the effect of triple excitations in coupled cluster calculations of raman scattering cross-sections

The effect of triple excitations in coupled clustercalculations of Raman scattering cross-sections

Magdalena Pecul a,b,*, Sonia Coriani c

a Department of Chemistry, University of Warsaw, ul. Pasteura 1, 02-093 Warsaw, Polandb Istituto di Chimica Quantistica ed Energetica Molecolare del C.N.R, Area della Ricerca di Pisa, via Moruzzi 1, I-56124, Pisa, Italy

c Dipartimento di Scienze Chimiche, Universit�aa degli Studi di Trieste, via L. Giorgieri 1, I-34127 Trieste, Italy

Received 19 November 2001; in final form 7 February 2002

Abstract

The frequency-dependent linear response approach for the hierarchy of coupled cluster methods CCS/CC2/CCSD/

CC3 has been used to evaluate geometric derivatives of polarizabilities and Q-branch Raman scattering cross-sections

of five diatomics: N2, CO, HF, HCl and Cl2. The triple excitation corrections are substantial for N2 and CO, where they

exceed 10%, and negligible for HF and HCl. The results at the CCSD and CC3 levels with correlation-consistent basis

sets have been used to obtain best estimates for the Raman scattering cross-sections. The agreement between the es-

timated values and experiment is satisfactory, considering the approximations inherent in the harmonic approach and

Placzek’s polarizability theory. � 2002 Published by Elsevier Science B.V.

1. Introduction

With the advancement of laser techniques,vibrational Raman spectroscopy has become astandard tool in chemistry, complementary to IRspectroscopy. Raman intensities are gaining inimportance, both as structural parameters [1], andfor analytical purposes [2]. As a consequence, thereis also considerable interest in developing quantumchemical computational methods capable of pro-viding accurate Raman intensities, possibly at lowcomputational cost [3–8]. For instance, variousimplementations of Density Functional Theory(DFT) for the calculation of Raman parameters

have been reported in recent years [3–6]. An ad-

vantage of DFT over conventional quantum

chemical methods based on a molecular orbital

approach lies in its scaling properties, making it

suitable for calculations of Raman intensities for

large molecules. On the other hand DFT ap-

proaches do not form a hierarchy, in contrast to,

for instance, coupled cluster methodologies, where

the exact non-relativistic values can be in principle

approached by systematically improving the de-

scription of the N-electron and one-electron space.

High-level ab initio results for Raman intensities

of simple molecules are therefore of interest also

for calibrating DFT functionals, especially since

the experimental values of Raman scattering cross-

sections in gases are usually burdened by large

uncertainities [9–11].

2 April 2002

Chemical Physics Letters 355 (2002) 327–338

www.elsevier.com/locate/cplett

* Corresponding author. Fax: +48-22-822-5996.

E-mail addresses: [email protected] (M. Pecul),

[email protected] (S. Coriani).

0009-2614/02/$ - see front matter � 2002 Published by Elsevier Science B.V.

PII: S0009-2614 (02 )00270-1

The first coupled cluster singles and doubles(CCSD) investigation of the Raman scatteringcross-sections of a few linear molecules has beenrecently presented [12], with special emphasis ondispersion and basis set effects. In this Letter, weextend the study to include a detailed investigationof correlation effects on Raman scattering cross-sections by using a hierarchy ofmodels consisting ofcoupled cluster singles (CCS), CC2 [13,14], CCSDand CC3 [14,15]. Special attention is paid to theeffect of triple excitations as described by the CC3linear response method [16]. Previous computa-tional studies of static [17] and frequency-dependent[16,18] molecular (hyper)polarizabilities show thattriple excitation corrections on these properties canbe considerable and that the CC3 response methodgives results close to coupled cluster singles, doublesand triples (CCSDT), though being computation-ally less expensive. As electron correlation in gen-eral tends to influence the geometric derivatives ofthe polarizability more than the polarizability itself[7], the effect of triple excitations on Raman scat-tering cross-sections – which are related to thegeometric derivatives of polarizability – is expectedto be even more significant than on polarizabilities.

