ab initio calculation of vibrational raman optical activity

Ab Initio Calculation of VibrationalRaman Optical Activity

MAGDALENA PECUL,1 KENNETH RUUD2

1Department of Chemistry, University of Warsaw, Pasteura 1, 02-093 Warsaw, Poland2Department of Chemistry, University of Tromsø, N-9037 Tromsø, Norway

Received 16 January 2005; accepted 14 February 2005Published online 15 April 2005 in Wiley InterScience (www.interscience.wiley.com).DOI 10.1002/qua.20601

ABSTRACT: We review recent developments in the ab initio calculation of Ramanoptical activity (ROA) and discuss the applications of modern ab initio methods to thecalculation of these properties, giving some examples of the kind of molecules andquestions that can now be addressed using ab initio methodology. We also give someperspectives on future developments and applications of theoretical methods as an aidin understanding and analyzing ROA spectra. © 2005 Wiley Periodicals, Inc. Int JQuantum Chem 104: 816–829, 2005

Key words: Raman optical activity; ab initio; London orbitals

1. Introduction

C hiral molecules play crucial roles in manyareas of chemistry and biochemistry. The de-

termination of the absolute configuration of chiralmolecules and the enantiomeric excess of a givenstereoisomer is therefore an important field of re-search. However, the a priori determination of ab-solute configuration is a difficult task, in most casesinvolving the use of empirical relations or elaboratesynthetic strategies: the first cannot be consideredreliable enough to allow for an unambiguous de-termination of the absolute configuration of a mol-ecule, and the second is a highly time-consuming

process that relies on the knowledge of the absoluteconfiguration of reference molecules which is notalways available.

Historically, optical rotation and circular dichro-ism have been the leading experimental methodsused to determine the presence of a given enantio-mer and, to some extent, also to determine theenantiomeric excess of a compound in a mixture [1].However, in general, the sign of an observed opticalrotation of plane-polarized light or the rotatorystrengths of the different electronic transitions ob-served in electronic circular dichroism spectros-copy cannot be easily associated with a given abso-lute configuration of a molecule. At most, the signof the optical rotation or the rotatory strengths isknown to be opposite for two enantiomers.

In recent years, a number of developments in theab initio calculation of optical rotation and circulardichroism has made it possible to reach a level

Correspondence to: K. Ruud; e-mail: [email protected] grant sponsor: Norwegian Research Council.Contract grant number: 154011/420 and 162746/V00.

International Journal of Quantum Chemistry, Vol 104, 816–829 (2005)© 2005 Wiley Periodicals, Inc.

where the absolute configuration of a molecule canbe determined by comparing experimental obser-vations with the results obtained from theoreticalcalculations, and more details about these recentdevelopments can be found, for example, in Ref. [2].However, limitations in the theoretical calcula-tions—arising for instance from the use of finitebasis sets [3], incomplete treatment of electron cor-relation effects [4, 5], and the neglect of vibrational[6] and solvent effects [7, 8]—as well as experimen-tal errors, make unambiguous determination of theabsolute configuration in many cases impossible.This problem is made more difficult by the fact thatonly a few observables are available for each mol-ecule. Limited experimental data obtained in opti-cal rotation or circular dichroism experiments fur-ther accentuate the problems of analyzing the datain the case of molecules with multiple chiral centers[9].

A much more powerful approach to determiningthe absolute configuration of chiral molecules isoffered by chiroptical spectroscopies in the infrared(IR) region of the electromagnetic spectrum, morespecifically techniques such as vibrational circulardichroism (VCD) [10–12] and vibrational Ramanoptical activity (ROA) [13, 14]. These spectroscopictechniques probe the differential absorption andscattering, respectively, of right- and left-circularlypolarized light due to molecular vibrations. Be-cause of the high number of vibrational modes inmost chiral molecules of interest, much more datacan be collected for each molecule, and thus a moredetailed comparison between theory and experi-ment can be hoped for. Moreover, VCD and ROAcan be applied to molecules lacking a suitable elec-tronic chromophore, and therefore inaccessible tocircular dichroism. Even though theoretical calcu-lations of ROA are hampered by the same limita-tions in accuracy as for optical rotation and circulardichroism calculations, one may hope that a suffi-ciently large number of circular intensity differ-ences (CIDs) have a large enough value that theycan be used to determine unambiguously the abso-lute configuration of a chiral molecule.

In contrast to optical rotation, and to some extentcircular dichroism as well, both VCD and ROA arerather new experimental methods. The theoreticalfoundations and the first experimental observationsof these phenomena were being done during thelate 1960s and 1970s: the first indisputable ROAspectra were measured by Barron, Bogaard, andBuckingham in 1973 [15] and by Hug et al. in 1975[16]. VCD has matured and become an important

research tool during the past 10–15 years, to a largeextent due to the availability of commercial instru-ments as well as the availability of theoretical meth-ods that allow the spectra to be analyzed. To someextent, ROA is still very much in its infancy; thefirst commercial instruments were introduced only2–3 years ago. Furthermore, whereas ab initio cal-culations of VCD spectra can now be performedrather routinely, the calculation of a full ROA spec-trum for all but the smallest molecules remains acomputational challenge.

Apart from the determination of the absoluteconfiguration, ROA is also gaining recognition as atool for structural studies, especially of biomol-ecules, since it allows chiral biomolecules to bestudied in their native water environment. Thisarea of application stimulates most of the currentdevelopments [17–20] (see also Ref. [21] for review).Advances in laser technology allow ROA spectra ofbiologically interesting molecules such as peptidesand nucleotides to be obtained [17, 18] despite thevery low intrinsic sensitivity of ROA. ROA hastherefore become a valuable tool for investigatingstructural characteristics of peptides and proteins,mainly by employing the vibrational amide I and IIbands (see Ref. [21], for a review of the efforts inthis field). ROA has a significant advantage instructural studies of biomolecules compared withnuclear magnetic resonance (NMR) because of themuch shorter time scale accessible in ROA. Thismakes it possible to investigate short-lived con-formers and, in contrast to crystallographic meth-ods, it is possible to study the molecules in anaqueous environment with the ROA technique.

