• ab initio calculations;
  • Raman optical activity;
  • chiral molecules


  1. Top of page
  2. Abstract
  8. Literature Cited

In this article, we will give a brief account of the different approaches that have been presented in the literature for calculating Raman optical activity (ROA) spectra by ab initio methods. We will also outline the general structure of a self-consistent-field-based approach for analytic calculations of ROA spectra, including also contributions from London orbitals. The use of London orbitals ensures that the relevant ROA parameters are gauge origin independent. We will also give an outlook on the future of ab initio calculations of Raman optical activity spectra. Chirality 21:E54–E67, 2009. © 2009 Wiley-Liss, Inc.


  1. Top of page
  2. Abstract
  8. Literature Cited

Raman optical activity (ROA)1–4 has during the last 35 years matured to become an important method for determining the absolute configuration of chiral molecules5 and for elucidating the structure of biomolecules in their native environment.6–9 This is to a very large extent due to recent advances in instrumentation,10, 11 but an equally important facet of this development has been the advances made in ab initio theory for the calculation of ROA spectra at an increasingly higher level of theory, thus obtaining much more reliable theoretical estimates of the ROA spectra.

In this minireview, we provide a brief account of the theoretical foundations of ROA, which will primarily serve to define the quantities that need to be determined in theoretical calculations of ROA spectra, and give a historical account of the theoretical developments that have been made in the calculation of ab initio ROA spectra. In addition to accounting for the important historical milestones in the development of ab initio methods for calculating ROA spectra, we will also try to give the reader a qualitative understanding of the strengths and weaknesses of the different methods and their theoretical basis. In the section “The ab initio calculation of Raman Optical activity spectra,” we will give the more detailed equations for how to analytically calculateROA circular intensity differences (CIDs). We will not here try to address all the different implementations, but we will instead give one special derivation, based on the quasi-energy derivative formulation of response theory,12–14 which we believe gives a transparent understanding of the necessary components of such analytic calculations. The theory will be applicable to any self-consistent field approach [which includes Hartree–Fock and density-functional theory (DFT)], and it includes London orbitals to ensure origin independent results.

In the section “Future challenges and outlook” section, we give some concluding remarks and an outlook, we give on the future challenges in the theoretical calculation of ROA spectra. We emphasize that we do not intend to provide a review of the many theoretical studies of ROA that have been published in the literature, referring instead the interested reader to recent reviews covering this topic.15, 16


  1. Top of page
  2. Abstract
  8. Literature Cited

We will in this chapter give a brief summary of the relevant quantities that determine the ROA CIDs. Our purpose is not here to give a complete account of the theory, which can be found elsewhere in different excellent presentations,1, 2, 17–19 but rather to give the key equations that allow us to define a notation for the quantities that need to be calculated using ab initio methods to determine a ROA spectrum theoretically.

In ROA, the key quantity is the differential scattering of right and left circularly polarized light. The general expression of scattered circular polarization experiments for a scattering angle ξ are given by the expression1

  • equation image(1)

The corresponding expressions for specific experimental setups, both for Raman and ROA experiments, are reported in Table 1.

Table 1. Expressions for the Raman and ROA intensities for different experimental setups
Experimental setupRamanROA
  1. All expressions should be multiplied by the factor equation image.

Polarized ( equation image)equation imageequation image
Depolarized ( equation image)equation imageequation image
Backward (ξ = π)equation imageequation image
Forward (ξ = 0)equation imageequation image

In eq. 1, we have introduced a number of Raman and ROA invariants, defined respectively as

  • equation image(2)
  • equation image(3)
  • equation image(4)
  • equation image(5)
  • equation image(6)

In these equations, Qi denotes the i'th mass-weighted vibrational normal mode of the molecule, ω the frequency of the incident light, ωi the frequency corresponding to the vibrational excitation of the i'th vibrational mode, and equation image an element of the Levi–Civita antisymmetric tensor. Implicit summation over repeated Greek indices has been used in all above mentioned equations. All derivatives are evalued at the molecular equilibrium geometry, indicated by the subscript Re.

equation image, equation image, and equation image are, respectively, the electric dipole–electric dipole, electric dipole–magnetic dipole, and electric dipole–electric quadrupole polarizabilities, defined in terms of sum-over-states expressions for exact wave functions as (using here, and for the remainder of the article, atomic units) [1]

  • equation image(7)
  • equation image(8)
  • equation image(9)

where we have introduced the electric dipole moment operator equation image, the traceless electric quadrupole operator equation image, and the magnetic dipole operator equation image as

  • equation image(10)
  • equation image(11)
  • equation image(12)

where the summations run over the nuclei K and electrons i, respectively, equation image is the α component of the momentum operator of electron i and ri is the corresponding position operator of the electron. The nuclear contribution to the magnetic dipole operator vanishes in the Born–Oppenheimer approximation, and has been omitted. Although the nuclear contributions to the dipole moment and quadrupole moment operators do not contribute directly to the polarizabilities in eqs. 7–9 within the Born–Oppenheimer approximation, we include these terms since they will contribute to the quantum-mechanical expressions for the polarizability gradients, as we will return to in the section “The ab initio calculation of Raman optical activity spectra”. It is also worth noting that the G′ tensor will vanish in the limit of a static field (ω = 0), as can be easily seen from eq. 8.

