convert pair styles srp to tersoff

doc/src/Eqs/pair_srp1.tex

@@ -1,9 +0,0 @@
-\documentclass[12pt]{article}
-
-\begin{document}
-
-\begin{eqnarray*}
- F^{SRP}_{ij} & = & C(1-r/r_c)\hat{r}_{ij} \qquad r < r_c \\
-\end{eqnarray*}                           
-
-\end{document}

doc/src/Eqs/pair_srp2.tex

@@ -1,10 +0,0 @@
-\documentclass[12pt]{article}
-
-\begin{document}
-
-\begin{eqnarray*}
- F_{i1}^{SRP} & = & F^{SRP}_{ij}(L) \\
- F_{i2}^{SRP} & = & F^{SRP}_{ij}(1-L)  
-\end{eqnarray*}                           
-
-\end{document}

doc/src/Eqs/pair_sw.tex

@@ -1,18 +0,0 @@
-\documentclass[12pt]{article}
-
-\begin{document}
-
-\begin{eqnarray*}
-  E & = & \sum_i \sum_{j > i} \phi_2 (r_{ij}) + 
-          \sum_i \sum_{j \neq i} \sum_{k > j} 
-          \phi_3 (r_{ij}, r_{ik}, \theta_{ijk}) \\
-  \phi_2(r_{ij}) & = & A_{ij} \epsilon_{ij} \left[ B_{ij} (\frac{\sigma_{ij}}{r_{ij}})^{p_{ij}} - 
-                    (\frac{\sigma_{ij}}{r_{ij}})^{q_{ij}} \right] 
-                    \exp \left( \frac{\sigma_{ij}}{r_{ij} - a_{ij} \sigma_{ij}} \right) \\
-  \phi_3(r_{ij},r_{ik},\theta_{ijk}) & = & \lambda_{ijk} \epsilon_{ijk} \left[ \cos \theta_{ijk} - 
-                    \cos \theta_{0ijk} \right]^2
-                    \exp \left( \frac{\gamma_{ij} \sigma_{ij}}{r_{ij} - a_{ij} \sigma_{ij}} \right)
-                    \exp \left( \frac{\gamma_{ik} \sigma_{ik}}{r_{ik} - a_{ik} \sigma_{ik}} \right)
-\end{eqnarray*}                           
-
-\end{document}

doc/src/Eqs/pair_tersoff_1.tex

@@ -1,24 +0,0 @@
-\documentclass[12pt]{article}
-
-\begin{document}
-
-\begin{eqnarray*}
-  E & = & \frac{1}{2} \sum_i \sum_{j \neq i} V_{ij} \\
-  V_{ij} & = & f_C(r_{ij}) \left[ f_R(r_{ij}) + b_{ij} f_A(r_{ij}) \right] \\
-  f_C(r) & = & \left\{ \begin{array} {r@{\quad:\quad}l}
-    1 & r < R - D \\
-    \frac{1}{2} - \frac{1}{2} \sin \left( \frac{\pi}{2} \frac{r-R}{D} \right) &
-      R-D < r < R + D \\
-    0 & r > R + D
-    \end{array} \right. \\
-  f_R(r) & = & A \exp (-\lambda_1 r) \\
-  f_A(r) & = & -B \exp (-\lambda_2 r) \\
-  b_{ij} & = & \left( 1 + \beta^n {\zeta_{ij}}^n \right)^{-\frac{1}{2n}} \\
-  \zeta_{ij} & = & \sum_{k \neq i,j} f_C(r_{ik}) g(\theta_{ijk})
-                   \exp \left[ {\lambda_3}^m (r_{ij} - r_{ik})^m \right] \\
-  g(\theta) & = & \gamma_{ijk} \left( 1 + \frac{c^2}{d^2} - 
-                  \frac{c^2}{\left[ d^2 + 
-		  (\cos \theta - \cos \theta_0)^2\right]} \right)
-\end{eqnarray*}
-
-\end{document}

doc/src/Eqs/pair_tersoff_2.tex

@@ -1,14 +0,0 @@
-\documentclass[12pt]{article}
-
-\begin{document}
-
-\begin{eqnarray*}
-\lambda_1^{i,j} &=& \frac{1}{2}(\lambda_1^i + \lambda_1^j)\\
-\lambda_2^{i,j} &=& \frac{1}{2}(\lambda_2^i + \lambda_2^j)\\
-A_{i,j} &=& (A_{i}A_{j})^{1/2}\\
-B_{i,j} &=& \chi_{ij}(B_{i}B_{j})^{1/2}\\
-R_{i,j} &=& (R_{i}R_{j})^{1/2}\\
-S_{i,j} &=& (S_{i}S_{j})^{1/2}\\
-\end{eqnarray*}                           
-
-\end{document}

