Exact numerical computation of a kinetic energy operator in curvilinear coordinates

Abstract

The hierarchical expansion of the kinetic energy operator in curvilinear coordinates extended to the vibrational configuration interaction method Hierarchical expansion of the kinetic energy operator in curvilinear coordinates for the vibrational self-consistent field methodThe conformation and dynamical behavior of molecular systems is very often advantageously described in terms of physically well-adapted curvilinear coordinates. It is rather easy to show that the numerous analytical expressions of the kinetic energy operator of a molecular system described in terms of n curvilinear coordinates can all be transformed into the following more usable expression:, and (q) are functions of the curvilinear coordinates qϭ(...,q i ,...). If the advantages of curvilinear coordinates are unquestionable, they do have a major drawback: the sometimes awful complexity of the analytical expression of the kinetic operator T for molecular systems with more than five atoms. Therefore, we develop an algorithm for computing T for a given value of the n curvilinear coordinates q. The calculation of the functions f 2 i j (q), f 1 i (q), and (q) only requires the knowledge of the Cartesian coordinates and their derivatives in terms of the n curvilinear coordinates. This coordinate transformation (curvilinear→Cartesian) is very easy to perform and is widely used in quantum chemistry codes resorting to a Z-matrix to define the curvilinear coordinates. Thus, the functions f 2 i j (q), f 1 i (q), and (q) can be evaluated numerically and exactly for a given value of q, which makes it possible to propagate wavepackets or to simulate the spectra of rather complex systems ͑constrained Hamiltonian͒. The accuracy of this numerical procedure is tested by comparing two calculations of the bending spectrum of HCN: the first one, performed by using the present numerical kinetic operator procedure, the second one, obtained in previous studies, by using an analytical kinetic operator. Finally, the ab initio computation of the internal rotation spectrum and wave functions of 2-methylpropanal by means of dimensionality reduction, is given as an original application.

Accurate Numerical Computation of Rovibrational G Matrices in Molecules of Arbitrary Size

“…In the general case, with arbitrary vibrational coordinates, analytical computation of these derivatives can be complicated and hardly generalizable. A more general approach is to resort to numerical computations [4,7] as long as the accuracy of the resulting data is known and appropriate for the problem at hand. Let us consider this problem.…”

“…Some of the previous works [1,2,4,9,16,17] are based in the early theoretical studies of Meyer [18,19]. This author developed a numerical treatment of internal motions in one and two dimensions, for non-linear molecules, as a tool to interpret spectroscopic molecular data.…”

“…More recently, Lauvergnat and Nauts [4] developed an algorithm for computing the molecular kinetic operator in curvilinear coordinates. In this program [25], the calculation of the Cartesian coordinate derivatives with respect to the internal coordinates, and the derivatives of the inertia tensor, Coriolis, deformation and rovibrational matrices are numerically evaluated.…”

“…In this context, several theoretical works have resorted to numerical evaluations of the kinetic energy operator, due to the complexity of analytical expressions in large systems or in problems taking into account several coupled vibrational coordinates. Most of these works deal with systems depending on large amplitude vibrational motions: one-dimensional studies of ring puckering in four-membered rings [1,2], methyl torsion in methanol [3] and 2-methylpropanal [4], inversion of ammonia [5,6], and bending in triatomic molecules [4,7]. Also two-dimensional studies have been carried out on several species such as the methyl + aldehydic torsions in propanal [8], the OH torsion + NH 2 wagging in hydroxylamine [9], the rotation + torsion of protonated methane [10], the methyl + phosphine torsions in ethylphosphine [11], or the methyl torsion + aldehydic hydrogen wagging of acetaldehyde [5,12,13] and acetone [12].…”

“…On the other hand, some numerical methods for computing the derivatives involved in the kinetic operator have been implemented in different software tools [2,4,6,10,12,13]. Thus, Niño and Muñoz-Caro [12] developed a program for the calculation of the rotational-vibrational G matrix [22].…”