Physical Chemistry , 1st ed.

The switch in order of writing the labels 1 and 2 implies that the two electrons
are exchanged. Electron 1 now has the coordinates of electron 2, and vice versa.
A wavefunction having this property is called antisymmetric.(By contrast, if
(1, 2) (2, 1), the wavefunction is labeled symmetric.) Particles having
half-integer spin (^12 ,^32 ,^52 ,.. .) are collectively called fermions.The Pauli principle
states that fermions must have antisymmetric wavefunctions with respect to
exchange of particles. Particles having integer spins, called bosons,are restricted
to having symmetric wavefunctions with respect to exchange.
Electrons are fermions (having spin ^12 ) and so according to the Pauli prin-
ciple must have antisymmetric wavefunctions. Consider, then, the two possi-
ble approximate wavefunctions for helium. They are

He,1



1

2 

[(1s 1 )(1s 2 ) (1s 1 )(1s 2 )] (12.7)

He,2



1

2 

[(1s 1 )(1s 2 ) (1s 1 )(1s 2 )] (12.8)

Are either of these antisymmetric? We can check by interchanging electrons 1
and 2 in the first wavefunction, equation 12.7, and get

(2, 1) 



1

2 

[(1s 2 )(1s 1 ) (1s 2 )(1s 1 )]

(Note the change in the subscripts 1 and 2.) This should be recognized as the
original wavefunction (1, 2), only algebraically rearranged. (Show this.)
However, upon electron exchange, the second wavefunction, equation 12.8,
becomes

(2, 1) 



1

2 

[(1s 2 )(1s 1 ) (1s 2 )(1s 1 )] (12.9)

which can be shown algebraically to be (1, 2). (Show this, also.)
Therefore, this wavefunction is antisymmetric with respect to exchange of
electrons and, by the Pauli principle, is a proper wavefunction for the spin
orbitals of the helium atom. Equation 12.8, but not equation 12.7, repre-
sents the correct form for a spin-orbital wavefunction of the ground state
of He.
The rigorous statement of the Pauli principle is that wavefunctions of elec-
trons must be antisymmetric with respect to exchange of electrons. There is a
simpler statement of the Pauli principle. It comes from the recognition that
equation 12.8, the only acceptable wavefunction for helium, can be written in
terms of a matrix determinant.
Recall that the determinant of a 2 2 matrix written as



ad

c b

is simply (a b) (c d), which is remembered mnemonically as



ad



c d
cb a^ b
The proper antisymmetric wavefunction, equation 12.8, for the helium atom
can also be written in terms of a 2 2 determinant:

He



1

2 

 ^1 s^11 s^1 

1 s 2 1 s 2 (12.10)

12.4 Spin Orbitals and the Pauli Principle 379

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