Affiliations: Theoretical and Physical Chemistry Institute, National Hellenic Research Foundation, Vasileos Constantinou 48, GR-11635, Athens, Greece. E-mail: nbacalis@eie.gr
Abstract: The study of excited states is already imperative especially as it concerns
reactions, after activation, of stable species, like CO2 or alkanes.
First principles studies can only be utilized in truncated Hilbert spaces.
Unfortunately, the standard methods of computing excited states in truncated
spaces, although perhaps adequate for the energy and for spectroscopy, may
yield incorrect wave functions (perhaps with correct energy), misleading for
desired proper excitations. Thus, a method is needed (such as the present
demonstrated) to yield excited state truncated wave functions that are not
veered away from the exact Hamiltonian eigenfunctions. The ability to extend
the variational principle to any excited state (without knowledge of the
lower-lying exact eigenfunctions) has long been proven to be an inherent
property of the Hamiltonian. The excited state truncated wave function based
on the standard method of the Hylleraas and Undheim/MacDonald (HUM)
theorem, is in principle incorrect in a more fundamental manner than just
being truncated: Its accuracy must be strictly less than the accuracy of the
ground state truncated approximant. On the other hand, an energy
minimization orthogonally to all lower approximants [``orthogonal optimization'' (OO)]
must lead to a wave function lying lower than, and veered away from, the
exact. A minimization principle for excited electronic states of a
non-degenerate Hamiltonian in any given symmetry type is presented, that
allows their computation to any desired accuracy, independently of the
accuracy of the lower-lying states of the same symmetry, therefore without
demanding orthogonality to known lower approximants, and, within a given
truncated wave function parameter space, can lead to a more correct than HUM
or OO approximation of the excited wave function. A demonstration is
presented for the first excited state of He1S (1s2s) using variationally
optimized, optionally state-specific, orbitals in Hylleraas coordinates,
while the standard truncated HUM answer, despite the correct energy, has
main orbitals 1s1s'$ instead. It is also demonstrated that the principle can be
used to identify a ``flipped root'' near an avoided crossing (useful to
guide MCSCF). Beyond the aforementioned demonstration, some results within
conventional configuration interaction based on similarly optimized Laguerre
type orbitals are also exhibited and compared to relevant literature.
