Orbital free ab initio molecular dynamics simulation study of some static and dynamic properties of liquid noble metals
Several static and dynamic properties of liquid Cu, Ag and Au at thermodynamic states near their respective melting points, have been evaluated by means of the orbital free ab-initio molecular dynamics simulation method. The calculated static structure shows good agreement with the available X-ray a...
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| Cite this: | Orbital free ab initio molecular dynamics simulation study of some static and dynamic properties of liquid noble metals / G.M. Bhuiyan, L.E. Gonz´lez, D.J. Gonz´lez // Condensed Matter Physics. — 2012. — Т. 15, № 3. — С. 33604:1-19. — Бібліогр.: 79 назв. — англ. |
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| citation_txt | Orbital free ab initio molecular dynamics simulation study of some static and dynamic properties of liquid noble metals / G.M. Bhuiyan, L.E. Gonz´lez, D.J. Gonz´lez // Condensed Matter Physics. — 2012. — Т. 15, № 3. — С. 33604:1-19. — Бібліогр.: 79 назв. — англ. |
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| description | Several static and dynamic properties of liquid Cu, Ag and Au at thermodynamic states near their respective melting points, have been evaluated by means of the orbital free ab-initio molecular dynamics simulation method. The calculated static structure shows good agreement with the available X-ray and neutron diffraction data. As for the dynamic properties, the calculated dynamic structure factors point to the existence of collective density excitations along with a positive dispersion for l-Cu and l-Ag. Several transport coefficients have been obtained which show a reasonable agreement with the available experimental data.
Оцiнено деякi статичнi та динамiчнi властивостi рiдких Cu, Ag i Au при термодинамiчних умовах, близьких до їх точок плавлення, за допомогою методу безорбiтальної першопринципної молекулярної динамiки.
Розрахована статична структура добре узгоджується з наявними даними, отриманими з рентгенiвської i нейтронної дифракцiй. Щодо динамiчних властивостей, то розрахованi динамiчнi структурнi фактори вказують на iснування колективних збуджень густини поряд з позитивною дисперсiєю для l–Cu i l–Ag.
Отримано деякi коефiцiєнти переносу, якi прийнятно узгоджуються з наявними експериментальними даними.
|
| first_indexed | 2025-12-07T18:10:02Z |
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Condensed Matter Physics, 2012, Vol. 15, No 3, 33604: 1–19
DOI: 10.5488/CMP.15.33604
http://www.icmp.lviv.ua/journal
Orbital free ab initio molecular dynamics simulation
study of some static and dynamic properties
of liquid noble metals
G.M. Bhuiyan1, L.E. González2, D.J. González2
1 Department of Theoretical Physics, University of Dhaka, Dhaka–1000, Bangladesh
2 Departamento de Física Teórica, Universidad de Valladolid, Valladolid, Spain
Received June 19, 2012, in final form July 31, 2012
Several static and dynamic properties of liquid Cu, Ag and Au at thermodynamic states near their respective
melting points, have been evaluated by means of the orbital free ab-initio molecular dynamics simulation
method. The calculated static structure shows good agreement with the available X-ray and neutron diffrac-
tion data. As for the dynamic properties, the calculated dynamic structure factors point to the existence of
collective density excitations along with a positive dispersion for l–Cu and l–Ag. Several transport coefficients
have been obtained which show a reasonable agreement with the available experimental data.
Key words: liquid noble metals, orbital free density functional theory, molecular dynamics simulations, static
structure, dynamic properties, transport coefficients
PACS: 61.25.Mv, 64.70.Fx, 71.15.Pd
1. Introduction
The d-electrons in the d-band metals are not so free as to justify a nearly free electron (NFE) approach
but, on the other hand, they are not so tightly bound as to be described by the tight bindingmethod (TBM)
or core electron theory. Indeed, the study of d -band metals poses difficult theoretical challenges although
some progress has been made towards their understanding, both in the solid and liquid phases [1–11].
From a theoretical point of view, accurate first principles electronic structure calculations of d-band
metals have been performed using the techniques such as the linearized augmented plane wave or the
linearized muffin-tin orbital (LMTO) methods [12]. Although it is possible to accurately calculate the in-
terionic forces within these schemes [13, 14], still the computational demand of such calculations has
so far prevented its use within the context of Molecular Dynamics (MD) simulations. As a consequence,
most realistic structural models for d -electron systems have been constructed by means of empirical or
semiempirical interatomic potentials [15–21].
In the particular case of the noble metals, the d -bands are completely filled but the sp-d hybridiza-
tion is still there [4, 22]. This sp-d hybridization effect can be accounted for by either changing the s, p, d
band occupancy number (in the case of ab-initio pseudopotential theory) or by using an effective valence
Z [23]. In this respect, it has already been found from the density functional based generalized pseu-
dopotential theory that the effective sp-electron valence lies [5–7] within the range 1.1 to 1.7, where this
non-integral number is mostly due to sp-d hybridization effects [5–7, 22, 23].
The structure of the liquid noble metals has been studied at several temperatures by Waseda [24]
using X-ray (XR) diffraction methods. Neutron diffraction has been also used in the case of Cu at two
temperatures [25] and Ag near melting [26]. As concerns their thermophysical properties, the situation
is different in the case of l–Cu and l–Ag on the one hand, and l–Au on the other hand. Two recent com-
pilations of thermophysical properties of liquid metals, due to Blairs [27] and Singh et al. [28], the latter
including several temperatures, analyze the previous experimental measurements of the adiabatic sound
© G.M. Bhuiyan, L.E. González, D.J. González, 2012 33604-1
http://dx.doi.org/10.5488/CMP.15.33604
http://www.icmp.lviv.ua/journal
G.M. Bhuiyan, L.E. González, D.J. González
velocity (cs), density (ρ), and specific heat at a constant pressure (CP) of the systems, and therefrom de-
duce several other magnitudes, such as the isothermal compressibility (κT) or the ratio of specific heats
at a constant pressure and at a constant volume (γ). Now, in the cases of l–Cu and l–Ag, several experi-
mental measurements were available, so an assessment was performed and recommended values were
given by the authors [27, 28]. On the contrary, only a single experiment is available to determine the
sound velocity of l–Au, within a wider study of the Au–Co alloy [29]. Therefore, one should consider that
the uncertainty in the thermophysical data of l–Au is larger that for l–Cu and l–Ag. Other transport prop-
erties of the liquid noble metals, such as self-diffusion coefficient (D), or shear viscosity (η) are readily
available [30, 31]. In particular, the self-diffusion coefficients of l–Cu over a wide temperature range, have
recently been determined by means of quasielastic neutron scattering measurements [32]. More specif-
ically, the experimental data were used to calculate the self intermediate scattering functions, Fs(q, t),
at several q-values, and the associated self-diffusion coefficients were evaluated from their decay rate at
small wavevectors.
Most theoretical studies on the liquid noble metals have focused on the static structural properties
and thermodynamic properties, usually characterizing the liquid system by effective interatomic poten-
tials constructed either empirically by fitting to some experimental data or derived from some approxi-
mate theoretical model. Therefrom, the liquid structure is determined by resorting to either liquid state
theories [33] or to classical molecular dynamics (CMD) simulations.
