The complex rotation method (CRM) for the description of resonance states is critically analyzed by noting that quantum mechanical wave functions and properties are not affected by a change in spatial coordinates, complex or otherwise. It is shown by means of the Cauchy-Coursat Theorem that equivalent approximate 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 approximate wave functions obtained by diagonalization of the latter matrix should satisfy the stationary principle as closely as possible. This objective can optimally be achieved by individually scaling the complex exponents in the basis functions. The nature of this approximation is investigated by means of explicit calculations which are based on diabatic RKR potentials for the B1Σ+-D′1Σ+ vibronic resonance states of the CO molecule.
