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CALTECH ASCI TECHNICAL REPORT 003 caltechASCI/2000.003

The Quantum Sutton-Chen Many-Body Potential for Properties of fcc Metals Y. Kimura, Y. Qi, T. Cagin, , W. Goddard III

The Quantum Sutton-Chen Many-Body Potential for Properties of fcc Metals Yoshitaka Kimura, Yue Qi, Tahir Cagn, and William A. Goddard IIIy Materials and Process Simulation Center, Beckman Institute (139-74), Division of Chemistry and Chemical Engineering, California Institute of Technology, Pasadena, California 91125

(July 13, 1998)

Abstract The simple Sutton-Chen [Philos. Mag. Lett. 61, 139 (1990)] (SC) type many-body force eld leads to an accurate description of many properties of metals and their alloys. We have modi ed SC to include quantum corrections (e.g., zero-point energy) in comparing properties to experiment, leading to the quantum Sutton-Chen, or Q-SC force eld. We have applied the Q-SC description to nine face-centered cubic (fcc) metals (Al, Ni, Cu, Rh, Pd, Ag, Ir, Pt, and Au). The Q-SC parameters were optimized to describe the lattice parameter, cohesive energy, bulk modulus, elastic constants, phonon dispersion, vacancy formation energy, and surface energy. These potentials were tested by calculating the equation of state, thermal expansion, and speci c heat. We nd generally good agreement with all properties, indicating that this Q-SC type force eld should be useful in molecular dynamics and Monte Carlo simulations of metallic alloys. To illustrate the application of these parameters, we show how they have been used for predicting the viscosity of liquid metal alloys, and alloy melting and solidi cation (to form crystal or glass). Typeset using REVTEX Permanent Address: Nippon Steel Corporation, 6-3, Otemachi 2-chome, Chiyoda-ku, Tokyo



100-71, Japan y

Author to whom correspondence should be addressed.

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#51

I. INTRODUCTION In recent years there has been a great deal of progress in developing accurate force elds (FF) for describing the structures and properties of organic1 systems (including polymers,2 proteins,3{5 and DNA3{5) and of nonmetallic inorganic systems6 (including semiconductors and ceramics7). Indeed there are new general procedures from obtaining the parameters directly from quantum mechanical calculations.8 These FF have been used in molecular dynamics (MD) and Monte Carlo (MC) computer simulations to predict structures and properties. However, for metallic systems the situation is less sanguine. Here the lack of well de ned covalent bonds requires a dierent paradigm to relate the energy to atomic spacings. It is well known that a simple two body or pair potential is not adequate, since it leads to the Cauchy equality (C12 = C44 elastic constants for cubic systems) and to poor values of vacancy formation energies. Recent developments in formulating empirical many-body potentials suitable for metallic systems include a local volume or density dependence to describe metallic binding for example: the eective medium theory,9 the embedded atom method,10;11;61 Finnis-Sinclair,13 Rosato-Guillope-Legrand,14 and Sutton-Chen15;16 type. An alternative formulation, the interstitial electron model,17;18 includes pseudo-electrons directly in the FF calculations. All of these approaches allow a proper description of the C12 and C44 elastic constants. These empirical density-dependent or many-body interatomic potentials have been used succesfully in many applications to transition metals and intermetallic alloys such as describing point defects (vacancies), extended defects (grain boundaries, dislocations), interfaces, surface properties, microclusters, and adatom diusion.19{36 In this paper we will focus on the Sutton-Chen (SC) FF, which has been used for MD and MC simulations on metallic systems.27{36 With SC the total potential energy of the metal is taken to have the form Eq. (1) 2 3 X X 4X 1 1=25 Utot = Ui =  (1) 2 V (rij ) ; ci i

i

j 6=i

2

Here V (rij ) is a pair potential de ned by Eq. (2);

!n a V (rij ) = r (2) ij accounting for the repulsion (Pauli orthogonality) between the i and j atomic cores and i is a local density accounting for cohesion associated with atom i de ned by Eq. (3); X X a !m i =  (rij ) = (3) j 6=i j 6=i rij In Eqs. (1)-(3), rij is the distance between atoms i and j , a is a length parameter scaling all spacings (leading to dimensionless V and ), c is a dimensionless parameter scaling the attractive terms,  sets the overall energy scale, and n, m are integer parameters such that n > m. Given the exponents (n,m), c is determined by the equilibrium lattice parameter and  is determined by the total cohesive energy (Ecoh ). Sutton and Chen restricted m to be greater than 6 and used the integral power indices giving the closest agreement with the bulk modulus (B ) rst and then elastic constants. Relaxing the condition that n and m be integral, the B and the Cauchy discrepancy or pressure (Pc = C12 ; C44) can be exactly t with the analytical solution Eqs. (4) and (5) ; s    n = 3 E B 2BP + 1 (4) coh c v  B  u u t  BE  m = 6u (5) + 1 2P coh

