Rates of convergence of the partial-wave expansions of atomic correlation energies

Abstract

The coefficients of the leading terms of the partial‐wave expansion of atomic correlation energies in powers of (l+1/2)⁻¹ are derived for the second‐ and third‐order perturbed energies in the 1/Z expansion for all possible states of two‐electron atoms, and for second‐order Mo/ller–Plesset (many‐body perturbation) theory for arbitrary n‐electron atoms. The expressions for these coefficients given in Table I involve simple integrals over the zeroth‐order wave functions (for the third order energies first‐order wave functions are also involved). The leading term of E⁽²⁾ goes as (l+1/2)⁻⁴ for natural parity singlet states, as (l+1/2)⁻⁶ for triplet states, and as (l+1/2)⁻⁸ for unnatural parity singlet states. There are no odd powers of (l+1/2)⁻¹ present in E⁽²⁾, and the coefficient of the (l+1/2)⁻⁶ term for natural parity singlet states of two‐electron systems in the 1/Z expansion is generally −5/4 times the coefficient of the (l+1/2)⁻⁴ term. In E⁽³⁾ there are terms that go as odd powers of (l+1/2)⁻¹; the leading term is expressible in terms of the zeroth‐order wave function and goes as (l+1/2)⁻⁵ for natural parity singlet states, as (l+1/2)⁻⁷ for triplet states, and as (l+1/2)⁻⁹ for unnatural parity singlet states. Numerical values of the coefficients for E⁽²⁾ in two‐electron atoms are given in Table II. The relation of our results to those obtained by Hill for variational calculations on the He ground state is discussed and generalizations of Hill’s formulas are conjectured. The unexpected results for unnatural parity singlet states are traced back to the behavior of their wave functions at the point of coalescence of two electrons. In terms of the relative coordinate of the two electrons they represent d waves, in contrast to s waves for natural parity singlet states and p waves for triplet states of either parity.