Vibrational spectroscopy by means of first-principles molecular dynamics simulations

Abstract

Vibrational spectroscopy is one of the most important experimental techniques for the characterization of molecules and materials. Spectroscopic signatures retrieved in experiments are not always easy to explain in terms of the structure and dynamics of the studied samples. Computational studies are a crucial tool for helping to understand and predict experimental results. Molecular dynamics simulations have emerged as an attractive method for the simulation of vibrational spectra because they explicitly treat the vibrational motion present in the compound under study, in particular in large and condensed systems, subject to complex intramolecular and intermolecular interactions. In this context, first-principles molecular dynamics (FPMD) has been proven to provide an accurate realistic description of many compounds. This review article summarizes the field of vibrational spectroscopy by means of FPDM and highlights recent advances made such as the simulation of Infrared, vibrational circular dichroism, Raman, Raman optical activity, sum frequency generation, and nonlinear spectroscopies.

This article is categorized under:

• Electronic Structure Theory > Ab Initio Electronic Structure Methods
• Theoretical and Physical Chemistry > Spectroscopy
• Molecular and Statistical Mechanics > Molecular Mechanics
• Electronic Structure Theory > Density Functional Theory

Graphical Abstract

First-principles molecular dynamics is a powerful approach to calculate vibrational spectra of gas and condensed phase systems.

1 INTRODUCTION

Spectroscopy, in general terms, is the study of the interaction of light with matter. This includes absorption and scattering effects observed when a light source is pointed at a sample consisting of a gas, liquid, solid, and mixtures thereof. Such experiments can take place in the whole range of the electromagnetic spectrum, ranging from low-frequency radio and microwave frequencies for nuclear magnetic resonance (NMR) and electron paramagnetic resonance (EPR) experiments, over the infrared region for vibrational spectra, the ultraviolet-visible (UV/VIS) range for electronic excitations, and the XUV/X-ray region for ionization experiments.

From a theoretical point of view, there are two main ways of simulating vibrational spectroscopic experiments: the static approach and the dynamic approach. In the static approach, the nuclear positions are first optimized at a fictitious temperature of 0 K. Then the coordinates are either displaced according to a finite differences scheme or perturbatively, while usually only the electron density response to the radiation is considered for the calculation of properties. Utilizing the harmonic approximation and computing the second derivative of the electronic energy with respect to nuclear positions (the Hessian) around a local minimum nuclear structure, the vibrational spectrum results from derivatives of the system's properties along the vibrational normal modes. In this picture, the eigenvectors $\left\{{\overrightarrow{Q}}_k\right\}$ of the mass-weighted Hessian matrix are the vibrational normal modes, while the corresponding eigenvalues give access to vibrational frequencies {ω_i}. A static calculation, where the nuclear positions are fixed, is usually an appropriate approach for many kinds of spectroscopy, such as X-ray and NMR. However, it can be advantageous to sample the vicinity of the equilibrium structure and then take the averages of the properties of interest retrieved from multiple calculations,¹ especially when simulating spectra of nonrigid and large molecules. Generally, for vibrational spectra and spectra probing reaction dynamics, the dynamic approach is superior due to being able to sample the potential energy surface (PES) at finite temperatures. In this approach, the nuclear positions are propagated according to classical or quantum dynamics.

Nuclear quantum effects, can have a large influence on computed spectra depending on the studied system, and have been used to achieved chemical accuracy for small and rigid molecules.^2-4 Approaches including nuclear quantum dynamics incur a high computational cost and are thus limited to small system sizes.⁵ Nuclear quantum effects, which can also be retrieved from path integral molecular dynamics,⁶ are in general most significant at low temperatures and in the presence of light atoms, such as hydrogen.

The first classical MD simulations, based on empirical force fields, have kicked off the field of (IR) spectra simulations via molecular dynamics.^7-10 Since then, more elaborate schemes have appeared to improve the force fields based on chemical intuition and available computational resources. Advances were not only made in force field development but also in methodology. For example, to extract the time-dependent Hamiltonian operator from classical MD trajectories to calculate IR spectra.¹¹

A higher accuracy compared to classical MD can usually be expected from first-principles molecular dynamics (FPMD). Vibrational spectroscopy via FPMD simulations is the focus of this review. We restrict this article to the discussion of vibrational spectra and do not consider rovibrational transitions. We cover the basic theory behind the constituents of molecular dynamics simulations, including the general ideas in the Born–Oppenheimer (BO) and Car–Parrinello approaches. For detailed derivations and explanations on the peculiarities of implementations we refer the reader to the literature.^{12, 13} We will focus on the advancements and applications of first principles-, especially density functional theory (DFT)-based, MD simulations for vibrational spectra, that were achieved in recent years by leveraging efficient implementations and ever-growing computational power of supercomputers around the world.

The great advantage of FPMD simulations compared to static calculations—where usually no sampling of the PES is carried out and quantum dynamics are neglected—is the possibility of relating spectroscopic signals with the evolution of the microscopic structure of the sample, including rearrangements and the natural inclusion of certain anharmonic effects. Additionally, complex interactions in floppy molecules and condensed phase systems, such as hydrogen-bonding networks, isomerizations, and solvent interactions are directly observable in MD simulations. Thus, especially for large systems, FPMD is the preferred method for the simulation of vibrational spectra. A frequently studied and yet not fully understood system is, for instance, liquid water.¹⁴ The complex hydrogen bonding network results in spectroscopic bands that can only be understood by combining experiments with (FP)MD simulations. Chemical reactions and other dynamic processes can also only be understood experimentally and theoretically in a time-dependent FPMD description of the model system. Such theoretical descriptions might also be combined with the solution of the (time-dependent) electronic structure problem via, for example, time-dependent DFT¹⁵ or density functional perturbation theory (DFPT)¹⁶ for properties needed to calculate specific vibrational spectra.

Our review is divided into two parts. First, we explain the theoretical framework of FPMD, within the BO approximation. Then we shortly explain how the use of time-correlation functions allows for calculating spectra from MD trajectories. In the second part of the review, we highlight applications of FPMD for infrared absorption (IR), vibrational circular dichroism (VCD), Raman, Raman optical activity (ROA), sum-frequency generation (SFG), and nonequilibrium spectroscopy. Finally, we present the conclusion and an outlook highlighting possible improvements and novel techniques that might be applied for vibrational spectroscopy via FPMD in the future.

2 THEORY

2.1 Constituents of first-principles molecular dynamics

2.1.1 General

MD simulations are used to simulate the time-resolved dynamics of molecular, liquid, and solid-state systems at finite temperatures. Time discretization into fixed steps δt is used to propagate the positions and velocities of the atoms in the system to integrate the (classical) equations of motion. The propagation can be governed solely by quantum chemical (QC) calculations of the PES, by a mix of QC and classical physics in semi-classical approaches, or purely by classical MD employing parameterized force fields. For the latter two, the leap-frog or velocity Verlet algorithm¹⁷ is used in standard calculations. In this case, at each (half) time step, either the positions or the forces and velocities are updated according to the chosen level of theory.¹⁸

For simulations of large systems and time scales, it is advantageous to use force fields fitted to experiments or data acquired from first principles. In this case, the forces are determined by considering bond lengths, bond angles and dihedral angles among other classifiers.^19-26 Such force fields can also include information on bond cleavage and formation to simulate reactive processes,^{27, 28} although the quality of the description of such processes depends on the fitting procedure and training data and is not generally transferable to other systems. Alternatively, the forces acting on the nuclei can be determined on the fly from QC calculations. QC methods are then employed to calculate the gradient of the PES to retrieve accurate forces based on the chosen level of theory. The quality of the forces is the most important factor determining the quality of the physical quantities that are extracted from the MD simulation.²⁹ While ideally, high-level QC methods would be employed, their computational cost—especially for large and condensed systems—makes their application difficult. The electronic structure method employed most often and very successfully is DFT in combination with BO-MD³⁰ or Car–Parrinello molecular dynamics (CPMD).¹³ While DFT-MD can be regarded as a semi-classical method because of the treatment of the nuclei by Newtonian dynamics, it is free of ad hoc parameterizations compared to force fields that are the result of an empirical fitting procedure.²⁹ Still, the choice of exchange-correlation functional and basis set can, among others, significantly impact the obtained results in Kohn–Sham DFT-MD. Temperature effects are taken into account by coupling to a heat bath via a thermostat. In this way, the temperature of the simulation can be defined and observed during the MD run.^31-33 The simulated MD trajectories should, in principle, be as long as possible to ensure sufficient sampling of the phase space. In practice, the calculated spectra are usually monitored to stop the MD run as soon as the spectrum does not change anymore, assuming sufficient sampling. The achieved time scales in MD simulations vary from picoseconds to microseconds depending on the level of theory and number of atoms, where the number of atoms or particles can go up to 10¹² when high-performance computing architectures are leveraged.^34-37

