Journal of Electron Spectroscopy and Related Phenomena 135 (2004) 119–125

Franck-Condon simulation of photoelectron spectroscopy of HOO−

and DOO−: including Duschinsky effects

Jun Lianga,c, Haiyang Lia,b,∗a Laboratory of Environment Spectroscopy, Anhui Institute of Optics and Fine Mechanics, Chinese Academy of Sciences,

P.O. BOX 1125, Hefei 230031, PR Chinab Dalian Institute of Chemical Physics, Chinese Academy of Science, Dalian 116023, PR China

c Department of Physics, Wuhu Normal College, Anhui Province, Wuhu 241008, PR China

Received 27 November 2003; received in revised form 12 February 2004; accepted 18 February 2004

Available online 9 April 2004

Abstract

A theoretical method to calculate multidimensional Franck-Condon (FC) factors including Duschinsky effects is described and used tosimulate the photoelectron spectra of HOO− and DOO− radicals. Geometry optimization and harmonic vibrational frequency calculationshave been performed on theX2A ′′ state of HOO/DOO andX1A ′ state of HOO−/DOO−. In addition, the equilibrium geometry parameter,R(OO) = 0.1493± 0.0005 nm, of theX1A ′ state of HOO−/DOO− are derived by employing an iterative Franck-Condon analysis (IFCA)procedure in the spectral simulation.© 2004 Elsevier B.V. All rights reserved.

Keywords: Franck-Condon factor; Duschinsky effect; Spectral simulation; Photoelectron spectra

1. Introduction

Recently, we have determined the geometries of the an-ions HNO− and DNO− by applying Franck-Condon (FC)analyses to their photoelectron spectra[1]. In the study,the ab initio force constants were used in FC analysisvia the reduced-mass-weighted atom displacement matrixas obtained from an ab initio frequency calculation. Withquantum chemical computing programs being readily avail-able, geometries and normal modes of small to mediumsize molecules in different electronic states can now becalculated 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 simulation ofvibronic spectra of polyatomic molecules can be regardedas a valuable test with respect to the quality of calculated

∗ Corresponding author. Tel.:+86-551-559-3204;fax: +86-551-559-1550.

E-mail addresses: liangjun@aiofm.ac.cn (J. Liang), hli@aiofm.ac.cn(H. Li).

geometries and as a starting point to obtain improvedstructures. In addition, spectral simulations of vibrationalstructure based on computed Franck-Condon factors (FCF)could provide fingerprint type identification of an observedspectrum, in terms of both the carrier and the electronicstates in the transition (see, for examples[8,9], and refer-ences therein). Also, it has been demonstrated that spectralsimulations can be very useful in establishing vibrationalassignments in an electronic spectrum observed with com-plex vibrational structure[1–4,6,9,10]. Moreover, even ifthe electronic spectrum is not rotationally resolved, if thegeometrical parameters of one of the electronic states iswell established by, for example, microwave spectroscopicmeasurements, the geometrical parameters of the otherstate can be determined by estimating the geometry changebetween the states by the iterative Franck-Condon analysis(IFCA) method[1–4,6–13].

In this paper, the theoretical methods to calculate multidi-mensional Franck-Condon factors are described. Geometryoptimization and harmonic vibrational frequency calcula-tions were performed on theX2A ′′ state of HOO/DOO andX1A ′ state of HOO−/DOO−. Franck-Condon analyses andspectral simulation were carried out on the first PE bandof HOO−/DOO− (i.e. X2A ′′–X1A ′ transition) [14,15]. In

0368-2048/$ – see front matter © 2004 Elsevier B.V. All rights reserved.doi:10.1016/j.elspec.2004.02.159

120 J. Liang, H. Li / Journal of Electron Spectroscopy and Related Phenomena 135 (2004) 119–125

addition, employing the iterative Franck-Condon analysisprocedure in the spectral simulation, the equilibrium geom-etry of theX1A ′ state of HOO−/DOO− is derived.

