We have presented the analytic expressions of thermodynamic quantities of cerium dioxide compounds with fluorite structure including the anharmonicity effects of thermal lattice vibratio[r]
(1)Investigation of thermodynamic properties of cerium dioxide by statistical moment method
Vu Van Hunga, Jaichan Lee a, K Masuda-Jindob,* aDepartment of Materials Science and Engineering, Center for Advanced Plasma Surface Technology,
Sungkyunkwan University, 300 Chunchun-dong, Jangan-gu, Suwon, 440-746, South Korea b
Department of Materials Science and Engineering, Tokyo Institute of Technology, Nagatsuta, Midori-ku, Yokohama 226-8503, Japan Received 21 April 2005; received in revised form September 2005; accepted 16 September 2005
Abstract
The thermodynamic properties of the cerium dioxide (CeO2) are studied using the statistical moment method, including the anharmonicity
effects of thermal lattice vibrations The free energy, linear thermal expansion coefficient, bulk modulus, specific heats at the constant volume and those at the constant pressure, CVand CP, are derived in closed analytic forms in terms of the power moments of the atomic displacements The
temperature dependence of the thermodynamic quantities of cerium dioxide is calculated using three different interatomic potentials The influence of dipole polarization effects on the thermodynamic properties and thermodynamic stability of cerium dioxide have been studied in detail
q2005 Elsevier Ltd All rights reserved
Keywords: A Inorganic compounds; D Anharmonicity; D Thermodynamic properties
1 Introduction
Cerium dioxide (CeO2) has been the subject of the recent
studies in catalysis, fuel cell electrolyte materials, gas sensors, optical coatings, high-Tc superconductor structures, high
storage capacity devices, and other related applications[1–6] For instance, owing to the remarkable redox properties ceria-based mixed oxides are active components of three-way automotive catalysts (TWC) [7,8], in which CeO2 play an
important role for enhancing the removal of carbon monoxide (CO), hydrocarbon (HC) and nitrogen oxide (NOx) pollutants
Ceria is able to store oxygen under lean conditions and to release it when the O2concentration in the gas phase becomes virtually
nil A large number of experimental and theoretical studies have been carried out on catalytic[10,11], lattice vibrational[12,13], and structural [14–16] properties of cerium dioxides Theo-retical study on the structure, stability and morphology of stoichiometric ceria crystallites has been done using the simulation method[9] Recently, Tsunekawa et al.[14]have
shown that the anomalous lattice expansion in the monodisperse ceria (CeO2Kx) nanoparticles are caused by the valence
reduction of Ce ions, i.e decrease of the electrostatic forces The reduction of the valance induces an increase in the lattice constant due to the decrease in electrostatic forces, but the increase in the valence does not always lead to lattice shrinkage This behavior is in contrast to a decrease of the lattice constant often observed in the metal nanoparticles with decreasing particle size In this respect, it is of great importance to take into account the electrostatic Coulomb interactions between the constituent ions in the calculations of stabilities and thermo-dynamic quantities of CeO2
Most of the previous theoretical studies, however, are concerned with the materials properties of cerium dioxides at absolute zero temperature, and temperature dependence of the thermodynamic quantities has not been studied in detail The purpose of the present article is to investigate the temperature dependence of the thermodynamic properties of cerium dioxide using the analytic statistical moment method (SMM)[17–19] The thermodynamic quantities are derived from the Helmholtz free energy, and the explicit expressions of the thermal lattice expansion coefficient, unit cell volume, specific heats at constant volume and those at the constant pressure CV and
CP, and elastic modulus are presented taking into account
the anharmonicity effects of the thermal lattice vibrations In the present study, the influence of dipole polarization effects
