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# Atomic dynamics in fluids: Normal mode analysis revisited

Jaeyun Moon  To whom correspondence should be addressed; E-mail:
<EMAIL_ADDRESS>Materials Science and Technology Division,
Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA  Lucas Lindsay
Materials Science and Technology Division,
Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA  Takeshi Egami
Materials Science and Technology Division,
Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA
Department of Materials Science and Engineering,
University of Tennessee, Knoxville, Tennessee 37996, USA
Department of Physics and Astronomy,
University of Tennessee, Knoxville, Tennessee 37996, USA

(August 28, 2024)

###### Abstract

Developing microscopic understanding of the thermal properties of liquids is
challenging due to their strong dynamic disorder, which prevents
characterization of the atomic degrees of freedom. There have been significant
research interests in the past few decades to extend the normal mode analysis
for solids to instantaneous structures of liquids. However, the nature of
normal modes that arise from these unstable structures is still elusive. In
this work, we explore the instantaneous eigenmodes of dynamical matrices of
various Lennard-Jones argon liquid/gas systems at high temperatures and show
that the normal modes can be interpreted as an interpolation of $T\to\infty$
(gas) and $T=0$ (solid) mode descriptions. We find that normal modes become
increasingly collisional and translational, recovering atomistic gas-like
behavior rather than vibrational with increase in temperature, suggesting that
normal modes in liquids may be described by both solid-like and gas-like
modes.

##  I Introduction

In contrast to solids and gases, the nature of atomic dynamics of liquids
remains elusive due to strong atomic correlations, leading to difficulties in
microscopically understanding materials properties of liquid. Among other
theoretical approaches that characterize atomic dynamics, normal mode analysis
is widely used in various physical [1] and life sciences [2]. In normal mode
analysis, one obtains normal mode/eigenmode frequencies and eigenvectors from
diagonalization of dynamical matrices built from second derivatives of the
interatomic potential of a given structure. Therefore, eigenmode frequencies
represent local potential curvatures and eigenvectors depict constituent
atomic motions relative to each other.

For solids, normal modes are physically understood as harmonic oscillators
about static equilibrium positions, with some exceptions including highly
anharmonic systems [3]. These harmonic oscillators are categorized as phonons,
propagons, diffusons, and locons based on translational symmetry and mode
interaction mechanisms. Normal mode analysis has been successful in describing
various thermodynamic and thermal properties of solids, including free energy
and heat capacity from a bottom-up perspective [1, 4, 5]. These properties can
now be predicted with high accuracy from routine lattice dynamics calculations
[6].

Encouraged by the success in describing vibrational properties of solids, many
efforts have been made in recent years to extend the normal mode analysis to
liquid systems where instantaneous structure snapshots are used instead of the
lattice structure [7, 8, 9, 10, 11, 12, 13, 14, 15]. These unstable snapshot
structures (non-zero net forces) give rise to modes with imaginary frequencies
in addition to modes with real frequencies. As such, a lot of research has
been conducted to understand the nature of imaginary modes and various aspects
of these have been used to describe some non-equilibrium processes including
atomic diffusion [9, 16], glass transition [17, 18], and melting [19], with
varying degrees of success. Despite these efforts, physical understanding of
imaginary modes and their relation to various materials properties remains
elusive. Real modes in liquids are often interpreted as extrapolation of real
modes in solids as harmonic oscillators with well-defined frequencies in a
local potential well. However, this assumption becomes particularly
questionable at high temperatures as the structures themselves change much
faster than most real mode periods. Accurate understanding of the nature of
real modes in liquids is, therefore, also lacking.

