[figure]style=plain,subcapbesideposition=top

# Dissecting the experimental vibrational density of states of liquids using
instantaneous normal mode theory

Sha Jin  School of Physics and Astronomy, Shanghai Jiao Tong University,
Shanghai 200240, China  Wilczek Quantum Center, Shanghai Jiao Tong University,
Shanghai 200240, China  Xue Fan  Shanghai National Center for Applied
Mathematics, Shanghai Jiao Tong University, Shanghai 200240, China  Materials
Genome Institute, Shanghai University, Shanghai 200444, China  Caleb Stamper
Institute for Superconducting and Innovative Materials, University of
Wollongong, Wollongong, NSW 2500, Australia  The Australian Nuclear Science
and Technology Organisation, Lucas Heights, NSW 2232, Australia  Richard A.
Mole  Institute for Superconducting and Innovative Materials, University of
Wollongong, Wollongong, NSW 2500, Australia  Yuanxi Yu  School of Physics and
Astronomy, Shanghai Jiao Tong University, Shanghai 200240, China  Liang Hong
School of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai
200240, China  Shanghai National Center for Applied Mathematics, Shanghai Jiao
Tong University, Shanghai 200240, China  Institute of Natural
Sciences,Shanghai Jiao Tong University, Shanghai 200240, China  Shanghai
Artificial Intelligence Laboratory, Shanghai 200232, China  Dehong Yu  Matteo
Baggioli  School of Physics and Astronomy, Shanghai Jiao Tong University,
Shanghai 200240, China  Wilczek Quantum Center, Shanghai Jiao Tong University,
Shanghai 200240, China

###### Abstract

Liquid dynamics play crucial roles in chemical and physical processes, ranging
from biological systems to engineering applications. Yet, the vibrational
properties of liquids are poorly understood when compared to the more familiar
case of crystalline solids. Here, we report experimental neutron-scattering
measurements of the vibrational density of states (VDOS) of water and liquid
Fomblin in a wide range of temperatures. In the liquid phase, we observe a
universal low-energy linear scaling of the experimental VDOS as a function of
the frequency, which persists at all temperatures. Importantly, in both
systems, we observe that the slope of this linear behavior grows with
temperature. We confirm this experimental behavior using molecular dynamics
simulations, and we explain it using instantaneous normal mode (INM) theory,
whose predictions are in good agreement with the experimental and simulation
data. Finally, we experimentally observe a sharp crossover at the melting
point of water, below which the standard Debye’s law is recovered. On the
contrary, in Fomblin, we observe a gradual and continuous crossover indicating
its glassy dynamics, and proving that the low-frequency power-law scaling of
the VDOS is a good probe for the nature of the solid-to-liquid transition. Our
results experimentally confirm the validity of INM theory and the success of a
normal mode approach to liquid dynamics, which could pave the way towards a
deeper understanding of liquids and their properties in general.

## Introduction

The vibrational density of states (VDOS) is a fundamental concept in solid
state physics, and plays a key role in determining vibrational, thermodynamic
and transport properties of a given material (e.g., heat capacity, thermal
conductivity, superconductivity, etc.). In crystalline solids with long-range
order, the definition and determination of the VDOS can be achieved using a
standard normal mode analysis, which ultimately leads to the construction of
the Debye model [1]. The latter has been very successful in describing the
VDOS, $g(\omega)$, of crystalline solids, and in particular its low frequency
regime, which displays a universal quadratic scaling
$g(\omega)\propto\omega^{2}$, known as Debye’s law.

On the other hand, the dynamics of liquids are profoundly different from
solids as they do not display any translational order, and are more complex
than gases due to their high density and strong particle interactions.
Insisting on a phonon-like harmonic description of liquids, based on normal
mode analysis, appears at first sight totally unjustified, and immediately
incurs several fundamental difficulties [2]. In solids, atomic motion is
entirely characterized by stable oscillations around well-defined potential
minima, which are well approximated by a quadratic function of the
coordinates. In liquids, or more in general disordered systems, there are many
minima contributing to the thermodynamics, together with negative curvature
regions and saddles connecting those minima (see Fig.1 for a cartoon). In the
end, these hopping processes across energy barriers are the responsible for
macroscopic diffusion, and cannot be neglected.

After acknowledging these facts, one possibility is to abandon the normal mode
route, and simply rely on the computation of the velocity auto-correlation
function [3], which can be easily obtained using molecular dynamics
simulations. However, its connection to actual vibrational modes is unclear. A
better approach is to still calculate the VDOS using the Hessian matrix, by
extending the concept of normal modes in solids to instantaneous normal modes
(INMs) (see [4] for a review on the topic), and proceed with a normal-mode
analysis for the liquid state [5, 6]. The main idea is that, for short time-
scales, a liquid is indeed not so different from a solid, or that in more
technical words, beyond the hydrodynamic regime, solids and liquids are alike
(e.g., they both support propagating shear waves [7, 8]). Maxwell [9]
suggested that the time-scale separating the liquid-like from the solid-like
regime corresponds to the stress relaxation time, derived from linear
viscoelasticity. As a concrete manifestation of that idea, Zwanzig [10] showed
that a normal mode analysis for liquids is still meaningful at short time-
scales, i.e., at each instantaneous snapshot.

In short, INMs are the eigenvalues of the force constant matrix at an instant
of time. Rhaman and collaborators [11] early realized that the diagonalization
of such a matrix in disordered systems would give rise not only to positive
eigenvalues, but also to negative ones which correspond to purely imaginary
frequencies, $\lambda_{\alpha}\equiv\omega_{\alpha}^{2}<0$, and which are
labelled as unstable INMs. The INMs density of states can be therefore split
into a stable part $g_{s}(\omega)$, and an unstable one $g_{u}(\omega)$. In
order to understand this distinction, it is illustrative to think of a liquid
as a collection of relatively stable local minima, around which the dynamics
is harmonic and solid-like, accompanied by structural relaxation in the form
of barrier crossing to neighbor wells, with a certain hopping frequency [12,
13] (see Fig.1).

