PyRAMD Scheme: A Protocol for Computing the Infrared Spectra of Polyatomic Molecules Using ab Initio Molecular Dynamics

Abstract

Here, we present a general framework for computing the infrared anharmonic vibrational spectra of polyatomic molecules using Born–Oppenheimer molecular dynamics (BOMD) with PyRAMD software. To account for nuclear quantum effects, we suggest using a simplified Wigner sampling (SWS) approach simultaneously coupled with Andersen and Berendsen thermostats. We propose a new criterion for selecting the parameter of the SWS based on the molecules’ harmonic vibrational frequencies and usage of the large-time-step blue shift correction, allowing for a decrease in computational expenses. For the Fourier transform of the dipole moment autocorrelation function, we propose using the regularized least-squares analysis, which allows us to obtain higher-frequency resolution than with the direct application of fast Fourier transform. Finally, we suggest the usage of the pre-parameterized scaling factors for the IR spectra from BOMD, also providing the scaling factors for the spectra at the BLYP-D3(BJ)/6-31G, PBE-D3(BJ)/6-31G, and PBEh-3c levels of theory.

1. Introduction

Molecular dynamics (MD) is the method of describing nuclear motions using classical mechanics [1,2,3,4]. If the potential energy surface (PES) for a simulation is taken from quantum-chemical calculations, such simulations are called ab initio MD (AIMD) [4]. AIMD is a powerful tool for simulating various physical observables of medium to large systems (from tens to a few hundreds of atoms) with nontrivial dynamic behavior  [3,4,5,6]. Examples of modeled observables include radial distribution functions in diffraction experiments [7,8,9,10], dynamic properties of solutions [6,11], and various types of spectra [5,12,13]. Calculating the spectral response in the case of electronic excitations, such as ultraviolet or X-ray spectroscopy, usually relies on sampling the conformational space of molecular systems [12,14]. In contrast, in the case of the vibrational spectra, i.e., infrared (IR) [4,5,10], Raman [13], vibrational circular dichroism [15], etc., the fluctuation–dissipation theorem (FDT) [16,17] has to be applied.

Bare AIMD simulations, by definition, model the motion of classical particles, thus producing a classical microcanonical ($NVE$) ensemble [18]. To emulate a canonical ensemble ($NVT$ or $NPT$), an artificial external system (thermostat and/or barostat) is added to MD simulations [18]. As real nuclei are quantum objects, the absent nuclear quantum effects (NQEs) can also be accounted for in MD using various approaches [19], such as path-integral MD [20,21], quantum thermostats [22], Wigner sampling [14,23,24], etc. [19].

In this paper, we provide a protocol for simulating the IR spectra of molecules with BOMoND software. This MD interface software can take energies and gradients from the quantum-chemical package ORCA 5 [25] or the semiempirical package xTB [26]. BOMoND is part of the PyRAMD packet for AIMD-based simulations, which is also capable of metadynamics [27] and MD-based mass-spectra calculations [28]. We provide examples of applications of BOMoND software, illustrating the various aspects of the MD procedure and subsequent analysis.

2. Methods

The results presented here were obtained using BOMoND software from the PyRAMD package [29]. This software can be obtained from the corresponding repository (Ref.  [29]). The density functional theory (DFT) calculations [30] at the BLYP-D3(BJ)/6-31G  [31,32,33,34], PBE-D3(BJ)/6-31G [33,34,35], and PBEh-3c [36] levels of theory and semiempirical calculations at the GFN2-xTB level of theory [37], including structure optimization and harmonic frequencies computation, were performed using ORCA 5 [25] and xTB software [26], respectively. BOMoND used the same program suits to obtain the gradients for the AIMD simulations. The dipole moment autocorrelation functions calculations and the subsequent Fourier transform (FT) [38] and regularized least-squares spectral analysis (rLSSA) [39] were conducted using the script from the PyRAMD package.

3. Simulation Protocol for Vibrational Spectra Using Molecular Dynamics

3.1. General Idea of Calculation of the IR Spectra from MD Trajectories

MD simulation produces an MD trajectory with $N_{\mathrm{steps}}$ simulation points, which span the dynamics in time t from $t=0$ to $N_{\mathrm{steps}}·\mathrm{\Delta }t$, where $\mathrm{\Delta }t$ is the time step in the simulation. Therefore, for various computed physical observables ($\mathcal{O}$, e.g., coordinates, velocities, energy, dipole moment, polarizability, etc.), we can determine their values ($\mathcal{O}\left(n·\mathrm{\Delta }t\right)={\mathcal{O}}_n$) for an nth given point in the simulation ($0\le n<N_{\mathrm{steps}}$), corresponding to time $t=n·\mathrm{\Delta }t$ [18].

