Hylleraas–Configuration Interaction calculations on the 1¹S ground state of helium atom

Abstract

Hylleraas–Configuration Interaction (Hy–CI) calculations on the ground 1¹S state of helium atom are presented using s-, p-, d-, and f-Slater orbitals of both real and complex form. Techniques of construction of adapted configurations, optimization of the orbital exponents, and structure of the wave function expansion are explored. A new method to evaluate the two-electron kinetic energy integrals occurring in the Hy–CI method has been tested in this work and compared with other methods. The non-relativistic Hy–CI energy values are ≈10 picohartree accurate, about 2.2 × 10⁻⁶ cm⁻¹. The Hy–CI calculations are compared with Configuration Interaction (CI) and Hylleraas (Hy) calculations employing the same orbital basis set, same computer code, and same computer machines. The computational required times are reported.

Keywords:

• Helium atom
• Hylleraas–Configuration Interaction
• Hylleraas
• Configuration Interaction
• Slater orbitals
• Kinetic energy integrals

Acknowledgements

This paper is dedicated to Rudi van Eldik, Professor of Inorganic Chemistry at the Department of Chemistry and Pharmacy of our University. Electronic Structure calculations counts among his fields of interest and attention. This work treats about the noble gas helium, the smallest of the inorganic substances. The author is deeply indebted to Peter Otto for his teaching on electron structure computer programing, support, and encouragement of this project. Finally, the author would like to thank James S. Sims for interesting discussions about the Hy–CI method and the helium atom, and also to James S. Sims and Stanley A. Hagstrom for providing the values of some matrix elements to check the program code.

Disclosure statement

No potential conflict of interest was reported by the author.

Notes

¹ Along this paper we are citing the ground state energy of helium atom calculated by the ICI method [16], worldwide best value: −2.9037 2437 7034 1195 9831 1159 2451 9440 4446 6969 924 865 E_h as exact value for chemical and physical purposes. For convenience, we write down in the tables only 20 decimal digits.

² As the Hy–CI wave function consists on only one r_ij per configuration and treating now helium atom with only two electrons, the terms of the Hamiltonian including and vanish.

³ The symmetry of the two-electron integrals should be taken into account. We write here only one case for each symmetry.

⁴ All the calculations of this paper have been carried out in a single computer without parallelization. The computer is an Intel Xeon E3-1240 V2 3.4 GHz machine with four processors. The compiler used is the Intel Fortran Composer XE 2013 for Linux. The program is compiled automatically to QP with the command ifort -r16 -i8 -o program.x program.f.