Correct small-truncated excited state wave functions obtained via minimization principle for excited states compared/ opposed to Hylleraas-Undheim and McDonald higher ``roots''
Affiliations: [a] Space Science and Technology Research Institute, Southeast University, Nanjing 21006, Jiangsu, China | [b] School of Economics and Management, Southeast University, Nanjing 210096, Jiangsu, China | [c] Korai 21, Halandri, GR-15233, Greece | [d] Theoretical and Physical Chemistry Institute, National Hellenic Research Foundation, Athens, GR-11635, Greece
Correspondence:
[*]
Corresponding author: N.C. Bacalis, Theoretical and Physical Chemistry Institute, National Hellenic Research Foundation, Athens, GR-11635, Greece. E-mail:[email protected]
Abstract: We demonstrate that, if a truncated expansion of a wave function is large, then the standard excited states
computational method, of optimizing one ``root'' of a secular equation, according to the theorem of
Hylleraas, Undheim and McDonald (HUM), tends to the correct excited wave
function, comparable to that obtained via our proposed minimization
principle for excited states [J. Comput. Meth. Sci. Eng. 8, 277
(2008)] (independent of orthogonality to lower lying approximants). However,
if a truncated expansion of a wave function is small - that would be
desirable for large systems - then the HUM-based methods may lead to an
incorrect wave function - despite the correct energy (: according to the HUM
theorem) whereas our method leads to correct, reliable, albeit small
truncated wave functions. The demonstration is done in He excited states,
using truncated series ``small'' expansions both in Hylleraas coordinates,
and via standard configuration-interaction truncated ``small'' expansions,
in comparison with corresponding ``large'' expansions. Beyond that, we give
some examples of linear combinations of Hamiltonian eigenfunctions that have
the energy of the 1st excited state, albeit they are orthogonal to it,
demonstrating that the correct energy is not a criterion of correctness of
the wave function.