Holender et al. [34] have used the embedded atom model (EAM) to obtain some effective interatomic
potentials which were later on used in CMD simulations aimed at evaluating the static structure of liquid
noble metals near melting. Bogicevic et al. [35] used the effective medium theory to obtain a many-body
potential which, combined with CMD simulations, provided information on the static properties and the
self-diffusion coefficient of l–Au at different temperatures. Their calculated pair distribution function,
g (r ), near melting has the main peak which is somewhat lower than experiment and the subsequent
oscillations are slightly out of phase.
Alemany et al. [36–39] used both the EAM and TBM to derive many-body potentials which were used
in CMD simulations so as to obtain information on various static and dynamic properties of l–Cu, l–Ag
and l–Au. Their calculated static structure factors, S(q), showed a good agreement with experiment ex-
cept for a somewhat smaller height of the main peak. They also obtained reasonable estimates for the
self-diffusion coefficients excepting l–Cu which was clearly underestimated. A similar approach was used
by Han et al. [40] to evaluate the self-diffusion and shear viscosity coefficients in liquid and undercooled
Cu. We also note that other workers [21, 41, 42] have resorted to integral equation-type liquid state the-
ories which, combined with semiempirical interatomic potentials, have lead to reasonable estimates for
several static and thermodynamic properties of a range of 3d , 4d and 5d liquid transition metals.
In principle, an accurate approach to the study of the static and dynamic properties of the liquid no-
ble metals, would be provided by ab-initio molecular dynamics (AIMD) simulation methods, which have
become widespread in the last twenty years or so. Most AIMD methods are based on the density func-
tional theory (DFT) [43, 44] which permits to calculate the ground state electronic energy of a collection
of atoms, for given nuclear positions, as well as yields the forces on the nuclei via the Hellmann-Feynman
theorem. It enables one to perform MD simulations in which the nuclear positions evolve according to
classical mechanics whereas the electronic subsystem follows adiabatically. The Kohn-Sham (KS) orbital
representation of the DFT (KS-AIMD method) has been the usual approach when performing AIMD sim-
ulations although it is acknowledged that this approach imposes heavy computational demands which
limit the size of the systems as well as the simulation times. These limitations are enhanced in the case of
d -electron systems such as the noble and transition metals because a large number of electronic orbitals
are needed. Nevertheless, and despite the above shortcomings, a few AIMD studies have already been
performed on the liquid noble metals [45–49].
The first AIMD calculation of l–Cu was performed by Pasquarello et al. [45], who studied some static
properties near melting using ultrasoft pseudopotentials [50] combined with a plane-wave expansion for
the electronic orbitals. The simulation used 50 atoms, lasted for 2 ps and results were obtained for the
pair distribution function, the self-diffusion coefficient and the electronic density of states. More recently,
Mitrohkin [46] has performed AIMD simulations to analyze the melting process in Cu. The study used 62
atoms, lasted for 3 ps and produced results for some static properties and diffusion coefficient of l–Cu
near melting. Two further AIMD studies of Cu [47, 48] focused on the possible appearance of icosahedral
33604-2
OF-AIMD simulation study of the liquid noble metals
arrangements of atoms in liquid and undercooled Cu, with sample sizes between 100 and 200 particles
and equilibrium simulation times from 1 to 5 ps. Pasturel et al. [49], within a wider study of Au–Si al-
loys, also performed AIMD simulations of liquid and undercooled Au, using 256 atoms and equilibrium
runs 6 ps long, and obtained results for the temperature variation of the self-diffusion coefficient and
icosahedral atomic arrangements. However, none of these AIMD calculations produced results for the
dynamical properties, because its evaluation requires in general larger systems, and, in particular, sub-
stantially longer simulation times.
This goal can be achieved by resorting to the orbital free ab-initiomolecular dynamics (OF-AIMD) sim-
ulation method [51–57]. It is based on the Hohenberg and Kohn version of the DFT theory [43] where the
electronic orbitals are replaced by the total valence electron density which now becomes the basic vari-
able. This procedure greatly reduces the number of variables describing electronic states and, therefore,
it enables one to study larger samples (a few thousand atoms) and for longer simulation times (tens of
picoseconds). Now, the interaction among the positive ions and the valence electrons is characterized by
means of a local pseudopotential which plays an important role in determining the ground state energy
and the realistic forces acting on ions.
This paper reports an OF-AIMD study of the static and dynamic properties of the liquid noble metals
(Cu, Ag, Au) at thermodynamic conditions close to their respective melting points. The layout of the paper
is as follows. In section 2 we briefly describe the OF-AIMD method and provide some technical details.
We also describe the local ionic pseudopotentials used in this calculations. Section 3 reports and discusses
the results of the ab-initio simulations for several static and dynamic properties which, moreover, are
compared with the available experimental data. We conclude this paper in section 4.
2. Theory
A simple liquid metal is modelled as a disordered array of N bare ions with valence Z , enclosed in a
volume V , and interacting with Ne = N Z valence electrons through an electron-ion potential v(r ). The
total potential energy of the system can be written, within the Born-Oppenheimer approximation, as the
sum of the direct ion-ion coulombic interaction energy and the ground state energy of the electronic
system under the external potential created by the ions, Vext(~r , {~Rl }) =
∑N
i=1 v(|~r −~Ri |) ,
E ({~Rl })=
∑
i< j
Z 2
|~Ri −~R j |
+Eg[ng(~r ),Vext(~r , {~Rl })] , (1)
where ng(~r ) is the ground state valence electron density and ~Rl are the ionic positions. According to DFT,
the ground state valence electron density, ng(~r ), can be obtained by minimizing an energy functional
E [n], which can be written
E [n(~r )]= Ts[n]+EH [n]+Exc[n]+Eext[n] , (2)
where the terms represent, respectively, the electronic kinetic energy, Ts[n], of a non-interacting system
of density n(~r ), the classical electrostatic energy (Hartree term), the exchange-correlation energy, Exc[n],
for whichwe used the local density approximation and finally the electron-ion interaction energy, Eext[n],
where the electron-ion potential is characterized by a local ionic pseudopotential,
Eext[n] =
∫
d~r n(~r )Vext(~r ) . (3)
For Ts[n] we used an explicit, albeit approximate, functional of the valence electron density. Several
expressions were proposed and in the present calculations we used an average density model [55, 56],
which provided a good description for a range of liquid simple metals, namely Ts[n] = TW[n]+Tα[n],
where
TW[n(~r )] =
1
8
∫
d~r |∇n(~r )|2
/
n(~r ) (4)
33604-3
G.M. Bhuiyan, L.E. González, D.J. González
is the well-known von Weizsäcker term, and
Tα[n] =
3
10
∫
d~r [n(~r )]5/3−2α[k̃(~r )]2 ,
k̃(~r ) = (2k0
F)3
∫
d~s k(~s)wα
(
2k0
F |~r −~s|
)
, (5)
where k(~r ) = (3π2)1/3 [n(~r )]α, k0
F
is the Fermi wavevector for a mean electron density ne = Ne/V , and
wα(x) is a weight function chosen so that both the linear response theory and Thomas-Fermi limits are
correctly recovered. Further details are given in reference [55, 56].