As said in the Introduction, the vibrational spectra from MD were computed with the help of FDT [16,17]. We first need to compute the autocorrelation function ($\langle \mathcal{O}\left(0\right)·\mathcal{O}\left(t\right)\rangle$) of the variable $\mathcal{O}$ that is responsible for the response to the external excitation. For the IR spectra, the observable is the total dipole moment of the molecule; for the Raman spectra, it is the molecular polarizability tensor; for the power spectrum, these are the mass-weighted coordinates [13]. For a real-valued observable, the autocorrelation function is defined as [4,5,13,40]

$$\langle \mathcal{O}\left(0\right)·\mathcal{O}\left(t\right)\rangle ={\int }_0^{\infty }\mathcal{O}\left(t^{\prime }\right)\mathcal{O}(t^{\prime }+t)d{t}^{\prime },$$

(1)

and it shows the repeatable periods in the time evolution of the given observable. The numerical calculation of the autocorrelation function for finite time series is available in various packages. For instance, in Python, one could use either the “correlate” method from the NumPy module [41] or the “signal.correlate” method from the SciPy module [42]. We can transfer the response characteristic from the time domain (t) to the frequency domain ($\nu$), thus obtaining the spectrum of a given process. By performing FT of $\langle \mathcal{O}\left(0\right)·\mathcal{O}\left(t\right)\rangle$ as [4,5,13,40], we obtain the spectrum

$$S\left(\nu \right)={\nu }^2{\int }_0^{\infty }\langle \mathcal{O}\left(0\right)·\mathcal{O}\left(t\right)\rangle ·exp(-i2\pi \nu )dt$$

(2)

For numerical reasons, a more advantageous computational strategy is to use the velocity of the observable ($\dot{\mathcal{O}}$) to obtain spectrum [5]. In this case, the expression to be used is [5]

$$S\left(\nu \right)={\int }_0^{\infty }\langle \dot{\mathcal{O}}\left(0\right)·\dot{\mathcal{O}}\left(t\right)\rangle ·exp(-i2\pi \nu )dt.$$

(3)

The resulting spectra from Equation (2) or (3) can be multiplied with the correction factors that account for the statistical properties of the different spectral ranges [5,43].

The FT procedure for the MD results (Equation (2) or (3)) is usually performed using a fast FT (FFT) approach [38,44], which places certain limitations on the frequency discretization of the resulting spectra. The frequency grid increment ($\mathrm{\Delta }\nu$) is determined by the total trajectory duration ${\tau }_{\mathrm{tot}}$ through the Nyquist–Shannon–Kotelnikov (NSK) theorem [45,46,47] as $\mathrm{\Delta }\nu =1/{\tau }_{\mathrm{tot}}$, while the time step $\mathrm{\Delta }t$ determines the upper boundary of the spectrum as ${\nu }_{max}=1/\mathrm{\Delta }t$. To increase the formal resolution, extending the trajectory for a given observable by an arbitrary number of steps with zeros is possible, which is called zero padding [48]. However, such direct extension causes artifacts in the resulting spectrum; thus, zero padding should be combined with either frequency filters or smoothing techniques [39,48].

An alternative procedure to FFT is regularized least-squares spectral analysis (rLSSA). The idea is to transform the time-dependent dataset

$$\{(t_1,y_1),(t_2,y_2),\dots ,(t_N,y_N)\}$$

(4)

composed of N time values $t_k$ ($1\le k\le N$) with corresponding observable values $y_k$ into the frequency domain with M points

$$\{({\nu }_1,f_1),({\nu }_2,f_2),\dots ,({\nu }_M,f_M)\},$$

(5)

where ${\nu }_l$ ($1\le l\le M$) is the frequency, and the $f_l$ is the corresponding spectral amplitude value. The rLSSA procedure is derived from the regularized weighted least-squares analysis (rwLSSA) procedure by choosing a unit weights matrix (see Refs. [39,49]). As a result, rLSSA is just a matrix transformation [39,49]

$$\mathbf{f}=\left|{\Sigma }_{\alpha }{\mathcal{S}}^{†}\mathbf{y}\right|,$$

(6)

