Concise Physical Chemistry

c17 JWBS043-Rogers September 13, 2010 11:28 Printer Name: Yet to Come

SLATER DETERMINANTS 279

exchange. Electrons are fermions. If both electrons in the 1sorbital had the same

spin, the negative (antisymmetric) combination above would integrate to zero and

there would be no probability of finding the electron. That is why electrons must have

opposite spins if they are to reside in the same space orbital.

17.7 SLATER DETERMINANTS

The linear combination selected above for ground state helium is the same as the ex-

pansion of a 2×2 determinant, which is the simplest example of aSlater determinant

(Section 15.3):

1 s(1)α 1 s(2)β− 1 s(1)β 1 s(2)α=

∣

∣

∣

∣

∣

1 s(1)α 1 s(1)β

1 s(2)α 1 s(2)β

∣

∣

∣

∣

∣

Larger Slater determinants also coincide with uniquely acceptable linear combina-

tions for fermions. In the generaln×ncase with some simplification of notation,

the Slater determinant is

ψ(1, 2 ,...,n)=

1

√

n!

∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣

φ 1 α 1 φ 1 β 1 ... φnα 1 φnβ 1

φ 1 α 2 φ 1 β 2 ...

... ...

φ 1 αn φ 1 βn ··· φnαn φnβn

∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣

where the premultiplier 1/

√

n! is a normalization constant. The many-electron wave

function for orthonormal basis functionsψi, in more concise notation, is now written

(Pople, Nobel Prize, 1998)

ψ=(n!)−

(^12)
det[(ψ 1 α)(ψ 1 β)(ψ 2 α)...]
The Slater determinant always produces a linear combination of one-electron orbitals
that allows for electron indistinguishability by giving equal weight to electrons of
opposing spins in each ofnorbitals. The Slater determinantal molecular orbital
andonlythe Slater determinant satisfies the two great generalizations of quantum
chemistry, the Heisenberg principle of uncertainty (which is why you can’t tell in
advance whether the first electron ionized from He will beαorβ) and the Fermi–Dirac
principle of antisymmetric fermion exchange.
In addition to his work on determinantal wave functions, Slater also fitted functions
to numerical compilations of SCF data and obtained orbitals in analytic form. These
Slater-type orbitals(STO) resemble hydrogen wave functions but have adjustable
parameters to account for field differences between hydrogen and many-electron