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S. Rashev and D.C. Moule
Our search/selection procedure, serves to select an optimally small however representative active space of basis states, that are most relevant to the particular vibrational calculation. A search/selection procedure is started from a particular basis
(feature) state |0, chosen to be the best zeroth-order representation of the vibrational levels to be calculated. The algorithm is symmetrically adapted to search and
select only such basis states whose symmetry coincides with the symmetry species
of the initial state |0. During the implementation of the search/selection algorithm,
many basis states are probed and each state that satisfies the criteria for sufficient
coupling strength is selected and consecutively added to the previously selected AS.
There are three parameters, determining the scope and the quality of the search, C,
f and R, whose values have to be fixed at the outset, that have been defined and
discussed in our previous work [26]. According to the values chosen for the three
parameters C, f and R, the search/selection procedure will select a varying number of basis states, i.e., include more and more weakly coupled basis states into the
selected AS, which will result in enhanced accuracy and convergence in the calculation of the desired molecular vibrational levels. All selected basis states are stored in
an array in computer core memory. Simultaneously the Hamiltonian matrix is being
built, containing the diagonal and nondiagonal matrix elements of all selected basis
states.
The Hamiltonian matrix H constructed in the course of the search/selection procedure, besides being optimal in size, is also quite sparse, because the algorithm
employed automatically discards the matrix elements that are too small according
to the criteria of the search. This makes our vibrational procedure both memory and
time efficient. For the tridiagonalization of H we employ a conventional Lanczos
iteration without reorthogonalization [31, 32], started again with the vector |0. We
diagonalize the obtained tridiagonal Lanczos matrix using the routine tqli() from
Numerical recipes [31], in slightly modified form.
8.3 Adjustment of the Original MLT PES to the Experimentally
Measured Frequencies
Prior to presenting our adjustment procedure and the obtained refined quartic potential field for formaldehyde, we shall have to discuss the recently ab initio computed
and refined field by Yachmenev et al. [24]. In fact, before starting our own work on
the refinement of the MLT field, we wanted to use the refined field of the authors [24]
for our large scale calculations on formaldehyde and its deuterated species [26–29].
For that purpose we have spent much effort to present in the required product form
the analytic expression for the PES, supplied in the supplementary material to [24],
in order to incorporate it into our vibrational code. Then we performed converged
variational calculations on the vibrational frequencies of S 0 H 2 CO, using both the
set of ab initio parameters as well as the refined set from [24]. Our experience with
the field [24] is briefly summarized below.
In Table 8.1 are presented the results from our calculations on the H 2 CO vibrational frequencies of A 1 symmetry (J = 0), using both the ab initio and refined
S. Rashev and D.C. Moule
Our search/selection procedure, serves to select an optimally small however representative active space of basis states, that are most relevant to the particular vibrational calculation. A search/selection procedure is started from a particular basis
(feature) state |0, chosen to be the best zeroth-order representation of the vibrational levels to be calculated. The algorithm is symmetrically adapted to search and
select only such basis states whose symmetry coincides with the symmetry species
of the initial state |0. During the implementation of the search/selection algorithm,
many basis states are probed and each state that satisfies the criteria for sufficient
coupling strength is selected and consecutively added to the previously selected AS.
There are three parameters, determining the scope and the quality of the search, C,
f and R, whose values have to be fixed at the outset, that have been defined and
discussed in our previous work [26]. According to the values chosen for the three
parameters C, f and R, the search/selection procedure will select a varying number of basis states, i.e., include more and more weakly coupled basis states into the
selected AS, which will result in enhanced accuracy and convergence in the calculation of the desired molecular vibrational levels. All selected basis states are stored in
an array in computer core memory. Simultaneously the Hamiltonian matrix is being
built, containing the diagonal and nondiagonal matrix elements of all selected basis
states.
The Hamiltonian matrix H constructed in the course of the search/selection procedure, besides being optimal in size, is also quite sparse, because the algorithm
employed automatically discards the matrix elements that are too small according
to the criteria of the search. This makes our vibrational procedure both memory and
time efficient. For the tridiagonalization of H we employ a conventional Lanczos
iteration without reorthogonalization [31, 32], started again with the vector |0. We
diagonalize the obtained tridiagonal Lanczos matrix using the routine tqli() from
Numerical recipes [31], in slightly modified form.
8.3 Adjustment of the Original MLT PES to the Experimentally
Measured Frequencies
Prior to presenting our adjustment procedure and the obtained refined quartic potential field for formaldehyde, we shall have to discuss the recently ab initio computed
and refined field by Yachmenev et al. [24]. In fact, before starting our own work on
the refinement of the MLT field, we wanted to use the refined field of the authors [24]
for our large scale calculations on formaldehyde and its deuterated species [26–29].
For that purpose we have spent much effort to present in the required product form
the analytic expression for the PES, supplied in the supplementary material to [24],
in order to incorporate it into our vibrational code. Then we performed converged
variational calculations on the vibrational frequencies of S 0 H 2 CO, using both the
set of ab initio parameters as well as the refined set from [24]. Our experience with
the field [24] is briefly summarized below.
In Table 8.1 are presented the results from our calculations on the H 2 CO vibrational frequencies of A 1 symmetry (J = 0), using both the ab initio and refined