Physical Chemistry , 1st ed.

Solution

The following table is taken from the Rh(3)character table:

EC iS 

 1 sDg(0) 1 1 1 1 1

 2 sDg(0) 1 1 1 1 1

EM opDu(1) 3 1 2 cos  3  1 2 cos 1

The product of any two rows must be equal to or contain the third, else

the integral must be exactly zero by symmetry arguments. It is easiest to de-

termine the product  1 s 2 s, which are all simply 1 1 for each symme-

try operation. Therefore, we are comparing the two sets of characters:

EC iS 

 1 s 2 s 1 1 1 1 1

EM op 3 1 2 cos  3  1 2 cos 1

Clearly, 1 does not equal 3 (for the character ofE). Clearly, 1 does not

equal 1 2 cos for all values of (which would be required); and so forth.

Therefore, by showing that  1 s 2 sdoes not equal EM op, we can say that

the integral

* 1 s(EM radiation operator)  2 sd

is exactly zero. We will find in a later chapter that this is what defines a for-

bidden transition.

Symmetry considerations are especially useful in spectroscopy, where the
operator can also be assigned some symmetry species of the point group.
Polarized light and magnetic fields can be assigned a symmetry species within
the point group of the molecule, and whether or not a spectroscopic transition
will occur can be determined by the idea embodied in equation 13.9. This is
the symmetry basis for selection rules. We will cover such topics in the next few
chapters.
For linear combinations of wavefunctions, the symmetry species of the in-
dividual wavefunctions is very important. One ramification of symmetry con-
siderations is that wavefunctions of different symmetry species do not combine
to, say, make bonds. Since we are seeking linear combinations of atomic wave-
functions, this allows us to conclude that the only useful combinations for
molecules will be of those atomic wavefunctions that belong to the same sym-
metry species of the molecule. The construction of symmetry-adapted linear
combinations utilizes this simplifying idea.

13.9 Symmetry-Adapted Linear Combinations

By keeping symmetry in mind, it is possible to construct appropriate combi-
nations of atomic orbital wavefunctions to approximate molecular orbital
wavefunctions that cover, or span,the entire molecule. The use of symmetry is
the first real restriction we have placed on linear combinations, but it makes
sense. After all, it serves no purpose to use a pxatomic orbital in a linear com-
bination of a molecular orbital when the chemical bond points in the zdirec-
tion.Symmetry-adapted linear combinations(SALCs) are more intuitively cor-
rect approximations than any random combination of atomic orbitals.

13.9 Symmetry-Adapted Linear Combinations 443