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Manuscript Title: Accurate crystal fields for embedded cluster calculations.
Authors: M. Klintenberg, S.E. Derenzo, M.J. Weber
Program title: Ewald
Catalogue identifier: ADME_v1_0
Distribution format: tar.gz
Journal reference: Comput. Phys. Commun. 131(2000)120
Programming language: C.
Computer: Silicon Graphics (Origin 2000 and Power challenge).
Operating system: IRIX 6.2/6.4/6.5.5.
RAM: 16M words
Word size: 32
Keywords: Embedded Cluster calculation, Crystal field potential, Ewald potential, Solid state physics.
Classification: 7.3.

Nature of 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.

Solution method:
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.

The algorithm can be applied to any ionic crystal where the unit cell information is available.

Unusual features:
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.

Running time:
Typical running time is less than 40 minutes on a R10000 processor, slightly longer on a PII 450MHz (Linux) system.

[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).