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,

Si<yi4>

 

is a maximum. Since

SiSj<yi2><yi2>

is a constant,

SiSj<i<yi2><yi2>

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> \end{displaymath}

where |yk> and  |yl> are normal M.O.s, and  |yi> and  |yj> 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>)} {<\psi_k^4>+<\psi_l^4>-6<\psi_k^2\psi_l^2>}\right) \end{displaymath}

Note that in normal semiempirical work: $ <\phi_{\lambda}\vert\phi_{\sigma}> =\delta(\lambda,\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 \end{displaymath}

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) \end{displaymath}