where $\mathbf{y}=(y_1,\dots ,y_N)$ is the N-dimensional vector of the time-domain values, $\mathbf{f}=(f_1,\dots ,f_M)$ is the M-dimensional vector of the frequency-domain spectrum amplitude values, $\mathcal{S}$ is the $N\times M$ matrix of elements $S_{kl}=exp(-\mathfrak{i}2\pi {\nu }_l{t}_k)$ with $\mathfrak{i}$ being an imaginary unit, and ${\mathcal{S}}^{†}$ is the conjugate transpose of matrix $\mathcal{S}$. The matrix ${\Sigma }_{\alpha }$ of size $M\times M$ is defined as ${\Sigma }_{\alpha }=\alpha \mathcal{E}+{\mathcal{S}}^{†}\mathcal{S}$, with $\mathcal{E}$ being a unit matrix of size $M\times M$ and the regularization parameter $\alpha$ being provided by the regularization criterion [39,49]

$$\alpha =tr\left({\mathcal{S}}^{†}\mathcal{S}\right)·{\left(M+{\displaystyle \frac{tr\left({\mathcal{S}}^{†}\mathcal{S}\right)·\left({\mathbf{y}}^{†}\mathbf{y}\right)}{M}}\right)}^{-1}.$$

(7)

To smooth the data even further, parameter $\alpha$ can be increased compared to that from Equation (7).

To illustrate this, we take the vibrational spectrum of carbon dioxide (${\mathrm{CO}}_2$), as shown in Figure 1. A single $NVT$-MD trajectory at the GFN2-xTB level of theory was obtained for a temperature of 300 K using a Berendsen thermostat and Maxwell–Boltzmann sampling of the initial conditions. The trajectory was 1 ps in length with a time step of 0.5 fs, and half of the trajectory was taken as the equilibration phase. The NSK theorem provides the frequency resolution from the dataset of total length ${\tau }_{\mathrm{tot}}=0.5$ ps as $\mathrm{\Delta }\nu ={\left(c·{\tau }_{\mathrm{tot}}\right)}^{-1}=66.8$ cm⁻¹, where $c=0.02998$ cm/ps is the speed of light. Therefore, upon applying the FFT procedure without zero padding with frequency filtering, we obtain quite coarse spectra. We can pad the trajectory with ∼$6.6\times {10}^4$ zeros to obtain the effective frequency resolution of $\mathrm{\Delta }\nu =1$ cm⁻¹. In this case, we obtain a spectrum with high-frequency oscillations around the main peaks. These oscillations can be thought of as the result of the convolution of the original spectrum with the Fourier image of the rectangular function, which represents the sudden cutoff of the real-time dataset. By applying the rLSSA approach to the same dataset with the requested frequency increase of $\mathrm{\Delta }\nu =1$ cm⁻¹, from Equation (7), we have $\alpha =1000$ and a more detailed spectrum (see Figure 1). Note that (1) the rLSSA procedure cannot provide new spectroscopic details and only produces a smoother representation of the same dataset; (2) the rLSSA spectrum also shows similar artifacts in the finite-length trajectory as the zero-padded FFT.

3.2. Cheapening Simulations by Using Large Integration Steps with Frequency Correction

The usual recommendation for computing molecules’ vibrational spectra is to use as small a time step $\mathrm{\Delta }t$ as possible. The reason for this is the artificial blue shift from the numerical integration error, which is especially prominent for high-frequency vibrational bands [50,51]. However, it is possible to correct this behavior by using an analytical formula that compensates for such a shift. In the case of the Verlet algorithm and its equivalent methods (velocity Verlet and leapfrog) [18], one needs to replace the observed frequencies from the FT ($\nu$) with the corrected ones (${\nu }_{\mathrm{corr}}$) using [40,52]

$${\nu }_{\mathrm{corr}}={\displaystyle \frac{\sqrt{2·\left(1-cos\left(2\pi ·\mathrm{\Delta }t·\nu \right)\right)}}{2\pi ·\mathrm{\Delta }t}}=\nu ·sinc\left(\pi ·\mathrm{\Delta }t·\nu \right),$$

(8)

where $sinc\left(x\right)=sin\left(x\right)/x$. This correction is applicable if the NSK-like limit $\mathrm{\Delta }t\le {\left(\pi {\nu }_{max}\right)}^{-1}$ is satisfied, where ${\nu }_{max}$ is the maximal vibrational frequency in the system. For the higher-order methods, similar correction schemes can be derived; see an example in the Appendix A.