c

where is the volume per atom. We are interested in predicting properties involving defects, surfaces, and interfaces which are not well described by the SC potential with the original parameterization (based only on the experimental lattice parameter, cohesive energy, and bulk modulus).38 In addition, the SC calculations did not include the quantum description of the phonons. We introduce here quantum corrections to take into account zero-point energy (ZPE). This allows the potential to be more useful in calculating the temperature dependence of properties. We use this quantum Sutton-Chen (Q-SC) class of potentials and optimize parameters to take into 3

account additional experimental properties: phonon frequencies (at the X point), vacancy formation energy, and surface energies. We nd that the new parameters describe defect and surface properties while retaining a good accuracy for elastic properties. We compare and contrast our results to the results obtained from the original parameterization and experiment for the fcc metals, Ni, Cu, Ag, Ir, Rh, Pd, Au, and Pt. Understanding the kinetics and thermodynamics of supercooled bulk metallic glass forming liquids is of critical importance in developing light weight high-performance amorphous metallic glasses60 Especially, determination of viscosity as a function of temperature and concentration and microscopic level studies on the kinetics of crystallization and glass formation are amenable through computer simulations. Here, we applied these new FF parameters in MD simulation of metals and alloys to 1. determine the shear viscosity of Au:Cu binary alloys61 as a function of temperature and concentration from nonequilibrium molecular dynamics (NEMD).62;63 2. study the role of atomic size in crystalization and glass formation processes in metallic alloys64 from equilibrium molecular dynamics (EMD). The method of calculation is explained in Section II. Section III details the new parameter sets for all the metals and compares the predictions of properties with experiment. In Section IV, we present MD applications using the new paramaters. First, we describe the calculation of shear viscosity of Cu as a function of temperature using NEMD method.62;63 Then, we brie y present a constant pressure constant temperature MD simulation investigating the role of size mismatch in metallic glass formers60 using Ag:Cu and Ni:Cu as two cases.

II. THE CALCULATIONAL METHODS Given a length parameter a, the optimum parameters c and  were determined by requiring that the optimum lattice parameters match experiment with the experimental cohesive energy. Given these optimum parameters, we calculated the elastic constants, the phonon 4

dispersion relations, vacancy energies, and surface energies. Such classical calculations omit the quantum eects arising from zero point vibrations. The total ZPE of the system reduces the cohesive energy and provides a pressure that increases the lattice parameters. Thus, to compare with the experimental properties at T = 0K, we must include ZPE. In addition, the temperature dependence of the phonon terms aects the higher temperature properties. Section A considers the classical approach and Section B includes the quantum eects. Hereafter we follow Ra i-Tabar and Sutton in restricting the lattice sums to jrij j  2a, where a is the lattice parameter.

A. Classical Calculations 1. Optimization Procedure

We want the parameters to lead to a lattice parameter exactly matching experiment at T = 0K. From Eq. (1) the virial is given by 2 3 X 4X 0 c X 0 5 

P = ; (6) 6 i j6=i Vij ; 1i =2 j6=i ij where the primes de ne the derivative operations:

f 0 = r @f @r

2 f 00 = r2 @@rf2

Setting Eq. (6) to 0 leads to the zero pressure equilibrium condition 2 3 X 4X 0 c X 0 5  P0 = ; 6

Vij ; 1=2 ij = 0 (7) i j6=i i j 6=i This leads directly to the lattice parameter a consistent with an external pressure P0 and hence, the equation of state. Our procedure for determining the parameters (, c, m, and n) of the potential is as follows: 5

1. Set a to be the experimental lattice parameter at 0K (Sutton and Chen set a as the lattice parameter at room temperature). The parameter c is determined by the zero pressure equilibrium condition Eq. (7) and the parameter  is chosen to obtain;

Utot = ;Ecoh

(8)

where Ecoh is the experimental cohesive energy at 0K. 2. Calculate the bulk modulus (B ) and the elastic constants (c11, c12, c44) at 0K using Eqs. (9)-(12) 8 N = X X c 1 6 00 0 0 A 7 @ ; 1=2 4 (ij ; ij ) ; 2 ij 5> (9) ; i j 6=i i j6=i 8 4 N 4 =   X X ; 1c=2 64 xr4ij 00ij ; 0ij ; 21 @ xr2ij 0ij A 75>; (10) i j 6=i ij i j6=i ij 8 N \m"

