Time-dependent formulation of Resonance Raman Optical Activity

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Spectroscopy and Excited States

Time-dependent formulation of Resonance Raman Optical Activity spectroscopy. Alberto Baiardi, Julien Bloino, and Vincenzo Barone J. Chem. Theory Comput., Just Accepted Manuscript • DOI: 10.1021/acs.jctc.8b00488 • Publication Date (Web): 03 Oct 2018 Downloaded from http://pubs.acs.org on October 7, 2018

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Journal of Chemical Theory and Computation

Time-dependent formulation of Resonance Raman Optical Activity spectroscopy. Alberto Baiardi,∗,†,‡ Julien Bloino,† and Vincenzo Barone† †Scuola Normale Superiore, piazza dei Cavalieri 7, I-56126 Pisa, Italy ‡Current address: Laboratorium f¨ ur Physikalische Chemie, ETH Z¨ urich, Vladimir-Prelog-Weg 2, 8093 Z¨ urich E-mail: [email protected]

Abstract In this work, we extend the theoretical framework recently developed for the simulation of resonance Raman (RR) spectra of medium-to-large sized systems to its chiral counterpart, namely resonance Raman Optical activity (RROA). The theory is based on a time-dependent (TD) formulation, with the transition tensors obtained as halfFourier transforms of the appropriate cross-correlation functions. The implementation has been kept as general as possible, supporting adiabatic and vertical models for the PES representation, both in Cartesian and internal coordinates, with the possible inclusion of Herzberg-Teller (HT) effects. Thanks to the integration of this TD-RROA procedure within a general-purpose quantum-chemistry program, both solvation and leading anharmonicity effects can be included in an effective way. The implementation is validated on one of the smallest chiral molecule (methyloxirane). Practical applications are illustrated with three medium-size organic molecules (naproxen-OCD3 , quinidine and 2-Br-Hexahelicene), whose simulated spectra are compared to the corresponding experimental data.

1

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1

Introduction

Molecular spectroscopy is one of the most commonly used tools for the characterization of molecular systems. 1 Among the various techniques now available, chiroptical spectroscopies have gained, in the last decades, a growing interest, due to the increasing relevance of chiral systems in numerous fields of fundamental and applied sciences. 2 Different experimental techniques are currently available, ranging from standard electronic (ECD) 3 and vibrational (VCD) 4,5 circular dichroism, to more advanced methods, such as those based on scattering effects, like Raman Optical Activity (ROA) 6 and its resonant form, resonance Raman Optical Activity (RROA). 7,8 The interpretation of experimental chiroptical spectra usually requires the support of refined quantum mechanical calculations, and simple selection rules derived from phenomenological models are only rarely sufficient. From a theoretical point of view, the simulation of chiroptical spectra requires the evaluation of high-order response properties, 9,10 and this task is far from being straightforward, especially for advanced electronic structure methods. Another challenge is associated with the inclusion of nuclear effects, which are still currently neglected in most applications, or included in approximate ways. As a matter of fact, the bandshape of an electronic chiral spectrum is determined by the convolution of all possible vibronic transitions between the two electronic states of interest, and this convolution usually leads to asymmetric bandshapes, which cannot be reproduced with symmetric distribution functions, such as Gaussian or Lorentzian functions. Furthermore, it has already been noted that for both ECD 11 and circularly polarized luminescence (CPL) 12,13 the inclusion of vibronic effects can lead to significant intensity redistributions in the computed spectrum, with the possibility of sign inversions. Thus, the development of vibronic models applied to chiral electronic spectra is mandatory to reliably support the interpretation of experimental spectra. In this work, we will focus on RROA, which is the chiral counterpart of resonance Raman (RR). In RR, vibrational transitions are studied from the light scattered from the sample 2


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subsequently to the irradiation with a laser, whose energy matches the excitation energy to an electronic state, usually referred to as intermediate state. The experimental setup of RROA is the same of RR, but circularly polarized light is used instead of the standard, linearly polarized one. Even if in both RR and RROA, transitions between vibrational levels of the same electronic state (usually the ground state) are actually observed, the intensity of the bands is determined by vibronic effects with the intermediate state. The experimental observables recorded in a RROA spectrum and computed quantities can be related through three transition tensors, namely the electric dipole-electric dipole, electric dipole-magnetic dipole and electric dipole-electric quadrupole tensors, which can be computed in different ways. For example, the so-called complex polarizability approach introduced by Schatz and co-workers 14,15 introduces a damping factor which prevents the breakdown of the standard formulation of those tensors, assumed for far-from-resonance conditions. However, as the incident laser is in resonance with one or several electronic states, the neglect of vibrational contributions is not satisfactory anymore and full vibronic models are required. The starting point for the derivation of full-vibronic models of RROA is the so-called sumover-state (SOS) formulation, 16,17 which expresses transition tensors as the sum of the single contributions associated to each vibronic transition. Within the so-called time-independent (TI) approaches, 18,19 the tensors are obtained by explicitly including all possible vibronic states. However, TI-based approaches suffer from several limitations. First of all, the number of vibronic transitions to be included in the sum is virtually infinite, thus an algorithm to select a-priori the most relevant transitions must be devised. Even if several schemes to perform this task have been proposed for the OP case, 20,21 this is not the case for RR. Furthermore, no analytical sum rules are available, which would enable checking if all the most relevant vibronic transitions have been included in the treatment. In the present work, an alternative, time-dependent (TD) strategy, developed for onephoton 22 and RR 23,24 has been extended to RROA. Our TD model is based on an harmonic

3



approximation of the potential energy surfaces (PESs) of both electronic states to express the transition tensors as the half-Fourier transforms of appropriate cross-correlation functions. The main advantage of the TD model is the possibility of computing the cross-correlation function from analytical relations. Thus, the only approximate step of the simulation is the integral evaluation. This is a major advantage over TI approaches since the convergence of TD calculations can be monitored by changing the graining of the integration grid. The derivation of the TD-RROA theory has been kept as general as possible, in order to support all the models already supported for TD-RR. In particular, the inclusion of the so-called Herzberg-Teller effects is presented, which are usually relevant in chiroptical spectroscopies. Furthermore, TD-RROA has been generalized to support internal coordinates, using a powerful framework recently applied to OP 21,25 and RR 26 spectroscopies. The paper is organized as follows: the next section will present the TD-RROA theory, with particular care devoted to the connection with the parallel, TD-RR formulation and to the gauge-invariance problem. Then, the theory is applied to the simulation of RROA spectra of three organic molecules. The first one is (1R)-methyloxirane, which will be used to test the reliability of our formulation. Then, the RROA of the methylester of naproxen, for which the experimental RROA spectrum is available, will be simulated. To show the reliability of our approach also for large molecules, the computed RROA spectrum of quinidine and 2-Br-Hexahelicene will be presented.

4


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2

Theory

2.1

General theory of RROA spectroscopy

The first formulation of the theory of ROA spectroscopy was proposed by Barron and coworkers, 4 within a semi-classical framework, where the light is described classically, whereas the molecular system is treated at the quantum mechanical level. Using the time-dependent second-order perturbation theory, the ROA intensity of a transition between molecular levels | Ψi i and | Ψf i can be related to the following five transition tensors: 4,17

αif,ρη = Gif,ρη = Gif,ρη = Aif,ρ,ηδ =

Aif,ρ,ηδ

1 X h Ψf ~ m6=i,f 1 X h Ψf ~ m6=i,f 1 X h Ψf ~ m6=i,f 1 X h Ψf ~ m6=i,f

| µρ | Ψm ih Ψm | µη | Ψi i h Ψf | µρ | Ψm ih Ψm | µη | Ψi i + ωmi − ωI + iγm ωmi + ωI + iγm

(1)

| µρ | Ψm ih Ψm | mη | Ψi i h Ψf | mρ | Ψm ih Ψm | µη | Ψi i + ωmi − ωI + iγm ωmi + ωI + iγm

| mρ | Ψm ih Ψm | µη | Ψi i h Ψf | µρ | Ψm ih Ψm | mη | Ψi i + ωmi − ωI + iγm ωmi + ωI + iγm

(2)

| µρ | Ψm ih Ψm | Θηδ | Ψi i h Ψf | Θηδ | Ψm ih Ψm | µρ | Ψi i + ωmi − ωI + iγm ωmi + ωI + iγm

(3)

(4) 1 X h Ψf | Θηδ | Ψm ih Ψm | µρ | Ψi i h Ψf | µρ | Ψm ih Ψm | Θηδ | Ψi i = + ~ m6=i,f ωmi − ωI + iγm ωmi + ωI + iγm (5)

where µ, m and Θ are the electric dipole, magnetic dipole and electric quadrupole operators, respectively. The molecular levels are labeled |Ψm i with lifetimes γm and energes Em . Finally, ωI is the frequency of the incident light and ωmi the excitation frequency

Em −Ei . ~

αif is the

electric dipole-electric dipole tensor, also referred to as the polarizability tensor, Gif and G if the electric dipole-magnetic dipole tensors and Aif and Aif the electric dipole-electric quadrupole tensors. In order to express the ROA intensity in terms of the five tensors given in equations 1-5, it is useful to introduce the following 13 invariants: 4,27,28 5



1 s s {αif,ηη }? α2 = < αif,ρρ 9 1 s s s s }? {αif,ηη }? − αif,ρρ {αif,ρη βs (α)2 = < 3αif,ρη 2 3 a a βa (α)2 = < 3αif,ρη }? {αif,ρη 2 1 s ? s αG = = αif,ρρ {Gif ηη } 9 1 s s s s βs (G)2 = = 3αif,ρη {Gif,ρη }? − αif,ρρ {Gif,ηη }? 2 3 a a βa (G)2 = = 3αif,ρη {Gif,ρη }? 2 1 s 0s 0 ? αG = = αif,ρρ {Gif,ηη } 9 1 s 0s 0s s βs (G 0 )2 = = 3αif,ρη }? }? − αif,ρρ {Gif,ηη {Gif,ρη 2 3 a 0a βa (G 0 )2 = = 3αif,ρη {Gif,ρη }? 2 ω s βs (A)2 = = iαif,ρη (ρδλ Aif,δ,λη )s ? 2 ω 2 a a βa (A) = = iαif,ρη (ρδλ Aif,δ,λη )a ? + iαif,ρη (ρλδ Aif,λ,δλ )a ? 2 s ? ω s ρδλ A0if,δ,λη βs (A0 )2 = = iαif,ρη 2 a ? a ? ω a a ρδλ A0if,δ,λη βa (A)2 = = iαif,ρη + iαif,ρη ρηδ A0if,λ,δλ 2