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0022-3697/$ - see front matter q 2005 Elsevier Ltd All rights reserved doi:10.1016/j.jpcs.2005.09.100
* Corresponding author Permanent address: Dept of Physics, Hanoi National Petagogic University, Hanoi, Vietnam
(2)on the thermodynamic properties have also been studied, using three different interatomic potentials Each set of potentials is based on a fully ionic description of the fluorite (CaF2) lattice
We will compare the results of the present calculations with those of the previous theoretical calculations as well as with the available experimental results
2 Theory
Let us consider a quantum system given by the following Hamiltonian:
^
H Z ^H0K X
i
aiV^i; (1)
where ^H0 denotes the lattice Hamiltonian in the harmonic approximation, and the second term is added due to the anharmonicity of thermal lattice vibrations denotes a
parameter characterizing the anharmonicity of lattice vibrations and ^Vi the related operator The Helmholtz free energy of the system given by Hamiltonian (1) is formally written as j Z j0K
X i
ð ai
0
h ^Viiadai; (2a)
with h ^ViiaZ
Kvj vai
; (2b)
where h ^Viia expresses the expectation value at the thermal equilibrium with the (anharmonic) Hamiltonian ^H The expectation values in the above Eq (2a) and (2b) are evaluated with the use of the density matrix formalism
h ^ViaZ Tr½ ^r ^V; (3)
where the density matrix ^r is defined, with qZkBT, as
^
r Z exp jK ^H q
: (4)
In the above Eq (2a), j denotes the free energy of the system at temperature T, and j0is the free energy of the harmonic lattice
corresponding to the Hamiltonian ^H0
We will present the SMM formulation for the oxide material with fluorite (CaF2) structure like CeO2, as schematically
shown inFig The concentrations of cerium and oxygen ions are simply denoted by CCeZ(N1/N), and COZ(N2/N),
respectively The free energy of cerium dioxide are then written by taking into account the configurational entropies Sc,
via the Boltzmann relation as
J Z CCeJCeCCOJOKTSc; (5)
where JCeand JOdenote the free energy of Ce and O ions,
respectively In the fluorite (CaF2) structure the Ce 4C
ions occupy an fcc sublattice and O2Kions occupy the tetrahedral interstitial sites forming a simple cubic sublattice of length a0/2 However, the present thermodynamic calculations will be
done, assuming the completely ordered phase of CeO2, and the
thermodynamic properties and the phase stability of non-stoichiometric CeO2 will be discussed in a forthcoming
publication
Firstly, we expand the potential energy of the system in terms of the atomic (ionic) displacements Uiof the atom i
U Z UCeCUOZN1
X i
4CeioðjriCuijÞ
C N2
2 X
i
4OioðjriCuijÞ Z N1
2 X
i
4CeioðjrijÞ
C
X a;b
v24Ceio vuiavuib
eq uiauib
C
X a;b;g
v34Ce io vuiavuibvuig
eq
uiauibuig
C 24
X a;b;g;h
v44Ce io vuiavuibvuigvuih
eq
uiauibuiguihC/ )
C N2
2 X
i
4OioðjrijÞ C
X a;b
v24Oio vuiavuib
eq uiauib (
C
X a;b;g
v34Oio vuiavuibvuig
eq
uiauibuig
C 24
X a;b;g;h
v44Oio vuiavuibvuigvuih
eq
uiauibuiguihC/ )
; (6)
where riis the equilibrium position of the ith atom, uiadenotes
a-Cartesian component of the atomic displacement of ith atom, and 4Ceio (or 4Oio) the effective interaction energy between the zeroth and ith atoms, respectively It should be reminded, however, that the expansion of the above Eq (6) is done for deriving the criterion of the thermodynamical equilibrium of the system, and in Eq (6), the subscript eq means the quantities calculated at the equilibrium state Using Eq (6), the thermal average of the crystalline potential energy of the system is given in terms of the power moments huni of the atomic displacements and the harmonic vibrational parameter k,
(3)and three anharmonic parameters b, g1, and g2as
hUi Z UCe0 CU O C3N1
kCe hu
2 Cei Cg
Ce hu
4 Cei Cg
Ce hu Cei C3N2 kO hu
Oi CbohuOihu Oi Cg
O 1hu
4 Oi Cg
O 2hu Oi C/ (7) where
kCeZ1
X i
v24Ceio vu2
ia
eq
; kOZ1
2 X
i
v24Oio vuO
ia
eq
(8a)
b0Z
X i
v34Oio vuiavuibvuig
eq
; (8b)
gCe1 Z 48
X i
v44Ceio vu4
ia
eq
; gO1Z 48
X i
v44Oio vu4
ia
eq
gCe2 Z 48
X i
v44Ce io vu2
iavu2ib !
