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Inner projection with and without perturbation theory: The anharmonic oscillator revisited and the quadratic Zeeman effect in ground‐state hydrogen
Author(s) -
Cizek J.,
Vrscay E. R.
Publication year - 1985
Publication title -
international journal of quantum chemistry
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.484
H-Index - 105
eISSN - 1097-461X
pISSN - 0020-7608
DOI - 10.1002/qua.560280602
Subject(s) - anharmonicity , ground state , hydrogen atom , eigenvalues and eigenvectors , mathematics , hamiltonian (control theory) , perturbation (astronomy) , quartic function , quantum mechanics , physics , mathematical analysis , mathematical physics , pure mathematics , mathematical optimization , group (periodic table)
This paper serves a twofold purpose. First, Löwdin's inner projection in both nonperturbative and perturbative forms is applied to the quartic anharmonic oscillator. Inner projection with perturbation theory yields rational approximations to Brillouin–Wigner‐type perturbation expansions. These lower bounds are compared with [ N − 1, N ] Padé approximants to the Rayleigh–Schrödinger perturbation series for this problem. These Padés are also expressible as the even convergents, w 2 N , of a Stieltjes‐type continued fraction. The latter representation has certain advantages with respect to its Padé counterpart. Inner projection without perturbation theory provides significantly better results than the perturbative version. The application of inner projection techniques to a perturbed hydrogen atom is not straightforward. The usual problems associated with the continuum spectrum of hydrogen are present. By means of a nonunitary “tilting” transformation associated with the Lie group SO (4, 2), these problems may be bypassed. In the SO (4, 2)‐reformulated eigenvalue problem, a reinterpretation of the basic variables, as developed by Silverstone and Moats, yields a new Hamiltonian that permits direct use of the inner projection method. This method has been applied to the ground state of the hydrogen atom in a magnetic field, using both four‐ and eight‐dimensional basis manifolds. This represents the first application of inner projection to this problem.