A New Method for Computation of Long Ranged ... - Springer Link

Journal of Low Temperature Physics, Vol. 139, Nos. 5/6, June 2005 (© 2005) DOI: 10.1007/s10909-005-5451-5

A New Method for Computation of Long Ranged Coulomb Forces in Computer Simulation of Disordered Systems E. Yakub1 and C. Ronchi2 1 European Commission, Joint Research Centre, Institute for Transuranium Elements, P.O. Box 2340, D-76125 Karlsruhe, Germany; 2 Computer Science Dept., Odessa State Economic University, Preobrazhenskaya 8, 65026, Odessa, Ukraine

E-mail: [email protected]

Applications of a new method for computation of Coulomb forces in Monte Carlo or molecular dynamics simulation of a wide class of disordered systems including plasmas, ionic fluids and amorphous solids is discussed. This method, based on angular averaging of Ewald sums over all orientations of the reciprocal lattice under conditions of computer simulation, eliminates periodicity artifacts imposed by conventional Ewald scheme and provides much faster computation of electrostatic energy in computer simulations of disordered condensed systems.

1.

Introduction

Computer simulations of disordered systems like plasmas, ionic fluids, amorphous solids, etc., require an accurate account for long ranged Coulomb forces which determine to the largest extent the correctness in the predicted stability of disordered condensed phases and strongly affect the predictions of their thermodynamic and transport properties. An essential problem both in conventional1 and ab initio 2 computer simulations of such systems3 is how to combine the accurate account of long ranged Coulomb forces with periodic boundary conditions (PBC). Despite the great progress achieved in the microscopic description of condensed matter since the pioneering works of Madelung4 and Ewald,5 the correct description of Coulomb forces under conditions of computer simulation still remains a topical issue.6 Usual Ewald summation procedure when applied to simulation of disordered Coulomb 633 0022-2291/05/0600-0633/0 © 2005 Springer Science+Business Media, Inc.

634

E. Yakub and C. Ronchi

systems invokes a non-isotropic electric field having the cubic symmetry of the crystalline lattice composed of main cells as elementary units. It results in an artificial “crystalline field” effects in simulation of fluids, amorphous solids and other spatially uniform condensed phases.7 The mean Coulomb interaction energy of two particles at given distance r in uniform systems is independent of orientation of vector r. This is obviously not the case in the Ewald scheme, and the example presented in Ref. 7 shows how important may be these effects of artificial “crystalline field” in real simulations. In ab initio computer simulations,3 the number of particles in the main cell is very limited, mainly by supercomputer facilities available. Periodicity artifacts imposed by conventional Ewald summation procedure may appear a major issue here. The absolute value of the periodicity artifacts in effective Coulomb interaction is almost negligible at small distances, even for a relatively small number N of charged particles in the cell, but becomes essential at larger distances. For instance, for N = 200, which is characteristic for up-to-date ab initio simulations, using the best available computing codes and facilities,9,10 the maximum value of the Ewald artifact is about 10% in energy in third coordination sphere and reaches ∼ 100% at the edge of the cell.7 Accurate computer simulations of biochemical and other systems having complex elemental composition require sometimes up to a million particles8 in the main Monte Carlo or molecular dynamics computation cell1 . Obviously, the larger the number of charged particles in the main cell, the more acute is the problem of the effective evaluation of the electrostatic contribution. Fast and accurate evaluation of electrostatic fields is important therewith for any size of the cell. Periodicity artifacts in this case are small but heavy processor load imposed by conventional Ewald summation procedure is crucial in such simulations. We apply here the recent modification of the Ewald scheme7 to a few simplest systems both spatially uniform and ordered. In Section 2. we outline the method, in Section 3. we present its generalization on one-component plasma (OCP) and new Monte Carlo simulation results on OCP and restricted primitive ionic model (RPIM) fluid, as well as calculations of Madelung constant for three ordered structures, in Section 4. we discuss possible applications of the method in study of a wide class of disordered Coulomb systems including highly compressed fluids and amorphous phases of cryogenic solids and present some concluding remarks.

1

Thereinafter simply called main cell.

Computation of Coulomb Forces in Disordered Systems

2.

