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# Derivative learning of tensorial quantities – Predicting finite temperature
infrared spectra from first principles

Bernhard Schmiedmayer<EMAIL_ADDRESS>University of
Vienna,
Faculty of Physics and Center for Computational Materials Sciences,
Kolingasse 14-16, 1090, Vienna, Austria  Georg Kresse
<EMAIL_ADDRESS>[  University of Vienna,
Faculty of Physics and Center for Computational Materials Sciences,
Kolingasse 14-16, 1090, Vienna, Austria

###### Abstract

We develop a strategy that integrates machine learning and first-principles
calculations to achieve technical accurate predictions of infrared spectra.
Specifically, the methodology allows to predict infrared spectra for complex
systems at finite temperatures. The method’s effectiveness is demonstrated in
challenging scenarios, such as the analysis of water and the organic-inorganic
halide perovskite MAPbI3, where our results consistently align with
experimental data. A distinctive feature of the methodology is the
incorporation of derivative learning, which proves indispensable for obtaining
accurate polarization data in bulk materials and facilitates the training of a
machine learning surrogate model of the polarization adapted to rotational and
translational symmetries. We achieve polarisation prediction accuracies of
about $1\text{\,}\mathrm{\char 37\relax}$ by training only on the predicted
Born effective charges.

††preprint: ???

##  I Introduction

Infrared (IR) spectroscopy is an indispensable methodology for discerning
local structures and the intricacies of functional groups within a material.
It finds pervasive use across various scientific disciplines, firmly
establishing itself as a key analytical instrument Stuart (2004). Notably,
within the realm of catalysis research, IR spectroscopy is an important
technique with the unique capability to elucidate surface structures and
mechanisms at the molecular scale in situ Vimont, Thibault-Starzyk, and Daturi
(2010); Chabal (1988); Ryczkowski (2001). Beyond catalysis, the application of
IR spectroscopy extends its reach into the domains of clinical and biomedical
analyses Barth (2007); Ng and Simmons (1999); Luypaert, Massart, and Vander
Heyden (2007). In the field of materials science, IR spectroscopy is also
highly valuable, as it provides indirect insight on the bonding properties of
the material Theophile (2012); Fernández-Carrasco  _et al._ (2012); Silva  _et
al._ (2012).

Although experimental IR spectroscopy is a mature technique, it is not always
a simple matter to relate an experimental IR spectrum to a specific local
structural feature. Prior knowledge is almost ubiquitously required to relate
experimental data to structures (fingerprints). For instance, the frequencies
of certain functional groups often relate to the oxidation state of the group
or the electronegativity of the adsorption site. Computer simulations,
particularly first-principles (FP) methods, have played a vital role in
establishing these relationships. However, such FP calculations are very often
limited to zero-temperature simulations Biczysko, Bloino, and Puzzarini
(2018); Jansen (2021); Beć, Grabska, and Huck (2021), whereas in-situ
observations of catalytic reactions almost always involve finite temperature.
In this study, we harness recent advancements in machine learning (ML)
techniques in combination with FP calculations to develop a method that yields
highly accurate computational IR spectra at finite temperatures.

In recent years, ML techniques have emerged as a potent new avenue in
computational materials sciences Behler and Parrinello (2007); Bartók  _et
al._ (2010, 2018); Bonati and Parrinello (2018); Morawietz  _et al._ (2016).
Our study uses an on-the-fly learning method Jinnouchi, Karsai, and Kresse
(2019); Jinnouchi  _et al._ (2019, 2020a, 2020b) to generate transferable
machine learned force fields (MLFFs). The first step is thus to harness MLFF
to attain the required nano-second-long high-quality molecular dynamics (MD)
trajectories. In the present work, we rely on a now fairly standard approach
for the MLFF that uses rotational and translationally invariant descriptors
and a kernel-based regression, although more refined approaches could be
readily adopted Drautz (2019); Batzner  _et al._ (2022); Batatia  _et al._
(2022). Learning the polarization, though, is a more challenging task. The
first approach to learn tensorial quantities dates back to $\lambda$-SOAP
Grisafi  _et al._ (2018); Wilkins  _et al._ (2019); Sommers  _et al._ (2020)
that we have also decided to adapt in the present work. Equivariant message-
passing networks would be equally suitable, but they are matter of fact
closely related to the tensorial descriptors in $\lambda$-SOAP.

The second key issue that we address in the present work is that the
polarization is not uniquely determined in bulk materials King-Smith and
Vanderbilt (1993); Vanderbilt and King-Smith (1993); Resta (1994, 1998). This
is in stark contrast to molecules, where the polarization can be determined by
appropriate integration of the density Gastegger, Behler, and Marquetand
(2017). Direct learning of the polarization hence requires some curation of
the polarization data e.g. deriving the polarization from Wannier-centres
imposing some continuity condition Zhang  _et al._ (2020) or manual
“alignment" of the polarization data.

