Computational Chemistry

this criterion and Lewis structures the C/C bond order in ethane is 1, in ethene 2, and
in ethyne 3, in accordance with the classical assignment of a single, a double, and a
triple bond, respectively. However, if a bond is a manifestation of the electron
density between two nuclei, then the bond order need not be an integer; thus the
C¼C bond in H 2 C¼CH–CHO might be expected to have a lower bond order than
the C¼C in H 2 C¼CH–CH 3 , because the C¼O group might drain electron density
away toward the electronegative oxygen. However, an attempt to calculate bond
order from electron density runs into the problem that in a polyatomic molecule, at
any rate, it is not clear how to define precisely the region “between” two atomic
nuclei (Fig.5.38b).
Assigning atom charges and bond orders involves calculating the number of
electrons “belonging to” an atom or shared “between” two atoms, i.e. the “popula-
tion” of electrons on or between atoms; hence such calculations are said to involve
population analysis. Earlier schemes for population analysis bypassed the problem
of defining the space occupied by atoms in molecules, and the space occupied by
bonding electrons, by partitioning electron density in a somewhat arbitrary way.
The earliest such schemes were utilized in the simple H€uckel or similar methods
[ 256 ], and related these quantities to the basis functions (which in these methods are
essentially valence, or even justp, atomic orbitals; see Section 4.3.4). The simplest
scheme used in ab initio calculations isMulliken population analysis[ 257 ].
Mulliken population analysis is in the general spirit of the scheme used in the
simple H€uckel method, but allows for several basis functions on an atom and does
not require the overlap matrix to be a unit matrix. In ab initio theory each molecular
orbital has a wavefunctionc(Section 5.2.3.6.1):

c 1 ¼c 11 f 1 þc 21 f 2 þc 31 f 3 þ(((cm 1 fm

c 2 ¼c 12 f 1 þc 22 f 2 þc 32 f 3 þ(((cm 2 fm

c 3 ¼c 13 f 1 þc 23 f 2 þc 33 f 3 þ(((cm 3 fm

...

cm¼c 1 mf 1 þc 2 mf 2 þc 3 mf 3 þ(((cmmfm

ð 5 : 209 ¼ 5 : 51 Þ

Here the chosen basis set {f 1 ,f 2 , ...,fm} engenders MOsc 1 ,c 2 , ...,cm.
Several basis functions can reside on each atom, socsiis the coefficient of basis
functionsin MOi(not, as in simple H€uckel theory, the sole coefficient of atomsin
MOi). For any MOci, squaring and integrating over all space gives

Z

jcij^2 dv¼ 1 ¼c 1 ic 1 iS 11 þc 2 ic 2 iS 22 þ(((

þ 2 c 1 ic 2 iS 12 þ 2 c 1 ic 3 iS 13 þ 2 c 2 ic 3 iS 23 þ(((

ð 5 : 210 Þ

The integral equals 1 because the probability that the electron issomewherein
the MO (which, strictly, extends over all space) is 1; theSii(bothf’s the same)

5.5 Applications of the Ab initio Method 345