# Evaluation of Molecular Polarizability and of Intensity Carrying Modes Contributions in Circular Dichroism Spectroscopies

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## Abstract

**:**

`polar`to compute the molecular polarizability complex tensors from quantum chemistry outputs, thus simulating straightforwardly UV-visible absorption (UV-Vis)/electronic circular dichroism (ECD) spectra, and infrared (IR)/vibrational circular dichroism (VCD) spectra. We validate the theory and the code by referring to literature data of a large group of chiral molecules, showing the remarkable accuracy of density functional theory (DFT) methods. We anticipate the application of this methodology to the interpretation of vibrational spectra in various measurement conditions, even in presence of metal surfaces with plasmonic properties. Our theoretical developments aim, in the long run, at embedding the quantum-mechanical details of the chiroptical spectroscopic response of a molecule into the simulation of the electromagnetic field distribution at the surface of plasmonic devices. Such simulations are also instrumental to the interpretation of the experimental spectra measured from devices designed to enhance chiroptical interactions by the surface plasmon resonance of metal nanostructures.

## 1. Introduction

`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.

`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.

## 2. Results

`polar`code.

- (i)
- to simulate UV-Vis absorption spectra in the absence of vibronic effects:$${w}_{0k}^{abs}\left(\omega \right)=\frac{1}{3}\frac{I\omega}{c{\u03f5}_{0}}\Im \left(\right)open="["\; close="]">Tr\left(\right)open="("\; close=")">{\mathit{\alpha}}^{ee}(-\omega ;\omega );$$
- (ii)
- to simulate electronic CD (ECD) spectra [24]:$${w}_{0k}^{CD}\left(\omega \right)={w}_{-,0k}\left(\omega \right)-{w}_{+,0k}\left(\omega \right)=-\frac{4}{3}\frac{I\omega}{{c}^{2}{\u03f5}_{0}}\Re \left(\right)open="["\; close="]">Tr\left(\right)open="("\; close=")">{\mathit{\alpha}}^{em}(-\omega ;\omega );$$
- (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]:$$g\left(\omega \right)=\frac{{w}_{0k}^{CD}\left(\omega \right)}{{w}_{0k}^{abs}\left(\omega \right)}=-\frac{4}{c}\frac{\Re \left(\right)open="["\; close="]">Tr\left(\right)open="("\; close=")">{\mathit{\alpha}}^{em}(-\omega ;\omega )}{}$$

## 3. Discussion

#### 3.1. Polarizability Due to Electronic Transitions and UV-Vis Absorption

`polar`implementation, we report in Figure 1 the results of its application on TDDFT outputs of the benzene molecule computed for increasing values of the number of excited states (N). The correctness of

`polar`implementation is checked against independent results obtained from frequency-dependent linear response theory (i.e., CPKS equations). The latter approach is implemented in Gaussian09 and computes the polarizability avoiding the sum over states. It can be noted that the convergence of

`polar`results to the CPKS results is smooth over a wide range of photon energies. No appreciable differences are observed when the number of states in the sum of Equation (5) is large enough (in our case N = 1000). The static limit of the polarizability $\Re \left[Tr\left({\mathit{\alpha}}^{ee}(0;0)\right)\right]$ is known to be proportional to the volume occupied by core and valence electrons, both for atoms [52] and molecules [53]. For this reason, to reach convergence with respect to CPKS results, one needs to include enough electronic excitations in the sum over states expansion of ${\mathit{\alpha}}^{ee}$, so to make sure that also contributions from core-excitations are present. As expected, convergence is faster when approaching resonance conditions (e.g., close to 7 eV in Figure 1a), for which single transitions dominate the sum over states expression of the polarizability.

`polar`against the reported values of the complex electric dipole polarizability of 4,4’-diaminoazobenzene [54]. A direct numerical comparison is difficult in this case due to the different quantum chemistry code used here (Gaussian09) and in [54] (ADF). To approach the computational conditions of Haghdani et al. [54], we adopted for 4,4’-diaminoazobenzene the PBE functional and the cc-pVTZ basis set. Our results, reported in Figure 2, nicely compare with those reported in Figure 3b of Ref. [54]: the maximum value of the imaginary part of $Tr\left({\mathit{\alpha}}^{ee}\right)/3$ is about 1000 bohr${}^{3}$ and the maximum value of the real part of $Tr\left({\mathit{\alpha}}^{ee}\right)/3$ is about 600 bohr${}^{3}$. We also note that the shapes and linewidths of both the real and imaginary parts of the isotropic electric polarizability in Figure 2 are very close to those reported in Figure 3b of Ref. [54].

