SOME CALCULATIONS ON THE GROUND AND LOWEST-TRIPLET STATE OF HELIUM INTHE FIXED-NUCLEUS APPROXIMATION

The series solution method developed by Pekeris [Phys. Rev. 112, 1649 (1958); 115, 1216 (1959)] for the Schrodinger equation for two-electro n atoms, as generalized by Frost et al. [J. Chem. Phys. 41, 482 (1964) ] to handle any three particles with a Coulomb interaction has been us ed. The wave function is expanded in a triple orthogonal set in three perimetric coordinates. From the Schrodinger equation an explicit recu rsion relation for the coefficients in the expansion is obtained, and the vanishing of the determinant of these coefficients provides the co ndition for the energy eigenvalues and for the eigenvectors. The Schro dinger equation is solved and the matrix elements are produced algebra ically by using the computer algebra system MAPLE. The substitutions f or a particular atom and diagonalization were performed by a program w ritten iu the C language. Since the determinant is sparse, it is possi ble to go to the order of 1078 as Pekeris did without using excessive memory or computer CPU time. By using a nonlinear variational paramete r in the expression used to remove the energy, nonrelativistic energie s, within the fixed-nucleus approximation, have been obtained. For the ground-state singlet 1 1S state this is of the accuracy claimed by Fr ankowski and Pekeris [Phys. Rev. 146, 46 (1966); 150, 366(E) (1966)] u sing logarithmic terms for Z from 1 to 10, and for the triplet 2(3)S s tate, energies have been obtained to 12 decimal places of accuracy, wh ich, with the exception of Z = 2, are lower than any previously publis hed, for all Z from 3 to 10.