Manu: saving work

Manuscript/eDFT.tex

@@ -198,17 +198,17 @@ where $\hat{n}(\br)$ is the density operator, $n_{\Psi}$ denotes the
 density of wavefunction $\Psi$, and
 $\bw\equiv\left(w^{(1)},w^{(2)},\ldots\right)$ is the collection of
 (decreasing) ensemble weights assigned to the excited states. Note that
-$w^{(0)}=1-\sum_{K>0}w^{(K)}\geq 0$.\\ 
-
-Ground-state theory:
-
+$w^{(0)}=1-\sum_{K>0}w^{(K)}\geq 0$. When $\bw=0$, the
+conventional ground-state universal functional is recovered,
 \beq
 F^{\bw=0}[n]=F[n]=\underset{\Psi\rightarrow n}{\rm min}
 \bra{\Psi}\hat{T}+\hat{W}_{\rm
-ee}\ket{\Psi}
+ee}\ket{\Psi},
 \eeq
-
-
+where the ensemble reduces to a single wavefunction. In the latter case,
+the HF-like expression (or a fraction of it, as usually done in
+practical calculations) for the Hx energy can be introduced rigorously
+into DFT by considering the following decomposition,   
 \beq\label{eq:generalized_KS-DFT_decomp}
 F[n]&=&
 \underset{\Phi\rightarrow n}{\rm min}
@@ -216,15 +216,17 @@ F[n]&=&
 ee}\ket{\Phi}+\overline{E}_{\rm c}[n]
 \nonumber\\
 &=&
-\underset{\Phi\rightarrow n}{\rm min}
+\underset{\bmg^\Phi\rightarrow n}{\rm min}
 \left\{{\rm
 Tr}\left[\bmg^\Phi{\bm t}\right]+W_{\rm
 HF}\left[{\bmg}^{\Phi}\right]\right\}+
 \overline{E}_{\rm c}[n]
+,
 \eeq
 where ${\bm t}$ is the matrix representation of the one-electron kinetic
 energy operator, $\bmg^\Phi$ is the one-electron reduced density
-matrix (just referred to as density matrix in the following) of $\Phi$,
+matrix (just referred to as density matrix in the following) obtained
+from $\Phi$,
 and     
 \beq
 W_{\rm
@@ -233,7 +235,7 @@ HF}\left[{\bmg}\right]\equiv\frac{1}{2} \Tr(\bmg \, \bG \, \bmg)
 is the conventional density-matrix functional HF Hartree-exchange
 energy. By analogy with Eq.~(\ref{eq:generalized_KS-DFT_decomp}), we
 decompose the ensemble universal functional as follows:    
-\beq
+\beq\label{eq:generalized_F_w}
 F^{\bw}[n]&=&
 \underset{\hat{\Gamma}^{{\bw}}\rightarrow n}{\rm min}\left\{{\rm
 Tr}\left[\hat{\Gamma}^{{\bw}}\hat{T}\right]
@@ -249,7 +251,7 @@ Hxc}[n]
 \left[{\bmg}^{\bw}{\bm t}\right]
 +W_{\rm HF}\left[{\bmg}^{\bw}\right]
 \right\}+
-\overline{E}^{\bw}_{\rm Hxc}[n]
+\overline{E}^{\bw}_{\rm Hxc}[n],
 \eeq
 where the minimization in Eq.~(\ref{eq:ens_LL_func}) has been restricted
 to density matrix operators 
@@ -257,42 +259,87 @@ to density matrix operators
 \hat{\Gamma}^{{\bw}}=\sum_{K\geq 0}w^{(K)}\vert\Phi^{(K)}\rangle\langle\Phi^{(K)}\vert=\sum_{K\geq 0}w^{(K)}\hat{\Gamma}^{(K)}
 \eeq 
 that are constructed from single Slater
-determinants $\Phi^{(K)}$.
-
-The complementary ensemble Hx energy removes the ghost-interaction
-errors introduced in $W_{\rm
-HF}\left[{\bmg}^{\bw}[n]\right]$:
+determinants $\Phi^{(K)}$. Note that the density matrices
+${\bmg}^{(K)}={\bmg}^{\Phi^{(K)}}$ are idempotent and diagonal in the
+same spin-orbital basis). On the other hand, the ensemble
+density matrix ${\bmg}^{{\bw}}=\sum_{K\geq 0}w^{(K)}{\bmg}^{(K)}$, which is a convex combination of the ${\bmg}^{(K)}$
+matrices, is {\it not} idempotent, unless ${\bw}=0$. Indeed,
 \beq
-\overline{E}^{{\bw}}_{\rm
-Hx}[n]&=&\sum_{K\geq0}w^{(K)}W_{\rm
-HF}\left[{\bmg}^{(K)}[n]\right]
--W_{\rm
-HF}\left[{\bmg}^{\bw}[n]\right].