The theoretical background for the ROA phe-nomenon was given by Barron and Buckingham in1971 [14]. An ab initio calculation of the moleculartensors contributing to the optical rotation was firstcarried out by Amos [22] at the Hartree–Fock (HF)level in the so-called static-limit approximation(vide infra) for a few simple, nonchiral molecules(CO, HF, HCl, and CH3F). An implementation andthe first calculations of the frequency-dependenttensors were presented for the water molecule byLazzeretti and Zanasi in 1986 [23]. The first com-plete theoretical study of the CIDs observed in ex-perimental ROA spectra were presented by Polav-arapu in 1990 [24], using the static approximation ofAmos [22]. The first correlated calculations werepresented using multiconfigurational self-consis-tent field (MCSCF) wave functions and Londonatomic orbitals 4 years later, although the systemstudied (CFHDT) was perhaps of little experimen-

AB INITIO CALCULATION OF VIBRATIONAL RAMAN OPTICAL ACTIVITY

INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY 817

tal interest [25]. Nowadays, other computationaltechniques are available, including time-dependentdensity functional theory (DFT) [26, 27]. However,the calculation of ROA spectra is still not a routinetask and, considering the potential importance ofROA as an experimental structural tool, in particu-lar in the study of biomolecules in water, the de-velopment of efficient means for calculating ROACIDs for realistic molecules is a task of great im-portance for ensuring the future success of ROA asan experimental tool.

In this study, we will review the basic principlesfor the ab initio calculation of ROA CIDs, withparticular emphasis on the challenges facing reli-able calculations of these properties, including therealistic simulation of ROA spectra of molecules intheir native environment. We will review the theo-retical efforts made in the field of ab initio calcula-tions of ROA spectra during the past 10–15 years.We will indicate the directions in which we believethe field should be developed, as well as the prob-lems that are of most immediate concern. Our focusis on theoretical investigations of the problems thataffect comparison of ab initio results and experi-mental data, such as basis set and electron correla-tion effects, as well as solvent effects. We will, bythe term ab initio, refer both to traditional electronicstructure theory methods such as Hartree–Fock andmulticonfigurational self-consistent field, as well asto the popular DFT approach. More typical appli-cations, such as structural studies (including theassignment of absolute configurations) are also dis-cussed. Since the subject of the review is ab initiocalculations, we will not discuss early semi-empir-ical calculations such as the bond-polarizabilitymodel of Barron [28], subsequently extended tosemi-empirical molecular orbital theory in the com-plete neglect of differential overlap approximation(CNDO) introduced by Polavarapu [29].

2. Origin of Vibrational RamanOptical Activity

We will not give a complete account of the phys-ical origins of ROA, as a very detailed and excellentpresentation was presented by Barron [13]. Instead,our focus will be on the ab initio calculation of thetensors involved in this quantity. However, forcompleteness, we will briefly outline the physicalorigins of ROA before turning to the its calculationusing modern ab initio techniques.

Vibrational ROA is manifested as differential Ra-man (inelastic) scattering of left and right circularlypolarized incident light by a chiral sample. TheROA effect is usually described by means of thedimensionless circular intensity difference (CID) ��

[14]:

�� I�

R � I�L

I�R � I�

L , (1)

where I�R and I�

L are the scattered intensities withlinear �-polarization for right- and left-circularlypolarized incident light, respectively.

Within a semi-classical treatment, the scatteredlight is considered to originate from the oscillatingelectric and magnetic multipole moments, inducedby the incident light. Far from resonance, the in-duced electric dipole moment �ind,�(t), the inducedmagnetic dipole moment mind,�(t), and the inducedquadrupole moment Qind,��(t) are [13]

�ind,��t� � ��E� � ��1E� � G��1B�

�13 A��E� � · · · , (2)

mind,��t� � ��B� � G��1E� � · · · , (3)

Qind,��t� � A��E� � · · · , (4)

where implicit summations over repeated indiceshave been used.

Using these expansions, expressions for CIDsassociated with different experimental configura-tions for analyzing the scattered light can be de-rived by considering the electric and magneticfields generated by the oscillating induced electricdipole moment, the induced magnetic dipole mo-ment, and the induced quadrupole moment, andevaluating the scattered intensities I�

R and I�L for left-

and right-circularly polarized incident light or, in amore general manner, by evaluating the four Stokesparameters of the light beam [13].

For a sample of randomly oriented molecules,the resulting expressions for CIDs of Rayleigh (elas-tic) scattering are the following:

1. Forward scattering:

��0°� �8�45��G��2 � ��G��2 � �� A�2�

2c�45��2 � 7��2�. (5)

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818 VOL. 104, NO. 5

2. Backscattering:

��180°� �

48��G��2 �13 �� A�2�

2c�45��2 � 7��2�. (6)

3. For right-angle scattering, two types of mea-surements can be made in which a linear po-larization analyzer is put either perpendicular(in the x direction) or parallel (in the z-direc-tion) to the scattering plane, leading to thepolarized and depolarized ROA spectrum, re-spectively. The associated CIDs, �x(90°) and�z(90°), can be calculated from the expres-sions

�x�90°� �2�45��G��2 � 7��G��2 � �� A�2�

c�45��2 � 7��2�,

(7)

