Superposition pattern

To find the coefficient of the Coulomb integral for two structures, superimpose their vector-bond patterns to form the superposition pattern (Fig. 1). The Coulomb coefficient is 2 " times the sum (— 1)E for the different patterns in which each orbit serves either as the head or as the tail... 

Fig. 1. The vector-bond diagrams for three structures of the canonical set of fourteen for n — 4, and some of their superposition patterns.

For canonical structures the sign is positive, the Coulomb coefficient being l/2n i. In drawing the superposition pattern for canonical structures the arrows may be replaced by lines. 

To find the coefficient of a given exchange integral in a matrix element, (I/F/PII), draw the vector-bond diagram for structure II, change it as indicated by the permutation diagram for P, and form the superposition pattern of I and PI I. The coefficient is then given, except for the factor (—l)p, by the above rules for the Coulomb coefficient that is, it is (— 1)F(— V)r2 in t>. 

FlG. 3. The permutation diagram for P abcdef=bcfdae, and the superposition pattern for I PII. 

For a set of superposition patterns obtained by combining a set for each of the three groups in all possible ways, the sums of the coefficients (for equal weights of the corresponding structures) are... 

It should be noted that the use of arrows (a — (3), rather than simple links, is in general necessary to ensure the correct phase ( ) of the result. Two superimposed arrows in a superposition pattern form an island, while an unlinked point counts as an O-chain. 

The matrix element rules (40) and (41), which apply to the superposition pattern for any two structures, 4>a for a singlet state, were first given by Pauling [14] in 1933. They were to form the basis of nearly all semi-empirical applications of VB theory to polyatomic molecules during the next few decades. 

Let us now discuss the energy integral for a particular valence-bond wave function, in order to justify our correlation of valence-bond distribution and wave function as given in Equation 46-18. The superposition pattern for a structure with itself, as shown by 11 in Figure 46-1, consists of n islands, each consisting of two bonded orbitals. We see that... 

The pattern contains two islands, and no arrow reversals are required thus = 2 and v a = 0, (0 0a) = 2" = To obtain the remaining coefficients in (7.3.8), we may compare the number of islands in the pattern for <0,J6 >, where x = P.>0i. with that in the pattern for (0K I 0 i). If this number is increased by one, the matrix element is multiplied by 2 , if decreased by one, the factor is 2. The sign may also be changed if rematching of arrow heads and tails requires an odd number of arrow reversals. Since Py simply crosses the arrows attached to positions i and j in structure 0a, the effect on the superposition pattern is always apparent. In the present example there are only three distinct types of interchange, leading to the following results ... 

These observations are completely general, for non-polar singlet structures, depending only on the topological relationship of each pair (ty) in the superposition pattern for 0 [ 0a)- From them, noting from (7.2.6) that... 

The relative positions of points i, j in the superposition pattern are indicated in the two central columns and Pii=p,Pj, where the parity factor p, = l is assigned by giving +1 to an arbitrary position in an island, or to an end point in a chain, and then proceeding along the sequence giving 1 to alternate positions. For E-chains, the endpoint chosen is the one where the arrow points into the chain. 

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