Manu: saving work

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Emmanuel Fromager 2019-09-10 17:10:12 +02:00
parent c1f4eb75df
commit 0d01c4376d

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@ -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
@ -228,77 +231,102 @@ 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]
\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,
\beq