�z�90°� �

12��G��2 �13 �� A�2�

6c��2 . (8)

The iso- and anisotropic tensor invariants �, �(G�),�(�)2, �(G�)2, and �(A)2 introduced in Eqs. (5)–(8)are defined as

��G�� 13 G��, (9)

� �13 ��, (10)

��2 �12 �3�� , (11)

��G��2 �12 �3��G�� G��, (12)

�� A�2 �12 ��A�,��, (13)

where �� is the Levi–Civita tensor. In the aboveequations, � is the linear polarization of the electricdipole moment by an electric field (the electric di-pole–electric dipole polarizability), G� is the linearpolarization of the electric dipole moment by themagnetic field component of the incident light (theelectric dipole–magnetic dipole polarizability), andA is the linear polarization of the electric dipolemoment by the electric field gradient of the electro-magnetic light wave (the electric dipole–electricquadrupole polarizability).

The tensors and CIDs given above will accountfor the Rayleigh scattering of the incident light.However, the richness of the ROA spectrum arisesfrom the Raman scattering, in which the scatteredlight have interacted with the molecular vibrations,inducing excitations or deexcitations in the differ-ent vibrational modes of the molecule. The CIDarising from the scattering of the light after vibra-tional (de)excitation is determined by the vibra-tional transition moments of the three differentmixed polarizabilities �, G� and A; that is, we areinterested in considering matrix elements of theform 0��1, where 0� and �1 denotes the vibra-tional ground and first excited state, respectively. Itis customary to apply the Placzek approximation[30] in evaluating these vibrational transition ma-trix elements, in which the vibrational wave func-tion is approximated by a harmonic oscillator wavefunction and the expansion of the molecular prop-erty with respect to displacements along the normalcoordinates is truncated after the terms linear in thenuclear displacements. Within this approximation,the different ROA invariants determining the ROACIDs for an excitation in the pth vibrational modecan be written as

0��1p1p��0 �1

2�p��

�Qp��

�Qp� , (14)

0��1p1p�G��0 �1

2�p��

�Qp��G��

�Qp� , (15)

0��1p1p��A��0 �1

2�p��

�Qp��A��

�Qp� ,

(16)

where Qp denotes the normal coordinate of vibra-tion p. By means of this approximation, ROA CIDscan be evaluated from expressions analogous toEqs. (5)–(7), with the difference that it is not theelectronic tensors themselves that determine theROA CIDs through the different tensor invariantsin Eqs. (11)–(13), but rather the geometric deriva-tives of these tensors.

It is customary to calculate the tensor derivativesin Eqs. (14)–(16) in Cartesian coordinates, and totransform them to the normal coordinate basisthrough a straightforward linear transformation.This opens for the possibility of calculating theforce field, and thus the normal coordinates, at adifferent level of theory than the geometric deriva-



tives of the mixed polarizabilities. In this way, theeffects of electron correlation can in part be in-cluded in the property gradients as well throughthe transformation of these gradients from the Car-tesian coordinate system to the (correlated) normalcoordinate basis. Several examples of the use of thisapproximation will be given in Section 4.

3. Calculations of Vibrational RamanOptical Activity

As seen in Section 2, the key quantities determin-ing ROA CIDs are the geometric derivatives of theelectric dipole polarizability �(�), the mixed electricdipole–magnetic dipole polarizability G�(�), andthe mixed electric dipole–electric quadrupole po-larizability A(�). A general framework for the cal-culation of frequency-dependent molecular proper-ties is response theory [31], and the first correlated,gauge-origin independent calculations of ROACIDs were presented using MCSCF linear responsetheory in 1994 [25]. The first ab initio calculations ofROA CIDs were presented by Polavarapu 4 yearsearlier [24], using a static-limit approximation toG�(�) as first proposed by Amos [22].

Formally, the geometric derivatives of the ten-sors �(�), G�(�), and A(�) correspond to quadraticresponse functions, implemented for a large rangeof molecular wave functions for one-electron oper-ators [31–35]. However, because the Gaussian basisfunctions normally used in molecular electronicstructure calculations are attached to the nuclearcenters, the geometry dependence of the basis setgives rise to what is often referred to as Pulay forces[36, 37], giving rise to two-electron contributions tothe perturbing operators. Whereas analytical ex-pressions have been presented for the frequency-dependent polarizability and mixed electric dipole–electric quadrupole polarizability [38–40], no suchextension has been presented for the mixed electricdipole–magnetic dipole polarizability, where addi-tional two-electron contributions arise because ofthe need to ensure that this contribution is originindependent (vide infra). For this reason, all abinitio calculations of ROA CIDs presented to datehave involved analytical calculations of the tensors�(�), G�(�), and A(�), with the geometry deriva-tives obtained as finite differences with respect todistortions of the nuclear framework.

3.1. RESPONSE THEORY APPLIED TO ROACALCULATIONS

The electric–dipole polarizability, mixed electricdipole–electric quadrupole polarizability, andmixed electric dipole–magnetic dipole polarizabil-ity can be expressed in terms of linear responsefunctions as [25]

��,�� ; ��, (17)

G��,�� ; m��, (18)

A�,�� ; Q��, (19)

where we have introduced the operator for theelectric dipole moment �� which in the formalismof second quantization [41, 42] is given as

�� pq

�r��pqEpq, (20)

where r� is the � component of the electron coor-dinate, the summation runs over all molecular or-bitals p and q, and Epq is a generator of the unitarygroup [41].

In a similar manner, the magnetic dipole opera-tor m� is defined as

m� � �12 �

pq

�l��pqEpq, (21)

where l� � ��r�p� is the orbital angular momen-tum of the electron, and the traceless quadrupolemoment has been defined as

Q�� 12 �

pq

�r�r� � r�r��Epq. (22)

In these equations and throughout this study,atomic units have been used.

