Abstract
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/
r
ij
-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.
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