Benchmark values for molecular three-center integrals arising in the Dirac equation

Previous papers by the authors report that they obtained compact, arbitrarily accurate expressions for two-center, one- and two-electron relativistic molecular integrals expressed over Slater-type orbitals. In the present study, accuracy limits of expressions given are examined for three-center nuclear attraction integrals, which are one-electron, three-center integrals with no analytically closed-form expression. In this work new molecular auxiliary functions are used. They are obtained via Neumann expansion of the Coulomb interaction. The numerical global adaptive method is used to evaluate these integrals for arbitrary values of orbital parameters and quantum numbers. Several methods, such as Laplace expansion of Coulomb interaction, single-center expansion, and the Fourier transformation method, have previously been used to evaluate these integrals considering the values of principal quantum numbers in the set of positive integer numbers. This study of three-center integrals places no restrictions on quantum numbers in all ranges of orbital parameters.

Figure 1

Depiction of the coordinates for motion of an electron in the field of three stationary Coulomb centers, namely, $A, B, C$, where $A=\left\{X_A,Y_A,Z_A\right\}, B=\left\{X_B,Y_B,Z_B\right\}$, and $C=\left\{X_C,Y_C,Z_C\right\}$, and $\left\{X,Y,Z\right\}$ are the axes of Cartesian coordinates.

Figure 2

CPU time for computation of ${\mathcal{J}}_{N_1{N}_2}^{L\mathrm{\Lambda },q}$ auxiliary function in Eq. (20) according to methods used for calculation of the Legendre polynomials, showing the mathematica built-function (MF), recurrence relation formula (RF), explicit formula (EF), and $L=3, \mathrm{\Lambda }=1, q=0, N_1=3, N_2=2, p_1=2.5, p_2=1.5$, with $p_1=\frac{1}{2}\left(\zeta +{\zeta }^{\prime }\right)R_{AB}, p_2=\frac{1}{2}\left(\zeta -{\zeta }^{\prime }\right)R_{AB}$.