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

Abstract

The Slater-type orbital basis with non-integer principal quantum numbers is a physically and mathematically motivated choice for molecular electronic structure calculations in both non-relativistic and relativistic theory. The non-analyticity of these orbitals at $r=0$, however, requires analytical relations for multi-center integrals to be derived. This is nearly insurmountable. Previous papers by the present authors eliminated this difficulty. Highly accurate results can be achieved by the procedure described in these papers, which place no restrictions on quantum numbers in all ranges of orbital parameters. The purpose of this work is to investigate computational aspects of the formulae given in the previous paper. It is to present a method which helps to increase computational efficiency. In terms of the processing time, evaluation of integrals over Slater-type orbitals with non-integer principal quantum numbers is competitive with those over Slater-type orbitals with integer principal quantum numbers.

Appendices

Appendix A

Integrating Eq. (30) by parts twice while $q=0$: in the first operation we use, for Eq. (35),

$$\begin{aligned} U=\left( \xi +\nu \right) ^{n_{2}}\mathrm{e}^{-p_{2}\xi -p_{3}\nu }, \quad \mathrm{d}V=\left( \xi -\nu \right) ^{n_{3}}\mathrm{d}\xi , \end{aligned}$$

for Eq. (36),

$$\begin{aligned} U=\left( \xi -\nu \right) ^{n_{3}}\mathrm{e}^{-p_{2}\xi -p_{3}\nu },\quad \mathrm{d}V=\left( \xi +\nu \right) ^{n_{2}}\mathrm{d}\xi ; \end{aligned}$$

thus we have

$$\begin{aligned} \mathcal {G}^{n_{1},0}_{n_{2}n_{3}}(p_{123}) &= {} \frac{p_{2}}{\left( n_{3}+1 \right) } \mathcal {G}^{n_{1},0}_{n_{2},n_{3}+1}(p_{123})\nonumber \\&\quad -\frac{n_{2}}{\left( n_{3}+1 \right) } \mathcal {G}^{n_{1},0}_{n_{2}-1,n_{3}+1}(p_{123}) -\frac{1}{\left( n_{3}+1 \right) } \mathcal {N}^{n_{1},0}_{n_{2},n_{3}+1}(p_{123}), \end{aligned}$$

(59)

$$\begin{aligned} \mathcal {G}^{n_{1},0}_{n_{2}n_{3}}(p_{123}) &= {} \frac{p_{2}}{\left( n_{2}+1 \right) } \mathcal {G}^{n_{1},0}_{n_{2}+1,n_{3}}(p_{123})\nonumber \\&\quad -\frac{n_{3}}{\left( n_{2}+1 \right) } \mathcal {G}^{n_{1},0}_{n_{2}+1,n_{3}-1}(p_{123}) -\frac{1}{\left( n_{2}+1 \right) } \mathcal {N}^{n_{1},0}_{n_{2}+1,n_{3}}(p_{123}). \end{aligned}$$

(60)

Performing the second operation on the first terms on the right-hand side of Eqs. (59, 60) with,

$$\begin{aligned} U=\left( \xi +\nu \right) ^{n_{2}}\left( \xi -\nu \right) ^{n_{3}+1}, \mathrm{d}V=\mathrm{e}^{-p_{3}\nu }\mathrm{d}\nu , \end{aligned}$$

$$\begin{aligned} U=\left( \xi +\nu \right) ^{n_{2}+1}\left( \xi -\nu \right) ^{n_{3}}, \mathrm{d}V=\mathrm{e}^{-p_{3}\nu }\mathrm{d}\nu , \end{aligned}$$

the Eqs. (35, 36) are obtained, respectively.