The coupled cluster hierarchy of modelsemployed here is defined upon truncation in the(time-dependent) cluster operator T ðtÞ and uponadditional approximations in the (time-dependent)cluster (constraint) equations hlje�T ðtÞHðtÞeT ðtÞjSCF i ¼ 0 [19]. The CCS model corresponds toT ¼ T1 – that is, only the single excitations out of theSCF reference are considered – and CCSD toT ¼ T1 þ T2. The hybrid methods CC2 and CC3 areapproximations to, respectively, CCSD andCCSDT based on a perturbational analysis for theconstraint equations. In CC2 the CCSD equationsfor the singles are retained in the original form, andthose for the doubles are approximated to first-or-der in the fluctuation potential, treating T1 as a zero-order operator. Analogously in CC3 the singles anddoubles equations of CCSDT are retained, whereasthe triples equations are approximated to second-order in the fluctuation potential. The hierarchyso-defined is particularly suited for frequency–de-pendent response calculation. See [14,19] for details.

The Letter is organized as follows. In Section 2,the equations relating the Q-branches Raman

scattering cross-sections of diatomics to the geo-metric derivatives of the polarizability are given,followed by the computational details. In the nextsection, the electron correlation effects within theCCS, CC2, CCSD and CC3 hierarchy are dis-cussed, and the core correlation effects at the CC3level are analysed. The CC3 results are then usedto estimate – within the limitations of harmonicapproximation and Placzek’s polarizability theory[20] – the most accurate up-to-date theoretical re-sults for Raman scattering cross-sections availablein literature. A short summary and main conclu-sions are given at the end.

As in [12], the molecules under investigation areN2, CO, HF, HCl and Cl2.

2. Definitions and computational details

2.1. Vibrational Raman spectra of diatomics

The vibrational Raman spectrum is character-ized primarily by the differential scattering cross-section ðdr=dXÞ, and the depolarization ratio ql

(or qn for non-polarized light). According toPlaczek’s polarizability theory [20] these parame-ters can be obtained from the geometrical deriva-tives of the electric dipole polarizability, evaluatedat the frequency of the incident light. For diatomicmolecules, in the harmonic approximation, the Q-branch (DJ ¼ 0) differential Raman cross-sectionðdr=dXÞQð~mm0Þ observed perpendicularly to the lin-early polarized incoming beam depends on thebond length derivatives of the isotropic molecularpolarizability að~mm0Þ and of the polarizability an-isotropy cð~mm0Þ according to

drdX

� �Q

ð~mm0Þ ¼ð2pÞ4

45

h8p2c~mm1l

ð~mm0 � ~mm1Þ4

1� expð�hc~mm1=kBT Þ

!

45dað~mm0ÞdR

!2

Re

24

þ 7

4

dcð~mm0ÞdR

!2

Re

35; ð1Þ

328 M. Pecul, S. Coriani / Chemical Physics Letters 355 (2002) 327–338

where ~mm0 is the laser wavenumber (in cm�1), ~mm1 thevibrational wavenumber (in cm�1), l the reducedmass, R the internuclear distance, and c the ve-locity of light. The polarizabilities in Eq. (1) areexpressed in cm3, while the Raman scatteringcross-section is expressed in cm2 sr�1.

The isotropic polarizability að~mm0Þ and the po-larizability anisotropy cð~mm0Þ are defined by

að~mm0Þ ¼1

32a?ð~mm0Þ

þ akð~mm0Þ; ð2Þ

cð~mm0Þ ¼ akð~mm0Þ

� a?ð~mm0Þ; ð3Þ

where a?ð~mm0Þ and akð~mm0Þ are the components of thepolarizability tensor perpendicular and parallel tothe molecular axis, respectively.

Under these conditions the depolarization ratioql is

qlð~mm1Þ ¼3ðcð~mm0Þ=dRÞ2Re

45ðdað~mm0Þ=dRÞ2Reþ 4ðdcð~mm0Þ=dRÞ2Re

: ð4Þ

For a non-polarized incoming beam the coeffi-cients 3 and 4 in the above equation should bereplaced by 6 and 7, respectively [20]. The depol-atization ratios will not be presented in this Letter,since few experimental data are available forcomparison. The computational aspects, on theother hand, are more easily discussed on the basisof the geometric derivatives of the polarizabilities,rather than Raman scattering cross-sections anddepolarization ratios.

2.2. Computational details

The dynamic polarizability tensors have beencalculated by means of the coupled cluster linearresponse CCS, CC2, CCSD and CC3 methods, forfrequencies corresponding to wavelengths of514.5, 435.8, 351.1 nm, and, for N2, 300.0 nm.Static dipole polarizabilities have also been eval-uated for comparison.