To illustrate this artificial blue shift of vibrational bands and the effectiveness of the correction in Equation (8), we ran four sets of ten $NVE$-MD trajectories of methane (${\mathrm{CH}}_4$) at the GFN2-xTB level of theory, with initial conditions sampled from the Maxwell–Boltzmann distribution at 300 K; each trajectory was 0.5 ps in length. The sets were differing by the time step $\mathrm{\Delta }t$, namely, with $\mathrm{\Delta }t=0.1,0.5,1.0,1.5$ fs. The mean vibrational spectra for each set of trajectories and their frequency-corrected counterparts are shown in Figure 2. As one can see, larger time steps lead to a noticeable shift in the C–H stretching band. In the reference spectrum with $\mathrm{\Delta }t=0.1$ fs, this peak is located at approximately 3015 cm⁻¹, whilst in simulations with $\mathrm{\Delta }t=1.5$ fs, this peak appears at approximately 3150 cm⁻¹. However, rescaling the frequency axis using Equation (8) restores the positions of the vibrational bands to their correct origin.

3.3. Simplified Wigner Sampling for Generating Initial Conditions

To include NQEs in MD simulations, we can use the simplified Wigner sampling (SWS) approach introduced in Ref. [14]. The sampling relies on simultaneously generating displacements along the coordinate and momentum axis in a given direction. The displacements are generated using a Gaussian distribution with standard deviations [14]

$${\sigma }_x^2={\displaystyle \frac{\hslash {\tau }_{\mathrm{SWS}}}{2m}}$$

(9)

for coordinate and

$${\sigma }_T^2=m{k}_{\mathrm{B}}{T}_{\mathrm{eff}}$$

(10)

for momentum. Here, $\hslash =1.05\times {10}^{-34}$ J$·$s is the Planck constant, m is the mass of the nucleus, $k_{\mathrm{B}}=1.38\times {10}^{-23}$ J/K is the Boltzmann constant, ${\tau }_{\mathrm{SWS}}$ is a free parameter with a dimension of time, and $T_{\mathrm{eff}}$ is the effective temperature, defined as

$$T_{\mathrm{eff}}(T,{\tau }_{\mathrm{SWS}})=T+{\displaystyle \frac{\hslash }{2k_{\mathrm{B}}{\tau }_{\mathrm{SWS}}}},$$

(11)

where $T\ge 0$ is the temperature of the system.

The main problem with the SWS is the choice of the free parameter ${\tau }_{\mathrm{SWS}}$. In [14], the authors proposed performing several MD simulations with different values of ${\tau }_{\mathrm{SWS}}$ and then taking the parameter value that minimizes the total energy or the mean temperature of the simulation. In some sense, this approach relates to the variational principle, adjusting the underlying Wigner distribution to fit the PES as much as possible. However, such an approach is quite slow and inefficient. Thus, a cheaper criterion for the choice of ${\tau }_{\mathrm{SWS}}$ is required.

To provide an alternative criterion, we may use the harmonic approximation as the guiding principle, as single-harmonic frequency calculation with analytical Hessian generally takes less computational time than running multiple MD trajectories. The total kinetic energy of the system is defined as [18]

$$\mathrm{KE}=\sum _{k=1}^{N_{\mathrm{at}}}{\displaystyle \frac{{\mathbf{p}}_k^2}{2m_k}},$$

(12)

where ${\mathbf{p}}_k$ is the momentum vector of the k-th atom, $m_k$ is the mass of the k-th atom, and $N_{\mathrm{at}}$ is the total number of atoms in the system. In the case of the distribution sampled at $T=0$ with SWS, the mean value of ${\mathbf{p}}_k^2$ is given as $\langle {\mathbf{p}}_k^2\rangle =3{\sigma }_0^2=3\hslash m_k/\left(2\tau \right)$ (see Equation (10)). This gives a total mean kinetic energy of $\langle \mathrm{KE}\rangle =3N_{\mathrm{at}}\hslash m_k/\left(4\tau \right)$, while for a given degree of freedom, the mean kinetic energy is $\langle {\mathrm{KE}}_1\rangle =\langle \mathrm{KE}\rangle /\left(3N_{\mathrm{at}}\right)=\hslash /\left(4\tau \right)$. In the harmonic approximation, for a given l-th mode ($l=1,2,\dots ,N_{\mathrm{f}}$, where $N_{\mathrm{f}}$ is the number of vibrational degrees of freedom) with vibrational frequency ${\nu }_l$, the kinetic energy is ${\mathrm{KE}}_l=h{\nu }_l/4$ [53], and the mean kinetic energy over all of the vibrational modes is

$${\mathrm{KE}}_{\mathrm{h}}=\sum _{l=1}^{N_{\mathrm{f}}}{\displaystyle \frac{h{\nu }_l}{4}}={\displaystyle \frac{N_{\mathrm{f}}h\langle \nu \rangle }{4}},$$

