Computational Chemistry

(«, not«^0 , as the energy will not depend on manipulation of a given set of basis
functions) where the matricesH, C, Sand « were defined inSection 4.3.4
(Eqs.4.55) andH^0 andS^0 are analogous toHandSwithf^0 in place offandC^0
is the matrix of coefficientsc^0 that satisfies the equation with the energy levelse(the
elements of«) being the same as in the original equationHC¼SC«. Since from
Eq.4.97S^0 ¼ 1 , the unit matrix (Section 4.3.3), Eq.4.98simplifies to

HC¼SC« """"!

Process

H^0 C^0 ¼C^0 « (4.99)

TheProcessthat effects the transformation is calledorthogonalization, since the
result is to make the basis functions orthogonal. The favored orthogonalization
procedure in computational chemistry, which I will now describe, is L€owdin
orthogonalization (after the quantum chemist Per-Olov L€owdin).
Define a matrixC^0 such that

C^0 ¼S^1 =^2 C i.e: C¼S"^1 =^2 C^0 (4.100)

(By multiplying on the left byS"1/2and noting thatS"1/2S1/2¼S^0 ¼ 1 ).
Substituting Eq.4.100intoHC¼SC«and multiplying on the left byS"1/2
we get

S"^1 =^2 HS"^1 =^2 C^0 ¼S"^1 =^2 SS"^1 =^2 C^0 e (4.101)

Let

S"^1 =^2 HS"^1 =^2 ¼H^0 (4.102)

and note thatS"^1 =^2 SS"^1 =^2 ¼S^1 =^2 S"^1 =^2 ¼ 1
Then we have from Eqs.4.101and4.102

H^0 C^0 ¼1C^0 e

i.e.

H^0 C^0 ¼C^0 e (4.103)

Thus the orthogonalizing process of Eq.4.99(or rather one possible orthogonal-
ization process, L€owdin orthogonalization) is the use of anorthogonalizing matrix
S"1/2to transformHby pre- and postmultiplication (Eq.4.102) intoH^0 .H^0 satisfies
the standard eigenvalue equation (Eq.4.103), so

4.4 The Extended H€uckel Method 157