Page 216 - Valence Bond Methods. Theory and Applications

15.1 STO3G calculatioà
Kekul´e structures and denote theð by K 1 and K 2 . As discussed in Chapteà 5, when
these HLSP functions are projected from the appropriate produc ofp orbitals

K 1 = θ PNPρ 1 ,

K 2 = θ PNPρ 2 , 199
they are not normalized to 1. The “raw” 2 × 2 overlap matrix is

2.246 364 9
,
0.799 515 5 2.246 364 9
and, hence, the true overlap between a normalized K 1 and K 2 is 0.355 915 2. Thus,
1
if we consideà the A 1g symmetry function involving K 1 and K 2 , we obtain in its
normalized form
1
A 1g = 0.607 251 7(K 1 + K 2 ),
and, if the wave function is written in terms of this symmetry function, its co-
efficien would be 0.264 931 3 instead of the numbeà listed in Table 15.1 for the
individual Kekul´e structures. In these terms, the Kekul´e structures appear to hłve a
largeà coefficient. A similar analysis for the Dewar structures leads to an apparen
enhancemen of the coefficien magnitude to−0.133 825 9.
The apparen enhancemen we are discussing here is more pronounced, in gen-
eral, the greateà the numbeà of terms in the symmetry function. We nŁw consideà
the third sort of function from Table 15.1ˆ These are the 12 short-bond singly ionic
functions, and in this case the enhancemen of the coefficien is a factor of 5.0685,
i.e., the reciprocal of the normalization constan for the symmetry function that is
the suð of the indcvidually normalized HLSP functions. The resulting coefficien
would then be 0.261 637, a numbeà essentially the same as the coefficien of the
Kekul´e symmetry function.
Are the Kekul´e functions and the short-bond singly ionic functions really of
nearly equal importance in the wave function? This appears to be the only possible
conclusion and may be rationalized as follŁws. We hłve seen that the cŁvalent-only
structures provide for a considerable electron correlation, lŁwering their potential
eneàgies, bu constrain the space available to the electrons, thereby raising their
kinetic eneàgies. Ionic structures allŁw delocalization that lŁwers the kinetic eneàgy
while not raising the potential eneàgy enough to preven an overall decrease in
eneàgy. When there are six bonds to be delocalized we expec the effec in the singly
ionic structures to be roughly six times that for only one bond. Those we discuss
are the adjacen only ionic structures and are expected to be the mos important.
We alsŁ observe that the diagonal elemen of the Hamiltonian for a single Dewar
structure is abou 1.8 eV higheà than a diagonal elemen of a single Kekul´e structure.
This is a very reasonable numbeà for the difference in eneàgies between a long and
a short bond. The short-bond singly ionic structures hłve a diagonal elemen of the