Figure 1: Schematic illustration of water molecules hopping across the
potential energy landscape. Regions with negative local curvature correspond
to unstable modes, while minima with positive curvature correspond to solid-
like stable modes performing a quasi-harmonic motion. The color-map indicates
the local value of the potential energy.

The imaginary frequency modes relate to this relaxational dynamics and, from a
potential landscape picture, they correspond to visiting regions of the
potential with locally negative curvature. In simpler words, unstable modes
are a measure of fluidity, as suggested by several authors [14, 15]. It then
comes as no surprise that unstable INMs bear a close relation to diffusion
[16, 17, 18, 19, 20, 21, 22] and several other properties of liquids [23, 24,
25, 26, 27, 28].

From a computational perspective, the INM density of states has been
investigated in several works using molecular dynamics simulations (e.g.,
water[29, 30, 31, 32], CS2[33], glass-forming liquids [34, 35], and even
proteins[36]). On the contrary, the experimental investigation of the
vibrational density of states of liquids at low energies is much less explored
due to the fact that many techniques used to study liquid vibrational
properties, such as infrared, Raman and nonlinear IR, do not have sufficient
energy resolution to probe the details at very low energies. Along with the
development of modern cold neutron spectrometers and intense inelastic X-ray
spectrometers based on synchrotron radiation facilities, this field has
recently become more active. Recent experimental results for the low energy
vibrational properties, using inelastic neutron scattering (INS), on several
liquid systems including water, liquid metal and polymer liquids has been
reported in [37] (see also [38] and [39] for previous studies using neutrons).

Interestingly, the VDOS of liquids displays a certain degree of universality,
which is reminiscent of the Debye law mentioned above. In particular, both in
simulations and in recent experiments, the VDOS is observed to exhibit a
universal linear scaling law, $g(\omega)\propto\omega$, in the low-energy
limit. Hence, one can distinguish liquids and solids simply by looking at
their VDOS at small frequency, and at the corresponding power law. From a
theoretical perspective, several explanations for this universal linear law in
liquids have been proposed in the past, based on different degrees of
simplification [18, 40, 41, 42, 43, 44, 45, 46, 47, 48]. In one way or
another, all of them attribute this novel scaling to the presence of unstable
modes. Notice that such a linear relation is valid also for the stable branch
of normal modes, casting serious doubts about the phonon theory of liquid
thermodynamics [49] in which the Debye formalism is assumed.

Within the framework of INM theory, Keyes [18] used a simplified theoretical
framework to predict the whole frequency behavior of the INM density of
states, which in first approximation is given by (see [48] for details):

$g(\omega)=a(T)\omega\exp\left[-\left(\frac{a_{2}(T)\omega}{\sqrt{T}}\right)^{a_{3}(T)}\right]\,.$
(1)

In the formula above, $a(T)$ represents the slope at low-frequencies.
Additionally, the $a_{2}$ parameter is expected to be approximately constant,
while the $a_{3}$ parameter is a measure of the strength of the energy barrier
fluctuations. $a_{3}=2$ corresponds to Arrhenius behavior, $a_{3}>2$ to super-
Arrhenius, and $a_{3}=4$ to the Zwanzig-Bassler dependence
$\exp(-E^{4}/T^{2})$ [50, 51]. The above theoretical prediction is in good
agreement with the data from simulations [18, 52], and in the low-frequency
regime reduces to the aforementioned linear law:

$g(\omega)=a(T)\omega+\dots$ (2)

Following the analogy with the Debye model for solids, where the coefficient
in front of the quadratic scaling is simply given by the speed of sound $v$,
$g_{\text{Debye}}(\omega)\propto\omega^{2}v^{-3}$, one could ask whether the
linear pre-factor $a(T)$ in liquids also exhibits some sort of universality.
Is its temperature dependence universal? And, which of the liquid properties
determine it?

At least from simulation data, the temperature dependence of the linear
coefficient $a(T)$ is not universal, and it does not depend on a simple
transport coefficient (as for the Debye’s coefficient in solids), but rather
on the details of the topology of the potential landscape and the associate
complex thermodynamic structure. In the literature, one can find simulation
systems in which such a coefficient decreases with temperature, and systems in
which it increases. In this respect, the emblematic examples are Lennard Jones
liquid [18] for the former, and CS2 for the latter [33]. There are no
available experimental data to ascertain if that is the case also for
realistic liquids. As we will see, at least for the two experimental systems
considered in this work, the linear slope $a(T)$ always grows with
temperature.

From the theoretical side, in a series of works [18, 33] (see [48] for a
review), Keyes and collaborators derived an analytical expression for the
temperature dependence of the linear slope which, after some simplifications,
is given by

$a(T)\propto\left[f_{u}^{\text{max}}-f_{u}(T)\right]\,e^{-\langle
E\rangle/k_{B}T}\,.$ (3)

Here, $f_{u}(T)$ is the fraction of unstable modes, which is simply defined as
the number of unstable modes divided by the total number of INM.
$f_{u}^{\text{max}}$ is a parameter which depends on the topology of the
potential landscape, and it roughly corresponds to the maximum value for
$f_{u}$, hence it is expected to be smaller than unity. Finally, $\langle
E\rangle$ is the average barrier height in the potential landscape, where
fluctuations have been neglected. All in all, the temperature dependence of
the linear coefficient is mostly determined by the number of unstable modes as
a function of temperature, and by the Boltzmann factor which takes into
account the thermally activated hopping across the barrier, and which is
directly connected to the hopping frequency and the diffusion constant.
Importantly, $\left[f_{u}^{\text{max}}-f_{u}(T)\right]$ decreases with
temperature since the fraction of unstable modes grows with it. On the
contrary, the Boltzmann term $e^{-\langle E\rangle/k_{B}T}$ grows with
temperature. Thus, using Eq.(3) one can in principle rationalize both
dependencies of $a(T)$ with respect to temperature, depending on which of the
two terms dominate. In the LJ case, such a growing behavior was not observed
probably because of the uniquely small barriers which render the Boltzmann
term negligible. Since in general $f_{u}$ grows as a power-law with $T$, we
expect the linear slope to grow with temperature exponentially and the second
term to dominate. As we will see, that is the case for the experimental liquid
systems considered in this manuscript.