Appendix B

Using the series expansion of exponential functions $\mathrm{e}^z$, where $z=-p_{3}\nu$ in the Eq. (30) while $q=0$ we have

$$\begin{aligned} \mathcal {G}^{n_{1},0}_{n_{2}n_{3}}(p_{123}) &= {} \frac{p_{1}^{n_{1}}}{\Gamma \left( n_{1}+1 \right) } \sum _{s=0}^{\infty }\left( -1 \right) ^{s}\frac{p_{3}^{s}}{\Gamma \left( s+1 \right) }\nonumber \\&\quad \times \int _{1}^{\infty }\int _{-1}^{1} \nu ^{s}\left( \xi +\nu \right) ^{n_{2}}\left( \xi -\nu \right) ^{n_{3}}\mathrm{e}^{-p_{3}\xi }\mathrm{d}\xi \mathrm{d}\nu . \end{aligned}$$

(61)

Dividing and multiplying the integral on the right-hand side, by $\xi ^{s}$ gives:

$$\begin{aligned} {^{\nu }\mathcal {G}^{n_{1},s,s}_{n_{2}n_{3}}(p_{123})} &= {} \frac{p_{3}^{s}}{\Gamma \left( s+1 \right) }\nonumber \\&= {} \int _{1}^{\infty }\int _{-1}^{1} \xi ^{-s} \left( \xi \nu \right) ^{s} \left( \xi +\nu \right) ^{n_{2}} \left( \xi -\nu \right) ^{n_{3}} \mathrm{e}^{-p_{2}\xi } \mathrm{d}\xi \mathrm{d}\nu . \end{aligned}$$

(62)

Equation (40) is obtained by analogously generalizing power functions $\xi ^{-s}$, $\left( \xi \nu \right) ^{s}$ to $\xi ^{-q_{1}}$, $\left( \xi \nu \right) ^{q_{2}}$, respectively.

Appendix C

The normalized form of incomplete beta functions is used:

$$\begin{aligned} \mathfrak {B}_{nn'}\left( z \right) = \frac{B_{nn'}\left( z \right) }{B_{nn'}}; \mathfrak {B}_{nn'}\left( z \right) =1-\mathfrak {B}_{n'n}\left( 1-z \right) . \end{aligned}$$

(63)

They are usually represented by “I”; however, we use “$\mathfrak {B}$” in order to avoid any confusion with the Bessel functions. Thus, the recurrence relations and the derivatives used in the present work have the following form:

$$\begin{aligned} \mathfrak {B}_{nn'}\left( z \right) =\mathfrak {B}_{n+1,n'-1}\left( z \right) +\frac{z^{n}\left( 1-z\right) ^{n'-1}}{nB_{nn'}}, \end{aligned}$$

(64a)

$$\begin{aligned} \mathfrak {B}_{nn'}\left( z \right) =\mathfrak {B}_{n-1,n'+1}\left( z \right) +\frac{z^{n-1}\left( 1-z\right) ^{n'}}{n'B_{n'n}}, \end{aligned}$$

(64b)

$$\begin{aligned} \frac{\partial \mathfrak {B}_{nn'}\left( z \right) }{\partial z} =\frac{\left( z\right) ^{n-1}\left( 1-z\right) ^{n'-1}}{B_{nn'}}, \end{aligned}$$

(64c)

$B_{nn'}=B_{n'n}$ and $z=\frac{\xi +1}{2\xi }$, $1-z=\frac{\xi -1}{2\xi }$.

Appendix D

Two-center overlap integrals over STO with integer n.

This defines the two-center overlap integrals for STOs. Thus, the binomial series expansion may be used for the power functions $\left( \xi +\nu \right) ^{n_{2}}$, $\left( \xi -\nu \right) ^{n_{3}}$:

$$\begin{aligned} {\mathcal {G}^{n_{1},q}_{n_{2}n_{3}}}(p_{123}) &= {} \frac{p_{1}^{n_{1}}}{\Gamma \left( n_{1}+1\right) } \sum _{s=0}^{n_{2}+n_{3}} F_{s}\left( n_{2},n_{3}\right) \nonumber \\&\quad \times \int _{1}^{\infty }\xi ^{n_{1}+n_{2}+q-s}\mathrm{e}^{-p_{2} \xi }\mathrm{d}\xi \int _{-1}^{1}\nu ^{q+s}\mathrm{e}^{-p_{3} \nu }\mathrm{d}\nu , \end{aligned}$$

