Manu: saving work

Manuscript/eDFT.tex

@@ -52,7 +52,7 @@
 
 % matrices/operator
 \newcommand{\br}{\bm{r}}
-\newcommand{\bw}{\bm{w}}
+\newcommand{\bw}{{\bm{w}}}
 \newcommand{\bG}{\bm{G}}
 \newcommand{\bS}{\bm{S}}
 \newcommand{\bGamma}[1]{\bm{\Gamma}^{#1}}
@@ -98,8 +98,8 @@
 \newcommand{\bmk}{\bm{\kappa}} % orbital rotation vector
 \newcommand{\bmg}{\bm{\Gamma}} % orbital rotation vector
 \newcommand{\bxi}{\bm{\xi}} 
-\newcommand{\bfx}{\bf{x}}
-\newcommand{\bfr}{\bf{r}}
+\newcommand{\bfx}{{\bf{x}}}
+\newcommand{\bfr}{{\bf{r}}}
 \DeclareMathOperator*{\argmin}{arg\,min}
 \newcommand{\blue}[1]{{\textcolor{blue}{#1}}}
 %%%% 
@@ -351,7 +351,35 @@ $\Phi^{(K)}$. We can then construct the ensemble density matrix
 {\bmg}^{{\bw}}=\sum_{K\geq 0}w_K{\bmg}^{(K)}
 \eeq
 and compute the ensemble density as follows:
-$n^{\bw}({\br})=$
+\blue{
+\beq
+n^{\bw}({\br})&=&\sum_{K\geq 0}\sum_{\sigma=\alpha,\beta}{\tt
+w}_Kn^{(K)}({\bfx})
+\nonumber\\
+&=&
+\sum_{K\geq 0}\sum_{\sigma=\alpha,\beta}{\tt
+w}_K\sum_{pq}\varphi_p({\bfx})\varphi_q({\bfx})\Gamma_{pq}^{(K)}
+\nonumber\\
+&=&
+\sum_{\sigma=\alpha,\beta}
+\sum_{K\geq 0}
+{\tt
+w}_K\sum_{p\in (K)}\varphi^2_p({\bfx})
+\nonumber\\
+&=&
+\sum_{\sigma=\alpha,\beta}
+\sum_{K\geq 0}
+{\tt
+w}_K
+\sum_{\mu\nu}
+\sum_{p\in (K)}c_{\mu p}c_{\nu p}\AO{\mu}({\bfx})\AO{\nu}({\bfx})
+\nonumber\\
+&=&\sum_{\sigma=\alpha,\beta}\sum_{\mu\nu}\AO{\mu}({\bfx})\AO{\nu}({\bfx}){\Gamma}^{\bw}_{\mu\nu}
+\eeq
+}
+\beq
+n^{\bw}({\br})=\sum_{\sigma=\alpha,\beta}\sum_{\mu\nu}\AO{\mu}({\br,\sigma})\AO{\nu}(\br,\sigma){\Gamma}^{\bw}_{\mu\nu}
+\eeq
 can be determined. 
 %%%%%%%%%%%%%%%
 %\subsection{Hybrid GOK-DFT}