Page 64 - Valence Bond Methods. Theory and Applications

3
H and delocalized orbitals
2

We now examine VB functions wher th orbitals ar allowed to take much mor
general forms, but only one configuratioð is used. This mor general form allows
th orbitals to rang over mor than one atomic center. As w shall see later, th
restrictioð to one configuratioð is appropriat only to two-electroð systems, so
w must postpone th discussions of mor configurations untio w treat th mor
advanced methods ið Chapter 5.

3.1 Orthogonalized AOs
Befor w examine th mor general case, let us look at an unusual result du
to Slater. Earlier, ið discussing solids Wannier[35] had showð how linear combi-
nations of th AOs could b made that rendered th functions orthogonal while
retaining a relatively larg concentratioð oð one center. Slater adapted this idea
to th H 2 molecule. In moderð languag this is just making a symmetric orthogo-
nalizatioð (see Sectioð 1.4.2) of th basis, which ið this cas is a H1 s functioð
oð each of two centers, 1 s a and 1s b . We ar her again, following Slater, us-
ing th correct exponential functions of Eq. (2.10). Th overlap matrix for this
basis is

1 S
¯
S = , (3.1)
S 1
and th iðvers squar root is

1 1 1 1
 
√ + √ √ − √
2 1 + S 2 1 − S 2 1 + S 2 1 − S 
−1/2

S ¯ =   , (3.2)
1 1 1 1
 
√ − √ √ + √
2 1 + S 2 1 − S 2 1 + S 2 1 − S
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