doc/src/Eqs/pair_tersoff_mod.tex

@@ -1,24 +0,0 @@
-\documentclass[12pt]{article}
-
-\begin{document}
-
-\begin{eqnarray*}
-  E & = & \frac{1}{2} \sum_i \sum_{j \neq i} V_{ij} \\
-  V_{ij} & = & f_C(r_{ij}) \left[ f_R(r_{ij}) + b_{ij} f_A(r_{ij}) \right] \\
-  f_C(r) & = & \left\{ \begin{array} {r@{\quad:\quad}l}
-    1 & r < R - D \\
-    \frac{1}{2} - \frac{9}{16} \sin \left( \frac{\pi}{2} \frac{r-R}{D} \right) - \frac{1}{16} \sin \left( \frac{3\pi}{2} \frac{r-R}{D} \right) &
-      R-D < r < R + D \\
-    0 & r > R + D
-    \end{array} \right. \\
-  f_R(r) & = & A \exp (-\lambda_1 r) \\
-  f_A(r) & = & -B \exp (-\lambda_2 r) \\
-  b_{ij} & = & \left( 1 + {\zeta_{ij}}^\eta \right)^{-\frac{1}{2n}} \\
-  \zeta_{ij} & = & \sum_{k \neq i,j} f_C(r_{ik}) g(\theta_{ijk})
-                   \exp \left[ \alpha (r_{ij} - r_{ik})^\beta \right] \\
-  g(\theta) & = & c_1 + g_o(\theta) g_a(\theta) \\
-  g_o(\theta) & = & \frac{c_2 (h - \cos \theta)^2}{c_3 + (h - \cos \theta)^2} \\
-  g_a(\theta) & = & 1 + c_4 \exp \left[ -c_5 (h - \cos \theta)^2 \right] \\
-\end{eqnarray*}
-
-\end{document}

doc/src/Eqs/pair_tersoff_mod_c.tex

@@ -1,10 +0,0 @@
-\documentclass[12pt]{article}
-\pagestyle{empty}
-
-\begin{document}
-
-\begin{eqnarray*}
-  V_{ij}  & =  & f_C(r_{ij}) \left[ f_R(r_{ij}) + b_{ij} f_A(r_{ij}) + c_0 \right]
-\end{eqnarray*}
-
-\end{document}

doc/src/Eqs/pair_tersoff_zbl.tex

@@ -1,33 +0,0 @@
-\documentclass[12pt]{article}
-
-\begin{document}
-
-\begin{eqnarray*}
-  E & = & \frac{1}{2} \sum_i \sum_{j \neq i} V_{ij} \\
-  V_{ij} & = & (1 - f_F(r_{ij})) V^{ZBL}_{ij} + f_F(r_{ij}) V^{Tersoff}_{ij} \\
-f_F(r_{ij}) & = & \frac{1}{1 + e^{-A_F(r_{ij} - r_C)}}\\
-  \\
-  \\
-  V^{ZBL}_{ij} & = & \frac{1}{4\pi\epsilon_0} \frac{Z_1 Z_2 \,e^2}{r_{ij}} \phi(r_{ij}/a) \\
-  a & = & \frac{0.8854\,a_0}{Z_{1}^{0.23} + Z_{2}^{0.23}}\\
-  \phi(x) & = & 0.1818e^{-3.2x} + 0.5099e^{-0.9423x} + 0.2802e^{-0.4029x} + 0.02817e^{-0.2016x}\\
-  \\
-  \\
-  V^{Tersoff}_{ij} & = & f_C(r_{ij}) \left[ f_R(r_{ij}) + b_{ij} f_A(r_{ij}) \right] \\
-  f_C(r) & = & \left\{ \begin{array} {r@{\quad:\quad}l}
-    1 & r < R - D \\
-    \frac{1}{2} - \frac{1}{2} \sin \left( \frac{\pi}{2} \frac{r-R}{D} \right) &
-      R-D < r < R + D \\
-    0 & r > R + D
-    \end{array} \right. \\
-  f_R(r) & = & A \exp (-\lambda_1 r) \\
-  f_A(r) & = & -B \exp (-\lambda_2 r) \\
-  b_{ij} & = & \left( 1 + \beta^n {\zeta_{ij}}^n \right)^{-\frac{1}{2n}} \\
-  \zeta_{ij} & = & \sum_{k \neq i,j} f_C(r_{ik}) g(\theta_{ijk})
-                   \exp \left[ {\lambda_3}^m (r_{ij} - r_{ik})^m \right] \\
-  g(\theta) & = & \gamma_{ijk} \left( 1 + \frac{c^2}{d^2} - 
-                  \frac{c^2}{\left[ d^2 + 
-		  (\cos \theta - \cos \theta_0)^2\right]} \right)
-\end{eqnarray*}                           
-
-\end{document}

doc/src/pair_srp.rst

@@ -56,19 +56,25 @@ Bonds of specified type *btype* interact with one another through a
 bond-pairwise potential, such that the force on bond *i* due to bond
 *j* is as follows
 