For neutral but dipolar systems, the definitions of the electric quadrupole and magnetic dipole operators depend on an arbitrarily chosen gauge origin O. Only in the limit of a complete basis set using variational wave functions* will the results calculated using these operators show the correct origin dependencies. If we, for instance, move the global gauge origin to the nucleus which we are differentiating with respect to, we get20

  • equation image(13)
  • equation image(14)

The use of so-called London atomic orbitals21 will ensure that the correct origin dependence can be obtained also in finite basis sets for variational wave functions. We will return to this point in the next section.


  1. Top of page
  2. Abstract
  8. Literature Cited

Compared to many other molecular properties, the number of theoretical developments of ab initio methods for the calculation of ROA spectra, and corresponding implementations, is rather limited. The reason for this is in part due to the limited experimental activity using this spectroscopic tool, but also the computational complexity of the relevant molecular properties that are needed for determining the ROA spectrum, and in particular for obtaining results with the correct origin dependence for the mixed electric dipole–magnetic dipole polarizability in eq. 8. However, with the advances made in linear and nonlinear response theories for the calculation of frequency-dependent molecular properties, many of these obstacles have been overcome, and there is an increasing number of ab initio programs that can currently calculate ROA spectra, most notably the Dalton,22 Gaussian23 and Turbomole24, 25 programs, but also the CADPAC program,26 with which the first ab initio ROA calculations were done.27, 28

Numerical Schemes

The first full ab initio spectra presented in the literature were computed at the Hartree–Fock level of approximation by Polavarapu and coworkers.27, 28 These calculations utilized the observation and implementation by Amos29 that even though the mixed electric dipole–magnetic dipole polarizability in eq. 8 vanishes the limit of a static field, thus preventing its calculation using conventional time-independent coupled-perturted Hartree–Fock (CPHF) methods, the quantity equation image does have a static limit

  • equation image(15)

In this equation, ΨB and ΨF denote the first-order perturbed wave functions with respect to an external magnetic field induction and an external electric field, respectively. Thus, even though more CPHF equations need to be solved to determine this tensor than is normally the case for second-order molecular properties (we normally have to determine the perturbed wave functions for only one of the applied perturbations), eq. 15 enables the calculation of the mixed electric dipole–magnetic dipole polarizability using conventional CPHF programs. It is interesting to note that even though the static limit of the equation image tensor was published already in 1982, it was not really used for calculations of chiroptical properties for almost a decade, when it was adopted by Polavarapu and coworkers, first for the calculation of ROA spectra,27, 28 and later for the optical rotation itself.30

In the calculation of the ROA spectra in Refs.27 and28, the geometrical derivatives of the different electronic tensors in eqs. 7–9 were obtained by numerical differentiation of the electronic tensors with respect to the Cartesian displacements of the nuclei, followed by a transformation of these geometrically differentiated tensors to the normal coordinate basis, defined in matrix form as

  • equation image(16)

where X is a vector collecting the mass-weighted Cartesiancoordinates of all the nuclei, and L−1 is the transformation matrix from the mass-weighted Cartesian coordinates X to the normal coordinate basis Q. L diagonalizes both the kinetic energy and harmonic electronic potential energy of the nuclei.

This decoupling of the calculation of the force field from the ROA CIDs opens for the obvious possibility of calculating the force field that determines the normal coordinates at a different, and in general a much higher, level of theory than the more computationally expensive polarizability gradients. This combination of different computational levels is the dominating mode of operation in ROA calculations, to a large extent motivated by the somewhat stronger importance of the quality of the force field in obtaining reliable ROA spectra compared with the quality of the polarizability gradients (although the number of thorough investigations of basis set requirements and correlation effects on the polarizability gradients are limited31–35). This hybrid computational approach was also used in early studies of vibrational circular dichroism (VCD), see e.g. Ref.36. However, in the case of VCD, the computational benefits are much more limited, since only six additional CPHF equations need to be solved in addition to the equations that need to be solved for the force field (which is three times the number of nuclei in the molecule). Thus, assuming the computational requirements to be the same for the axial and polar tensors as for the force field, there is limited computational advantage of such an approach for VCD in comparison with ROA calculations. We will return briefly to this point in the next section in relation to recent analytic implementations of the polarizability gradients.

