Quantum mechanics with the basis set guided by Ehrenfest trajectories: Theory and application to spin-boson model

2009 ◽  
Vol 130 (24) ◽  
pp. 244101 ◽  
Author(s):  
Dmitrii V. Shalashilin
Keyword(s):  
Quantum Mechanics ◽  
Basis Set ◽  
Boson Model

scholarly journals Current Developments in the Relativistic Quantum Mechanics of Atoms and Molecules

1986 ◽  
Vol 39 (5) ◽  
pp. 649 ◽  
Author(s):  
IP Grant
Keyword(s):  
Electronic Structure ◽  
Quantum Mechanics ◽  
Atomic Structure ◽  
Quantum Electrodynamics ◽  
Relativistic Quantum ◽  
Finite Basis ◽  
Basis Set ◽  
Structure Theory ◽  
Relativistic Quantum Mechanics ◽  
Atoms And Molecules

Current work in relativistic quantum mechanics by the author and his associates focusses on four topics: atomic structure theory using the GRASP package (Dyall 1986); extension of GRASP to handle electron continuum processes; the relation of quantum electrodynamics and relativistic quantum mechanics of atoms and molecules; and development of methods using finite basis set expansions for studying electronic structure of atoms and molecules. This paper covers only the last three topics, giving emphasis to growing points and outstanding difficulties.

scholarly journals Bound states of the D-dimensional Schrödinger equation for the generalized Woods–Saxon potential

2019 ◽  
Vol 34 (14) ◽  
pp. 1950107 ◽  
Author(s):  
V. H. Badalov ◽  
B. Baris ◽  
K. Uzun
Keyword(s):  
Quantum Mechanics ◽  
Energy Spectrum ◽  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Wave Functions ◽  
Basis Set ◽  
Saxon Potential ◽  
Radial Wave ◽  
Energy Eigenvalues ◽  
Approximate Analytical Solutions

The formal framework for quantum mechanics is an infinite number of dimensional space. Hereby, in any analytical calculation of the quantum system, the energy eigenvalues and corresponding wave functions can be represented easily in a finite-dimensional basis set. In this work, the approximate analytical solutions of the hyper-radial Schrödinger equation are obtained for the generalized Wood–Saxon potential by implementing the Pekeris approximation to surmount the centrifugal term. The energy eigenvalues and corresponding hyper-radial wave functions are derived for any angular momentum case by means of state-of-the-art Nikiforov–Uvarov and supersymmetric quantum mechanics methods. Hence, the same expressions are obtained for the energy eigenvalues, and the expression of hyper-radial wave functions transforming each other is shown owing to these methods. Furthermore, a finite number energy spectrum depending on the depths of the potential well [Formula: see text] and [Formula: see text], the radial [Formula: see text] and [Formula: see text] orbital quantum numbers and parameters [Formula: see text], [Formula: see text], [Formula: see text] are also identified in detail. Next, the bound state energies and corresponding normalized hyper-radial wave functions for the neutron system of the [Formula: see text]Fe nucleus are calculated in [Formula: see text] and [Formula: see text] as well as the energy spectrum expressions of other higher dimensions are revealed by using the energy spectrum of [Formula: see text] and [Formula: see text].

Relaxation constants in electron thermalization: Comparison of WKB and SWKB eigenvalues with exact results

1990 ◽  
Vol 68 (10) ◽  
pp. 1213-1219 ◽  
Author(s):  
B. Shizgal
Keyword(s):  
Quantum Mechanics ◽  
Average Energy ◽  
Planck Equation ◽  
Exact Result ◽  
Quantization Condition ◽  
Supersymmetric Quantum Mechanics ◽  
Inert Gases ◽  
Basis Set ◽  
Nonequilibrium Distribution ◽  
Semiclassical Quantization

The relaxation to equilibrium of a nonequilibrium distribution of electrons in gases is governed by a linear–Planck equation. The decay in time of the electron average energy as well as other transport properties can be expressed as a sum of exponential terms with each term characterized by an eigenvalue of the Fokker–Planck operator. This eigenvalue problem can be transformed into a Schrödinger equation with a potential function for which the Hamiltonian factors and belongs to the class of potentials encountered in supersymmetric quantum mechanics. The eigenvalues are calculated with the standard Wentzel–Kramers–Brillouin (WKB) semiclassical quantization condition as well as with a modified semiclassical quantization condition based on supersymmetric quantum mechanics (SWKB). The eigenvalues calculated in these ways are compared with the exact values obtained by the diagonalization of the operator in a large basis set. The applications considered are for the relaxation of electrons in the inert gases for which the electron–atom momentum transfer cross sections are available. The SWKB quantization condition gives results in much better agreement with the exact result (often within much less than 1%) than the WKB approximation.

Quantum mechanics studies of cellobiose conformations

2006 ◽  
Vol 84 (4) ◽  
pp. 603-612 ◽  
Author(s):  
Alfred D French ◽  
Glenn P Johnson
Keyword(s):  
Quantum Mechanics ◽  
Hydrogen Bonds ◽  
Crystal Structures ◽  
Central Region ◽  
Single Point ◽  
Unconstrained Minimization ◽  
Basis Set ◽  
Energy Structure ◽  
B3lyp Method ◽  
Carbohydrate Conformation

Three regions of the conformation space that describes the relative orientations of the two glucose residues of cellobiose were analyzed with quantum mechanics. A central region, in which most crystal structures are found, was covered by a 9 × 9 grid of 20° increments of the linkage torsion angles ϕ and ψ. Besides these 81 constrained minimizations, we studied two central subregions and two regions at the edges of our maps of complete ϕ,ψ space with unconstrained minimization, for a total of 85 target geometries. HF/6-31G(d) and single-point HF/6-311+G(d) calculations were used to find the lowest energies for each geometry. B3LYP/6-31G+G(d) and single point B3LYP/6-11+G(d) calculations were also used for all unconstrained minimizations. For each target, 181 starting geometries were tried (155 for the unconstrained targets). Numerous different starting geometries resulted in the lowest energies for the various target structures. The starting geometries came from five different sets that were based on molecular mechanics energies. Although all five sets contributed to the adiabatic map, use of any single set resulted in discrepancies of 3–7 kcal/mol (1 cal = 4.184 J) with the final map. For most of the targets, the starting geometry that gave the lowest energy depended on the basis set and whether the HF or B3LYP method was used. However, each of the above four calculations gave the same overall lowest energy structure that was found previously by Strati et al. This global minimum, stabilized by highly cooperative hydrogen bonds, is in a region that is essentially not populated by crystal structures. HF/6-31G(d) energy contours of the mapped central region were compatible with the observed crystal structures. Observed structures that lacked O3···O5′ hydrogen bonds were about 1 kcal/mol above the map's minimum, and observed structures that have a pseudo twofold screw axis ranged from about 0.4 to 1.0 kcal/mol. The HF/6-311+G(d) map accommodated the observed structures nearly as well.Key words: cellulose, carbohydrate, conformation, energy, flexibility, folding, helix, shape.

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