Although not explicitly indicated in the aboveequations, both the magnetic dipole operator andthe electric quadrupole operator are origin depen-dent for polar molecules. Therefore, the tensorsG��,�(�) and A�,�� are both origin dependent,whereas the tensor invariants given in Eqs. (11)–(13) are origin independent if tensors G��,�(�) andA�,�� both have the correct origin dependence. Inapproximate ab initio calculations, using finite basissets, the magnetic dipole operator does in generalnot display the correct origin dependence when

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820 VOL. 104, NO. 5

contributing to (mixed) second-order electric ormagnetic polarizabilities, and special care musttherefore be taken to ensure unambiguous resultsin finite basis set calculations of ROA CIDs. Weshall discuss one solution to this problem in thenext section.

The linear response functions in Eqs. (17)–(19)can be written for exact states in terms of a sum-over-states expression

A; B� � �n�0

� 0�A�nn�B�0

�n0 � ��

0�B�nn�A�0

�n0 � � � .

(23)

If both operators A and B are real, we may rewritethis expression as

A; B� � 2 �n�0

�0n

0�A�nn�B�0

�0n2 � �2 . (24)

For the case in which one operator is purely imag-inary, e.g., for the mixed electric dipole–magneticdipole polarizability, Eq. (23) can be rewritten as

A; B � 2� �n�0

0�A�nn�B�0

�n02 � �2 , (25)

where it is easy to see that the mixed electric dipole–magnetic dipole polarizability will vanish for staticelectromagnetic fields for which � � 0.

By taking advantage of the fact that it is not G�(�)that contributes to the ROA CIDs, but rather (1/�)G�(�) [see Eqs. (2) and (3)], we note that we candefine a static-limit approximation

lim�30

�1/��G�� 2 �n�0

0�A�nn�B�0

�0n2 � �

B� �F.

(26)

This approximation was first introduced by Amosin 1982 [22] and was used in the first ab initio studyof ROA by Polavarapu [24]. In this equation, B

and F denote the first-order perturbed wave func-tions due to an external magnetic field inductionand an external electric field, respectively. Becauseof the dispersion of G�, this expression is obviouslyonly approximate and, since it provides no compu-tational advantages—in particular, in the formgiven on the right-hand side of Eq. (26), it does not

exploit the 2n � 1 rule of energy derivative theory[43]—there is no particular reason to use this ap-proximation. It is still widely used, however [44,45].

3.2. ROA FOR APPROXIMATE WAVEFUNCTION MODELS AND GAUGE ORIGINPROBLEM

Since Eqs. (17)–(19) express the electronic tensorsthat contribute to the ROA invariants as linear re-sponse functions, these quantities can be calculatedat any level of theory for which expressions for theresponse functions have been derived. Indeed,ROA calculations have been presented at the Har-tree–Fock [24], MCSCF [25], and density functional[26] levels of theory. In all these calculations, thegeometry derivatives of the second-order electroniccontributions in Eqs. (17)–(19) are obtained by nu-merical differentiation with respect to the nucleardisplacements.

The need for a numerical differentiation of sec-ond-order molecular responses with respect to nu-clear distortions is probably part of the reason thatno ROA calculations have been performed at afairly highly correlated level, using for instance sec-ond-order Møller–Plesset (MP2) or coupled-cluster(CC) wave functions, even though response func-tions have been presented for highly correlatedwave functions [35]. However, in many cases suchhighly correlated wave functions do not givegauge-origin independent CIDs. The angular mo-mentum operator and the quadrupole moment op-erators in Eqs. (21) and (22) have a dependence onan arbitrary gauge origin arising from the represen-tation of the external magnetic field induction by amagnetic vector potential, which in the commonlyused Coulomb gauge can be chosen as

A �12 B � rO. (27)

In this equation, the gauge origin O can be chosenarbitrarily without changing the magnetic field,since

B � � � A. (28)

Thus, the gauge origin does not contribute to themagnetic field induction. However, this arbitrari-ness causes problems in approximate calculations,as the calculated properties in general are not in-



variant to the choice of gauge origin [46]. Only inthe case of variational wave functions in the limit ofcomplete basis sets can strict gauge origin indepen-dence be achieved [46]. For nonvariational wavefunctions, such as MP2 and truncated coupled-clus-ter wave functions, gauge origin independence can-not be achieved, even in the limit of a completebasis set [47, 48].

The calculation of magnetic properties was to alarge extent revolutionized by the work of Wolin-ski, Hinton, and Pulay, who demonstrated that theuse of London atomic orbitals ��(B, RM) [49]

��B, RM� � exp��iAMO � r��RM�, (29)

where � in most modern applications is a Gauss-ian basis function, gives truly gauge-origin inde-pendent nuclear shielding constants as well as fastbasis set convergence. The magnetic vector poten-tial appearing in the complex phase factor is givenas

AMO �12 B � RMO �

12 B � �RM � O�, (30)

and the effect of this phase factor is to move theglobal gauge origin O to the best local gauge originfor each individual basis function, namely theatomic center M to which the basis function isattached.