The solution presented in this study to overcome this challenge involves
employing derivative learning Chmiela  _et al._ (2017). Specifically, we
utilize the derivative of the polarization with respect to the ionic
positions, the Born effective charge tensors. The hypothesis is that by
computing this derivative information across numerous structures (along any
relevant “adiabatic" pathway), it becomes feasible to determine the anti-
derivative, i.e., the bulk polarization. This methodology offers a second
crucial advantage: akin to the construction of MLFF where learning the forces
proved pivotal, the Born effective charges are significantly more expressive
yet nearly as computationally efficient to calculate in solid-state codes as a
single polarization.

In the next section, we summarize our methodological approach, then
demonstrate the feasibility of derivative-based learning for the water dimer,
and discuss results for liquid water, where we find excellent agreement with
the experiment only for a functional including van der Waals corrections.
Lastly, we demonstrate very good agreement for the IR spectrum of an organic
perovskite with experimental data. We finish with discussions and conclusions,
and mention points worthwhile an improvement.

##  II Method

###  II.1 General remarks:

In general, the energy of a molecule or material in the presence of an
electric field is described by

$E(\mathbf{x},\mathcal{E})=E_{\rm
KS}(\mathbf{x})-\mathcal{E}\cdot\mathbf{P}(\mathbf{x},\mathcal{E})$ (1)

where $\mathcal{E}$ is the electric field, $\mathbf{P}$ the polarization, and
$E_{\rm KS}$ is the Kohn-Sham energy at zero field for the atoms at the
position $\mathbf{x}$ Umari and Pasquarello (2002). There is an implicit
dependence of the Kohn-Sham energy on the electronic field, as the orbitals
need to be determined by minimizing the energy in the presence of the field,
but thanks to the variational properties and the Hellman-Feynman theorem,
variations of the orbitals can be neglected for first derivatives. To
calculate second derivatives, only the first derivatives of the orbitals are
required. Obviously, the first derivative of the energy with respect to the
electric field yields the polarization $\mathbf{P}$. The second derivative of
the energy with respect to the field corresponds to the electronic
polarizability, and the second mixed derivative with respect to the electric
field and the positions yields the Born effective charge tensor:

$\mathbf{Z}^{*}=\frac{\partial^{2}E(\mathbf{x},\mathcal{E})}{\partial\mathbf{x}\,\partial\mathcal{E}}.$
(2)

This is a second-rank cartesian tensor. In the present work, we set out to
learn the polarization as a function of the positions and neglect the
dependence of the polarization on the electric field (implicitly assuming zero
electric field). If we were to evaluate the energy using Eq. (1) this would
only be correct to linear order in the field.

###  II.2 Green-Kubo relation:

Using the Green-Kubo formalism, the ionic contribution to the polarizability,
denoted as $\chi(\omega)$, is directly proportional to the Fourier transform
of the autocorrelation function of the polarization $\mathbf{P}$ and its time
derivative and can be expressed as follows (SI units)Kubo (1957); Zwanzig
(1965); Sangalli, Marini, and Debernardi (2012):

$\chi_{\mu,\nu}(\omega)=\frac{\beta}{V\epsilon_{0}}\int_{0}^{T}\langle\mathbf{P}_{\mu}(0)\dot{\mathbf{P}}_{\nu}(t)\rangle
e^{-i(\omega-i\delta)t}\mathrm{d}t.$ (3)

Here $\mu$ and $\nu$ represent Cartesian indices, $V$ is the volume, $\beta$
is the inverse temperature, $\epsilon_{0}$ the vacuum permittivity, $\omega$
the vibrational frequency, and $\delta$ denotes a complex shift causing a
Lorenzian broadening. It is necessary that the product $T\delta$ is small to
avoid truncation artefacts. The time derivative of the polarization
$\dot{\mathbf{P}}$ can be written as

$\frac{\partial\mathbf{P}}{\partial
t}=\frac{\partial\mathbf{P}}{\partial\mathbf{x}}\frac{\partial\mathbf{x}}{\partial
t}=\mathbf{Z}^{*}\dot{\mathbf{x}}.$ (4)

To determine Eq. (3) accurately three prerequisites exist: first, accurate
velocities $\dot{\mathbf{x}}$ i.e. high-quality MD trajectories to describe
the time evolution of the system, second, long simulation times $T$, and
third, a reliable method to model the polarization and the Born effective
charge tensors.

