Tab III
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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 2020-04-07 11:01:45 +0200
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%% Created for Pierre-Francois Loos at 2020-04-07 20:33:37 +0200
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%% Saved with string encoding Unicode (UTF-8)
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@article{Slater_1951,
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Author = {J. C. Slater},
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Date-Added = {2020-04-07 19:53:52 +0200},
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Date-Modified = {2020-04-07 19:54:32 +0200},
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Doi = {10.1103/PhysRev.81.385},
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Journal = {Phys. Rev.},
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Pages = {385},
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Title = {A Simplification of the Hartree-Fock Method},
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Volume = {81},
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Year = {1981}}
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@book{Slater_1974,
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Date-Added = {2020-04-07 19:48:23 +0200},
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Date-Modified = {2020-04-07 19:52:27 +0200},
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Publisher = {McGraw-Hill, New-York},
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Title = {The Self-Consistent Field for Molecular and Solids, Quantum Theory of Molecular and Solids},
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Volume = {4},
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Year = {1974}}
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@article{Gilbert_2008,
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Author = {A. T. B. Gilbert and N. A. Besley and P. M. W. Gill},
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Date-Added = {2020-04-07 15:46:23 +0200},
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Date-Modified = {2020-04-07 15:47:19 +0200},
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Journal = {J. Phys. Chem. A},
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Pages = {13164},
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Title = {Self-consistent field calculations of excited states using the Maximum Overlap Method (MOM)},
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Volume = {112},
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Year = {2008}}
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@article{Loos_2020,
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Author = {P. F. Loos and E. Fromager},
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Date-Added = {2020-04-07 10:59:44 +0200},
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@ -307,57 +307,89 @@ Numerical quadratures are performed with the \texttt{numgrid} library using 194
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This study deals only with spin-unpolarised systems, \ie, $\n{\uparrow}{} = \n{\downarrow}{} = \n{}{}/2$ (where $\n{\uparrow}{}$ and $\n{\downarrow}{}$ are the spin-up and spin-down electron densities).
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Moreover, we restrict our study to the case of a two-state ensemble (\ie, $\nEns = 2$) where both the ground state ($I=0$) and the first doubly-excited state ($I=1$) are considered.
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Although we will sometimes ``violate'' this variational constraint, we should have $0 \le \ew{} \le 1/2$ to ensure the GOK variational principle.
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However, the limit $\ew{} = 1$ is of particular interest as it corresponds to a genuine saddle point of the KS equations, and match perfectly the results obtained wit the maximum overlap method developed by Gilbert, Gill and coworkers. \cite{Gilbert_2008,Barca_2018a,Barca_2018b}
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Moreover, the limits $\ew{} = 0$ and $\ew{} = 1$ are the only two weights for which there is no ghost-interaction error.
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Results}
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\label{sec:res}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\subsection{Weight-independent local exchange-correlation functionals}
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\label{sec:S51}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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First, we compute the ensemble energy of the \ce{H2} molecule using the weight-independent Slater's local exchange functional, \cite{Dirac_1930}
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which is explicitly given by
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First, we compute the ensemble energy of the \ce{H2} molecule using the aug-cc-pVTZ basis set and the weight-independent Slater local exchange functional, \cite{Dirac_1930, Slater_1951} which is explicitly given by
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\begin{align}
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\e{\ex}{\text{S51}}(\n{}{}) & = \Cx{} \n{}{1/3},
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\e{\ex}{\text{S}}(\n{}{}) & = \Cx{} \n{}{1/3},
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&
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\Cx{} & = -\frac{3}{4} \qty(\frac{3}{\pi})^{1/3}.
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\end{align}
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The ensemble energy $\E{}{w}$ is depicted in Fig.~\ref{fig:Ew-H2} as a function of the weight $0 \le \ew{} \le 1$.
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The ensemble energy $\E{}{w}$ is depicted in Fig.~\ref{fig:Ew_H2} as a function of the weight $0 \le \ew{} \le 1$.
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Because this exchange functional does not depend on the ensemble weight, there is no contribution from the ensemble derivative term [last term in Eq.~\eqref{eq:dEdw}].