(13)

where $\langle \nu \rangle =\left({\sum }_{l=1}^{N_{\mathrm{f}}}{\nu }_l\right)/N_{\mathrm{f}}$ is the mean vibrational frequency of the system. By taking $\langle {\mathrm{KE}}_1\rangle ={\mathrm{KE}}_{\mathrm{h}}/N_{\mathrm{f}}$, we arrive at the following expression:

$${\tau }_{\mathrm{SWS}},\mathrm{h}={\displaystyle \frac{1}{2\pi \langle \nu \rangle }}={\left(2\pi \stackrel{\langle \nu \rangle }{\overbrace{{\displaystyle \frac{1}{N_{\mathrm{f}}}}\sum _{l=1}^{N_{\mathrm{f}}}{\nu }_l}}\right)}^{-1}.$$

(14)

Equation (14) provides a criterion for the choice of $\tau$ from the harmonic vibrational frequencies of the system. To show its applicability, we ran a set of $NVE$-MD calculations for molecules XH ($\mathrm{X}=\mathrm{F},\mathrm{Cl},\mathrm{Br},\mathrm{I}$), XH₂ ($\mathrm{X}=\mathrm{O},\mathrm{S},\mathrm{Se},\mathrm{Te}$), XH₃ ($\mathrm{X}=\mathrm{N},\mathrm{P},\mathrm{As},\mathrm{Sb}$), and XH₄ ($\mathrm{X}=\mathrm{C},\mathrm{Si},\mathrm{Ge},\mathrm{Sn}$) at the GFN2-xTB level of theory. The MD trajectories were obtained with a 1 fs time step for 0.5 ps in total. The ${\tau }_{\mathrm{SWS}}$ for the initial SWS conditions was scanned from 1 to 10 fs with a 1 fs step, and the MD simulation for a given ${\tau }_{\mathrm{SWS}}$ contained ten individual trajectories. The optimal ${\tau }_{\mathrm{SWS}}$ was chosen as the one minimizing the total mean temperature of the MD simulation. The results are given in Figure 3. As one can see, both criteria correlate with each other (Pearson’s correlation coefficient is 0.71). Therefore, we can apply the Equation (Figure 3) to other molecular systems.

3.4. Thermostats Incorporating Simplified Wigner Sampling

The SWS approach, despite being quite crude, has its merits. Unlike standard Wigner sampling, which can be used only at the beginning of the MD simulation, as the displacements are connected to the equilibrium reference structure, the SWS allows a resampling of the displacements during the simulation. This property makes the SWS compatible with canonical ensemble simulations, particularly with Andersen [54] and Berendsen [55] thermostats. Here, we describe this process for both of these thermostats.

The canonical Andersen thermostat works by reassigning the velocity (or of one of the three components) of a randomly chosen atom according to the Maxwell–Boltzmann distributions at random points of the MD simulation. The probability of reassignment is given as $p_{\mathrm{A}}=\mathrm{\Delta }t/{\tau }_{\mathrm{A}}$, where ${\tau }_{\mathrm{A}}$ is the time of the reassignment. At each time point, a random number $0\le p_{\mathrm{trial}}<1$ is generated from the uniform distribution, and if condition $p_{\mathrm{trial}}<p_{\mathrm{A}}$ is fulfilled, the velocity reassignment procedure is initiated. Typically, ${\tau }_{\mathrm{A}}$ should be greater than the characteristic periods of motions in the system, so the Andersen thermostat does not significantly disrupt the dynamics of a molecule. To combine the Andersen thermostat with SWS, one has to replace the Maxwell–Boltzmann distribution resampling stage with the SWS routine, sampling both the momentum along gthe given degrees of freedom and the displacement of the coordinate.

The Berendsen thermostat relies on the soft resampling of velocities at each step of an MD simulation by multiplying them with a scale factor [18,55]

$$s=1+{\displaystyle \frac{\mathrm{\Delta }t}{{\tau }_{\mathrm{B}}}}·\left({\displaystyle \frac{T_{\mathrm{d}}}{T\left(t\right)}}-1\right),$$

(15)

where ${\tau }_{\mathrm{B}}$ is the relaxation time, a free parameter of the Berendsen thermostat; $T_{\mathrm{d}}$ is the desired temperature of the MD simulation; and $T\left(t\right)$ is the instant temperature of the nuclei in the MD simulations, defined as $T\left(t\right)=2·\mathrm{KE}\left(t\right)/N_{\mathrm{f}}$, with $\mathrm{KE}$ being the total kinetic energy of the molecule at time t, and $N_{\mathrm{f}}$ being the number of degrees of freedom (i.e., $N_{\mathrm{f}}=3\times N_{\mathrm{at}}-N_{\mathrm{constr}}$, where $N_{\mathrm{at}}$ is the number of atoms in the system, and $N_{\mathrm{constr}}$ is the number of constraints). By replacing $T_{\mathrm{d}}$ with the effective desired temperature according to Equation (11), we make the Berendsen thermostat compatible with SWS sampling.