#### 3.2. The Electric Dipole–Magnetic Dipole Polarizability and ECD

`polar`for this kind of calculations, we address in Figure 3 the case of hexahelicene. This is a representative molecule of the class of helicenes, which are $\pi $-conjugated systems consisting of ortho-fused benzene rings and have many possible applications [55]. They are characterized by inherent helical chirality, and possess interesting spectroscopic properties [56,57,58,59,60,61].

`polar`on the basis of TDDFT calculations nicely match with the sign and relative magnitude of the rotatory strengths ${R}_{0k}$ reported by Gaussian09 within the same TDDFT calculation. In comparison with hexahelicene, we have considered the hexamer model of a helically coiled graphene nanoribbon of recent synthesis [62] (HGNR-6), which is also a $\pi $-conjugated system with inherent chirality (see Figure 4a). Interestingly, if we compare hexahelicene with HGNR-6 (for the same P-helicity), we obtain the same sign of the rotatory strength for the optically more active low energy $\pi \to {\pi}^{*}$ transition (computed at 391 nm in HGNR-6, see Figure 4b). Notably, being $\pi $-conjugation more extended in HGNR-6 than in hexahelicene, this causes a significant red-shift of the wavelength of such transition, and higher values of its rotatory strength. This is evident by comparing Figure 3b and Figure 4b. Furthermore, following the assignment of this low energy $\pi \to {\pi}^{*}$ transition in hexahelicene to a helical sense-responsive feature (H-type band) [60], and by observing that the associated rotatory strengths have the same sign in P-hexahelicene and P-HGNR-6, we infer that such low-energy transition (computed at 391 nm) is of H-type also in HGNR-6.

#### 3.3. Vibrational Polarizabilities and IR/VCD Spectroscopies

`polar`, we begin by considering ${\mathit{\alpha}}^{ee,v}$ in the static limit. Such a quantity is straightforwardly provided by the Gaussian code in standard vibrational frequency calculations, and for small molecules (CH${}_{4}$, CF${}_{4}$, and CCl${}_{4}$) it is also available from published independent calculations [70] and experiments [71]. As shown in Table 1, the numerical data provided by

`polar`have the same quality as those reported by Gaussian, and compare reasonably with other theoretical data and experimental results.

`polar`, we decided to consider rigid molecules, for the sake of simplicity. We selected four chiral molecules with unique conformations, which offers a more straightforward comparison with the experimental IR and VCD spectra. The application of

`polar`to molecules with many conformations is a conceptually straightforward task, which could be considered in future versions of the code. The experimental VCD and IR spectra of (1S)-Fenchone ((1S) -FEN), (1S)-Camphor ((1S)-CAM), (1S)-2-Methylenefenchone ((1S)-MEFEN) and (1S)-2-Methylenecamphor ((1S)-MECAM) were investigated in [79], and the main features of the spectra where assigned with the help of DFT calculations. The satisfactory comparison between frequency-dependent polarizabilities and experimental IR and VCD spectra of the four molecules is presented in Figure 6 and Figure 7.

`polar`with the corresponding experimental quantity measured for 1S-CAM, taken as a representative reference. Both the order of magnitude and the spectral shape of the computed $g\left(\omega \right)$ match reasonably well with the experimental results, which we take as a solid benchmark of the implementation of the

`polar`code.

`polar`, we have computed the transition dipole moments and the g-ratios of methyloxirane (Figure 9), a well-known small and rigid chiral molecule which was investigated in the past by three of us [82]. The above quantities have been compared with the corresponding g-ratios that can be straightforwardly determined from the dipole and rotatory strengths extracted (in cgs units) from the typical output of a VCD calculation made by Gaussian. We collect in Table 2 the satisfactory comparison between the g-ratios computed by

`polar`and those determined from the Gaussian output.

