## 1. Introduction

The steady and exciting development of quantum chemistry and, notably, density functional theory (DFT) methods has greatly enhanced our capabilities in understanding molecular spectra [

1]. Current quantum chemistry computer codes can be used very reliably to assign the features observed in molecular spectra to electronic or vibrational transitions, with increasingly accurate levels of theory [

2,

3,

4,

5,

6,

7,

8,

9]. Notable advances in tools available for computational spectroscopy have appeared in the recent literature [

10,

11,

12,

13,

14,

15,

16,

17,

18,

19]. These are commonly based on existing quantum chemistry engines, and their development often enhances the accuracy of the existing approaches. For instance, the harmonic approximation can be improved to deal with anharmonic effects [

20,

21,

22,

23]. Within the rich and evolving scenario of computational spectroscopy, it is interesting to discuss approaches complementary to those most often employed to simulate spectra, which have been in use for a long time, and basically consist in weighted sums of selected lineshape functions [

24] (ultimately, the weights of such sums are related to transition moments computed by quantum chemistry—see, for instance, [

25]). In fact, one of the aims of the developments of our work is the embedding of the quantum-mechanical details of the chiroptical spectroscopic response of a molecule (which requires its complex electric dipole–magnetic dipole polarizability) within the simulation of the electromagnetic field distribution at the surface of plasmonic devices. Such simulations are instrumental to the interpretation of the experimental spectra measured from devices designed to enhance chiroptical interactions by the surface plasmon resonance of metal nanostructures [

26]. The role of molecular polarizability is well recognized in the field of linear and non-linear spectroscopy, and led to the development of insightful computational tools for non-linear optics (e.g., [

27,

28,

29]). Interestingly, polarizability was also introduced in the context of electronic CD aiming to calculate CD spectra of large molecules from the polarizability characterization of fragments [

30].

We have investigated polarizability from the principles of quantum mechanics and light–matter interaction [

31] adopting the notation used in [

24]. The paper is organized as follows. We first illustrate the connection of CD with the polarizability tensor to obtain the relevant equations which relate the quantum mechanical transition rate to the trace of molecular polarizability (i.e., Equation (

9)). Thereafter we show the use of the

`polar` code, which has been written to implement such equations, for both electronic and vibrational transitions, based on the transition matrix elements obtained from quantum chemistry codes such as Gaussian [

5]. For the benchmark of the electronic polarizability we consider a set of rigid molecules, thus avoiding conformational issues (which are irrelevant to the present study). Hence,

$\pi $-conjugated molecules offer a rich playground, with reliable reference theoretical and experimental data available for comparison. For vibrational polarizability, we have considered another set of small and rigid chiral molecules, which have been very well characterized in the VCD literature, both experimentally and theoretically. In the final part of this work, we show how the Intensity Carrying Modes, which were pioneered by Torii et al. in the context of vibrational polarizability and IR spectroscopy [

32,

33], can be smoothly extended to VCD spectroscopy as well.

Before describing our results, let us introduce the needed complex polarizabilities

${\mathit{\alpha}}^{ee}$,

${\mathit{\alpha}}^{em}$, and

${\mathit{\alpha}}^{mm}$. This notation resembles the one adopted in theoretical investigations of antennas (e.g., [

34]). It is also a notation that fits computer coding, and has fostered the development of the

`polar` program. In the context of chiroptical spectroscopy, the

${\mathit{\alpha}}^{em}$ tensor is often denoted

$\tilde{\mathit{G}}$ [

35]. Similarly, the

${\mathit{\alpha}}^{mm}$ tensor represents the magnetic susceptibility

$\tilde{\mathit{\chi}}$ [

35,

36]; the electric polarizability

${\mathit{\alpha}}^{ee}$ is typically written as

$\tilde{\mathit{\alpha}}$ [

35]. We remark that the electric dipole–magnetic dipole polarizability

${\mathit{\alpha}}^{em}$ was repeatedly written as sum over states in the literature [

35,

37,

38,

39,

40], even though with different purposes and different notation. In the present work, we neglect the dipole–quadrupole contribution (denoted

$\tilde{\mathit{A}}$ [

35]), which is relevant in presence of anisotropic systems and/or gradients of the electric field. Such contribution, which can be relevant, e.g., in plasmonics, can be evaluated in perspective by a natural extension of the proposed approach.

