Computing chiroptical properties with first‐principles theoretical methods: Background and illustrative examples^†

Abstract

This “tutorial style” review outlines the theoretical foundation for computations of chiroptical properties for optically active molecules. The formalism covers electronic and vibrational CD, optical rotation, and Raman optical activity. The focus is on first‐principles methods. A dedicated section highlights the strengths and weaknesses of currently popular time‐dependent density functional methods. The article also contains a section with input examples and results for a small molecule (trans‐2,3‐dimethyloxirane). Several representative applications of computational methods roughly from 2001 to mid‐2009 are reviewed in detail. Chirality 21:E116–E152, 2009. © 2009 Wiley‐Liss, Inc.

INTRODUCTION

Chirality, and its manifestation in chiroptical effects such as optical rotation or electronic and vibrational circular dichroism, is of paramount importance in chemistry, biochemistry, and neighboring disciplines. An overwhelming majority of biologically important molecules are chiral.1, 2 Chiral organic molecules and metal complexes are also of considerable interest in many other areas of application, for example, in materials science or in solar cell research. In addition to its important applications, chirality and optical activity are of interest for fundamental chemical and physical research as well.1, 3-5

In Djerassi's 1960 book “Optical Rotatory Dispersion”,6 Albert Moscowitz provided a beautifully written theoretical chapter on optical rotatory dispersion (ORD) curves, the relation of ORD to circular dichroism (CD), and applications in organic chemistry. Moscowitz placed a lot of emphasis on the Kramers–Kronig (KK) transforms7, 8 which intimately relate optical rotation (OR) and its dispersion to CD, and vice versa, and were of considerable help in optical activity research for the following decade.9 At that time, however, reliable first‐principles calculations of either chiroptical property were not possible. Even the comparatively straightforward numerical KK transformation of experimental data had to be performed on what was then probably considered a supercomputer.

Computational chemistry based on first‐principles theory has made stunning progress during the past two decades and moved from being a highly specialized discipline into the mainstream. Recently, Polavarapu has identified a “renaissance” in chiroptical methods10 driven by the availability of correlated yet efficient first‐principles methods that can be used to predict, confirm, and assign experimental data. Predominantly, these are chiroptical response methods based on density functional theory (DFT) and time‐dependent DFT (TD‐DFT) as well as on coupled cluster wave functions.

This article is meant as an introduction to the theoretical and computational aspects of optical activity. Illustrative applications will emphasize density functional theory (DFT) and TD‐DFT for two reasons: To draw attention to its strengths and current weaknesses, and the fact that the reader is most likely to carry out and encounter computational studies that have made use of these first‐principles methods. Because of the high importance of TD‐DFT for this field of research, a separate section provides a synopsis of TD‐DFT linear response theory which is underlying the majority of chiroptical property calculations presently carried out. The interested reader will also find in this article a section with a more practical guide to calculations, followed by a section where some of the basic definitions for several of the important chiroptical parameters are collected. In some sense in the above mentioned Section, we will try to follow the footsteps of Moscowitz but the author hopes that it will provide a fresh view on the topic and, together with the remainder of the article provide useful details about the more practical aspects of modeling optical activity using first principles theory.

The less patient reader might initially want to skim the theoretical background and move on to the examples section where for a small molecule (trans‐dimethyloxirane) samples of input files and representative results for electronic CD, OR, as well as vibrational CD (VCD) and Raman optical activity (ROA) are provided. Calculations of CD and OR with different quantum chemistry packages are discussed in this section. One of them is the Gaussian code and for comparison we also provide results obtained with the ADF package and with NWChem. Finally, we will briefly review representative applications of first principles computational methods to give the reader an idea about the applicability of the available software and what to expect in terms of agreement of the computed data with experiment. A summary of benchmark studies and a discussion of general accuracy consideration can also be found at the end of the section following this introduction.

COMPUTING CHIROPTICAL PROPERTIES: PRACTICAL CONSIDERATIONS

The Computational Model, and How to Approach the Theoretical Simulation of Chiroptical Properties

The computation of a chiroptical property is often the last step in a series of computations leading to the simulation of the property with the help of theoretical methods. For a flow chart see Figure 2 further below.

• a

  The computational model; conformational searches: Before an attempt is made to compute a chiroptical property for a molecule, it is first necessary to determine its molecular structure. Nowadays, this typically involves geometry optimizations at some first principles level of theory (often with Density Functional Theory, DFT) with a good quality polarized basis set, ideally followed by a confirmation of the stationary points as local minima on the potential energy surface (PES) by calculations of the harmonic force constants. If the molecule is flexible and has a large conformational space this task is not entirely trivial and is most conveniently preceded by a conformational search. This is often done with semiautomated tools at a lower level of theory to save computation time. Various force fields and semiempirical methods, along with first‐principles theory based methods, are available in a number of commercial software packages to perform conformational searches semiautomatically. There are dedicated computational chemistry textbooks available to guide the practical chemist through the selection of an electronic structure method and basis set for the optimizations as well as the various steps of such calculations; for an example see Ref.11.

  The modeling of the chemical environment of a molecule can also be very important, for instance, if experimental data in polar solvents are to be reproduced. Here, the researcher has a choice of computationally efficient and easy to apply continuum solvent models which attempt to model the solvent as some kind of polarizable continuum that forms a nonuniformly charged surface around the solute. Popular examples are the various flavors of polarizable continuum models (PCMs)12, 13 or the conductor‐like screening model (COSMO).14 Discrete solvent models that treat the solvent molecules as individual entities are also available—in the simplest case they are based on point dipoles or point charges.15 However, because of the discrete nature of the solvent and the fact that any dissymmetry in the solvent–solute arrangement and among the solute molecules may contribute to a chiroptical property of the system, their application generally involves carefully benchmarked molecular dynamics (MD) simulations or some type of Monte Carlo (MC) sampling. The most demanding approach, from a computational viewpoint as well as in terms of human work hours spent, would be to perform dynamic (or MC) simulations of the solute embedded in a large number of solvent molecules, all treated with a first‐principles quantum chemical method. Although this might appear as the most desirable model (apart from computing and human time spent), the most affordable first principles methods may not have desirable features needed for MD or MC simulations. For instance, if the method does not yield realistic intermolecular potentials, the simulation is perhaps better performed with a well‐parametrized semiempirical method geared toward large‐scale dynamics in solution. For organic molecules and standard solvents (especially water), there are suitable force fields available in popular MD packages such as GROMACS.16 The outlook for metal complexes, or for nonstandard solvents, is less favorable and despite its computational cost adopting an ab‐initio MD method may be an easier choice than to develop new force fields (as long as the dynamic sampling can be accomplished with a relatively short simulation and not too many solvent molecules).

• b

  Conformational averaging: Consider molecular structure in the “old–fashioned” static picture of local minima, transition states, etc., which remains closest to how chemists traditionally think of structure. If various local minima with comparable energies W_i, i = 1…N are found, the chiroptical property P (e.g., optical rotation or CD intensity at a given frequency/wavelength) can be computed from a Boltzmann average of the property P_i calculated for each conformer, i.e.,

  (1)

  The W_i are here in energy units per mol. The W_i should ideally be the free energies of the system, but often optimized internal energies with or without zero‐point corrections, or enthalpies, are used instead depending on the size of the system and other factors limiting the ability to calculate entropy terms, zero‐point energies, and finite temperature enthalpy corrections. When a dynamic or statistical sampling of structure is employed, the conformational space accessible in a single short MD (or MC) simulation might cover PES regions with the most important local minima, or not. Depending on the technique and extent of conformational sampling, MD/MC averages might be derived without or with averaging of separate simulations on different conformers or different sets of conformers. Boltzmann factors derived from the average energies of separate simulations are not always employed when the results are averaged.

  Quite frequently, different conformers of a molecule have very different chiroptical responses (sometimes even different signs of these properties). Every computation involves approximations. An averaging for conformationally flexible molecules introduces uncertainties in the computed results in addition to those typically found for rigid systems. If chiroptical property data are available for a solid state instead, computations performed to assign or confirm these data may take advantage of the fixed geometries adopted in the solid state.17 However, the reduced conformational space comes at the expense of having to model the crystal environment.

• c

  Vibrational corrections: Even for a rigid molecule in the gas phase, averaging over the nuclear vibrational motions is desirable although at present not routinely performed. In calculations of optical rotation, this includes zero‐point and finite‐temperature vibrational averages. Consider nuclear motion along one of the vibrational modes as shown qualitatively in Figure 1. To lowest order, the average of a calculated ground state property P such as the optical rotation involves a term from the anharmonicity of the potential which causes the molecule to adopt an effective geometry that is different from the optimized equilibrium geometry, and a term from the curvature of the property surface that would cause vibrational corrections even if the potential were completely harmonic. Quantitatively,18-23

  (2)

  The term involving the cubic force constants k_ijj represents the anharmonicity correction (left panel of Fig. 1) whereas the last term (right panel of Fig. 1) represents the property curvature correction. In eq. 2, ω_i is the frequency of normal mode i. P_e is the property calculated at the equilibrium geometry, T is the temperature, k is the Boltzmann constant, and the q_i are dimensionless normal coordinates. For kT ≫ ℏω_i, the hyperbolic cotangent factor increases linearly with temperature. As the temperature approaches zero K, the coth factors decrease to 1, and the zero‐point vibrational average is obtained.

  If various conformers are well‐separated in energy it is possible to include vibrational corrections in the Boltzmann average, eq. 1, for each calculated property value P_i and for each energy separately. A TD‐DFT study of optical rotation in rigid organic molecules reported an average zero‐point vibrational correction on the order of 20%22 with no systematic sign. This is certainly a significant correction and should be kept in mind when assessing how well a computation agrees with experiment. Solvent effects might be of similar importance (see later.) Using vibrational averaging, Mort and Autschbach also computed the temperature dependence of optical rotation in rigid organic molecules from eq. 2,24, 25 which will be discussed in more detail in the Section on Selected Case Studies. See also Kongsted et al.26, Ruud and Zanasi,27 and Crawford et al.28. If MD or MC simulations are performed, an average over vibrationally distorted structures can be obtained automatically. However, no zero‐point corrections are included unless the nuclear motion is treated quantum mechanically (which would be a very demanding computation for obtaining a response property average). Not a lot of work has yet been performed along these lines.

  Another aspect of nuclear vibrational motion is the modeling of vibrational progressions and vibronic coupling in electronic absorption spectra.29 Some attempts have been made at extending simulations of CD spectra to include vibronic corrections on a first‐principles level,30 with rather promising results for trans‐(2,3)‐dimethyloxirane. A more recent TD‐DFT study using a different approach for calculating the vibronic terms has been carried out by Lin et al.31 for the 3‐methylcyclopentanone molecule. For vibrational CD, including vibronic coupling is essential to obtain the electronic contributions to the rotatory strengths. The calculation of Raman optical activity bears similarities to the way that vibrational averages of electronic properties are calculated. See the Section on Vibrational Optical Activity for an outline of the formalism.

• d

  Computing chiroptical properties—putting it all together —electronic structure methods: Assuming that one or several optimized structures for the system of interest, or a set of configurations from MD/MC simulations, are available the next task is then to compute the chiroptical response property. See Figure 2 for a flow chart of computational tasks leading up to the point of chiroptical computations. Although semiempirical methods are available for certain specialized tasks (e.g., to simulate the CD of polypeptides32, 33), let us focus on more generally applicable methods that are derived from first‐principles. Among those currently popular are DFT and time‐dependent DFT (TD‐DFT), as well as Hartree–Fock (HF, TD–HF) and correlated post‐Hartree–Fock wave function based methods. As a general rule, chiroptical properties such as CD and OR are sensitive to the approximations made in the electronic structure calculation. Therefore, it is advisable to use the most accurate method that one can afford.

  HF theory is generally not recommended for computing chiroptical response because it lacks treatment of the electron correlation. Post‐HF wave function methods scale unfavorably with the size of the system (typically counted by the number of basis functions). Also, increasingly large basis sets are needed to bring about the effects from electron correlation which pushes the computational demand even further than indicated by the formal scaling of the method (i.e., for example when going from CCSD to CCSDT not only does the formal scaling with the number of basis functions increase but ideally one should also increase the size of the basis set). Most theoretical studies of chiroptical methods therefore presently employ TD‐DFT which has a comparatively low scaling with system size and typically also has somewhat lower demands on the basis set. Note, however, that for smaller molecules it is now also possible to perform correlated wave function based computations routinely, for instance with fast flavors of coupled cluster (CC) methods such as CC234, 35 (an approximate CC singles‐doubles) or CC with localized orbitals.36, 37 Going beyond doubles in the cluster expansion appears to be too demanding at present for routine applications, but might be required in selected cases.35

  DFT and TD‐DFT are umbrella acronyms for a large variety of different levels of approximation each of which has its advantages and drawbacks. Because of its importance for modeling chiroptical properties of molecules this article contains a separate section outlining the TD‐DFT “linear response” formalism. That section also contains a description of why TD‐DFT is applicable for modeling chiroptical properties and a summary of various known issues with popular exchange‐correlation potentials that one should be aware of. Among the properties discussed here, VCD does not require the calculation of dynamic (frequency‐dependent) response and is therefore not affected by specific shortcomings of TD‐DFT. As VCD requires the knowledge of magnetically and geometrically perturbed wave functions or orbitals, however, it can be affected by possible shortcomings of the potentials used in DFT (or wave function approximations in post‐HF methods).

• e

  Basis sets: A general guide as to what method and basis set should be used is difficult to provide since each system poses its own challenges. Generally, it is useful to check if a computational benchmark study on systems with similar features (similar chromophores, for example) has already been performed. For organic molecules, optimizations with a standard hybrid or non‐hybrid density functional (hybrids: B3LYP, PBE0, or similar; non‐hybrids: BP86, PBE, or similar) with polarized basis sets of 6‐31G* or SV(P) or DZ(P) quality or preferably better are well‐established. For the calculations of chiroptical properties, in particular those that depend on dynamic response, a good quality basis including polarization functions and, especially for smaller molecules, diffuse functions should be adopted. Basis sets with flexibility, such as aug‐cc‐pVDZ, or perhaps 6‐311++G**, appear to offer a good compromise between accuracy and computational cost for small to medium sized molecules. Economic polarized basis sets with diffuse functions, such as the popular basis set aug‐cc‐pVDZ and other members of this basis set family, can be conveniently downloaded at the PNNL basis set exchange portal.38 Slater‐type basis sets suitable for molecular chiroptical response calculations are available at http://www.scm.com. Large molecules can often be treated reasonably well with nonaugmented basis sets of triple‐zeta plus polarization quality, which is certainly appropriate if valence excitations are dominating the response. For instance, if the electronic CD of valence transitions is of interest it is often sufficient to employ a basis without diffuse functions. For optical rotation, unless the molecule is quite large, diffuse functions are usually beneficial. Regarding the electronic structure, standard functionals with a somewhat higher fraction of exact exchange (such as Becke's Half&Half functionals) are often beneficial for molecular response property calculations. Coupled cluster (CC) methods have also been applied extensively for calculations of OR (and CD), with good success. Standard DFT and CC methods rely on a single reference configuration which restricts their use to systems that are not inherently of multireference character, see also Ref.39. For TD‐DFT, range separated hybrid functionals appear to be quite promising and will become available in major quantum chemistry packages during the coming years if not already implemented (see Section on TD‐DFT Linear Response for further remarks on functionals, and for selected references).

Figure 1

Lowest‐order effects of nuclear motion along some vibrational mode q on a calculated molecular property P (here: optical rotation or some other ground state electronic chiroptical property) corresponding to the first and the second term after P_e on the right‐hand side of eq. 2 .

Figure 2

Simple flow chart for computations of chiroptical properties of a molecule. The important steps are: (i) obtaining structures, (ii) modeling the environment (usually solvent, if necessary), and (iii) computing chiroptical response (CD, ORD, vibrational optical activity).

How do the Calculations Usually Compare with Experiment?

From a computational viewpoint, comparisons between theory and experiments in the UV/Vis energy range are best made with chiroptical properties represented as a function of energy, frequency, or wavenumbers, instead of wavelengths because of the inverse relationship between the wavelength and energy. For instance, the excitation energies in electronic spectra calculated with TD‐DFT are often too low in energy but can sometimes be shifted to obtain reasonable agreement with experiment. Deviations of 0.5 eV or even larger with respect to accurate correlated wave function calculations, or with respect to experiment, are not uncommon with standard density functionals. The deviation on the wavelength scale varies, depending on the excitation energy. In particular for very low lying electronic excitations, the wavelength shifts resulting from typical errors can be substantial, but are difficult to avoid.