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As anticipated, $\E{}{w}$ is far from being linear, which means that the excitation energy obtained via the derivative of the local energy varies greatly with the weight (see Fig.~\ref{fig:Om-H2}).
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Note that the exact xc correlation ensemble functional should yield a perfectly linear energy and the same excitation energy independently of $\ew{}$.
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As anticipated, $\E{}{w}$ is far from being linear, which means that the excitation energy obtained via the derivative of the local energy varies greatly with the weight of the double excitation (see Fig.~\ref{fig:Om_H2}).
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Note that the exact xc correlation ensemble functional would yield a perfectly linear energy and the same value of the excitation energy independently of $\ew{}$.
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\begin{figure}
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\includegraphics[width=\linewidth]{Ew_H2}
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\caption{
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\ce{H2} at equilibrium bond length: deviation from linearity of the ensemble energy $\E{}{\ew{}}$ (in hartree) as a function of the weight of the double excitation $\ew{}$ for various functionals and the aug-cc-pVTZ basis set.
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\label{fig:Ew_H2}
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}
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\end{figure}
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As a first example, we compute the ensemble energy of the \ce{H2} molecule as a function of the weight $\ew{}$ using the SVWN5 local functional which corresponds to the combination of Slater's local exchange functional \cite{Dirac_1930} and the VNW5 local correlation functional. \cite{Vosko_1980}
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The SVWN5 xc functional is explicitly given by
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\begin{figure}
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\includegraphics[width=\linewidth]{Om_H2}
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\caption{
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\ce{H2} at equilibrium bond length: error (with respect to FCI) in excitation energy (in eV) as a function of the weight of the double excitation $\ew{}$ for various functionals and the aug-cc-pVTZ basis set.
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\label{fig:Om_H2}
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}
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\end{figure}
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Second, in order to remove this spurious curvature of the ensemble energy (which is partly due to the ghost-interaction error, but not only), one can easily reverse-engineer (for this particular system and basis set) a local exchange functional to make $\E{}{\ew{}}$ as linear as possible for $0 \le \ew{} \le 1$.
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Doing so, we have found that the present weight-dependent exchange functional (denoted as MSFL in the following), represented in Fig.~\ref{fig:Cx_H2},
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\begin{equation}
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\e{\xc}{\text{SVWN5}}(\n{}{}) = \e{\ex}{\text{S51}}(\n{}{}) + \e{\co}{\text{VWN5}}(\n{}{}),
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\e{\ex}{\ew{},\text{MSFL}}(\n{}{}) = \Cx{\ew{}} \n{}{1/3},
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\end{equation}
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with
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\begin{equation}
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\label{eq:Cxw}
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\Cx{\ew{}} = \Cx{} \qty{ 1 - \ew{} (1 - \ew{})\qty[ \alpha + \beta (\ew{} - 1/2) + \gamma (w - 1/2)^2 ]}
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\end{equation}
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and
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\begin{align}
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\e{\ex}{\text{S51}}(\n{}{}) & = \Cx{} \n{}{1/3},
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\alpha & = + 0.575\,178,
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&
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\Cx{} & = -\frac{3}{4} \qty(\frac{3}{\pi})^{1/3}.
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\beta & = - 0.021\,108,
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&
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\gamma & = - 0.367\,189,
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\end{align}
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makes the ensemble much more linear (see Fig.~\ref{fig:Ew_H2}), and the excitation energy much more stable and closer to the full configuration interaction (FCI) reference of $28.75$ eV \cite{Barca_2018a} (see Fig.~\ref{fig:Om_H2})
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As readily seen from Eq.~\eqref{eq:Cxw}, $\Cx{\ew{}}$ reduces to $\Cx{}$ for $\ew{} = 0$.
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Note that we are not only using data from $\ew{} = 0$ to $\ew{} = 1/2$, but we also consider $1/2 < \ew{} \le 1$.
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We ensure that the weight-dependent functional does not affect the two ghost-interaction-free limit at $\ew{} = 0$ and $1$.
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\begin{figure}
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\includegraphics[width=0.8\linewidth]{Cx_H2}
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\caption{
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$\Cx{\ew{}}/\Cx{\ew{}=0}$ as a function of $\ew{}$ [see Eq.~\eqref{eq:Cxw}].
