Calculations of potential energy surfaces in the complex plane

Procedures are developed and tested for the numerical analytic continuation of the difference between two potential energy surfaces of the same symmetry in the vicinity of their complex-valued intersection. Rational fractions are used for curve-fitting ΔE as a function of either one or two independent, complex nuclear coordinates. The rational fractions are constructed from discrete values of ΔE(R, r) to exhibit the branch-point structure explicit in the complex square root function. For analytic continuation to values of the nuclear coordinates with small imaginary parts only real-valued input points are required. In order to analytically continue ΔE farther off the real-axis a few complex-valued input points must be used in addition to the real-valued data. The rational-fraction methods are tested for two systems : (a) the energy difference between the 3σ and 4σ states of HeH⁺⁺ and (b) the energy difference between the two lowest singlet states of H₃ ⁺ at collinear geometries. In both cases, the rational fractions accurately represent the actual potentials on the real axis and when analytically continued into the complex nuclear coordinate plane. In (b), the first derivatives of the rational fractions are calculated and found to be sufficiently accurate for semiclassical calculations on molecular collisions.