3.5. Scaling of Vibrational Spectra from Molecular Dynamics

Scaling the harmonic vibrational frequencies and zero-point vibrational energies is a simple yet powerful tool for increasing the accuracy of IR spectra and thermodynamic properties calculations [56]. For vibrational frequencies, multiplication by a precomputed scale factor for a given method accounts for systematic errors due to the quantum-chemical approximation quality and the absence of anharmonic effects [56,57]. Therefore, it seems reasonable to suggest to also allow for scaling of the anharmonic spectra from MD simulations.

Here, we propose a procedure for obtaining the scale factors ($\gamma$) for IR spectra from MD and provide a set of precomputed values for three quantum-chemical approximations: BLYP-D3(BJ)/6-31G, PBE-D3(BJ)/6-31G, and PBEh-3c. For this, we selecteed molecules: water (${\mathrm{H}}_2\mathrm{O}$), ammonia (${\mathrm{NH}}_3$), methane (${\mathrm{CH}}_4$), ethane (${\mathrm{C}}_2{\mathrm{H}}_6$), methylamine (${\mathrm{CH}}_3{\mathrm{NH}}_2$), and methanol (${\mathrm{CH}}_3\mathrm{OH}$). For each of these molecules for each of the quantum-chemical methods, three trajectories were computed using a combination of Andersen and Berendsen thermostats with ${\tau }_{\mathrm{A}}=100$ fs and ${\tau }_{\mathrm{B}}=50$ fs and SWS sampling with ${\tau }_{\mathrm{SWS}}$ taken according to the criterion from Equation (14). Each trajectory was driven at $T=0$ K with a time step of $\mathrm{\Delta }t=1$ fs for a total duration of 1 ps. The spectra were computed using the FFT procedure without ignoring the equilibration phase and with the application of the frequency correction from Equation (8). The experimental gas-phase IR spectra of the same molecular species were taken from the NIST Chemistry WebBook [58].

It is not straightforward to compare theoretical spectra with a given frequency increment with their experimental counterparts since they have different frequency increments. Moreover, when the frequency axis is scaled for the spectra from MD, the frequency increment also changes. Therefore, the following procedure was applied to determine the scale factor of the MD spectra for a given molecule. Let us assume that the MD spectrum is given as a set of M points $\{({\nu }_1^{\left(\mathrm{MD}\right)},I_1^{\left(\mathrm{MD}\right)}),({\nu }_2^{\left(\mathrm{MD}\right)},I_2^{\left(\mathrm{MD}\right)}),\dots ({\nu }_M^{\left(\mathrm{MD}\right)},I_M^{\left(\mathrm{MD}\right)})\}$ and the experimental spectrum is a set of N points $\{({\nu }_1^{\left(\exp \right)},I_1^{\left(\exp \right)}),({\nu }_2^{\left(\exp \right)},I_2^{\left(\exp \right)}),\dots ({\nu }_N^{\left(\exp \right)},I_N^{\left(\exp \right)})\}$, where ${\nu }_i^{\left(X\right)}$ and $I_i^{\left(X\right)}$ are the i-th frequency and intensity obtained with method X. To remove the rotational substructure of the bands for small molecules, such as water or methane, the experimental data were also smoothed by a convolution with the Gaussian function. The frequency scale factor $\gamma$ is obtained by maximizing the following expression:

$$C\left(\gamma \right)={\left({\mathbf{I}}^{\left(\mathrm{MD}\right)}\right)}^{†}\mathcal{D}\left(\gamma \right){\mathbf{I}}^{\left(\exp \right)}\to max,$$

(16)

where ${\mathbf{I}}^{\left(\mathrm{MD}\right)}$ and ${\mathbf{I}}^{\left(\exp \right)}$ are the vectors composed of the intensity values of N and M dimensions each, and $\mathcal{D}\left(\gamma \right)$ is an $M\times N$ matrix composed of elements

$$D_{ij}\left(\gamma \right)=exp\left(-{\displaystyle \frac{{\left(\gamma ·{\nu }_i^{\left(\mathrm{MD}\right)}-{\nu }_j^{\left(\exp \right)}\right)}^2}{2{\sigma }^2}}\right)$$

(17)

with $1\le i\le M$, $1\le j\le N$, and $\sigma =min(\gamma ·\mathrm{\Delta }{\nu }^{\left(\mathrm{MD}\right)},\mathrm{\Delta }{\nu }^{\left(\exp \right)})$ with $\mathrm{\Delta }\nu$ being the frequency increments in the respective datasets. The scale factor $\gamma$ was searched in the interval $0.8\le \gamma \le 1.2$.