The linear slope in Eq.(2) has been confirmed using Inelastic Neutron
Scattering (INS) experiments in [37] but only at one specific temperature.
Eq.(3) has never been confronted with real experimental data. Here, we perform
a full scan of the experimental VDOS of water and Fomblin oil in a wide range
of temperatures, below and above the melting transition, and with particular
emphasis in the liquid phase. Our goal is to study the frequency and
temperature dependences of the low-energy experimental VDOS and to explain
those features using theoretical models, more specifically, instantaneous
normal mode theory. Additionally, we will investigate how much information
about the relaxational dynamics, the topology of the potential energy
landscape, and the collective vibrational modes are contained in the low-
frequency regime of the experimental VDOS. Based on experimental data and
simulations, we will test the validity of INM theory and reveal several
universal features of liquids which were disclosed before only for some
specific simulation models.

## Experimental results

Using inelastic neutron scattering (INS), we measured the vibrational density
of states (VDOS) of bulk water and Fomblin oil
$CF_{3}O[-CF(CF_{3})CF_{2}O-]_{x}(-CF_{2}O-)_{y}CF_{3}$ at different
temperatures. In Fig.2a, we present the inelastic neutron scattering data of
the bulk water for temperatures ranging from $250$ K to $360$ K. For water, in
the energy range up to $150$ meV, we observe two major peaks in the VDOS
spectrum. As shown in Fig.2a, the first peak appears around $\approx 6.5$ meV.
This peak corresponds to the hydrogen-bond bending, perpendicular to the
hydrogen bond (O-H-O) [53]. The large peak at around $65$ meV is attributed to
the librational motion due to intermolecular coupling [54]. In the region
between $20-35$ meV, the VDOS is characterized by a flat band, which
represents the weak hydrogen bond stretching modes in line with the hydrogen
bond [55]. Additionally, we observe that the librational band shifts toward
lower energies the higher the temperature of the liquid water is.

As shown in the inset of Fig.2a, the VDOS exhibits a universal linear in
frequency behavior at small frequencies above the melting temperature, $T>273$
K, i.e., in the liquid phase. This observation confirms the validity of the
universal linear scaling $g(\omega)\propto\omega$ in the whole liquid phase,
independently of the temperature, and it expands the results of [37]. Below
the melting temperature, in the solid phase, the low-energy VDOS is
drastically modified. First, the zero frequency value disappears, as a
confirmation that the self-diffusion constant vanishes in the solid phase.
Second, the low-frequency scaling is suddenly modified to recover the standard
quadratic Debye law, $g(\omega)\propto\omega^{2}$. For temperatures below
$273$ K, the system is in the solid ice phase and the linewidth of the peak
corresponding to the hydrogen-bond bending is significantly smaller. Moreover,
the amplitude of the two major peaks is larger than in the liquid phase. As
shown in the inset of Fig.2a, upon the phase transition from solid to liquid,
the well-defined peak around $6.5$ meV becomes significantly broadened with
much high-intensity signal shifted to low energies. Around $330$ K, in the
liquid phase, the lowest peak $\approx 6.5$ meV becomes completely overdamped
and the VDOS becomes flat up to the next broadened and red-shifted peak at
$\approx 65$ meV. This tells us that the dynamics become very unstable with a
wide distribution in frequency. This observation already fits into the INM
picture in which many unstable modes with a wide frequency range dominate the
relaxation processes in the liquid state.

In Fig.2b, we show the VDOS measured by INS for the Fomblin oil
$CF_{3}O[-CF(CF_{3})CF_{2}O-]_{x}(-CF_{2}O-)_{y}CF_{3}$. The considered sample
has (on average) x $\approx$ 8, y $\approx$ 28\. For the Fomblin oil, the
lowest band in the VDOS has been suggested to correspond to the low-frequency
modes caused by the torsion of fluoromethyl ($O-CF_{3}$) groups at the end of
the chain, and C-C torsions of the chain. In the range of $25-45$ meV, the
deformational in-plane modes ($\delta[O-C-C],\delta[O-C-F]$ and
$\delta[C-O-C]$) represent the dominant contribution to the VDOS spectra[56].
Importantly, as shown in the inset of Fig.2b, the VDOS of Fomblin at low
frequency is also universally linear above $200$ K. Finally, let us anticipate
that for both liquids the slope of the linear region grows by increasing the
temperature as displayed in the insets of Fig.2a  and Fig.2b. We will return
to this point later in the manuscript.

Figure 2: The experimental vibrational density of states (VDOS), measured by
INS for different temperatures for (a) water (b) and Fomblin oil. The VDOS
curves have been normalized by the total area. The insets zoom on the low
frequency region and illustrate the linear scaling $g(\omega)\propto\omega$.
In the left inset, the sharp change between the linear behavior to the
quadratic solid-like one (blue curves) is evident.