V S>\m"

where Ahkl is area per atom on the (hkl) surface and V S > \m" denotes that the atom sites whose index is larger than \m" are vacant. This is referred to as the rigid surface energy. We calculated a relaxed surface energy using the RQM in conjunction with a MD code. In this scheme we considered all atoms within a cylinder of radius 8a and of height 8a as the basic particles. This is about 6500 atoms (depending slightly upon surface direction). Of these we considered the movable particles to be those (840 atoms) within a cylinder of radius 4a and of height 4a and the density-changeable particles as those within a cylinder of 8

radius 6a and height 6a (2800 atoms). RQM allowed us to calculate an accurate relaxed surface energy without using periodic boundary conditions.

B. Quantum Corrections In order to include quantum eects, we start with the Helmholtz free energy of the metal 3 2 ~ X 4 h  ! ( k ) (19) F (T ) = Utot + Fvib(T ) = Utot + kB T ln 2 sinh 2kj T 5 B ~k;j At zero temperature,

F0K = Utot + Fvib(0K ) = Utot +

X1 ~ h !j (k) ~k;j 2

(20)

where kB is the Boltzmann constant, T is the temperature, h is Planck's constant, and !j (~k) is the phonon (angular) frequency of mode j and wave vector ~k. Changing the distance parameter a, changes the free energy explicitly by the change in Ui and implicitly by the change in !j (~k). Thus, we determine the parameters c and  so that the free energy is minimized and equal to ;Ecoh when the lattice parameter is equal to the experimental value at 0K. To determine the equilibrium lattice parameter we must add to Eq. (6) the eect of the ZPE on the pressure (generally it increases the lattice parameter). To include ZPE eects, we calculate the phonons40;41 and sum over the BZ. We also include the phonons in calculating the termal expansion and thermodynamic properties with temperature.42{45 1. Procedure

To optimize the FF parameters, we start with the n and m optimized for the classical approach in Section II.A. We then 6. Keep the parameter a as the experimental lattice parameter at 0K. 9

7. Calculate the phonon modes at the lattice parameter for 0K and obtain the total ZPE by summing over the BZ. We optimize the parameters c and  so that

F0K = ;Ecoh

(21)

simultaneously with the quantum zero pressure equilibrium condition;

where

P0 + Pvib = 0

(22)

0 1 X @X 0 X 0 A @Fvib  1 P0 + Pvib = ; 6

Vij ; c 1=2 ij ; @

i j6=i i j 6=i

(23)

8. Calculate the phonon dispersion curve using the room temperature lattice parameter to compare with experiment as in step 3. The entropy, enthalpy, speci c heat, and free energy as a function of temperature are calculated using the equilibrium lattice parameter at each temperature. 9. Calculate other physical properties (vacancy formation energy, surface energy, equation of state, thermal expansion, and speci c heat) as a function of temperature using the phonon corrections from step 8. 10. Choose the best set of parameters for predicting the properties. To calculate the speci c heat we used the calculated phonon modes and assume that each mode can be described as an independent harmonic oscillator (the quasi-harmonic approximation40;41). This leads to 2 3 2 3 2F ! " @ ! #2 ~ X 4 h !j (~k) 52 h  ! ( k ) @ j 2 4 5 Cp = kB (24) 2k T cosech 2k T + T @ 2 @T ~k;j

B

B

T

P

Using the above formulae, we calculated the properties for each fcc metal element.

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III. CALCULATIONAL RESULTS Table I presents the optimum new parameter sets for the quantum corrected SC results, denoted (q-sc) and for the classical approach (without quantum corrections), denoted (c-sc), original parameters from Sutton and Chen are denoted as (sc). In Tables II and III, various physical properties from previous works are compared. We compared these results to the following potentials: embedded atom method10;11 (denoted ea), RGL potential14 (denoted rgl), improved RGL potential by Cleri and Rosato26 (denoted irgl), and modi ed embedded atom method by Baskes61 (denoted mea). Table IV shows the fractional root mean square errors (25) using these potentials. 2 N 3 calc ; P expt !2 2 X 1 P i i 5 RMS = 4 N (25) expt P i i=1 Here Picalc and Piexpt are calculated and experimental values, respectively.

A. Elastic Properties The calculated and experimental B and elastic constants (Cij ) at 0K are presented in Table II. The RMS deviation from experiment is presented in Table IV. The best potential for elastic properties is irgl. The SC form leads to poorer accuracy for elastic constants, but this level of accuracy is retained in c-sc and q-sc while tting other properties.

