From fd285764421f68600670e533939a80604e636eb3 Mon Sep 17 00:00:00 2001 From: Emmanuel Fromager Date: Wed, 11 Sep 2019 12:49:00 +0200 Subject: [PATCH] Manu: saving work --- Manuscript/eDFT.tex | 109 +++++++++++++++++++++++++++++++------------- 1 file changed, 78 insertions(+), 31 deletions(-) diff --git a/Manuscript/eDFT.tex b/Manuscript/eDFT.tex index 60e8107..c584288 100644 --- a/Manuscript/eDFT.tex +++ b/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\{