PROGRAM SUMMARY
Title of program:
Ewald
Catalogue identifier:
ADME
Ref. in CPC:
131(2000)120
Distribution format: tar gzip file
Operating system: IRIX 6.2, 6.4 and 6.5.5
High speed store required:
16MK words
Number of bits in a word:
32
Number of lines in distributed program, including test data, etc:
2359
Keywords:
Embedded Cluster calculation, Crystal field potential,
Ewald potential, Solid state physics.
Programming language used: C
Computer:
Silicon Graphics (Origin 2000 and Power challenge).
Nature of physical problem:
The electrostatic potential, due to the crystal field (CF) in a periodic
system can be calculated using the Ewald summation method [1]. To model
local electronic structure defects in ionic crystals a periodic approach
should not be used because a periodic approach can not handle defects
with charge or dipole moment. Instead the system is modelled as a
quantum cluster (QC) embedded in an array of point charges. In this
paper we describe how to choose these point charges to obtain Ewald-like
electrostatic potentials in the interior of the QC.
Method of solution
The method of solution can be divided into 5 steps. 1) An array of
typically 1.0e4 point charges is generated using 2Nx2Nx2N unit cells.
2) Three zones are identified. Zone 1 is defined by the QC atoms. Zone
2 consists of approximately 500 point charges that embed the QC
spherically. These point charges have their ionic charges. Zone 3 is
the remainder of the array. These point charge values are later treated
as parameters. 3) The Ewald potentials are calculated for all sites in
zone 1 and 2. 4a) If the system has any dipole moment this is removed
using a random Deltaq approach. (Note that this dipole moment could be
removed directly in the system of linear equations but we have found
that better solutions are usually obtained if the dipole moment is
removed before solving the system of equations.) 4b) Solving a system
of linear equations give the charge values (Deltaq's) for the zone 3
sites so that the direct sum electrostatic potential in zone 1 and 2
become equal to the correspoinding Ewald values. 5) Using 1000 randomly
chosen points in the interior of the QC the result is checked and
Deltarms is computed.
Deltarms = (Sigma(i-Nr)[VEwald(ri) - VDS(ri)]**2/Nr)**(1/2),
Nr is the number of random points in the QC VEwald(ri) and VDS(ri) are
the Ewald and direct sum potential at ri, respectively. Deltarms is
usually < 1muV using our algorithm.
Restrictions on the complexity of the program
The algorithm can be applied to any ionic crystal where the unit cell
information is available.
Typical running time
Typical running time is less than 40 minutes on a R10000 processor,
slightly longer on a PII 450MHz (Linux) system.
Unusual features of the program
The numerical libraries CLAPACK and CBLAS [2] are used for the linear
algebra operations. dgelsx computes the minimum norm least squares
solution to an over- or under-determined system of linear equations
using a complete orthogonal factorization. A pseudo-random number
generator (randome.c) [3] is also provided and needed for the code.
References
[1] P.P. Ewald. Ann. Phys. 64, 253 (1921).
[2] E. Anderson, Z. Bai, C. Bischof, J. Demmel, J. Dongarra, J. Du Croz,
A. Greenbaum, S. Hammarling, A. McKenney, S. Ostrouchov,
D. Sorensen. LAPACK Users' Guide, Society for Industrial and
Applied Mathematics, 1995, Philadelphia, PA, ISBN 0-89871-345-5.
http://www.netlib.org/clapack/
[3] P. L'Ecuyer. Comm. of the ACM, 31, 742-749 (1988).