The geometrical derivatives of the polarizabilityhave been calculated numerically using the threepoint formula, the stability of which has beenverified against the five point formula.

The basis sets employed for all five moleculesare the correlation consistent valence sets aug-cc-

pVTZ and d-aug-cc-pVTZ of Dunning and co-workers [21,22], plus the core-valence sets aug-cc-pCVTZ [23] for N2, HF and CO. Core-electronshave been kept frozen in all calculations performedwith the valence sets (frozen 1s approximation forC, N, O and frozen 1s2s2p for Cl), whereas allelectrons have been correlated when using thecore-valence sets.

The dipole polarizabilities and their geometricderivatives have been evaluated at the followingequilibrium geometries: 2.074 a.u. for N2, 2.132a.u. for CO, 1.732879 a.u. for HF, 2.408645 a.u.for HCl, and 3.755 a.u. for Cl2. The Boltzmanfactor in Eq. (1) has been calculated for roomtemperature 298 K. The reduced masses l havebeen calculated for most abundant isotopes ofeach species and the vibrational wavenumbers ~mm1have been taken from experiment [9].

The SCF, CCS, CC2 and CCSD calculationshave been carried out using a recent release of theDALTON program [24], which contains the linearresponse coupled cluster code of [25,26]. The CC3calculations have been performed with a programbased on the ACESII code [27] which includes thelinear response coupled cluster implementation ofChristiansen, Gauss and Stanton [16].

3. Results and discussion

3.1. The electron correlation effects on dynamicpolarizabilities and their geometrical derivatives

3.1.1. The comparison of SCF and coupled clusterCCS–CC3 sequence

The behaviour of the calculated polarizabilitiesand their geometrical derivatives when the wave-function model is improved is illustrated in Figs.1–5, for N2, CO, HF, HCl and Cl2, respectively.The displayed data have been obtained using thed-aug-cc-pVTZ basis set. Note that in the CCS/CC2/CCSD/CC3 sequence the wavefunctionmodel is extended in a hierarchical manner. Linearresponse SCF, in which the orbitals are relaxedwith respect to the external time-dependent fields,does not belong to this hierarchy but it has beenincluded for comparison. The discussion of theresults will focus on the geometric derivatives of

M. Pecul, S. Coriani / Chemical Physics Letters 355 (2002) 327–338 329

the molecular polarizabilities, since they are theonly relevant quantities for the calculation ofRaman intensities. Electron correlation effects onthe polarizabilities have been already discussedextensively in the literature [17].

The trends observed for the polarizabilities andtheir derivatives when the computational level isextended are practically the same for all wave-lengths. The only exception is represented by thebehaviour of polarizabilities and their geometricderivatives for Cl2 at the lowest wavelength(k0 ¼ 351:1 nm). We shall return to this later.

The behaviour of the calculated polarizabilitiesand their derivatives when improving the wave-function model varies substantially from onemolecule to another, but certain trends are sharedby all of them. Namely, the diversity of the resultsfor geometric derivatives of polarizabilities withrespect to the method used is much larger than forthe polarizabilities themselves. In many cases, theoverall behaviour of the isotropic polarizability

and its geometrical derivative (or of the aniso-tropic counterparts) is similar, but the derivativetends to vary much more from one method toanother.

The changes of the calculated polarizabilityderivatives with respect to the wavefunction modelare non-monotonic. In many cases the extremumvalue is obtained on the CC2 rather than CCSlevel, but there are also instances where the CC2model appears to be a reasonable approximationto CCSD, in the sense that the results are similar.On the basis of our results, we conclude that theCC2 model tends to overestimate the electroncorrelation effects on the Raman scattering cross-sections. We suggest that it is used with caution inthis field, and more to get a qualitative estimate ofthe correlation effects rather than the quantitativeresults.

For HF, HCl and Cl2 (for the latter with theexception of the shortest laser wavelength) thecalculated geometric derivatives of the polariz-

Fig. 1. N2. The isotropic polarizability a, the geometric derivative of the isotropic polarizability ðda=dRÞRe, the polarizability

anisotropy c and the geometric derivative of the polarizability anisotropy ðdc=dRÞReas a function of the wavefunction model. Basis set

d-aug-cc-pVTZ. Atomic units.


abilities converge rather smoothly with thewavefunction model, i.e. the triple excitationcorrection (as given by the difference between theCC3 and CCSD values) is significantly smallerthan the difference between CCSD and CC2 val-ues. This resembles the trends previously observedfor excitation energies [28,29], although the con-vergence is less regular here. In the case of N2 andof ðdc=dRÞRe

in CO the (CC3)CCSD) differenceis of similar magnitude as the (CCSD)CC2) one.For these two cases, it would be interesting toverify the convergence with respect to the wave-function model by calculations at the full CCSDTlevel.

