Analytical evaluation of relativistic molecular integrals: III. Computation and results for molecular auxiliary functions

Bağcı, A.; Hoggan, P. E.

doi:10.1007/s12210-020-00953-3

Analytical evaluation of relativistic molecular integrals: III. Computation and results for molecular auxiliary functions

Research Paper
Published: 22 September 2020

Volume 31, pages 1089–1103, (2020)
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Abstract

This work describes the fully analytical method for the calculation of the molecular integrals over Slater-type orbitals with non-integer principal quantum numbers. These integrals are expressed through relativistic molecular auxiliary functions derived in our previous paper (Bağcı and Hoggan in Phys Rev E 91(2):023303, 2015). The procedure for computation of the molecular auxiliary functions is detailed. It applies both in relativistic and non-relativistic electronic structure theory. It is capable of yielding highly accurate molecular integrals for all ranges of orbital parameters and quantum numbers.

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Acknowledgements

More details of the method of computation and the results presented are available in an electronic form upon request to AB at the e-mail address: ali.bagci@yahoo.com.tr. In this study, the author A. B. was supported by the Scientific Research Coordination Unit of Pamukkale University under the project number 2020BSP011.

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Department of Physics, Faculty of Arts and Sciences, Pamukkale University, Çamlaraltı, Kınıklı Campus, 20160, Denizli, Turkey
A. Bağcı
Institute Pascal, UMR 6602 CNRS, University Blaise Pascal, 24 avenue des Landais, BP 80026, 63177, Aubiere Cedex, France
P. E. Hoggan

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Vectorized form for the m auxiliary functions given in Sects. 3 and IV

$m^{n_{1}}_{n_{2}}\left( p\right) $ auxiliary function computation requires special attention. These functions contain hypergeometric functions. Reliable and fast computation with finite-precision for these functions is a subject studied in computer science. Here we avoid calculating the hypergeometric functions in every step of computation. The details for these functions are given in this section.

In vector forms for $m_{1}$ and $m_{2}$ auxiliary functions, we have for $p_{3}=0$,

$$\begin{aligned} \begin{array}{c} m_{1}[s_{5}+1]=\frac{1}{4}c_{11}m_{11}[s_{5}]+\frac{1}{2}c_{12}m_{12}[s_{5}] \\ m_{2}[s_{5}+1]=\frac{1}{4}c_{21}m_{21}[s_{5}]+\frac{1}{2}c_{22}m_{22}[s_{5}] \\ m_{12}[s_{5}+1]=m_{1}[s_{5}] \\ m_{22}[s_{5}+1]=m_{2}[s_{5}] \\ M_{1}[s_{4},s_{1}]=m_{1}[s_{5}+2]/f[s_{4}] \\ M_{2}[s_{4},s_{p}]=m_{2}[s_{5}+2]/f[s_{4}], \end{array} \end{aligned}$$

(49)

where f is used to represent the factorials in Eq. (22), $2s_{1}\le s_{5} \le N+2s_{1}$, $s_{p}=q-s_{1}$, N is the upper limit of summation, respectively. $s_{6}$ defines the size of the list that contain all the required values of m auxiliary functions.

For $p_{3} \ne 0$,

$$\begin{aligned} m_{1}[s_{4},1]=\,\,& {} 4r_{11}[s_{4},1]m_{1}[s_{4}-2,1]\nonumber \\&+r_{12}[s_{4},1]m_{1}[s_{4}-1,1] \end{aligned}$$

(50)

$$\begin{aligned} m_{1}[s_{4},2]=\,\,& {} 4r_{11}[s_{4},2]m_{1}[s_{4}-2,2]\nonumber \\&+r_{12}[s_{4},2]m_{1}[s_{4}-1,2] \end{aligned}$$

(51)

$$\begin{aligned} m_{2}[s_{4},1]=\,\,& {} 4r_{21}[s_{4},1]m_{2}[s_{4}-2,1]\nonumber \\&+r_{22}[s_{4},1]m_{2}[s_{4}-1,1] \end{aligned}$$

(52)

$$\begin{aligned} m_{2}[s_{4},2]=\,\,& {} 4r_{21}[s_{4},2]m_{2}[s_{4}-2,2]\nonumber \\&+r_{22}[s_{4},2]m_{2}[s_{4}-1,2] \end{aligned}$$

(53)

$$\begin{aligned}&\begin{array}{rl} m_{1}[s_{4},s_{6}]=\frac{1}{4}c_{11}[s_{4},s_{6}]m_{1}[s_{4},s_{6}-2] \\ +c_{12}[s_{4},s_{6}]m_{1}[s_{4},s_{6}-1] \end{array} \end{aligned}$$

(54)

$$\begin{aligned}&\begin{array}{rl} m_{2}[s_{4},s_{6}]=\frac{1}{4}c_{21}[s_{4},s_{6}]m_{2}[s_{4},s_{6}-2] \\ +c_{22}[s_{4},s_{6}]m_{2}[s_{4},s_{6}-1] \end{array} \end{aligned}$$

(55)

with $r_{11},r_{12},r_{21},r_{22}$ and $c_{11}, c_{12}, c_{21},c_{22}$, e.g., used to represent the coefficients for the recurrence relationships given in Eqs. (25–35), respectively. “r” and “c” letters indicate the row and column elements of the matrices. The numbers used together with these letters such as $m_{1}$, $m_{2}$ are for the first and second auxiliary functions $l^{n_{1},q_{1}}_{n_{2}n_{3}}\left( p_{123}\right) $ arising in Eq. (24).

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Bağcı, A., Hoggan, P.E. Analytical evaluation of relativistic molecular integrals: III. Computation and results for molecular auxiliary functions. Rend. Fis. Acc. Lincei 31, 1089–1103 (2020). https://doi.org/10.1007/s12210-020-00953-3

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Received: 30 May 2020
Accepted: 29 August 2020
Published: 22 September 2020
Issue Date: December 2020
DOI: https://doi.org/10.1007/s12210-020-00953-3

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Analytical evaluation of relativistic molecular integrals: III. Computation and results for molecular auxiliary functions

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Vectorized form for the m auxiliary functions given in Sects. 3 and IV

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Analytical evaluation of relativistic molecular integrals: III. Computation and results for molecular auxiliary functions

Abstract

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Similar content being viewed by others

Conceptual density functional theory: status, prospects, issues

Permanent electric dipole moment of diatomic molecules using relativistic extended–coupled–cluster method

Molecular Mechanics: Principles, History, and Current Status

References

Acknowledgements

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Corresponding author

Additional information

Publisher's Note

Vectorized form for the m auxiliary functions given in Sects. 3 and IV

Vectorized form for the m auxiliary functions given in Sects. 3 and IV

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