181 lines
14 KiB
TeX
181 lines
14 KiB
TeX
In this memoir, I have presented succinctly some of the research projects we have been pursuing in the last ten years.
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Moreover, I have discussed two of our current research topics in the final chapter.
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As concluding remarks, I would like to talk further about various research projects we would like to work on in the years to come.
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This is a non-exhaustive list in no particular order but I hope it will give to the reader a feeling about what we are trying to achieve.
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\paragraph{What is the exact Green function?}
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As mentioned earlier in the memoir, exactly solvable models are particularly valuable for testing theoretical approaches.
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Here, we would like to use the electrons-on-a-sphere model presented above to unveil the form of the exact Green function \cite{Berger14}.
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Green function-based methods allows an explicit incorporation of the electronic correlation effects through a summation of Feynman diagrams.
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Important properties such as total energies, ionisation potentials, electron affinities as well as photo-emission spectra can be obtained directly from the Green function.
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A particularly successful variant of these methods in electronic-structure calculations is the so-called GW approximation, which consists in evaluating the self-energy $\Sigma$ starting from the Green function $G$ using a sequence of self-consistent steps (see Fig.~\ref{fig:pentagon}).
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Thanks to this toy system, our preliminary results show that we might be able to compute self-consistently $\Sigma$.
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This will help us understand approximation such as G$_0$W$_0$ where one eschews the iterative process.
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More importantly, we might be able to obtain the closed-form expression of the vertex correction $\Gamma$ --- a quantity hardly accessible in real systems --- and complete the entire five-step self-consistent process (see Fig.~\ref{fig:pentagon}).
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This could shed lights on how to approximate wisely the vertex function in real systems.
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\begin{figure}
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\centering
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\includegraphics[width=0.4\linewidth]{../Conclusion/fig/pentagon}
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\caption{
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\label{fig:pentagon}
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Hedin's pentagon \cite{Hedin65}.}
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\end{figure}
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\paragraph{GLDA correlation functional}
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In Sec.~\ref{sec:ExGLDA}, we have presented an exchange functional based on FUEGs.
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In order to create the associated correlation functional, one would need to compute accurate energies of FUEGs for various number of electrons and densities.
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One of the method of choice to achieve this would probably be DMC.
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Sadly, DMC on a curved manifold (like a $D$-sphere) is not as easy as one would have thought.
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What are the modifications required to the VMC and DMC algorithms to perform calculations of electrons in curved manifolds?
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While the modifications required for the VMC algorithm are fairly straightforward \cite{SGF14, Nodes15, LowGlo15}, the modifications one has to do in the DMC algorithm are much more subtle.
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DMC on a curved manifold has received very little attention in the past.
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To the best of our knowledge, the only work related to this has been done by Ortiz and coworkers to calculate the energy of composite fermions on the Haldane sphere \cite{Melik97}.
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On a curved manifold, when moving the electrons, one has to make sure that the move keeps the electrons on the surface of the sphere.
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In VMC, the only required modification is the Metropolis acceptance probability \cite{Nodes15, LowGlo15}.
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On a curved manifold, the DMC algorithm requires modifications in the diffusion and drift processes.
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The branching process is not affected by the curved nature of the manifold.
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The major difference appears in the calculation of the short-time approximation of the Green function
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This ``curved'' DMC method could be efficiently implemented in the local QMC software package (QMC=Chem) developed by Scemama, Caffarel and coworkers \cite{Scemama13b}.
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This would allow to perform DMC calculations and obtain the near-exact energies of FUEGs required to build the three-dimensional version of the GLDA correlation functional, similarly to what we have done in the one-dimensional case \cite{gLDA14, Wirium14}.
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\paragraph{Symmetry-broken LDA}
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Within DFT, the LDA correlation functional is typically built by fitting the difference between the near-exact and HF energies of the UEG, together with analytic perturbative results from the high- and low-density regimes.
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Near-exact energies are obtained by performing accurate DMC calculations, while HF energies are usually assumed to be the Fermi fluid HF energy.
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However, it has been known since the seminal work of Overhauser \cite{Overhauser59, Overhauser62} that one can obtain lower, symmetry-broken (SB) HF energies at any density \cite{Trail03, Delyon08, Bernu08, Bernu11, Baguet13, Baguet14, Delyon15}.
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Recently, we have computed the SBHF energies of the one-dimensional UEG \cite{1DEG13, ESWC17} and constructed a SB version of the LDA (SBLDA) from these results \cite{SBLDA16}.
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The newly designed functional, which we have named SBLDA, has shown to surpass the performance of its LDA parent in providing better estimates of the correlation energy in one-dimensional systems \cite{gLDA14, Wirium14, 1DChem15, Ball15, Leglag17}.
