Date of Award


Publication Type


Degree Name




First Advisor

G.W.F. Drake

Second Advisor

J. Rau

Third Advisor

J.W. Gould


Eigenvalue, Helium, Precision eigenvalues, Variational method, Quantum mechanical three-body problem, Higher-lying Rydberg states




The aim of this work is to solve the quantum mechanical three-body problem for helium, and to obtain high precision eigenvalues for the higher-lying Rydberg states where previous methods have been of limited accuracy. A variational method in correlated Hylleraas coordinates is used involving three distinct distance scales, called a triple basis set. The eigenvalues and matrix elements of other operators are computed for P states of helium up to n = 15 using the varational method with a triple basis set in Hylleraas coordinates. The construction of the wave functions, as well as the behaviour of the asymptotic, intermediate and short range nonlinear scale parameters is discussed. The convergence and accuracy of both the eigenvalues and the matrix elements of other operators is compared to those obtained using a double basis set. It is shown that the accuracy is improved by at least two orders of magnitude for basis sets of comparable size. The accuracy of the eigenvalues is compared to the quasi-exponential method developed by Azbanaev et al [1] requiring up to 100 digit arithmetic. The comparison verifies the accuracy of the present results for the low-lying states of helium, and demonstrates that the results for the higher-lying states up to n = 15 are by far the most accurate in the literature.