eq ;
gO2 Z 48
X i
v44O io vu2
iavu2ib !
eq
(8c)
with asbZx,y, or z U0Ce and U0O represent the sum of effective pair interaction energies for Ce ion and Oxgen ion, respectively,
U0CeZ N1
2 X
i
4CeioðjrijÞ; U O Z N2 X i
4OioðjrijÞ: (9)
The Helmholtz free energy JCe for Ce ions can be derived
from the functional form of the potential energy of the above Eq (7) through the straightforward analytic integrations I1and
I2, with respect to the two anharmonicity ‘variables’ g1and g2
Firstly, for Ce ions I1and I2are written in an integral form as
I1Z ð gCe hu2 Cei 2j gCe 1Z0dg
Ce
2 ; I2Z ð gCe hu4 Ceidg Ce
1 : (10)
Using moment expansion formulas[17–19], one can find the low-order moments hu2i and hu4i (see Appendix), and the final
expression of the partial Helmholtz free energy JCeof Ce ion
sites in cerium dioxide is given by JCeZ U0CeC3N1qẵx C ln1Ke
K2xị
C3N1 q2 k2
Ce gCe2 x
2
coth2xK2g Ce a1
C 2q2 k4 Ce 3ðg Ce Þ
a1x coth xK2ððg Ce Þ
2
C2gCe1 gCe2 Þa1ð2a1K1Þ
;
(11)
where
a1Z C xcoth x
2; x Z
ZuCe
2q h Z
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi kCe=mCe p
2q : (12)
Similar derivation can be also done for finding the partial Helmholtz free energy JO of oxygen ions by performing
analytic integrations I1, I2and I3as
I1Z ð gO
2
0 hu2
Oi2jbOZ0;gO1Z0dg
O
2; I2Z ð gO
1
0 hu4
OijbOZ0dg O 1; and
I3Z ð bO
0 huOihu
2
OidbO: ð13Þ
The final expression of the partial Helmholtz free energy JOof
the oxygen ions is given by JOZ UO0 C3N2q½x C lnð1Ke
K2x Þ
C3N2 q
2 k2 O
gO2x2coth2xK2g O a1
C 2q2 k4 O 3ðg O
2Þ2a1xcothxK2ððgO1Þ2C2gO1gO2Þa1ð2a1K1Þ
Cq bO 6KgO kO KK1 C q2bO
K
2gO 3K3a1
1=2
K bOa1
9K2 "
C bOkOa1
9K3 C
bO 6KkO
ðx coth xK1Þ
(14) where
x ZZuO 2q h Z
ffiffiffiffiffiffiffiffiffiffiffiffiffi kO=mO p
2q ; (15a)
gOZ 4ðgO1CgO2Þ; K Z kOKb2O
3gO : (15b)
In the above Eqs (11) and (14), the harmonic contributions to the Helmholtz free energies JCeand JOare derived using the
‘Einstein’ approximation Therefore, for more quantitative thermodynamic calculations at the low temperatures, one can use the lattice dynamical model[20]by evaluating the matrix elements of the dynamical matrix, Dab(q), in other words the
Fourier transform of the force constants between the neighbor-ing atomic sites The diagonal matrix elements (on-site energies in the language of TB electronic theories[21]) are given by the ‘harmonic force constants’, and related to the maximum frequencies wu2max=2 Therefore, the present formulation gives the exact treatments in the moments expansion of all orders coming from the harmonic phonon Hamiltonian The anharmonic contributions to the free energies are treated within the fourth order moment approximation
(4)the partial free energies JCe and JO, given by Eqs (11) and
(14), respectively Then, the specific heat at constant volume of the cerium dioxide is given by
CVZ CCeC Ce V CCOC
O
V: (16)
We assume that the average nearest-neighbor distance of cerium dioxide at temperature T can be written as
r1ðTÞ Z r1ð0Þ C CCey Ce
0 ðTÞ C COy O
0ðTÞ; (17)
in which yCe0 ðTÞ and yO0ðTÞare the atomic displacements of Ce and O ions from the equilibrium position in the fluorite lattice
[22–24], and r1(0) is the distance r1at zero temperature The
average nearest-neighbor distance at TZ0 K can be deter-mined from the minimum condition of the potential energy of the system composed of N1Ce ions and N2Oxygen ions:
vU vr1 ZvU Ce vr1 C vU0O
vr1 ZN1
2 v vr1
X i
4CeioðjrijÞ ! C N2 v vr1 X i
4OioðjrijÞ !
