380 lines
23 KiB
TeX
380 lines
23 KiB
TeX
%****************************************************************
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\section{
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\label{sec:ExGLDA}
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Exchange functionals based on finite uniform electron gases}
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%****************************************************************
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In this section, we show how to use these finite UEGs (FUEGs) to create a new type of exchange functionals applicable to atoms, molecules and solids \cite{ExGLDA17}.
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We have successfully applied this strategy to one-dimensional systems \cite{1DChem15, SBLDA16, Leglag17, ESWC17}, for which we have created a correlation functional based on this idea \cite{gLDA14, Wirium14}.
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%****************************************************************
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\subsection{Theory}
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%****************************************************************
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Within DFT, one can write the total exchange energy as the sum of its spin-up ($\sigma=$ $\upa$) and spin-down ($\sigma=$ $\dwa$) contributions:
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\begin{equation}
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E_\text{x} = E_{\text{x},\upa} + E_{\text{x},\dwa},
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\end{equation}
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where
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\begin{equation}
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E_{\text{x},\sigma} = \int e_{\text{x},\sigma}(\rho_\sigma,\nabla \rho_\sigma, \tau_\sigma, \ldots) \, \rho_\sigma(\br) \, d\br,
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\end{equation}
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and $\rs$ is the electron density of the spin-$\sigma$ electrons.
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Although, for sake of simplicity, we sometimes remove the subscript $\sigma$, we only use spin-polarised quantities from hereon.
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The first-rung LDA exchange functional (or D30 \cite{Dirac30}) is based on the IUEG \cite{WIREs16} and reads
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\begin{equation}
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\exsLDA(\rs) = \CxLDA \rs^{1/3},
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\end{equation}
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where
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\begin{equation}
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\CxLDA = - \frac{3}{2} \qty( \frac{3}{4\pi} )^{1/3}.
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\end{equation}
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A GGA functional (second rung) is defined as
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\begin{equation}
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\label{eq:GGA}
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\exsGGA(\rs,\xs)
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= e_{\text{x},\sigma}^\text{LDA}(\rs) \FxGGA(\xs),
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\end{equation}
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where $\FxGGA$ is the GGA enhancement factor depending only on the reduced gradient
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\begin{equation}
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x = \frac{\abs{\nabla \rho}}{\rho^{4/3}},
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\end{equation}
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and
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\begin{equation}
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\lim_{x \to 0} \FxGGA(x) = 1,
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\end{equation}
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i.e.~a well thought-out GGA functional reduces to the LDA for homogeneous systems.
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Similarly, motivated by the work of Becke \cite{Becke00} and our previous investigations \cite{gLDA14, Wirium14}, we define an alternative second-rung functional (see Fig.~\ref{fig:Jacob-rev}) that we call generalised LDA (GLDA)
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\begin{equation}
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\label{eq:GLDA}
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\exsGLDA(\rs,\as) = \exsLDA(\rs) \FxGLDA(\as).
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\end{equation}
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%%% FIGURE 1 %%%
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\begin{figure}
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\centering
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\includegraphics[width=0.5\linewidth]{../Chapter3/fig/Jacob-rev}
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\caption{
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\label{fig:Jacob-rev}
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Jacob's ladder of DFT revisited.}
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\end{figure}
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%%%
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By definition, a GLDA functional only depends on the electron density and the curvature of the Fermi hole (see Fig.~\ref{fig:Jacob-rev}):
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\begin{equation}
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\label{eq:eta-def}
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\alpha = \frac{\tau - \tau_\text{W}}{\tau_\text{IUEG}} = \frac{\tau}{\tau_\text{IUEG}} - \frac{x^2}{4 \Cf},
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\end{equation}
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which measures the tightness of the exchange hole around an electron \cite{Becke83, Dobson91}.
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In Eq.~\eqref{eq:eta-def},
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\begin{equation}
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\tau_\text{W} = \frac{\abs{\nabla\rho}^2}{4\,\rho}
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\end{equation}
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is the von Weizs{\"a}cker kinetic energy density \cite{vonWeizsacker35}, and
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\begin{equation}
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\tau_\text{IUEG} = \Cf \rho^{5/3}
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\end{equation}
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is the kinetic energy density of the IUEG \cite{WIREs16}, where
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\begin{equation}
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\Cf = \frac{3}{5} (6\pi^2)^{2/3}.