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(6) (7) (8) (9) (10) (11) (12) (13) (14) (15) (16) (17) (18)

where the Einstein conventions for the summation over the repeated indexes has been employed. Furthermore, the complex conjugate is represented by the ? superscript, whereas the s and a superscripts indicate the symmetric and the antisymmetric components of a bidimensional tensor, defined as, Tif,ρη + Tif,ηρ 2 Tif,ρη − Tif,ηρ = 2

Tsif,ρη = Taif,ρη

(19)

where T can be any tensor among the ones given in Eqs. 1-5. It should be noted that ρδλ Aif,δ,λη and ρλδ Aif,λ,δλ are two-dimensional tensors, with indexes ρ and η, since a con6


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traction over λ and δ is implicitly done in their definition. Several scattering geometries can be used experimentally to record a ROA spectrum. 28–30 Raman optical activity is measured as the difference between Raman scatterings associated to right (R) and left (L) circularly polarized light. However, the measurement setup is not unique, as either the incident (incident circular polarization, ICP) or the scattered (SCP) radiation beams can be modulated to get the right and left circular polarization states. Alternatively, both beams can be simultaneously modulated (dual circular polarization, DCP), either in-phase (DCPI ) or out-of-phase (DCPII ). In addition to the polarization, also the angle between the directions of the incident and the scattered lights can be set. In most experiments, the relative angle is set to 0◦ (forward scattering), 90◦ (right angle scattering) or 180◦ (backward). A third parameter which can be modified is the angle between the transmission axis of the linear polarization analyzer and the plane of the scattered light. In this case, four choices are commonly used: 1. 0◦ , corresponding to the depolarized scattering (referred to in the following with a z subscript) 2. 90◦ , corresponding to the polarized scattering (referred to in the following with a x subscript) 3. the so-called magic angle scattering (which corresponds to 54.74◦ ), which allows the removal of the contributions from the A and A tensors (referred to in the following with a ∗ subscript) 4. the so-called unpolarized scattering, where all possible angles are averaged out (referred to in the following with a u subscript) Depending on the setup, the invariants given in Eq. 6-18 must be combined differently to compute the differential cross section of the scattering processes. In order to derive a general formulation, supporting any combination of the three parameters described above, the following relation can be employed: 7



where

dσ dΩ if

dσ dΩ

if

2 π ωif 4 NA = IROA 0 2πc 90

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(20)

is the differential cross-section for the transition between the states |Ψi i and

| Ψf i, 0 is the vacuum permittivity constant, c is the speed of light and NA is the Avogadro constant. IROA , also known as the ROA activity, is a combination of different invariants, which depends on the experimental setup for the ROA experiment. The definition of IROA for different setups can be found in Ref. 28 and is reported in Tab. 1. In order to simulate ROA spectra, a first step is to compute the tensors given in Eqs. 15. The five transition tensors can be expressed in the same formulation, using the general framework previously introduced for non-resonant 30 and resonant 31 ROA spectroscopy. The definitions given in Eqs. 1-5 can be merged in a single formulation defining the general tensor Tif as follows:

Tif,ρη

1 X = ~ m6=i,f

A B h Ψf | PB h Ψf | PA ρ | Ψm ih Ψm | Pη | Ψi i ρ | Ψm ih Ψm | Pη | Ψi i + ωmi − ωI + iγm ωmi + ωI + iγm

!

(21)

where P A and P B are general operators representing the electric dipole, the magnetic dipole or the electric quadrupole operator. The equivalence rules given in Tab. 2 can be used to recover the definition of all transition tensors given in Eqs. 1-5. For the electric dipole-electric quadrupole tensors (Aif and Aif ), the index η in Eq. 21 correspond to the couple of indexes (ηδ) in Eqs. 4 and 5. Within the theoretical framework described up to this point, no particular assumption has been made on the value of the frequency of the incident light (ωI ), and therefore the theory holds for both non-resonant and resonant conditions. In the latter case, that is when the energy of the incident light matches the transition to an electronic excited state (usually known as intermediate state, labeled as m in the following), the previous theory can be simplified. Indeed, the second term of Eq. 21 is usually much smaller than the first one and 8


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can be neglected. This term could be included within the so-called near-resonance (NR) formulation of RR and RROA spectroscopy, 32 but in the present work strong resonance conditions will be assumed in all cases. In order to further simplify Eq. 21, the BornOppenherimer approximation is used and the Eckart conditions are assumed met so that v | Ψm i can be factorized as | φm ψn(m) i, where φm is the electronic wavefunction of the v intermediate state, and ψn(m) are vibrational levels of this state. Assuming that there are

no other nearby electronic states, so only transitions involving m will contribute, the sum can be restricted to the vibrational levels of this intermediate electronic state, and Eq. 21 can be expressed as, A,e B,e v v v v 1 X h ψf (0 ) | Pm0 | ψn(m) ih ψn(m) | Pm0 | ψi(0 ) i Tif = ~ ωmi − ωI − iγ

(22)

n(m)

A,e B,e where Pm0 = h φm | P A | φ0 i and Pm0 = h φm | P B | φ0 i are the electronic components v of the transition properties between the ground and intermediate states and | ψi(0 ) i and

| ψfv(0 ) i are the vibrational levels involved in the transition. For the sake of clarity, in the following, the electronic transition moments will be simply specified with the “m” subscript instead of “m0”. Furthermore, the “e” superscript will be dropped out. Both PmA and PmB are functions of the nuclear coordinates, and this dependence is usually approximated as a Taylor series in powers of the normal coordinates Q of one of the electronic states:

PmX (Q)

=

PmX (Qeq )

+

N X ∂P X m

k=1

= PmX,0 +

N X k=1

∂Qk

eq

Qk + Q2

PmX,k Qk + Q

(23)

2

where X can be either A or B. The zero-th order term is usually referred to as the FranckCondon (FC) term, 33,34 and the first-order one as the Herzberg-Teller (HT) term. 35 Furthermore, the affine transformation proposed by Duschinsky 36 will be employed to relate the normal modes of the initial (Q) and intermediate (Q) states:

9



Q = JQ + K

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(24)

where J is the Duschinsky matrix and K the shift vector. The definitions of J and K changes with the harmonic model used for the description of the intermediate-state PES. Two different classes of models are usually employed, the adiabatic ones, where the intermediatestate PES is expanded about its equilibrium geometry, and the vertical ones, where the expansion is performed about the equilibrium geometry of the initial, ground state. When the full transformation given in Eq. 24 is employed, the Adiabatic Hessian (AH) and Vertical Hessian (VH) models are obtained. The corresponding simplified Adiabatic Shift (AS) and Vertical Gradient (VG) models are obtained by neglecting mode-mixing effects (and therefore assuming that J is the identity matrix I) in AH and VH, respectively. It should be noted that the cost of the computations needed to generate the input data for AS and VG models is lower than for their AH and VH counterparts, since the calculation of the harmonic frequencies of the intermediate state, which is usually the most expensive step, can be avoided. The Duschinsky transformation given in Eq. 24 is effective for semi-rigid systems, but is usually ill-suited for flexible systems. As shown in Refs. 25,26, Eq. 24 can be generalized to support curvilinear internal coordinates, and this generalization improves the description of flexible systems in both one-photon (OP) 25,37–40 and RR 26 spectroscopies. Using Eq. 23, it is possible to write the sum-over-states expression given in Eq. 22 in terms of overlap integrals between vibrational levels of the ground and intermediate electronic states v v v v | ψi(0 (hψn(m) ) i and hψn(m) | ψf (0 ) i), usually referred to as Franck-Condon integrals. In the so-

called time-independent (TI) approaches, the Franck-Condon integrals are computed, within the harmonic approximation, using either analytical 41,42 or recursive 43,44 expressions, and the tensors are computed directly by evaluating the sum in Eq. 22. Several TI-based approaches have been proposed in the literature, 45–47 including the one recently proposed by some of the present authors and supporting both RR 19,24 and RROA 48 spectroscopies. However, the reliability of this model is hindered by some critical issues. First of all, the number of 10


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vibrational levels of the intermediate state is in principle infinite, thus the sum in equation 22 is virtually unbound. Therefore, to compute the transition tensors, an algorithm to select, preferably a-priori, the most relevant integrals is necessary. In our previous works, 19,48 the class-based prescreening scheme developed for one-photon spectroscopies 20 has been used to select the integrals to be included in the sum-over-state expression (Eq. 22). However, the efficiency of this scheme for RR spectroscopies is difficult to assess. It should be noted that this prescreening scheme is more efficient when the probability of transition, which can be related to the FC integrals, and the transition intensities are correlated. This condition is rarely met for chiroptical spectroscopies like ECD or CPL 13,49 and is expected to worsen for RROA, which involves additional transition tensors. Another hurdle, not present for ECD and CPL, is the impossibility to use analytic sum rules to assess precisely the reliability of the prescreening since each term of the sum in Eq. 22 is weighed by the relative energies of the vibrational states involved in the transition through the denominator. The only possibility is then to treat the denominator of Eq. 22 as a constant, which nevertheless introduces an approximation of unknown validity, especially under conditions of strong resonance, which significantly limits the interest of using this reference value. In order to overcome those limitations in the TI approach, a time-dependent (TD) model, which will be described in the next section, is better suited to perform the computations, especially to treat large-size systems. It should be noted that TD can then provide a mean to check the validity of TI, which can then be used to evaluate the enhancement effects from the individual transitions involving each intermediate vibrational state.

2.2

Time-Dependent Resonance Raman Optical Activity

Before describing the extension of the TD formalism to RROA spectroscopy, it is useful to recall its application to RR. 50–52 As already discussed above, among the five tensors given in Eqs. 1-5, only αif is needed in RR. By using the Fourier transform properties, the sumover-state formula given in Eq. 1 can be expressed in terms of time-dependent quantities as 11



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follows: 23,26,53–55

αif,ρη

i = 2 ~

Z 0

+∞

ˆ

−it(ωad −ωI −iγ) v dt h ψfv(0 ) | µem,ρ e−iHm t/~ µem,η | ψi(0 ) ie

(25)

ˆ

v h ψfv(0 ) | µem,ρ e−iHm t/~ µem,η | ψi(0 ) i is called cross-correlation function, and depends on v v the vibrational states involved in the RR transition | ψi(0 ) i and | ψf (0 ) i. Several strategies

can be implemented to compute αif,ρη starting from Eq. 25. The time-evolution of the cross-correlation function can be simulated using full quantum dynamics approaches, by solving explicitly the time-dependent Schrödinger equation. 56,57 However, the computational cost of those approaches is usually high, since they often require an accurate sampling of the intermediate-state PES. A significant reduction of the computational efforts can be achieved by assuming the PESs of both initial and intermediate electronic states to be harmonic. Under this hypothesis, the Feynmann path-integral theory can be employed to derive analytical, closed-form expressions for the cross-correlation functions. It is then possible to compute the polarizability tensor through a numerical evaluation of the Fourier integral of Eq. 25. While simplified vibronic models are often employed to compute the crosscorrelation function, 58,59 the TD framework can be straightforwardly extended to support, at the harmonic level, both mode mixing and Herzberg-Teller effects, 23,55,60,61 at the same level of approximation. The TD and TI formulations are formally equivalent, but the reliability of the TD approach is significantly higher due to the automatic inclusion of all transitions, so no prescreening schemes are needed to obtain the RR spectrum. On the other hand, the integral of Eq. 25 must be evaluated numerically, but this can be done efficiently as discussed in Refs. 23,26. It should be also noted that, for OP spectroscopies, the TI and TD approaches are complementary, the latter giving access to the overall band-shape with a limited computational cost, whereas the former provides information for the assignment of vibronic bands observed in the spectrum. 22,62 This is not true for RR spectroscopy, where both approaches provide the same data for each individual band, so that the TD approach is usually more suitable than its TI counterpart due to the higher reliability. 12


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The TD approaches described above are usually limited to RR spectroscopy, and an extension to RROA has been proposed, to the best of our knowledge, only in the so-called short-time limit, 14,15 which is well suited for pre-resonance conditions, but shows limitations in the presence of strong resonances, since the shape of the PES of the intermediate state is not fully taken into account. The TD formulation has been used in Ref. 31 to derive formal properties of the different transition tensors, but the computations were performed at the TI level, using the independent mode harmonic oscillator model (IMDHO, equivalent to the VG model described above). 54,55,58 Here, both mode-mixing and HT effects in Cartesian and in internal coordinates will be taken into account. The TD equivalent of Eq. 22 can be derived starting from Eq. 25 and using the generic transition properties PmA and PmB instead of the electric dipole transition moment:

Tif,ρη

i = ~

Z 0

+∞

ˆ

v −it(ωad −ωI −iγ) −iHm t/~ B dt h ψfv(0 ) | PA Pm,η | ψi(0 ) ie m,ρ e

(26)

Eq. 26 can be expressed in a more compact way by neglecting temperature effects, so v that the initial state | ψi(0 ) i is the vibrational ground state, labeled in the following as | 0 i.

Furthermore, only fundamental transitions will be considered, so that in the final state only one mode, of index k, is excited. Using the Dirac notation, the final states will be labeled as | 1k i. Finally, for the moment, the Franck-Condon model will be used to approximate PA m,ρ B,0 A,0 and PB m,η as constants, labeled as Pm,ρ and Pm,η (the inclusion of HT effects will be discussed

in the next subsection). Those approximations can be used to rewrite Eq. 26 as follows,

T0k,ρη

i B0 = PA0 m,ρ Pm,η ~

Z 0

+∞

ˆ

dt h 1k | e−iHm t/~ | 0 i e−it(ωad −ωI −iγ)

(27)

where subscripts i and f have been substituted with the vibrational states involved in the ˆ

transition, 0 and k. The cross-correlation function h1k | e−iHm t/~ | 0i, labeled in the following as χRR 0k (t), can be expressed in terms of the so-called Franck-Condon autocorrelation function ˆ

χFC (t, T = 0), defined as h 0 | e−iHm t/~ | 0 i. 23 The time-dependent autocorrelation function 13



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plays a key role in the TD formulation of OP spectroscopy, since its Fourier transform is proportional to the OP vibronic spectrum. Within the approximations outlined above, the following analytical expression for χFC (t, T = 0) can be derived, as discussed in Refs. 22,63– 65

χFC (t, T = 0) = F (t) e−K

T ΓK+v T D −1 v

,

(28)

Γ being the diagonal matrix of the reduced frequencies of the initial state. Furthermore, the normalization factor F (t), as well as vector v and matrix D are time-dependent quantities, whose complete definition is given in the Appendix. The cross-correlation function χRR 0k (t) can be expressed in terms of χFC (t, T = 0) based on the properties of the eigenfunctions of the harmonic oscillator Hamiltonian. 23 By defining η = D −1 v + D −1 v † , the relation is the following (see the Appendix for details on the derivation): s χRR 0k (t) =

Γk 2

N

1 X Jkl ηl + Kk −√ 2 l=1

! χFC (t, T = 0)

(29)

Combining Eq. 27 and Eq. 29, the element (ρσ) of the general transition operator Tm ρς takes the form,

s T0k,ρσ

i = ~

Γk A,0 B,0 P P 2 m,ρ m,σ

Z

+∞

dt

0

1 K − √ Jη 2

χFC (t, T = 0) e−it(ωad −ωI −iγ)

(30)

k

It should be noted that evaluating the previous integral takes the same time as in TD OP, since η is already needed to compute χFC (t, T = 0). Furthermore, the multiplication by the Duschinsky matrix, that would, in principle, increase the computational time, can be performed after the integration, since J is a time-independent quantity. In Eq. 30, the transition tensor T0k,ρσ is factorized as the product between a purely elecB,0 RR tronic term (PA,0 m,ρ ×Pm,η ), and the Fourier transform of the cross-correlation function χ0k (t),

14


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that is determined only by nuclear properties. A direct consequence of this factorization is that only a limited subset of the invariants among the ones given in Eqs. 6-18 are actually needed to compute the RROA spectrum. In fact, by choosing, without loss of generality, the z axis of the reference system along the electric dipole moment µ, the only non-null element of αif is αif,zz . As a consequence, αaif = 0 and βa (G) = βa (G) = βa (A) = βa (A) = 0. Furthermore, βs (A) = 0, since, in Eq. 15, αif,ρη 6= 0 only for ρ = η = z and, for the same reason, Asif,λ,γη 6= 0 only for λ = z, but in those cases zzγ = 0, and therefore βs (A) = 0. For a similar reason, βs (A) = 0. The same relations have been derived by Nafie and co-workers in Refs. 7 and 32 within the so-called single electronic state limit (SES), where mode-mixing and frequency change effects are neglected and only one resonant electronic state is considered. However, the previous discussion shows that the same relations hold also if mode-mixing and frequency changes effects are included, provided that the Franck-Condon approximation is used. When multiple excited states are included in the treatment, the RROA spectrum can be computed from the relations given in Eqs. 6-18, and the final transition tensors are obtained as a sum of the single component involving one intermediate state at a time over all the Nres resonant states:

Tif,ρη =

Nres X

(m)

Tif,ρη

(31)

m=1 (m)

where Tif,ρη is the transition tensor obtained for the m-th resonant state. 19,31 In this case, the transition dipole moments for each transition has, in general, a different orientation, and therefore it is not possible to choose a coordinate system where the transition dipole moments are aligned along the same axis. Therefore, the simplified relations derived for the FC case do not hold for multiple resonant excited states. 48,66

15



2.3

Page 16 of 72

Inclusion of Herzberg-Teller effects

Several generalizations of the TD approach to support also Herzberg-Teller effects both for one-photon 67 and for RR spectroscopy 61 have been proposed in the literature. As already done for FC, the general framework already developed for OP 22 and RR 23 spectroscopies will be employed here. In practice, if HT terms are included in Eq. 26, additional contributions to the cross-correlation function appear, which can be divided in three classes, listed in the following: ˆ

−iHm t/~ Ql | 0 i χRR,l 0k (t) = h 1k | e ˆ

−iHm t/~ χRR |0i 0k,l (t) = h 1k | Ql e

(32)

ˆ

−iHm t/~ χRR,l Ql | 0 i 0k,j (t) = h 1k | Qj e

Following the approach previously developed for the RR case, 22,23 the cross-correlation functions introduced in Eq. 32 can be expressed in terms of the following quantities, ˆ

χl (t, T = 0) = h 0 | e−iHm t/~ Ql | 0 i ˆ

χl (t, T = 0) = h 0 | Ql e−iHm t/~ | 0 i ˆ

χlj (t, T = 0) = h 0 | Qj e−iHm t/~ Ql | 0 i

(33)

ˆ

χjl (t, T = 0) = h 0 | Qj Ql e−iHm t/~ | 0 i ˆ

χljm (t, T = 0) = h 0 | Qj Qm e−iHm t/~ Ql | 0 i which can then be computed following the procedure described in the Appendix. The final relations are,

16


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χRR,l 0k (t)

l

= Kk χ (t, T = 0) +

χRR 0k,l (t) = Kk χl (t, T = 0) + l χRR,l 0k,j (t) = Kk χj (t, T = 0) +

Nvib X i=1 N vib X i=1 N vib X

Jki χli (t, T = 0) Jki χil (t, T = 0)

(34)