The proposed method to simulate molecular spectra is essentially a post-processing of outputs from standard quantum chemistry packages (e.g., Gaussian in the present case, but interfaces to other packages can be introduced straightforwardly). This approach is general, and it is meant to support the simulation of spectra in selected experimental conditions without the need of re-computing the molecular response from quantum chemistry. This also alleviates modifications at level of the source code of the adopted quantum chemistry engine. Moreover, the calculation of the polarizability elements is a crucial step in the implementation of effective chiral media within nano-optical electromagnetic simulations.

## 2. Results

The theoretical description of CD phenomena entails both electric and magnetic effects. In a detailed review [

24], Schellman presented the theoretical description of circular dichroism and optical rotation for molecules in solution state. The central role is played by the perturbation operator

${V}_{\pm}$ associated to left-handed (−) and right-handed (+) circularly polarized light, which is defined through the electric (

$\mathit{\mu}$) and the magnetic (

$\mathit{m}$) dipole operators. By making use of the results illustrated in Ref. [

24], for a given transition between states 0 and

k, the expectation value of such operators is the following:

where

${E}_{0}$ and

${B}_{0}$ are the electric and magnetic field amplitudes of the impinging light wave,

${\mathit{\mu}}_{0k}=\langle 0|\mathit{\mu}|k\rangle $ is the electric transition dipole moment, and

${\mathit{m}}_{0k}=\langle 0|\mathit{m}|k\rangle $ is the magnetic transition dipole moment.

I represents the light intensity,

c is the speed of light, and

$(a,b)$ indicates the complex number

$\tilde{z}=a+ib$. Schellman’s review [

24] adopts cgs units. For the reasons discussed by McWeeny [

41], in this work we prefer using atomic units for the numerical values of polarizabilities and we consider the SI system for the description of the electromagnetic field. For the sake of simplicity, we disregard solvent effects and adopt electromagnetic waves

in vacuum. Therefore, the intensity of the electromagnetic wave required by Schellman’s treatment (Equation (6) in Ref. [

24]) is given by

$I=\frac{1}{2}c{\u03f5}_{0}{\left|{E}_{0}\right|}^{2}$, where the magnetic (

${B}_{0}$) and electric field (

${E}_{0}$) amplitudes are related one another by the relation

${B}_{0}={E}_{0}/c$ [

42]. The electromagnetic energy absorbed per unit time as a consequence of the

$0\to k$ transitions promoted by the electromagnetic perturbation at frequency

$\omega $ is given by [

24]:

where

${\rho}_{k}\left(\omega \right)$ is the density of states for the

$0\to k$ transition, and it is normalized, i.e.,

$\int d\omega \phantom{\rule{0.166667em}{0ex}}{\rho}_{k}\left(\omega \right)=1$. By substituting Equation (

1) into Equation (

2), one obtains:

To connect Equation (

3) with the molecular polarizability tensors, it is instrumental to write the inner products among the transition dipoles (Equation (

1)) as traces of outer products:

We recall that the tensor (outer) product $\mathit{a}\otimes \mathit{b}$ of two vectors $\mathit{a}$ and $\mathit{b}$ has matrix elements ${c}_{ij}={a}_{i}{b}_{j}$. Therefore, the dot product $\mathit{a}\xb7\mathit{b}$ can be seen as $\mathit{a}\xb7\mathit{b}={\sum}_{i}{a}_{i}{b}_{i}={\sum}_{i}{c}_{ii}=Tr(\mathit{a}\otimes \mathit{b})$.

