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.