Manu: saving work
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@ -180,7 +180,7 @@ KS scheme, a HF-like Hartree-exchange energy is employed. This
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formulation is in principle exact and applicable to higher dimensions.
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formulation is in principle exact and applicable to higher dimensions.
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Let us start from the analog for ensembles of Levy's universal
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Let us start from the analog for ensembles of Levy's universal
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functional,
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functional,
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\beq
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\beq\label{eq:ens_LL_func}
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F^{\bw}[n]&=&
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F^{\bw}[n]&=&
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\underset{\hat{\gamma}^{{\bw}}\rightarrow n}{\rm min}\left\{{\rm
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\underset{\hat{\gamma}^{{\bw}}\rightarrow n}{\rm min}\left\{{\rm
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Tr}\left[\hat{\gamma}^{{\bw}}\left(\hat{T}+\hat{W}_{\rm
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Tr}\left[\hat{\gamma}^{{\bw}}\left(\hat{T}+\hat{W}_{\rm
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@ -188,17 +188,19 @@ ee}\right)\right]\right\}
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\eeq
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\eeq
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where ${\rm
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where ${\rm
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Tr}$ denotes the trace. The minimization over trial ensemble density matrix operators
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Tr}$ denotes the trace. The minimization over trial ensemble density matrix operators
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$\hat{\gamma}^{{\bw}}=\sum^M_{K=0}w^{(K)}\vert\Psi^{(K)}\rangle\langle\Psi^{(K)}\vert$
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$\hat{\gamma}^{{\bw}}=\sum_{K\geq0}w^{(K)}\vert\Psi^{(K)}\rangle\langle\Psi^{(K)}\vert$
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is performed under the following density constraint:
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is performed under the following density constraint:
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\beq
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\beq
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{\rm
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{\rm
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Tr}\left[\hat{\gamma}^{{\bw}}\hat{n}(\br)\right]=\sum^M_{K=0}w^{(K)}n_{\Psi^{(K)}}(\br)=n(\br),
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Tr}\left[\hat{\gamma}^{{\bw}}\hat{n}(\br)\right]=\sum_{K\geq0}w^{(K)}n_{\Psi^{(K)}}(\br)=n(\br),
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\eeq
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\eeq
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where $\hat{n}(\br)$ is the density operator, $n_{\Psi}$ denotes the
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where $\hat{n}(\br)$ is the density operator, $n_{\Psi}$ denotes the
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density of wavefunction $\Psi$, and
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density of wavefunction $\Psi$, and
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$\bw\equiv\left(w^{(1)},w^{(2)},\ldots\right)$ is the collection of
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$\bw\equiv\left(w^{(1)},w^{(2)},\ldots\right)$ is the collection of
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(decreasing) ensemble weights assigned to the excited states. Note that
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(decreasing) ensemble weights assigned to the excited states. Note that
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$w^{(0)}=1-\sum^M_{K>0}w^{(K)}\geq 0$.
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$w^{(0)}=1-\sum_{K>0}w^{(K)}\geq 0$.\\
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Ground-state theory:
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\beq
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\beq
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F^{\bw=0}[n]=F[n]=\underset{\Psi\rightarrow n}{\rm min}
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F^{\bw=0}[n]=F[n]=\underset{\Psi\rightarrow n}{\rm min}
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@ -206,7 +208,8 @@ F^{\bw=0}[n]=F[n]=\underset{\Psi\rightarrow n}{\rm min}
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ee}\ket{\Psi}
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ee}\ket{\Psi}
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\eeq
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\eeq
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\beq
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\beq\label{eq:generalized_KS-DFT_decomp}
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F[n]&=&
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F[n]&=&
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\underset{\Phi\rightarrow n}{\rm min}
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\underset{\Phi\rightarrow n}{\rm min}
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\bra{\Phi}\hat{T}+\hat{W}_{\rm
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\bra{\Phi}\hat{T}+\hat{W}_{\rm
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@ -228,77 +231,102 @@ W_{\rm
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HF}\left[{\bmg}\right]\equiv\frac{1}{2} \Tr(\bmg \, \bG \, \bmg)
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HF}\left[{\bmg}\right]\equiv\frac{1}{2} \Tr(\bmg \, \bG \, \bmg)
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\eeq
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\eeq