In infrared reflectivity or transmission experiments, an absorbance $\alpha$
is measured and specified (the absorbance is proportional to $\omega$ times
the spectral function of the polarizability). We report the product of the
absorbance $\alpha(\omega)$ and the refractive index $\eta(\omega)$,
isotropically averaged over the three diagonal components of the
polarizability tensor:

$\alpha(\omega)\eta(\omega)=\frac{\beta}{3Vc\epsilon_{0}}\mathfrak{Re}\int_{0}^{T}\langle\dot{\mathbf{P}}(0)\cdot\dot{\mathbf{P}}(t)\rangle
e^{-i(\omega-i\delta)t}\mathrm{d}t.$ (5)

In this equation, $c$ is the speed of light. We note that one can also
autocorrelate $\mathbf{P}(0)$ with $\mathbf{P}(t)$, or $\mathbf{P}(0)$ with
$\dot{\mathbf{P}}(t)$, adding simultaneously factors $\omega^{2}$ and
$-i\omega$, respectively. We have tested all three approaches and found
identical results for the three versions.

###  II.3 Molecular dynamics simulations:

For disordered materials, the required long simulation times are unattainable
using first principles simulations, and hence surrogate models are required.
To obtain FP data for constructing the MLFF, we employed the Vienna Ab initio
Simulation Package (VASP) Kresse and Furthmüller (1996a, b); Kresse and
Joubert (1999). Both the training of the MLFF and its subsequent application
were conducted using the ML framework integrated in the VASP code Jinnouchi
_et al._ (2019); Jinnouchi, Karsai, and Kresse (2019); Jinnouchi  _et al._
(2020a). It is essential to emphasize that the quality of an MLFF is not only
dependent on the underlying ML algorithm but also the quality and
representativeness of the training dataset concerning the problem at hand Li,
Jiang, and Guo (2013); Botu and Ramprasad (2015). Ideally, the training
dataset should comprehensively cover all relevant regions of the potential
energy surface, and this coverage should be compact to minimize the necessity
for computationally expensive FP calculations. To accomplish this, we have
employed the on-the-fly learning scheme integrated within VASP.

The training dataset was curated from a collection of multiple MD
trajectories. These trajectories were obtained using Langevin thermostats
Allen and Tildesley (2017); Hoover, Ladd, and Moran (1982); Evans (1983) with
a fiction coefficient of $10\text{\,}{\mathrm{ps}}^{-1}$. To comprehensively
cover all phases of interest, a series of heating and cooling runs were
executed as discussed in the supplementary. Notably, the temperature range
considered was slightly wider than the region of interest, a choice aimed at
enhancing the stability of the MLFF.

To obtain an IR spectrum, we used multiple MD trajectories within a micro-
canonical ensemble to avoid artefacts caused by thermostats. The initial
configurations and velocities for these individual MD trajectories were drawn
from an isothermal-isobaric ensemble. The final IR spectrum was then computed
as the average of the individual IR spectra obtained from these trajectories,
providing a statistically accurate representation of the system’s vibrational
modes. We note that IR spectra are only sensitive to the long-range
vibrational modes, i.e. at zero temperature the calculations can be performed
using the unit cell. However, at finite temperature, supercells are required
to account for the finite temperature disorder. We checked carefully that the
IR spectra are cell-size converged for all cases reported here.

###  II.4 Polarization model

As highlighted in the introduction, obtaining the polarization for bulk
systems presents inherent challenges. This is due to an undetermined modulo
resulting from the absence of a unique phase origin or reference point King-
Smith and Vanderbilt (1993); Vanderbilt and King-Smith (1993); Resta (1994,
1998). To address this problem, we use derivative learning. Specifically, the
polarization $\mathbf{P}$ of a system can be understood as the antiderivative
of the Born effective charge tensor $\mathbf{Z}^{*}$ Wang, Yu, and Krakauer
(1996). The Born effective charge tensor for ion $i$ is mathematically
expressed as

$\mathbf{Z}^{*}_{i,\alpha\beta}=Z_{J(i)}\delta_{\alpha,\beta}+V\frac{\partial\mathbf{P}_{\alpha}^{\text{elec}}}{\partial\mathbf{x}_{i\beta}}\Bigg{|}_{\mathcal{E}=0}.$
(6)

Here, $Z_{J(i)}$ represents the bare ionic charge of the $i$th ion and $J(i)$
signifies the atomic species, $V$ stands for the volume of the unit cell, and
$\mathbf{P}_{\alpha}^{\text{elec}}$ corresponds to the Cartesian component
$\alpha$ of the macroscopic electronic polarization. In the equation,
$\mathbf{x}_{i\beta}$ denotes the position of the $i$th ion along the
Cartesian component $\beta$.

