# Orientational Disorder and Molecular Correlations in Hybrid Organic-
Inorganic Perovskites: From Fundamental Insights to Technological Applications

Carlos Escorihuela-Sayalero  Group of Characterization of Materials,
Departament de Física, Universitat Politècnica de Catalunya, Campus Diagonal-
Besòs, Av. Eduard Maristany 10–14, 08019 Barcelona, Spain  Research Center in
Multiscale Science and Engineering, Universitat Politècnica de Catalunya,
Campus Diagonal-Besòs, Av. Eduard Maristany 10–14, 08019 Barcelona, Spain
Ares Sanuy  Group of Characterization of Materials, Departament de Física,
Universitat Politècnica de Catalunya, Campus Diagonal-Besòs, Av. Eduard
Maristany 10–14, 08019 Barcelona, Spain  Research Center in Multiscale Science
and Engineering, Universitat Politècnica de Catalunya, Campus Diagonal-Besòs,
Av. Eduard Maristany 10–14, 08019 Barcelona, Spain  Luis Carlos Pardo
<EMAIL_ADDRESS>Group of Characterization of Materials, Departament
de Física, Universitat Politècnica de Catalunya, Campus Diagonal-Besòs, Av.
Eduard Maristany 10–14, 08019 Barcelona, Spain  Research Center in Multiscale
Science and Engineering, Universitat Politècnica de Catalunya, Campus
Diagonal-Besòs, Av. Eduard Maristany 10–14, 08019 Barcelona, Spain  Claudio
Cazorla<EMAIL_ADDRESS>Group of Characterization of Materials,
Departament de Física, Universitat Politècnica de Catalunya, Campus Diagonal-
Besòs, Av. Eduard Maristany 10–14, 08019 Barcelona, Spain  Research Center in
Multiscale Science and Engineering, Universitat Politècnica de Catalunya,
Campus Diagonal-Besòs, Av. Eduard Maristany 10–14, 08019 Barcelona, Spain

###### Abstract

Hybrid organic-inorganic perovskites (HOIP) have emerged in recent years as
highly promising semiconducting materials for a wide range of optoelectronic
and energy applications. Nevertheless, the rotational dynamics of the organic
components and many-molecule interdependencies, which may strongly impact the
functional properties of HOIP, are not yet fully understood. In this study, we
quantitatively analyze the orientational disorder and molecular correlations
in the archetypal perovskite CH3NH3PbI3 (MAPI) by performing comprehensive
molecular dynamics simulations and entropy calculations. We found that, in
addition to the usual vibrational and orientational contributions, rigid
molecular rotations around the C–N axis and correlations between neighboring
molecules noticeably contribute to the entropy increment associated with the
temperature-induced order-disorder phase transition in MAPI, $\Delta S_{t}$.
Molecular conformational changes are equally infrequent in the low-$T$ ordered
and high-$T$ disordered phases and have a null effect on $\Delta S_{t}$.
Conversely, the couplings between the angular and vibrational degrees of
freedom are substantially reinforced in the high-$T$ disordered phase and
significantly counteract the phase-transition entropy increase resulting from
other factors. Furthermore, the tendency for neighboring molecules to be
orientationally ordered is markedly local, consequently inhibiting the
formation of extensive polar nanodomains both at low and high temperatures.
This theoretical investigation not only advances the fundamental knowledge of
HOIP but also establishes physically insightful connections with contemporary
technological applications like photovoltaics and solid-state cooling.

##  I Introduction

Hybrid organic-inorganic perovskites (HOIP) are solids with the chemical
formula ABX3, where A and B–X represent organic and inorganic ions,
respectively. HOIP exhibit cubic, or pseudocubic, crystalline symmetry at room
and higher temperatures, analogous to oxide perovskites. HOIP have emerged as
a promising family of optoelectronic and energy materials, with significant
potential for realizing photovoltaic and light-emitting devices photo1 ;
photo2 ; photo3 ; photo4 , field-effect transistors transistor1 ; transistor2
; transistor3  and solid-state refrigerators cool1 ; cool2 ; cool3 ; cool4 ,
among other technological applications. Methylammonium lead iodide, CH3NH3PbI3
(MAPI), is an archetypal HOIP that has been extensively investigated in the
fields of solar cells and quantum dots solar1 ; solar2 ; solar3 ; qd1 ; qd2 .

A characteristic physical trait of HOIP is that, upon increasing temperature,
they undergo order-disorder phase transitions involving orientational
molecular disorder, which on average gives rise to the high-symmetry cubic
lattice rot1 . Interestingly, the orientational dynamics of the organic
cations may profoundly affect the functional properties and lattice dynamics
of HOIP review1 ; rot2 ; rot3 . To cite few examples, the contribution of the
molecular CH3NH3+ (MA+) rotations to the static dielectric response of MAPI
has been estimated to be as large as $\sim 40$% diel1 . The origin of the
hysteresis frequently observed in photocurrent-voltage measurements of MAPI-
based solar devices, which offers promise for memristors and nonvolatile
memory applications berruet22 , is also thought to be related to the
rotational dynamics of the molecular cations hyste1 . Additionally, the
contribution of the orientational MA+ degrees of freedom to the caloric
response of MAPI, as driven by external bias like hydrostatic pressure and
electric field shifts, has been shown to be substantial liu16 ; cazorla24 .

Despite these advancements, there remains a fundamental lack of understanding
regarding the correlations between organic–organic and organic–inorganic ions,
and how these correlations may impact the molecular orientational dynamics
and, in turn, the functional properties of HOIP. For instance, it has been
experimentally shown for MAPI that, even at temperatures well below the order-
disorder phase transition point, the organic cations remain highly disordered
and mobile question1 ; question2 . Conversely, MA+ reorientational motion is
largely inhibited in mixed-halide hybrid perovskites such as MAPbIBr2 and
MAPbI2Br question3 . The role of molecular correlations in the potential
formation of ordered molecular domains also remains debatable diel1 ; hyste1 ;
zhang23 . Similarly, previous studies assessing the existence of caloric
effects in HOIP have mostly considered organic cations as independent
entities, thus neglecting the likely influence of many-molecule
interdependencies on the observed solid-state cooling figures of merit cool1 ;
cool2 ; cool3 ; cool4 ; cazorla24 .

In this study, we delve into the analysis of molecular disorder and likely
organic–organic and organic–inorganic ionic correlations in HOIP by conducting
extensive molecular dynamics (MD) simulations for MAPI. Our quantitative
investigations focus on evaluating the entropy change associated with the
temperature-induced order-disorder phase transition, $\Delta S_{t}$, which is
a physically interpretable and readily measurable quantity. We also address
through MD simulations the question about the possible formation of ordered MA
cations nanodomains at finite temperatures. Additionally, we establish
physically insightful connections between our fundamental findings and
technological applications of current interest (e.g., photovoltaics and solid-
state cooling).