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\end{equation}
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The dimensionless parameter $\alpha$ has two characteristic features: i) $\alpha=0$ for any one-electron system, and ii) $\alpha=1$ for the IUEG.
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Some authors call $\alpha$ the inhomogeneity parameter but we will avoid using this term as we are going to show that $\alpha$ can have distinct values in homogeneous systems.
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For well-designed GLDA functionals, we must ensure that
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\begin{equation}
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\label{eq:limGLDA}
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\lim_{\alpha \to 1} \FxGLDA(\alpha) = 1,
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\end{equation}
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i.e.~the GLDA reduces to the LDA for the IUEG.\footnote{While some functionals only use the variable $\tau$ \cite{Ernzerhof99, Eich14}, we are not aware of any functional only requiring $\alpha$.}
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Although any functional depending on the reduced gradient $x$ and the kinetic energy density $\tau$ is said to be of MGGA type, here we will define a third-rung MGGA functional as depending on $\rho$, $x$ and $\alpha$:
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\begin{equation}
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\exsMGGA(\rs,\xs,\as) = \exsLDA(\rs) \FxMGGA(\xs,\as),
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\end{equation}
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where one should ensure that
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\begin{equation}
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\label{eq:limMGGA}
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\lim_{x \to 0} \lim_{\alpha \to 1} \FxMGGA(x,\alpha) = 1,
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\end{equation}
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i.e.~the MGGA reduces to the LDA for an infinite homogeneous system.
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The Fermi hole curvature $\alpha$ has been shown to be a better variable than the kinetic energy density $\tau$ as one can discriminate between covalent ($\alpha=0$), metallic ($\alpha \approx 1$) and weak bonds ($\alpha \gg 0$) \cite{Kurth99, TPSS, revTPSS, MS0, Sun13a, MS1_MS2, MVS, SCAN, Sun16b}.
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The variable $\alpha$ is also related to the electron localisation function (ELF) designed to identify chemical bonds in molecules \cite{Becke83, ELF}.
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Moreover, by using the variables $x$ and $\alpha$, we satisfy the correct uniform coordinate density-scaling behaviour \cite{Levy85}.
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In conventional MGGAs, the dependence in $x$ and $\alpha$ can be strongly entangled, while, in GGAs for example, $\rho$ and $x$ are strictly disentangled as illustrated in Eq.~\eqref{eq:GGA}.
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Therefore, it feels natural to follow the same strategy for MGGAs.
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Thus, we consider a special class of MGGA functionals (rung $2.9$ in Fig.~\ref{fig:Jacob-rev}) that we call factorable MGGAs (FMGGAs)
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\begin{equation}
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\exsFMGGA(\rs,\xs,\as) = \exsLDA(\rs) \FxFMGGA(\xs,\as),
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\end{equation}
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where the enhancement factor is written as
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\begin{equation}
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\FxFMGGA(x,\alpha) = \FxGGA(x) \FxGLDA(\alpha).
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\end{equation}
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By construction, $\FxFMGGA$ fulfills Eq.~\eqref{eq:limMGGA} and the additional physical limits
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\begin{subequations}
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\begin{align}
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\lim_{x \to 0} \FxFMGGA(x,\alpha) & = \FxGLDA(\alpha),
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\\
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\lim_{\alpha \to 1} \FxFMGGA(x,\alpha) & = \FxGGA(x).
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\end{align}
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\end{subequations}
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The MVS functional designed by Sun, Perdew and Ruzsinszky is an example of FMGGA functional \cite{MVS}.
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Unless otherwise stated, all calculations have been performed self-consistently with a development version of the Q-Chem4.4 package \cite{qchem4} using the aug-cc-pVTZ basis set \cite{Dunning89, Kendall92, Woon93, Woon94, Woon95, Peterson02}.
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To remove quadrature errors, we have used a very large quadrature grids consisting of 100 radial points (Euler-MacLaurin quadrature) and 590 angular points (Lebedev quadrature).
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As a benchmark, we have calculated the (exact) unrestricted HF (UHF) exchange energies.