The central idea is that the surrogate ML model describes the polarization
$\mathbf{P}_{\alpha}^{\text{elec}}$. However, we aim to avoid training this
model on the polarization, but instead train the model on derivative data
$\mathbf{Z}^{*}$. Clearly, the polarization must transform like a vectorial
quantity under rotations. To this end, we employ ridge regression, with a
linear kernel function $K$, constructed using the covariance of 3-dimensional
descriptors $D^{\mu}_{n}(\mathcal{X})$. Here, $\mathcal{X}$ represents an
atomic environment, $\mu$ corresponds to a Cartesian component, and $n$ is the
feature dimension. The linear kernel function is defined as follows:

$K_{\mu\nu}(\mathcal{X},\mathcal{X}^{\prime})=\sum_{n}D^{\mu}_{n}(\mathcal{X})D^{\nu}_{n}(\mathcal{X}^{\prime}).$
(7)

This equation captures the similarity between atomic environments
$\mathcal{X}$ and $\mathcal{X}^{\prime}$. To describe the atomic environment,
we utilized the $\lambda$-SOAP (Smooth Overlap of Atomic Potentials)
descriptors developed by Grisafi et al. Grisafi  _et al._ (2018). These
descriptors conserve the rotational symmetry of tensorial quantities:

$\hat{S}D^{\mu}_{n}(\mathcal{X})=D^{\mu}_{n}(\hat{S}\mathcal{X}),$ (8)

where $\hat{S}$ is a generalized symmetry operator of the $SO(3)$ group. The
descriptor is tailored to describe the surroundings of an atom. In the present
work we use two- and three-body descriptors and Bessel functions as radial
basis sets, as in the original MLFF implementation of VASP Jinnouchi, Karsai,
and Kresse (2019); Jinnouchi  _et al._ (2019). To ensure smoothness in
derivatives and avoid abrupt discontinuities, we incorporated a Behler and
Parrinello cutoff function Behler and Parrinello (2007). The radial cutoffs
are typically set to $5.5\text{\,}\mathrm{\text{\r{A}}}$.

The polarization $\mathbf{P}$ of a configuration, characterized by atomic
environments $\mathcal{X}_{j}$, is determined using descriptors
$D^{\nu}_{n}(\mathcal{X}_{I_{\text{ref}}})$ of reference atomic environments
$\mathcal{X}_{I_{\text{ref}}}$ following the equation:

$\mathbf{P}_{\alpha}=\sum_{\nu
I_{\text{ref}}}\omega_{I_{\text{ref}}}^{\nu}\sum_{jn}D^{\alpha}_{n}(\mathcal{X}_{j})D^{\nu}_{n}(\mathcal{X}_{I_{\text{ref}}}).$
(9)

Here $\omega_{I_{\text{ref}}}^{\nu}$ represents weights that are paired with
each reference descriptor $D^{\nu}_{n}(\mathcal{X}_{I_{\text{ref}}})$. These
weights are determined through derivative learning of the Born effective
charge tensor, as given by the equation:

$\mathbf{Z}^{*}_{\alpha\beta}(i)=\sum_{\nu
I_{\text{ref}}}\omega_{I_{\text{ref}}}^{\nu}\sum_{jn}\frac{\partial
D^{\alpha}_{n}}{\partial\mathbf{x}_{i\beta}}(\mathcal{X}_{j})D^{\nu}_{n}(\mathcal{X}_{I_{\text{ref}}}).$
(10)

The weights are obtained by utilizing a linear regression model using the
least squares method, allowing for simultaneous calculation of
$\omega_{I_{\text{ref}}}^{\nu}$ across all training configurations. We use
sparse regression, i.e. reduce the number of kernel-basis functions
$\mathcal{X}_{I_{\text{ref}}}$ using farthest point sampling.

Our investigation revealed a substantial improvement in the quality of the fit
by specially treating the diagonal elements of the Born effective charge
tensor. To achieve this improvement, we applied a preprocessing step that
involved subtracting the mean value of the diagonal elements of the Born
effective charge tensor for each atomic species $\bar{Z}^{*}_{J}$ before the
training process. Here, $J$ represents the atomic species. Subsequently, we
add back to the polarization $\mathbf{P}$ the following term:

$\bar{\mathbf{P}}_{\alpha}=\sum^{N_{\text{atom}}}_{i=1}\bar{Z}^{*}_{J(i)}\mathbf{x}_{i\alpha}.$
(11)

The computation of Born effective charges was carried out using density
functional perturbation theory (DFPT) Wu, Vanderbilt, and Hamann (2005);
Gajdoš  _et al._ (2006); Pérez-Osorio  _et al._ (2015) by computing the static
ion-clamped dielectric matrix, as discussed by Baroni and Resta Baroni and
Resta (1986) and Gajdoš et al. for the projector-augmented wave (PAW) method
Gajdoš  _et al._ (2006). It is noteworthy that the calculation of the Born
effective charge tensor is computationally three times more intensive than a
comparable Density Functional Theory (DFT) groundstate calculation.
Essentially, it requires calculating the orbitals’ first derivative with
respect to the three directions of the external fields $\mathcal{E}$, which
suffices to obtain the mixed derivatives as defined in Eq. (2). Alternatively,
one could also determine the first derivative of the orbitals with respect to
the positions, and then predict the mixed second derivatives, however, this
scales linearly with system size and is thus computationally more involved
(the results for the Born effective charges are independent of the order of
differentiation). To the best of our knowledge, any electronic structure code
can determine the Born-effective charges via the orbital derivative with
respect to the field. The present approach is therefore applicable to most
first-principles codes. The inclusion of derivatives provides considerably
more information, in particular, an additional $3N_{\text{atom}}$ of data, so
that only a small number of FP calculations are required to achieve a high
degree of accuracy as demonstrated below.