#### 3.4. Introducing Intensity-Carrying Modes in VCD

`polar`, for both IR and VCD spectroscopy, considering the $\mathcal{M}$ matrices given by Equations (18) and (20). The eigenvalues resulting from the solution of Equation (17) are collected in Table 3 (for IR) and in Table 4 (for VCD). We observe that the number of non-zero eigenvalues of ${\mathcal{M}}^{IR}$ is three (as previously found by Torii [33]), while it is six for ${\mathcal{M}}^{VCD}$. As pointed out by Luber et al. in discussing Raman and ROA ICMs [86], the rank of the $\mathcal{M}$ matrix indeed equals the number of independent components of the polarizability. Similarly, in VCD, we have three independent components of the electric and magnetic dipoles, which justifies the rank of ${\mathcal{M}}^{VCD}$ being six. Furthermore, since the sum of the rotatory strengths is zero [47] (i.e., ${\sum}_{a}{R}_{{0}_{a}{1}_{a}}=0$), this justifies the fact that the sum of the eigenvalues is approximately zero, within the numerical accuracy affordable by the selected computational method.

## 4. Materials and Methods

`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.

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

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

`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):

`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.

## 5. Conclusions

`polar`code makes the simulation of CD spectra straightforward through the evaluation of the trace of the frequency-dependent electric dipole–magnetic dipole molecular polarizability, ${\mathit{\alpha}}^{em}\left(\omega \right)$.

`polar`allows simulating both ECD and VCD spectra based on the quantum chemical outputs routinely produced by the Gaussian code [5]. These calculations allow for the validation of

`polar`thanks to the available literature data.

`polar`are of relevance to spectroscopic applications in the presence of controlled distributions of electromagnetic fields that can be realized via nano-optical engineering in order to enhance the chiroptical response [87,88,89]. In this framework, the molecular calculations can serve both the purpose of introducing the chiral medium into the electromagnetic simulations via effective medium approaches and of evaluating the weight of the quadrupolar contributions in the CD spectra because of the presence of specific field gradients.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

AAT | Atomic Axial Tensor |

APT | Atomic Polar Tensor |

CD | circular dichroism |

CPKS | coupled perturbed Kohn Sham equations |

DFT | density functional theory |

ECD | electronic circular dichroism |

TDDFT | time-dependent density functional theory |

VCD | vibrational circular dichroism |

## Appendix A. Analysis of Polarizabilities in Resonance Condition

## Appendix B. Polarizability as a Sum over Vibrational States

## Appendix C. Relation with Lambert–Beer law

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**Figure 1.**The real part of the isotropic invariant of the electric dipole polarizability of benzene ($\frac{1}{3}\Re \left[Tr\left({\mathit{\alpha}}^{ee}\right)\right]$) computed by

`polar`from a time-dependent density functional theory(TDDFT) calculation at B3LYP/6-31G(d,p) level: (

**a**) far from the resonance range; and (

**b**) close to the resonance range. The dots in both panels have been computed by the CPKS scheme available in Gaussian09 for applied electric fields at given frequencies. The results from the sum over states implemented by

`polar`(Equation (5)) are shown with solid lines for increasing values of the number of states N included in the sum. The damping used in the

`polar`calculations is $\Gamma $ = 0.001 eV.

**Figure 2.**The real and imaginary parts of the isotropic invariant of the electric dipole polarizability of 4,4’-diaminoazobenzene ($\frac{1}{3}Tr\left({\mathit{\alpha}}^{ee}\right)$) computed by

`polar`from a TDDFT calculation at PBE/cc-pVTZ level: (

**a**) the chemical structure of 4,4’-diaminoazobenzene and the graphical representation of the computed equilibrium structure, which is essentially planar, with the -NH${}_{2}$ groups slightly pyramidalized; and (

**b**) the results obtained by the

`polar`code for two choices of N, demonstrating good convergence at $N=100$ states in this photon energy range. To approach the conditions used in [54], the damping $\Gamma $ for

`polar`calculations is to 0.24 eV.

**Figure 3.**(

**a**) Graphical representation of the equilibrium structure of P-hexahelicene (data from B3LYP/6-31G(d,p) DFT calculation); and (

**b**) comparison of the experimental UV-Vis absorption of hexahelicene and the electronic circular dichroism(ECD) spectra of P-hexahelicene and M-hexahelicene (from [60]) vs. the corresponding spectra simulated by

`polar`($\Gamma $ = 0.5 eV) based on results from TDDFT calculations (200 states, B3LYP/6-31G(d,p)).

