Localization Theory

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,



is a maximum. Since


is a constant,


must be a minimum.

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

\begin{displaymath}\vert\psi_i> =a\vert\psi_k> +b\vert\psi_l> \end{displaymath}

\begin{displaymath}\vert\psi_j> =-b\vert\psi_k> +a\vert\psi_l>

where |ψk> and  |ψl> are normal M.O.s, and  |ψi> and  |ψj> are the LMOs.

The ratio a/b is given by

\begin{displaymath}a/b = \frac{1}{4}\arctan\left(\frac{4(<\psi_k\psi_l^3>-<\psi_k^3\psi_l>)}

Note that in normal semiempirical work: $ <\phi_{\lambda}\vert\phi_{\sigma}>

From this it follows that, given $\psi_k = \sum_{\lambda}C_{\lambda k}\phi_{\lambda}$,

\begin{displaymath}<\psi_k\psi_l^3> = \sum_{\lambda}C_{\lambda k}C_{\lambda l}^3

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

\begin{displaymath}<\psi_k^4> = \sum_A(\sum_{\lambda\in A}C_{\lambda k}^2)^2 \end{displaymath} ,

\begin{displaymath}<\psi_k^3\psi_l> = \sum_A(\sum_{\lambda\in A}C_{\lambda k}^2)
\sum_{\lambda\in A}C_{\lambda k}C_{\lambda l}\end{displaymath} ,


\begin{displaymath}<\psi_k^2\psi_l^2> = \sum_A(\sum_{\lambda\in A}C_{\lambda k}^2)(\sum_{\lambda\in A}C_{\lambda l}^2)

Number of Centers

The value of 1/<ψ4>  is a direct measure of the number of centers involved in the MO: thus, the value of 1/<ψ4>  is 2.0 for H2, 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:

 Ψ = Σiciφi = Σi10φi

The value of <Ψ4> would be

4> = Σijkl<ciφicjφjckφkclφl>

which, because of orthonormality of the atomic orbitals in semiempirical methods would simplify to

4> = Σikci2iφi>ck2kφk>


4> = Σik10-1x10-1δ(i,k) = 0.1

So the LMO would involve 10.0 centers.

Bonding contribution of each M.O.

The RHF bond-order matrix, Pλσ, is constructed using the occupied set of M.O.s:

Pλσ, = 2Σi (occ) cλicσi

so the atomic orbital contribution of localized M.O. "j" to bonding, Bjj, of each LMO is given by:

Bjj = 2ΣλΣσcλjcσji (occ) cλicσi

Where λ and σ in the first two summations exclude all matrix elements involving the same atom.  (Remember - bonds involve pairs of atoms.)

This summation can be re-cast in simpler form as:

Bjj = 2ΣλΣσcλjcσjPλσ

Note 1: The sum of the bonding contributions over all M.O.s (occupied plus virtual) is zero.

Note 2: A bonding M.O. will contribute ca. 2.0 to the bond-orders; a lone-pair will contribute ca. 0.0.