A key activity of the group's research is to develop theoretical models for the simulation of a wide range of spectroscopies. They are implemented as components of a modular platform called the virtual spectrometer,1,12 designed to have the following features:
Under those conditions, one of the main activities of the group is to derive new, more general theories for the simulation of molecular spectra, which may also lead to possibilities of extensions to new, more complex spectroscopies or systems. For the latter, this may be achieved by proposing more refined or less expensive models, depending on the complexity and size of the molecules to be supported, or by developing entirely new approaches. Indeed, as the system to be studied becomes larger and the experimental setup more complex, a direct comparison between theoretical results and measurements requires the capability to support large-dimensional systems and to account directly for the conditions of experiment. Regarding the latter, one can mention the temperature, the presence of solvent or the possibility of mixture of several conformers.
In order to build a proper platform, aimed at supporting the interpretation and prediction of experimental spectra, a preliminary work was dedicated to the conception of a modular structure. Indeed, the higher flexibility provided in this way makes also easier the inclusion of new spectroscopies on demand. A schematic representation of the tool, also called virtual multifrequency spectrometer (VMS), is shown in the figure below.
Work is currently focused on three modules of this spectrometer:
A version of the virtual spectrometer is available within the suite of quantum chemical programs Gaussian.
Note: The modules currently available work within the Born-Oppenheimer approximation, assuming that the electronic and nuclear components of the wave functions are fully separated.
Historically the first module of the spectrometer in its current form, its primary aim is the efficient simulation of electronic spectra with proper account for the vibrational effects. Indeed, most calculations are still done by only considering the electronic transition dipole moments, ignoring entirely the vibrational structure in the UV-vis band-shapes. This is particularly ill-suited when dealing with high-resolution spectra clearly showing a fine structure, but single electronic transitions successively broadened with distribution functions to match experimental results cannot reproduce the asymmetry of low-resolution band-shapes.
Several strategies can be followed to account for the vibronic structure of UV-vis spectra. A straightforward way is to calculate the band-shape as a sum of individual transitions between vibrational states belonging to different electronic states involved in the transition (sum-over-states). The complexity of the resulting integrals can be alleviated at different degrees based on the approximations made. For medium-large molecular systems, the most common ones are the harmonic approximation and the fulfilment of the Eckart conditions. Other important aspects of the theoretical framework are:
As a final remark on the general theoretical setting, it should be noted that the minima of the potential energy surfaces (PESs) coincide rarely and the corresponding shift can reveal the limitation of their harmonic representations. Depending on one's needs (accuracy of the most intense peaks' shapes and heights or of the overall band-shape), it may be more appropriate to describe the PESs about the same region corresponding to the equilibrium geometry of the initial state (vertical model or vertical Hessian, VH) or at their respective minima (adiabatic model or adiabatic Hessian, AH). The module supports both description, as well as their approximated versions, in which the PESs are assumed equal, either corresponding to the initial-state one or to the less expensive calculation, often the ground state. These models are called vertical gradient (VG), also known as linear coupling model, and adiabatic shift (AS).
The setup presented previously make band-shape simulations a rather straightforward and cost-effective task. However, hurdles remain due to the nature of the sum-over-states (or time-independent, TI) approach itself. First, because of the potentially infinite summation over all initial and final vibrational states, it is necessary to devise a way to choose, preferably a priori, which transitions must be taken into account.13,14 Next, the intricacy of the analytic formulas, especially with a high number of quanta involved in the tr""ansition, to compute the vibrational overlap integrals makes them more readily treatable recursively, but this poses problems of optimization to obtain a computationally efficient program, especially when temperature effects are taken into proper account. An alternative way is to use a time-dependent (TD) formulation, obtained from the TI one by switching from the frequency the time domain using the Feynmann path integral theory.9 It is assimilable to a simplified quantum dynamical simulation on a model system with harmonic surfaces. An appealing feature with respect to the time-independent framework is the automatic inclusion of all vibrational initial and final states, which eliminates the need for prescreening but also simplifies the inclusion of temperature effects since there is no increase in the computational cost. However, the lack of information regarding the most intense vibronic transitions makes band assignment impossible and numerical instabilities can be observed in critical cases. For those reasons, both TI and TD frameworks are complementary and are available within the spectrometer, so they can be combined for instance by using TD to get the overall band-shape of a spectrum and then TI to get additional information about the contributing transitions in a specific region.
