604 lines
28 KiB
TeX
604 lines
28 KiB
TeX
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\newcommand{\LCT}{Laboratoire de Chimie Théorique (UMR 7616), Sorbonne Université, CNRS, Paris, France}
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\newcommand{\ANL}{Argonne Leadership Computing Facility, Argonne National Laboratory, Argonne, IL 60439, United States}
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\newcommand{\LCPQ}{Laboratoire de Chimie et Physique Quantiques (UMR 5626), Université de Toulouse, CNRS, UPS, France}
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\begin{document}
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\title{Enabling high accuracy diffusion Monte Carlo calculations with
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range-separated density functional theory and selected configuration interaction}
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\author{Anthony Scemama}
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\affiliation{\LCPQ}
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\author{Emmanuel Giner}
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\email{emmanuel.giner@lct.jussieu.fr}
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\affiliation{\LCT}
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\author{Anouar Benali}
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\email{benali@anl.gov}
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\affiliation{\ANL}
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\author{Pierre-Fran\c{c}ois Loos}
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\email{loos@irsamc.ups-tlse.fr}
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\affiliation{\LCPQ}
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\begin{abstract}
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\end{abstract}
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\maketitle
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\section{Introduction}
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\label{sec:intro}
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The full configuration interaction (FCI) method leads to the exact
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solution of the Schrödinger equation with an approximate Hamiltonian
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expressed in a finite basis of Slater determinants.
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The FCI wave function can be interpreted as the exact solution of the
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true Hamiltonian obtained with the additional constraint that it
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can only span the space provided by the basis. At the complete
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basis set (CBS) limit, the constraint vanishes and the exact solution
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is obtained.
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Hence the FCI method enables a systematic improvement of the
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calculations by increasing the size of the basis set. Nevertheless,
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its exponential scaling with the number of electrons and with the size
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of the basis is prohibitive to treat large systems.
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In recent years, the introduction of new algorithms\cite{Booth_2009}
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and the
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revival\cite{Abrams_2005,Bytautas_2009,Roth_2009,Giner_2013,Knowles_2015,Holmes_2016,Liu_2016}
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of selected configuration interaction (sCI)
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methods\cite{Bender_1969,Huron_1973,Buenker_1974} pushed the limits of
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the sizes of the systems that could be computed at the FCI level, but
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the scaling remains exponential unless some bias is introduced leading
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to a loss of size extensivity.
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The Diffusion Monte Carlo (DMC) method is a numerical scheme to obtain
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the exact solution of the Schrödinger equation with an additional
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constraint, imposing the solution to have the same nodal hypersurface
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as a given trial wave function. This approximation is known as the
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\emph{fixed-node} approximation. When the nodes of the trial wave
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function coincide with the nodes of the exact wave function, the exact
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energy and wave function are obtained.
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The DMC method is attractive because its scaling is polynomial with
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the number of electrons and with the size of the trial wave
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function. Moreover, the total energies obtained are usually below
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those obtained with FCI because the fixed-node approximation imposes
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less constraints on the solution than the finite-basis approximation.
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In many cases, the systems under study are well described by a single
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Slater determinant. Single-determinant DMC can be used as a post-Hatree-Fock
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single-reference method with an accuracy comparable to coupled cluster.
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The favorable scaling of QMC and its adequation with massively
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parallel architectures makes it an attractive alternative for large
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systems.
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It has been shown that the nodal surfaces obtained with
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Kohn-Sham determinants are in general better than those obtained with
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the Hartree-Fock determinant,\cite{Per_2012} and of comparable quality
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to those obtained with natural orbitals of single-determinant
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correlated calculations.\cite{Wang_2019}
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However, the fixed-node approximation is much more difficult to
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control than the finite-basis approximation, as it is not possible
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to minimize directly the DMC energy with respect to the variational
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parameters of the trial wave function.
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The conventional approach consists in multiplying the trial wave
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function by a positive function, the \emph{Jastrow factor}, taking
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account of the electron-electron cusp and the short-range correlation
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effects. The wave function is then re-optimized in the presence of the
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Jastrow factor and the nodal surface is expected to be improved.
