From db5360d99fd8e7bebd312dac97b1b4fd8f637b30 Mon Sep 17 00:00:00 2001 From: Emmanuel Fromager Date: Mon, 9 Sep 2019 16:11:21 +0200 Subject: [PATCH] Manu: copy paste from the SI (to be polished) --- Manuscript/eDFT.tex | 184 ++++++++++++++++++++++++++++++++++++++++++++ 1 file changed, 184 insertions(+) diff --git a/Manuscript/eDFT.tex b/Manuscript/eDFT.tex index eeebb47..6d916a5 100644 --- a/Manuscript/eDFT.tex +++ b/Manuscript/eDFT.tex @@ -93,6 +93,10 @@ \newcommand{\manu}[1]{{\textcolor{blue}{ Manu: #1 }} } \newcommand{\beq}{\begin{eqnarray}} \newcommand{\eeq}{\nonumber\end{eqnarray}} +\newcommand{\bmk}{\bm{\kappa}} % orbital rotation vector +\newcommand{\bmg}{\bm{\gamma}} % orbital rotation vector +\newcommand{\bfx}{\bf{x}} +\newcommand{\bfr}{\bf{r}} %%%% \begin{document} @@ -170,6 +174,186 @@ Atomic units are used throughout. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \alert{Manu, you might want to add general details about the eDFT here.} +\manu{Yes. Copy paste from the SI. Will polish the all thing.} + +\beq +F^{\bw}_{\rm HF}[n]&=& +\underset{\hat{\gamma}^{{\bw}}\rightarrow n}{\rm min}\left\{{\rm +Tr}\left[\hat{\gamma}^{{\bw}}\hat{T}\right]+W_{\rm +HF}\left[{\bmg}^{\bw}\right]\right\} +\nonumber\\ +&=&{\rm +Tr}\left[\hat{\gamma}^{{\bw}}[n]\hat{T}\right]+W_{\rm +HF}\left[{\bmg}^{\bw}[n]\right] +\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: + +\beq +F^{\bw}[n]= F^{\bw}_{\rm HF}[n]+\overline{E}^{{\bw}}_{\rm +Hx}[n]+\overline{E}^{{\bw}}_{\rm c}[n] +\eeq + +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 +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 +\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 ?} + +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 +HF}\left[{\bmg}^{\bw}\right] ++ +\overline{E}^{{\bw}}_{\rm +Hxc}\left[n_{\hat{\gamma}^{{\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 + +Note that, if we use orbital rotations, the gradient of the DFT energy +contributions can be expressed as follows, +\beq +\left.\dfrac{\partial +\overline{E}^{{\bw}}_{\rm Hxc}\left[n^{{\bw}}({\bmk})\right] +}{\partial \kappa_{lm}} +\right|_{{\bmk}=0}=\int d{\br}\dfrac{\delta \overline{E}^{{\bw}}_{\rm +Hxc}\left[n^{{\bw}}\right]}{\delta +n({\br})}\left.\dfrac{\partial n^{{\bw}}({\bmk},{\br})}{\partial \kappa_{lm}} +\right|_{{\bmk}=0}, +\eeq +where +\beq +n^{{\bw}}({\bmk},{\br})=\sum_\sigma\sum_{pq}\varphi_p({\bmk},{\bfx})\varphi_q({\bmk},{\bfx})\gamma_{pq}^{\bw} +\eeq +thus leading to +\beq +&&\left.\dfrac{\partial +\overline{E}^{{\bw}}_{\rm Hxc}\left[n^{{\bw}}({\bmk})\right] +}{\partial \kappa_{lm}} +\right|_{{\bmk}=0}= +\sum_{pq}\gamma_{pq}^{\bw} +\nonumber\\ +&&\times\left.