To build a better understanding of normal mode behaviors, particularly for
non-static structures (i.e., liquids and gases), we study normal mode
characteristics of molecular dynamics (MD) snapshots of various argon systems
from solid to gas at extremely high temperatures, up to $10^{8}$ K, at
constant volume. In particular, we explore the high temperature limit
($T\to\infty$) of normal modes such that physical interpretations of normal
modes at finite, intermediate temperatures can potentially become an
interpolation between solid and gas rather than an extrapolation from solid
behavior alone. We find that as temperature increases, normal modes involving
diatomic units become a dominant feature in the normal mode spectra. Many of
these pairs of atoms have positive potential energies and large kinetic
energies such that normal modes with eigenvectors that are parallel with the
‘bond’ can be described as collisional rather than vibrational. Further, we
also observe modes progressively being projected to a single atom, mimicking
single atom translating in space. Our findings, therefore, set the scale of
the spectrum of eigenmodes of dynamical matrices where in one end, we have
solid-like vibrational normal modes and in the other end we have gas-like
collisional and translational modes.

##  II Methods

Crystalline argon of 1372 atoms in the FCC structure and with a lattice
parameter of 5.27 Å was used as the initial input structure for MD simulations
using LAMMPS [20] to generate equilibrated atomic structures at various
temperatures at constant volume. Interatomic interactions are described by the
Lennard-Jones potential [21, 22, 23]. The parameters used were $\epsilon$ =
0.0104 eV and $\sigma$ = 3.4 Å. The cutoff distance was 2.5$\sigma$ and the
potential was shifted to eliminate discontinuity. Timesteps of 0.1 fs (up to
106 K) and 0.01 fs (107 K and above) were used. At each temperature, all
systems except that of the glass system were equilibrated for $5\times 10^{6}$
timesteps in the canonical ensemble (NVT) prior to data recording of another
$5\times 10^{6}$ timesteps under the same ensemble. The equilibrated atomic
structures (snapshots) generated from MD were used to perform lattice dynamics
calculations using GULP [24] to obtain eigenfrequencies and eigenvectors of
the $\Gamma$ point dynamical matrices with the entire domain considered as a
unit cell. For better statistics, ten such structures at each temperature from
MD simulations were used for the lattice dynamics calculations.

Temperatures considered were 1 K for crystal and glass systems and ranging
from $10^{3}$ to $10^{8}$ K in increments of factors of ten for high
temperature liquids and gases. In this work, we consider all systems at 106 K
and above as gas states as their specific heat is $\frac{3}{2}Nk_{B}$ within
numerical errors where $N$ is the number of atoms. Even though the physical
density is kept constant, the effective atomic radius, where the potential
energy equals $k_{B}T$, thus the effective packing fraction, becomes much
reduced at high temperatures, as in gas. Thus, for simplicity, we loosely
refer to all other high temperature systems below 106 K as liquids. The glass
system was created by a standard melt-quench technique (1000 K melting
followed by an instantaneous quench to 1 K). This fast cooling rate was
required due to typical poor glass formability of monatomic systems. Quenched
glass was then equilibrated for $2\times 10^{7}$ timesteps before data
recording for $5\times 10^{6}$ timesteps.

To characterize the phases of the various structures, the pair distribution
function for each system (shown in Fig. 1) was calculated by

$g(r)=\frac{1}{4\pi
Nnr^{2}}\sum_{i,j}\langle\delta(r-|\bm{r}_{i}-\bm{r}_{j}|)\rangle$ (1)

where $N$, $n$, and $\bm{r}_{i}$ are number of atoms, number density, and the
atomic position of atom $i$, respectively. Sharp peaks are observed for the 1
K crystal and broadened peaks and valleys are observed for the 1 K glass
demonstrating short and medium range orders are still present in the
structure. For the liquid systems, peaks and valleys become broader with
increase in temperature and eventually become step-functions similar to hard-
sphere or dilute gases observed in the literature [25]. The effective
diameters of argon atoms decrease with increase in temperature as high kinetic
energy enables atoms to approach close to each other, overcoming more of the
repulsive portion of the potential, thus mimicking a dilute gas.