Jki χjil (t, T = 0)

i=1

It should be noted that the formulation presented here, even if formally equivalent to that introduced in Ref. 23, is more compact and clearly highlights the connection between the quantities computed in one-photon and resonance Raman spectroscopies. The general transition tensor T defined in Eq. 21 can be expressed, using the definitions introduced in Eq. 26, as,

T0k,ρσ

i = 2 ~

+∞

Z

h i FCHT,1 FCHT,2 HT dt χFC + χ + χ + χ 0k,ρσ 0k,ρσ 0k,ρσ 0k,ρσ

(35)

0

FCHT,2 where the three Herzberg-Teller cross-correlation functions (χFCHT,1 and χHT 0k,ρσ ) are 0k,ρσ , χ0k,ρσ

defined as, A,0 B,0 RR χFC 0k,ρσ (t) = Pm,ρ Pm,σ χ0k (t)

χFCHT,1 0k,ρσ (t) χFCHT,2 0k,ρσ (t)

= =

χHT 0k,ρσ (t) =

N X l=1 N X l=1 N X

RR,l B,l PA,0 m,ρ Pm,σ χ0k (t)

PA,l m,ρ

PB,0 m,σ

χRR 0k,l (t)

(36)

RR,j B,l PA,j m,ρ Pm,σ χ0k,l (t)

jl=1

2.4

Gauge Invariance

As is well known, for a given molecular structure, the definition of the transition properties µm , mm and Θm is not univocal, since it changes with the origin of the reference axes used in the simulation. 15,48,68–70 This effect is usually known as gauge-dependence. Let us

17



Page 18 of 72

consider, for example, two reference systems with origin in X0 and X0 + a, respectively. The transition properties in the first (µm (X0 ), mm (X0 ), Θm (X0 )) and in the second reference system (µm (X0 + a), mm (X0 + a), Θm (X0 + a)) are related by the following transformations: µm (X0 + a) = µm (X0 ) 1 mm (X0 + a) = mm (X0 ) + a × pm 2 3 3 Θm,ρη (X0 + a) = Θm,ρη (X0 ) − aρ µm,η − aη µm,ρ + a · µm δρη 2 2

(37)

where δρη is the Kronecker delta. Eq. 37 can be used to derive analytical relations between the tensors computed in the two different reference systems. For the sake of simplicity, we will use the Franck-Condon approximation to derive those relations, and we will include only a single resonant state. Under those approximations, the transition tensors become 48 αif (X0 + a) = αif (X0 ) i Gif,ρη (X0 + a) = Gif,ρη (X0 ) + ωad ηγλ aγ αif,ρλ 2 i Gif,ρη (X0 + a) = Gif,ρη (X0 ) + ωad ργλ aγ αif,λη 2 3 3 Aif,ρ,ηλ (X0 + a) = Aif,ρ,ηλ (X0 ) − aη αif,λρ − aγ αif,ρη + ληλ aη αif,ρη 2 2 3 3 Aif,ρ,ηλ (X0 + a) = Aif,ρ,ηλ (X0 ) − aη αif,ρλ − aγ αif,ηρ + ληλ aη αif,ηρ 2 2

(38)

Despite their origin-dependence, once introduced in Eq. 38 to compute the invariants, the terms depending on the origin shift a cancel out. However, the transformation rules given in Eq. 37 are valid only if the transition properties are obtained using an electronic wavefunction expressed in a complete basis set. When a finite-size basis set is employed, the relation between mm (X0 + a) and mm (X0 ) does not hold anymore, 71 and therefore the gauge-invariance of the RROA intensities is not guaranteed. This limitation can be overcome within the so-called gauge-independent atomic orbitals (GIAO, also known as London orbitals), 72,73 in which a magnetic field-dependent phase factor is included in order

18


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to remove the gauge dependence of the transition magnetic dipole moment also for finite basis sets. Another solution is to compute the electric transition dipole and quadrupole moments using the so-called velocity representation (labeled in the following as µpm and Θpm , respectively). 71,74 The relation between µpm and Θpm and the their length gauge counterparts (indicated with a r superscript) is,

µrm

Nat X i~ i~ =− h φm | p i | φ0 i = − µp Em − E0 i=1 Em − E0 m

Nat X i~ i~ r h φm | p i ⊗ r i | φ 0 i = − Θm = − Θp Em − E0 i=1 Em − E0 m

(39)

where E0 and Em are the electronic energies of the ground and intermediate electronic states, respectively, computed at the same geometry used for the calculation of the transition properties. In this case as well, the transition tensors are origin-dependent, but this dependence cancels out for the invariants. 48 However, at variance with the length representation, in this case, the gauge-invariance holds also for finite basis sets, and therefore stands for practical quantum-chemical calculations. For this reason, in the following, all transition properties have been computed in the velocity representation. The previous discussion is valid only at the FC level and including only a single intermediate state. As discussed in Ref. 48, further inclusion of HT effects requires the transformation of the first-order derivatives of the transition properties from length to velocity representations. This can be done by differentiation of both sides of Eq. 39 with respect to the normal mode Qk , 48

iµpm ∂ωm i ∂µpm ∂µrm = 2 − ∂Qk ωm ∂Qk ωm ∂Qk r p ∂Θm iΘm ∂ωm i ∂Θpm = 2 − ∂Qk ωm ∂Qk ωm ∂Qk where ωm =

Em −E0 ~

and

∂ωm ∂Qk

can be expressed as

19

1 ~

(40) (41)

gk − gk , where g and g are the gradients



Page 20 of 72

of the initial- and intermediate-state PESs computed at the same geometry as for the transition properties. This reference geometry is different whether adiabatic or vertical models are used. For vertical models, all quantities are computed at the equilibrium geometry of the electronic ground state, and therefore g = 0 and g is the gradient of the intermediate-state PES which used in the definition of the Duschinsky relation. 49 Finally, the energy separation ωm is the vertical excitation energy. The calculation is less trivial for adiabatic models, since the transition properties are computed at the equilibrium geometry of the intermediate-state PES. For this nuclear configuration, g = 0, whereas g is in general non null, but unneeded for the Duschinsky transformation. In order to avoid an additional gradient computation, g can be extrapolated using the harmonic approximation of the initial state PES as follows,

g=

1 eq H X − X eq ~

(42)

where H is the Hessian of the initial state, whereas X eq and X eq are the equilibrium geometries of the initial and intermediate electronic states, respectively. The energy separation ωm can be computed using the same technique, T 1 eq 1 eq eq eq E−E+ X −X H X −X ωs = ~ 2

(43)

In Ref. 48 it is shown that, even if the derivatives of the transition properties are computed within the velocity representation using the transformation given in Eq. 40, the gauge invariance of the RROA intensities cannot be enforced. In fact, in order for invariants βs (G)2 , βa (G)2 , βs (A)2 , βa (A)2 , as well as their counterparts with G and A, to be gauge invariant, the polarizability tensor must be symmetric, and this is not the case when HT effects are included. To conclude, let us remark that the issue of the origin-dependence is even more problematic when multiple intermediate states are included in the simulation. In fact, in this case an additional gauge-dependent term appears in the definition of the βs (G)2 and βs (G)2 invariants. A possible solution to this problem has been proposed in Ref. 48, by assuming

20


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that all resonant states have the same excitation energy, which is set to be the average of the excitation energies of the resonant states.

21



3

Computational details

All computations have been performed with a modified version of the Gaussian 16 suite of quantum chemical programs 75 extended to support the theoretical simulation of RROA spectra. Calculations were performed at the density functional theory (DFT) level and its time-dependent extension (TD-DFT) using the B3LYP exchange-correlation functional 76 in conjunction with the SNSD basis set, 77 obtained by adding to the 6-31G(d,p) basis set a minimal number of core-valence (one s function on all non-hydrogen atoms) and diffuse (s,p,d on non-hydrogen atoms and p on hydrogen atoms) functions optimized specifically for global hybrid functionals. 78,79 For 2-Br-Hexahelicene, electronic structure calculations were performed with the 6-311G** basis sets on all atoms. 80,81 If not specified otherwise, the standard parameters of the class-based prescreening scheme, which can be found, for example, in Refs. 25 and 21 have been used for all TI vibronic simulations. For TI RROA, a modified version of the program described in Ref. 19 was used. TD calculations on methyloxirane were performed by sampling the cross-correlation function in 212 points for a total propagation time of 10−13 s. For naproxen-OCD3 and quinidine, a more refined grid of 218 points was used in order to ensure convergence of the numerical integration. An higher number of points is needed to converge the integration especially when the incident wavelength is different from the energy of the 0-0 transition. In those cases, the oscillating factor present in Eq. 27, makes the variation of the cross-correlation function more rapid, and for this reason a more refined grid is required for its correct representation. To designate the vibronic model employed in the simulation, the approximation for the PES (VG, VH, AS, AH) and the electronic transition moments (FC, FCHT) will be combined in a single label (as, for instance, AH|FC). If not specified otherwise, the damping factor γm was set to 100 cm−1 for all transitions, accordingly to Ref. 18. It should be noted that, in principle, the damping constant is related to the homogeneous broadening to be applied to match the theoretical spectrum with its experimental counterpart. Due to the lack of sufficiently accurate experimental results, this 22