In the calculation of the mixed electric dipole–magnetic dipole polarizability in the work of Polavarapu and coworkers,27, 28 conventional basis sets were used. This has the consequence that the calculated tensor invariants in eqs. 4–6 are origin dependent for all practically realizable calculations. In 1995, Helgaker et al. presented the first gauge-origin-independent calculations of ROA spectra at the Hartree–Fock and multiconfigurational self-consistent field (MCSCF) level of theory.31 Gauge-origin independence was ensured through the use of London atomic orbitals,21 also commonly referred to as Gauge-including atomic orbitals (GIAOs).37 These orbitals are defined by multiplying each individual atomic basis function by a magnetic field-dependent phase factor

  • equation image(17)

χμ is here an ordinary Cartesian or spherical Gaussian basis function. The effect of the magnetic vector potential appearing in the complex phase factor

  • equation image(18)

is to move the global gauge origin O to the “best” local gauge origin for each individual basis function, which is chosen to be the center to which the basis function is attached. It can be shown that this choice of local gauge origin is optimal in the sense that for a one-electron, one-center system, the London orbital is correct to first order in the external magnetic field, whereas the conventional Gaussian basis function is only correct to zeroth order.38 A consequence of the introduction of the London orbitals is that any reference to the external gauge origin vanishes in the one- and two-electron integrals, making even the G′ tensor itself seemingly origin independent. However, as we see from eq. 8, the equation image tensor is not origin independent, rather the problem is that the correct origin dependence is not recovered in approximate calculations.

Helgaker et al. introduced in Ref.31 a definition of the angular momentum operator in a London atomic orbital basis which has the correct origin dependence also for finite basis sets. This was achieved by requiring that the perturbed molecular orbitals (MOs) are as similar as possible to the unperturbed molecular orbitals, by defining a natural connection between the orthonormal perturbed MOs and the original orthonormal unperturbed MOs.39 We will not discuss in more detail this natural connection. We will see that this origin dependence alternatively can be considered as being due to the time dependence of the atomic basis set, following the generalization of the London orbitals to time-dependent magnetic fields as proposed by Krykunov and Autschbach.40

Another extension introduced in the work of Helgaker et al. relative to that of Polavarapu and coworkers, is that the static-limit approximation was lifted, the different mixed polarizabilities now being calculated from response functions12, 13, 41, 42 solved at the frequency of the applied laser light. Although the static-limit approximation has been used in several applications, see for instance Refs.43 and44, we believe that all current calculations have abandoned, and indeed should abandon, the static-limit approximation, using instead time-dependent Hartree–Fock (TDHF) or time-dependent Kohn–Sham (TDKS) methodologies, in which the frequency is taken into account correctly. This switch from the static-limit approximation to a formalism in which the frequency is taken into account, has also been strongly advocated in the case of optical rotation calculations.45

Interestingly, Ref.31 represents to date one out of only two studies33 of ROA CIDs using correlated wave functions (excluding DFT). This is due in part to the fact that chiral molecules often have no symmetry and are in general too large for being studied using correlated wave functions, but also due to the problems faced by non-variational wave functions in achieving gauge-origin independent results for frequency-dependent properties even when using London atomic orbitals.46–48 Still, the MCSCF calculations in Refs.31 and33 cannot really treat dynamic correlation effects, and the set of model systems used in these studies (H2O2, CHFDT, and CHDTOH) are not very representative of chiral molecules nor representative in terms of expected electron correlation effects. There is thus clearly a need for benchmark calculations on representative and realistic chiral systems, for instance at the coupled-cluster level of theory,48 against which more approximate theoretical approaches can be benchmarked.

An important step forward in terms of improving the quality of calculated ROA CIDs was made in 2001, when Ruud et al. presented the first calculations of ROA CIDs at the DFT level of theory.32 This is an extention of the approach presented in Ref.31, building on the implementation of linear response theory at the DFT level49, 50 and the corresponding extension of the calculation of optical rotations to the DFT level using London atomic orbitals.51, 52 Within a Kohn–Sham formalism, assuming the adiabatic approximation,53, 54 the implementation of ROA CIDs at the DFT level represents a rather minor modification of an existing TDHF code, assuming routines for numerical integration of the relevant exchange-correlation kernels are present. Indeed, building on the implementation of the equation image tensor at the DFT level of theory using London atomic orbitals,51 ROA CIDs are currently available also as part of the Gaussian 03 program, even though the implementation has not been documented. However, applications using this implementation have been reported in the literature.55–57

Very recently, Luber and Reiher implemented the mixed electric dipole–electric quadrupole polarizability into the Turbomole program25 enabling faster calculations of ROA spectra by utilizing the efficiency of the Turbomole integral package. Earlier use of the Turbomole program for ROA calculations had been limited to the polarizability and mixed electric dipole–magnetic dipole polarizability alone,58, 59 and these studies focussed instead to some extent on the limited importance of the mixed electric dipole–electric quadrupole polarizability for depolarized right-angle scattering and backscattering ROA experiments, so that this contribution could be neglected without any loss of accuracy. However, as there is no computational overhead of calculating the mixed electric dipole–electric quadrupole polarizability in a numerical differentiation scheme, since it only requires the contraction of the electric-field-perturbed density matrix with the one-electron quadrupole integrals, there really is no need to introduce this approximation in numerical calculations. In the case of analytical calculations, only minor savings can be achieved when using the 2n + 1 rule, whereas the computational cost can be reduced by approximately a factor of two when using the n + 1 rule (vide infra).

A feature of the implementation of ROA in the Turbomole package is that all calculations are done in the so-called velocity gauge,60 in which one utilizes the relation

  • equation image(19)

for the dipole-length operator, and the relation

  • equation image(20)

for the quadrupole operator. These velocity forms of the dipole and quadrupole operators ensure that origin-independent results are obtained also using finite basis sets. Although the velocity forms of the operators in general display somewhat slower basis set convergence than the length forms, these concerns are in general small for the basis sets conventionally used in current quantum-chemical calculations. However, when using rarified basis sets optimized for ROA CIDs,34 some care regarding the basis set completeness for these operators need to be taken into account when using the velocity gauge.

