clean up results
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%% This BibTeX bibliography file was created using BibDesk.
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%% http://bibdesk.sourceforge.net/
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%% Created for Pierre-Francois Loos at 2019-11-25 22:20:38 +0100
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%% Created for Pierre-Francois Loos at 2020-02-22 09:50:18 +0100
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%% Saved with string encoding Unicode (UTF-8)
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@article{QP2,
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Author = {Y. Garniron and K. Gasperich and T. Applencourt and A. Benali and A. Fert{\'e} and J. Paquier and B. Pradines and R. Assaraf and P. Reinhardt and J. Toulouse and P. Barbaresco and N. Renon and G. David and J. P. Malrieu and M. V{\'e}ril and M. Caffarel and P. F. Loos and E. Giner and A. Scemama},
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Date-Added = {2020-02-22 09:29:06 +0100},
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Date-Modified = {2020-02-22 09:29:06 +0100},
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Doi = {10.1021/acs.jctc.9b00176},
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Journal = {J. Chem. Theory Comput.},
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Pages = {3591},
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Title = {Quantum Package 2.0: A Open-Source Determinant-Driven Suite Of Programs},
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Volume = {15},
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Year = {2019},
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Bdsk-Url-1 = {https://doi.org/10.1021/acs.jctc.9b00176}}
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@article{Perdew_1981,
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Author = {J. P. Perdew and A. Zunger},
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Date-Added = {2020-02-22 09:26:50 +0100},
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Date-Modified = {2020-02-22 09:27:38 +0100},
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Doi = {10.1103/PhysRevB.23.5048},
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Journal = {Phys. Rev. B},
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Pages = {5048},
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Title = {Self-interaction correction to density-functional approximations for many-electron systems},
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Volume = {23},
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Year = {1981},
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Bdsk-Url-1 = {https://doi.org/10.1103/PhysRevB.23.5048}}
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@article{Avery_1993,
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Author = {J. Avery},
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Date-Added = {2019-11-23 21:56:19 +0100},
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@ -6025,20 +6049,6 @@
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Title = {QMC=Chem},
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Year = 2017}
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@misc{QP,
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Author = {A. Scemama and T. Applencourt and Y. Garniron and E. Giner and G. David and M. Caffarel},
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Date-Added = {2018-10-24 22:38:52 +0200},
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Date-Modified = {2018-10-24 22:38:52 +0200},
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Doi = {10.5281/zenodo.200970},
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Month = {Dec},
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Note = {\url{https://github.com/LCPQ/quantum_package}},
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Publisher = {Zenodo},
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Title = {Quantum Package v1.0},
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Url = {https://github.com/LCPQ/quantum_package},
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Year = {2016},
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Bdsk-Url-1 = {https://github.com/LCPQ/quantum_package},
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Bdsk-Url-2 = {http://dx.doi.org/10.5281/zenodo.200970}}
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@article{Ralphs_2013,
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Author = {K Ralphs and G Serna and L R Hargreaves and M A Khakoo and C Winstead and V McKoy},
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Date-Added = {2018-10-24 22:38:52 +0200},
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@ -224,11 +224,11 @@ is the density matrix operator, $\Det{I}{\bw}$ are single-determinant wave funct
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& = \E{\Ha}{}[\n{}{}] + \E{\xc}{\bw}[\n{}{}]
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\\
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& = \frac{1}{2} \iint \frac{\n{}{}(\br{}) \n{}{}(\br{}')}{\abs{\br{}-\br{}'}} d\br{} d\br{}'
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+ \int \e{\xc}{\bw}[\n{}{}(\br{})] \n{}{}(\br{}) d\br{}.
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+ \int \e{\xc}{\bw}(\n{}{}(\br{})) \n{}{}(\br{}) d\br{}.
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\end{split}
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\end{equation}
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is the ensemble Hartree-exchange-correlation (Hxc) functional.
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Note that the weight-independent Hartree functional $\E{\Ha}{}[\n{}{}]$ causes the infamous ghost-interaction error (GIC) \cite{Gidopoulos_2002, Pastorczak_2014, Alam_2016, Alam_2017, Gould_2017} in eDFT, which is supposed to be cancelled by the weight-dependent xc functional $\E{\xc}{\bw}[\n{}{}]$.
