HARTREE–FOCK WAVE FUNCTIONS FOR EXCITED STATES: II. SIMPLIFICATION OF THE ORBITAL EQUATIONS

1966 ◽  
Vol 44 (12) ◽  
pp. 3227-3240 ◽  
Author(s):  
Maurice Cohen ◽  
Paul S. Kelly
Keyword(s):  
Total Energy ◽  
Excited States ◽  
Wave Functions ◽  
Lower State ◽  
Hartree Fock ◽  
Orthogonality Constraint ◽  
State Functions

Hartree–Fock wave functions have been calculated for a number of excited states of the helium sequence, the wave functions being constrained to be orthogonal to all lower state functions. The effect of choosing the inner 1s orbital so that the orthogonality constraint is satisfied automatically has been examined, and it is shown that such a choice has a very small effect on the total energy. An extension to heavier systems is proposed.

Stationary properties of approximate wave functions

1969 ◽  
Vol 47 (21) ◽  
pp. 2355-2361 ◽  
Author(s):  
A. R. Ruffa
Keyword(s):  
Wave Function ◽  
Total Energy ◽  
Helium Atom ◽  
Wave Functions ◽  
Quantum Mechanical ◽  
Mechanical Wave ◽  
Expectation Value ◽  
Charge Densities ◽  
Hartree Fock ◽  
Approximate Wave

The accuracy of quantum mechanical wave functions is examined in terms of certain stationary properties. The most elementary of these, namely that displayed by the class of wave functions which yields a stationary value for the total energy of the system, is demonstrated to necessarily require few other stationary properties, and none of these appear to be particularly useful. However, the class of wave functions which yields both stationary energies and charge densities has very important stationary properties. A theorem is proven which states that any wave function in this class yields a stationary expectation value for any operator which can be expressed as a sum of one-particle operators. Since the Hartree–Fock wave function is known to possess these same stationary properties, this theorem demonstrates that the Hartree–Fock wave function is one of the infinitely many wave functions of the class. Methods for generating other wave functions in this class by modifying the Hartree–Fock wave function without changing its stationary properties are applied to the calculation of wave functions for the helium atom.

HARTREE–FOCK WAVE FUNCTIONS FOR EXCITED STATES: III. DIPOLE TRANSITIONS IN THREE-ELECTRON SYSTEMS

1967 ◽  
Vol 45 (5) ◽  
pp. 1661-1673 ◽  
Author(s):  
Maurice Cohen ◽  
Paul S. Kelly
Keyword(s):  
Excited States ◽  
Wave Functions ◽  
Electron Systems ◽  
Matrix Elements ◽  
Isoelectronic Sequence ◽  
Hartree Fock ◽  
Transition Matrix Elements ◽  
Simplified Procedure ◽  
Good Agreement ◽  
Dipole Length

Hartree–Fock wave functions for a number of S, P, and D states of the lithium isoelectronic sequence have been calculated, using a simplified procedure described in an earlier paper. Transition matrix elements for all permitted dipole transitions between these states have been computed using both the dipole length and the dipole velocity formulations. The results are in good agreement with earlier calculations.

HARTREE–FOCK WAVE FUNCTIONS FOR EXCITED STATES: IV. OSCILLATOR STRENGTHS IN THE HELIUM ISOELECTRONIC SEQUENCE

1967 ◽  
Vol 45 (6) ◽  
pp. 2079-2090 ◽  
Author(s):  
Maurice Cohen ◽  
Paul S. Kelly
Keyword(s):  
Matrix Element ◽  
Excited States ◽  
Transition Matrix ◽  
Transition Matrix Element ◽  
Wave Functions ◽  
Oscillator Strengths ◽  
Isoelectronic Sequence ◽  
Hartree Fock ◽  
Good Agreement

Orbital wave functions for a number of singlet and triplet S, P, and D states of the helium sequence through C+4 have been calculated using an approximation described earlier. The wave functions have been employed to calculate the oscillator strengths for all allowed dipole transitions between these states, using both the length and velocity forms of the transition matrix element. Our results are in good agreement with the most accurate values available.

ON THE SPHERICAL OSCILLATOR NUCLEUS

1959 ◽  
Vol 37 (8) ◽  
pp. 944-964 ◽  
Author(s):  
T. D. Newton
Keyword(s):  
Binding Energy ◽  
Model Potential ◽  
Wave Functions ◽  
Potential Well ◽  
Free Nucleon ◽  
Hartree Fock ◽  
Self Consistent ◽  
Low Energies ◽  
Spherical Oscillator ◽  
State Functions

The degree of consistency of an oscillator model of a nucleus is examined by means of a type of Hartree–Fock calculation based on a simple form of internucleon potential valid at low energies. An effective mass equal to 0.757 times the mass of a free nucléon is used. The oscillator wave functions are found to be not far from self-consistent and the oscillator frequency derived is physically reasonable, but the bound on the binding energy is not good. It is also shown that the oscillator wave functions are a good approximation for the state functions of particles bound in a finite potential well having the shape of a cutoff oscillator so that the Hartree–Fock calculation can be used to prescribe a shell model potential.

