‘tripleint_cc’: A program for 2-centre variational leptonic Coulomb potential matrix elements using Hylleraas-type trial functions, with a performance optimization study

Abstract

We present a program used to calculate intricate three-particle integrals for variational calculations of solutions to the leptonic Schrödinger equation with two nuclear centres in which inter-leptonic distances (electron–electron and positron–electron) are included directly in the trial functions. The program has been used so far in calculations of He – H ¯ interactions and positron H 2 scattering, however the precisely defined integrals are applicable to other situations. We include a summary discussion of how the program has been optimized from a ‘legacy’-type code to a more modern high-performance code with a performance improvement factor of up to 1000. Program summary Program title: tripleint.cc Catalogue identifier: AEEV_v1_0 Program summary URL: http://cpc.cs.qub.ac.uk/summaries/AEEV_v1_0.html Program obtainable from: CPC Program Library, Queen's University, Belfast, N. Ireland Licensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.html No. of lines in distributed program, including test data, etc.: 12 829 No. of bytes in distributed program, including test data, etc.: 91 798 Distribution format: tar.gz Programming language: Fortran 95 (fixed format) Computer: Modern PC (tested on AMD processor) [1], IBM Power5 [2] Cray XT4 [3], similar Operating system: Red Hat Linux [1], IBM AIX [2], UNICOS [3] Has the code been vectorized or parallelized?: Serial (multi-core shared memory may be needed for some large jobs) RAM: Dependent on parameter sizes and option to use intermediate I/O. Estimates for practical use: 0.5–2 GBytes (with intermediate I/O); 1–4 GBytes (all-memory: the preferred option). Classification: 2.4, 2.6, 2.7, 2.9, 16.5, 16.10, 20 Nature of problem: The ‘tripleint.cc’ code evaluates three-particle integrals needed in certain variational (in particular: Rayleigh–Ritz and generalized-Kohn) matrix elements for solution of the Schrödinger equation with two fixed centres (the solutions may then be used in subsequent dynamic nuclear calculations). Specifically the integrals are defined by Eq. (16) in the main text and contain terms proportional to r i k × r j k / r i j , i ≠ j , i ≠ k , j ≠ k , with r i j the distance between leptons i and j. The article also briefly describes the performance optimizations used to increase the speed of evaluation of the integrals enough to allow detailed testing and mapping of the effect of varying non-linear parameters in the variational trial functions. Solution method: Each integral is solved using prolate spheroidal coordinates and series expansions (with cut-offs) of the many-lepton expressions. 1-d integrals and sub-integrals are solved analytically by various means (the program automatically chooses the most accurate of the available methods for each set of parameters and function arguments), while two of the three integrations over the prolate spheroidal coordinates ‘ λ’ are carried out numerically. Many similar integrals with identical non-linear variational parameters may be calculated with one call of the code. Restrictions: There are limits to the number of points for the numerical integrations, to the cut-off variable itaumax for the many-lepton series expansions, and to the maximum powers of Slater-like input functions. For runs near the limit of the cut-off variable and with certain small-magnitude values of variational non-linear parameters, the code can require large amounts of memory (an option using some intermediate I/O is included to offset this). Unusual features: In addition to the program, we also present a summary description of the techniques and ideology used to optimize the code, together with accuracy tests and indications of performance improvement. Running time: The test runs take 1–15 minutes on HPCx [2] as indicated in Section 5 of the main text. A practical run with 729 integrals, 40 quadrature points per dimension and itaumax = 8 took 150 minutes on a PC (e.g., [1]): a similar run with ‘medium’ accuracy, e.g. for parameter optimization (see Section 2 of the main text), with 30 points per dimension and itaumax = 6 took 35 minutes.