The use of London atomic orbitals in the calcu-lation of nuclear shielding constants not only en-forces gauge origin-independent results, throughthe introduction of different local gauge origin foreach individual atomic orbital, but also leads tomuch improved basis sets convergence [50]. Later,Olsen et al. [51] presented a magnetic dipole oper-ator using London atomic orbital that has the cor-rect origin dependence even for finite basis sets,and it has been proved that gauge origin-indepen-dent ROA CIDs can be obtained using this operator[25]. However, the hypervirial relations needed toprove that the magnetic dipole operator definedusing London atomic orbitals [51] has the correctorigin dependence also for finite basis sets is onlyfulfilled for variational wave functions, and thusthe use of this London orbital magnetic dipole op-erator will not give gauge origin-independent re-sults for nonvariational wave functions. Indeed,since in general the convergence of the Londonatomic orbital magnetic dipole operator is onlyslightly better than the conventional magnetic di-

pole operator in Eq. (21), except for very small basissets, no additional benefits in nonvariational wavefunction calculations can be expected from the useof the London orbital magnetic dipole operator [52].

3.3. FURTHER APPROXIMATIONS IN ROACALCULATIONS

The previous section has outlined the generalcomputational schemes needed for the calculationof ROA CIDs through the numerical differentiationof analytically calculated linear response functions.However, the computational cost of such calcula-tions has necessitated the investigation of more ap-proximate schemes in the calculation of ROA CIDs.

The most common approximation introduced isbased on the experience gained in the theoreticalstudy of VCD [53], where the harmonic force field iscalculated at a high level of theory, using, for in-stance, MP2 or DFT/B3LYP with basis sets of po-larized triple zeta quality. The generalized polariz-ability tensor derivatives in Eqs. (14)–(16) are thencalculated at a lower level of theory, in most casesusing a Hartree–Fock wave function with a rathersmall basis set. However, the adequacy of this ap-proach has been verified numerically only to a verylimited extent [26, 54], although the two studies thathave addressed this question indicate that this isindeed a very cost-effective approach to the calcu-lation of ROA CIDs.

A further simplification in the calculation of theelectronic tensors �, G�, and A was presented byBour [27, 55]. In his general sum-over-states ap-proach, the generalized polarizability tensors ex-pressed as a sum over excited states in Eq. (23) isused directly, calculating the energy denominatorsas orbital energy differences and the transition mo-ments as the transition overlap matrix elementsbetween different Kohn–Sham determinants. Wenote that this approach does not include orbitalrelaxation, which may be important in many casesfor an accurate representation of the molecularproperty.

4. Computational Studies ofVibrational Raman Optical Activity

In this section we will provide a brief summaryof ab initio studies of vibrational Raman opticalactivity. The number of studies of this property isstill rather scarce, but we hope to show that throughthe current understanding of the computational re-

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822 VOL. 104, NO. 5

quirements for accurate and reliable ROA calcula-tions, combined with the computational advancessummarized in the previous sections, we can expectan increase in the number of theoretical studies ofROA in the future. First, we discuss methodologicalstudies, and next some of the applications for chem-ical systems are reviewed.

4.1. BASIS SET INVESTIGATIONS

The proper choice of a basis set is essential forthe ab initio calculation of any molecular property,and ROA intensities present some unique chal-lenges in this respect.

The ROA spectra of a few small, vibrationallychiral molecules, e.g., H2O2, CHDTF, and CHD-TOH, have been calculated by Pecul and Rizzo [56]using a series of correlation-consistent basis setsand linear response SCF and MCSCF theory toinvestigate basis set and electron correlation effectson the ROA spectra. The same wave function ansatzwas used for both the molecular Hessian and theoptical tensors. Some of the results obtained (forH2O2 and CHDTF) are shown in Table I. It was

concluded that the use of basis sets augmented withdiffuse functions is essential in the calculation ofROA spectra, and if possible doubly augmentedsets should be employed, though these sets in gen-eral will remain too large for practical applications.In Ref. [56], the convergence of the ROA parameterswith the basis set size was also shown to in generalbe slower than that of the Raman intensities. Itseems that at least a d-aug-cc-pVTZ basis set shouldbe used to avoid an error larger than 10% in any ofthe vibrational modes with respect to the estimatedbasis set limit (the largest basis set used was aug-cc-pVQZ). However, the calculated ROA CIDs ofthe stretching vibrations are less sensitive to thesize of the basis set, and the aug-cc-pVDZ basis setseems to be sufficiently large for qualitative studiesin cases where the CIDs are large in magnitude.Basis set effects at the SCF and MCSCF levels werefound to be somewhat different: as a rule, ROACIDs tend to be affected more by the basis set sizewhen calculated at the MCSCF level, especially inthe case of the bending vibrations.

The performance of basis sets smaller than theaug-cc-pVDZ set has been investigated by Zuber

TABLE I ______________________________________________________________________________________________ROA (in Å4 amu�1 10�2) parameters of H2O2 and CHDTF calculated at the HF level using various correlation-consistent basis sets.*

Modea cc-pVDZ cc-pVTZcc-

pVQZaug-cc-pVDZ

aug-cc-pVTZ

aug-cc-pVQZ

d-aug-cc-pVDZ

d-aug-cc-pVTZ

H2O21 �658 �528.2 �484.7 �450.5 �444.9 �439.7 �433.2 �439.02 501.5 407 377.9 349.3 348.2 346.2 342.1 347.73 �3.225 �8.473 �8.148 �5.543 �2.438 �1.725 �0.7325 �1.1304 11.95 15.06 11.875 11.103 7.55 6.733 6.495 6.3655 �8.78 �8.595 �8.388 �8.608 �8.913 �9.073 �9.72 �9.3786 74.32 50.26 35.79 22.59 19.81 18.44 16.82 18.14