###  II.5 Results and Discussion

To demonstrate the versatility of the developed methodology, we have applied
it to three distinct systems. First, we consider the water dimer 2(H2O). In
the case of molecules, the polarization is a well-defined property. Second, we
examine water. Lastly, we extend our analysis to a complex solid state system,
focusing on an organic perovskite MAPbI3, since it exhibits sizable
anharmonicities.

###  II.6 Water dimer

We start our analysis with a water dimer 2(H2O). Given that molecules possess
a well-defined polarization, we can make a comparative assessment between the
derivative learning approach and the conventional method of directly learning
the polarization. This provides a valuable benchmark for assessing the
reliability of the methodology.

The training and test configurations were extracted from an MD trajectory. We
commenced with a water dimer placed in a simulation box with ample vacuum
space. One of the oxygen atoms was constrained using selective dynamics to
anchor the system. To simulate thermalization, we executed a heating run,
spanning temperatures from $10\text{\,}\mathrm{K}320\text{\,}\mathrm{K}$,
utilizing a Langevin thermostat. Importantly, the MD run was enhanced with on-
the-fly machine learning to speed up the computational process. For the FP
calculations, we opted for a revised Perdew-Burke-Ernzerhof (RPBE) functional
Hammer, Hansen, and Nørskov (1999) with van der Waals (vdW) dispersion energy
corrections of Grimme et al. with zero-damping function Grimme  _et al._
(2010) (RPBE-D3). This choice ensured that our calculations account for vdW
interactions.

During the MD, the mass of the hydrogen atom was increased to
$8\text{\,}\mathrm{u}$, to allow for larger time steps of
$1.5\text{\,}\mathrm{fs}$. Out of a total of $200\,000$ MD time steps, we
selected $1000$ uniformly distributed configurations for the computation of
the polarization. For calculating the Born effective charge tensor $150$
configurations were chosen. To determine polarization, we integrated the total
(electronic and ionic) charge times the position operator by switching on
dipole corrections in VASP (see e.g. Refs. Makov and Payne (1995); Neugebauer
and Scheffler (1992)). To improve the accuracy of the Born effective charge
tensor, we used a strict convergence criterion of
$1\text{\times}{10}^{-7}\text{\,}\mathrm{eV}$ for the electronic self-
consistency loop and large cells. To confirm the internal consistency between
polarization and Born effective charges, we also calculated numerical
derivatives of the polarization. These derivatives showed an excellent
agreement with the Born charges with a root mean square error (RMSE) less than
$5\text{\times}{10}^{-6}\text{\,}\mathrm{|}\mathrm{\text{$e$}}\mathrm{|}$,
confirming the reliability of the resulting database.

Figure 1: Learning curves for the polarization ($P$) and Born effective
charges ($Z^{*}$) of a water dimer. Errors on the test set are expressed in
root mean square percentage errors (RMSPE), where the RMSE is normalized by
the standard deviation ($\sigma$) of the test set to yield a dimensionless
quantity (note that $10^{-1}$ and $10^{-2}$ at the y-axis correspond to
$10\text{\,}\mathrm{\char 37\relax}$ and $1\text{\,}\mathrm{\char 37\relax}$,
respectively). The blue line (circle) denotes training solely on polarization
($P$), and the red line (triangle) represents training solely on Born
effective charges ($Z^{*}$). The yellow line (square) demonstrates combined
learning of polarization and Born effective charges ($P+Z^{*}$).

To construct the learning curves, we used a dataset consisting of $1000$
polarization calculations. The dataset was split in half by alternately
selecting configurations for training and validation. Additionally, we
selected $50$ Born effective charge calculations as validation data, chosen to
be uniformly distributed across the dataset of $150$ Born effective charge
tensor calculations. The remaining $100$ Born effective charge calculations
served as the training dataset. During the combined training process, the
polarization data was weighted ten times higher than the Born effective charge
data. To determine the optimal number of fitting parameters, we selected the
number of kernel functions that minimized the error in polarization
prediction. It is worth noting that, as proposed by Cortes et al. for
regression Cortes  _et al._ (1993), there should be a relationship between the
RMSE and the number of training data, characterized by a power-law decay. The
learning curves, which provide insights into the model’s performance, are
presented in Fig. 1.

