Cite this article as:

Nechaev V. V., Ziganshina O. D., Suchkova N. K. Calculation of Atomic Integrals with Exponentialy Correlated Functions. Izvestiya of Saratov University. New series. Series Physics, 2012, vol. 12, iss. 1, pp. 18-25.

Heading: 
Physics
UDC: 
539.182/.184, 519.677
Language: 
Russian

Calculation of Atomic Integrals with Exponentialy Correlated Functions

Authors: 
Nechaev Vladimir Vladimirovich, Yuri Gagarin State Technical University of Saratov, 77, Politechnicheskaya Str., Saratov, 410054, Russia, vl-nechaev@ya.ru
Ziganshina Olga Dmitrievna, Saratov State Technical University named after Yuri Gagarin, 77, Politechnicheskaya str., Saratov, 410054, Russia, ziganshinaod@front.ru
Suchkova Natalia K., Saratov State University, 410012, Russia, Saratov, Astrakhanskaya street, 83, rinoa_27@mail.ru
Abstract

A new type of correlation atomic integrals occurring in variation energy calculations of three-particle Coulomb systems is studied. A integrand in them along with an interparticle distance linear term under an exponent additionally contains a quadratic term. It is demonstrated that these integrals are analytically expressed through Faddeeva function of a pure imaginary argument and its derivatives. A stable and fast algorithm for calculation of Faddeeva function derivatives to the twentieth order is developed. The test values of the studied special functions are provided.

Key words: 
atomic integrals, three-particle Coulomb systems, derivatives of Faddeeva function, stable algorithm for calculation
References

1. Shershakov D. A., Nechaev V. V., Berezin V. I. Exponential basis functions with quadratic dependence on interelectron distance for variational calculations of two-electron atoms // J. Phys. B. 2000. Vol. 33, № 1. P. 123–130.

2. Calais J.-L., Löwdin P. O. A simple method of treating atomic integrals containing function of r12 // J. Mol. Spectr. 1962. Vol. 8, № 3. P. 203–211.

3. Pekeris C. L. Ground state of two-electron atoms // Phys. Rev. 1958. Vol. 112, № 5. P. 1649–1658.

4. Sack R. A., Roothan C. C. J., Kolos W. Recursive evalution of some atomic integrals // J. Math. Phys. 1967. Vol. 8, № 5. P. 1093–1094.

5. Эфрос В. Д. Задача трех тел. Обобщенное экспоненциальное разложение, произвольные состояния в коррелированном базисе и энергия связи мезомо- лекул // Журн. эксперим. и теорет. физ. 1986. Т. 90, № 1. С. 10–24.

6. Ley-Koo E., Bunge C. F., Jauregui R. Evalution of relativistic atomic integrals using perimetric coordinates // Intern. J. Quant. Chem. 1997. Vol. 63, № 1. P. 93–97.

7. Фаддеева В. Н., Терентьев Н. М. Таблицы значений функции 2 2 0 ( ) (1 2 / ) z z t w z e i e dt = + π ∫ от комплексно- го аргумента. М., 1954. 268 с.

8. Абрамовиц М., Стиган И. Справочник по специ- альным функциям. М., 1979. 832 с.

9. Gautschi W. Efficient computation of the complex error function // SIAM J. Numer. Anal. 1970. Vol. 7, № 1. P. 187–198. 

10. Poppe G. P. M., Wijers C. M. More efficient computation of the complex error function // ACM Trans. Math. Soft. 1990. Vol. 16, № 1. P. 38–46.

11. Maplesoft, «Maple», Version 15, Waterloo Maple Inc. (2012). URL: http://www.maplesoft.com (дата обращения: 01.07.2012).

12. Wolfram Research, Inc., «Mathematica», Version 8.0, Champaign, IL (2012). URL: http://www.wolfram.com (дата обращения: 01.07.2012).

13. Джоунс У., Трон В. Непрерывные дроби / пер. с англ. М., 1985. 414 с.

14. Cody W. J. Rational Chebyshev approximations for the error function // Math. Comput. 1969. Vol. 23, № 107. P. 631–637.

Full text (in Russian):
download
(downloads: 79)