Figure 1. Non-Coulombic part of the electron-ion interaction for liquid Cu, Ag and Au.
Another basic ingredient in the above formalism, is the local ionic pseudopotential, vps(r ), that de-
scribes the ion-electron interaction. The AIMD simulations based on KS-AIMD method usually employ
non-local pseudopotentials [58] obtained by fitting to some properties of the free atom [59–61]. However,
in the present OF-AIMD approach, the valence electron density is the basic variable, and non-local pseu-
dopotentials cannot be used. Therefore, the interaction among the valence electrons and the ions must be
described using a local pseudopotential which is usually chosen so as to include an accurate description
of the electronic structure in the physical state of interest. Bhuiyan et al. [57] developed a local pseudopo-
tential model which in conjunction with the OF-AIMD method has provided a good description of several
static and dynamic properties of l–Sn near melting [57]. Specifically, it is defined as
vps(r )=
{
A+B exp(−r /a), r < RC ,
−Z /r, r > RC ,
where A and B are contants, RC is a core radius and a is the softness parameter. Aiming to reduce the
number of free parameters, we impose the condition that the logarithmic derivatives of the potential
inside and outside the core are exactly the same at the core radius. This permits to eliminate B as a
parameter, and the successfulness of this approach can be judged by the capability of recovering the
available experimental data. The other parameters A, a and RC, and the effective valence Z , have been
chosen so that the OF-AIMD simulation reproduces the experimental static structure factor. The values
obtained herein are given in table 1, where we notice that for a given system the parameters remain
constant for both thermodynamic states. In figure 1 we depicted the non-Coulombic part of the ionic
pseudopotential for l–Cu, l–Ag and l–Au. It shows that in the long wavelength limit (q → 0), the value of
33604-4
OF-AIMD simulation study of the liquid noble metals
Table 1. Input parameters used in the calculations; temperature T , ionic number density ρ, amplitude in
the core A, softness parameter a, core radius RC and the effective ionic valence Z .
System T (K) ρ(Å−3) A (au) a (au) RC (au) Z
Cu 1423 0.0755 0.05 0.3 1.40 1.35
1773 0.0728 0.05 0.3 1.40 1.35
Ag 1273 0.0517 0.05 0.3 1.55 1.35
1673 0.0496 0.05 0.3 1.55 1.35
Au 1423 0.0525 0.05 0.2 1.50 1.35
1773 0.0517 0.05 0.2 1.50 1.35
vps(q) is the largest for l–Au and the smallest for l–Cu. Note also that the phase of oscillations is different
for each system.
We stress that the combination of the OF-AIMD method with local ionic pseudopotentials has already
provided accurate descriptions of several static and dynamic properties for a range of bulk liquid simple
metals and binary alloys [55, 56, 62–64].
3. Results and discussion
OF-AIMD simulations have been performed for l–Cu, l–Ag and l–Au at two thermodynamic states near
their respective triple points. Those states were chosen due to the availability of experimental XR diffrac-
tion data [24]. Table 1 gives additional information about thermodynamic states and other input param-
eters used for the simulation.
The simulations were carried out using 500 particles in a cubic cell with periodic boundary conditions
and whose size was appropriate for the corresponding experimental ionic number density. Given the
ionic positions at time t , the electronic energy functional is minimized with respect to n(~r ) represented
by a single effective orbital, ψ(~r ), defined as n(~r ) = [ψ(~r )]2. The orbital is expanded in plane waves which
are truncated at a cutoff energy, ECut = 20.0 Ryd. The energy minimization with respect to the Fourier
coefficients of the expansion is performed every ionic time step using a quenching method which results
in the ground state valence electron density and energy. The forces on the ions are obtained from the
electronic ground state via the Hellman-Feynman theorem, and the ionic positions and velocities are
updated by solving Newton’s equations, using the Verlet leapfrog algorithmwith a timestep of 6.0·10−3 ps.
Equilibration in the simulations lasted 10 ps. and the calculation of properties was made by averaging
over 150 ps.
In this study, we have evaluated several liquid static properties (pair distribution function and static
structure factor) as well as various dynamic properties, both single-particle ones (velocity autocorrelation
function, mean square displacement) and collective ones (intermediate scattering functions, dynamic
structure factors, longitudinal and transverse currents). The calculation of the time correlation functions
(CF) was performed by taking time origins every five time steps. Several CF have also a dependence on
the wave vectors ~q which depend only on q ≡ |~q| because our system is isotropic.
3.1. Static Properties
3.1.1. Liquid Cu
The OF-AIMD simulation permits to directly evaluate the static structure factor, S(q), and its real
space counterpart, i.e., the pair distribution function g (r ). Figure 2 (a) shows the calculated S(q) for
l–Cu at two different thermodynamic states characterized by temperatures T = 1423 and 1773 K. For both
states, the main peak is located at qp ≈ 2.88 Å−1. Comparison with the XR data [24] shows an overall good
agreement for both the positions and phases of the oscillations, although the present OF-AIMD results
slightly overestimate the height of the main peak. Note, however, that the height of the main peak in
33604-5
G.M. Bhuiyan, L.E. González, D.J. González
Figure 2. (a) Static structure factors and (b) pair correlation functions for l–Cu at two thermodynamic
states. Solid lines are the OF-AIMD results and the open circles stand for the XR diffraction data.
the neutron data of Eder et al. at 1393 K (not shown) is substantially higher than in the XR data, being
in better agreement with our results. A similar overestimation of the height of the main peak of S(q) in
l–Cu, as compared to XR measurements, was also obtained in CMD studies carried out using EAM-based
potentials [17, 38, 65]. The KS-AIMD of Ganesh andWidom [47] at 1398 K also yield a structure factor with
a height of the main peak similar to our data and to the neutron measurements. The agreement of our
high temperature results with experiment is the same, while in this case, the XR structure factor at 1773 K
and the corresponding neutron data at 1833 K agree better with each other than at lower temperatures.
The long wavelength limit of the static structure factor, S(q → 0), is linked with thermodynamics
through the relationship S(q → 0) = ρkB T κT where kB is Boltzmann’s constant and κT is isothermal
compressibility. A least squares fit of S(q) = s0 + s2q2 + s4q4 to the calculated S(q) for small q-values
yields an estimate κT,OF−AIMD = 0.90±0.03 (in units of 10−11 N−1 m2 ) for T = 1423 K, underestimating
the experimental value of 1.49 [27, 66], or 1.41 [28]. For T = 1773 K we find κT = 1.09±0.03, while the
experimental value is 1.74 (in the same units) [28].
