Computational Chemistry

The Slater determinant for a two-electron closed-shell molecule is

C¼

1

ffiffiffiffi

2!

p

c 1 ð 1 Það 1 Þ c 1 ð 1 Þbð 1 Þ

c 1 ð 2 Það 2 Þ c 1 ð 2 Þbð 2 Þ

(1)

consisting of one spatial MO (c 1 ), or two spin MOs (c 1 aandc 1 b), each of which is
populated alternately with electron 1 and with electron 2. When expanded accord-
ing to the usual rule this gives

1 =

p

2 !½c 1 ð 1 Það 1 Þ:c 1 ð 2 Þbð 2 Þ#c 1 ð 1 Þbð 1 Þ:c 1 ð 2 Það 2 ފ (2)

The expansion presentsCas a sum of products. Realizing that the second term in
(2) can be derived from the first by switching the coordinates of electrons 1 and
2 and replacingþby#leads to the idea of writingCas a sum of “switched” or
permuted terms:

C¼ 1 =

p

2!

X

ð# 1 ÞPP^½c 1 ð 1 Það 1 Þ:c 1 ð 2 Þbð 2 ފ (3)

where the sum is over all possible permutations (two) of the two spin orbitals which
can be obtained by switching the electron coordinates. The permutation operatorP^
has the effect of switching electron coordinates. As a check on this (ignoring the
1/

pffi
2! normalization factor):
Permutation 1 leads to (#1)^1 ½c 1 ð 2 Það 2 Þ:c 1 ð 1 Þbð 1 ފ¼c 1 ð 1 Þbð 1 Þ:c 1 ð 2 Það 2 ފ,
the second term in (2).
Permutation 2 (acting on the result of permutation 1) leads to

ð# 1 Þ#^2 ½c 1 ð 1 Það 1 Þ:c 1 ð 2 Þbð 2 ފ¼c 1 ð 1 Það 1 Þ:c 1 ð 2 Þbð 2 Þ;

the first term in (2).
Particularly forCwith more than two spin orbitals the permutation operator
formulation [1] is less transparent than the determinant one.

Reference

1. E.g. (a) Levine IN (2000) Quantum chemistry, 5th edn. Prentice Hall, Upper Saddle River,
   NJ, pp 286–287. (b) Cook DB (2005) Handbook of computational quantum chemistry.
   Dover, Mineola, NY, section 1.6. (c) Pople JA, Beveridge DL (1970) Approximate molec-
   ular orbital theory. McGraw-Hill, New York, sections 1.7, 2.2. (d) Hehre WJ, Radom L,
   Schleyer PvR, Pople JA (1986) Ab initio molecular orbital theory. Wiley, New York,
   section 2.4

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