Hamiltonian matrix representations for the determination of approximate wave functions for molecular resonances
Wave functions obtained using a standard complex Hamiltonian matrix diagonalization procedure are square integrable and therefore constitute only approximations to the corresponding resonance solutions of the Schrödinger equation. The nature of this approximation is investigated by means of explicit calculations using the above method which employ accurate diabatic potentials of the B 1Σ+ - D’ 1Σ+ vibronic resonance states of the CO molecule. It is shown that expanding the basis of complex harmonic oscillator functions gradually improves the description of the exact resonance wave functions out to ever larger internuclear distances before they take on their unwanted bound-state characteristics. The justification of the above matrix method has been based on a theorem that states that the eigenvalues of a complex-scaled Hamiltonian H (ReiΘ) are associated with the energy position and linewidth of resonance states (R is an internuclear coordinate and Θ is a real number). It is well known, however, that the results of the approximate method can be obtained directly using the unscaled Hamiltonian H (R) in real coordinates provided a particular rule is followed for the evaluation of the corresponding matrix elements. It is shown that the latter rule can itself be justified by carrying out the complex diagonalization of the Hamiltonian in real space via a product of two transformation matrices, one of which is unitary and the other is complex orthogonal, in which case only the symmetric scalar product is actually used in the evaluation of all matrix elements. There is no limit on the accuracy of the above matrix method with an un-rotated Hamiltonian, so that exact solutions of the corresponding Schrödinger equation can in principle be obtained with it. This procedure therefore makes it unnecessary to employ a complex-scaled Hamiltonian to describe resonances and shows that any advantages which have heretofore been claimed for its use are actually non-existent.