Manu: saving work

Manuscript/eDFT.tex

@@ -180,7 +180,7 @@ KS scheme, a HF-like Hartree-exchange energy is employed. This
 formulation is in principle exact and applicable to higher dimensions.
 Let us start from the analog for ensembles of Levy's universal
 functional,   
-\beq
+\beq\label{eq:ens_LL_func}
 F^{\bw}[n]&=&
 \underset{\hat{\gamma}^{{\bw}}\rightarrow n}{\rm min}\left\{{\rm
 Tr}\left[\hat{\gamma}^{{\bw}}\left(\hat{T}+\hat{W}_{\rm
@@ -188,17 +188,19 @@ ee}\right)\right]\right\}
 \eeq
 where ${\rm
 Tr}$ denotes the trace. The minimization over trial ensemble density matrix operators
-$\hat{\gamma}^{{\bw}}=\sum^M_{K=0}w^{(K)}\vert\Psi^{(K)}\rangle\langle\Psi^{(K)}\vert$
+$\hat{\gamma}^{{\bw}}=\sum_{K\geq0}w^{(K)}\vert\Psi^{(K)}\rangle\langle\Psi^{(K)}\vert$
 is performed under the following density constraint:
 \beq
 {\rm
-Tr}\left[\hat{\gamma}^{{\bw}}\hat{n}(\br)\right]=\sum^M_{K=0}w^{(K)}n_{\Psi^{(K)}}(\br)=n(\br),
+Tr}\left[\hat{\gamma}^{{\bw}}\hat{n}(\br)\right]=\sum_{K\geq0}w^{(K)}n_{\Psi^{(K)}}(\br)=n(\br),
 \eeq
 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
-$w^{(0)}=1-\sum^M_{K>0}w^{(K)}\geq 0$. 
+$w^{(0)}=1-\sum_{K>0}w^{(K)}\geq 0$.\\ 
+
+Ground-state theory:
 
 \beq
 F^{\bw=0}[n]=F[n]=\underset{\Psi\rightarrow n}{\rm min}
@@ -206,7 +208,8 @@ F^{\bw=0}[n]=F[n]=\underset{\Psi\rightarrow n}{\rm min}
 ee}\ket{\Psi}
 \eeq
 
-\beq
+
+\beq\label{eq:generalized_KS-DFT_decomp}
 F[n]&=&
 \underset{\Phi\rightarrow n}{\rm min}
 \bra{\Phi}\hat{T}+\hat{W}_{\rm
@@ -228,76 +231,101 @@ W_{\rm
 HF}\left[{\bmg}\right]\equiv\frac{1}{2} \Tr(\bmg \, \bG \, \bmg) 
 \eeq
 is the conventional density-matrix functional HF Hartree-exchange
-energy. By analogy with Eq.~(\ref{}),   
-
+energy. By analogy with Eq.~(\ref{eq:generalized_KS-DFT_decomp}), we
+decompose the ensemble universal functional as follows:    
 \beq
-F^{\bw}_{\rm HF}[n]&=&
+F^{\bw}[n]&=&
 \underset{\hat{\Gamma}^{{\bw}}\rightarrow n}{\rm min}\left\{{\rm
-Tr}\left[\hat{\Gamma}^{{\bw}}\hat{T}\right]+W_{\rm
+Tr}\left[\hat{\Gamma}^{{\bw}}\hat{T}\right]
++W_{\rm
 HF}\left[{\bmg}^{\bw}\right]\right\}
+ +\overline{E}^{{\bw}}_{\rm
+Hxc}[n]
 \nonumber\\
-&=&{\rm
-Tr}\left[\hat{\Gamma}^{{\bw}}[n]\hat{T}\right]+W_{\rm
-HF}\left[{\bmg}^{\bw}[n]\right]
+&=&
+\underset{{\bmg}^{{\bw}}\rightarrow n}{\rm min}
+\left\{
+{\rm Tr}
+\left[{\bmg}^{\bw}{\bm t}\right]
++W_{\rm HF}\left[{\bmg}^{\bw}\right]
+\right\}+
+\overline{E}^{\bw}_{\rm Hxc}[n]
 \eeq
-where
-$\hat{\Gamma}^{{\bw}}=\sum^M_{K=0}w^{(K)}\vert\Phi^{(K)}\rangle\langle\Phi^{(K)}\vert=\sum^M_{K=0}w^{(K)}\hat{\Gamma}^{(K)}$ is an ensemble density matrix operator constructed
-from Slater determinants, the ensemble 1RDM elements are $\Gamma_{pq}^{\bw}={\rm
-Tr}\left[\hat{\Gamma}^{{\bw}}\hat{a}^\dagger_p\hat{a}_q\right]$,
-and $W_{\rm
-HF}\left[{\bmg}\right]=\frac{1}{2}\sum_{pqrs}\langle \varphi_p\varphi_q\vert\vert
-\varphi_r\varphi_s\rangle
-%\times
-\Gamma_{pr}\Gamma_{qs}$.\\
-
-In-principle-exact decomposition:
-
+where the minimization in Eq.~(\ref{eq:ens_LL_func}) has been restricted
+to density matrix operators 
 \beq
-F^{\bw}[n]= F^{\bw}_{\rm HF}[n]+\overline{E}^{{\bw}}_{\rm
-Hx}[n]+\overline{E}^{{\bw}}_{\rm c}[n]
-\eeq
+\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 $W_{\rm
 HF}\left[{\bmg}^{\bw}[n]\right]$:
 \beq
 \overline{E}^{{\bw}}_{\rm
-Hx}[n]=\sum^M_{K=0}w^{(K)}W_{\rm
+Hx}[n]&=&\sum_{K\geq0}w^{(K)}W_{\rm
 HF}\left[{\bmg}^{(K)}[n]\right]
 -W_{\rm
-HF}\left[{\bmg}^{\bw}[n]\right],
-\eeq
-which gives in the canonical orbital basis
-\beq
-&&\overline{E}^{{\bw}}_{\rm
-Hx}[n]=
-\dfrac{1}{2}\sum_{pq}
-\langle \varphi^{{\bw}}_p[n]\varphi^{{\bw}}_q[n]\vert\vert
-\varphi^{{\bw}}_p[n]\varphi^{{\bw}}_q[n]\rangle
+HF}\left[{\bmg}^{\bw}[n]\right].
 \nonumber\\
-&&\times\left[\sum^M_{K=0}w^{(K)}\nu^{(K)}_p \left(\nu^{(K)}_q
--\sum^M_{L=0}w^{(L)} \nu^{(L)}_q\right)\right]
-.\eeq
-\manu{I would guess that, in a uniform system, the GOK-DFT and our
-canonical orbitals are the same. This is nice since we can construct
-in a clean way density-functional approximations for both $\overline{E}^{{\bw}}_{\rm
-Hx}[n]$ and $E^{{\bw}}_{\rm c}[n]$ functionals. Am I right ?}
+&=&
+{\rm
+Tr}\left[\hat{\Gamma}^{{\bw}}[n]\hat{W}_{\rm ee}\right]-W_{\rm
+HF}\left[{\bmg}^{\bw}[n]\right]
+\eeq
+
+Note that $\overline{E}^{{\bw}=0}_{\rm
+Hx}[n]=0$.\\
+
+Ensemble correlation energy:
 