2.1.2 Born–Oppenheimer approximation

In the adiabatic approximation one assumes that the PESs corresponding to the nuclear wavefunctions do not overlap.^{30, 38} In the BO approximation,³⁰ terms in the Hamiltonian operator which depend on the atomic masses are neglected, corresponding to the picture of nuclei moving vanishingly slow compared to the electronic degrees of freedom. After decoupling electronic and nuclear degrees of freedom, the propagation of the nuclei may be approximated by classical dynamics, describing the nuclei as point-like particles. Then, the quantum mechanical operators arising from the nuclei are assumed to be classical variables that can be propagated according to Newtonian mechanics on the PES generated by the electrons in the system. Generally, the PES and the associated energy eigenstates E_k are parameterized by the electronic wave function Ψ, the positions of the nuclei $\left\{{\overrightarrow{R}}_J\right\}$ and any external constraints {α_ν} that are imposed on the system (e.g., the temperature or volume of the simulation cell)³⁹ as $E_k=E_k\left({\mathrm{\Psi }}_k,\left\{{\overrightarrow{R}}_J\right\},\left\{{\alpha }_{\nu }\right\}\right)$. The electronic energies and corresponding wave functions result from the time-independent electronic Schrödinger equation, which is given by

$$\mathcal{H}\left(\left\{{\overrightarrow{r}}_j\right\},\left\{{\overrightarrow{R}}_J\right\},\left\{{\alpha }_{\nu }\right\}\right){\mathrm{\Psi }}_k\left(\left\{{\overrightarrow{r}}_j\right\},\left\{{\overrightarrow{R}}_J\right\}\right)=E_k\left({\mathrm{\Psi }}_k,\left\{{\overrightarrow{R}}_J\right\},\left\{{\alpha }_{\nu }\right\}\right){\mathrm{\Psi }}_k\left(\left\{{\overrightarrow{r}}_j\right\},\left\{{\overrightarrow{R}}_J\right\}\right),$$ (1)

where, Ψ_k and E_k denote the wave function and energy of the kth electronic state, respectively, the indices j, J, and ν go over the number of electrons, nuclei, and external constraints, respectively. $\mathcal{H}$ is the Hamiltonian operator according to the chosen level of theory. In the BO approximation, the total wave function for state k, Φ_k, is factored into a nuclear part χ_k and the electronic part Ψ_k as

$${\mathrm{\Phi }}_k\left(\left\{{\overrightarrow{r}}_j\right\},\left\{{\overrightarrow{R}}_J\right\}\right)={\mathrm{\Psi }}_k\left(\left\{{\overrightarrow{r}}_j\right\},\left\{{\overrightarrow{R}}_J\right\}\right)\cdot {\chi }_k\left(\left\{{\overrightarrow{R}}_J\right\}\right),$$ (2)

where the dependence of the electronic wave functions {Ψ_k} on the nuclear positions is only parametric. With the solution of these static (i.e., time-independent) equations available on the fly, the equations of motion affecting the classically treated nuclei can be obtained by solving the Euler–Lagrange equations resulting from the Lagrangian.¹²

$$\begin{array}{l}{\mathcal{L}}_{\mathrm{BO}}\left({\mathrm{\Psi }}_k,\left\{{\overrightarrow{r}}_j\right\},\left\{{\overrightarrow{R}}_J\right\},\left\{{\alpha }_{\nu }\right\}\right)=\frac{1}{2}\sum _J{M}_J{\overrightarrow{v}}_J^2-\underset{\left\{{\mathrm{\Psi }}_k\right\}}{\min }\left\{E\left[\left\{{\mathrm{\Psi }}_k\right\},\left\{{\overrightarrow{R}}_J\right\},\left\{{\alpha }_{\nu }\right\}\right]\right\}\\ +\sum _{\mathit{ij}}{\mathrm{\Lambda }}_{\mathit{ij}}\left(⟨{\phi }_i|{\phi }_j⟩-{\delta }_{\mathit{ij}}\right),\end{array}$$ (3)

where M_J is the mass of the Jth nucleus, while ${\overrightarrow{v}}_J$ are the nuclear velocities. $E\left[\left\{{\mathrm{\Psi }}_k\right\},\left\{{\overrightarrow{R}}_J\right\},\left\{{\alpha }_{\nu }\right\}\right]$ is the energy functional depending parametrically on the nuclear positions and the wave functions and Λ_ij is the Lagrangian multiplier matrix enforcing orthonormality of the electronic orbitals {φ}. Solving for the equations of motion then yields

$$M_J\frac{{\partial }^2{\overrightarrow{R}}_J}{\partial t^2}=-{\nabla }_J\underset{\left\{{\mathrm{\Psi }}_k\right\}}{\min }\left\{E\left[\left\{{\mathrm{\Psi }}_k\right\},\left\{{\overrightarrow{R}}_J\right\}\right]\right\},$$ (4)

where the expectation value of the energy is minimized with respect to the electronic wave functions Ψ_k in a variational manner under the condition of orthonormal molecular orbitals.¹² FPMD comes in a second flavor, called CPMD, where the Lagrangian (Equation (3)) additionally includes the electronic degrees of freedom as

$$\frac{1}{2}\mu \sum _k⟨{\dot{\phi }}_k|{\dot{\phi }}_k⟩,$$ (5)

where μ is a fictitious mass parameter and $\left\{{\dot{\phi }}_k\right\}$ are the orbital velocities. We refer the reader to one of the many reviews on BOMD and CPMD for more details.⁴⁰ The solution of the differential equations of motion is usually carried out by numerical integration. The negative gradients of the energy (i.e., the forces) are determined at each geometry, starting from the initial configuration. At each time step, the forces, velocities, and positions of the nuclei are propagated in time, and then the electronic structure problem is again solved. The time step has to be chosen such that the vibrational frequencies of the nuclear motion are resolved accurately, while the total simulation length needs to be long enough to sample the configuration space adequately. Usually values around δt = 0.5 fs are used. The Velocity Verlet algorithm¹⁷ propagates the nuclear positions ${\overrightarrow{R}}_J$ and velocities ${\overrightarrow{v}}_J$ at separate steps, mixing the potential energies at timesteps t and t + δt according to

$${\overrightarrow{R}}_J\left(t+\mathit{\delta t}\right)={\overrightarrow{R}}_J\left(t\right)+{\overrightarrow{v}}_J\left(t\right)\mathit{\delta t}+\frac{1}{2M_J}{\nabla }_{J}E\left(t\right)\delta t^2$$ (6)

and

$$\overrightarrow{v}\left(t+\mathit{\delta t}\right)=\overrightarrow{v}\left(t\right)+\frac{1}{2M_J}{\nabla }_{J}E\left(t\right)\mathit{\delta t}+\frac{1}{2M_J}{\nabla }_{J}E\left(t+\mathit{\delta t}\right)\mathit{\delta t},$$ (7)

where we dropped the parametric dependencies of the PES E(t) on the coordinates and constraints. While in principle an exact solution of Equation (1) would be desirable, in practice further approximations have to be employed to make the solution tractable. Among the possible approximations for the time independent Schrödinger equation are tight-binding,^41-44 the Hartree–Fock method,⁴⁵ DFT,^{46, 47} Moller–Plesset perturbation theory,^48-50 approximations to relativistic quantum mechanics,^{51, 52} coupled cluster,^53-55 configuration interaction,⁵⁶ as well as combinations of methods either by embedding (e.g., hybrid quantum mechanics/molecular mechanics QM/MM⁵⁷) or by applying additional electronic structure methods to trajectories obtained from DFT-MD.

MD calculations are carried out at finite temperatures. A thermostat has to be applied to maintain the ensemble during an MD run. There are multiple choices for thermostats.^{33, 39, 58-62} Furthermore, one must choose a statistical ensemble according to which to propagate the system in question. In the microcanonical ensemble, the number of particles, the volume, and the energy are kept constant (NVE).⁶³ In the canonical ensemble, instead of the total energy, the temperature is conserved (NVT).⁶³ A special case of the Gibbs canonical ensemble keeps the pressure constant instead of the volume (NPT).⁶⁴ And in the grand canonical ensemble, which poses the greatest difficulties in numerical simulations, the chemical potential μ is kept constant instead of the particle number (μVT).⁶³ For the simulation of vibrational spectra via FPMD, the NVT ensemble is usually first applied for equilibration, and the spectra are then collected in the NVE ensemble.

2.1.3 Beyond the BO approximation

While the BO approximation holds for many systems, it is in certain cases advantageous to propagate the system not only on the ground state adiabatic PES but also on multiple PESs belonging to different electronic states. The consideration of additional electronic states is crucial in systems where the electronic states are not well separated in energy. In such cases, very fast chemical processes like photodissociation, photoisomerization, and internal conversion can take place.^{65, 66}

A BO treatment of nuclei neglects not only the couplings of energetically close electronic states but also the quantum nature of light atoms, and the vibrational zero-point energy (ZPE).⁶⁷ Especially for high-frequency motions of groups involving hydrogen atoms, the ZPE might be large compared to the classical vibrational energy at room temperature, leading to frequency shifts of calculated stretching modes compared to experimental results.⁴⁶ Furthermore, nuclear quantum effects can be included in MD simulations by employing the path-integral approach of Feynman.^{2, 68-70} We will not go into the details of the treatment of nuclear quantum effects in this review and instead refer the reader to the literature.^{2, 71}

It has been shown in a combined gas-phase and bulk-phase study,⁷² that while corrections like applying a Boltzmann factor to static calculations can significantly improve the match with experimental results,⁷³ a treatment using FPMD methods is desirable for theoretical IR spectroscopy, especially considering flexible molecules.⁷² In FPMD simulations, quantum correction factors are often employed to correct the line shapes resulting from time-correlation functions for vibrational spectra. When the nuclei are propagated according to classical dynamics instead of quantum dynamics, important symmetry properties of the time-correlation functions and their Fourier transforms are not generally fulfilled. To remedy this problem, quantum correction factors that depend on frequency and temperature are often applied to enforce the so-called detailed balance condition. The choice of the correction factor is not unique, and multiple ones have been proposed in the literature. By comparing the lineshapes obtained from centroid MD to the lineshapes obtained from FPMD, it has been argued that the correction factor that performs best in most cases is the so-called harmonic approximation factor given by^{74, 75}

$$Q_{\mathrm{HA}}\left(\omega \right)=\frac{\beta \mathrm{\hslash }\omega }{1-\exp \left(-\beta \mathrm{\hslash }\omega \right)},$$ (8)

where ω is the frequency, β is the reciprocal of the Boltzmann constant multiplied by the temperature, and $\mathrm{\hslash }$ is the reduced Planck constant. We will omit this factor henceforth for the sake of brevity.