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is the conventional density-matrix functional HF Hartree-exchange
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is the conventional density-matrix functional HF Hartree-exchange
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energy. By analogy with Eq.~(\ref{}),
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energy. By analogy with Eq.~(\ref{eq:generalized_KS-DFT_decomp}), we
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decompose the ensemble universal functional as follows:
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\beq
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\beq
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F^{\bw}_{\rm HF}[n]&=&
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F^{\bw}[n]&=&
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\underset{\hat{\Gamma}^{{\bw}}\rightarrow n}{\rm min}\left\{{\rm
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\underset{\hat{\Gamma}^{{\bw}}\rightarrow n}{\rm min}\left\{{\rm
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Tr}\left[\hat{\Gamma}^{{\bw}}\hat{T}\right]+W_{\rm
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Tr}\left[\hat{\Gamma}^{{\bw}}\hat{T}\right]
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+W_{\rm
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HF}\left[{\bmg}^{\bw}\right]\right\}
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HF}\left[{\bmg}^{\bw}\right]\right\}
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+\overline{E}^{{\bw}}_{\rm
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Hxc}[n]
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\nonumber\\
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\nonumber\\
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&=&{\rm
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&=&
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Tr}\left[\hat{\Gamma}^{{\bw}}[n]\hat{T}\right]+W_{\rm
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\underset{{\bmg}^{{\bw}}\rightarrow n}{\rm min}
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HF}\left[{\bmg}^{\bw}[n]\right]
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\left\{
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{\rm Tr}
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\left[{\bmg}^{\bw}{\bm t}\right]
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+W_{\rm HF}\left[{\bmg}^{\bw}\right]
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\right\}+
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\overline{E}^{\bw}_{\rm Hxc}[n]
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\eeq
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\eeq
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where
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where the minimization in Eq.~(\ref{eq:ens_LL_func}) has been restricted
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$\hat{\Gamma}^{{\bw}}=\sum^M_{K=0}w^{(K)}\vert\Phi^{(K)}\rangle\langle\Phi^{(K)}\vert=\sum^M_{K=0}w^{(K)}\hat{\Gamma}^{(K)}$ is an ensemble density matrix operator constructed
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to density matrix operators
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from Slater determinants, the ensemble 1RDM elements are $\Gamma_{pq}^{\bw}={\rm
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Tr}\left[\hat{\Gamma}^{{\bw}}\hat{a}^\dagger_p\hat{a}_q\right]$,
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and $W_{\rm
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HF}\left[{\bmg}\right]=\frac{1}{2}\sum_{pqrs}\langle \varphi_p\varphi_q\vert\vert
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\varphi_r\varphi_s\rangle
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%\times
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\Gamma_{pr}\Gamma_{qs}$.\\
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In-principle-exact decomposition:
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\beq
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\beq
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F^{\bw}[n]= F^{\bw}_{\rm HF}[n]+\overline{E}^{{\bw}}_{\rm
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\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)}
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Hx}[n]+\overline{E}^{{\bw}}_{\rm c}[n]
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\eeq
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\eeq
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that are constructed from single Slater
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determinants $\Phi^{(K)}$.
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The complementary ensemble Hx energy removes the ghost-interaction
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The complementary ensemble Hx energy removes the ghost-interaction
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errors introduced in $W_{\rm
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errors introduced in $W_{\rm
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HF}\left[{\bmg}^{\bw}[n]\right]$:
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HF}\left[{\bmg}^{\bw}[n]\right]$:
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\beq
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\beq
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\overline{E}^{{\bw}}_{\rm
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\overline{E}^{{\bw}}_{\rm
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Hx}[n]=\sum^M_{K=0}w^{(K)}W_{\rm
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Hx}[n]&=&\sum_{K\geq0}w^{(K)}W_{\rm
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HF}\left[{\bmg}^{(K)}[n]\right]
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HF}\left[{\bmg}^{(K)}[n]\right]
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-W_{\rm
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-W_{\rm
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HF}\left[{\bmg}^{\bw}[n]\right],
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HF}\left[{\bmg}^{\bw}[n]\right].