Despite the fact that TD‐DFT excitation energies for organic molecules are often too low, it should be noted that an overestimation of OR magnitudes is not a general trend in TD‐DFT computations (which one might expect from KK relations or sum‐over‐states equations for the OR; see Section on Chiroptical Response as a Function of Frequency). The following discussion has been adapted from a more extensive review of optical rotation calculations66: TD‐DFT benchmark data from Refs.40-42 indicate that the magnitude of the OR among the members of the test set is overestimated about equally often as it is underestimated, when compared with experiment. Statistical errors have also been noted in Ref.43. Given the numerous approximations made in the computations (finite basis sets, finite integration grids, approximate functionals, neglect of chemical environment, temperature, and vibrational corrections, etc.) which all lead to different signs and magnitudes of errors, statistical deviations from experiment may be expected for a large diverse test set of molecules. Density functionals with no or a small fraction of HF exchange and without asymptotic corrections have a tendency to underestimate the energies, e.g., of Rydberg and certain π‐to‐π* excitations, which may lead to more systematic errors in the optical rotation among a set of compounds with no chromophores or with π‐bonds in similar chemical environments. In cases in which gas‐phase optical rotations are available an overestimation of the OR magnitude seems to be more common than underestimations, but too few systematic experimental data are available to draw definite conclusions at this time. Exceptions of the tendency of TD‐DFT to underestimate excitation energies are are known, e.g., low‐lying transitions in cyanine dyes or ligand field transitions in 3d metal complexes. Overall, the available benchmark studies show that best agreement with experiment (usually solution OR data) is obtained with polarized basis set augmented with one or two sets of diffuse functions which highlights the importance of low‐lying Rydberg‐type excitations. As most of the Kohn–Sham potentials used so far for the benchmarks have an incorrect asymptotic behavior it is likely that there is a error trade‐off from truncating the basis, approximating the potential, and neglecting specific solvent effects on high‐lying excited states, which on average leads to quite good agreement with experimental OR data obtained from solution. Coupled‐cluster benchmark data will be discussed in the Section on Selected Case Studies.

Stephens et al. have considered a set of 65 conformationally rigid organic molecules with specific rotations [α]_D of less than 100 degree/[dm (g/cm⁻³)].43 At the B3LYP/aug‐cc‐pVDZ//B3LYP/6‐31G* level of theory a RMS deviation of σ = 28.9 degree/[dm (g/cm⁻³)] was obtained between computed and experimental [α]_D values. For eight molecules, an incorrect sign of the OR was determined by the computations. The authors suggested that to assign the absolute configuration based on a single‐wavelength optical rotation the following criterion should be fulfilled: The unsigned difference between the experimental [α]_D (unknown configuration) and the computed [α]_D for one of the enantiomers has to be smaller than 2σ whereas at the same time the unsigned difference between experiment and computed [α]_D for the other enantiomer has to be larger than 2σ. In this case, the absolute configuration can be assigned to the enantiomer with the smaller deviation between calculated [α]_D and experiment with 95% confidence. The authors of Ref.43 also noted that for conformationally flexible molecules, due to additional uncertainties in the computed mole fractions, the criterion should be larger than 2σ. It was pointed out that a criterion of one σ offers only a slight advantage over simply guessing the configuration. The test set used for the study included 17 alkanes, 16 alkenes, 19 ketones, and 13 miscellaneous molecules (mostly hetero three‐membered ring and oxa‐bicyclic structures). More recently, McCann and Stephens computed the OR and electronic CD of 26 conformationally rigid alkenes with [α]_D of magnitude up to 500 degree/[dm (g/cm⁻³)] using the same computational protocol.44 Two conformationally flexible alkenes were also investigated. The RMS deviation was found to be 28.7 degree/[dm (g/cm⁻³)], which is surprisingly consistent with the RMS error from Ref.43. Results for the CD spectra using this level of theory were of “modestly good to poor” quality.

The statistical errors found in the benchmark studies mentioned in the previous paragraph are based on a particular computational model which does not include solvent effects or vibrational corrections. Approximations in the density functional are likely to be one of the main sources of error. Unfortunately, it was recently found that for a test set of 13 organic molecules the agreement with experiment is not improved on average when using coupled cluster (CC) methods instead of TD‐DFT.45 Likewise, we found earlier for a set of 20 organic molecules that adding vibrational corrections to TD‐DFT data did not reduce the average deviations from experimental data (solution data), i.e., the vibrationally averaged ORs scatter about the same around the experimental values although in individual cases improvements may be obtained.22 Solvent effects on ORs were investigated by Mennucci et al.46 using dynamic computations of β(ω) with TD‐DFT and a polarizable continuum model (PCM) for the solvent. The authors did not obtain consistent improvement from the PCM model over gas phase computed values in comparison with experiment. This deficiency was attributed to the electrostatic nature of the PCM model which was argued to work well only for solvents where electrostatic interactions dominate. Kundrat and Autschbach have computed the OR and OR dispersion of naturally occurring amino acids in aqueous solution.47, 48 In this case, the zwitterionic form is not even stable in gas phase which makes the application of a solvent model mandatory. It was found that the combination of a continuum model with hybrid TD‐DFT computations performed reasonably well in reproducing the OR and ORD of individual conformers of amino acids in their protonated, zwitterionic, and deprotonated forms. However, solvation energies were only qualitatively reproduced and the authors noted problems related to the balance between intra‐ and intermolecular hydrogen bonding. Kongsted et al. reported that continuum models were not able to reproduce solvent‐induced trends for the OR of methyloxirane correctly.26 A refined treatment of solvent effects including explicit solvation at the quantum‐mechanical level is clearly desirable.

A route to an improved description of solvent effects (as well as temperature effects) is to average computed ORs along the trajectory of a molecular dynamics (MD) simulation. Such computations have been performed, for instance, on (S)‐methyloxirane to model the effect from aqueous solution49 (BP86/aug‐cc‐pVDZ). On solvation, the long‐wavelength limit of the OR changes sign. The resulting MD‐averaged ORD between 350 and 650 nm agreed well with experimental data. The solvent influence was mainly attributed to a preferred coordination of water on the side of the three‐membered ring opposite to the methyl group and the effect thereof on the electronic structure of the solute. Gas‐phase and computations with a continuum model yielded a sign change of the ORD around 400 nm which is in disagreement with experimental data from aqueous solution. Kundrat and Autschbach have performed combined MD/TD‐DFT calculations on conformationally flexible amino acids and showed that both the conformational averaging as well as the solvent influence are challenging to deal with, but that the MD protocol is indeed very promising.50, 51 However, the improved accuracy of treating solvent effects by MD comes with a price tag: the ORD was obtained from averaging large numbers (ideally thousands) of configurations instead of the single computation for one or a few optimized structures necessary for gas‐phase or a continuum model.

At present it appears that to systematically improve the predictive power of computations of OR it would be necessary to consider a high‐level electronic structure method and include solvent effects and vibrational corrections, preferably in some type of dynamic treatment. It should be pointed out that the predictive power of OR calculations for assignments of absolute configurations can be improved considerably by calculating the dispersion of the OR (ORD) instead of relying on a single frequency calculation.52, 53

The quality of CD spectra is influenced by errors in the excitation energies and by errors in the calculated rotatory strengths. From the available benchmark data it appears that electronic CD spectra in the experimentally accessible energy range tend to be qualitatively correct, but deviations of a factor of two between theory and experiment are not untypical for the intensities, and larger deviations are possible. The excitation energies, and the number and the nature of the excitations, appear to be critical because popular linear response TD‐DFT models, for instance, may significantly under‐ or overestimate certain types of excitation energies or yield spurious excitations. See the section on TD‐DFT Linear Response for a detailed discussion and references. If the excitations are systematically too low or too high with respect to experiment, this is easily enough corrected for by shifting the calculated spectrum accordingly (which is commonly done).

DFT tends to yield good quality force fields and therefore the experience with modeling vibrational VCD and ROA has been very positive. See the section on Vibrational Optical Activity for a description of the fundamental parameters, and selected references. Harmonic force fields are not always suitable to obtain a close match between computed and measured vibrational frequencies. Anharmonic effects have been considered in vibrational optical activity calculations. See, for instance, Ref.54 for an application to trans‐succinic‐d₂ anhydride.

In the case that electronic CD spectra have been computed without explicit consideration of vibrational effects (which is the common situation at present), the computed results will lead to a “stick spectrum” (delta function peaks) as shown later in Figure 6. In VCD/ROA, the calculated excitations are also obtained in the form of stick spectra. For comparison with experiment the stick spectra are usually broadened, for instance with Lorentzian (especially for vibrational optical activity) or with Gaussian line shape functions (in particular for electronic CD). The line widths represent empirical parameters that are typically chosen within reasonable limits to obtain best agreement of the resulting CD bands with experiment. Although a certain degree of “fudging” is possible by changing the line width parameters, in our experience different sets of line widths chosen within reasonable limits normally will not, and should not, lead to qualitatively different interpretations of the data. For electronic CD, Gaussian broadening parameters on the order of 0.2 eV (two thousand wavenumbers) appear to be suitable in most cases.55 For VCD and ROA spectra, the broadening may be on the order of 10 to 20 cm⁻¹.

COMPUTING CHIROPTICAL PROPERTIES: THEORETICAL BACKGROUND

Chiroptical Properties: Basic Definitions

The basic molecular parameters that govern the chiroptical response of a molecule come in pairs, one part absorptive and one part refractive. For instance, circular dichroism (CD) is the differential absorption between left‐ and right‐handed circular polarized light. The optical rotation (OR) of the chiral medium, on the other hand, is caused by the difference in the refractive indices for left‐ and right‐handed circular polarized light.9, 56 This article will be concerned mainly with the formalism and applications of electronic CD and of OR caused by electronic transitions. For completeness, there is also a section of vibrational optical activity (VCD and ROA), and a few case studies will be discussed in the section on Selected Case Studies. For additional information see also Refs.9,56–58.

The chiroptical analog of the “regular” electronic excitation spectrum (dipole‐allowed excitations, typically in the UV‐Vis range) is the circular dichroism spectrum. The chiroptical analog of the isotropic line strength of UV‐Vis spectroscopy is the rotatory strength R_j. For an excitation number j out of the ground state,

(3)

Here, and are the electric and magnetic transition dipole vectors, respectively, with the dipole moment operator and the magnetic dipole moment operator . We ignore for the moment any nuclear motion which would require to consider the corresponding nuclear operators. For electronic CD the dipole operators for the N‐electron system are used. Usually, for the rotatory strength c.g.s. units are employed* in which the electric and magnetic dipole moment have the same units of esu cm. The factor of c⁻¹ ≈ 1/137 au in the magnetic moment operator indicates the relative order of magnitude of CD as compared to regular absorption spectra. Typically, †R –values are quoted in units of 10⁻⁴⁰ esu² cm².

For interpretations of CD spectra, it is often helpful to consider distortions from an idealized non‐chiral structure or locally symmetric chromophore causing the transition moments to be nonzero or changing their angle. The rotatory strength is zero if either d_j = 0 or m_j = 0, or if the angle between d_j and m_j is 90 degrees. One of these condition is fulfilled for each of the transitions in a nonchiral molecule. Qualitatively, one may consider if charge is displaced in a way that creates a net electric dipole from the transition density (leading to d_j ≠ 0) or a “charge rotation” (rotating charges are associated with magnetic moments, and thus m_j ≠ 0). The concept has been nicely illustrated in Snatzke and Woody's chapter59 in Ref.60. For example, consider the common CO chromophore. See Figure 3 for isosurface plots of the main orbitals involved in its low‐lying electronic excitations. The leading terms in the transition densities are approximated by products of occupied with unoccupied orbitals. For the n‐to‐π* transition, in a nonchiral environment the electric transition dipole moment is zero whereas the magnetic dipole is nonzero. The transition is electric‐dipole forbidden and magnetic‐dipole allowed. To lowest order, a rotatory strength is induced by disssymmetries causing d_j not to vanish identically. A π‐to‐π* transition is electric‐dipole allowed, i.e., d_j ≠ 0. For a CO or CC chromophore in a symmetric environment, for this transition m_j = 0. The excitation is magnetic‐dipole forbidden. The leading term to the rotatory strength of a π‐to‐π* transition in a chiral environment results from a nonzero m_j. Finally, for the moderatly UV‐Vis intense n‐to‐σ* transition of a CO chromophore neither d_j nor m_j are zero, but in a nonchiral environment the two vectors are at a right angle and the rotatory strength vanishes. To first order, a nonzero rotatory strength is caused by dissymmetries leading to this angle not being exactly π/2.

Figure 3

Some of the orbitals that are important for the excitations of a CO chromophore. Shown are natural bond orbitals for formaldehyde, calculated at the BP/TZP density functional level (isosurface values: 0.03 au) and three products approximating the transition densities for the n‐to‐π*, π‐to‐π*, and n‐to‐σ* excitations (iso: 0.004 au), along with textbook–style graphs representing the angular behavior of the functions.

Unlike “regular” excitation spectra, the bands in a CD spectrum can have positive and negative signs, and in exact theory the sum rule

(4)

holds for molecules without a permanent electron orbital magnetic moment.‡ The sum is to be understood somewhat symbolically because it includes the infinitely many bound and continuum states of a molecule.

A common measure of the CD intensity is the differential molar absorption coefficient Δε =ε_L − ε_R for the absorption of left‐ and right‐handed circular polarized light. See Figure 4 for an illustration. For a CD band corresponding to an isolated transition, the relation between the rotatory strength of this excitation and the isotropic CD intensity Δε is

(5)

Figure 4

Left: Circular dichroism measured as the difference of absorption coefficients Δε for left‐ and right‐handed circular polarized light of frequency ω (amplitudes and wavelengths not to scale). Δε is typically two orders of magnitude smaller than the absorption coefficient. Right: Optical rotation in an absorbing medium: The outgoing wave not only has its plane of polarization rotated but also exhibits an amount of ellipticity, which for small ellipticities is proportional to the intensity of the CD. The magnitude of the CD and the ellipticity is exaggerated and the box size has been chosen for illustrative purposes only. For animations see Refs.61 and62.

The numerical value of the conversion factor in the last equation is approximately 22.97 · 10⁻⁴⁰ c.g.s. units assuming that Δε is given in the usual units of l /(mol cm). It is also not uncommon to report dimensionless reduced rotatory strengths, which are obtained from the R values in 10⁻⁴⁰ esu² cm² by multiplication with 1.078.9 Another frequently used quantity to express the intensity of the circular dichroism is the molar ellipticity [θ], with units of degree cm²/dmol. The conversion from Δε is [θ] = (18,000ln(10)/4π)Δε. The numerical value of the conversion factor from Δε in l/(mol cm) to [θ] in degree cm²/dmol is approximately 3298.2.

The optical rotation (OR) is due to nonabsorptive interactions of an electromagnetic wave with the charged particles of chiral molecules (or with an otherwise circularly birefringent medium), i.e., the oscillating electric and magnetic fields of the electromagnetic wave interact with the charged particles in the molecules constituting the medium. At the molecular level, the optical rotation is quantified by a rank‐2 OR tensor β which describes the linear (first order) response of the electric (d) and magnetic (m) dipole moments of a single chiral molecule to the presence of time‐dependent electric ( ) and magnetic () fields. For instance, in the time domain we have for the linear response of the dipole moments57

(6a)

(6b)

and similarly for the other components. The dots indicate the time derivatives of the fields. The physical significance of the tensor β is that it determines the magnitude of an induced electric dipole moment on a molecule by a time‐dependent magnetic field. The equations are symmetric in the sense that β also yields the magnitude of a magnetic dipole moment induced by a time dependent electric field.

In a nutshell, the relation of the tensor β with the observable optical rotation is established as follows [see Condon's review56 for details]: (i) Consider the electric field vector of a linear polarized electromagnetic (EM) wave as a superposition of two circular polarized waves with equal amplitudes and the same frequency as the plane polarized wave.61, 62 (ii) The plane of polarization is defined by the phase between the circular polarized components.61 (iii) Consider now the indices of refraction of the medium, n_L,n_R which are related to how fast each of the circular components travels through the medium. The refractive index is related to the electric polarization induced by the EM waves. Down at the molecular level, this means one needs to know d⁽¹⁾ caused by left‐ and right‐hand circular polarized EM waves. (iv) If n_l − n_R ≠ 0 the phase between the L and R component of the linear polarized EM wave changes as it passes through the medium. As a consequence, the plane of polarization turns. (v) The tensors α, β, etc. enter the expression for d⁽¹⁾ via a multipole expansion of the EM wave. The leading term is from the polarizability tensor α which approximates the EM wave as a homogeneous electric field that changes its amplitude over time. Truncating the multipole expansion at this level yields n_L − n_R = 0. One needs to go one step further in the multipole expansion and include the electric‐dipole–magnetic field term, and also an electric quadrupole–electric field gradient term, to account for the field's inhomogeneity. If β or the quadrupole tensor are nonzero, these terms do not cancel in n_L − n_R and an optical rotation can occur. (vi) The quadrupole term will not be discussed here because its isotropic average is zero (for rapidly rotating molecules in gas phase and solution). It needs to be calculated to predict correct optical activity tensor elements. For a brief discussion see the end of the section on Selected Case Studies.