We have already mentioned that the electroncorrelation effects are larger on the geometric de-rivatives of the polarizabilities than on the polar-izabilities. In accordance with that, the tripleexcitation corrections on the geometric derivativesof the polarizabilities are, on average, significantlylarger than those on the polarizabilities. They

range from below 1% of the CCSD value for HFand HCl, through approximately 4% for ðdc=dRÞRe

in Cl2 (for large wavelengths), up to 10% and morefor N2 and ðdc=dRÞRe

of CO.The geometric polarizability derivatives of Cl2

calculated for the shortest laser wavelength, 351.1nm, exhibit an unusual behaviour when thewavefunction model is improved. In particular, thetriple excitation correction is very substantial,much larger than for the longer wavelengths. Thisis a consequence of the fact that k0 ¼ 351:1 nm(corresponding to the wavenumber 28 482 cm�1)lies in the pre-resonance region. The calculationsof the first vertical singlet excitation energy resultin the values of 31 895 cm�1 at the CCSD/d-aug-cc-pVTZ level and 31 260 cm�1 at the CC3/d-aug-cc-pVTZ level. The substantial triple excitationcorrection propagates into the results of the cal-culations of geometric derivatives of polarizabilityand, ultimately, into the Raman scattering cross-sections.

Fig. 2. CO. The isotropic polarizability a, the geometric derivative of the isotropic polarizability ðda=dRÞRe, the polarizability an-

isotropy c and the geometric derivative of the polarizability anisotropy ðd=c=dRÞReas a function of the wavefunction model. Basis set d-

aug-cc-pVTZ. Atomic units.


3.1.2. Core correlation and double augmentationeffects

We collect in Table 1 the CC3 results obtained(a) at the CC3(frozen core)/aug-cc-pVTZ level (b)when correlating all electrons in conjunction withthe use of a core-valence basis set, i.e. a basis setenriched with tight functions, and (c) when addinga second set of diffuse functions (in the frozen coreapproximation). Core correlation is only analyzedfor N2, CO and HF, since core-valence basis setsare not available for chlorine. Only the resultsobtained at k0 ¼ 351:1 nm are reported in the ta-ble, but the calculations have been carried out forall four frequencies (five for N2).

With the exception of c and its derivative in HF,core correlation leads to a slight contraction of theresults. The effect is at any rate very small, at mostabout 2% for the geometric derivatives polariz-abilities for N2. This supports the conclusiondrawn in [12] on the basis of CCSD results that thefrozen core approximation does not introducesignificant errors in the calculations of Raman

intensities for the molecules composed of secondrow elements. The effects are equally small for HCl(less than 1%) and only slightly larger for Cl2(max. 4%), as the CCSD results indicate [12].

The effect of double augmentation is morepronounced than that of core-valence correlation.That is why the d-aug-cc-pVTZ basis set has beenused in the studies of convergence within thecoupled cluster hierarchy described in Section3.1.1. Still, the effect of additional diffuse functionsis at most 5% as far as the geometrical derivativesare concerned. The effect in percentage on thecross-sections ranges from 2.3% (CO) to 9% (HF).The unusually large basis set effect on the Ramanscattering cross-section of HF was already ob-served at the CCSD level [12].

3.2. Best estimates and comparison with literaturedata

The results presented so far, together with theCCSD(frozen core)/d-aug-cc-pVQZ values from

Fig. 3. HF. The isotropic polarizability a, the geometric derivative of the isotropic polarizability ðda=dRÞRe, the polarizability




[12], are used to obtained best estimates for thegeometrical derivatives of the isotropic and an-isotropic polarizability and the Raman scatteringcross-section. The triple excitation correctionsand the core-valence correlation effects are addedto the CCSD values, according to

Estimate ¼ CCSDðfrozen coreÞ=d-aug-cc-pVQZ

þ ½CC3ðfrozen coreÞ=d-aug-cc-pVTZ– CCSDðfrozen coreÞ=d-aug-cc-pVTZ þ ½CC3ðall-electronsÞ=aug-cc-pCVTZ– CC3ðfrozen coreÞ=aug-cc-pVTZ :

ð5ÞFor HCl and Cl2, for which all-electron CC3 cal-culations have not been carried out, the last termin the expression above is replaced by the differ-ence between the all-electron CCSD results [12],obtained with the fully uncontracted aug-cc-pVTZbasis set, and the CCSD(frozen-core)/aug-cc-pVTZresults.

