performed employing the Gaussian 03 suite of pro-grams [27] on the SGI workstation at the Computing Cen-ter of the Hefei Institute of Physical Science.

Different methods [28–38] have been proposed to calcu-late multidimensional Franck–Condon integrals. We chosethe multidimensional generating function methoddescribed by Sharp and Rosenstock [29] for the integrals.The FCFs are easily produced using the algebraic expres-sions given in our previous paper (see Ref. [1]). FCF calcu-lations on the ~X1A1 � ~X2B1 photo-detachment werecarried out, employing the ab initio force constants andgeometries initially (see later text) for the two electronic

Table 1Summary of some computed and experimental geometrical parameters and vilevels of calculation

Method re (nm) he (deg

QCISD/6-311+G(2d,p) 0.12603 117.89CCSD/6-311+G(2d,p) 0.12556 117.74QCISD(T)/6-311+G(2d,p) 0.12843 117.22CCSD(T)/6-311+G(2d,p) 0.12816 117.092R CISD/DZPa 0.1271 116.2CCSD(T)/DZPb 0.1287 116.8CCSDT/DZPb 0.1286 116.7CCSD(T)/ANOc 0.1273 117.1CASSCF/DZPd 0.1296 116.5B-CCD(T)/DZPd 0.1288 116.8CISD[TQ]/DZPd 0.1281 116.7CASSCF/cc-pVQZ+(1s1p)e 0.12781 116.95icMRCI/cc-pVQZ+(1s1p)e 0.12701 116.96icMRCI+Q/cc-pVQZ+(1s1p)e 0.12743 116.86Expt. 0.12717f 116.78

a Ref. [14].b Ref. [15].c Ref. [16].d Ref. [17].e Ref. [18].f Ref. [19].g Ref. [20].

states involved in the transition. The harmonic oscillatormodel was employed and Duschinsky rotation wasincluded in the FCF calculations. The computed FCFswere then used to simulate the vibrational structure ofthe ~X1A1 � ~X2B1 photo-detachment spectrum of O3

�,employing a Gaussian line-shape and a full-width-at-half-maximum (FWHM) of 200 cm�1 for the ~X1A1 � ~X2B1

detachment.In order to obtain a reasonable match between the sim-

ulated and observed spectra, the iterative Franck–Condonanalysis procedure [38] was also carried out, where theground state geometrical parameters of the O3

� moleculeswere fixed to the experimental values for the ~X1A1 � ~X2B1

photo-detachment processes, while the ground state geo-metrical parameters of the anion O3

� were varied systemat-ically. Thus, the ground state geometrical parameters ofO3� were varied until a best match between the simulated

and observed spectra was obtained.

3. Results and discussions

3.1. Geometry optimization and frequency calculations

The optimized geometric parameters and computedvibrational frequencies for the ~X1A1 states of O3 and~X2B1 states of O3

� as obtained in this work are listed inTables 1 and 2. The theoretical and/or experimental valuesavailable in the literatures are also included for compari-son. The bending vibration is denoted x2, according tothe convention for triatomic molecules.

From Table 1, for the state ~X1A1 of O3, the computedbond lengths and angles obtained at different levels of

brational frequencies (cm�1) of the ~X1A1 state of O3 obtained at different

) x1 (a1) x2 (a1) x3 (b2)

1 1229.59 736.26 931.198 1256.99 751.08 1243.367 1119.08 693.64 918.824 1128.74 702.85 1014.30

1234 745 13521129 703 9761141 705 10771153 718 10531098 689 9891127 703 9761166 716 1138

1125.16 718.79 1105.28f 1135g 716g 1089g

Fig. 1. Simulated photoelectron spectrum and the stick diagram oftheoretical FCFs for the ð~X1A1 � ~X2B1Þ detachment process invoking theCCSD/6-311+G(2d,p) geometries for the two combining states (Tables 1and 2) with vibrational assignments provided. The FWHM used for thesimulated spectrum is 200 cm�1.

