Character tables and the GOT are crucial elements in determining a proper
SALC of atomic orbitals. The first step is to identify the correct point group of
the molecule, because the wavefunction that describes any molecular orbital
must also have this symmetry. The second step is to identify all of the atomic
orbitals that are being considered for the molecular orbitals. Typically, the nec-
essary atomic orbitals are those that are occupied in the atoms themselves. We
must not forget that there exist three independent porbitals, five independent
dorbitals, and so on, that can contribute to the bonding. All such orbitals must
be considered. For example, in considering the atomic orbitals that make up
the molecular orbitals of H 2 O, we include the 1satomic orbital of hydrogen 1,
the 1sorbital of hydrogen 2, the 1sorbital of O, the 2sorbital of O, and the
2 px,2py, and 2pzorbitals of the O atom. A total ofsevenindividual atomic or-
bitals are used.
It should be understood that only atomic orbitals that belong to the same
symmetry species will combine. At first glance, it can be challenging to deter-
mine which atomic orbitals belong to the same symmetry species, and it may
also be challenging to identify all of the atomic orbitals of any one irreducible
representation. It is sometimes easier to include all atomic orbitals in an initial
treatment. They will separate themselves into the proper groupings when the
symmetry requirements are imposed. Therefore, in the following examples all
of the atomic orbitals will be considered. In the end, we will see how the atomic
orbitals are separated by their symmetry species. (With practice, separation of
atomic orbitals by symmetry species becomes obvious; however, in the begin-
ning, it may seem nonsensical. This is why the more complete treatment is
introduced here.) Determination of SALCs follows a sort of recipe. We will not
go into the specific group theory derivation of the procedure, but there is
mathematical justification for the following steps.- Determine which atomic orbitals will be used.
- Make a table that has each individual atomic orbital listed on one side
(say, the left) and the symmetry operations listed on a perpendicular
side. List the symmetry operations individually,not by class. There
are hsymmetry operations in the point group, where his the order of
the group. There should therefore be hentries for the symmetry op-
erations. - Operate on each individual atomic orbital with each symmetry operation
and write the result in the table. The result will be either (1) the orbital
itself or the negative of the orbital itself or, rarely, some fraction of itself,
or (2) some other orbital or the negative of some other orbital or, rarely,
some fraction of another orbital, or (3) not correspond to the position
of any other orbital in the molecule, in which case the result is zero. The
table should be completely filled when finished. - Consult the character table of the point group of the molecule. (Note
that in step 2 we have constructed our own table to mimic the structure
of the character table.) Consider each irreducible representation in the
character table. For each individual symmetry operation (you may have
to separate classes of symmetry operations), multiply the character by the
result of the corresponding symmetry operation in each row of your
table. - Add all of these products in each column,giving you hsums. Multiply
each sum by a normalization factor of 1/h(horder of group). The re-
sulting expressions are the SALCs, which will have the symmetry prop-
erties of that irreducible representation.
444 CHAPTER 13 Introduction to Symmetry in Quantum Mechanics