Numerical Generation of Optimized Discrete Variable Representations

Abstract

We develop a procedure for calculating an optimized Discrete Variable Representation DVR optimized for a given potential. The method leads to an e cient and compact way to obtain numerical solutions of quantum mechanical problems. The procedure is applied to several physical problems. To illustrate the strength of the algorithm in dealing with multidimensional calculations, we obtain accurate levels up to 19; 000 cm ,1 for the vibrational energies of the water molecule.The e ciency of quantum mechanical numerical calculations rely in great part on the right c hoice of basis functions. The solution of multidimensional problems requires the manipulation of large matrices and a bad choice of basis functions could easily turn the problem unfeasible. We should also be concerned with the matrix structure and the computer time for the computation of the matrix elements. The calculation of the elements of a matrix makes use of Gaussian quadratures and an excessive n umb e r o f p o i n ts may lead to prohibitive time consuming codes. This paper shows a procedure for optimizing all these aspects. The strength of the approach is easily seen when we deal with threedimensional problems.There are several ways of expanding the quantum mechanical wave function. The more traditional procedure is the expansion in terms of global basis functions 1-3 . Generally, w e c hoose the functions that diagonalize the kinetic energy operator and integrate numerically the potential using Gaussian quadratures. Another family of procedures are the nite element methods 4-12 . The basis functions are local and we h a ve to discretize the space in many elements to implement the calculation 6 . Finally, w e m a y c hoose a set of basis functions that have the feature of diagonalizing the potential. The great advantage in this case is that the kinetic energy operator may b e i n tegrated analytically, leading to a great e ciency during the evaluation of the matrix elements. The last procedure is called the Discrete Variable Representation" DVR 13-23 . The Discrete Variable Representation has been used extensively by Light and collaborators 13-16 who performed accurate calculations of the ro-vibrational energies of several triatomic molecules. Choi and Light 24 also used this procedure to study Van der Waals molecules. Le Quere and Leforestier 25 used a time dependent approach and a DVR implementation for studying the photodissociation of Ozone. More recently, the method for expanding the multidimensional quantum mechanical wave function 26-32 has appeared in other calculations. This paper develops a procedure for obtaining a Numerically Generated Discrete Variable Representation NG-DVR . All publications mentioned before, except the work of Echave and Clary 26 , do not use a numerically optimized Discrete Variable Representation for the potential of the system under consideration. In this way, the number of basis functions is not optimized for the particular problem to be solved. The procedure proposed by E c have and Clary 26 does not calculate e...

Theoretical study of tetrahydrofuran: Comparative investigation of spectroscopic and structural properties between gas and liquid phases

“…• Determination of the PECs; • Analytical description of the PECs; • Calculation of rovibrational energies by the discrete variable representation method (DVR) [12].…”

Applications of Discrete Singular Convolution Algorithm in One-Electron Systems: Hydrogen Atom

“…In the DVR, 3,8,16,[19][20][21] the generalized Gauss quadrature rule is adapted and results in nonuniformly discretized grid points, which are dense near the origin Table 1. Eigenenergies (a.u.)…”

Revisiting the generator coordinate approximation for calculating the ro‐vibrational energies of H\documentclass{article}\pagestyle{empty}\begin{document}$_{3}^{+}$\end{document}

“…A Finite Element Method (FEM) basis set was employed in the mentioned work. We try in our present work a special type of Discrete Variable Representation (DVR) called the Numerically Generated-Discrete Variable Representation (NG-DVR) 8 for one of the bases instead. The present work also proposes a different matrix contraction scheme for the problem.…”

“…The NG-DVR was previously shown to be highly precise. 8 Therefore, we test it to see if the NG-DVR improves on the results obtained using the FEM R j , r j = R i , r i cos α ij − sin α ij sin α ij cos α ij ,…”

“…8 The NG-DVR has been shown to be a highly efficient and accurate basis representation. 8 For the x, y hyperangles basis, b l (x, y), we choose to use the basis of the p-version of the FEM also because it has been shown to be efficient and accurate. 14 The three-dimensional integrals over q, x, and y are done by numerical quadratures.…”