The organization of this article is as follows. First, we present a detailed
description of the simulation approach employed for the evaluation of $\Delta
S_{t}$. Next, we report our computational results along with some discussions,
and subsequently provide a summary of the main conclusions. The technical
details of our classical MD simulations and entropy calculation approach can
be found in the Methods section.

Figure 1: MAPI unit cell and MA molecule angular coordinates. (a) Sketch of
MAPI in the high-temperature cubic perovskite phase. MA molecules are
orientationally disordered. (b) Angles definition for describing the
orientation of MA molecules. Angles $\theta$ and $\phi$ describe the
orientation of the molecular C–N axis. Angle $\psi$ refers to the rotation of
the molecule around its C–N axis. (c) Dihedral angle describing the
conformations of a MA molecule. Hydrogen, carbon and nitrogen atoms are
represented with white, green and blue spheres, respectively.

##  II Computational Approach

###  II.1 Molecular dynamics simulations

Classical molecular dynamics (MD) simulations in the $NpT$ ensemble (i.e.,
considering fixed number of particles, pressure and temperature) have been
conducted for bulk MAPI (Fig.1a) using the atomistic force field developed by
Mattoni and collaborators mattoni1 ; mattoni2 . This classical interatomic
potential is based on a hybrid formulation of the Lennard–Jones and Buckingham
pairwise interaction models, and describes the MA molecules through harmonic
bonded interactions. Long-range electrostatic interactions are also
appropriately accounted for by this atomistic force field. The high accuracy
of this classical interatomic potential in providing correct MA+ orientational
configurations, as benchmarked by quantum first-principles simulations, has
been previously demonstrated cazorla24 ; kim23 . The technical details of our
MD–$NpT$ simulations can be found in the Methods section.

###  II.2 Reference systems and angles definition

The orientation of each individual MA molecule has been determined in a fixed
lab reference system through the Euler angles $\theta_{\rm lab}$, $\phi_{\rm
lab}$ and $\psi_{\rm lab}$ (Fig.1b). The angles $\theta_{\rm lab}$ and
$\phi_{\rm lab}$ determine the orientation of each C–N molecular bond in the
fixed reference frame. The angle $\psi_{\rm lab}$ describes the rotation of
each molecule around its C–N molecular axis. In order to analyse possible
molecular orientational correlations (see next sections), the relative
orientations between MA molecules need to be tracked. For this end, a co-
mobile reference system is set on an arbitrary molecule and the orientation of
the other molecules are referred to it through the angles $\theta_{\rm rel}$,
$\phi_{\rm rel}$ and $\psi_{\rm rel}$. The origin of this co-mobile reference
system is set at half the distance between the C–N atoms, with the $z$ axis
pointing to the carbon atom, the $y$ axis being perpendicular to the plane
defined by the atoms N–C–H, and the $x$ axis being perpendicular to the $y$
and $z$ axes. In addition, a dihedral angle, $\epsilon_{\rm dih}$, formed by
the intersecting planes containing the H–C–N and C–N–H∗ atoms (where H belongs
to the molecular methyl group and H∗ to the ammonia group, Fig.1c), is used to
monitor possible MA+ conformational changes.

###  II.3 Entropy calculations

The entropy of the low-$T$ ordered and high-$T$ disordered phases of MAPI were
determined as a function of temperature, $S_{\rm total}(T)$, using the
relation:

$S_{\rm total}(T)=S_{\rm vib}(T)+S_{\rm ang}(T)+S_{\rm vib-ang}(T)~{},$ (1)

where $S_{\rm vib}$ is the entropy contribution resulting from the atomic
vibrations, $S_{\rm ang}$ from the molecular angular degrees of freedom (i.e.,
orientational and conformational), and $S_{\rm vib-ang}$ accounts for possible
couplings between the vibrational and molecular angular degrees of freedom.

Likewise, the molecular angular entropy was assumed to be correctly evaluated
with the expression:

$S_{\rm ang}(T)=S_{\rm ori}(T)+S_{\rm conf}(T)+S_{\rm ori-ori}(T)~{},$ (2)

where $S_{\rm ori}$ is the entropy contribution resulting from the
orientational degrees of freedom of a single and noninteracting MA molecule,
$S_{\rm conf}$ from the conformational changes of a single and noninteracting
MA molecule, and $S_{\rm ori-ori}$ accounts for possible orientational
correlations between different molecules.

In this study, we primarily have focused on a comparative analysis of the
entropy of the low-$T$ ordered and high-$T$ disordered phases of MAPI at the
corresponding $T$-induced phase transition point (other possible external
fields like hydrostatic pressure and electric bias are set to zero). Thus, the
physical quantity that is quantitatively analyzed in detail here corresponds
to the phase-transition entropy change, defined as:

$\Delta S_{t}=\Delta S_{\rm vib}+\Delta S_{\rm ori}+\Delta S_{\rm conf}+\Delta
S_{\rm ori-ori}+\Delta S_{\rm vib-ang}~{},$ (3)

where $\Delta S_{\rm x}\equiv S_{\rm x}^{\rm disord}-S_{\rm x}^{\rm ord}$ and
all the entropy terms are evaluated at the phase-transition temperature
$T_{t}$. The main improvement of this $\Delta S_{t}$ definition, compared to
that in a previous work cazorla24 , is the inclusion of molecular
conformational changes ($\Delta S_{\rm conf}$), molecular orientational
correlations ($\Delta S_{\rm ori-ori}$), and possible couplings between the
vibrational and molecular orientational degrees of freedom ($\Delta S_{\rm
vib-ang}$).

####  II.3.1 Vibrational entropy

The vibrational density of states $\rho(\omega)$, where $\omega$ represents a
lattice vibration frequency, provides information on the phonon spectrum of a
crystal and allows to estimate key thermodynamic quantities like the
vibrational free energy, $F_{\rm vib}$, and vibrational entropy, $S_{\rm
vib}\equiv-\frac{\partial F_{\rm vib}}{\partial T}$. A possible manner of
calculating $\rho(\omega)$ from the outputs of MD–$NpT$ simulations consists
in estimating the Fourier transform of the velocity autocorrelation function
(VACF) cazorla24 ; sagotra19 ; cibran23 , defined like:

$\text{VACF}(t)=\left\langle\mathbf{v}(0)\cdot\mathbf{v}(t)\right\rangle=\frac{1}{N}\sum_{i}^{N}\mathbf{v_{i}}(0)\cdot\mathbf{v_{i}}(t)~{},$
(4)

where $\mathbf{v_{i}}(t)$ represents the velocity of the $i$-th particle and
$\langle\cdots\rangle$ statistical average performed in the $NpT$ ensemble.
Subsequently, the vibrational density of states can be estimated like:

$\rho(\omega)=\int_{0}^{\infty}\text{VACF}(t)~{}e^{i\omega t}~{}dt~{},$ (5)

which fulfills the normalization condition:

$\int_{0}^{\infty}\rho(\omega)~{}d\omega=3N~{},$ (6)

$3N$ being the number of real and positively defined phonon frequencies of the
crystal.