The learning curves demonstrate the power-law relation between the RMSE and
the number of training configurations. However, it is important to note that
with a large number of training data points, the improvement in the mean
squared errors seems to plateau somewhat, indicating that our linear
regression with two- and three-body descriptors will likely yield some
residual (but acceptable) model errors. Overall, the most favourable results
are achieved through combined learning, with percentage errors below
$0.5\text{\,}\mathrm{\char 37\relax}$ and $3\text{\,}\mathrm{\char 37\relax}$
for the polarization and the Born effective charges, respectively. Crucially,
learning only Born charges from $100$ training configurations also attains a
high relative accuracy of approximately $1\text{\,}\mathrm{\char 37\relax}$
for the polarization, and is as accurate as combined learning for the Born
effective charges. We also note that there is no noticeable offset in the ML
polarization compared to the FP calculations. Likely this is so since the
molecule is free to rotate (and does rotate during the MD) and the surrogate
model reliably determines the offset. On the other hand, attempting to train
on the polarization data only requires more training data (blue line), but
even with $500$ training data points, the Born effective charges still show
twice the errors as combined training does using $100$ training data points.

###  II.7 Liquid water

For our second proof-of-concept, we selected liquid H2O at room temperature.
While the IR spectrum of water has been extensively studied and is well
understood Falk and Ford (1966); Draegert  _et al._ (1966); Walrafen (1972);
Hasted  _et al._ (1985); Bertie and Lan (1996); Schmidt and Miki (2007),
theoretical interpretations of these spectra remain challenging Madden and
Impey (1986); Bansil  _et al._ (1986); Guillot (1991); Corongiu (1992);
Silvestrelli, Bernasconi, and Parrinello (1997); Auer and Skinner (2008); Xu
_et al._ (2019); Medders and Paesani (2015). Even with the use of modern DFT
methods, accurately reproducing the experimental properties of water is a
complex task and often yields results that are far from accurate Gillan, Alfe,
and Michaelides (2016). As a result, this system serves as an ideal test case
for validating the reliability and effectiveness of the methodology as well as
gleaning some insight on the underlying dynamics of water.

We chose the training configurations from an MD trajectory using an on-the-fly
learning scheme. The simulation was conducted within a cubic box with periodic
boundary conditions, containing a total of $64$ H2O molecules. The lattice
constant of the cubic box was adjusted to attain a density of approximately
$997\text{\,}\mathrm{kg}\text{\,}{\mathrm{m}}^{-3}$, closely resembling the
density of water at room temperature Tanaka  _et al._ (2001). Data was
collected during multiple heating and cooling runs conducted within a
canonical MD ensemble. We employed various temperatures, spanning from
$270\text{\,}\mathrm{K}420\text{\,}\mathrm{K}$. The MLFF was trained using FP
data obtained from DFT calculations. Specifically, we again employed the
RPBE-D3 functional but found it necessary to use hard PAW potentials and a
cutoff of $800\text{\,}\mathrm{eV}$ to obtain accurate stretch frequencies de
Hijes  _et al._ (2023). Additionally, we trained an MLFF using SCAN Sun,
Ruzsinszky, and Perdew (2015) to determine whether SCAN offers a reasonable
description of the water dynamics.

Figure 2:  Scatter plot of the trained and predicted Born effective charges
for water.

The training configurations for the tensorial machine learning framework were
once more chosen from a canonical MD ensemble. This ensemble was maintained at
a constant temperature of $298.2\text{\,}\mathrm{K}$ and controlled by a
Langevin thermostat. A total of $10\,000$ MD steps, accelerated by the MLFF,
were executed. From the MD trajectory, $100$ configurations were uniformly
selected as the training dataset for the Born effective charges. A scatter
plot of the trained and predicted Born effective charges is shown in Fig. 2.

Figure 3:  Experimental and computational IR spectra for liquid water. The
experimental reference data is from Ref. Bertie and Lan (1996)

.

The computational IR spectrum presented alongside experimental data in Fig. 3
was computed by averaging the results over $20$ individual IR spectra. Each of
these individual spectra was calculated from a micro-canonical MD trajectory.
For each MD run, we initiated the simulation from an uncorrelated starting
configuration, ensuring the appropriate average corresponding to room
temperature. In each run, we performed a total of $100\,000$ MD steps, with a
time step of $0.25\text{\,}\mathrm{fs}$. This strategy of conducting multiple
calculations from uncorrelated starting configurations was employed to enhance
the statistical accuracy of our results. Instead of Lorenzian broadening, a
Gaussian filter was applied before performing the Fourier transform. This
improves the feature sharpness in the computed IR spectra.

As illustrated in Fig. 3, the present methodology allows the computation of
the IR spectrum of liquid water with a remarkable level of agreement with
experimental data. There are two important conclusions to draw from this
result. That the intensity follows so closely the experimental data, is
related to an accurate description of the Born effective charges. Since we
train on the Born effective charges, the good agreement is likely not
astonishing. Second, RPBE+D3 yields an excellent description of the dynamics
of liquid water, both for the high-frequency stretch, but also for the medium-
frequency bending motions. Most important are the lower frequency modes that
are related to intra-molecular motions. It is important to note that the
results for the SCAN functional are significantly worse for these modes.
Specifically, the low-frequency mode around $500\text{\,}{\mathrm{cm}}^{-1}$
is wrong by almost $40\text{\,}\mathrm{\char 37\relax}$ indicating serious
deficiencies in the description of the molecular motion of water molecules en-
caged by the four surrounding water molecules.

