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The Wave Mechanics of an Atom with a Non-Coulomb Central Field. Part II. Some Results and Discussion

Published online by Cambridge University Press:  24 October 2008

D. R. Hartree
D. R. Hartree
Affiliation:
St John's College.
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Abstract

The methods of solution of the wave equation for a central field given in the previous paper are applied to various atoms. For the core electrons, the details of the interaction of the electrons in a single nk group are neglected, but an approximate correction is made for the fact that the distributed charge of an electron does not contribute to the field acting on itself (§2).

For a given atom the object of the work is to find a field such that the solutions of the wave equation for the core electrons in this field (corrected as just mentioned for each core electron) give a distribution of charge which reproduces the field. This is called the self-consistent field, and the process of finding it is one of successive approximation (§ 3).

Approximations to the self-consistent field have been found for He (§ 4), Rb+ (§ 5), Na+, Cl− (§ 9). For He the energy parameter for the solution of the wave equation for one electron in the self-consistent field of the nucleus and the other corresponds to an ionisation potential of 24·85 volts (observed 24·6 volts); this agreement suggests that for other atoms the values of the energy parameter in the self-consistent field (corrected for each core electron) will probably give good approximations to the X-ray terms (§4).

The most extensive work has been carried out for Rb+. The distribution of charge given by the wave functions in the self-consistent field is compared with the distribution calculated by other methods (§ 6). The values of X-ray and optical terms calculated from the self-consistent field show satisfactory agreement with those observed (§ 7).

The wave mechanical analogue of the case in which on the orbital model an internal and an external orbit of the same energy are possible is discussed (§ 8).

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 24 , Issue 1 , January 1928 , pp. 111 - 132
Copyright
Copyright © Cambridge Philosophical Society 1928

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References

* Corresponding work for the orbital atomic model has been done from the. first point of view by Fues, E., Zeit. f. Phys., Vol. XI, p. 364 (1922); Vol. XII, p. 1; Vol. XIII, p. 211 (1923); Vol. XXI, p. 265 (1924);CrossRefGoogle ScholarHartree, D. R., Proc. Camb. Phil. Soc., Vol. XXI, p. 265 (1923)Google Scholar and Sugiura, Y. and Urey, H. C., Det Kongel. Danske Videnskab. Selskab., Math. Phys. Medd., Vol. VII, No. 13 (1926)Google Scholar, and from the second by Lindsay, R. B., Publ. Mass. Inst. Technology, Series II, No. 20 (1924).Google Scholar

† For the orbital atomic model the writer has tried to obtain such results, but without success.Google Scholar

* Sugiura, Y., Phil. Mag., Ser. 7, Vol. IV, p. 495 (1927).CrossRefGoogle Scholar

† The determination of the distribution of charge directly from a potential or field given as a function of the radius involves numerical differentiation, which is an unsatisfactory process, especially in this case, when it is possible to make a small increase of the field at one radius and a small decrease at another without appreciably affecting the fit between calculated and observed term values.Google Scholar

‡ See, for example, Hund, F., Linienspektren, p. 114et seq.Google Scholar

* Thomas, L. H., Proc Camb. Phil. Soc., Vol. XXIII, p. 542 (1927). It is necessary to extrapolate the field empirically beyond the range to which Thomas' results apply.CrossRefGoogle Scholar

* The process of successive approximation by taking the initial field always equal to the final field of the previous approximation is not always convergent, though perhaps it usually is. Even when it is, a more rapid convergence to the self-consistent field is obtained by the rule given here.Google Scholar

† The large difference between Z and Zp is not always realised; for Rb the maximum value of Z/Zp is over 2 and that of Z − Zp is over 13, so that Z and Zp cannot be considered as even approximately equal for a non-Coulomb field.Google Scholar

‡ The reason is that Z is known at equal intervals of r, and for the direct integration unduly small intervals would have to be used in some regions, in order to keep down the higher orders of difference due to the r −2 factor; the formula for the mean value for , when Z is known at equal intervals of r not of 1/τ, is more trouble to use than the differential equation.Google Scholar

* Actually the work was done in the inverse order; the application of the method to helium as an experiment was suggested by the good agreement between values of e so calculated and the observed X-ray term values for more complicated atoms.Google Scholar

* In Figs. 1, 2, 3 a different scale of r is used for r < 1 and r > 1. A scale open enough to show the detail of the curves for large r is unnecessarily open for larger r; the use of two uniform scales has seemed preferable to the continuous distortion introduced, for example, by a logarithmic scale of r.+1.+A+scale+open+enough+to+show+the+detail+of+the+curves+for+large+r+is+unnecessarily+open+for+larger+r;+the+use+of+two+uniform+scales+has+seemed+preferable+to+the+continuous+distortion+introduced,+for+example,+by+a+logarithmic+scale+of+r.>Google Scholar

† The charge density is calculated from the wave functions from which the final field is built up, not by numerical differentiation of Z.Google Scholar

* loc cit.Google Scholar

† For simplicity it has been assumed that all orbits of the same n have the same inner apsidal distance and that all of the same k have the same outer apsidal distance, both of which assumptions are approximately but not accurately true. The orbital model has not been worked oat completely for half integer values of k, but the curve is drawn from data estimated from the work with integer values.Google Scholar

* Pauling, L., Proc. Roy. Soc., Vol. CXIV, p. 181 (1927).CrossRefGoogle Scholar

* With the half integer values of k there are no circular orbits. The curve for the orbital model is really much more broken than as shown in Fig. 3 (see second footnote on page 119).Google Scholar

† The effect of Pauling's correcting factor ΔSs is small in this case, so that the errors lie in the values of S20.Google Scholar

† Fowler, A., Report on the Series in Line Spectra, p. 104. The first d term has been given the principal quantum number n=4 in accordance with § 8 of the present paper. The doublet separation for this term has not been observed and the value given is estimated from the value observed for the second d term.Google Scholar

‡ Integer values of k were used in these calculations. The writer has tried some work with half integral values of k, but without any very marked improvement in the general agreement between observed and calculated values.Google Scholar

† In the case of He (terms other than s terms) the core consists of a hydrogenlike system, for which the second order Stark effect, on which the polarisability depends, can be worked out exactly for a uniform perturbing field. This has already been treated on the wave mechanics by Waller, I., Zeit. f. Phys., Vol. XXXVIII, p. 635 (1926); see particularly § 3.CrossRefGoogle Scholar

* See Wentel, G., Zeit. f. Phys., Vol. XIX, p. 52 (1923).Google Scholar