Coefficients of Fractional Parentage in J-Representation

The article investigates coefficients of fractional parentage in j-representation, compulsory for the relativistic approach in the theory of multielectron atoms or multiply charged ions, in which the electrons are relativistic. The LS-coupling is broken in the relativistic approximation - the concepts of orbital and spin quantum numbers lose their meaning, that is, they become bad quantum numbers. The method of coefficients of fractional parentage is more convenient than the method of determinant functions, although it is more complex at first glance. The coefficients of fractional parentage in the j - representation allow us to obtain antisymmetric functions of the states of any N - electron j - subshell of equivalent electrons with any number of N-p and p electrons in these subshells. In principle, this makes it possible to study the matrix elements of an operator with any number of electrons (one-electron operator, two-electron operator, three-electron operator, etc.). The study of coefficients of fractional parentage is conducted in the formalism of the method of irreducible tensor operators using the method of secondary quantization. Simple relations have been received between coefficients of fractional parentage with p-number of split electrons and submatrix elements of p operators of the change in the number of electrons acting in double space - the space of angular momentum and quasispin. Working in double space allows obtaining analytical expressions for coefficients of fractional parentage in cases of subshells with two and three electrons. In cases of a larger number of electrons, simple recurrent relations between the coefficients of fractional parentage with any p-number of split electrons were obtained. The acquired recurrence relations allow us to find the values of the coefficients even in complex cases without resorting to the general procedure for finding them. Particularly, simple recurrence relations are obtained between the genealogical coefficients for partially and almost filled j-subshell. Simple recurrence relations are also gained in cases where the numbers of precedence are separately distinguished.

j-представление, electron creation operator, electron annihilation operator, secondary quantization, quasispin, j-representation, coefficients of fractional parentage, antisymmetric wave function, seniority number, Wigner-Eckart theorem

1. Arvanitaki A., Huang J., and Van Tilburg K. Searching for dilaton dark matter with atomic clocks // Phys. Rev. D. - 2015. - Vol. 91. - № 015015. - pp. 1-17.

2. Osin D., Gillaspy J. D., Reader J. and Ralchenko Yu. EUV magnetic-dipole lines from highly-charged high-Z ions with an open 3d shell // Eur. Phys. J. D. - 2012. - Vol. 66, - № 286. - pp. 1-10.

3. Ficek F., Kimball D. F. J., Kozlov M. G., Leefer N., Pustelny S., and Budker D. Constraints on exotic spin-dependent interactions between electrons from helium fine-structure spectroscopy // Phys. Rev. A. - 2017. - Vol. 95. - № 032505. - pp. 1-9.

4. Roberts B. M., Blewitt G., Dailey C., Pospelov M., Rollings A., Sherman J., Williams W., and Derevianko A. Search for domain wall dark matter with atomic clocks on board global positioning system satellites // Nat. Commun. - 2017. - Vol. 8. - № 1195. - pp. 1-9.

5. Zhao Z. L., Wang K., Li S., Si R., Chen C.Y., Chen Z. B., Yan J., and Ralchenko Yu. Multiconfiguration Dirac-Hartree-Fock calculations of forbidden transitions within the 3dk ground configurations of highly charged ions (Z=72-83) // At. Data Nucl. Data Tables. - 2018. - Vol. 119, 314.

6. Froese Fisher C., Gaigalas G., and Jönsson P. Core Effects on Transition Energies for Configurations in Tungsten Ions // Atoms. - 2017. - Vol. 5. - № 7. - pp. 1-34.

7. Дирак П. А. М. Собрание научных трудов. Т. 2. Квантовая теория (научные статьи 1924-1947), М., ФИЗМАТЛИТ, 2003. 848 с. - (Классики науки) - ISBN 5-9221-0381-4 (Т. II)

8. Берестецкий В. Б., Лифшиц Б. М., Питаевский Л. П. Релятивистская квантовая теория: ч.1. - М.: Наука, 1968, - 480 с.

9. Собельман И. И. Введение в теорию атомных спектров. - М.: Физматгиз, 1963. - М.: Книга по требованию, 2013. - 642 с.

10. Юцис А. П., Савукинас А. Ю. Математические основы теории атома. - Вильнюс: Минтис, 1973. - 480 с.

11. Кычкин И. С., Сивцев В. И. Оператор Брейта энергии межэлектронного взаимодействия // Успехи современной науки. - 2017. - Т. 7. - № 3. - С. 56-61.

12. Кычкин И. С., Сивцев В. И. Релятивистские матричные элементы оператора энергии электростатического взаимодействия в случае одной подоболчки эквивалентных электронов // Вестник СВФУ. - 2018. - № 6. - С. 86-93.

13. Кычкин И. С., Сивцев В. И. Операторы рождения и уничтожения электронов - двойной неприводимый тензорный оператор // Вестник СВФУ. - 2019. - № 2. - C. 39-50.

14. Юцис А. П., Бандзайтис А. А. Теория момента количества движения в квантовой механике. - Вильнюс: Мокслас, 1977. - 470 с.