Approximate wave functions for carboranes parametrized from self-consistent field model calculations

1970 ◽  
Vol 9 (12) ◽  
pp. 2743-2748 ◽  
Author(s):  
William N. Lipscomb ◽  
Thomas F. Koetzle
Keyword(s):  
Field Model ◽  
Wave Functions ◽  
Model Calculations ◽  
Consistent Field ◽  
Self Consistent ◽  
Self Consistent Field ◽  
Approximate Wave

SCATTERING FACTORS FOR SOME OF THE HEAVIER ATOMS

1959 ◽  
Vol 37 (9) ◽  
pp. 967-969 ◽  
Author(s):  
Beatrice H. Worsley
Keyword(s):  
Wave Functions ◽  
Radial Wave ◽  
X Ray ◽  
University Of Toronto ◽  
Consistent Field ◽  
Self Consistent ◽  
Self Consistent Field ◽  
Atomic Scattering ◽  
Approximate Wave ◽  
The University

A program for calculating X-ray atomic scattering factors from the radial wave functions has been written for the IBM 650 installation at the University of Toronto. It has been applied to the results of self-consistent field calculations previously performed at this University on the FERUT computer. Results are given for Ne, V++, Kr, Ag+, and Pb+++. The results for Ne and V++ are compared with those calculated by Freeman using Allen's wave functions for Ne and Hartree's approximate wave functions for V++.

APPROXIMATE WAVE FUNCTIONS OF Pb+++ BY THE METHOD OF SELF-CONSISTENT FIELD WITHOUT EXCHANGE

1959 ◽  
Vol 37 (9) ◽  
pp. 983-988 ◽  
Author(s):  
J. F. Hart ◽  
Beatrice H. Worsley
Keyword(s):  
Common Point ◽  
Wave Functions ◽  
The Self ◽  
Decimal Digit ◽  
Hartree Fock ◽  
The Core ◽  
Consistent Field ◽  
Self Consistent ◽  
Self Consistent Field ◽  
Approximate Wave

The FERUT program previously described for calculating Hartree–Fock wave functions by the method of the self-consistent field has been adapted to the configuration Pb+++. Although the exchange factors were omitted, the program was extended beyond its original scope in other respects, and an assessment of the difficulties so encountered is made. It might be noted, however, that, except in the case of the 4ƒ wave function, it was possible to begin all the integrations at a common point. Initial estimates were made from the Douglas, Hartree, and Runciman results for thallium. The estimates for the core functions were not assumed to be satisfactory. The errors in the final wave functions are considered to be no more than one or two units in the second decimal digit.

scholarly journals Self-consistent field, with exchange, for beryllium

1935 ◽  
Vol 150 (869) ◽  
pp. 9-33 ◽  
Keyword(s):  
Wave Functions ◽  
The Other ◽  
Numerical Work ◽  
Spin Effects ◽  
Consistent Field ◽  
Self Consistent ◽  
Self Consistent Field ◽  
Average Distribution ◽  
Approximate Wave ◽  
The Individual

Except for the lightest atoms, most calculations of approximate wave functions and fields for many-electron atoms have been carried out by the method of the “self-consistent field,” of which the principle is, shortly, the determination of a set of one-electron wave functions such that each represents a stationary state of an electron in the field of the nucleus and the Schrödinger charge distribution of the electrons occupying the other wave functions of the set. This method has been found quite practicable for numerical work, even for the heaviest atoms. As so far applied, it involves three main approximations, namely, ( a ) neglect of relativity and spin effects, ( b ) neglect of exchange effects, and ( c ) treatment of the wave function of the whole atom as built up of functions of the co-ordinates of the individual electrons only, its depen­dence on the mutual distances between every pair of electrons being neglected; or, in other words, each electron is replaced by a statistical average distribution, in calculating its effect on the other electrons on the atom.

scholarly journals Self-consistent field, including exchange and superposition of configurations, with some results for oxygen

1939 ◽  
Vol 238 (790) ◽  
pp. 229-247 ◽  
Keyword(s):  
Wave Functions ◽  
Radial Wave ◽  
Consistent Field ◽  
Self Consistent ◽  
Self Consistent Field ◽  
Approximate Wave ◽  
The One ◽  
The Individual ◽  
Single Configuration ◽  
Good Agreement

