added LD potential and wrote html-style doc

doc/src/Eqs/pair_local_density_energy.tex

@@ -0,0 +1,11 @@
+\documentclass[12pt]{article}
+
+\begin{document}
+
+$$
+ U_{LD} = \sum_i F(\rho_i)
+$$
+
+
+\end{document}
+~                  

doc/src/Eqs/pair_local_density_energy_implement.tex

@@ -0,0 +1,9 @@
+\documentclass[12pt]{article}
+
+\begin{document}
+
+$$
+U_{LD} = \sum_k U_{LD}^{(k)} = \sum_i \left[ \sum_k a_\alpha^{(k)} F^{(k)} \left(\rho_i^{(k)}\right) \right] 
+$$
+
+\end{document}

doc/src/Eqs/pair_local_density_energy_multi.tex

@@ -0,0 +1,9 @@
+\documentclass[12pt]{article}
+
+\begin{document}
+
+$$
+U_{LD} = \sum_i a_\alpha F(\rho_i)
+$$
+
+\end{document}

doc/src/Eqs/pair_local_density_indicator_func.tex

@@ -0,0 +1,16 @@
+\documentclass[12pt]{article}
+\usepackage[utf8]{inputenc}
+\usepackage{amsmath}
+\usepackage{amsfonts}
+
+\begin{document}
+\[
+  \varphi(r) = 
+    \begin{cases}
+       1 & r \le R_1 \\
+       c_0 + c_2r^2 + c_4r^4 + c_6r^6  & r \in (R_1, R_2) \\
+       0 & r \ge R_2
+  \end{cases}
+\]
+
+\end{document}

doc/src/Eqs/pair_local_density_ld.tex

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+\documentclass[12pt]{article}
+
+\begin{document}
+
+
+$$
+\rho_i = \sum_{j \neq i} \varphi(r_{ij})
+$$
+
+\end{document}

doc/src/Eqs/pair_local_density_ld_implement.tex

@@ -0,0 +1,10 @@
+\documentstyle[12pt]{article}
+
+\begin{document}
+
+
+$$
+\rho_i^{(k)} = \sum_j b_\beta^{(k)} \varphi^{(k)} (r_{ij})
+$$
+
+\end{document}

doc/src/Eqs/pair_local_density_ld_multi.tex

@@ -0,0 +1,10 @@
+\documentclass[12pt]{article}
+
+\begin{document}
+
+
+$$
+\rho_i = \sum_{j \neq i} b_\beta \varphi(r_{ij})
+$$
+
+\end{document}