2.2 Analysis of FPMD trajectories for the simulation of vibrational spectra

At the core of MD simulations is the PES, a 3-N-dimensional hypersurface in the configuration space of the investigated system, where N is the number of atoms. The configuration space needs to be sampled such that the statistical ensemble is approximated in order to be able to apply statistical analyses to trajectories resulting from an MD run. From a given trajectory, it is impossible to determine whether all parts of the configuration space are sampled sufficiently.⁷⁶ In many cases, enhanced sampling techniques like well-tempered metadynamics⁷⁷ or umbrella sampling⁷⁸ can be used to mitigate this problem, but they do not apply to vibrational spectroscopy. Although it is impossible to determine whether the time average of a physical observable has converged to the ensemble average, the length of an FPMD run is in any case limited by the available computational resources. Because the employed time step needs to be small enough to resolve vibrational motion, the total run time cannot be extended easily. In the case of vibrational spectroscopy, the change observed in the simulated spectra after each time step can still serve as a signal for the convergence of the simulation, in particular, if no large conformational changes are expected in the simulation. This issue can be further alleviated by taking averages not only of a single MD run but averaging multiple MD runs that have been started from different initial conditions. These initial conditions, in turn, can be found from separate MD runs at the quantum or classical level. In this way probabilistic statements can be made about the simulations so that researchers can provide errors bars and standard deviations when plotting results.^{21, 79-82}

2.2.1 Sampling of the configuration space

The averaging over many points in configuration space is the most important advantage of MD spectroscopy over static calculations. In the harmonic approximation, the PES around the equilibrium geometry is approximated by a harmonic potential. This approximation does, of course, not hold in the limit of dissociation reactions but can also be a source of error in vibrational spectra governed by anharmonic effects.⁸³ Corrections to vibrational modes that are obtained from the Hessian matrix at a local minimum on the PES have been published for different wave function methods, basis sets, and DFT functionals^84-86 in the form of empirical linear scaling factors. A nonempirical way of approximating the anharmonicity of vibrational modes and frequencies is the evaluation of higher-order derivatives of the PES by perturbation theory,⁸⁷ vibrational configuration interaction, and vibrational self-consistent field (VSCF) approaches.^88-90 Without perturbation theory, the anharmonic effect due the shape of the PES is accessible in a nonempirical manner via FPMD. Nuclear quantum effects are still not accounted for in this approach⁹¹ as are overtones, mode coupling, combination bands, and Fermi resonances,⁴⁷ reducing the overall accuracy compared to high-level methods based on VSCF. An additional problem of FPMD, and especially DFT-MD, is the difficulty to systematically increase the accuracy of simulations, or to even quantify the error introduced by the choice of functional, basis set, and other approximations. Approaches based on the solution of the vibrational Schrödinger equation, such as the multiconfigurational time-dependent Hartree method,⁹² do not suffer from this problem, but are not applicable to large systems at all.

In the widely used static approach, where usually only a single geometry on the PES is used for obtaining a vibrational spectrum, IR, and Raman intensities are calculated from the derivatives of the electric dipole moment and electric dipole–electric dipole polarizability, respectively, along the vibrational normal modes. This approach is only tractable for gas-phase molecules or small bulk systems. In the presence of solvent molecules in condensed phase systems, finding minima of the PES is often challenging, or the number of local minima can be vast because of the complex interactions of hydrogen bonds and flexible solvent molecules that can undergo large rearrangements along a shallow PES. In such cases, the calculation of vibrational normal modes becomes increasingly demanding with the size of the system. Additionally, the problem of choosing an appropriate point in configuration space for calculating the Hessian matrix arises. Moreover, using a temperature of 0 K as done in the static approach does not provide a realistic description of the system studied at finite temperature. It is possible to use a Boltzmann distribution to average the contributions of different conformers, corresponding to local minima on the PES, for the calculation of vibrational spectra.⁹³ Geometry optimized snapshots from FPMD trajectories can generally be used to calculate multiple spectra from the Hessian matrix that are averaged afterwards,⁹⁴ but we limit the discussion in this review to spectra obtained from time-correlation functions.

2.2.2 Time-correlation functions

In FPMD simulations, the derivatives of the electric dipole moments with respect to the normal coordinates are not directly accessible, and all vibrational modes are excited simultaneously. The picture of vibrational normal modes as obtained from the Hessian matrix in the harmonic approximation does not apply to MD simulations. To determine vibrational frequencies, one can rely on the power spectrum, which is given by the autocorrelation function of particle velocities and would correspond to the vibrational density of states (VDOS) in the harmonic approximation.⁹⁵ Efforts to assign vibrational modes to the bands in IR spectra have been published as well.⁹⁶ For the calculation of IR and Raman intensities, the autocorrelation functions of the electric dipole moment and electric dipole–electric dipole polarizability are employed, respectively, and similar relations hold for other types of vibrational spectroscopy such as VCD^97-99 and ROA.^{100, 101} Generally, the autocorrelation of a time-dependent quantity f(t) with respect to a lag-time τ is given by,¹⁰²

$${⟨f\left(\tau \right)f\left(t+\tau \right)⟩}_{\tau }={\int }_{-\infty }^{\infty }f\left(\tau \right)f\left(t+\tau \right)\mathit{d\tau }=\frac{1}{2\pi }{\int }_{-\infty }^{\infty }{\left|{\int }_{-\infty }^{\infty }f\left(t\right)e^{-\mathit{i\omega t}}\mathit{dt}\right|}^2{e}^{\mathit{i\omega t}}\mathit{d\omega },$$ (9)

where the second equality reduces the computational complexity of the calculation from $\mathcal{O}\left(n^2\right)$ to $\mathcal{O}\left(n\log n\right)$, where n is the number of data points. The two necessary Fourier transformations are usually implemented using the fast Fourier transform (FFT) algorithm.¹⁰³ Similarly, the cross-correlation of two signals f(t) and g(t) is given by the convolution

$$\left(f\star g\right)\left(t\right)={\int }_{-\infty }^{\infty }f\left(\tau \right)g\left(t-\tau \right)\mathit{d\tau }.$$ (10)

To find the vibrational frequencies in the collective motion, one can generate the power spectrum P(ω) by decomposing the autocorrelation function of nuclear positions via a Fourier transformation.⁴⁷

$$P\left(\omega \right)={\omega }^2\int {\int }_{-\infty }^{\infty }{⟨\overrightarrow{r}\left(\tau \right)\overrightarrow{r}\left(t+\tau \right)⟩}_{\tau }{e}^{-\mathit{i\omega t}}\mathit{dt}=\int {\int }_{-\infty }^{\infty }{⟨\overrightarrow{v}\left(\tau \right)\overrightarrow{v}\left(t+\tau \right)⟩}_{\tau }{e}^{-\mathit{i\omega t}}\mathit{dt},$$ (11)

where the brackets denote the autocorrelation functions of the positions $\overrightarrow{r}$ or the velocities $\overrightarrow{v}$ respectively. The second equality is due to the properties of the Fourier transformation for time derivatives. Using the velocity autocorrelation function is numerically advantageous to the position autocorrelation function.⁴⁷

2.2.3 Mode assignment

The assignment of spectral bands to vibrational motions can be challenging in the MD approach because the power spectrum assigns frequencies to the system's collective motion, which does not necessarily correspond to well-defined stretching and bending modes. Also, the spectral bands in power spectra are usually rather broad, while the eigenfrequencies of the Hessian matrix are discrete, leading to stick spectra. As such, the power spectrum is not equivalent to the eigenmodes obtained from the diagonalized Hessian matrix, which can be visualized individually. Still, approaches have been presented to localize the power spectrum and make a qualitative interpretation possible. For example, Gaigeot et al. proposed to construct the modes $\left\{q_k\right\}$ as linear combinations of the coordinates describing the dynamics of the system. The power spectrum of the kth mode is then given by^{96, 104}

$$P_k^q\left(\omega \right)=\int ⟨{\overrightarrow{\dot{q}}}_k\left(0\right){\overrightarrow{\dot{q}}}_k\left(t\right)⟩e^{\mathit{i\omega t}}\mathit{dt}$$ (12)

and can be localized by minimizing the spread functional

$${\mathrm{\Omega }}^{\left(n\right)}=\sum _k\left[\frac{1}{2\pi k_{B}T}\int \mathit{d\omega }\left|{\omega }^{2n}{P}_k^q\left(\omega \right)\right|-{\left(\frac{1}{2\pi k_{B}T}\int \mathit{d\omega }\left|{\omega }^n{P}_k^q\left(\omega \right)\right|\right)}^2\right],$$ (13)

where n is a free parameter, usually set to n = 2, and the index k runs over the number of modes. By enforcing the orthogonality of the vibrational modes during the minimization, the localized modes are obtained. This approach was, for example, applied to the QM/MM spectrum of bacteriorhodopsin in solution¹⁰⁵ and the DFT-MD spectrum of bare CH₅^+.75 An approach taking also into account symmetries and multiple conformations of a molecule was presented by Baer and Mathias.^{106, 107}

The needed molecular properties are usually determined on the fly using an electronic structure method. While the energy contributions, temperature, and nuclear velocities are always accessible during the MD run, the analysis of spectroscopic observables can be carried out either during the FPMD run, using the same electronic structure code for the calculation of properties, or after the trajectory has been generated, by using specialized software^{108, 109} or other electronic structure codes, taking the nuclear structures of the FPMD run as input for property calculations.

2.2.4 Condensed phase systems

FPMD can provide more accurate results than static calculations, in the gas-phase, by taking into account the effects of the PES shape, temperature, and applied pressure¹¹⁰ on the frequencies, intensities, and associated line shapes.^{73, 111} The more important application though lies in the investigation of more complex systems. In the case of solvent/solute systems, it is well known that the simulations have to include a sufficient number of solvent molecules. In the QM/MM method, the solvent molecules are usually treated explicitly using a force field, while the solute is treated by a QC method. Alternatively, a solvent continuum model offers a computationally cheap treatment of such systems by including the dielectric constant of the solvent.¹¹² However, this approach usually cannot replicate the crucial dynamics of hydrogen bonding networks of the solvent,^113-115 although extensions of continuum models to include such interactions have been published.¹¹⁶

In the treatment of liquids and solid state compounds, periodic boundary conditions can be introduced to avoid artifacts arising from surface effects at the edges of the simulation cell. This poses an additional difficulty because the periodic boundary conditions can introduce unphysical jumps of interatomic distances in the postprocessing of trajectories and the position operator is ill-defined (see Section 3.1 for more details).

