Vibrational Raman optical activity (ROA) has long been a focus of interest for theoretical chemists. The theoretical background for the ROA phenomenon was given by Barron and Buckingham in 1971,1 prior to the first indisputable measurements by Barron et al. in 19732 and by Hug et al. in 1975.3 The first complete ab initio simulation of a ROA spectrum was presented in 1990 by Polavarapu,4 who used the static approximation of Amos.5 The first correlated calculations using MCSCF wave functions and London atomic orbitals were carried out 4 years later,6 and frequency-dependent optical tensors were used. Currently, most of the computational techniques available for modeling of ROA spectra rely on time-dependent density functional theory.7–9

In the last few years, the increasing activity in calculations of vibrational ROA using quantum chemical methods can be observed, reflecting the growing number of experimental works using ROA. In particular, ROA has become a valuable tool in the investigation of structural types of peptides and proteins, mainly by using the vibrational amide bands I and II (Ref.10 for the review), and many of the computational works are geared toward supporting this branch of research.

The present mini review outlines new developements and remaining problems in computational ROA spectroscopy. We want to focus less on the applications of ROA calculations for standard chemical problems such as determination of absolute configuration, since those have been comprehensively reviewed for example by Polavarapu,11, 12 and more on the remaining challenges in theoretical prediction of ROA spectra, and attempts to meet them. Thus, we will discuss only briefly and only the most recent calculations of ROA spectra geared toward establishing absolute configuration of natural and synthetic organic molecules and high-order structure of peptides and proteins. This review is focused more on modeling of solvent effects in ROA spectra, of anharmonic effects, and of resonance and pre-resonance ROA. Perspectives for relativistic calculations of ROA spectra are also discussed. Particular attention is paid to computational treatment of surface enhanced Raman optical activity (SEROA). Finally, some concluding remarks are given.

THEORETICAL INTRODUCTION

The ROA effect is described by means of the absolute difference between intensities of the scattered light with the incident light circularly polarized left and right, , where I are the scattered intensities with linear k polarization for right- (R) and left- (L) circularly polarized incident light, and k denotes the Cartesian component (incident circular polarization Raman optical activity, ICP-ROA). Alternatively, ROA can be measured as a small circularly polarized component in the scattered light using incident light of fixed polarization, including unpolarized light (scattered circular polarization Raman optical activity, SCP-ROA). These two approaches are equivalent in the far-from resonance limit. The differential scattering intensity between right and left circularly polarized light for the Z-polarized backward scattering is given by1, 13:

where ω_{rad} is the radiation angular frequency, ϵ_{kjl} is the unit third rank antisymmetric tensor, and the other quantities are defined later.

Most of the computations of the ROA spectra is carried out within the double harmonic approximation (see section “Modeling of Anharmonicity in ROA” for a review of works going beyond that). Within the double harmonic approximation, the other quantities in eqs. 2 and 3, defined as products of vibrational transition moments, can be described by means of geometric derivatives of optical tensors.

The tensors in eqs. 4 and 5 are the electric dipole–electric dipole polarizability α, the imaginary part of the electric dipole–magnetic dipole polarizability G′, and the real part of the electric-dipole–electric quadrupole polarizability A.13 Q is the normal coordinate of the vibration under study. The subscript 0 indicates that the quantities are calculated at the equilibrium geometry. The second equality in eqs. 4 and 5 is valid only within the harmonic approximation.

The α, G′ and A tensors in the notation of modern response theory can be written as14

Expressions 〈〈A;B〉〉_{ω} in the above equations denote linear response functions.

CALCULATIONS OF ROA SPECTRA OF AMINO ACIDS AND PEPTIDES

The most important field of experimental applications of ROA is structural investigation of proteins and their aggregates. As a result, most of the calculations of ROA spectra nowadays are carried out for amino acids and peptides. It is not our aim to review all those calculations, but to highlight some of the most interesting, from our point of view, applications published after 2004. For a review of the former (pre-2004) computational investigations of ROA spectra of amino acids and peptides, we refer the reader to the review by Pecul and Ruud.15 Other reviews of interest in this topic are those by Jalkanen et al.16, 17