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Note that the weight-independent Hartree functional $\E{\Ha}{}[\n{}{}]$ causes the infamous ghost-interaction error (GIE) \cite{Gidopoulos_2002, Pastorczak_2014, Alam_2016, Alam_2017, Gould_2017} in eDFT, which is supposed to be cancelled by the weight-dependent xc functional $\E{\xc}{\bw}[\n{}{}]$.
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From the GOK-DFT ensemble energy expression in Eq.~\eqref{eq:Ew-GOK}, we obtain \cite{Gross_1988b,Deur_2019}
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\begin{equation}
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@ -262,7 +262,7 @@ where $\hHc(\br{}) = -\nabla^2/2 + \vext(\br{})$, and
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& = \fdv{\E{\Ha}{\bw}[\n{}{}]}{\n{}{}(\br{})} + \fdv{\E{\xc}{\bw}[\n{}{}]}{\n{}{}(\br{})}
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\\
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& = \frac{1}{2} \int \frac{\n{}{}(\br{}')}{\abs{\br{}-\br{}'}} d\br{}'
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+ \left. \pdv{\e{\xc}{\bw{}}(\n{}{})}{\n{}{}} \right|_{\n{}{} = \n{}{}(\br{})} \n{}{}(\br{}) + \e{\xc}{\bw{}}[\n{}{}(\br{})]
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+ \left. \pdv{\e{\xc}{\bw{}}(\n{}{})}{\n{}{}} \right|_{\n{}{} = \n{}{}(\br{})} \n{}{}(\br{}) + \e{\xc}{\bw{}}(\n{}{}(\br{}))
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\end{split}
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\end{equation}
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is the Hxc potential.
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@ -481,7 +481,8 @@ Consequently, in the following, we name this weight-dependent xc functional ``eL
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Also, we note that, by construction,
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\begin{equation}
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\label{eq:dexcdw}
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\left. \pdv{\be{\xc}{\ew{}}[\n{}{}]}{\ew{}}\right|_{\n{}{} = \n{}{\ew{}}(\br)} = \be{\xc}{(1)}[\n{}{\ew{}}(\br)] - \be{\xc}{(0)}[\n{}{\ew{}}(\br)].
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\left. \pdv{\be{\xc}{\ew{}}[\n{}{}]}{\ew{}}\right|_{\n{}{} = \n{}{\ew{}}(\br)}
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= \be{\xc}{(1)}(\n{}{\ew{}}(\br)) - \be{\xc}{(0)}(\n{}{\ew{}}(\br)).
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\end{equation}
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This embedding procedure can be theoretically justified by the generalised adiabatic connection formalism for ensembles (GACE)
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@ -498,7 +499,7 @@ In the case of a homogeneous system (or equivalently within the LDA), substituti
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%%% TABLE I %%%
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\begin{table*}
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\caption{
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Total energies (in hartree) and excitation energies (in \titou{hartree}) of \ce{H2} with $\RHH = 1.4$ bohr for various methods with the STO-3G minimal basis.
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Total energies (in hartree) and excitation energies (in hartree) of \ce{H2} with $\RHH = 1.4$ bohr for various methods with the STO-3G minimal basis.
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\label{tab:Energies}
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}
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\begin{ruledtabular}
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@ -514,7 +515,7 @@ Total energies (in hartree) and excitation energies (in \titou{hartree}) of \ce{
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accurate\fnm[1] & & & & \\
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\end{tabular}
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\end{ruledtabular}
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\fnt[1]{FCI/aug-cc-pV5Z excitation energies computer with QUANTUM PACKAGE. \cite{QP2}}
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\fnt[1]{FCI/cc-pV5Z excitation energies computed with QUANTUM PACKAGE. \cite{QP2}}
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\end{table*}
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%%% %%% %%% %%%
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@ -556,8 +557,8 @@ Ensemble energies (in hartree) of \ce{H2} with $\RHH = 1.4$ bohr as a function o
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Here, we consider as testing ground the minimal-basis \ce{H2} molecule.
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We select STO-3G as minimal basis, and study the behaviour of the total energy of \ce{H2} as a function of the internuclear distance $\RHH$ (in bohr).
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This minimal-basis example is quite pedagogical as the molecular orbitals are fixed by symmetry.
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We have then access to the individual densities of the ground and doubly-excited states (which is not usually possible in practice).
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Therefore, thanks to the spatial symmetry and the minimal basis, the individual densities extracted from the ensemble density are equal to the \textit{exact} individual densities.