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\eeq
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which gives in the canonical orbital basis
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\beq
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&&\overline{E}^{{\bw}}_{\rm
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Hx}[n]=
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\dfrac{1}{2}\sum_{pq}
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\langle \varphi^{{\bw}}_p[n]\varphi^{{\bw}}_q[n]\vert\vert
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\varphi^{{\bw}}_p[n]\varphi^{{\bw}}_q[n]\rangle
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\nonumber\\
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\nonumber\\
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&&\times\left[\sum^M_{K=0}w^{(K)}\nu^{(K)}_p \left(\nu^{(K)}_q
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&=&
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-\sum^M_{L=0}w^{(L)} \nu^{(L)}_q\right)\right]
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{\rm
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.\eeq
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Tr}\left[\hat{\Gamma}^{{\bw}}[n]\hat{W}_{\rm ee}\right]-W_{\rm
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\manu{I would guess that, in a uniform system, the GOK-DFT and our
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HF}\left[{\bmg}^{\bw}[n]\right]
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canonical orbitals are the same. This is nice since we can construct
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\eeq
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in a clean way density-functional approximations for both $\overline{E}^{{\bw}}_{\rm
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Hx}[n]$ and $E^{{\bw}}_{\rm c}[n]$ functionals. Am I right ?}
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Note that $\overline{E}^{{\bw}=0}_{\rm
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Hx}[n]=0$.\\
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Ensemble correlation energy:
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Variational expression for the ensemble energy:
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\beq
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\beq
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E^{{\bw}}=\underset{\hat{\Gamma}^{{\bw}}}{\rm min}\Big\{
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\overline{E}^{{\bw}}_{\rm
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&&{\rm
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c}[n]&=&
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Tr}\left[\hat{\Gamma}^{{\bw}}\hat{T}\right]+W_{\rm
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{\rm
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Tr}\left[\hat{\gamma}^{{\bw}}[n]\left(\hat{T}+\hat{W}_{\rm
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ee}\right)\right]
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\nonumber\\
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&&-
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{\rm
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Tr}\left[\hat{\Gamma}^{{\bw}}[n]\left(\hat{T}+\hat{W}_{\rm
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ee}\right)\right]
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\eeq
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Variational expression of the ensemble energy:
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\beq
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E^{{\bw}}=\underset{{\bmg}^{{\bw}}}{\rm min}\Big\{
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{\rm
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Tr}\left[{\bmg}^{{\bw}}{\bm h}\right]+W_{\rm
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HF}\left[{\bmg}^{\bw}\right]
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HF}\left[{\bmg}^{\bw}\right]
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+
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+
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\overline{E}^{{\bw}}_{\rm
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\overline{E}^{{\bw}}_{\rm
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Hxc}\left[n_{\hat{\Gamma}^{{\bw}}}\right]
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Hxc}\left[n_{{\bmg}^{{\bw}}}\right]
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%+E^{{\bw}}_{\rm c}\left[n_{\hat{\Gamma}^{{\bw}}}\right]
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%+E^{{\bw}}_{\rm c}\left[n_{\hat{\Gamma}^{{\bw}}}\right]
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\nonumber\\
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&&
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+\int d{\br}\;v_{\rm ext}({\bfr})n_{\hat{\Gamma}^{{\bw}}}({\bfr})
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\Big\}
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\Big\}
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\eeq
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\eeq
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For $K>0$
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\alert{
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\beq
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\dfrac{\partial E^{{\bw}}}{\partial w^{(K)}}&=&{\rm
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Tr}\left[{\bmg}^{(K)}{\bm h}\right]-{\rm
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Tr}\left[{\bmg}^{(0)}{\bm h}\right]
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\nonumber\\
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&&+\Tr(\bmg^{(K)} \, \bG \, \bmg^{\bw})
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-\Tr(\bmg^{(0)} \, \bG \, \bmg^{\bw})
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+...
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\eeq
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\beq
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E^{(I)}&=&E^{{\bw}}+\sum_{K>0}\left(\delta_{IK}-w^{(K)}\right)\dfrac{\partial E^{{\bw}}}{\partial w^{(K)}}
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\nonumber\\
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&=&
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...+\dfrac{1}{2}\Tr(\bmg^{\bw} \, \bG \, \bmg^{\bw})
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+...
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\eeq
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}
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Note that, if we use orbital rotations, the gradient of the DFT energy
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Note that, if we use orbital rotations, the gradient of the DFT energy
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contributions can be expressed as follows,
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contributions can be expressed as follows,
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\beq
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\beq
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