635

PRE-AVERAGED EFFECTIVE POTENTIALS

Let us consider a standard cubic main cell of edge L and volume V = L3 , P containing N = M α=1 Nα charged particles of M sorts. The electrostatic forces acting between the i-th and the j-th particles obey the Coulomb law: Fij =

Qi Qj 2 . 4πε0 rij

Here rij = |r i −r j | is the distance between i-th and j-th charged particles and Qi is the value of the i-th point charge of type α : Q(α) = {Q(1) , . . . , Q(M ) }. We shall here assume that the electro-neutrality condition N X

Qi =

M X

Nα Q(α) = 0

(1)

α=1

i=1

is satisfied, and the standard PBC are imposed as described in Refs. 1,11. The total Coulomb energy of N charges in the main cell is: (C)

UN =

X

Qi ϕ(r i ) ,

1≤i≤N

where ϕ(r i ) is the electrostatic potential at the position r i of i-th charge. According to Ewald,5 this contribution is, in turn, the sum of one- and twoparticle terms: N 1X ϕ(r i ) = ϕ1 (r i ) + ϕ2 (r i , r j ) . 2 j6=i In the absence of external field, the unary potential is a constant: Qi ϕ1 = 4πε0 L

"

#

1 X 1 π 2 n2 δ exp − −√ 2 2 2π n>0 n δ π

!

(2)

and the binary contribution can be written as follows:5,11 ϕ2 (r ij ) = +

Qj 4πε0

(

erfc(δ(rij /L)) rij "

#

1 X 1 π 2 n2 2π exp − cos n · r ij πL n>0 n2 δ2 L 

)

.

(3)

Here δ/L is the conventional Ewald parameter,5,11 n/L is the three-dimensional reciprocal lattice site vector (n = |n|), and erfc(x) is the complementary error function.

636

E. Yakub and C. Ronchi

Keeping in mind that all orientations of the main cell in an isotropic media should be equivalent, we can average both sides of (3) over all directions of the vector n at a fixed distance rij . If the brackets h. . .i =

1 4π

Z +1

d(cos ϑ)

−1

Z π

dψ . . . ,

−π

indicate averaging, whereas ψ, ϑ are the polar and azimuthal angles defining the direction of the vector n (n · r = nr cos ϑ), we can determine the preaveraged (effective) potential as ϕ2 (rij ) ≡ hϕ2 (r ij )i. Integration of (3) over all orientations of the vector n gives immediately: ϕ2 (rij ) = +

Qj rij erfc δ 4πε0 rij L 



"



#

  1 X 1 π 2 n2 2π exp − sin nr ij 2π 2 n>0 n3 δ2 L

)

.

(4)

The pre-averaged charge–charge potential (4) is a continuous function of the inter-particle distance rij and can be expanded in converging power series in terms of this distance. Since both erfc(x) − 1 and sin(x) are odd functions   X Qj  2k+1  ϕ2 (rij ) = 1+ Ck rij . (5) 4πε0 rij k≥0

The coefficients Ck in (5) are found in Ref. 7 by direct expansion of (4) in a MacLaurin’s series. The procedure is straightforward: by applying the Euler–MacLaurin formula — generalized for summation over three-dimensional integers n,12 — the following result holds: C0 =

"

#

1 X 1 π 2 n2 2δ exp − 2 − √ , 2 π n>0 n δ π

2π , 3L3 = 0, for k > 1 .

C1 = Ck

By taking into account the electro-neutrality condition (1), it can be seen that the term in (5), which is independent of distance (proportional to C0 ) and the one-particle contribution (2) cancel one another. This implies that the total Coulomb energy of N charged particles in the main cell can be described by the sum: (C)

UN = −

N X i=1

N X N 3Q2i 1X ˜ ij ) , + φ(r 16πε0 rm 2 i=1 j=1 j6=i

(6)

Computation of Coulomb Forces in Disordered Systems

637

˜ ij ) is an effective potential defined by: where φ(r        2  Qi Qj 1 + 1 r r − 3 4πε0 r 2 rm rm ˜ φ(r) = 

0

r < rm

(7)

r ≥ rm

and rm = (3/4π)1/3 L is the radius of the volume-equivalent sphere of the 3 = L3 . main cell: 43 πrm The pair effective (pre-averaged) potential (7) has the following properties: • It tends to the pure Coulomb pair potential at small distances. • It is zero at r = rm and remains zero at r > rm . • Its first derivative is zero at r = rm .

