scholarly journals USING LEBEDEV GRIDS, SINE SPHERICAL HARMONICS, AND MONOMER CONTRACTED BASIS FUNCTIONS TO CALCULATE BENDING ENERGY LEVELS OF HF TRIMER

2003 ◽  
Vol 02 (04) ◽  
pp. 599-608 ◽  
Author(s):  
XIAO-GANG WANG ◽  
TUCKER CARRINGTON
Keyword(s):  
Spherical Harmonics ◽  
Energy Levels ◽  
Finite Basis ◽  
Chem Phys ◽  
Previous Method ◽  
Basis Functions ◽  
Bending Energy ◽  
Quadrature Scheme ◽  
Finite Basis Representation ◽  
Quadrature Grid

We calculate energy levels of a six-dimensional bending Hamiltonian for HF trimer using a finite basis representation (FBR) in conjunction with the Lanczos eigensolver. We improve on our previous method [J. Chem. Phys.115, 9781 (2001)] using three techniques: (1) Lebedev's quadrature scheme is used to reduce the size of quadrature grid by a factor of 3.4. (2) Since the barrier separating the two equivalent versions of HF trimer is high and wide, it is a good approximation to confine the bending motion to one well by using sine spherical harmonics basis functions (this reduces the size of the basis by a factor of 8). (3) The sine spherical harmonic basis is contracted for each monomer to generate a very efficient basis. It is shown that the best approach is to combine all the three techniques.

scholarly journals Analytical angular solutions for the atom–diatom interaction potential in a basis set of products of two spherical harmonics: two approaches

2021 ◽  
Author(s):  
Mariusz Pawlak ◽  
Marcin Stachowiak
Keyword(s):  
Spherical Harmonics ◽  
Interaction Potential ◽  
Chem Phys ◽  
Basis Set ◽  
Matrix Elements ◽  
Trigonometric Functions ◽  
Variational Theory ◽  
Molecular Collision ◽  
The Matrix ◽  
Analytical Expressions

AbstractWe present general analytical expressions for the matrix elements of the atom–diatom interaction potential, expanded in terms of Legendre polynomials, in a basis set of products of two spherical harmonics, especially significant to the recently developed adiabatic variational theory for cold molecular collision experiments [J. Chem. Phys. 143, 074114 (2015); J. Phys. Chem. A 121, 2194 (2017)]. We used two approaches in our studies. The first involves the evaluation of the integral containing trigonometric functions with arbitrary powers. The second approach is based on the theorem of addition of spherical harmonics.

ChemInform Abstract: The Bending Energy Levels of C4H2

ChemInform ◽  
2010 ◽  
Vol 23 (49) ◽  
pp. no-no
Author(s):  
E. ARIE ◽  
J. W. C. JOHNS
Keyword(s):  
Energy Levels ◽  
Bending Energy

Energy Spectrum of a Particle Within a Confined Double Well Potential

2006 ◽  
Vol 3 (2) ◽  
pp. 257-262
Author(s):  
J. L. Marin ◽  
G. Campoy ◽  
R. Riera
Keyword(s):  
Energy Levels ◽  
Numerical Approach ◽  
Symmetric Case ◽  
Basis Functions ◽  
Hidden Symmetry ◽  
The Matrix ◽  
Jahn Teller ◽  
Free Case ◽  
Jahn Teller Effect ◽  
Double Well Potential

The energy levels of a particle within a confined double well potential are studied in this work. The spectrum of the particle can be obtained by solving the corresponding Schrödinger equation but, for practical purposes, we have used a numerical approach based in the diagonalization of the matrix related to the Hamiltonian when the wavefunction is represented as an expansion in terms of "a particle-in-a-box" basis functions. The results show that, in the symmetric confining case, the energy levels are degenerate and a regular pairwise association between them is observed, similarly as it occurs in the free case. Moreover, when the confining is asymmetric, the degeneration is partially lifted but the pairwise association of the energy levels becomes irregular. The lifting of the degeneration in the latter case is addressed to the lack of symmetry or distortion of the system, namely, to a sort of Jahn-Teller effect which is common in the energy levels of diatomic molecules, to which a double well potential can be crudely associated. In the symmetric case, the states with nodes at the origin are recognized to be the same as those of the harmonic oscillator confined by two impenetrable walls, in such a way that the system presented in this work would be interpreted as half the solution of the problem of a particle within a confined four well potential. The latter suggests the existence of a sort of hidden symmetry which remains to be studied in a more detailed way.

CALCULATION OF SELECTED HIGHLY EXCITED VIBRATIONAL STATES OF POLYATOMIC MOLECULES BY THE DAVIDSON ALGORITHM

2003 ◽  
Vol 02 (04) ◽  
pp. 609-620 ◽  
Author(s):  
FABIENNE RIBEIRO ◽  
CHRISTOPHE IUNG ◽  
CLAUDE LEFORESTIER
Keyword(s):  
Energy Levels ◽  
Normal Modes ◽  
Chem Phys ◽  
Polyatomic Molecules ◽  
Zero Order ◽  
Basis Set ◽  
Vibrational States ◽  
Diagonalization Method ◽  
Davidson Algorithm ◽  
High Overtone

We described an improved version of a modified Davidson scheme previously introduced (F. Ribeiro, C. Iung and C. Leforestier, Chem. Phys. Lett.362, 199 (2002)), aimed at computing highly excited energy levels of polyatomic molecules. The key ingredient is a prediagonalization-perturbation step performed on a subspace of a curvilinear normal modes basis set (including diagonal anharmonicities). The efficiency of the method is demonstrated by computing the lowest 350 vibrational states of A′ symmetry of the HFCO molecule. Also shown is the possibility to restrict the calculation to selected energy levels, based on their zero-order description. This State Filtered Diagonalization method is illustrated on a high overtone (7ν5) of the OCF bend, and on the few energy levels (20) which have been experimentally assigned up to 5000 cm -1 of excitation energy.

scholarly journals On the Use of B-Splines as Ritz Variational Basis Functions to Solve the Schrodinger Equation (TISE) for a constrained free Quantum Particle

2021 ◽  
Vol 20 (1) ◽  
Author(s):  
Anandaram Mandyam N
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Quantum Particle ◽  
Bernstein Polynomials ◽  
Energy Levels ◽  
Basis Functions ◽  
Spline Collocation ◽  
Stationary Schrödinger Equation ◽  
B Splines ◽  
Domain Length

B-Splines as piecewise adaptation of Bernstein polynomials (aka, B-polys) are widely used as Ritz variational basis functions in solving many problems in the fields of quantum mechanics and atomic physics. In this paper they are used to solve the 1-D stationary Schrodinger equation (TISE) for a free quantum particle subject to a fixed domain length by using the Python software SPLIPY with different sets of computation parameters. In every case it was found that over 60 percent of energy levels had excellent accuracy thereby proving that the use of B-spline collocation is a preferred method.

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