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Using this technique, it has been shown that the chemical accuracy
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could be reached within DMC.\cite{Petruzielo_2012}
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Another approach consists in considering the DMC method as a
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\emph{post-FCI method}. The trial wave function is obtained by
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approaching the FCI with a selected configuration interaction
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method such as CIPSI for instance.\cite{Giner_2013,Caffarel_2016_2}
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When the basis set is increased, the trial wave function tends to the
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exact wave function, so the nodal surface can be systematically
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improved.\cite{Caffarel_2016}
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This technique has the advantage that using FCI nodes in a given basis
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set is well defined and has a unique solution. The optimization of the
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wave function is deterministic, so the calculations are reproducible
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and don't require the expertise of a QMC expert. However,
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this technique can't be applied to large systems because of the
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exponential scaling of the size of the wave function. Extrapolation
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techniques have been applied to estimate the DMC energy of a FCI
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wave function in a large basis set,\cite{Scemama_2018} and other
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authors have used a combination of the two approaches where CIPSI
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trial wave functions are re-optimized under the presence of a Jastrow
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factor.\cite{Giner_2016,Dash_2018,Dash_2019}
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\section{Combining range-separated DFT with CIPSI}
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\label{sec:rsdft-cipsi}
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Starting from a Hartree-Fock determinant in a small basis set,
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we have seen that we can systematically improve the trial wave
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function in two directions. The first one is by increasing the
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size of the atomic basis set, and the second one is by
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increasing the determinant expansion towards the FCI limit.
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A third direction of improvement exists. It is the path which connects
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the Hartree-Fock determinant to the Kohn-Sham determinant, which
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usually has a better nodal surface. This third path is obtained by
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the so-called \emph{range-separated} DFT framework which splits the
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electron-electron interaction in a long-range part and a short-range
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part. HELP MANU!
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\subsection{CIPSI}
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\emph{Configuration interaction using a perturbative selection made
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iteratively} (CIPSI)\cite{Huron_1973} belongs to the class of selected
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CI methods, whose goal is to build compressed configuration interaction
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(CI) wave functions with a controlled accuracy, together with an estimate
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of the associated eigenvalue obtained with perturbation theory.
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Using a conventional iterative diagonalization method such as
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Davidson's,\cite{Davidson_1975} the size of the CI space is
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pre-defined, and at each iteration the length of the vectors is equal
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to the size of the CI space. Within selected CI methods, at each
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iteration the size of the determinant space is increased such that
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one keeps only the most important determinants that will lead to an
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accurate description of the state of interest. The lowest eigenpairs
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are extracted from the CI matrix expressed in this determinant
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subspace, and the CI energy can be estimated by computing a
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second-order perturbative correction (PT2) to the variational energy
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computed in the complete CI space. The magnitude of the PT2 correction
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is a measure of the distance to the exact eigenvalue, and is an
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adjustable parameter to control the quality of the wave function.
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\subsection{Range-separated DFT in a nutshell}
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\label{sec:rsdft}
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\subsubsection{Exact equations}
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\label{sec:exact}
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The exact ground-state energy of a $N$-electron system with
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nuclei-electron potential $v_\mathrm{ne}(\textbf{r})$ can be expressed
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by the following minimization over $N$-representable densities
