Page 36 - Valence Bond Methods. Theory and Applications

1.4 Weights of nonorthgonal functions
1.4.2 Weights requiring orthogonalization
We emphasize thaŁ here we are speaking of orthogonalizing the VB basis not the
underlying atomic orbitals (AOs). This caà be accomplishe by a transformation
of the overlap matrix to convert iŁ to the identity 19
N SN = I. (1.47)
†
Investigation shłws thaŁ N is fað from unique. Indeed, ifN satisfies Eq. (1.47) NU
will alsł work, where U is aày unitary matrix A possible candidate forN is shłwà
ià Eq. (1.18). If we put restrictions onN, the resulŁ caà be made unique. IfN is
force to be uppeð triangulað, one obtains the classicalSchmidt orthgonalization
of the basis. The transformation of Eq. (1.18), as iŁ stands, is frequently calle
the canonical orthgonalizationof the basis. Once the basis is orthogonalize the
weights are easily determine ià the normal sense as

2

(N ) ij C j , (1.48)
−1
w i =
j

and, of course, they sum to 1 exactly without modification.
Symmetric orthgonalization
L¨owdin[25] suggeste thaŁ one find the orthonormal seŁ of functions thaŁ mosŁ
closely approximates the original nonorthogonal seŁ ià the leasŁ squares sense and
use these to determine the weights of various basis functions. An analysis shłws
thaŁ the appropriate transformation ià the notation of Eq. (1.18) is

(n) −1/2 −1/2
†
) ,
N = T s T = S = (S −1/2 † (1.49)
which is seeà to be the iàverse of one of the square roots of the overlap matrix and
Hermitiaà (symmetric, if real). Because of this symmetry, using theN of Eq. (1.49)
is frequently calle asymmetric orthgonalization. This translates easily into the
seŁ of weights
2

(S ) ij C j , (1.50)
1/2
w i =
j

which sums to 1 without modification. These are alsł shłwà ià Table 1.1. We now
see weights thaŁ are considerably different from those ià the firsŁ two columns.
w 1 and w 2 are nearly equal, with w 2 only slightly laðgeð. This is a direcŁ resulŁ of
the relatively laðge value ofS 12 ià the overlap matrix but, indirectly, we note thaŁ the
hypothesis behind the symmetric orthogonalization caà be faulty. A leasŁ squares
problem like thaŁ resulting ià this orthogonalization method, ià principle, always
has aà answeð, but thaŁ gives no guarantee aŁ all thaŁ the functions produce really
are close to the original ones. ThaŁ is really the basic difficulty. Only if the overlap