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%****************************************************************
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\subsection{GLDA exchange functionals}
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%****************************************************************
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As stated in the previous section, 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{Avery, Avery93}.
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We confine our attention to ferromagnetic (i.e.~spin-polarised) systems in which each orbital with $\ell = 0, 1, \ldots , L_{\sigma}$ is occupied by one spin-up or spin-down electron.
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As mentioned in the previous section, this yields an electron density that is uniform over the surface of the sphere.
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Note that the present paradigm is equivalent to the jellium model \cite{WIREs16} for $L_{\sigma} \to \infty$.
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We refer the reader to Ref.~\cite{Glomium11} for more details about this paradigm.
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The number of spin-$\sigma$ electrons is
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\begin{equation}
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\ns = \frac{1}{3} (\Ls+1)(\Ls+3/2)(\Ls+2),
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\end{equation}
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and their one-electron uniform density around the 3-sphere is
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\begin{equation}
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\rs = \frac{\ns}{V} = \frac{(\Ls+2)(\Ls+3/2)(\Ls+1)}{6 \pi^2 R^3},
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\end{equation}
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where $V = 2 \pi^2 R^3$ is the surface of a 3-sphere of radius $R$.
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Moreover, using Eq.~\eqref{eq:eta-def}, one can easily derive that \cite{gLDA14, Wirium14}
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\begin{equation}
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\label{eq:alpha}
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\as = \frac{\Ls(\Ls+3)}{\qty[ (\Ls+1)(\Ls+3/2)(\Ls+2) ]^{2/3}},
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\end{equation}
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which yields
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\begin{align}
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\lim_{\ns \to 1 } \as & = 0,
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&
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\lim_{\ns \to \infty } \as & = 1.
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\end{align}
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We recover the results that $\alpha = 0$ in a one-electron system (here a one-electron FUEG), and that $\alpha = 1$ in the IUEG.
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In particular, we have shown that the exchange energy of these systems can be written as \cite{Glomium11, Jellook12}
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\begin{equation}
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\label{eq:Ex}
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E_{\text{x},\sigma}(\Ls) = \Cx(\Ls) \int \rs^{4/3} d\bm{r}.
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\end{equation}
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where
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\begin{equation}
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\label{eq:CxGLDA}
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\Cx(L)
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= \CxLDA \frac{\frac{1}{2} \qty( L+\frac{5}{4}) \qty(L+\frac{7}{4}) \qty[\frac{1}{2} H_{2 L+\frac{5}{2}} + \ln 2]
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+ \qty(L+\frac{3}{2})^2 \qty(L^2+3 L+\frac{13}{8})}
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{\qty[(L+1) \qty(L+\frac{3}{2}) (L+2)]^{4/3}}
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\end{equation}
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and $H_{k}$ is an harmonic number \cite{NISTbook}.
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Therefore, thanks to the one-to-one mapping between $\Ls$ and $\as$ evidenced by Eq.~\eqref{eq:alpha}, we have created the \gX~functional
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\begin{equation}
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\label{eq:FxGLDA}
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\FxgX(\alpha) = \frac{\CxGLDA(0)}{\CxGLDA(1)}
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\\
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+ \alpha \frac{c_0+c_1\,\alpha}{1+(c_0+c_1-1)\alpha} \qty[ 1 - \frac{\CxGLDA(0)}{\CxGLDA(1)} ],
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\end{equation}
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where $c_0 = +0.827 411$, $c_1 = -0.643 560$, and
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\begin{align}
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\CxGLDA(1) & = \CxLDA = - \frac{3}{2} \qty( \frac{3}{4\pi} )^{1/3},
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\\
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\CxGLDA(0) & = -\frac{4}{3} \qty(\frac{2}{\pi})^{1/3}.
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\end{align}
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The parameters $c_0$ and $c_1$ of the \gX~enhancement factor \eqref{eq:FxGLDA} have been obtained by fitting the exchange energies of these FUEGs for $ 1 \le L \le 10$ given by Eq.~\eqref{eq:Ex}.
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$\FxgX$ automatically fulfils the constraint given by Eq.~\eqref{eq:limGLDA}.
