Ab initio calculations of potential energy surfaces in the complex plane. II. Comparison with different methods of analytic continuation

Abstract

A comparison is presented between four methods for the numerical analytic continuation of potential energy surfaces to their values corresponding to complex nuclear coordinates. The four methods are: (a) ab initio calculations in the complex plane, (b) point method, (c) moment method. and (d) polynomial least squares fit. Method (a) involves the actual diagonalization of the electronic hamiltonian for complex nuclear coordinates and is the most rigorous method for numerical analytic continuation. Methods (b)–(d) involve the curvefitting of ab initio calculations on the real axis to yield a potential energy function which is analytically continued into the complex plane. The example chosen for study is the function Δ E( R) = E 4σ( R)- E 3σ(R), where E 4σ and E 3σ correspond to the 4σ and 3σ states of HeH 2+ and R is the internuclear distance. The zeroes (branch points) of ΔE( R) occur only for complex R, and the position of a particular branch point along with a conformal mapping of Δ E( R) for lines of constant R r and R i (R = R r + i R i) are presented for each method. These results for the four methods are compared and discussed, and optimal procedures for the analytic continuation of potential energy surfaces are suggested. Such procedures are hybrids of the above methods.