The experimental data for water and Fomblin presented in Fig.2 are taken at
different temperatures, and go beyond the solid-liquid phase transition for
the two systems. Now, we are interested to understand how the universal linear
scaling in the liquid phase is modified by reducing the temperature and
entering into the solid phase. In order to analyze this feature, we fit the
low-frequency regime of the experimental VDOS with the following expression:

$g(\omega)=a(T)\,\omega^{b(T)}+c(T)\,.$ (4)

The constant in frequency parameter $c(T)$ relates to the self-diffusion
constant in the liquid phase and vanishes in solids. More precisely, when the
density of states is properly normalized, we have

$g(0)\equiv c(T)=\frac{2m\,D}{k_{B}T},$ (5)

where $D$ is the self-diffusion constant, which grows with temperature
(usually following Arrhenius law). Here, we are mostly interested in the
power-law $b(T)$. In the liquid phase, as already discussed, we always find
that $b(T)=1$. On the contrary, in a crystalline solid with long-range order,
we have $b(T)=2$, as predicted by Debye’s law. By simple extrapolation, we
then expect such a power-law to interpolate between these two values by
decreasing the temperature.

In Fig.3, we show the behavior of the power-law $b(T)$ as a function of the
temperature for water and Fomblin. For water, we observe a sharp jump of the
power-law between the liquid value $1$ to the solid value $2$ at around $273$
K, which coincides exactly with the solidification temperature. The behavior
is very drastic and it reflects the first-order nature of the liquid-solid
phase transition in water. The case of Fomblin, Fig.3b, is more interesting.
There, we observe a deviation from the liquid-like scaling at $\approx 260$ K,
which is a much higher temperature than the reported pour temperature for
Fomblin, $238$ K. Differently from the water-ice case, Fig.3a, the transition
to the Debye scaling in Fomblin is not sharp, but rather continuous. At least
close to this “critical” temperature, our data follow a power law behavior
$b(T)-1\propto\left(T^{*}-T\right)^{0.648}$. This behavior is reminscent of
the dynamics of the order parameter across a continuous (second-order) phase
transition, e.g., Curie’s law for magnetic materials, and it deserves further
investigation. Moreover, it correlates with the different structure changes of
the two systems. As shown in SI Fig.7, water (using D2O, as H2O does not give
diffraction peaks due to the dominant incoherent neutron scattering cross
sections) has a first order phase transition from a liquid state at $286$ K,
represented by a broad peak in the structure factor, to a crystallized
structure at $260$ K with well defined sharp peaks in the structure factor. In
contrast, in the structure factor of Fomblin, there is no sharp peak but a
similar broad feature appears across the whole temperature range covered. This
indicates that the Fomblin structure changes gradually as a function of
temperature from a liquid state (low viscosity) to a short range ordered or
glassy state (high viscosity), at least in the temperature range studied here.

Figure 3: The power-law $b(T)$ of the low-frequency experimental VDOS as a
function of the temperature for (a) liquid water and (b) Fomblin oil. The data
are extracted by fitting the experimental VDOS with Eq.(4). The horizontal
gray dashed lines indicate the liquid-like power, $b(T)=1$. The red dashed
line is the fit to $b(T)\propto\left(T^{*}-T\right)^{0.648}$, where
$T^{*}=260$ K.

## Instantaneous normal mode analysis

In order to confirm our experimental results, we resort to the instantaneous
normal mode (INM) analysis. Previous normal mode analysis for supercooled
water can be found in [32, 30, 31, 29]. Unfortunately, for Fomblin we have not
been able to carry out an INM analysis.

In our INM study, we carried out a $1$ ns molecular dynamics simulation of the
TIP3P model for liquid water. The derivation of the INM spectrum is performed
by diagonalizing the dynamical matrix for $100$ different liquid
configurations generated at $5$ ps intervals during the last $500$ ps of the
full simulation using GROMACS. The resulting INM density of states after
average is shown in Fig.4a. We have followed the standard notation and plotted
the imaginary frequencies, corresponding to the unstable modes, on the
negative frequency axis. As evident from Fig.4a, both the stable and unstable
parts display a clear linear-in-frequency behavior at low frequency. Moreover,
the slope is the same for the two parts [48]. Mathematically, it is easy to
show that both spectra are identical at low-frequency and start to differ only
for larger frequencies. The density of states for stable INMs can be
decomposed into two regimes. The region below $50$ meV involves mostly modes
related to the translational motion. On the contrary, the region above $50$
meV is governed by rotational modes [32]. By increasing temperature, the
intensity of the spectrum of imaginary frequencies increases, and so does the
number of unstable modes. Additionally, despite both parts of the INM VDOS
displaying a clear peak at low frequency, the one for stable modes appear to
be insensitive to the temperature $T$, while the one for the unstable modes
grows in intensity with temperature.

In Fig.4b, we show the comparison of the experimental VDOS for water at $310$
K and the VDOS of stable INMs at the same temperature. The two curves have
been normalized to the first peak. As evident from Fig.4b, there is a crucial
difference between the two curves since the INM density of states does not
contain the diffusion component, and therefore $g(0)=0$, while the
experimental curve clear shows a finite value at zero frequency. As explained
above, the zero frequency value is given by the self-diffusion coefficient $D$
and decreases with temperature. In the SI, we show a more in depth comparison
between the two curves at different temperatures. Since $g(0)$ diminishes with
temperature, the two curves resemble each other more near the solid-phase.
Indeed, in a solid, we expect the two curves to be identical (see [26] for a
similar observation). Moving towards finite frequency, we observe that in the
low-energy regime, below approximately $20$ meV, the two curves are similar to
each other. Both of them show a clear linear-in-frequency regime, with
approximately the same slope, and then a sharp peak with a similar linewidth.
At larger frequencies, the two curves differ considerably and the INM density
of states is not able anymore to capture the salient features of the
experimental data.

Figure 4: The instantaneous normal mode (INM) analysis for liquid water. (a)
The INM density of states (plotted with the imaginary frequency branch on the
negative $\omega$ axis) for different temperatures. (b) Experimental data for
the VDOS obtained by INS (red), and INM density of states (blue) at $310$ K.
(c) The fraction of unstable modes $f_{u}(T)$ as a function of temperature.
The dashed line indicates the result of a linear fit. (d) The slope of the INM
DOS $a(T)$ as a function of the temperature. The lines indicate the fits to
INM theory, Eq.(3) and its simplified version given by the sole Boltzmann
term. The background colored regions illustrate the uncertainties of the
numerical data.

