p Electron systems

D6h ¼ D6 Ci¼ fD6g  IfD6g, (3) where

D6¼ fE 2C62C3C23C20

3C200g: (4)

Refer to AppendixA3for the character table of D6h. Some conventions used for benzene are illustrated in Figure 6.3. The C2 axes normal to the principal axis fall into two geometrically distinct sets. Those passing through pairs of opposite atoms are given precedence and are called C20, and those that bisect pairs of opposite bonds are named C200. Consequently, the set of three vertical planes that bisect pairs of opposite bonds are designated 3
d(because they bisect the angles between the C20 axes), while those that contain the C200axes are called 3
v. Note the sequence of classes in the character table: the classes in the second half of the table are derived from I(ck), where ckis the corresponding class in the first half. Thus 2S3precedes 2S6because IC^þ₆ ¼ S
₃, and 3
dprecedes 3
v

because IC20¼
dbut IC200¼
v. Notice that the characters of the u representations in the first set of classes (those of D6) repeat those of the g representations in the top left corner of the table, but those of the u representations in the bottom right quarter have the same magnitude as those for the corresponding g representations in the top right quarter, but have the opposite sign. Some authors only give the character table for D6, which is all that is strictly necessary, since the characters for D6hcan be deduced from those for D6using the properties explained above. The systematic presentation of character tables of direct product groups in this way can often be exploited to reduce the amount of arithmetic involved, particularly in the reduction of representations.

To find the MOs for benzene, we choose a basis comprising a 2pzAO on each carbon atom and determine the characters of
, the reducible representation generated by

6

Figure 6.3. (a) Numbering scheme used for the six C atoms in the carbon skeleton of benzene. Also shown are examples of the locations of the C20and C200axes, and of the
dand
vplanes of symmetry.

The C₂⁰ axes are given precedence in naming the planes. (b) Partial projection diagram for D_6h showing that IC20¼
dand IC200¼
v.

T^hj ¼ h⁰j ¼ hj
, (5) where T2 D6handh j stands for the basis set h 1 ₂ ₃ ₄ ₅ ₆j; ris a 2pzAO on the r th carbon atom. Although we could determineh ⁰j from the effect of the function operator Tˆ on each rin turn, it is not necessary to do this. A much quicker method is to use the rotation of the contour of the function r¼ 2pzon atom r under the symmetry operator T to determine ⁰_r, and then recognize that ⁰_ronly contributes to the character of
(T ) when it transforms intor. A positive sign contributesþ1 to the character of
(T ); a negative sign contributes
1; the contribution is zero if r0¼ s, s6¼ r. The character system of


may thus be written down by inspection, without doing any calculations at all. In this way we find that

ð
Þ ¼ f6 0 0 0
2 0 0 0 0
6 0 2g: (6) In benzene, T2 {C6C3C2I S3S6C200
_d} sends each rinto s6¼rso that there are no non-zero diagonal entries in
 for these operators and consequently (T )¼ 0. For the C20

operators, the 2pzorbitals on one pair of carbon atoms transform into their negatives, so that

(C20)¼
2. For
h, each of the six atomic orbitals r transforms into
r, so that

(
h)¼
6. For the
v operators, the pair of 2pz orbitals in the symmetry plane are unaffected, while the other four become 2pz orbitals on different atoms, so (
v)¼ þ2.

Finally, for the identity operator each rremains unaffected, so
(E) is the 6 6 unit matrix and (E)¼ 6. Note that
is a reducible representation,

¼P

j

c^j
_j,  ð
Þ ¼P

j

c^j_j, (7)

where

c^j¼ g
¹P

k

ck _jðckÞ^ ðckÞ, (8)

and (ck) is the character for the kth class in the reducible representation.

(6), (8)
¼ A2u B2g E1g E2u: (9)

For example,

cðA_2uÞ ¼ ð1=24Þ½1ð1Þð6Þ þ 3ð
1Þð
2Þ þ 1ð
1Þð
6Þ þ 3ð1Þð2Þ ¼ 1:

In (
), the characters in the second half of the character system do not reproduce those in the first half (or reproduce their magnitudes with a change in sign). If this had been so,


would have been a direct sum of g IRs (or u IRs). Here we expect the direct sum to contain both g and u representations, which turns out to be the case. The basis functions for these IRs may now be obtained by using the projection operator ^P^j(eq. (5.2.10)),

j¼ NjP

T

_jðT Þ^T ;^ (10)

6.2 p Electron systems 111

 can be any arbitrary function defined in the appropriate subspace, which here is a subspace of functions for which the six AOs {r} form a basis. Chemical intuition tells us that a sensible choice would be ¼ 1.

Srsis called the overlap integral because the integrand is only significant in regions of space where the charge distributions described by the AOs rand soverlap. When either

_ror sis very small, the contribution to the integral from that volume element is small and so there are only substantial contributions from those regions of space where rand s

overlap. A useful and speedy approximation is to invoke the zero overlap approximation (ZOA) which sets

S_rs¼ 0, r 6¼ s: (13)

The ZOA is based more on expediency than on it being a good approximation; in fact, the value of Srsis about 0.2
0.3 (rather than zero) for carbon 2pzorbitals on adjacent atoms.

When s is not joined to r, it is much more reasonable. Nevertheless, it is customary to use the ZOA at this level of approximation since it yields normalization constants without performing any calculations. One should remark that it affects only the N^j, the ratio of the coefficients being given by the group theoretical analysis. Using the ZOA,

a2u¼¹= ffiffip6

½1þ 2þ 3þ 4þ 5þ 6: (14) In eq. (14) we have followed the usual practice of labeling the MO by the IR (here A2u) for which it forms a basis, but using the corresponding lower-case letter instead of the capital letter used for the IR in Mulliken notation. It is left as a problem to find the MOs that form bases for the other IRs in the direct sum, eq. (9). In the event of lj-fold degeneracy, there are ljlinearly independent (LI) basis functions, which we choose to make mutually orthogonal.

So for lj¼ 2, we use the projection operator ^P^jagain, but with a different function ¼ 2. For E1g, for example, 1 and 2 give 1(E1g) and 2(E1g), which are LI but are not orthogonal. Therefore we combine them in a linear combination to ensure orthogonality while preserving normalization. Usually this can be done by inspection, although the systematic method of Schmidt orthogonalization (see, for example, Margenau and Murphy

(1943)) is available, if required. Remember that can always be multiplied by an arbitrary phase factor without changing the charge density, or any other physical property, so that it is common practice to multiply by
1 when this is necessary to ensure that the linear combination of atomic orbitals (LCAO) does not start with a negative sign.

6.2.1 Energy of the MOs

the second equality following from the fact that Hˆ is an Hermitian operator. For p electron systems there are useful approximations due to Hu¨ckel. If

s¼ r, Hrr¼ ,

(s$ r means ‘‘s joined to r’’). The effective energy of a bound electron in a carbon 2pz

atomic orbital is given by ; the delocalization energy comes from .

(17), (15) E^j¼ jNjj²h P

Substituting for the coefficients (see eq. (14) and Problem6.2) and evaluating E^j from eq. (18) gives the energy-level diagram shown in Figure6.4. Only the energies depend on

b_2g

Figure 6.4. Energy-level diagram for the molecular orbitals of benzene evaluated in the Hu¨ckel approximation.

6.2 p Electron systems 113

the Hu¨ckel approximations. The orbitals are correctly given within the ZOA, which only affects Nj, the ratios of the coefficients being completely determined by the symmetry of the molecule.