From the snapshot structures of all systems, we perform lattice dynamics
calculations to obtain eigenmode frequencies ($\omega_{n}$) and eigenvectors
($\bm{e}_{n}$) from $\Gamma$ point dynamical matrices ($\bm{D}$) as

$\omega^{2}_{n}\cdot\bm{e}_{n}=\bm{D}\cdot\bm{e}_{n}$ (2)

where subscript $n$ denotes a particular mode from 1 to $3N$ ($N=1372$ for our
systems). Dynamical matrices are related to the potential ($U$) by

$D_{\alpha\beta,ij}=\frac{1}{\sqrt{m_{i}m_{j}}}\frac{\partial^{2}U_{ij}}{\partial
u_{\alpha,i}\partial u_{\beta,j}}$ (3)

where $m_{i}$ is the mass of atom $i$ and Greek subscripts denote Cartesian
directions. There are no phase factors in Eq. 3 as the wavevector under
consideration is zero and there are no lattice vectors for our non-periodic
systems. All eigenvectors are real and are orthonormal such that
$\bm{e}_{n}^{T}\cdot\bm{e}_{n^{\prime}}=\delta_{n,n^{\prime}}$. As dynamical
matrices are Hermitian, all eigenvalues ($\omega_{n}^{2}$) are real, while
eigenfrequencies ($\omega_{n}$) can be real or imaginary. We denote imaginary
frequencies as negative values in figures below. Overall distributions of
available eigenmode populations known as the density of states $g(\omega)$,
can then be found by

$g(\omega)=\sum_{n=1}^{3N}\delta(\omega-\omega_{n})$ (4)

Using the mode eigenvectors, we can further characterize each mode by its
inverse participation ratio (IPR) and phase quotient (PQ). IPRs ($p_{n}^{-1}$)
are found by [26]

$p_{n}^{-1}=\sum_{i}\Big{(}\sum_{\alpha}e_{i\alpha,n}^{*}e_{i\alpha,n}\Big{)}^{2}$
(5)

where $e_{i\alpha,n}$ is the eigenvector component for atom $i$ in direction
$\alpha$ for the mode $n$. IPR gives a measure of how many atoms participate
appreciably in the motion of a particular mode; therefore, IPR has been used
to qualitatively provide information about the degree of localization of modes
for various complex systems [27, 28] and even scattering behaviors of acoustic
and optic phonons in binary compound crystals [29]. The IPR is conventionally
interpreted such that $1/N$ describes all atoms participating in the mode
(delocalized) and 1 if the mode motion is completely localized to a single
atom.

Phase quotients have also been useful in characterizing normal modes in
various materials, especially disordered solids [30, 31, 32] and is defined as
[30]

$PQ_{n}=\frac{\sum_{i,j\in
m}\sum_{\alpha}e_{i\alpha,n}e_{j\alpha,n}}{\sum_{i,j\in
m}\Big{|}\sum_{\alpha}e_{i\alpha,n}e_{j\alpha,n}\Big{|}}$ (6)

where the sum over $m$ represents counting only the nearest neighbor pairs
among atoms $i$ and $j$. PQ is a measure of the phase relationship of an atom
and its neighbors for a given mode. A PQ value of 1 then represents cluster of
atoms all moving in-phase, which is the case for Goldstone translational
modes, and a PQ value of -1 means that all atoms participating in the mode are
moving in the opposite phase/direction with their nearest neighbors. For our
systems, we define nearest neighbors to be within $d_{c}=2r_{c}$ where
$\frac{4}{3}\pi r_{c}^{3}=\frac{V}{N}$; however, we find that our results are
not sensitive to the choice of $d_{c}$.