Additionally, the wavelet theory approach allows for generating time-resolved spectra. In this scheme, the time-derivative of the generalized mode velocity autocorrelation function C_α(t) is expanded by the continuous wavelet transformation

$$W_{\alpha }\left(a,b\right)=\int C_{\alpha }\left(t\right){\psi }_{a,b}\left(t\right)\mathit{dt},$$ (14)

where a and b parameterize the wavelet ψ_a,b(t). In this way, the frequencies hidden in the time-dependent signal are extracted and localized in the time domain.¹¹⁷

3 TYPES OF SPECTROSCOPY

3.1 Infrared absorption

IR spectroscopy relies on measuring the absorption of infrared radiation at different wavelengths by matter and is among the most adopted experimental techniques in vibrational spectroscopy. It probes the molecular structure as well as the interactions of functional groups within molecules and with their environment. In this way, IR spectra can serve as a fingerprinting technique to identify functional groups or whole compounds and monitor the time evolution of the structure. The IR intensity I_i of the ith vibrational normal mode, in the double-harmonic approximation, is given as

$$I_i\propto \left(\frac{\partial \overrightarrow{\mu }}{\partial Q_i}\right)\cdot \left(\frac{\partial \overrightarrow{\mu }}{\partial Q_i}\right),$$ (15)

where we omit proportionality factors and quantum corrections, Q_i is the ith normal coordinate, and $\overrightarrow{\mu }$ is the value of the system's dipole moment given by

$$\overrightarrow{\mu }=-e\sum _j{\overrightarrow{r}}_j+\sum _J{Z}_J\overrightarrow{R_J},$$ (16)

where e is the elementary charge, ${\overrightarrow{r}}_j$ is the position of the j-th electron, ${\overrightarrow{R}}_J$ is the position of the Jth nucleus and Z_J the corresponding core charge.¹¹⁸

Analogous to Equation (11), the IR spectrum can be calculated using the electric dipole moment itself or its time derivative as

$$I\left(\omega \right)\propto \int {⟨\overrightarrow{\dot{\mu }}\left(\tau \right)\overrightarrow{\dot{\mu }}\left(t+\tau \right)⟩}_{\tau }{e}^{-\mathit{i\omega t}}\mathit{dt},$$ (17)

where it is convenient to use the time derivative because constant offsets of the electric dipole moment are eliminated in this way.⁴⁷ The definition of $\overrightarrow{\mu }$ is only well-defined for isolated molecules in the absence of periodic boundary conditions. In bulk systems the Berry phase approach to polarization^{119, 120} and maximally localized Wannier functions (MLWFs)^121-123 have been successfully applied to overcome this problem.

Applications of FPMD for IR spectra range from gas-phase molecules to solid/water interfaces. Marx et al. have studied the THz range of IR spectra focusing on water in bulk^{124, 125} and simpler environments. Instead of investigating liquid bulk water, it is possible to consider microsolvated systems such as Zundel cations¹²⁶ or HCl–water aggregates including up to six water molecules¹²⁷ shedding light onto the influence of individual ions on the IR spectra, while keeping the computational cost low. The protonated methane ion CH₅⁺ is another prototypical system for FPMD since it exhibits a very anharmonic PES compared to the harmonic PES of methane. Studies in gas-phase^{75, 128} have elucidated the structure and spectroscopic signature thereof. The difficulty of describing the protonated methane ion arises from a flat PES that allows for motions with large amplitudes. In a microsolvated system, the effects of tagging the methane ion with up to three H₂ molecules have been elucidated, and individual vibrational modes have been identified.¹²⁹

It has been shown that it can be beneficial to average trajectories from different initial configurations to increase coverage of the PES further. The (R)-butan-2-ol molecule has nine local minima on the PES, corresponding to different conformations. By averaging nine gas-phase trajectories corresponding to the possible monomer conformations, IR spectra in excellent agreement with experiments are obtained.¹³⁰ The sampling of the PES, parameterized by two dihedral angles, is reproduced from the study of Kirchner et al. and shown in Figure 1.

FIGURE 1

Conformation plot indicating whether all nine minima of the PES of (R)-butan-2-ol molecule are visited. The plot shows the dihedral angle τ₁: C2−C1−C3−C4 against the dihedral angle τ₂: H2−O1−C1−H1. Gas-phase calculations are shown on the left, and bulk-phase calculations are shown on the right. The plot on the left is the average of nine different gas-phase trajectories. Reprinted with permission from Ref. [130]

In the mentioned study of (R)-butan-2-ol¹³⁰ a temperature of 400 K has been applied to enhance the sampling, and the study compares results for the bulk and gas-phase, showcasing the influence of intermolecular interactions on IR and VCD spectra. Such an exploration of the conformation space is most important in gas-phase calculations, where no intermolecular energy exchange can mediate conformational changes, which are naturally present in bulk calculations. As a result, an FPMD run in gas-phase will tend only to sample a single conformer.¹³⁰ The group of Gaigeot studied flexible polypeptides in the gas-phase extensively, computing, among others, IR spectra of a protonated polypeptide. It was shown via FPMD that these peptides do not possess well-defined structures and instead explore the conformational space in contrast to the more rigid butan-2-ol molecule mentioned earlier.¹³¹ The spectrum retrieved for the Ala-Ala-H⁺ ions, obtained by means of a CPMD simulation and from experiment is shown in Figure 2. The simulated spectrum qualitatively matches the experimental one and a mode assignment of the bands was carried out by analyzing the obtained VDOS.¹³¹ Other studies focused on the isomerization dynamics of biological systems in gas-phase by applying CPMD.^132-134 Further proving a point often made, it has been shown by Gupta et al. that a microsolvated system does not replace a fully periodic treatment in the bulk-phase because even solvation shells far away from the solute can influence the spectroscopic signature in the THz region.¹³⁵

FIGURE 2

Shown is the IR spectrum of the gas-phase Ala-Ala-H+ ion peptide in trans conformation. Black curve: R-IRMPD experiment, red curve: Calculation from CPMD at 300 K. The band assignments above are deduced from a VDOS analysis. Reprinted with permission from Ref. [131]; permission conveyed through Copyright Clearance Center, Inc

FPMD has also been applied to study the influence of the solvation shells on THz spectra around small ions¹³⁶ as well as small molecules such as glycine^{124, 137, 138} and trimethylamine N-oxide (TMAO) in solution.¹³⁹ Furthermore, the influence of the pressure exerted on liquid water was studied recently in this context by applying pressure as an external constraint in THz spectra calculations.¹²⁵ Further, Marx et al. studied the liquid bulk water THz spectrum¹⁴⁰ including solvated Na⁺ and Cl⁻ ions.¹⁴¹ The group of Gaigeot studied flexible polypeptides in solution,^{142, 143} demonstrating that DFT-MD is an appropriate tool for reproducing and understanding experimental results. A combination of static and dynamical IR spectroscopy¹⁴⁴ was also successfully applied to the silica/water interface, characterizing the nature of water in the hydration layer on the surface.¹⁴⁵

One often-used way of investigating the origin of spectroscopic signals are localized MOs (e.g., Pipek–Mezey localized functions) in case of nonperiodic systems. The counter part for periodic systems are MLWFs. By localizing the Wannier functions and computing the expectation values of the position operator for each Wannier center, positions and thus also velocities can be assigned to the orbitals. Via the classical connection of the electric dipole moment with positions and charges, local dipole moments can be assigned to each Wannier function.¹⁴⁶ Marx et al., for example, decomposed the total electronic contributions to the electric dipole moment of zwitterions into contributions from molecular fragments via MLWFs. In this way, they were able to split the dipole–dipole correlation function into autocorrelation and cross-correlation terms, elucidating which modes are coupled and which modes are independent.¹²⁷ The same approach has been applied for the mode assignment in the H₂ tagged CH₅⁺ ion using MLWFs and was called the maximum entropy method.¹²⁹

Another way of separating the spectroscopic signals was applied by Luber and is based on periodic subsystem DFT. In this approach, the ill-definition of the position operator in periodic systems can be circumvented by defining subsystems in the total investigated system efficiently using atom-centered basis functions. The electron density is then separated, and a formula for corresponding subsystem electric dipoles was derived. This approach allows for a computationally cheaper treatment while offering the separation of dipole moments and even IR intensities into intra-atomic and inter-atomic contributions as long as the subsystem DFT embedding performs reasonably.¹⁴⁶ A recent extension of the formula of local electric dipole moments along this line is based on following the trajectories of Wannier centers during an FPMD run instead of subsystem DFT.¹⁴⁷

3.2 Vibrational circular dichroism

The chiral equivalent of IR spectroscopy is VCD^148-150 which relies on the different absorption of left and right circularly polarized light by chiral molecules. Its origin lies in the angular momentum of a photon and the interaction thereof with the sample.

As it is very sensitive to structural differences, it is suitable for distinguishing structures of chiral compounds. In a static picture, the rotational strengths associated with VCD spectra depend on the atomic axial tensors (AATs). The rotational strength corresponding to the ith normal coordinate is given by¹⁵¹

$$R_i=\mathrm{Im}\left[\left(\frac{\partial \overrightarrow{\mu }}{\partial Q_i}\right)\cdot \left(\frac{\partial \overrightarrow{m}}{\partial Q_i}\right)\right],$$ (18)

where $\overrightarrow{\mu }$ is the electric dipole moment and $\overrightarrow{m}$ is the magnetic dipole moment consisting of the electronic and nuclear contributions given by

$$\overrightarrow{m}=-\frac{e}{2\mathit{mc}}\overrightarrow{r}\times \overrightarrow{p}+\sum _J\frac{Z_{J}e}{{2M}_{J}c}{R}_J\times P_J,$$ (19)

the electron's angular momentum, and P_J is the atom's angular momentum. In the correlation function formalism, the VCD spectrum can be retrieved from FPMD trajectories as¹⁵²

$$\mathrm{\Delta }\alpha \left(\omega \right)=\frac{4\beta \mathrm{\hslash }\omega }{2\pi }\mathrm{Im}\left[\int d{\mathit{te}}^{-\mathit{i\omega t}}⟨\overrightarrow{\mu }\left(0\right)\cdot \overrightarrow{m}\left(t\right)⟩\right]$$ (20)

$$=\frac{4\beta \mathrm{\hslash }}{2\pi }\mathrm{Im}\left[\int d{\mathit{te}}^{-\mathit{i\omega t}}⟨\overrightarrow{\dot{\mu }}\left(0\right)\cdot \overrightarrow{m}\left(t\right)⟩\right],$$ (21)

where all quantities have the same meaning as before. The second equality corresponds to the velocity form of the electric dipole-magnetic dipole correlation function and is found by using the properties of the Fourier transformation, as explained in the previous section.