Z 0:
(18)
From the definition of the linear thermal expansion coefficient, it is easy to derive the formula
aTZ CCea Ce T CCOa
O
T; (19)
where aCeT Z
kB r1ð0Þ
vyCe0
vq ; a
O T Z
kB r1ð0Þ
vyO0
vq (20)
The bulk modulus of cerium dioxide is derived from the free energy of Eq (5) as
BTZKV0 vP vV
T
ZKV0
v2J vV2
T
Z CCeB Ce T CCOB
O T (21)
where P denotes the pressure, V0 the lattice volume at zero
temperature, and the bulk moduli BCeT and BOT are given by BCeT ZK
kB 3aCe
T
v2JCe vVvq
; BOT ZK
kB 3aO
T
v2JO vVvq
: (22)
Due to the anharmonicity of thermal lattice vibrations, the heat capacity at a constant pressure, CP, is different from the heat
capacity at a constant volume, CV The relation between CPand
CVis
CPZ CVKT vV vT
2
P vP vV
T
Z CVC9a2TBTVT: (23)
3 Results and discussions
To calculate the thermodynamic quantities of CeO2 with
fluorite structure, we will use three different potentials [9], which include the electrostatic Coulomb interactions and two body terms to describe the short-range interactions The two body terms arise from the electronic repulsion and attractive van der Waals forces, and they are described by a Buckingham potential form The internal energy of CeO2 compound can
then be written in the following form U ZN1
2 X
i
4CeioðriÞ C N2
2 X
i
4OioðriÞ; (24)
where 4CeioðriÞ Z
qCeqi ri
CACeKiexp K ri BCeKi
K CCeKi
r6i ; (25a)
and 4OioðriÞ Z
qOqi ri
CAOKiexp K ri BOKi
K COKi
r6i : (25b)
Here qi(iZCe or O) denotes the ionic charge of ion i, rithe
distance between them and Aij, Bij and Cij are the potential
parameters between the ions i and j The ionic Coulomb terms in the above Eqs (25a) and (25b) can be summed explicitly using the Ewald method In Eqs (25a) and (25b), the exponential term corresponds to electron cloud overlap and the Cij/r6term originates from attractive Van der Waal’s force
Potential parameters Aij, Bij and Cij are taken from Ref [9],
cation–cation interactions are assumed to be purely ionic Coulomb and the cation–anion interaction is considered to be the form of Eqs (25a) and (25b) The parameters of the three potentials are presented inTable
Firstly, we calculate the lattice parameters and bulk moduli of CeO2at 293 K (room temperature) using the three different
potentials In Table 2, we compare the calculation results of lattice parameters and bulk moduli of CeO2obtained by using
the SMM analytic formulae with the simulation results (absolute zero temperature) of Ref [9] Here, it should be noted that the three potentials are fitted to reproduce the experimental lattice parameter 5.411 A˚ of CeO2compound at
the absolute zero temperature, without including quantum mechanical zero point vibrations It is well known that the inclusion of thermal lattice vibrations (zero point vibrations) leads to the lattice expansion of crystals at lower temperatures All three sets of potentials give reasonable lattice constants of CeO2, near the experimental 5.411 A˚ even when we take into
accounts the contributions of thermal lattice vibrations In
Table 2, we also compare the bulk moduli BTcalculated by the
present SMM with those by the simulation method [9] The bulk modulus of Ref [9](with dipole polarization effects) is larger than the experimental values while the present SMM results of bulk modulus are smaller than the experimental ones The reason for the lower bulk modulus is due to the larger lattice spacings of CeO2compound in the present calculations,