**Figure 4.**(

**a**) Graphical representation of the equilibrium structure of P-HGNR-6 (data from B3LYP/6-31g(d,p) DFT calculation); and (

**b**) electronic absorption and circular dichroism of P-HGNR-6 simulated by

`polar`($\Gamma $ = 0.5 eV) based on results from TDDFT calculations (B3LYP/6-31G(d,p); $N=100$ and $N=400$ states).

**Figure 5.**(

**a**) The optimized structure of C${}_{84}$ used to compute the electric dipole–magnetic dipole polarizability ${\mathit{\alpha}}^{em}$ reported in the top part of (

**c**); and (

**b**) the (${}^{f}A$)-${D}_{2}$-C${}_{84}$ configuration of C${}_{84}$ which was assigned in [69] to the experimental CD spectrum represented in the bottom part of (

**c**). The color scheme of (

**a**,

**b**) helps in distinguishing pentagons from hexagons in the fullerene. The experimental CD spectrum is taken from Ref. [69]. To simulate ${\mathit{\alpha}}^{em}(-\omega ,\omega )$ ($\Gamma $ = 0.35 eV), we fed

`polar`with data from a TDDFT calculation (B3LYP/6-31G(d,p) level, 400 states).

**Figure 6.**(

**a**,

**c**) The equilibrium structure of (1S)-FEN and (1S)-CAM, respectively; and (

**b**,

**d**)

`polar`-simulated IR and VCD spectra of (1S)-FEN and (1S)-CAM ($\Gamma $ = 10 cm${}^{-1}$), respectively, compared to their experimental counterparts [79]. The vertical black bars in the top panels are proportional to the rotatory strengths computed by Gaussian (${R}_{0k}$). To ease the comparison with the experimental data, the wavenumber axis of the simulated polarizabilities has been uniformly scaled by the empirical factor 0.98. DFT calculations were carried out at the B3LYP/aug-cc-pVTZ level.

**Figure 7.**(

**a**,

**c**) The equilibrium structure of (1S)-MEFEN and (1S)-MECAM, respectively; and (

**b**,

**d**)

`polar`-simulated IR and VCD spectra of (1S)-MEFEN and (1S)-MECAM ($\Gamma $ = 10 cm${}^{-1}$), respectively, compared to their experimental counterparts [79]. The vertical black bars in the top panel are proportional to the rotatory strengths computed by Gaussian (${R}_{0k}$). To ease the comparison with the experimental data, the wavenumber axis of the simulated polarizabilities has been uniformly scaled by the empirical factor 0.98. DFT calculations were carried out at the B3LYP/aug-cc-pVTZ level.

**Figure 8.**Comparison between the experimental g-ratio of 1S-CAM and the g-ratio calculated by

`polar`based on Equation (12) ($\Gamma $ = 10 cm${}^{-1}$). DFT calculations on 1S-CAM have been carried out at B3LYP/aug-cc-pVTZ level; for ease of comparison with experimental data, the wavenumbers computed by DFT were scaled by 0.98.

**Figure 9.**Representation of the equilibrium structure of (R)-methyloxirane computed by DFT at the B3LYP/aug-cc-pVQZ level of theory.

**Table 1.**Vibrational electric polarizabilities of CH${}_{4}$, CF${}_{4}$, and CCl${}_{4}$ (data are in atomic units [41]; B3LYP/aug-cc-pVTZ DFT calculations for Columns 1 and 2—present work). The minor discrepancy between Columns 1 and 2 is due to slight differences in the numerical values of the physical constants involved in the calculations (i.e.,

`polar`in Column 1 vs. Gaussian in Column 2). This is proved by the ratios of the values in Columns 1 and 2 being 1.0072, independently of the molecule.