-.. image:: Eqs/pair_srp1.jpg
+.. math::
-   :align: center
 
-where *r* and *rij* are the distance and unit vector between the two
+   F^{SRP}_{ij} & = C(1-r/r_c)\hat{r}_{ij} \qquad r < r_c
-bonds.  Note that *btype* can be specified as an asterisk "\*", which
+
-case the interaction is applied to all bond types.  The *mid* option
+
-computes *r* and *rij* from the midpoint distance between bonds. The
+where *r* and :math:`\hat{r}_{ij}` are the distance and unit vector
-*min* option computes *r* and *rij* from the minimum distance between
+between the two bonds.  Note that *btype* can be specified as an
-bonds. The force acting on a bond is mapped onto the two bond atoms
+asterisk "\*", which case the interaction is applied to all bond types.
-according to the lever rule,
+The *mid* option computes *r* and :math:`\hat{r}_{ij}` from the midpoint
+distance between bonds. The *min* option computes *r* and
+:math:`\hat{r}_{ij}` from the minimum distance between bonds. The force
+acting on a bond is mapped onto the two bond atoms according to the
+lever rule,
+
+.. math::
+
+   F_{i1}^{SRP} & = F^{SRP}_{ij}(L) \\
+   F_{i2}^{SRP} & = F^{SRP}_{ij}(1-L)  
 
-.. image:: Eqs/pair_srp2.jpg
-   :align: center
 
 where *L* is the normalized distance from the atom to the point of
 closest approach of bond *i* and *j*\ . The *mid* option takes *L* as
@@ -80,7 +86,7 @@ the data file or restart file read by the :doc:`read_data <read_data>`
 or :doc:`read_restart <read_restart>` commands:
 
 * *C* (force units)
-* *rc* (distance units)
+* :math:`r_c` (distance units)
 
 The last coefficient is optional. If not specified, the global cutoff
 is used.
@@ -114,7 +120,7 @@ Pair style *srp* turns off normalization of thermodynamic properties
 by particle number, as if the command :doc:`thermo_modify norm no <thermo_modify>` had been issued.
 
 The pairwise energy associated with style *srp* is shifted to be zero
-at the cutoff distance *rc*\ .
+at the cutoff distance :math:`r_c`.
 
 
 ----------
@@ -127,7 +133,7 @@ This pair styles does not support mixing.
 This pair style does not support the :doc:`pair_modify <pair_modify>`
 shift option for the energy of the pair interaction. Note that as
 discussed above, the energy term is already shifted to be 0.0 at the
-cutoff distance *rc*\ .
+cutoff distance :math:`r_c`.
 
 The :doc:`pair_modify <pair_modify>` table option is not relevant for
 this pair style.

doc/src/pair_sw.rst

@@ -39,12 +39,23 @@ Description
 The *sw* style computes a 3-body :ref:`Stillinger-Weber <Stillinger2>`
 potential for the energy E of a system of atoms as
 
-.. image:: Eqs/pair_sw.jpg
+.. math::
-   :align: center
 
-where phi2 is a two-body term and phi3 is a three-body term.  The
+   E & =  \sum_i \sum_{j > i} \phi_2 (r_{ij}) + 
-summations in the formula are over all neighbors J and K of atom I
+          \sum_i \sum_{j \neq i} \sum_{k > j} 
-within a cutoff distance = a\*sigma.
+          \phi_3 (r_{ij}, r_{ik}, \theta_{ijk}) \\
+  \phi_2(r_{ij}) & =  A_{ij} \epsilon_{ij} \left[ B_{ij} (\frac{\sigma_{ij}}{r_{ij}})^{p_{ij}} - 
+                    (\frac{\sigma_{ij}}{r_{ij}})^{q_{ij}} \right] 
+                    \exp \left( \frac{\sigma_{ij}}{r_{ij} - a_{ij} \sigma_{ij}} \right) \\
+  \phi_3(r_{ij},r_{ik},\theta_{ijk}) & = \lambda_{ijk} \epsilon_{ijk} \left[ \cos \theta_{ijk} - 
+                    \cos \theta_{0ijk} \right]^2
+                    \exp \left( \frac{\gamma_{ij} \sigma_{ij}}{r_{ij} - a_{ij} \sigma_{ij}} \right)
+                    \exp \left( \frac{\gamma_{ik} \sigma_{ik}}{r_{ik} - a_{ik} \sigma_{ik}} \right)
+
+
+where :math:`\phi_2` is a two-body term and :math:`\phi_3` is a
+three-body term.  The summations in the formula are over all neighbors J
+and K of atom I within a cutoff distance :math:`a `\sigma`.
 