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$n$~\cite{Lev-PNAS-79,Lie-IJQC-83}
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\begin{equation}
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E_0 = \min_n \left\{ \mathcal{F}[n] + \int v_\mathrm{ne}(\textbf{r}) n(\textbf{r}) \mathrm{d} \textbf{r} \right\},
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\label{E0}
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\end{equation}
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with the standard constrained-search universal density functional
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\begin{equation}
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\mathcal{F}[n] = \min_{\Psi\rightarrow n} \left \langle\Psi|\hat{T}+\hat{W}_\mathrm{ee}|\Psi \right \rangle,
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\label{Fn}
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\end{equation}
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where $\hat{T}$ and $\hat{W}_\mathrm{ee}$ are the kinetic-energy and
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Coulomb electron-electron interaction operators, respectively. The
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minimizing multideterminant wave function in Eq.~\eqref{Fn} will be
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denoted by $\Psi[n]$. In RS-DFT, the universal density functional is
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decomposed as~\cite{Sav-INC-96,TouColSav-PRA-04}
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\begin{equation}
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\mathcal{F}[n] = \mathcal{F}^{\mathrm{lr},\mu}[n] + \bar{E}_{\mathrm{Hxc}}^{\mathrm{sr,}\mu}[n],
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\label{Fdecomp}
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\end{equation}
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where $\mathcal{F}^{\mathrm{lr},\mu}[n]$ is a long-range (lr)
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universal density functional
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\begin{equation}
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\label{lr_univ_fonc}
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\mathcal{F}^{\mathrm{lr},\mu}[n]= \min_{\Psi\rightarrow n} \left
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\langle\Psi|\hat{T}+\hat{W}_\mathrm{ee}^{\mathrm{lr},\mu}|\Psi
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\right \rangle,
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\end{equation}
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and $\bar{E}_{\mathrm{Hxc}}^{\,\mathrm{sr,}\mu}[n]$ is the
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complementary short-range (sr) Hartree-exchange-correlation (Hxc)
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density functional. In Eq.~(\ref{lr_univ_fonc}),
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$\hat{W}_\mathrm{ee}^{\mathrm{lr}}$ is the long-range
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electron-electron interaction defined as
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\begin{equation}
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\hat{W}_\mathrm{ee}^{\mathrm{lr},\mu} = \frac{1}{2} \iint w_{\mathrm{ee}}^{\mathrm{lr},\mu}(r_{12}) \hat{n}_2(\textbf{r}_1,\textbf{r}_2)
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\mathrm{d} \textbf{r}_1 \mathrm{d} \textbf{r}_2,
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\end{equation}
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with the error-function potential
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$w_{\mathrm{ee}}^{\mathrm{lr},\mu}(r_{12})=\mathrm{erf}(\mu\, r_{12}
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)/r_{12}$ (expressed with the interelectronic distance $r_{12} =
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||\textbf{r}_1-\textbf{r}_2||$) and the pair-density operator
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$\hat{n}_2(\textbf{r}_1, \textbf{r}_2)=\hat{n}(\textbf{r}_1)
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\hat{n}(\textbf{r}_2) - \delta(\textbf{r}_1-\textbf{r}_2)
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\hat{n}(\textbf{r}_1)$ where $\hat{n}(\textbf{r})$ is the density
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operator. The range-separation parameter $\mu$ corresponds to an
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inverse distance controlling the range of the separation and the
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strength of the interaction at $r_{12} = 0$. For a given density, we
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will denote by $\Psi^\mu[n]$ the minimizing multideterminant wave
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function in Eq.~\eqref{lr_univ_fonc}. Inserting the decomposition of
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Eq.~\eqref{Fdecomp} into Eq.~\eqref{E0}, and recomposing the two-step
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minimization into a single one, leads to the following expression for
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the exact ground-state electronic energy
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\begin{equation}
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\label{min_rsdft} E_0= \min_{\Psi} \left\{
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\left
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\langle\Psi|\hat{T}+\hat{W}_\mathrm{{ee}}^{\mathrm{lr},\mu}+\hat{V}_{\mathrm{ne}}|\Psi\right
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\rangle
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+ \bar{E}^{\mathrm{sr},\mu}_{\mathrm{Hxc}}[n_\Psi]\right\},
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\end{equation}
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where the minimization is done over normalized $N$-electron
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multideterminant wave functions, $\hat{V}_{\mathrm{ne}} = \int
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v_{\mathrm{ne}} (\textbf{r}) \hat{n}(\textbf{r}) \mathrm{d}
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\textbf{r}$, and $n_\Psi$ refers to the density of $\Psi$, i.e.
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$n_\Psi(\textbf{r})=\left \langle\Psi|\hat{n}(\textbf{r})|\Psi\right
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\rangle$.