Previous works, such as those by Sommers et al. Sommers  _et al._ (2020),
Zhang et al. Zhang  _et al._ (2020), and Gastegger et al. Gastegger, Behler,
and Marquetand (2017), have undertaken similar approaches utilizing MD
trajectories in conjunction with symmetry-preserving machine learning
frameworks to obtain Raman spectra for water and IR spectra for molecules. In
these prior studies, the focus has been on learning the polarization directly.
This required significantly more training data and a careful calculation of
the polarization in order to avoid any discontinuities.

###  II.8 Anharmonic solids

Our final test system is the organic-inorganic halide perovskite,
methylammonium lead iodide (MAPbI3). This material has been the subject of
numerous experimental and theoretical studies, including state-of-the-art
vibrational studies Weber (1978); Onoda-Yamamuro, Matsuo, and Suga (1990);
Tanaka  _et al._ (2001); Kawamura, Mashiyama, and Hasebe (2002); Stoumpos,
Malliakas, and Kanatzidis (2013); Baikie  _et al._ (2013); Bokdam  _et al._
(2016); Schuck  _et al._ (2018); Jinnouchi  _et al._ (2019). Notably, MAPbI3
undergoes three entropy-driven phase transitions: from an orthorhombic phase
to a tetragonal phase at $160\text{\,}\mathrm{K}$, and from the tetragonal
phase to a cubic phase at $330\text{\,}\mathrm{K}$. The thermodynamic nature
of this material makes it challenging to model the IR spectra of the
tetragonal phase using traditional $0\text{\,}\mathrm{K}$ methods like DFPT
Wu, Vanderbilt, and Hamann (2005); Gajdoš  _et al._ (2006); Pérez-Osorio  _et
al._ (2015). Therefore, MAPbI3 serves as a valuable test case for validating
the reliability of the scheme for strongly anharmonic solids.

Figure 4:  Experimental and computational IR spectra for the orthorhombic and
tetragonal phases of MAPbI3. The vibrational frequencies of the computational
IR spectra have been red-shifted by $1.5\text{\,}\mathrm{\char 37\relax}$ for
alignment with experimental data. The experimental reference data is from Ref.
Schuck  _et al._ (2018).

The training process for the MLFF applied to MAPbI3 closely mirrored the
methodology employed for water and previous studies of MAPbI3 Jinnouchi  _et
al._ (2019). Multiple heating runs, encompassing the two phases— orthorhombic,
and tetragonal —were conducted using Langevin thermostat-driven MD
simulations. The temperature range spanned from
$80\text{\,}\mathrm{K}430\text{\,}\mathrm{K}$. Initially, fixed cell volumes
were used, followed by simulating an isothermal-isobaric ensemble using the
Parrinello-Rahman method Parrinello and Rahman (1980, 1981). The training was
performed using a strongly-constrained and appropriately normed (SCAN) meta-
gradient corrected functional Sun, Ruzsinszky, and Perdew (2015). We note that
we found in previous studies that SCAN is better suited for the simulation of
MAPbI3 Bokdam  _et al._ (2017) than say RPBE-D3, as RPBE-D3 does not account
for screening of the vdW interactions by the strongly polarizable cage atoms.

In Fig. 4, we present the IR spectra for the orthorhombic phase at
$107\text{\,}\mathrm{K}$ and the tetragonal phase at $228\text{\,}\mathrm{K}$,
alongside experimental results for comparison. The spectrum of the well-
ordered orthorhombic phase and the tetragonal phase was calculated using a
$4\times 4\times 4$ supercell for better statistics and to allow for some
reordering of the MA molecules. The starting configurations for the individual
MD runs were chosen from an isothermal-isobaric ensemble. Aside from using
starting configurations with varying cell vectors, the procedure for
calculating the IR spectrum closely mirrors the one described for water.

The analysis of the computed IR spectra reveals excellent agreement with the
experimental data but also some small discrepancies, particularly, in the
intensities of the individual peaks. Notably, the peaks around
$900\text{\,}{\mathrm{cm}}^{-1}$ appear with higher intensity. The
experimental reference spectra, obtained from a single crystal Schuck  _et
al._ (2018), may also be influenced by surface effects and crystal structure
orientation, contributing to variations in intensity between computational and
experimental results.

