scholarly journals Energy Expectation Values and the Integral Hellmann–Feynman Theorem: H2+ Molecule

1968 ◽  
Vol 49 (3) ◽  
pp. 1284-1287 ◽  
Author(s):  
Stuart M. Rothstein ◽  
S. M. Blinder
Keyword(s):  
Expectation Values ◽  
Energy Expectation ◽  
H2 Molecule ◽  
Feynman Theorem

scholarly journals Application of the quantum mechanical hypervirial theorems to even-power series potentials

2020 ◽  
Vol 5 ◽  
pp. 104
Author(s):  
T. E. Liolios ◽  
M. E. Grypeos
Keyword(s):  
Power Series ◽  
Excited States ◽  
Energy Operator ◽  
Quantum Mechanical ◽  
Expectation Values ◽  
Gaussian Potential ◽  
Energy Eigenvalues ◽  
The Mean ◽  
Potential Parameters ◽  
Feynman Theorem

The class of the even-power series potentials:V(r)=-D+ Σ_k^{\infty} V_kλ^kr^{2k+2}, Vo=ω^2>0, is studied with the aim of obtaining approximate analytic ex­pressions for the energy eigenvalues, the expectation values for the potential and the kinetic energy operator, and the mean square radii of the orbits of a particle in its ground and excited states. We use the Hypervirial Theorems (HVT) in conjunction with the Hellmann-Feynman Theorem (HFT) which provide a very powerful scheme especially for the treatment of that type of potentials, as previous studies have shown. The formalism is reviewed and the expressions of the above mentioned quantities are subsequently given in a convenient way in terms of the potential parameters and the mass of the particle, and are then applied to the case of the Gaussian potential and to the potential V(r)=-D/cosh^2(r/R). These expressions are given in the form of series expansions, the first terms of which yield in quite a number of cases values of very satisfactory accuracy.

Exact formula for the dipole moment of a one-electron diatomic molecule

1985 ◽  
Vol 401 (1821) ◽  
pp. 373-392 ◽  
Keyword(s):  
Dipole Moment ◽  
Electronic State ◽  
Diatomic Molecule ◽  
Recursion Relation ◽  
Energy Eigenvalue ◽  
Exact Formula ◽  
Electronic Energy ◽  
Moment Function ◽  
Expectation Values ◽  
Feynman Theorem

It is shown that the dipole moment function, μ ( R , Z a , Z b ), for an arbitrary bound electronic state of a one-electron diatomic molecule, with inter-nuclear distance R and atomic numbers Z a , Z b may be expressed exactly in terms of the separation eigenconstant C and the electronic energy eigenvalue W of the Schrödinger equation by means of the Hellmann-Feynman theorem and a new recursion relation. The formula is used to investigate the behaviour of μ in the vicinity of the united atom and when the nuclei are far apart. The generalization required to extend the relation to other expectation values is derived in an appendix.

scholarly journals On the variational principle for the non-linear Schrödinger equation

2019 ◽  
Vol 58 (1) ◽  
pp. 340-351
Author(s):  
Zsuzsanna É. Mihálka ◽  
Ádám Margócsy ◽  
Ágnes Szabados ◽  
Péter R. Surján
Keyword(s):  
Variational Principle ◽  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Energy Functional ◽  
Numerical Calculations ◽  
Eigenvalue Equation ◽  
Expectation Value ◽  
Energy Expectation ◽  
Feynman Theorem ◽  
Energy Expectation Value

AbstractWhile variation of the energy functional yields the Schrödinger equation in the usual, linear case, no such statement can be formulated in the general nonlinear situation when the Hamiltonian depends on its eigenvector. In this latter case, as we illustrate by sample numerical calculations, the points of the energy expectation value hypersurface where the eigenvalue equation is satisfied separate from those where the energy is stationary. We show that the variation of the energy at the eigensolution is determined by a generalized Hellmann–Feynman theorem. Functionals, other than the energy, can, however be constructed, that result the nonlinear Schrödinger equation upon setting their variation zero. The second centralized moment of the Hamiltonian is one example.