A crucial quantity in the INM analysis is given by the fraction of unstable
modes $f_{u}$, which is defined as the ratio of the number of unstable modes
to the total 3N modes. Fig.4c  displays the fraction of unstable modes $f_{u}$
as a function of temperature. As expected, the fraction increases with
increasing temperature and for the case of water exhibits an evident linear in
$T$ dependence. Interestingly, this functional form coincides exactly with the
low-temperature prediction of the random energy model [47] (e.g., Eq.13.33 in
[48]). To analyze further the INM VDOS, we fit the low-frequency region using
Eq.(1). We find that the theoretical formula, derived in [18], provides an
excellent fit for the numerical data, at least below $\approx 13$ meV. Details
about this result are presented in the SI. More precisely, we find that the
parameter $a_{2}(T)$ is a number very close to unity and it slowly increases
with temperature, following an approximate linear scaling. On the contrary,
the parameter $a_{3}(T)$ ranges between $2.58$ close to the melting
temperature, and up to $1.95$ for the highest temperature measured. For the
simulated LJ liquid [18, 33, 52], $a_{2}$ was found to be independent of
temperature, while $a_{3}(T)$ interpolated between $4$ and $2$, respectively
at low and high temperature. On the contrary, for CS2 [33], $a_{3}(T)$ has
been found to be not too far from $2$ for a wide range of temperatures,
decreasing a small amount with increasing temperature. Here, for water, we
find that $a_{2}\approx 1$ is not independent of temperature, but slowly
increasing with $T$, and $a_{3}$ is close to $2$ and decreasing with
temperature, similarly to the case of CS2. This analysis (see SI for details)
confirms the validity of the simplified Arrhenius-like formula in Eq.(3)
(which coincides exactly with $a_{3}=2$), which will be used in the rest of
the manuscript. The low temperature value of $a_{3}$ appears to be slightly
greater than $2$ and it suggests potential super-Arrhenius dynamics for water
close to the freezing temperature.

Let us now focus on the the linear slope $a(T)$, which is plotted in Fig.4d
as a function of the temperature. First, and most importantly, the behavior of
the linear slope is consistent with the result from the experiments, as it
increases monotonically with the temperature. In order to provide a more
quantitative understanding of the slope, we resort to an INM analysis and to
Eq.(3) presented above. As shown in Fig.4d, the theoretical formula is in
perfect agreement with data from simulations. Moreover, we find that the
temperature dependence is dominated by the Boltzmann term in Eq.(3), and the
pre-factor depending on the fraction of unstable modes contributes only in a
minor way and can be therefore neglected. We find that the value of the
average barrier height extracted from the numerical fit is given by:

$0.54k_{B}T<\langle E\rangle<0.74k_{B}T\,,$ (6)

depending whether the fit is performed with the full Eq.(3) or just with the
reduced form including only the Boltzmann factor. Here, and in the rest of the
manuscript, when using $k_{B}T$ as unit of energy, we always take room
temperature as a reference, i.e., $k_{B}T\approx 25.7$ meV. In any case, we
find that the obtained value is very reasonable and in good agreement with the
analysis performed in [32] (see Fig.9 therein). Notice that a value $\langle
E\rangle=k_{b}T/2$ would correspond to a perfectly harmonic motion. Therefore,
the obtained value represents a clear indication of the anharmonicity of the
potential surface, consistent with large red-shifts in the VDOS versus
temperature.

## Matching experiments with theory

We return to the experimental data for the VDOS of water and Fomblin displayed
in Fig.2. Here, we are mostly concerned with the temperature dependence of the
linear slope which has been observed to be universal in the whole liquid
phase. The data, obtained by fitting the experimental results with Eq.(4), are
shown in Fig.5. For Fomblin, the slope is extracted by fitting the
experimental data from INS normalized by the total area. This normalization is
justified by the fact that around $\approx 120$ meV, where our numerical data
stops, the spectral weight is already very small (see Fig.2b), and the
remaining tail is negligible. For water (see Fig.2a), this is not the case, as
around $\approx 140$ meV, the VDOS is still large. As a consequence, a
normalization of the VDOS by the area of the curves up to that experimental
cutoff would lead to uncontrollable results. Therefore, for water, we have
normalized all the curves by their value at zero frequency, using the
experimental and simulation data for the self-diffusion constant $D(T)$ [57],
which is known precisely at all temperatures, together with the theoretical
expression, Eq.(5). More details are presented in the Methods and the
Supplementary Information.

Figure 5: The temperature dependence of the linear slope $a(T)$ for the
experimental VDOS of water (a) and Fomblin (b). The fits are the predictions
from instantaneous normal mode theory, Eq.(3). For water, the two lines
overlap with each other demonstrating the dominance of the Boltzmann term. The
background shaded regions indicate the uncertainties of the experimental data.

As already anticipated, for both systems, the slope of the experimental VDOS
increases monotonically with temperature. Additionally, we find that the
formula Eq.(3), derived from INM theory [18, 33], provides an excellent fit
for both liquids. As for the simulated data reported in the previous section,
we observed that the Boltzmann term in Eq.(3) constitutes the dominant
contribution to the temperature dependence of the slope $a(T)$. From the fits
of the experimental data, we obtain the value for the average barrier height
as:

$\text{water:}\quad\langle
E\rangle=2.44k_{B}T\,,\qquad\text{Fomblin:}\quad\langle
E\rangle=0.97k_{B}T\,.$ (7)

Let us discuss in detail these results. In the case of water, from the
experimental data, we obtain an average barrier which is $\approx 3-4$ times
larger than the one obtained by fitting the INM VDOS. This is not surprising,
since the INM analysis cannot capture the highly-energetic libration and
bending modes of water. This is evident from the comparison between the INM
VDOS and the experimental VDOS provided in Fig.4b. Above $\approx 20$ meV, the
INM VDOS quickly decreases while the experimental one displays a large portion
of the spectral weight all the way up to $120$ meV. In other words, the INM
analysis cannot capture those highly energetic barriers, and therefore it
predicts a much lower value for $\langle E\rangle$. It is nevertheless
important to observe that the predictions from experiments and INM are about
the same order of magnitude. Interestingly, the value obtained experimentally
for water is not far from the activation energy for the hydrogen bond, which
is approximately $4k_{B}T$ [58].

