Density-kernel variation

In section 7.3 the electron number gradient with respect to the density-kernel, which is here denoted ${\mit\Delta}$, is given as

$$\frac{\partial N}{\partial K^{\alpha \beta}}= 2 S_{\beta \alpha}={\mit\Delta}_{\alpha \beta} .$$ (7.75)

The density-kernel along this search direction ${\mit\Delta}$ is parameterised by $\lambda$ as

$$K(\lambda) = K(0) + \lambda {\mit\Delta}$$ (7.76)

where $K(0)$ denotes the initial density-matrix. The electron number, given by $N = 2 {\rm Tr}(KS)$ thus behaves linearly:

$$N(\lambda) = N(0) + 2 \lambda {\rm Tr}({\mit\Delta}S)$$

$$= N(0) + 4 \lambda {\rm Tr}(S^2)$$ (7.77)

and it is a trivial matter to calculate the required value of $\lambda$ to return the electron number to its correct value.

In general during the minimisation, the search direction is ${\mit\Lambda}$, and again this search can be parameterised by a single parameter $\lambda$:

$$K(\lambda) = K(0) + \lambda {\mit\Lambda} .$$ (7.78)

We wish to project out from ${\mit\Lambda}$ that component which is parallel to ${\mit\Delta}$. The modified search direction ${\tilde {\mit\Lambda}}$ can be written as

$${\tilde {\mit\Lambda}} = {\mit\Lambda} - \omega {\mit\Delta} .$$ (7.79)

The variation of the electron number along this modified direction is

$$N(\lambda) = N(0) + 2 \lambda {\rm Tr}({\mit\Lambda}S) - 2 \omega \lambda {\rm Tr}({\mit\Delta}S)$$ (7.80)

and we wish the coefficient of the linear term in $\lambda$ to vanish, which defines the required value of $\omega$ to be

$$\omega = \frac{{\rm Tr}({\mit\Lambda}S)}{{\rm Tr}({\mit\Delta}S)} = \frac{{\rm Tr}({\mit\Lambda}S)}{2{\rm Tr}(S^2)} .$$ (7.81)

Since the electron number depends linearly upon the density-kernel, after this projection, the electron number is constant along the modified search direction, and the electron number need not be corrected after a trial step is taken.