Résumé
We present the efficiency of a new modified Davidson scheme which yields selectively one high-energy vibrationally excited eigenstate or a series of eigenstates. The calculation of a highly vibrationally excited state
ψ
located in a dense part of the spectrum requires a specific prediagonalization step before the Davidson scheme. It consists in building a small active space
P
containing the zero-order states which are coupled with the zero-order description of the eigenstate of interest. We propose a general way to define this active space
P
which plays a crucial role in the method. The efficiency of the method is illustrated by computing and analyzing the high-energy excited overtones of the out-of-plane mode
∣
6
,
8
,
10
ν
6
⟩
in HFCO. These overtone energies correspond to the 234th, 713th, and 1774th energy levels in our reference basis set which contains roughly 140000 states. One of the main advantages of this Davidson scheme comes from the fact that the eigenstate and eigenvalue convergence can be assessed during the iterations by looking at the residual
∥
(
H
−
E
M
)
ψ
M
∥
. The maximum value
ϵ
allowed for this residual constitutes a very sensitive and efficient parameter which sets the accuracy of the eigenvalues and eigenstates, even when the studied states are highly excited and are localized in a dense part of the spectrum. The physical analysis of the eigenstates associated with the 5th, 7th, and 9th out-of-plane overtones in HFCO provides some interesting information on the energy localization in this mode and on the role played by the in-plane modes. Also, it provides some ideas on the numerical methods which should be developed in the future to tackle higher-energy excited states in polyatomics.