doc/src/pair_local_density.txt

@@ -0,0 +1,207 @@
+"LAMMPS WWW Site"_lws - "LAMMPS Documentation"_ld - "LAMMPS Commands"_lc :c
+
+:link(lws,http://lammps.sandia.gov)
+:link(ld,Manual.html)
+:link(lc,Commands_all.html)
+
+:line
+
+pair_style local/density command :h3
+
+[Syntax:]
+
+pair_style style arg :pre
+
+style = {local/density}
+arg = name of file containing tabulated values of local density and the potential :ul
+
+[Examples:]
+
+pair_style local/density benzene_water.localdensity.table :pre
+
+pair_style hybrid/overlay table spline 500 local/density 
+pair_coeff * * local/density  benzene_water.localdensity.table :pre
+
+[Description:]
+
+The local density (LD) potential is a new potential style that is, in some 
+sense,a generalization of embedded atom models (EAM). The name "local density 
+potential" arises from the fact that it assigns an energy to an atom depending 
+on the number of neighboring atoms of given type around it within a predefined 
+spherical volume (i.e., within a cutoff). The bottom-up coarse-graining (CG)
+literature sugggests that such potentials can be widely useful  in capturing 
+effective multibody forces in a computationally efficient manner so as to 
+improve the quality of CG models of implicit solvation"(Sanyal1)"_#Sanyal1 and 
+phase-segregation in liquid mixtures"(Sanyal2)"_#Sanyal2, and provide guidelines 
+to determine the extent of manybody correlations present in a CG 
+model."(Rosenberger)"_#Rosenberger The LD potential in LAMMPS is primarily 
+intended to be used as a corrective potential over traditional pair potentials 
+in bottom-up CG models, i.e., as a hybrid pair style with 
+other explicit pair interaction terms (e.g., table spline, Lennard Jones, etc.). 
+Because the LD potential is not a pair potential per se,  it is implemented 
+simply as a single auxiliary file with all specifications that will be read 
+upon initialization.
+
+NOTE: Thus when used as the only interaction in the system, there is no 
+corresponding pair_coeff command and when used with other pair styles using the 
+hybrid/overlay option, the corresponding pair_coeff command must be supplied
+*  * as placeholders for the atomtypes.
+
+:line
+
+[System with a single CG atom type:]
+
+A system of a single atom type (e.g., LJ argon) with a single local density (LD)
+potential would have an energy given by:
+
+:c,image(Eqs/pair_local_density_energy.jpg)
+
+where rho_i is the LD at atom i and F(rho) is similar in spirit to the 
+embedding function used in EAM potentials. The LD at atom i is given by the sum
+
+:c,image(Eqs/pair_local_density_ld.jpg)
+
+where phi is an indicator function that is one at r=0 and zero beyond a cutoff 
+distance R2. The choice of the functional form of phi is somewhat arbitrary, 
+but the following piecewise cubic function has proven sufficiently general: 
+"(Sanyal1)"_#Sanyal1, "(Sanyal2)"_#Sanyal2 "(Rosenberger)"_#Rosenberger
+
+:c,image(Eqs/pair_local_density_indicator_func.jpg)
+
+The constants {c} are chosen so that the indicator function smoothly 
+interpolates between 1 and 0 between the distances R1 and R2, which are 
+called the inner and outer cutoffs, respectively. Thus phi satisfies 
+phi(R1) = 1, phi(R2) = dphi/dr @ (r=R1) =  dphi/dr @ (r=R2) = 0. The embedding 
+function F(rho) may or may not have a closed-form expression. To maintain 
+generality, it is practically represented with a spline-interpolated table 
+over a predetermined range of rho. Outside of that range it simply adopts zero 
+values at the endpoints.
+
+It can be shown that the total force between two atoms due to the LD potential 
+takes the form of a pair force, which motivates its designation as a LAMMPS 
+pair style. Please see "(Sanyal1)"_#Sanyal1 for details of the derivation. 
+
+:line
+
+[Systems with arbitrary numbers of atom types:]
+
+The potential is easily generalized to systems involving multiple atom types:
+
+:c,image(Eqs/pair_local_density_energy_multi.jpg)
+
+with the LD expressed as
+
+:c,image(Eqs/pair_local_density_ld_multi.jpg)
+
+where alpha gives the type of atom i, beta the type of atom j, and the 
+coefficients a and b filter for atom types as specified by the user. a is 
+called the central atom filter as it determines to which atoms the 
+potential applies; a_alpha = 1 if the LD potential applies to atom type alpha 