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procedure could not be followed in the present work. Regarding internal coordinates, primitive internal coordinates (PICs) have been generated from the molecular topology as described in Ref. 25. The redundant Wilson B matrices have been computed using analytical expressions for bond lengths, valence, out-of-plane and dihedral angles. 82? In the calculation of the single value decomposition (SVD) of the redundant B matrix to generate delocalized internal coordinates (DICs), a threshold of 10−5 has been used to select the singular vectors associated to non-negligible singular values. Solvation effects were simulated by mean of the polarizable continuum model (PCM) 83,84 in its integral-equation formulation (IEF-PCM). 85 The solute cavity was built by using a set of interlocking spheres centered on the atoms with the following radii (in ˚ A): 1.443 for hydrogen, 1.926 for carbon and 1.750 for oxygen, each scaled by a factor of 1.1, which is the default value in Gaussian. The solvent static and dynamic dielectric constant used here are = 2.228 and ∞ = 2.132 for CCl4 , = 78.355 and ∞ = 1.778 for H2 O and = 4.7113 and ∞ = 2.091 for H2 O. As mentioned in our previous works, 19,23 different solvation regimes can be chosen for the simulation of resonance spectra. In this work, we will use as reference the equilibrium regime for the adiabatic models and the fixed-cavity regime for the vertical ones. Anharmonic calculations for naproxen-OCD3 have been performed using second-order vibrational perturbation theory (VPT2). The necessary third and semi-diagonal fourth derivatives of the potential energy were obtained by numerical differentiation of the analytical harmonic force constants along the mass-weighted normal coordinates (Q) with the √ Gaussiandefault step (δQ = 0.01 ˚ A× amu). 86,87 The so-called generalized VPT2 (GVPT2) model 21,88 was used to treat resonances. Fermi and Darling-Dennison resonances were identified through two-step procedures based on the difference of energy between the resonant states and the deviation of the VPT2 term from a model variational system (for the Fermi ones) or the magnitude of the variational correction term itself (for the Darling-Dennison ones). Fermi resonant terms were removed from the VPT2 calculations and reintroduced

23



subsequently through a variational treatment together with the Darling-Dennison ones, as discussed in Ref. 21,88 Naproxen-OCD3 displays several low-frequency, large-amplitude modes (LAMs) that are poorly described at a perturbative level based on a quartic force field. In order to get reliable results, a reduced-dimensionality (RD) scheme 88,89 has been adopted, where the anharmonicity of the modes with wavenumbers below 300 cm−1 , as well as their coupling with other modes through the cubic and quartic force constants, was neglected. Excited-state harmonic frequencies were computed using analytical TD-DFT secondorder derivatives of excitation energies 90,91 as implemented inside Gaussian. 92,93

24


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4

Results

4.1

(1R)-methyloxirane

Oxiranes are amongst the smallest chiral molecules and, due to their small size, they have been widely used to test the reliability of theoretical approaches for the simulation of vibrational 94–98 and vibronic chiroptical properties. 13,99–102 (1R)-methyloxirane (structure reported in Fig. 1) is a standard chiral prototype, used here to test the reliability of the implementation by comparing our results with those obtained using the TI implementation presented in Ref. 48. As already discussed in the previous section, the TD and TI approaches are formally equivalent when based on the same approximations (i.e. harmonic models of the PESs, approximation of the transition properties). Therefore, in the limit case where all vibronic transitions are included in the simulation (for the TI approach) and if a sufficiently accurate integration grid is used (for the TD approach), the same final spectrum must be obtained. As already recalled above, due to the unavailability of a reliable prescreening scheme to select the relevant vibronic transitions and of analytical sum-rules to check the convergence of the TI spectrum, it is difficult to evaluate the reliability of those simulations. Here, the prescreening developed for one-photon spectroscopies (including their chiral versions) 20,21,103 has been employed, using first the default parameters in Gaussian and the convergence of the band-shape has been checked by further increasing those parameters. Fig. 2 shows a comparison of the RROA SCP(180◦ ) spectrum of methyloxirane in full resonance with the S1 electronic state obtained using the TD and TI approaches with different vibronic models. This comparison shows that, for both vertical and adiabatic models, the TI and TD spectra are superimposable. This indicates that the TI simulation reached convergence with respect to the vibrational levels of the intermediate electronic state, and also confirms the reliability of our TD implementation. The results also show that the choice of the vibronic model has only a minor effect on the computed RROA spectrum in this case. This is a direct consequence of the rigidity of methyloxirane upon electronic excitation.

25



In fact, for rigid systems, the difference in the equilibrium geometries of the ground and intermediate electronic states is usually small, and in those cases vertical and adiabatic models provide nearly equivalent results. This also explains the very good performance of the prescreening for TI since a small number of low-quanta transitions are expected to contribute to each band of the resonance spectra. In order to highlight the full support of HT effects, the RROA SCP(180◦ ) spectra computed at the VG|FCHT level using both TI and TD approaches are reported in the left panel of Fig. 3. The two spectra are again superimposable, showing that the prescreening system is reliable enough to select the most relevant transitions in this case too. Comparing the VG|FCHT and the VG|FC spectra, it is evident that the FC term is dominant, and the further inclusion of the HT terms has a detectable effect only in the range between 800 and 1000 cm−1 , where the relative intensities of the bands increase with respect to the FC spectrum. In order to isolate the HT contributions, the spectra have been simulated by including only first-order terms in the Taylor expansion of the transition properties (results are labelled as VH|HT). Here too, the TI and TD spectra are equivalent. Unlike for VG|FC, the VG|HT spectrum displays both positive and negative bands, the most intense one being negative, and therefore with an opposite sign with respect to those of the VG|FC spectrum. Those results are a direct consequence of the breakdown of the single electronic state (SES) approximation 7,32 (which predicts the RROA spectrum to be monosignated) due to the inclusion of HT effects. It should be also noted that, even if the inclusion of HT effects increases the relative intensity of the bands in the 800-1000 cm−1 range, the sign of those bands is negative in the VG|HT spectrum. This does not contradict the theory presented above since the VG|FCHT spectrum is not simply the sum of the VG|FC and VG|HT ones, but includes also cross terms, related to the product between zeroth- and first-order contributions.

26


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4.2

Naproxen

The second test case is the methyl-D3 ester of (S)-(+)-6-methoxy-α-methyl-2-naphtaleneacetic acid, denoted in the following as naproxen-OCD3 (structure reported in Fig. 1), which is one of the first organic molecules for which an RROA spectrum has been reported, together with other ibuprofen derivatives, by Nafie and co-workers. 8 From a computational point of view, the only theoretical analysis of the RROA spectrum of Naproxen-OCD3 has been performed by Reiher and co-workers, 31 and has been recently extended to the free acid forms of ibuprofen and naproxen. 104 The first work uses a sum-over-states theory based on the Kramers-Krönig transform 105,106 in the framework of a VG|FC vibronic model. This analysis highlights the relevance of interference effects between the S1 and S2 states in the final computed spectrum. The second study, on naproxen and ibuprofen, is based on a different theoretical formulation, relying on the complex polarizability approach introduced by Schatz and co-workers, 14,15 where only electronic contributions are included. Those analyses will be complemented here by our TD implementation. In particular, both vertical and adiabatic models will be employed, checking also the effects of the choice of the coordinate system on the spectral bandshape, with the possible inclusion of HT effects. The experimental RROA spectrum has been recorded with an incident frequency of 512 nm, thus significantly distant from the first intense absorption band, which is detected around 340 nm. 8 However, the experimental spectrum is monosignated, and this has been interpreted as a signature of resonance enhancement. Indeed, as already discussed in the previous section, RROA spectra are expected to have only one sign under the SES limit. At variance, this enhancement is not detected in the ROA spectrum of ibuprofen, whose electronic spectrum is blue-shifted with respect to naproxen-OCD3 . 8,104 In addition to the band at about 320 nm, a second, more intense band, is present at higher energy, but its maximum cannot be determined from the experimental spectrum since its intensity is higher than the upper limit of the spectrum. As discussed in Ref. 31, simulations performed at the TD-DFT level predict two bands, one at about 320 nm and a second one at about 280 nm. However, 27



the computed intensity of the lowest-energy transition is higher than the other one, in disagreement with experimental findings. For this reason, in the calculations presented in Ref. 31, adiabatic excitation energies were shifted in order to match the experimental spectrum. With the aim of extending this analysis, the excitation energies and oscillator strengths of the two lowest-energy excitations have been computed using different exchange-correlation functionals, and the results are reported in Tab. S1. The trend noticed in Ref. 31 is confirmed, since, in all cases the first transition has an higher oscillator strength than the second one. Moreover, transitions with high oscillator strength are present at about 250 nm. Even if those transitions are outside the energy range of the experimental spectrum, they might have a non-negligible effect in the resonance enhancement, and will be considered in the following analysis. It is also noteworthy that excitation energies are significantly overestimated by all long-range corrected (CAM-B3LYP and LC-ωPBE) as well as Minnesota (M06-2X and MN15) functionals tested. For this reason, B3LYP and PBE0 hybrid functionals will be considered in the following. The experimental 8 and theoretical OPA and ECD spectra, computed at the VG|FC level, are reported in Figs. 4 and 5. As expected, the computed intensity of the high-energy tail of the OPA spectrum is lower than the experimental one. However, the increase of the intensity in the experimental spectrum at about 300 nm can be ascribed to the overlap of this band with the tail of the intense transitions at about 250 nm. For this reason, in contrast with the analysis reported in Ref. 31, the vertical excitation energies have not been shifted. The discrepancy between theoretical and experimental results is less evident for the ECD spectrum, reported in Fig. 5, even if in this case as well the intensity of the second band is slightly underestimated with respect to experiment. Those results seem to confirm the reliability of the magnetic dipole transition moments computed at the B3LYP/SNSD level, which are then used for the calculation of the G and G tensors. Thus, provided that the quadrupole transition moments are described correctly, a reliable reproduction of the RROA spectrum is expected.

