VariationPerturbation Method for Excited States
Abstract
The firstorder wave function, X_{1}, in the perturbation method can be obtained by a variational principle instead of summing the usual infinite series with a large continuum contribution. For a ground state or the lowest state of a given symmetry suitable trial functions, X~_{1}, are chosen to attain E_{2}, the secondorder contribution to the energy, as a minimum. This method is extended here to any excited state, m, regardless of its symmetry. To obtain X_{1}^{m}, the expression E~_{2}^{m}={2<Φ_{0}^{m}, (H_{1} E_{1}^{m})X~_{1}^{m}>+<X~_{1}^{m}, (H_{0} E_{0}^{m})X~_{1}^{m}>}>=E_{2}^{m}, is to be minimized with X~_{1}^{m} in the form X~_{1}^{m}=X̄_{1}^{m}+k=0m 1Φ_{0}^{k}<Φ_{0}^{k}, H_{1}Φ_{0}^{m}>(E_{0}^{m}E_{0}), with X̄_{1}^{m} orthogonal to the known unperturbed functions of the states lower than m. The X_{1} gives also the thirdorder energy. The method may be applied to such excited states as (1s2s) ^{1}S of Helike ions and to the similar electron pairs that arise in the writer's theory of a manyelectron atom or molecule.
 Publication:

Physical Review
 Pub Date:
 April 1961
 DOI:
 10.1103/PhysRev.122.491
 Bibcode:
 1961PhRv..122..491S