Essentials of Computational Chemistry

4.5 MANY-ELECTRON WAVE FUNCTIONS 125
where the prefactor (N!) −1/2 is implicit. Furthermore, if two spin orbitals differ only in
the spin eigenfunction (i.e., together they represent a doubly filled orbital) this is typically
represented by writing the spatial wave function with a superscript 2 to indicate double
occupation. Thus, if χ1 and χ2 represented α and β spins in spatial orbital ψ1, one would
write
SD = ψ 2 1 χ3 ···χN〉 (4.48)
Slater determinants have a number of interesting properties. First, note that every electron
appears in every spin orbital somewhere in the expansion. This is a manifestation of the
indistinguishability of quantum particles (which is violated in the Hartree-product wave
functions). A more subtle feature is so-called quantum mechanical exchange. Consider the
energy of interelectronic repulsion for the wave function of Eq. (4.43). We evaluate this as
3SD 1 3
SDdr1dω1dr2dω2
r12
= 1
|ψa(1)|
2
2 2 1
|α(1)| |ψb(2)|
r12
2 |α(2)| 2 dr1dω1dr2dω2
2 1
− 2 ψa(1)ψb(1) |α(1)| ψb(2)ψa(2) |α(2)|
r12
2 dr1dω1dr2dω2
+ |ψa(2)| 2 2 1
|α(2)| |ψb(1)|
r12
2 |α(1)| 2
dr1dω1dr2dω2
= 1
2 1
|ψa(1)| |ψb(2)|
2
r12
2 dr1dr2
− 2 ψa(1)ψb(1) 1
ψb(2)ψa(2)dr1dr2
r12
2 1
+ |ψa(2)| |ψb(1)|
r12
2
dr1dr2
= 1
Jab − 2 ψa(1)ψb(1)
2
1
ψa(2)ψb(2)dr1dr2 + Jab
r12
= Jab − Kab
(4.49)
Equation (4.49) indicates that for this wave function the classical Coulomb repulsion between
the electron clouds in orbitals a and b is reduced by Kab, where the definition of this integral
may be inferred from comparing the third equality to the fourth. This fascinating consequence
of the Pauli principle reflects the reduced probability of finding two electrons of the same
spin close to one another – a so-called ‘Fermi hole’ is said to surround each electron.
Note that this property is a correlation effect unique to electrons of the same spin. Ifwe
consider the contrasting Slater determinantal wave function formed from different spins
SD = 1
√ 2 [ψa(1)α(1)ψb(2)β(2) − ψa(2)α(2)ψb(1)β(1)] (4.50)