J. Liang et al. / Journal of Molecular Structure: THEOCHEM 814 (2007) 99–103 101

calculation seem to be highly consistent. For re(OO)and he(O–O–O), the largest deviations between calculatedand experimental bond lengths and angles are less than0.002 nm and 1.1�, respectively (see Table 1). The estimatedvalues based on ab initio theory at the CCSD(T)/6-311+G(2d,p) level, are 0.12816 nm and 117.094�. Thedifferences between calculated and experimental valuesare only 0.00096 nm and 0.294� for re and he, respectively.Both the optimized geometric parameters and the vibra-tional frequencies calculated at CCSD(T)/6-311+G(2d,p)level gave the best agreement with the correspondingavailable experimental values, and were therefore utilizedin subsequent iterative FC analyses and spectralsimulation.

From Table 2, for the state ~X2B1 of O3�, the computed

bond lengths and angles obtained at different levels of cal-culation seem to be highly consistent. However, the differ-ences between the computed bond angle and experimentalone are bigger. It is expected that the geometrical parame-ters obtained at the higher levels of calculation should bethe more reliable. Regarding the computed vibrational fre-quencies, for the state ~X2B1 of O3

�, the values obtained atthe various levels are reasonably consistent. The estimatedvalues at the CCSD(T)/6-311+G(2d,p) level, are 1016.82,573.99 and 963.73 cm�1. The differences between calculatedand experimental values are only 47.82, 23.99 and83.73 cm�1 for the symmetric stretching, bending and anti-symmetric stretching modes, respectively. The CCSD(T)/6-311+G(2d,p) results are the best overall agreement to thecorresponding available experimental and theoretical val-ues, and were therefore utilized in subsequent iterativeFC analyses and spectral simulations.

Table 2Summary of some computed and experimental geometrical parameters and vibrational frequencies (cm�1) of the ~X2B1 state of O�3 obtained at differentlevels of calculation

Method re (nm) he (deg) x1 (a1) x2 (a1) x3 (b2)

QCISD/6-311+G(2d,p) 0.13517 115.363 1033.26 586.99 576.58CCSD/6-311+G(2d,p) 0.13453 115.249 1072.49 603.71 926.22QCISD(T)/6-311+G(2d,p) 0.13677 115.322 990.28 563.00 960.87CCSD(T)/6-311+G(2d,p) 0.13628 115.251 1016.82 573.99 963.73CASSCF/D95+(d)a 0.1374 115.63CASPT2/ D95+(d)a 0.1376 115.86B3LYP/6-311+G(d)a 0.1352 115.64 1051 599 874CASPT2/ANOa 0.1361 115.4 989 556 870CASSCF/DZPb 0.1385 115.4 976 552CCSD(T)/cc-pVDZc 0.1361 115.4 1025 569 900CCSD(T)/cc-pVTZc 0.1358 115.3 1044 587 911Expt. 0.134(0.003)d 112.6(2.0)d 982(30)e 550(50)e

Expt.f 0.136(0.003) 111.8(2.0) 975(50) 550(50) 880(50)

a Ref. [21].b Ref. [22].c Ref. [23].d Ref. [24].e Ref. [25].f Ref. [26].

102 J. Liang et al. / Journal of Molecular Structure: THEOCHEM 814 (2007) 99–103

3.2. Franck–Condon simulations

The simulated photoelectron spectra of theO3ð ~X1A1Þ �O�3 ð~X2B1Þ photo-detachment using dataobtained from the CCSD/6-311+G(2d,p) and CCSD(T)/6-311+G(2d,p) calculations are shown in Figs. 1 and 2,respectively. Vibrational assignments for the symmetricstretching x1 and bending x2 modes of the neutral mole-cule O3 are also provided, respectively, with the label(n, 0,0–0,0,0) and (n, 1,0–0,0,0) corresponding to the(x1,x2,0–0,0,0) transition. From the harmonic calcula-tion, it was found that the FCFs for transitions involvingthe asymmetric stretching mode x3 are negligibly smalland therefore the x3 mode is not included in the assign-ments. In the spectral simulation, a FWHM of 200 cm�1

was utilized with Gaussian band envelopes for each vibra-tional component. At first glance, these two spectra lookquite different. Yet the major features due to the twoprogressions of the bending and symmetric stretchingmodes can be identified. The large difference between thetwo spectra indicates that the calculated vibrational inten-sity distributions are very sensitive to the geometries uti-

Fig. 2. Simulated photoelectron spectrum and the stick diagram oftheoretical FCFs for the ð~X1A1 � ~X2B1Þ detachment process invoking theCCSD/6-311+G(2d,p) geometry for the ~X1A1 state of O3 (Table 1) andthe CCSD(T)/6-311+G(2d,p) one for the ~X2B1 state of O3

� (Table 2) withvibrational assignments provided. The FWHM used for the simulatedspectrum is 200 cm�1.

lized. Small changes in the structural parameters lead toa mark variation in the theoretical FCFs. This implies thatthe FC method employed is a very sensitive one in obtain-ing reliable geometric parameters for the system understudy.