##  III Results

###  III.1 Normal modes in solids

Density of states. – Eigenmode densities of states (DOS) for all systems are
shown in Fig. 2. Modes with real and imaginary frequencies are plotted on the
positive and negative sides of the horizontal axis, respectively. For solids,
all modes have real frequencies as shown in Fig. 2A. Crystalline DOS computed
from integration of wavevector points using the primitive unit cell (one atom
per unit cell with primitive lattice vectors:
$a_{1}=(0.5,0.5,0),a_{2}=(0,0.5,0.5),$ and $a_{3}=(0.5,0,0.5)$) are also shown
as reference. Sharp kinks in the densities of states for crystals are referred
to as van Hove singularities where $\frac{d\omega}{dk}=0$, mostly near zone-
boundaries [33]. Crystalline argon has three acoustic branches with
frequencies ranging up to 2 THz. There are no optical branches due to the
primitive unit cell having only one atom. For the rest of the work, we utilize
supercell argon crystal normal modes for analysis for more appropriate
comparisons with the other non-periodic systems of the same size. For the
glass system, there are no clear Brilliouin zone-boundaries nor translational
symmetry; thus, it has a much broader density of states, though still has
three Goldstone modes with zero frequency. The frequency ranges up to 3 THz,
larger than that of crystalline argon likely due to some bonds being shorter
than the mean equilibrium bond lengths.

Inverse participation ratio. – Crystalline modes are expected to be spatially
delocalized whereas glasses are expected to have some localized modes due to
structural disorder. Inverse participation ratios for both are shown in Fig.
3A and give a measure of the degree of localization for each mode. In the
literature, a normal mode has been considered to be localized when less than
10 to 20% of atoms are participating in the mode motion [31, 34, 35]. For our
purposes here, modes with IPR values over IPRloc = 0.007 (corresponding to 10%
atom participation) are considered to be localized. At 0 THz, we have the
Goldstone modes with IPR = $1/N$ for both solids where all atoms are
participating in the purely translational motion. As expected, all modes for
crystalline argon have IPR values less than IPRloc, demonstrating that all
modes are delocalized. For our glass system, most low frequency modes are also
delocalized with some exceptions known as quasi-localized or resonant modes.
Prior works have proposed that some localization at low frequencies is due to
finite size effects [36]. At high frequencies, we observe a significant number
of modes with IPR values greater than IPRloc, similar to that observed
previously in another single element glass (amorphous silicon) [31]. These
localized vibrational modes, often referred to as locons in the glass
literature, form a small fraction of the total number of modes, about 2 to 3%.
Locons typically have low thermal diffusivities and conductivities compared to
extended modes [31, 37, 38]. Maximum IPR values found in our glass system are
around 0.3.

Phase quotient. – Modal phase quotients (PQ) for our solid argon systems that
characterize relative mode motion of an atom with its neighbors are shown in
Fig. 4A (PQ=1 for fully in-phase and PQ=-1 for fully out-of-phase). For the
crystal system, some modes, including the 0 THz Goldstone modes, have PQ
$\sim$ 1 at low frequencies demonstrating that atoms are moving nearly in
phase with their neighbors. With increase in frequency, PQ decreases and
eventually appears to fluctuate between PQ = 0 and PQ = -0.5 and approaches PQ
$\sim$ -1. While crystalline argon does not have optical modes (typical out-
of-phase vibrations in compound crystals), away from 0 THz acoustic modes
develop phase differences among neighboring atoms which underlies the decrease
in PQ values. For the argon glass, we observe a monotonic decrease in PQ
values with increase in frequency away from 0 THz Goldstone modes that have PQ
= 1, similar to other glasses reported in the literature [31, 32]. The lowest
PQ values found in the glass system are around -0.9, close to the fully out-
of-phase value of PQ = -1. Prior works utilizing PQs in glasses have proposed
that modes with PQ $>$ 0 and PQ $<$ 0 represent acoustic-like and optical-like
modes, respectively [31, 32]. Using this notation, an acoustic to optical like
mode transition occurs around 1.2 THz for argon glass.

With this normal mode analysis for solids as background context, we next
characterize normal modes in the same way for various liquid and gas systems
at higher temperatures.

