dr at the end

Manuscript/eDFT.tex

@@ -206,7 +206,7 @@ The GOK ensemble energy~\cite{Gross_1988a,Oliveira_1988,Gross_1988b} is defined
 \eeq
 where the $K$th energy level $\E{}{(K)}$ [$K=0$ refers to the ground state] is the eigenvalue of the electronic Hamiltonian $\hH = \hh + \hWee$, where 
 \beq
-	\hh = \sum_{i=1}^\nEl \qty[ -\frac{1}{2} \nabla_{\br{i}}^2 + \vne(\br{i}) ]
+	\hh = \sum_{i=1}^\nEl \qty[ -\frac{1}{2} \nabla_{i}^2 + \vne(\br{i}) ]
 \eeq
 is the one-electron operator describing kinetic and nuclear attraction energies, and $\hat{W}_{\rm ee}$ is the electron repulsion operator. 
 The (positive) ensemble weights $\ew{K}$ decrease with increasing index $K$. They are normalized, \ie,
@@ -251,10 +251,10 @@ where, according to the {\it variational} ensemble energy expression of Eq.~\eqr
 	\pdv{\E{}{\bw}}{\ew{K}} 
 	& = \mel*{\Det{(K)}}{\hh}{\Det{(K)}}-\mel*{\Det{(0)}}{\hh}{\Det{(0)}}
 	\\
-	& + \Bigg\{\int d\br{}\,\fdv{\E{Hx}{\bw}[\n{}{}]}{\n{}{}(\br{})} \qty[ \n{\Det{(K)}}{}(\br{}) - \n{\Det{(0)}}{}(\br{}) ]
+	& + \Bigg\{\int \fdv{\E{Hx}{\bw}[\n{}{}]}{\n{}{}(\br{})} \qty[ \n{\Det{(K)}}{}(\br{}) - \n{\Det{(0)}}{}(\br{}) ] d\br{}
 	+ \pdv{\E{Hx}{\bw} [\n{}{}]}{\ew{K}}
 	\\
-	& + \int d\br{} \fdv{\E{c}{\bw}[n]}{\n{}{}(\br{})} \qty[ \n{\Det{(K)}}{}(\br{}) - \n{\Det{(0)}}{}(\br{}) ]
+	& + \int \fdv{\E{c}{\bw}[n]}{\n{}{}(\br{})} \qty[ \n{\Det{(K)}}{}(\br{}) - \n{\Det{(0)}}{}(\br{}) ] d\br{} 
 	+ \pdv{\E{c}{\bw}[n]}{\ew{K}}
 	\Bigg\}_{\n{}{} = \n{\opGam{\bw}}{}}.
 \end{split}
@@ -275,8 +275,7 @@ from the exact expression in Eq.~\ref{eq:exact_ens_Hx} that
 \beq
 	\E{Hx}{\bxi}[\n{}{\bxi,\bxi}] = \sum_{K \geq 0} \xi_K \mel*{\Det{(K),\bxi}}{\hWee}{\Det{(K),\bxi}}
 \eeq  
-with $\xi_0 = 1 - \sum_{K>0}\xi_K$
-and
+with $\xi_0 = 1 - \sum_{K>0}\xi_K$, and
 \beq
 	\E{Hx}{\bw}[\n{}{\bw,\bxi}] = \sum_{K \geq 0} \ew{K} \mel*{\Det{(K),\bxi}}{\hWee}{\Det{(K),\bxi}},
 \eeq  
@@ -288,11 +287,11 @@ thus leading, according to Eqs.~\eqref{eq:deriv_Ew_wk} and \eqref{eq:_deriv_wk_H
 	- \mel*{\Det{(0)}}{\hH}{\Det{(0)}}
 	\\
 	& + \qty{
-	\int d\br{} \fdv{\E{c}{\bw}[\n{}{}]}{\n{}{}({\br{}})} 
+	\int \fdv{\E{c}{\bw}[\n{}{}]}{\n{}{}({\br{}})} 
 	\qty[ \n{\Det{(K)}}{}(\br{}) - \n{\Det{(0)}}{}(\br{}) ]
 	+
 	\pdv{\E{c}{\bw} [\n{}{}]}{\ew{K}} 
-	}_{\n{}{} = \n{\opGam{\bw}}{}}.
+	}_{\n{}{} = \n{\opGam{\bw}}{}} d\br{}.
 \end{split}
 \eeq
 Since the ensemble energy can be evaluated as follows:
@@ -307,8 +306,8 @@ we finally recover from Eqs.~\eqref{eq:KS_ens_density} and
 	\E{}{(I)} 
 	& = \mel*{\Det{(I)}}{\hH}{\Det{(I)}} + \E{c}{{\bw}}[\n{\opGam{\bw}}{}]
 	\\
-	& + \int d\br{} \fdv{\E{c}{\bw}[\n{\opGam{\bw}}{}]}{\n{}{}(\br{})}
-	\qty[ \n{\Det{(I)}}{}(\br{}) - \n{\opGam{\bw}}{}(\br{}) ]
+	& + \int \fdv{\E{c}{\bw}[\n{\opGam{\bw}}{}]}{\n{}{}(\br{})}
+	\qty[ \n{\Det{(I)}}{}(\br{}) - \n{\opGam{\bw}}{}(\br{}) ] d\br{}
 	\\
 	&+
 	\sum_{K>0} \qty(\delta_{IK} - \ew{K} )
@@ -379,8 +378,8 @@ rewritten as follows:
 	+ \frac{1}{2} \Tr[\bGam{(I)} \bG \bGam{(I)}]
 	+ \E{c}{{\bw}}[\n{\bGam{\bw}}{}]
 	\\
-	& + \int d\br{} \fdv{\E{c}{\bw}[\n{\bGam{\bw}}{}]}{\n{}{}(\br{})}
-	\qty[ \n{\bGam{(I)}}{}(\br{}) - \n{\bGam{\bw}}{}(\br{}) ]
+	& + \int \fdv{\E{c}{\bw}[\n{\bGam{\bw}}{}]}{\n{}{}(\br{})} 
+	\qty[ \n{\bGam{(I)}}{}(\br{}) - \n{\bGam{\bw}}{}(\br{}) ] d\br{} 
 	\\
 	& + \sum_{K>0} \qty(\delta_{IK} - \ew{K})
 	\left. \pdv{\E{c}{\bw}[\n{}{}]}{\ew{K}}\right|_{\n{}{} = \n{\bGam{\bw}}{}}
@@ -389,7 +388,7 @@ rewritten as follows:
 \eeq
 where 
 \beq
-	\bh \equiv h_{\mu\nu} = \int d\br{} \AO{\mu}(\br{}) \qty[-\frac{1}{2} \nabla_{\br{}}^2 + \vne(\br{}) ]\AO{\nu}(\br{})
+	\bh \equiv h_{\mu\nu} = \int \AO{\mu}(\br{}) \qty[-\frac{1}{2} \nabla_{\br{}}^2 + \vne(\br{}) ]\AO{\nu}(\br{}) d\br{} 
 \eeq
 denote the one-electron integrals matrix.
 The individual Hx energy is obtained from the following trace
@@ -528,7 +527,7 @@ where $\bS \equiv \eS{\mu\nu} = \braket*{\AO{\mu}}{\AO{\nu}}$ is the metric and
 with
 \beq
 	\eh{\mu\nu}{\bw} 
-	= \eh{\mu\nu}{} + \int d\br{} \AO{\mu}(\br{}) \fdv{\E{c}{\bw}[\n{\bGam{\bw}}{}]}{\n{}{}(\br{})} \AO{\nu}(\br{}).
+	= \eh{\mu\nu}{} + \int \AO{\mu}(\br{}) \fdv{\E{c}{\bw}[\n{\bGam{\bw}}{}]}{\n{}{}(\br{})} \AO{\nu}(\br{}) d\br{}.
 \eeq
 