The final results are collected in Table 2 andcompared with theoretical [3–8,12] and experi-mental [9,11,30,31] values from the literature. Thelast term in Eq. (5) does not introduce significantcorrection and the difference between the esti-mated values and CCSD values from [12] stemspredominantly from the the triple excitation cor-rections.

3.2.1. The geometric derivatives of polarizabilitiesThe largest differences with respect to the

CCSD values from [12] are observed for the bondlength derivatives of the polarizability of CO andN2, since for these two molecules the triple exci-tation corrections are the most significant. Notalways the corrections improve the agreement withexperiment. The estimated values are closer toexperiment than the CCSD ones for ðda=dRÞRe

inCO and ðdc=dRÞRe

in N2, but farther for ðda=dRÞRe

in N2 and ðdc=dRÞRein CO. For HF the correc-

tions on the polarizability derivatives are negligi-

Fig. 4. HCl. The isotropic polarizability a, the geometric derivative of the isotropic polarizability ðda=dRÞRe, the polarizability




ble. In HCl ðda=dRÞReis improved slightly by the

triple excitation corrections – but the discrepancywith experiment remains substantial – whereas theestimate for ðdc=dRÞRe

is slightly farther from ex-periment than the CCSD result. In the case of Cl2the triple excitation correction shifts the calculatedstatic ðda=dRÞRe

value further from experiment[31], but it remains still well within the experi-mental error bar. The dynamic polarizability de-rivatives seem to be underestimated whencompared with experiment [31] but different ex-perimental results [31,32] are also discrepant. Evenif the agreement with experiment is generally notperfect, it should be kept in mind that the calcu-lations have been carried out only for the elec-tronic polarizability and in a harmonicapproximation. Moreover, the experimental re-sults are in most cases rather dated and no errorbars are given. At any rate, our estimates signifi-cantly improve over the SOPPA numbers from1982 [7], to which most of the up-to-date results inthe literature have been compared.

3.2.2. The Raman scattering cross-sectionsFor the Raman scattering cross-sections of N2

our estimates seem to be too high, whereas there is aremarkable good agreement between DFT [3] andexperimental [9] results, and even better betweenCCSD [12] and experimental results. This can beintepreted as a cancellation of errors originatingfrom approximations in Placzek’s polarizabilitytheory and triple excitation (and core-valence cor-relation) effects. There is a noticeable discrepancybetween the theoretical and experimental value forthe Raman scattering cross-section of N2 fork0 ¼ 300 nm, even after correcting for triple exci-tation effects. Since the experimental number is atvariance not only with theory, but also with theexperimental data collected at the other wave-lengths [9] it may be worthwhile to repeat thismeasurement.

The estimated Raman scattering cross-sectionsfor CO improve on the CCSD results and arewithin the error bars suggested for the experi-mental numbers. This is not the case for HF, but

Fig. 5. Cl2. The isotropic polarizability a, the geometric derivative of the isotropic polarizability ðda=dRÞRe, the polarizability




for this molecule only one experimental number isavailable for the comparison and its error bar isnot given. Our theoretical estimates for HCl seemto be in good agreement with experiment. For Cl2the theoretical estimate is not far from the exper-imental number for k0 ¼ 514:5 nm, but the differ-ence between theoretical and experimental result ishuge for k0 ¼ 351:1 nm. This originates from thepre-resonance effects, not accounted for by Plac-zek’s polarizability theory and already discussed inSection 3.1.

Our coupled cluster estimates confirm the goodperformance of the DFT approach from [3] for thecalculations of Raman intensities. Similarly as forthe geometric derivatives of the polarizabilities,our estimated results improve significantly on theSOPPA results [7].

4. Summary

We have presented a linear response coupledcluster investigation of the bond length polariz-ability derivatives and the Raman scattering cross-

sections for the linear molecules N2, CO, HF, HCland Cl2. The convergence of the results in a hier-archy of coupled cluster methods CCS/CC2/CCSD/CC3 has been investigated. The CCSD andCC3 results have been used for estimation pur-poses.