+\left[{\bmg}^{{\bw}}\right]^2&=&\sum_{K,L\geq
+0}w^{(K)}w^{(L)}{\bmg}^{(K)}{\bmg}^{(L)}
+\nonumber\\
+&=&\sum_{K\geq
+0}\left(w^{(K)}\right)^2{\bmg}^{(K)}+\sum_{K\neq L\geq
+0}w^{(K)}w^{(L)}{\bmg}^{(K)}{\bmg}^{(L)}
 \nonumber\\
 &=&
+{\bmg}^{{\bw}}+\sum_{K,L\geq
+0}w^{(K)}\left(w^{(L)}-\delta_{KL}\right){\bmg}^{(K)}{\bmg}^{(L)}
+\nonumber\\
+&=&{\bmg}^{{\bw}}+w^{(0)}{\bmg}^{(0)}\times\sum_{K>0}w^{(K)}\left(2{\bmg}^{(K)}-1\right)
+\nonumber\\
+&&+\sum_{K, L >0
+}w^{(K)}\left(w^{(L)}-\delta_{KL}\right){\bmg}^{(K)}{\bmg}^{(L)}
+\nonumber\\
+&\neq&{\bmg}^{{\bw}}
+.
+\eeq
+This is of course expected since using an ensemble is, in this context,
+analogous to assigning
+fractional occupation numbers (which are determined from the ensemble
+weights) to the KS orbitals.\\ 
+
+Another issue with the use of
+ensembles in DFT is the introduction of spurious ghost-interaction errors
+(i.e. unphysical interactions between different states) into the
+ensemble energy when inserting ${\bmg}^{{\bw}}$ into the HF
+density-matrix functional Hx energy $W_{\rm
+HF}\left[\bmg\right]$. This type of errors is specific to ensembles
+which explains why, in constrast to ground-state DFT [see
+Eq.~(\ref{eq:generalized_KS-DFT_decomp})], a complementary ensemble Hx
+energy is needed to recover a ghost-interaction-free energy: 
+\beq
+\overline{E}^{{\bw}}_{\rm
+Hx}[n]&=&
 {\rm
 Tr}\left[\hat{\Gamma}^{{\bw}}[n]\hat{W}_{\rm ee}\right]-W_{\rm
 HF}\left[{\bmg}^{\bw}[n]\right]
+\nonumber\\
+&=&
+\sum_{K\geq0}w^{(K)}W_{\rm
+HF}\left[{\bmg}^{(K)}[n]\right]
+-W_{\rm
+HF}\left[{\bmg}^{\bw}[n]\right],
 \eeq
-
-Note that $\overline{E}^{{\bw}=0}_{\rm
-Hx}[n]=0$.\\
-
-Ensemble correlation energy:
-
+where ${\bmg}^{\bw}[n]$ is the minimizing ensemble density matrix in
+Eq.~(\ref{eq:generalized_F_w}) and, by construction, $\overline{E}^{{\bw}=0}_{\rm
+Hx}[n]=0$. Consequently, the ensemble correlation functional can be
+expressed as follows [see Eq.~(\ref{eq:generalized_F_w})]:
 \beq
 \overline{E}^{{\bw}}_{\rm
 c}[n]&=&
-{\rm
+\overline{E}^{\bw}_{\rm Hxc}[n]-\overline{E}^{{\bw}}_{\rm
+Hx}[n]
+\nonumber\\
+&=&{\rm
 Tr}\left[\hat{\gamma}^{{\bw}}[n]\left(\hat{T}+\hat{W}_{\rm
 ee}\right)\right]
-\nonumber\\
-&&-
+%\nonumber\\
+%&&
+-
 {\rm
 Tr}\left[\hat{\Gamma}^{{\bw}}[n]\left(\hat{T}+\hat{W}_{\rm
 ee}\right)\right]
+\nonumber\\
+&=&
+\sum_{K\geq 0}w^{(K)}\Bigg(\bra{\Psi^{(K)}[n]}\hat{T}+\hat{W}_{\rm
+ee}\ket{\Psi^{(K)}[n]}
+\nonumber\\
+&&-\bra{\Phi^{(K)}[n]}\hat{T}+\hat{W}_{\rm
+ee}\ket{\Phi^{(K)}[n]}\Bigg)
 \eeq
-
+where $\hat{\gamma}^{{\bw}}[n]$ and $\hat{\Gamma}^{{\bw}}[n]$ are the minimizing density matrix
+operators in Eqs.~(\ref{eq:ens_LL_func}) and (\ref{eq:generalized_KS-DFT_decomp}), respectively.
 Variational expression of the ensemble energy:
 \beq
 E^{{\bw}}=\underset{{\bmg}^{{\bw}}}{\rm min}\Big\{