The calculated pair distribution functions, g (r ), are depicted in figure 2 (b) along with the correspond-
ing XR data [24]. The main peak is located at rp = 2.53 Å and 2.55 Å for T = 1443 and 1773 K, respectively,
which agrees with the corresponding experimental data. A similar good agreement is found for the po-
sitions and the phase of oscillations of the subsequent peaks. The only noticeable discrepancy concerns
the height of the main peak which is slightly underestimated by the present calculations. Nevertheless,
we note that a similar disparity is also reported in KS-AIMD studies [45, 47, 48]. The average number of
nearest neighbors, also known as coordination number (CN), is obtained by integrating the radial distri-
bution function (RDF), 4πr 2ρg (r ), up to a distance rm which is usually identified as the position of the
first minimum in either the RDF or the g (r ) [67, 68]. Both choices often lead to rather similar results and
in what follows we report the results obtained by integrating up to the first minimum of the RDF which
was found at rm ≈ 3.42 and 3.44 Å for T = 1443 and 1773 K, leading to values CN ≈ 12.9 and 12.6, respec-
tively. For comparison, we note that the KS-AIMD studies at 1500 K produce CN ≈ 12.5 [45], 12.3 [47], and
12.9 [48] using a bit different integration limits.
3.1.2. Liquid Ag
The calculated S(q) for l–Ag at two different states are plotted in figure 3 (a) where they are com-
pared with the corresponding XR data [24]. The calculated position of the main peak are at qp = 2.57
and 2.59 Å−1 for T = 1273 K and 1673 K, respectively. For a lower temperature, T = 1273 K, we observe
that the calculated height of the main peak is a bit bigger than that of the XR data [24]; indeed, a similar
disparity has also been reported in other CMD studies for l–Ag [17, 38]. Note also that the neutron S(q)
of Bellisent et al. at 1323 K (not shown) has the height of the main peak of 2.85, which is much higher
than Waseda’s data, and is more in line with our result. On the other hand, the positions and phase of
33604-6
OF-AIMD simulation study of the liquid noble metals
Figure 3. (a) Static structure factors and (b) pair correlation functions for l–Ag at two thermodynamic
states. Solid lines are the OF-AIMD results and the open circles are the XR diffraction data.
oscillations of the subsequent peaks are found to be in very good agreement with experiment. We have
also calculated the isothermal compressibility of l–Ag at T = 1273 K and we have obtained κT = 1.94±0.08
(in units of 10−11 N−1 m2) to be compared with the experimental data of 2.11 [66], 1.92 [27], or 1.80 [28].
For T = 1673 K, we have obtained κT = 2.19±0.05, while experiment yields 2.21 [28].
The calculated pair correlation functions, g (r ), for l–Ag are depicted in figure 3 (b) for T = 1273 K and
1673 K where we observe a good agreement with the respective XR data [24]. Integrating up to the first
minima of the RDF, found at rm = 3.86 Å and 3.82 Å for T = 1273 and 1673 K, respectively, we obtain the
values CN ≈ 12.6 and 11.7, respectively.
3.1.3. Liquid Au
The calculated S(q) for l–Au at T = 1423 and 1773 K are depicted in figure 4 (a) along with the corre-
sponding XR data [24]. For both states, the main peak is located at qp = 2.60 Å−1 and a good agreement
with experiment is observed for the positions andmagnitudes of the main and subsequent peaks. The cal-
culated isothermal compressibility has yielded values κT = 1.61±0.07 (in units of 10−11 N−1 m2) at 1423 K,
to compare with 1.31 [27] or 1.27 [28], and κT = 2.06±0.06 at 1773 K, where Singh et al. report 1.61 [28].
The g (r ) for l–Au at T = 1423 K and 1773 K are depicted in figure 4 (b). For both states, the main peak
is located at rp = 2.80 Å, which coincides with the experimental value, although the height of the main
Figure 4. (a) Static structure factors and (b) pair correlation functions for l–Au at two thermodynamic
states. Solid lines are the OF-AIMD results and the open circles are the XR diffraction data.
33604-7
G.M. Bhuiyan, L.E. González, D.J. González
peak is somewhat underestimated, especially for the lower temperature. The RDF has a first minimum at
rm = 3.86 Å and 3.82 Å which yields values of CN ≈ 12.7 and 12.2 for T = 1423 K and 1773 K, respectively.
3.2. Dynamic properties: Single particle dynamics
Relevant information concerning the single particle dynamics can be derived from several magni-
tudes and here we report our results obtained for some of those magnitudes.
The self-intermediate scattering function, Fs(q, t), provides a detailed information on the single par-
ticle dynamic properties over different length scales going from hydrodynamic (q → 0) to free particle
(q →∞) limits. This is defined as
Fs(q, t)=
1
N
〈
N∑
j=1
exp
[
i~q ~R j (t + t0)
]
exp
[
−i~q~R j (t0)
]
〉
,
where 〈. . .〉 denotes the average over time origins and wavevectors with the same module. Closely con-
nected to the Fs(q, t), is the velocity autocorrelation function (VACF) of a tagged ion in the fluid, Z (t),
which can be obtained as the q → 0 limit of the first-order memory function of the Fs(q, t) although in
the present simulations it was calculated from its definition
Z (t)= 〈~v1(t)~v1(0)〉
/
〈v2
1〉 , (6)
which stands for the normalized VACF. It provides information on the motion of an atom inside the cage
created by the shell of nearest neighbors. Besides, its time integral leads to the self-diffusion coefficient,
D , namely
D =
1
βm
∞∫
0
Z (t)dt , (7)
where β= 1/(kBT ). D can also be obtained from the slope of the mean square displacement δR2(t) of a
tagged ion in the fluid, as
D = lim
t→∞
δR2(t)/6t = lim
t→∞
1
6
d δR2(t)
dt
, δR2(t) ≡ 〈|~R1(t)−~R1(0)|2〉 . (8)
In the present OF-AIMD calculations, both routes have led to practically the same D value.
3.2.1. Liquid Cu
Figure 5 (a) shows, for several q-values, the calculated Fs(q, t) for l–Cu at T = 1423 K. We observe
the typical monotonous, non-linear decrease with time which becomes faster with increasing q-values;
moreover, comparison with the simple liquid metals near their respective melting points shows that at
similar q/qp values, the Fs(q, t) has a comparable rate of decay [55, 56, 69–72].
The calculated Z (t) for l–Cu are shown in figure 5 (b). The main features in the Z (t) are comparable
to those obtained for simple liquid metals near melting [55, 56, 69], namely a first minimum about 0.30
deep and a subsequent maximum with a rather weak amplitude. We recall that the negative values of
Z (t) represent a backscattering effect induced by the cage effect; moreover, with increasing temperature
(and decreasing density) the cage effect becomes less relevant, i.e., the first minimum in Z (t) is shallower
while the subsequent oscillations are less marked.