scholarly journals An economic prediction of the finer resolution level wavelet coefficients in electronic structure calculations

2015 ◽  
Vol 17 (47) ◽  
pp. 31558-31565 ◽  
Author(s):  
Szilvia Nagy ◽  
János Pipek
Keyword(s):  
Electronic Structure ◽  
Prediction Method ◽  
Electronic Structure Calculations ◽  
Wave Functions ◽  
Wavelet Coefficients ◽  
Expectation Values ◽  
Resolution Level ◽  
Energy Expectation ◽  
Fine Resolution ◽  
Structure Calculations

A highly economic prediction method for fine resolution wavelet coefficients of wave functions and energy expectation values is presented.

The Correlated Basis Function Method and its Application to Liquid 3He, Nuclear Matter and Neutron Matter

1975 ◽  
Vol 30 (8) ◽  
pp. 923-936
Author(s):  
J. Nitsch
Keyword(s):  
Correlation Function ◽  
Nuclear Matter ◽  
Lagrange Equation ◽  
Expansion Method ◽  
Liquid Structure ◽  
Energy Functional ◽  
Basis Functions ◽  
Neutron Matter ◽  
Expectation Values ◽  
Energy Expectation

Abstract The method of correlated basis functions is studied and applied to the Fermi systems: liquid 3 He, nuclear matter and neutron matter. The reduced cluster integrals xijkl... and so the sub-normalization integrals Iijkl... are generalized to coinciding quantum numbers out of the set {i, j, k, I,...}. This generalization has an important consequence for the radial distribution function g (r) (and then for the liquid structure function) ; g(r) has no contributions of the order O (A-1). For 3 He the state-independent two-body correlation function g(r) is calculated from the Euler-Lagrange equation (in the limit of r → 0) for the unrenormalized two-body energy functional. For nuclear matter and neutron matter we adopt the three-parameter correlation function of Bäckman et al. Then the energy expectation values are calculated by including up to the three-body terms in the unrenormalized and renormalized version of the correlated basis functions method. The experimental ground-state energy and density of liquid s He can be well reproduced by the present method with the Lennard-Jones-(6 -12) potential. The same method is applied to the nuclear matter and neutron matter calculations with the OMY-potential. The results of the energy expectation values indicate a practical superiority of the unrenormalized cluster expansion method over the renormalized one.

INVESTIGATION OF ELECTRONIC STRUCTURE OF A QUANTUM DOT USING SLATER-TYPE ORBITALS AND QUANTUM GENETIC ALGORITHM

2007 ◽  
Vol 18 (01) ◽  
pp. 61-72 ◽  
Author(s):  
BEKİR ÇAKIR ◽  
AYHAN ÖZMEN ◽  
ÜLFET ATAV ◽  
HÜSEYİN YÜKSEL ◽  
YUSUF YAKAR
Keyword(s):  
Genetic Algorithm ◽  
Quantum Dot ◽  
Exact Result ◽  
Slater Type Orbitals ◽  
Slater Type ◽  
Expectation Values ◽  
Orbital Exponent ◽  
Quantum Genetic Algorithm ◽  
Energy Expectation ◽  
Low Dimensional

In this study, electronic properties of a low-dimensional quantum mechanical structure have been investigated by using Genetic Algorithm (GA). One- and two-electron Quantum Dot (QD) systems with an on-center impurity are considerable by assuming the confining potential to be infinitely deep and spherically symmetric. Linear combinations of Slater-Type Orbitals (STOs) were used for the description of the single electron wave functions. The parameters of the wave function of the system were used as individuals in a generation, and the corresponding energy expectation values were used for objective functions. The energy expectation values were determined by using the Hartree-Fock-Roothaan (HFR) method. The orbital exponent ζi's and the expansion coefficient ci's of the STOs were determined by genetic algorithm. The obtained results were compared with the exact result and found to be in a good agreement with the literature.

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