Moving to Fomblin, we find that the average barrier height is approximately
half of that of water. In order to understand this difference, we need to
emphasize that the kinematic viscosity of Fomblin $\nu$ is extremely high,
$\approx 276$ cSt at $20^{\circ}$ C, which is almost $300$ times larger than
the value for water. Assuming that the hopping rate between the different
minima in the potential is well approximated by the inverse of the Maxwell
time, $\tau_{M}\propto\nu$, this implies an extremely long average jump time.
In other words, we do expect that the dynamics of Fomblin are much more
localized around the local minima of the potential where the vibrational
dynamics are more solid-like and, hence, harmonic. These more harmonic
dynamics explain a smaller value for $\langle E\rangle$ and the difference
with the larger value for water.

## Conclusions

In this work, we studied the vibrational density of states of liquid water and
Fomblin oil, combining experimental neutron scattering techniques (INS), MD
simulations and instantaneous normal mode theory. Our focus is the low-energy
regime of the VDOS and in particular (I) its scaling with frequency, and (II)
its temperature dependence. In such a regime, and independently of the value
of the temperature (from the melting temperature to $\approx 340$ K), we have
experimentally verified that both liquids display a universal linear in
frequency scaling $g(\omega)=a(T)\omega+\dots$, which was previously
experimentally observed for only one value of temperature in [37], and derived
from INM theory in [40].

By analyzing the temperature dependence of the linear slope $a(T)$, we
observed that $a(T)$ grows monotonically with temperature over a wide range of
temperatures in the liquid phase. In order to rationalize this behavior, we
have resorted to INM theory [18]. Our INM simulations show a good qualitative
agreement with the experimental data and predict the same temperature
dependence of the slope values as was observed in the INS experiments. More
importantly, we have verified the theoretical framework established by Keyes
and collaborators [18, 33, 48] and proved experimentally that the temperature
dependence of the linear slope is dominated by a Boltzmann factor
$\exp(-\langle E\rangle/k_{B}T)$, where $\langle E\rangle$ is the average
barrier height in the potential landscape.

Finally, we have tracked the power-law scaling of the low-frequency portion of
the experimental VDOS across the melting temperature. For water, we have
observed a sharp transition between the liquid-like linear scaling to the
quadratic Debye law, which appears around the expected melting temperature. On
the contrary, for Fomblin, the transition from the liquid behavior to the
Debye’s law is smooth and continuous, as expected from its polymeric glassy
structure. These findings imply that the low-frequency temperature dependent
behavior of the experimental VDOS is able not only to capture the solid-liquid
transition but also its nature, whether a sharp first-order thermodynamic
transition or a continuous glassy-like transition.

Let us emphasize that our results, and in general a deeper understanding of
the liquid VDOS and the pre-factor in its linear in frequency scaling, could
also be of great benefit for revealing the origin of the boson peak excess in
supercooled liquids and glasses [59], where non-phononic quasi-localized
vibrations might directly appear from a reconstruction of a liquid-like DOS
[60, 61]. Not surprisingly, an intimate relation between the VDOS of liquids
and the anharmonic soft potential model for glasses has already observed in
the past [41].

In conclusion, we have shown that the low-energy regime of the experimental
VDOS of liquids contain a great amount of information about their potential
energy landscape, their relaxational dynamics and even the nature of their
transition to the solid phase at low temperature. We have proved that a normal
mode analysis, based on the concept of INMs, is able to predict both the
frequency scaling of the experimental VDOS and also the temperature dependence
of its linear coefficient, providing a theoretical framework comparable with
Debye’s theory for solids. We expect our findings to have a pivotal role in
establishing the usefulness and validity of INM theory to understand the
vibrational and thermodynamic properties of real liquids, beyond the
simulation playground. We hope that further effort in this direction, guided
by our analysis, could finally provide long-sought answers to to these
questions.

## Methods

Samples.
Two samples have been studied with INS. General-purpose laboratory grade of
deionized water with a mass of $0.5$ g is filled inside an annular aluminum
can having $0.1$ mm gap. A similar can with $0.5$ mm gap is filled with $4$ g
of Fomblin oil (YL $25/6$, Fomblin® Y LVAC $25/6$ Solvay) [62];
CF3O[-CF(CF3)CF2O-]x(-CF2O-)yCF3. The considered sample has (on average) x
$\approx$ 8, y $\approx$ 28\. These aluminum cans have been specially designed
with high strength aluminum to maintain the integrity of the sample can under
high vapor pressure of water at high temperatures. A half dozen of sample cans
have been destroyed during the test. The estimated transmission is $90\%$ for
these samples and this ensures the $10\%$ level of scattering to minimize
multiple scatterings.

