Computational Chemistry

6.2.3 The Complete Neglect of Differential Overlap (CNDO)

Method

The first semiempirical SCF-type method to go beyond justpelectrons was the
complete neglect of differential overlap method (ca. 1966) [ 15 ]. This was a general-
geometry method, since it is not limited to planarpsystems (molecules with con-
jugatedpelectron systems, like benzene, are usually planar). Like the other early
general-geometry method, the extended H€uckel method, which appeared in 1963
(Section4.4), CNDO calculations use a minimal valence basis set of Slater-type
orbitals, using just the valence electrons and the conventional atomic orbitals
of each atom. The Fock matrix elements are calculated from Eq.6.1¼5.82; for a
CNDO calculationHrscorerepresents the nuclei plus all core electrons,Ptuis calcu-
lated from the coefficients of the valence AOs, and the two-electron repulsion
integrals refer to valence electrons.
There are two versions of CNDO, CNDO/1 and an improved version, CNDO/2.
First look at CNDO/1. Consider the core integralsHcorerArA,wherebothorbitalsare
the same (i.e. the same orbital occurs twice in the integral frð 1 ÞjH^corerr jfrð 1 Þ
)
and are on the same atom A. Recall the example of an ab initio calculation on
HHe+(Section5.2.36e). Consider, say, element (1,1) of thatHcorematrix. From
Eq. 5.116:

Hcore 11 ¼ f 1 ð 1 ÞjT^jf 1 ð 1 Þ

þ f 1 ð 1 ÞjV^Hjf 1 ð 1 Þ

þ f 1 ð 1 ÞjV^Hejf 1 ð 1 Þ

¼ f 1 ð 1 ÞjT^þV^Hjf 1 ð 1 Þ

þ f 1 ð 1 ÞjV^Hejf 1 ð 1 Þ

(6.9)

Equation6.9can be generalized to a matrix element (r,r) and a molecule with

HcorerArA¼ frAð 1 ÞjT^þV^AjfrAð 1 Þ

þ frAð 1 ÞjV^BjfrAð 1 Þ

þ frAð 1 ÞjV^CjfrAð 1 Þ

þ)))

¼Urrþ

X

B 6 ¼A

frAð 1 ÞjV^BjfrAð 1 Þ

¼UrrþVAB ð 6 : 10 Þ

wherefrAis a basis function on atom A. TheUrrterm in Eq. 7.0 is regarded as the
energy of an electron in the AO on A corresponding to the functionfrA, and is
taken as the negative of the valence-state ionization energy of such an electron. The
integrals in theVABterm are simply calculated as the potential energy of a valence
sorbital in the electrostatic field of the core of atom A, B, etc., e.g.

frAð 1 ÞjV^BjfrAð 1 Þ

¼ SAð 1 Þ

CB

r1B

(^)
(^) SAð 1 Þ

(6.11)

whereCBis the charge on the core of atom B, i.e. the atomic number minus the
number of core (non-valence) electrons, and the variabler1Bis the distance of the 2s
electron from the center of the core (from the atomic nucleus). The core integrals
with different orbitalsfrandfs, on the same atom (A¼B; one-center integrals) or

398 6 Semiempirical Calculations