2. Theoretical method

Different methods[16–22] have been proposed to calcu-late multidimensional Franck-Condon integrals. We chosethe multidimensional generating function method describedby Sharp and Rosenstock[17] for the integrals. Firstly, theDuschinsky effect, or mode mixing of the initial and finalelectronic states, is expressed as

Q′ = JQ + K (1)

where theQ andQ′ are the normal coordinates of the twoelectronic states, respectively, and the matrixJ and columnvectorK define the linear transformation. To obtain the ma-trix J and vectorK in Eq. (1), in terms of Cartesian coordi-nates displacements[23–25], the general linear transforma-tion of an arbitrary distortionX can be written as

X′ = ZX + R (2)

whereX andX′ are distortions expressed as Cartesian dis-placements from the equilibrium geometries of the anionand neutral, respectively.R = ZReq − R′

eq is the change inequilibrium geometry between the anion and the neutral inCartesian coordinates centered on the molecular center ofmass, andZ is a rotation matrix that removes contributionsto the Franck-Condon factors due to overall rotation of themolecule, which can be derived by using a static analogueto the Eckhart conditions (see[26,27]). In a conventionalnormal–mode analysis and in terms of the Gaussian03 (g03)output for the two electronic states, finally,J and K aregiven by

J = [M(g03′)(V′)−1/2]†Z(g03)V−1/2 and

K = [M(g03′)(V ′)−1/2]†R (3)

(see[1] for the definitions of these symbols used above).Having then computedJ andK, the Franck-Condon factorsare easily produced by using the more general algebraicexpressions given in[1,28].

3. Computational details

Geometry optimization and harmonic vibrational fre-quency calculations were carried out on theX2A ′′ state ofthe neutral molecule HOO/DOO, andX1A ′ state of the neg-ative ion HOO−/DOO− at the B3LYP, QCISD, QCISD(T),CCSD and CCSD(T) levels with the 6-311+ G(d, p) basissets. The B3LYP, QCISD, QCISD(T), CCSD and CCSD(T)calculations were performed employing the Gaussian03suite of programs[29] on the SGI workstation at the Com-puting Center of the Hefei Institute of Physical Science.

FCF calculations on theX2A ′′–X1A ′ photo-detachmentwere carried out, employing CCSD(T)/6-311+ G(d, p)force constants and geometry for the two electronic statesinvolved in the transition. The theoretical method usedin the FCF calculations has been described inSection 2.Briefly, the harmonic oscillator model was employed andDuschinsky rotation was included in the FCF calculations.The computed FCFs were then used to simulate the vi-brational structure of theX2A ′′–X1A ′ photo-detachmentspectra of HOO−/DOO−, employing a Gaussian line-shapeand a full-width-at-half-maximum (FWHM) of 300 cm−1

for the X2A ′′–X1A ′ detachment.In order to obtain a reasonable match between the sim-

ulated and observed spectra, the iterative Franck-Condonanalysis procedure[8,9] was also carried out, where theground state geometrical parameters of the HOO/DOOmolecule were fixed to the experimental values for theX2A ′′–X1A ′ photo-detachment processes, while the groundstate geometrical parameters of the HOO−/DOO− werevaried systematically. Thus, the ground state geometricalparameters of the HOO−/DOO− were varied until a bestmatch between the simulated and observed spectra wasobtained.

4. Results and discussions

4.1. Geometry optimization and frequency calculations

The optimized geometric parameters and computed vibra-tional frequencies for theX2A ′′ states of HOO and DOO,andX1A ′ state of HOO− and DOO− as obtained in this workare listed inTables 1–4. The theoretical and/or experimen-tal values available in the literatures are also included forcomparison. The bending vibration is denotedω2, accordingto the convention for triatomic molecules. The other vibra-tional modes are listed in decreasing order of size, whichdesignates the H–O (and D–O) stretch asω1and the O–Ostretch asω3.