 %%%%%%%%%%%%%%%
@@ -673,7 +672,7 @@ according to Eq.~\eqref{eq:exact_ind_ener_rdm}.
 Turning to the density-functional ensemble correlation energy, the
 following eLDA will be employed:  
 \beq\label{eq:eLDA_corr_fun}
-	\E{c}{\bw}[\n{}{}] = \int d\br{} \n{}{}(\br{}) \e{c}{\bw}[\n{}{}(\br{})],
+	\E{c}{\bw}[\n{}{}] = \int \n{}{}(\br{}) \e{c}{\bw}[\n{}{}(\br{})] d\br{},
 \eeq
 where the correlation energy per particle is {\it weight-dependent}. Its
 construction from a finite uniform electron gas model is discussed
@@ -686,14 +685,14 @@ within eLDA:
 	\E{}{(I)} 
 	& \approx \Tr[\bGam{(I)} \bh] + \frac{1}{2} \Tr[\bGam{(I)} \bG \bGam{(I)}]
 	\\
-	& + \int d\br{} \e{c}{\bw}[\n{\bGam{\bw}}{}(\br{})] \n{\bGam{(I)}}{}(\br{})
+	& + \int \e{c}{\bw}[\n{\bGam{\bw}}{}(\br{})] \n{\bGam{(I)}}{}(\br{}) d\br{}
 	\\
 	&
-	+ \int d\br{} \n{\bGam{\bw}}{}(\br{}) \qty[ \n{\bGam{(I)}}{}(\br{}) - \n{\bGam{\bw}}{}(\br{}) ]
-	\left. \pdv{\e{c}{{\bw}}(\n{}{})}{\n{}{}} \right|_{\n{}{} = \n{\bGam{\bw}}{}(\br{})}
+	+ \int \n{\bGam{\bw}}{}(\br{}) \qty[ \n{\bGam{(I)}}{}(\br{}) - \n{\bGam{\bw}}{}(\br{}) ]
+	\left. \pdv{\e{c}{{\bw}}(\n{}{})}{\n{}{}} \right|_{\n{}{} = \n{\bGam{\bw}}{}(\br{})} d\br{}
 	\\
-	& + \int d\br{} \sum_{K>0} \qty(\delta_{IK} - \ew{K} ) \n{\bGam{\bw}}{}(\br{})
-	\left. \pdv{\e{c}{\bw}(\n{}{})}{\ew{K}} \right|_{\n{}{}=\n{\bGam{\bw}}{}(\br{})}.
+	& + \int \sum_{K>0} \qty(\delta_{IK} - \ew{K} ) \n{\bGam{\bw}}{}(\br{})
+	\left. \pdv{\e{c}{\bw}(\n{}{})}{\ew{K}} \right|_{\n{}{}=\n{\bGam{\bw}}{}(\br{})} d\br{}.
 \end{split}
 \eeq