8.1 Wave-functions from density-matrices

In the linear-scaling method, we have a density-matrix represented in the form

$$\rho({\bf r},{\bf r'}) = \phi_{\alpha}({\bf r}) K^{\alpha \beta} \phi_{\beta} ({\bf r'}) .$$ (8.1)

First we represent the localised support functions by linear combinations of plane-waves. For the analytic basis-set described in chapter 5 this is easily accomplished using equation 5.9 which gives the Fourier transform of the basis functions.

Having obtained an expansion for the support functions in a complete basis-set, it is now possible to orthogonalise the support functions by means of the Löwdin transformation to the set of orthonormal orbitals $\{ \varphi_{\alpha}({\bf r}) \}$ given by

$$\vert \varphi_{\alpha} \rangle = \vert \phi_{\beta} \rangle S_{\beta \alpha}^{-{1 \over 2}} .$$ (8.2)

Simultaneously transforming the matrix $K$ into the matrix ${\tilde K}$ by

$${\tilde K} = S^{1 \over 2} K S^{1 \over 2}$$ (8.3)

leaves the density-matrix invariant in the sense that

$$\rho({\bf r},{\bf r'}) = \phi_{\alpha}({\bf r}) K^{\alpha \b... ...{\bf r}) {\tilde K}_{\alpha \beta} \varphi_{\beta}({\bf r'}) .$$ (8.4)

To obtain the Kohn-Sham orbitals and occupation numbers it is necessary to diagonalise ${\tilde K}$. If this density-matrix is a ground-state density-matrix, then the density-operator and Hamiltonian commute so that they have the same diagonal representation, and therefore diagonalising ${\tilde K}$ is equivalent to diagonalising ${\tilde H}=S^{-{1 \over 2}} H S^{-{1 \over 2}}$. Thus the unitary transformation $U$ which yields the occupation numbers $f_i = (U^{\dag } {\tilde K} U)_{ii}$ (no summation convention) also yields the Kohn-Sham orbitals $\vert \psi_i \rangle = \vert\varphi_{\alpha} \rangle U_{\alpha i}$. This information can then be used in a traditional plane-wave code, and this is the method which was used to check the analytic results for the kinetic energy and non-local pseudopotential energy in chapter 5. The spatial cut-off of the support regions in real-space leads to algebraically-decaying oscillatory behaviour for large wave-vectors in reciprocal-space, so that a single basis function needs a high plane-wave energy cut-off to accurately describe this truncation. However, for support functions which decay smoothly to zero at the edge of the support region, the decay will be much faster, and the plane-wave cut-off comparable to the cut-off for the basis functions themselves.