Upon determination of $\rho$, the vibrational free energy can be
straightforwardly estimated with the formula cazorla17 :

$\displaystyle F_{\rm vib}(p,T)$ $\displaystyle=$ $\displaystyle k_{B}T\times$
(7)
$\displaystyle\int_{0}^{\infty}\ln{\left[2\sinh{\left(\frac{\hbar\omega}{2k_{B}T}\right)}\right]}\rho(\omega,p)~{}d\omega~{},$

where $k_{B}$ is the Boltzmann’s constant. Consistently, the vibrational
entropy adopts the expression:

$\displaystyle S_{\rm vib}(p,T)$ $\displaystyle=$
$\displaystyle-\int_{0}^{\infty}k_{B}\ln{\left[2\sinh{\left(\frac{\hbar\omega}{2k_{B}T}\right)}\right]}\rho(\omega,p)~{}d\omega+$
(8)
$\displaystyle\int_{0}^{\infty}\frac{\hbar\omega}{2T}\tanh^{-1}{\left(\frac{\hbar\omega}{2k_{B}T}\right)}\rho(\omega,p)~{}d\omega~{}.$

####  II.3.2 Molecular orientational entropy

For a continuous random variable $x$ with probability density $f(x)$, its
entropy is defined as informationtheory :

$S=-\int_{X}f(x)\log{f(x)}~{}dx~{},$ (9)

where the integral runs over all possible values of $x$. If $x$ represents a
microstate characterizing a particular thermodynamic macrostate, then the
following Gibbs entropy can be defined for the system of interest pathria1972
:

$S_{G}=-k_{B}\int_{X}f(x)\log{f(x)}~{}dx~{}.$ (10)

In an orientationally disordered crystal, molecules reorient in a quasi-random
manner. By assuming the MA molecules in the HOIP crystal to be independent one
from the other, one may estimate a probability density function for their
orientation, $f(\theta_{\rm lab},\phi_{\rm lab},\psi_{\rm lab})$, from the
atomistic trajectories generated during long MD–$NpT$ simulations. In this
case, the following three-dimensional molecular orientational entropy can be
defined, under the implicit assumption that the length of the MA molecules
remains fixed:

$\displaystyle S_{\rm ori}(T)$ $\displaystyle=$ $\displaystyle-k_{B}\int
f(\theta_{\rm lab},\phi_{\rm lab},\psi_{\rm lab})\log{[f(\theta_{\rm
lab},\phi_{\rm lab},\psi_{\rm lab})]}\times$ (11) $\displaystyle
d\cos(\theta_{\rm lab})~{}d\phi_{\rm lab}~{}d\psi_{\rm lab}~{}.$

In practice, the calculation of $S_{\rm ori}$ entails the construction of
histograms for which the continuous polar variables are discretised,
$\\{\theta_{\rm lab},\phi_{\rm lab},\psi_{\rm lab}\\}\to\\{\theta_{{\rm
lab},i},\phi_{{\rm lab},i},\psi_{{\rm lab},i}\\}$. Accordingly, one may define
the bin probabilities:

$\displaystyle p_{i}(\theta_{{\rm lab},i},\phi_{{\rm lab},i},\psi_{{\rm
lab},i})\approx f(\theta_{{\rm lab},i},\phi_{{\rm lab},i},\psi_{{\rm
lab},i})\times$ $\displaystyle\Delta\cos(\theta_{\rm lab})~{}\Delta\phi_{\rm
lab}~{}\Delta\psi_{\rm lab}~{},$ (12)

where $\Delta\cos(\theta_{\rm lab})~{}\Delta\phi_{\rm lab}~{}\Delta\psi_{\rm
lab}$ is the volume of a histogram bin (selected to be constant in this
study). Consistently, one can rewrite $S_{\rm ori}$ in the discretised form
informationtheory :

$\displaystyle S_{\rm ori}(T)$ $\displaystyle=$ $\displaystyle-
k_{B}\sum_{i}p_{i}\log{p_{i}}+$ (13) $\displaystyle
k_{B}\log{[\Delta\cos(\theta_{\rm lab})~{}\Delta\phi_{\rm
lab}~{}\Delta\psi_{\rm lab}]}~{}.$

####  II.3.3 Molecular conformational entropy

At finite temperatures, MA molecules may undergo conformational changes that
contribute to the total entropy, $S_{\rm conf}$. The dihedral angle formed by
the intersecting planes containing the H–C–N and C–N–H∗ atoms (where H belongs
to the molecular methyl group and H∗ to the ammonia group, Fig.1c),
$\epsilon_{\rm dih}$, may be used to monitor such conformational molecular
changes. Analogously to the case of the orientational entropy, a probability
density function can be estimated for this molecular dihedral angle from the
atomistic trajectories generated during long MD–$NpT$ simulations,
$f(\epsilon_{\rm dih})$. Likewise, a bin probability can be defined,
$p_{i}(\epsilon_{{\rm dih},i})\approx f(\epsilon_{{\rm
dih},i})~{}\Delta\epsilon_{\rm dih}$, leading to the conformational entropy
expression:

$S_{\rm conf}(T)=-k_{B}\sum_{i}p_{i}\log{p_{i}}+k_{B}\log{(\Delta\epsilon_{\rm
dih})}~{},$ (14)

where $\Delta\epsilon_{\rm dih}$ is the fixed length of a histogram bin.

####  II.3.4 Molecular correlation entropy

Equation (13) would provide the total orientational entropy for the MA
molecules if these were completely independent one from the other. However,
assuming the absence of molecular orientational correlations in HOIP might not
be a good approximation, so that first these correlations need to be assessed.
In this study, we account for the effects of molecular correlations in the
entropy, $S_{\rm ori-ori}$, by considering up to second-order terms in the
many-body expansion of the full orientational entropy karplus96 ; pardo16 . To
this end, we monitor the relative orientation between pairs of molecules in a
co-mobile reference system (Sec.II.2), and calculate the molecular correlation
entropy difference between the disordered and ordered phases like karplus96 ;
pardo16 :

$\Delta S_{\rm ori-ori}=\frac{1}{2}k_{B}\sum_{i=1}^{n_{\rm
shell}}n_{i}~{}\Delta S^{\rm rel}_{{\rm ori},i}~{},$ (15)

where $n_{i}$ is the number of molecules in the $i$-th spherical coordination
shell (e.g., $n_{1}=6$ and $n_{2}=12$), and $n_{\rm shell}$ is the number of
considered coordination shells (equal to $5$ in the present study). The
expression for the individual entropy terms $S^{\rm rel}_{{\rm ori},i}$ is
equivalent to that shown in Eq.(13), making the angular substitutions
$\\{\theta_{\rm lab},\phi_{\rm lab},\psi_{\rm lab}\\}\to\\{\theta_{\rm
rel},\phi_{\rm rel},\psi_{\rm rel}\\}$.