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Moreover, because $ 1\le \FxgX \le 1.233$, it breaks only slightly the tight Lieb-Oxford bound \cite{Lieb81, Chan99, Odashima09} $\Fx < 1.174$ derived by Perdew and coworkers for two-electron systems \cite{Perdew14, Sun16a}.
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This is probably due to the non-zero curvature of these FUEGs.
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Albeit very simple, the functional form \eqref{eq:FxGLDA} is an excellent fit to Eq.~\eqref{eq:CxGLDA}.
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In particular, $\FxgX$ is linear in $\alpha$ for small $\alpha$, which is in agreement with Eq.~\eqref{eq:CxGLDA} \cite{Glomium11}.
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Also, Eq.~\eqref{eq:CxGLDA} should have an infinite derivative at $\alpha=1$ and approached as $\sqrt{1-\alpha} \ln(1-\alpha)$.
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Equation \eqref{eq:FxGLDA} does not behave that way.
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However, it has a marginal impact on the numerical results.
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As one can see in Fig.~\ref{fig:FxGLDA}, albeit being created with FUEGs, the \gX~functional has a fairly similar form to the common MGGA functionals, such as MS0 \cite{MS0}, MS1 \cite{MS1_MS2}, MS2 \cite{MS1_MS2}, MVS \cite{MVS}, and SCAN \cite{SCAN} for $ 0 \le \alpha \le 1$.
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This is good news for DFT as it shows that we recover functionals with similar physics independently of the paradigm used to design them.
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However, around $\alpha \approx 1$, the behaviour of $\FxgX$ is very different from other MGGAs (except for MVS) due to the constraint of the second-order gradient expansion (which is not satisfied in our case) \cite{Ma68}.
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For $ 0 \le \alpha \le 1$, it is also instructive to note that the \gX~functional is an upper bound of all the MGGA functionals.
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Taking into account the inhomogeneity of the system via the introduction of $x$ should have the effect of decreasing the MGGA enhancement factor (at least for $0 \le \alpha \le 1$).
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Unlike other functionals, we follow a rather different approach and guide our functional between $\alpha=0$ and $1$ using FUEGs.
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For example, the MS0 functional uses the exact exchange energies of non-interacting hydrogenic anions to construct the functional from $\alpha = 0$ to $1$ \cite{Staroverov04, MS0}, while revTPSS has no constraint to guide itself for this range of $\alpha$ \cite{revTPSS}.
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Nonetheless, because these uniform systems only give valuable information in the range $0 \le \alpha \le 1$, we must find a different way to guide our functional for $\alpha > 1$.\footnote{Except for one- and two-electron systems, any atomic and molecular systems has region of space with $\alpha_\sigma > 1$, as discussed in details by Sun et al.\cite{Sun13a}}
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To do so, we have extended the \gX~functional beyond $\alpha = 1$ using a simple one-parameter extrapolation:
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\begin{equation}
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\label{eq:FxGMVS}
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\FxGX(\alpha) =
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\begin{cases}
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\FxgX(\alpha), & 0 \le \alpha \le 1,
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\\
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1 + (1-\alpha_\infty) \frac{1-\alpha}{1+\alpha}, & \alpha > 1,
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\end{cases}
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\end{equation}
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where $\alpha_\infty$ is an adjustable parameter governing the value of $\FxGX$ when $\alpha \to \infty$.
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For large $\alpha$, $\FxGX$ converges to $\alpha_\infty$ as $\alpha^{-1}$, similarly to the MVS functional \cite{MVS}.
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Far from claiming that this choice is optimal, we have found that the simple functional form \eqref{eq:FxGMVS} for $\alpha > 1$ yields satisfactory results (see below).
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%%% FIG 2 %%%
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\begin{figure*}
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\centering
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\includegraphics[width=0.4\linewidth]{../Chapter3/fig/fig2a}
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\includegraphics[width=0.4\linewidth]{../Chapter3/fig/fig2b}
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\caption{
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\label{fig:FxGLDA}
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Enhancement factors $\FxGLDA(\alpha)$ or $\FxMGGA(x=0,\alpha)$ as a function of $\alpha$ for various GLDA and MGGA exchange functionals.