Figure 4 shows an example of a result of the procedure described above. In this figure, the unscaled and scaled spectra of methane obtained from the MD simulations at the PBEh-3c level of theory are shown, as well as the corresponding raw and smoothed experimental data from the NIST Chemistry WebBook. As one can see, the scaling indeed improves the positions of the vibrational bands in the spectra. The scale factors obtained from all the training set molecules were averaged to produce the estimated scale factor for the given level of theory. The results are given in

Table 1. Tabulated scale factors $\gamma$ for MD-based vibrational spectra at various levels of theory.

Method	Scale Factor $\mathit{\gamma }$
BLYP-D3(BJ)/6-31G	$1.046\pm 0.040$
PBE-D3(BJ)/6-31G	$1.041\pm 0.046$
PBEh-3c	$0.968\pm 0.006$

. In the cases of the BLYP- and PBE-based spectra, ammonia had to be removed from the dataset, as the most intense low-frequency vibrational band, corresponding to the valence band vibrations, was too far from the experimental position.

4. Discussion

In the previous section (Section 3), the protocol for IR spectra simulations from AIMD using PyRAMD was introduced. In an orderly fashion, the protocol looks as follows:

• First, we need to optimize the structure of the molecule at the given level of theory and compute the harmonic vibrational frequencies. Then, using Equation (14), we can calculate the ${\tau }_{\mathrm{SWS}}$ parameter to define the SWS sampling routine.
• Then, we can set the Berendsen and Andersen thermostats for simultaneous usage in an $NVT$-MD simulation. The combination of the two acts as a friction and random force in more sophisticated thermostats, such as the Langevin-based models [59] (including the color noise generalized Langevin equation [60]) and the Bussi–Donadio–Parrinello thermostat [61]. This requires setting the two free parameters: relaxation time ${\tau }_{\mathrm{B}}$ for the Berendsen thermostat and the resampling time ${\tau }_{\mathrm{A}}$ for the Andersen thermostat. SWS compatibility is assured by using the effective temperature (Equation (11)) for the Berendsen thermostat. In the case of the Andersen thermostat, Maxwell-Bolztmann resampling is replaced with the SWS procedure.
• Then, a single or a few MD trajectories are collected with reasonably large time steps. The choice criterion is dictated by the integration method and the corresponding frequency correction (Equation (8), see also Appendix A). In the cases of Verlet, velocity Verlet, and leapfrog integration schemes, the limit is given as $\mathrm{\Delta }t\le {\left(\pi {\nu }_{\max }\right)}^{-1}$, where ${\nu }_{\max }$ is the maximal vibrational frequency of the system. If we take the H-F stretching frequency in hydrofluoric acid (${\nu }_{\mathrm{HF}}=4138$ cm⁻¹ [62]), we obtain the maximal allowed time step of $\mathrm{\Delta }t=2.6$ fs. Therefore, time steps of around 1 fs are possible for most chemical systems. The total dipole moment of the molecular system is stored at every time step of the MD simulation.
• After collection of the trajectory, the vibrational spectrum is computed as the FT of the dipole moment (Equation (2)) or its velocity’s (Equation (3)) autocorrelation function (Equation (1)). The initial part of the trajectory is usually disregarded as the equilibration phase. The frequency resolution of the FT is given as $\mathrm{\Delta }\nu =1/{\tau }_{\mathrm{tot}}$, where ${\tau }_{\mathrm{tot}}$ is the total duration of the trajectory (without the equilibration phase). An alternative way to transfer the autocorrelation function from the time domain to the frequency domain with an arbitrary frequency increment is the rLSSA routine (Equation (6)), although this procedure is much more computationally expensive than FFT; thus, it makes sense to use it only for short (∼${10}^3$ steps) trajectories.
• Finally, the frequency correction (Equation (8)) is applied by transforming the frequency axis. Afterward, a tabulated scale factor for the corrected spectrum can be applied to account for the systematic errors in the quantum-chemical approximation.

