Minimization principle for non degenerate excited states (independent of orthogonality to lower lying known approximants)

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 He<sup>1</sup>S (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.