Power series expansion of the roots of a secular equation containing symbolic elements: Computer algebra and Moseley's law

We use computer algebra to-expand the Pekeris secular determinant for two-e lectron atoms symbolically, to produce an explicit polynomial in the energy parameter epsilon, with coefficients that are polynomials in the nuclear c harge Z. Repeated differentiation of the polynomial, followed by a simple t ransformation, gives a series for epsilon in decreasing powers of Z. The le ading term is linear, consistent with well-known behavior that corresponds to the approximate quadratic dependence of ionization potential on atomic n umber (Moseley's law). Evaluating the 12-term series for individual Z gives the roots to a precision of 10 or more digits for Z greater than or equal to 2. This suggests the use of similar tactics to construct formulas for ro ots vs atomic, molecular, and variational parameters in other eigenvalue pr oblems, in accordance with the general objectives of gradient theory. Matri x elements can be represented by symbols in the secular determinants, enabl ing the use of analytical expressions for the molecular integrals in the di fferentiation of the explicit polynomials. The mathematical and computation al techniques include modular arithmetic to handle matrix and polynomial op erations, and unrestricted precision arithmetic to overcome severe digital erosion. These are likely to find many further applications in computationa l chemistry. (C) 2001 American Institute of Physics.