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The TPSS functional is represented as a dot-dashed line, the MS family of functionals (MS0, MS1 and MS2) are represented as dashed lines, while the MVS and SCAN functionals are depicted with solid lines.
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The new functionals \gX~and \PBEGX~are represented with thick black lines.
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Note that $\FxgX(\alpha) = \FxPBEGX(0,\alpha)$ for $ 0 \le \alpha \le 1$.
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For $\FxPBEGX$, $\alpha_\infty = +0.852$.}
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\end{figure*}
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%%% %%%
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Following the seminal work of Sham \cite{Sham71} and Kleinman \cite{Kleinman84, Antoniewicz85, Kleinman88} (see also Ref.~\cite{Svendsen96}), it is also possible, using linear response theory, to derive a second-order gradient-corrected functional.
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However, it does not provide any information for $\alpha >1$.
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The performance of the \GX~functional is illustrated in Table \ref{tab:atoms}.
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Although \GX~is an improvement compared to LDA, even for one- and two-electron systems, we observe that the \GX~functional cannot compete with GGAs and MGGAs in terms of accuracy.
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%%% TABLE 1 %%%
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\begin{table*}
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\centering
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\caption{
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\label{tab:atoms}
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Reduced (i.e.~per electron) mean error (ME) and mean absolute error (MAE) (in kcal/mol) of the error (compared to UHF) in the exchange energy of the hydrogen-like ions, helium-like ions and first 18 neutral atoms for various LDA, GGA, GLDA, FMGGA and MGGA functionals.
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For the hydrogen-like ions, the exact density has been used for all calculations.}
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\begin{tabular}{llcccccc}
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\hline
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& & \mc{2}{c}{hydrogen-like ions} & \mc{2}{c}{helium-like ions} & \mc{2}{c}{neutral atoms} \\
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\cline{3-4} \cline{5-6} \cline{7-8}
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& & ME & MAE & ME & MAE & ME & MAE \\
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\hline
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LDA & D30 & $153.5$ & $69.7$ & $150.6$ & $69.5$ & $70.3$ & $9.1$ \\
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GGA & B88 & $9.5$ & $4.3$ & $9.3$ & $4.7$ & $2.8$ & $0.5$ \\
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& G96 & $4.4$ & $2.0$ & $4.4$ & $2.2$ & $2.1$ & $0.5$ \\
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& PW91 & $19.4$ & $8.8$ & $19.1$ & $9.3$ & $4.5$ & $0.8$ \\
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& PBE & $22.6$ & $10.3$ & $22.3$ & $10.7$ & $7.4$ & $0.6$ \\
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GLDA & \GX & $61.8$ & $123.5$ & $61.0$ & $122.0$ & --- & --- \\
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FMGGA & MVS & $0.0$ & $0.0$ & $0.3$ & $0.2$ & $2.7$ & $0.9$ \\
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& \PBEGX & $0.0$ & $0.0$ & $0.7$ & $0.4$ & $1.0$ & $1.1$ \\
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MGGA & M06-L & $44.4$ & $88.8$ & $12.0$ & $24.0$ & $4.2$ & $2.9$ \\
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& TPSS & $0.0$ & $0.0$ & $0.7$ & $0.4$ & $0.7$ & $1.1$ \\
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& revTPSS & $0.0$ & $0.0$ & $0.5$ & $0.3$ & $3.5$ & $2.5$ \\
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& MS0 & $0.0$ & $0.0$ & $0.4$ & $0.2$ & $1.3$ & $2.4$ \\
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& SCAN & $0.0$ & $0.0$ & $0.3$ & $0.2$ & $1.2$ & $1.6$ \\
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\hline
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\end{tabular}
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\end{table*}
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%%%
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%****************************************************************
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\subsection{FMGGA exchange functionals}
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%****************************************************************
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One of the problem of GLDA functionals is that they cannot discriminate between homogeneous and inhomogeneous one-electron systems, for which we have $\alpha = 0$ independently of the value of the reduced gradient $x$.
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For example, the \GX~functional is exact for one-electron FUEGs, while it is inaccurate for the hydrogen-like ions.
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Unfortunately, it is mathematically impossible to design a GLDA functional exact for these two types of one-electron systems.