Molecule | $\frac{1}{3}\mathit{Tr}\left({\mathit{\alpha}}^{\mathit{ee},\mathit{v}}\right)$ | From Gaussian09 Output | From Ref. [70] | Expt. from Ref. [71] |
---|---|---|---|---|

CH${}_{4}$ | 0.2371 | 0.2354 | 0.2299 | 0.2025 (0.2025) |

CF${}_{4}$ | 7.3981 | 7.3450 | 8.0347 | 5.3988 (7.6259) |

CCl${}_{4}$ | 8.1828 | 8.1241 | 6.3524 | 5.6688 (6.4111) |

**Table 2.**Comparison of the results from a VCD calculation carried out with Gaussian09 on (R)-methyloxirane at the B3LYP/aug-cc-pVQZ level, and the results obtained by a corresponding

`polar`calculation based on the numerical data contained in the

`*.fchk`file associated with the Gaussian calculation. Column (a) is the wavenumbers computed by Gaussian09 and

`polar`are not distinguishable at the shown precision. The APTs and AATs stored in the

`*.fchk`file are used by

`polar`to compute the dipole and rotatory strengths (Columns (b) and (c)) in atomic units (see Equation (14) and Section 4 for details), from which we computed the dissymmetry ratios ${g}_{0k}=4{R}_{0k}/\left(c{D}_{0k}\right)$ listed in Column (d)—please note the use of SI for the electromagnetic field, which implies the factor $c=137.035999074$ in atomic units [41]. The dipole and rotatory strengths computed by Gaussian09 (Columns (e) and (f)) are used to compute the dissymmetry ratios ${g}_{0k}=4{R}_{0k}/{D}_{0k}$ (cgs system) reported in Column (g).

(a) | (b) | (c) | (d) | (e) | (f) | (g) |
---|---|---|---|---|---|---|

${\overline{\mathit{\nu}}}_{\mathit{k}}$ | ${\mathbf{10}}^{\mathbf{4}}\xb7{\mathit{D}}_{\mathbf{0}\mathit{k}}$ | ${\mathbf{10}}^{\mathbf{7}}\xb7{\mathit{R}}_{\mathbf{0}\mathit{k}}$ | ${\mathbf{10}}^{\mathbf{4}}\xb7{\mathit{g}}_{\mathbf{0}\mathit{k}}$ | ${\mathbf{10}}^{\mathbf{40}}\xb7{\mathit{D}}_{\mathbf{0}\mathit{k}}$ | ${\mathbf{10}}^{\mathbf{44}}\xb7{\mathit{R}}_{\mathbf{0}\mathit{k}}$ | ${\mathbf{10}}^{\mathbf{4}}\xb7{\mathit{g}}_{\mathbf{0}\mathit{k}}$ |

(cm${}^{-\mathbf{1}}$) | (at. un.) | (at. un.) | (dimensionless) | (esu${}^{\mathbf{2}}$ cm${}^{\mathbf{2}}$) | (esu${}^{\mathbf{2}}$ cm${}^{\mathbf{2}}$) | (dimensionless) |