The appearance of the inverse of the excitation energy in front of the dipole velocity forms of the operators in eqs. 19 and 20 raises the question whether additional TDKS equations need to be solved to use the velocity forms. However, for pure exchange-correlation functionals, the electronic Hessian is diagonal and only involves the KS orbital energy differences. Additional inverse orbital energy differences do therefore not pose any significant implementational problems. The situation differs in the case of hybrid functionals, in which the Coulomb and exchange corrections to the orbital differences require the solution of additional sets of linear equations when using the velocity form. The approach is therefore less suitable for hybrid functionals such as the commonly used B3LYP functional.45, 61, 62

Jensen et al. have utilized the complex polarization propagator approach63, 64 to calculate resonance Raman optical activity spectra.65 The necessary geometry derivatives of the different mixed polarizabilities were also in this approach obtained using numerical differentiation. However, to calculate the polarizabilities at resonance, a dampingfactor γ is introduced to avoid divergencies at resonance. The damping factor can be related to a phenomenological lifetime τ of the excited states of the molecule through τ = 1/γ, and the conventional infinite-lifetime approximation thus corresponds to γ = 0.

With the introduction of this complex damping factor, the mixed electric dipole–magnetic dipole polarizability is given as (assuming the same damping factor for all excited states)

  • equation image(21)

Similar expressions apply to the other polarizabilities. We note that the polarizability in eq. 21 reduces to eq. 8 when γ = 0. The imaginary part of the polarizability in eq. 21 corresponds to the scattering process, whereas the real part corresponds to the absorption process. The calculation of the complex tensor equation image can thus provide simultaneously both the optical rotatory dispersion and the circular dichroism of a molecule.63, 66

Analytic Schemes for ROA CIDs

An approximate scheme for analytic calculations of ROA CIDs was presented by Bouř67 using the sum-over-states (SOS) expression in combination with Hartree–Fock wave functions or TDKS states. In this approach, the SOS summation is restricted to single excitations from the occupied to the unoccupied orbitals. The energy differences in the denominators in eqs. 7–9 are approximated by the corresponding orbital energy differences in the case of TDKS with pure exchange-correlation functionals, corrected for exchange and Coulomb contributions in the case of Hartree–Fock (or hybrid DFT) states. The transition moments were calculated by evaluating the corresponding matrix elements of the orbitals involved. Although in principle more efficient than the purely numerical schemes, it remains an approximation to the exact polarizability derivatives and have in practice not been much utilized. Bouř has also presented a scheme in which this SOS expression is used for the individual polarizabilities, but then in combination with numerical differentiation with respect to the nuclear distortions.68

The first fully consistent analytic implementations of ROA CIDs have started to appear very recently. To date, only one implementation has been published at the Hartree–Fock level of theory,69 building on work in the group of Champagne and coworkers on analytic calculations of first- and second-order geometrical derivatives of electric (hyper)polarizabilities.70, 71 A limitation of this implementation is that conventional basis sets are used, and thus that the results will be origin dependent.

Two additional analytic approaches have been developed by the present authors and by Cheeseman and coworkers (personal communication). The latter bears some resemblance to the implementation of Liegeois et al., but includes two important extensions relative to Ref.69, namely that London atomic orbitals are used, and that the implementation also is applicable to KS-DFT, a feature that will no doubt make this a very attractive approach. The implementation of Thorvaldsen et al. is, as Ref.69, also limited to HF wave functions and conventional basis sets, but is formulated fully in the atomic orbital basis, making it in principle ready for exploiting advances made in linear-scaling SCF methodology (such as HF and KS-DFT).72–75 The implementation is based on a very general framework for the calculation of higher-order molecular properties in the atomic orbital basis using basis sets that may be both time and perturbation dependent.14

We also make a small comment here regarding the so-called n + 1 and 2n + 1 rules, used in the calculation of molecular response functions.76, 77 These rules state that one from the perturbed density matrix of order n can calculate energy corrections of order n + 1 and 2n + 1, respectively. These rules are utilized for instance in the calculation of the dipole gradients that determine infrared intensities. As we only need first-order perturbed density matrices to determine these second-order energy corrections, we only solve TDKS equations for the perturbation that has the fewest components, that is, the electric field perturbation (unless we are also determining the molecular force field in the same calculation).

Clearly, the 2n + 1 rule is in general the strongest of these rules, as only first-order perturbed wave functions are needed to calculate the polarizability gradients determining the ROA CIDs. However, we need to determine these perturbed densities for all perturbations, electric field, electric field gradient, magnetic field, and nuclear displacements, in total 12 + 3 × N, where N is the number of nuclei. The n + 1 rule instead requires the solution of second-order perturbed densities, but only for two of the applied perturbations. Thus, the n + 1 rule requires the solution of 45 response equations [12 first-order and 33 second-order equations (6 electric field–electric field, 9 electric field–magnetic field, and 18 electric field-electric field gradient)], but none with respect to the nuclear displacements.