Calculations for L-Alanine and Its Oligomers

The system for which the largest number of calculations of ROA spectra has been carried out is L-alanine and its oligomers. The first theoretical work on this subject was published as early as 1991,18 and since then there has been many approaches to modeling of the ROA spectrum of the simplest chiral amino acid. Recently, vibrational spectra of L-alanine and N-acetyl L-alanine N′-methyl amide (alanine dipeptide) in aqueous solution have been carried out by Jalkanen et al.19 Calculations of the ROA spectra for the same dipeptide have been presented by Mukhopadhyay et al.20 In this case, Monte-Carlo simulations in water environment have been used to obtain the structures. In another work of interest,21 the band shapes of ROA spectrum of L-alanine zwitterion has been modeled. It has been found that that the shapes of the spectral bands are to a large extent determined by the internal rotation of the NH, CO, and CH_{3} groups.

ROA spectra of N-acetyl-(L)-alanine N-methyl amide, and trialanine isomers containing L and S alanine enantiomers have been calculated by Herrmann et al.22 to clarify the problem of relative contributions from the conformations of amino acids side chains and peptide backbone to the ROA bands of amide vibrations. ROA spectra of helical deca-alanine of several enantiomeric compositions has been studied previously by the same group,23 and the results for all-L deca-alanine have been compared with experiment.

Calculations for L-Proline and Its Oligomers

Many calculations of the ROA have also been carried out for L-proline24 and proline-containing peptides.21, 25–27 These studies have been motivated by the role of proline, a cyclic amino acid, in forming important structural elements (such as turns) in proteins. The ROA spectra of proline tripeptides with different ring conformations have been calculated to analyze experimental Raman and ROA spectra of polyproline.25 It has been found that two conformers of the proline ring are almost equally populated in polyproline, but that only one helical conformation prevails. This finding is of particular interest since the helical conformation of polyproline (polyproline II, also abbreviated as PPII) constitutes an important structural motif in proteins.

The comparison of performance of nuclear magnetic resonance and ROA spectroscopy in conformational analysis for model proline-containing dipeptides (Pro-Gly, Gly-Pro, Pro-Ala, and Ala-Pro) has been carried out by Budĕs̆íský et al.,26 and the calculated ROA spectra (with COSMO model of the aqueous environment) have been compared with experiment. The agreement was far from perfect, but the main features of the experimental spectrum in the 700–1600 cm^{−1} region of Stokes shift have been reproduced by means of the calculations. The ROA spectra (experimental and theoretical) of the same dipeptides have been analyzed in terms of the dependence of the line shape on the conformational dynamics.21 Further studies by the same group included simulations of the ROA spectrum of D-Ala-L-Pro-Gly-D-Ala peptide, focused on conformations of the Pro-Gly motif.27

Calculations for Other Amino Acids and Peptides

ROA spectra of amino acids other than alanine or proline (or peptides containing them) are calculated less frequently. Correlation of ROA intensitiy of a stretching vibration of an indole ring in tryptophan with the conformation of the aromatic ring with respect to the remaining part of a tryptophan-containing peptide has been investigated by Jacob et al.28 by means of DFT calculations (β(G ′) derivatives only) for N-acetyl-(S)-tryptophan-N-methylamide. Another small peptide for which ROA spectrum has been calculated is tri-L-serine.29 The ROA spectra have also been modeled by means of DFT and HF calculations for L-serine and L-cysteine, but only for neutral gas phase-like conformations.30L-Histidine has also attracted attention of computational chemists: vibrational spectra (IR, VCD, Raman, ROA) of its zwitterion forms have been computed by Deplazes et al.31 to compare the conformational information derived from different types of vibrational spectroscopy.

Local Versus Nonlocal Effects in ROA Spectra of Peptides

Some of the previously mentioned works are focused on the problem of separation of local and nonlocal structural effects in the ROA spectra of peptides in the amide I, II and III regions (e.g., see Refs.23 and22). It has been suggested in Ref.23 that the backbone conformation dominates the ROA patterns for delocalized backbone vibrations, in accordance with the findings for helical conformations of synthetic polymers.32, 33 However, Ref.22 partly contradicts it, at least for short peptides with relatively flat (nonhelical) backbone structure, pointing out to the importance of side chain conformation. Similar conclusions (comparable contributions from backbone helicity and side chain conformations) have been drawn for polyproline.25 In this context, it should be mentioned that some qualitative conclusions on the dependence of ROA bands of amide vibrations can also be drawn from simulations on a simple system like nonplanar N-methylacetanide, as shown in Ref.34.