B. Phonon Dispersion The calculated phonon dispersion curves for Ni, Cu, Pt and Au are presented in Figure 1. The experimental data are for room temperature, and hence we calculated the phonon dispersions using the equilibrium lattice parameters at room temperature. The phonon frequencies at the X- and L-points of the BZ are tabulated in Table II. The RMS deviations from experiment are presented in Table IV. The c-sc and q-sc potentials lead to the best descriptions of phonons, with a greatly increased accuracy over SC. 11

C. Vacancy Formation Energy The calculated vacancy formation energies at 0K (unrelaxed and relaxed) are compared in Table III. The RMS deviations from experiment are presented in Table IV. The q-sc potential leads to the best description of vacancy energies even though we did not explicitly include this in the optimization (as did eam and rgl). This shows that the SC form of the potential is suitable for describing vacancies in the fcc.

D. Surface Energy The calculated surface energies at 0K are compared in Table III. The RMS deviations from experiment are presented in Table IV. The q-sc potential leads to the best agreement with experiment. In all cases the theory leads to 110 > 100 > 111. This is consistent with simple ideas of bonding but the experimental data to test these trends is not available.

E. Phonon Density of States The predicted phonon density of states for Cu from q-sc at room temperature is given in Figure 2. This was constructed by choosing 106 random points over the BZ and using a bin size of 0.01 THz. Similar results are obtained for all fcc systems. Using Figure 2 we can calculate the free energy, entropy, speci c heat, and other properties as a function of temperature.

F. Equation of State Using Eq. (22) we calculated the pressure as a function of lattice spacing, leading to the equation of state (at room temperature). The results are presented in Table V for a compression by V=V0 = 0:80. We also show the universal equation of state by Rose, et al,46 which accurately describes uniform scaling of the cell for all metals. The q-sc parameter set 12

generally lead to better agreement with experiment than SC (except Cu for which there is equal accuracy).

G. Thermal Expansion The thermal expansion was calculated using the phonons states as described in (Eq. 13) (the quasi-harmonic approximation).47 At each temperature the optimum lattice spacing was calculated by minimizing the Helmholtz free energy (Eq. 19) in terms of the lattice spacing. The values at zero temperature and room temperature (300K) are presented in Table II. Generally the predicted thermal expansion rate agrees with experiment for T < 50K but is too large for higher T . However, the maximum error at room temperature is at most 0:22% (for Ag and Au). This excessive thermal expansion might be due to the quasi-harmonic approximation. For every metal, the q-sc parameters give better agreement with experiment than does SC.

H. Speci c Heat The speci c heat was calculated using Eq. (24) at each temperature. Here the speci c heat at 100K for each metal is presented in Table VI. The reason why we select 100K is that speci c heat increases rapidly from zero becomes saturated to the same value around room temperature. Thus, the best test point for FF in this properties is at 100K. The new parameter set leads to better agreement with experiment than SC for every metal except for Rh, Pd, and Ag, which lead to equal accuracy.

IV. MOLECULAR DYNAMICS RESULTS In this section, we present an outline of applications of the new FF to various model systems using EMD and NEMD methods. MD simulations were carried on model systems 13

with 500 to 1000 atoms per periodic cell. In the EMD simulations we used the ParinelloRahman-Nose-Hoover formalism.65 The interaction cut-o range is chosen to be the twice of the lattice parameter. The integration time step is chosen as either 2 fs (for low temperatures) or 1 fs for higher temperature simulations. In the NEMD simulations we used the SLLOD algorithm with a gaussian isokinetic thermostat.62 These calculations used 500 atom model systems with a constant integration time step of 1 fs for all temperature and shear rate ranges. We have used Lees-Edwards boundary condition.62 In both EMD and NEMD simulations, we integrated the equations of motion using a 6-value Gear Predictor Corrector algorithm.

A. Viscosity of Liquid Cu The NEMD method facilitates a direct way to evaluate the shear viscosity of liquids. Our objective is to investigate the applicability of the new FF in estimating transport properties of liquid metals and alloys. Thus, we chose liquid Cu as a case study. The results of simulations at various shear rates and temperatures are given in Figure 6. Extrapolating these results to zero shear rate, we obtain 3:68, 2:82, and 2:23 mPa s as indicate the conversion to cp theoretical values at 1500, 1750, and 2000 K, respectively. These values are in very good agreement with the experimental values,66 3:68, 2:82 and 2:23 mPa.s. The application is described in detail elsewhere where we study the viscosity of the binary alloy Au:Cu as a function of concentration and temperature.

