Physical Chemistry , 1st ed.

where the individual hydrogens andtheir electrons are labeled as either 1 or 2.
(It is important to keep track of the labels 1 and 2. They will independently re-
fer to the atom and the electron.) The wavefunction of the complete system
(that is, the two individual hydrogen atoms) is the product of the two indi-
vidual wavefunctions:

systemH1H2 1 sH1(1)  1 sH2(2) (13.10)

where we are assuming that each hydrogen atom has its electron in the 1s
atomic orbital. When the two atoms come together to make a hydrogen mol-
ecule, it is assumed that the wavefunction of the molecule also has this sort of
wavefunction. But in the molecule, the individual electrons aren’t tied down to
one particular nucleus. Another possible product of wavefunctions might be
1 sH1(2)  1 sH2(1), where each electron is now associated with the other hydro-
gen atom. As done previously with multiple possible wavefunctions, we con-
sider that the best wavefunction is a linear combination of the individual wave-
functions:

system

1

2

[1sH1(1)  1 sH2(2)  1 sH1(2)  1 sH2(1)] (13.11)

where 1/ 2
is a normalization factor for the linear combination. Equation
13.11 actually represents two possible wavefunctions, one being the sum and
the other the difference of the two product wavefunctions. There are two ad-
ditional concerns, however: spin and the Pauli principle. Spin functions must
be included with spatial functions for a complete wavefunction. The Pauli
principle also requires that the complete wavefunction be antisymmetric
upon the exchange of the two electrons [that is,(1, 2) (2, 1)]. So in
each of the wavefunctions in equation 13.11, a spin function must be in-
cluded andthe spin function must be of the proper form so that the com-
plete wavefunction is antisymmetric with respect to exchange of the two
electrons.
There are several possible forms for the spin function of the molecule.
Because there are two electrons, we are going to have to number the spin func-
tions with the electron number to keep track of which electron has what spin
function. One possible molecular spin function is for both electrons to have
the spin function. The molecular spin function is thus (1)(2). Or, they
could both have spin functions: the molecular spin function is then (1)(2).
Or, one electron can have the function and the other the function, but
which? Remember that individual electrons are indistinguishable from each
other. Just as for the spatial wavefunction in equation 13.11, there are two
possible combinations ofand spin functions,(1)(2) and (2)(1). The
most appropriate wavefunction is the linear combination of the two:
(1/ 2
)[(1)(2) (2)(1)]. Again, note that the sign means that there
are two separate functions here. These are the four possibilities for the spin
part of the total wavefunction. These possibilities must be combined with the
spatial wavefunctions in equation 13.11 to make antisymmetric wavefunctions
for H 2.
Since symmetric and antisymmetric properties follow the same rules of
multiplication as odd/even or positive/negative multiplication (that is, sym 
sym sym, sym antisym antisym, antisym antisym sym), all we
need to do is identify the symmetry properties of the spatial and spin wave-
functions and combine antisymmetric parts with symmetric parts to get an

13.10 Valence Bond Theory 447