In the adiabatic approximation, the ground state electron density is real-valued, whereas the magnetic dipole operator $\overrightarrow{m}$ is imaginary and thus vanishes in standard calculations. First-principles methods for retrieving the AATs or magnetic dipoles for the calculation of VCD spectra are the magnetic field perturbation (MFP), which has only been applied in static calculations employing the harmonic approximation^{150, 153-155} up to now, and only for nonperiodic systems, and nuclear velocity perturbation (NVP) which has also been applied in FPMD simulations.^156-160

VCD implementations should take care of the origin dependence due to the definition of the magnetic dipole operator, which results in the definition of a gauge.¹⁶¹ In the common origin gauge, the same origin of the magnetic dipole operator is applied in all response calculations. In the distributed origin gauge, on the other hand, the origin of the magnetic dipole operator is shifted to the nuclear coordinate in each response calculation. In practice, both gauge choices result in identical VCD spectra, and the AATs calculated in one gauge can be translated to the other.¹⁵⁵ In the MFP approach, this problem has been solved by utilizing gauge-including atomic orbitals (GIAO), which has resulted in gauge independent rotational strengths in static DFT calculations.^{155, 162} In the NVP approach, two strategies have been employed to approach this issue.^{152, 161} MLWFs lead to the definition of local magnetic dipole moments and are related closely to the modern theory of polarization via the definition of the Wannier centers in terms of the Berry phase.¹⁵² Jähnigen et al. instead obtained an explicitly gauge-invariant expression for time-correlation VCD spectra by enforcing the nearest image convention on the time-derivative of the electric dipole moment evaluated in the different periodic images of the simulation cell.¹⁶¹

Scherrer et al. implemented the NVP as proposed by Nafie¹⁵¹ in DFT using a plane-wave basis in the CPMD code.¹⁶⁰ They subsequently applied the implementation to the FPMD trajectory of a series of model systems, including (S)-d₂-oxirane and (R)-propylene-oxide in the gas-phase and solution.¹⁵² In this way, it is possible to treat solvent effects such as the chirality transfer from a chiral molecule to an achiral solvent, taking into account the dynamics of the system.¹⁵²

In another work, the molecular crystal of l-alanine was studied, showing that the coherent vibrations of the solid-state structure enhance the VCD signal, especially in the carbonyl region, reproducing the experimental results well.¹⁶³ The solvent effects on (S)-1-indanol in nonpolar CCl₄ and polar DMSO have been investigated by Barbu-Debus et al. in a combined static and dynamic study.¹⁶⁴ An assessment of cluster models, static calculations, classical MD and FPMD was carried out using the flexible (1S,2S)-trans-1-amino-2-indanol molecule in DMSO solution.¹⁶⁵ The NVP approach was also extended to fully periodic calculations and applied to the molecular crystal of (1S,2S)-CHDO¹⁶¹ and used to investigate the hydrogen bonding network in an aqueous solution of lactic acid.¹⁶⁶

Another way to calculate VCD spectra is to use the classical definition of the magnetic dipole moment based on fixed partial charges.^{97, 167, 168} Thomas and Kirchner have applied this classical approach by using the time-dependent electronic density $\rho \left(\overrightarrow{r},t\right)$ available from DFT simulations combined with radical Voronoi tessellation, using the following definition of the magnetic dipole moment¹⁶⁹:

$$\overrightarrow{m}\left(t\right)=\frac{1}{2}\int \overrightarrow{r}\times \overrightarrow{j}\left(\overrightarrow{r},t\right)d\overrightarrow{r},$$ (22)

where $\overrightarrow{j}\left(\overrightarrow{r},t\right)$ is the electric current density which fulfills the continuity equation

$$\frac{\partial \rho \left(\overrightarrow{r},t\right)}{\partial t}+\nabla \cdot \overrightarrow{j}\left(\overrightarrow{r},t\right)=0.$$ (23)

The reference point has been set to the system's center of mass to reduce the origin dependence. In this approach, the electron density is stored at each time step so that finite differences with respect to the time propagation can be applied afterwards.¹⁶⁹ They applied their method to the IR and VCD spectra of (R)-2-butanol solution in carbon disulfide^{130, 169} and glucose in an ionic liquid.¹⁷⁰

3.3 Raman

Raman spectroscopy relies on the inelastic scattering of light in the near-infrared, visible, and near-ultraviolet regions of the electromagnetic spectrum. Light excites the molecule into a higher energy state, and then it decays back to the ground state, carrying information on the vibrational modes through its frequency shift. Core to the theoretical treatment of Raman scattering is the electric dipole–electric dipole polarizability tensor α and its higher-order counterparts, that is, the second-rank hyperpolarizability tensor β and the third-rank second hyperpolarizability tensor γ. In the presence of an external electric field $\overrightarrow{E}$, an electric dipole moment $\overrightarrow{\mu }$ is induced according to¹⁷¹

$${\mu }_{\rho }={\alpha }_{\mathit{\rho \sigma }}{E}_{\sigma }+\frac{1}{2}{\beta }_{\mathit{\rho \sigma \tau }}{E}_{\sigma }{E}_{\tau }+\frac{1}{6}{\gamma }_{\mathit{\text{ρστυ}}}{E}_{\sigma }{E}_{\tau }{E}_{\upsilon }+\dots ,$$ (24)

where summation over repeated indices is implied. Raman spectra can be retrieved in both static and dynamic calculations. The intensities will generally depend on the polarization of the incident light and the detected polarization. In a static approach, considering the random orientation of molecules in a sample, the averaged Raman intensity measured along the polarization of the incident light is given by⁴⁷

$$I_{\parallel }\left(\nu \right)\propto \frac{{\left({\nu }_{\text{in}}-{\nu }_i\right)}^4}{{\nu }_k}\frac{45a_k^2+4{\gamma }_k^2}{45},$$ (25)

where ν_in and ν_i are the wavenumbers of the incident light and the ith normal coordinate, respectively, a_k is the isotropic polarizability invariant

$$a_k=\frac{1}{3}\left(\frac{\partial {\alpha }_{\mathit{xx}}}{\partial Q_k}+\frac{\partial {\alpha }_{\mathit{yy}}}{\partial Q_k}+\frac{\partial {\alpha }_{\mathit{zz}}}{\partial Q_k}\right)$$ (26)

and γ_k is the anisotropic polarizability invariant

$${\gamma }_k^2={\left(\frac{\partial {\alpha }_{\mathit{xx}}}{\partial Q_k}-\frac{\partial {\alpha }_{\mathit{yy}}}{\partial Q_k}\right)}^2+{\left(\frac{\partial {\alpha }_{\mathit{yy}}}{\partial Q_k}-\frac{\partial {\alpha }_{\mathit{zz}}}{\partial Q_k}\right)}^2+{\left(\frac{\partial {\alpha }_{\mathit{zz}}}{\partial Q_k}-\frac{\partial {\alpha }_{\mathit{xx}}}{\partial Q_k}\right)}^2$$ (27)

$$+3{\left(\frac{\partial {\alpha }_{\mathit{xy}}}{\partial Q_k}\right)}^2+3{\left(\frac{\partial {\alpha }_{\mathit{yz}}}{\partial Q_k}\right)}^2+3{\left(\frac{\partial {\alpha }_{\mathit{zx}}}{\partial Q_k}\right)}^2.$$ (28)

In the dynamic approach, employing FPMD, the derivatives with respect to the normal coordinates are replaced by autocorrelation functions so that the Raman intensity for the measurement of the Raman signal polarized parallel to the incident light reads.⁴⁷

$$I_{\parallel }\left(\omega \right)\propto \frac{{\left({\omega }_{\text{in}}-\omega \right)}^4}{\omega }\int {⟨{\dot{\alpha }}_{\mathit{xx}}\left(\tau \right){\dot{\alpha }}_{\mathit{xx}}\left(t+\tau \right)⟩}_{\tau }{e}^{-\mathit{i\omega t}}\mathit{dt}$$ (29)

and similar expressions result for the other polarization directions. The polarizability tensors can be calculated by employing perturbation theory,¹⁷² although it was also reported in the literature that a finite differences formula from the induced electric dipole moments under an external electric field can be employed in combination with MLWFs to retrieve molecular polarizabilities.⁴⁷ Another means to calculate the polarizability is given by real-time propagation (RTP) techniques such as real-time time-dependent DFT (RT-TDDFT).^173-177 The main advantage of RTP methods is that they naturally determine the whole frequency range of spectra, because the spectra are connected to the time evolution via a Fourier transformation, while perturbative approaches target specific energy ranges.¹⁷⁸ The full Raman excitation spectrum obtained by this approach for liquid (S)-methyl-oxirane is shown in Figure 3. Additionally, the RTP method exhibits a favorable scaling with the system size and also converges in the presence of a high density of electronic states. Real-time propagation is the only approach that has allowed for the calculation of (pre-)resonance Raman spectra with FPMD.^{175, 177} Studies suggest that the polarizability does not have to be computed at every single time step during the MD run, making it possible to save computational time.^{179, 180}

FIGURE 3

Raman excitation profile of liquid (S)-methyl-oxirane obtained from a combined FPMD and RT-TDDFT simulation. The whole frequency spectrum is obtained by Fourier transformations along the RTP time axis and the FPMD time axis. Reprinted with permission from Ref. [177]. Copyright 2021 American Chemical Society

Under periodic boundary conditions, the position operator and thus the electric dipole moment operator are ill-defined. When calculating the electric dipole–electric dipole polarizability tensor, it is necessary to adapt the computational scheme to account correctly for the periodicity of the system. One such approach is the Berry phase formalism,¹⁸¹ which was recently applied to calculate the polarizability tensor in a study of molten salts via Raman spectra from FPMD.¹⁸¹ but also the velocity representation of the electric dipole moment operator can be used. The velocity representation is found by making use of the relation.¹⁷⁷

$$\left[\widehat{\overrightarrow{r}},\widehat{H}\right]=\frac{i\mathrm{\hslash }}{m_e}\widehat{\overrightarrow{p}}+\left[\widehat{\overrightarrow{r}},{\widehat{V}}^{\mathrm{nl}}\right],$$ (30)

where ${\widehat{V}}^{\mathrm{nl}}$ collects the nonlocal potentials from the Hamiltonian operator.

