donw 1st step cleaning

Manuscript/eDFT.tex

@@ -38,6 +38,7 @@
 \newcommand{\hT}{\Hat{T}}
 \newcommand{\vne}{v_\text{ne}}
 \newcommand{\hWee}{\Hat{W}_\text{ee}}
+\newcommand{\WHF}{W_\text{HF}}
 
 % functionals, potentials, densities, etc
 \newcommand{\eps}{\epsilon}
@@ -262,7 +263,7 @@ 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 $\xi_0=1-\sum_{K>0}\xi_K$
+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}},
@@ -317,12 +318,12 @@ ensemble KS spinorbitals [from which the latter are constructed] in an atomic or
 then the density matrix elements obtained from the   
 determinant $\Det{(K)}$ can be expressed as follows in the AO basis:
 \beq
-	\bmg^{(K)} \equiv \eGam{\mu\nu}{(K)} = \sum_{\SO{p}{} \in (K)} \cMO{\mu p}{} \cMO{\nu p}{},
+	\bGam{(K)} \equiv \eGam{\mu\nu}{(K)} = \sum_{\SO{p}{} \in (K)} \cMO{\mu p}{} \cMO{\nu p}{},
 \eeq
 where the summation runs over the spinorbitals that are occupied in $\Det{(K)}$. 
 Note that the density of the $K$th KS state reads
 \beq
-	\n{\bmg^{(K)}}{}(\br{}) = \sum_{\mu\nu} \AO{\mu}(\br{}) \eGam{\mu\nu}{(K)} \AO{\nu}(\br{}).
+	\n{\bGam{(K)}}{}(\br{}) = \sum_{\mu\nu} \AO{\mu}(\br{}) \eGam{\mu\nu}{(K)} \AO{\nu}(\br{}).
 \eeq 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 % Manu's derivation %%%
@@ -347,29 +348,29 @@ p}}c^\sigma_{{\nu p}}
 We can then construct the ensemble density matrix
 and the ensemble density as follows:
 \beq
-	\bmg^{\bw} 
-	= \sum_{K\geq 0} \ew{K} \bmg^{(K)}
+	\bGam{\bw} 
+	= \sum_{K\geq 0} \ew{K} \bGam{(K)}
 	\equiv \eGam{\mu\nu}{\bw}
 	= \sum_{K\geq 0} \ew{K} \eGam{\mu\nu}{(K)}
 \eeq
 and
 \beq
-	\n{\bmg^\bw}{}(\br{}) = \sum_{\mu\nu} \AO{\mu}(\br{}) \eGam{\mu\nu}{\bw} \AO{\nu}(\br{}),
+	\n{\bGam{\bw}}{}(\br{}) = \sum_{\mu\nu} \AO{\mu}(\br{}) \eGam{\mu\nu}{\bw} \AO{\nu}(\br{}),
 \eeq
 respectively. The exact energy level expression in Eq.~\eqref{eq:exact_ener_level_dets} can be
 rewritten as follows:
 \beq\label{eq:exact_ind_ener_rdm}
 \begin{split}
 	\E{}{(I)} 
-	& =\Tr[\bmg^{(I)} \bh]
-	+ \frac{1}{2} \Tr[\bmg^{(I)} \bG \bmg^{(I)}]
-	+ \E{c}{{\bw}}[\n{\bmg^{\bw}}{}]
+	& =\Tr[\bGam{(I)} \bh]
+	+ \frac{1}{2} \Tr[\bGam{(I)} \bG \bGam{(I)}]
+	+ \E{c}{{\bw}}[\n{\bGam{\bw}}{}]
 	\\
-	& + \int d\br{} \fdv{\E{c}{\bw}[\n{\bmg^{\bw}}{}]}{\n{}{}(\br{})}
-	\qty[ \n{\bmg^{(I)}}{}(\br{}) - \n{\bmg^{\bw}}{}(\br{}) ]
+	& + \int d\br{} \fdv{\E{c}{\bw}[\n{\bGam{\bw}}{}]}{\n{}{}(\br{})}
+	\qty[ \n{\bGam{(I)}}{}(\br{}) - \n{\bGam{\bw}}{}(\br{}) ]
 	\\
 	& + \sum_{K>0} \qty(\delta_{IK} - \ew{K})
-	\left. \pdv{\E{c}{\bw}[\n{}{}]}{\ew{K}}\right|_{\n{}{} = \n{\bmg^{\bw}}{}}
+	\left. \pdv{\E{c}{\bw}[\n{}{}]}{\ew{K}}\right|_{\n{}{} = \n{\bGam{\bw}}{}}