The self-diffusion coefficient was calculated according to equations (7)–(8), leading to values D =
0.39 Å2/ps ( T = 1423 K) and 0.58 Å2/ps (T = 1773 K). The reference experimental value at T = 1423 K
is 0.40 Å2/ps [30, 31], while the recent measurements of Meyer yielded D = 0.37 Å2/ps at 1420 K, which
are both in excellent agreement with our estimate. Other CMD studies yielded the values D = 0.31 and
0.27 Å2/ps [38, 39] and D = 0.36 Å2/ps for T = 1400 K [40]. The AIMD studies at 1500 K produced diffusion
coefficients of D = 0.28 [45], which clearly underestimates the experimental data, presumably due to a
small number of particles (50 atoms) used in the simulation, and 0.40±0.05 [48] in good agreement with
experiment. The higher temperature is outside the range of measurements by Meyer [32] (up to 1620 K).
33604-8
OF-AIMD simulation study of the liquid noble metals
Figure 5. (a) Self-intermediate scattering function of l–Cu at T = 1423 K. Full line: 0.6 Å−1, dashed line:
1.5 Å−1, dotted line: 2.5 Å−1, dotted-dashed line: 3.1 Å−1 and double dotted-dashed line: 4.3 Å−1 (b) Nor-
malized velocity autocorrelation function for l–Cu at 1423 K (full line) and 1773 K (dashed line).
However, he found that the measured data could be well described through an Arrhenius formula. The
value obtained with this expression for T = 1773 K is D = 0.65±0.05 Å2/ps, which is in reasonable agree-
ment with our result. For this temperature, the CMD study of Han et al. [40] has reported D = 0.59 Å2/ps
for T = 1700 K, which is also similar to our present estimate.
3.2.2. Liquid Ag
Figure 6 (a) shows the calculated Fs(q, t), at several q-values, for l–Ag at T = 1273 K and we observe
the features very similar to those already found in l–Cu. The normalized VACF for l–Ag at T = 1273 and
1673 K are depicted in figure 6 (b) where we observe the typical cage effect; now the variation with tem-
perature is more marked than in l–Cu because the relative change in the ionic density is greater. Now the
Z (t) of l–Ag becomes negative for longer times and this is because the backscattering associated with the
cage effect in l–Ag is reduced by the combination of two factors, namely, a smaller ionic number density
Figure 6. (a) Self-intermediate scattering function of l–Ag at T = 1273 K. Full line: 0.59 Å−1, dashed line:
1.5 Å−1, dotted line: 2.7 Å−1, dotted-dashed line: 3.3 Å−1 and double dotted-dashed line: 4.2 Å−1 (b) nor-
malized velocity autocorrelation function for l–Ag at 1273 K (full line) and 1673 K (dashed line).
33604-9
G.M. Bhuiyan, L.E. González, D.J. González
and a greater atomic mass. The calculated self-diffusion coefficients are D = 0.29 Å2/ps and 0.55 Å2/ps for
T = 1273 K and 1673 K, respectively, which is very close to the experimental data of D = 0.28 Å2/ps and
0.58 Å2/ps [30]
3.2.3. Liquid Au
The calculated Fs(q, t), at several q-values, for l–Au at T = 1423 K is depicted in figure 7 (a). As for the
normalized VACF, figure 7 (b) shows the calculated Z (t) for T = 1423 and 1773 K. Notice that in compari-
son with the previous results for l–Ag, the Z (t) for l–Au take longer to become negative and this is due to a
weaker backscattering effect induced by the greater atomicmass of the Au ions. Comparisonwith the Z (t)
of Bogicevic et al. [35] shows that these authors obtain a Z (t) with a narrower and shallower first mini-
mum along with weaker oscillations. The calculated self-diffusion coefficients are D = 0.27 and 0.46 Å2/ps
Figure 7. (a) Self-intermediate scattering function of l–Au at T = 1473 K. Full line: 0.59 Å−1, dashed line:
1.39 Å−1, dotted line: 2.41 Å−1, dotted-dashed line: 3.2 Å−1 and double dotted-dashed line: 4.2 Å−1 (b) nor-
malized velocity autocorrelation function for l–Au at 1423 K (full line) and 1773 K (dashed line).
at T = 1423 and 1773 K, respectively, to be compared with an experimental value [66] of 0.24 Å2/ps for
l–Au at T = 1423 K. The calculations of Bogicevic et al. [35] yielded somewhat bigger values, namely
D = 0.31 and 0.60 Å2/ps at T = 1423 and 1773 K, respectively, whose origin can be traced back to the less
marked cage effect in their Z (t). On the other hand, wemention that CMD simulations using two different
EAM potentials for l–Au at T = 1423 K yielded D = 0.26 Å2/ps [43,48]. The AIMD simulations of Pasturel et
al. [49] produced a value of D = 0.153 Å2/ps at 1400 K, somewhat smaller than experiment, and 0.30 Å2/ps
at 1700 K. To our knowledge, no experimental data are availble for the self-diffusion coefficient of l–Au
at T = 1773 K.
3.3. Dynamic properties: Collective dynamics
Regarding the collective dynamics, the most important magnitude is the intermediate scattering func-
tion, F (q, t), which provides information on the collective dynamics of density fluctuations. It is defined
as
F (q, t) =
1
N
〈
N∑
j=1
exp
[
iq̄ ~R j (t + t0)
] N∑
l=1
exp
[
−i~q ~Rl (t0)
]
〉
. (9)
Its space Fourier transform (FT) produces the van Hove correlation function whereas its time FT results
in the dynamic structure factor, S(q,ω), which is the magnitude measured in the inelastic XR (or neutron)
scattering experiments.
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OF-AIMD simulation study of the liquid noble metals
Another interesting magnitude associated with the density fluctuations is current due to the overall
motion of particles, i.e.,
~j (q, t) =
N∑
j=1
~v j (t) exp[i~q ·~R j (t)] , (10)
which is usually split into a longitudinal component, ~jL(q, t) parallel to ~q, and a transverse component,
~jT(q, t), perpendicular to ~q . Therefrom, the longitudinal, JL(q, t), and transverse JT(q, t), current corre-
lation functions are obtained as
JL(q, t) =
1
N
〈~jL(q, t) ·~j∗L (q,0)〉, JT(q, t) =
1
2N
〈~jT(q, t) ·~j∗T (q,0)〉 . (11)
The corresponding time FT produce the associated spectra, JL(q,ω) and JT(q,ω), respectively, with
JL(q,ω) =ω2S(q,ω). The transverse current correlation function JT(q, t), is not associated with any mea-
surable quantity and can only be determined bymeans of MD simulations. It provides information on the
shear modes, and its shape evolves from a Gaussian, in both q and t , for the free particle limit towards a
Gaussian in q and exponential in t for the hydrodynamic limit (q → 0), namely
JT(q → 0, t) =
1
βm
exp
[
−q2η|t |
/
(mρ)
]
, (12)
where η is the shear viscosity. On the other hand, for intermediate q-values, the JT(q, t) shows a compli-
cated behaviour, because it may oscillate signaling the propagation of shear waves. From the calculated
JT(q, t) it is possible to obtain the shear viscosity coefficient η as follows. The memory function represen-
tation of JT(q, t),
J̃T(q, z) =
1
βm
[
z +
q2
ρm
η̃(q, z)
]−1
, (13)
where the tilde denotes the Laplace transform, introduces a generalized shear viscosity coefficient η̃(q, z).