###  III.2 Normal modes in liquids and gases

Density of states. – Normal mode densities of states for all liquid and gas
systems are shown in Fig. 2B. Similar to the solids, densities of states are
scaled such that the area under the curve is 1. Only the low frequency region
$|\omega|$ $<$ 10 THz is shown for ease of comparisons. Perhaps, the most
notable difference in the normal mode densities of states of liquids and gases
compared to those of solids is the significant presence of imaginary modes. We
see a clear temperature dependence: the imaginary part of the spectra becomes
more prominent with increase in temperature as measured by the area. However,
as noted previously [39], the rate at which this happens slows down at high
temperatures as demonstrated in Fig. 2B. We have previously proposed that
equal number of imaginary modes and real modes marks a transition to a gas-
like phase [39]. With increase in temperature, also the frequency ranges
increase significantly up to 104 THz and down to -2500 THz ($i\omega$) for the
108 K gas but with very small probability.

Following Zwanzig’s approach to relate self-diffusion coefficients to normal
modes [40], some prior works have tried to express self-diffusion coefficient
with the fractions of imaginary modes and real modes over all modes [10, 9,
41, 19, 12]. Some other works have tried to relate normal modes in non-solids
to their specific heats [42, 39] and glass transitions [17, 18]. Extrapolating
the physical interpretation of real modes in solids to non-solids, it is often
assumed that for real modes, liquid atoms vibrate as harmonic oscillators
within a local potential well determined from the cage of nearest neighbor
atoms [19, 42, 14]. However, physical interpretation of both real and
imaginary modes in liquids is not clear and motivated our study.

We aim below to characterize the high temperature limit of eigenmodes of
dynamical matrices ($T\to\infty$) under the same density as those of solids
such that interpretation of eigenmodes of liquids may potentially become an
interpolation from both $T=0$ and $T\to\infty$ eigenmodes rather than an
extrapolation only from the $T=0$ crystalline modes that we are familiar with.

Inverse participation ratio. – Similar to solids, we characterize eigenmodes
of liquids and gases via inverse participation ratios and phase quotients.
Inverse participation ratio spectra at select high temperatures ($10^{3}$ K,
$10^{5}$ K, and $10^{8}$ K) are shown in Fig. 3B-D. At $10^{3}$ K, we see an
asymmetric U shape with high real and imaginary frequency modes being strongly
localized and low frequency modes being more extended. This IPR behavior
mimics the glass spectral features, although the liquid system has a large
component of imaginary modes. Similar trends have been previously shown in
other liquid systems [16]. In the limit of high frequency, we seem to approach
IPR = 0.5. With increase in temperature, low frequency modes have higher IPR
values, filling up the empty region in the U shapes found at low temperatures.
For our highest temperature system, we have a cross-like shape with the
vertical limits dictated by the lowest IPR ($1/N$) from the pure translational
modes and highest IPR approaching $\sim$ 0.9 and with high frequency modes
having IPRs exactly equal to 0.5. At $10^{8}$ K, nearly all modes (over 98% of
the mode populations) are ‘localized’ in the conventional interpretation
having IPR $>$ IPRloc.

Phase quotient. – Phase quotients for liquids and gases are shown in Fig.
4B-D. For all systems, the phase quotient spectra appear nearly symmetric for
real and imaginary modes with the exception of the long tail in the real
frequency spectrum. A monotonic decrease in PQs is apparent moving away from 0
THz, similar to the glass system. One notable trend observed for liquids and
gases vs. solids is the increasing presence of modes with PQ = -1 with
increase in temperature. In both metrics (IPR and PQ), we do not observe
distinct differences between real mode and imaginary mode characters.

Examining the gas modes more closely, we find some modes with exact values of
IPR = 0.5 and/or PQ = -1, modes that are absent in the solid systems.
Bivariate probability distributions of the IPRs and PQs for argon at select
temperatures are plotted in Fig. 5 and demonstrate a strong trend: going from
solids to gases, modes with both IPR = 0.5 and PQ = -1 become the dominant
population. In the limit that $T\to\infty$, we imagine that this trend
continues as atoms become effectively smaller and this character will govern
the normal mode spectra. As our interest lies in using normal mode analysis to
understand liquids in terms of higher T gas phases, we aim to fully understand
these modes.