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Based on the same methodology, we would like to design of new exchange and correlation functionals for two- and three-dimensional systems for which SBHF calculations have already been performed \cite{Trail03, Bernu11, Baguet13, Baguet14}.
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\paragraph{Resolution of geminals}
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The main idea behind the ``resolution of geminals'' (RG) is a generalisation of the resolution of the identity (RI).
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To explain what we mean by this, let us state the RI identity in its two-electron version, i.e.
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\begin{equation}
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\label{eq:2eRI}
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\delta(\ree) = \sum_{\mu}^\infty \dyad{\chi_\mu(\br_1)}{\chi_\mu(\br_2)},
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\end{equation}
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where $\delta(r)$ is the Dirac delta function and the one-electron basis set $\chi_\mu(\br)$ is formally complete.
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In other words, we have just ``resolved'' the Dirac delta function, i.e. we wrote a two-electron function as a sum of products of one-electron functions.
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One can show that most of the usual two-electron operators used in quantum chemistry can be written in a similar form.
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For example, one can write a Gaussian geminal, using a Gauss-Hermite quadrature, as \cite{Limpanuparb11b}
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\begin{equation}
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G_{12} = \sum_{n \ell m} \dyad{\phi_{n \ell m}(\br_1)}{\phi_{n \ell m}(\br_2)},
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\end{equation}
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with
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\begin{equation}
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\phi_{n \ell m}(\br) = \sqrt{8 \pi^{1/2}b_n}\,\beta_n\,j_{\ell} (2\sqrt{\lambda}\,\beta_n\,r) Y_{\ell m}(\bm{r}),
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\end{equation}
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where $\beta_n$ and $b_n$ are the (positive) Hermite roots and weights, and $j_n(r)$ is a spherical Bessel function of the first kind \cite{NISTbook}.
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A similar expression can be found for the long-range Coulomb operator, the Slater geminal and many others.
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In this way, as one inserts the RI ``in-between'' two operators, one can now resolve operators!
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This is illustrated diagrammatically in Fig.~\ref{fig:RIvsRG} for some of the three-electron integrals involved in explicitly-correlated methods.
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This procedure generates new integrals as one has to deal with spherical Bessel functions now.
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However, these are only one- and two-electron integrals.
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%%% FIGURE 1 %%%
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\begin{figure*}
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\centering
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\begin{minipage}[b]{0.3\textwidth}
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\includegraphics[width=0.8\textwidth]{../Conclusion/fig/fig1a}
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\scriptsize $I_{ijk}^{lmn} \overset{\text{RI}} \approx \sum_{\mu} G_{i\mu}^{jm} F_{\mu l}^{kn}$
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\end{minipage}
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\begin{minipage}[b]{0.3\textwidth}
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\includegraphics[width=0.8\textwidth]{../Conclusion/fig/fig1b}
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\scriptsize $I_{ijk}^{lmn} = \left< ijk \left| r_{12}^{-1}\,f_{13} \right| lmn \right>$
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\end{minipage}
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\begin{minipage}[b]{0.3\textwidth}
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\includegraphics[width=0.8\textwidth]{../Conclusion/fig/fig1c}
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\scriptsize $I_{ijk}^{lmn} \overset{\text{RG}} \approx \sum_{\mu}G_{il,\mu}^{jm} S_{kn,\mu} $
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\end{minipage}
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%
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\begin{minipage}[b]{0.3\textwidth}
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\includegraphics[width=0.8\textwidth]{../Conclusion/fig/fig2a}
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\scriptsize $J_{ijk}^{lmn} \overset{\text{RI}} \approx \sum_{\mu} F_{i\mu}^{jm} F_{\mu l}^{kn}$
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\end{minipage}
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\begin{minipage}[b]{0.3\textwidth}
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\includegraphics[width=0.8\textwidth]{../Conclusion/fig/fig2b}
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\scriptsize $J_{ijk}^{lmn} = \left< ijk \left| f_{12}\,f_{13} \right| lmn \right>$
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\end{minipage}
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\begin{minipage}[b]{0.3\textwidth}
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\includegraphics[width=0.8\textwidth]{../Conclusion/fig/fig2c}
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\scriptsize $J_{ijk}^{lmn} \overset{\text{RG}} \approx \sum_{\mu\nu} S_{il,\mu\nu} S_{jm,\mu} S_{kn,\nu}$
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\end{minipage}
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%
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\begin{minipage}[b]{0.3\textwidth}
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\includegraphics[width=0.8\textwidth]{../Conclusion/fig/fig4a}
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\scriptsize $L_{ijk}^{lmn} \overset{\text{RI}} \approx \sum_{\mu \nu \lambda} F_{i\mu}^{j\nu} G_{\nu m}^{k \lambda} F_{\mu l}^{\lambda n}$
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\end{minipage}
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\begin{minipage}[b]{0.3\textwidth}
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\includegraphics[width=0.8\textwidth]{../Conclusion/fig/fig4b}
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\scriptsize $L_{ijk}^{lmn} = \left<ijk \left| f_{12}\,r_{23}^{-1}\,f_{13} \right| lmn \right>$
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\end{minipage}
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\begin{minipage}[b]{0.3\textwidth}
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\includegraphics[width=0.8\textwidth]{../Conclusion/fig/fig4c}
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\scriptsize $L_{ijk}^{lmn} \overset{\text{RG}} \approx \sum_{\mu \nu} S_{il,\mu\nu} G_{jm,\mu}^{kn,\nu}$
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\end{minipage}
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\caption{RI (left) vs RG (right) for the three-electron integrals $I_{ijk}^{lmn}$, $J_{ijk}^{lmn}$ and $L_{ijk}^{lmn} $.}
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\label{fig:RIvsRG}
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\end{figure*}
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%************************************************************************
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\paragraph{A zero-variance hybrid MP2 method}
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The stochastic MP2 algorithm developed by Hirata and coworkers \cite{Willow12, Willow14, Johnson16, Gruneis17} has been shown to be particularly promising due to its low computational scaling and its independence towards the correlation factor \cite{Johnson17}.