 Only a single pair\_coeff command is used with the *sw* style which
 specifies a Stillinger-Weber potential file with parameters for all
@@ -86,24 +97,25 @@ and three-body coefficients in the formula above:
 * element 1 (the center atom in a 3-body interaction)
 * element 2
 * element 3
-* epsilon (energy units)
+* :math:`\epsilon` (energy units)
-* sigma (distance units)
+* :math:`\sigma` (distance units)
 * a
-* lambda
+* :math:`\lambda`
-* gamma
+* :math:`\gamma`
-* costheta0
+* :math:`\cos\theta_0`
 * A
 * B
 * p
 * q
 * tol
 
-The A, B, p, and q parameters are used only for two-body
+The A, B, p, and q parameters are used only for two-body interactions.
-interactions.  The lambda and costheta0 parameters are used only for
+The :math:`\lambda` and :math:`\cos\theta_0` parameters are used only
-three-body interactions. The epsilon, sigma and a parameters are used
+for three-body interactions. The :math:`\epsilon`, :math:`\sigma` and
-for both two-body and three-body interactions. gamma is used only in the
+*a* parameters are used for both two-body and three-body
-three-body interactions, but is defined for pairs of atoms.
+interactions. :math:`\gamma` is used only in the three-body
-The non-annotated parameters are unitless.
+interactions, but is defined for pairs of atoms.  The non-annotated
+parameters are unitless.
 
 LAMMPS introduces an additional performance-optimization parameter tol
 that is used for both two-body and three-body interactions.  In the
@@ -141,9 +153,9 @@ are usually defined by simple formulas involving two sets of pair-wise
 parameters, corresponding to the ij and ik pairs, where i is the
 center atom. The user must ensure that the correct combining rule is
 used to calculate the values of the three-body parameters for
-alloys. Note also that the function phi3 contains two exponential
+alloys. Note also that the function :math:`\phi_3` contains two exponential
 screening factors with parameter values from the ij pair and ik
-pairs. So phi3 for a C atom bonded to a Si atom and a second C atom
+pairs. So :math:`\phi_3` for a C atom bonded to a Si atom and a second C atom
 will depend on the three-body parameters for the CSiC entry, and also
 on the two-body parameters for the CCC and CSiSi entries. Since the
 order of the two neighbors is arbitrary, the three-body parameters for
@@ -152,8 +164,8 @@ parameters for entries SiCC and CSiSi should also be the same.  The
 parameters used only for two-body interactions (A, B, p, and q) in
 entries whose 2nd and 3rd element are different (e.g. SiCSi) are not
 used for anything and can be set to 0.0 if desired.
-This is also true for the parameters in phi3 that are
+This is also true for the parameters in :math:`\phi_3` that are
-taken from the ij and ik pairs (sigma, a, gamma)
+taken from the ij and ik pairs (:math:`\sigma`, *a*\ , :math:`\gamma`)
 
 
 ----------

doc/src/pair_tersoff.rst

@@ -50,12 +50,28 @@ Description
 The *tersoff* style computes a 3-body Tersoff potential
 :ref:`(Tersoff\_1) <Tersoff_11>` for the energy E of a system of atoms as
 