IPR = 0.5 & PQ = -1 modes. – IPR of 0.5 indicates that two atoms are
participating in the mode and PQ = -1 describes eigenvectors among neighboring
atoms that are fully out-of-phase. From the simulations we have identified
pairs of atoms that govern modes with IPR = 0.5 and PQ = -1 and examined the
roles of these atoms in all modes of the system. As a representative example,
Fig. 6A gives ($x$-direction) eigenvector components for such a pair (atom 1:
blue and atom 2: red) for all modes at $10^{8}$ K. This pair of atoms
contributes negligibly to most modes (near zero amplitudes) as nearly all
modes are localized to other atoms (large IPR values). However, for modes with
finite and sometimes large $x$-components (see inset) for this pair, the
eigenmotion of the two atoms are the same (overlapping components), while
three modes have identical amplitudes in opposite direction. Thus, of the
multitudes of other mode motions, their relative motions are defined by three
modes with opposite directions (IPR = 0.5 and PQ = -1). $y$ and $z$
eigenvector components have similar features (not shown to avoid redundancy).

Interestingly, the three modes that govern the motion of the pairs of atoms
with IPR = 0.5 and PQ = -1 can be decomposed into one real mode and two
imaginary modes. Visualization of these three modes projected on a pair of
atoms is representatively shown in the inset of Fig. 6B. The real modes have
eigenvectors along the ‘bond’ (green arrows), while atomic motion for the
other two imaginary modes are orthogonal to the bond. In essence, the
repulsive force along the bond directs the atoms to a more stable position
while there is no energy loss in the motion of the other two modes, similar to
rotational modes.

In bulk dense solids, each atom sits in an effective potential that is the
summation of all neighboring interatomic potential interactions. However, for
the pairs of atoms dominating the eigenmotions for IPR = 0.5 and PQ = -1
modes, the behavior is simulating independent two-atom motion governed by a
single pair potential ($U_{i,j}(r)$) that defines the mode frequencies, while
other interactions are negligible. This leads to three Goldstone modes (two
atoms moving in-phase for three Cartesian degrees of freedom) and three out-
of-phase modes with opposite eigenvector directions with one being along the
bond. Mode frequency along the bond is strictly determined by
$\frac{\partial^{2}U_{i,j}(r)}{\partial r^{2}}$. Along-the-bond mode
frequencies with IPR = 0.5 and PQ = -1 as a function of distance between the
atomic pairs are shown in Fig. 6B for both liquid and gas systems. The nearly
perfect power law behavior demonstrates that these mode frequencies are purely
determined by the second derivative of the pair potential, similar to truly
independent two-atom interactions. We also observe that there is a temperature
dependence in the bond distance and mode frequencies. There is a large
distribution of bond lengths and frequencies at high temperature while the
distribution is smaller for low temperatures. While dynamical matrices are
built based purely on the potential of a static structure, kinetic energies
play a dominant role in determining the atomic locations that feed into the
dynamical matrices, especially at high temperatures (liquids and gases). The
larger the kinetic energies that atoms have, the closer they can approach each
other, leading to higher frequencies for these modes.

Collisons. – Normal modes are typically used to describe phonon quasi-
particles or harmonic oscillators in solids. Even in gases, there can be
internal degrees of freedom for non-monoatomic species that lead to vibrations
(e.g., H2 and O2). However, the results described above show that along-the-
bond modes with IPR = 0.5 and PQ = -1 should be considered as collisional
rather than vibrational. Even with absence of kinetic energy, if pairs of
argon atoms are initially placed at distances shown in Fig. 6B and their
motions are allowed to evolve with time, their potential energy is positive
($U_{LJ}(r<3.4$ Å$)>0)$ such that they dissociate after participating in the
along-the-bond mode motion with kinetic energies determined by energy
conservation. We, thereby, characterize these modes as two-body collisional
modes or ‘collisons’.