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We have then access to the individual densities of the ground and doubly-excited states (which is not straightforward in practice).
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Moreover, thanks to the spatial symmetry and the minimal basis, the individual densities extracted from the ensemble density are equal to the \textit{exact} individual densities.
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In other words, there is no density-driven error and the only error that we are going to observe is the functional-driven error (and this is what we want to study).
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The bonding and antibonding orbitals of the \ce{H2} molecule are given by
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@ -568,7 +569,7 @@ The bonding and antibonding orbitals of the \ce{H2} molecule are given by
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\MO{2}{}(\br{}) & = \qty[ \AO{A}(\br{}) - \AO{B}(\br{}) ]/\sqrt{2(1 - S_{AB})},
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\end{align}
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\end{subequations}
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where $\AO{A}$ and $\AO{B}$ are the two contracted Gaussian basis functions centred on each of the nucleus, and $S_{AB} = \braket{\AO{A}}{\AO{B}}$.
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where $\AO{A}$ and $\AO{B}$ are the two contracted Gaussian basis functions centred on each of the nucleus, and $S_{AB} = \braket{\AO{A}}{\AO{B}}$ is the overlap between these two basis functions.
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The HF energies of the ground state and the doubly-excited states are
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\begin{subequations}
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@ -591,7 +592,10 @@ with
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\end{align}
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\end{subequations}
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Note that, in the HF case, there is no self-interaction error as $\eJ{pp} = \eK{pp}$.
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We also define the HF excitation energy as $\Ex{\HF}{(1)} = \E{\HF}{(1)} - \E{\HF}{(0)}$.
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We also define the HF excitation energy as
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\begin{equation}
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\Ex{\HF}{(1)} = \E{\HF}{(1)} - \E{\HF}{(0)}.
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\end{equation}
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%The HF orbital energies are
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%\begin{subequations}
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%\begin{align}
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@ -601,7 +605,7 @@ We also define the HF excitation energy as $\Ex{\HF}{(1)} = \E{\HF}{(1)} - \E{\H
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%\end{align}
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%\end{subequations}
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As reference results, we consider CID (configuration interaction with doubles) computed in the same (minimal) basis set.
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It is also instructive to consider the CID (configuration interaction with doubles) excitation energies computed in the same (minimal) basis set.
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The CID energies of the ground state and doubly-excited states are provided by the eigenvalues of the following CID matrix:
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\begin{equation}
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\bH_\CID =
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@ -609,7 +613,7 @@ The CID energies of the ground state and doubly-excited states are provided by t
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\E{\HF}{(0)} & \eK{12}
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\\
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\eK{12} & \E{\HF}{(1)}
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\end{pmatrix},
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\end{pmatrix}.
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\end{equation}
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These CID energies are explicitly given by
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\begin{subequations}
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@ -624,23 +628,23 @@ and the CID excitation energy reads
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\Ex{\CID}{(1)} = \sqrt{\qty(\E{\HF}{(1)} - \E{\HF}{(0)})^2 + 4 \eK{12}^2} \ge \Ex{\HF}{(1)}.
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\end{equation}
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At the (ground-state) LDA level (\ie, we only consider ground-state functionals), these energies reads
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At the (ground-state) LDA level (\ie, we only consider ground-state functionals), the individual energies reads
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\begin{subequations}
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\begin{align}
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\label{eq:LDA0}
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\E{\LDA}{(0)} & = 2 \eHc{1} + 2 \eJ{11} + \int \e{\xc}{\LDA}[\n{}{(0)}(\br{})] \n{}{(0)}(\br{}) d\br{},
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\E{\LDA}{(0)} & = 2 \eHc{1} + 2 \eJ{11} + \int \e{\xc}{\LDA}(\n{}{(0)}(\br{})) \n{}{(0)}(\br{}) d\br{},
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\\
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\label{eq:LDA1}
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\E{\LDA}{(1)} & = 2 \eHc{2} + 2 \eJ{22} + \int \e{\xc}{\LDA}[\n{}{(1)}(\br{})] \n{}{(1)}(\br{}) d\br{},
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\E{\LDA}{(1)} & = 2 \eHc{2} + 2 \eJ{22} + \int \e{\xc}{\LDA}(\n{}{(1)}(\br{})) \n{}{(1)}(\br{}) d\br{},
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\end{align}
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\end{subequations}
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with
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\begin{align}
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\n{}{(0)}(\br{}) & = 2 [\MO{1}{\ew{}}(\br{})]^2,
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\n{}{(0)}(\br{}) & = 2 \MO{1}{2}(\br{}),
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&
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\n{}{(1)}(\br{}) & = 2 [\MO{2}{\ew{}}(\br{})]^2,
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\n{}{(1)}(\br{}) & = 2 \MO{2}{2}(\br{}),
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\end{align}
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Note that, contrary to the HF case, self-interaction is present in LDA.