B. Phase Transformations in Ni:Cu and Ag:Cu In this application the objective is to investigate the role of disparity in metallic radii (a microscopic parameter) on glass formation in metallic alloys. Good metallic glass formers generally involve constituents with disparate sizes. To test this we considered Ag:Cu and Ni:Cu, and selected two benchmark systems: 14

1. Ag:Cu which dier by 15% in atom radius size, and 2. Ni:Cu which dier by only 1% in atomic radius. We expect Ni:Cu to have a small barrier to crystallization whereas Ag:Cu should form a glass at high quenching rates. Starting from well equilibrated liquids, we cooled at various rates ranging dT=dt = 2  1012K=s to 4  1014K=s. For Ag:Cu, each cooling rate led to a glass. To determine the glass transition temperature, we used the Wendt-Abraham parameter which is the ratio of the distances at which the rst minimum, rmin , and the rst maximum, rmax, of radial distribution function occur, Figure 7. In contrast, for Ni:Cu and pure Cu, even the highest quenching rates used resulted in crystallization.

V. SUMMARY We developed a quantum extension of the Sutton-Chen type of many-body potential suitable for investigating various of physical properties. We report parameters for nine fcc metals. For all metals the Q-SC description leads to improved values for all properties except elastic constants. Thus, at some slight cost in the accuracy in elastic properties, the Q-SC potential is most suitable for investigating surface, interface, and defect properties. The Q-SC potential provides an accurate physical description for various mechanical and thermal properties. The SC parameters n and m remain in a narrow range, changing in a somewhat regular way over the columns or rows of the periodic table. This allows us to de ne combination rules for treating alloys. This is essential since the experimental data is insucient to determine independent parameters for alloys. We illustrate the use of these new parameters for simulations of various alloys of these systems.

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ACKNOWLEDGMENTS The research was funded by Nippon Steel Corporation and by NSF (CHE 95-22179 and ASC 92-17368). The facilities of the MSC are also supported by grants from DOEBCTR, Asahi Chemical, Chevron Petroleum Technology, Owens Corning, Saudi Aramco, Beckman Institute, Exxon Corp., Chevron Chemical Co., Asahi Glass, Chevron Research and Technology Co., Hercules, Avery Dennison, and BP Chemical.

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66 Handbook

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FIGURES FIG. 1. Phonon dispersion curves for the Q-SC (a) Ni, (b) Cu, (c) Rh, (d) Pd, (e) Ag, (f) Ir, (g) Pt, and (h) Au. The bold lines are the dispersion curves of Q-SC, the dotted lines are for the original Sutton-Chen parameter set. The dots present the experimental data (Cu, Pd, Pt, and Au from reference 51, Ag from reference 54, and Ir from reference 55). We found no experimental data for Rh. FIG. 2. Phonon density of states for Cu at room temperature (using Q-SC). FIG. 3. Equation of state: (a) Ni, (b) Cu, (c), Rh, (d) Pd, (e) Ag, (f) Ir, (g) Pt, and (h) Au. The bold lines are calcualted using Q-SC, the dashed-dotted lines using SC-new, and the dotted lines using SC. The dashed line is the universal equation of state by Rose, et al.44 The dots present experimental data (Ni, Cu, and Ag from reference 56, Pd from reference 57, Pt from reference 58, and Au from reference 59). No experimental data is available for Rh and Ir. FIG. 4. Thermal expansion for Q-SC: (a) Ni, (b) Cu, (c) Rh, (d) Pd, (e) Ag, (f) Ir, (g) Pt, and (h) Au. The bold lines are calculated using Q-SC, the dashed-dotted lines use SC-new, and the dotted lines use the original SC parameters. The dashed lines are from the experimental data,50 except for Ir where only a discrete set of points (black dots) were available. FIG. 5. Speci c heat for Q-SC: (a) Ni, (b) Cu, (c) Rh, (d) Pd, (e) Ag, (f) Ir, (g) Pt, and (h) Au. The solid lines are calculated using Q-SC, and the dotted lines use the original SC. The dots represent the experimental data.60 FIG. 6. The calculated shear viscosity of Copper at various temperatures and shear rates. Open symbols represent the calculated values, lled symbols are the zero shear rate experimental values.

22

FIG. 7. Glass transition temperature as a function of cooling rate for Ag6Cu4 . The cooling rates are B = 4  1012K=s, C = 2  1012K=s, and D = 4  1014K=s. The variation of Wendt-Abraham parameter, R = rmin =rmax, displays a discontinuity at T = Tg , is used to determine the Tg , TgD = 700K , TgB = 500K , and TgC = 450K .

