Physical Chemistry Third Edition

I Matrix Representations of Groups 1299

If the 3dxz,3dyz,3dxy,3dx (^2) −y 2 , and 3dz 2 orbitals are used as a basis for theC 2 v
group (in that order), we get the representation (withR(̂E) not displayed, since it is
just the identity matrix):

R(̂C 2 )

⎡ ⎢ ⎢ ⎢ ⎢ ⎣

−1 0000

0 − 1000

0 0100

0 0010

0 0001

⎤ ⎥ ⎥ ⎥ ⎥ ⎦

(I-20)

R(̂σxz)

⎡ ⎢ ⎢ ⎢ ⎢ ⎣

1 0 000

0 −1 000

00 − 100

0 0 010

0 0 001

⎤ ⎥ ⎥ ⎥ ⎥ ⎦

(I-21)

R(̂σyz)

⎡ ⎢ ⎢ ⎢ ⎢ ⎣

−10 000

01 000

00 − 100

00 010

00 001

⎤ ⎥ ⎥ ⎥ ⎥ ⎦

(I-22)

I.5 Applications of Group Theory to Molecular

Orbitals

We can determine something about the molecular orbitals in a polyatomic molecule

by the use of group theory. In Chapter 21 we first described the bonding in the

water molecule in terms of hybrid orbitals on the oxygen atom in order to have

only LCAOMOs containing two basis functions. In Hartree–Fock–Roothaan calcu-

lations, the LCAOMOs are linear combinations of the entire set of basis functions.

In Section 21.8 we defined basis orbitals for the water molecule that were eigenfunc-

tions of the symmetry operators belonging to the molecule. These basis functions have

symmetry properties like those of the irreducible representations and produce a secular

determinant that is in block-diagonal form, simplifying the calculation. If a basis orbital

has the same symmetry properties as a representation labeled A 1 , it is labeled with the

subscripta 1 , and so on. This label is called itssymmetry speciesand identifies the irre-

ducible representation to which it corresponds. The two basis orbitals of Eqs. (21.8-1)

and (21.8-2) were

ψa 1 ψ 1 sHa+ψ 1 sHb (I-23)

ψb 2 ψ 1 sHa−ψ 1 sHb (I-24)

These linear combinations are calledsymmetry-adapted basis functions. Pitzer and

Merrifield carried out a Hartree–Fock–Roothaan calculation on H 2 O using a minimal

basis set of Slater-type orbitals and obtained the orbitals displayed in Table 21.2. The

a 1 orbitals are numbered from lower to higher energy, as are theb 1 andb 2 orbitals.

Only basis orbitals of the same symmetry species enter in any one molecular orbital.

Thea 1 basis function can combine with the 1s,2s, and 2pzfunctions on the oxygen.