scholarly journals Adapted Gaussian basis sets for atoms from Li through Xe generated with the generator coordinate Hartree-Fock method

2001 ◽  
Vol 73 (4) ◽  
pp. 511-517 ◽  
Author(s):  
EUSTÁQUIO V. R. DE CASTRO ◽  
FRANCISCO E. JORGE
Keyword(s):  
Total Energy ◽  
Wave Functions ◽  
Basis Sets ◽  
Gaussian Basis Sets ◽  
Hartree Fock ◽  
Generator Coordinate ◽  
Discretization Technique

The generator coordinate Hartree-Fock method is used to generate adapted Gaussian basis sets for the atoms from Li (Z=3) through Xe (Z=54). In this method the Griffin-Hill-Wheeler-Hartree-Fock equations are integrated through the integral discretization technique. The wave functions generated in this work are compared with the widely used Roothaan-Hartree-Fock wave functions of Clementi and Roetti (1974), and with other basis sets reported in the literature. For all atoms studied, the errors in our total energy values relatively to the numerical Hartree-Fock limits are always less than 7.426 mhartree.

THE ORTHOGONALITY ASSUMPTION IN THE HARTREE–FOCK APPROXIMATION

1967 ◽  
Vol 45 (1) ◽  
pp. 7-12 ◽  
Author(s):  
Charlotte Froese
Keyword(s):  
Excited States ◽  
Lower State ◽  
Hartree Fock ◽  
Orthogonality Constraints

Several procedures are considered for maintaining orthogonality between one-electron functions of the nsn's1S excited states as well as orthogonality to a lower state. The results are compared with those obtained without any orthogonality constraints.

The orthogonality assumption in the Hartree–Fock approximation for 2pn3p configurations

1968 ◽  
Vol 46 (21) ◽  
pp. 2435-2440 ◽  
Author(s):  
Charlotte Froese Fischer
Keyword(s):  
Total Energy ◽  
Hartree Fock ◽  
Orthogonality Constraints ◽  
Orthogonality Constraint

Hartree–Fock (HF) results for Na+ 2p53p1S and Ne+ 1s2p53p2S are reported. In order for the SCF process to converge, additional orthogonality constraints to the corresponding 2p6 states had to be introduced. The resulting 2p functions were nodeless. Similar calculations were performed for O 2p33p2P and Ne+ 2p43p2P for which standard HF results were available. A comparison of the two methods indicates that, unlike the He 1s2s1S case, the additional orthogonality constraint did not improve the total energy although it removed the extra node from the 2p function.

HARTREE–FOCK WAVE FUNCTIONS FOR EXCITED STATES: THE 2 1S STATE OF HELIUM

1965 ◽  
Vol 43 (10) ◽  
pp. 1867-1881 ◽  
Author(s):  
Maurice Cohen ◽  
Paul S. Kelly
Keyword(s):  
Excited State ◽  
Ground State ◽  
Wave Functions ◽  
State Function ◽  
Hartree Fock ◽  
Singlet States ◽  
Low Energies ◽  
Orthogonality Constraint ◽  
Satisfactory Approximation ◽  
Excited Singlet States

Hartree–Fock wave functions for the first-excited singlet states of several members of the helium isoelectronic sequence have been calculated. It is shown that unphysically low energies result if the excited-state function is not constrained to be orthogonal to the ground-state eigenfunction, and the excited-state orbitals are chosen orthogonal.If the excited-state orbitals are not chosen orthogonal, the effect of the overall constraint is small, raising the total energy very slightly; however, the effect of imposing both constraints simultaneously is appreciable. It is concluded that the most satisfactory approximation involves nonorthogonal excited-state orbitals, together with an overall orthogonality constraint towards the ground state.

scholarly journals Valence: A Massively Parallel Implementation of the Variational Subspace Valence Bond Method

2018 ◽  
Author(s):  
Graham Fletcher ◽  
Colleen Bertoni ◽  
Murat Keçeli ◽  
Michael D'Mello
Keyword(s):  
Excited States ◽  
Software Package ◽  
Parallel Implementation ◽  
Valence Bond ◽  
Wave Functions ◽  
Massively Parallel ◽  
Hartree Fock ◽  
External Software ◽  
Single Configuration ◽  
Bond Method

This work describes the software package, Valence, for the calculation of molecular<br>energies using the variational subspace valence bond (VSVB) method. VSVB is a highly scalable ab initio electronic structure method based on non-orthogonal orbitals. Important features of practical value include: Valence bond wave functions of Hartree–Fock quality can be constructed with a single determinant; excited states can be modeled with a single configuration or determinant; wave functions can be constructed automatically by combining orbitals from previous calculations. The opensource software package includes tools to generate wave functions, a database of generic orbitals, example input files, and a library build intended for integration with other packages. We also describe the interface to an external software package, enabling the computation of optimized molecular geometries and vibrational frequencies.

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