Expressions based on the c.g.s. unit system are used here in which the speed of light appears as the coupling parameter between the electric dipole moment and the magnetic field. As already pointed out, c⁻¹ indicates the relative smallness of the chiroptical effects compared with, say, the electric polarizability and dipole‐allowed electronic UV‐Vis absorptions. The Fourier components of the mixed electric–magnetic linear response describe linear chiroptical properties in the frequency domain. The isotropic part of the tensor, β(ω), is called the OR parameter, written here as a function of frequency ω. Equation6a,b also contain the electric polarizability tensor α and the magnetizability tensor χ. As outlined above, a nonzero real part of the OR parameter causes a difference in the refractive indices of left‐ and right‐handed circular polarized light, which manifests itself in a rotation of the plane of polarization of linear polarized light as it passes through the medium (Fig. 4). A nonzero imaginary part quantifies the differential absorption (Δε or [θ]).

Typically, experimental optical rotations are reported as specific rotation [α] in units of degree/[dm (g/cm⁻³)] at a given wavelength λ or wavenumber or frequency ω = 2πc/λ. In atomic units (au), the frequency ω and the energy E of the light wave have the same value. Frequency units are converted as follows: A frequency ω of 1 au corresponds to 27.211 eV (electron Volt) in energy units. The eV to wavelength conversion is E/eV ∼ 1239.8/(λ/nm), and to wavenumbers it is E/eV ∼ 8065.5 /cm⁻¹. The specific rotation can be obtained from β as

(7)

where M is the molecular weight in g/mol, and ω is in s⁻¹. N_A is Avogadro's number. Equation 7 excludes local field corrections, concentration effects, and other interactions between a solute molecule for which the computation is carried out and its environment. The optical rotation parameter β is here in c.g.s. units of cm⁴ per molecule. For conversions from β calculated in atomic units for a single molecule (as it is typically computed in quantum chemistry programs), for the light frequency given in cm⁻¹ the numerical conversion

(8)

is useful. The equations apply to a molecule in vacuum (refractive index n = 1). In a condensed medium with refractive index n, the rotation values of the last equations should be multiplied with the Lorentz field factor (n² + 2)/3. However, it has been demonstrated convincingly that the agreement of TD‐DFT computations with experimental data from solution deteriorates upon inclusion of this factor40, 46 and therefore we will omit it from the equations.

The molar rotation

(9)

is another frequently used quantity. With [α] in the usual units of degree/[dm (g/cm⁻³)], the molar rotation obtained from the conversion in Eq.9 is in units of degree cm²/dmol, which is the same unit as that for the molar ellipticity [θ].§

Apart from the optical rotation tensor β, the gyration tensor G′ is also frequently used as the basis of computing optical rotations because it is straightforward to define working equations for it in the frequency domain. The relation to the OR tensor is

(10)

Notation‐wise, G′ shall indicate the isotropic average of the G′ tensor, G′ = (G′_xx + G′_yy+ G′_zz)/3, which is the tensor invariant relevant to measurements in solutions and gas phase where the molecules are rapidly rotating. The gyration tensor is the imaginary part of the mixed electric–magnetic dipole response, relating the perturbed dipole moments of eq. 6 in the frequency domain directly to the field, not the time derivative. The relation with β becomes clear when considering, for instance, the time‐dependent magnetic field amplitude in eqs. 6a and 6b: the Fourier components of the time‐derivative are which is the origin of the factor of ω in eq. 10. For static fields (ω = 0), G′ = 0 but the OR tensor β has a non vanishing static limit. Before the availability of dynamic linear response methods the static limit of β has sometimes been used for computations of optical rotations1, 63-65; however, this approach is now deprecated because the dispersion of β tends to be significant and the correct dynamic values may disagree considerably with the static limit. Figure 5 shows the OR parameter for the fenchone molecule calculated over a range of frequencies including the lowest energy n‐to‐π* excitation and the next higher transition. The n‐to‐π* is at approximately 4 eV or 3 · 10² nm. No damping was used in these calculations (vide infra); therefore, the OR parameter diverges around the excitation frequencies. The Na D‐line wavelength of 589.3 nm corresponds to a frequency of 0.07732 au. where for this particular system the dispersion of β is not yet particularly pronounced and a reasonable estimate of the OR may also be obtained from the static limit. The OR parameter in Figure 5 has been calculated in two different ways. One has been the route taken in many implementations where the G′ tensor is computed first. The other route that one may take is to determine the OR tensor β directly from the overlap of the perturbed MOs determined for a magnetic and an electric field, respectively, with one of the fields being time dependent and the other one static.66 As the observable rotation angles are proportional to the frequency squared times the OR parameter, i.e., negative frequency times G′, one may consider both approaches as equally useful. At very low frequencies, the direct calculation of β is potentially more stable numerically because it has a finite zero‐frequency limit. Typically, the OR tensor has large diagonal elements with opposing sign that cancel to a large extent.42 Therefore, it is a numerically sensitive property that requires well‐converged wavefunctions/densities and well‐converged linear response equations.

Figure 5

OR parameter β of fenchone, calculated directly and from −G′/ω. PBE0 hybrid functional, 6‐31+G* basis.66 The singularity indicates an electronic excitation.

Chiroptical Response as a Function of Frequency or Wavelength

CD and OR are linear response properties of a molecule, i.e., they arise from the terms linear in the field amplitudes when considering the interaction of a molecule with an electromagnetic wave as in Eqs. 6a and 6b. Both the CD intensity and the OR are dependent on the frequency ω of the field. An important fact to know about linear response functions is that causality, along with a general requirement of mathematical “well‐behavedness,” require that they are complex functions of ω.69 The real and the imaginary part of the complex response function are intimately related to each other in the sense that they form a so‐called Kramers–Kronig transform pair.9, 67, 68 The real and the imaginary part of a complex linear response function f(ω) are related as follows:¶

(11a)

(11b)

These are the Kramers–Kronig (KK) relations. Note that at each frequency point ω the integration over μ includes the singularity, where μ = ω. To properly take into account the singularity, “PV” indicates that the principal value of the integral must be taken.

In optical activity, the complex linear chiroptical response function may be written in the form of a complex quantity such as the complex molar rotation. Its real part [ϕ] describes the refractive process, i.e., the observable optical rotation, whereas its imaginary part describes the absorption process, i.e., the molar ellipticity [θ] measuring the strength of the CD. See Figure 6 for an illustration. In Figure 4 if the outgoing wave is elliptically polarized, the angle between the long axis of the ellipse and the polarization plane of the incoming wave indicates the rotation whereas the magnitude of the short axis compared with the long axis can be used as a measure of the differential absorption. If no broadening of the absorption is considered, the frequency‐dependent OR parameter is purely real except at the excitation frequencies ω_k. The real part becomes singular at the excitations. Right at the excitation frequencies, the response then switches from purely real to purely imaginary. The CD spectrum would be obtained in the form of δ‐peaks. For explicit expressions see the section on Sum‐Over‐States Expressions. The singular behavior makes it impractical to compute β(ω) and other dynamic response properties right at the excitations without using some form of absorption broadening along with the corresponding damping of the rotatory dispersion. A broadening/damping can be introduced in the formalism, for example, via finite lifetimes for excited states, or more pragmatically via semi‐empirical damping constants.70, 71 In this case, one obtains nonsingular complex response functions instead.

Figure 6

Top and center: Qualitative behavior of a linear chiroptical response function in the vicinity of an excitation with frequency ω_k. Re = real part; Im = imaginary part. Top: no absorption line broadening. The imaginary part would be a line‐spectrum (δ, functions) with corresponding singularities in the real part at ω_k. Center: Imaginary part with a Gaussian line shape, and corresponding real part.67 Bottom: Several excitations. Electronic absorptions shown as peaks in the imaginary part of the optical rotation parameter. Real part of β showing several poles in the vicinity of the excitations. The qualitative behavior for the [ϕ], [θ] pair is similar.9, 67, 68

Using KK transformations in chemical applications of ORD and CD have been of considerable interest in the 1950s and 1960s.6, 9 For instance, before CD spectrometers became commercially available rotatory strengths for CD transitions from the CO chromophore have been computed from measured OR dispersion curves (see Moscowitz' chapter in Ref.6). With the advent of CD spectroscopy it is obviously much easier to extract rotatory strengths directly from the spectra and the ORD has become less important. Now that concerted computational studies of CD and ORD can be performed at correlated levels of theory, there has been a renewed interest in the complex nature of the chiroptical response function, as well as in the use of KK transformations.67, 68, 72, 73 For instance, systematic deviations between experimental ORD and KK‐transformed CD, or vice versa, may provide useful information about high lying excited states not directly accessible spectroscopically (see Polavarapu, Ref.73). In other cases, in which experimental ORD is not available, KK‐transformations of experimental CD spectra may be useful to allow for an assessment of the performance of ORD computations.

The complex nature of the OR parameter and the OR tensor shall be denoted by a tilde, i.e., and similar for G′ and the individual tensor elements. Poles of β(ω) occur around each excitation frequency, as shown qualitatively in Figure 6 (bottom). As optical rotation parameters can be computed using various approximate quantum chemistry methods, this is therefore also a possible route to obtaining excitation energies. It is important to note that this way the excitation energies are computed from a frequency‐dependent perturbation of the ground state. Hence this approach bypasses the need for calculating wave functions or electron densities for excited states. Usually, if only the CD spectrum is of interest, a scan of β(ω) or other linear response functions (polarizabilities, mainly) as a function of frequency, as in Figure 6, is not performed explicitly. Instead, an expression is derived for a dynamic linear response property such as β(ω) within linear response theory. Broadening of the absorptions is deliberately excluded which means the equation for β should afford singularities at the excitation energies. The equation is then written in a form that makes it obvious where it becomes singular as a function of frequency. The locations of the singularities on the real frequency axis and the residues of the response function at these frequencies are typically computed directly in the form of an eigenvalue–type problem yielding the “stick spectrum” of Figure 6. This is a “linear‐response route” to computing excitation energies. The majority of such computations are now performed with TD‐DFT; CC methods are also quite frequently applied if the systems are small enough.

Sum‐Over‐States (SOS) Expressions

Suppose we knew the ground state of a molecule, i.e., its wave function Ψ₀ and energy E₀, and all its excited state wave functions ψ_j, energies E_j, and the excited states life time parameters Γ_j. In practice we never know all of them (or even a single one), but pretending so may lead to illustrative and useful expression of the OR and other response properties in terms of excited states contributions. One can then express the perturbation of the molecule's ground state by the electric and magnetic field component of linear polarized light waves in the “basis” of the excited states to arrive at a sum‐over‐states (SOS) expression for the complex gyration tensor elements (u,v ∈ {x,y,z})

(12a)

(12b)

with rotatory strength tensor elements

(13)

and the line shape functions

(14a)

(14b)

where ω_j = E_j − E₀. Equation 12a is the real part and 12b the imaginary part of the gyration tensor. The OR tensor elements are given by −ω⁻¹ times eqs. 12a and 12b. From now on it should be assumed that the wave functions are real.** As the magnetic moment operator is imaginary, the contributions from Re[R] vanish if the wave functions are real. The isotropic rotatory strength R_j of eq. 3 is given by R_j = Im(R + R + R) From Eqs. 12a and 12b the sign and the magnitude of the rotatory strength tensor and the isotropic rotatory strength are seen to govern the sign and magnitude both of the real and the imaginary part of the complex optical rotation.

The SOS expression emphasizes the significance of the rotatory strengths both for CD and for optical rotations. For analysis purposes one may consider a transition with a particularly intense CD and calculate its partial rotation, i.e., its contribution to the optical rotation from the SOS. Unfortunately, recent research has shown that for optical rotations, partial SOSs including an growing number of excitations tend to be strongly oscillating as a function of the number of states included, instead of smoothly converging to the full response result.74, 75 This behavior limits the interpretative value of the SOS and a chemically motivated analysis might be better achieved my other means. Nonetheless, because both CD and optical rotation are parts of the complex chiroptical linear response function it is important to consider their relationship when attempting to analyze these properties. Apart from interpretation purposes, the SOS (or a KK transformation) can have many useful applications not the least important being that it allows to quantify if excitations outside of the observable frequency range may contribute to the optical rotation of a molecule or not.

When including dephasing constants Γ_j which are related to the finite life time of the excited states, the derivation leads to the intrinsic Lorentzian line shape of the imaginary part in (12b). The dephasing constants are only partially responsible for the line broadening observed experimentally (as qualitatively shown in Figure 6). Response equations with a common lifetime parameter γ replacing the Γ_i for all excited states have been applied to model optical rotation in the vicinity of resonances. Typically, in such computations, the parameter γ represents more of an empirical broadening parameter than a common excited states lifetime parameter because there are other broadening mechanisms that may dominate over the lifetime broadening. See Refs.52 and68 for case studies and related references. The Lorentzian line shape dictated by the form of the response equations can be easily replaced with Gaussian shapes that resemble broad featureless absorption and CD bands of large organic molecules in solution more closely.67

Assuming real wave functions, the elements of the real part of the complex OR tensor read in SOS notation

(15a)

and the elements of the imaginary part of the OR tensor read

(15b)

In the limit of infinite lifetimes (i.e., in the absence of damping), the real part simplifies to

(16)

An equivalent expression has been given, for instance, in the textbook by Kauzmann.57 For the imaginary part, we note that in the limit of vanishing damping,

(17)

which is one of the many representations of the delta “function”. Thus, the imaginary part of the OR tensor elements reads in the absence of damping (assuming real wave functions)

(18)

This is the “line spectrum” implied in Figure 6 for ω > 0, whereas the divergence of the real part without damping at the excitation energy is easily seen to be caused by the ω − ω² denominator in eq. 16.

With the delta functions in the imaginary part β′, i.e., in the limit of vanishing broadening, the KK transformation (11a) from the imaginary to the real part becomes easy to perform. Recall that for a reasonably well‐behaved function f(μ). Thus, the imaginary‐to‐real KK transformation (11a) will generate eq. 16 from the first delta function term in eq. 18 whereas the second term does not contribute at positive frequencies ω. We note in passing that the real part of is an even function of ω whereas the imaginary part is an odd function. These symmetry properties are required for the KK transformation eq. 11a and 11b to be valid. The functions generated by , i.e., the molar rotation [ϕ] and the molar ellipticity [θ], also constitute a KK transform pair.

Origin Dependence of Chiroptical Properties

The electric and magnetic dipole moment operators entering the rotatory strength tensor, eq. 13, are dependent on the origin. Suppose we shift the origin from O to O′ = O − a, with each electron coordinate becoming r_i′ = r_i + a. The electric dipole moment causes no problems, but the derivative term in does. In a nutshell, in computations with a finite atomic‐orbital (AO) basis, computed ORs and rotatory strengths and other response properties involving are generally origin dependent, i.e., they may change if an origin shift a is applied to the coordinates of the molecule. The origin dependence is a manifestation of a more general so‐called “gauge dependence” problem of magnetic properties in finite basis set calculations.76

Currently, the two most common ways to circumvent the origin dependence are (i) the use of magnetic‐field dependent “gauge‐including AOs” (GIAOs, also known as “London‐orbitals”)65, 77, 78 and (ii) the use of the “velocity‐gauge” for the electric dipole moment integrals.79 The GIAO basis depends explicitly on the magnetic field amplitude which requires the computation of a number of additional terms when solving response equations.78 Such a basis eliminates the origin dependence and also tends to moderately improve convergence of calculated optical rotations with respect to the basis set. However, GIAOs do the trick only in variational perturbation methods based on DFT/TD‐DFT and certain classes of variational wave functions79 in which the orbitals are fully optimized (including HF and MCSCF). Using the “velocity gauge” is conceptually simpler because it does not introduce a dependence of the basis set on the perturbing fields. Formally, the electric transition dipole moment in Eq.3 can be replaced by its velocity (momentum) form by considering that formally

(19)

where and are the one‐electron position and momentum operators for the N‐electron system. There are similar relations for nuclear wave functions. It can be shown that isotropic optical rotations and rotatory strengths independent of the choice of the coordinate origin can be calculated in their velocity form on replacing dipole moment matrix elements in the linear response computation by those of the momentum operator and division of the result by ω. With incomplete basis sets eq. 19 is not exactly satisfied but the origin independence of the results is retained. As was demonstrated by Pedersen at al.79, for optical rotations calculated in a finite basis this approach leads to sometimes large values of G′(ω = 0) although G′ has to vanish for zero frequency. The “modified velocity” gauge for the optical rotation tensor79 involves the subtraction of G′(ω = 0) from the result calculated for nonzero frequency, both computed at the same level of theory using the velocity form of the electric dipole integrals. This procedure yields origin‐independent results that compare well with the dipole‐length form for large basis sets, and to G′ obtained with GIAOs. It was suggested to use the difference between length and velocity results as a convenient measure for estimating the basis set errors on computed optical rotations in TD‐DFT computations.80

Vibrational Optical Activity

Optical activity is not restricted to transitions between electronic states. For instance the circular dichroism of transitions between vibrational energy levels of a molecule's ground state is also an extremely useful source of information about a chiral molecule. A rotatory strength for a vibrational transition can be defined in the same way as eq. 3 but now considering vibrational transitions, typically in the electronic ground state. This leads to vibrational CD (VCD), the chiroptical analog of “regular” vibrational (infrared, IR) spectroscopy. The VCD strength is the difference in the infrared absorption for left‐ and right‐hand circular polarized IR light. What is needed, accordingly, is the electric and magnetic transition dipole moments for the vibrational transitions. The vibrational frequencies and normal modes are calculated with standard techniques that are also used for routine simulations of regular vibrational spectra, i.e., the difference between simulating VCD and regular vibrational spectra is in the calculation of the intensities.