CHDTF1 0.0592 0.0352 0.0584 0.024 0.0272 — 0.02 0.02242 �0.2936 �0.38 �0.3264 �0.24 �0.248 — �0.2416 �0.24563 0.1888 0.4936 0.4688 0.3208 0.316 — 0.3488 0.33444 8.692 8.895 7.126 4.796 4.587 — 4.773 4.7655 �11.153 �11.402 �9.154 �6.14 �5.878 — �6.142 �6.1206 0.8848 0.8624 0.744 0.5216 0.4904 — 0.5192 0.49767 4.228 3.549 2.772 1.862 1.716 — 1.847 1.7978 �2.079 �1.53 �1.193 �0.8592 �0.7312 — �0.8456 �0.76169 �0.528 �0.5216 �0.4264 �0.2824 �0.2808 — �0.2776 �0.2896

* From Pecul and Rizzo [56].a Vibrational modes in H2O2 in order of decreasing frequency: 1, �as(OOH); 2, �s(OOH); 3, �as(OOH); 4, �s(OOH); 5, �(OOO); 6,�(HOOOOOH). Vibrational modes in CHDTF order of decreasing frequency: 1, �(COH); 2, �(COD); 3, �(COT); 4, 5, �(COH); 6,�(COF); 7–9, �(COD), �(COT).



and Hug [57] for a range of small chiral molecules.In this case, the force field and the electronic tensors�, G�, and A were treated separately, the formercalculated by means of DFT/B3LYP with the6-311��G** basis set, the latter using Hartree–Fockwave functions with different small- and medium-size basis sets. The results have been benchmarkedagainst results obtained with the aug-cc-pVDZ [58–61] and the Sadlej Pol [62] basis sets. The authorsconcluded that for hydrogen-containing molecules,a proper description of the gradients of the elec-tronic tensors (and therefore the ROA intensities)requires moderately diffuse p-functions on the hy-drogen atoms. To render reliably the ROA intensi-ties of the stretching vibrations of XOH bonds,diffuse functions with valence angular quantumnumbers (i.e., diffuse d functions for second rowatoms) are required. Provided these requirementsare fulfilled, the sign of the ROA intensities can bereproduced using basis sets as small as a 3-21Gaugmented by an appropriate diffuse function, anda p exponent of 0.2 was recommended for the hy-drogen atoms. The importance of diffuse functionsin the calculations of gradients of optical tensorshas also been observed by Bour [27].

We note that the importance of diffuse func-tions on the hydrogen atoms of chiral moleculesalso has been observed by Wiberg et al. [63], whodemonstrated that the correct dependence of theoptical rotation, which is determined by the G�tensor, with respect to the change in the dihedralangle only could be obtained if the basis set con-tained diffuse basis functions on the hydrogenatoms. Indeed, the importance of diffuse p func-tions in the calculation of magnetic properties, inparticular using London atomic orbitals, was ob-served already in 1994 [64, 65]. It is worth notingthat the popular 6-31G* basis set, used exten-sively in ROA calculations presented in the liter-ature (vide infra) does not contain diffuse polar-izing functions on the hydrogen atoms, and thusit appears that these basis sets may not be suffi-ciently accurate for studies of ROA CIDs.

4.2. ELECTRON CORRELATION EFFECTS

Since ROA calculations are very time-consum-ing, even at the Hartree–Fock level of theory, therehave been very few studies of electron correlationeffects on the CIDs by other methods than DFT. Thefirst correlated study of ROA was presented in 1994by Helgaker et al. [25], using MCSCF wave func-tions. The model system used, CHFDT, allowed for

modest-size active spaces to be used and demon-strated the importance of electron correlation ef-fects, though no attempts were made at distinguish-ing between the contributions arising fromcorrelation effects on the harmonic force field andthe correlation effects on the optical tensors. Thisstudy was also the first study to show the impor-tance of using London atomic orbitals in ROA cal-culations when small basis sets are used to calculatethe G� tensor.

Another MCSCF study was presented by Peculand Rizzo [56], who calculated the ROA spectra ofthree small vibrationally chiral molecules: H2O2,CHDTF and CHDTOH using the same methodol-ogy as in Ref. [25], that is, using linear responseHartree–Fock and MCSCF methods (for both theoptical tensors and the force field). From the datareproduced in Table II, we note that the CIDs cal-culated at the HF level are in most cases consistentwith the RASSCF results, at least as far as the signis concerned. The exceptions are the stretching vi-brations of CHDTF. It seems that when the CIDs arelarge, the HF method predicts the correct sign forthe CID, and can therefore be a useful and compu-tationally inexpensive tool for in assigning absoluteconfigurations of molecules (vide infra).

4.3. DENSITY FUNCTIONAL CALCULATIONS

The most widespread method in use for the cal-culation of ROA spectra is, apart from the Hartree–Fock method, DFT. However, even such calcula-tions are fairly rare to date and in most casesinvolving additional approximations in the evalua-tion of the property tensors. A brief summary of theDFT approaches presented in the literature will bepresented in this section.

Bour [27] has combined sum-over-states DFT cal-culations (including hybrid functionals) of opticaltensors with numerical differentiation with respectto nuclear displacements. The author proposed theBecke–three-parameter–Lee–Yang–Parr (B3LYP)hybrid functional [66] with the excited electronicstates replaced by Kohn–Sham Slater determinants(and excitation energies approximated as a differ-ence between orbital energies) as a reasonablemodel balancing the need for accuracy to the com-putational cost. The methodology has been testedfor �-pinene and helical peptides. A similar SOSapproach has also been presented previously by thesame author for the Hartree–Fock method [55].

The first analytical DFT calculations of ROA spec-tra using linear response theory and London atomic

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824 VOL. 104, NO. 5

orbitals was presented by Ruud, Helgaker, and Bour[67]. In this study, the validity of using DFT forcefields with Hartree–Fock optical tensors was investi-gated and shown to be a rather good approximationin most cases compared with a pure DFT approach,although a few weak modes might show differencesin sign. However, the use of a high-level force fieldwith a lower-level method for the calculation of theoptical tensors were shown to be adequate to deter-mine fairly unambiguously the absolute configura-tion of �-pinene and trans-pinane. The DFT-SOS ap-proach of Bour [27] was also shown to yield verygood agreement with the full DFT results.