In one of the earliest FPMD studies of Raman scattering, Putrino et al. applied DFPT¹⁸² and the Berry phase approach for polarization to the trajectory of ice at different high pressures. In this way, they were able to reproduce the Raman spectra associated with the phase transitions of water in the GPa range, also showing the effects of anharmonicity.¹⁸³ Similar studies were carried out with a naphthalene crystal, including additionally van der Waals corrections.¹⁸⁴

The first Raman FPMD study of the dynamics of liquid water by Wan et al., containing 64 heavy water molecules at 400 K, employed the MLWF approach to decompose the calculated Raman intensities. In this way, the differences of intramolecular and intermolecular contributions and the vibrational signature of intermolecular charge fluctuations were made apparent.¹⁷⁹ Luber et al. applied DFPT to liquid (S)-methyl-oxirane and derived a decomposition of the Raman signal into intramolecular and intermolecular contributions exploiting atom-centered basis functions with the mixed Gaussian and plane-wave approach.¹⁸⁰ In a study by Mattiat and Luber, a first Raman excitation profile, including the nonresonant and resonant parts for liquid (S)-methyl-oxirane, was calculated in one go by making use of RTP.¹⁷⁷ Brehm and Thomas applied RT-TDDFT in a bulk study of uracil in aqueous solution, calculating the electric dipole derivatives with respect to time by Voronoi integration.¹⁷⁵

An FPMD study of surface-enhanced Raman scattering employing linear-response TDDFT has also been reported by Aprà et al. The authors set up multiple FPMD trajectories of two aromatic thiols adsorbed to a silver substrate. For each of the trajectories, the molecule was tilted by a different angle (in steps of 20°) and first optimized. By running FPMD simulations starting from each of the structures and computing the corresponding Raman spectra, the authors showed that they could extract the orientation of the molecule relative to the surface, matching the results of X-ray experiments.¹⁸⁵

The Wannier polarizability method propagates the Wannier centers together with the nuclei during the simulation. The positions of the Wannier center can then be used to calculate the polarizability α in a classical way. This approach allows for on-the-fly calculations of the polarizability by decomposing the total polarizability α as¹⁸⁶

$$\mathit{\alpha }=\sum _i^{\mathrm{WF}}{\mathit{\alpha }}_i^{\mathrm{WF}}\simeq \sum _i^{\mathrm{WF}}\beta S_i^3,$$ (31)

where the sum goes over all Wannier centers, β is a constant, and S_i is the spread of the ith Wannier function. In this way, the perturbative or RTP treatment of the system is avoided.¹⁸⁶ Aside from that, Voronoi tessellation of the electron density has been reported as an alternative to the MLWF approach, which is computationally cheaper, where the polarizability is computed by finite differences applied to the classical definition of electric dipole moments.¹⁸⁷

3.4 Raman optical activity

The chiral variant of Raman spectroscopy is ROA spectroscopy. The calculation of vibrational ROA spectra requires in the general case three polarizabilities: the electric dipole–electric dipole polarizability tensor α, the electric dipole–electric quadrupole polarizability tensor A, and the electric dipole–magnetic dipole polarizability tensor G^′.

The ROA intensity difference for the backscattering experimental setup is, for instance, given by¹⁰¹

$$\left(I_R-I_L\right)\propto \beta {\left({\mathit{G}}^{\prime }\right)}^2+\frac{1}{3}\beta {\left(\mathit{A}\right)}^2,$$ (32)

where the anisotropic ROA invariants are obtained as

$$\beta {\left({\mathit{G}}^{\prime }\right)}^2=\sum _{\mathit{\alpha \beta }}\frac{1}{2}\left(3{\alpha }_{\mathit{\alpha \beta }}{G}_{\mathit{\alpha \beta }}^{\prime }-{\alpha }_{\mathit{\alpha \alpha }}{G}_{\mathit{\beta \beta }}^{\prime }\right)$$ (33)

$$\beta {\left(\mathit{A}\right)}^2=\sum _{\mathit{\text{αβγδ}}}\frac{1}{2}{\alpha }_{\mathit{\alpha \beta }}{\mathrm{ϵ}}_{\mathit{\alpha \gamma \delta }}{A}_{\gamma ,\mathit{\delta \beta }},$$ (34)

where ϵ_αγδ is the anti-symmetric Levi-Civita tensor, α_αβ and $G_{\mathit{\alpha \beta }}^{\prime }$ are the components of the electric dipole–dipole tensor and the electric dipole-magnetic dipole tensor respectively. A_γ,δβ are the components of the electric dipole-electric quadrupole polarizability tensor. In static calculations, the derivatives of the ROA tensors with respect to the normal coordinates are required. In FPMD simulations, the time correlation formulations for the invariants are given by¹⁰¹

$${\left(\beta {\left({\mathit{G}}^{\prime }\right)}^2\right)}_{\mathrm{MD}}\propto \sum _{\mathit{\alpha \beta }}\int d{\mathit{te}}^{-\mathit{i\omega t}}\left[⟨3{\alpha }_{\mathit{\alpha \beta }}\left(0\right)G_{\mathit{\alpha \beta }}^{\prime }\left(t\right)⟩-⟨{\alpha }_{\mathit{\alpha \alpha }}\left(0\right)G_{\mathit{\beta \beta }}^{\prime }\left(t\right)⟩\right]$$ (35)

$${\left(\beta {\left(\mathit{A}\right)}^2\right)}_{\mathrm{MD}}\propto \sum _{\mathit{\text{αβγδ}}}\int \mathrm{d}{\mathit{te}}^{-\mathit{i\omega t}}{\mathrm{ϵ}}_{\mathit{\alpha \gamma \delta }}⟨{\alpha }_{\mathit{\alpha \beta }}\left(0\right)A_{\gamma ,\mathit{\delta \beta }}\left(t\right)⟩.$$ (36)

Different methods for the calculation of ROA spectra based on FMPD have been proposed. The first FPMD implementation of ROA used the DFPT approach and was reported by Luber and applied to an S-methyl-oxirane molecule in the gas-phase¹⁰¹ (see Figure 4). Gauge independent results were obtained by using the velocity representation for the electronic part of the electric dipole moment.¹⁰¹

FIGURE 4

The backscattering ROA spectra of an (S)-methyloxirane molecule in gas-phase obtained from FPMD simulations. Reprinted with permission from Ref. [101]. Copyright 2017 American Chemical Society

A more approximate and computationally straightforward approach, proposed by Brehm and Thomas, relies on the classical approximation of these tensors via Voronoi tessellation and a finite differences derivative with respect to an external electric field for obtaining the ROA invariants. The method was applied to a bulk system of liquid (R)-propylene oxide. In order to minimize the gauge dependency, the origin of the electric moments was set, as an approximation, to each molecule's center of mass in this study.¹⁰⁰

3.5 Sum frequency generation

Sum frequency generation (SFG) spectroscopy is a powerful technique for characterizing interfaces at the molecular level. It is a second-order process, where two photons with frequencies ω₁ and ω₂ are absorbed by the sample, resulting in the generation of a third photon with frequency ω₃ = ω₁ + ω₂, conserving the energy. The origin of the effect lies in an induced polarization P_α.^{188, 189}

$$\begin{array}{l}{P}_{\alpha }\left({\omega }_3\right)=\sum _{\mathit{\beta \gamma }}{ϵ}_0{\chi }_{\mathit{\alpha \beta \gamma }}^{\left(2\right)}\left({\omega }_1,{\omega }_2\right)E_{\beta }\left({\omega }_1\right)E_{\gamma }\left({\omega }_2\right)\\ +\sum _{\mathit{\beta \gamma \delta }}{\chi }_{\mathit{\text{αβγδ}}}^{\left(3\right)}\left({\omega }_1,{\omega }_2\right)E_{\beta }\left({\omega }_1\right){\nabla }_{\gamma }{E}_{\delta }\left({\omega }_2\right)+\sum _{\mathit{\beta \gamma \delta }}{\chi }_{\mathit{\text{αβγδ}}}^{\left(3\right)}\left({\omega }_2,{\omega }_1\right)E_{\beta }\left({\omega }_2\right)\left({\nabla }_{\gamma }{E}_{\delta }\left({\omega }_1\right)\right)+\dots \end{array}$$ (37)

where ${ϵ}_0$ is the vacuum permittivity, χ⁽²⁾ is the second-order nonlinear susceptibility rank-3 tensor, χ⁽³⁾ is the rank-4 quadrupole susceptibility tensor, $\overrightarrow{E}$ are the electric fields associated with the incoming photons, and the indices α, β, γ, δ go over the Cartesian coordinates. The special case of ω₁ = ω₂ is called second harmonic generation (SHG). SFG processes are forbidden in centrosymmetric systems in the electric-dipole approximation. At interfaces, this symmetry is usually broken, making SFG an interface-specific method.¹⁸⁸ If higher-order moments such as electric quadrupoles and magnetic moments are considered as well, additional bulk contributions arise in the measured spectra.¹⁸⁸ Experimentally, these contributions cannot be distinguished from the interface signal.¹⁸⁸ The correlation expressions for the susceptibility tensors employed in the case of FPMD, using the electric dipole-polarizability correlation, are given by¹⁹⁰

$${\chi }_{\mathit{\alpha \beta \gamma }}^{\left(2\right)}\left(\omega \right)=\mathit{i\omega \beta }{\int }_0^{\infty }d{\mathit{te}}^{\mathit{i\omega t}}⟨{\alpha }_{\mathit{\alpha \beta }}\left(t\right){\mu }_{\gamma }\left(0\right)⟩,$$ (38)

where α_αβ and μ_k are the components of the polarizability tensor and electric dipole moment of the whole system, respectively, and β is the reciprocal of the Boltzmann constant multiplied by the system's temperature. The phase-resolved SFG spectra can have both positive and negative peaks, corresponding to the transition dipole moment between vibrational states pointing up into the air or down to the bulk, respectively.¹⁴