Since we are concerned with nuclear motion, the electric and magnetic dipole moment operators now explicitly contain electronic and and nuclear and contributions. Consider the Born–Oppenheimer separation of electron and nuclear motion in which lowest order to the total wavefunction is a simple product of electronic Ψ(τ) and nuclear wave functions for an electronic reference state “0” (ground state) and an associated vibrational state “ν”. Here, τ is the set of electronic coordinates and q a set of vibrational coordinates. Consequently, we have for a transition dipole moment of the vibrational transition 0 → 1 (written symbolically for one of the vibrational transitions)

(20)

where is either the electric or the magnetic dipole moment operator and 〈Ψ|Ψ〉=1 has been used. The approximation in the last line involves, to zeroth order, complete neglect of the dependence of the electronic contribution on the nuclear coordinates. As the ground and excited state nuclear wavefunctions are orthogonal, the electronic magnetic moment contribution then vanishes, and the only term left would be the purely nuclear term . However, this treatment is insufficient for VCD simulations.

Thus, it appears necessary to consider averaging the electronic part of the transition moments over the nuclear wave functions as in the term of eq. 20. For infrared intensities, this is readily achieved in the so‐called double harmonic (electrically and geometrically harmonic) approximation.81 However, for closed shell molecules, the contribution from the magnetic moment vanishes because irrespective of geometry, and therefore there would still be no electronic magnetic moment contributions to the VCD intensity. It has turned out that the VCD effect is subtle and one needs to include coupling between electron and nuclear motion to obtain meaningful results.

Defining nuclear displacement coordinates relative to the equilibrium positions in terms of the normal modes q_p with the help of a matrix–vector S_p,A,

(21)

in the harmonic approximation the nuclear terms work out to be (length gauge)82

(22)

(23)

Here the ω_p are the harmonic vibrational frequencies.

The inclusion of electronic terms in the formalism can be achieved by considering a normal‐coordinate q_p dependence of the electronic wave function including coupling with electronically excited states in the form83-85 (see also Refs.86 and87)

(24)

The coefficients in the last term are obtained from first order perturbation theory as

(25)

with the perturbation operator

(26)

The superscripts Ψ and Φ indicate the function on which the operator acts. The set of nuclear wave functions used here are harmonic oscillator functions which is consistent with the way the vibrational frequencies are obtained.

Using this ansatz, the electronic contributions in eq. 20 to the electric and magnetic vibrational transition dipole moments can be obtained as84-86, 88

(27a)

(27b)

for normal mode p with frequency ω_p, using the approximation that E_0ν − E_jν′ ≈ E₀ − E_j. The wave function derivatives with respect to the normal coordinates are already calculated when the vibrational frequencies are determined; therefore, the electric dipole part can be readily evaluated from dipole moment derivatives. The magnetic part as written in eq. 27b involves excited states. However, apart from the factor the sum over states in (27b) is in fact a SOS expansion of the derivative of Ψ with respect to a magnetic dipole field. Thus, in Stephens' magnetic field perturbation formalism84 the magnetic moment term (27b) is written as

(28)

The VCD intensities are therefore readily computed once the first‐order perturbed wave functions (or orbitals) with respect to an external magnetic field and nuclear displacements are available, the normal modes and frequencies are known, and the nuclear contributions to the VCD have been evaluated. Standard methodology can be employed for the task of computing the perturbed wavefunctions/orbitals, because these quantities are already necessary on the one hand in calculations of harmonic force constants and IR frequencies (which are required for a simulation of a VCD spectrum), and on the other hand for NMR or ESR or magnetizability computations. Once the rotatory strengths are available, VCD spectra can be simulated by applying empirical broadening functions, similar to what is usually done for electronic CD.

Yet another very promising technique is Raman optical activity (ROA).89-92 Like other chiroptical methods, ROA measures a differential intensity with respect to left‐ and right‐hand circular polarized light. However, like in regular Raman spectroscopy, this is not a simple absorption process but an inelastic scattering involving the absorption of a photon into a ‘virtual’ state and subsequent emission of a photon of different energy as the molecule returns to the electronic ground state but a different vibrational state as from which the absorption occurred. In a back scattering setup, the direction in which the scattered beam is observed at 180°, opposite to the direction of the incident laser beam. The corresponding isotropic ROA intensities are1, 93

(29)

where c is the speed of light, β(G′) the anisotropic invariant of the product of the electric dipole‐magnetic dipole polarizability transition tensor and the electric dipole–dipole polarizability transition tensor, and β(A) the anisotropic invariant of the product of the electric dipole‐electric quadrupole polarizability transition tensor with the electric dipole–dipole polarizability transition tensor. The parameter K_p is independent of the experimental setup but depends on both the incident and scattered frequencies. It is given by

(30)

where and are the wave numbers of the incident light and of the p'th vibrational mode, respectively. The Raman intensity units here are in cm²/s. As in the case of VCD, the harmonic vibrational frequencies and normal modes required for the computation are determined in the usual way whereas the new aspect of the calculation is to determine the ROA intensity.

The tensor invariants needed for eq. 29 involve contractions of different combinations of the dipole–dipole polarizability transition tensor, α^p, the gyration transition tensor G′^p, and the electric dipole‐ electric quadrupole polarizability transition tensor, A^p, namely,

(31a)

and

(31b)

Repeated indices imply summation. ε is the Levi–Civita tensor. For further details and alternative expressions for a forward scattering geometry see Ref.93. Adopting the approximation E_0ν − E_jν′ ≈ E₀ − E_j, as it is also done when deriving the VCD expressions, these tensors can be calculated from derivatives of the electronic response functions along the normal modes as follows93

(32a)

(32b)

(32c)

where q_p is the normal coordinate for the p′ th vibration, α_uv is an element of the electronic electric dipole–dipole polarizability tensor, G′_uv is an element of the gyration tensor and A_wtv an element of the electric dipole–quadrupole polarizability tensor.

The normal mode derivatives can be calculated either by numerical or analytical differentiation. The calculation of “regular” Raman intensities requires only the knowledge of the electric dipole polarizability derivatives; otherwise, Raman and ROA computations are quite similar. Although somewhat more involved than VCD, the calculation of ROA intensities is reasonably straightforward and consequently has been implemented in a number of program packages at various first‐principles levels of theory including TD‐DFT. The implementation by Jensen et al93 incorporates damping techniques which allow also for calculations of some of the leading terms of Resonance ROA (RROA) and simulations of surface‐enhanced ROA (SERROA).

TD‐DFT LINEAR RESPONSE: A SYNOPSIS

TD‐DFT is an extremely active field of research. Rather than surveying the literature, this section is intended to provide a synopsis of the method and to summarize some of the issues that are important for researchers who want to use TD‐DFT for computations of electronic response properties that are important for optical activity (mainly for electronic CD, OR, and ROA). For additional details, the reader is referred to a number of books and articles. General DFT references:94 and95. TD‐DFT: 96–99. We will pay attention to some of the shortcomings of approximate functionals that are frequently encountered as sources of error in TD‐DFT computations. A list of “take‐home” messages can be found at the end of this section. For a more detailed overview of TD‐DFT by the author see Ref.99 from which this section has been adapted.

The Kohn–Sham DFT ground state energy expression is

(33)

where ρ = ∑_i n_i φ φ_i is the electron density, φ_i a Kohn–Sham molecular orbital (MO) with occupation n_i, is approximately the kinetic energy, is the Coulomb repulsion energy for the electrons, the attraction between electrons and nuclei (nuclear charges Z_A), and is the internuclear repulsion. E_NN is not an electronic term but it is usually included in the total energy. Furthermore, r_A is an electron–nucleus distance and R_AB the distance between two nuclei. The DFT–specific part is E_XC, the exchange‐correlation (XC) energy of the molecule written here as a functional of the densities for α and β spin. It accounts for all electron correlation effects and eliminates the artificial self‐repulsion of the electrons contained in E_C (self‐interaction). It also accounts for “exchange” which enters the energy expression when considering the Pauli principle. E_XC should ideally also contain corrections for the kinetic energy T which is not exact in eq. 33. The form of T and the orbital parameterization of the electron density ρ was introduced by Kohn and Sham (KS) to circumvent problems with approximate kinetic energy functionals of the density. These days, DFT is often considered synonymous with KS DFT although it is worthwhile to emphasize that in KS DFT the kinetic energy is not computed from the density. Orbital‐free approaches are also being researched.

The functional E_XC is not known and therefore approximations have to be constructed. These approximations define a hierarchy of functionals with increasing complexity and, one hopes, increasing accuracy.

In ground state DFT, the MOs satisfy the time‐independent KS equations

(34)

(spin indices are not used to keep the notation simple). Here, is the KS Fock operator and ε_i is the orbital energy. The equation appears like a Schrödinger equation for one electron, with playing the role of the Hamilton operator. The Fock operator is an “effective” one‐electron operator that consists of the kinetic energy term , the electron–nucleus attraction V_Ne, and the electronic Coulomb potential . Furthermore, the Fock operator contains the DFT‐specific exchange‐correlation potential V_XC which is obtained from E_XC by taking the functional derivative with respect to ρ(r). The V_XC term is itself a functional of the density. The two terms V_C and V_XC account for the presence of other electrons in the system.

As an example, one of the simplest XC functionals is called Xα. For this functional, the XC energy may be written as where C is a constant, which leads to an XC potential V_XC = −Cρ^1/3. One of the problems with approximate functionals is an unphysical electron self‐interaction. Consider the hydrogen atom as an example. The correct Fock operator is the one‐electron Hamiltonian, i.e., where r is the electron‐nucleus distance. In KS DFT, the Fock operator is with V_C and V_XC defined in the previous paragraph. The presence of V_C for this one‐electron system represents an unphysical electrostatic self‐repulsion of the electron. In Hartree‐Fock and other wavefunction theories, this term is exactly canceled by an (equally unphysical) self‐exchange term. Only partial cancellation is obtained if the approximate DFT potential V_XC = −Cρ^1/3 is used, leading to a self‐interaction error. Such errors are generally present in approximate DFT, not just in one‐electron systems, and in all functionals that do not consider full exact exchange or otherwise avoid self‐interaction explicitly. Self‐interaction may introduce significant errors in computed chiroptical properties of molecules.

There is another deficiency of common XC potentials that affect TD‐DFT computations. Consider the KS Fock operator for a many‐electron system. The effective potential V = V_Ne + V_C + V_XC that is “felt” by an electron should be that of the sum of nuclear charges plus the N − 1 other electrons. Further away from the system, asymptotically, the potential term in the Fock operator for a neutral atom or molecule therefore becomes similar to that of a singly positively charged molecular core (like the potential for the hydrogen atom) and should decay as − 1/r where r is the distance of the electron from the positive core. The description of diffuse states obviously depends critically on this behavior of the potential. Consider, again, as an example the XC potential V_XC = −Cρ^1/3. This, and many other approximate local KS potentials vanish rapidly at large distances because the electron density vanishes exponentially at large r. At the same time, at large r the sum V_Ne + V_C approaches zero for a neutral system since V_C contains the Coulomb potential from all electrons. As a consequence, the asymptotic behavior for local XC potentials does not contain the −1/r term. Many DFT XC potentials suffer from an incorrect asymptotic behavior which leads to errors in particular in the excitation energies computed for diffuse valence state and Rydberg states. The term “local” refers here to a KS potential at some position r in space that depends on the density, the density gradient and potentially higher density derivatives, and optionally the kinetic energy density, at that very same position r. For example, the Xα potential at r is given by the density value at this position to the power 1/3. Sometimes, in older DFT literature, density gradient terms in the functional and in the KS potential have been refereed to as “nonlocal” in the sense that the gradients terms may be understood as sampling density values from nearby points (in the sense of a power series expansion of a function in terms of the function and its derivatives evaluated at a nearby point). This type of “nonlocality” is very different from the nonlocality of the Coulomb potential or the Hartree‐Fock exchange potential which is the type of nonlocality referred to in this article.

Shape corrections to V_XC have been developed that restore the correct behavior at long range and lead to significant improvements of computed excitation energies for such states. See for example, Tozer and Handy100, and the shape corrected XC potentials developed by Baerends and coworkers such as SAOP and GRAC.101, 102 In Hartree‐Fock theory because the self‐repulsion of the electrons in V_C is canceled by the self‐exchange, the potential affords a −1/r asymptotic term. With hybrid functionals such as B3LYP which afford a certain fraction of exact exchange, a fraction of the −1/r term is recovered which leads to some improvements. Obviously, using Hartree‐Fock theory instead is not desirable either because the treatment of electron correlation would be lost.

In the time‐dependent case,96, 97 the KS equation resembles a single‐electron time‐dependent Schrödinger equation, now with the KS Fock operator and the orbitals being time‐dependent,§§

(35)

( ) and the time‐dependent electron density is

(36)

In , the kinetic energy operator is the same as in Eq.34. The V_C and V_Ne expressions are also the same but they are evaluated with the time‐dependent density. Furthermore, may contain a time‐dependent external potential, such as from an oscillating electric field. The Fock operator also contains an XC potential . It is time‐dependent but it is not derived from a functional derivative of the energy which is not well defined in time‐dependent potentials.

Conceptually, V_XC may be defined via eqs. 36 and 35 as the XC potential that yields the exact time‐dependent density of the molecule.103 In practice, approximations are used. In the simplest case, the same approximate functionals commonly applied in stationary ground‐state DFT are employed, but evaluated with the time‐dependent density. For example, consider an instantaneous Xα potential . In this case, the XC potential functional itself is not explicitly time‐dependent, leading to the adiabatic approximation. In this approximation, the XC potential at a given time is independent of the state of the system at past times, i.e., memory effects are not accounted for. To illustrate the case, consider a molecule under the influence of a static electric field. If the field is gradually, slowly (adiabatically) switched off, oscillations in ρ will dampen very slowly and the adiabatic V_XC will gradually adapt to the correct form for the stationary ground state. If, on the other hand, the external field is suddenly switched off, an adiabatic V_XC has no memory of the prior presence of the field nor of its disappearance, and therefore it represents an unsuitable approximation. One can imagine that similar considerations apply to the density changes induced by a rapidly oscillating field.

The vast majority of applications of TD‐DFT make use of response theory instead of solving time‐dependent equations explicitly. Imagine a molecule in its ground state, and an external field to which the molecule's electron density “responds.” One is often interested in the response as a function of frequency of the applied field. For not too strong fields the most convenient strategy is to evaluate the response as a perturbation of the stationary ground state in powers (linear, quadratic, cubic, and so on) of the field amplitude in the frequency domain. Perturbation expansions of eqs. 35 and 36 are used to calculate the perturbations of the orbitals and the density. The relation to the polarizability and the optical rotation parameter has been made previously in eqs. 6a and 6b, i.e. the terms linear in the field strengths have to be evaluated to compute most of the important chiroptical properties, in particular, electronic CD and optical rotation. To summarize, for calculating the linear response of the MOs, usually a first–order expansion of eq. 35 is solved for a specified frequency of the field. The solutions are used to calculate the linear response of the density or density matrix, and in turn this density/density matrix is used to calculate the response tensors α, G′, β, and so on. The interested reader is referred to Refs.86,99, and104 in which details of the formalism are provided.