The implementations by Liegeois et al. and Cheeseman et al. both utilize the 2n + 1 rule. Such an approach is particularly beneficial if the ROA CIDs are determined at the same level of theory as the force field, since we then already have determined the perturbed densities for the nuclear distortions. As HF force fields are considered to not be of sufficient accuracy for a reliable approximation to the true force field, this advantage cannot be utilized in the approach by Liegeos et al., whereas the approach of Cheeseman and coworkers will allow these computational benefits to be exploited. However, this advantage can only be utilized if we assume that the basis set requirements for the force fields are similar to those of the ROA CIDs. Should these turn out not to be similar, the approach of Thorvaldsen et al. will be computationally most efficient, requiring only the solution of 45 response equations independent of the molecular size. Considering that highly efficient small basis sets, designed for the calculations of ROA CIDs of high quality, has been developed,34 this would appear to give some support for this latter approach. We will return to the details of the formalism in the next section. Furthermore, in combination with mode-selection methods,78, 79 the approach of Thorvaldsen et al. will have clear advantages.

Other Methodological Developments

Hess, Reiher and coworkers have introduced a numerical approach in which only selected vibrational modes are targeted, or only vibrations in a specific frequency window.78, 79 The approach can start from trial modes and obtain the true normal modes using Davidson diagonalization techniques.80 These normal modes can then be used to determine the polarizability derivatives numerically without ever having to construct the full quantum-mechanical force field. For very large molecules, this has obvious advantages, as characteristic vibrational frequencies can be targeted, or only the relevant vibrational frequencies within the frequency window accessible to the ROA spectrometer. The method has allowed for ROA calculations on very large systems, involving more than 100 atoms in the largest calculations.79

The scheme of Reiher and coworkers has so far only been utilized in combination with numerical schemes, differentiating the different polarizabilities directly in the optimized normal coordinates.35, 79, 81 However, assuming that an efficient way of obtaining the polarizability gradients analytically can be achieved, for instance using the n + 1 rule as in our scheme, the combination of local-mode optimization with analytical calculations of the ROA CIDs may prove to be a computationally very efficient approach for the study of ROA spectra of large molecules.

An important technique for the rapid application of ab initio methodology to ROA calculations on large molecules has been the tensor transfer technique of Bour̆ et al.82 In this approach, the exact origin dependence of the different polarizabilities is used to translate the polarizability gradients calculated for a molecular fragment in a given orientation into a larger molecular complex. In this way one can “synthesize” the full ROA spectrum from the polarizability gradients calculated for the different fragements, translated and rotated into the full molecular structure. The method has proven to give very good approximations to the full ROA spectra,83 and have enabled calculations on large molecules using ab initio methodology.84, 85 It is worth noting that the harmonic force field fulfills similar translational and rotational relations, and that the tensor transfer technique thus also can be used to generate accurate force fields for large molecules from high-quality force fields calculated for smaller fragments.84, 86

Bouř, Bamruk and Hanzlíkova have extended the polarization model of Freedman and Nafie20 to the ab initio level.87 In this model, the exact origin dependence of the mixed electric dipole–magnetic dipole polarizability and the mixed electric dipole–electric quadrupole polarizability is used to move the global gauge origin to the nucleus which is being differentiated (see eqs. 13 and 14). By setting the first term on the right-hand side of both equations to zero as an approximation, the calculation of the electric dipole–magnetic dipole and electric dipole–electric quadrupole polarizabilities comes at no additional costs compared with the calculation of the polarizability gradients and very little programming effort. Although used succesfully in some applications,85, 88 the availability of codes for calculating the different mixed polarizabilities with little or no computational overhead will in the long run make this approximation of limited use.

A somewhat different set of developments are the description of the effects of a solvent on the calculation of ROA spectra. Several studies have combined force fields calculated in the presence of a dielectric continuum, most often the polarizable continuum model (PCM)89–92 or the COSMO model.93 To the best of our knowledge, the only study in which the PCM has been applied to all quantities determining the ROA spectrum is the study by Pecul et al.,94 in which a nonequilibrium scheme for the interaction between the solvent and the solute90, 95, 96 is included also in the calculation of the ROA CIDs. The study also takes into account local field effects on the interaction tensors.97, 98

The solvent most commonly used in ROA experiments is water, for which explicit intermolecular interactions can be expected to be important. These specific solute–solvent interactions cannot be expected to be well modeled by continuum approaches. Jalkanen and coworkers have devoted much attention to the description of specific solute-water interactions in the modeling of ROA spectra, optimizing the structure of the dominating conformation of the solute with the innermost coordination sphere.99–103 It is beyond the scope of this review to discuss the efforts in this direction, but this clearly remains one of the major challenges to computational ROA spectroscopy.