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\label{fig:Cx_H2}
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}
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\end{figure}
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In a third time, we add up correlation via the VWN5 local correlation functional. \cite{Vosko_1980}
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For the sake of clarity, the explicit expression of the VWN5 functional is not reported here but it can be found in Ref.~\onlinecite{Vosko_1980}.
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The combination of the Slater and VWN5 functionals (SVWN5) yield a highly non-linear ensemble energy (as shown in Fig.~\ref{fig:Ew_H2}), while the combination of the MSFL and VWN5 functionals exhibit a small curvature and improved excitation energies, especially at small weights.
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%%%%%%%%%%%%%%%%%%
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%%% FUNCTIONAL %%%
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%%%%%%%%%%%%%%%%%%
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\section{Functional}
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\label{sec:func}
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The present work deals with the explicit construction of the (reduced) LDA xc functional $\e{\xc}{\bw}[\n{}{}]$ defined in Eq.~\eqref{eq:exc_def}.
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The generalisation to a larger number of states (in particular the inclusion of the first singly-excited state) is trivial and left for future work.
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%\section{Functional}
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%\label{sec:func}
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%The present work deals with the explicit construction of the (reduced) LDA xc functional $\e{\xc}{\bw}[\n{}{}]$ defined in Eq.~\eqref{eq:exc_def}.
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%The generalisation to a larger number of states (in particular the inclusion of the first singly-excited state) is trivial and left for future work.
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The construction of these two functionals is described below.
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Extension to spin-polarised systems will be reported in future work.
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%The construction of these two functionals is described below.
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%Extension to spin-polarised systems will be reported in future work.
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To build our weight-dependent xc functional, we propose to consider the singlet ground state and the first singlet doubly-excited state of a two-electron FUEG which consists of two electrons confined to the surface of a 3-sphere (also known as a glome).\cite{Loos_2009a,Loos_2009c,Loos_2010e}
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Fourth, in the spirit of our recent work \cite{Loos_2020}, we have designed a weight-dependent correlation functional.
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To build this weight-dependent correlation functional, we consider the singlet ground state and the first singlet doubly-excited state of a two-electron finite UEGs which consists of two electrons confined to the surface of a 3-sphere (also known as a glome).\cite{Loos_2009a,Loos_2009c,Loos_2010e}
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Notably, these two states have the same (uniform) density $\n{}{} = 2/(2\pi^2 R^3)$, where $R$ is the radius of the 3-sphere onto which the electrons are confined.
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Indeed, the orbitals for an electron on a 3-sphere of unit radius are the normalised hyperspherical harmonics $Y_{\ell\mu}$, where $\ell$ is the principal quantum number and $\mu$ is a composite index of the remaining two quantum numbers. \cite{AveryBook, Avery_1993}
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As mentioned above, we confine our attention to paramagnetic (or unpolarised) systems, and in particular to the simple two-electron system in which the orbital with $\ell = 0$ is doubly-occupied by one spin-up and one spin-down electron, thus yielding an electron density that is uniform on the 3-sphere.
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@ -365,8 +397,8 @@ Note that the present paradigm is equivalent to the IUEG model in the thermodyna
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We refer the interested reader to Refs.~\onlinecite{Loos_2011b,Loos_2017a} for more details about this paradigm.
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\subsection{Weight-dependent exchange functional}
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\label{sec:Ex}
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%\subsection{Weight-dependent exchange functional}
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%\label{sec:Ex}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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The reduced (\ie, per electron) Hartree-Fock (HF) energies for these two states are
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@ -377,54 +409,54 @@ The reduced (\ie, per electron) Hartree-Fock (HF) energies for these two states
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\e{\HF}{(1)}(\n{}{}) & = \frac{3\pi^{2}}{2} \qty(\frac{\n{}{}}{\pi})^{2/3} + \frac{176}{105} \qty(\frac{\n{}{}}{\pi})^{1/3}.