TD‐DFT methodology for computations of electronic CD spectra105, 106 and OR40-42, 65, 107, 108 has been reported starting around the year 2000. The first TD‐DFT computations of metal complex CD spectra started to appear in the literature in 2003.109, 110 TD‐DFT was previously found to reproduce the excitation spectra of a number of chiral Ru complexes quite well111, but no CD computations had been carried out. VCD methodology has been around for quite some time82, 84, 112-115 and older Hartree–Fock codes have been readily extended to incorporate density functional methods.116 As already mentioned, dynamic response methods are not needed for VCD. TD‐DFT based ROA programs have also been developed recently93, 117-119 with some of them having their origin in previous Hartree–Fock codes.115, 120-122

As already pointed out, the fact that α, G′, β, etc. become singular at the excitation energies can be used to obtain an equation that will lead to the excitation spectrum along with the transition dipole moments directly. This is the linear response route to calculating the excitation spectrum (UV/Vis, CD) from a perturbation of the ground state, without the need to calculate excited states wave functions or densities.***

Apart from self‐interaction and the asymptotic behavior of the potential, there is another important aspect of approximate TD‐DFT with popular functionals deserves attention in the context of modeling chiroptical properties (in particular for electronic CD in large molecules, in metal complexes, or in molecular complexes): Consider a charge‐transfer (CT) excitation where a substantial amount of electronic charge is transferred over a significant distance from a donor (D) to an acceptor (A) moiety. The CT excited state can be approximated by a pair of point charges at a distance R. The CT excitation energy (in au) is then roughly

(37)

Here, IP^D is the ionization potential of the donor and EA^A is the electron affinity of the acceptor moiety. In the limit R → ∞ the energy approaches E_CT ≈ IP^D − EA^A, i.e., the energy it takes to remove the electron from the donor minus the energy gained from attaching it to the acceptor.

Let the transition density be proportional to φ_iφ_a, where φ_i is an occupied orbital localized on the donor and φ_a an unoccupied orbital localized on the acceptor, with orbital energies ε, ε. In linear response TD‐DFT or time‐dependent Hartree–Fock (TDHF), apart from a 2[ai|r|ia] term for singlet transitions (small at large R), for the two‐orbital situation described here the excitation energy is approximately†††123

(38)

where f_XC is a frequency‐dependent exchange‐correlation (XC) kernel. When the adiabatic approximation and a local XC potential are adopted, f_XC(r₁,r₂,ω) = f_XC(r₁) · δ(r₂ − r₁). This applies to the majority of TD‐DFT computations currently performed as far as the pure DFT XC component of the response kernel is concerned.

At separations R where the orbitals of D and A do not significantly overlap the product φ_iφ_a is close to zero. With nonoverlapping orbitals φ_a,φ_i and a local kernel f_XC the [ai|f_XC|ia] term in eq. 38 for TD‐DFT vanishes exponentially and E_CT approaches (ε − ε). The asymptotic − 1/R term is not obtained. Moreover, in DFT based on approximate local XC potentials the orbital energy difference (ε −ε) is not an approximation for IP^D − EA^A. As a consequence substantial errors in excitation energies may occur for CT excitations. On the other hand, in TDHF the − 1/R term at large separations is obtained from the − [ii|r₁₂⁻¹|aa] Coulomb integral in Eq. (38) and per Koopmans' theorem (ε − ε) is an approximation for IP^D − EA^A. Thus, TDHF has the correct qualitative long range behavior. However, as already mentioned TDHF lacks correlation and is clearly not the method of choice for calculating excitation spectra.

Using hybrid functionals which contain some Hartree‐Fock exchange might alleviate the CT errors of DFT somewhat. If the charge separation is not complete, which is the case for many excitations formally labeled as CT, the errors from the functional may not appear, or only to a certain degree. Thus, it can be difficult to predict in which situation a computed excitation energy affords significant errors due to the CT problem and when not. It is clear, however, that spatial separation of the orbitals plays an important role. Neugebauer et al. proposed a test for CT errors based on the spatial separation of φ_a and φ_i along with a simple correction for the KS potential.124, 125 Recently, Tozer and coworkers proposed a similar criterion that might be able to warn users of TD‐DFT programs about possible CT issues.126

Functionals have been proposed that address both the CT issues as well as the asymptotic behavior of the KS potential leading to poor description of diffuse states. One possibility is to separate the XC potential into a short‐range part which is reasonably well‐described by DFT and takes care of short‐range correlation and a long‐range part that is reasonably well‐described by Hartree‐Fock exchange.127 We mention two such range‐separated functionals (long–range corrected functionals), one by Tawada et al.128 and the CAM‐B3LYP functional by Yanai et al.129, which were indeed shown to yield improved response properties when compared with standard GGA and hybrid functionals. A more recently developed range‐separated functional is BNL.130 Somewhat related, conceptually, but at present computationally more demanding are local hybrids in which the mixing ratio between DFT components and HF exchange is a function of space and is evaluated locally from the kinetic energy density.131

To summarize this section:

• Linear response TD‐DFT is a computationally efficient first‐principles quantum chemistry method that can be applied to quite large molecules. The performance for valence excitations is generally good and a large number of excitations can be obtained allowing for the simulation of spectra over a large frequency/wavelength range.

• TD‐DFT often refers to static or frequency‐dependent linear and higher order response calculations, instead of an explicit time‐dependent treatment, although the acronym TD‐DFT certainly does not exclude explicit time dependent methods.

• The poles of response functions on the real frequency axis indicate the excitation energies of an atom or molecule. Linear response TD‐DFT can be used to obtain these excitation energies and transition moments by computing the location and residues of the poles directly. See also the section on Chiroptical Response as a Function of Frequency or Wavelength.

• Many approximate functionals afford self‐interaction which may cause pronounced errors in computed response properties and excitation spectra.

• In TD‐DFT, using an adiabatic functional can be a severe approximation if the potential and the electron density vary rapidly in time. Also, adiabatic TD‐DFT does not describe double excitations. Problems of adiabatic TD‐DFT describing excited state surfaces in case of homolytic bond dissociation have also been noted.132 Nonadiabatic TD‐DFT codes are not widely available.

• Charge‐transfer excitation energies may afford significant errors with standard functionals. This can be a serious problem which requires careful checking of all TD‐DFT results (no “black box” use). Improved functionals and XC potentials have been developed that alleviate such problems.

• The nonhybrid part of standard KS potentials has the wrong asymptotic behavior which causes significant errors for excitation energies into diffuse valence states and Rydberg states. These errors can be alleviated, for instance by using shape corrected XC potentials, or by functionals that separate short‐ and long‐range exchange and treat the latter using a large fraction of Hartree–Fock exchange.

AN EXAMPLE: CHIROPTICAL PROPERTIES OF TRANS‐DIMETHYLOXIRANE

This section shall provide some examples for calculations that can be repeated in a relatively short time to gain some experience with first‐principles chiroptical computations. For practical reasons a small molecule, trans‐(2,3)‐dimethyloxirane (DMO), was chosen, with focus on DFT and TD‐DFT calculations. The geometry was optimized for the (−)‐(2S,3S) isomer at the B3LYP/6‐31G* level; the coordinates are listed explicitly in Input Example 1. For this molecule, optimizations with larger basis sets do not strongly alter the results shown here. Calculations have been performed with this enantiomer. Where experimental data were reported for the (+)‐(2R,3R) optical antipode, the sign of the calculated chiroptical response was changed accordingly. To prepare inputs and to postprocess the results the following numerical conversions between energy, frequency, and wavelength of an electromagnetic wave are useful:

(39)

Table . Input Example 1 Calculating the optical rotation of DMO at 589.3 nm (0.07732 au) using a density functional method (here: the PBE0 functional, PBE1PBE keyword) and the aug‐cc‐pVDZ basis. Input for the Gaussian program

#PBE1PBE/aug‐cc‐pVDZ Polar=OptRot CPHF=RdFreq
dmo optical rotation
0 1
C	0.0000000	0.7357250	−0.0925540
O	0.0000000	0.0000000	1.1435730
C	0.0000000	−0.7357250	−0.0925540
H	0.9521050	−1.2309100	−0.3020970
C	−1.2311140	−1.5551300	−0.3882150
H	−1.2469430	−1.8669100	−1.4396770
H	−1.2515180	−2.4588320	0.2320180
H	−2.1387810	−0.9798110	−0.1799270
H	−0.9521050	1.2309100	−0.3020970
C	1.2311140	1.5551300	−0.3882150
H	1.2515180	2.4588320	0.2320180
H	2.1387810	0.9798110	−0.1799270
H	1.2469430	1.8669100	−1.4396770
0.07732

Software packages that are capable of optical activity computations are available commercially and under noncommercial licenses. A list of selected software packages capable of DFT/TD‐DFT chiroptical response computations is (in no particular order): Gaussian133 (“G03” and the new 2009 release. Commercial. CD, OR without damping, VCD, ROA, GIAO and velocity gauge functionality), Dalton 2.x134 (non–commercial. CD, OR with and without damping, VCD, ROA, and others, GIAO and velocity gauge functionality); Turbomole135 (commercial. CD, OR without damping, ROA add–on available, velocity gauge functionality), Amsterdam Density Functional136 (“ADF,” commercial. CD, OR with and without damping, VCD, ROA, and others, GIAO and velocity gauge functionality); ORCA137 (noncommercial. CD spectra with TD‐DFT and multireference ab‐initio methods); Psi3138 (non–commercial. CD and OR, velocity gauge functionality, HF and post‐HF methods, TD‐DFT). Optical activity implementations in NWChem139 (noncommercial) are under way. Presently there is TD‐DFT functionality available in NWChem for OR with and without damping available66 (velocity gauge functionality), which was used for instance to create Figure 5 and part of the data used for Figure 8, and for CD spectra. For details please contact the author.

The first example is a single‐wavelength computation of the optical rotation. See Input Example 1. The isotropic OR parameter β for this PBE0/aug‐cc‐pVDZ GIAO level of theory is printed in the output as follows:

w = 0.077320 a.u., Optical Rotation Beta = −0.1458 au.

Using a molar mass of 72.1 g/mol, the code also provides the specific rotation in the next output line Molar Mass = 72.1066 g/mole, [Alpha] (5892.8 A) = −78.17 degree. which is the same result in degree/[dm(g/cm⁻³)] as obtained from the conversion from atomic units provided in eq. 8 with cm⁻¹. The result compares well to a B3LYP/aug‐cc‐pVDZ specific rotation of 78.6 reported in Ref.40 for the optical antipode (i.e., the sign is changed to obtain a PBE0 result for (+)‐(2R,3R)‐trans‐DMO of +78.2). Both these TD‐DFT values are too large in magnitude when compared with the experimental specific rotation of 58.8 but the deviation is within expected limits for this level of theory. For comparison, the OR was also calculated with the shape‐corrected SAOP functional and a diffuse Slater‐type basis set using the ADF input of Example 3 but replacing the freqrange input line by frequency 1 5893 Angstrom. The calculated length gauge OR parameter of −0.101 au translates into [α]_D = −54.1 degree/[dm (g/cm⁻³)]. With the modified velocity gauge the specific rotation was calculated to be −36.3 degree/[dm (g/cm⁻³)] instead, which for a basis set of this size can be considered an unusually large effect. This indicates, however, that high lying excited states that are not correctly described by this basis set may have a significant impact on the OR which cautions against premature assessments of which basis combination and XC potential is the “best” for a system like this. Note that for the UV‐Vis excitations in this small molecule charge‐transfer states are unlikely to be a major concern, but contributions from Rydberg states to the ORD and the CD are very important which benefit from the use of the asymptotically corrected XC potential.

Next, consider the electronic CD spectrum for DMO. See Input Example 2. Figure 7 displays the results in comparison with an experimental gas‐phase spectrum. The calculated rotatory strengths are plotted as the “stick” spectrum, along with a simulated broadened CD intensity. Each excitation was broadened with a normalized Gaussian function centered at the vertical excitation energy and subsequently scaled such that eq. 5 is fulfilled, with a width parameter of σ = 0.2 eV.‡‡‡ The simulated spectrum shown is the sum of these Gaussian functions. Overall, the qualitative behavior of the spectrum seems to be reproduced reasonably well, apart from shifts in some of the simulated CD bands and a too high intensity at the high energy cutoff of the experimental spectrum. For assigning the absolute configuration and for assigning the nature of the lowest excitation the quality of the spectrum appears to be good enough. However, from the data it is unclear if the opposing signs of the SAOP and the experimental spectrum between 7.5 and 8 eV are due to errors in the calculated vertical excitations. Likewise, there are uncertainties about the position of the higher energy PBE0 CD bands, even after blue shifting the spectrum by 0.2 eV to obtain an overall reasonable match. It is clear that the experimental spectrum has a well‐resolved vibrational structure which should be modeled as well. We will return to this spectrum in the section on Selected Case Studies and address the quality of the spectra shown here.

Figure 7

Vertical electronic CD spectra of (−)‐(2S,3S)‐trans‐DMO. Left: calculation with the SAOP potential and a diffuse Slater type basis set. Right: PBE0/aug‐cc‐pVDZ (calculated PBE0 spectrum blue shifted by 0.2 eV). Gaussian broadening with σ = 0.15 eV. Experimental gas‐phase spectral data from Carnell et al., Ref.140.

Table . Input Example 2 Calculation of the electronic CD with two different programs. Calculations with G03 using the PBE0 hybrid functional and the aug‐cc‐pVDZ Gaussian type basis. See Input Example 1. Calculation with ADF using the SAOP shape–corrected potential and an even–tempered diffuse STO basis

For the G03 program, change the first line of Example (1) to

#PBE1PBE/aug‐cc‐pVDZ TD=(NStates=15)

and remove the frequency input. The calculation will determine the lowest 15 excitations. With the ADF code, a corresponding input would read

Basis

Type ET/ET‐QZ3P‐1DIFFUSE

Core None

End

Atoms

< The coordinate section from

Example 1 can be used here. >

End

XC

model SAOP

End

Excitations

cdspectrum

allowed

lowest 15

End

Next, we shall investigate the OR dispersion (ORD) of this molecule. A sample input for the ADF code is provided in Input Example 3 where we use the shape corrected SAOP XC potential again to obtain a good description of the low lying Rydberg states of this molecule (see Section on TD‐DFT Linear Response: A Synopsis). The ORD is shown in Figure 8. For comparison, an ORD curve was also calculated with the PBE0 functional using the author's NWChem implementation.66 The overestimation of the D‐line OR with the PBE0 functional is seen to be a likely consequence of the lower lying excitations. Refer to Figure 6 for the qualitative behavior of the ORD expected in the vicinity of electronic transitions. This behavior is clearly reproduced in the calculations. The first transition in the CD calculated below 7 eV is causing the strong anomalous dispersion (Cotton effect) in the ORD, with the sign change of the ORD close to where the excitation is located. The ORD then starts to oscillate above 7 eV in correspondence to the denser CD spectrum which affords transitions with varying signs of their rotatory strengths, as seen in Figure 7. Recall that the Na D‐line rotation for this enantiomer is negative, which is also seen in the inset. In terms of the SOS Eq. 15a or the KK transformation 11a, the long‐wavelength OR of DMO must be dominated by the intense negative CD bands at higher energy. The equations indicate that on the one hand low‐lying excitations may dominate the low‐frequency/long wavelength OR because of the relatively smaller denominator in the SOS equation or the KK transformation. On the other hand, the magnitude of the rotatory strength in the numerator is just as important and it may outbalance a large denominator. This appears to be the case here for DMO. However, as the frequency approaches that of the first CD band, the sign of its rotatory strength begins to dominate and the OR changes sign from negative to positive as the frequency increases. The resulting bisignate ORD at low energy is a good indicator of such a CD pattern even if the transitions are too high in energy to be easily recorded in a spectrum. It is clear that if an ORD curve such as the one calculated for DMO can be matched with an experimental one (ideally including the anomalous dispersion around an electronic transition) this would lead to a rather confident assignment of the molecule's absolute configuration even if the calculated long‐wavelength OR differs substantially from experiment. See Refs.52 and53 for examples. In Ref.52, we discussed another small three‐ring molecule (dimethylcyclopropane, along with several larger molecules) and compared the bisignate ORD with experimental data. See also the Selected Case Studies.

Figure 8

Calculated optical rotatory dispersion for (−)‐(2S,3S)‐trans‐DMO, using the SAOP potential and a Slater type basis with diffuse functions (solid line). For comparison, results obtained with the PBE0 hybrid functional and the aug‐cc‐pVDZ basis are also shown (dashed line, NWChem program). A global damping constant of 0.2 eV was applied in both sets of calculations. Molar rotation [ϕ] in degree cm²/dmol.

