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 Self-consistent field, with exchange, for beryllium - II—The (2 s ) (2 p ) 3 P and 1 P excited states

1936 ◽  
Vol 154 (883) ◽  
pp. 588-607 ◽  
Keyword(s):  
Excited States ◽  
Normal State ◽  
Wave Functions ◽  
The Self ◽  
Energy Separation ◽  
Radial Wave ◽  
Consistent Field ◽  
Exchange Effects ◽  
Self Consistent ◽  
Self Consistent Field

In a recent paper we gave an account of the method and results of the solution of Fock’s equations of the self-consistent field, including exchange effects, for the normal state of neutral beryllium. The present paper is concerned with the extension of the calculations to the first two excited states, (1 s ) 2 (2 s ) (2 p ) 3 P and 1 P, of the same atom. This extension was undertaken for two reasons. Firstly, before going on to attempt the solution of Fock’s equations for a heavier atom, we wished to get some experience of the process of solution of Fock’s equations for a configuration involving wave functions which overlap to a greater extent than the wave functions (1 s ) and (2 s ) of the normal state, and for which exchange effects might be expected to be corre­spondingly greater; and secondly, for an atom with more than one electron outside closed ( nl ) groups, so that a given configuration gives rise to more than one term, the equations of the self-consistent field, when exchange effects are included, are no longer the same for the different terms, and it seemed likely to be of interest to examine the consequent difference between the radial wave functions for the different terms (here 3 P and 1 P), and the effect of this difference on the calculated energy separation between the terms.

scholarly journals Self-consistent field, with exchange, for Cu +

1936 ◽  
Vol 157 (892) ◽  
pp. 490-502 ◽  
Keyword(s):  
Complete Solution ◽  
Point Of View ◽  
Wave Functions ◽  
The Self ◽  
Radial Wave ◽  
Electron Atom ◽  
Consistent Field ◽  
Exchange Effects ◽  
Self Consistent ◽  
Self Consistent Field

Solutions of Fock’s equations for the self-consistent field of a many-electron atom, including exchange effects, have already been carried out for several atoms by Fock and Petrashen and the present authors. The heaviest atom for which results of such calculations have previously been published is Cl - ; Cu + was selected as the next atom for which to attempt the solution of Fock’s equations, for the following reasons. As already pointed out in Paper IV, the results of the solution of Fock’s equations are most interesting for atoms for which exchange effects are large; the self-consistent field without exchange, which is an almost necessary preliminary to the solution of Fock’s equation, had already been worked out for Cu + , and from this work it was known that the (3 d ) 10 group of Cu + is very sensitive to the atomic field, so that it is likely to be con­siderably affected by the inclusion of exchange terms in the equations. Further, in view of the interest of Cu from the point of view of metal theory, it is desirable to have as good wave functions for Cu + as possible, particularly for the outer groups, which are those likely to be most affected by the inclusion of exchange terms in the equation from which they are determined. The number of radial wave functions involved in the normal con­figuration of Cu + is perhaps almost the largest for which a complete solution of Fock’s equations is practicable, for the following reason.

scholarly journals Self-consistent field, with exchange, for nitrogen and sodium

1948 ◽  
Vol 193 (1034) ◽  
pp. 299-304 ◽  
Keyword(s):  
Ionization Potential ◽  
Normal State ◽  
Wave Functions ◽  
The Self ◽  
Electron Atom ◽  
Consistent Field ◽  
Self Consistent ◽  
Self Consistent Field

Wave functions for the normal configurations of neutral nitrogen and N - have been calculated by the method of the self-consistent field with exchange (Fock’s equations). To the accuracy of the approximation represented by these equations, the N - ion would be unstable and liable to auto-ionization, but it is estimated that a better approximation of the treatment of a many-electron atom would give a small positive ionization potential for N - . Revised wave functions for Na + and the normal state of neutral Na have also been calculated. Tables of results are given.

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.

Wave functions for the normal states of Ne+3 and Ne+4

1957 ◽  
Vol 53 (3) ◽  
pp. 663-668 ◽  
Author(s):  
Charlotte Froese ◽  
D. R. Hartree
Keyword(s):  
Atomic Number ◽  
Wave Functions ◽  
The Self ◽  
Atomic Wave ◽  
Consistent Field ◽  
Self Consistent ◽  
Self Consistent Field ◽  
Normal States ◽  
Mathematical Laboratory

ABSTRACTSolutions of Fock's equations for the self-consistent field with exchange have been carried out for Ne+4, using the EDSAC at the Mathematical Laboratory, Cambridge. The initial estimates for the calculation were made by a simplified version of a method previously suggested for interpolating atomic wave functions with respect to atomic number. This gave good estimates for Ne+4, and it is probable that estimates for Ne+3, obtained similarly, are already accurate enough for practical use. Such wave functions for Ne+3 are given. The results have application in astrophysical contexts.

scholarly journals Results of calculations of atomic wave functions. I.—Survey, and self-consistent fields for Cl - and Cu +

1933 ◽  
Vol 141 (844) ◽  
pp. 282-301 ◽  
Keyword(s):  
Radial Wave Function ◽  
Wave Functions ◽  
The Self ◽  
Radial Wave ◽  
Atomic Structures ◽  
Consistent Field ◽  
Self Consistent ◽  
Self Consistent Field ◽  
Electron Group ◽  
Working Out

An approximation to the structure of a many-electron atom can be obtained by considering each electron to be a stationary state in the field of the nucleus and the Schrodinger charge distribution of the other electrons, and rather more than five years ago I gave a method of working out atomic structures based on this idea, and called the field of the nucleus and distribution of charge so obtained the “self-consistent field.” The method of working out the self-consistent field for any particular atom involves essentially ( a ) the estimation of the contributions to the field from the various electron groups constituting the atom in question; ( b ) the solution of the radial wave equation for an electron in the field of the nucleus and other electrons, this solution being carried out for each of the wave functions sup­posed occupied by electrons in the atomic state considered; and ( c ) the calculation of the contribution to the field from the Schrodinger charge dis­tribution of an electron group with each radial wave function. The estimates of the contributions to the field have to be adjusted by trial until the agreement between the contributions finally calculated and those estimated is considered satisfactory.

scholarly journals Self-consistent field with exchange for potassium and argon

1938 ◽  
Vol 166 (927) ◽  
pp. 450-464 ◽  
Keyword(s):  
Wave Function ◽  
Sufficient Accuracy ◽  
Wave Functions ◽  
The Self ◽  
Consistent Field ◽  
Self Consistent ◽  
Self Consistent Field ◽  
The Difference

As already mentioned in a previous paper on the calculation of the self- consistent field with exchange for calcium (D. R. and W. Hartree 1938), wave functions more accurate than those calculated without exchange are required both for K + and for Ar, and the calculations for calcium were carried out partly in the hope that it would be possible to interpolate with sufficient accuracy the difference between the wave functions calculated with and without exchange for K + and Ar from the corresponding differences for Cl - and Ca ++ . The results showed that for the (I s ), (2 s ) and (2 p ) wave function this interpolation would probably be satisfactory, but for the (3 s ) and (3 p ) wave functions it was not as straightforward as had been hoped, though even for the latter wave functions, estimates of the differences could be made which, while rather uncertain, would probably give appreciably better wave functions than those calculated without exchange and taken without modification.

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