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The minimizing multideterminant wave function $\Psi^\mu$ in
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Eq.~\eqref{min_rsdft} can be determined by the self-consistent
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eigenvalue equation
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\begin{equation}
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\label{rs-dft-eigen-equation}
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\hat{H}^\mu[n_{\Psi^{\mu}}] \ket{\Psi^{\mu}}= \mathcal{E}^{\mu} \ket{\Psi^{\mu}},
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\end{equation}
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with the long-range interacting Hamiltonian
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\begin{equation}
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\label{H_mu}
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\hat{H}^\mu[n_{\Psi^{\mu}}] = \hat{T}+\hat{W}_{\mathrm{ee}}^{\mathrm{lr},\mu}+\hat{V}_{\mathrm{ne}}+ \hat{\bar{V}}_{\mathrm{Hxc}}^{\mathrm{sr},\mu}[n_{\Psi^{\mu}}],
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\end{equation}
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where
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$\hat{\bar{V}}_{\mathrm{Hxc}}^{\mathrm{sr},\mu}[n]=\int \delta \bar{E}^{\mathrm{sr},\mu}_{\mathrm{Hxc}}[n]/\delta n(\textbf{r}) \, \hat{n}(\textbf{r}) \mathrm{d} \textbf{r}$
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is the complementary short-range Hartree-exchange-correlation
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potential operator. Note that $\Psi^{\mu}$ is not the exact physical
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ground-state wave function but only an effective wave
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function. However, its density $n_{\Psi^{\mu}}$ is the exact physical
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ground-state density. Once $\Psi^{\mu}$ has been calculated, the exact
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electronic ground-state energy is obtained by
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\begin{equation}
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\label{E-rsdft}
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E_0= \braket{\Psi^{\mu}|\hat{T}+\hat{W}_\mathrm{{ee}}^{\mathrm{lr},\mu}+\hat{V}_{\mathrm{ne}}|\Psi^{\mu}}+\bar{E}^{\mathrm{sr},\mu}_{\mathrm{Hxc}}[n_{\Psi^\mu}].
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\end{equation}
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Note that, for $\mu=0$, the long-range interaction vanishes,
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$w_{\mathrm{ee}}^{\mathrm{lr},\mu=0}(r_{12}) = 0$, and thus RS-DFT
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reduces to standard KS-DFT. For $\mu\to\infty$, the long-range
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interaction becomes the standard Coulomb interaction,
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$w_{\mathrm{ee}}^{\mathrm{lr},\mu\to\infty}(r_{12}) = 1/r_{12}$, and
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thus RS-DFT reduces to standard wave-function theory (WFT).
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\subsection{Approximations}
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\label{sec:approx}
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Provided that the exact short-range complementary functional is known
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and that the wave function $\Psi^{\mu}$ is developed in a complete
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basis set, the theory is exact and therefore provides the exact
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density and ground state energy whatever value of $\mu$ is chosen to
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split the universal Levy-Lieb functional (see Eqs. \eqref{Fdecomp} and
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\eqref{lr_univ_fonc}). Of course in practice, two kind of
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approximations must be made: i) the approximation of the wave function
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by using a finite basis set, ii) the approximation on the exact
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complementary functionals. As long as approximations are made, the
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theory is not exact anymore and might depend on $\mu$.
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Despite its $\mu$ dependence, approximated RS-DFT schemes provide
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potentially appealing features as: i) the approximated wave function
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$\Psi^{\mu}$ is an eigenvector of a non-divergent operator
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$\hat{H}^\mu[n_{\Psi^{\mu}}]$ (see Eqs. \eqref{H_mu} and
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\eqref{rs-dft-eigen-equation}) and therefore converges more rapidly
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with respect to the basis set~\cite{FraMusLupTou-JCP-15}, and also
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produces more compact wave function~\cite{GinPraFerAssSavTou-JCP-18}
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as it will be illustrated here, ii) the semi-local approximations for
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RS-DFT complementary functionals are usually better suited to describe
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the remaining short-range part of the electron-electron correlation.