4.4. APPLICATIONS TO STRUCTURALSTUDIES

4.4.1. Determination of AbsoluteConfiguration

Probably the most successful use of theoreticalcalculations of ROA CIDs in the determination ofthe absolute configuration of a molecule was thework by Polavarapu and coworkers [68,69]. Theseinvestigators confirmed the absolute configura-tion of bromochlorofluoromethane on the basis ofa comparison of DFT/B3LYP calculations of theoptical rotation and ROA spectra to experimentalobservations. ROA CIDs are particularly valuablein this case, as it has been proved difficult todetermine the absolute configuration of this mol-

ecule using other chiroptical techniques. The first[69] assignment of (S)-(�) and (R)-(�) bromo-chlorofluoromethane was later confirmed by cal-culations using larger basis sets in Ref. [68].

Another example of the application of ab initiocalculations of ROA for this purpose is the work byZuber and Hug [9], where the absolute configura-tions of (4S)-4-methylisochromane and (4S)-isomersof 1,3,4,6,7,8-hexahydro-4,6,6,7,8,8-hexamethyl-indeno[5,6-c]pyran were considered. In this case,the task was complicated by the presence of differ-ent conformers for the molecules of interest and bythe fact that they have several stereogenic centers.Despite these potential complications in the theo-retical analysis and the small basis set used, resultsin good agreement with the experimental data for(4S)-4-methylisochromane were obtained, and theabsolute configuration of 1,3,4,6,7,8-hexahydro-4,6,6,7,8,8-hexamethylindeno[5,6-c]pyran wasalso confirmed.

4.4.2. Conformational Dependence

Most ROA measurements carried out currentlyare aimed at obtaining structural information onpeptide conformation. Not surprisingly, this is alsothe focus of most applications of quantum mechan-ical methods to the calculation of ROA spectra.However, because of the large computational effortrequired for ROA calculations, most applications in

TABLE II ______________________________________________________________________________________________ROA (in Å4 amu�1 10�2) parameters of H2O2 and CHDTF calculated at the HF and RAS–SCF levels with aug-cc-pVTZ basis set.*

H2O2 CHDTF

Modea HF RAS–SCF Modeb HF RAS–SCF

1 �444.9 �486.2 1 0.0272 �0.85122 348.2 372.6 2 �0.248 0.43523 �2.438 �9.29 3 0.316 1.10644 7.55 17.19 4 4.587 6.0625 �8.913 �9.17 5 �5.878 �7.2846 19.81 18.32 6 0.4904 0.5208— — — 7 1.716 1.692— — — 8 �0.7312 �0.5024— — — 9 �0.2808 �0.3136

* From Pecul and Rizzo [56].a Vibrational modes in order of decreasing frequency: 1, �as(OOH); 2, �s(OOH); 3, �as(OOH); 4, �s(OOH); 5, �(OOO); 6,�(HOOOOOH).b Vibrational modes in order of decreasing frequency: 1, �(COH); 2, �(COD); 3, �(COT); 4, 5, �(COH); 6, �(COF); 7–9, �(COD);�(COT).



this field are limited to single amino acids [44, 70] ordipeptides [71–73].

l-Alanine is the species for which ROA spectrahave been most frequently simulated. At the sametime that the experimental measurements of thebackscattering ROA spectra of l-alanine in aqueoussolutions of different pH were completed, theoret-ical calculations of ROA CIDs were carried out fora single conformer of l-alanine, using the static-limit approximation for SCF optical tensors [70] andthe 6-31G* basis set, with the common gauge originat the center of mass of the molecule (London or-bitals were not used). With improvements in abinitio methodology, the calculations have been re-peated in several other investigations [44, 45].

Yu et al. [44] calculated the ROA spectrum ofseveral isotopomers of deuterated l-alanine, usingthe static SCF approximation for G� and the 6-31G*basis set. A reaction field model with a sphericalcavity was used to simulate the aqueous solventeffects on the geometry and the force field, but noaccount of the effects of the dielectric medium onthe optical tensors was included.

The use of approximate computational schemesallows calculations of ROA spectra to be extendedto molecules of the size of small peptides. To dem-onstrate this, the ROA spectra for the models offour standard peptide conformations (�-helix, 310-helix, coil, and �-sheet) have been calculated byBour [27].

The ROA spectrum of cyclo(l-Pro-l-Pro) hasbeen simulated by Bour et al. [72] using the SOS-DFT method with the B3LYP functional and the6-31��G** basis set, for three conformations of thepeptide. By comparison of the experimental andcalculated ROA spectra, it was concluded that themost stable conformer is present in large excess inthe solution. However, it was still found that theNMR spectrum was generally more informative.Another study carried out by the same group forl-alanyl-l-alanine [71] focused on the effects of anaqueous environment and is discussed in more de-tail in the next section.

The structural studies on small peptides, includ-ing those involving calculation of ROA spectra, arereviewed in Ref. [17]. One of the first of the ROAcalculations in this cycle, carried out by Jalkanen’sgroup, was the study of N-acetyl-l-alanine N�-methylamide [73], in which the molecular confor-mations and the corresponding Raman, VCD, andROA spectra were investigated for a complex ofN-acetyl-l-alanine N�-methylamide with four watermolecules.