Supporting multiple spectroscopies on a single, unified platform can be challenging in view of the diversity of properties, which may be involved, in particular for chiroptical spectroscopies. To facilitate maintenance and future extensions, much effort has been made to develop and implement generalized formulations of the underlying theory. This word made possible the definition of a single formula to support one-photon absorption (OPA) and emission (OPE) spectroscopies, as well as their chiral counterparts, electronic circular dichroism (ECD) and circularly polarized luminescence (CPL), respectively.9,13 This first version has been later extended to support resonance Raman (RR) and more recently resonance Raman optical activity (RROA).7,8 Like OPA and OPE (and ECD and CPL for chiral molecules), RR carries information about the excited electronic state(s) but, since the bandwidth depends on the ground state, the resolution of RR spectra is better. Thanks to the resonance enhancement, RR intensities are far stronger than the non-resonant ones, making the latter negligible. This gives the possibility to isolate a specific molecule among others, which is of particular interest for the study of biological macromolecules. Finally, as a vibrational spectroscopy, it is much easier to extract structural information about the system with respect to OPA and OPE for instance. At variance with most approaches to simulate RR spectra, which rely on simplified models where either the excited electronic state is not explicitly taken into account or the excited-state potential energy surface is assumed to be the same as the ground-state one, the version implemented here can support different PES in the initial and intermediate state.
The overall framework has been successfully applied to understand spectroscopic phenomena in a wide variety of systems (e.g. π-stacking interactions of anisole dimer, optical properties of organic dyes, electronic spectra of free radicals. Simulations also on larger-size systems are currently performed, representing a challenging task due to the high computational cost to obtain the data needed to simulate electronic spectra, in particular for the excited state. To render calculations more feasible, several paths have been explored. If only a region of the system plays a role in the electronic transition and can be isolated from the rest, it is possible to apply multi-scale techniques where the former is treated at a higher level of theory than the latter. An alternative way, which can also be complementary to the previous one, is to use approximated vibronic models, reducing considerably the quantity of data needed to simulate the spectra.
Part of the recent activity on this field has been extending the basic framework beyond the Franck-Condon principle, and support beter non-rigid systems. Indeed, harmonic models based on Cartesian coordinates are often unsuited to cases where the molecular structure changes significantly upon the electronic transitions. This can be ascribed to two important factors. First, the description based on normal modes stemming from Cartesian coordinates results in strong couplings among them, which make simulations unreliable already for a qualitative interpretation. Those couplings can be significantly reduced with improvement in the calculated band-shapes by using curvilinear, internal coordinates, even at the harmonic level. Moreover, flexible systems are usually characterized by highly anharmonic, large-amplitude motions, and an appropriate description of those modes requires the development of vibronic anharmonic models. Using internal coordinates, it is possible to focus anharmonic effects on a reduced set of degrees of freedom (even a single mode for the simplest deformations) and, in this way, hybrid schemes can be developed, where modes are treated at different levels of theory.4 In the last years, a hybrid Discrete Variable Representaton (DVR)-harmonic model, based on the Reaction Path Hamiltonian model, has been developed within the group and successfully applied to several molecular systems.2
The objective of this second research line is to develop and implement methods to simulate vibrational spectra at the anharmonic level, still aimed at medium-to-large systems. A common approach in the simulation of vibrational spectra is to study them is to extract peak positions and compare them to calculated vibrational energies, most of the time obtained at the harmonic level of theory. Correction of harmonic frequencies to match experimental results can be achieved by applying either constant or range-specific scaling factors derived from statistical samples. A more robust approach lies in the proper inclusion of anharmonic effects by either perturbative or variational methods, which have become more accessible in the recent years due to their implementation in widely used, general-purpose programs and the improvements in hardware performance and algorithms efficiency. However, even at this level of theory, direct comparison of computed spectra with their experimental counterparts often relies on harmonic intensities.
