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

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.

Figure 1

The comparison of numerical and series representation calculations of ${\mathcal{P}}_{100100100}^{0,100}(p_1,12,9.6)$ auxiliary functions depending on $p_1$. The solid black line and the dashed black line is the numerical and series representation formulas, respectively.

Figure 2

The comparison of numerical and series representation calculations of ${\mathcal{P}}_{505050}^{0,50}(p_1,12,9.6)$ auxiliary functions depending on upper limits of summation of series representation formulas with different values of $p_1$.