Sulpizi et al. have shown that the vibrational SFG spectra can be decomposed using MLWFs to extract the contributions of individual molecular electric dipole moments and polarizability tensors.^{190, 191} For this purpose, the polarizability and electric dipole moments are calculated (classically) from the positions of the Wannier centers, applying a finite differences formula to a weak external field. In their study, they investigated the air–water interface and were able to qualitatively reproduce the experimental results reported for this system—despite a relatively small system size of 128 water molecules, short MD trajectory of 3 ps and neglect of cross-correlation terms between the individual molecules. The computational cost of such studies makes a wide application still very challenging because long trajectories have to be considered while also applying an explicit electric field to the model when using finite differences.¹⁹⁰ Liang et al. devised another scheme for reducing the computational cost of SFG simulations of interfacial water. The polarizability tensor was decomposed into molecular polarizabilities of individual water molecules using MLWFs and CCSD calculations. For this purpose, the system was split into a single water molecule and point charges at the Wannier centers of the remaining water molecules. The process was then repeated for each water molecule, and the molecular polarizability was computed at the CCSD level for each water molecule, in an approach akin to embedding.¹⁹² The resulting SFG spectrum is shown in Figure 5.

FIGURE 5

Shown are the simulated IR, Raman, and SFG_xxz(yyz) spectra of interfacial water at the water–air interface. The libration and bending regions (left) and the OH stretch region (right) of the SFG spectrum for interfacial water are highlighted in the bottom plot. Reprinted with permission from Ref. [192]. Copyright 2019 American Chemical Society

Instead of relying on a finite differences formula including an applied external electric field, DFPT can be applied to calculate SFG spectra.¹⁹³ This was demonstrated for a FPMD SFG calculation of a solid–gas interface, calculating the polarizability tensor using DFPT. Another advantage is the higher accuracy and reduced computational cost because more calculations are needed when applying finite differences.¹⁹³ DFPT has been applied to a layer of acetonitrile molecules adsorbed to a rutile (110) TiO₂ surface, investigating the asymmetric methyl stretching vibration.¹⁹³ Wan et al. employed the DFPT treatment for the SFG simulation of a periodic ice system,¹⁷⁹ also going beyond the standardly used electric dipole approximation by determining the electric quadrupole contributions using MLWFs and decomposing the spectrum into bulk and interfacial contributions.¹⁹⁴

In the quest for understanding water surfaces at the molecular level in even more detail, the partitioning of water surfaces into the binding interfacial layer (BIL), the diffuse layer (DL), and the bulk offers another way of understanding the complex hydrogen bonding network.¹⁹⁵ Pezzotti et al. have provided an unambiguous definition of these layers in simulations from the intrinsic structural properties of the water model. The DL structure is adopted by liquid water influenced by interfacial charges, while the BIL is specific to each interface. For this reason it is advantageous to define the SFG spectrum in terms of two contributions, that is, ${\chi }^{\left(2\right)}\left(\omega \right)={\chi }_{\mathrm{BIL}}^{\left(2\right)}\left(\omega \right)+{\chi }_{\mathrm{DL}}^{\left(2\right)}\left(\omega \right)$. This approach was applied to bulk water, the air/water interface including KCl, and a α-quartz/water interface¹⁹⁶ and the resulting SFG spectra are reproduced in Figure 6. In the same study, the third-order susceptibility χ⁽³⁾ was additionally calculated from the electric field resulting from the surface charge. The approach was extended to extract information on the surface potential and charge, the isoelectric point, electric double layer formation, and the relationship between pH and surface hydroxylation state in α-quartz.¹⁹⁷ The controversial silica/water interface was studied using the same formalism^{198, 199} also including THz spectroscopy as an additional probe into the structure of the interface.²⁰⁰

FIGURE 6

The simulated phase resolved SFG spectrum for aqueous solutions of KCl (left: 0.4 M, right: 2 M) at the interface with (0001) α-quartz is shown for the BIL, DL, and bulk. The top-most plot is the sum of all contributions, while the decomposition into the individual contributions is shown in the bottom plots. Reprinted with permission from Ref. [196]; permission conveyed through Copyright Clearance Center, Inc

To reduce the computational cost of the polarizability calculation further, Ohto and co-workers proposed a different scheme based on velocity–velocity correlations, which allow for skipping the calculation of the electric dipole–electric dipole polarizability altogether.²⁰¹ Because the OH-stretch vibrations have been shown to dominate the SFG spectrum of the water–air interface due to dangling bonds towards the interface, the correlation function termed surface-specific velocity–velocity autocorrelation function (ssVVAF) is given in terms of OH bonds ${\overrightarrow{r}}_j^{\mathrm{OH}}$ by

$${\chi }_{\mathit{\alpha \alpha \gamma }}^{\left(2\right)}\left(\omega \right)=\frac{{\mu }_{\mathrm{str}}^{\prime }{A}_{\mathrm{str}}^{\prime }}{i{\omega }^2}{\int }_0^{\infty }d{\mathit{te}}^{-\mathit{i\omega t}}〈\sum _{i,j}{g}_t(r_{\mathit{ij}}\left(0\right);r_t){\dot{r}}_{\gamma ,j}^{\mathrm{OH}}\left(0\right)\frac{{\overrightarrow{\dot{r}}}_j^{\mathrm{OH}}\left(t\right)\cdot {\overrightarrow{r}}_j^{\mathrm{OH}}\left(t\right)}{\left|{\overrightarrow{\dot{r}}}_j^{\mathrm{OH}}\left(t\right)\right|}〉,$$ (39)

where the indices i and j go over all water molecules, ${\overrightarrow{A}}^{\prime }={\alpha }_{\mathit{xx}}/r^{\mathrm{OH}}$, assuming that α_xx = α_yy = α_zz, and analogous for ${\overrightarrow{\mu }}^{\prime }$, g_t is a cross-correlation cutoff term using the cutoff radius r_t and it is assumed that only the diagonal elements of the polarizability tensor α^OH contribute to the surface-specific vibrational SFG spectrum and can be parameterized.²⁰¹ An additional coupling factor can be included to account for the intramolecular and intermolecular coupling between OH-stretch motions. This approach results in a speed-up of an order of magnitude compared to the explicit calculation of the polarizability while still reproducing the experimental features of the air–water interface SFG spectrum quite well.²⁰¹ The same approach was applied to the α-Al₂O₃ (0001) surface with adsorbed D₂O.²⁰² In a similar study employing the same ssVVAF method, the aqueous TMAO-solution/air interface was characterized. Helped by a vibrational analysis at the VQDPT2 level of theory to assign vibrational bands, the influence of alkyl chain length on the surface SFG spectra was demonstrated.²⁰³ Similar studies were also carried out on the fluorite/water interface²⁰⁴ and extended to the bending region of the spectrum by projecting the atom velocities onto the molecular normal modes defined via collective variables.^{205, 206}

Similar to the calculation of Raman spectra,¹⁸⁶ the Wannier polarizability method has been applied for on-the-fly simulations of vibrational SFG spectra of the air/water interface.²⁰⁷ While the authors in the studies mentioned above computed the time-averaged vibrational SFG spectra, it is also possible to calculate time-resolved SFG spectra via a pump-probe scheme,²⁰⁸ although to our knowledge, this has not been carried out in a fully first-principles framework.

3.6 Nonequilibrium molecular dynamics

Among the third-order infrared spectroscopic methods, there are two-dimensional (2D)-IR (pump-probe) spectra and stimulated photon echo signals. The theoretical foundation of such spectroscopies is the third-order polarization contribution given by²⁰⁹

$${\overrightarrow{P}}^{\left(3\right)}\left(t_3,t_2,t_1\right)={\int }_0^{\infty }d{t}_3{\int }_0^{\infty }d{t}_2{\int }_0^{\infty }d{t}_{1}R\left(t_3,t_2,t_1\right)$$ (40)

$$\cdot \overrightarrow{E}\left(t-t_3\right)\overrightarrow{E}\left(t-t_3-t_2\right)\overrightarrow{E}\left(t-t_3-t_2-t_1\right),$$ (41)

where R(t₃, t₂, t₁) is the third-order response function in terms of time-delays t₁, t₂, and t₃, and the electric field pulses are given by²⁰⁹

$$E\left(t\right)=\mathrm{ϵ}\left(t\right)\exp \left(-\mathit{i\omega t}\right)\exp \left(-i\overrightarrow{k}\cdot \overrightarrow{r}\right)+\mathrm{c}.\mathrm{c}.,$$ (42)

where ω, $\overrightarrow{k}$, and ϵ(t) are the carrier frequency, wave vector, and envelope function of the pulse, respectively. The third-order response function is given as²⁰⁹

$$R\left(t_1,t_2,t_3\right)=i^3\mathrm{Tr}\left\{\mu \left(t_1+t_2+t_3\right)\left[\mu \left(t_1+t_2\right),\left[\mu \left(t_1\right),\left[\mu \left(t_0\right),{\rho }_{\mathrm{eq}}\right]\right]\right]\right\},$$ (43)

where μ are the time-dependent electric dipole moments and ρ_eq is the equilibrium density matrix. Pump–probe spectroscopy measures the third-order polarization, ${\overrightarrow{P}}^{\left(3\right)}\left(t_3,t_2,t_1\right)$, which is expressed by the convolution of the third-order response function, R(t₁, t₂, t₃), with electric field pulses.²¹⁰

By applying more than one pulse to the investigated system, 2D and time-resolved spectra can be observed. In the frequency domain, a tunable narrow-band IR pump pulse excites the vibrational states, while another broadband IR probe pulse can then be used to record a spectrum for each pump frequency.^{211, 212} In this way, absorption spectra can be resolved in two dimensions, allowing to resolve the diagonal peaks at equal pump and probe frequency—corresponding to the 1D-IR spectrum—as well as cross-peaks due to coupling of different vibrational modes.^{211, 212} Sequences with more than two pulses have been used to retrieve even more detailed information, including the time evolution of vibrational dynamics on ultra-fast time scales.^{213, 214} A first pulse creates vibrational coherence in the system while the subsequent pulses after tunable delays disturb the coherence and finally reveal information on occurring coherence, population transfers, and energy relaxations.²¹⁵ In complex biological systems such as peptides, the 2D-IR peaks also make it possible to extract structural information and to distinguish conformers.²¹² A procedure for transient vibrational spectral analysis, combining FPMD and time-integrated modes obtained by means of TDDFT, has been presented, allowing to study the temporal evolution of vibrational signatures that follow from electronic excitations.²¹⁶ This approach was also applied to transient Raman spectroscopy,²¹⁷ providing a direct route to assigning vibrational motion to photodynamics.