Translatons. – As evident from Fig. 3, the highest IPR values which are
located near 0 THz also depend on temperature. Traditional interpretation of
IPRs that IPR is a measure of the number of atoms participating in the mode
does not strictly work here as there are many IPRs between $\frac{1}{2}$ (2
atoms participating) and 1 (1 atom participating). However, we find that these
high IPR modes appear to have concentrated eigenvector amplitude on a single
atom as demonstrated in Fig. 7.

It is, therefore, expected that under different conditions there exist modes
with IPR approaching 1 where mode motion is projected to a single atom
translating through space with small frequency. This is consistent with what
we know about dilute gas eigenmodes where nearly all modes have zero
eigenfrequencies due to lack of potential interactions. As we progressively
reach lower densities, atomic interactions will become less frequent and more
uninterrupted translational motion will appear, which would have $\sim$ zero
eigenfrequency. In the same token, in the limit of $T\to\infty$ where atoms
become point particle-like and overall collisions become less frequent, we
would expect modes that purely describe translational motion, or what we may
characterize as ‘translatons’, to appear. We have done 1012 K IPR calculations
(not shown) and found that IPR approaches as high as 0.97 vs. 0.85 observed in
our 108 K system.

##  IV Concluding remarks

Our results on the normal mode analysis on heat capacities [39], pair
distribution functions, and mode characters by IPRs and PQs, suggest that in
the limit of $T\to\infty$, particles will effectively behave as point-
particles and two-body collisional modes dominate the particle dynamics, even
when the volume is kept constant at that of a solid. This is consistent with
real gas collisions where two-atom collisions become statistically dominant
with increase in temperature over many-body collisions [43]. In glasses,
eigenmodes are vibrational within a local harmonic well and have been
categorized into propagons, diffusons, and locons depending on the degree of
localization [44, 31]. Although there exist different ways to categorize these
modes [34, 37, 45], they often rely on the normal mode analysis, i.e.,
eigenvector characters. Similarly, we categorize eigenmodes at $T\to\infty$
into collisons and translatons from their eigenvector characters and their
local atomic environment.

With these insights into the behavior of high temperature (gas) eigenmodes of
dynamical matrices coupled with the vast body of research regarding T = 0
eigenmodes (solid), we propose that eigenmodes in liquids may be best
considered as a collection of both solid-like vibrational and gas-like
collisional and translational modes as summarized in Fig. 8. While the
probability of having two-body collisional modes decreases with decreasing
temperature, we still observe these modes even in the lowest temperature
liquid argon considered here. In real liquid systems at lower densities, where
potential interactions are weaker, we expect that normal mode spectra will
still consist of some gas-like modes. In this work, a simple Lennard-Jones
pair potential was used throughout. We anticipate that our high temperature
limit results will not depend on the choice of interatomic potential (two-
body, many-body, etc) as all systems will become an atomic gas where details
of the nature of the interactions become less important.

In this work, we explored the high temperature limits of eigenmodes of
dynamical matrices through the lens of inverse participation ratios and phase
quotients in the context of $T\sim 0$ K eigenmodes. In crystals and glasses,
normal modes are considered to be vibrational in nature with most being
delocalized in space. With increase in temperature we find that an increasing
number of eigenmodes become gas-like collisional and translational modes. Our
work, therefore, sets the scale of spectra of different eigenmodes such that
normal modes of liquids can be described in terms of evolving characters,
rather than extrapolation from our understanding of normal modes in solids
alone as often done in literature.

## ACKNOWLEDGEMENT

Authors thank Simon Thébaud for helpful discussions on the nature of normal
modes. This research was supported by the U.S. Department of Energy, Office of
Science, Basic Energy Sciences, Materials Sciences and Engineering Division.
This work used the Extreme Science and Engineering Discovery Environment
(XSEDE) Expanse under Allocation No. TG-MAT200012. This research used
resources of the National Energy Research Scientific Computing Center (NERSC),
a U.S. Department of Energy Office of Science User Facility located at
Lawrence Berkeley National Laboratory, operated under Contract No. DE-
AC02-05CH11231 using NERSC award BES-ERCAPERCAP0020503.