+else zero. On the other hand, b is called the neighbor atom filter because it 
+specifies which atom types to use in the calculation of the LD; b_beta = 1 if 
+atom type beta contributes to the LD and zero otherwise.
+
+NOTE: Note that the potentials need not be symmetric with respect to atom types, 
+which is the reason for two distinct sets of coefficients a and b. An atom type 
+may contribute to the LD but not the potential, or to the potential but not the 
+LD. Such decisions are made by the user and should (ideally) be motivated on 
+physical grounds for the problem at hand.
+
+:line
+
+[General form for implementation in LAMMPS:]
+
+Of course, a system with many atom types may have many different possible LD 
+potentials, each with their own atom type filters, cutoffs, and embedding 
+functions. The most general form of this potential as implemented in the 
+pair_style local/density is:
+
+:c,image(Eqs/pair_local_density_energy_implement.jpg)
+
+where, k is an index that spans the (arbitrary) number of applied LD potentials 
+N_LD. Each LD is calculated as before with:
+
+:c,image(Eqs/pair_local_density_ld_implement.jpg)
+
+The superscript on the indicator function phi simply indicates that it is 
+associated with specific values of the cutoff distances R1(k) and R2(k). In 
+summary, there may be N_LD distinct LD potentials. With each potential type (k), 
+one must specify:
+
+the inner and outer cutoffs as R1 and R2
+the central type filter a(k), where k = 1,2,...N_LD
+the neighbor type filter b(k), where k = 1,2,...N_LD
+the LD potential function F(k)(rho), typically as a table that is later spline-interpolated :ul
+
+:line
+
+[Tabulated input file format:]
+
+Line 1:             comment or blank (ignored)
+Line 2:             comment or blank (ignored)
+Line 3:             N_LD N_rho (# of LD potentials and # of tabulated values, single space separated)
+Line 4:             blank (ignored)
+Line 5:             R1(k) R2(k) (lower and upper cutoffs, single space separated)
+Line 6: 	        central-types (central atom types, single space separated)
+Line 7: 	        neighbor-types (neighbor atom types single space separated)
+Line 8: 	        rho_min rho_max drho (min, max and diff. in tabulated rho values, single space separated)
+Line 9: 	        F(k)(rho_min + 0.drho)
+Line 10:	        F(k)(rho_min + 1.drho)
+Line 11:	        F(k)(rho_min + 2.drho)
+............ 
+Line 9+N_rho:       F(k)(rho_min + N_rho . drho)
+Line 10+N_rho:      blank (ignored) :ul
+
+Block 2 :ul
+
+Block 3 :ul
+
+Block N_LD :ul
+
+Lines 5 to 9+N_rho constitute the first block. Thus the input file is separated 
+(by blank lines) into N_LD blocks each representing a separate LD potential and 
+each specifying its own upper and lower cutoffs, central and neighbor atoms, 
+and potential.  In general, blank lines anywhere are ignored. 
+
+:line
+
+[Mixing, shift, table, tail correction, restart, info]:
+This pair style does not support automatic mixing. For atom type pairs alpha,
+beta and alpha != beta, even if LD potentials of type (alpha, alpha) and 
+(beta, beta) are provided, you will need to explicitly provide LD potential 
+types (alpha, beta) and (beta, alpha) if need be (Here, the notation (alpha,
+beta) means that alpha is the central atom to which the LD potential is applied
+and beta is the neighbor atom which contributes to the LD potential on alpha).
+
+This pair style does not support the "pair_modify"_pair_modify.html
+shift, table, and tail options.
+
+The local/density pair style does not write its information to "binary restart
+files"_restart.html, since it is stored in tabulated potential files.
+Thus, you need to re-specify the pair_style and pair_coeff commands in
+an input script that reads a restart file.
+
+:line
+
+[Restrictions:]
+
+The local/density pair style is a part of the USER-MISC package. It is only
+enabled if LAMMPS was built with that package.  See the "Build
+package"_Build_package.html doc page for more info.
+
+[Related commands:]
+
+"pair_coeff"_pair_coeff.html
+
+[Default:] none
+
+:line
+
+
+:link(Sanyal1)
+[(Sanyal1)] Sanyal and Shell, Journal of Chemical Physics, 2016, 145 (3), 034109.
+:link(Sanyal2)
+[(Sanyal2)] Sanyal and Shell, Journal of Physical Chemistry B, 122 (21), 5678-5693.
+
+:link(Rosenberger)
+[(Rosenberger)] Rosenberger, Sanyal, Shell and van der Vegt,  Journal of Chemical Physics, 2019, 151 (4), 044111.