Projecting plane-wave eigenstates onto support functions

The plane-wave eigenstates are denoted $\vert \psi_i \rangle$ and the support functions are denoted $\vert\phi_{\alpha}\rangle$. The states obtained by projecting the plane-wave eigenstates onto the space spanned by the support functions are denoted $\vert \xi_{\alpha} \rangle$. As in section 4.6 we also introduce the dual states $\vert \phi^{\alpha} \rangle$ and $\vert \xi^{\alpha} \rangle$ with the properties outlined below.

$$S_{\alpha \beta} = \langle \phi_{\alpha} \vert \phi_{\beta} \rangle \qquad \qquad \Sigma_{\alpha \beta} = \langle \xi_{\alpha} \vert \xi_{\beta} \rangle$$ (8.5)

$$\vert \phi^{\alpha} \rangle = \vert \phi_{\beta} \rangle S_{\beta \alpha}^{-1} \qquad \qquad \vert \xi^{\alpha} \rangle = \vert \xi_{\beta} \rangle \Sigma_{\beta \alpha}^{-1}$$ (8.6)

$$\langle \phi^{\alpha} \vert \phi_{\beta} \rangle = \langle \phi_{\alpha} \vert \phi^{\beta} \rangle = \delta_{\alpha}^{\beta} \qquad \qquad \langle \xi^{\alpha} \vert \xi_{\beta} \rangle = \langle \xi_{\alpha} \vert \xi^{\beta} \rangle = \delta_{\alpha}^{\beta}$$ (8.7)

The projection operator onto the subspace spanned by the support functions is defined by

$${\hat P} = \vert \phi_{\alpha} \rangle \langle \phi^{\alpha}... ...ha} \rangle S_{\alpha \beta}^{-1} \langle \phi_{\beta} \vert .$$ (8.8)

A spilling parameter ${\cal S}$ can be defined to measure how much the subspace spanned by the plane-wave eigenstates falls outside the subspace spanned by the support functions. Minimising this quantity is one method of optimising the choice of support functions, and is described for the case of the spherical-wave basis (chapter 5) in section 8.2.3.

$${\cal S} = \frac{1}{N_{\mathrm b}} \langle \psi_i \vert \bigl( 1 - \hat P \bigr) \vert \psi_i \rangle$$ (8.9)

where $N_{\mathrm b}$ is the number of bands (labelled $i$) considered. The density-operator is then defined by

$$\hat \rho = \sum_{\alpha}^{\mathrm{occ}} \vert \xi_{\alpha} \rangle \langle \xi^{\alpha} \vert$$ (8.10)

where the sum is taken over occupied bands only. Substitution of the results given above then yields the following expression for the density-kernel:

$$K^{\alpha \beta} = \langle \phi^{\alpha} \vert \hat \rho \ve... ...} \langle \psi_j \vert \phi_{\mu} \rangle S_{\mu \beta}^{-1} .$$ (8.11)

Defining the rectangular matrix $L$ as

$$L_{\lambda i} = \langle \phi_{\lambda} \vert \psi_i \rangle$$ (8.12)

we give an expression for the matrix $\Sigma$ in terms of $L$ and $S$:

$$\Sigma_{\alpha \beta} = \langle \xi_{\alpha} \vert \xi_{\bet... ...nu}^{-1} S_{\nu \beta} = (L^{\dag } S^{-1} L)_{\alpha \beta} .$$ (8.13)

We can thus minimise the spilling parameter ${\cal S}$ to optimise our choice of support functions, and then calculate $K$ to obtain all of the information required to start a linear-scaling calculation.