## Data Availability

All data are included in the text.

Figure 1: Pair distribution functions of various argon phases from 1 K to 108
K under constant volume. 1 K crystal (yellow curve) has very sharp peaks, some
exceeding $g(r)=30$ (not shown). Broadened peaks and valleys are shown for 1 K
glass (black curve) demonstrating short and medium range order. From liquid to
gas, progressive disappearance of peaks and valleys are observed with increase
in temperature. Temperature ranges are from 103 (blue curve) to 108 K (red
curve) with increments of a factor of 10.

Figure 2: Densities of states of normal modes for (A) 1 K solids and (B) high
temperature liquids and gases from 103 K (blue shade) to 108 K (red shade)
with increments of a factor of 10. Negative frequencies denote imaginary
modes. We observe more dominant real modes at low temperatures and increasing
importance of imaginary mode populations as measured by the area with increase
in temperature. Density of states for glass argon is broadened (black shade)
and has a wider frequency range compared to that of crystal argon (orange
curve).

Figure 3: Inverse participation ratios (measure of localization of modes) for
(A) 1 K solids (yellow circles for crystal and black circles for glass) and
(B-D) select high temperature liquids and gases at 103 K (blue circles), 105 K
(purple circles), and 108 K (red circles). IPR = $1/N$ corresponds to fully
delocalized (all atoms involved), IPR = 1 corresponds to localized to a single
atom. Negative frequencies denote imaginary modes. Dashed black line is at IPR
= 0.5 (two-body modes) and solid brown line represent IPR = IPRloc. Modes with
IPR $>$ IPRloc are considered to be localized. Inset plot in (D) x-axis is
scaled the same as (B) for comparisons.

Figure 4: Phase quotients (measure of motion of atoms compared to neighbors)
for (A) 1 K solids (yellow circles for crystal and black circles for glass)
and (B-D) select high temperature liquids and gases at 103 K (blue circles),
105 K (purple circles), and 108 K (red circles). PQ = 1 corresponds to in-
phase, while PQ = -1 corresponds to out-of-phase. Negative frequency denotes
imaginary modes. Inset plot in (D) x-axis is scaled the same as (B) for
comparisons.

Figure 5: Bivariate (inverse participation ratio and phase quotient)
probability distributions (A) 1 K solids (yellow for crystal and black for
glass) and (B-D) select high temperature liquids and gases at 103 K (blue),
105 K (purple), and 108 K (red). We observe a peak appearing at IPR = 0.5 and
PQ = -1 with increase in temperature. Figure 6: (A) Eigenvector components
($x$-direction for a pair of atoms identified to have IPR = 0.5 and PQ = -1 )
for all modes at 108 K. Blue and red colors represent atom 1 and atom 2 in the
pair. Inset shows a zoomed-in view near 0 THz. All eigenvectors are in-phase
between the two atoms except three which are fully out-of-phase. All pairs of
atoms with these features are identified and their along-the-bond mode (green
arrows in inset) frequencies and their distances are shown in (B). Clear
relations to the LJ pair potential (black dashed line) is demonstrated.

Figure 7: Eigenvector maps of all atoms for (A) -0.3 THz mode with IPR = 0.82
and (B) 0.1 THz mode with IPR = 0.74, among the highest IPRs for $10^{8}$ K
systems. Eigenvector amplitudes are multiplied by a factor of 10 for easier
visualization, same as Fig. 6B. We observe that for these modes, eigenvector
magnitudes are concentrated to a single atom. Figure 8: Spectrum of eigenmodes
of dynamical matrices from $T=0$ K to $T\to\infty$ under constant volume. At
$T=0$ limit, we have vibrational modes (phonons, propagons, diffusons, and
locons depending on disorder and localization). At the high temperature limit,
both real and imaginary modes are expected to be gas-like, which we
characterize as collisons and translatons. Eigenmodes for finite, intermediate
temperature structures could potentially be described through these two
limits.

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