\dfrac{\partial} +{\partial \kappa_{lm}} +\Big[\left\langle\varphi_p(\bmk)\middle\vert\hat{\overline{v}}^{{\bw}}_{\rm +Hxc} +\middle\vert \varphi_q(\bmk)\right\rangle +\Big] +\right|_{{\bmk}=0}. +\eeq + +In conclusion, the minimizing canonical orbitals fulfill the following +hybrid HF/GOK-DFT equation, +\beq +&&\left(-\frac{\nabla_{\bfr}^2}{2}+v_{\rm +ext}({\bfr})+\hat{u}_{\rm HF}\left[\gamma^{\bw}\right] ++\dfrac{\delta \overline{E}^{{\bw}}_{\rm Hxc}\left[n^{{\bw}}\right]}{\delta +n({\br})}\right)\varphi^{{\bw}}_p({\bfx}) +\nonumber +\\ +&&=\varepsilon^{{\bw}}_p\varphi^{{\bw}}_p({\bfx}). +\eeq + + +Since $\partial \gamma_{pq}^{\bw}/\partial +w^{(I)}=\gamma_{pq}^{(I)}-\gamma_{pq}^{(0)}$, it comes + +\manu{just for me ... +\beq +&&+\dfrac{1}{2} +\sum_{pqrs}\langle \varphi_p\varphi_q\vert\vert +\varphi_r\varphi_s\rangle +%\times +\left(\gamma_{pr}^{(I)}-\gamma_{pr}^{(0)}\right)\gamma^{\bw}_{qs} +\nonumber\\ +&&+\dfrac{1}{2}\sum_{pqrs}\langle \varphi_q\varphi_p\vert\vert +\varphi_s\varphi_r\rangle +%\times + \gamma^{\bw}_{pr}\left(\gamma_{qs}^{(I)}-\gamma_{qs}^{(0)}\right) +\nonumber\\ +&&= +\sum_{pqrs}\langle \varphi_p\varphi_q\vert\vert +\varphi_r\varphi_s\rangle +%\times +\left(\gamma_{pr}^{(I)}-\gamma_{pr}^{(0)}\right)\gamma^{\bw}_{qs} +\nonumber\\ +&&= +\sum_{pr}\left[\hat{u}_{\rm HF}\left[\gamma^{\bw}\right]\right]_{pr}\left(\gamma_{pr}^{(I)}-\gamma_{pr}^{(0)}\right) +\nonumber\\ +&&= +\sum_p\left[\hat{u}_{\rm +HF}\left[\gamma^{\bw}\right]\right]_{pp}\left(\nu_p^{(I)}-\nu_p^{(0)}\right) +\eeq +} + +\beq +\dfrac{dE^{\bw}}{dw^{(I)}}=\sum_p\varepsilon^{{\bw}}_p\left(\nu_p^{(I)}-\nu_p^{(0)}\right)+\left.\dfrac{\partial \overline{E}^{{\bw}}_{\rm +Hxc}\left[n\right]}{\partial w^{(I)}}\right|_{n=n^{{\bw}}}. +\eeq + +LZ shift in this context: $\varepsilon^{{\bw}}_p\rightarrow +\overline{\varepsilon}^{{\bw}}_p=\varepsilon^{{\bw}}_p+\overline{\Delta}_{\rm +LZ}^{{\bw}}$ where + +\beq +N\overline{\Delta}_{\rm +LZ}^{{\bw}}&=&\overline{E}^{{\bw}}_{\rm Hxc}\left[n^{{\bw}}\right] +-\int d{\br}\dfrac{\delta \overline{E}^{{\bw}}_{\rm +Hxc}\left[n^{{\bw}}\right]}{\delta +n({\br})}n^{{\bw}}({\bfr}) +\nonumber\\ +&& +-W_{\rm +HF}\left[{\bmg}^{\bw}\right] +\eeq + +such that +\beq +E^{{\bw}}=\sum^M_{K=0}w^{(K)}\sum_p\nu_p^{(K)}\overline{\varepsilon}^{{\bw}}_p. +\eeq + +Thus we conclude that individual energies can be expressed in principle +exactly as follows, + +\beq +E^{(K)}=\sum_p\nu_p^{(K)}\overline{\varepsilon}^{{\bw}}_p+\sum^M_{I>0}\left(\delta_{IK}-w^{(I)}\right)\left.\dfrac{\partial \overline{E}^{{\bw}}_{\rm +Hxc}\left[n\right]}{\partial w^{(I)}}\right|_{n=n^{{\bw}}}. +\eeq %In eDFT, the ensemble energy %\begin{equation}