23

TABLES TABLE I. Parameters sets for the Sutton-Chen many-body potential for fcc transition metals. Here (q-sc) denotes QSC, the new parameters using quantum corrections; (c-sc) denotes SC-new, new parameters developed using the classical approach; (sc) denotes the original parameters by Sutton and Chen15 as revised by Ra i-Tabar and Sutton.16 n m (eV) c a( A) Ni (q-sc) 10 5 7.3767E-3 84.745 3.5157 (c-sc) 10 5 7.5144E-3 83.073 3.5157 (sc) 9 6 1.5714E-2 39.756 3.5200 Cu (q-sc) 10 5 5.7921E-3 84.843 3.6030 (c-sc) 10 5 5.9066E-3 83.073 3.6030 (sc) 9 6 1.2351E-2 39.756 3.6100 Rh (q-sc) 13 5 2.4612E-3 305.499 3.7984 (c-sc) 13 5 2.5027E-3 299.946 3.7984 (sc) 12 6 4.9371E-3 145.658 3.8000 Pd (q-sc) 12 6 3.2864E-3 148.205 3.8813 (c-sc) 12 6 3.3401E-3 145.658 3.8813 (sc) 12 7 4.1260E-3 108.526 3.8900 Ag (q-sc) 11 6 3.9450E-3 96.524 4.0691 (c-sc) 11 6 4.0072E-3 94.948 4.0691 (sc) 12 6 2.5330E-3 145.658 4.0900 Ir (q-sc) 13 6 3.7674E-3 224.815 3.8344 (c-sc) 13 6 3.8060E-3 222.348 3.8344 (sc) 14 6 2.4524E-3 337.831 3.8400 Pt (q-sc) 11 7 9.7894E-3 71.336 3.9163 (c-sc) 11 7 9.8721E-3 70.743 3.9163 (sc) 10 8 1.9768E-2 34.428 3.9200 Au (q-sc) 11 8 7.8052E-3 53.581 4.0651 (c-sc) 11 8 7.8863E-3 53.082 4.0651 (sc) 10 8 1.2896E-2 34.428 4.0800

24

TABLE II. Comparison of calculated and experimental (exper) values for cohesive energy (Ecoh ), lattice constant (a) , and elastic constants (Cij ), and phonon modes ( ). Same notation as in Table I. The cohesive energy is for zero temperature, including vibrational energy (experimental values are from reference 48). The experimental lattice parameters at room temperature are from reference 49. The values at zero temperature were corrected by integrating the data for thermal expansion.50 The bulk modulus (B ) and elastic constants are at zero temperature (experimental data are from reference 51). Here Xi and Li are the phonon frequencies at the X-point and L-point, respectively, at room temperature (experimental data are from reference 52). i = T and L denote transverse and longitudinal polarization, respectively.

a0K

Al (exper) (q-sc) (sc) (irgl) Ni (exper) (q-sc) (sc) (ea) (rgl) (irgl) Cu (exper) (q-sc) (sc) (ea) (rgl) (irgl) Rh (exper) (q-sc) (sc) (rgl) (irgl) Pd (exper) (q-sc) (sc) (ea) (rgl) (irgl) Ag (exper) (q-sc) (sc) (ea) (rgl) (irgl) Ir (exper)

( A) 4.032 4.032 4.066 3.516 3.516 3.529

3.603 3.603 3.620

3.798 3.798 3.806 3.881 3.881 3.897

4.069 4.069 4.099

3.834

a300K Ecoh ( A) 4.050 4.054 4.123 4.050 3.524 3.529 3.544 3.520 3.520 3.523 3.615 3.622 3.641 3.615 3.610 3.615 3.804 3.808 3.816 3.800 3.803 3.891 3.897 3.913 3.890 3.890 3.887 4.086 4.095 4.123 4.090 4.090 4.085 3.839

(eV) 3.39 3.39 3.37 3.34 4.44 4.44 4.41 4.45 4.44 4.44 3.49 3.49 3.46 3.54 3.50 3.54 5.75 5.75 5.72 5.75 5.75 3.89 3.89 3.86 3.91 3.94 3.94 2.95 2.95 2.93 2.85 2.96 2.96 6.94

B

(GPa) 88.19 81.52 82.30 81.00 187.60 179.74 190.12 180.33 195.67 189.00 142.03 131.16 138.34 138.33 136.67 142.00 268.63 240.44 263.22 274.67 288.67 195.47 168.60 192.61 195.33 189.33 196.00 108.72 101.69 107.69 103.67 105.33 108.67 370.37

c11

(GPa) 122.96 100.15 92.45 95.00 261.20 225.37 226.05 233.00 257.00 257.00 176.20 164.47 164.51 167.00 182.00 176.00 422.10 314.97 334.56 384.00 392.00 234.12 214.32 239.89 218.00 236.00 232.00 131.49 126.98 136.92 129.00 132.00 132.00 599.47