Inelastic neutron Scattering.
The inelastic neutron scattering (INS) experiments are performed on the time-
of-flight cold-neutron spectrometer, PELICAN, at Australian Nuclear Science
and Technology Organisation (ANSTO)[63]. Incident neutron wavelength of $4.69$
Å, ($3.72$ meV), is used and this gives an energy resolution of $0.135$ meV at
the elastic line. A cryo-furnace capable of delivering temperature from $1.5$
K to $800$K is used for sample temperature control. The corresponding empty
can is measured in the same conditions as for the samples for background
subtraction. A standard vanadium sample is also measured for detector
efficiency normalization and energy resolution determination. The data
reduction, including background subtraction and detector normalization are
carried out using the Large Array Manipulation Program (LAMP)[64]. The VDOS is
determined through the experimental dynamic scattering function $S(k,\omega)$
which is determined from the energy gain side of the scattering process.
According to the incoherent one-phonon approximation, the VDOS for a Bravais
powder sample (isotropic system) is given by [65]:

$\displaystyle
g(\omega)\,=\,C\int\frac{\hslash\omega}{{k}^{2}}\,S(k,\omega)\,\left(1-e^{-\frac{\hslash\omega}{k_{B}T}}\right)dk,$
(8)

where $C$ is a factor containing the atomic mass and Debye-Waller factor,
$\exp(-2W)$, which is taken as unity for all samples here, $k_{B}$ is the
Boltzmann constant. The integration over $k$ covers the experiment-accessible
range of momentum transfer. No correction on multiple scatterings has been
considered.

For a non-Bravais sample, we measure a neutron-weighted phonon density of
states:

$g_{NW}(\omega)=\sum_{i}f_{i}\frac{\sigma_{i}}{M_{i}}g_{i}(\omega),$ (9)

where the sum over $i$ includes all elements in the sample. $f_{i}$, $M_{i}$,
$\sigma_{i}$ and $g_{i}(\omega)$ are the concentration, atomic mass, total
neutron bound cross section and partial VDOS of the element $i$, respectively.
For water, the VDOS is dominated by hydrogen as the $\sigma/M$ ratio for H
($82$) is much larger than $O$ ($0.26$). For the Fomblin sample, the fluorine
is the main contributor because of its large concentration as compared to
other elements.

Molecular dynamics (MD) simulations.
The MD simulations were performed on GROMACS[66] to simulate liquid water with
the TIP3P[67] interaction potential at different temperatures. We used $392$
flexible water molecules in our liquid system which was equilibrated in NVT
and NPT ensembles, using the Nosé-Hoover thermostat and Parrinello-Rahman
barostat to control the temperature and pressure. We carried a $1$ ns
production MD simulation for data collection at atmospheric pressure and
constant temperature. During the whole simulation, the periodic boundary
conditions were employed. Using the Leapfrog-Verlet algorithm, the time step
was set to $0.2$ fs. The long-range electric interactions were calculated by
using the particle mesh Ewald (PME) method. The self-diffusion coefficient of
the system can be determined through the mean squared displacement (MSD) via
the Einstein relation:

$\displaystyle
D=\frac{1}{6}\frac{d}{dt}\left<\left|\mathbf{r}_{i}(t)-\mathbf{r}_{i}(0)\right|^{2}\right>,$
(10)

where $\mathbb{r}_{i}(t)$ is the position vector of the ith particle at time
t.

Instantaneous normal mode analysis.
The instantaneous normal modes are calculated by diagonalization of the
dynamical Hessian matrix. For each liquid configuration, the Hessian matrix is
a $3N\times 3N$ matrix, evaluating the second derivatives of the potential
energy. Its elements are constructed as follows:

$\displaystyle
H_{i\mu,j\nu}(\mathbf{R})=\frac{1}{\sqrt{m_{i}m_{j}}}\frac{\partial^{2}V}{\partial
r_{i,\mu}\partial r_{j,\nu}},$ (11)

where $i,j=1,...,N,\mu,\nu=x,y,z$.
$\mathbf{R}\equiv{\mathbf{r}_{1},...,\mathbf{r}_{N}}$ represents each liquid
configuration and $\mathbb{r}_{i}$ is the position of the ith atom. V is the
potential energy and $r_{i,\mu}$ represents the $\mu$-coordinate of the ith
atom. The instantaneous normal mode frequencies $\omega_{i}$ are the square
roots of the eigenvalues of the dynamical matrix. The INM spectrum is obtained
by:

$\displaystyle\left<\rho(\omega)\right>=\left<\frac{1}{3N}\sum_{i}^{3N}\delta(\omega_{i}-\omega)\right>.$
(12)

At each temperature, we performed instantaneous normal mode analysis for $100$
different liquid configurations generated at $5$ ps intervals during the last
$500$ ps of the full simulation. The INM spectrum was averaged over the
analyzed liquid configurations.

## Data availability

The datasets generated and analysed during the current study are available
upon reasonable request by contacting the corresponding authors.

## Code availability

The code that supports the findings of this study is available upon reasonable
request by contacting the corresponding authors.

## Acknowledgements

We would like to thank H. Xu, J. Douglas, Y. Feng and especially T. Keyes for
fruitful discussions and related collaborations on the topic of liquids. We
are grateful to Tom Keyes for comments and suggestions on a preliminary
version of this manuscript. M.B. acknowledges the support of the Shanghai
Municipal Science and Technology Major Project (Grant No.2019SHZDZX01) and the
sponsorship from the Yangyang Development Fund. D. Y., C. S. and R. M.
acknowledge the beam time awarded from ANSTO for the access to Pelican
instrument (P13964).

## Author contributions

D.Y., C.S. and R.M. performed the experimental measurements; M.B., D.Y and
L.H. conceived the idea of this work; S.J, X.F. implemented the MD simulations
and the INM analysis; S.J., X.F., Y.Y. and C.S. performed the analysis of the
experimental and simulation data; M.B. and S.J. wrote the manuscript with the
help of D.Y. and L.H.

## Competing interests

The authors declare that no competing interests exist.

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## Supplementary information

### Fitting the experimental data

The linear fit of the experimental VDOS of Fomblin oil, used to extract the
slope presented in the main text, is plotted in Fig.6a. The VDOS curve is
normalized by the total area under the curve. On the other hand, Fig.6b
presents the power law fit of the low frequency VDOS of Fomblin. Since the
value of the power extracted from the fit is not affected by the normalization
of the data, the VDOS data for $T<200$ K are also shown but they are not
normalized to the total area because the experimental data stops around $20$
meV.