• Its range is related to the size of the main cell: rm = 0.62035 L. The last property entails some inconsistency with the initial main cell configuration. If the range of interaction does not exceed L/2 each particle in the cell contributes (or not) to the sum of interactions with the selected one just one time: either as original object (inside the main cell) or as one of its “images.” This is not the case for potential (7) because rm > L/2. The subsequent modification of the simulation algorithm was described in Ref. 7. Two different zones exist within the effective sphere surrounding an arbitrary charge in the main cell.7 The first zone contains charged particles (inside the main cell) or their images (outside it) which contribute to the sum of pair interactions of the chosen charge just one time. The second zone contains those charged particles which contribute twice to the interaction energy with the given charge — both as original charges and as their additional “phantom” images.7

3.

MONTE CARLO SIMULATION 3.1.

One-Component Plasma

One-component plasma (OCP), i.e. a system of point ions placed in neutralizing background is a text-book example of classical Coulomb system for which the standard Ewald approach was successfully applied.13,14 OCP may be regarded as a limiting case of a two-component system when the number of e.g. negative ions increases and the charge per ion decreases, thus maintaining the electroneutrality. Separating the interactions of the negative (background) charge distribution in (6) and replacing summation over negative ions by integration over uniformly distributed background charge, one

638

E. Yakub and C. Ronchi

gets the following generalization of (6) for OCP: (OCP)

UN

= −0.9

N X N N Q2 1X ˜ ij ) , + φ(r 4πε0 rm 2 i=1 j=1

(8)

j6=i

where N is the number of positive ions. It should be noted that both terms in (8) are size-dependent but not directly proportional to N . The first contribution in (8) at given temperature T may be expressed in terms of plasma parameter Γ = Q2 N 1/3 (4πε0 rm kT )−1 as −0.9kT Γ N 2/3 . (OCP) Table 1. Comparison of predicted Monte Carlo internal energies −UMC /N kT

for one-component plasma.

Γ

N = 64 (This work)

N = 216 13

1

0.599 ± 0.010

0.580

43.073 ± 0.019

43.094

141.00 ± 0.016

140.890

10

7.895 ± 0.020

50 100

87.555 ± 0.033

160 200 a) b)

176.62 ± 0.012

N = 686 14

7.996 87.480

87.52a) 141.72 176.77b)

Best fit value Fluid initial conditions

We performed two series of Monte Carlo simulations using the method outlined above and the algorithm described in Ref. 7 to prove that (8) gives correct estimations of the OCP energy at relatively small N and the mean (OCP) value of UN /N converges for large N . Detail of Monte Carlo simulation procedure is the same as reported earlier.16 The results are presented in Table 1 for Γ = 1 . . . 200 at fixed N = 64 and compared in Fig. 1 to results obtained within standard Ewald scheme by Hansen,14 and Stringfellow, DeWitt and Slattery14 at fixed Γ = 100 for wide range of N .

3.2.

Restricted Primitive Ionic Model

Another simple model system suitable for testing of the method is the Restricted Primitive Ionic Model (RPIM), i.e. two-component fluid build from oppositely charged hard spheres of one size (diameter σ). We performed an additional set of Monte Carlo simulations for RPIM fluid at

Computation of Coulomb Forces in Disordered Systems

639

(MC)

Thermal contribution to internal energy ∆E (th) /N kT = (UN − of one-component plasma at fixed Γ = 100 and different sizes of the main cell. Comparison of our Monte Carlo simulation results (open circles) with data of Hansen13 (solid square), and the “best fit” value recommended in Ref. 14 (solid triangle). Cubic spline interpolation of data presented in Table 3.3. was util(bcc) ized to calculate the Madelung bcc-contribution UN .

Fig. 1.

(bcc) UN )/N kT

one reduced temperature T ∗ = 4πε0 kT σQ−2 and different packing fractions η = πN σ 3 (6V )−1 for wide range of N to test the convergence for large N . In Fig. 2 we compare our calculations with results of Larsen.15 3.3.

