8 A Refined Quartic Potential Surface for S 0 Formaldehyde
143
8.2 Variational Vibrational Calculation Procedure for
Formaldehyde
In our calculations we use the expression of Handy for the kinetic energy of
formaldehyde [25], in terms of his curvilinear coordinates q k (three bond stretches,
two interbond angles and one dihedral “book” angle [22, 25]). For the PES of S 0
formaldehyde, we use a quartic expansion in terms of the shifts from equilibrium of
the above internal curvilinear coordinates, exactly in the form, given by MLT [22].
This surface does not give spectroscopic accuracy, but it is quite realistic as a starting point of the fitting procedure and very well adapted to our vibrational method,
allowing the calculations to access very high vibrational excitation energies (E v ).
Our 6D basis functions are products of 6 1D basis functions Ψ i =
χ n k (q k ). The
1D basis functions are chosen to resemble most closely the lower excited molecular vibrational eigenfunctions so that the nondiagonal Hamiltonian matrix elements
be as small as possible. For the three stretching coordinates of the C–H 1 , C–H 2
and C–O bonds, we employ Morse oscillator eigenfunctions χ n k (q k ), k = 1, 2, 3
n k = 0, 1, . . . , n k0 that are optimally adapted to the relevant molecular motions, by
setting appropriately the two parameter values of the Morse oscillators. For the out
of plane bend, we employ harmonic oscillator eigenfunctions χ n 4 (q 4 ) [q 4 = ϕ—the
out-of-plane bend (“book”) angle]. Finally, for the two O–C–H(θ ) bends (coordinates q 5 = cos θ 1 , q 6 = cos θ 2 ), we use a set of normalized associated Legendre
polynomials P 2
n (cos θ), n = 2, 3, . . . , that cancel the singularities in the KE operator. However since they have no free parameters to adjust and are not well adapted
to the molecular vibrations, for them we apply a prediagonalization of the 1D basis
(using a simple 1D Hamiltonian) in order to obtain suitable 1D basis functions as
linear combinations of the original wavefunctions. This procedure was described in
detail in our previous work [26].
Our specific search/selection procedure for constructing the Hamiltonian matrix
H in a vibrational calculation, designed for selection of a characteristic and representative active space (AS) of basis vectors from a huge primitive space, involves
the intermediate calculation of a great number of Hamiltonian matrix elements (that
are employed to test whether a state should be selected or not), greatly exceeding
the final number of elements in H . Therefore we need a very fast method for calculation of matrix elements that does not include numerical integrations. For this
purpose, prior to each actual vibrational calculation, we compute a number of 2D
arrays P
i,α i
m i ,n i = =χ m i (q i )|F α i (q i )|χ n i (q i ), m i , n i = 0, 1, 2, . . . , n i0 , corresponding
to all vibrational coordinates q i and each function or operator F α i (q i ), occurring
in either KE or PES expressions, using either Gauss-Hermite, Gauss-Laguerre or
Gauss-Legendre numerical integrations [30], where n i0 is the number of basis functions employed for the vibrational coordinate q i . All computed n i0 × n i0 arrays are
stored in computer core memory, ready to use in the subsequent matrix elements
calculations. As a result of this and of the separable forms of the KE and PES, each
matrix element is obtained as the sum of products of the appropriate values, thus
reducing the actual calculation to a number of multiplications and summations and
no integrations, which greatly accelerates the calculation of matrix elements.
143
8.2 Variational Vibrational Calculation Procedure for
Formaldehyde
In our calculations we use the expression of Handy for the kinetic energy of
formaldehyde [25], in terms of his curvilinear coordinates q k (three bond stretches,
two interbond angles and one dihedral “book” angle [22, 25]). For the PES of S 0
formaldehyde, we use a quartic expansion in terms of the shifts from equilibrium of
the above internal curvilinear coordinates, exactly in the form, given by MLT [22].
This surface does not give spectroscopic accuracy, but it is quite realistic as a starting point of the fitting procedure and very well adapted to our vibrational method,
allowing the calculations to access very high vibrational excitation energies (E v ).
Our 6D basis functions are products of 6 1D basis functions Ψ i =
χ n k (q k ). The
1D basis functions are chosen to resemble most closely the lower excited molecular vibrational eigenfunctions so that the nondiagonal Hamiltonian matrix elements
be as small as possible. For the three stretching coordinates of the C–H 1 , C–H 2
and C–O bonds, we employ Morse oscillator eigenfunctions χ n k (q k ), k = 1, 2, 3
n k = 0, 1, . . . , n k0 that are optimally adapted to the relevant molecular motions, by
setting appropriately the two parameter values of the Morse oscillators. For the out
of plane bend, we employ harmonic oscillator eigenfunctions χ n 4 (q 4 ) [q 4 = ϕ—the
out-of-plane bend (“book”) angle]. Finally, for the two O–C–H(θ ) bends (coordinates q 5 = cos θ 1 , q 6 = cos θ 2 ), we use a set of normalized associated Legendre
polynomials P 2
n (cos θ), n = 2, 3, . . . , that cancel the singularities in the KE operator. However since they have no free parameters to adjust and are not well adapted
to the molecular vibrations, for them we apply a prediagonalization of the 1D basis
(using a simple 1D Hamiltonian) in order to obtain suitable 1D basis functions as
linear combinations of the original wavefunctions. This procedure was described in
detail in our previous work [26].
Our specific search/selection procedure for constructing the Hamiltonian matrix
H in a vibrational calculation, designed for selection of a characteristic and representative active space (AS) of basis vectors from a huge primitive space, involves
the intermediate calculation of a great number of Hamiltonian matrix elements (that
are employed to test whether a state should be selected or not), greatly exceeding
the final number of elements in H . Therefore we need a very fast method for calculation of matrix elements that does not include numerical integrations. For this
purpose, prior to each actual vibrational calculation, we compute a number of 2D
arrays P
i,α i
m i ,n i = =χ m i (q i )|F α i (q i )|χ n i (q i ), m i , n i = 0, 1, 2, . . . , n i0 , corresponding
to all vibrational coordinates q i and each function or operator F α i (q i ), occurring
in either KE or PES expressions, using either Gauss-Hermite, Gauss-Laguerre or
Gauss-Legendre numerical integrations [30], where n i0 is the number of basis functions employed for the vibrational coordinate q i . All computed n i0 × n i0 arrays are
stored in computer core memory, ready to use in the subsequent matrix elements
calculations. As a result of this and of the separable forms of the KE and PES, each
matrix element is obtained as the sum of products of the appropriate values, thus
reducing the actual calculation to a number of multiplications and summations and
no integrations, which greatly accelerates the calculation of matrix elements.