From Tables 1 and 2, for the X2A ′′ state of HOO andDOO, the computed bond lengths and angles obtained at dif-ferent levels of calculation seems to be highly consistent. ForR(HO)/R(DO),R(OO) and∠(HOO)/∠(DOO), the largest de-viations between calculated and experimental bond lengthsand angles are less than 0.0006, 0.00085 nm and 1.848◦, re-spectively (seeTables 1 and 2). The geometrical parametersobtained at the higher levels of calculation should be themore reliable. The estimated values based on the ab initiotechniques at CCSD(T)/6-311+ G(d, p) level, are 0.09705,0.13329 nm and 105.0097◦. The differences between calcu-lated and experimental values are only 0.00055, 0.00038 nmand 0.37◦ for R(HO)/R(DO), R(OO) and∠(HOO)/∠(DOO),respectively. As for the vibrational frequencies, for theX2A ′′state of HOO and DOO, the CCSD(T)/6-311+ G(d, p) re-sults compared with the corresponding available experimen-tal values and other theoretical values are also reasonably

J. Liang, H. Li / Journal of Electron Spectroscopy and Related Phenomena 135 (2004) 119–125 121

Table 1Summary of some computed and experimental geometrical parameters and vibrational frequencies (cm−1) of the X2A ′′ state of HOO obtained at differentlevels of calculation

R(HO) (nm) R(OO) (nm) ∠(HOO) (◦) ω1 (H–O) ω2 (bend) ω3 (O–O)

B3LYP/6-311+ G (d, p) 0.09771 0.13282 105.8684 3600.7 1423.0 1155.6QCISD/6-311+ G(d, p) 0.09710 0.13346 104.7071 3711.4 1441.6 1112.1QCISD(T)/6-311+ G(d, p) 0.09735 0.13376 104.5830 3671.4 1425.5 1105.9CCSD/6-311+ G(d, p) 0.09705 0.13261 105.0097 3719.4 1458.6 1158.7CCSD(T)/6-311+ G(d, p) 0.09705 0.13329 104.3911 3700.6 1435.2 1137.0MP2/6-311+ G(d)a 0.09695 0.13109 104.8MP2/6-311G(d, p)a 0.09695 0.13093 104.819 3735 1461 1247QCISD/6-311G(d, p)a 3734 1446 1125MP4/6-311G(d, p)a 0.09709 0.13246 104.519B3LYP/6-311G(d, p)b 0.0976 0.1328 105.5 3533 1418 1162CASSCF/CCIc 0.0971 0.1330 104.3GVB/PPd 0.0981 0.1364 103.2 3611 1457 1094GVB-CId 0.0983 0.1367 103.3 3612 1438 1110POL-CId 0.0991 0.1369 103.3 3655 1416 1181QCISD(T)/cc-pVDZe 0.0978 0.1344 103.7QCISD(T)/aug-cc-pVDZe 0.0979 0.1351 103.8CCSD(T)/aug-cc-pVDZe 0.0978 0.1348 103.9Experiment 0.0976f 0.13291f 104.02f 3415.1g 1391.75h 1097.63i

a [30].b [14].c [31].d [32].e [33].f [34].g [35].h [36].i [15].

good agreement (seeTables 1 and 2), and were thereforeutilized in subsequent FC analyses and spectral simulations.

From Tables 3 and 4, for the X1A ′ state of HOO− andDOO−, the computed bond lengthsR(HO)/R(DO) andR(OO) obtained by different theoretical methods are seen tobe highly consistent. However, the computed bond angle ofthe X1A ′ state appears to be more sensitive to the levels ofcalculation. FromTables 3 and 4, it can be seen that the com-puted angles converge towards smaller values when the lev-els of calculation are improved from B3LYP/6-311+ G(d,p) to QCISD/6-311+ G(d, p) or CCSD/6-311+ G(d, p),and then to QCISD(T)/6-311+ G(d, p) or CCSD(T)/6-311+ G(d, p). This shows the effects of higher-order electron