The area under the normalized JT(q, t) gives βm J̃T(q, z = 0), from which η̃(q, z = 0) ≡ η̃(q) can be ob-
tained and when extrapolated to q → 0 it produces a usual shear viscosity coefficient η. This is per-
formed [73] by exploiting the property that inversion is a symmetry in the system and, therefore, η̃(q)
should be an even function of q which permits to approximate (when q → 0) η̃(q) = η(1−αq2).
3.3.1. Liquid Cu
Figure 8 (a) shows the calculated F (q, t)/F (q, t = 0) at several q-values for l–Cu at T = 1423 K. The
F (q, t) exhibit an oscillatory behaviour at low q-values with the oscillations becoming weaker for in-
creasing q ’s until they finally disappear at q ≈ 2.2 Å−1 ≈ 0.75qp. We stress that this behaviour is very
similar to the one found for the simple liquid metals near melting [55, 56, 69].
From the calculated F (q, t), we performed a time FT (with an appropriate window to smooth out
the truncation effects) leading to the associated dynamic structure factor, S(q,ω). The obtained results
are depicted in figure 8 (b) for several q-values. We notice that up to q ≈ 0.75qp , the calculated S(q,ω)
exhibit well defined side-peaks which are indicative of collective density excitations. The FT of the longi-
tudinal current correlation function, JL(q,ω), shows, however, side-peaks for all wavevectors, and from
the positions of these side-peaks,ωL(q), a dispersion relation of the density fluctuations was obtained and
plotted infigure 9. In the hydrodynamic region (small q) the slope of the dispersion relation curve is the q-
dependent adiabatic sound velocity cs(q)= vth
√
γ/S(q), with vth =
√
kBT /m being the thermal velocity,
γ the ratio of specific heats and kB Boltzmann’s constant. In the q → 0 limit, the cs(q) reduces to the bulk
adiabatic sound velocity cs. Using the smallest q-value achieved by the simulations, qmin = 0.334 Å−1, we
get an estimate cs(qmin) ≈ 3880 m/s, which is clearly above the experimental hydrodynamic value cs ≈
3481 [27] or 3449 m/s [28]. In figure 10 we have plotted the cs(q) which clearly points to the existence of
some positive dispersion in l–Cu. This behaviour qualitatively agrees with an estimate of 4230 m/s [74]
obtained with inelastic XR scattering data. For the higher temperature T = 1773 K, we get a smaller value
cs ≈ 3802 m/s, whereas the experimental hydrodynamic sound velocity is 3266 m/s [28], signaling the
persistence of the positive dispersion at these higher temperatures.
33604-11
G.M. Bhuiyan, L.E. González, D.J. González
(a) (b)
Figure 8. (a) Intermediate scattering function of l–Cu at T = 1423 K. Full line: 0.58 Å−1 , dashed line:
1.5 Å−1, dotted line: 2.22 Å−1, dotted-dashed line: 3.1 Å−1, and double dotted-dashed line: 4.31 Å−1 (b)
Same as above but for the dynamic structure factor.
Figure 11 (a) shows the results for the normalized JT(q, t) of l–Cu at T = 1423 K at several q values.
Notice that for small q ’s, the corresponding JT(q, t) decrease slowly but it becomes faster with increasing
q values. The corresponding spectrum JT(q,ω) is depicted in figure 11 (b) and for some intermediate
q-range we observe an inelastic peak at nonzero frequency. This peak, which reflects the propagation of
shear waves in the liquid, does not appear at the smallest value reached by the simulation (q = 0.334 Å−1)
but it already shows up for q = 0.473 Å−1 ≈ 0.16qp , and remains up to q ≈ 2.5qp. The associated peak
frequency increases with q , takes a maximum value at q ≈ qp, and then decreases with increasing q as
JT(q,ω) evolves towards a gaussian shape. In fact, we recall that a similar behaviour has already been
reported for the alkali metals [69] where the inelastic peak appears for q Ê 0.07qp. On the other hand,
from the position of the peaks in the JT(q,ω) we can derive an associated transverse dispersion relation,
ωT(q), which is plotted in figure 16. The ωT(q) shows, for small q-values, a linear behaviour which
Figure 9. Dispersion relations from the peak posi-
tions of the calculated CL(q,ω) = ω2S(q,ω) for l–
Cu, l–Ag and l–Au at T = 1423, 1273 and 1423 K,
respectively.
Figure 10. q-dependent adiabatic sound velocity
for l–Cu, l–Ag and l–Au at T = 1423, 1273 and
1423 K, respectively. The full lines on the y-axis
stand for the respective hydrodynamic sound ve-
locities.
33604-12
OF-AIMD simulation study of the liquid noble metals
Figure 11. (a) Transverse current correlation function of l–Cu at T = 1423 K. (b) Same as above but for the
JT(q,ω).
can be approximated by ωT(q) ≈ ct(q − qc ), where qc is the value at which the JT(q,ω) starts showing a
maximum and ct is the velocity of propagation of the shear waves in the liquid metal. In the case of l–Cu
at 1473 K, we obtained an estimate ct ≈ 3000±200 m/s.
From the previous JT(q, t) and using the above mentioned procedure, we evaluated the shear viscos-
ity for l—Cu. Thiswas performed by calculating the generalized shear viscosity coefficient η̃(q) for a range
q É 0.8 Å−1 and fitting it to the expression η̃(q) = η(1−αq2). Herein we estimated η ≈ 3.63 ·10−3 kg/ms
(for T = 1423 K) and 2.25 ·10−3 kg/ms (T = 1773 K) which compare rather well with the corresponding
experimental data of 3.98 ·10−3 kg/ms and 2.39 ·10−3 kg/ms, respectively [30].
3.3.2. Liquid Ag
Figure 12 (a) shows the calculated F (q, t)/F (q, t = 0) at several q-values for l–Ag at T = 1273 K. Again,
for small q-values we observe an oscillatory behaviour which is gradually dampened with increasing
q-values until it finally disappears at q ≈ 0.75qp. The corresponding S(q,ω) are plotted in figure 12 (b)
Figure 12. (a) Intermediate scattering function of l–Ag at T = 1273 K. Full line: 0.59 Å−1 , dashed line:
1.5 Å−1, dotted line: 2.1 Å−1, dotted-dashed line: 3.0 Å−1, and double dotted-dashed line: 4.2 Å−1 (b)
Same as above but for the dynamic structure factor.
33604-13
G.M. Bhuiyan, L.E. González, D.J. González
and we observe side-peaks for a range of small q-values, namely up to q ≈ 0.75qp. From the position of
the peaks of JL(q,ω) we obtain the corresponding dispersion relation, ωL(q), plotted for T = 1273 K in
figure 9. Using the smallest q-value provided due to the simulations, qmin = 0.294 Å−1, we get an estimate
cs(qmin) ≈ 2930m/s, which is somewhat greater than the hydrodynamic values of cs = 2751 [27], 2710 [75]
or 2797 m/s [28], and suggests the existence of a small positive dispersion effect. At a higher temperature,
T = 1673 K, our calculation predicts a value of 2720 m/s, whereas the experimental adiabatic sound
velocity is 2663 m/s [28]. We are not aware of any inelastic XR or neutron scattering experiments for
liquid Ag to compare with.