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Note that, contrary to the HF case, self-interaction is present in LDA. \cite{Perdew_1981}
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%The KS orbital energies are given by
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%\begin{subequations}
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%\begin{align}
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@ -657,10 +661,10 @@ At the eLDA, we have
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\begin{subequations}
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\begin{align}
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\label{eq:eLDA0}
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\E{\eLDA}{(0)} & = 2 \eHc{1} + 2 \eJ{11} + \int \be{\xc}{(0)}[\n{}{(0)}(\br{})] \n{}{(0)}(\br{}) d\br{},
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\E{\eLDA}{(0)} & = 2 \eHc{1} + 2 \eJ{11} + \int \be{\xc}{(0)}(\n{}{(0)}(\br{})) \n{}{(0)}(\br{}) d\br{},
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\\
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\label{eq:eLDA1}
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\E{\eLDA}{(1)} & = 2 \eHc{2} + 2 \eJ{22} + \int \be{\xc}{(1)}[\n{}{(1)}(\br{})] \n{}{(1)}(\br{}) d\br{},
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\E{\eLDA}{(1)} & = 2 \eHc{2} + 2 \eJ{22} + \int \be{\xc}{(1)}(\n{}{(1)}(\br{})) \n{}{(1)}(\br{}) d\br{},
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\end{align}
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\end{subequations}
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with $\be{\xc}{(0)}(\n{}{}) \equiv \e{\xc}{\LDA}(\n{}{})$ and $\be{\xc}{(1)}(\n{}{}) = \e{\xc}{\LDA}(\n{}{}) + \e{\xc}{(1)}(\n{}{}) - \e{\xc}{(0)}(\n{}{})$.
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@ -669,20 +673,11 @@ Interestingly here, there is a strong connection between the LDA and eLDA excita
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\begin{equation}
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\begin{split}
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\Ex{\eLDA}{(1)}
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& = \Ex{\LDA}{(1)} + \int \qty( \e{\xc}{(1)} - \e{\xc}{(0)} )[\n{}{(1)}(\br{})] \n{}{(1)}(\br{}) d\br{}.
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& = \Ex{\LDA}{(1)} + \int \qty[ \e{\xc}{(1)}(\n{}{(1)}(\br{})) - \e{\xc}{(0)}(\n{}{(1)}(\br{})) ] \n{}{(1)}(\br{}) d\br{}.
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\\
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& = \Ex{\LDA}{(1)} + \int \left. \pdv{\e{\xc}{\ew{}}[\n{}{}]}{\ew{}} \right|_{\n{}{} = \n{}{(1)}(\br{})} \n{}{(1)}(\br{}) d\br{}.
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& = \Ex{\LDA}{(1)} + \int \left. \pdv{\e{\xc}{\ew{}}(\n{}{})}{\ew{}} \right|_{\n{}{} = \n{}{(1)}(\br{})} \n{}{(1)}(\br{}) d\br{}.
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\end{split}
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\end{equation}
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The KS orbital energies are given by
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%\begin{subequations}
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%\begin{align}
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% \eps{1}{\eLDA} & = \eHc{1} + 2\eJ{11} + \ldots,
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% \\
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% \eps{2}{\eLDA} & = \eHc{2} + 2\eJ{12} + \ldots.
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%\end{align}
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%\end{subequations}
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These equations can be combined to define three ensemble energies
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\begin{subequations}
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@ -698,14 +693,16 @@ These equations can be combined to define three ensemble energies
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\end{align}
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\end{subequations}
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which are all, by construction, linear with respect to $\ew{}$.
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Excitation energies can be easily extracted from these formulae via differenciation with respect to $\ew{}$.
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Excitation energies can be easily extracted from these formulae via differentiation with respect to $\ew{}$.
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Note that this is not how one would do in a ``practical'' ensemble calculation as one does not have (usually) access to the individual densities.
