Calculation of two-center integrals involving a rapidly oscillating free electron wave function

Abstract

Quantum Monte Carlo calculations of the dissociation energies of three-electron hemibonded radical cationic dimersOptical potentials are used in a quantum mechanical treatment of loss processes, e.g., ionization, where the loss of flux is described by the imaginary part. We present a numerical method for calculating two-center two-electron integrals necessary to construct the imaginary part of the optical potential. By introducing Slater-type orbitals with complex-valued exponents ͑CSTOs͒, we are able to represent the free electron wave with a limited number of CSTOs. For the representation of free electron wave functions with many oscillations, i.e., in a large r range or for a high kinetic energy, these new CSTOs form a more natural set of basis functions. The introduction of CSTOs is inevitable for the calculation of integrals concerning collisions in the mK energy range, where the interaction acts over large internuclear distances. Extensive numerical checks show that the final imaginary part of the optical potentials can be calculated with an accuracy better than 2%.

Formulas and numerical table for the radial part of overlap integrals with the same screening parameters of Slater-type orbitals

“…The radial part of overlap integrals with the same screening parameters can be obtained by repeated application of recurrence relations (Eqs. 16,17,18,19,20,21,22,23) and an analytical formula (Eq. 2).…”

“…STOs are still used in most semiempirical methods, usually as minimal valence sets, sometimes including d functions [15]. In ab initio theory, GTOs are now widely established, except for atomic ab initio calculations, which often employ STO bases [16,17,18] and some benchmark STO calculations on small molecules [19,20,21,22]. A comparison of STO and GTO bases needs about twice the size of a STO basis to obtain comparable accuracy [20,21]; thus, STOs bases are still very attractive.…”

“…[23,24,25,26]. A variety of approaches for the evaluation of molecular two-electron multicenter integrals in STO bases also exists, for example, one-center expansions [27, 28,29], recurrence schemes [30], integral transformation methods [31,32], and other methods [22,33].…”

Electric and magnetic multipole integrals in STO basis sets: an analytical approach

“…Equation 11 diers from the corresponding equation in the treatment of Zheng and Zerner [39] only in that it does need master formulae. Their one-centre results are special cases of Eq.…”

“…denote coordinate functions with respect to A. It has been shown that it is always possible to express a real spherical harmonic centred in P in terms of Cartesian monomials [11,39,40]:…”