The study of the performance of the CCS/CC2/CCSD/CC3 hierarchy brings out the conclusionthat the bond length derivatives, and therefore theRaman scattering cross-sections are more sensitiveto the higher-order electron correlation effects thanthe polarizabilities themselves. The trends in thesequence of the wavefunction models are in somecases similar for the isotropic polarizability and itsgeometric derivative, and for the anisotropiccomponents, but the derivatives tends to varymuch more with the method employed. The tripleexcitation corrections on the polarizability deriv-atives are substantial for N2 and CO (more than10%), and practically negligible for HF and HCl(less than 1%). Cl2 is an intermediate case in thisrespect, at least when the frequency at which thecalculations are carried out is far from the reso-nance region. Closer to the resonance region the

Table 1

Comparison of the isotropic polarizability a, the polarizability anisotropy c, their geometric derivatives ðda=dRÞReand ðdc=dRÞRe

and

the Q-branch Raman scattering cross-sections ðdr=dXÞQ calculated at the CC3(frozen core)/aug-cc-pVTZ, CC3(all electrons)/aug-cc-

pCVTZ and CC3(frozen core)/d-aug-cc-pVTZ levels. Wavelength k0 ¼ 351:1 nm

a (a.u.) c (a.u.) ðda=dRÞRe(a.u.) ðdc=dRÞRe

(a.u.) ðdr=dXÞQ (cm2 sr�1 1031)

N2

aug-cc-pVTZ 12.343 5.008 6.869 8.362 29.46

d-aug-cc-pVTZ 12.384 4.997 6.989 8.250 30.40

aug-cc-pCVTZ 12.314 5.007 6.855 8.353 29.35

CO

aug-cc-pVTZ 14.020 3.585 6.018 8.729 26.44

d-aug-cc-pVTZ 14.069 3.534 6.099 8.641 27.06

aug-cc-pCVTZ 13.989 3.584 6.004 8.689 26.30

HF

aug-cc-pVTZ 5.574 1.465 3.066 4.778 20.20

d-aug-cc-pVTZ 5.860 1.119 3.223 4.578 22.00

aug-cc-pCVTZ 5.562 1.469 3.065 4.782 20.20

HCl

aug-cc-pVTZ 18.211 1.651 4.581 8.297 74.02

d-aug-cc-pVTZ 18.695 1.464 4.747 8.198 78.65

Cl2aug-cc-pVTZ 32.720 18.235 13.340 8.978 251.65

d-aug-cc-pVTZ 33.345 18.282 13.370 8.564 252.39


Table 2

Comparison of the estimated values for the geometric derivatives of the polarizabilities and Raman scattering cross-sections with

theoretical and experimental results in literature

k0 (nm) ðda=dRÞRe(a.u.) ðdc=dRÞRe

(a.u.) ðdr=dXÞQ ðcm2 sr�1 1031)

Estim. Other calc. Exp. Estim. Other calc. Exp. Estim. Other calc. Exp.a

N2

1 6.33 4.73b; 5.73c 7.43 6.12b; 6.70c

514.5 6.61 4.95b; 5.95c 7.80 6.42b; 6.96c 4.98 2.82b; 4.03c;

4.11d4.32 � 0.1

435.8 6.73 5.05b; 6.04c 6.25e 7.95 6.54b; 7.07c 8.2e 10.89 6.19b; 8.77c 9.2 � 0.9

351.1 6.97 5.23b; 6.23c 8.26 6.78b; 7.28c 30.21 17.2b; 24.10c;

24.88d24.3 � 3.3

300.0 7.23 6.43c 8.61 7.52c 64.33 50.76c 97

CO

1 5.32 6.79b; 5.67c 7.80 11.62b; 9.03c

514.5 5.63 7.23b; 5.95c 8.15 12.41b; 9.45c 4.30 7.29b; 4.88c 4.1

435.8 5.77 7.42b; 6.08c 5.34f 8.29 12.74b; 9.61c 10.06f 9.43 16.1b; 7.365d;

10.64c8.5

351.1 6.06 7.82b; 6.34c 8.54 13.39b; 9.91c 26.67 45.9b; 20.93d;