28


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The computed RR spectra involving the S1 and S2 electronic states, with different vibronic models and coordinate systems are reported in Fig. S1 and Fig. S2 of the Supporting Information (SI). For S1 , the AH|FC results in Cartesian and internal coordinates are nearly superimposable, and only minor differences are present between 800 and 1200 cm−1 . This similarity suggests that the geometry deformation upon electronic excitation is small, since this is the only condition for which the results obtained in the two different representations are equivalent. This is further confirmed by the fact that adiabatic and vertical results are equivalent as well. Indeed, the choice of the reference geometry of the harmonic expansion is expected to have a minor impact on the spectrum only for small geometrical deformations. For the S2 state, the similarity disappears. For instance, significant differences are present between the AH|FC spectra in Cartesian and DICs (left panel), especially in the lower-energy region of the spectrum, below 1300 cm−1 . The graphical representation of the equilibrium geometries of the S0 and S2 states, reported in Fig. 6, shows that the electronic excitation is accompanied by a torsion along the dihedral angle between the naphtyl moiety and the carboxyl group. Because of this large amplitude motion (LAM), Cartesian and internal coordinates give different results. It is noteworthy that the spectra computed using vertical models (right panel of Fig. S2) are equivalent to the adiabatic ones in internal coordinates. As discussed in our recent papers, 25,26 this equivalence, that has been already noticed for other systems, 39 is caused by the fact that, within the simpler VG model, Cartesian and internal coordinates are equivalent. Based on those observations, the full RR spectrum, obtained by properly including resonance effects between the two excited states is reported in Fig. 7. The AH|FC spectrum provides a satisfactory reproduction of the band-shape on a qualitative level, especially in the region about 1400 cm−1 . The agreement worsens at about 1600 cm−1 , where the experimental spectrum displays two bands, whereas in the theoretical results only a single band is present. In order to improve the accuracy of the theoretical results, the anharmonic fundamental wavenumbers of the ground (S0 ) state were used in place of the harmonic ones. As expected, this causes a red-shift of most bands,

29



and a significant improvement in the position of the band at about 1400 cm

Page 30 of 72

−1

. However,

no improvement is detectable at about 1600 cm−1 , where the intense band present in the experimental spectrum is not present in the theoretical one. This band is actually detectable in the spectrum computed at the AH|FCHT level, thus indicating that FC simulations are not sufficient to reproduce correctly the experimental results. However, the inclusion of HT effects worsens the agreement between theoretical and experimental results in the region at about 1400 cm−1 , where the relative intensity of the bands becomes underestimated. As shown in Fig. S2, the RR spectrum obtained in resonance with the S2 state displays an intense band at about 1400 cm−1 thus, as already noticed in Ref. 31, the discrepancy might be caused by an incorrect reproduction of the relative dipole strengths of the two excited states, leading to an incorrect reproduction of the overall bandshape. Based on the previous results, the RROA spectrum has been computed at the AH|FC level including both S1 and S2 electronic states in the treatment. The results are reported in Fig. 8. As expected, the RROA spectrum computed at the AH|FC level is proportional to the RR one, displaying two intense bands, one at about 1400 cm−1 and a second one at about 1600 cm−1 . The RROA spectrum is negative, and thus has opposite sign with respect to the ECD one, in agreement with the analysis in Ref. 8. Similarly to what has been already noticed for RR, inclusion of HT effects increases the intensity of the band at about 1650 cm−1 , leading to a significant overestimation of the experimental value. Finally, the effects of higher excited states have been evaluated by simulating the full RROA spectrum obtained by including also the S3 , S4 and S5 excited states in the simulations. In fact, as already shown in Fig. 4 and Fig. 5, even if the separation between the excitation energies of those states and ωI is larger than with the S1 and S2 states, they have an higher rotatory strength, and therefore their resonance effect could be non-negligible. The RROA spectra of the S3 , S4 and S5 excited states were computed at the VG level, in order to avoid geometry optimizations of such highly excited electronic states. The results reported in Fig. 9 show that the inclusion of those states does not change significantly the bandshape

30


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computed at the FC level, but lowers the intensity of the band at about 1600 cm−1 , leading to a better agreement with the experimental results. Another important aspect, not considered in the present study, is that the non-resonant terms for those states could also give non-negligible contributions to the overall spectrum.

4.3

Quinidine

The third test-case studied in the present work is quinidine (structure reported in Fig. 1), a stereoisomer of quinine that has been widely studied due to its pharmaceutical interest. 107,108 Several spectroscopic investigations of quinidine are available in the literature, ranging from standard one-photon absorption and emission 109 to vibrational circular dichroism 110 and pump-probe 111 spectroscopies. This system has been also used to detect resonance effects in the ROA spectrum in the pioneering work of Nafie and co-workers. 112 Unlike more recent experiments, in which the resonance ROA spectrum is recorded using the SCP(180◦ ) scattering configuration, 113,114 the RROA spectrum of quinidine was recorded in the DCPII (180◦ ) setup. In this scattering configuration, the RROA spectrum is proportional to the difference between the ICPu (180◦ ) and DCPI (180◦ ) ones. As already noticed in the literature, 112,115 the RROA DCPII (180◦ ) scattered intensity vanishes in the far-from-resonance approximation, thus a non-null DCPII usually indicates the presence of resonance effects. 112,115 Using the symmetry relations derived in the theoretical section, it can be proven that the DCPII (180◦ ) spectrum is null also in the single electronic state limit. For this reason, simulations including only one resonant state and based on the FC approximation are not sufficient to simulate DCPII spectra. Here, the relevance of both HT and resonance effects will be assessed by comparing the theoretical results with the experimental ones. 112 Before computing the DCPII (180◦ ) spectra, the ICP(180◦ ) and DCPI (180◦ ) RROA spectra of quinidine computed at the AH|FC and AH|FCHT levels in resonance with the first bright excited state (S2 ) of quinidine are reported in Fig. 10. As expected, the spectra obtained at the AH|FC level with the two scattering geometries are equal, and for this reason 31



the DCPII spectrum will be null. The inclusion of HT effects does not alter significantly the bandshape, since the S2 ← S0 electronic transition is fully allowed. The impact of HT effects is however not equivalent for ICP(180◦ ) and DCPI (180◦ ) spectra. This is particularly evident for the band at about 1700 cm−1 , whose enhancement due to HT effects is higher for the ICP(180◦ ) configuration/setup. Thus, the DCPII spectrum is expected to have an intense band in this region. A similar analysis is reported in figure S4 of SI for the RROA spectrum computed in resonance with the S1 electronic state. Even if the oscillator strength of the S1 state is significantly lower than that of the S2 state (0.038 and 0.141 respectively at the B3LYP/SNSD level), its excitation energy is closer to the energy of the incident laser used in the experiment (512 nm). For this reason, it is expected to have detectable interference effects. Similarly to the S2 state, the AH|FC ICP(180◦ ) and DCPI (180◦ ) spectra are equivalent. With respect to the spectrum of the S2 , intense bands are also present between 800 and 1200 cm−1 . On overall, HT effects are however less relevant than for the S2 state. The full theoretical RROA spectrum, computed at the AH|FC and AH|FCHT levels using the DCPII (180◦ ) scattering geometry by taking into proper account interference effects between the S1 and S2 electronic states are reported in Fig. 11, together with the experimental one. 112 In order to reproduce the experimental conditions, ωI was set to correspond to a wavelength of 512 nm. As expected, the most intense band of the experimental spectrum lies at about 1600 cm−1 , where HT effects have been shown to be more relevant, in good agreement with the experimental findings. 112 In the region between 1000 and 1400 cm−1 , the band-shape is significantly affected by the approximation used for the transition properties. In fact, the AH|FC spectrum is characterized by a second band, at about 1400 cm−1 , and the spectrum is near-null elsewhere. The AH|FCHT spectrum is more complex, and provides, in general, a better agreement with the experimental findings, especially in the region below 1300 cm−1 , where only the inclusion of HT effects allows for reproducing the pattern of low-intensity, negative bands that are present in the experimental spectrum. The agreement worsens for the bands at about 800 cm−1 , where the experimental spectrum shows an intense,

32


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positive band whose intensity is underestimated in the theoretical counterpart. However, it should be noted that the sign change is correctly reproduced by simulations. To conclude, let us remark that, due to the low-resolution of the experimental spectrum, 112 it is not trivial to distinguish between noise and real transitions. For this reason, more accurate experimental data would be necessary to assess more reliably the quality of the theoretical results.

4.4

2-Br-Hexahelicene

The last molecules studied in this work is 2-Br-Hexahelicene (structure reported in Fig. 1). Helicene and its derivatives have been extensively studied with different chiroptical spectroscopic techniques, including VCD, 116,117 ECD, 118 and CPL. 119 In the last years, RROA spectra of two helicene derivatives, 2-Br-Hexahelicene 120 and BisIron Ethynylcarbo[6]helicene, 121 have been reported in the literature. In the present work, we will focus on the first compound, 2-Br-Hexahelicene, for which a complete characterization of the RROA spectrum is still lacking. In fact, the original experimental work 120 already addressed the possibility of a resonance enhancement of the ROA spectrum. Nevertheless, spectra were interpreted based on off-resonance DFT calculations. Here, RROA spectra will be computed by assuming strong resonance conditions based on different vibronic models, to check if the inclusion of resonance effects improves the agreement with experimental results. The energy of the 5 lowest-energy excited states, together with their respective oscillator and rotatory strengths computed at the B3LYP/6-311G** level, are reported in Table 2 of the Supporting Information. In the single electronic state limit and at the FC level, the electric dipole-magnetic dipole contribution to the RROA intensity is proportional to the rotatory strength R. Hence, electronic states with an high value of R will be the ones giving the largest contribution to the RROA spectrum. As shown in Table 2 of the Supporting Information, the third and the fourth electronic states in ascending order of energy, labeled in the following as S3 and S4 , are the ones with the largest value of R. The other three states 33



Page 34 of 72

(S1 , S2 and S5 ) have a rotatory strength about 20 times smaller than S3 and S4 , thus they will be neglected in all simulations. The SCP(180◦ ) RROA spectra of 2-Br-Hexahelicene computed at the VG|FC and VG|FCHT level for full-resonance with the S3 and S4 states are compared in Fig. 12 to the experimental data, taken from Ref. 120. As expected, the bandshape is mainly determined by the S3 state, which has a negative value of R and thus gives a positive RROA spectrum. The intense, positive band at about 1400 cm−1 is correctly reproduced by the S3 RROA spectrum, while the second, positive band at 1500 cm−1 is shifted towards higher energy by ≈ 100 cm−1 . However, the experimental spectrum displays, at about 1400 and 1600 cm−1 , two weak negative peaks. The intensity of the spectrum obtained for the S4 state is not high enough to explain this change of sign with interference effects. As discussed in the theoretical section, another sign changes in RROA spectra can arise also in presence of strong HT effects. As shown in the right panel of Fig. 12, the inclusion of HT effects increases the intensity of the RROA spectrum associated to the S4 state compared to that of S3 . To assess the impact of mode-mixing effects, the RROA spectra were computed also at the VH|FC level, and the results are reported in the left panel of Fig. 13. The VH|FC spectrum is nearly unchanged if compared with the VG|FC one (reported in Fig. 12). The only difference is a shift of the most intense band of the spectrum associated to the S4 state towards higher energies which might be assigned to the negative band of the experimental spectrum at about 1400 cm−1 . However, due to the low intensity of the RROA spectrum associated to the S4 state, the negative band is not present in the full spectrum, where resonance effects between S3 and S4 states are considered. As for the VG spectra, the inclusion of HT effects increases the relative intensity of the S4 spectrum. The main effect of this gain of intensity in the overall spectrum, where both the S3 and the S4 states are included, is the presence of a negative band below 1600 cm−1 , which is present also in the experimental spectrum, even if shifted at higher energies compared to the theoretical spectrum. Moreover, the negative band at 1400 cm−1 is still not present in the VH|FCHT spectrum. The presence of a negative band

