Optimising the support functions

As mentioned above, we can optimise our choice of support functions by minimising the spilling parameter ${\cal S}$. We describe this process here when the support functions are themselves described in terms of a localised basis:

$$\vert {\phi}_{\alpha} \rangle = \sum_{n \ell m} c^{n \ell m}_{(\alpha)} \vert \chi_{\alpha , n \ell m} \rangle .$$ (8.17)

The spilling parameter can be written in terms of the matrices $L$ and $S$ by:

$${\cal S} = \frac{1}{N_{\mathrm b}} \langle \psi_i \vert \bigl( 1 - \hat P \b... ...e \psi_i \vert \phi_{\alpha} \rangle \langle \phi^{\alpha} \vert \psi_i \rangle$$

$$= 1 - \frac{1}{N_{\mathrm b}} L_{i \alpha}^{\dag } S^{-1}_{\alpha \beta} L_{\beta i} = 1 - \frac{1}{N_{\mathrm b}} {\rm Tr}[L^{\dag } S^{-1} L]$$ (8.18)

and we wish to obtain the gradients of ${\cal S}$ with respect to the expansion coefficients $\{ c^{n \ell m}_{(\alpha)} \}$.

$$\frac{\partial {\cal S}}{\partial c^{n \ell m}_{(\alpha)}} =... ...c{\partial L_{ki}}{\partial c^{n \ell m}_{(\alpha)}} \right] .$$ (8.19)

We obtain the derivative of the inverse matrix by differentiating $S^{-1} S = 1$ i.e.

$$\frac{\partial (S^{-1}S)_{\alpha \beta}}{\partial x} = \frac... ... \gamma}^{-1} \frac{\partial S_{\gamma \beta}}{\partial x} = 0$$ (8.20)

which can be rearranged to give

$$\frac{\partial S_{\alpha \beta}^{-1}}{\partial x} = - S_{\al... ...partial S_{\gamma \delta}}{\partial x} S_{\delta \beta}^{-1} .$$ (8.21)

Therefore (no summation over $\alpha$)

$$\frac{\partial {\cal S}}{\partial c^{n \ell m}_{(\alpha)}} = - \frac{1}{N_{\mathrm b}} \left[ \langle \psi_i \vert \chi_{\alph... ... \ell m} \rangle \delta_{\beta \gamma} \right) S_{\gamma k}^{-1} L_{ki} \right.$$

$$\qquad \left. + L_{ij}^{\dag } S_{j \alpha}^{-1} \langle \chi_{\alpha , n \ell m} \vert \psi_i \rangle \right]$$

$$= - \frac{2}{N_{\mathrm b}} {\rm Re} \left[ L_{\alpha \beta}^{\dag ... ...hi_{\beta , n \ell m} \rangle S_{\beta \gamma}^{-1} L_{\gamma \alpha} \right] .$$ (8.22)

In the case of the set of basis functions introduced in chapter 5, the overlap between plane-wave eigenstates and localised basis functions, e.g. $\langle \psi_{\alpha} \vert \chi_{\beta , n \ell m} \rangle$, can be calculated using the expression for the basis function Fourier transform (5.9).

We can use these gradients to minimise the spilling parameter (by the conjugate gradients method) to obtain the set of optimal coefficients $\{ c^{n \ell m}_{(\alpha)} \}$ which define the set of support functions which best span the space of the occupied plane-wave orbitals. The final minimum spilling parameter value also gives an estimate of the quality of the basis-set being used.