Table
The parameters of CeO2used in potentials 1, and Butler potential Interaction A/eV B/A˚ C/eVA˚6
O2–O2K 9547.92 0.2192 32.00 Potential 1 Ce4C–O2K 1809.68 0.3547 20.40
O2–O2K 9547.92 0.2192 32.00 Potential 2 Ce4C–O2K 2531.5 0.335 20.40
(5)compared to those of Ref [9], because in the present calculations we take into account the contributions of thermal lattice vibrations The dipolar and quadrupolar parts of the interatomic potentials of CeO2 are derived from ab initio
density functional calculations and available in the literature When both dipole and quadrupole effects (C8/r8) are added the
SMM calculations of the bulk modulus of cerium dioxide give a larger value and a much better agreement with experiments In Fig 2, we compare the temperature dependence of the lattice parameters, a0, and unit cell volumes of cerium
dioxide calculated by using the potentials and 2, with
the experimental results for temperature range
300 K&T&1300 K The calculated lattice parameters and unit cell volumes by the present theory are slightly larger than the experimental values, but overall features are in good
agreement with experimental results[25,26] One can see in
Fig 2(a) and (b) that the calculated lattice parameter and unit cell volume by potentials and are very similar The small difference between the two calculations simply comes from the difference in cerium-oxygen interaction potentials, since the ionic Coulomb contribution and the oxygen-oxygen potential are the same for potentials and InFig 3(a) and (b), we show the lattice parameters of CeO2with and without including
the dipole polarization effects, respectively for wider tempera-ture range of 100 K&T&2500 K The calculated lattice parameters by potentials and are almost identical, while the Butler potential gives somewhat larger values, shifted upwards about 1% at lower temperatures Therefore, tempera-ture dependence of the lattice parameters by three potentials are similar for lower temperature region, but at higher
Table
Calculated lattice constants and bulk moduli of CeO2
a0(A˚ ) BT(GPa)
Method Pot Pot Butler Pot Pot Butler Ref
Simulation 5.411 5.411 5.411 267.9 289.4 263.6
SMM (with dipolar) 5.4106 5.4156 5.4531 170.070 192.519 154.159
SMM (without dipolar) 5.4057 5.4108 5.4472 184.163 207.065 166.312
Expt 5.411 236 20
Expt 230G10 21
Fig Comparison of calculated lattice parameters a (in A˚ ) and unit cell volumes (in A˚3) of CeO2with the experimental results for temperature range 293 K&T&1300 K
(6)temperatures near the melting temperature, Butler potential gives nonlinear increase of the lattice constants
In contrast, the calculated bulk moduli depend sensitively on the parameters of the potentials In Fig 4, we show the calculated bulk moduli BT of the cerium dioxide, with and
without dipole polarization effects, as a function of the temperature T We have found that the bulk modulus BT
depends sensitively on the potential parameters and it is decreasing function of the temperature T The decrease of BT
