Merge pull request #920 from ye-luo/manual-J3

Update three body Jastrow manual section

manual/bibliography.bib

@@ -28,6 +28,18 @@
 	url = {http://dx.doi.org/10.1103/PhysRevLett.45.566}
 }
 
+@article{Drummond2004,
+  author = {Drummond, N. D. and Towler, M. D. and Needs, R. J.},
+  doi = {10.1103/PhysRevB.70.235119},
+  issn = {10980121},
+  journal = {Physical Review B - Condensed Matter and Materials Physics},
+  number = {23},
+  pages = {1--11},
+  title = {{Jastrow correlation factor for atoms, molecules, and solids}},
+  volume = {70},
+  year = {2004}
+}
+
 @article{Burkatzki07,
    author = "Burkatzki, M. and Filippi, C. and Dolg, M.",
    title = "Energy-consistent pseudopotentials for quantum Monte Carlo calculations",

manual/jastrow.tex

@@ -371,6 +371,46 @@ For both the Yukawa and Gaskell RPA Jastrows, the default value for $r_s$ is $r_
 
 
 \subsection{Three-body Jastrow functions}
-Explicit three body correlations can be included in the wavefunction via the three-body
-Jastrow factor.
+Explicit three-body correlations can be included in the wavefunction via the three-body Jastrow factor.
+The three-body electron-electron-ion correlation function ($u_{\sigma\sigma'I}$) currently used in \qmcpack is identical to the one proposed in \cite{Drummond2004}:
+\begin{eqnarray}
+u_{\sigma\sigma'I}(r_{\sigma I},r_{\sigma'I},r_{\sigma\sigma'}) &= \sum_{\ell=0}^{M_{eI}}\sum_{m=0}^{M_{eI}}\sum_{n=0}^{M_{ee}}\gamma_{\ell mn} r_{\sigma I}^\ell r_{\sigma'I}^m r_{\sigma\sigma'}^n \\
+   &\times \left(r_{\sigma I}-\frac{r_c}{2}\right)^3 \Theta\left(r_{\sigma I}-\frac{r_c}{2}\right) \nonumber \\
+   &\times \left(r_{\sigma' I}-\frac{r_c}{2}\right)^3 \Theta\left(r_{\sigma' I}-\frac{r_c}{2}\right) \nonumber
+\end{eqnarray}
+Here $M_{eI}$ and $M_{ee}$ are the maximum polynomial orders of the
+electron-ion and electron-electron distances, respectively,
+$\{\gamma_{\ell mn}\}$ are the optimizable parameters (modulo
+constraints), $r_c$ is a cutoff radius, and $r_{ab}$ are the distances
+between electrons or ions $a$ and $b$. i.e. The correlation function
+is only a function of the interparticle distances and not a more
+complex function of the particle positions, $\mathbf{r}$. As indicated by the
+$\Theta$ functions, correlations are set to zero beyond a distance of
+$r_c/2$ in either of the electron-ion distances and the largest
+meaningful electron-electron distance is $r_c$.  This is the
+highest-order Jastrow correlation function currently implemented.
 
+
+Today, solid state applications of \qmcpack usually utilize one and
+two-body B-spline Jastrow functions, with calculations on heavier
+elements often also using the three-body term described above.
+
+\paragraph{Example use case}
+Here is an example of H2O molecule. After optimizing one and two body Jastrow factors, add the following block in the wavefunction.
+The coefficients will be filled zero automatically if not given.
+\begin{lstlisting}[language=xml]
+<jastrow name="J3" type="eeI" function="polynomial" source="ion0" print="yes">
+  <correlation ispecies="O" especies="u" isize="3" esize="3" rcut="10">
+    <coefficients id="uuO" type="Array" optimize="yes"> </coefficients>
+  </correlation>
+  <correlation ispecies="O" especies1="u" especies2="d" isize="3" esize="3" rcut="10">
+    <coefficients id="udO" type="Array" optimize="yes"> </coefficients>
+  </correlation>
+  <correlation ispecies="H" especies="u" isize="3" esize="3" rcut="10">
+    <coefficients id="uuH" type="Array" optimize="yes"> </coefficients>
+  </correlation>
+  <correlation ispecies="H" especies1="u" especies2="d" isize="3" esize="3" rcut="10">
+    <coefficients id="udH" type="Array" optimize="yes"> </coefficients>
+  </correlation>
+</jastrow>
+\end{lstlisting}