8 A Refined Quartic Potential Surface for S 0 Formaldehyde
147
might be some formal errors in the supplied electronic file for the set of refined parameters or that the authors [24] might have serious problems with their fitting code.
Anyway, we were not able to confirm the spectroscopic quality of the refined field in
Ref. [24] and we decided to carry out our own refinement of the MLT quartic field.
For the adjustment procedure, we employed the Marquardt’s method for nonlinear parameter estimation through the “chi-square” minimization (chi-square is equal
to the sum of squares of the differences between calculated and experimental values
for all 29 frequencies, involved in the fitting process), essentially implemented in the
routine mrqmin( ) from [30]. We incorporated our vibrational calculation code into
the least squares fitting routine mrqmin( ) [30]. We fitted the calculated 29 frequencies to the corresponding experimentally measured values (indicated in Table 8.3
by an asterisk), all of them of the same symmetry type A 1 , in order to be able to
obtain all required frequencies in a single calculation. However most of these 29 vibrational energy levels represented combinations and overtones of all six molecular
vibrational modes, which meant that all molecular modes of all symmetries were
involved into the refinement process. We proceeded to carry out the fitting process
by varying all 80 force constants sequentially, in groups of 10. We first started with
harmonic force constants, next cubic and finally quartic force constants were varied.
The whole sequence was repeated a second time. The three equilibrium parameters
(two bond lengths and one interbond angle) were left unchanged, equal to the values
used by MLT themselves [22]. In the vibrational calculations for the fitting routine,
the state 1 1 (A 1g symmetrized combination of the one quantum excitation in the C–
H stretching mode) was used as the initial state of the search |0. The convergence
of the results obtained in this calculation was better than 0.01 cm −1 for the frequencies up to ∼6000 cm −1 and 0.1 cm −1 for the higher excited ones, as discussed in
detail in our recent work [26, 27]. The final chi-square value achieved in the fitting
process was 9.27 for the 29 frequencies employed.
8.4 Results and Discussion
The set of harmonic, cubic and quartic force constants, that were obtained as a result of our fitting process, are displayed in Table 8.2. In the Supplement to this work
we have provided a C++ code, for calculation of the potential energy for arbitrary
input values of the deviations of all six vibrational coordinates from equilibrium. In
Table 8.2 are displayed the resulting vibrational energy levels of various symmetry
types for S 0 H 2 CO, up to ∼6000 cm −1 , that were obtained from a series of calculations with the final adjusted set of force constants (Table 8.2). The vibrational
energy levels in Table 8.3, calculated using our newly determined set of force constants (Table 8.2), are compared with the experimentally measured values [6], as
well as with the vibrational level energies calculated by the authors, for two earlier
refined fields: Ref. [12] and Ref. [14]. As it is seen from Table 8.3, the agreement
between the calculated and the experimentally measured frequencies as well as with
the calculated results by the authors [12, 14], up to ∼6000 cm −1 is quite satisfactory (with very few exceptions). In addition we have also performed calculations
147
might be some formal errors in the supplied electronic file for the set of refined parameters or that the authors [24] might have serious problems with their fitting code.
Anyway, we were not able to confirm the spectroscopic quality of the refined field in
Ref. [24] and we decided to carry out our own refinement of the MLT quartic field.
For the adjustment procedure, we employed the Marquardt’s method for nonlinear parameter estimation through the “chi-square” minimization (chi-square is equal
to the sum of squares of the differences between calculated and experimental values
for all 29 frequencies, involved in the fitting process), essentially implemented in the
routine mrqmin( ) from [30]. We incorporated our vibrational calculation code into
the least squares fitting routine mrqmin( ) [30]. We fitted the calculated 29 frequencies to the corresponding experimentally measured values (indicated in Table 8.3
by an asterisk), all of them of the same symmetry type A 1 , in order to be able to
obtain all required frequencies in a single calculation. However most of these 29 vibrational energy levels represented combinations and overtones of all six molecular
vibrational modes, which meant that all molecular modes of all symmetries were
involved into the refinement process. We proceeded to carry out the fitting process
by varying all 80 force constants sequentially, in groups of 10. We first started with
harmonic force constants, next cubic and finally quartic force constants were varied.
The whole sequence was repeated a second time. The three equilibrium parameters
(two bond lengths and one interbond angle) were left unchanged, equal to the values
used by MLT themselves [22]. In the vibrational calculations for the fitting routine,
the state 1 1 (A 1g symmetrized combination of the one quantum excitation in the C–
H stretching mode) was used as the initial state of the search |0. The convergence
of the results obtained in this calculation was better than 0.01 cm −1 for the frequencies up to ∼6000 cm −1 and 0.1 cm −1 for the higher excited ones, as discussed in
detail in our recent work [26, 27]. The final chi-square value achieved in the fitting
process was 9.27 for the 29 frequencies employed.
8.4 Results and Discussion
The set of harmonic, cubic and quartic force constants, that were obtained as a result of our fitting process, are displayed in Table 8.2. In the Supplement to this work
we have provided a C++ code, for calculation of the potential energy for arbitrary
input values of the deviations of all six vibrational coordinates from equilibrium. In
Table 8.2 are displayed the resulting vibrational energy levels of various symmetry
types for S 0 H 2 CO, up to ∼6000 cm −1 , that were obtained from a series of calculations with the final adjusted set of force constants (Table 8.2). The vibrational
energy levels in Table 8.3, calculated using our newly determined set of force constants (Table 8.2), are compared with the experimentally measured values [6], as
well as with the vibrational level energies calculated by the authors, for two earlier
refined fields: Ref. [12] and Ref. [14]. As it is seen from Table 8.3, the agreement
between the calculated and the experimentally measured frequencies as well as with
the calculated results by the authors [12, 14], up to ∼6000 cm −1 is quite satisfactory (with very few exceptions). In addition we have also performed calculations