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We propose to investigate two potential improvements of So Hirata's stochastic MP2 algorithm.
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First, following the philosophy of the multireference perturbation theory algorithm we have published recently \cite{PT2}, we would like to introduce a small deterministic orbital window.
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It would allow to catch the largest fraction of the MP2 correlation energy using a small number of MOs around the Fermi level.\footnote{This idea is somewhat related to the semi-stochastic FCIQMC algorithm developed by Umrigar and coworkers \cite{Petruzielo12b}.}
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This would significantly reduce the statistical error of the stochastic part because the magnitude of the statistical error in a Monte Carlo calculation is proportional to the magnitude of the actual expectation value one is actually computing.
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Second, we would like to propose a zero-variance version of Hirata's stochastic algorithm \cite{Assaraf99}.
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Indeed, one of the weak point of his algorithm comes from the choice of the probability distribution function.
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Surely, a zero-variance algorithm would significantly decrease the statistical error on the MP2 correlation energy.
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%************************************************************************
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\paragraph{Chemistry without Coulomb singularity}
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One of the most annoying feature of the Coulomb operator is its divergence as $\ree \to 0$.
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Indeed, the exact wave function must have a well-defined cusp at electron coalescences so that the infinite Coulomb interaction is exactly cancelled by an opposite divergence in the kinetic term.
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Removing such a divergence would have one very important consequence: accelerating the rate of convergence of the energy with respect to the one-electron basis set (see, for example, Ref.~\cite{Franck15}).
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A well-known procedure to remove the Coulomb singularity is to use a range-separated operator \cite{Savin96}:
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\begin{equation}
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\frac{1}{\ree} = \frac{\erfc(\omega \ree)}{\ree} + \frac{\erf(\omega \ree)}{\ree},
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\end{equation}
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where $\erf(x)$ is the error function and $\erfc(x) = 1 - \erf(x)$ its complementary version.
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The parameter $\omega$ controls the separation range.
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For $\omega = 0$, the long-range interaction vanishes while, for $\omega \to \infty$, the short range disappears
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In range-separated methods, two different methods are usually used for the short-range and long-range parts.
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For example, in range-separated DFT, one usually used DFT for the short-range interaction and a wave function method (such as MP2) to model the long-range part \cite{Toulouse04}.
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Here, we propose something slightly different.
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What if we approximate the Coulomb operator by its long-range component only, i.e.
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\begin{equation}
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\frac{1}{\ree} \approx \frac{\erf(\omega \ree)}{\ree},
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\end{equation}
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with $\omega$ large enough to be chemically meaningful?
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The variational energy would definitely be altered by such a choice \cite{Prendergast01}.
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But what about the nodes? At the end of the day, the nodes are the only things which matters in DMC!
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Are the nodes obtained with this attenuated Coulomb operator worse than the ones obtained with the genuine, singular Coulomb operator?
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As stated above, the singularity of the Coulomb operator gives birth to Kato's cusp.
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However, for a same-spin electron pair, the node in the wave function at $\ree = 0$ is produced by the antisymmetry of the wave function (Pauli exclusion principle).
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For an opposite-spin electron pair, Kato taught us that the wave function is non-zero at $\ree = 0$.
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In other words, there can't be a node!
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Therefore, one could argue that removing the Coulomb singulary would have a marginal effect on the nodal surface of the electronic wave function.
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This is what we would like to investigate in the future.
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Amother possiblity would be to create a local or non-local pseudopotential for the electron-electron interaction.
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