Figure 2: Vibrational density of states (VDOS) and vibrational entropy
($S_{\rm vib}$) of MAPI calculated at different temperatures. VDOS results
were obtained (a) for the ordered phase at $T=240$ K and (b) for the
disordered phase at $T=245$ K. (c) Vibrational entropy of MAPI expressed as a
function of temperature.

##  III Results and Discussion

In a previous work cazorla24 , we already estimated the phase-transition
entropy change for MAPI through MD–$NpT$ simulations by considering
vibrational and molecular orientational degrees of freedom. Nevertheless, in
this study, we present several critical improvements to our original $\Delta
S_{t}$ calculation, which can be generalized to other HOIPs and solids
exhibiting molecular orientational disorder (e.g., plastic crystals plastic1 ;
plastic2 ; plastic3 ; plastic4 ; plastic5 ). These critical computational
improvements include considering entropy contributions from molecular
conformational changes ($\Delta S_{\rm conf}$), molecular orientational
correlations ($\Delta S_{\rm ori-ori}$), and possible couplings between the
vibrational and molecular orientational degrees of freedom ($\Delta S_{\rm
vib-ang}$) (Sec.II.3). Next, we present and discuss the calculation of $\Delta
S_{t}$ term-by-term, comment on the potential formation of ordered MA+
nanodomains, and make insightful connections with technological applications.
Our main entropy numerical findings are summarized in Table 1.

Figure 3: Bivariate angular probability density function (pdf) for MA
molecules in the low-$T$ ordered phase. Results were obtained at $T=240$ K in
the (a) lab-fixed (“lab”) and (b) molecule-mobile (“rel”) reference systems.
For the “rel” case, the six MA cations within the first coordination shell
were considered. Red, green and blue colours represent high-probability,
medium-probability and low-probability configurations, respectively. Figure 4:
Bivariate angular probability density function (pdf) for MA molecules in the
high-$T$ disordered phase. Results were obtained at $T=260$ K in the (a) lab-
fixed (“lab”) and (b) molecule-mobile (“rel”) reference systems. For the “rel”
case, the six MA cations within the first coordination shell were considered.
Red, green and blue colours represent high-probability, medium-probability and
low-probability configurations, respectively.

###  III.1 Vibrational entropy

The simulation results presented in this section have been mostly reported in
a previous work of ours cazorla24 . These numerical findings are shown here
for coherence and completeness.

Figure 2 shows the vibrational density of states (VDOS) and partial organic
and inorganic contributions estimated for MAPI at temperatures slightly above
and below the simulated order-disorder phase transition point (i.e., $245$ and
$240$ K). The VDOS contribution corresponding to the inorganic part, namely,
the PbI3 octahedra, is clearly dominant in the low-frequency range
$0\leq\nu\leq 5$ THz (Figs. 2a–b). This result follows from the fact that the
lighter atoms, which typically vibrate at higher frequencies, entirely reside
in the organic molecules. Consistently, the range of moderate and high
frequencies, $\nu\geq 5$ THz, is mostly governed by the MA cation vibrations.

Albeit the VDOS enclosed in Figs. 2a–b may seem quite similar at first glance,
there are significant differences among them. In particular, the high-$T$
disordered phase accumulates more phonon modes in the low-frequency range
$0\leq\nu\leq 2$ THz than the low-$T$ ordered phase. This effect has a strong
influence on the vibrational entropy of the system, $S_{\rm vib}$, which is
significantly larger for the high-$T$ disordered phase.

Figure 2c shows the $S_{\rm vib}$ estimated as a function of temperature. A
clear surge in vibrational entropy is observed at the phase-transition point,
amounting to $\Delta S_{\rm vib}=+21.7$ J K-1 kg-1. The positive sign
indicates that the vibrational entropy of the high-$T$ disordered phase is
larger than that of the low-$T$ ordered phase. The primary contribution to
this vibrational entropy increase comes from the molecular cations, which is
equal to $+13.9$ J K-1 kg-1 and almost twice as large as that calculated for
the inorganic part (namely, $+7.8$ J K-1 kg-1). Thus, although the low-
frequency range in the VDOS is dominated by the inorganic anions, the organic
MA cations have a larger influence on the vibrational entropy change
associated with the order-disorder phase transition, $\Delta S_{\rm vib}$.

Figure 5: Dynamical description of the MA cation orientation around its C–N
axis and molecular conformations for the low-temperature ordered phase. Time
evolution of (a) the azimuthal angle $\psi_{\rm lab}$ describing the MA+
rotation around its C–N axis and (b) the dihedral angle $\epsilon_{\rm dih}$
describing molecular conformation (Fig.1).

###  III.2 Molecular orientational entropy

To estimate the orientational entropy associated with the order-disorder phase
transition of MAPI, $\Delta S_{\rm ori}$, we have extended the computational
approach introduced in a previous work of ours cazorla24  to include the three
Euler angles that fully determine the orientation of an arbitrary (and rigid)
MA cation, namely, $\\{\theta,\phi,\psi\\}$ (Fig.1b). To calculate $\Delta
S_{\rm ori}$, we initially assume the MA cations to be independent of each
other. Consequently, the three Euler molecular angles should be referred to
the stationary lab reference system, namely, $\\{\theta_{\rm lab},\phi_{\rm
lab},\psi_{\rm lab}\\}$ (Sec.II.2). The complete molecular orientation maps
thus correspond to three-dimensional probability density functions (pdf)
considering those three angles.

Figure 6: Dynamical description of the MA cation orientation around its C–N
axis and molecular conformations for the high-temperature disordered phase.
Time evolution of (a) the dihedral angle $\epsilon_{\rm dih}$ describing
molecular conformation, and the rest of MA angles (b) $\phi_{\rm lab}$, (c)
$\theta_{\rm lab}$ and (d) $\psi_{\rm lab}$ describing molecular orientation
(Fig.1).