25

c12

(GPa) 70.80 72.20 77.19 74.00 150.80 156.92 172.15 154.00 165.00 155.00 124.94 114.51 125.26 124.00 114.00 125.00 191.90 203.17 227.55 220.00 237.00 176.14 145.74 168.97 184.00 166.00 178.00 97.33 89.05 93.08 91.00 92.00 97.00 255.82

c44

XT

XL

LT

LL

(GPa) (THz) (THz) (THz) (THz) 30.90 9.64 5.66 9.53 4.04 40.98 8.67 5.77 8.70 3.71 20.02 5.70 3.73 5.74 2.37 37.00 131.70 8.55 6.17 8.88 4.24 97.30 8.47 5.67 8.48 3.68 77.46 7.45 5.03 7.44 3.27 128.00 9.90 6.78 9.77 4.52 93.00 136.00 9.88 6.78 9.80 4.49 81.77 7.19 5.08 7.40 3.37 71.01 6.94 4.66 6.95 3.03 56.39 6.07 4.11 6.06 2.68 76.00 7.58 5.02 7.49 3.12 68.00 82.00 7.75 5.21 7.71 3.34 194.00 N/A N/A N/A N/A 146.44 8.11 5.52 8.11 3.66 140.34 7.91 5.42 7.89 3.60 165.00 199.00 71.17 6.70 4.56 6.86 3.21 89.92 6.17 4.23 6.15 2.82 90.20 6.10 4.23 6.06 2.84 65.00 5.91 4.09 5.88 2.66 69.00 73.00 5.77 4.02 5.72 2.67 51.09 4.94 3.45 5.05 2.25 51.13 4.64 3.18 4.62 2.10 57.46 4.93 3.39 4.90 2.26 57.00 5.22 3.59 5.06 2.29 40.00 51.00 4.80 3.31 4.75 2.17 268.82 N/A N/A N/A N/A

(q-sc) (sc) (rgl) (irgl) Pt (exper) (q-sc) (sc) (ea) (rgl) (irgl) Au (exper) (q-sc) (sc) (ea) (rgl) (irgl)

3.834 3.844 3.916 3.916 3.925

4.065 4.065 4.086

3.842 3.852 3.840 3.839 3.924 3.928 3.941 3.920 3.920 3.924 4.079 4.088 4.117 4.080 4.080 4.079

6.94 6.91 6.93 6.93 5.84 5.84 5.82 5.77 5.86 5.85 3.81 3.81 3.80 3.93 3.78 3.78

339.51 361.06 367.00 414.67 288.40 264.49 271.03 283.00 261.00 295.67 180.32 175.53 156.24 167.00 174.67 165.00

437.89 471.69 505.00 554.00 358.00 321.93 308.10 303.00 329.00 341.00 201.63 207.06 177.63 183.00 192.00 187.00

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290.32 305.75 298.00 345.00 253.60 235.77 252.50 273.00 227.00 273.00 169.67 159.76 145.54 159.00 166.00 154.00

188.98 207.92 207.00 261.00 77.40 112.72 72.41 68.00 102.00 91.00 45.44 59.87 41.78 45.00 39.00 45.00