For the linear fit of the low-frequency experimental VDOS of liquid water
($T\geq 273$ K), the normalization by the area of the curves could lead to
uncontrollable results due to the large weight of the spectral at the
experimental data cutoff, $140$ meV (see Fig.2a  in the main text). In order
to normalize the VDOS properly, we resorted to a different method and used the
value of the VDOS at zero frequency, which is theoretically given by the self-
diffusion constant (see Eq.(5) in the main text). Since the exact zero
frequency value of the VDOS is not experimentally accessible, first, we fitted
the low-frequency VDOS of liquid water using

$g(\omega)=a(T)\,\omega+\tilde{c}(T),$ (13)

and extracted the constant term $\tilde{c}(T)$. By multiplying the
experimental VDOS with the ratio $c(T)/\tilde{c}(T)$, where $c(T)$ is given in
terms of the self-diffusion constant $D$ as in Eq.(5), we have normalized all
the curves by their value at zero frequency. The self-diffusion coefficient is
obtained from the MD simulation, as presented in the Methods. The linear fit
of the low-frequency normalized VDOS of liquid water is plotted in Fig.6c.
Finally, Fig.6d  presents the power law fit of the low frequency VDOS of
water. For $T<273$ K, the VDOS is normalized by the area as the self-diffusion
constant vanishes in solids.

Figure 6: Fits of the experimental data. The dashed lines indicate the result
of the fits. (a) The linear fit of the area normalized experimental VDOS of
Fomblin oil. (b) The power law fit of the experimental VDOS of Fomblin oil.
For $T\geq 200$ K, the VDOS is normalized by the area. For $T<200$ K, the data
is not normalized because the experimental data stops around $20$meV. (c) The
linear fit of the normalized experimental VDOS of liquid water ($T\geq 273$
K). (d) The power law fit of the experimental VDOS of liquid water. For $T\geq
273$ K, the VDOS is normalized by the value at zero frequency. For $T<273$ K,
the VDOS is normalized by the area.

### Verification of the crystalline, liquid and glassy phases

In Fig.7, we show the measured structure factor of Fomblin and water
respectively. For Fomblin, the change of structure factors in a wide
temperature range is shown in Fig.7a. For water, the structure factor of D2O
is presented, as H2O does not give clear diffraction peaks due to the
dominating incoherent neutron scattering cross sections. Fig.7b  shows a clear
first order phase transition from a crystallized structure at $260$ K, with
well defined sharp peaks in the structure factor, to a liquid state at $286$
K, with a broad peak in the structure factor. In contrast, the structure
factor of Fomblin does not exhibit any sharp peak for all the temperatures
considered, but displays a similar broad peak shifting toward lower $k$ with
increasing temperature. The continuous behavior of the structure factor of
Fomblin, upon moving to the low-temperature liquid state, verifies that
fomblin is not a is not a crystalline solid but it rather exhibits a short-
range glassy structure with high viscosity at low temperature. This is
consistent with the drastically different behavior of the low-frequency power-
law of the VDOS reported for the two systems in Fig.3 in the main text.

Figure 7: The experimental structure factor $S(k)$ of Fomblin oil (a) and D2O
(b).

### MD simulations and INM analysis

In Fig.8a, we show a snapshot of the structure of the bulk water sample used
in the MD simulation. The INM DOS was fitted by Eq.(4) for further analysis.
Fig.8b  displays the fit of the INM DOS at $290$ K using Eq.(4). The dashed
line in the figure shows the fit up to $13$ meV. Eq.(4) provides an excellent
fit for the numerical data. Fig.8c  and Fig.8d  show the fitting parameters
$a_{2}(T)$ and $a_{3}(T)$ as a function of the temperature. Different fitting
regions have been considered. The values and the evolution of fitting
parameters did not differ much. We have found that imposing the fitting cutoff
at $\approx 13$ meV minimizes the fitting error. Using such a procedure, we
find that $1.23<a_{2}(T)<1.55$ and grows approximately linearly with
temperature. On the contrary, $1.95<a_{3}(T)<2.58$ and decreases monotonically
with temperature. More discussion about the physical meaning of these outcomes
can be found in the main text.

Figure 8: The fit of the INM DOS using eq.(1). (a) A snapshot of the structure
of the bulk water sample used in the MD simulation. (b) The fit of the INM
spectrum at $290$ K using Eq.(1). The dashed line marks the fit up to 13 meV.
(c) The fitting parameter $a_{2}(T)$ in Eq.(1) as a function of temperature.
(d) The fitting parameter $a_{3}(T)$ in Eq.(1) versus temperature. The
background colored region illustrates the uncertainties of the numerical data.

### Experimental VDOS versus INM DOS

In Fig.9, we show the comparison of the experimental VDOS for water and the
VDOS of stable INM at different temperatures. The comparisons at $280$ K,
$290$ K, $300$ K and $320$ K are shown respectively in Fig.9a, Fig.9b, Fig.9c
and Fig.9d. The two DOS curves have been normalized to the first peak. The two
curves are quite similar in the low frequency region. As explained before,
there is a difference between the two DOS, as the INM DOS does not contain the
diffusion component while the experimental VDOS shows a finite value at zero
frequency. As expected, the two curves are closer to each other in the low
frequency region for smaller temperatures. This is simply because $g(0)$,
which is not captured by the normal mode analysis, diminishes with
temperature. Above approximately $20$ meV, the two curves differ and the INM
DOS shows much flatter and weaker bands, representing the stretching modes and
librational motion. For all the temperatures considered, the slope of the
linear low frequency regime is approximately the same in both VDOS curves,

Figure 9: The comparison between experimental DOS and INM DOS at (a) 280 K
(purple), (b) 290 K (blue), (c) 300 K (orange), (d) 320 K (red).