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Exact numerical computation of a kinetic energy operator in curvilinear coordinates

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Lauvergnat, D Nauts, André [UCL]

The 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: (T) over cap=Sigma(ij)f(2)(ij)(q)partial derivative(2)/partial derivativeq(i)partial derivativeq(j)+Sigma(i)f(1)(i)(q)partial derivative/partial derivativeq(i)+nu(q), where f(2)(ij)(q), f(1)(i)(q), and nu(q) are functions of the curvilinear coordinates q=(...,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) over cap for molecular systems with more than five atoms. Therefore, we develop an algorithm for computing (T) over cap for a given value of the n curvilinear coordinates q. The calculation of the functions f(2)(ij)(q), f(1)(i)(q), and nu(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)(ij)(q), f(1)(i)(q), and nu(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 Hamiltonian). 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. (C) 2002 American Institute of Physics.

Document type Article de périodique (Journal article) – Article de recherche
Publication date 2002
Language Anglais
Journal information "Journal of Chemical Physics" - Vol. 116, no. 19, p. 8560-8570 (2002)
Peer reviewed yes
Publisher Amer Inst Physics (Melville)
issn 0021-9606
e-issn 1089-7690
Publication status Publié
Affiliation UCL - SC/PHYS - Département de physique
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Bibliographic reference Lauvergnat, D ; Nauts, André. Exact numerical computation of a kinetic energy operator in curvilinear coordinates. In: Journal of Chemical Physics, Vol. 116, no. 19, p. 8560-8570 (2002)
Permanent URL http://hdl.handle.net/2078.1/41977