To sum up this part, the number of computational simulations of ROA spectra of peptides is growing, as ROA spectroscopy becomes more and more widespread method of structural investigations. It is also worth mentioning that the calculations are not limited to quantum chemical studies of the whole molecule: the recent work by Choi and Cho35 has shown that the fragment approximation method (commonly used in calculations of electronic circular dichroism spectra) can also be sucessfully applied to simulate ROA spectra of peptides. One of the still remaining problems in this type of calculations is taking into account solvent effects, discussed in one of the next subsections.

CALCULATIONS OF ROA SPECTRA AS A TOOL FOR DETERMINATION OF ABSOLUTE CONFIGURATION

Calculations of ROA spectra for nonbiological molecules are to a large extent devoted to the determination of absolute configuration. Quantum chemical prediction of ROA spectra of small systems is now reliable enough for this purpose, especially if there are no or little solvent effects to account for (that is, when experiment is carried out for a neat aprotic liquid) and conformational flexibility is limited. Recently, there has been several such applications.36–39 We mention them only briefly and only the newest works, since the subject has been adequately covered recently by Polavarapu.11

Zuber and Hug have assigned absolute configuration for (4S)-4-methylisochromane on the basis of its experimental and calculated ROA spectra. Absolute configuration has also been assigned on this basis for juniouone (a natural cyclobutane monoterpenoid).38 The study of chiral deuterated neopentane37 is particularly interesting, since vibrational optical activity has been, in this case, the only way to determine absolute configuration: the molecule does not contain heavy atoms allowing for its investigation by means of crystaleographic methods, and its “electronic” chirality (natural optical rotation, electronic circular dichroism) is expected to be too weak to be measured. Finally, we would like to mention the work of Gheorghe et al.39 on the assignment of absolute configuration to a quaternary ammonium salt with a chiral alkyl substituent, methyl-(R)-(1-methylpropyl)di(n-propyl)ammonium iodide. Similarly as prediction of ROA spectra for peptides, applications of ROA spectroscopy for the determination of absolute configuration would certainly benefit from more accurate liquid phase models, capable of accurate rendering of the ROA spectra of neat liquids, including those with hydrogen bonds, and of aqueous solutions.

MODELING OF SOLVENT EFFECTS IN ROA

Modeling of solvent effects is essential for rendering realistic ROA spectra of molecules in solution, especially in the case of hydrated species. There are basically three approaches to this problem: (a) the supermolecular approach, in which solvent molecules are treated quantum mechanically (usually at the same level of theory as the solvated molecule); (b) a family of continuum models, in which the solvent is modeled as a macroscopic continuum dielectric medium (assumed homogeneous and isotropic) characterized by a scalar dielectric constant, and the solute, described at quantum mechanics level, is placed in a cavity in a dielectric medium; and (c) the hybrid model, merging the other two, and treating, for example, the first solvation shell quantum mechanically and the remaining solvent as polarizable continuum. The most widespread approaches of the family of continuum models are the Integral Equation Formalism (IEF)-PCM method by Cancés et al.40 which uses a molecule-shaped cavity to define the boundary between the solute and the solvent, and the COSMO(COnductorlike Screening MOdel) method by Klamt and coworkers,41–43 in which the surrounding medium is modeled as a conductor instead of a dielectric. Most of the calculations carried out for amino acids and peptides outlined in the first subsection of this chapter have been carried out with some variety of a solvent model, so here we will concentrate only on those focused specifically on the developement and investigation of performance of the solvent models.

Extensive studies on hydration of alanine have been carried out by Jalkanen et al.19 by means of supermolecular calculations and the hybrid model. The authors compare the results obtained by means of Born-Oppenheimer molecular dynamics simulations with 20 solvating water molecules and the rest of the solvent modeled by means of various polarizable continuum models with the previous results obtained by the same group with simpler aqueous environment models (four water molecules in Ref.44). The authors conclude that the hybrid model is the best approach currently available, and advocate its use for further studies of amino acids and peptides. However, it seems that some features can be captured even by the use of polarizable continuum model alone, as indicated in the calculations carried out by Pecul et al. (Manuscript in preparation) for ROA spectra of hydroxyproline in zwitterion, anionic, and cationic conformations.