211 | 1.07 | 7.15 | 1.94 | 7.30 | 3.46 | 1.89 |

368 | 6.49 | −26.54 | −1.19 | 41.96 | −12.44 | −1.19 |

411 | 6.62 | −10.04 | −0.44 | 42.62 | −4.81 | −0.45 |

771 | 6.01 | 27.05 | 1.31 | 38.79 | 12.75 | 1.31 |

842 | 32.85 | 8.60 | 0.08 | 212.20 | 4.06 | 0.08 |

909 | 1.79 | 53.14 | 8.68 | 11.55 | 25.05 | 8.68 |

973 | 9.67 | −73.10 | −2.21 | 62.50 | −34.46 | −2.21 |

1044 | 5.10 | 13.38 | 0.77 | 32.94 | 6.31 | 0.77 |

1131 | 3.63 | −15.52 | −1.25 | 23.42 | −7.31 | −1.25 |

1159 | 1.32 | 10.50 | 2.33 | 8.50 | 4.94 | 2.33 |

1169 | 1.62 | −26.62 | −4.78 | 10.50 | −12.55 | −4.78 |

1191 | 0.49 | 1.34 | 0.80 | 3.15 | 0.63 | 0.80 |

1295 | 3.21 | −20.81 | −1.89 | 20.73 | −9.81 | −1.89 |

1409 | 1.28 | 4.71 | 1.07 | 8.30 | 2.21 | 1.07 |

1440 | 9.06 | 30.39 | 0.98 | 58.53 | 14.33 | 0.98 |

1485 | 2.26 | 3.73 | 0.48 | 14.58 | 1.76 | 0.48 |

1499 | 2.53 | −2.97 | −0.34 | 16.32 | −1.40 | −0.34 |

1530 | 2.35 | 11.85 | 1.47 | 15.19 | 5.59 | 1.47 |

3028 | 3.63 | 2.77 | 0.22 | 23.48 | 1.31 | 0.22 |

3080 | 5.46 | 0.96 | 0.05 | 35.24 | 0.45 | 0.05 |

3085 | 3.78 | −3.99 | −0.31 | 24.41 | −1.88 | −0.31 |

3088 | 1.77 | −15.01 | −2.47 | 11.45 | −7.08 | −2.47 |

3110 | 7.31 | 14.04 | 0.56 | 47.20 | 6.62 | 0.56 |

3164 | 5.28 | −11.24 | −0.62 | 34.10 | −5.30 | −0.62 |

**Table 3.**IR Intensity Carrying Modes (ICM)s of the four chiral compounds for which the corresponding simulated and experimental spectra are reported in Figure 6 and Figure 7. The APTs and AATs required for the evaluation of ICMs have been computed at the B3LYP/aug-cc-pVTZ level of theory. All numerical data are given in the atomic units of the ${\mathcal{M}}^{IR}$ matrix.

Molecule | ${\mathit{\lambda}}_{1}$ | ${\mathit{\lambda}}_{2}$ | ${\mathit{\lambda}}_{3}$ | $\sum {\mathit{\lambda}}_{\mathit{i}}$ |
---|---|---|---|---|

(1S)-FEN | 3.29 | 1.52 | 0.30 | 5.11 |

(1S)-CAM | 4.06 | 1.18 | 0.32 | 5.57 |

(1S)-MEFEN | 0.52 | 0.41 | 0.32 | 1.25 |

(1S)-MECAM | 0.57 | 0.40 | 0.30 | 1.27 |

**Table 4.**VCD Intensity Carrying Modes (ICM)s of the four chiral compounds for which the corresponding simulated and experimental spectra are reported in Figure 6 and Figure 7. The APTs and AATs required for the evaluation of ICMs have been computed at the B3LYP/aug-cc-pVTZ level of theory. All numerical data are given in the atomic units of the ${\mathcal{M}}^{VCD}$ matrix.

Molecule | ${\mathit{\lambda}}_{1}$ | ${\mathit{\lambda}}_{2}$ | ${\mathit{\lambda}}_{3}$ | ${\mathit{\lambda}}_{4}$ | ${\mathit{\lambda}}_{5}$ | ${\mathit{\lambda}}_{6}$ | $\sum {\mathit{\lambda}}_{\mathit{i}}$ |
---|---|---|---|---|---|---|---|

(1S)-FEN | 3.67 | 1.96 | 0.58 | −0.60 | −2.07 | −3.44 | 0.09 |

(1S)-CAM | 4.25 | 1.93 | 0.64 | −0.66 | −1.76 | −4.26 | 0.13 |

(1S)-MEFEN | 0.83 | 0.71 | 0.48 | −0.51 | −0.72 | −0.77 | 0.01 |

(1S)-MECAM | 0.87 | 0.76 | 0.47 | −0.49 | −0.77 | −0.87 | −0.03 |

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**MDPI and ACS Style**

Zanchi, C.; Longhi, G.; Abbate, S.; Pellegrini, G.; Biagioni, P.; Tommasini, M.
Evaluation of Molecular Polarizability and of Intensity Carrying Modes Contributions in Circular Dichroism Spectroscopies. *Appl. Sci.* **2019**, *9*, 4691.
https://doi.org/10.3390/app9214691

**AMA Style**

Zanchi C, Longhi G, Abbate S, Pellegrini G, Biagioni P, Tommasini M.
Evaluation of Molecular Polarizability and of Intensity Carrying Modes Contributions in Circular Dichroism Spectroscopies. *Applied Sciences*. 2019; 9(21):4691.
https://doi.org/10.3390/app9214691

**Chicago/Turabian Style**

Zanchi, Chiara, Giovanna Longhi, Sergio Abbate, Giovanni Pellegrini, Paolo Biagioni, and Matteo Tommasini.
2019. "Evaluation of Molecular Polarizability and of Intensity Carrying Modes Contributions in Circular Dichroism Spectroscopies" *Applied Sciences* 9, no. 21: 4691.
https://doi.org/10.3390/app9214691