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\end{align}
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\end{subequations}
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These two energies can be conveniently decomposed as
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\begin{equation}
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\e{\HF}{(I)}(\n{}{}) = \kin{s}{(I)}(\n{}{}) + \e{\Ha}{(I)}(\n{}{}) + \e{\ex}{(I)}(\n{}{}),
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\end{equation}
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with
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\begin{subequations}
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\begin{align}
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\kin{s}{(0)}(\n{}{}) & = 0,
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&
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\kin{s}{(1)}(\n{}{}) & = \frac{3\pi^{2}}{2} \qty(\frac{\n{}{}}{\pi})^{2/3},
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\\
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\e{\Ha}{(0)}(\n{}{}) & = \frac{8}{3} \qty(\frac{\n{}{}}{\pi})^{1/3},
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&
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\e{\Ha}{(1)}(\n{}{}) & = \frac{352}{105} \qty(\frac{\n{}{}}{\pi})^{1/3},
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\\
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\e{\ex}{(0)}(\n{}{}) & = - \frac{4}{3} \qty(\frac{\n{}{}}{\pi})^{1/3},
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&
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\e{\ex}{(1)}(\n{}{}) & = - \frac{176}{105} \qty(\frac{\n{}{}}{\pi})^{1/3}.
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\end{align}
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\end{subequations}
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In analogy with the conventional Dirac exchange functional, \cite{Dirac_1930} we write down the exchange functional of each individual state as
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\begin{equation}
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\e{\ex}{(I)}(\n{}{}) = \Cx{(I)} \n{}{1/3},
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\end{equation}
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and we then obtain
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\begin{align}
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\Cx{(0)} & = - \frac{4}{3} \qty( \frac{1}{\pi} )^{1/3},
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&
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\Cx{(1)} & = - \frac{176}{105} \qty( \frac{1}{\pi} )^{1/3}.
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\end{align}
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We can now combine these two exchange functionals to create a weight-dependent exchange functional for a two-state ensemble
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\begin{equation}
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\label{eq:exw}
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\e{\ex}{\ew{}}(\n{}{})
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= (1-\ew{}) \e{\ex}{(0)}(\n{}{}) + \ew{} \e{\ex}{(1)}(\n{}{})
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= \Cx{\ew{}} \n{}{1/3}
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\end{equation}
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with
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\begin{equation}
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\Cx{\ew{}} = (1-\ew{}) \Cx{(0)} + \ew{} \Cx{(1)}.
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\end{equation}
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Conveniently, the weight dependence of the exchange functional can be transferred to the $\Cx{}$ coefficient; an expected feature from a theoretical point of view, yet a nice property from a more practical aspect.
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%These two energies can be conveniently decomposed as
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%\begin{equation}
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% \e{\HF}{(I)}(\n{}{}) = \kin{s}{(I)}(\n{}{}) + \e{\Ha}{(I)}(\n{}{}) + \e{\ex}{(I)}(\n{}{}),
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%\end{equation}
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%with
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%\begin{subequations}
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%\begin{align}
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% \kin{s}{(0)}(\n{}{}) & = 0,
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% &
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% \kin{s}{(1)}(\n{}{}) & = \frac{3\pi^{2}}{2} \qty(\frac{\n{}{}}{\pi})^{2/3},
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% \\
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% \e{\Ha}{(0)}(\n{}{}) & = \frac{8}{3} \qty(\frac{\n{}{}}{\pi})^{1/3},
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% &
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% \e{\Ha}{(1)}(\n{}{}) & = \frac{352}{105} \qty(\frac{\n{}{}}{\pi})^{1/3},
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% \\
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% \e{\ex}{(0)}(\n{}{}) & = - \frac{4}{3} \qty(\frac{\n{}{}}{\pi})^{1/3},
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% &
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% \e{\ex}{(1)}(\n{}{}) & = - \frac{176}{105} \qty(\frac{\n{}{}}{\pi})^{1/3}.
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%\end{align}
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%\end{subequations}
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%
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%In analogy with the conventional Dirac exchange functional, \cite{Dirac_1930} we write down the exchange functional of each individual state as
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%\begin{equation}
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% \e{\ex}{(I)}(\n{}{}) = \Cx{(I)} \n{}{1/3},
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%\end{equation}
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%and we then obtain
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%\begin{align}
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% \Cx{(0)} & = - \frac{4}{3} \qty( \frac{1}{\pi} )^{1/3},
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% &
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% \Cx{(1)} & = - \frac{176}{105} \qty( \frac{1}{\pi} )^{1/3}.