Table 2Summary of some computed and experimental geometrical parameters and vibrational frequencies (cm−1) of the X2A ′′ state of DOO obtained at differentlevels of calculation

R(DO) (nm) R(OO) (nm) ∠(DOO) (◦) ω1 (D–O) ω2 (bend) ω3 (O–O)

B3LYP/6-311+ G (d, p) 0.09773 0.13282 105.8646 2630.4 1047.4 1169.6QCISD/6-311+ G(d, p) 0.09710 0.13346 104.7368 2712.4 1057.5 1127.2QCISD(T)/6-311+ G(d, p) 0.09738 0.13376 104.5313 2680.9 1044.4 1122.4CCSD/6-311+G(d, p) 0.09705 0.13261 105.0097 2718.4 1074.3 1170.1CCSD(T)/6-311+G(d, p) 0.09736 0.13347 104.6489 2683.2 1050.6 1138.8Experiment 0.0976a 0.13291a 104.02a 2529.2b 1027.3b 1124.7b

a [34].b [35].

correlation on the computed bond angle of theX1A ′ stateof HOO−/DOO−. For the X1A ′ state of HOO−/DOO−,because no experimental geometric values are available forcomparison, 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 theX1A ′ state of HOO−/DOO−, the valuesobtained at the various levels are reasonably consistent.The CCSD(T)/6-311+ G(d, p) results compared with thecorresponding available experimental values and other the-oretical values are also reasonably good agreement (seeTables 3 and 4), and were therefore utilized in subsequentFC analyses and spectral simulation.

122 J. Liang, H. Li / Journal of Electron Spectroscopy and Related Phenomena 135 (2004) 119–125

Table 3Summary of some computed and experimental geometrical parameters and vibrational frequencies (cm−1) of the X1A ′ state of HOO− obtained atdifferent levels of calculation

R(HO) (nm) R(OO) (nm) ∠(HOO) (◦) ω1 (H–O) ω2 (bend) ω3 (O–O)

B3LYP/6-311+ G(d, p) 0.09625 0.15219 99.3628 3782.3 1122.9 768.4QCISD/6-311+ G(d, p) 0.09589 0.15139 98.9490 3854.5 1141.6 763.3QCISD(T)/6-311+ G(d, p) 0.09613 0.15359 97.8880 3822.3 1097.5 705.1CCSD/6-311+ G(d, p) 0.09585 0.15085 99.1571 3861.6 1156.0 787.8CCSD(T)/6-311+ G(d, p) 0.09610 0.15345 97.9325 3826.7 1101.9 708.4GVB/PPa 0.0966 0.1557 97.6 3539 1205 755GVB-CIa 0.0968 0.1543 97.1 3583 1155 777POL-CIa 0.0972 0.1529 98.5 3662 1153 821QCISD(T)/6-311+ + G(2df,pd)b 0.09619 0.15270 97.34 3827.0 1099.0 745.0CCSD(T)/aug-cc-pVTZc 0.09593 0.15219 97.64 3810.4 1120.8 758.3CCSD(T)/aug-cc-pVQZd 0.09596 0.15204 97.77 3804.6 1131.4 758.2CCSD-T/aug-cc-pVQZd 0.09594 0.15200 97.79 3807.5 1132.9 758.8Experiment 775 (250)e

a [32].b [37].c [38].d [39].e [32].