There have been two main approaches to calculate such spectra in theoretical studies: the vibrational modes of interest can be excited in a targeted, albeit “classical” way. In this scheme, additional kinetic energy contributions are applied to atoms involved in the vibrational modes of interest, modeling the excitation arising from a pump pulse without having to explicitly treat the different pulses necessary in experimental setups.²¹⁸

Alternatively, the third-order response functions can be calculated from first-principles by evaluating Equation (43). FPMD studies of 2D-IR spectroscopy have only emerged in the past decade, while previous studies sometimes combined classical trajectories with first-principles methods to calculate electronic properties. The classical MD studies have focused mainly on spectra obtained from bulk water.^{11, 215, 219} The nonlinear response functions, which are functions of the time delays and pulse frequencies, are 2D-Fourier transformed to retrieve the frequency correlation spectra.²²⁰ Only recently, DFT-MD for 2D-IR spectroscopy has been applied for photon echo spectroscopy. The relevant observables are then determined from FPMD trajectories combined with the previously mentioned wavelet approach, while the explicit calculation of the third-order response function gives the time dependence of the observables.²²⁰ This approach was applied to elucidate the hydrogen bonding networks in neat D₂O,²²⁰ solvation shells of ions²²¹ and small molecules in solution^{222, 223} including the temperature dependence of the dynamics.²²⁴

Nagata et al. applied the reverse nonequilibrium molecular dynamics (RNEMD) in an FPMD framework²¹⁸ for studying the dynamics of bulk water. In this scheme, the O–H stretch vibrations are excited, and then the excess velocity correlation function is determined from the difference of the RNEMD and equilibrium MD trajectories. In this way, the effect of delocalized OH stretch modes on the spectral diffusion has been studied, as well as the contributions of intramolecular and intermolecular energy transfers by tracking specific OH chromophore subgroups.²²⁵ In the same NE-FPMD framework, the interfacial OH stretching modes on a water-covered α-alumina (0001) structure were excited, elucidating the coupling to surface phonons and the water adlayer.²²⁶ The corresponding VDOS at different time delays is shown in Figure 7. Furthermore, the air–water interface²²⁷ and the difference of OH stretching energy relaxation in neat bulk water and deuterated bulk water were investigated in the NEMD framework.²²⁸ Similarly, by classically exciting the vibrational states, Lesnicki and Sulpizi investigated the energy relaxation via FPMD in bulk water²²⁹ and at the CaF₂/water interface.²³⁰

FIGURE 7

Shown are the vibrational densities of states (VDOS) for a water-covered alumina surface. (a) The VDOS from an equilibrium FPMD simulation (black) is compared to the curves obtained from NE-FPMD, with time delays Δt = 1 ps (blue) and Δt = 10 ps (red), respectively. (b) VDOS spectra from NE-FPMD simulations at various delay times are shown. The red spectrum corresponds to the VDOS just before the excitation and is similar to the spectrum shown on the left. Reprinted with permission from Ref. [226]; permission conveyed through Copyright Clearance Center, Inc

4 CONCLUSION AND OUTLOOK

We reviewed the theoretical foundations of FPMD and various vibrational spectroscopies from FPMD, among them, 1D absorption spectroscopy in the non-chiral and chiral variant (IR and VCD), chiral and non-chiral 1D scattering spectroscopies (Raman and ROA) as well as 2D (SFG) and nonequilibrium spectroscopies (pump-probe).

The essential components for FPMD spectroscopy are, the power spectrum obtained from trajectories that need to correctly and sufficiently sample the configuration space using accurate force calculations, and the intensities that usually result from the same electronic structure calculation as used for the forces. The growing availability of high-performance computing facilities and the improvement of consumer-grade hardware in recent years have enabled more groups to collect FPMD trajectories. The analysis of power spectra has been aided by identifying vibrational modes and assigning them to spectral bands,^{96, 104} whereas MLWFs,⁴⁷ subset embedding approaches,¹⁴⁶ and other local (polarization) methods¹⁸⁰ are powerful tools for the in-depth analysis of obtained intensities.

While the power spectra and vibrational absorption spectra can be generated rather cheaply, advances in theoretical methods have opened the possibilities for investigating more kinds of spectroscopies and systems. This includes the recently published implementation of FPMD NVPT for VCD,¹⁵⁹ the application of perturbation theory and real-time propagation techniques for (resonance) Raman and ROA spectra,^175-177 and the application of perturbation theory and decomposition techniques for SFG spectra.^{190, 192, 193} Nonequilibrium MD simulations, including 2D IR spectroscopy, pump-probe schemes, and stimulated photon echo studies, have been studied only recently.^{218, 223} The electronic properties beyond the electric dipole moment have been obtained mainly by applying simple finite difference formulas, DFPT, or real-time and linear response TD-DFT. In principle, more advanced techniques or other electronic structure methods besides DFT can still be employed to retrieve accurate electric and magnetic properties of the studied systems. Worth mentioning is also the simplified Tamm–Dancoff (TD)-DFT formulation that allows for a speed-up of calculations by several orders of magnitude in some cases.²³¹

In the other direction, approximative or more classical methods such as Voronoi tessellation^{100, 169, 187} or the treatment of Wannier centers as point charges^{146, 147, 152, 186} have been applied to reduce the computational cost. Additionally, specialized analysis methods have emerged that circumvent some of the complexity arising from high-level QC calculations. Among those is the Wannierization approach¹⁸⁶ for the calculation of polarizabilities in the context of Raman spectra.

The wavelet approach allows for the calculation of time-dependent spectra.^{117, 232} This approach has already proved invaluable in other fields of science and offers great advantages also in the context of FPMD. A more empirical but useful approach lies in leveraging our understanding of vibrational modes in many materials to focus analyses on the main contributors to spectroscopic signals. Ohto et al. specified the widely used time-correlation functions to the OH stretching motions at water surfaces and coined the term of surface-specific velocity–velocity autocorrelation functions (ssVVAF) for this purpose in SFG simulations.²⁰¹

Many improvements can still be made. This includes the calculation of higher-order response functions for multidimensional spectroscopy²²⁰ as well as tweaks to the BO-FPMD scheme as, for example, Nagata et al. have proposed for their reverse nonequilibrium molecular dynamics (RNEMD), that allows for the excitation of vibrational states in a direct way.^{218, 233} The acceleration of FPMD simulations is also an important goal since it will allow to study larger systems, for longer trajectories that ensure better convergence of the accumulated spectra and will make it possible to apply more expensive electronic structure methods for retrieving properties. Polack et al. for instance, recently published an efficient method for extrapolating the density matrix guess between MD steps for faster convergence, especially with tight convergence requirements.²³⁴ Other extrapolation techniques have been proposed, for instance, by Herr and Steele.²³⁵ Another example for accelerating FPMD simulations is the use of multiple time steps.²³⁶ Second-generation CPMD is another approach that combines aspects of CPMD and BOMD to extend the size of possible simulations,³ and linear scaling DFT offers an important way of investigating systems that are orders of magnitude larger than standard calculations.²³⁷ Hybrid functionals for DFT have rarely been employed for calculating vibrational spectra via PFMD because of the large computational cost associated with evaluating the exchange-correlation potential. Advances such as the auxiliary density matrix method (ADMM),²³⁸ the resolution of identity approach²³⁹ or machine learning could provide speed-ups making the use of these functionals viable.

Machine learning is a field of research that has attracted significant interest recently, also for applications to vibrational spectroscopy. Machine learning approaches have been applied to calculate electric dipoles efficiently for the calculation of IR spectra^240-243 also in bulk-phase systems.²⁴⁴ In combined schemes, the electric dipole and molecular polarizabilities have been obtained^{243, 245-247} and machine learning can generally be applied to accelerate the MD simulations themselves (see, e.g., Refs [242, 248, 249]) and to generate accurate force fields.²⁵⁰ Since, to the best of our knowledge, machine learning has only been used for IR and Raman FPMD simulations, there is still plenty of room to advance other FPMD spectroscopies as well.

AUTHOR CONTRIBUTIONS

Edward Ditler: Methodology (equal); visualization (lead); writing – review and editing (equal). Sandra Luber: Methodology (equal); resources (lead); supervision (lead); visualization (supporting); writing – review and editing (equal).

ACKNOWLEDGMENTS

This work is supported by the University of Zurich. Open Access Funding provided by Universitat Zurich.

CONFLICT OF INTEREST

The authors have declared no conflicts of interest for this article.

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DATA AVAILABILITY STATEMENT

Data sharing is not applicable to this article as no new data were created or analyzed in this study.