The variations of geometries of the molecules betweenthe electronic states using the iterative FC analysis (IFCA)method would yield better matches between the simulatedand observed spectra than that obtained with the ab initiogeometries. Since the experimental geometry of the ~X1A1

state of O3 is available, the IFCA method was carriedout on the ~X2B1 state of O3

�. The simulated spectrum,which matches best with observation, is shown in Fig. 3bwith the experimental observed photoelectron spectrumshown in Fig. 3a. It was found that the computed photo-electron spectrum of O3

� for the ~X1A1 � ~X2B1 detachmentis almost identical to the experimental spectra. Thissuggests that the computed geometry changes upondetachment by IFCA method are highly accurate, andthe harmonic model seems to be reasonably adequate.However, Discrepancies between simulation and observa-tion become larger for peaks with higher quantum

Fig. 3. (a) The experimental photoelectron spectrum of O3� (from Ref.

[26]) and (b) the simulated spectrum invoking the experimental geometryfor the ~X1A1 state of O3 and the IFCA one for the ~X2B1 state of O3

� withvibrational assignments provided for the ð~X1A1 � ~X2B1Þ detachmentprocess. The FWHM used for the components of the simulated spectra is200 cm�1.

J. Liang et al. / Journal of Molecular Structure: THEOCHEM 814 (2007) 99–103 103

numbers. This is mainly due to anharmonicity effects notincluded in the FCF calculation. With vibrational quantumnumbers increasing, the stronger the anharmonicity effect isand the greater the influence on the simulated spectra.

By using the IFCA method, reasonably reliable bondlength re and bond angle he for the ground state ~X2B1 stateof O�3 can be deduced based on the experimental structuralparameters of O3 in the ~X1 A1 state and the computedgeometry changes upon photo-detachment at theCCSD(T)/6-311+G(2d,p) level. For the ~X1A1 � ~X2B1

photo-detachment spectrum, the latter values were shownto be reliable from the excellent agreement between thesimulated and observed spectra (see Fig. 3). Hence, byusing the iterative Franck–Condon analysis method, thebest IFCA bond length re and bond angle he obtained forthe ground state ~X2B1 state of O3

�, employing theCCSD(T)/6-311+G(2d,p) force constants, are 0.1355 ±0.0005 nm and 113.5 ± 0.5�, respectively.

Comparison of the O3� geometry obtained from this

analysis with other experiments and theoretical studiesfinds reasonable agreement in many cases and a slight dis-agreement in others. Excellent agreement is found with theresults of a vibrationally resolved O3

� photo-detachmenttotal cross-section measurement [re = 0.134 ± 0.003 nm;he = 112.6 ± 2.0�] by Wang et al. [24]. Ab initio calcula-tions predict O3

� bond lengths in agreement with the pres-ent results but predicts a larger bond angle that determinedfrom the iterative FC analysis (see Table 2).

4. Conclusion

In the present study, attempts have been made to simu-late the observed O3ð~X1A1Þ � O�3 ð~X2B1Þ detachment pho-toelectron spectrum using a harmonic model and includingDuschinsky effects. In the case of the photoelectron spec-trum of the ~X1A1 � ~X2B1 detachments, it seems that theharmonic model is reasonably adequate. The rather reliablebond length re(OO) and bond angle he(O–O–O) wereobtained, through the IFCA procedure. Based on the sen-sitivity of the relative intensities towards the variation ofthe bond length and bond angle, the uncertainties in there(OO) and he(O–O–O) are probably around ±0.0005 nmand ±0.5�, respectively.

Acknowledgements

This work was supported by the National NaturalScience Foundation of China (No.10674002), the Programfor Innovative Research Team in Anhui Normal Univer-sity, and the Doctoral Research Foundation of AnhuiNormal University.

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