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.

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: V. THE 3dδ STATE OF HeH++

1967 ◽  
Vol 45 (8) ◽  
pp. 2749-2754 ◽  
Author(s):  
M. Cohen ◽  
R. P. McEachran ◽  
Sheila D. McPhee
Keyword(s):  
Wave Function ◽  
Perturbation Theory ◽  
Variational Methods ◽  
High Accuracy ◽  
Molecular Orbitals ◽  
Molecular Properties ◽  
Simple Criterion ◽  
Molecular Wave Function ◽  
Schrödinger Perturbation

The techniques of Rayleigh–Schrödinger perturbation theory and variational methods have been used to obtain an approximate molecular wave function for the lowest δ state of the HeH++ ion. Its accuracy may be judged by a simple criterion proposed in an earlier paper, and molecular properties computed using it should have high accuracy. The main conclusions of this series of papers are reviewed briefly.

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.

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.

The electrostatic calculation of molecular energies - I. Methods of calculating molecular energies

1954 ◽  
Vol 226 (1165) ◽  
pp. 170-178 ◽  
Keyword(s):  
Complex Systems ◽  
Conventional Method ◽  
Hydrogen Molecule ◽  
Electronic States ◽  
Wave Functions ◽  
Hydrogen Molecular Ion ◽  
New Methods ◽  
Electrostatic Method ◽  
Conventional Calculation ◽  
Approximate Wave

The calculation of molecular energies from assumed approximate wave functions is discussed. It is shown that the conventional method, based on the Hamiltonian integral, is but one of several possible approximations, and that two other methods, the virial method and the electrostatic method, avoid the most serious difficulties encountered in a conventional calculation. The mathematical simplicity of the new methods makes them especially suitable for non-empirical calculations on complex systems. The electrostatic method is exemplified by detailed calculations on various electronic states of the hydrogen molecule and the hydrogen molecular ion.

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.

Ab initio calculations of the ground state potential function of the hydroxyl anion by means of the many-body Rayleigh-Schrödinger perturbation theory

1981 ◽  
Vol 46 (11) ◽  
pp. 2595-2599
Author(s):  
Ivan Kozák ◽  
Vladimír Špirko ◽  
Petr Čársky
Keyword(s):  
Perturbation Theory ◽  
Ground State ◽  
Negative Ions ◽  
Spectroscopic Constants ◽  
Many Body ◽  
Hydroxyl Anion ◽  
Wide Range ◽  
State Potential ◽  
The Many ◽  
Schrödinger Perturbation

Many-body Rayleigh-Schrödinger perturbation theory (MB-RSPT) up to third order applied to OH- in the range of interatomic distances from 0.0815 to 0.1175 nm. The energy data obtained are combined with the experimental RKR (ground state) potential of HF, and, a ground state potential of OH- is constructed (over a wide range of internuclear distances) within the framework of the reduced potential curve method. With the use of this potential the corresponding rotation-vibration Schrödinger equation is solved for 16OH-. The computed spectroscopic constants are compared with best reported calculations and available experimental evidence. The comparison indicates that MB-RSPT may be used as an adequate (and convenient) tool for the study of negative ions.

scholarly journals A quantum mechanical discussion of the cohesive forces and thermal expansion coefficients of the alkali metals

1937 ◽  
Vol 158 (893) ◽  
pp. 97-110 ◽  
Keyword(s):  
Thermal Expansion ◽  
Alkali Metals ◽  
Wave Functions ◽  
Quantum Mechanical ◽  
Atomic Sphere ◽  
Cohesive Forces ◽  
Thermal Expansion Coefficients ◽  
Expansion Coefficients ◽  
Approximate Wave ◽  
Exact Calculations

1—The first successful calculation of the cohesive forces in metals was made by Wigner and his collaborators, who have obtained wave functions for electrons in metallic Li and Na by numerical integration. The purpose of this paper is twofold: firstly, we shall show that good approximate wave functions for the alkali metals can be obtained by simple analytical methods, the results agreeing well with those of the more exact calculations and with experiment; and secondly, we shall use these wave functions to calculate for the alkali metals the thermal expansion coefficient, a quantity which has not previously been derived from the theory. 2—Following Wigner and Seitz, we surround each atom by a polyhedron, which we replace by a sphere (the atomic sphere) of radius r 1 defined by 4π/3 r 1 3 = atomic volume.

The electrostatic calculation of molecular energies - II. Approximate wave functions and the electrostatic method

1954 ◽  
Vol 226 (1165) ◽  
pp. 179-192 ◽  
Keyword(s):  
Conventional Method ◽  
Hydrogen Molecule ◽  
Electronic States ◽  
Wave Functions ◽  
Chemical Bonds ◽  
Variational Procedure ◽  
Charge Densities ◽  
Electrostatic Method ◽  
Approximate Wave ◽  
Consistent Application

The electrostatic method of calculating molecular energies introduced in the first paper of this series is investigated more closely. It is shown that, for wave functions obtained by a consistent application of the Ritz variational procedure (floating functions), the electrostatic method is equivalent to the conventional method in terms of the Hamiltonian integral. Such floating functions are used to investigate various electronic states of the hydrogen molecule and the hydrogen molecular ion, and to explain certain anomalies in previous calculations. A method for estimating charge densities in localized chemical bonds is outlined.

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