In order to overcome this limitations, a general theoretical framework has been employed for the simulation of fully-anharmonic spectra of large-size molecular systems. The framework is based on second-order perturbation theory (VPT2), which offers a straightforward and efficient mean to account for the anharmonicity of the potential energy surface.
In order to get accurate results, VPT2 energies3,15,16 and transition moments of the electric dipole5,11 are needed to generate fully anharmonic infrared spectra, where both property and wave functions are treated beyond the harmonic approximation. This significantly increases the reliability of the simulations since, contrary to the harmonic approximation, transitions to overtones and combination bands have non-null intensities at the anharmonic level, which results in a richer pattern of peaks.
Extension to chiroptical spectroscopies, such as vibrational circular dichroism (VCD), requires an additional development for properties function of the momenta, instead of the normal coordinates like the electric dipole. This provides a general formulation to compute transition integrals to fundamental bands, overtones and combination bands of properties function of the normal coordinates or their conjugate momenta, leading to the possibility of simulating various vibrational spectra at the anharmonic level, such as IR, VCD, Raman scattering and Raman Optical Activity with only one equation per transition.6
The work done for the transition moments has also been useful to derive general equations of anharmonic vibrational averages for a wide range of electric-field, magnetic-field and frequency-dependent properties.
One strong drawback of VPT2 lies in the potential presence of singularities, which give unphysical results. To avoid those effects, terms contributing excessively to the anharmonic correction need to be identified. This is generally done by applying criteria based on empirical thresholds to discard the unsuited terms, which can be treated variationally instead. While the criteria have shown to give very satisfactory results for spectroscopic studies, this direct dependence on thresholds can raise issues whenever one has to consider a series of force fields for a given system, or a series of structures along a reaction path, and the list of resonant terms may vary on this ensemble. For this reason, an alternative model has been devised and implemented, called hybrid degeneracy-corrected perturbative theory (HDCPT2) to overcome this situation and have a smoother progression over a sample of case studies.10 While originally intended for thermodynamic studies, where high accuracy is less critical, the method has shown to be quite satisfactory to compute vibrational spectra with limited errors.
Even though perturbation theory allows the inclusion of an anharmonic correction in a cost-effective way, generation of the necessary data can be expensive. For instance, third and fourth derivatives of the potential energy are obtained through numerical differentiation of the harmonic force constants at displaced geometries along the normal coordinates. This means that the number of force constant matrices to compute grows proportionally with the number of normal modes (more precisely twice this number). For this reason, alternative and more affordable schemes have been explored. A possible way to reduce easily the computational cost is to carry out the numerical differentiation to obtain the anharmonic data only along a subset of normal modes, labeled active in contrast with the other ones treated at the harmonic level. The selected modes can be chosen based on features of interest on the spectrum under investigation or a specific region on the whole system, such as a chromophore. Another possibility is to use different levels of theory for the harmonic and anharmonic components, with the latter treated at a higher, more expensive level of theory than the former. This model is based on the assumption that most of the discrepancies of the lower-level method with respect to experiment lies in the harmonic part. To correct this, a higher-level method can be used instead. In order to simplify the use of such a scheme, an important work is dedicated to the automatization of the validation process regarding the coherence of the two sets of data (from the higher and lower levels), so that users can simply provide the necessary data and the program takes care of this analysis before doing the calculation. Thanks to this, the approach can be routinely employed without the need for the user to carry out manually this cumbersome task. It should be noted that this multi-scale approach can be combined to the reduced-dimensionality scheme described previously where the anharmonic correction is applied to a subset of normal modes, but also to more classical electronic multi-scale methods.