Some calculations on the ground and lowest-triplet state of helium in the fixed-nucleus approximation

The series solution method developed by Pekeris [Phys. Rev. 112, 1649 (1958); 115, 1216 (1959)] for the Schrödinger equation for two-electron atoms, as generalized by Frost et al. [J. Chem. Phys. 41, 482 (1964)] to handle any three particles with a Coulomb interaction, has been used. The wave function is expanded in triple orthogonal set in three perimetric coordinates. From the Schrödinger equation an explicit recursion relation for the coefficients in the expansion is obtained, and the vanishing of the determinant of these coefficients provides the condition for the energy eigenvalues and for the eigenvectors. The Schrödinger equation is solved and the matrix elements are produced algebraically by using the computer algebra system m a p l e. The substitutions for a particular atom and diagonalization were performed by a program written in the c language. Since the determinant is sparse, it is possible to go to the order of 1078 as Pekeris did without using excessive memory or computer CPU time. By using a nonlinear variational parameter in the expression used to remove the energy, nonrelativistic energies, within the fixed-nucleus approximation, have been obtained. For the ground-state singlet 1 ${}_{}{}^1{}_{}{}^{}$S state, this is of the accuracy claimed by Frankowski and Pekeris [Phys. Rev. 146, 46 (1966); 150, 366(E) (1966)] using logarithmic terms for Z from 1 to 10, and for the triplet 2 ${}_{}{}^3{}_{}{}^{}$S state, energies have been obtained to 12 decimal places of accuracy, which, with the exception of Z=2, are lower than any previously published, for all Z from 3 to 10.