Various methods of localizing M.O.s have been
proposed [61,62,63]. The method
described here is a modification of Von Niessen's
technique, and is ideally suited for semiempirical methods.

For a set of LMOs,

Σ_{i}<ψ_{i}^{4}>

is a maximum. Since

Σ_{i}Σ_{j}<ψ_{i}^{2}><ψ_{i}^{2}>

is a constant,

Σ_{i}Σ_{j<i}<ψ_{i}^{2}><ψ_{i}^{2}>

must be a minimum.

The operation to localize M.O. consists of a series of binary unitary
transforms of the type:

where |ψ_{k}>
and |ψ_{l}>
are normal M.O.s, and |ψ_{i}>
and |ψ_{j}>
are the LMOs.

The ratio a/b is given by

Note that in normal semiempirical work:
.

From this it follows that, given
,

In order to preserve rotational invariance, all contributions on each atom must be
added together. This gives:

,

,

and

Number of Centers

The value of 1/<ψ^{4}>
is a direct measure of the number of centers involved
in the MO: thus, the value of
1/<ψ^{4}>
is 2.0 for H_{2}, 3.0 for a three-center bond and 1.0 for a lone pair.
There is no upper limit to the number of centers that can be in a localized
M.O., although there are seldom more than 3 in any system. To
understand this, consider a hypothetical system of 10 atoms that forms a perfect
decagon, and each atom has only one atomic orbital, and the system has only one
M. O. occupied. Okay, this is a ridiculous system, but it is being used
for illustration only. The LMO and M.O., Ψ,
would be the same: