@article {4328,
	title = {The Quadrature Discretization Method (QDM) in comparison with other numerical methods of solution of the Fokker-Planck equation for electron thermalization},
	journal = {Journal of Mathematical Chemistry},
	volume = {24},
	number = {4},
	year = {1998},
	note = {ISI Document Delivery No.: 165EGTimes Cited: 7Cited Reference Count: 67},
	pages = {291-319},
	type = {Article},
	abstract = {The determination of the relaxation of electrons in atomic gases continues to be an important physical problem. The main interest is the determination of the time scale for the thermalization of electrons in different moderators and the nature of the time-dependent electron energy distribution. The theoretical basis for the study of electron thermalization is the determination of the electron distribution function from a solution of the Lorentz-Fokker- Planck equation. The present paper considers a detailed comparison of different numerical methods of solution of the Lorentz-Fokker- Planck equation for the electron distribution function. The methods include a pseudospectral method referred to as the Quadrature Discretization Method (QDM) which is based on non-standard polynomial basis sets, a finite-difference method, and a Lagrange interpolation method. The Fokker-Planck equation can be transformed to a Schrodinger equation, and methods developed for the solution of either equation apply to the other.},
	keywords = {ACCELERATION, BOLTZMANN-EQUATION, DISCRETE-ORDINATE, ENERGY-DISTRIBUTION, FIELD DEPENDENCE, method, MULTITERM CALCULATIONS, QUANTUM-MECHANICS, RARE-GAS MODERATORS, STOCHASTIC, TRANSPORT-COEFFICIENTS, VELOCITY DISTRIBUTION FUNCTION},
	isbn = {0259-9791},
	url = {<Go to ISI>://000078506600001},
	author = {Leung, K. and Shizgal, B. D. and Chen, H. L.}
}
@article {4130,
	title = {The quadrature discretization method (QDM) in the calculation of the rotational-vibrational transitions in rare gas dimers},
	journal = {Theochem-Journal of Molecular Structure},
	volume = {391},
	number = {1-2},
	year = {1997},
	note = {ISI Document Delivery No.: WU757Times Cited: 3Cited Reference Count: 38},
	month = {Feb},
	pages = {131-139},
	type = {Article},
	abstract = {A discretization method referred to as the Quadrature Discretization Method (QDM) is employed for the determination of the rotational-vibrational states of the inert gas dimers. The QDM is based on the discretization of the wave function on a grid of points that coincide with the points of a quadrature. The quadrature is based on a set of non-classical polynomials orthogonal with respect to a weight function determined by the potential function in the Schroedinger equation. In this paper, the weight functions used in the solution of the Schroedinger equation are related to the ground state wavefunctions of the Morse potential that approximates the potential of interest. Calculations are carried out for vibrational energy levels for HF modelled with a Morse potential. Excellent agreement is obtained with the known spectrum of the Morse potential. The rate of convergence of the eigenvalues and the eigenfunctions of the Schroedinger equation is very rapid with this approach. The vibrational and rotational transitions for the rare gas dimers are calculated and compared with the results reported recently by Ogilvie and Wang (J. Mol. Struc., 291 (1993) 313). (C) 1997 Elsevier Science B.V.},
	keywords = {AR-XE, COEFFICIENTS, DISCRETE-VARIABLE REPRESENTATION, NE-KR, ORDINATE METHOD, POTENTIALS, quadrature discretion method, QUANTUM-MECHANICS, rare gas dimer, rotation-vibration, SCHRODINGER-EQUATION, Schroedinger equation, SPECTRA, STATES, TRANSITION},
	isbn = {0166-1280},
	url = {<Go to ISI>://A1997WU75700014},
	author = {Shizgal, B. D.}
}
@article {3811,
	title = {The quadrature discretization method (QDM) in the solution of the Schrodinger equation with nonclassical basis functions},
	journal = {Journal of Chemical Physics},
	volume = {104},
	number = {11},
	year = {1996},
	note = {ISI Document Delivery No.: TY728Times Cited: 34Cited Reference Count: 66},
	month = {Mar},
	pages = {4137-4150},
	type = {Article},
	abstract = {A discretization method referred to as the Quadrature Discretization Method (QDM) is introduced for the solution of the Schrodinger equation. The method has been used previously for the solution of Fokker-Planck equations. The Fokker-Planck equation can be transformed to a Schrodinger equation with a potential of the form that occurs in supersymmetric quantum mechanics, For this class of potentials, the groundstate wave function is known. The QDM is based on the discretization of the wave function on a grid of points that coincide with the points of a quadrature. The quadrature is based on a set of nonclassical polynomials orthogonal with respect to a weight function determined by the potential function in the Schrodinger equation. For the Fokker-Planck operator, the weight function that provides rapid convergence of the eigenvalues are the steady distributions at infinite time, that is, the ground state wave functions. In the present paper, the weight functions used in an analogous solution of the Schrodinger equation are related to the ground state wave functions if known, or some approximate form. Calculations are carried out for a model systems, the Morse potential, and for the vibrational levels of O-2 and Ar-Xe with realistic pair potentials. For O-2, the wave functions are used to calculate the vibrationally inelastic transition amplitudes for a Morse potential and compared with exact analytic results. The eigenvalues of a two-dimensional Schrodinger equation with the Henon-Heiles potential are also calculated. The rate of convergence of the eigenvalues and the eigenfunctions of the Schrodinger equation is very rapid with this approach. (C) 1996 American Institute of Physics.},
	keywords = {BOUND-STATES, COEFFICIENTS, DISCRETE-ORDINATE METHOD, EIGENVALUE PROBLEMS, FOKKER-PLANCK EQUATION, ORTHOGONAL POLYNOMIALS, POTENTIALS, QUANTUM-MECHANICS, RECURRENCE, ROVIBRATIONAL STATES},
	isbn = {0021-9606},
	url = {<Go to ISI>://A1996TY72800030},
	author = {Shizgal, B. D. and Chen, H.}
}