For a molecule with eigenstates

$|k\rangle $ and transition energies

$\hslash {\omega}_{k}$ from the ground state

$|0\rangle $, the electric polarizability can be written as a sum over states, with opposite sign damping

$\Gamma $ in the resonance term (first term) and off-resonance term (second term) [

35,

43,

44,

45]:

By recognizing in the second line of Equation (

5), the numerators as the outer products of the transition electric dipoles (namely,

${\mathit{\mu}}_{0k}\otimes {\mathit{\mu}}_{k0}$ and

${\mathit{\mu}}_{k0}\otimes {\mathit{\mu}}_{0k}$), one can compactly write the polarizability tensor as follows:

By introducing the formal substitution (

${\mu}_{i}\to {\mu}_{i}$,

${\mu}_{j}\to {m}_{j}$) in the expression of the electric polarizability given by Equation (

5), one can devise the expression of the complex electric dipole–magnetic dipole polarizability

${\mathit{\alpha}}^{em}$:

With a similar reasoning, the following expression can be written for the magnetic dipole polarizability tensor

${\mathit{\alpha}}^{mm}$:

Near resonance conditions with the

$0\to k$ transition, Equations (

6)–(

8) can be used to relate the trace of molecular polarizabilities to the transition rate given by Equation (

3) (see

Appendix A for details):

As well-known, we recall that the imaginary part of

${\mathit{\alpha}}^{ee}$ relates to absorption phenomena, and the real part to dispersive ones. Because of the presence of the purely imaginary magnetic dipole transition moment in the expression of

${\mathit{\alpha}}^{em}$, the absorption and dispersive terms in

${\mathit{\alpha}}^{em}$ are the real and the imaginary parts of

${\mathit{\alpha}}^{em}$, respectively (see also Equation (

A7) in

Appendix A). When neglecting vibronic effects, the

$0\to k$ transitions introduced in Equations (

6)–(

8) are taken as vertical electronic transitions, and they can be conveniently computed by, e.g., TDDFT. This is the approach adopted by the

`polar` code.

In summary, by the calculation of frequency-dependent polarizabilities, Equation (

9) can be used:

- (i)
to simulate UV-Vis absorption spectra in the absence of vibronic effects:

- (ii)
to simulate electronic CD (ECD) spectra [

24]:

and

- (iii)
to simulate the dissymmetry factor

g, which is the ratio of CD (

$\Delta \u03f5\propto {w}^{CD}$) to ordinary absorption (

$\u03f5\propto {w}^{abs}$) [

24]:

To simulate IR and VCD spectra, one has to evaluate the molecular polarizabilities by considering the

$0\to k$ transitions as vibrational. If one focuses on the vibrational transitions in the mid-IR range, within the double harmonic approximation, the sum over states is restricted to one-quantum transitions

${0}_{a}\to {1}_{a}$ over the 3N-6 vibrational modes of the system, based on mechanical harmonicity. Therefore, by making the formal change

$(0\to k)\iff ({0}_{a}\to {1}_{a})$ in Equations (

6)–(

8), one directly obtains the desired result:

where

$\hslash {\omega}_{a}$ is the vibrational quantum of the

a-th normal mode). Moreover, the matrix elements of the electric and magnetic dipole over one-quantum vibrational transitions can be evaluated according to the following expressions [

46,

47], based on electrical harmonicity assumption:

In Equation (

14),

${L}_{ia}=\partial {x}_{i}/\partial {q}_{a}=\partial {\dot{x}}_{i}/\partial {\dot{q}}_{a}$ is the transformation matrix relating the Cartesian nuclear displacements

${x}_{i}$ to the normal coordinate

${q}_{a}$. Wilson’s

$\mathit{L}$ matrix [

48] is often named

$\mathit{S}$ in VCD literature, and individual indexes are adopted to label the atom

$\lambda $ and the

ith Cartesian component of the nuclear displacement (

$i=1,2,3$). This double index notation is cumbersome when writing loops in programs. For this reason, we prefer to merely list the Cartesian nuclear displacements along a vector with

$3N$ components:

$\mathit{x}=\left({x}_{1x},{x}_{1y},{x}_{1z},{x}_{2x},{x}_{2y},{x}_{2z},...,{x}_{Nx},{x}_{Ny},{x}_{Nz}\right)$.