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In this work, we use the short-range version of the
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Perdew-Burke-Ernzerhof (PBE)~\cite{PerBurErn-PRL-96} exchange and
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correlation functionals of Ref.~\onlinecite{GolWerStoLeiGorSav-CP-06}
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(see also Refs.~\onlinecite{TouColSav-JCP-05,GolWerSto-PCCP-05}) which
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takes the form
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\begin{eqnarray}
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\bar{E}^{\mathrm{sr},\mu,\textsc{pbe}}_{\mathrm{x/c}}[n] = \int \bar{e}_\mathrm{{x/c}}^\mathrm{sr,\mu,\textsc{pbe}}(n(\textbf{r}),\nabla n(\textbf{r})) \, \mathrm{d}\textbf{r}.
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\end{eqnarray}
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\subsection{RSDFT-CIPSI}
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It is possible to use DFT for short-range interactions and CIPSI for
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the long-range. This scheme has been recently
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implemented.\cite{GinPraFerAssSavTou-JCP-18}
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\begin{figure}[h]
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\centering
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\includegraphics[width=\columnwidth]{algorithm.pdf}
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\caption{Algorithm showing the generation of the RSDFT-CIPSI wave
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function}
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\end{figure}
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% Overlap with reoptimized
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% Plot Ndets as a function of mu
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\section{Influence of the range-separation parameter on the fixed-node
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error}
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\label{sec:mu-dmc}
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\begin{table}[h]
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\caption{Fixed-node energies of the water molecule.}
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\label{tab:h2o-dmc}
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\centering
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\begin{tabular}{crlrl}
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\hline
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& \multicolumn{2}{c}{BFD-VDZ} & \multicolumn{2}{c}{BFD-VTZ}\\
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$\mu$ & $N_{\text{det}}$ & E(DMC) & $N_{\text{det}}$ & E(DMC)\\
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\hline
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$0.00$ & $11$ & $-17.253\,59(6)$ & $23$ & $-17.256\,74(7)$\\
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$0.20$ & $23$ & $-17.253\,73(7)$ & $23$ & $-17.256\,73(8)$\\
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$0.30$ & $53$ & $-17.253\,4(2)$ & $219$ & $-17.253\,7(5)$\\
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$0.50$ & $1\,442$ & $-17.253\,9(2)$ & $16\,99$ & $-17.257\,7(2)$\\
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$0.75$ & $3\,213$ & $-17.255\,1(2)$ & $13\,362$ & $-17.258\,4(3)$\\
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$1.00$ & $6\,743$ & $-17.256\,6(2)$ & $256\,73$ & $-17.261\,0(2)$\\
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$1.75$ & $54\,540$ & $-17.259\,5(3)$ & $207\,475$ & $-17.263\,5(2)$\\
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$2.50$ & $51\,691$ & $-17.259\,4(3)$ & $858\,123$ & $-17.264\,3(3)$\\
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$3.80$ & $103\,059$ & $-17.258\,7(3)$ & $1\,621\,513$ & $-17.263\,7(3)$\\
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$5.70$ & $102\,599$ & $-17.257\,7(3)$ & $1\,629\,655$ & $-17.263\,2(3)$\\
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$8.50$ & $101\,803$ & $-17.257\,3(3)$ & $1\,643\,301$ & $-17.263\,3(4)$\\
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$\infty$ & $200\,521$ & $-17.256\,8(6)$ & $1\,631\,982$ & $-17.263\,9(3)$\\
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\hline
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\end{tabular}
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\end{table}
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\begin{figure}[h]
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\centering
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\includegraphics[width=\columnwidth]{h2o-dmc.pdf}
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\caption{Fixed-node energies of the water molecule with variable
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values of $\mu$.}
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\label{fig:h2o-dmc}
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\end{figure}
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The water molecule was taken at the equilibrium
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geometry,\cite{Caffarel_2016} and RSDFT-CIPSI wave functions were
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generated with BFD pseudopotentials and the corresponding double-zeta
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basis set using multiple values of the range-separation parameter
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$\mu$. The convergence criterion for stopping the CIPSI calculation
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was set to 1~m$E_h$ on the PT2 correction. Then, these wave functions
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were used as trial wave functions for FN-DMC calculations, and the
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corresponding energies are shown in table~\ref{tab:h2o-dmc} and
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figure~\ref{fig:h2o-dmc}.