In the previous work cazorla24 , the $\psi_{\rm lab}$ angle, which accounts
for the rotation of the MA molecules around their C–N molecular axis (Fig.1b),
was not considered, thus potential entropy contributions resulting from this
angular degree of freedom were totally neglected. Results for the full MA
orientational pdf calculated for the low-$T$ ordered and high-$T$ disordered
phases of MAPI, as projected onto two different planes, are shown in Figs.3a
and 4a, respectively.

As shown in Fig.3a, in the low-$T$ ordered phase, the MA cations do not
reorient since the equilibrium molecular orientations, represented by bright
spots in the $\cos(\theta_{\rm lab})$–$\phi_{\rm lab}$ pdf, are disconnected
(i.e., angular transition paths connecting them are absent). This outcome was
anticipated and is consistent with our previous results reported in cazorla24
. On the other hand, as expected, in the high-$T$ disordered phase, the
molecular C–N axis are free to reorient, as indicated by the regions of
nonzero probability appearing between the equilibrium molecular orientations
(Fig.4a). For this phase, and also in agreement with our previous findings
cazorla24 , there a six possible equilibrium molecular orientations: two
“apical” [$\cos(\theta_{\rm lab})=\pm 1$] and four “equatorial”
[$\cos(\theta_{\rm lab})=0$, $\phi_{\rm lab}=\pm 60,120^{\circ}$].

Figures 3a and 4a also show the pdf associated with the azimuthal angle
$\psi_{\rm lab}$, which describes the rigid rotation of the MA cations around
their C–N axis (Fig.1b). Interestingly and surprisingly, it is found that for
both the low-$T$ ordered and high-$T$ disordered phases of MAPI, there are
rotational paths connecting high probability regions (i.e., equilibrium
molecular orientations). This result implies that, close to the transition
temperature, the molecules rotate around their C–N axis both in the ordered
and disordered states. It is worth noting that this type of azimuthal
orientational disorder does not break the symmetry of the crystal (i.e., the
corresponding equilibrium configurations are equivalent among them) and hence
cannot be resolved in diffraction experiments azi1 ; azi2 ; azi3 ; azi4 . To
the best of our knowledge, this is the first time that such a class of
molecular disorder, present in both the low-$T$ ordered and high-$T$
disordered phases, has been reported for MAPI.

The orientational phase-transition entropy change obtained from the two-
dimensional pdf associated with the two angles $\\{\cos(\theta_{\rm
lab}),\phi_{\rm lab}\\}$ amounts to $+10.7$ J K-1 kg-1 cazorla24 . However,
when considering the full orientational pdf associated with the three Euler
angles $\\{\cos(\theta_{\rm lab}),\phi_{\rm lab},\psi_{\rm lab}\\}$, it is
found that $\Delta S_{\rm ori}=+13.5$ J K-1 kg-1. Therefore, by including the
molecular azimuthal degree of freedom, the orientational entropy change
increases by approximately $25$%, thus representing a substantial correction.
It is noted in passing that the degree of disorder associated with the
molecular azimuthal angle $\psi_{\rm lab}$ is larger for the high-$T$ phase.

Figure 7: Angular probability density function (pdf) for the molecular
dihedral angle $\epsilon_{\rm dih}$ describing the conformation of MA
molecules. Results are represented for the low-$T$ ordered (red circles) and
high-$T$ disordered (black squares) phases. Solid lines are guides to the eye.

It is instructive to compare the $\Delta S_{\rm ori}$ value obtained from the
atomistic MD–$NpT$ simulations with that estimated straightforwardly from
symmetry and thermodynamic arguments. From the $\cos(\theta_{\rm
lab})$–$\phi_{\rm lab}$ pdf shown in Figs.3a and 4a, it is observed that the
MA cations can adopt two possible orientations in the low-$T$ ordered phase
and five in the high-$T$ disordered phase. This elementary counting leads to
an orientational entropy change of $k_{B}\left(\ln{5}-\ln{2}\right)$, which
equals $+12.3$ J K-1 kg-1 and shows reasonably good agreement with the
corresponding fully atomistic calculation. We may thus conclude that such a
rough estimation appears to be a valid initial guess for $\Delta S_{\rm ori}$.

###  III.3 Molecular conformational entropy

In the previous section, we asserted that in both the low-$T$ ordered and
high-$T$ disordered phases of MAPI, the MA cations perform rigid rotations
around their molecular C–N axis, which do not affect the symmetry of the
crystalline lattice. However, changes in the azimuthal angle $\psi_{\rm lab}$
could, in principle, also be due to conformational changes of the MA cation
associated with disconnected rotations of the methyl (CH3) and/or ammonia
(NH3) groups at the ends of the molecule. To check this possibility, we plot
in Figs.5 and 6 the time evolution of the molecular angles $\\{\psi_{\rm
lab},\epsilon_{\rm dih}\\}$ for the low-$T$ ordered and high-$T$ disordered
phases, respectively, where $\epsilon_{\rm dih}$ is the dihedral angle formed
by the intersecting planes containing the H–C–N and C–N–H∗ atoms (H belonging
to the molecular methyl group and H∗ to the ammonia group, Fig.1c). For
completitude purposes, we also represent the dynamical evolution of the angles
$\theta_{\rm lab}$ and $\phi_{\rm lab}$ for the high-$T$ disordered phase in
Fig.6.

For the low-$T$ ordered phase (Fig.5), it is clearly observed that the
sequence of $\psi_{\rm lab}$ and $\epsilon_{\rm dih}$ changes occurring over
time are uncorrelated. For instance, the dihedral angle changes twice its
value within the time interval of $80$ ps, from $-60^{\circ}$ to $+60^{\circ}$
and viceversa (Fig.5b), while the azimuthal angle changes several tens of
times its value, $-180^{\circ}\leq\psi_{\rm lab}\leq+180^{\circ}$, within the
same time interval (Fig.5a). A very similar behaviour is also appreciated for
the high-$T$ disordered phase (Figs.6a,d). Consequently, it can be safely
concluded that, owing to the absence of time correlations between the angles
$\psi_{\rm lab}$ and $\epsilon_{\rm dih}$ in both phases, changes in the
azimuthal angle correspond to rigid molecular rotations around the C–N axis.
We note in passing that the dynamics of the dihedral angle is extremely slow
in both the low-$T$ ordered and high-$T$ disordered phases, which denotes high
MA+ rigidity.

Figure 7 shows the pdf estimated for the molecular dihedral angle
$\epsilon_{\rm dih}$ for both the low-$T$ ordered and high-$T$ disordered
phases of MAPI, close to the phase-transition temperature. It is observed that
$\epsilon_{\rm dih}$ essentially adopts the equilibrium values $\pm
60^{\circ}$ and $\pm 180^{\circ}$ in both phases. Additionally, the estimated
pdf are very similar for the two phases. As a consequence, the phase-
transition entropy variation associated with molecular conformational changes
(Eq.14) turns out to be practically null for MAPI, namely, $\Delta S_{\rm
conf}\approx 0$. It can be concluded then that the MA cation presents high
rigidity in both the low-$T$ ordered and high-$T$ disordered phases, and that
close to room temperature molecular conformational disorder is strongly
inhibited in MAPI.