The real vectors

$\partial \mathit{\mu}/\partial {x}_{i}={\mathit{P}}_{i}$ collect the Atomic Polar Tensors (APTs) [

49,

50,

51]. The

${P}_{ik}$ matrix element represents the change of the

kth Cartesian component of the electric dipole caused by a change in nuclear position along the

${x}_{i}$ coordinate (

$i=1.3N$). The real vectors

${\mathit{M}}_{i}=\partial \mathit{m}/\partial {\dot{x}}_{i}$ collect the Atomic Axial Tensors (AATs) [

49,

50,

51]. The

${M}_{ik}$ matrix element represents the change in the

kth Cartesian component of the magnetic dipole caused by a change in the nuclear velocity

${\dot{x}}_{i}$. By making use of Equation (

14), one obtains the following expressions for the dipole and rotatory strengths of the

${0}_{a}\to {1}_{a}$ vibrational transition:

Further details about the evaluation of Equation (

13) are given in

Appendix B. The relation of molecular polarizabilities with Lambert-Beer’s law and the concept of cross section is given in

Appendix C.

## 4. Materials and Methods

All the polarizability calculations reported here were carried out through the `polar` code developed in this work. `polar` allows the numerical evaluation of the three polarizability tensors ${\mathit{\alpha}}^{ee}$, ${\mathit{\alpha}}^{em}$, ${\mathit{\alpha}}^{mm}$ as a function of the photon frequency $\omega $. Both electronic and vibrational polarizabilities can be computed. `polar` is written in the C++ language, and adopts complex numbers for better code readability.

Electronic polarizabilities are evaluated as sum over vertical electronic transitions (Equations (

6)–(

8)). The required input data are the excitation energies

$\hslash {\omega}_{k}$ of a selected set of excited states

$|k\rangle $, and the associated electric (magnetic) transition dipoles

${\mathit{\mu}}_{0k}$ (

${\mathit{m}}_{0k}$). The output file of an ordinary TDDFT calculation made by Gaussian code [

5] can be parsed by

`polar` to gather all information required to operate. By writing additional parsers,

`polar` can be straightforwardly interfaced to other quantum chemistry packages.

The calculation of vibrational polarizabilities by Equation (

13) requires the quantities computed by Gaussian in a standard VCD job. These are the Hessian of the energy vs. the Cartesian nuclear displacements, and the derivatives of the electric and magnetic dipole vs. the Cartesian nuclear displacements (the latter two quantities are also, respectively, named APTs and AATs, see for instance [

49,

50,

51]). By using the Hessian matrix,

`polar` solves the secular equation to obtain the vibrational frequencies (

${\omega}_{k}$) and the normal modes, which are required to compute the dipole derivatives vs. normal coordinates (

${q}_{k}$) from APTs and AATs. The Hessian, APTs, and AATs are stored by Gaussian in the formatted checkpoint file (

`*.fchk`) and are read by

`polar` before addressing the evaluation of vibrational polarizabilities. For convenience, since Gaussian checkpoint and output files report computed data in atomic units, we adopt the same convention in

`polar`. The quite useful and neat paper by McWeeny [

41] can help the reader in converting all the quantities reported here from atomic to SI units. We merely remind here the following identities (where

${E}_{0}={e}^{2}/\left({\kappa}_{0}{a}_{0}\right)$ is the symbol for the Hartree,

${a}_{0}$ the symbol for the Bohr, and

${\kappa}_{0}=4\pi {\u03f5}_{0}$ is the electric permittivity):

Since the atomic units of the electric and magnetic fields are

$\left[\mathcal{E}\right]={E}_{0}/\left(e{a}_{0}\right)$ and

$\left[\mathcal{B}\right]=\hslash /\left(e{a}_{0}^{2}\right)$ [

41], it is straightforward to show that

$\left[\mathcal{E}{\alpha}^{ee}\mathcal{E}\right]=\left[\mathcal{E}{\alpha}^{em}\mathcal{B}\right]=\left[\mathcal{B}{\alpha}^{mm}\mathcal{B}\right]={E}_{0}$.

The DFT and time-dependent density functional theory(TDDFT) calculations reported in this work have been carried out with Gaussian09 Rev. D.01 [

5]. Because of the scattered nature of the literature reference data employed in this work to validate

`polar`, we had to chose individual functionals and basis sets, as best suited to carry out the required comparison. Therefore, in the caption of each figure or table presenting computational data, we report the selected functional and basis set.