Violation and Conservation of Symmetries in Atomic Systems

• Název práce v češtině: Narušení a zachování symetrií v atomových systémech
• Název v anglickém jazyce: Violation and Conservation of Symmetries in Atomic Systems
• Klíčová slova: Hartree-Fock|Dirac-Hartree-Fock|metoda vázaných klastrů|konfigurační interakce|elektronová korelace|narušení symetrie|adaptace k symetrii
• Klíčová slova anglicky: Hartree-Fock|Dirac-Hartree-Fock|coupled cluster method|configuration interaction|electron correlation|symmetry breaking|adaptation to symmetry
• Akademický rok vypsání: 2017/2018
• Typ práce: disertační práce
• Jazyk práce: angličtina
• Ústav: Katedra chemické fyziky a optiky (32-KCHFO)
• Vedoucí / školitel: doc. Mgr. Jaroslav Zamastil, Ph.D.
• Řešitel: skrytý - zadáno a potvrzeno stud. odd.
• Datum přihlášení: 27.09.2018
• Datum zadání: 27.09.2018
• Datum potvrzení stud. oddělením: 08.10.2018
• Datum a čas obhajoby: 22.03.2023 10:00
• Datum odevzdání elektronické podoby: 03.02.2023
• Datum odevzdání tištěné podoby: 06.02.2023
• Datum proběhlé obhajoby: 22.03.2023
• Oponenti: doc. RNDr. Martin Čížek, Ph.D.

Zásady pro vypracování

This project will be a purely theoretical research. Numerical (and partially symbolic) calculations will be carried out in Matlab, analytical work in Maple.

We intend to solve the project in the following steps:
- Calculate the atomic integrals according to a method previously developed by the supervisor (Phys. Rev. A 79, 0225026 (2007)). This method calculates the integrals via recurrence relations that follow from the symmetries of the hydrogen-like functions.
- Program the HF method based on the above mentioned efficient method of calculation of atomic integrals in its conventional and perturbation forms (i.e. as an expansion in 1/Z, Z being the charge of the nucleus).
- Search for singular points of the HF energies by a method also previously developed by the supervisor (J. Phys. A 38,4009 (2005)): Given a sufficiently large number (10-20) of sufficiently accurate perturbation energies, one can deduce the position and character of the singularity closest to the point of the expansion.
- Formulate the stability conditions following the ones given in Phys. Rev. A 2, 2268 (1970).

In addition, if there is enough time left after the completion of the above listed goals, we would like to further focus on the following:
- derivation of the pertinent coupled cluster equations for atomic systems with broken symmetry HF wave function;
- programming of the correlation methods (coupled cluster and configuration interaction) for atomic systems with broken-symmetry HF wave function.

Seznam odborné literatury

[1] D. J. Thouless, Nucl. Phys. 21, 225 (1960).
[2]
J. Čížek and J. Paldus, J. Chem. Phys. 47, 3976 (1967);
J. Paldus, and J. Čížek, Phys. Rev. A 2, 2268 (1970);
J. Čížek, and J. Paldus, Phys. Rev. A 3, 525 (1971);
J. Paldus, and J. Čížek, J. Chem. Phys. 52, 2919 (1970);
J. Čížek, and J. Paldus, J. Chem. Phys. 53, 821 (1970);
J. Paldus, and J. Čížek, J. Chem. Phys. 54, 2293 (1971);
J. Paldus, and J. Čížek, Can. J. Chem. 63, 1803 (1985).
[3] H. Fukutome, Int. J. Quantum Chem. 20, 955 (1981).
[4] J. Stuber and J. Paldus, in Fundamental World of Quantum Chemistry: A Tribute Volume to the
Memory of Per-Olov Lowdin, vol. 1, ed. by R. J. Brandas, E. S. Kryachko (Kluwer, Dordrecht, 2003).
[5] X. Li and J. Paldus, J. Chem. Phys. 130, 084110 (2009).
[6] X. Li and J. Paldus, J. Chem. Phys. 126, 224304 (2007).
[7] K. Kowalski and K. Jankowski, Phys. Rev. Lett. 81, 1195 (1998).
[8] C. M. Bender and T. T. Wu, Phys. Rev. 184, 1231 (1969).
[9] J. Zamastil and F. Vinette, J. Phys. A 38, 4009 (2005).
[10] G. Alvarez and H. J. Silverstone, Phys. Rev. A 50, 4679 (1994).
[11] H. Cartarius et al, Phys. Rev. A 79, 053408 (2009).
[12] J. Zamastil et al, Phys. Rev. A 81, 032118 (2010);
C. S. Estienne et al, Phys. Rev. Lett. 112, 173001 (2014);
N. L. Guevara and A. V. Turbiner, Phys. Rev. A 84, 064501 (2011).

Předběžná náplň práce v anglickém jazyce

Symmetry breaking is one of the most important themes in contemporary physics of elementary particles and solid state. Symmetry breaking on Hartree-Fock level of approximation in atomic and molecular physics has been known for a long time since pioneering work of Čížek and Paldus. However, its consequences remain to be largely unexplored. The purpose of the proposed PhD. thesis is firstly complete classification of singularities of Hartree-Fock equations in atomic systems along with elucidation of its physical significance. Secondly, to elucidate question what happens with symmetry broken solutions when electron correlation is taken into account and how to account for the electron correlation in most efficient way.