###  III.4 Molecular correlation entropy

Figures 3b and 4b show the pdf estimated for the three Euler angles
$\\{\theta_{\rm rel},\phi_{\rm rel},\psi_{\rm rel}\\}$ describing the relative
orientation between closest MA cations (i.e., within the first coordination
shell) in a co-mobile molecular reference system (Sec.II.2) for the low-$T$
ordered and high-$T$ disordered phases of MAPI, respectively. Using these
orientational probability maps, and others involving successive coordination
shells, it is possible to quantify molecular ordering and the phase-transition
entropy change associated with the molecule-molecule correlations, as
explained in Sec.II.3.4.

In the low-$T$ ordered phase (Fig.3b), the relative orientation between the
closest molecules is most likely when $\cos(\theta_{\rm rel})=\pm 1$, which
translates into parallel and antiparallel C–N bond arrangements or,
equivalently, parallel and antiparallel molecular dipoles. Interestingly, it
is observed that these maximum probability regions differ in shape. The reason
for this asymmetry is that, out of the six closest MA cations, two are
parallel [$\cos(\theta_{\rm rel})=+1$], located in the “tail” and “head”
positions of the central MA+, and four are antiparallel [$\cos(\theta_{\rm
rel})=-1$], positioned in the equatorial plane of the central MA+. The
molecules parallel to the reference MA cation slightly tilt their C–N axis
relative to the central one to change their $\psi_{\rm rel}$ orientation,
causing $\cos(\theta_{\rm rel})$ to slightly depart from $+1$. This behaviour,
however, is not observed for the antiparallel molecules.

In the high-$T$ disordered phase (Fig.4b), strong orientational correlations
between closest MA molecules are evident, as otherwise, much more uniform and
symmetric probability maps would have been obtained. For example, molecular
parallel arrangements are strongly suppressed at high temperatures, as shown
by the very low probability estimated for $\cos(\theta_{\rm rel})=+1$. In this
case, the most likely relative MA+ orientations correspond to three different
molecular arrangements resulting in $\cos(\theta_{\rm rel})\approx 0.4$, which
compensate for the total dipole moment deriving from the also highly probable
antiparallel configuration [$\cos(\theta_{\rm rel})=-1$]. It is worth noting
that the number of equilibrium relative orientations is the same for the
low-$T$ ordered and high-$T$ disordered phases, totaling four. The most
significant molecular orientation difference between the two phases is a shift
of the maximum probability from $\theta_{\rm rel}\approx 0^{\circ}$ at low
temperatures to $\theta_{\rm rel}\approx 66^{\circ}$ at high temperatures.

In view of the fact that the reorientational MA+ motion in MAPI is correlated,
we calculated the entropy difference between the low-$T$ ordered and high-$T$
disordered phases resulting from the molecular correlations at the
corresponding phase-transition temperature, $\Delta S_{\rm ori-ori}$
(Sec.II.3.4). A positive (negative) $\Delta S_{\rm ori-ori}$ value would
indicate larger (smaller) molecular correlation entropy in the high-$T$
disordered phase, and consequently less (more) correlated molecular
orientational dynamics in that phase. Adding consecutive entropy terms up to
the fifth coordination shell (i.e., a maximum radial distance of $14.7$ Å and
$56$ molecules), we estimated $\Delta S_{\rm ori-ori}=3.8$ J K-1 kg-1. It was
found that this molecular correlation entropy was already numerically
converged to within $0.1$ J K-1 kg-1 at the fourth coordination shell, which
involves a maximum radial distance of $13.2$ Å and $32$ molecules. The
calculated $\Delta S_{\rm ori-ori}$ value is positive, suggesting more
substantial correlations in the molecular orientational dynamics of the
low-$T$ phase. Moreover, $\Delta S_{\rm ori-ori}$ is comparable in magnitude,
for instance, to the phase-transition entropy gain associated with the
azimuthal angle $\psi_{\rm lab}$ (Sec.III.2) and hence is not negligible.

Figure 8: Probability density function (pdf) for molecular dipole orientation
in the co-mobile (“rel”) reference system obtained across successive
coordination shells. Results obtained for the (a) first, (b) second, (c) third
and (d) fourth coordination shells. Orange and green bars represent results
obtained for the low-temperature ordered and high-temperature disordered
phases, respectively.

###  III.5 Molecular dipoles ordering

To further elucidate the molecular orientational disorder in MAPI and evaluate
the possible formation of ordered MA+ nanodomains at finite temperatures, we
represent in Fig.8 the pdf estimated for the angular quantity
$\cos(\theta_{\rm rel})$ across successive coordination shells (i.e., from the
first up to the fourth). Although our primary focus is on the molecular
structure of the high-$T$ disordered phase, we have also included pdf results
for the low-$T$ ordered phase in Fig.8 to facilitate the interpretation and
comparison of our findings.

It is clear from Fig.8 that the relative orientation between molecules in the
high-$T$ disordered phase follows a fairly uniform probability distribution
within all the analyzed coordination shells, implying that there is no
particularly preferred MA+ orientation. For instance, the average value of the
angular pdf within the first coordination shell (Fig.8a) amounts to
$<\cos(\theta_{\rm rel})>=0.02$, with similar values obtained for the other
consecutive coordination shells. Conversely, the pdf of the low-$T$ ordered
phase exhibits well-defined peaks within all the coordination shells at
$\cos(\theta_{\rm rel})=\pm 1$, as expected (Sec.III.4), leading to general
antipolar ordering.

Figure 9: Probability distribution function (pdf) for the number of molecules
in the first coordination shell of an arbitrary MA molecule that are parallel
and antiparallel. Results obtained for (a) parallel and (b) antiparallel
relative molecular dipole orientations (black and red boxes). The crosses in
the figures represent analogous results obtained for a random distribution of
relative molecular dipole orientations.

Interestingly, a very subtle alternation from marginally parallel to
marginally antiparallel relative molecular arrangements is observed for the
high-$T$ disordered phase as the radial distance increases. However, this
slight variation of relative MA+ orientations developing through successive
coordination shells does not induce the appearance of nanoregions with a well-
defined polarization. This outcome is illustrated in Fig.9, where we compare
the pdf estimated for the number of parallel and antiparallel MA cations to a
chosen one within the first coordination shell as directly obtained from our
MD–$NpT$ simulations and by considering completely random and uncorrelated
relative molecular orientations. No appreciable differences are observed
between the two pdf. It can then be concluded that the appearance of polar
nanodomains in MAPI is precluded since the MA molecules tend to orient
homogeneously throughout the crystal. A similar behaviour has been recently
reported for the plastic crystal pentachlronitrobenzene gebbia23 .