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However, for pedagogical purposes and to study the magnitude of the ghost-interaction error, it is interesting to defined them.
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Similar energies than the ones given in Eqs.~\eqref{eq:EwHF}, \eqref{eq:EwLDA} and \eqref{eq:EweLDA} can also be obtained directly from the ensemble density
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Alternatively to Eqs.~\eqref{eq:EwHF}, \eqref{eq:EwLDA} and \eqref{eq:EweLDA}, one can obtain ensemble energies directly from the ensemble density
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\begin{equation}
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\n{}{\ew{}} = (1-\ew{}) \n{}{(0)} + \ew{} \n{}{(1)}.
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\end{equation}
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(This is what one would do in practice, \ie, by performing a KS ensemble calculation.)
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We will label these energies as $\tE{}{\ew{}}$ to avoid confusion.
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We will label these energies as $\tE{}{\ew{}}$ to avoid confusion with the expressions reported in Eqs.~\eqref{eq:EwHF}, \eqref{eq:EwLDA} and \eqref{eq:EweLDA}.
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\begin{widetext}
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For HF, we have
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\begin{equation}
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@ -729,13 +726,13 @@ In the case of the LDA, it reads
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& = \Ts{\ew{}}[\n{}{\ew{}}(\br{})]
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+ \int \vext(\br{}) \n{}{\ew{}}(\br{}) d\br{}
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+ \iint \frac{\n{}{\ew{}}(\br{})\n{}{\ew{}}(\br{}')}{\abs{\br{} - \br{}'}} d\br{} d\br{}'
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+ \int \e{\xc}{\LDA}[\n{}{\ew{}}(\br{})] \n{}{\ew{}}(\br{}) d\br{}
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+ \int \e{\xc}{\LDA}(\n{}{\ew{}}(\br{})) \n{}{\ew{}}(\br{}) d\br{}
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\\
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& = 2 (1-\ew{}) \eHc{1} + 2 \ew{} \eHc{2}
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+ 2(1-\ew{})^2 \eJ{11} + 2\ew{}^2 \eJ{22} + 4 (1-\ew{})\ew{} \eJ{12}
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\\
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& + (1-\ew{}) \int \e{\xc}{\LDA}[\n{}{\ew{}}(\br{})] \n{}{(0)}(\br{}) d\br{}
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+ \ew{} \int \e{\xc}{\LDA}[\n{}{\ew{}}(\br{})] \n{}{(1)}(\br{}) d\br{},
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& + (1-\ew{}) \int \e{\xc}{\LDA}(\n{}{\ew{}}(\br{})) \n{}{(0)}(\br{}) d\br{}
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+ \ew{} \int \e{\xc}{\LDA}(\n{}{\ew{}}(\br{})) \n{}{(1)}(\br{}) d\br{},
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\end{split}
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\end{equation}
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which is also clearly quadratic with respect to $\ew{}$ because the (weight-independent) LDA functional cannot compensate the ``quadraticity'' of the Hartree term.