Figure 13. (a) Transverse current correlation function of l–Ag at T = 1273 K. (b) Same as above but for the
JT(q,ω).
Figure 13 shows the calculated JT(q, t) and their Fourier Transforms for l–Ag at T = 1423 K and for
several q values. Themain features for JT(q, t) and JT(q,ω) are similar to those found in l–Cu. The JT(q,ω)
shows the peaks fromwhich the corresponding dispersion relationωT(q) was calculated and it is depicted
in figure 16. Again, a linear fit of the small q values yields an estimate of ct ≈ 1950± 150 m/s for the
velocity of the associated shear waves. The calculation of the shear viscosity yields η = 3.48 ·10−3 kg/ms
and 2.16 ·10−3 at T = 1273 K and 1673 K, which is in good agreement with the respective experimental
data η= 3.69 ·10−3 and 2.23 ·10−3 kg/ms [30].
3.3.3. Liquid Au
Figure 14 (a) shows the calculated F (q, t)/F (q, t = 0) for several q-values. Their main features are
similar to those found in l–Cu and l–Ag, namely the existence of oscillations up to q ≈ 0.75qp. Figure 14 (b)
depicts, for several q-values, the corresponding S(q,ω) which show clear side-peaks up to q ≈ 0.75qp. The
dispersion relation of the longitudinal currents is plotted in figure 9 for T = 1423 K. From the smallest
q-value in the simulations, qmin = 0.296 Å−1, we get an estimate cs(qmin) ≈ 2030 m/s, which is clearly
below the hydrodynamic adiabatic value of cs = 2567 m/s for 1337 K [27, 31] and 2513 m/s at 1423 K [28].
This points towards the presence of negative dispersion in the dispersion relation. Although negative
dispersion has been indeed found and explained in some systems, such as supercritical fluids [76], as
a consequence of an increased ratio between the high-frequency sound velocity and the adiabatic one,
driven mainly by a decreased value of the density, the present case of liquid Au near melting certainly
does not fit into this category of liquids. Two scenarios could possibly explain the negative dispersion
obtained in our calculation. The first one is related to the value of the isothermal speed of sound, cT.
Blairs’ data [27] are consistent with a value of γ = 1.50 at 1337 K, which yields a value of cT = 2096 m/s,
so that at 1423 K, the isothermal sound velocity would be somewhat smaller, i.e., quite similar to our
result for cs(qmin). Singh et al.’s data [28] give, however, a value of γ = 1.36 at 1336 K, and γ = 1.40 at
33604-14
OF-AIMD simulation study of the liquid noble metals
Figure 14. (a) Intermediate scattering function of l–Au at T = 1423 K. Full line: 0.59 Å−1, dashed line:
1.39 Å−1, dotted line: 2.0 Å−1, dot-dashed line: 3.2 Å−1 and double dot-dashed line: 4.2 Å−1 (b) Same as
above but for the dynamic structure factor.
1423 K, producing an isothermal sound velocity of 2124 m/s at 1423 K, which is also similar, although still
larger, than our cs(qmin) We could, therefore, argue that we are in a wavenumber domain where sound
propagation is isothermal in nature. This effect was indeed found in l-Ni [74, 77] and it is connected
with the existence of an intermediate isothermal domain standing between hydrodynamic and high-
frequency domains [78]. The second scenario is simply a scenario of either inaccuracies in the theoretical
method that lead to a simulation adiabatic sound velocity which is too small compared with experiment,
or inaccuracies in the experimental data which would report too high a value of the real sound velocity.
In either case, the negative dispersion that we find would just be an artifact produced by the wrong value
of the hydrodynamic sound velocity. In this respect, it is worth recalling that just one measurement of the
speed of sound in liquid Au exists [29]. It is also worthmentioning that very few theoretical calculations of
this property can be found in the literature. For instance, the only KS-AIMD simulations of liquid Au [49]
did not address the collective dynamics. Concerning CMD, we have only found one reference where a
value of cs is mentioned [79], which was obtained using a glue model interatomic potential, successfully
used previously to study several properties of solid Au; the value of cs was 3700 m/s at 1360 K, which is
notoriously high as compared to our result and the experimental value.
Figure 15. (a) Transverse current correlation function of l–Au at T = 1423 K. (b) Same as (a) but for the
JT(q,ω).
33604-15
G.M. Bhuiyan, L.E. González, D.J. González
Figure 16. Transverse dispersion relations for l–Cu, l–Ag and l–Au at T = 1423, 1273 and 1423 K, respec-
tively.
Figure 15 shows the calculated JT(q, t) and JT(q,ω) for l–Au at T = 1423 K and several q values.
The corresponding transverse dispersion relation is plotted in figure 16. Its low q behaviour leads to
an estimate ct ≈ 1380± 150 m/s for the velocity of the corresponding shear waves. The calculation of
the shear viscosity has yielded values ≈ 4.05 · 10−3 kg/ms and 3.304 · 10−3 kg/ms for T = 1423 K and
1773 K, respectively; these are close to the corresponding experimental data of 4.34 · 10−3 kg/ms and
3.33 ·10−3 kg/ms [30].
4. Conclusion
We have reported several static and dynamic properties of the liquid noble metals (Cu, Ag, Au), each
at two thermodynamic states near their respective triple points. This was carried out by using the orbital
free ab-initiomolecular dynamic simulation method which has already shown its capability for yielding
accurate estimates of the same properties for a range of simple metals and alloys [55, 56].
The static structure of the three systems at the two temperatures is globally very well described. Only
the low-q part of the structure factor differs from the values obtained from thermodynamic data in l–Cu,
where S(0) is underestimated and in l–Au where it is overestimated. The close similarities between the
structures of l–Ag and l–Au near melting are indeed reproduced within our model. For both systems, the
main peaks in their respective g (r ) and S(q) are located at very similar positions. This can be explained
by noting that the static structure is mostly determined by the repulsive part of the interionic interaction
and density of ions. Figure 1 shows that the repulsive part of the non-Coulombic part of the electron
ion interaction for l–Ag and l–Au are practically coincidental whereas the experimental ionic number
densities near melting are 0.0551 Å−3 (l–Ag) and 0.0525 Å−3 (l–Au).
As for the dynamic properties, we begin by noting that the calculated Z (t) show the characteristic
shape of high density systems [55, 56, 69] (i.e., the simple liquid metals near melting), which can be
explained in terms of the so-called cage effect, namely, a tagged particle is enclosed in a cage formed
by its adjacent neighbors. Results have also been reported for the selfdiffusion coefficients, adiabatic
sound velocities and shear viscosities. The calculated dynamic structure factors, S(q,ω), show side-peaks
up to q ≈ 0.75qp , which is similar, albeit a bit larger than that of the simple liquid metals [55, 56, 69].