We start with a comparison for the orthorhombic low-temperature phase
($107\text{\,}\mathrm{K}$). The agreement between the DFPT and the finite
temperature simulation is very good, but there are some marked improvements in
the finite temperature data. The first peak around
$900\text{\,}{\mathrm{cm}}^{-1}$ shows a double peak in both the DFPT and FT
simulation, in agreement with the experiment. The peak at
$970\text{\,}{\mathrm{cm}}^{-1}$ is completely missing in the DFPT data but
visible in the finite temperature data. It is a result of anharmonic
interactions. The shoulder at $1420\text{\,}{\mathrm{cm}}^{-1}$ is pronounced
using DFPT but washed out at finite temperature. This peak corresponds to the
CH3-bending motions Pérez-Osorio  _et al._ (2015). In the supplementary, we
show that using the ionic charges only, this peak is visible in the spectrum,
but the intensity of the peak is overestimated. Calculating the electronic
contribution (el) only shows an almost identical peak, however, the phase of
the electronic polarization is opposite to the ionic contribution, so when
autocorrelating the sum of the electronic and ionic dipoles, the peak is
strongly suppressed. This brings better overall agreement with the experiment,
where the peak is also weak, but likely our Born effective charges are not
quite sufficiently accurate (linear regression). The physics behind the
vanishing of the peak is fairly simple: in this particular mode the H atoms
move orthogonal to their bond direction, a direction in which the total Born
effective charges $\mathbf{Z}^{*}$ are small to start with (electronic
contribution cancels ionic one). Furthermore, the movement of three hydrogen
atoms is concerted being close to a helicopter motion which reduces the IR
intensity further.

In the experiment, we see quite some intensity between
$1450\text{\,}{\mathrm{cm}}^{-1}$ and $1600\text{\,}{\mathrm{cm}}^{-1}$, which
is also nicely reproduced by the FT simulations. We note that this frequency
range becomes even more populated in the tetragonal phase in the FT
simulations, and this population is a result of the strongly anharmonic
rattling motion of the molecules in the cage in turn affecting the bending
motion of the hydrogens.

The tetragonal phase spectrum ($228\text{\,}\mathrm{K}$) is also in excellent
agreement with the experimental spectrum. We note that the DFPT spectrum now
shows many deficiencies, with a complete lack of peaks at
$950\text{\,}{\mathrm{cm}}^{-1}$ and a tiny peak around
$1250\text{\,}{\mathrm{cm}}^{-1}$, as well as sharp features around
$1580\text{\,}{\mathrm{cm}}^{-1}$. The FT data resolves these issues, albeit
the two main peaks around $900\text{\,}{\mathrm{cm}}^{-1}$ and
$1480\text{\,}{\mathrm{cm}}^{-1}$ are somewhat too broad and washed out, and
again the peak around $1480\text{\,}{\mathrm{cm}}^{-1}$ is suppressed by the
electronic contribution.

##  III Conclusion

The present study showcases the effectiveness of a computational framework
combining first-principles simulations with machine-learning methodologies.
Machine-learned force fields can be used to access timescales that were
previously very difficult to attain. This allows us to obtain vibrational
spectra with excellent statistical accuracy, which would be very expensive to
calculate without force fields. The main advance of the present work is,
however, that we learn the polarization from its derivative the Born effective
charges. As mentioned in the main text, in VASP— but likely so in any plane
wave-based code —the calculation of the Born-effective charges via the first
derivative of the orbitals with respect to external fields is only roughly
three times more costly than a groundstate Kohn-Sham calculation. Learning of
force fields using first-principle nuclear derivatives is now ubiquitous, and
learning the polarization from its nuclear derivative is a natural extension
to this idea. Crucially, we have shown that the inclusion of polarization
data, which is difficult to determine without some arbitrary modulus, is not
required. As the polarization is the anti-derivative of the Born-effective
charges, it can be directly calculated from the machine-learning model as long
as derivative data is supplied along all relevant adiabatic pathways. We
successfully applied this approach to challenging scenarios, including water
and the organic-inorganic halide perovskite MAPbI3. In both cases, our method
demonstrates excellent agreement with experimental results, highlighting its
capacity to capture the vibrational properties of diverse materials.

We finish with a few comments on further developments. In the present work, we
have only used linear regression with two-body and three-body descriptors.
Although the prediction accuracies are good (condensed matter) to excellent
(molecules), we feel that the inclusion of higher body-order terms, or non-
linear kernel-based regression might improve the predicted Born-effective
charges further. Furthermore, the subtraction of a diagonal component from the
Born-effective charges, albeit not particularly cumbersome, seems somewhat
unsatisfactory, and one would prefer to avoid it.

Overall, the present methodology is already very robust and can be readily
applied to many relevant problems. Further research could be directed towards
infrared simulations of more complex adsorbates on surfaces or of water
interacting with electrodes.

##  IV Acknowledgements

This research was funded in whole by the Austrian Science Fund (FWF)
10.55776/F81. For open access purposes, the author has applied a CC BY public
copyright license to any author accepted manuscript version arising from this
submission. The computational results presented have been achieved in part
using the Vienna Scientific Cluster (VSC).

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