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@ -747,24 +744,24 @@ For eLDA, the ensemble energy can be decomposed as
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& = \Ts{\ew{}}[\n{}{\ew{}}(\br{})]
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+ \int \vext(\br{}) \n{}{\ew{}}(\br{}) d\br{}
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+ \iint \frac{\n{}{\ew{}}(\br{})\n{}{\ew{}}(\br{}')}{\abs{\br{} - \br{}'}} d\br{} d\br{}'
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+ \int \be{\xc}{\ew{}}[\n{}{\ew{}}(\br{})] \n{}{\ew{}}(\br{}) d\br{}
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+ \int \be{\xc}{\ew{}}(\n{}{\ew{}}(\br{})) \n{}{\ew{}}(\br{}) d\br{}
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\\
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& = 2 (1-\ew{}) \eHc{1} + 2 \ew{} \eHc{2}
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+ 2(1-\ew{})^2 \eJ{11} + 2\ew{}^2 \eJ{22} + 4 (1-\ew{})\ew{} \eJ{12}
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\\
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& + (1-\ew{})^2 \int \be{\xc}{(0)}[\n{}{\ew{}}(\br{})] \n{}{(0)}(\br{}) d\br{}
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+ \ew{}^2 \int \be{\xc}{(1)}[\n{}{\ew{}}(\br{})] \n{}{(1)}(\br{}) d\br{}
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& + (1-\ew{})^2 \int \be{\xc}{(0)}(\n{}{\ew{}}(\br{})) \n{}{(0)}(\br{}) d\br{}
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+ \ew{}^2 \int \be{\xc}{(1)}(\n{}{\ew{}}(\br{})) \n{}{(1)}(\br{}) d\br{}
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\\
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& + (1-\ew{})\ew{} \int \be{\xc}{(0)}[\n{}{\ew{}}(\br{})] \n{}{(1)}(\br{}) d\br{}
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+ \ew{}(1-\ew{}) \int \be{\xc}{(1)}[\n{}{\ew{}}(\br{})] \n{}{(0)}(\br{}) d\br{}
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& + (1-\ew{})\ew{} \int \be{\xc}{(0)}(\n{}{\ew{}}(\br{})) \n{}{(1)}(\br{}) d\br{}
|
||||
+ \ew{}(1-\ew{}) \int \be{\xc}{(1)}(\n{}{\ew{}}(\br{})) \n{}{(0)}(\br{}) d\br{}
|
||||
\\
|
||||
& = 2 (1-\ew{}) \eHc{1} + 2 \ew{} \eHc{2}
|
||||
+ (1-\ew{})^2 \qty[ 2\eJ{11} + \int \be{\xc}{(0)}[\n{}{\ew{}}(\br{})] \n{}{(0)}(\br{}) d\br{} ]
|
||||
+ \ew{}^2 \qty[ 2\eJ{22} + \int \be{\xc}{(1)}[\n{}{\ew{}}(\br{})] \n{}{(1)}(\br{}) d\br{} ]
|
||||
+ (1-\ew{})^2 \qty[ 2\eJ{11} + \int \be{\xc}{(0)}(\n{}{\ew{}}(\br{})) \n{}{(0)}(\br{}) d\br{} ]
|
||||
+ \ew{}^2 \qty[ 2\eJ{22} + \int \be{\xc}{(1)}(\n{}{\ew{}}(\br{})) \n{}{(1)}(\br{}) d\br{} ]
|
||||
\\
|
||||
& + 2 (1-\ew{})\ew{} \qty[ 2\eJ{12}
|
||||
+ \frac{1}{2} \int \be{\xc}{(0)}[\n{}{\ew{}}(\br{})] \n{}{(1)}(\br{}) d\br{}
|
||||
+ \frac{1}{2} \int \be{\xc}{(1)}[\n{}{\ew{}}(\br{})] \n{}{(0)}(\br{}) d\br{} ],
|
||||
+ \frac{1}{2} \int \be{\xc}{(0)}(\n{}{\ew{}}(\br{})) \n{}{(1)}(\br{}) d\br{}
|
||||
+ \frac{1}{2} \int \be{\xc}{(1)}(\n{}{\ew{}}(\br{})) \n{}{(0)}(\br{}) d\br{} ],
|
||||
\end{split}
|
||||
\end{equation}
|
||||
which \textit{could} be linear with respect to $\ew{}$ if the weight-dependent xc functional compensates exactly the quadratic terms in the Hartree term.
|
||||
|
|
@ -772,7 +769,7 @@ This would be, for example, the case with the exact xc functional.
|
|||
|
||||
Extracting excitation energies from Eqs.~\eqref{eq:bEwHF}, \eqref{eq:bEwLDA} and \eqref{eq:bEweLDA} is more tricky.
|
||||
To do so, we will employ Eq.~\eqref{eq:dEdw}.