To illustrate this procedure in action, we will consider a case of protonated methane (${\mathrm{CH}}_5^{+}$), with the vibrational spectrum obtained in Ref. [63]. For that, three $NVT$-MD trajectories at PBEh-3c level of theory with Andersen (${\tau }_{\mathrm{A}}=150$ fs) and Berendsen (${\tau }_{\mathrm{B}}=50$ fs) thermostats were computed. Each trajectory was 3 ps in length with 1 fs time step, and the ${\tau }_{\mathrm{SWS}}=4.2$ fs was taken from the harmonic frequencies at the same level of theory. After the MD simulations ended, for each of the trajectories, an rLSSA-computed spectrum was obtained with the first 1 ps of each trajectory disregarded for the equilibration phase. The spectra were smoothed by a convolution with Gaussian function with full width at half maximum (FWHM) of 50 cm⁻¹. For this, before performing the rLSSA procedure, the dipole velocity autocorrelation function was multiplied by a Fourier image of the frequency smoothing Gaussian, i.e., with another Gaussian distribution in the time domain with FWHM inversely proportional to the one in the frequency domain. Finally, the frequency correction and scale factor were applied. However, in addition to that, the total intensity in the spectrum was corrected by multiplying with the function of the form [14]

$$f\left(\nu \right)=\left\{\begin{array}{c}0,\mathrm{if}\nu <D,\hfill \\ 1-{\displaystyle \frac{D}{\nu }},\mathrm{if}\nu \ge D,\hfill \end{array}\right.$$

(18)

where $\nu$ is the frequency in cm⁻¹, and $D=383$ cm⁻¹ is the reaction energy needed for the signal to be detected. This type of correction was introduced in Ref. [14] and is not specific to MD spectra but to action spectroscopic methods in general. As we can see from Figure 5, the MD routine already produces a decent-looking spectrum. However, the application of the frequency correction, scale factor, and intensity correction (Equation (18)) brings the theoretical spectrum quite close to the experimental one.

${\mathrm{CH}}_5^{+}$ is an exceptionally flexible molecular system [63]; thus, comparison with scaled harmonic frequency calculations is essentially meaningless [14]. Therefore, to illustrate the effect of the tabulated scaling factor for MD-based IR spectra, we chose two molecules with the experimental spectra available from the NIST Chemistry WebBook [58]: acetic acid (${\mathrm{CH}}_3\mathrm{COOH}$) and indene (${\mathrm{C}}_9{\mathrm{H}}_8$). We computed the harmonic and MD-based spectra for these molecules at the PBEh-3c level of theory. The harmonic spectra included the fundamental bands, overtones, and combination bands, using the NearIR routine of ORCA 5 [64,65]. The MD spectra were computed with the same settings as for the spectra for ${\mathrm{CH}}_5^{+}$. The unscaled and scaled theoretical spectra are compared with the gas-phase experimental counterparts in Figure 6. As one can see, due to the presence of anharmonic effects, the unscaled MD spectra match the experimental ones better than the unscaled harmonic band positions. Upon scaling, both harmonic and MD-based spectra became much more similar in band positions to the experimental data. Arguably, the scaled harmonic spectra fit the experimental spectrum slightly better than the scaled IR spectrum from the MD simulations, which was probably due to the quality of the scale factors themselves. While the scale factors for the harmonic calculations were obtained based on a dataset of 441 molecules of various sizes [57], the scale factors for MD introduced here were based on a training set of only six molecules. Thus, there might be room for improvement in the scale factors for the IR spectra from MD simulations. In particular, we can think of extending the training set of molecules and producing scale factors for classical MD simulations without the SWS routine.

5. Conclusions

We presented a general approach (a “PyRAMD scheme”) for computing the IR spectra of polyatomic systems using the AIMD approach implemented in PyRAMD software. The approach involves the following essential components: (1) inclusion of the NQEs via the usage of the SWS approach coupled to thermostats; (2) frequency correction, allowing for larger time steps in MD simulations; (3) pre-parametrized scaling factors, similar to those used for harmonic frequency calculations. The synergistic effect of these components was demonstrated using test cases of protonated methane, acetic acid, and indene.

Despite this scheme being theoretically inferior to the more direct and expensive approaches, such as a combination of path integral MD for inclusion of the NQEs and of smaller time steps in combination with highly accurate methods to obtain the PES, it was demonstrated that the approach proposed here could produce reasonable results. In this sense, the “PyRAMD scheme” can serve as an intermediate stage between classical MD and path integral MD, combining the simplicity and cost efficiency of the former with the inclusion of the NQEs of the latter. Therefore, the presented algorithm is recommended for providinig theoretical counterparts to experimental measurements.