closed-shell system. We need two spatial molecular orbitals, since each can hold a
maximum of two electrons; each spatial orbitalc(spatial) is used to make two spin
orbitals,c(spatial)aandc(spatial)b(alternatively, each spatial orbital could be
thought of as a composite of two spin orbitals, which we are separating and using to
build the determinant). Along the first (top) row of a determinant we write succes-
sively the firstaspin orbital, the firstbspin orbital, the secondaspin orbital, and
the secondbspin orbital, using up our occupied spatial (and thus spin) orbitals.
Electron 1 is then assigned to all four spin orbitals of the first row – in a sense it is
allowed to roam among these four spin orbitals [ 10 ]. The second row of the
determinant is the same as the first, except that it refers to electron 2 rather than
electron 1; likewise the third and fourth rows refer to electrons three and four,
respectively. The result is the determinant of Eq.5.10.
C¼
1
ffiffiffiffi
4!pc 1 ð 1 Það 1 Þ c 1 ð 1 Þbð 1 Þ c 2 ð 1 Það 1 Þ c 2 ð 1 Þbð 1 Þ
c 1 ð 2 Það 2 Þ c 1 ð 2 Þbð 2 Þ c 2 ð 2 Það 2 Þ c 2 ð 2 Þbð 2 Þ
c 1 ð 3 Það 3 Þ c 1 ð 3 Þbð 3 Þ c 2 ð 3 Það 3 Þ c 2 ð 3 Þbð 3 Þ
c 1 ð 4 Það 4 Þ c 1 ð 4 Þbð 4 Þ c 2 ð 4 Það 4 Þ c 2 ð 4 Þbð 4 Þ(^)
(^)
$ð 5 : 10 Þ
(The 1/√4! factor ensures that the wavefunction is normalized, i.e. thatjCj^2 inte-
grated over all space¼1[ 11 ]). ThisSlater determinantensures that there are no more
than two electrons in each spatial orbital, since for each spatial orbital there are only
two 1-electron spin functions, and it ensures thatCis antisymmetric since switching
two electrons amounts to exchanging two rows of the determinant, and this changes
its sign (Section 4.3.3). Note that instead of assigning the electrons successively to
row 1, row 2, etc., we could have placed them in column 1, column 2, etc.:C^0 of
Eq.5.11¼Cof Eq.5.10. Some authors use the row format for the electrons, others
the column format.
C^0 ¼
1
ffiffiffiffi
4!pc 1 ð 1 Það 1 Þ c 1 ð 2 Það 2 Þ c 1 ð 3 Það 3 Þ c 1 ð 4 Það 4 Þ
c 1 ð 1 Þbð 1 Þ c 1 ð 2 Þbð 2 Þ c 1 ð 3 Þbð 3 Þ c 1 ð 4 Þbð 4 Þ
c 2 ð 1 Það 1 Þ c 2 ð 2 Það 2 Þ c 2 ð 3 Það 3 Þ c 2 ð 4 Það 4 Þ
c 2 ð 1 Þbð 1 Þ c 2 ð 2 Þbð 2 Þ c 2 ð 3 Þbð 3 Þ c 2 ð 4 Þbð 4 Þ(^)
(^)
ð 5 : 11 Þ
Slater determinants enforce the Pauli exclusion principle, which forbids any two
electrons in a system to have all quantum numbers the same. This is readily seen for
an atom: if the three quantum numbersn,landmmofc(x,y,z) (Section 4.2.6) and
the spin quantum numbermsofaorbwere all the same for any electron, two rows
(or columns, in the alternative formulation) would be identical and the determinant,
hence the wavefunction, would vanish (Section 4.3.3).
For 2nelectrons (we are limiting ourselves for now toeven-electron species, as
the theory for these is simpler) the general form of a Slater determinant is clearly the
2 n' 2 ndeterminant
5.2 The Basic Principles of the ab initio Method 183