34


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at about 1400 cm−1 in the S4 spectrum suggests that the main source of inaccuracy might lie in the underestimation of the rotatory strength of the S4 state. To assess this, higher-level electronic structure methods would be needed, but this goes beyond the scope of the present work. Another possible source of the accuracy is the incident frequency employed in the simulation. In all cases, full resonance conditions with the S3 state were employed, and the same incident frequency was used also to compute the S4 spectra. However, the wavelength of the laser employed in the experiment 120 is 512 nm, while the excitation energy to the S3 state, as computed at the B3LYP/6-311G** level, is 342 nm. Calculations done under pre-resonance conditions 32 would then probably improve the accuracy of the calculation.

35



5

Conclusions

In the present work, a new approach for simulating resonance Raman optical activity (RROA) spectra has been presented. The theory is based on the time-dependent formulation of vibronic spectroscopy 122–124 and relies on the general framework previously introduced for simulating one-photon 22 and resonance Raman 23,24 spectroscopy. The transition tensors, giving the RROA scattering intensity, are computed as Fourier transform of the appropriate crosscorrelation functions, which can in turn be computed analytically. The main advantage of the TD formulation with respect to the alternative, time-independent (TI) one 31,48,66 is that no infinite summations are present, and thus no prescreening schemes are needed to select the vibronic states to be included in a sum-over-states expression. In this way, even large-size and flexible systems can be tackled without a steep increase of the computational effort. The integration of the TD-RROA theory within our general tool for simulating different kinds of spectra 21,125 allowed the automatic support of both adiabatic and vertical vibronic models 126,127 in Cartesian and internal coordinates. 25,26 Furthermore, even HT effects, that are usually critical to correctly reproduce the sign of chiroptical spectra, 49,66,128 can be included in an effective way. The reliability of the implementation of TD-RROA has been verified using a small chiral prototypical system (methyloxirane), for which well-converged reference data can be obtained at the TI level. Furthermore, the theoretical formulation has been used to compute the spectra of three larger-size systems (naproxen-OCD3 , quinidine and hexahelicene) for which experimental results are available. The work presented here can be improved in several respects. First of all, the present approach is well suited to simulate RROA spectra with strong resonance effects. However, in several cases, RROA spectra are recorded in pre-resonance regime 113,129 and the complex polarizability model introduced by Schatz and co-workers 14,15 might be better suited in those cases. For this reason, a systematic comparison of this approach with our TD-RROA implementation might be useful to assess the reliability and limits of both approaches, as already done for standard RR. 130 Furthermore, coupling TD-RROA with the method recently 36


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developed for including anharmonic effects for a single large-amplitude mode (LAM) 21,131 would lead to more accurate spectra for larger-size flexible systems. 132 Finally, this work paves the route towards the extension of our TD-based formulation to higher-order RR spectroscopies, such as hyper resonance Raman (HRR). 133,134

37



6

Page 38 of 72

Appendix

Following the derivation reported in our previous work, 22 it can be shown that the FranckCondon autocorrelation function can be expressed in terms of two, Nvib -dimensional Gaussiantype integrals as follows:

s χ(t, T = 0) =

det(Γ a) T Γ K exp −K (2πi~)2N

Z

Z dU

√ 1 dZ exp − Z T D Z − 2v T Z 2 1 T × exp − U C U 2

(44) where Z and U are the symmetric and antisymmetric linear combinations of the normal modes of the initial and intermediate states (Q and Q, respectively). Furthermore, the diagonal matrices a, c and d are defined as, ωi δij sinh ~τ ω i ω ωi i cij = coth (~τ ωi ) + δij ~ ~ ω ωi i dij = tanh (~τ ωi ) + δij ~ ~ aij =

(45)

and, C = c + J T cJ

(46)

T

D = d + J dJ It should be noted that, when mode-mixing effects are included in the simulations, matrices C and D are not diagonal. By using the properties of multidimensional Gaussian integrals, 135 the integral given in Eq. 44 can be expressed as, s χ(t, T = 0) =

det(Γ a) T T −1 × e−K Γ K+v D v 2N (i~) det(C D)

38


(47)

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The inclusion of HT effects requires the calculation of integrals similar to the one reported in Eq. 44 where the exponential part is scaled by powers of the position operator, which can be expressed in terms of Z and U . 22,23 Using recursive expressions for Gaussian-type integrals, those terms can be expressed as derivatives of χFC (t) with respect to the elements of D and v,

1 ∂χ(t, T = 0) 2 ∂vj 2 ∂χ(t, T = 0) 1 ∂ χ(t, T = 0) χkj (t, T = 0) = + ∂Cjk 4 ∂vk ∂vl 3 1 ∂ χ(t, T = 0) 1 ∂ 2 χ(t, T = 0) 1 ∂ 2 χ(t, T = 0) 1 ∂ 2 χ(t, T = 0) χkjl (t, T = 0) = + − − 8 ∂vj ∂vk ∂vl 2 ∂Cjk ∂vl 2 ∂Cjl ∂vk 2 ∂Ckl ∂vj (48) χj (t, T = 0) = χj (t, T = 0) = −

By direct differentiation of Eq. 47, the final results are,

1 χj (t, T = 0) = χj (t, T = 0) = − ηk χ(t, T = 0) 2 −1 −1 1 −1 χkj (t, T = 0) = ηj ηk + Djk + Dkj − 2 Ckj 4 (49) −1 −1 1 −1 −1 k + Dlj + Dlk + ηk Djl χjl (t, T = 0) = − ηj ηk ηl + ηj Dkl 8 1 +ηl {Djk }−1 + {Dkj }−1 − {Cjk }−1 ηl − {Cjk }−1 ηk − {Ckl }−1 ηj 4

39



7

Figures

Figure 1: Graphical representation of the molecules studied in the present work: (R)methyloxirane (a), naproxen-OCD3 (b), quinidine (c) and 2-Br-Hexahelicene.

40


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1.0

1.0

VG|FC TD

AS|FC TD

VG|FC TI

0.6

0.4

0.2

0.0

AS|FC TI

0.8 Normalized CID

0.8 Normalized CID

0.6

0.4

0.2

0

500

1000 Wavenumbers [cm−1 ]

1500

2000

0.0

0

500


1500

2000

1.0

AH|FC TD AH|FC TI

0.8 Normalized CID

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


0.6

0.4

0.2

0.0

0

500


1500

2000

Figure 2: RROA SCP(180◦ ) spectra of (1R)-methyloxirane computed at the B3LYP/SNSD level using the TI (green, dashed lines) and TD (red, solid line) approaches within the VG|FC (upper left panel), AS|FC (upper right panel) and AH|FC (lower panel) models. Lorentzian functions with half-widths at half-maximum (HWHMs) of 10 cm−1 have been used to simulate the broadening effects.

41



1.0

0.2

VG|FCHT TD

VG|HT TD

VG|FCHT TI

VG|HT TI 0.0

0.8

−0.2 Normalized CID

0.6 Normalized CID

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 42 of 72

−0.4

0.4 −0.6

0.2 −0.8

0.0

0

500


1500

2000

−1.0

0

500


1500

2000

Figure 3: RROA SCP(180◦ ) spectra of (1R)-methyloxirane computed at the B3LYP/SNSD level using the TI (green, dashed lines) and TD (red, solid line) approaches within the VG|FCHT (left panel) and VG|HT (right panel) models. Lorentzian functions with HWHMs of 10 cm−1 have been used to simulate the broadening effects.

42


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S1 S2 S3 S4 S5 Absorbance [au]

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


200

Total Exp.

250

300 Energy [nm]

350

400

Figure 4: Experimental 8 (dashed black line) and theoretical (solid black line) OPA spectra of naproxen-OCD3 , computed at the VG|FC level and including the five lowestenergy excited states. The contributions of the single excited states are also reported (solid red line for the S1 state, solid green line for the S2 state and solid blue line for the S3 state). Electronic structure calculations were performed at the B3LYP/SNSD level. Gaussian distribution functions with HWHMs of 1000 cm−1 were used to simulate the broadening effects. Solvent effects (CCl4 ) were included by means of PCM.

43



S1 S2 S3 S4 S5 Absorbance [au]

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 44 of 72

200

Total Exp.

250

300 Energy [nm]

350

400

Figure 5: Experimental 8 (dashed black line) and theoretical (solid black line) ECD spectra of naproxen-OCD3 , computed at the VG|FC level and including the five lowestenergy excited states. The contributions of the single excited states are also reported (solid red line for the S1 state, solid green line for the S2 state and solid blue line for the S3 state). Electronic structure calculations were performed at the B3LYP/SNSD level. Gaussian distribution functions with HWHMs of 1000 cm−1 were used to simulate broadening effects. Solvent effects (CCl4 ) were included by means of PCM.

44


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Figure 6: Graphical representation of the equilibrium geometries of the ground (red) and 1 La (blue) states of naproxen-OCD3 computed at the B3LYP/SNSD level.

45



1.0

0.8

S1-S2 FC S1-S2 Anh. FC S1-S2 Anh. FCHT Exp.

