scholarly journals Results of calculations of atomic wave functions IV—Results for F - , Al +3 , and Rb +

1935 ◽  
Vol 151 (872) ◽  
pp. 96-105 ◽  
Author(s):  
Douglas Rayner Hartree
Keyword(s):  
Energy Parameter ◽  
Wave Functions ◽  
Numerical Accuracy ◽  
Radial Wave ◽  
Atomic Wave ◽  
Consistent Field ◽  
Self Consistent Field ◽  
The One ◽  
High Degree ◽  
Radial Wave Equation

In three previous papers, results of calculations of atomic wave functions, carried out by the method of the self-consistent field to a fairly high degree of numerical accuracy (for work of this kind), have been given for a number of atoms. The present paper gives further results of this kind for F - , Al +3 , and Rb + . Similar calculations are in progress for Ag + , and the results of the preliminary stages of the calculation have been given by Miss Black. The results here given are presented in the same form as in previous papers, namely:― (1) Unnormalized radial wave functions P, and the values of the normalization integral ∫ 0 ∞ P 2 dr , and of the energy parameter ε in the one-electron radial wave equation, for each function P; also values of P/ r l + 1 for small r .

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 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.

Accurate Numerical Hartree–Fock Self-Consistent-Field Wave Functions for La+, Tm+, and Yb+

1972 ◽  
Vol 50 (7) ◽  
pp. 708-709 ◽  
Author(s):  
K. M. S. Saxena
Keyword(s):  
Rare Earth ◽  
Virial Theorem ◽  
Wave Functions ◽  
Rare Earth Ions ◽  
Radial Wave ◽  
Hartree Fock ◽  
Consistent Field ◽  
Self Consistent ◽  
Self Consistent Field ◽  
Field Wave

Accurate numerical Hartree–Fock (HF) self-consistent-field (SCF) wave functions have been obtained for La+(4ƒ16S)3F and 1F, Tm+(4ƒ136S)3F and 1F, and Yb(4ƒ146S)2S rare-earth ions. In general, the total energy values have an accuracy of seven figures, the virial theorem is satisfied to seven significant digits, and the radial wave functions are self-consistent and without tail oscillations to three decimals. Several Hartree–Fock parameters are also evaluated with these functions.

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++.

Atomic wave functions for gold and thallium

1955 ◽  
Vol 51 (3) ◽  
pp. 486-503 ◽  
Author(s):  
A. S. Douglas ◽  
D. R. Hartree ◽  
W. A. Runciman
Keyword(s):  
Wave Function ◽  
Absorption Edge ◽  
Radial Wave Function ◽  
Wave Functions ◽  
Radial Wave ◽  
Consistent Field ◽  
Self Consistent ◽  
Self Consistent Field ◽  
Independent Variable ◽  
Independent Calculation

Before the war, self-consistent field calculations for the Au+ ion had been carried out by W. Hartree but were left still unpublished at his death (see prefatory note in (5)). These results have been used by Brenner and Brown (1) in a relativistic calculation of the K-absorption edge for gold, and they were also used in obtaining initial estimates for the partial self-consistent field calculations for thallium of which results are given in §§3–5 of the present paper. In the meantime an independent calculation for Au+ has been carried out by Henry (6), and his results agree closely with those of W. Hartree. However, it still seems desirable to publish the latter, since they give directly the radial wave function P(nl; r) at exact values of r, whereas Henry used log r as independent variable, as had been done for similar calculations for Hg(4), and has tabulated r½P(nl; r) which is the natural dependent variable to use with log r as independent variable (2); in some applications it is more convenient to have the radial wave functions themselves.

Time-dependent variational principle and self-consistent field equations

1985 ◽  
Vol 98 (2) ◽  
pp. 373-379 ◽  
Author(s):  
V. U. Nazarov
Keyword(s):  
Variational Principle ◽  
Wave Functions ◽  
Time Dependent ◽  
Quantum Mechanical ◽  
Field Equations ◽  
Consistent Field ◽  
Self Consistent Field ◽  
The One ◽  
The Given ◽  
Derivatives Of

In this paper a quantum-mechanical variational principle is proposed, in which the functional varied is the precision, with which the time-dependent Schrödinger equation is satisfied by the wave functions of the given class. Another distinctive feature of our approach is that the independently varied functions are the time derivatives of the one-particle wave functions, of which the wave function of the system is constructed.

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.

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.

Radial wave functions with exchange for Va 2+ , Kr and Ag +

1958 ◽  
Vol 247 (1250) ◽  
pp. 390-399 ◽  
Keyword(s):  
Wave Functions ◽  
Initial Slope ◽  
Radial Wave ◽  
University Of Toronto ◽  
Hartree Fock ◽  
Consistent Field ◽  
Self Consistent ◽  
Physical Approximation ◽  
Self Consistent Field ◽  
The University

A generalized program for calculating atomic radial wave functions with exchange has been prepared for the Ferranti computer (FERUT) at the University of Toronto, and is described in a separate paper. This program has now been applied to V 2+ , Kr and Ag + . The wave functions for these atoms, together with the energy and initial slope parameters, are presented to the accuracy justified by the physical approximation of the Hartree–Fock formulation. The configurations of Kr and Ag + are considerably larger than any which have previously been treated by the self-consistent field process with exchange.

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.

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