###  III.6 Vibrational–orientational correlation entropy

There is an additional source of potential entropy variation in MAPI related
to the correlations between the molecular angular and vibrational degrees of
freedom, $\Delta S_{\rm vib-ang}$ [Eq.(3)], which mainly results from the
interdependencies between the organic and inorganic ions. Unlike the rest of
terms in Eq.(3), there is neither an analytical expression nor a computational
recipe to directly evaluate $\Delta S_{\rm vib-ang}$. Nevertheless, this
particular entropy variation can be indirectly estimated at the order-disorder
phase-transition point using simple thermodynamic arguments and MD–$NpT$
simulations, as we explain below.

At the order-disorder phase-transition temperature, the Gibbs free energy
($G\equiv H-TS$, where $H$ stands for enthalpy) of the low-$T$ ordered and
high-$T$ disordered phases of MAPI should be equal. Therefore, it follows that
$\Delta S_{t}=\Delta H_{t}/T_{t}$. Since our MD–$NpT$ simulations are
performed at zero pressure, the enthalpy difference between the high-$T$
disordered and low-$T$ ordered phases are equal to their internal energy
difference at the phase-transition temperature, namely, $\Delta H_{t}=\Delta
U_{t}$. This latter quantity can be directly obtained with great precision
from the MD–$NpT$ simulations and, consequently, $\Delta S_{t}$ as well.
Knowing the value of all the entropy terms in Eq.(3) except $\Delta S_{\rm
vib-ang}$, it follows that:

$\Delta S_{\rm vib-ang}=\Delta S_{t}-\Delta S_{\rm vib}-\Delta S_{\rm
ori}-\Delta S_{\rm conf}-\Delta S_{\rm ori-ori}~{}.$ (16)

|  $\Delta S_{\rm vib}$ |  $\Delta S_{\rm ori}$ |  $\Delta S_{\rm conf}$ |  $\Delta S_{\rm ori-ori}$ |  $\Delta S_{\rm vib-ang}$ |  $\Delta S_{t}$
---|---|---|---|---|---|---
(J K-1kg-1) | $+21.7$ | $+13.5$ | $\approx 0$ | $+3.8$ | $-12.6$ | $+26.4$
(%) | $+82$ | $+51$ | $\approx 0$ | $+15$ | $-48$ | $+100$
Table 1: Contributions to the total entropy change associated with the
$T$-induced order-disorder phase transition in MAPI, $\Delta S_{t}$. $\Delta
S_{\rm x}\equiv S_{\rm x}^{\rm disord}-S_{\rm x}^{\rm ord}$ and all the
entropy terms are evaluated at the phase-transition temperature, $T_{t}$.
$\Delta S_{\rm vib}$ represents contributions from the lattice vibrations,
$\Delta S_{\rm ori}$ from the MA orientational degrees of freedom, $\Delta
S_{\rm conf}$ from the MA conformations, $\Delta S_{\rm ori-ori}$ from the
molecule-molecule orientational correlations, and $\Delta S_{\rm vib-ang}$
from the couplings between the vibrational and molecular angular degrees of
freedom.

From our MD–$NpT$ atomistic simulations, we obtained $\Delta S_{t}=+26.4$ J
K-1 kg-1 for the $T$-induced order-disorder phase transition occurring in
MAPI. This value is in fairly good agreement with an equivalent estimation
reported in work cazorla24 , $\Delta S_{t}^{\rm CC}=+28.4$ J K-1 kg-1, which
was based on the Clausius-Clapeyron (CC) method and is therefore more prone to
numerical errors. Using Eq.(16) and the phase-transition entropy change values
reported in previous sections, we conclude that $\Delta S_{\rm vib-ang}=-12.6$
J K-1 kg-1 (Table 1). The size of this vibrational–orientational correlation
entropy is roughly $50$% of the total phase-transition entropy change, which
is surprisingly large and comparable in absolute value to the entropy change
contribution $\Delta S_{\rm ori}$ resulting from the individual MA+
orientational degrees of freedom (Sec.III.2). This source of entropy change in
MAPI, and arguably in HOIP in general, has not been considered in previous
studies but, given its large magnitude, should not be neglected.

Interestingly, the sign of the estimated $\Delta S_{\rm vib-ang}$ is negative,
unlike the rest of entropy terms in Eq.(3). Consequently, the crossed
vibrational–orientational contribution decreases the total phase-transition
entropy change. From a physical point of view, the negative sign of $\Delta
S_{\rm vib-ang}$ can be understood in terms of the couplings between the
vibrational and molecular angular degrees of freedom, which are substantially
reinforced in the high-$T$ disordered phase (i.e., a smaller entropy is
identified here with a higher degree of concertation). This reasoning is
consistent with the large surge in vibrational entropy observed across the
order-disorder phase transition (Sec.III.1), as extended atomic correlations
can induce the appearance of phonon modes with very large effective masses and
consequently low vibrational frequencies plastic1 .

###  III.7 Connections with technological applications

Table 1 reports the value of the different contributions to the total entropy
change associated with the $T$-induced order-disorder phase transition
occurring in MAPI, as well as their relative percentage. The three entropy
variations $\Delta S_{\rm vib}$, $\Delta S_{\rm ori}$ and $\Delta S_{\rm vib-
ang}$ are found to be the largest in absolute value and similar in size, with
the peculiarity that only the sign of the crossed vibrational–orientational
entropy change is negative.

Solid-state cooling is an energy efficient and ecologically friendly technique
with potential for solving the environmental problems posed by conventional
refrigeration technologies relying on compression cycles of greenhouse gases
cazorla19 ; lloveras21 . Upon moderate magnetic, electric or mechanical field
variations, auspicious caloric materials experience large adiabatic
temperature variations ($|\Delta T|\sim 1$–$10$ K) due to phase
transformations entailing large isothermal entropy changes ($|\Delta S|\sim
10$–$100$ J K-1 kg-1). HOIP have shown great promise as caloric materials
because their order-disorder phase transition can be driven near room
temperature under the influence of various external fields cool1 ; cool2 ;
cool3 ; cool4 ; zhang23 ; ren20 .

