A program for accurate solutions of two-electron atoms

Abstract

We present a comprehensible computer program capable of treating non-relativistic ground and excited states for a two-electron atom having infinite nuclear mass. An iterative approach based on the implicitly restarted Arnoldi method (IRAM) is employed. The Hamiltonian matrix is never explicitly computed. Instead the action of the Hamiltonian operator on discrete pair functions is implemented. The finite difference method is applied and subsequent extrapolations gives the continuous grid result. The program is written in C and is highly optimized. All computations are made in double precision. Despite this relatively low degree of floating point precision (48 digits are not uncommon), the accuracy in the results can reach about 10 significant figures. Both serial and parallel versions are provided. The parallel program is particularly suitable for shared memory machines such as the Sun Starcat series. The serial version is simple to compile and should run on any platform. Program summary Title of program: corr2el Catalogue identifier: ADUX Program summary URL: http://cpc.cs.qub.ac.uk/summaries/ADUX Program obtainable from:CPC Program Library, Queen's University of Belfast, N. Ireland Distribution format: tar.gz Computer for which the program is designed and others on which it has been tested: Computers: Sun Fire 15K StarCat, Sun Ultra SPARC III, PC Operating systems or monitors under which the program has been tested: Sun Solaris 9, Linux Programming language used: ANSI C Memory required to execute with typical data: 3 Mwords or more No. bits in a word: 32 No. processors used: arbitrary Has the code been vectorized or parallelized: parallelized Number of lines in distributed program, including test data, etc.:5885 Number of bytes in distributed program, including test data, etc.: 26 199 Nature of physical problem: The Schrödinger equation for two-electron atoms is solved using finite differences. Method of solution: An iterative eigenvalue-solver that requires only the action of the Hamiltonian on a trial function is applied. The two-electron wave function is expanded in a sum of partial waves. The finite difference method is then applied to approximate the derivatives of the pair functions. The total action of the Hamiltonian on the partial waves, including correlation effects, is computed using highly optimized routines. Restriction on the complexity of the problem: The Hamiltonian employed here does not take relativistic or finite nuclear mass effects into account. The amount of computing time may become unreasonable for excited states far above the ground state. The use of double precision puts a limit on the accuracy obtainable. Typical running time: This ranges from half a minute (to obtain 10 significant figures for the s-limit of the Helium ground state) to perhaps a day for advanced examples depending on the level of parallelization. Unusual features of the program: The implicitly restarted Arnoldi method used to obtain the eigenvalues is implemented by using the ARPACK/PARPACK program library [ http://www.netlib.org/arpack]. This package also depends on the standard numerical libraries BLAS and LAPACK [ http://www.netlib.org/lapack]. Good performance is obtained by using Sun's optimized performance libraries [ http://www.sun.com]. By using a 64-bit environment (Ultra SPARC III and Solaris 9), memory limitations are non-problematic. Shared memory is used in the parallel version. Fast communication between the nodes is made over shared memory using Sun's implementation of MPI.