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Series wave functions for the helium atom
Author(s) -
Davis C. L.,
Maslen E. N.
Publication year - 2009
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.560240828
Subject(s) - multipole expansion , eigenfunction , power series , series (stratigraphy) , recursion (computer science) , wave function , series expansion , mathematics , mathematical analysis , helium atom , separable space , differential equation , function (biology) , quantum mechanics , physics , eigenvalues and eigenvectors , helium , paleontology , algorithm , evolutionary biology , biology
Series methods for solving differential equations have been applied to the Schrödinger equation for helium. Formally exact solutions have been obtained for singlet S states. The solutions are expressed as multipole expansions in the angle between the electron position vectors r 1 and r 2 at the nucleus. The radial functions cannot be expressed as power series in r 1 and r 2 , but as series involving powers of r 1 , r 2 , log r 1 , and log r 2 . Recurrence relations, together with conditions for smoothness at r 1 = r 2 and proper behavior at the nucleus, determine most of the expansion coefficients. Simple, exact expressions for infinitely many of the coefficients, namely those for terms involving r i 1 r j 2 with i + j = 0, 1, and 2, have been determined. Coefficients not determined in the recursion process (of which there are infinitely many for each multipole component) can have arbitrary values in the formal solution. They are determined by the additional requirement that the wave function should have a finite square integral. An observation that the eigenfunction should be asymptotically separable in r 1 and r 2 for large values of these variables leads to approximate relations among the arbitrary coefficients, reducing the number of undetermined parameters to one per multipole component. Estimates for these remaining coefficients can be made easily. All approximations can be improved systematically to obtain arbitrarily high accuracy. The possibility of obtaining the exact solutions is considered.