Physical Chemistry Third Edition

19.2 The Self-Consistent Field Method 797

We describe the application of the method to the ground state of the helium atom.

If electron 2 were fixed at locationr 2 the Schrödinger equation would be

−

h ̄^2

2 m

∇^21 ψ(1)−

Ze^2

4 πε 0 r 1

ψ 1 (1)+

e^2

4 πε 0 r 12

ψ 1 (1)E 1 ψ 1 (1) (19.2-1)

wherer 12 is the distance between the fixed position of electron 2 and the variable posi-

tion of electron 1. We now seek to construct a one-electron equation without assuming

that electron 2 is fixed. We assume that electron 2 occupies the normalized orbitalψ 2 (2)

so that its probability of being found in the volume elementd^3 r 2 is

(Probability of electron 2 being ind^3 r 2 )ψ 2 (2)∗ψ 2 (2)d^3 r 2 |ψ 2 (2)|^2 d^3 r 2

(19.2-2)

We replace the electron–electron repulsion term in the Hamiltonian of Eq. (19.2-1)

by a term containing an average over all positions of electron 2, using the probability

density of Eq. (19.2-2):

−

h ̄^2

2 m

∇^21 ψ 1 (1)−

Ze^2

4 πε 0 r 1

ψ 1 (1)+

[∫

e^2

4 πε 0 r 12

|ψ 2 (2)|^2 d^3 r 2

]

ψ 1 (1)E 1 ψ 1 (1)

(19.2-3)

This is anintegrodifferential equation. It contains both the derivative of an unknown

function and an integral containing an unknown function. In the case of the helium atom

ground state, both electrons occupy the same space orbital, soψ 1 andψ 2 are the same

unknown function and there is a single integrodifferential equation. Ifψ 1 (1) andψ 2 (2)

were different orbitals, we would have two simultaneous integrodifferential equations.

For an atom with several electrons, there is an integrodifferential equation for each

occupied space orbital.

The integrodifferential equation is solved by iteration (successive approximations)

as follows: The first approximation is obtained by replacing the orbital under the integral

by the zero-order function or some other known function, which we denote byψ(0)(1s)(2).

Equation (19.2-3) for the ground state of the helium atom now becomes

−

h ̄^2

2 m

∇ 12 ψ

(1)

1 s(1)−

Ze^2

4 πε 0 r 1

ψ

(1)

1 s(1)+

[∫

e^2

4 πε 0 r 12

|ψ

(0)

1 s(2)|

(^2) d (^3) r 2

]

ψ

(1)

1 s

E

(1)

1 sψ

(1)

1 s(1) (19.2-4)

The integral overr 2 contains only known functions. It cannot be integrated mathemati-

cally, but can be evaluated numerically for various values ofr 1 , giving a table of values

for different values ofr 1. Equation (19.2-4) becomes a differential equation that can

be solved numerically, using standard methods of numerical analysis.^6 The solution

ψ(1(1)s)(1) is independent ofθ 1 andφ 1 and is represented by a table of values ofψ(1) 1 s(1)

as a function ofr 1.

The next iteration (repetition) is carried out by replacingψ(0) 1 s(2) under the integral

sign byψ(1)(1s)(2), obtaining an equation for the next approximation, denoted byψ 1 (2)s(1).

(^6) R. L. Burden, J. D. Faires, and A. C. Reynolds,Numerical Analysis, 2nd ed., Prindle, Weber, & Schmidt,
Boston, 1981, p. 505ff.