(65)

where $F_{s}{\left( n,n' \right) }$: generalized binomial coefficients (Guseinov 1970; Pople and Beveridge 1970),

$$\begin{aligned} F_{s}{\left( n,n' \right) } =\sum _{s'=\frac{1}{2}\left[ \left( s-n \right) +\vert s-n \vert \right] }^{\mathrm{min}\left( s,n' \right) }(-1)^{s'}F_{s-s'}(n)F_{s'}(n'), \end{aligned}$$

(66)

where the coefficients $F_{s}(n)$ are the binomial coefficients indexed by n, s which is usually written as $\left( \begin{array}{cc} n \\ s \end{array} \right)$ with,

$$\begin{aligned} \left( \begin{array}{cc} n \\ s \end{array} \right) =\frac{\Gamma \left( n+1\right) }{\Gamma \left( s+1\right) \Gamma \left( n-s+1\right) }. \end{aligned}$$

(67)

If the orbital parameters $\left\{ \zeta , \zeta ' \right\}$ are equal $\left( p_{2}=p,p_{3}=0 \right)$, the only integral to be computed takes the form:

$$\begin{aligned} A_{\alpha }\left( p\right) =\int _{1}^{\infty } \tau ^{\alpha }\mathrm{e}^{-p \tau }\mathrm{d}\tau , \end{aligned}$$

(68)

and for $p>0$ are easily and stably generated by up-ward recursion in $\alpha$ for all positive p.

If the orbital parameters are different $\left( p_{2}=p,p_{3}=p t \right)$, then an additional and much more difficult integral arises:

$$\begin{aligned} B_{\alpha }\left( p \right) =\int _{-1}^{1} \tau ^{\alpha }\mathrm{e}^{-p \tau }\mathrm{d}\tau . \end{aligned}$$

(69)

A down-ward recursive procedure,

$$\begin{aligned} B_{\alpha }\left( p t \right) =\frac{1}{p}\left[ \alpha B_{\alpha -1}\left( p t \right) +\left( -1\right) ^{\alpha }\mathrm{e}^{p t}-\mathrm{e}^{-p t} \right] , \end{aligned}$$

(70)

for these integrals, stable for all $\alpha$ and pt was given in Corbató (1956). It uses modified Bessel functions. Bessel functions are first generated by Eq. (70), after which the $B_{\alpha }$ are given as linear combinations of them. This procedure, however, requires more computational effort than an optimal use of up- and down-ward recursion directly in $B_{\alpha }$. By representing the starting values of down-ward recursion formula as incomplete gamma functions, the behavior of the $B_{\alpha }$ integrals was also investigated in Harris (2004). In another study (Guseinov and Mamedov 2007), in order to calculate the $B_{\alpha }$ integrals for large values of pt, the following sum (Mulliken et al. 1949) was used:

$$\begin{aligned} B_{\alpha }\left( p t \right) =\left( -1\right) ^{\alpha +1}A_{\alpha }\left( -p t \right) -A_{\alpha }\left( p t \right) . \end{aligned}$$

(71)

For small values of pt, the finite series representations of $A_{\alpha }$ integrals,

$$\begin{aligned} A_{\alpha }\left( p t \right) =\mathrm{e}^{-p t}\sum _{s=1}^{\alpha +1}\frac{\alpha !}{\left( p t\right) ^{s}\left( \alpha -s+1 \right) !} \end{aligned}$$

(72)

by replacing s with $\alpha -s+1$; $0 \le s \le \alpha$, were used first. Afterwards, the infinite series representations of exponential functions were used in the resulting expression. Improvements on this formulae were made according to comments in Barnett (2002) on results in Guseinov and Mamedov (1999) (in fact, the formulae given in Guseinov and Mamedov (2002) were more carefully coded, not expressed with any significant change). Regardless of the method, an infinite sum that needs to be accurately calculated, is obtained in fine.