Benchmark values for molecular two-electron integrals arising from the Dirac equation

Abstract

The two-center two-electron Coulomb and hybrid integrals arising in relativistic and nonrelativistic ab initio calculations on molecules are evaluated. Compact, arbitrarily accurate expressions are obtained. They are expressed through molecular auxiliary functions and evaluated with the numerical Global-adaptive method for arbitrary values of parameters in the noninteger Slater-type orbitals. Highly accurate benchmark values are presented for these integrals. The convergence properties of new molecular auxiliary functions are investigated. The comparison for two-center two-electron integrals is made with results obtained from single center expansions by translation of the wave function to a single center with integer principal quantum numbers and results obtained from the Cuba numerical integration algorithm, respectively. The procedures discussed in this work are capable of yielding highly accurate two-center two-electron integrals for all ranges of orbital parameters.

Analytical evaluation of relativistic molecular integrals. I. Auxiliary functions

“…The attempts in this regard failed due to the absence of benchmark values. The authors in their previous work [2,61,62] first focused on this issue e.g., obtaining results for molecular integrals with unquestionable precision via numerical techniques. They presented benchmark values for two-and three-center integrals.…”

“…Besides, performing this operation in advance may block analogously generalization of relativistic molecular auxiliary functions [2], P n 1 ,q n 2 n 3 n 4 (p 123 ) Q n 1 ,q n 2 n 3 n 4 (p 123 )…”

“…given. A discussion was made on the relativistic molecular auxiliary functions introduced previously [2]. They are used when the principal quantum numbers in a Slater-type orbital basis (STO) [3] are free from any restriction [4],…”

Analytical evaluation of relativistic molecular integrals. II: Method of computation for molecular auxiliary functions involved

“…Accordingly, these approximations applied for solution of various problems [86][87][88][89][90][91][92][93][94]. Finally, the benchmark results for two−, three−center one− and two−electron molecular integrals which have been demanded for years in order to test the accuracy of any analytical method to be derived, obtained via numerical global-adaptive method through Gauss-Kronrod numerical integration extension [95] by us [96][97][98] with 35 correct decimals. A new molecular auxiliary functions introduced [97].…”

“…Finally, the benchmark results for two−, three−center one− and two−electron molecular integrals which have been demanded for years in order to test the accuracy of any analytical method to be derived, obtained via numerical global-adaptive method through Gauss-Kronrod numerical integration extension [95] by us [96][97][98] with 35 correct decimals. A new molecular auxiliary functions introduced [97]. Note that, the analytical evaluation of these functions involve some challenges namely, power functions with non-integer exponents, incomplete gamma functions and their multiplications have no explicit closed-form relations.…”