Cross section [au]

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 46 of 72

0.6

0.4

0.2

0.0 1100

1200

1300

1400 Wavenumbers [cm−1]

1500

1600

1700

Figure 7: Experimental 8 and computed RR DCPI (180◦ ) spectra of naproxen-OCD3 computed by including the S1 and S2 electronic states. Simulations were performed at the AH|FC (red line) and AH|FCHT (blue line) levels using harmonic (dashed line) and anharmonic (solid lines) wavenumbers for the ground state. Lorentzian functions with HWHMs of 10 cm−1 were used to reproduce the broadening effects. An incident wavenumber of 28572 cm−1 was used, and γ was set to 100 cm−1 . Solvent effects (CCl4 ) were included by means of PCM.

46


Page 47 of 72

0.4

0.2

S1-S2 Anh. FC S1-S2 Anh. FCHT Exp.

0.0 Normalized CID

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


−0.2 −0.4 −0.6 −0.8 −1.0 1100

1200

1300


1500

1600

1700

Figure 8: Experimental 8 and computed RROA DCPI (180◦ ) spectra of naproxen-OCD3 computed by including the S1 and S2 electronic states. Simulations were performed at the AH|FC (red line) and AH|FCHT (blue line) levels using anharmonic wavenumbers for the ground state. Lorentzian functions with HWHMs of 10 cm−1 were used to reproduce the broadening effects. An incident wavenumber of 28572 cm−1 was used, and γ was set to 100 cm−1 . Solvent effects (CCl4 ) were included by means of PCM.

47



0.4

0.2

S1-S5 Anh. FC S1-S5 Anh. FCHT Exp.

0.0 Normalized CID

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 48 of 72

−0.2 −0.4 −0.6 −0.8 −1.0 1100

1200

1300


1500

1600

1700

Figure 9: Experimental 8 and computed RROA DCPI (180◦ ) spectra of naproxen-OCD3 computed by including the five lowest-energy (S1 -S5 ) electronic states. Simulations were performed at the AH level for the S1 and S2 states and at the VG level for the S3 , S4 and S5 ones, using the FC (solid red line) and FCHT (solid green line) approximations. Frequencies of the ground state were computed at the anharmonic level. Lorentzian functions with HWHMs of 10 cm−1 were used to reproduce the broadening effects. An incident wavenumber of 28572 cm−1 was used, and γ was set to 100 cm−1 . Solvent effects (CCl4 ) were included by means of PCM.

48


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0.0

0.4

ICP(180) DCP1(180)

ICP(180) DCP1(180)

0.2

−0.2

Normalized CID

0.0 Normalized CID

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


−0.4

−0.6

−0.2 −0.4 −0.6

−0.8

−1.0

−0.8

600

800

1000

1200 1400 Wavenumbers [cm−1]

1600

1800

−1.0

2000

600

800

1000

1200 1400 Wavenumbers [cm−1]

1600

1800

2000

Figure 10: Theoretical RROA ICP(180◦ ) (red line) and DCPI (180◦ ) (green line) spectra for resonance with the S2 state of quinidine, computed using the AH|FC (left panel) and AH|FCHT (right) models. Lorentzian functions with HWHMs of 10 cm−1 have been used to simulate the broadening effects. Solvent effects (H2 O) were included by means of PCM. The incident wavenumber was set to 27705 cm−1 .

49



1.0

S1+S2 AH|FC S1+S2 AH|FCHT Exp. 0.5 Normalized CID

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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0.0

−0.5

−1.0

1000

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1600

1800

2000

Figure 11: Theoretical and experimental 112 RROA DCPII (180◦ ) (green line) spectra of quinidine computed at the VG|FCHT level by including interference effects from the S1 and S2 electronic states. Lorentzian functions with HWHMs of 10 cm−1 have been used to simulate the broadening effects. The incident wavenumber was set to 27705 cm−1 for both electronic states. Solvent effects (CHCl3 ) were included by means of PCM.

50


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1.0

1.0

VG|FC S3

VG|FCHT S3

VG|FC S4

0.8

VG|FCHT S4

0.8

VG|FC S3+S4

VG|FCHT S3+S4

Exp.

Exp. 0.6 Normalized CID

0.6 Normalized CID

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


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1800

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2000

1000

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1600

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Figure 12: Theoretical and experimental 120 SCP(180◦ ) RROA spectra of 2-BrHexahelicene computed at the VG|FC (left panel) and VG|FCHT (right panel) for the S3 (red line) and S4 (blue line) electronic states. Lorentzian functions with HWHMws of 10 cm−1 have been used to simulate the broadening effects. The incident wavenumber was set to 27558 cm−1 for the S3 and S3 +S4 spectra, and to 29113 cm−1 for the S4 spectrum. Solvent effects (CHCl3 ) were included by means of PCM.

51



1.0

1.0

VH|FC S3

VH|FCHT S3

VH|FC S4

0.8

VH|FCHT S4

0.8

VH|FC S3+S4

VH|FCHT S3+S4

Exp.

Exp. 0.6 Normalized CID

0.6 Normalized CID

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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0.4

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1000

1200


1600

1800

−0.2 800

2000

1000

1200


1600

1800

2000

Figure 13: Theoretical and experimental 120 SCP(180◦ ) RROA spectra of 2-BrHexahelicene computed at the VG|FC (left panel) and VG|FCHT (right panel) for the S3 (red line) and S4 (blue line) electronic states. Lorentzian functions with HWHMws of 10 cm−1 have been used to simulate the broadening effects. The incident wavenumber was set to 27477 cm−1 for the S3 and S3 +S4 spectra, and to 28165 cm−1 for the S4 spectrum.

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8

Tables

53



54

Table 1: Equivalence table to obtain the definition of IROA in Eq. 20 in terms of linear combinations of ROA invariants for different experimental setups.

8 [45αG + 7βs (G)2 + 5βa (G)2 + βs (A)2 − βa (A)2 − 45αG + 5βs (G)2 − 5βa (G)2 − 3βs (A)2 + βa (A)2 ] 4 [45αG + 7βs (G)2 + 5βa (G)2 + βs (A)2 − βa (A)2 ] 8 [3βs (G)2 + 5βa (G)2 − βs (A)2 + βa (A)2 ] 40 [9αG + 2βs (G)2 + 2βs (G)2 ] 3 4 [45αG + 13βs (G)2 + 15βa (G)2 − βs (A)2 + βa (A)2 ] 2 8 [45αG + 7βs (G) + 5βa (G)2 + βs (A)2 − βa (A)2 + 45αG − 5βs (G)2 + 5βa (G)2 + 3βs (A)2 − βa (A)2 ] 8 [−45αG − 7βs (G)2 − 5βa (G)2 + βs (A)2 + βa (A)2 + 45αG − 5βs (G)2 + 5βa (G)2 − 3βs (A)2 − βa (A)2 ] 4 [−45αG − 13βs (G)2 − 15βa (G)2 − βs (A)2 − βa (A)2 ] 4 [−3βs (G)2 − 5βa (G)2 − βs (A)2 − βa (A)2 ] 8 [−3βs (G)2 − 5βa (G)2 − βs (A)2 − βa (A)2 ] 40 [−9αG − 2βs (G)2 − 2βa (G)2 ] 3 8 [−45αG − 7βs (G)2 − 5βa (G)2 + βs (A)2 + βa (A)2 + −45αG + 5βs (G)2 − 5βa (G)2 + 3βs (A)2 + βa (A)2 ] 8 [45αG + βs (G)2 + 5βa (G)2 − βs (A)2 − βa (A)2 + −45αG − βs (G)2 − 5βa (G)2 − βs (A)2 − βa (A)2 ] 2 [45αG + 13βs (G)2 + 15βa (G)2 − βs (A)2 + βa (A)2 + −45αG − 13βs (G)2 − 15βa (G)2 − βs (A)2 − βa (A)2 ] 16 [3βs (G)2 + βs (A)2 − 3βs (G)2 + βs (A)2 ] 16 [3βs (G)2 + βs (A)2 + 3βs (G)2 − βs (A)2 ] 2 2 2 [45αG + 13βs (G) + 15βa (G) − βs (A)2 + βa (A)2 + +45αG + 13βs (G)2 + 15βa (G)2 + βs (A)2 + βa (A)2 ] 8 [45αG + βs (G)2 + 5βa (G)2 − βs (A)2 − βa (A)2 + +45αG + βs (G)2 + 5βa (G)2 + βs (A)2 − βa (A)2 ]

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

ICP(0◦ ) ICPx (90◦ ) ICPz (90◦ ) ICP? (90◦ ) ICPu (90◦ ) ICP(180◦ ) SCP(0◦ ) SCPx (90◦ ) SCPz (90◦ ) SCP? (90◦ ) SCPu (90◦ ) SCP(180◦ ) DCPI (0◦ ) DCPI (90◦ ) DCPI (180◦ ) DCPII (0◦ ) DCPII (90◦ ) DCPII (180◦ )

Journal of Chemical Theory and Computation Page 54 of 72

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Table 2: Equivalence table to build the five transition tensors required in ROA spectroscopy from Eq. 21. T PA PB

α G G µ µ m µ m µ

A µ Θ

G Θ µ

Acknowledgment We are thankful for the computer resources provided by the high performance computer facilities of the SMART Laboratory (http://smart.sns.it/). We acknowledge funding from the Italian MIUR (PRIN 2015 Grant Number 2015XBZ5YA).

Supporting information The supporting information (SI) document contains: the RR spectra of the S1 and S2 states of naproxen-OCD3 obtained with different vibronic models and coordinate systems, the RROA spectrum of quinidine in resonance with the S1 electronic state computed at the AH|FC and AH|FCHT level, the excitation energies and oscillator strengths for the S1 , S2 and S3 electronic states of naproxen-OCD3 computed with different DFT exchange-correlation functionals, the excitation energies, oscillator and rotatory strengths of the S1 , S2 , S3 , S4 and S5 states of 2-Br-Hexahelicene computed with different DFT exchange-correlation functionals.

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Time-dependent formulation of Resonance Raman Optical Activity

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