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To cure this problem, we couple the \GX~functional designed above with a GGA enhancement factor to create a FMGGA functional.
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We have chosen a PBE-like GGA factor, i.e.
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\begin{equation}
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\FxPBEGX(x,\alpha) =\Fx^\text{PBE}(x) \FxGX(\alpha),
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\end{equation}
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where
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\begin{equation}
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\label{eq:FxPBEGX}
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\Fx^\text{PBE}(x) = \frac{1}{1+\mu\,x^2}.
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\end{equation}
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Similarly to various MGGAs (such as TPSS \cite{TPSS}, MVS \cite{MVS}, or SCAN \cite{SCAN}), we use the hydrogen atom as a ``norm'', and determine that $\mu = +0.001 015 549$ reproduces the exact exchange energy of the ground state of the hydrogen atom.
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Also, we have found that $\alpha_\infty = +0.852$ yields excellent exchange energies for the first 18 neutral atoms.
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Unlike \GX, \PBEGX~is accurate for both the (inhomogeneous) hydrogen-like ions and the (homogeneous) one-electron FUEGs, and fulfils the negativity constraint and uniform density scaling \cite{SCAN, Perdew16}.
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The right graph of Fig.~\ref{fig:FxGLDA} shows the behaviour of the MGGA enhancement factor for $x = 0$ as a function of $\alpha$.
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Looking at the curves for $\alpha > 1$, we observe that TPSS has a peculiar enhancement factor which slowly raises as $\alpha$ increases.
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All the other functionals (including \PBEGX) decay more or less rapidly with $\alpha$.
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We note that \PBEGX~and MVS behave similarly for $\alpha > 1$, though their functional form is different.
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%%% FIG 3 %%%
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\begin{figure}
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\centering
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\includegraphics[width=0.6\linewidth]{../Chapter3/fig/fig3}
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\caption{
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\label{fig:FxGGA}
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Enhancement factors $\FxGGA(x)$ or $\FxMGGA(x,\alpha=1)$ as a function of $x$ for various GGA, FMGGA and MGGA exchange functionals.
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The GGA functionals are represented in solid lines, while MGGAs are depicted in dashed lines.
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The new functional \PBEGX~is represented with a thick black line.}
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\end{figure}
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%%% %%%
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Figure \ref{fig:FxGGA} evidences a fundamental difference between GGAs and MGGAs: while the enhancement factor of conventional GGAs does increase monotonically with $x$ and favour inhomogeneous electron densities, $\FxMGGA$ decays monotonically with respect to $x$.
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This is a well-known fact: the $x$- and $\alpha$-dependence are strongly coupled, as suggested by the relationship \eqref{eq:eta-def}.
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Therefore, the $x$-dependence can be sacrificed if the $\alpha$-dependence is enhanced \cite{MS0, MVS, SCAN}.
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Similarly to $\FxPBEGX$, $\Fx^\text{MVS}$ and $\Fx^\text{SCAN}$ decay monotonically with $x$ (although not as fast as \PBEGX), while earlier MGGAs such as TPSS and MS0 have a slowly-increasing enhancement factor.
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We have observed that one needs to use a bounded enhancement factor at large $x$ (as in Eq.~\eqref{eq:FxPBEGX}) in order to be able to converge self-consistent field (SCF) calculations.
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Indeed, using an unbounded enhancement factor (as in B88 \cite{B88} or G96 \cite{G96}) yields divergent SCF KS calculations.
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Finally, we note that, unlike TPSS, \PBEGX~does not suffer from the order of limits problem \cite{regTPSS}.
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%%% TABLE 2 %%%
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\begin{table}
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\centering
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\caption{
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\label{tab:molecules}
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Reduced (i.e.~per electron) mean error (ME) and mean absolute error (MAE) (in kcal/mol) of the error (compared to the experimental value) in the atomisation energy ($E_\text{atoms} - E_\text{molecule}$) of diatomic molecules at experimental geometry for various LDA, GGA and MGGA exchange-correlation functionals.