  1. Top of page
  2. Abstract
  8. Literature Cited

The purpose of this section is to give an introduction to the basic formalism of ROA calculations. We will not give an account of all approaches that have been utilized in the past for the calculation of ROA CIDs, discussed qualitatively in the previous section, but rather present our most recent approach for the calculation of ROA CIDs. This choice is made in part because we believe our approach has certain advantages in terms of its potential applicability for ROA calculations on large molecules. However, we also believe the formulation of the relevant ROA parameters as derivatives of a so-called quasi-energy provides a rather intuitive picture of the strategy utilized in different ab initio programs for calculating ROA spectra. Where appropriate and possible, we will point out the differences to other approaches for calculating ROA CIDs. The approaches used to calculate force fields can be found elsewhere.104–106

The formalism we will utilize here is based on the very general framework derived for an atomic orbital-based response theory, in which the basis functions can be both time- and perturbation dependent.14 The formulation uses the elements of the density matrix as the basic parameters, and thus differs from most other implementations of ROA calculations, analytic69 as well as numerical25, 31, 32 differentiation schemes, where instead the molecular orbital coefficients are used as the variational parameters. By working in the atomic-orbital basis without explicitly using molecular orbitals, the approach is designed with modern linear-scaling methods in mind,72–75 thus making it well suited for use on very large molecules. The theory will be described elsewhere, but we illustrate here the basic elements of the approach, including also the use of London atomic orbitals21 to obtain origin-independent ROA CIDs, as recently presented by Cheeseman and coworkers using a molecular orbital-based formalism.

To find suitable expressions for the calculation of the geometrical derivatives of the frequency-dependent dipole polarizability, the mixed electric dipole–electric quadrupole polarizability and the electric dipole–magnetic dipole polarizability appearing in eqs. 7–9, we take as our starting point the molecular gradient of the quasi-energy Qg defined for an atomic orbital basis which is time dependent and which depends explicitly on the externally applied perturbation g (in this case corresponding to a change of the nuclear positions: g = RRe)107

  • equation image(22)

In this equation, equation image indicates that we have performed a time averaging over a full period of the applied electromagnetic field (or equivalently: over all time). We have also introduced the density matrix D in the atomic orbital basis, the generalized (time-dependent) self-consistent field (SCF) energy of the system, equation image, defined as

  • equation image(23)

The expression for the quasienergy gradient is then

  • equation image(24)

where we have defined an antisymmetric time-differentiated overlap matrix

  • equation image(25)

and the nuclear repulsion energy hnuc. h is the conventional one-electron operator integrals, containing the kineticenergy and nuclear attraction contributions (in atomic units)

  • equation image(26)

and G(D) is the two-electron interaction, which in the atomic orbital basis can be written as

  • equation image(27)

with the two-electron integrals being defined as

  • equation image(28)

where xi comprises a spin- and a spatial coordinate. vnuc in eq. 23 is the potential energy between the nuclei and the external fields, while V describes the interaction between the electrons and the external fields. In the specific case of ROA, using the electric quadrupole approximation, the field-molecule interation operator entering into our Hamiltonian is given by

  • equation image(29)

in which the monochromatic electromagnetic wave is represented as an oscillating inhomogeneous electric field and an oscillating homogeneous magnetic field.

  • equation image(30)

which in turn are expressed in terms of complex-valued frequency component vectors f, q, and b

  • equation image(31)
  • equation image(32)
  • equation image(33)

which we treat as perturbations (perturbation strengths). These are determined by the direction, intensity, phase, and polarization of the applied radiation. Since the magnetic moment operator equation image carries an imaginary phase factor, −i has been extracted from b to avoid imaginary perturbed integrals Vb = (d/db)V and properties (quasi-energy derivatives).

In the expression for the quasienergy gradient eq. 22, we have also introduced the derivative of the overlap matrix

  • equation image(34)

and the energy-frequency-weighted density matrix

  • equation image(35)

where we have introduced the generalized (time-dependent) Fock matrix in the AO basis equation image defined as the partial derivative of the generalized SCF energy in eq. 23 with respect to the density matrix transposed

  • equation image(36)

All these matrices are defined in the atomic-orbital basis,and provide a very general starting point for frequency-dependent response functions, where contributions from both the perturbation dependence and time dependence of the atomic orbital basis is included. We note that the gradient defined in eq. 22 can be considered as a generalization of the expression for the conventional geometrical gradient of a molecular system as introduced by Pulay108 to time-dependent basis sets.

As shown in Refs.14 and107, we can obtain higher-order, frequency-dependent molecular properties by term-by-term differentiation of eq. 24. Since the electric-field perturbations f and q only appear in the operator equation image (which gives rise to equation image and V), and not in the atomic orbitals, derivatives of hnuc, h, T, G, and S involving these perturbations will vanish. Differentiating eq. 22 with respect to f*, we thus get the (negative) dipole gradient

  • equation image(37)

where we have used that equation image. Note that we consider f and f* as being independent perturbations when differentiating. When pertubing a time-independent system, this greatly simplifies the equations since all derivatives will be of the form

  • equation image(38)

with M being time independent.

Differentiating the expression for the dipole gradient in eq. 37 a second time, with respect to f or q, respectively, we obtain formulas for the gradients of α in eq. 7, equation image, and the A tensor in eq. 9, equation image

  • equation image(39)
  • equation image(40)

where we have used that f and q only appear linearly in vnuc and V.

