mirror of https://github.com/QMCPACK/qmcpack.git
118 lines
5.0 KiB
TeX
118 lines
5.0 KiB
TeX
\subsubsection{Spline form}
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\label{sec:onebodyjastrowspline}
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The one-body spline Jastrow function is the most commonly used one-body Jastrow for solids. This form
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was first described and used in \cite{EslerKimCeperleyShulenburger2012}.
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Here $u_{ab}$ is an interpolating 1D B-spline (tricublc spline on a linear grid) between zero distance and $r_{cut}$. In 3D periodic systems
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the default cutoff distance is the Wigner Seitz cell radius. For other periodicities, including isolated
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molecules, the $r_{cut}$ must be specified. The cusp can be set. $r_i$
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and $R_I$ are most commonly the electron and ion positions, but any particlesets that can provide the
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needed centers can be used.
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\paragraph{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}{Correlation 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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ElementType & Text & Name & See below & Classical particle target \\
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SpeciesA & Text & Name & See below & Classical particle target \\
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SpeciesB & Text & Name & See below & Quantum species target \\
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Size & Integer & $> 0$ & (Required) & Number of coefficients \\
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Rcut & Real & $> 0$ & See below & Distance at which the correlation goes to 0 \\
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Cusp & Real & $\ge 0$ & 0 & Value for use in Kato cusp condition \\
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Spin & Text & Yes or no & No & Spin-dependent Jastrow factor \\
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\hline
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\multicolumn{5}{l}{Elements}\\ \hline
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& Coefficients & & & \\ \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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Additional information:
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\begin{itemize}
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\item \ixml{elementType, speciesA, speciesB, spin}. For a spin-independent Jastrow factor (spin = ``no''),
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elementType should be the name of the group of ions in the classical particleset to which the quantum
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particles should be correlated. For a spin-dependent Jastrow factor (spin = ``yes''), set speciesA to the
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group name in the classical particleset and speciesB to the group name in the quantum particleset.
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\item \ixml{rcut}. The cutoff distance for the function in atomic units (bohr).
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For 3D fully periodic systems, this parameter is optional, and a default of the Wigner
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Seitz cell radius is used. Otherwise this parameter is required.
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\item \ixml{cusp}. The one-body Jastrow factor can be used to make the wavefunction
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satisfy the electron-ion cusp condition\cite{kato}. In this case, the derivative of the Jastrow
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factor as the electron approaches the nucleus will be given by
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\begin{equation}
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\left(\frac{\partial J}{\partial r_{iI}}\right)_{r_{iI} = 0} = -Z .
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\end{equation}
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Note that if the antisymmetric part of the wavefunction satisfies the electron-ion cusp
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condition (for instance by using single-particle orbitals that respect the cusp condition)
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or if a nondivergent pseudopotential is used, the Jastrow should be cuspless at the
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nucleus and this value should be kept at its default of 0.
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\end{itemize}
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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}{Coefficients 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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Id & Text & & (Required) & Unique identifier \\
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Type & Text & Array & (Required) & \\
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Optimize & Text & Yes or no & Yes & if no, values are fixed in optimizations \\
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\hline
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\multicolumn{5}{l}{Elements}\\ \hline
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(None) & & & \\ \hline
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\multicolumn{5}{l}{Contents}\\ \hline
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(No name) & Real array & & Zeros & Jastrow coefficients \\ \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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\paragraph{Example use cases}
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\label{sec:1bjsplineexamples}
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Specify a spin-independent function with four parameters. Because rcut is not
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specified, the default cutoff of the Wigner Seitz cell radius is used; this
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Jastrow must be used with a 3D periodic system such as a bulk solid. The name of
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the particleset holding the ionic positions is ``i."
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\begin{lstlisting}[style=QMCPXML]
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<jastrow name="J1" type="One-Body" function="Bspline" print="yes" source="i">
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<correlation elementType="C" cusp="0.0" size="4">
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<coefficients id="C" type="Array"> 0 0 0 0 </coefficients>
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</correlation>
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</jastrow>
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\end{lstlisting}
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Specify a spin-dependent function with seven up-spin and seven down-spin parameters.
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The cutoff distance is set to 6 atomic units. Note here that the particleset holding
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the ions is labeled as ion0 rather than ``i,'' as in the other example. Also in this case,
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the ion is lithium with a coulomb potential, so the cusp condition is satisfied by
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setting cusp=``d."
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\begin{lstlisting}[style=QMCPXML]
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<jastrow name="J1" type="One-Body" function="Bspline" source="ion0" spin="yes">
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<correlation speciesA="Li" speciesB="u" size="7" rcut="6">
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<coefficients id="eLiu" cusp="3.0" type="Array">
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0.0 0.0 0.0 0.0 0.0 0.0 0.0
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</coefficients>
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</correlation>
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<correlation speciesA="C" speciesB="d" size="7" rcut="6">
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<coefficients id="eLid" cusp="3.0" type="Array">
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0.0 0.0 0.0 0.0 0.0 0.0 0.0
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</coefficients>
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</correlation>
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</jastrow>
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\end{lstlisting}
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