with increasing temperature arises from the thermal lattice expansion and the effects of the vibrational entropies The potential gives the largest values of the bulk modulus, while the Butler potential gives the lowest The low bulk modulus predicted by the Butler potential is due to the choice of the oxygen–oxygen interaction potentials The calculated lattice constants by the SMM with dipole polarization effects are greater than those without including dipole polarization effects The present SMM calculations of the bulk modulus, including the dipole polarization effects, are smaller compared to the experimental values The lattice constants are increasing by including dipole interactions (C/r6) for all three potentials, and accordingly the bulk modulus becomes smaller In the present calculations of the bulk moduli of CeO2, we have seen that
there is a clear correlation between the lattice spacing and calculated bulk modulus, and one can predict the relative
magnitudes of the bulk modulus However, due to the nature of the anharmonicity of the potential, the bulk modulus can be calculated accurately by including the anharmonicity theory of lattice vibrations (SMM)
The linear thermal expansion coefficients of CeO2are also
calculated using the three potentials InTable 3, we compare the calculated linear thermal expansion coefficients of CeO2, with
and without dipole polarization effects, at room temperature (293 K) and for higher temperature range (293–1300 K) Each potential gives reasonable values of thermal expansion coefficients compared with the experimental results InFig 5, we also show the temperature dependence of linear thermal expansion coefficients of CeO2, with and without the dipole
polarization effects The calculated results by the present theory are in good agreement with experimental results[25,27,28]and the agreement is better for the SMM calculations with potentials and 2, rather than the calculations by Butler potential Again, the different results obtained by potentials and from those by the Butler potential arise from the parameterization of oxygen– oxygen potentials The thermal lattice expansion coefficients calculated by including the dipole polarization effects are larger (w10% at lower temperatures and w20% at higher tempera-tures near the melting point) than those values without the dipole polarization effects
The calculated specific heats at constant volume Cv and
those at constant pressure Cpof CeO2compound are presented
in Fig As shown in Fig 6, the specific heat CV depends
weakly on the temperature for the whole temperature region, while the specific heat Cpdepends strongly on the temperature,
and becomes nonlinearly larger with increasing the tempera-ture It is surprising that the specific heats at the constant volume Cv are almost independent of the nature of the
potentials used for CeO2for whole temperature range up to the
melting temperature The three potentials (1, and Butler type) give the almost identical specific heats at constant volume CV,
while the Butler potential gives much larger CPthan those by
potentials and at higher temperatures The stronger nonlinear increase of CP values obtained by Butler potential
near the melting temperature is related to the stronger anharmonicity of thermal lattice vibrations described by its
Table
Calculated thermal expansion coefficients of CeO2 aT(10
K6 KK1)
Method T (K) 293 293–1300
SMM(with dipolar)
CPotential 11.391 11.391–13.510 CPotential 11.163 11.163–13.399 CButler 12.991 12.991–16.522 SMM(without dipolar)
CPotential 10.322 10.322–11.826 CPotential 10.187 10.187–11.820 CButler 11.731 11.731–14.329
Expt 11a 12.4b
11.2c a Ref.[26].
b Ref.[22]. c Ref.[23].
