Support function variation

The electron number gradient with respect to the support functions is given in section 7.3 and denoted by $\{ \zeta^{\alpha}({\bf r})\}$:

$$\frac{\delta N}{\delta \phi_{\alpha}({\bf r})}= 2 K^{\alpha \beta} \phi_{\beta}({\bf r}) = \zeta^{\alpha}({\bf r}) .$$ (7.82)

The support function variation is again parameterised by the parameter $\lambda$:

$$\phi_{\alpha}({\bf r};\lambda) = \phi_{\alpha}({\bf r};\lambda=0) + \lambda \zeta^{\alpha}({\bf r})$$ (7.83)

and this results in the following quadratic variation of the overlap matrix

$$S_{\alpha \beta}(\lambda) = S_{\alpha \beta}(0) + \lambda \langle \phi_{\alpha}(\lambda=0) \v... ...ambda=0) \rangle + \lambda^2 \langle \zeta^{\alpha} \vert \zeta^{\beta} \rangle$$

$$= S_{\alpha \beta}(0) + \lambda S'_{\alpha \beta} +\lambda^2 S''_{\alpha \beta}$$ (7.84)

which defines the matrices $S'$ and $S''$. The variation of the electron number is therefore also quadratic

$$N(\lambda) = N(0) + 2 \lambda {\rm Tr}(KS') + 2 \lambda^2 {\rm Tr}(KS'')$$ (7.85)

and the roots of this expression can be found to correct the electron number.

We consider a general search direction for the localised functions denoted $\{ \xi^{\alpha}({\bf r}) \}$ and modify this search direction to obtain the direction which maintains the electron number to first order:

$${\tilde \xi}^{\alpha}({\bf r}) = \xi^{\alpha}({\bf r}) - \omega \zeta^{\alpha}({\bf r}) .$$ (7.86)

Now varying the localised functions according to

$$\phi_{\alpha}({\bf r};\lambda) = \phi_{\alpha}({\bf r};\lambda=0) + \lambda {\tilde \xi}^{\alpha}({\bf r})$$ (7.87)

results in the following variation of the electron number:

$$N(\lambda) = N(0) + 2 \lambda {\rm Tr}(K{\tilde S}') - 2 \lambda \omega {\rm Tr}(KS') + {\cal O}(\lambda^2)$$ (7.88)

where ${\tilde S}'_{\alpha \beta} = \langle \phi_{\alpha}(\lambda=0) \vert \xi^{\beta} \rangle + \lambda \langle \xi^{\alpha} \vert \phi_{\beta}(\lambda=0) \rangle$. Thus to maintain the electron number to first order, we choose

$$\omega = \frac{{\rm Tr}(K{\tilde S}')}{{\rm Tr}(KS')} .$$ (7.89)

In this case, the electron number is not constant along the search direction, but will still vary quadratically, so that it is necessary to correct the density-matrix before evaluating the total functional.