6.76 7.08

4.65 4.88

6.75 7.06

3.11 3.28

5.80 5.17 3.96

3.84 3.56 2.76

5.85 5.14 3.93

2.90 2.37 1.85

4.95 4.61 3.59 2.87

3.94 2.75 2.51 2.01

4.87 4.70 3.56 2.83

2.25 1.86 1.70 1.36

3.20

1.51

3.24

2.27

TABLE III. Comparison of calculated and experimental values for vibrational frequencies, vacancy energies, and surface energies. Same notation as in Table I. Evf is the vacancy formation energy at zero temperature (experimental data are from reference 53). hkl is the surface energy at zero temperature for the (hkl) plane. The experimental surface energies are averaged over all surfaces (from reference 54). The calculated values for Evf and hkl are both with and without relaxation. Vacancy Energies (eV) Surface Energies (mJ=m2) Unrelaxed Relaxed Unrelaxed Relaxed 100 110 111 100 110 111 Al (exper) 0.60-0.77 1140 1140 1140 (q-sc) 1.11 1.06 1181 1247 1125 1174 1235 1119 (sc) 0.43 0.38 577 630 521 543 560 497 (mea) 0.67 897 969 618 Ni (exper) 1.45-1.80 (2380) (2380) (2380) (q-sc) 1.55 1.49 2105 2225 2003 2100 2213 1997 (sc) 1.19 1.14 1560 1677 1446 1533 1625 1425 (ea) 1.63 1580 1730 1450 (rgl) 1.46 1440 1530 1190 (mea) 1.46 2435 2384 2036 Cu (exper) 1.04-1.31 1790 1790 1790 (q-sc) 1.22 1.17 1577 1667 1500 1572 1657 1495 (sc) 0.93 0.89 1166 1254 1081 1145 1214 1065 (ea) 1.28 1280 1400 1170 (rgl) 1.20 1100 1200 940 (mea) 1.14 1651 1642 1409 Rh (exper) N/A 2659 2659 2659 (q-sc) 2.26 2.21 2478 2622 2344 2475 2615 2341 (sc) 1.99 1.95 2039 2187 1886 2027 2162 1877 (rgl) 2.49 1330 1410 1080 (mea) 2.70 2902 2921 2598 Pd (exper) N/A 2003 2003 2003 (q-sc) 1.36 1.33 1337 1434 1237 1328 1416 1229 (sc) 1.21 1.19 1129 1225 1017 1109 1190 1003 (ea) 1.44 1370 1490 1220 (rgl) 1.00 850 900 680 (mea) 1.40 1659 1670 1381 Ag (exper) 1.09-1.19 1246 1246 1246 (q-sc) 0.97 0.95 890 955 824 882 940 818 (sc) 1.03 0.99 903 969 836 897 957 830 (ea) 0.97 705 770 620 (rgl) 0.78 590 620 490 (mea) 1.02 1271 1222 1087 Ir (exper) N/A 3048 3048 3048 (q-sc) 2.52 2.48 2499 2680 2309 2490 2657 2300 (sc) 2.61 2.55 2530 2713 2334 2522 2693 2326

27

Pt

Au

(rgl) (mea) (exper) (q-sc) (sc) (ea) (rgl) (mea) (exper) (q-sc) (sc) (ea) (rgl) (mea)

3.50 1.66 1.09 1.30 0.89 0.71 0.90

2.70 1.15-1.60 1.62 1.02 1.68 1.28 0.89-1.00 0.87 0.66 1.03 0.60

2907

3058

2835

1580 1116

1717 1236

1423 961

2167

2131

1656

793 673

872 746

691 580

1084

1115

886

1980

2130

1650

2489 1551 1035 1650 1310

2489 1660 1093 1750 1400

2489 1404 913 1440 1080

1506 757 623 918 525

1506 808 658 980 550

1506 670 550 790 525

TABLE IV. Fractional root mean square (RMS) deviation from experimental values for various potentials. The sampling number N indicates the available data points for each item. Type Elastic Constants Phonon Frequencies EV F Surface Energies N 9 4 5 6 (q-sc) 0.306 0.163 0.120 0.565 (sc) 0.279 0.277 0.253 0.792 (ea) 0.135 0.185 0.132 0.629 (rgl) 0.234 0.223 0.910 (irgl) 0.096 0.196 0.179 TABLE V. Pressure (GPa) in compression (V=Vo = 0:80) for fcc metals. The rst row (exper) presents experimental data (Ni, Cu, Ag from referenced 55, Pd from reference 56, Pt from reference 57, Au from reference 58). No experimental data is available for Rh and Ir. The second row (u) presents calculated value from the universal equation of state by Rose et al.46 Al Ni Cu Rh Pd Ag Ir Pt Au (exper) 27.6 71.5 55.0 N/A 74.7 44.3 N/A 120.4 78.5 (u-eos) 33.1 71.5 54.5 110.6 83.2 44.4 186.6 123.4 78.9 (q-sc) 32.6 75.6 53.7 116.0 77.0 42.9 159.3 121.0 77.2 (sc) 22.9 77.5 54.7 124.2 89.4 47.4 168.3 118.0 63.5

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TABLE VI. Speci c heat (in kB units) at 100K for fcc metals . The rst row (exper) presents experimental data from reference 59. The second row (q-sc) is our work and the third row (sc) is calculated by the Sutton-Chen original parameters. Al Ni Cu Rh Pd Ag Ir Pt Au (exper) 1.64 1.64 1.93 1.82 2.15 2.43 2.09 2.35 2.59 (q-sc) 1.65 1.70 2.00 1.76 2.15 2.48 2.03 2.36 2.68 (sc) 2.46 1.88 2.17 1.79 2.15 2.42 1.97 2.59 2.83

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