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Experimental geometries are taken from Ref.~\cite{HerzbergBook}.}
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\begin{tabular}{lllcc}
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\hline
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& \mc{2}{c}{functional} & \mc{2}{c}{diatomics} \\
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\cline{2-3} \cline{4-5}
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& exchange & correlation & ME & MAE \\
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\hline
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LDA & D30 & VWN5 & $1.8$ & $3.7$ \\
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GGA & B88 & LYP & $0.6$ & $1.2$ \\
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& PBE & PBE & $0.7$ & $1.2$ \\
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MGGA & M06-L & M06-L & $0.4$ & $0.7$ \\
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& TPSS & TPSS & $0.6$ & $1.1$ \\
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& revTPSS & revTPSS & $0.6$ & $1.2$ \\
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& MVS & regTPSS & $0.5$ & $0.9$ \\
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& SCAN & SCAN & $0.4$ & $0.7$ \\
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& \PBEGX & PBE & $0.6$ & $1.2$ \\
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& \PBEGX & regTPSS & $0.6$ & $1.1$ \\
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& \PBEGX & LYP & $0.6$ & $1.1$ \\
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& \PBEGX & TPSS & $0.7$ & $1.3$ \\
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& \PBEGX & revTPSS & $0.8$ & $1.5$ \\
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& \PBEGX & SCAN & $0.6$ & $1.0$ \\
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|
\hline
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|
\end{tabular}
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|
\end{table}
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|
%%%
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%%% FIG 4 %%%
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|
\begin{figure}
|
|
\centering
|
|
\includegraphics[width=0.6\linewidth]{../Chapter3/fig/fig4}
|
|
\caption{
|
|
\label{fig:error}
|
|
Reduced (i.e.~per electron) error (in kcal/mol) in atomic exchange energies of the first 18 neutral atoms of the periodic table for the B88 (red), TPSS (blue), MVS (orange), SCAN (purple) and \PBEGX~(thick black) functionals.}
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|
\end{figure}
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|
%%% %%%
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|
|
How good are FMGGAs?
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|
This is the question we would like to answer here.
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In other word, we would like to know whether or not our new simple FMGGA functional called \PBEGX~is competitive within MGGAs.
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|
Unlike GGAs and some of the MGGAs (like M06-L), by construction, \PBEGX~reproduces exactly the exchange energy of the hydrogen atom and the hydrogenic ions (\ce{He^+}, \ce{Li^2+}, \ldots) due to its dimensional consistency (see Table \ref{tab:atoms}).
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|
\PBEGX~also reduces the error for the helium-like ions (\ce{H^-}, \ce{He}, \ce{Li^+}, \ldots) by one order of magnitude compared to GGAs, and matches the accuracy of MGGAs.
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|
For the first 18 neutral atoms (Table \ref{tab:atoms} and Fig.~\ref{fig:error}), \PBEGX~is as accurate as conventional MGGAs with a mean error (ME) and mean absolute error (MAE) of $1.0$ and $1.1$ kcal/mol.
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|
From the more conventional MGGAs, the TPSS and SCAN functionals are the best performers for neutral atoms with MEs of $0.7$ and $1.2$ kcal/mol, and MAEs of $1.1$ and $1.6$ kcal/mol.
|
|
\PBEGX~lies just in-between these two MGGAs.
|
|
|
|
We now turn our attention to diatomic molecules for which errors in the atomisation energy ($E_\text{atoms} - E_\text{molecule}$) are reported in Table \ref{tab:molecules} for various combinations of exchange and correlation functionals.
|
|
In particular, we have coupled our new \PBEGX~exchange functional with the PBE \cite{PBE}, regTPSS \cite{regTPSS} (also called vPBEc) and LYP \cite{LYP} GGA correlation functionals, as well as the TPSS, \cite{TPSS} revTPSS \cite{revTPSS} and SCAN \cite{SCAN} MGGA correlation functionals.
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|
|
|
Although very lightly parametrised on atoms, \PBEGX~is also accurate for molecules.
|
|
Interestingly, the results are mostly independent of the choice of the correlation functional with MEs ranging from $0.6$ and $0.8$ kcal/mol, and MAEs from $1.0$ and $1.5$ kcal/mol.
|
|
\PBEGX~is only slightly outperformed by the SCAN functional and the highly-parametrized M06-L functional, which have both a ME of $0.4$ kcal/mol and a MAE of $0.7$ kcal/mol.
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|