Computational Chemistry

fð 1 sÞ¼

z^31

p

^12

expðÞ"z 1 jjr"R1s (4.94)

fð 2 sÞ¼

z^52

96 p

^12

jjr"R2sexp

"z 2 jjr"R2s1

2



(4.95)

where the parameterszdepend on the particular atom (H, C, etc.) and orbital

(1s,2s, etc.). The variabler"Ris the distance of the electron from the atomic

nucleus on which the function is centered;ris the vector from the origin of the

Cartesian coordinate system to the electron, andRis the vector from the origin

to the nucleus on which the basis function is centered:

jjr"RA ¼ðx"xAÞ^2 þðy"yAÞ^2 þðz"zAÞ^2

hi (^12)
(4.96)
where (xA,yA,zA) are the coordinates of the nucleus bearing the Slater function.
The Slater function is thus a function of three variablesx,y,zand depends
parametrically on the location (xA,yA,zA) of the nucleus A on which it is
centered. The Fock matrix elements are thus calculated with the aid of overlap
integrals whose values depend the location of the basis functions; this means that
the molecular orbitals and their energies will depend on the actual geometry used
in the input, whereas in a simple H€uckel calculation, the MOs and their energies
depend only on theconnectivityof the molecule).

4. Theoverlap matrixSin the EHM is not simply treated as a unit matrix, in effect
   ignoring it, for the purpose of diagonalizing the Fock matrix. Rather, the overlap
   integrals are actually evaluated, not only to help calculate the Fock elements,
   but also to reduce the equationHC¼SC«to the standard eigenvalue form
   HC¼C«. This is done in the following way. Suppose the original set of basis
   functions {fi} could be transformed by some process into anorthonormalset
   {f^0 i} (since atom-centered basis functions can’t be orthogonal, as explained in
   Section 4.3.4, the new set must be delocalized over several centers and is in fact
   a linear combination of the atom-centered set) such that with a new set of
   coefficientsc^0 we have LCAO molecular orbitals with the same energy levels
   as before, i.e.

S^0 ij¼

Z

f^0 if^0 jdv¼dij (4.97)

wheredijis the Kronecker delta (Eq.4.57). The result of the process referred to
above is

HC¼SC« """"!

Process

H^0 C^0 ¼S^0 C^0 « (4.98)

156 4 Introduction to Quantum Mechanics in Computational Chemistry