From Eq.6.6and the definition of the two-electron integralðrsjtuÞ¼ZZ
f(rð 1 Þfsð 1 Þf(tð 2 Þfuð 2 Þ
r 12dv 1 dv 2 (6.7 = 5.73)it follows that (1) forr 6 ¼s,(rs|tu)¼0, and (2) forr¼sandt¼u,(rs|tu)¼(rr|tt).
Both cases are taken into account by writing
ðrsjtuÞ¼drsdtuðrrjttÞ (6.8)where thed’s are Kronecker deltas (¼1 if the subscripts are the same, zero
otherwise). Thus the four-center (i.e. (rs|tu)) and three-center (i.e. (rr|tu)) two-
electron integrals are ignored, but not the two-center (i.e. (rr|tt)) and one-center
(i.e. (rr|rr)) two-electron integrals. The one-center integrals (rr|rr) are taken as the
difference between the valence-state ionization energy and the electron affinity of
the atom bearingfr(these valence-state parameters refer to a hypothetical isolated
atom in the same hybridization state as in the molecule, and can be found spectro-
scopically).The two-center integrals (rr|tt) are estimated from (rr|rr) and (tt|tt) and
the distance between thefrandftatoms.
Although the overlap integralshfr|fsiare actually calculated for the evaluation of
Hcorers (Eq.6.5¼5.82), the overlap matrix is taken as a unit matrix as far as the matrix
Roothaan-Hall equationsFC¼SCego; thusFC¼CeorF¼CeC"^1 and the Fock
matrix is diagonalized, without transforming it with an orthogonalizing matrix, to
give the MO coefficients and energy levels. That the overlap matrix is a unit matrix
is a corollary of the ZDO approximation of Eq.6.6, from which it follows that the
off-diagonal matrix elements are zero; the diagonal elements are of course unity if
normalized AO basis functions are used. PPP energies arepelectron electronic
energiesESE, or electronic energies plus core–core repulsions,EtotalSE, ifVCCis added
(Eq.6.2).
The PPP method has been used to calculate the UV spectra of conjugated
compounds, especially dyes [ 14 ], a task it performs fairly well. The accuracy
of these calculations can be improved by incorporating electron correlation
(Section5.4), using the configuration interaction (CI) method. The calculations
were usually done at a fixed geometry, although an empirical bond length-bond
order relation permits optimization ofbond length. The classical PPP method is
not much used now, having evolved into other neglect of differential overlap
(NDO) methods, especially those parameterized for spectra, like INDO/S and the
very successful ZINDO/S (below). Wenow look at a hierarchy NDO methods
that, unlike the PPP approach, are not limited to planar arrays of p-orbitals, but
instead permit calculations on molecules of general geometry. In order of increas-
ing sophistication these are complete neglect of differential overlap CNDO),
intermediate neglect of differential overlap (INDO), and neglect of diatomic
differential overlap (NDDO).
6.2 The Basic Principles of SCF Semiempirical Methods 397