ROA spectra are sensitive to environment, and this field of application will certainly be growing. In particular, further applications of molecular dynamics to model the first solvation shell, possibly coupled with polarizable continuum model of the remaining solvent would be of interest. Another problem is that in most of the applications so far solvent effect was taken into account only for geometry optimization and in calculations of the force field, but not for optical tensors. The information on relative role of these contributions is partly contradictory, and this issue requires further investigation.

MODELING OF ANHARMONICITY IN ROA

A vast majority of the calculations of ROA spectra is carried out in double harmonic approximation, although the theory13 allows for a more general approach. The only work going beyond the double harmonic approach we are aware of is that of Danĕc̆ek et al.,24 in which ROA spectra (together with IR and Raman spectra) of alanine and proline zwitterions (obtained using COSMO-PCM solvent model) have been calculated by means of vibrational self consistent field, vibrational configuration interactions (VCI), and degeneracy-corrected perturbation calculations. Anharmonic effects have been evaluated both for vibrational frequencies and for ROA intensities. The authors concluded that the VCI method performs best in the case of the ROA and Raman spectra, and that the most important corrections on intensities originate from the force field (third and fourth energy derivatives in their model). The anharmonic corrections on spectral intensities stemming from second intensity tensor derivatives, although more important for Raman and ROA spectra than for IR, have been found to be relatively minor, and in most cases probably below experimental noise.

ROA OF TRANSITION METAL COMPLEXES

Applications of ROA spectroscopy are not limited, in principle, to organic chemistry and biochemistry. Although, as far as we know, there are no experimental ROA spectra collected for chiral transition metal complexes, a computational study of them was presented by Luber and Reiher.45 The authors used nonrelativistic DFT and tested several exchange-correlation potentials.

Further developements in this area would require inclusion of relativistic effects, at least by quasi-relativistic methods. There seem to be two relativistic approaches capable of handling ROA calculations. One is 4-component Dirac-Hartree-Fock and Dirac-Kohn-Sham approach implemented in Dirac,46 and the other is 2-component zeroth order regular approximation (ZORA) density functional theory approach47, 48 implemented in ADF.49, 50 Both approaches are already developed for a wide range of magnetic and optical properties, and calculations of ROA spectra using either of these approaches are in principle feasible, although, in the case of 4-component calculations, very expensive. It is to be hoped that at least 2-component calculations of the ROA spectra will become available in the near future.

RESONANCE ROA

Performing ROA measurements at the wavelength corresponding to electronic transition, thus using the resonance signal enhancement of vibrations of the group involved in the electronic transition, has been considered a natural extension of ROA, allowing to increase the sensitivity and selectivity of the method. However, the appearance of the resonance ROA (RROA) spectrum is very different than in the case of nonresonant ROA. The theory of the phenomenon, presented by Nafie51 in single electronic state limit, predicts that all resonance-enhanced ROA bands should have the same sign, opposite to the sign of the rotatory strength of the electronic transition. This has been verified experimentally for (+)-naproxen, its deutered methyl ester and (−)-naproxen52 and has been shown to be indeed the case.

A more general theoretical approach to RROA has been presented and implemented within time-dependent density functional theory approach by Jensen et al.53 The approach uses linear response theory incorporating a damping factor to account for finite lifetime of the electronic excited state. The authors used it to predict resonance vibrational ROA spectra of dihydrogen dioxide and (S)-methyloxirane, and they also obtained uniform sign for all vibrational transitions.

The lastest developement in the theory of RROA has been recently presented by Nafie54 in the form of near-resonance theory, which allows to treat the cases where far-from-resonance ROA theory breaks down and vibronic structure of the excited states becomes important (with the consequence of ICP and SCP forms of ROA becoming nonidentical). However, no numerical applications of this approach have been presented so far.