However, the entropy changes associated with the order-disorder phase
transition in HOIP ($|\Delta S|\sim 10$ J K-1 kg-1) are about an order of
magnitude smaller than those measured in plastic crystals ($|\Delta S|\sim
100$ J K-1 kg-1) plastic1 ; plastic2 ; plastic3 ; plastic4 ; plastic5 , a
related family of materials constituted exclusively of organic molecules. The
physical reason for this phase-transition entropy change difference between
HOIP and plastic crystals may partly originate from the $\Delta S_{\rm vib-
ang}$ contributions in the former materials, which, as revealed in this study,
tend to reduce $\Delta S_{t}$. Consequently, a possible rational design
strategy to improve the caloric figures of merit of HOIP could be to minimize
the crossed vibrational–orientational couplings through chemical and
structural engineering design1 .

In HOIP, lattice vibrations are expected to be intrinsically coupled with the
cation orientational motion, and both phenomena are deemed to influence the
optoelectronic performance of the material review1 ; rot2 ; rot3 ; diel1 ;
hyste1 . For instance, phonons are thought to play a significant role in the
modulation of the band gap, charge transport and exciton dynamics elphon1 ;
elphon2 . Likewise, cation orientational dynamics is assumed to affect, among
others, the ferroelectric, ion transport and dielectric properties of HOIP
func1 ; func2 ; func3 . In solar cells, the dielectric function of the light-
absorbing materials, $\epsilon$, is of the upmost importance since it directly
impacts the exciton binding energy, $E_{\rm bind}$ (i.e., in the Wannier-Mott
model $E_{\rm bind}\propto 1/\epsilon^{2}$ cong23 ). In turn, $\epsilon$ is
strongly affected by the phonons (i.e., the lattice dielectric contribution,
which generally adopts positive values dielphon ) and cation rotations.

In this study, we have theoretically proven that the couplings between the
phonons and cation orientational dynamics in HOIP are sizable and pivotal for
assessing the entropy change associated with their $T$-induced order-disorder
phase transition. Additionally, through the calculation of the entropy
difference term $\Delta S_{\rm vib-ang}$, we have ascertained that such
vibrational–orientational couplings are much stronger in the high-$T$
disordered phase than in the low-$T$ ordered phase. Therefore, a possible way
to enhance the binding energy of excitons in light-absorbing HOIP, to mitigate
to a certain extent detrimental electron-hole recombinations, may be to weaken
the vibrational–orientational couplings in the disordered phase, typically
stabilized near room temperature, through chemical doping and nanostructuring
techniques design1 . By doing this, one may expect to reduce the total
$\epsilon$ of the material, making the lattice dielectric contribution
smaller, and consequently increase $E_{\rm bind}$.

##  IV Conclusions

A comprehensive analysis of the orientational disorder and molecular
correlations in the archetypal hybrid organic-inorganic perovskite (HOIP) MAPI
has been presented, relying on atomistic MD–$NpT$ simulations and advanced
entropy calculations. The main findings of our computational research are the
following. Both in the low-$T$ ordered and high-$T$ disordered phases, there
is dynamical orientational disorder associated with rigid MA+ rotations around
the molecular C–N axis. This previously overlooked orientational degree of
freedom positively contributes to the total phase-transition entropy change,
$\Delta S_{t}$. The correlations between MA cations are substantial,
especially in the low-$T$ ordered phase, and have a sizeable augmenting effect
on $\Delta S_{t}$. Conformational molecular changes, on the other hand, are
relatively infrequent both in low-$T$ ordered and high-$T$ disordered phases
and do not appreciably contribute to the phase-transition entropy change.
Interestingly, the couplings between the vibrational and orientational degrees
of freedom are strengthened in the high-$T$ disordered phase and have a
substantial decreasing effect on $\Delta S_{t}$. Lastly, molecular
correlations in the high-$T$ disordered phase are predominant but markedly
local, thus the formation of nanoregions with a well-defined polarization is
strongly suppressed. These fundamental outcomes may be tentatively generalized
to other HOIP, and have important ramifications for energy and optoelectronic
applications relying on this family of materials.

## Methods

Molecular dynamics simulations. We used the LAMMPS simulation code lammps  to
perform systematic classical molecular dynamics simulations in the $NpT$
ensemble for bulk MAPI using the force field developed by Mattoni and co-
workers mattoni1 ; mattoni2 . The average temperature was set using a Nosé-
Hoover thermostat with a mean fluctuation of $5$ K. The simulation box
contained $3072$ atoms (equivalent to $256$ MAPI unit cells) and periodic
boundary conditions were applied along the three Cartesian directions. The
long-range electrostatic interactions were calculated by using a particle-
particle particle-mesh solver to compute Ewald sums up to an accuracy of
$10^{-4}$ kcal mol-1 Å-1 in the atomic forces. The cutoff distance for the
evaluation of the potential energy was set to $12$ Å. To determine the phase-
transition temperature of MAPI at zero pressure, we conducted comprehensive
$NpT$-MD simulations in the temperature range $180\leq T\leq 340$ K, taken at
intervals of $10$ K. In our MD–$NpT$ simulations, the temperature was steadily
increased up to a targeted value over $1$ ns. Subsequently, the system was
equilibrated at that selected temperature for $4$ ns. The production runs then
lasted for about $100$ ps with $\Delta t=0.5$ fs, from which the velocities of
the atoms and other key quantities (e.g., the potential energy and volume of
the system) were extracted. From the production MD–$NpT$ runs, a total of
$1,000$ equispaced configurations were retrieved to obtain uncorrelated
structural data and generate accurate probability density functions (pdf).

Molecular angular and entropy analysis. The MA+ angular degrees of freedom
have been retrieved from the atomic configurations generated during the
MD–$NpT$ simulations with the help of the freely available and open-source
software ANGULA angula . ANGULA is designed to automatically and
unsupervisedly determine the angles defining the orientational structure of
molecular disordered crystals from data files containing their atomic
positions, both in the fixed “lab” and co-mobile “rel” reference systems.
Among its many capabilities, ANGULA can generate angular probability density
maps and directional radial distribution functions directly from sequences of
molecular configurations. In this study, the angular molecular entropy terms
$S_{\rm ori}$, $S_{\rm ori-ori}$ and $S_{\rm conf}$ have been directly
computed from the outputs of our MD–$NpT$ simulations with ANGULA angula .

## Acknowledgments

C.C. acknowledges support from the Spanish Ministry of Science, Innovation and
Universities under the fellowship RYC2018-024947-I and grants
PID2020-112975GB-I00 and grant TED2021-130265B-C22. The authors also
thankfully acknowledge technical support and computational resources at
MareNostrum4 provided by Barcelona Supercomputing Center (FI-2023-1-0002,
FI-2023-2-0004, FI-2023-3-0004, FI-2024-1-0005 and FI-2024-2-0003). This work
is part of Maria de Maeztu Units of Excellence Programme CEX2023-001300-M
funded by MCIN/AEI (10.13039/501100011033).

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