Approximate Wave Functions for Molecules and Crystals Using Perturbation Theory

1967 ◽  
Vol 39 (1) ◽  
pp. 203-227 ◽  
Author(s):  
JEREMY I. MUSHER
Keyword(s):  
Perturbation Theory ◽  
Wave Functions ◽  
Approximate Wave

APPROXIMATE MOLECULAR ORBITALS: III. THE 1sσ AND 2pπ STATES OF HeH++

1967 ◽  
Vol 45 (7) ◽  
pp. 2231-2238 ◽  
Author(s):  
M. Cohen ◽  
R. P. McEachran ◽  
Sheila D. McPhee
Keyword(s):  
Perturbation Theory ◽  
Molecular Orbitals ◽  
Wave Functions ◽  
Variational Techniques ◽  
Approximate Wave ◽  
Schrödinger Perturbation

A combination of Rayleigh–Schrödinger perturbation theory and variational techniques, previously used to calculate the wave functions of the lowest σ and π states of H2+ has been applied to the 1sσ and 2pπ states of HeH++. The accuracy of the resulting approximate wave functions is demonstrated by comparing a number of quantities calculated with them with the corresponding exact values.

APPROXIMATE MOLECULAR ORBITALS: IV. THE 3dδg AND 4fδu STATES OF H2+

1967 ◽  
Vol 45 (8) ◽  
pp. 2533-2542 ◽  
Author(s):  
M. Cohen ◽  
R. P. McEachran ◽  
Sheila D. McPhee
Keyword(s):  
Perturbation Theory ◽  
Variational Methods ◽  
Excellent Agreement ◽  
Hydrogen Molecule ◽  
Wave Functions ◽  
Simple Criterion ◽  
Wide Range ◽  
Approximate Wave ◽  
Exact Calculations ◽  
Schrödinger Perturbation

Properties of the lowest even and odd δ states of the hydrogen molecule–ion have been calculated using approximate wave functions. These were derived using a combination of Rayleigh–Schrödinger perturbation theory and variational methods, which have been applied previously to calculate the corresponding wave functions of the lowest σ and π states. Our total molecular energies are in excellent agreement with the recent exact calculations of Hunter and Pritchard (1967). A simple criterion is suggested for judging the accuracy of the approximate orbitals, which indicates that all the molecular properties calculated will be accurate over a wide range of internuclear separations.

Hypervirial theorems and perturbation theory in quantum mechanics

1965 ◽  
Vol 283 (1393) ◽  
pp. 229-237 ◽  
Keyword(s):  
Wave Function ◽  
Perturbation Theory ◽  
Wave Functions ◽  
First Order ◽  
Optimum Energy ◽  
Arbitrary Operator ◽  
Approximate Wave Function ◽  
Hypervirial Theorem ◽  
Variational Wave Functions ◽  
Approximate Wave

Previous ideas about the way in which hypervirial theorems might be used to improve approximate wave functions are discussed. To provide a firmer foundation for these ideas, a link is established between hypervirial theorems and perturbation theory. It is proved that if the first-order perturbation correction to the expectation value of an arbitrary operator vanishes, then the approximate wave function used satisfies a certain hypervirial theorem. Conversely, if an arbitrary hypervirial theorem is satisfied by the wave function, then it is proved that the expectation values of certain operators have vanishing first-order corrections. Some consequences of the theory as applied to variational wave functions with optimum energy are developed. The results are illustrated by the use of a simple approximate wave function for the ground state of the helium atom.

A 1/ Z expansion study of the 2s2p 1 P and 3 P autoionizing resonances of the helium isoelectronic sequence

1971 ◽  
Vol 320 (1543) ◽  
pp. 549-560 ◽  
Keyword(s):  
Perturbation Theory ◽  
Expansion Method ◽  
Wave Functions ◽  
Basis Set ◽  
Isoelectronic Sequence ◽  
Autoionizing State ◽  
Screening Parameter ◽  
Approximate Wave

The calculation of approximate wave functions describing autoionizing resonances is formulated in terms of Hylleraas-Scherr-Knight 1/ Z expansion perturbation theory. The optimization of a screening parameter is discussed and it is shown that erroneous results may be obtained if the screening parameter is improperly chosen. The solution is expanded in a finite, correlated basis set and results obtained for the 2s2p 1 P and 3 P resonances of the helium isoelectronic sequence. The 1/ Z expansion method uniquely identifies which of the N roots arising from the diagonalization of the Hamiltonian in an N dimensional basis set corresponds to a particular autoionizing state.

scholarly journals Quantum square-well with logarithmic central spike

2018 ◽  
Vol 33 (02) ◽  
pp. 1850009 ◽  
Author(s):  
Miloslav Znojil ◽  
Iveta Semorádová
Keyword(s):  
Perturbation Theory ◽  
Boundary Conditions ◽  
Closed Form ◽  
Wave Functions ◽  
Change Of Variables ◽  
State Dependent ◽  
Square Well ◽  
Strength Formula ◽  
Schrödinger Perturbation ◽  
Dependent Interaction

Singular repulsive barrier [Formula: see text] inside a square-well is interpreted and studied as a linear analog of the state-dependent interaction [Formula: see text] in nonlinear Schrödinger equation. In the linearized case, Rayleigh–Schrödinger perturbation theory is shown to provide a closed-form spectrum at sufficiently small [Formula: see text] or after an amendment of the unperturbed Hamiltonian. At any spike strength [Formula: see text], the model remains solvable numerically, by the matching of wave functions. Analytically, the singularity is shown regularized via the change of variables [Formula: see text] which interchanges the roles of the asymptotic and central boundary conditions.

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