The calculated dispersion relations suggest the existence of some positive dispersion in l–Cu and, to a
smaller extent, in l–Ag; in the case of l–Au, some negative dispersion appears to happen, but we could
not ellucidate whether it is a real feature or it is due to inaccuracies in the experimental data or in the
theoretical model.
We conclude, as far as the agreement with the available experimental data is concerned the present
OF-AIMD results for static and dynamic properties are very good. Most importantly, the results also show
the capability and reliability of our approach in handling very complicated d -electron systems in liq-
33604-16
OF-AIMD simulation study of the liquid noble metals
uid phase from the perspective of ab-initio studies. Additional OF-AIMD calculations combined with our
model for the pseudopotential are already in progress for several liquid transition metals. Preliminary
results are very encouraging and those will be reported in due course.
Acknowledgements
This workwas supported byMICINN in conjunction with the EU FEDER funds (project FIS2011–22957)
and by Junta de Castilla y Leon (project VA104A11–2). DJG additionally acknowledges financial support
from MECD (project PR2011–0019). GMB is grateful to the Universidad de Valladolid for a fellowship
which allowed him to carry out this work. He also gratefully acknowledges the hospitality provided by
Professor D. J. González and L. E. González during his stay at the Universidad de Valladolid.
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OF-AIMD simulation study of the liquid noble metals
Дослiдження деяких статичних та динамiчних
властивостей рiдких благородних металiв методом
безорбiтальної першопринципної молекулярної динамiки
Ґ.М. Бгуян1, Л.Е. Ґонзалес2, Д.Дж. Ґонзалес2
1 Факультет теоретичної фiзики, унiверситет м. Дака, Дака–1000, Бангладеш
2 Факультет теоретичної фiзики, унiверситет м. Вальядолiд, Вальядолiд, Iспанiя
Оцiнено деякi статичнi та динамiчнi властивостi рiдких Cu, Ag i Au при термодинамiчних умовах, близьких
до їх точок плавлення, за допомогою методу безорбiтальної першопринципної молекулярної динамiки.
Розрахована статична структура добре узгоджується з наявними даними, отриманими з рентгенiвської
i нейтронної дифракцiй. Щодо динамiчних властивостей, то розрахованi динамiчнi структурнi фактори
вказують на iснування колективних збуджень густини поряд з позитивною дисперсiєю для l–Cu i l–Ag.
Отримано деякi коефiцiєнти переносу, якi прийнятно узгоджуються з наявними експериментальними да-
ними.
Ключовi слова: рiдкi благороднi метали, безорбiтальна теорiя функцiоналу густини, моделювання
методом молекулярної динамiки, статична структура, динамiчнi властивостi, коефiцiєнти переносу
33604-19
Introduction
Theory
Results and discussion
Static Properties
Liquid Cu
Liquid Ag
Liquid Au
Dynamic properties: Single particle dynamics
Liquid Cu
Liquid Ag
Liquid Au
Dynamic properties: Collective dynamics
Liquid Cu
Liquid Ag
Liquid Au
Conclusion
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| id | nasplib_isofts_kiev_ua-123456789-120175 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1607-324X |
| language | English |
| last_indexed | 2025-12-07T18:10:02Z |
| publishDate | 2012 |
| publisher | Інститут фізики конденсованих систем НАН України |
| record_format | dspace |
| spelling | Bhuiyan, G.M. Gonz´lez, L.E. Gonz´lez, D.J. 2017-06-11T11:11:07Z 2017-06-11T11:11:07Z 2012 Orbital free ab initio molecular dynamics simulation study of some static and dynamic properties of liquid noble metals / G.M. Bhuiyan, L.E. Gonz´lez, D.J. Gonz´lez // Condensed Matter Physics. — 2012. — Т. 15, № 3. — С. 33604:1-19. — Бібліогр.: 79 назв. — англ. 1607-324X PACS: 61.25.Mv, 64.70.Fx, 71.15.Pd DOI:10.5488/CMP.15.33604 arXiv:1210.2183 https://nasplib.isofts.kiev.ua/handle/123456789/120175 Several static and dynamic properties of liquid Cu, Ag and Au at thermodynamic states near their respective melting points, have been evaluated by means of the orbital free ab-initio molecular dynamics simulation method. The calculated static structure shows good agreement with the available X-ray and neutron diffraction data. As for the dynamic properties, the calculated dynamic structure factors point to the existence of collective density excitations along with a positive dispersion for l-Cu and l-Ag. Several transport coefficients have been obtained which show a reasonable agreement with the available experimental data. Оцiнено деякi статичнi та динамiчнi властивостi рiдких Cu, Ag i Au при термодинамiчних умовах, близьких до їх точок плавлення, за допомогою методу безорбiтальної першопринципної молекулярної динамiки.
 Розрахована статична структура добре узгоджується з наявними даними, отриманими з рентгенiвської i нейтронної дифракцiй. Щодо динамiчних властивостей, то розрахованi динамiчнi структурнi фактори вказують на iснування колективних збуджень густини поряд з позитивною дисперсiєю для l–Cu i l–Ag.
 Отримано деякi коефiцiєнти переносу, якi прийнятно узгоджуються з наявними експериментальними даними. This work was supported by MICINN in conjunction with the EU FEDER funds (project FIS2011–22957) and by Junta de Castilla y Leon (project VA104A11–2). DJG additionally acknowledges financial support from MECD (project PR2011–0019). GMB is grateful to the Universidad de Valladolid for a fellowship which allowed him to carry out this work. He also gratefully acknowledges the hospitality provided by Professor D. J. González and L. E. González during his stay at the Universidad de Valladolid. en Інститут фізики конденсованих систем НАН України Condensed Matter Physics Orbital free ab initio molecular dynamics simulation study of some static and dynamic properties of liquid noble metals Дослiдження деяких статичних та динамiчних властивостей рiдких благородних металiв методом безорбiтальної першопринципної молекулярної динамiки Article published earlier |
| spellingShingle | Orbital free ab initio molecular dynamics simulation study of some static and dynamic properties of liquid noble metals Bhuiyan, G.M. Gonz´lez, L.E. Gonz´lez, D.J. |
| title | Orbital free ab initio molecular dynamics simulation study of some static and dynamic properties of liquid noble metals |
| title_alt | Дослiдження деяких статичних та динамiчних властивостей рiдких благородних металiв методом безорбiтальної першопринципної молекулярної динамiки |
| title_full | Orbital free ab initio molecular dynamics simulation study of some static and dynamic properties of liquid noble metals |
| title_fullStr | Orbital free ab initio molecular dynamics simulation study of some static and dynamic properties of liquid noble metals |
| title_full_unstemmed | Orbital free ab initio molecular dynamics simulation study of some static and dynamic properties of liquid noble metals |
| title_short | Orbital free ab initio molecular dynamics simulation study of some static and dynamic properties of liquid noble metals |
| title_sort | orbital free ab initio molecular dynamics simulation study of some static and dynamic properties of liquid noble metals |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/120175 |
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