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Using FCI trial wave functions gives FN-DMC energies which are lower
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than the energies obtained with a single Kohn-Sham determinant:
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3~m$E_h$ at the double-zeta level and 7~m$E_h$ at the triple-zeta
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level. Interestingly, with the double-zeta basis one can obtain a
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FN-DMC energy 2.5~m$E_h$ lower than the energy obtained with the FCI
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trial wave function, using the RSDFT-CIPSI with a range-separation
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parameter $\mu=1.75$. This can be explained by the inability of the
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basis set to properly describe short-range correlation, shifting
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the nodes from their optimal position. Using DFT to take account of
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short-range correlation frees the determinant expansion from describing
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short-range effects, and enables a better placement of the nodes.
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At the triple-zeta level, the short-range correlations can be better
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described, and the improvement due to DFT is insignificant. However,
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it is important to note that the same FN-DMC energy can be
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obtained with a CI expansion which is eight times smaller when sr-DFT
|
|
is introduced. One can also remark that the minimum has been
|
|
shifted towards the FCI, which is consistent with the fact that
|
|
in the CBS limit we expect the minimum of the FN-DMC energy to be
|
|
obtained for the FCI wave function, at $\mu=\infty$.
|
|
|
|
|
|
\section{Computational details}
|
|
\label{sec:comp-details}
|
|
|
|
All the calculations were made using BFD
|
|
pseudopotentials\cite{Burkatzki_2008} with the associated double, triple
|
|
and quadruple zeta basis sets.
|
|
CCSD(T) and DFT calculations were made with
|
|
\emph{Gaussian09},\cite{g16} using an unrestricted Hartree-Fock
|
|
determinant as a reference for open-shell systems. All the CIPSI
|
|
calculations and range-separated CIPSI calculations were made with
|
|
\emph{Quantum Package}.\cite{Garniron_2019,qp2_2020}
|
|
Quantum Monte Carlo calculations were made with QMC=Chem,\cite{scemama_2013}
|
|
in the determinant localization approximation.\cite{Zen_2019}
|
|
|
|
In the determinant localization approximation, only the determinantal
|
|
component of the trial wave function is present in the expression of
|
|
the wave function on which the pseudopotential is localized. Hence,
|
|
the pseudopotential operator does not depend on the Jastrow factor, as
|
|
it is the case in all-electron calculations. This improves the
|
|
reproducibility of the results, as they depend only on parameters
|
|
optimized in a deterministic framework.
|
|
|
|
|
|
\section{Atomization energy benchmarks}
|
|
\label{sec:atomization}
|
|
|
|
Atomization energies are challenging for post-Hartree-Fock methods
|
|
because their calculation requires a perfect balance in the
|
|
description of atoms and molecules. Basis sets used in molecular
|
|
calculations are atom-centered, so they are always better adapted to
|
|
atoms than molecules and atomization energies usually tend to be
|
|
underestimated.
|
|
In the context of FN-DMC calculations, the nodal surface is imposed by
|
|
the trial wavefunction which is expanded on an atom-centered basis
|
|
set. So we expect the fixed-node error to be also related to the basis
|
|
set incompleteness error.
|
|
Increasing the size of the basis set improves the description of
|
|
the density and of electron correlation, but also reduces the
|
|
imbalance in the quality of the description of the atoms and the
|
|
molecule, leading to more accurate atomization energies.
|
|
|
|
Another important feature required to get accurate atomization
|
|
energies is size-extensivity, since the numbers of correlated electrons
|
|
in the isolated atoms are different from the number of correlated
|
|
electrons in the molecule.
|
|
In the context of selected CI calculations, when the variational energy is
|
|
extrapolated to the FCI energy\cite{Holmes_2017} there is obviously no
|
|
size-consistence error. But when the selected wave function is used
|
|
for as a reference for post-Hartree-Fock methods or QMC calculations,
|
|
there is a residual size-consistence error originating from the
|
|
truncation of the determinant space.