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%\end{align}
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%We can now combine these two exchange functionals to create a weight-dependent exchange functional for a two-state ensemble
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%\begin{equation}
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%\label{eq:exw}
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% \e{\ex}{\ew{}}(\n{}{})
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% = (1-\ew{}) \e{\ex}{(0)}(\n{}{}) + \ew{} \e{\ex}{(1)}(\n{}{})
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% = \Cx{\ew{}} \n{}{1/3}
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%\end{equation}
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%with
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%\begin{equation}
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% \Cx{\ew{}} = (1-\ew{}) \Cx{(0)} + \ew{} \Cx{(1)}.
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%\end{equation}
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%Conveniently, the weight dependence of the exchange functional can be transferred to the $\Cx{}$ coefficient; an expected feature from a theoretical point of view, yet a nice property from a more practical aspect.
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\subsection{Weight-dependent correlation functional}
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\label{sec:Ec}
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%\subsection{Weight-dependent correlation functional}
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%\label{sec:Ec}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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Based on highly-accurate calculations, \cite{Loos_2009a,Loos_2009c,Loos_2010e} one can write down, for each state, an accurate analytical expression of the reduced correlation energy \cite{Loos_2013a, Loos_2014a} via the following Pad\'e approximant
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@ -500,7 +532,7 @@ Combining these, we build a two-state weight-dependent correlation functional:
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%%% %%% %%% %%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\subsection{LDA-centered functional}
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%\subsection{LDA-centered functional}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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Our intent is to incorporate into standard functionals (which are ``universal'' in the sense that they do not depend on the number of electrons) information about excited states that will be extracted from finite systems (whose properties may depend on the number of electrons).
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@ -576,54 +608,50 @@ $\pdv{\E{\xc}{(1-\xi,\xi)}[\n{}{}]}{\xi}$ dans l'intégrale, non ? Can it be :
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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 hartree) of \ce{H2} with $\RHH = 1.4$ bohr for various methods with the STO-3G minimal basis.
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Excitation energies (in eV) of \ce{H2} with $\RHH = 1.4$ bohr for various methods and basis sets.
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\label{tab:Energies}
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}
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\begin{ruledtabular}
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\begin{tabular}{lddddddd}
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Method & \tabc{$\E{}{(0)}$} & \tabc{$\E{}{(1)}$} & \tabc{$\E{}{(1)} - \E{}{(0)}$}
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& \tabc{$\tE{}{\ew{} = 0}$} & \tabc{$\left. \pdv{\tE{}{\ew{}}}{\ew{}} \right|_{\ew{} = 0}$}
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& \tabc{$\tE{}{\ew{} = 1/2}$} & \tabc{$\left. \pdv{\tE{}{\ew{}}}{\ew{}} \right|_{\ew{} = 1/2}$} \\
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\begin{tabular}{llccccc}
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\mc{2}{c}{xc functional} \\
