Hamiltonian matrix representations for the determination of approximate wave functions for molecular resonances
2018-01-22
Buenker
Robert J.
Prof. Dr.
Buenker, Robert J.
aut
en
<p>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 <sup><font size="1">1</font></sup>Σ <sup><font size="1">+</font></sup> - D’ <sup><font size="1">1</font></sup>Σ<sup><font size="1">+</font></sup> 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 (Re<sup><font size="1">iΘ</font></sup>) 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 <i>symmetric</i> 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
unrotated 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 eliminates any advantages that have
heretofore been claimed for its use.
2018-01-22T10:45:53.681Z
2018-01-22T11:43:00.073Z
published
Pub