(7)potential The anomalous nonlinear increase in the thermo-dynamic quantities near the melting temperature are observed in the strongly anharmonic solids like solid Xe and solid Ar
[17] However, in the lack of the reliable experimental results of specific heats of stoichiometric CeO2 compound near
the melting temperature, it is difficult to draw definite conclusions on the validity of using the potentials 1, or Butler potential for the calculations of CP
Summarizing the above-mentioned thermodynamic calcu-lations done by the SMM, we would like to emphasize that it is particularly useful for the systematic and fundamental under-standing of the thermodynamic properties of cerium oxide We have presented the analytic expressions of thermodynamic quantities of cerium dioxide compounds with fluorite structure including the anharmonicity effects of thermal lattice vibrations, which are straightforward and useful for the realistic numerical calculations The present calculations of thermo-dynamic quantities of CeO2 (inorganic compound with the
fluorite structure) are much more efficient and straightforward than those by using the simulation method, e.g Monte Carlo or quantum Monte Carlo simulations The simulation methods without including the thermal (anharmonic) vibration effects are insufficient and often useless even for the calculations of the low temperature (room temperature) thermodynamic quan-tities In addition to the thermodynamic calculations, the present SMM can also be used for the parameterization of the new quantum mechanical potentials derived from the fitting to the ab initio density functional calculations, i.e improved version of the (empirically derived) interatomic potentials used in the present study, for high temperature thermodynamic quantities of the fluorite compounds
4 Conclusions
We have presented an analytic formulation for obtaining the thermodynamic quantities of the inorganic compounds (cerium dioxide) with fluorite structure using the statistical moment method The present formalism takes into account the higher-order anharmonic vibrational terms in the Helmholtz free energy and it enables us to derive the various thermodynamic quantities in closed analytic forms The thermodynamic quantities, i.e the unit cell volume, linear thermal expansion coefficient, elastic modulus and the specific heats Cvand Cp, of
the cerium dioxide are calculated and compared with the available experimental results The interatomic potentials and
Fig Temperature dependence of linear thermal expansion coefficients of CeO2with and without dipole polarization effects: (a) potential 1, (b) potential and (c) Butler potential
(8)2 used in the present study give almost identical results in thermodynamic quantities of CeO2 (except for the bulk
modulus), while the Butler potential gives the considerably different results This is originating from the different parameter values used for the oxygen–oxygen interactions in the potentials 1, and Butler potential In the present study, the dipole polarization effects are taken into account, and the inclusion of the dipolar terms is essential in order to understand the characteristic thermodynamic properties of CeO2
Acknowledgements
This work was supported by the Korea Science and Engineering Foundation (KOSEF) through the Center for Advanced Plasma Surface Technology (CAPST) and National Research Laboratory (NRL) program, and the Core Technol-ogy Development Program for Fuel Cells, the Ministry of Science and Technology, Korea
Appendix: Moments formulae in SMM
The present statistical moment method (SMM) has the virtue of treating exactly the correlations, deviations from the simple mean-field approximation, in the thermal averages huni of the atomic displacements For instance, (intra atomic) correlations appearing in the lower order moments are given by the following equations:
huiauigiaZhuiaiahuigiaCq vhuiaia
vag C
Zdag 2mucoth
Zu 2q
K qdag mu2;
(A1) and
huiauiguihiaZ huiaiahuigiahuihiaCqPaghhuiaia vhuigia
vah Cq2 vhuiaia
vagvah C
Zhuihiadag
2mu coth
Zu 2q
Kqhuihiadag
mu2 ; ðA2Þ
where Paghis (aZgZh) or (otherwise) depending on a, g
and h (Cartesian component) and u is the atomic vibration frequency defined by Eqs (12) and (15a), (uCe or uO), in the
text Here it is important to note that the root mean square atomic displacements hu2ii are evaluated exactly by using Eq (A1) and appropriate (ab initio) energetic for the interatomic potentials in the solids
However, the second moments, i.e root mean square atomic displacement hu2ii on atomic site i is different from the root mean square relative atomic displacements (second cumulants) which include the interatomic displacement–displacement correlations as given by the following equation
s2jTị Z hẵujKu0ị R
i Z hu2 ji C hu
2
0iK2huju0i; (A3) where, u0and ujare the atomic displacements of 0-th and j-th
sites from their equilibrium positions, respectively ðR is an unit
vector at the 0-th site pointing towards the j-th site, and the brackets denote the thermal average Here, simple decoupling scheme huju0izhujihu0i gives reasonable approximation The
root mean square relative displacements are important ingredients in the theory of EXAFS (X-ray-absorption fine structure)[29,30], the treatments of melting transition of solids by Lindemann’s criterion, and other related thermodynamic calculations of solids Using the exact moment formulae of Eqs (A1) and (A2), we find the second and fourth order moments, which include both harmonic and anharmonic contributions For the evaluation of free energies, we only need moments (not cumulants) up to fourth order in the present SMM formalism
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