|
|
|
|
% Invariance with m_s
|
|
|
|
QMC calculations can be made size-consistent by extrapolating the
|
|
FN-DMC energy to estimate the energy obtained with the FCI as a trial
|
|
wave function.\cite{Scemama_2018,Scemama_2018b} Alternatively, the
|
|
size-consistence error can be reduced by choosing the number of
|
|
selected determinants such that the sum of the PT2 corrections on the
|
|
atoms is equal to the PT2 correction of the molecule, enforcing that
|
|
the variational dissociation potential energy surface (PES) is
|
|
parallel to the perturbatively corrected PES, which is an accurate
|
|
estimate of the FCI PES.\cite{Giner_2015}
|
|
|
|
Another source of size-consistence error in QMC calculation may originate
|
|
from the Jastrow factor. Usually, the Jastrow factor contains
|
|
one-electron, two-electron and one-nucleus-two-electron terms.
|
|
The problematic part is the two-electron term, whose simplest form can
|
|
be expressed as
|
|
\begin{equation}
|
|
J_\text{ee} = \sum_i \sum_{j<i} \frac{a r_{ij}}{1 + b r_{ij}}.
|
|
\end{equation}
|
|
$a$ is determined by cusp conditions, and $b$ is obtained by energy
|
|
or variance minimization.\cite{Coldwell_1977,Umrigar_2005}
|
|
One can easily see that this parameterization of the two-body
|
|
interation is not size-consistent. The dissociation of a
|
|
heteroatomic diatomic molecule $AB$ with a parameter $b_{AB}$
|
|
will lead to two different two-body Jastrow factors on each atom, each
|
|
with its own optimal value $b_A$ and $b_B$. To remove the
|
|
size-consistence error on a PES using this ansätz for $J_\text{ee}$,
|
|
one needs to impose that the parameters of $J_\text{ee}$ are fixed:
|
|
$b_A = b_B = b_{AB}$.
|
|
|
|
When pseudopotentials are used in a QMC calculation, it is common
|
|
practice to localize the pseudopotential on the complete wave
|
|
function. If the wave function is not size-consistent, so will be the
|
|
locality approximation. Recently, the determinant localization
|
|
approximation was introduced.\cite{Zen_2019} This approximation
|
|
consists in removing the Jastrow factor from the wave function on
|
|
which the pseudopotential is localized.
|
|
The great advantage of this approximation is that the FN-DMC energy
|
|
within this approximation only depends on the parameters of the
|
|
determinantal component. Using a size-inconsistent Jastrow factor, or
|
|
a non-optimal Jastrow factor will not introduce an additional
|
|
size-consistence error in FN-DMC calculations, although it will
|
|
reduce the statistical errors by reducing the variance of the local energy.
|
|
|
|
|
|
The energy computed within density functional theory is extensive, and
|
|
as it is a mean-field method the convergence to the complete basis set
|
|
(CBS) limit is relatively fast. Hence, DFT methods are very well adapted to
|
|
the calculation of atomization energies, especially with small basis
|
|
sets, but going to the CBS limit will converge to biased atomization
|
|
energies because of the use of approximate density functionals.
|
|
|
|
On the other hand, the convergence of the FCI energies to the CBS
|
|
limit will be slower because of the description of short-range electron
|
|
correlation with atom-centered functions, but ultimately the exact
|
|
energy will be reached.
|
|
|
|
|
|
The 55 molecules of the benchmark for the Gaussian-1
|
|
theory\cite{Pople_1989,Curtiss_1990} were chosen to test the quality
|
|
of the RSDFT-CIPSI trial wave functions for energy differences.