Table . Input Example 3 ADF input for optical rotatory dispersion between 2 and 8 eV (approximately 620 to 155 nm). A damping constant of 0.00735 a.u. corresponding to 0.200 eV or about 1.61·10³ cm⁻¹ is applied. This keyword block replaces the Excitations … End block in Input Example (2)

AOResponse

freqrange 2.0 8.0 25 eV

lifetime 0.00735

opticalrotation

End

AllPoints

Next, consider a simulation of the VCD spectrum. Figure 9 shows calculations of the VCD spectrum of DMO in comparison with experiment. One calculation was performed with the aug‐cc‐pVDZ Gaussian‐type basis set and the B3LYP hybrid functional. A similar spectrum is also obtained for this system with the smaller 6–31G* basis. Another calculation was performed with a triple‐zeta polarized Slater‐type basis set and the nonhybrid BP86 functional. The BP86 calculation did not employ a diffuse basis but it should be noted that the Slater‐type basis extends further into the asymptotic region than an unaugmented Gaussian type basis. The calculated B3LYP frequencies were scaled by 0.971 as recommended by Sinha et al.142 although this appears to lead to a systematic slight redshift of the calculated spectrum. The BP86 frequencies were left unscaled. In both cases, the agreement between theory and experiment is obviously very good. Similarly good agreement can be found in many VCD studies. One of the reasons for this is that, as already pointed out, DFT yields good quality force fields and VCD does not require dynamic response. It also appears that the electronic contributions entering the VCD intensity (Section on Vibrational Optical Activity) are quite well‐described by standard functionals (hybrid DFT as well as nonhybrid functionals). The B3LYP calculation was performed with the G03 code by removing the frequency input and replacing the first line of the input in Input Example 1 with

• #B3LYP/cc‐pVDZ FREQ=VCD.

Figure 9

Computed (left: B3LYP/aug‐cc‐pVDZ; right: BP86/TZP) versus experimental VCD for (+)‐(2R,3R)‐trans‐DMO. Experimental VCD spectral data from Polavarapu and coworkers, Ref.141.

The Slater‐type basis calculation was performed with the VCD implementation in the developer's version of the ADF code143 using the keywords

• AnalyticalFreq

• End

• VCD

to trigger the computation of the VCD spectrum. The rotatory strengths are printed in the outputs (typically in units of 10⁻⁴⁴ esu² cm² for VCD) along with the frequencies and the IR intensities. Equation 5 was used with a Lorentzian broadening of 10 cm⁻¹ to generate the simulated spectrum shown in Figure 9.

A recurring problem in calculations involving nuclear vibrations can be hindered rotations from alkyl and other groups which show up in the calculations as large amplitude low‐frequency vibrations resembling a rotational motion. In particular, when optimizing the system with a basis set different from the one used for the response calculation, and unless very good quality numerical quadrature grids are used in the DFT/TD‐DFT calculations, these hindered rotations may be calculated with imaginary frequencies near zero (instead of real frequencies near zero). This was the case for the B3LYP calculation because the optimization was based on the 6–31G* basis. A visual inspection of the normal modes will easily identify such a situation and for VCD simulations such hindered rotations should normally not cause any particular problems. For further comments on hindered rotations see Section on Selected Case Studies. In general, it is advisable to use a consistent level of theory both for the optimization and for the spectra calculations because otherwise the optimized structure is unlikely to be a stationary point at the theory level used to compute the vibrational spectrum and the VCD intensities.§§§

For completeness, we also provide a simulation of a ROA spectrum in Figure 10. The simulation of the spectrum is based on ADF calculations at the BP86/TZP level using the code described in Ref.93. The corresponding input keywords for the ROA part were

• AOResponse

• newpolcode

• VROA

• frequency 1 0.0934 Hartree

• FitAOderiv

• EL_DIPOLE_EL_DIPOLE VELOCITY

• EL_DIPOLE_EL_QUADRUPOLE VELOCITY

• EL_DIPOLE_MAG_DIPOLE VELOCITY

• End

where the laser wavelength of 488 nm is specified as a frequency in atomic units, the required linear response quantities (see Section on Vibrational Optical Activity) are requested in modified velocity gauge, and for accurate derivatives the computation of density–fit related terms are specified (FitAOderiv), see Refs.78 and93. The main keyword for the vibrational Raman optical activity computation is ‘VROA’ (the keyword “newpolcode” will likely be obsolete in future releases of the program). A Lorentzian broadening of 20 cm⁻¹ has been applied to the computed “stick spectrum”. The agreement with experiment is similarly as good as the VCD spectrum in terms of reproducing the sign pattern and the major spectral features. The relative intensities are not in quite as good of an agreement. This may be rationalized by the fact that the dynamic response tensors needed for the ROA simulations (Section on Vibrational Optical Activity) can be rather sensitive to approximations in the electronic structure (basis or functional). Overall, the calculated and experimental spectrum agree quite well.

Figure 10

Simulated and experimental ROA spectrum for (−)‐(2S,3S)‐trans‐DMO. Simulation based on a laser wavelength of 488 nm and a depolarized 90° scattering. Calculations with BP86/TZP; Lorentzian broadening of 20 cm⁻¹. Experimental depolarized ROA spectral data from Polavarapu et al., Ref.121.

SELECTED CASE STUDIES

This section reviews a number of benchmark studies and applications, with an emphasis on TD‐DFT and CC methods. For additional benchmark data see also the section on How do the Calculations Usually Compare with Experiment? Most of the data discussed in that section will not be considered again here. Full coverage of the literature would also be beyond the scope of this “tutorial style” review because many of the available computational methods are quite mature at this stage and have been applied widely. Various specialized applications will be reviewed in which methodology has been used that is not yet in widespread application but has the potential to become part of the standard toolbox of chemists for computational studies in the optical activity field. Before providing specific examples it is appropriate to point to some theoretical work based on first‐principles methods preceding the “renaissance”10 in chiroptical methods, which has been reviewed in 2002.64 See, for example, theoretical work on electronic CD and OR carried out by Grimme and coworkers,140, 144, 145 by Polavarapu and coworkers,64, 146-148 by Bak et al.77, by Beratan, Wipf and coworkers,149-151 by Hansen et al.,152, 153 by Amos and Rice,63, 81, 154; early benchmark studies for optical rotation40, 65; and benchmark studies cited earlier in the section on How do the Calculations Usually Compare with Experiment? As already pointed out, first principles calculations of vibrational optical activity also date back to the 1980s and 1990s.54, 84, 112, 114-116, 121, 141, 115 For additional information not covered here, the reader is referred to some of the available review articles,64, 66, 104, 151, 156, 157 A list of references concerning semiempirical method developments and applications for CD spectra has recently been compiled as part of a review on CD calculations in organic chemistry (Ref.39).

Among the first TD‐DFT calculations of electronic CD that were reported in the literature were calculations of the CD spectra of [n]‐helicenes, which were obtained in a combined theoretical experimental study.105 These fascinating molecules do not possess asymmetric carbons but are chiral because of the overall helical conformation adopted by the fused benzene rings. With the BP86 nonhybrid functional and the SV(P) Gaussian type basis augmented with a set of diffuse functions at the ring centers, excellent agreement with experiment was obtained. The term “excellent” deserves some explanation: the main spectral features were correctly reproduced, and overall the broadened vertical CD spectra simulated from the calculations matched the experimental spectra quite well after a modest blue shift of 0.45 eV which can be attributed to a tendency of TD‐DFT to underestimate excitation energies.158 In Figure 11 is shown a simulation of the (M)‐hexahelicene spectrum. A similar spectrum was used to validate a TD‐DFT implementation106 of electronic CD in a code based on Slater‐type basis functions (ADF). Below 5.5 eV, the calculated spectrum in Figure 11 is almost identical to the one published in Ref.105 which is therefore not shown. The high‐energy cut‐off for the calculated transition is important for the sign and magnitude of the simulated CD band around 6 eV; the spectra published in Refs.105 and106 were based on a smaller number of excitations.

Figure 11

CD spectrum of (M)–hexahelicene. TD‐DFT calculations with BP86, a DZP Slater‐type basis, and a Gaussian broadening (σ = 0.15 eV), using the code described in Ref.106. Calculated excitation energies blue shifted by 0.45 eV. Experimental data for the CD spectrum taken from Ref.105.

Optical rotations have been computed for the helicenes as well.41, 42 Since the calculated transitions in the UV‐Vis range from TD‐DFT are too low in energy compared with experiment, one may expect an overestimation of the OR magnitude which is indeed what was found in the computations. As an example, for (M)‐hexahelicene the best BP86 value from Ref.42 was −5259 degree/[dm (g/cm⁻³)] compared with an experimental specific rotation of −3640. The B3LYP result by Grimme of −4887 agreed somewhat better with experiment although still too large in magnitude.41 For heptahelicene we reported in Ref.42 a specific rotation of −8293 degree/[dm (g/cm⁻³)] compared with −6200 for the experimental OR. The overestimation of the OR magnitude is systematic for the [n]‐helicene series with n = 5–7 and can be attributed mainly to the red shift in the excitation spectra.

Benchmark studies of electronic CD calculated with TD‐DFT methods soon followed where a more diverse test set of molecules was included. In Ref.106 in addition to penta‐ and hexahelicene, a set of three‐ring organic molecules (including DMO and a few aziridines) were investigated, along with substituted cyclohexanones for which it was shown that qualitative predictions from the octant rule as well as experimental rotatory strengths can be reproduced by the calculations. Diedrich and Grimme benchmarked TD‐DFT based CD for a set of nine organic molecules and one organometallic system (Fe) in comparison with TDHF, CC2, and multireference CI and MP2 data.110 The authors concluded that the quality of the TD‐DFT CD spectra was quite strongly dependent on the functional and that none of the methods performed reliably for all molecules in the set. Despite much progress on the theoretical front it is fair to state that this general assessment still holds 6 years later although newer functionals are able to significantly reduce the risk of catastrophic results (see the section on TD‐DFT Linear Response). For instance, perturbatively corrected excitation energies using recently proposed double‐hybrid functionals yielded better CD band positions than standard hybrid functionals and also afforded fewer “ghost” states (i.e., spurious excited states).159 Returning to the benchmark data of Ref.110, on the positive side most of the simulated spectra were deemed suitable to assign the absolute configuration with high confidence.

Indeed, an important application of electronic CD calculations is the assignment of absolute configurations of organic molecules., i.e., if a convincing match of a simulated spectrum for one enantiomer (or one particular isomer in case of several stereo centers that need to be assigned) with experiment can be obtained, this can be the basis of an assignment of the absolute configuration. Applications of this type involving first‐principles calculations of vertical CD spectra utilizing one or several optimized structures from a conformational search are now commonplace and have been reviewed a number of times.10, 39, 64, 151 Sometimes a particular focus is on conformational analysis which can be particularly well‐quantified via orientation‐dependent exciton CD between strongly absorbing chromophores; see the review by Berova et al.160 As an example for a combined experimental–theoretical study aimed at assigning absolute configurations and conformational behavior with the help of computed CD, Mori et al. recently investigated axially chiral 2,2′‐, 3,3′‐, and 4,4′‐biphenol ethers.161 CD spectra calculations for different rotamers exhibited opposite sign patterns, resulting in low‐intensity but characteristic CD spectra for the ensemble in solution. The simulated spectra reproduced the experimental solution spectra quite well (apart from the absolute intensities). The work highlighted the need for high quality calculated Boltzmann factors to obtain the conformer distribution correctly, which is obviously critical for a successful modeling of the averaged spectra. A follow‐up publication has focused on the CD spectra of chiral cyclophanes (mostly donor‐acceptor systems) using a similar protocol.162 VCD spectra were also modeled and compared with experiment. Regarding the geometries, the need for considering dispersive interactions for these π‐stacked systems was emphasized. At the DFT level with simple functionals they are missing.¶¶¶ For optimizations at the DFT level it is possible to augment the energy expression with suitably scaled semiempirical dispersion terms.163 Alternative to taking Boltzmann averages of the spectra, as discussed earlier, such computations can also be performed using a number of MD/MC configurations. As an example, a combination of MD with TD‐DFT calculations was applied to simulate CD spectra of bicyclic β‐lactam antibiotics.164

CD computations have provided additional support for the assignment of the structure of the C₈₄ fullerene.165, 166 The structure was determined to be D₂ symmetric, which is one of the chiral point groups. The agreement between a simulated BP86/SVP CD spectrum and experiment was very good [similar to what is shown in Figure 11 for (M)‐hexahelicene], except for the fact that the computed intensities had to be scaled by a factor of 1/14 to obtain reasonable agreement with experiment. It was correctly noted that such deviations between simulation and experiment are untypical. Factors of 2–4 for intensity deviations are more typical. It is likely that the discrepancy is due to uncertainties about the concentrations in the experiment, as very small quantities were used [Crassous, Private Communication].

In the section on Chiroptical Properties of trans‐dimethyloxirane, it was pointed out that the agreement between the simulated vertical CD spectrum and the vibrationally resolved experimental spectrum for DMO was qualitative at best. A number of research groups have developed efficient computational methods to obtain Franck–Condon (FC) factors for excitation spectra, in some cases along with additional Herzberg–Teller type vibronic coupling terms and Duschinsky rotations, to arrive at more realistic simulations of absorption spectra. For a selection of recently reported methods and applications, see, e.g., Refs.29,30,167-169. Some time ago we have used such an approach to CD spectra and revisited the case of DMO.30 Using calculated FC factors, the simulation of the spectrum was significantly improved, as seen in Figure 12. Note, in particular, the broadening by over 1 eV of excitation 2B. In the vertical spectrum, this excitation is fairly intense and leads to a negative CD in the range between 7.5 and 8 eV whereas in the vibrationally broadened spectrum it is overpowered by nearby transitions with positive rotatory strength. As a result the spectrum now agrees much better with experiment between 7.5 and 8 eV. The vibrational fine structure of the first CD band is also very well‐reproduced by the calculation. The reason for the low CD intensity in the experimental spectrum above 8 eV is still somewhat unclear; for a follow up investigation see Ref.167. More recently, Lin et al. have applied the methodology reported in Ref.168 to calculate the vibronic fine structure in the CD spectrum of (R)‐(+)‐3‐methylcyclopentanone,31 also leading to a convincing agreement between simulation and experiment. The two approaches are somewhat complementary in the sense that the one used for DMO constructs approximate harmonic excited states potential surfaces from the ground state force field and excited states gradients calculated at the ground state equilibrium geometry. The other approach requires optimized geometries (and ideally also force fields) for the ground state and each electronic excited state. As usual, with each method there is a trade‐off between accuracy and computational effort and both have their drawbacks and merits.

Figure 12

Computation of the CD spectrum of (−)‐(2S,3S)‐trans‐dimethyloxirane using the SAOP potential and a diffuse Slater‐type basis set. Vibrational fine structure of the CD bands calculated from Franck–Condon factors. Ref.30. Left: overall simulation; some excitations were slightly shifted for best agreement with experiment. Right: State‐resolved spectrum. Note in particular the broadening of 2B.

VCD spectra, as already pointed out, are somewhat more “benign” molecular properties and appear to be quite well, and very efficiently, modeled at most DFT levels. For a selection of DFT benchmark studies and applications to assign absolute configurations of organic molecules we refer to the following citations:54, 114, 116, 170-174. The MD route for averaging properties has also been used extensively in simulations of VCD and ROA. We cite two recent examples: Kaminsky et al used force‐field based MD simulations along with density functional computations of Raman and ROA spectra for the D‐gluconic acid anion.175 The resulting averaged simulated spectra reproduced most of the features seen in the experimental spectra quite well. Bour, Keiderling and coworkers recently (and in a number of related studies) applied a similar force‐field MD + DFT protocol to model IR and VCD spectra for a cyclic hexapeptide.176 It was pointed out that the MD simulations had to be performed over a quite long time at high temperatures for conformations other than the one obtained via simulated annealing to appear. Overall, the simulated spectra agreed modestly well with experiment.