Similarly, the formula for the gradient of the G′ tensor, equation image, is obtained by differentiating eq. 37 with respect to b

  • equation image(41)

Compared to eqs. 39 and 40, this formula has six additional contributions, which are all due to the fact that the integrals depend on b through the London atomic orbitals. We note that since, in the case of ROA, g perturbs the geometry of a field-free, time-independent reference system, the integrals Vg and Tg both vanish ( equation image, equation image). We have however kept these terms in the derivation, as their derivatives Vgb and Tgb are nonzero.

The first of these formulas, eq. 39, has previously been derived in the context of Coherent anti-Stokes Raman scattering107 and analytic calculations of pure vibrational contributions to nonlinear optical properties,109 whereas the formulas for the two polariability gradients in eqs. 40 and 41 are new and enters in addition when calculating ROA CIDs.

We note that the additional integrals required in passing from calculating the electric polarizability gradients to the calculation of the mixed electric dipole–electric quadrupole and electric dipole–magnetic dipole polarizability gradients, are the one-electron integrals Vgq (implemented in Ref.69), Vgb, hgb, Tgb, and Sgb, as well as the two-electron integrals Ggb.110 In addition, in the equations to be solved for the perturbed densities Db, the one-electron integrals hb, Vb, and Tb, and two-electron integrals Gb are required.

The required first- and second-order perturbed densities equation image, Dq, Db, equation image, equation image, and equation image can be determined by solving the corresponding derivatives of the time-dependent SCF equation and idempotency constraint14

  • equation image(42)
  • equation image(43)

where the time-dependent Fock matrix equation image is given by eq. 36 and equation image.

Equations 42 and 43 have been derived previously,14 and this derivation will not be repeated here. We will however outline the procedure for solving (response) equations for perturbed AO density matrices, i.e., derivatives of eqs. 42 and 43.

In the first step of the procedure, all known (lower-order) terms are collected in a right-hand-side N, and from the perturbed idempotency constraint a set of equations are obtained for the higher-order perturbed densities, which can then be evaluated. For a second-order equation for equation image, this is

  • equation image(44)
  • equation image(45)

in which perturbed overlap matrices equation image and equation image are zero since the basis set is independent of the electric field. An equation of the form of eq. 44 has the general solution

  • equation image(46)

where we call equation image the “particular” component of the solution, and the “homogeneous” component equation image can be any solution of the corresponding homogeneous equation

  • equation image(47)

which must be determined by the corresponding perturbed TDSCF equation.

If transformed to a basis of Hartree–Fock molecular orbitals C, the two solution components equation image and equation image would have diagonal and off-diagonal block structure, respectively

  • equation image(48)
  • equation image(49)
  • equation image(50)

that is, the particular component has occupied–occupied and virtual–virtual blocks, whereas the homogeneous component has virtual–occupied and occupied–virtual blocks.

In the second step of the procedure, all known terms (lower-order and equation image) in the perturbed TDSCF equation are collected on the right-hand-side as M, which can be evaluated. For instance, the equation for equation image is given by

  • equation image(51)
  • equation image(52)
  • equation image(53)

where, again since the basis set is independent of f* which only enters vnuc and V linearly, equation image, equation image, equation image, equation image, equation image, equation image, and equation image are all zero. In addition, in the case of ROA CIDs, equation image, thus equation image is also zero.

To bring eq. 51 to the more familiar form of an SCF response equation in the AO basis,72, 111equation image can be expressed in terms of another matrix X as an S-commutator equation image

  • equation image(54)

which brings eq. 51 to the form

  • equation image(55)
  • equation image(56)
  • equation image(57)

where equation image and equation image are the generalized electronic Hessian and metric operators. Equation 55 is then solved iteratively, either in the AO basis,72 or after transformation to the MO basis42, 112

  • equation image(58)

Using the 2n + 1 Rule

Calculation of the molecular force field (geometrical second derivatives of the energy, i.e., Egg), which determines the vibrational modes, requires solving first-order geometry-perturbed response equations (for Dg). Therefore, if ROA CIDs are wanted at the same level of theory as the vibrational modes, it is computationally advantageous to make use of the computed Dg by using the 2n + 1 rule instead of the n + 1 rule used earlier. This avoids solving 36 second-order response equations (for equation image, equation image, and equation image above).

However, the 2n + 1 rule is not directly applicable to the quasi-energy gradient eq. 22, since it is not variational ( equation image is not zero). Moreover, as explained in Ref.14, it is not straightforward to formulate the (undifferentiated) quasi-energy Q in terms of the density matrix D. Instead we make eq. 22 variational by augmenting it with a Lagrange multiplier for each equation satisfied by D: The TDSCF equation eq. 42 multiplied by Xg, and the idempotency constraint eq. 43 by Lg

  • equation image(59)
  • equation image(60)
  • equation image(61)

We note here that Xg and Lg are not derivatives of some matrices X and L, but carry subscripts to denote relation to Qg. As derived in Ref.14, Lg and Xg are both expressible in terms of the perturbed density matrix Dg and the corresponding perturbed Fock matrix equation image

  • equation image(62)
  • equation image(63)

and the multipliers are thus obtained by solving (ordinary) response equations for Dg, followed by some matrix algebra.

