From 3bf0fca3ad97c6b58824fc24bfb9015cef158e24 Mon Sep 17 00:00:00 2001 From: Emmanuel Fromager Date: Tue, 11 Feb 2020 10:11:16 +0100 Subject: [PATCH] Manu: saving work --- Manuscript/eDFT.tex | 22 +++++++++++++++++----- 1 file changed, 17 insertions(+), 5 deletions(-) diff --git a/Manuscript/eDFT.tex b/Manuscript/eDFT.tex index d3c5be7..131338e 100644 --- a/Manuscript/eDFT.tex +++ b/Manuscript/eDFT.tex @@ -333,14 +333,26 @@ _{n=n_{\opGamma{\bw}}}. \eeq For implementation purposes, we will use in the rest of this work -(one-electron reduced) density matrices rather than Slater determinants -as basic variables. If we denote ${\bmg}^{(K)}$ - +(one-electron reduced) density matrices +as basic variables, rather than Slater determinants. If we expand the +(spin-) orbitals [from which the latter are constructed] in an atomic +orbital (AO) basis, \beq -\Gamma_{\mu\nu}^{(K)}=\sum_{p\in (K)}c_{\mu p}c_{\nu p} +\varphi_p({\bfx})=\sum_{\mu}c_{{\mu p}}\AO{\mu}(\bfx), +\eeq +then the density matrix elements obtained from the +determinant $\Phi^{(K)}$ can be expressed as follows in the AO basis: +\beq +\Gamma_{\mu\nu}^{(K)}=\sum_{p\in (K)}c_{\mu p}c_{\nu p}, \eeq where the summation runs over the spin-orbitals that are occupied in -$\Phi^{(K)}$. +$\Phi^{(K)}$. We can then construct the ensemble density matrix +\beq +{\bmg}^{{\bw}}=\sum_{K\geq 0}w_K{\bmg}^{(K)} +\eeq +and compute the ensemble density as follows: +$n^{\bw}({\br})=$ +can be determined. %%%%%%%%%%%%%%% %\subsection{Hybrid GOK-DFT} %%%%%%%%%%%%%%%