Ordered Solid Structures

Perfect crystalline structures are the worst application case for a preaveraged effective potential essentially devised for spatially uniform systems. Nevertheless, the test on the perfect NaCl structure presented in Ref. 7 proves the ability of the method to predict accurate values of the electrostatic energy even in ordered condensed systems (see Table 1 in Ref. 7). In this work we performed additional tests of the method using three other cubic crystalline structures. The values of the Madelung constant for bcc-structure of Coulomb crystal formed in OCP, CsCl lattice and fluorite (CaF2 ) structure are computed from (6)–(7) and presented in Tables 3.3., 3.3. and 3.3. at different numbers of charged particles in the main cell.

640

E. Yakub and C. Ronchi

Fig. 2. Deviations of Monte Carlo internal energy from data of Larsen:15 ∆E/kT = (MC)

(MC)

(UN − UN )/N kT at fixed temperature T ∗ = 4πε0 kT σQ−2 = 0.595 and four different packing fractions plotted against the size of the main cell: 1 – η = 0.005023 (squares), 2 – η = 0.02058 (triangles), 3 – η = 0.04842 (circles), 4 – η = 0.09525 (diamonds).

4.

CONCLUDING REMARKS

Pre-averaging the Ewald sums over all spatial directions leads to an energy formulation which eliminates periodicity artifacts imposed by conventional Ewald scheme and provides a fast method for computation of electrostatic contribution to the energy of disordered dense systems in Monte Carlo or molecular dynamics computer simulations.7 Additional calculations and Monte Carlo simulation results presented in Section 3. prove the ability of the method to produce accurate simulation results at low computational cost. Monte Carlo simulations of one-component plasma (small cell size N = 64) presented in Table 1 are in good agreement with other simulation data.13,14 At the same time the comparison presented in Fig. 1 shows that it is very likely coincidental. The convergence to a constant value at large N is nonmonotonous. Predicted thermal contribution to internal energy ∆E (th) /N kT at Γ = 100 shown in Fig. 1 is in accordance with data of Hansen13 but is higher than predicted in Ref. 14. This rather small (about 0.5% of the total energy) difference is well beyond the total estimated error of both simulations. This discrepancy may be due to periodicity artifacts and requires additional study because it may affect the location of OCP crystallization point.

Computation of Coulomb Forces in Disordered Systems

641

Table 2. Predicted values of the Madelung constant for OCP Coulomb crystal (bcc lattice). L/a is the ratio of the cell size to the lattice constant, in the second and third columns are numbers of charged particles in the cell and in the equivalent sphere. The fourth column show their difference (the actual net charge of the equivalent sphere). The last two columns present the predicted Madelung constants and their deviations from the exact value. L/a 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Nr of particles in: Cell Sphere 16 15 54 59 128 137 250 259 432 411 686 701 1024 965 1458 1459 2000 1989 2662 2685 3456 3479 4394 4477 5488 5577 6750 6735

Deviation ∆(N ) −1 5 9 9 −21 15 −59 1 −11 23 23 83 89 −15 Exact =

Madelung constant This work ∆,% −0.89230 −0.41 −0.90361 +0.86 −0.89985 +0.44 −0.89921 +0.37 −0.89595 +0.00 −0.89523 −0.08 −0.89411 −0.20 −0.89510 −0.09 −0.89591 +0.00 −0.89656 +0.07 −0.89648 +0.06 −0.89633 +0.04 −0.89621 +0.03 −0.89579 −0.02 −0.89593

Our simulation of restricted primitive ionic model fluid presented in Fig. 2 is in good agreement with the results of Larsen.15 The smaller the packing fraction the faster is the convergence. As it was found in Ref. 7, the pre-averaged effective potential originally devised for spatially uniform systems provides surprisingly fast convergence of predicted energy to exact values in ordered crystalline structures, despite the inevitable non-zero net charge inside the equivalent sphere. New calculations of Madelung constant provided in Tables 3.3.–3.3. confirm this feature for three other periodic structures. Note that the calculated electrostatic energy oscillates upon approaching the exact value. The non-monotonous convergence to the large-N limit is also characteristic for OCP and RPIM models. This feature must be taken into account in precision calculations for obtaining results in this limit. We believe the implementation of pre-averaged potentials in existing and upcoming conventional and ab initio simulation packages may essentially speed up the simulations and decrease the ab initio simulation errors caused by periodicity artifacts, especially near the phase transition points.