Few works have investigated the conformationaleffects on the ROA spectra of molecules other thanamino acids or small peptides. The ROA spectra ofdifferent conformations of glyceraldehyde, lacticacid, and the lactate anion have been calculated byPecul et al. [74] by means of linear response SCFtheory with the aug-cc-pVDZ basis set. The ROAspectra of these molecules have been found to bevery sensitive to conformational changes. The ROAparameters of the stretching vibration of the car-bonyl group seems to be a promising parameter forstructural investigations, since it changes sign whena short internal hydrogen bond OH . . . OC isformed. However, at this point it should be notedthat no solvent effects were accounted for, and thatonly internal hydrogen bonds were analyzed.

4.5. SOLVENT EFFECTS

It is well known from experiments that bothoptical rotations and ECD spectra are very sensitiveto solvent effects [75, 76]. In some cases, a change ofthe solvent can even lead to a change of sign of theoptical rotatory strength or of the optical rotation[75, 76], even for rigid molecules such as methylox-irane [76]. This can therefore also be expected to bethe case for ROA spectra, although we are notaware of any systematic investigation of solventeffects on them, as such spectra are usually col-lected in neat liquids or in aqueous solution in largeconcentrations. A reliable model of the aqueousenvironment is particularly important for the inter-pretation of experimental ROA spectra of biomol-ecules. A relatively large effort has been devoted tothe simulation of the influence of water on the ROAspectra of l-alanine [44, 45] and dipeptide models[71].

The experimental ROA spectra of l-alanine inaqueous solutions have been collected by Barron etal. [70, 77]. In the same work [70], attempts weremade to reproduce these experimental spectra byab initio calculations for isolated zwitterionic struc-tures. Later, a polarizable Onsager model with aspherical cavity was used for the force field in thecalculation of the ROA spectrum of l-alanine by Yuet al. [44], although with limited success.

Jalkanen et al. [45] simulated the solvent effectson the ROA spectra of alanine using a force fieldobtained employing the Onsager model (and withthe optical tensors calculated for an isolated mole-cule), and also by adding water molecules explicitlyin order to model the first solvation shell. The cal-culations were carried out using the DFT/B3LYP

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826 VOL. 104, NO. 5

model with the 6-31G* basis set for the force field.The authors concluded that the combination of ex-plicit solvent molecules and the Onsager modelpermits better interpretation of the experimentalROA spectrum.

To reproduce the ROA spectrum of zwitteri-onic l-alanyl-l-alanine [71], different continuummodels have been used to simulate the aqueousenvironment: a dipolar Onsager model with aspherical cavity, an ionic model, and a conductor-like screening model (COSMO). In all cases, theoptical tensors calculated in vacuum were used.The COSMO model was found to be the mostuseful model, whereas the Onsager model wasshown to fail.

To date, all calculations of solvent effects onROA CIDs using dielectric continuum modelshave taken the effect of the environment intoaccount only through the changes to the molecu-lar geometry and force field, but not on the opti-cal tensors. It is worth noting that the polarizablecontinuum model [7, 78 – 80] has been used tostudy dielectric continuum effects on both forcefields and the optical tensors in the modeling ofthe solvent effects on Raman spectra [81] and hasalso been used for the studies of optical rotation[7] and electronic CD spectra [8]. We can expectthat this approach will be extended to ROA spec-tra in the near future.

5. Concluding Remarks and Outlook

We have briefly reviewed the theoretical foun-dations for the ab initio calculation of vibrationalROA, the computational requirements needed foraccurate theoretical studies of ROA, and the appli-cation of theoretical methods to understand therelationship between the structure of chiral mole-cules and experimentally recorded ROA spectra.

The bottleneck in these calculations remains theneed to calculate geometrical derivatives of the sec-ond-order molecular tensors �, G�, and A. Cur-rently, these geometric derivatives are calculatedusing numerical differences of the electronic ten-sors with respect to nuclear distortions, requiring atleast the calculation of 6N � 1 property calculationsfor a molecule containing N atoms. Clearly, theroutine calculation of ROA CIDs will strongly ben-efit from an analytical implementation of the prop-erty derivatives. However, the need to ensure thatthe results are not only gauge origin independent,but accurate and reliable when very small basis sets

are used necessitates the use of London atomicorbitals, which complicates the evaluation of ana-lytical geometry derivatives of the second-orderelectronic tensors. As a result, no analytical imple-mentation of ROA CIDs has been presented to date.

An alternative approach to speed up the calcu-lation of ROA CIDs would be to take advantage ofmodern computer architectures that allow for mas-sively parallel calculations. Indeed, the problem offinite numerical differentiation is intrinsically par-allel, and thus the calculation of the electronic ten-sors at each of the distorted molecular geometriescan be calculated on separate computers, signifi-cantly speeding up the wall-time used for the cal-culation of ROA CIDs [54].

In many cases, not all vibrational modes areneeded, for instance, only vibrational modes thatare expected to be signature bands for specificstructural characteristics of for instance biomol-ecules, such as the amide I and II bands are often ofinterest. In these cases, it should be possible to takeadvantage of the recent developments of local op-timizations of normal modes without a full calcu-lation of the molecular force field as presented byReiher and coworkers [82, 83], and work alongthese directions are in progress [84]. Such localforce field approaches may also prove valuable inthe study of solvent effects in which explicit solventmolecules are included in the model, but where theintermolecular vibrations and the vibrationalmodes of the solvent molecules are not of interest.

In summary, the theoretical calculation of ROACIDs is still a challenging task. However, there arealso many directions for improvement in the theo-retical models used to calculate ROA spectra, andwe expect the range of applicability of theoreticalmethods to help analyze experimental spectra tocontinue to increase in the future, and thus helpmake ROA an important structural tool in the fu-ture.

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ab initio calculation of vibrational raman optical activity

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