Using a pruned basis, a non-product quadrature grid, and the exact Watson normal-coordinate kinetic energy operator to solve the vibrational Schrödinger equation for C2H4

2011 ◽  
Vol 135 (6) ◽  
pp. 064101 ◽  
Author(s):  
Gustavo Avila ◽  
Tucker Carrington
Keyword(s):  
Kinetic Energy ◽  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Energy Operator ◽  
Kinetic Energy Operator ◽  
Normal Coordinate ◽  
Quadrature Grid

Ab Initio Calculations

1998 ◽  
Author(s):  
Anne-Marie Sapse
Keyword(s):  
Quantum Mechanics ◽  
Kinetic Energy ◽  
Schrödinger Equation ◽  
Potential Energy ◽  
Schrodinger Equation ◽  
Energy Operator ◽  
Hamiltonian Operator ◽  
Kinetic Energy Operator ◽  
Electronic Distribution ◽  
Positive Nucleus

Various difficulties of classical physics, including inadequate description of atoms and molecules, led to new ways of visualizing physical realities, ways which are embodied in the methods of quantum mechanics. Quantum mechanics is based on the description of particle motion by a wave function, satisfying the Schrodinger equation, which in its “time-independent” form is: ((−h2/8mπ2)⛛2+V)Ψ=E Ψ or, for short: HΨ = EΨ In this equation, H, the Hamiltonian operator, is defined by H = − ((h2/8mπ2)⛛2+V, where h is Planck’s constant (6.6 10−34 Joules), m is the particle’s mass, ⛛2 is the sum of the partial second derivative with x,y, and z, and V is the potential energy of the system. As such, the Hamiltonian operator is the sum of the kinetic energy operator and the potential energy operator. (Recall that an operator is a mathematical expression which manipulates the function that follows it in a certain way. For example, the operator d/dx placed before a function differentiates that function with respect to x.) E represents the total energy of the system and is a number, not an operator. It contains all the information within the limits of the Heisenberg uncertainty principle, which states that the exact position and velocity of a microscopic particle cannot be determined simultaneously. Therefore, the information provided by Ψ(nit) is in terms of probability: Ψ2(x,t) is the probability of finding the particle between x and x + dx, at time t. The Schrödinger equation applied to atoms will thus describe the motion of each electron in the electrostatic field created by the positive nucleus and by the other electrons. When the equation is applied to molecules, due to the much larger mass of nuclei, their relative motion is considered negligible as compared to that of the electrons (Born-Oppenheimer approximation). Accordingly, the electronic distribution in a molecule depends on the position of the nuclei and not on their motion. The kinetic energy operator for the nuclei is considered to be zero. For a many-electron molecule, the Hamiltonian operator can thus be written as the sum of the electrons’ kinetic energy term, which in turn is the sum of individual electrons’ kinetic energies and the electronic and nuclear potential energy terms.

scholarly journals On the position-dependent mass Schrödinger equation for Mie-type potentials

2021 ◽  
Vol 2090 (1) ◽  
pp. 012165
Author(s):  
G Ovando ◽  
J J Peña ◽  
J Morales ◽  
J López-Bonilla
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Energy Operator ◽  
Constant Mass ◽  
Point Canonical Transformation ◽  
Mass Distributions ◽  
Exactly Solvable ◽  
Reference Problem ◽  
Dependent Mass ◽  
Position Dependent Mass

Abstract The exactly solvable Position Dependent Mass Schrödinger Equation (PDMSE) for Mie-type potentials is presented. To that, by means of a point canonical transformation the exactly solvable constant mass Schrödinger equation is transformed into a PDMSE. The mapping between both Schrödinger equations lets obtain the energy spectra and wave functions for the potential under study. This happens for any selection of the O von Roos ambiguity parameters involved in the kinetic energy operator. The exactly solvable multiparameter exponential-type potential for the constant mass Schrödinger equation constitutes the reference problem allowing to solve the PDMSE for Mie potentials and mass functions of the form given by m(x) = skx s-1/(xs + 1))2. Thereby, as a useful application of our proposal, the particular Lennard-Jones potential is presented as an example of Mie potential by considering the mass distribution m(x) = 6kx 5/(x 6 + 1))2. The proposed method is general and can be straightforwardly applied to the solution of the PDMSE for other potential models and/or with different position-dependent mass distributions.

Kinetic-energy operator in the effective shell-model interaction

1992 ◽  
Vol 46 (6) ◽  
pp. 2333-2339 ◽  
Author(s):  
L. Jaqua ◽  
M. A. Hasan ◽  
J. P. Vary ◽  
B. R. Barrett
Keyword(s):  
Kinetic Energy ◽  
Shell Model ◽  
Energy Operator ◽  
Kinetic Energy Operator ◽  
Model Interaction

An exact kinetic energy operator for (HF)3 in terms of local polar and azimuthal angles

2001 ◽  
Vol 79 (2-3) ◽  
pp. 623-639 ◽  
Author(s):  
X -G Wang ◽  
T Carrington Jr.
Keyword(s):  
Kinetic Energy ◽  
Energy Operator ◽  
Kinetic Energy Operator

To facilitate exploiting the symmetry of (HF)3 we propose using local polar and azimuthal angles to specify the orientation of the HF units with respect to the frame of the trimer. We present and discuss the derivation of a kinetic energy operator in local polar and azimuthal angles and Pekeris–Jacobi coordinates. PACS No.: 31.15-P

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