The calculation of approximate wave functions for the normal configurations of the ions O +++, O ++, O +, and neutral O, and the calculation of energy values from the wave functions, was carried out some years ago by Hartree and Black (1933)- In this work, the one-electron radial wave functions were calculated by the method of the selfconsistent field without exchange, but exchange terms were included in the calculation of the energy from these radial wave functions. In the energy calculations, the same radial wave functions were taken for each of the spectral terms arising from a single configuration; * consequently the ratios between the calculated intermultiplet separations were exactly those given by Slater’s (1929) theory of complex spectra, f The ratios between the observed intermultiplet separations, however, depart considerably from these theoretical values (for example, we have for 0 ++ ( 1 D - 1 S) / ( 3 P - 1 D), calc. 3 : 2, obs. 1.04 :1), although the energies of the individual terms, and particularly the intermultiplet separation between the lower terms, show quite a good agreement with the observed values.

scholarly journals Results of calculations of atomic wave functions. II.—Results for K + and Cs +

1934 ◽  
Vol 143 (850) ◽  
pp. 506-517 ◽  
Keyword(s):  
Wave Functions ◽  
The Self ◽  
Numerical Accuracy ◽  
Atomic Wave ◽  
Consistent Field ◽  
Self Consistent ◽  
Self Consistent Field ◽  
Approximate Wave ◽  
High Degree

In a recent papers I presented the results of calculations of approximate wave functions of two atoms, based on the method of the “self-consistent field”, these calculations having been carried out to a fairly high degree of numerical accuracy (for work of this kind) as regards both precision of the work and the approximation to the self-consistent field attained, in order that the results published should be quite dependable. I also gave a survey of the situation which led to such calculations being undertaken, and mentioned other atoms for which they were being made. This paper presents a second instalment of such results, namely, those for the atoms K + and Cs + . Of these atoms, Cs is the heaviest for which calculations of the self-consistent field have so far been completed, though work on a still heavier atom, namely, mercury, has been started, and it is hoped that rough results, at least, will be available before long.

scholarly journals The relativistic self-consistent field

1935 ◽  
Vol 152 (877) ◽  
pp. 625-649 ◽  
Keyword(s):  
Wave Equation ◽  
Field Equation ◽  
Wave Functions ◽  
The Self ◽  
Magnetic Interactions ◽  
Consistent Field ◽  
Exchange Effects ◽  
Self Consistent ◽  
Self Consistent Field ◽  
Schrödinger's Equation

1—The method of the self-consistent field for determining the wave functions and energy levels of an atom with many electrons was developed by Hartree, and later derived from a variation principle and modified to take account of exchange and of Pauli’s exclusion principle by Slater* and Fock. No attempt was made to consider relativity effects, and the use of “ spin ” wave functions was purely formal. Since, in the solution of Dirac’s equation for a hydrogen-like atom of nuclear charge Z, the difference of the radial wave functions from the solutions of Schrodinger’s equation depends on the ratio Z/137, it appears that for heavy atoms the relativity correction will be of importance; in fact, it may in some cases be of more importance as a modification of Hartree’s original self-nsistent field equation than “ exchange ” effects. The relativistic self-consistent field equation neglecting “ exchange ” terms can be formed from Dirac’s equation by a method completely analogous to Hartree’s original derivation of the non-relativistic self-consistent field equation from Schrodinger’s equation. Here we are concerned with including both relativity and “ exchange ” effects and we show how Slater’s varia-tional method may be extended for this purpose. A difficulty arises in considering the relativistic theory of any problem concerning more than one electron since the correct wave equation for such a system is not known. Formulae have been given for the inter-action energy of two electrons, taking account of magnetic interactions and retardation, by Gaunt, Breit, and others. Since, however, none of these is to be regarded as exact, in the present paper the crude electrostatic expression for the potential energy will be used. The neglect of the magnetic interactions is not likely to lead to any great error for an atom consisting mainly of closed groups, since the magnetic field of a closed group vanishes. Also, since the self-consistent field type of approximation is concerned with the interaction of average distributions of electrons in one-electron wave functions, it seems probable that retardation does not play an important part. These effects are in any case likely to be of less importance than the improvement in the grouping of the wave functions which arises from using a wave equation which involves the spins implicitly.

scholarly journals A Hybrid Monte Carlo Self-Consistent Field Model of Physical Gels of Telechelic Polymers

2018 ◽  
Vol 14 (12) ◽  
pp. 6532-6543 ◽  
Author(s):  
J. Bergsma ◽  
F. A. M. Leermakers ◽  
J. M. Kleijn ◽  
J. van der Gucht
Keyword(s):  
Monte Carlo ◽  
Field Model ◽  
Telechelic Polymers ◽  
Hybrid Monte Carlo ◽  
Consistent Field ◽  
Self Consistent ◽  
Self Consistent Field
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