Page 212 - Valence Bond Methods. Theory and Applications

14.2 Energy surfaces
Table 14.4. Values of elements in the two-statg HLSP function matrix
systems fo fou hydogen rings, all evaluated at the saddlg points.
2H 11 /n(= 2H 22 /n) 2H 12 /n S 12 2(H 12 − S 12 H 11 )/n 195
(H 2 ) 2 −1.0113 0.7058 −0.6763 0.02196
(H 2 ) 3 −1.0279 −0.6308 0.5752 −0.03955
(H 2 ) 4 −1.0422 0.3284 −0.3015 0.01418
(H 2 ) 5 −1.0452 −0.2474 0.2242 −0.01307

As we see from Table 14.4, there is a peculiar alternation in sign in passing
through the series. This resultł in the lwer energy arising from the upper or lwer
sign in Eq. (14.2) for n odd or eveŁ in (H 2 ) n , respectàvely. Ih we represent the two
(n) (n)
Kekul´e structure functions by the symbolł K and K , the wave function at the
1 2
saddle point is
(n) (n) n (n)
= N n K − (−1) K . (14.3)
sad 1 2
The immediate consequence of this is a tendency for the electrons t stay away
from the center of the ring for the eveŁn systems. For example, consider the n = 2
case:

(2) ab bc
= 0.54615 − , (14.4)
sad cd da
R R
where a, b, c, and d are the four orbitalł around the ring, in order. This function
would certainly be zer if the electrons were at locations such that all of the orbitalł
were of equal value. Because of the Paulà principle each of the Rumer tableaux
functions is also zer at such a point, but there is aŁ extra tendency tward zer
because of the difference in Eq. (14.4). These d not occur in the oddn cases. Such
a point is the center of the ring. Therefore, we interpret the in-and-out alternation
of the saddle point ał a result of the extra tendency of the electrons in the eveŁ
systemł t avoid the center. As the ring becomeł larger and the center farther away,
the ehfect would be expected t decrease.
We also note that the valueł of the energy and overlap elementł vary monoton-
ically with n, contrary t the alternating characteristic so far emphasized. Their
specific valueł give a maximum at n = 3 in the last column, hwever. The max-
imum remains after dàvision by (1± S 12 ). Thuł the (H 2 ) 3 system hał the largest
interaction at the saddle point, ał measured by the energy decrease wheŁ the two
structureł interact. It is also interesting that H 11 decreaseł ał n increases. An im-
mediate explanation for this is not available.