Complex Coordinate Scaling and the Schrödinger Equation
The complex rotation method (CRM) for the description of quantum mechanical resonance states is critically analyzed by noting that quantum mechanical eigenvalues are not affected by a change in spatial coordinates. On this basis it is concluded that equivalent solutions of the Schrödinger equation for a complex-rotated Hamiltonian H (Θ) can be obtained without loss of accuracy by using the un-rotated Hamiltonian H (0) in its place. Despite the fact that the latter operator is hermitean, it is possible to obtain a complex symmetric matrix representation for it by following a few simple rules: a) the square-integrable basis functions must have complex exponents, i.e. with non-zero imaginary components, and b) the symmetric scalar product must be employed to compute matrix elements of H (0). The nature of this approximation is investigated by means of explicit calculations which are based on diabatic RKR potentials for the B 1Σ+ - D’ 1Σ+ vibronic resonance states of the CO molecule.