and in all cases the determinant is exactly zero, implying that the overall wave-
function does not exist.
On this basis, one consequence of the Pauli principle is that no two electrons
in any system can have the same set of four quantum numbers.(This statement
is sometimes used in place of the original statement of the Pauli principle.)
This means that each and every electron must have its own unique spin or-
bital, and since there are only two possible spin functions for an electron, each
orbital can be assigned only two electrons. Therefore, an ssubshell can ac-
commodate two electrons maximum; each psubshell, with three individual p
orbitals, can hold a maximum of six electrons; each dsubshell, with five dor-
bitals, can hold ten electrons; and so on. Because this consequence of the Pauli
principle excludes spin orbitals from having more than one electron, Pauli’s
statement is commonly referred to as the Pauli exclusion principle.
Example 12.4
Show for each row of the Slater determinants for Li in equation 12.12 that
the wavefunction represented by the determinant violates the Pauli exclusion
principle.Solution
By row, the set of four quantum numbers for each spin orbital is listed:1 s 1 1 s 1 1 s 1 (1, 0, 0,^12 ) (1, 0, 0,^12 ) (1, 0, 0,^12 )
1 s 2 1 s 2 1 s 2 (1, 0, 0,^12 ) (1, 0, 0,^12 ) (1, 0, 0,^12 )
1 s 3 1 s 3 1 s 3 (1, 0, 0,^12 ) (1, 0, 0,^12 ) (1, 0, 0,^12 )1 s 1 1 s 1 1 s 1 (1, 0, 0,^12 ) (1, 0, 0,^12 ) (1, 0, 0,^12 )
1 s 2 1 s 2 1 s 2 (1, 0, 0,^12 ) (1, 0, 0,^12 ) (1, 0, 0,^12 )
1 s 3 1 s 3 1 s 3 (1, 0, 0,^12 ) (1, 0, 0,^12 ) (1, 0, 0,^12 )
In each case, two of the three entries in each row have the same set of four
quantum numbers and so the wavefunction is not allowed by the Pauli ex-
clusion principle.Wavefunctions written in terms of a Slater determinant have a normaliza-
tion factor of 1/n!,where nis the number of rows or columns in the deter-
minant (and equals the number of electrons in the atom). This is because the
expanded form of the wavefunction has n! terms. In constructing Slater de-
terminants, we will follow the custom of writing the individual spin orbitals
going across, two spatial wavefunctions with an and a spin wavefunction
each, and listing the electrons sequentially going down. That is:
spin orbitals→
electron 11 s 1 s 2 s 2 s ...electron 2 1 s 1 s 2 s 2 s ...
electron 3 ... ... ... ... ...
.
.
.Going across the determinant, the spin part alternates: , , , ,....You also
have to keep track of the n,, and mquantum numbers to make sure each
shell and subshell is represented in the proper order and number. The follow-
ing example illustrates the use of this idea.
12.4 Spin Orbitals and the Pauli Principle 381