4.2. FC analyses and spectral simulations

The simulated photoelectron spectra of HOO− and DOO−using data obtained from the CCSD(T)/6-311+ G(d, p) cal-culations are shown inFigs. 1 and 2, respectively. The sim-ulated spectra invoking the experimental geometries for theX2A ′′ state of HOO/DOO and the IFCA geometries for theX1A ′ state of HOO−/DOO−, which match best with ob-servation, are shown inFigs. 3(b) and 4(b)with the ex-perimental observed photoelectron spectra[14,15] shownin Figs. 3(a) and 4(a), respectively. Vibrational assignmentsfor the stretching modeω3 of the neutral molecule HOOand DOO are also provided, respectively, with the label (0,0, n – 0,0,0) corresponding to (0, 0,ω3–0, 0, 0) transi-tion. The computed FCFs for the H–O/D–O stretchingω1and the H–O–O/D–O–O bendingω2 modes were found tobe negligibly small, therefore theω1 andω2 mode are not

Table 4Summary of some computed and experimental geometrical parameters and vibrational frequencies (cm−1) of the X1A ′ state of DOO− obtained atdifferent levels of calculation

R(DO) (nm) R(OO) (nm) ∠(DOO) (◦) ω1 (D–O) ω2 (bend) ω3 (O–O)

B3LYP/6-311+ G(d, p) 0.09625 0.15219 99.3724 2765.1 754.4 843.4QCISD/6-311+ G(d, p) 0.09589 0.15139 99.0261 2814.9 754.6 852.1QCISD(T)/6-311+ G(d, p) 0.09617 0.15353 97.9083 2788.9 697.6 820.6CCSD/6-311+ G(d, p) 0.09585 0.15085 99.1571 2821.9 777.3 863.9CCSD(T)/6-311+ G(d, p) 0.09610 0.15338 97.9514 2792.5 701.1 823.7QCISD(T)/6-311+ + G(2df,pd)a 0.09619 0.15270 97.34CCSD(T)/aug-cc-pVTZb 0.09593 0.15219 97.64 2776.0 740.3 843.4CCSD(T)/aug-cc-pVQZc 0.09596 0.15204 97.77 2771.7 742.8 848.4CCSD-T/aug-cc-pVQZc 0.09594 0.15200 97.79 2773.9 743.5 849.3Experiment 900 (250)d

a [37].b [38].c [39].d [32].

included in the assignment. In spectral simulation, a FWHMof 300 cm−1 was utilized with Gaussian band envelopes.

Comparing the simulated spectraFigs. 1 and 2with theobserved onesFigs. 3(a) and 4(a), respectively. It can beseen that the differences between the simulated spectra andthe experimental ones are rather large. The relative intensi-ties of the simulated vibrational peaks did not match thoseobserved. This suggest that for both HOO and DOO, thecomputed geometry changes upon photo-detachment at theCCSD(T)/6-311+ G(d, p) level are not highly accurate. Thevariations of geometries of the molecule between the elec-tronic states using the iterative FC analysis (IFCA) methodwould yield better matches between the simulated and ob-served spectra than that obtained with the ab initio geome-tries. Since the experimental geometry of theX2A ′′ state ofHOO/DOO is available, the IFCA method was carried outon theX1A ′ state of HOO−/DOO−. The simulated spectra,

J. Liang, H. Li / Journal of Electron Spectroscopy and Related Phenomena 135 (2004) 119–125 123

Fig. 1. Simulated photoelectron spectrum of HOO− invoking theCCSD(T)/6-311+ G(d, p) geometries for the two states (Tables 1 and 3)with vibrational assignments provided for the (X2A ′′–X1A ′) detachmentprocess. The FWHM used for the simulated spectrum is 300 cm−1.

which match best with observation, are shown inFigs. 3(b)and 4(b), respectively. It was found that the computed pho-toelectron spectra of HOO−/DOO− for the X2A ′′–X1A ′detachment are almost identical to the experimental spectra.This suggests that the computed geometry changes upondetachment by the IFCA method are rather accurate, and theharmonic model seems to be reasonably adequate. However,a perfect match between simulation and observation overthe whole spectra for theX2A ′′–X1A ′ detachment is notpossible. This is mainly due to anharmonicity effects not

Fig. 2. Simulated photoelectron spectrum of DOO− invoking theCCSD(T)/6-311+ G(d, p) geometries for the two states (Tables 2 and 4)with vibrational assignments provided for the (X2A ′′–X1A ′) detachmentprocess. The FWHM used for the simulated spectrum is 300 cm−1.