-Variational expression for the ensemble energy:
 \beq
-E^{{\bw}}=\underset{\hat{\Gamma}^{{\bw}}}{\rm min}\Big\{
-&&{\rm
-Tr}\left[\hat{\Gamma}^{{\bw}}\hat{T}\right]+W_{\rm
+\overline{E}^{{\bw}}_{\rm
+c}[n]&=&
+{\rm
+Tr}\left[\hat{\gamma}^{{\bw}}[n]\left(\hat{T}+\hat{W}_{\rm
+ee}\right)\right]
+\nonumber\\
+&&-
+{\rm
+Tr}\left[\hat{\Gamma}^{{\bw}}[n]\left(\hat{T}+\hat{W}_{\rm
+ee}\right)\right]
+\eeq
+
+Variational expression of the ensemble energy:
+\beq
+E^{{\bw}}=\underset{{\bmg}^{{\bw}}}{\rm min}\Big\{
+{\rm
+Tr}\left[{\bmg}^{{\bw}}{\bm h}\right]+W_{\rm
 HF}\left[{\bmg}^{\bw}\right]
 +
 \overline{E}^{{\bw}}_{\rm
-Hxc}\left[n_{\hat{\Gamma}^{{\bw}}}\right]
+Hxc}\left[n_{{\bmg}^{{\bw}}}\right]
 %+E^{{\bw}}_{\rm c}\left[n_{\hat{\Gamma}^{{\bw}}}\right]
-\nonumber\\
-&&
-+\int d{\br}\;v_{\rm ext}({\bfr})n_{\hat{\Gamma}^{{\bw}}}({\bfr})
 \Big\}
 \eeq
+
+For $K>0$
+
+\alert{
+\beq
+\dfrac{\partial E^{{\bw}}}{\partial w^{(K)}}&=&{\rm
+Tr}\left[{\bmg}^{(K)}{\bm h}\right]-{\rm
+Tr}\left[{\bmg}^{(0)}{\bm h}\right]
+\nonumber\\
+&&+\Tr(\bmg^{(K)} \, \bG \, \bmg^{\bw})
+-\Tr(\bmg^{(0)} \, \bG \, \bmg^{\bw})
++...
+\eeq
+\beq
+E^{(I)}&=&E^{{\bw}}+\sum_{K>0}\left(\delta_{IK}-w^{(K)}\right)\dfrac{\partial E^{{\bw}}}{\partial w^{(K)}}
+\nonumber\\
+&=&
+...+\dfrac{1}{2}\Tr(\bmg^{\bw} \, \bG \, \bmg^{\bw})
++...
+\eeq
+}
  
 Note that, if we use orbital rotations, the gradient of the DFT energy
 contributions can be expressed as follows,