Small molecules such as DMO which we investigated in Section on Chiroptical Properties of Trans‐Dimethyloxirane are often the target of benchmark studies to push the limits of the computational model.**** Methyloxirane has also been a particularly popular target. It has even fewer atoms than DMO, and apart from the hindered methyl rotation it is rigid and has a sizable specific rotation on the order of −2×10¹ degree/[dm (g/cm⁻³)] for the S‐enantiomer in solution at 589.3 nm. The ORD is bisignate. At the shorter wavelength of 355 nm, the OR was determined experimentally in gas phase as +7.49±0.30.177 An extrapolated solution phase OR in cyclohexane was estimated to be −26.4 degree/[dm(g/cm⁻³)] at 355 nm. TD‐DFT calculations with the B3LYP functional yield a positive optical rotation (e.g., +27.5 degree/[dm (g/cm⁻³)] in Ref.27), the sign being in agreement with the gas‐phase measurement but it has been argued repeatedly that the DFT calculations benefit from error cancellation. For instance, when vibrational corrections are included the OR calculated at the B3LYP/aug‐cc–pVTZ level rises to +75.6.27 Basis set effects are very pronounced: The B3LYP/aug‐cc–pVDZ level yielded a vibrationally averaged OR of only 30.1 deg/[dm (g/cm⁻³)] compared with +75.6 with the triple–ζ basis. Ruud and Zanasi obtained a more reliable theoretical estimate by combining an approximate coupled cluster triples (CC3) equilibrium OR of −23.2 degree/[dm (g/cm⁻³)] with the B3LYP/aug‐cc–pVTZ vibrational corrections to arrive at 24.9 degree/[dm (g/cm⁻³)] which happens so be close to the B3LYP/aug‐cc–pVTZ equilibrium value of +27.5. It is consensus that at a highly correlated CC level the equilibrium OR of (S)‐methyloxirane in gas phase is negative at 355 nm and that the B3LYP equilibrium value is positive because of an underestimation of the lowest excitation energies.26, 178 At 633 nm, CC calculations including vibrational corrections yielded −10.8 degree/[dm (g/cm⁻³)] compared with a gas phase experimental value of −8.38 ± 0.2026, 177 whereas SAOP DFT calculations yielded a positive vibrationally averaged OR (the equilibrium value was negative). Recall from the section on Chiroptical Response as a Function of Frequency and from the DMO example that the bisignate ORD when going from long to shorter wavelengths is most likely caused by one or more low‐lying transitions with positive rotatory strength which, if underestimated in energy, will dominate the OR everywhere in the transparent region and prevent the bisignate ORD from occurring.

CC calculations have also been applied to larger systems and have shown good promise. As an example, the ORD of (P)‐(+)‐[4]triangulane in the transparent frequency range was computed with CC and TD‐DFT methods.37 Although both methods yielded equilibrium ORs that agreed well with experiment, those obtained with the CC method were convincingly close enough to experiment to term the agreement quantitative (assuming negligible vibrational corrections). The authors of Ref.37 noted, however, that the calculations required a large amount of computer time. Rudolph and Autschbach have recently considered the triangulane system along with other organic molecules to demonstrate that well‐resolved ORD curves can be obtained with very few frequency points if additionally a few excitations and their rotatory strengths can be computed.67 In the cases studied, this led to a reduction by about an order of magnitude in the required computational effort.

Mort and Autschbach considered zero‐point vibrational corrections on ORs at 589.3 nm for a test set of 22 rigid organic molecules at the B3LYP/aug‐cc–pVDZ level.22 The median correction was on the order of 20% of the OR calculated at the equilibrium geometry but, as mentioned in the section on How do the Calculations Usually Compare with Experiment? the agreement with experiment was not improved on average. The magnitude of vibrational corrections confirmed expectations built on computations on methyloxirane and trans‐2,3‐dimethylthiirane performed by Ruud et al. which indicated that zero‐point vibrational corrections (ZPVCs) might be highly significant, possibly ranging on the order of 20–40% of the computed equilibrium value.179 Equation 2 was subsequently used to model the OR as a function of temperature in the six rigid bicyclic organic molecules α‐pinene, β‐pinene, cis–pinane, camphene, camphor, and fenchone.24, 25 T‐dependent ORs had previously been measured at three different wavelengths by Wiberg et al.180 The trends with temperature were attributed to the intrinsic temperature–dependence of the vibrational corrections to the OR. For four of these molecules, the computations supported this conclusion whereas for two of the systems, β‐pinene and cis‐pinane, the computations were inconclusive. Solvent effects might play a larger than expected role for cis‐pinane. Also, intrinsic errors of the TD‐DFT treatment (mainly for β‐pinene) might be too large to allow for a reliable comparison between theory and experiment for these two systems. For a successful case where T–dependent vibrational corrections appear to be the cause for the experimentally observed trends see the fenchone molecule, in Figure 13.

Figure 13

Temperature dependence of the specific rotation of fenchone. Experimental data from Wiberg et al.180 Computational data from Ref.24. The slopes are given in degree/[dm (g/cm⁻³)] per K.

Treating hindered rotations in vibrational averages (and in other cases where nuclear vibrations are considered) can be a challenge. Hindered rotations might induce large changes in calculated ORs and therefore it is desirable to treat them as rotations, not as large amplitude vibrations, especially at higher temperatures. However, for methyloxirane a computational study181 has shown that within a temperature range at which the molecule is likely to be stable the treatment of the hindered rotation as a vibration is a reasonable approximation. Counter examples are known.181 See Figure 14 for an illustration.†††† The computations were performed for (R)‐methyloxirane. It is clear that most of the temperature dependence of the OR is due to the hindered methyl rotation, caused by the very large oscillation of the OR as a function of the rotation angle.

Figure 14

Averaging the OR of (R)–methyloxirane (589.3 nm) over the hindered rotation of the methyl group and vibrational modes. B3LYP/aug‐cc‐pVDZ computations, Ref.181. (a) Relaxed hindered rotor potential. (b) Eigenvalue spectrum showing the transition from oscillator‐to free rotor‐like behavior. (c) Specific rotation as a function of the rotation angle. (d) Comparison of vibrational averaging (VA) to the vibrational average excluding the hindered rotation (VA*), and to VA* with the rotational average (RA) added.

Regarding the pronounced differences between solution and gas phase ORs for methyloxirane, a MD averaging based TD‐DFT study by Mukhopadhyay et al.49 was already mentioned earlier. For the polar solvent water, the simulations based on BP86/aug‐cc‐pVDZ TD‐DFT calculations to obtain the ORs were successful to reproduce the OR at wavelengths longer than 400 nm. In aqueous solution, the bisignate ORD is suppressed; between 355 and 650 nm the OR remains positive for (S)‐methyloxirane. Given the difficulties with reproducing, the gas phase ORs for methyloxirane, as discussed in previous paragraphs, there are several possible reasons for the good agreement of the computational data with experiment, among those: (i) the calculations in Ref.49 benefited from a particularly fortuitous error cancellation, or (ii) in the polar solvent the priority of difficulties with modeling the OR changes from the need to describe diffuse excited states accurately to describing the solvent solute interactions very well, or (iii) more likely, a combination thereof.

A subsequent MC based study, using otherwise similar techniques, took aim at the possibility of creating a chiral imprint in a nonchiral solvent, and possible contributions to the OR from this chiral solvent pocket.151 See also a highlight article, Ref.182. If the solvent has low‐lying excitations a chiral configuration of several solvent molecules can potentially cause a sizable OR. Reference151 considered methyloxirane dissolved in benzene. It was previously found that a continuum solvent model, otherwise very popular and generally quite successful, does not reproduce solvent effects on the OR of methyloxirane.35 In Ref.151, an average over a large number of MC configurations yielded reasonable agreement with the monosignate experimental ORD between 350 and 600 nm when the explicit solvent was used, whereas a bisignate ORD was obtained with an implicit solvent model. Based on comparisons with computations on the solvent configurations with the solute removed, the authors of Ref.151 argued that for the methyloxirane–benzene system most of the OR actually results from the solvent imprint. It remains to be seen if this conclusion will gain additional support with the help of new experimental data. In the context of solvent effects, or more generally, condensed phase effects (e.g., when an optical rotation is to be computed for a liquid) we also cite a study of 2,3‐hexadiene and 2,3‐pentadiene by Wiberg et al. who used MC simulations in an attempt to explain large differences between the gas‐phase ORs and the ORs for the neat liquids.183

In keeping with the topic of small chiral molecules, the textbook example for an asymmetric carbon center, the molecule CHFClBr, has been studied with ROA to reaffirm the assignment of the S–enantiomer as the one having a positive OR at 589.3 nm.118 ORD calculations were also performed. The DFT and HF calculations confirmed previous assignments of the structure.184 The ROA computations employed numerical differentiation of B3LYP based dynamic linear response tensors (see Section on Vibrational Optical Activity) using polarized diffuse basis sets. Matching signs of the calculated and experimental ROA intensities were obtained for between six to eight of the nine vibrational modes depending on the method and basis set used. At 589.3 nm, the OR of this molecule is only 1.8 degree/[dm(g/cm⁻³)] and therefore it would be difficult to assign its absolute configuration based on a single–wavelength OR calculation even though sign and magnitude of the calculations tend to agree with experiment for this molecule. The B3LYP/aug‐cc‐pVDZ ORD calculations of Ref.118 also reproduced the magnitude of the dispersion quite well lending additional support to the configurational assignment. Previously calculated ORs at 589.3 nm ranged between +1.3 and +3.7 degree/[dm (g/cm⁻³)]41, 106). When vibrational corrections are included the averaged B3LYP/aug‐cc‐pVDZ OR at 589.3 nm is 1.5 degree/[dm(g/cm⁻³)].22

It is well known that enormous intensity enhancements for Raman spectra can be obtained under resonance conditions or when the electromagnetic field is amplified in the vicinity of a strongly curved metal surface. Resonance enhancement mechanisms have also been proposed for ROA (resonance ROA = RROA) based on theoretical considerations.185 Experimental evidence for RROA has been obtained in a study of (+)‐naproxene ((S)‐6‐methoxy‐α‐naphthalenacetic acid)186 although the experimental data were collected not in full resonance but at 514.5 nm in the tail of an intense 325 nm absorption. When considering an excited state that is energetically well‐separated from other excited states, a single state approximation185 predicts that under resonance the RROA spectrum should become monosignate, with the sign being determined by the negative rotatory strength of the excited state with the frequency at which the laser is tuned. For naproxene, this was indeed the observed behavior. Jensen et al. have recently developed a RROA TD‐DFT program.93 As the only available experimental data were not for a clear–cut resonance situation a prediction of a RROA spectrum using BP86/TZ2P was made instead for (S)‐methyloxirane which the reader will by now recognize as among the theoreticians' most favorite benchmark systems for chiroptical response.‡‡‡‡ The results are displayed in Fig. 15. The spectra under full resonance condition are indeed monosignate, with the expected sign, which confirms that the single state approximation provides a suitable description of the dominant factors leading to the monosignate RROA. The profile shows that the situation may be more complicated between resonances. Also, from the comparison of the S1 and S2 RROA spectra it can be seen that the enhancements are dependent on which electronic and vibrational state one focuses on. Therefore, the technique appears promising if one were able to selectively tune into different resonances in which case the loss of sign information on the ROA bands is compensated by a gain in information from the different selective enhancements.

Figure 15

Resonance ROA spectra for (S)‐methyloxirane calculated at the BP86/TZP level of theory. Left‐ and Center: simulated RROA spectra for the S1 and S2 excited state. Right: RROA profile along with relative enhancement factors for a range of wavelengths. Lorentzian broadening of 20 cm⁻¹. See Ref.93 for details.

Comparatively few first‐principles theory studies of optical rotation have so far dealt with conformationally flexible molecules. For examples, see Refs.20,44,187,188 and189. Amino acids represent a challenging case due to the fact that different rotamers may have large ORs of opposite sign, and due to the influence of solvent solute interactions on the conformer mole fractions and on their optical rotation.47, 48 For natural amino acids, the zwitterionic form adopted in aqueous solution at neutral pH is not even stable in gas phase which makes the application of a solvent model mandatory. Kundrat and Autschbach found that a continuum model along with hybrid TD‐DFT performed reasonably well in reproducing the OR and ORD of individual conformers of amino acids in their protonated, zwitterionic, and deprotonated forms.47, 48 However, for amino acids and other conformationally flexible molecules with the ability for internal hydrogen bonding in some of the low‐energy conformers, there is a competition between inter‐ and intra‐molecular hydrogen bonding which complicates the correct determination of the mole fraction and requires, in principle, a high‐level explicit solvent model. Other TD‐DFT work on amino acids has been carried out by Pecul et al.190, 191 where the issue of conformational flexibility has also been emphasized. More recently, Kundrat et al. have adopted a MD‐based modeling of amino acid optical activity.50, 51 In the MD, the treatment of the solvent is of paramount importance. When the MD averaging of optical rotation was performed at the TD‐DFT level it turned out that using a point‐charge water model had some merits for two reasons: First, the computational cost of a point charge model is negligible, and second spurious charge‐transfer excitations, likely the source of errors in explicit water‐based models with TD‐DFT, are avoided.51 The model was benchmarked against CC2 calculations with explicit and point charge solvent. A conformational averaging is also straightforwardly obtained in the MD and offers interesting comparisons with static Boltzmann–factor based averaging.50

The effects of conformational flexibility on the OR of the benchmark system (R)‐epichlorohydrin was studied by Polavarapu et al.192 using TD‐DFT. The two lowest‐energy conformers have almost equal magnitudes of the optical rotation (on the order of 2 × 10² degree/[dm (g/cm⁻³)] for [α]_D) but opposite sign. Depending on the relative stability in different solvents a very small net OR and a quite strong solvent dependence is obtained. Tam and Crawford have re‐investigated the epichlorohydrin case with coupled‐cluster (CC) and TD‐DFT methods193 and found that CC offers some improvement which was attributed to an underestimation of the excitation energies in the TD‐DFT calculations. Both methods were able to reproduce the solvent‐induced trends in the OR as long as experimentally derived conformer populations were used.

An interesting case that is somewhat related to the “flexibility” issue was studied by Goldsmith et al. in Ref.194: The equilibrium constant for the formation of dimers of pantolactone in CCl₄ solution is known experimentally. Goldsmith et al. used the equilibrium constant along with TD‐DFT computations of the OR for the monomer and the dimer to predict the concentration dependence of the optical rotation of the solution. The computed results followed the concentration dependent measurements showing an increasingly negative OR at larger concentrations. The magnitude of the calculated ORs was only about half that of the measured rotations. The calculations indicated large negative contributions from the hydrogen bonds in the dimer which helps to rationalize the measured trend.

The pH dependence of the OR of natural amino acids in solution has been subject of a recent computational study.74 It has been known for a century that the OR of amino acids in their natural absolute configuration becomes more positive when going from neutral pH to a strongly acidic medium [Clough–Lutz–Jirgensons (CLJ) rule]. We have proposed that such an effect could be exploited to predict absolute configurations of flexible molecules more reliably. i.e., instead of relying on a match of absolute experimental and calculated ORs for a molecule of interest one would model the change of the OR between closely related species and also compare the trends with experimental data (if available, of course). For the amino acids, although the sign of the OR is difficult to reproduce (in part due to the small magnitudes) the correct sign and order of magnitude of the CLJ effect was reproduced for all amino acids considered. The origin of the CLJ effect was explained as follows: (i) The lowest energy transitions cause an exaggerated CLJ effect when used in the KK transformations or a SOS formula. Thus, for this particular case, the explanation can be reduced, approximately, to explaining the trends in the CD. (ii) The trends in the CD spectra can be rationalized with sector maps considering relatively simple electrostatic perturbations of the carboxyl and carboxylate chromophores. For the study of Ref.74, the sector maps were in fact computed from first principles. (iii) The change of overall charge on protonation at the carboxylate group seems to induce a change from octant to antioctant behavior. These studies are on‐going.

The bulk of applications of computational methods for chiroptical properties has been in the general area of organic and natural product chemistry. However, chirality is ubiquitous also in inorganic chemistry, in materials chemistry, and many other areas. Let us first take a look at metal complexes. As far as first‐principles chiroptical property computations on metal complexes are concerned, VCD studies have been reported first. As pointed out earlier, among the chiroptical properties considered in this article, VCD is the only one that does not require the evaluation of dynamic linear response. Among the first examples is a study by He et al.195 who employed DFT along with the magnetic field perturbation formalism of VCD. The VCD of the complex Zn(sp)₂Cl₂, with sp = 6R,7S,9S,11S–(−)‐Sparteine, has been investigated experimentally and by DFT computations (B3LYP). The theoretical results were in good agreement with the experimental spectra. The same authors have also subsequently studied the VCD spectrum of Λ‐Co(en)₃³⁺ (en = ethylenediamine), with similar success.196 The role of the anion for stabilizing the various conformers has been of particular interest. The computations predicted the ob₃ conformer to be most stable for the isolated cation, whereas previous theoretical estimates197 and experimental data suggest a predominance of lel conformations.

