Physical Chemistry Third Edition

782 18 The Electronic States of Atoms. II. The Zero-Order Approximation for Multielectron Atoms

Antisymmetrization

The orbital wave function of Eq. (18.6-3) can be antisymmetrized by writing a sum

of the six terms corresponding to each possible order of the spin orbitals, keeping the

particle labels in a fixed order. Each term that is generated from the first term by one

exchange (“permutation”) of a pair of indexes must be given a negative sign, and each

term that is generated by two permutations of pairs of indexes must be given a positive

sign. The antisymmetrized function is

Ψ

1

√

6

[ψ 1 (1)ψ 2 (2)ψ 3 (3)−ψ 2 (1)ψ 1 (2)ψ 3 (3)−ψ 1 (1)ψ 3 (2)ψ 2 (3)

−ψ 3 (1)ψ 2 (2)ψ 1 (3)+ψ 3 (1)ψ 1 (2)ψ 2 (3)+ψ 2 (1)ψ 3 (2)ψ 1 (3)] (18.6-5)

Exercise 18.6

Show that the function produced by exchanging particle labels 1 and 3 in Eq. (18.6-5) is the

negative of the original function. Choose another permutation and show the same thing.

If a spin orbital occurs more than once in each term in the wave function, the wave

function vanishes, in agreement with the Pauli exclusion principle.

EXAMPLE18.5

If orbitalsψ 1 andψ 3 are the same function, show that the wave function of Eq. (18.6-5)

vanishes.

Solution

Ψ

1

√

6

[ψ 1 (1)ψ 2 (2)ψ 1 (3)−ψ 2 (1)ψ 1 (2)ψ 1 (3)−ψ 1 (1)ψ 1 (2)ψ 2 (3)

−ψ 1 (1)ψ 2 (2)ψ 1 (3)+ψ 1 (1)ψ 1 (2)ψ 2 (3)+ψ 2 (1)ψ 1 (2)ψ 1 (3)]

The first and fourth terms cancel, the third and fifth terms cancel, and the second and sixth

terms cancel, so that the wave function vanishes.

The Slater determinant is named for
John C. Slater, 1900–1976, a prominent
American physicist who made various
contributions to atomic and molecular
quantum theory.

Slater Determinants

There is another notation that can be used to write the antisymmetrized wave function

of Eq. (18.6-5). It is known as aSlater determinant. A determinant is a quantity derived

from a square matrix by a certain set of multiplications, additions, and subtractions.

There is a brief introduction to matrices and determinants in Appendix B. If the elements

of the matrix are constants, the determinant is equal to a single constant. If the elements

of the matrix are orbitals, the determinant of that matrix is a single function of all of the

coordinates on which the orbitals depend. The wave function of Eq. (18.6-5) is equal

to the determinant:

Ψ

1

√

6

∣ ∣ ∣ ∣ ∣ ∣ ∣

ψ 1 (1) ψ 1 (2) ψ 1 (3)

ψ 2 (1) ψ 2 (2) ψ 2 (3)

ψ 3 (1) ψ 3 (2) ψ 3 (3)

∣ ∣ ∣ ∣ ∣ ∣ ∣

(18.6-6)