29.74c21.9

HF

1 2.94 3.10g; 2.94c 4.27 4.68g; 4.32c

514.5 3.05 3.23g; 3.05c 3.6h 4.41 4.83g; 4.45c 3.14 2.46d; 3.40g;

3.14c3.8

435.8 3.10 3.28g; 3.10c 4.46 4.89g; 4.51c 7.32 7.96g; 7.33c

351.1 3.20 3.39g; 3.20c 4.57 5.01g; 4.61c 21.66 23.59g; 21.67c

HCl

1 4.15 4.43b; 4.43g;

4.18c7.33 7.54b; 8.30g;

7.39c

514.5 4.39 4.72b; 4.68g;

4.41c7.72 7.95b; 8.78g;

7.78c11.80 13.5b; 12.2d;

13.6g; 11.91c12.1

435.8 4.49 4.85b; 4.80g;

4.51c3.60i 7.89 8.12b; 8.98g;

7.94c8.53i 26.67 30.7b; 30.9g;

26.88c27.8

351.1 4.70 5.11b; 5.05g;

4.72c8.20 8.44b; 9.30g 77.41 91.0b; 81.5d;

90.7g; 77.84c67.2

Cl21 7.03 7.29b; 7.16c 7.21� 0.79j 12.35 19.8b; 12.95c

514.5 7.87 8.72b; 7.94c –k 13.67 21.89b; 14.31c 20.10 18.45d; 20.63c 15.9

435.8 8.41 9.86b; 8.40c 14.00 22.08b; 14.73c 44.98 45.41c

351.1 13.28 11.30c 9.12 12.75c 249.39 186.27c 66

aCompilation by Schr€ootter and Kl€oockner [9]. The error bars, when not explicitly given, are on the average approximately 10% for

visible light (k0 ¼ 514:5 nm and 435.8 nm) and 15% for UV (k0 ¼ 351:1 and 300 nm).b SOPPA results from [7].c CCSD/d-aug-cc-pVQZ results from [12].dDFT/LB94 (van Leeuwen-Baerends potential) results from [3].e Experimental results from [11].f Experimental results from Ref [30].g Sum-over states TDGI (time dependent gauge-invariant) calculations of [8].h Experimental result from [10].i Experimental result from [30].j Experimental result from [31].k Experimental result is (9.89� 0.86) a.u. for k0 ¼ 543:5 nm [31] and 7.22 a.u. for excitation by various lines of Ar/Kr laser [32].


triple excitation correction is sizeable, which is aconsequence of a substantial triple excitation cor-rection at the first singlet excitation energy. Thecore-valence correlation effects are small in allmolecules for which they have been calculated (N2,CO and HF), as it was already found at the CCSDlevel [12].

The calculated triple excitation and core-va-lence correlation corrections have been used inconjunction with previous CCSD results in a large(d-aug-cc-pVQZ) basis set [12] to obtain ‘best’ es-timates for the bond-length polarizability deriva-tives and Raman scattering cross-sections. Theagreement of the resulting values with experimentis in general satisfactory, considering the approx-imations inherent in the harmonic approach andPlaczek’s polarizability theory and the large errorbars of some of the experimental numbers. An-harmonic effects are usually moderate in stronglybound molecules as those under investigation.Therefore, the error associate with their neglect isexpected to be less significant than the errors dueto the approximations introduced by Placzek’spolarizability theory. The latter are most probablysizeable, in all likelihood surpassing the errorsassociated with the approximate treatment of tri-ple and higher-order excitations inherent in theCC3 approach. Nevertheless the final estimatedresults can be used as reference values for thecalibration purposes, since they represent, inprinciple, the most accurate theoretical values inthe literature.

Acknowledgements

We are deeply grateful to O. Christiansen, J.Gauss, and J. Stanton for allowing us to use theirdevelopment version of the ACESII code and toA. Rizzo for helpful discussions. This work wassupported by the European Research and TrainingNetwork: ‘Molecular Properties and MolecularMaterials’ (MOLPROP), Contract No. HPRN-CT-2000-00013 and Grant No. 120–501/68/BW-1522/17/2001 (M.P.). M.P. is a recipient ofDomestic Grant for Young Researchers foundedby Foundation for Polish Science. S.C. acknowl-edges support from the Danish Research Council

(Grant No. 9600856) and from MURST,Programmi di Ricerca Cofinanziata di InteresseNazionale (ex 40 %).