|
||||
The two first terms are simply $\Eps{0}{\ew{}} = 2 \eps{1}{\ew{}}$, $\Eps{1}{\ew{}} = 2 \eps{2}{\ew{}}$, and the HF, LDA and eLDA weight-dependent orbital energies are
|
||||
The two first terms are simply $\Eps{0}{\ew{}} = 2 \eps{1}{\ew{}}$, $\Eps{1}{\ew{}} = 2 \eps{2}{\ew{}}$, and the HF, LDA and eLDA weight-dependent orbital energies are, respectively,
|
||||
\begin{subequations}
|
||||
\begin{align}
|
||||
\eps{1}{\ew{},\HF}
|
||||
|
|
@ -788,12 +785,12 @@ The two first terms are simply $\Eps{0}{\ew{}} = 2 \eps{1}{\ew{}}$, $\Eps{1}{\ew
|
|||
\eps{1}{\ew{},\LDA}
|
||||
& = \eHc{1} + 2(1-\ew{}) \eJ{11} + 2\ew{} \eJ{12}
|
||||
+ \frac{1}{2} \int \qty{ \left. \pdv{\e{\xc}{\LDA}(\n{}{})}{\n{}{}} \right|_{\n{}{} = \n{}{\ew{}}(\br{})} \n{}{\ew{}}(\br{})
|
||||
+ \e{\xc}{\LDA}[\n{}{\ew{}}(\br{})] } \n{}{(0)}(\br{}) d\br{},
|
||||
+ \e{\xc}{\LDA}(\n{}{\ew{}}(\br{})) } \n{}{(0)}(\br{}) d\br{},
|
||||
\\
|
||||
\eps{2}{\ew{},\LDA}
|
||||
& = \eHc{2} + 2(1-\ew{}) \eJ{12} + 2 \ew{} \eJ{22}
|
||||
+ \frac{1}{2} \int \qty{ \left. \pdv{\e{\xc}{\LDA}(\n{}{})}{\n{}{}} \right|_{\n{}{} = \n{}{\ew{}}(\br{})} \n{}{\ew{}}(\br{})
|
||||
+ \e{\xc}{\LDA}[\n{}{\ew{}}(\br{})] } \n{}{(1)}(\br{}) d\br{},
|
||||
+ \e{\xc}{\LDA}(\n{}{\ew{}}(\br{})) } \n{}{(1)}(\br{}) d\br{},
|
||||
\end{align}
|
||||
\end{subequations}
|
||||
|
||||
|
|
@ -802,20 +799,22 @@ The two first terms are simply $\Eps{0}{\ew{}} = 2 \eps{1}{\ew{}}$, $\Eps{1}{\ew
|
|||
\eps{1}{\ew{},\eLDA}
|
||||
& = \eHc{1} + 2(1-\ew{}) \eJ{11} + 2\ew{} \eJ{12}
|
||||
+ \frac{1}{2} \int \qty{ \left. \pdv{\be{\xc}{\ew{}}(\n{}{})}{\n{}{}} \right|_{\n{}{} = \n{}{\ew{}}(\br{})} \n{}{\ew{}}(\br{})
|
||||
+ \be{\xc}{\ew{}}[\n{}{\ew{}}(\br{})] } \n{}{(0)}(\br{}) d\br{},
|
||||
+ \be{\xc}{\ew{}}(\n{}{\ew{}}(\br{})) } \n{}{(0)}(\br{}) d\br{},
|
||||
\\
|
||||
\eps{2}{\ew{},\eLDA}
|
||||
& = \eHc{2} + 2(1-\ew{}) \eJ{12} + 2\ew{} \eJ{22}
|
||||
+ \frac{1}{2} \int \qty{ \left. \pdv{\be{\xc}{\ew{}}(\n{}{})}{\n{}{}} \right|_{\n{}{} = \n{}{\ew{}}(\br{})} \n{}{\ew{}}(\br{})
|
||||
+ \be{\xc}{\ew{}}[\n{}{\ew{}}(\br{})] } \n{}{(1)}(\br{}) d\br{},
|
||||
+ \be{\xc}{\ew{}}(\n{}{\ew{}}(\br{})) } \n{}{(1)}(\br{}) d\br{}.
|
||||
\end{align}
|
||||
\end{subequations}
|
||||
respectively.
|
||||
|
||||
The derivative discontinuity is modelled by the last term of the RHS of Eq.~\eqref{eq:dEdw}.
|
||||
Note that this contribution is only non-zero in the case of an explicitly weight-dependent functional [see Eq.~\eqref{eq:dexcdw}].
|
||||
\end{widetext}
|
||||
|
||||
The derivative discontinuity is modelled by the last term of the right-hand-side of Eq.~\eqref{eq:dEdw}.
|
||||
Note that this contribution is only non-zero in the case of an explicitly weight-dependent functional [see Eq.~\eqref{eq:dexcdw}].
|
||||
|
||||
Numerical results are reported in Table \ref{tab:Energies}.
|
||||
|
||||
|
||||
%%%%%%%%%%%%%%%%%%
|
||||
%%% CONCLUSION %%%
|
||||
%%%%%%%%%%%%%%%%%%
|
||||
|
|
|
|||
Loading…
Reference in New Issue
Block a user