## Extract

1. The theoretical determination of the energies of the stationary states of an atomic system is bound up with the solution of the many-body problem— in particular, with the determination of wave functions of many-electron atoms. An exact solution is not known, but approximations to it have been made by Hartree, Slater, Fock and Lennard-Jones.§ The method adopted is to replace the physical problem by an artificial one which admits of a solution, e. g., Hartree replaces the actual many-body problem by a one-body problem with a central field for each electron. Generally, the Schrodinger equation for an atom of nuclear charge N is {^{N}Ʃ _{i = 1} (-1/2∇*i*^{2}-N/*r** _{i}*) +

^{N}Ʃ

_{i>j = 1/rij}-E} Ψ = 0, using atomic units11 and the usual notation. The artificial system replacing (1.1) has the equation {

^{N}Ʃ

_{i = 1}(-1/2∇

*i*

^{2}-

*v*

_{i}) -E} ψ = 0,

*V*

_{i}being a function of the co-ordinates of the

*i*th. electron only. Equation (1.2) is separable, and reduces to equations of the type {-1/2∇

*i*

^{2}-

*v*

_{i}) -E

_{i}} ψ = 0, in the space co-ordinates of the -

*i*th electron alone. If the solutions of equations (1. 3) are Ψ(α∣1), Ψ(π|p), where the Greek letter is the label of the wave function, and the numeral or Roman letter indicates the electron whose co-ordinates are substituted, then a solution of (1. 2) is ψ = Ψ(α∣1) Ψ (β∣2)....Ψ(π|p). The form of wave function which must be assumed in order to satisfy Pauli’s Exclusion Principle and be antisymmetric in the co-ordinates of all pairs of electrons, is the determinantal form Ψ = ∣ψ = Ψ(α∣1) Ψ (α∣2)....Ψ(α|

*p*) ∣ ∣ψ = Ψ(β∣1) Ψ (β∣2)....Ψ(β|

*p*) ∣ .................................................. ∣ψ = Ψ(π∣1) Ψ (π∣2)....Ψ(π|

*p*) ∣ which is the sum of the expressions obtained by permuting the co-ordinates 1, 2,.........,

*p*in the product (1. 4) and taking account of the signs of the permutations. Thus we obtain an approximate wave function for the whole atom in terms of the one-electron wave functions.

## Footnotes

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- Received July 19, 1932.

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