mirror of https://github.com/QMCPACK/qmcpack.git
120 lines
4.8 KiB
TeX
120 lines
4.8 KiB
TeX
\section{Jastrow Factors}
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\label{sec:jastrow}
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Jastrow factors are among the simplest and most effective ways of including
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dynamical correlation in the trial many body wavefunction. The resulting many body
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wavefunction is expressed as the product of an antisymmetric (in the case
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of Fermions) or symmetric (for Bosons) part and a correlating jastrow factor
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like so:
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\begin{equation}
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\Psi(\vec{R}) = \mathcal{A}(\vec{R}) \exp\left[J(\vec{R})\right]
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\end{equation}
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In this section we will detail the types and forms of Jastrow factor used
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in QMCPACK. Note that each type of Jastrow factor needs to be specified using
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its own individual jastrow XML element. For this reason, we have repeated the
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specification of the jastrow tag in each section, with specialization for the
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options available for that given type of jastrow.
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\subsection{One-body Jastrow functions}
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\label{sec:onebodyjastrow}
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The one-body Jastrow factor is a form that allows for the direct inclusion
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of correlations between particles that are included in the wavefunction with
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particles that are not explicitly part of it. The most common example of
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this are correlations between electrons and ions.
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The jastrow function is specified within a \texttt{wavefunction} element
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and must contain one or more \texttt{correlation} elements specifying
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additional parameters as well as the actual coefficients. Section
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\ref{sec:1bjsplineexamples} gives examples of the typical nesting of
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\texttt{jastrow}, \texttt{correlation}, and \texttt{coefficient} elements.
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\subsubsection{Input Specification}
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\begin{table}[h]
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\begin{center}
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\begin{tabular}{l c c c l }
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\hline
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\multicolumn{5}{l}{Jastrow element} \\
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\hline
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\bfseries name & \bfseries datatype & \bfseries values & \bfseries defaults & \bfseries description \\
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\hline
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name & text & & (required) & Unique name for this Jastrow function \\
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type & text & One-body & (required) & Define a one-body function \\
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function & text & Bspline & (required) & BSpline Jastrow \\
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& text & pade2 & & Pade form \\
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& text & \ldots & & \ldots \\
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source & text & name & (required) & name of attribute of classical particle set \\
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print & text & yes / no & yes & jastrow factor printed in external file?\\
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\hline
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\multicolumn{5}{l}{elements}\\ \hline
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& Correlation & & & \\ \hline
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\multicolumn{5}{l}{Contents}\\ \hline
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& (None) & & & \\ \hline
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\end{tabular}
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%\end{tabular*}
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\end{center}
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\end{table}
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To be more concrete, the one-body jastrow factors used to describe correlations
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between electrons and ions take the form below
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\begin{equation}
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J1=\sum_I^{ion0}\sum_i^e u_{ab}(|r_i-R_I|)
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\end{equation}
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where I runs over all of the ions in the calculation, i runs over the electrons
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and $u_{ab}$ describes the functional form of the correlation between them.
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Many different forms of $u_{ab}$ are implemented in QMCPACK. We will detail
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two of the most common ones below.
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\include{jastrow_one_body_spline}
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\include{jastrow_one_body_pade}
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\subsection{Two-body Jastrow functions}
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The two-body Jastrow factor is a form that allows for the explicit inclusion
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of dynamic correlation between two particles included in the wavefunction. It
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is almost always given in a spin dependent form so as to satisfy the Kato cusp
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condition between electrons of different spins\cite{kato}.
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The two body jastrow function is specified within a \texttt{wavefunction} element
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and must contain one or more correlation elements specifying additional parameters
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as well as the actual coefficients. Section \ref{sec:2bjsplineexamples} gives
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examples of the typical nesting of \texttt{jastrow}, \texttt{correlation} and
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\texttt{coefficient} elements.
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\subsubsection{Input Specification}
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\begin{table}[h]
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\begin{center}
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\begin{tabular}{l c c c l }
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\hline
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\multicolumn{5}{l}{Jastrow element} \\
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\hline
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\bfseries name & \bfseries datatype & \bfseries values & \bfseries defaults & \bfseries description \\
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\hline
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name & text & & (required) & Unique name for this Jastrow function \\
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type & text & Two-body & (required) & Define a one-body function \\
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function & text & Bspline & (required) & BSpline Jastrow \\
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print & text & yes / no & yes & jastrow factor printed in external file?\\
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\hline
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\multicolumn{5}{l}{elements}\\ \hline
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& Correlation & & & \\ \hline
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\multicolumn{5}{l}{Contents}\\ \hline
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& (None) & & & \\ \hline
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\end{tabular}
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%\end{tabular*}
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\end{center}
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\end{table}
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The two-body jastrow factors used to describe correlations between electrons take the form
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\begin{equation}
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J2=\sum_i^{e}\sum_{j>i}^{e} u_{ab}(|r_i-r_j|)
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\end{equation}
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The most commonly used form of two body jastrow factor supported by the code is a splined
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jastrow factor, with many similarities to the one body spline jastrow.
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\include{jastrow_two_body_spline}
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\subsection{Three-body Jastrow functions}
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Explicit three body correlations can be included in the wavefunction via the three-body
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jastrow factor.
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