A B-spline Hartree-Fock program

A B-spline version of a Hartree-Fock program is described. The usual differential equations are replaced by systems of non-linear equations and generalized eigenvalue problems of the form (H-a epsilon B-aa)P-a = 0, where a designates the orbital. When orbital a is required to be orthogonal to a fixed orbital, this form assumes that a projection operator has been applied to eliminate the Lagrange multiplier. When two orthogonal orbitals are both varied, the energy must also be stationary with respect to orthogonal transformations. At such a stationary point, the matrix of Lagrange multipliers, epsilon(ab) = (P-b vertical bar H-a vertical bar P-a), is symmetric and the off-diagonal Lagrange multipliers may again be eliminated through projection operators. For multiply occupied shells, convergence problems are avoided by the use of a single-orbital Newton-Raphson method. A self-consistent field procedure based on these two possibilities exhibits excellent convergence. A Newton-Raphson method for updating all orbitals simultaneously has better numerical properties and a more rapid rate of convergence but requires more computer processing time. Both ground and excited states may be computed using a default universal grid. Output from a calculation for Al 3s(2)3p P-2 shows the improvement in accuracy that can be achieved by mapping results from low-order splines on a coarse grid to splines of higher order onto a refined grid. The program distribution contains output from additional test cases. Program summary Program title: SPHF version 1.00 Catalogue identifier: AEIJ_v1_0 Program summary URL: http://cpc.cs.qub.ac.uk/summaries/AEIJ_v1_0.html Program obtainable from: CPC Program Library, Queen's University, Belfast, N. Ireland Licensing provisions: Standard CPC icense, http://cpc.cs.qub.ac.uk/licence/licence.html No. of lines in distributed program, including test data, etc.: 13 925 No. of bytes in distributed program, including test data. etc.: 714 254 Distribution format: tar.gz Programming language: Fortran 95 Computer: Any system with a Fortran 95 compiler. Tested on Intel Xeon CPU X5355, 2.66 GHz Operating system: Any system with a Fortran 95 compiler Classification: 2.1 External routines: LAPACK (http://www.netlib.org/lapack/) Nature of problem: Non-relativistic Hartree-Fock wavefunctions are determined for atoms in a bound state that may be used to predict a variety atomic properties. Solution method: The radial functions are expanded in a B-spline basis W. The variational principle applied to an energy functional that includes Lagrange multipliers for orthonormal constraints defines the Hartree-Fock matrix for each orbital. Orthogonal transformations symmetrize the matrix of Lagrange multipliers and projection operators eliminate the off-diagonal Lagrange multipliers to yield a generalized eigenvalue problem. For multiply occupied shells, a single-orbital Newton-Raphson (NR) method is used to speed convergence with very little extra computation effort. In a final step, all orbitals are updated simultaneously by a Newton-Raphson method to improve numerical accuracy. Restrictions: There is no restriction on calculations for the average energy of a configuration. As in the earlier HF96 program [2], only one or two open shells are allowed when results are required for a specific LS coupling. These include: 1. (n1)(N)n's, where l = 0. 1. 2. 3 2. (np)Nn'l, where l = 0. 1.2.3.... 3. (nd)(n' f) Unusual features: Unlike HF96, the present program is a Fortran 90/95 program without the use of COMMON. It is assumed that Lapack libraries are available. Running time: For Ac 7s(2)7p P-2 the execution time varied from 6.9 s to 9.1 s depending on the iteration method. References: [1] C. Froese Fischer, Adv. At. Mol. Phys. 55 (2008) 235. [2] G. Gaigalas, C. Froese Fischer, Comput. Phys. Common. 98 (1996) 255. (C) 2011 Elsevier B.V. All rights reserved.