Essentials of Computational Chemistry

124 4 FOUNDATIONS OF MOLECULAR ORBITAL THEORY
(the reader is urged to verify that 3 SD does indeed satisfy the Pauli principle; for the ‘SD’
subscript, see next section). Note that if we integrate | 3 SD| 2 over all space we have
3
SD
2 dr1dω1dr2dω2 = 1
2
|ψa(1)| 2 |α(1)| 2 |ψb(2)| 2 |α(2)| 2 dr1dω1dr2dω2
− 2 ψa(1)ψb(1) |α(1)| 2 ψb(2)ψa(2) |α(2)| 2 dr1dω1dr2dω2
+
|ψa(2)| 2 |α(2)| 2 |ψb(1)| 2 |α(1)| 2 dr1dω1dr2dω2
= 1
(1 − 0 + 1)
2
= 1 (4.44)
where ω is a spin integration variable, the simplification of the various integrals on the
r.h.s. proceeds from the orthonormality of the MOs and spin functions, and we see that the
prefactor of 2 −1/2 in Eq. (4.43) is required for normalization.
4.5.4 Slater Determinants
A different mathematical notation can be used for Eq. (4.43)
3
SD = 1
√
2
ψa(1)α(1)
ψa(2)α(2)
ψb(1)α(1)
ψb(2)α(2)
(4.45)
where the difference of MO products has been expressed as a determinant. Note that the
permutation operator P applied to a determinant has the effect of interchanging two of
the rows. It is a general property of a determinant that it changes sign when any two
rows (or columns) are interchanged, and the utility of this feature for use in constructing
antisymmetric wave functions was first exploited by Slater (1929). Thus, the ‘SD’ subscript
used in Eqs. (4.43)–(4.45) stands for ‘Slater determinant’. On a term-by-term basis, Slaterdeterminantal
wave functions quickly become rather tedious to write down, but determinantal
notation allows them to be expressed reasonably compactly as, in general,
SD = 1
χ1(1) χ2(1) ··· χN(1)
χ1(2) χ2(2) ··· χN(2)
√
N!
.
(4.46)
. . .. .
χ1(N) χ2(N) ··· χN(N)
where N is the total number of electrons and χ is a spin-orbital, i.e., a product of a
spatial orbital and an electron spin eigenfunction. A still more compact notation that finds
widespread use is
SD = |χ1χ2χ3 ···χN〉 (4.47)