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\cline{1-2}
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exchange & correlation & Basis & GOK($\ew{} = 0$) & GOK($\ew{} = 1/2$) & LIM & MOM \\
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\hline
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HF & -1.11671 & 0.460576 & 1.57729 & -1.11671 & 2.49694 & -0.0981563 & 1.57729 \\
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LDA & -1.12120 & 0.379745 & 1.50095 & -1.12120 & 1.49536 & -0.370725 & 1.50565 \\
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eLDA & -1.12120 & 0.175337 & 1.29654 & -1.12120 & 1.31995 & -0.462421 & 1.30839 \\
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HF & & aug-cc-pVDZ & 38.52 & 30.86 & 34.55 & 28.65 \\
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& & aug-cc-pVTZ & 38.58 & 35.82 & 35.68 & 28.65 \\
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& & aug-cc-pVQZ & 39.12 & 35.94 & 35.64 & 28.65 \\
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\\
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S & & aug-cc-pVDZ & 38.40 & 27.35 & 23.54 & 26.60 \\
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& & aug-cc-pVTZ & 39.21 & 27.42 & 23.62 & 26.67 \\
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& & aug-cc-pVQZ & 39.78 & 27.42 & 23.62 & 26.67 \\
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\\
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S & VWN5 & aug-cc-pVDZ & 38.10 & 27.76 & 24.40 & 27.10 \\
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& & aug-cc-pVTZ & 38.54 & 27.81 & 24.46 & 27.17 \\
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& & aug-cc-pVQZ & 38.81 & 27.81 & 24.46 & 27.17 \\
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\\
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MSFL & & aug-cc-pVDZ & 26.83 & 26.51 & 26.53 & 26.60 \\
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& & aug-cc-pVTZ & 26.88 & 26.59 & 26.61 & 26.67 \\
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& & aug-cc-pVQZ & 26.82 & 26.60 & 26.62 & 26.67 \\
|
||||
\\
|
||||
MSFL & VWN5 & aug-cc-pVDZ & 28.54 & 26.94 & 27.48 & 27.10 \\
|
||||
& & aug-cc-pVTZ & 28.66 & 27.00 & 27.56 & 27.17 \\
|
||||
& & aug-cc-pVQZ & 28.64 & 27.00 & 27.56 & 27.17 \\
|
||||
\\
|
||||
MSFL & MSFL & aug-cc-pVDZ & 28.78 & 27.10 & 27.56 & 27.27 \\
|
||||
& & aug-cc-pVTZ & 28.90 & 27.16 & 27.64 & 27.34 \\
|
||||
& & aug-cc-pVQZ & 28.89 & 27.16 & 27.65 & 27.34 \\
|
||||
\\
|
||||
B88 & LYP & aug-mcc-pV8Z & & & & 28.42\fnm[1] \\
|
||||
B3 & LYP & aug-mcc-pV8Z & & & & 27.77\fnm[1] \\
|
||||
HF & LYP & aug-mcc-pV8Z & & & & 29.18\fnm[1] \\
|
||||
HF & & aug-mcc-pV8Z & & & & 28.65\fnm[1] \\
|
||||
HF & FCI & aug-mcc-pV8Z & & & & 28.75\fnm[1] \\
|
||||
\end{tabular}
|
||||
\end{ruledtabular}
|
||||
\fnt[1]{Reference \onlinecite{Barca_2018a}.}
|
||||
\end{table*}
|
||||
%%% %%% %%% %%%
|
||||
|
||||
%%% FIG 1 %%%
|
||||
\begin{figure}
|
||||
% \includegraphics[width=\linewidth]{fig/GSetDES_exact_HF_LDA_eLDA}
|
||||
\caption{
|
||||
Total energies (in hartree) of \ce{H2} as a function of $\RHH$ (in bohr) for various methods with the STO-3G minimal basis.
|
||||
%\label{fig:Energies}
|
||||
}
|
||||
\end{figure}
|
||||
%%% %%% %%% %%%
|
||||
|
||||
%%% FIG 2 %%%
|
||||
\begin{figure}
|
||||
% \includegraphics[width=\linewidth]{fig/ExcitationEnergyExact_wHF_wLDA_weLDA_w=0etw=0.5}
|
||||
\caption{
|
||||
Excitation energies (in hartree) of \ce{H2} as a function of $\RHH$ (in bohr) for various methods with the STO-3G minimal basis.
|
||||
%\label{fig:Energies}
|
||||
}
|
||||
\end{figure}
|
||||
%%% %%% %%% %%%
|
||||
|
||||
%%% FIG 3 %%%
|
||||
\begin{figure}
|
||||
% \includegraphics[width=\linewidth]{fig/EnsembleEnergy_wHF_wLDA_weLDA_wHFbarre_wLDAbarre_weLDAbarre_R=1.4}
|
||||
\caption{
|
||||
Ensemble energies (in hartree) of \ce{H2} with $\RHH = 1.4$ bohr as a function of the weight $\ew{}$ for various methods with the STO-3G minimal basis.
|
||||
%\label{tab:Energies}
|
||||
}
|
||||
\end{figure}
|
||||
%%% %%% %%% %%%
|
||||
|
||||
|
||||
%%%%%%%%%%%%%%%%%%
|
||||
%%% CONCLUSION %%%
|
||||
%%%%%%%%%%%%%%%%%%
|
||||
|
|
|
|||
Loading…
Reference in New Issue
Block a user