References

[1] P.R. Carey, Biochemical Applications of Raman and

Resonance Raman Spectroscopies, Academic Press, New

York, 1982.

[2] K. Kneipp, H. Kneipp, I. Itzkan, R.R. Dasari, M.S. Feld,

Chem. Rev. 99 (1999) 2957.

[3] S.J.A. van Gisbergen, J.G. Snijders, E.J. Baerends, Chem.

Phys. Lett. 259 (1996) 599.

[4] B.G. Johnson, J. Florian, Chem. Phys. Lett. 347 (1995)

120.

[5] A. Stirling, J. Chem. Phys. 104 (1996) 1254.

[6] B.S. Jursic, J. Mol. Struct. (Theochem) 394 (1997) 15.

[7] J. Oddershede, E.N. Svendsen, Chem. Phys. 64 (1982)

359.

[8] M. M�eerawa, A. Dargelos, J. Mol. Struct. (Theochem) 528

(2000) 37.

[9] H. Schr€ootter, H.W. Kl€oockner, in: A. Weber (Ed.), Raman

Spectroscopy of Gases and Liquids, Topics in Current

Physics, Springer, Berlin, Heidelberg New York, 1979, pp.

123–166 (Chapter 4).

[10] Y. Le Duff, W. Holzer, J. Chem. Phys. 60 (1974) 2175.

[11] T. Yoshino, H.J. Bernstein, J. Mol. Struct. 2 (1958) 213.

[12] M. Pecul, A. Rizzo, J. Chem. Phys. 116 (2002) 1259.

[13] O. Christiansen, H. Koch, P. Jørgensen, Chem. Phys. Lett.

243 (1995) 409.

[14] T. Helgaker, P. Jørgensen, J. Olsen, Molecular Electronic-

Structure Theory, Wiley, Chichester, 2000.

[15] H. Koch, O. Christiansen, P. Jørgensen, A.S. de Mer�aas, T.

Helgaker, J. Chem. Phys. 106 (1997) 1808.

[16] O. Christiansen, J. Gauss, J.F. Stanton, Chem. Phys. Lett.

292 (1998) 437.

[17] O. Christiansen, C. H€aattig, J. Gauss, J. Chem. Phys. 109

(1998) 4745.

[18] O. Christiansen, J. Gauss, J.F. Stanton, Chem. Phys. Lett.

305 (1999) 147.

[19] O. Christiansen, C. H€aattig, P. Jørgensen, Int. J. Quantum.

Chem. 68 (1998) 1.

[20] G. Placzek, in: E. Marx (Ed.), Handbuch der Radiologie,

vol. 6, Akademische Verlagsgesellschaft, Leipzig, 1934,

p. 205.

[21] T.H. Dunning Jr., J. Chem. Phys. 90 (1989) 1007.

[22] D.E. Woon, T.H. Dunning Jr., J. Chem. Phys. 98 (1993)

1358.

[23] D.E. Woon, T.H. Dunning Jr., J. Chem. Phys. 103 (1995)

4572.

[24] T. Helgaker, et al., DALTON, an ab initio electronic

structure program, Release 1.2, 2001. Available from

http://www.kjemi.uio.no/software/dalton/dalton.html.

[25] O. Christiansen, A. Halkier, H. Koch, P. Jørgensen, T.

Helgaker, J. Chem. Phys. 108 (1998) 2801.


[26] C. H€aattig, O. Christiansen, P. Jørgensen, J. Chem. Phys.

107 (1997) 2801.

[27] J.F. Stanton, J. Gauss, J.D. Watts, W.J. Lauderdale, R.J.

Bartlett, Int. J. Quantum. Chem. Quantum. Chem. Symp.

26 (1992) 879.

[28] O. Christiansen, H. Koch, P. Jørgensen, J. Olsen, Chem.

Phys. Lett. 256 (1996) 185.

[29] M. Pecul, M. Jaszu�nnski, H. Larsen, P. Jørgensen, J. Chem.

Phys. 112 (2000) 3671.

[30] W.F. Murphy, W. Holzer, H.J. Bernstein, Appl. Spectry.

23 (1969) 211.

[31] U. Hohm, U. Tr€uumper, J. Raman Spectrosc. 26 (1995)

1059.

[32] B. Fontal, T.G. Spiro, Spectrochim. Acta A 33 (1977) 507.


the effect of triple excitations in coupled cluster calculations of raman scattering cross-sections

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