Fig. 3. (a) The experimental photoelectron spectrum of HOO− (from [14])and (b) the simulated spectrum invoking the experimental geometry forthe X2A ′′ state of HOO and the IFCA one for theX1A′ state of HOO−with vibrational assignments provided for the (X2A ′′–X1A ′) detachmentprocess. The FWHM used for the components of the simulated spectrais 300 cm−1.

included in the FCF calculation. With vibrational quantumnumbers increasing, the stronger the anharmonicity effectis and the greater the influence on the simulated spectra.The best IFCA bond lengthR(OO) obtained for theX1A ′state of HOO−/DOO−, employing the CCSD(T)/6-311+ G(d, p) force constants, is 0.1493 ± 0.0005 nm. Theuncertainty given reflects the sensitivity of the pre-dicted relative vibrational intensities toward the structuralchanges.

It should be noted that since no H–O/D–O stretchingand H–O–O/D–O–O bending structures (i.e. structures inω1 and ω2), are observed in the first PE band (in the ex-perimental spectra of[14,15], respectively) for both statesof interest, the computedR(HO)/R(DO) bond length and∠(HOO)/∠(DOO) bond angle changes from the CCSD(T)calculations were assumed in the IFCA procedure. In viewof the near-perfect match between the best-simulated spec-trum and the observed spectrum, we anticipate the CCSD(T)R(HO)/R(DO)and∠(HOO)/∠(DOO) changes to be close tothe real ones.

124 J. Liang, H. Li / Journal of Electron Spectroscopy and Related Phenomena 135 (2004) 119–125

Fig. 4. (a) The experimental photoelectron spectrum of DOO− (from [15])and (b) the simulated spectrum invoking the experimental geometry forthe X2A ′′ state of DOO and the IFCA one for theX1A ′ state of DOO−with vibrational assignments provided for the (X2A ′′–X1A ′) detachmentprocess. The FWHM used for the components of the simulated spectrais 300 cm−1.

5. Conclusion

In the present study, attempts have been made to simulatethe observedX2A ′′–X1A ′ detachment photoelectron spectraof HOO−/DOO−, using a harmonic model and includingDuschinsky effects. In the case of the photoelectron spectraof the X2A ′′–X1A ′ detachment, it seems that the harmonicmodel is reasonably adequate. A rather reliable bond lengthR(OO) of HOO−/DOO− was obtained for the first time,through the IFCA procedure. Based on the sensitivity of therelative intensities towards the variation of the bond length,the uncertainty in theR(OO) distance is probably around±0.0005 nm. Nevertheless, focusing on the relative intensi-ties of the vibrational peaks with higher state quantum num-bers, the FC simulated spectra are not reasonably consis-tent with the experimental spectra. Clearly, anharmonicityshould be incorporated into the model for computing FCFsin order to achieve better agreement between simulation andobservation. A simple way to include anharmonicity in an

FCF calculation is to express the anharmonic vibrations aslinear combinations of the products of harmonic oscillatorfunctions. In this way, the anharmonic vibrational FCF canbe reduced to a sum of harmonic overlap integrals, whichcan be evaluated readily with available analytical formu-las. Botschwina et al. are one of the few research groupsperforming anharmonic FCF calculations in this way[40].Recently, details of a few other types of anharmonic FCFmethods have been published[3,6,41–43]. Franck-Condonfactor calculations that consider anharmonicity effects willbe our future work.

Acknowledgements

This work was supported by the National Natural Sci-ence Foundation of China (No. 20073042) and the NaturalScience Foundation of Anhui Province (No. 2001kj263zc),and the Director Research Grants (2003) of Hefei Instituteof Physical Science and Anhui Institute of Optics and FineMechanics, Chinese Academy of Science.

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