642

E. Yakub and C. Ronchi

Table 3. Predicted values of the Madelung constant for CsCl lattice. Columns are the same as in the Table 2, except for the deviations in number of ions and net charge (columns four and five) differ. L/a 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Numb. inside: Cell Sphere 16 15 54 59 128 137 250 259 432 411 686 701 1024 965 1458 1459 2000 1989 2662 2685 3456 3479 4394 4477 5488 5577 6750 6735

Deviations of: Number Charge −1 −1 5 −5 9 25 9 −7 −21 −5 15 21 −59 5 1 19 −11 53 23 29 23 7 83 −67 89 25 −15 79

Madelung constant This work ∆,% 1.7568 −0.33 1.7081 −3.10 1.8137 +2.89 1.7419 −1.18 1.7687 +0.34 1.7719 +0.52 1.7527 −0.57 1.7711 +0.48 1.7612 −0.08 1.7604 −0.13 1.7669 +0.24 1.7577 −0.28 1.7636 +0.05 1.7633 +0.03

Exact =

1.76267

Table 4. Predicted values of the Madelung constant for the fluorite structure. Columns 3 to 6 show the deviations in number of ions in equivalent spheres centered on Ca2+ and F− ions, and the net charges within that spheres. L/a

Ncell

2 4 6 8 10 12 14 16 18 20 22

12 96 324 768 1500 2592 4116 6144 8748 12000 15972

Nsphere − Ncell Ca2+ F− −3 −1 3 13 −29 −17 −39 −43 −1 5 31 19 93 61 −19 −61 −17 −89 21 37 5 83

Sphere net Ca2+ −6 30 −34 18 94 38 18 154 −34 −30 298

charge F− 1 −25 5 −5 −53 −7 23 −35 89 −1 −227

Exact =

Madelung constant This work ∆, % 4.8351 −4.06 5.1141 +1.47 5.0442 +0.09 5.0223 −0.35 5.0424 +0.05 5.0531 +0.26 5.0454 +0.11 5.0370 −0.06 5.0379 −0.04 5.0422 +0.05 5.0407 +0.02 5.0398

Computation of Coulomb Forces in Disordered Systems

643

REFERENCES 1. R.W. Hockney and J.W. Eastwood, Computer Simulation Using Particles, McGraw-Hill (1981). 2. R. Car and M. Parrinello, Phys. Rev. Lett. 55, 2471 (1985). 3. See, e.g., S.A. Bonev, B. Militzer, and G. Galli, Phys. Rev. B 69, 01 4101 (2004), and references therein. 4. E. Madelung, Phys. Z. 19, 524 (1918). 5. P.P. Ewald, Ann. Phys.(Leipzig) 64, 253 (1921). 6. G.J. Martyna and M.E. Tuckerman, J. Chem. Phys. 110, 2810 (1999); Z.-H. Duan and R. Krasny, J. Chem. Phys. 113, 3492 (2000). 7. E. Yakub and C. Ronchi, J. Chem. Phys. 119, 11 556 (2003). 8. H.-Q. Ding, N. Karasawa, and W.A. Goddard, J. Chem. Phys. 97, 4309 (1992). 9. F. Gygi, Ab-initio Molecular Dynamics Code GP 1.20.0, Lawrence Livermore National Laboratory (1999–2003). 10. W. Andreoni and A. Curioni, Parallel Computing 26, 819 (2000). 11. D.C. Rappaport, The Art of Molecular Dynamics Simulation, Cambridge University Press (1995). 12. E.M. Lifshitz and L.P. Pitaevskii, Electrodynamics of Continuous Media. Course of Theoretical Physics Vol. 8. 2. ed., Butterworth-Heinemann, Oxford... (2000). 13. J.P. Hansen, Phys. Rev. A 8, 3110 (1973). 14. G.S. Stringfellow, H.E. DeWitt, and W.L. Slattery, Phys. Rev. A 41, 1105 (1990). 15. B. Larsen, Chem. Phys. Lett. 27, 47 (1974). 16. E.S. Yakub, J. Low Temp. Phys. 122, 559 (2001).