SURFACE–ENHANCED ROA

Another approach expected to enhance intensity of ROA spectra is to use the phenomenon of so-called surface-enhanced Raman scattering (SERS) effect: a very large increase of the Raman cross-section for molecules placed in resonators formed from some (e.g., Ag, Au, Cu) metal nanoclusters (such resonators can be found, for example, at electrochemically roughened metal electrodes or in metal sols). The efficiency of SERS scattering can be in some cases 15 orders of magnitude higher in comparison with the normal Raman scattering,55 allowing for measurement of Raman spectra of samples at very low concentrations, and, in some cases, permitting the observation of Raman spectra even of a single molecule.56–58 Unquestionable SEROA spectra have been measured only in 200659 (the measurements of Kneipp et al.60 for achiral adenine are controversial), but a lot of attention has been devoted to the SEROA effect by theoretical chemists.

The first works on the effect a metal surface has on ROA spectrum were those of Efrima61, 62 on the electromagnetic effects for subtrates fixed with respect to the laboratory frame. His analysis predicted significant enhancement of ROA intensity of metal surface, provided large local inhomogenity of the electric field, phase difference between the field gradient and the field itself, and existence of imaginary part of molecular polarizability tensor. The author argued that those conditions are met in SERS-active systems. The theory of SEROA from molecules attached to a surface fixed with respect to the laboratory frame has been further explored by Hecht and Barron,63, 64 who have shown that lack of orientational averaging leads to an increase of ROA intensity and that the enhancement depends on the electric dipole polarizability tensor.

The theory of electromagnetic enhancement in SEROA on freely rotating substrates (such as metal nanoparticles in soles) has been developed by Janesco and Scuseria.65 They have found that for dipolar substrate the enhancement of SEROA is very weak, since the terms contributing to the first order for fixed orientation average out to zero. Larger enhancements are to be expected for quadrupolar substrates. They have implemented the methods for calculating SEROA electromagnetic enhancement for molecules adsorbed on substrates modeled as dipolar and quadrupolar spheres, and then carried out the calculations for (R)-bromochlorofluoromethane. The numerical results supported the theoretical predictions: the SEROA enhancement on a dipolar sphere is very weak, but it is larger on a quadrupolar sphere.

The calculations for dipolar spheres have been repeated by Bour̆66 in matrix formulation. The results obtained for (R)-bromochlorofluoromethane on a single dipolar sphere are in agreement with those of Janesco and Scuseria.65 Interestingly, the spectrum has been found to change very much for a molecule in between two or more conductive dipolar spheres: the intensity is much larger and ROA signals can even change their signs. Thus, it seemed that SEROA spectra, similarly as SERS spectra, originate mostly from the so-called “hot spots” (sites with particularly high SERS activity) present on the metal surface, and are very sensitive to the size and shape of metal nanoparticles. This is consistent with experimental findings.67

It may be worth mentioning that the theoretical predictions about the SEROA enhancement being larger on substrates fixed with respect to the laboratory frame than on freely rotating ones have been confirmed to some extent by experiment. Although the measurements in silver sols yield small surface enhancement,59, 67, 68 the SEROA enhancement factor has been estimated to be at least 3–4 orders of magnitute for electrochemically roughened solid silver substrate (Pecul and Kudelski, submitted).

According to our knowledge, there are no theoretical studies of “chemical” effect on SEROA spectra. Such calculations, involving explicit quantum mechanical treatment of a small metal cluster and adsorbed molecule, would be of considerable interest. We refer the reader to the recent review by Jensen et al.69 for a more complete review of theoretical approaches to SERS, including SEROA.

CONCLUDING REMARKS

Much progress has been done recently in quantum chemical modeling of ROA spectra. On one hand, even larger systems are becoming accessible due to developement of computational technique, in particular due to introduction of analytical differentiation of optical tensors with respect to nuclear coordinates.9, 70 On the other hand, new theoretical works have appeared, allowing to handle nonstandard calculations of ROA spectra: significant advances have been made in the prediction of resonance and pre-resonance ROA spectra and in theory of SEROA spectra. There are also approaches developed allowing to calculate ROA spectra beyond double harmonic approximation. What still remains a challenge is including relativistic effects in the calculations of ROA spectra of transition metal complexes, and there is still a wide scope for improvements in the modeling of solvent effects.