Since Qg in eq. 59 is variational, a formula for e.g. equation image is obtained by differentiating the quasienergy gradient Lagrangian in eq. 59 twice, followed by discarding second-order perturbed matrices equation image, equation image, and equation image (according to case equation image of the 2n + 1 rule), as well as the first-order perturbed multipliers equation image, equation image, equation image and equation image, since these multiply first-order perturbed response equations Yf, equation image, Pf, and equation image, respectively, which are zero. Aftereliminating additional zero terms equation image, equation image, equation image, Tg, equation image, equation image, equation image, equation image, equation image, T, equation image, and equation image, we reach the following polarizability gradient formula, equivalent to eq. 39

  • equation image(64)

Alternatively, one could start from the formula for the quasi-energy derivative with respect to f*, i.e. the (negative) equation image frequency component of the molecular dipole moment

  • equation image(65)

then differentiate with respect to g and f, discard second-order and zero terms, to arrive at yet another formula equivalent to eq. 39

  • equation image(66)

Although more compact, this formula resembles that used in the analytical calculation of Raman intensities in Ref.70 (eqs. 3 and 35 therein), and in the calculation of the first analytical ROA CIDs in Ref.69. Although eqs. 64 and 66 are equivalent and require solving the same response equations ( equation image and Dg), the former will be preferred for all but the smallest molecules, because it requires fewer matrix multiplications.


  1. Top of page
  2. Abstract
  8. Literature Cited

We have in this review given a brief account of the foundations for ab initio calculations of ROA CIDs, as well as given a historical account of the developments in this field. Although for a long time being hampered by very large computational costs, the improvements in computer hardware has made ROA calculations increasingly more feasible, and mixed analytic-numerical calculations on very large systems have been presented in the literature.79

However, the last 2–3 years represent a shift in the applicability of ab initio methodology to the analysis of experimental ROA spectra. With the introduction of several analytic schemes for the calculation ROA CIDs,69 the major obstacle preventing routine calculations of ROA CIDs has been overcome. Although the computational cost will remain high, the combination of analytic techniques with tools such as tensor transfer techniques82, 83 or mode tracking algortihms78 will lead to a boost in ab initio studies of ROA, in particular as experimental apparatus also becomes more widespread. ROA is at a stage where it will have the potential to develop into a method competitive in its worldwide use to that seen for VCD.

Still, several challenges remain, but these challenges are no longer unique to ROA calculations, but are shared with most chiroptical spectroscopies. We would in particular highlight two major challenges to the routine application of ab initio studies of ROA spectra: Conformational flexibility and the effects of a solvent on both the frequencies and the ROA CIDs.

Most molecules are flexible and can exist in several conformations. These different conformations can have very different chiroptical properties, ROA CIDs15, 113–116 as well a for instance optical rotations.117–119 The total ROA spectrum is a weighted sum over the individual stable conformers, the weight in most cases calculated according to the Boltzmann distribution. Because of the potentially very large differences in the chiroptical properties of the different conformers, the conformationally averaged spectrum can become very dependent on the relative weights of the different conformers. Indeed, it is in many cases more difficult for ab initio methods to determine the relative energies of the different conformations accurately than to obtain reliable estimates for a molecular property at a given geometry. Added complexity comes from the possibility of using either electronic energies or the free energies of the molecules.120 Whereas the situation is clear in the gas phase, it is currently not obvious that the quality of e.g. continuum models is high enough to make the use of free energies reliable for molecules in solution. Finally, as the molecule grows larger, the number of possible conformations grow rapidly, and finding efficient and reliable ways of locating the relevant conformations becomes a challenging task.

One of the great advantages of Raman optical activity is the possibility it offers for measuring the ROA signatures of biomolecules in their native water environment. At the same time, water is one of the hardest solvents to model reliably using ab initio methods due to it's affinity for making hydrogen bonds. Specific solute-solvent interactions are poorly handled by continuum solvent approaches, and in most cases, some explicit water molecules need to be included. Jalkanen and coworkers have been particularly active in this field by optimizing solute-solvent stabilized structures obtained from molecular dynamics simulations.99–103 However, it is little explored to what extent these minimized structures are a good representation of the solvent dynamics occuring in a water solution. Considering the importance of correct conformational searches, it is also not clear that currently used force fields are of high enough quality to provide accurate sampling of the relevant solute-solvent conformational space.

Although conformation and solvent effects remain important and challenging tasks for the modelling of ab initio ROA spectra, the development in recent years of a wide range of ab initio methods for the efficient calculation of ROA CIDs clearly has lifted a major computational obstacle. We therefore believe that there will be a continued increasing interest in ab initio calculations of ROA spectra in the coming years as a result of these recent computational advances, and that these studies will lead to the development of reliable protocols for calculating ROA spectra that are of high enough quality to aid in the analysis of experimental ROA spectra.

  • *

    Hartree–Fock and multiconfigurational self-consistent field wave functions are both variational. Methods such as MP2, truncted configuration interaction (CI), and coupled cluster (CC) are not variational. DFT also belongs to the class of variational methods.

Literature Cited

  1. Top of page
  2. Abstract
  8. Literature Cited