In a recent study, Nicu et al. used VCD computations to show that in the [Co(en)₃]³⁺ complex the VCD intensities of axial N–H stretching modes are strongly enhanced when the complex is closely cooordinated by chloride anions.143 The proposed mechanism involves charge–transfer from chloride to N–H. In the IR spectrum this mechanism would enhance A₂ symmetric modes, while in the VCD the enhancement is rather selective for E modes for which the electric and magnetic transition dipoles lie in the plane perpendicular to the NH bond directions.

Because CD bands can have positive and negative signs, electronic CD spectra of chiral metal complexes often reveal more detail about the excitation spectrum than the typically quite broad UV–Vis absorption spectra. Furthermore, just as in organic chemistry computed CD spectra and ORs can be instrumental in assigning absolute configurations of chiral metal complexes via direct comparison of computations with experiment. TD‐DFT computations of the CD spectra of several tris–bidentate Co^III complexes (including [Co(en)₃]³⁺) as well as Λ–[Rh(en)₃]³⁺ were reported by Autschbach et al.109 The series [M(phen)₃]²⁺ with M = Fe, Ru, Os was studied in Ref.198. Two representative simulated spectra from Refs.109,198 are shown in Figure 16 in direct comparison with experiment. The authors of Ref.198 also investigated the bipy₃ series (unpublished) during the course of the same study and found that the CD spectra were of similar quality as those for the phen₃ systems. Figure 16 shows quite satisfactory agreement between theory and experiment regarding the overall spectral features. However, for the Λ–[Co(en)₃]³⁺ spectrum, the d‐to‐d excitation energies were found to be strongly overestimated and the LMCT excitations were somewhat underestimated in energy. To allow for an easier visual comparison with experiment, the d‐to‐d ligand–field transitions (CD bands α,β,γ) were red‐shifted by 0.74 eV in Figure 16. The whole calculated spectrum for Λ–[Os(phen)₃]²⁺ was blue‐shifted by 0.20 eV. The errors in the calculated ligand‐field transition energies for the Co complex can be attributed in part to self‐interaction errors in the XC response kernel.109 Furthermore, deficiencies in the XC potential cause an overestimation of the covalent metal‐d‐ligand interaction which causes too large HOMO–LUMO gaps, particularly for 3d metals. For [Rh(en)₃]³⁺ as well as a range of other Rh complexes the ligand‐field transition energies were rather well‐reproduced by TD‐DFT.109, 199, 200 Likewise, problems with excitations involving metal d‐orbitals were noted for the Λ–[Fe(phen)₃]²⁺ complex but not for the Ru and Os analogs.198 Overall, the agreement with experiment can be considered satisfactory for the Co complex (and similar for other chiral complexes199, 200 and excellent for the Os complex. For the +3 charged Co complex inclusion of solvent effects is important whereas they do not have much of an impact on Λ−[Os(phen)₃]²⁺ and its Fe, Ru analogs. In any case, use of a continuum model is recommended for charged complexes in polar solvents. In passing we note a CD/solvation study of the [Co(en)₃]³⁺ complex with a discrete solvent model and molecular dynamics.201 Unfortunately, the improved solvent treatment did not lead to much better agreement with experiment than what is shown in Figure 16, leaving the XC potential and kernel most likely as the main source of error for this system.

Figure 16

CD spectra of Λ–[Co(en)₃]³⁺ (ob₃ –conformer, nonrelativistic computations) and Λ–[Os(phen)₃]²⁺ (scalar relativistic computations), from Refs.109 and199. Simulated CD spectra (TD‐DFT) using an empirical Gaussian broadening. Left: Gaussian line width parameter σ proportional to .109 Right: σ = 0.13 eV. Calculated vertical excitation energies and rotatory strengths indicated as line spectrum. Computations used a continuum solvation model (water). For easier comparison with experiment some of the excitation energies were shifted, see text for details.

The strong ligand exciton coupling CD in phen₃ and bpy₃ complexes appears to be generally well reproduced by TD‐DFT computations. For the Λ−[Os(phen)₃]²⁺ complex of Figure 16 this CD mechanism is responsible for the intense pair of CD bands around 35 to 40 × 10³cm⁻¹. Telfer et al. have reported interesting cases of exciton CD where the coupling occurs not only at the same metal center, but also between ligands bound to different metal centers in multicenter complexes.202 Computations were performed with a semiempirical method (ZINDO) which supported the internuclear exciton CD assignment of the spectra.

The majority of CD computations have been performed on closed shell systems—for good reasons since the single reference nature of DFT and coupled cluster methods is difficult to reconcile with a treatment of open shell systems of true multireference character. Occasionally, TD‐DFT computations of the CD of open‐shell metal complexes have been reported. See, for instance, computational studies of [Co^II (bipy)₃]²⁺203 and spin‐triplet ground state bis(biuretato) Co^III complexes.204 A t high spin case is reasonably straightforward to tackle. Recently, Fan et al. considered a set of trigonal dihedral Cr(III)L₃ complexes with bidentate ligands L = en, acac, ox, mal, and thiox and compared their calculated CD spectra with those of the corresponding closed shell Co(III) systems205, using spin‐unrestricted TD‐DFT. For the Cr(III) complexes the agreement with experiment turned out to be similar to the Co complexes (see Fig. 16 for an example), which is satisfactory to a degree that would allow to assign the absolute configuration with high confidence and to assign the bands in the spectra.

The origin of the circular dichroism in the ligand field transitions and in the ligand–to–metal CT excitations in trigonal dihedral complexes has been analyzed in great detail,109, 199, 200, 206, 207 by linking the orbitals of the chiral complexes to linear combinations of the orbitals in an achiral O_h parent symmetry and considering small perturbations. For instance, Jorge et al.199 found that the splitting and the energetic ordering of the ligand field transitions, formally d‐to‐d HOMO–LUMO, is determined by the sign and magnitude of the elongation or compression of the six ligand atoms along the three–fold symmetry axis which lowers the symmetry from O_h to D_3d and splits the degeneracy of the HOMOs. The sign and magnitude of an angular distortion, further lowering the symmetry to D₃, determines the sign and magnitude of the rotatory strengths R(A₂) and R(E) for these transitions. The angular distortion mixes even (if classified by O_h –symmetry) metal orbitals with odd combinations of orbitals of the directly coordinating ligand atoms which has to vanish in the O_h parent symmetry. As an example for the relevance of older model theories, the contributions from combinations of even with odd metal orbitals (such as n d with (n + 1) p) in the TD‐DFT computations provided less than 10% of the optical activity whereas this would be the only possible contribution based on crystal field theory.

As already pointed out, an important application of CD spectra computations is to determine the absolute configuration of a molecule. As a recent example from organometallic chemistry, Coughlin et al.208 applied TD‐DFT to compute UV‐Vis and CD spectra of Fe, Ru, and Zn complexed with the enantiopure hemicage ligand (−)‐(5R,5′R,5″R,7R, 7″R,7″R,8S,8′S,8″S)‐8,8′,8″‐[(2,4,6‐trimethyl‐1,3,5‐benzenetriyl) tris(methylene)] tris[5,6,7,8‐tetrahydro‐6,6‐dimethyl‐3‐(2‐pyridinyl)‐5,7‐methano isoquinoline]. The computed excitation energies and intensities agreed well with experimental data. The authors found that the computed spectra were accurate enough to assign the configuration around the metal center with high confidence. Another interesting study as been reported in Ref.209. The absolute configuration of extended metal atom chains of the type Ni₃ [(C₅H₅N)₂N]₄Cl₂ was established with the help of VCD, electronic CD, OR dispersion, and accompanying density functional computations.

Recently, enantiopure complexes with transition metals and derivatized helicenes have been prepared.210 The combined experimental–theoretical study showed that for the CD spectrum of a Pd‐bis(phosphole‐aza‐hexahelicene) complex hybrid TD‐DFT computations were able to yield good agreement between theory and experiment. The computations helped to assign the configuration around the metal center and in the phosphole groups, and allowed an assignment of the electronic nature of the various CD bands (see Fig. 17). The CD bands in the UV‐Vis range appear free helicene–like, with some exciton coupling and additional contributions from the phosphole group. However, the long wavelength tail of the spectrum involves charge transfer between the metal center and the ligand and is likely responsible for the large molar rotation of 23,100 degree cm²/dmol. A related Cu complex afforded a much smaller OR along with a less intense CD. The computational investigations are ongoing.

Figure 17

CD spectrum of a Pd bis(phosphole aza‐hexahelicene) complex. Computation (BHLYP/SV(P)) versus experiment. For details see Ref.210.

Chiral gold clusters have in recent years attracted the interest of researchers. Garzón et al. have theoretically investigated structure and circular dichroism of bare and passivated chiral gold clusters.211, 212 CD spectra were also modeled, but not yet with a first‐principles method.213 Goldsmith et al. have recently investigated chirality that may arise in symmetric gold clusters when perturbed dissymmetrically by adsorbates.214 TD‐DFT computations of CD spectra of Au₁₄ (R‐methylthiirane) yielded qualitative agreement with a simple point‐charge perturbed particle in a box model for the gold cluster. The authors suggested that chiral signatures should be observable in the IR/NIR spectral region for an ensemble of gold clusters passivated with chiral adsorbates.

Computations of OR of metal complexes or chiral clusters have not yet been widely reported. For studies of optical activity of metal complexes, the complex [Co(en)₃]³⁺ has traditionally served as a prototype system for a chelate complex with saturated ligands.60 The ORD for the Λ and the Δ enantiomer in the visible range were already reported in the 1930s (Mathieu 1934, Jaeger 1937, see Ref.215). The anomalous OR dispersion between 450 and 550 nm is caused by the ligand field d‐to‐d transitions. Figure 18 shows a comparison of the experimental ORD with non‐hybrid and hybrid TD‐DFT data.66 The computations used global damping of 0.007 au (0.2 eV). The hybrid functional yields lower d‐to‐d excitation energies which is in better agreement with experiment but still too high. As mentioned earlier, Hartree–Fock exchange in a hybrid XC potential can compensate for some of the self‐interaction errors. The long‐wavelength OR (above 600 nm) agrees better with experiment for the non‐hybrid functional, but from Figure 18 it is clear that this is partially the result of an error compensation, as the ligand field transitions are overestimated in energy. For the PBE0 functional, the ORD is very close to experiment near the Na D‐line wavelength. The calculated specific rotation of [α]_D = −105 degree/[dm (g/cm⁻³)] at this wavelength is in good agreement with an experimental value of −123 reported for a solution of the tris‐bromide monohydrate salt.215

Figure 18

Molar rotation of Δ−[Co(en)₃]³⁺, ob₃ conformer. Straight line: Experimental data for a solution of Λ−[Co(en)₃]³⁺ (bromide salt) from Ref.215, inverted. Computations: revPBE nonhybrid functional, Slater‐type basis set (circles, data from Ref.52), and PBE0 hybrid functional, Gaussian‐type basis (triangles, data from Ref.66). A global damping constant of 0.007 au was used in the computations to obtain a nonsingular anomalous dispersion in the vicinity of the ligand‐field transitions.

In the section on Computing Chiroptical Properties: Theoretical Background it was pointed out that the OR and rotatory strengths are rank‐2 tensors. Experiments in gas phase or solution where molecules rotate rapidly measure the isotropic parts of these tensors. During the 1990s experimental techniques for a determination of the OR tensor in crystals have been devised.216-218 For an overview see Claborn et al., Ref.219. Spectroscopic methods for measurements of anisotropic CD have also been reported.220-222 The CD measurements were performed in liquid crystals. It is conceivable that OR tensors can be measured in a similar way. A measurement of the OR tensor at λ = 670 nm has been reported for pentaerithrol which crystallizes in the achiral point group S₄.223 TD‐DFT computations were also performed on a single pentaerithrol molecule in the geometry that it adopts in the single crystal. The B3LYP/aug‐cc‐pVDZ computations yielded good agreement with experiment for the orientation of the OR tensor. TD‐DFT computations of anisotropic origin‐independent OR both for the nonresonant and the resonant case (using a damping technique) were also implemented.224 Because of the lack of data for non‐ionic crystals at the time when the study was carried out the imaginary part of the origin‐independent OR tensor elements were compared to the experimental n‐to‐π* CO chromophore CD tensor of a steroid molecule recorded in a liquid crystal environment.221 The sign and magnitude of the isotropic CD that were obtained from the imaginary part of the linear‐response OR tensor, and signs and relative magnitudes of the anisotropic CD tensor elements, agreed quite well with experiment. Anisotropic CD has also been computed using TD‐HF225, 225 and coupled cluster response theory.227-229

CONCLUDING REMARKS

The practical chemist has a variety of computational tools available to study CD, OR, VCD, ROA, and other natural and induced chiroptical properties with the help of first principles theory. The quality of the computational model is of critical importance. This not only includes the choice of a basis set and electronic structure method, but also careful consideration of the molecule's environment, or the inclusion of vibrational effects. In studies in which such effects have been taken into consideration, or specifically been ruled out as important influences, the agreement with experimental is generally very encouraging. Undoubtedly, this research area will remain highly active as new computational methods are being developed.

Acknowledgements

The author would like to thank present and past members of his research group as well as many dear collaborators. Special thanks to Mark Rudolph for preparing some of the computational data in the section An Example: Chiroptical Properties of Trans‐Dimethyloxirane, to Mark Rudolph, Lucia Nitsch, and Ben Pritchard for proofreading the manuscript, to Prof. L. Jensen for providing a program used to convert units for the ROA example, to Dr. J. Crassous for information regarding C₈₄ and for the exciting collaboration on metal–helicene complexes, and to Prof. P. Polavarapu for exchanging some thoughts on unit conversions for optical rotations.

* In some of the equations we will use Hartree atomic units (au) in which the electron mass m_e = 1, unit of charge e = 1, the proportionality constant in Coulomb's law 1/(4πε₀) = 1, Planck's constant h = 2π, and speed of light c ≈ 137.036 atomic units. Factors of one (e, m_e, 1/(4πε₀), h/(2π)) can therefore be dropped in these units. Results are generally computed in these dimensionless atomic units (au) and converted to the desired units afterwards. Masses are in electron masses, lengths in bohr radii (52.918 pm), and energies in Hartree (2625.5 kJ/mol).

† In the optical activity research field SI units are not always employed.

‡ If ∑_i R_i goes over the complete set of molecular wavefunctions, one can use the completeness relation 1 = ∑_j |Ψ_i〉〈Ψ_i| to obtain eq. 4.

§ Using a molar mass in the usual units of g/mol and a specific rotation in degree/[dm (g/cm⁻³)], the conversion9 does not seem to yield degree cm²/dmol but degree cm²/(10,000 dmol) instead, which might be confusing. The conversion is for the values of specific and molar rotation in their respective standard units. The conventions used for [α] of solutions involve a standard concentration measured as grams of solute in 100 cm³ solution (i.e., a concentration value in percent).59 The corresponding reference concentration would be the amount of substance (in mol) per 100 cm³, also expressed as a percent value, leading to the conversion [Eq. (9)] for the values of the rotations.

¶ Here and elsewhere we assume that the real and imaginary part of f go to zero as ω → ±∞.

** This assumption can be made without loss of generality as long as spin–orbit coupling does not play a role, i.e., in the nonrelativistic limit or within a spin‐free relativistic framework.

§§ The KS equations are here in canonical form.

*** Adiabatic TD‐DFT is not able to describe simultaneous excitations of two (or more) electrons. However, the full exact linear response must include singularities from all excitations, including double excitations.

††† Assuming no orbital relaxation occurs upon excitation, and assuming a small coupling matrix. Notation used: . Note that f_XC may not be the same for singlet and triplet excitations. The orbital energies and the orbital shapes are different in DFT and Hartree–Fock.

‡‡‡ The software used to broaden the spectra and to prepare the plots is available from the author's WWW page.

§§§ For the simulation of vertical electronic spectra, using different basis sets for the optimizations and the spectra computations is accepted practice. However, when vibronic effects are to be included the question of whether stationary points and force fields obtained for some level of theory can be used for spectra calculations at a different level of theory also becomes an important question. Similar considerations apply to calculations of vibrational corrections of optical rotations, polarizabilities, and other response properties.

¶¶¶ Refer to Section on TD‐DFT Linear Response: A Synopsis: Dispersion is a long range correlation effect, while standard correlation functionals depend on the density and its derivatives which vanish exponentially with distance between two subsystems.

**** Computations have also frequently been performed on twisted ethene, and on H₂O₂ in a gauche conformation, for instance to benchmark computational models. We will forgo a discussion of such computations and focus on more realistic cases.

†††† Never mind the stability of the molecule at 1000+ K; the purpose of the graph is to show at which T, reasonable or not, the purely vibrational average and the rotational‐vibrational average start to differ.

‡‡‡‡ DMO has a few more atoms but the computational saving gained from its C₂ symmetry makes up for the additional basis functions, making it an almost equally popular benchmark system.