 	,
 \end{split}
 \eeq
@@ -380,20 +381,18 @@ where
 denote the one-electron integrals matrix.
 The individual Hx energy is obtained from the following trace
 \beq
-	\Tr(\bmg^{(K)} \bG \bmg^{(L)}) 
-	= \sum_{\mu\nu\lambda\omega}\sum_{\sigma=\alpha,\beta}\sum_{\tau=\alpha,\beta}G_{\mu\nu\lambda\omega}^{\sigma\tau}
-\eGam{\mu\nu}{(K)\sigma} \eGam{\lambda\omega}{(L)\tau}
-\nonumber\\
+	\Tr[\bGam{(K)} \bG \bGam{(L)}]
+	= \sum_{\mu\nu\la\si} \eGam{\mu\nu}{(K)} \eG{\mu\nu\la\si} \eGam{\la\si}{(L)},
 \eeq
 where the two-electron Coulomb-exchange integrals read
 \beq
-	G_{\mu\nu\lambda\omega} = 
-	\dbERI{\mu\nu}{\la\si} 
+	G_{\mu\nu\la\omega} 
+	= \dbERI{\mu\nu}{\la\si} 
 	= \ERI{\mu\nu}{\la\si} - \ERI{\mu\si}{\la\nu},
 \eeq
 with
 \beq
-\ERI{\mu\nu}{\la\si} = \iint \frac{\AO{\mu}(\br{1}) \AO{\nu}(\br{1}) \AO{\la}(\br{2}) \AO{\si}(\br{2})}{\abs{\br{1} - \br{2}}} d\br{1} d\br{2}.
+	\ERI{\mu\nu}{\la\si} = \iint \frac{\AO{\mu}(\br{1}) \AO{\nu}(\br{1}) \AO{\la}(\br{2}) \AO{\si}(\br{2})}{\abs{\br{1} - \br{2}}} d\br{1} d\br{2}.
 \eeq
 %Note that, in Sec.~\ref{sec:results}, the theory is applied to (1D) spin
 %polarized systems in which $\eGam{\mu\nu}{(K)\beta}=0$ and
@@ -493,18 +492,15 @@ As Hartree and exchange energies cannot be separated in the
 one-dimension systems considered in the rest of this work, we will substitute the Hartree--Fock
 density-matrix-functional interaction energy,
 \beq\label{eq:eHF-dens_mat_func}
-W_{\rm
-HF}\left[{\bmg}\right]=\frac{1}{2} \Tr(\bmg \, \bG \, \bmg), 
+	\WHF[\bGam{}] = \frac{1}{2} \Tr[\bGam{} \bG \bGam{}], 
 \eeq
 for the Hx density-functional energy in the variational energy
 expression of Eq.~\eqref{eq:var_ener_gokdft}:  
 \beq
-	{\bmg}^{\bw} 
+	\bGam{\bw} 
 	\approx \argmin_{\bgam{\bw}} 
 	\qty{ 
-	\Tr[\bgam{\bw} \bh ] 
-	+ W_{\rm HF}[ \bgam{\bw}]
-	+ \E{c}{\bw}[\n{\bgam{\bw}}{}]
+	\Tr[\bgam{\bw} \bh ] + \WHF[ \bgam{\bw}] + \E{c}{\bw}[\n{\bgam{\bw}}{}]
 	}.
 \eeq
 The minimizing ensemble density matrix fulfills the following
@@ -514,12 +510,12 @@ stationarity condition
 \eeq
 where $\bS \equiv \eS{\mu\nu} = \braket*{\AO{\mu}}{\AO{\nu}}$ is the metric and the ensemble Fock-like matrix reads
 \beq
-	\eF{\mu\nu}{\bw} = \eh{\mu\nu}{\bw} + \sum_{\lambda\si} \eG{\mu\nu\la\si} \eGam{\la\si}{\bw}
+	\eF{\mu\nu}{\bw} = \eh{\mu\nu}{\bw} + \sum_{\la\si} \eG{\mu\nu\la\si} \eGam{\la\si}{\bw}
 \eeq 
 with
 \beq
 	\eh{\mu\nu}{\bw} 
-	= \eh{\mu\nu}{} + \int d\br{} \AO{\mu}(\br{}) \fdv{\E{c}{\bw}[\n{\bmg^\bw}{}]}{\n{}{}(\br{})} \AO{\nu}(\br{}).
+	= \eh{\mu\nu}{} + \int d\br{} \AO{\mu}(\br{}) \fdv{\E{c}{\bw}[\n{\bGam{\bw}}{}]}{\n{}{}(\br{})} \AO{\nu}(\br{}).
 \eeq
 