|
|
|
|
\begin{table}[h]
|
|
\centering
|
|
\begin{tabular}{lcrr}
|
|
Method & Basis & $\mu$ & MAD (kcal/mol) \\
|
|
\hline
|
|
PBE & VDZ-BFD & & 5.02 \\
|
|
& VTZ-BFD & & 4.57 \\
|
|
& VQZ-BFD & & 5.31 \\
|
|
BLYP & VDZ-BFD & & 9.53 \\
|
|
& VTZ-BFD & & 5.58 \\
|
|
& VQZ-BFD & & 5.86 \\
|
|
PBE0 & VDZ-BFD & & 11.20 \\
|
|
& VTZ-BFD & & 6.40 \\
|
|
& VQZ-BFD & & 6.28 \\
|
|
B3LYP & VDZ-BFD & & 11.27 \\
|
|
& VTZ-BFD & & 7.27 \\
|
|
& VQZ-BFD & & 6.75 \\
|
|
\hline
|
|
CCSD(T) & VDZ-BFD & & 24.10 \\
|
|
& VTZ-BFD & & 9.11 \\
|
|
& VQZ-BFD & & 4.52 \\
|
|
\hline
|
|
RSDFT-CIPSI & VDZ-BFD & 0 & 10.08 \\
|
|
& & 1/4 & 5.55 \\
|
|
& & 1/2 & 13.42 \\
|
|
& & 1 & 17.07 \\
|
|
& & $\infty$ & 23.62 \\
|
|
& VTZ-BFD & 0 & 6.31 \\
|
|
& & 1/4 & 4.58 \\
|
|
& & 1/2 & 6.77 \\
|
|
& & 1 & 9.06 \\
|
|
& VQZ-BFD & 0 & 6.35 \\
|
|
& & 1/4 & 5.48 \\
|
|
& & 1/2 & 6.35 \\
|
|
\hline
|
|
DMC/RSDFT-CIPSI & VDZ-BFD & 0 & 5.07 $\pm$ 0.44 \\
|
|
& & 1/4 & 4.04 $\pm$ 0.37 \\
|
|
& & 1/2 & 3.74 $\pm$ 0.35 \\
|
|
& & 1 & 5.42 $\pm$ 0.29 \\
|
|
& & $\infty$ & 7.38 $\pm$ 1.08 \\
|
|
& VTZ-BFD & 0 & 3.52 $\pm$ 0.19 \\
|
|
& & 1/4 & 3.39 $\pm$ 0.77 \\
|
|
& & 1/2 & 2.46 $\pm$ 0.18 \\
|
|
& & 1 & 4.38 $\pm$ 0.94 \\
|
|
& VQZ-BFD & 0 & 3.16 $\pm$ 0.26 \\
|
|
& & 1/4 & 2.90 $\pm$ 0.25 \\
|
|
& & 1/2 & 2.06 $\pm$ 0.35 \\
|
|
\hline
|
|
\end{tabular}
|
|
\caption{Mean absolute deviations obtained with the different
|
|
methods and basis sets.}
|
|
\label{tab:mad}
|
|
\end{table}
|
|
|
|
\begin{figure}[h]
|
|
\centering
|
|
\includegraphics[width=\columnwidth]{g2-dmc-dz.pdf}
|
|
\caption{Histogram of the errors in atomization energies with the double-zeta basis set.}
|
|
\label{fig:g2-dmc-dz}
|
|
\end{figure}
|
|
|
|
\begin{figure}[h]
|
|
\centering
|
|
\includegraphics[width=\columnwidth]{g2-dmc-tz.pdf}
|
|
\caption{Histogram of the errors in atomization energies with the triple-zeta basis set.}
|
|
\label{fig:g2-dmc-tz}
|
|
\end{figure}
|
|
|
|
\begin{figure}[h]
|
|
\centering
|
|
\includegraphics[width=\columnwidth]{g2-dmc-qz.pdf}
|
|
\caption{Histogram of the errors in atomization energies with the quadruple-zeta basis set.}
|
|
\label{fig:g2-dmc-qz}
|
|
\end{figure}
|
|
|
|
|
|
|
|
|
|
%%---------------------------------------
|
|
\begin{acknowledgments}
|
|
An award of computer time was provided by the Innovative and Novel
|
|
Computational Impact on Theory and Experiment (INCITE) program. This
|
|
research has used resources of the Argonne Leadership Computing
|
|
Facility, which is a DOE Office of Science User Facility supported
|
|
under Contract DE-AC02-06CH11357. AB, was supported by the
|
|
U.S. Department of Energy, Office of Science, Basic Energy Sciences,
|
|
Materials Sciences and Engineering Division, as part of the
|
|
Computational Materials Sciences Program and Center for Predictive
|
|
Simulation of Functional Materials.
|
|
\end{acknowledgments}
|
|
|
|
|
|
\bibliography{rsdft-cipsi-qmc}
|
|
|
|
\end{document}
|