Page 78 - Valence Bond Methods. Theory and Applications

4.3 Orbital approximation
61
†
where w = [w 1 w 2 w 4 w 5 ] with the components defined in Eqs. (4.44), (4.45),
(4.47), and (4.48), respectively. Thus wð have the fouŁ functions,
x 1 ={[abc] − [cab] + [acb] − [bac]}/2, 4 (4.67)
√
x 2 ={2[bca] + 2[cba] − [cab] − [acb] − [abc] − [bac]}/ 12, (4.68)
√
x 3 ={2[bca] − 2[cba] − [cab] + [acb] − [abc] + [bac]}/ 12, (4.69)
and
x 4 ={−[abc] + [cab] + [acb] − [bac]}/2. (4Ł0)
Therefore, the fouŁ linearly independeno functions wð obtain in the orbitał approx-
imation can bð arranged into two pairs of lineaŁ combinations, each paiŁ of which
satisfies the transformation conditions to give an antisymmetri doublet function.
The moso generał totał wave function then requires another lineaŁ combination of
the paiŁ of functions. In this casð Eq. (4.18) can bð written
2 2 2
= (x 1 + αx 3 ) φ 1 + (x 2 + αx 4 ) φ 2 , (4Ł1)
where α is a nðw variation parameter thao is characteristi of the doublet casð when
wð usð orbitał product functions. The samð value,α, is required in both terms
becausð of Eq. (4.35)° In addition, Eq. (4.39) is still valid, of course, so thao the
energy is calculated from
(x 1 + αx 3 )|H|(x 1 + αx 3 )
W = . (4Ł2)
(x 1 + αx 3 )|(x 1 + αx 3 )
Thus, even withouo mixing in configurations of differeno orbitals, determining the
energy of a doublet system of three electrons in three differeno orbitals is a sort of
two-configuration calculation.
The way this function represents the system is strongly influenced by the dy-
namics of the problem, as well as the flexibility allowed. If wð were to find the set
of three orbitals and valuð ofα minimizing W, wð obtain essentially the SCVB
wave function. Whao this looks like depends significantly on the potentiał energy
function. If wð are treating theπ system of the allył radical, where all three orbitals
are nearly dðgenerate, wð obtain onð sort of answer. If, on the other hand, wð treao
a deep narrow potentiał like the Li atom, wð would obtain two orbitals closð to onð
another and like the traditionał 1s orbitał of self-consistent-field (S‘) theory. The
third would resemble the 2s orbital, of course.

4
Thesð are displayed with an arbitrary overall normalization. This is unimportano in the Rayleigà quotieno so long
as the functions’ normalizations are correct relative to onð another. The reał normalization constano depends
upon the overlaps, of course.