 %%%%%%%%%%%%%%%
@@ -649,19 +645,17 @@ optimized from a non-local exchange potential [rather than a local one,
 as expected from Eq.~\eqref{eq:var_ener_gokdft}] is applicable to real
 (three-dimension) systems. As readily seen from
 Eq.~\eqref{eq:eHF-dens_mat_func}, \textit{ghost interactions}~\cite{Gidopoulos_2002, Pastorczak_2014, Alam_2016, Alam_2017, Gould_2017}
-and curvature~\cite{} will be
-introduced in the Hx energy:
+and curvature~\cite{} will be introduced in the Hx energy:
 \beq
 \begin{split}
-	W_{\rm HF}[{\bmg}^\bw] 
-	& = \frac{1}{2}\sum_{K\geq 0} \ew{K}^2
-	\Tr(\bmg^{(K)} \bG \bmg^{(K)})
+	\WHF[\bGam{\bw}] 
+	& = \frac{1}{2} \sum_{K\geq 0} \ew{K}^2 \Tr[\bGam{(K)} \bG \bGam{(K)}]
 	\\
-	& + \sum_{L>K\geq 0} \ew{K} \ew{L}\Tr(\bmg^{(K)} \bG \bmg^{(L)}).
+	& + \sum_{L>K\geq 0} \ew{K} \ew{L}\Tr[\bGam{(K)} \bG \bGam{(L)}].
 \end{split}
 \eeq
 These errors will be removed when computing individual energies
-according to Eq.~\eqref{eq:exact_ind_ener_rdm}.\\    
+according to Eq.~\eqref{eq:exact_ind_ener_rdm}.    
 
 Turning to the density-functional ensemble correlation energy, the
 following eLDA will be employed:  
@@ -677,17 +671,16 @@ within eLDA:
 \beq
 \begin{split}
 	\E{}{(I)} 
-	& \approx \Tr[\bmg^{(I)} \bh]
-	+ \frac{1}{2} \Tr(\bmg^{(I)} \bG \bmg^{(I)})
+	& \approx \Tr[\bGam{(I)} \bh] + \frac{1}{2} \Tr[\bGam{(I)} \bG \bGam{(I)}]
 	\\
-	& + \int d\br{} \e{c}{\bw}(\n{\bmg^{\bw}}{}(\br{})) \n{\bmg^{(I)}}{}(\br{})
+	& + \int d\br{} \e{c}{\bw}[\n{\bGam{\bw}}{}(\br{})] \n{\bGam{(I)}}{}(\br{})
 	\\
 	&
-	+ \int d\br{} \n{\bmg^{\bw}}{}(\br{}) \qty[ \n{\bmg^{(I)}}{}(\br{}) - \n{\bmg^{\bw}}{}(\br{}) ]
-	\left. \pdv{\e{c}{{\bw}}(\n{}{})}{\n{}{}} \right|_{\n{}{} = n{\bmg^{\bw}}{}(\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 d\br{} \sum_{K>0} \qty(\delta_{IK} - \ew{K} ) \n{\bmg^{\bw}}{}(\br{})
-	\left. \pdv{\e{c}{\bw}(\n{}{})}{\ew{K}} \right|_{\n{}{}=\n{\bmg^{\bw}}{}(\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{})}.
 \end{split}
 \eeq