-.. image:: Eqs/pair_tersoff_1.jpg
+.. math::
-   :align: center
 
-where f\_R is a two-body term and f\_A includes three-body interactions.
+  E & = \frac{1}{2} \sum_i \sum_{j \neq i} V_{ij} \\
-The summations in the formula are over all neighbors J and K of atom I
+  V_{ij} & = f_C(r_{ij}) \left[ f_R(r_{ij}) + b_{ij} f_A(r_{ij}) \right] \\
-within a cutoff distance = R + D.
+  f_C(r) & = \left\{ \begin{array} {r@{\quad:\quad}l}
+    1 & r < R - D \\
+    \frac{1}{2} - \frac{1}{2} \sin \left( \frac{\pi}{2} \frac{r-R}{D} \right) &
+      R-D < r < R + D \\
+    0 & r > R + D
+    \end{array} \right. \\
+  f_R(r) & =  A \exp (-\lambda_1 r) \\
+  f_A(r) & =  -B \exp (-\lambda_2 r) \\
+  b_{ij} & =  \left( 1 + \beta^n {\zeta_{ij}}^n \right)^{-\frac{1}{2n}} \\
+  \zeta_{ij} & =  \sum_{k \neq i,j} f_C(r_{ik}) g(\theta_{ijk})
+                   \exp \left[ {\lambda_3}^m (r_{ij} - r_{ik})^m \right] \\
+  g(\theta) & =  \gamma_{ijk} \left( 1 + \frac{c^2}{d^2} - 
+                  \frac{c^2}{\left[ d^2 + (\cos \theta - \cos \theta_0)^2\right]} \right)
+
+
+where :math:`f_R` is a two-body term and :math:`f_A` includes three-body
+interactions.  The summations in the formula are over all neighbors
+J and K of atom I within a cutoff distance = R + D.
 
 The *tersoff/table* style uses tabulated forms for the two-body,
 environment and angular functions. Linear interpolation is performed
@@ -104,22 +120,24 @@ above:
 * element 2 (the atom bonded to the center atom)
 * element 3 (the atom influencing the 1-2 bond in a bond-order sense)
 * m
-* gamma
+* :math:`\gamma`
-* lambda3 (1/distance units)
+* :math:`\lambda_3` (1/distance units)
 * c
 * d
-* costheta0 (can be a value < -1 or > 1)
+* :math:`\cos\theta_0` (can be a value < -1 or > 1)
 * n
-* beta
+* :math:`\beta`
-* lambda2 (1/distance units)
+* :math:`\lambda_2` (1/distance units)
 * B (energy units)
 * R (distance units)
 * D (distance units)
-* lambda1 (1/distance units)
+* :math:`\lambda_1` (1/distance units)
 * A (energy units)
 
-The n, beta, lambda2, B, lambda1, and A parameters are only used for
+The n, :math:`\beta`, :math:`\lambda_2`, B, :math:`\lambda_1`, and A
-two-body interactions.  The m, gamma, lambda3, c, d, and costheta0
+parameters are only used for
+two-body interactions.  The m, :math:`\gamma`, :math:`\lambda_3`, c, d,
+and :math:`\cos\theta_0`
 parameters are only used for three-body interactions. The R and D
 parameters are used for both two-body and three-body interactions. The
 non-annotated parameters are unitless.  The value of m must be 3 or 1.
@@ -149,7 +167,8 @@ SiCC entry.
 The parameters used for a particular
 three-body interaction come from the entry with the corresponding
 three elements.  The parameters used only for two-body interactions
-(n, beta, lambda2, B, lambda1, and A) in entries whose 2nd and 3rd
+(n, :math:`\beta`, :math:`\lambda_2`, B, :math:`\lambda_1`, and A)
+in entries whose 2nd and 3rd
 element are different (e.g. SiCSi) are not used for anything and can
 be set to 0.0 if desired.
 
@@ -165,16 +184,24 @@ it reduces to the form of :ref:`Albe et al. <Albe>` when beta = 1 and m = 1.
 Note that in the current Tersoff implementation in LAMMPS, m must be
 specified as either 3 or 1.  Tersoff used a slightly different but
 equivalent form for alloys, which we will refer to as Tersoff\_2
-potential :ref:`(Tersoff\_2) <Tersoff_21>`. The *tersoff/table* style implements
+potential :ref:`(Tersoff\_2) <Tersoff_21>`.
+The *tersoff/table* style implements
 Tersoff\_2 parameterization only.
 
 LAMMPS parameter values for Tersoff\_2 can be obtained as follows:
-gamma\_ijk = omega\_ik, lambda3 = 0 and the value of
+:math:`\gamma_{ijk} = \omega_{ik}`, :math:`\lambda_3 = 0` and the value of
 m has no effect.  The parameters for species i and j can be calculated
 using the Tersoff\_2 mixing rules:
 
-.. image:: Eqs/pair_tersoff_2.jpg
+.. math::
-   :align: center
+
+   \lambda_1^{i,j} & = \frac{1}{2}(\lambda_1^i + \lambda_1^j)\\
+   \lambda_2^{i,j} & = \frac{1}{2}(\lambda_2^i + \lambda_2^j)\\
+   A_{i,j} & = (A_{i}A_{j})^{1/2}\\
+   B_{i,j} & = \chi_{ij}(B_{i}B_{j})^{1/2}\\
+   R_{i,j} & = (R_{i}R_{j})^{1/2}\\
+   S_{i,j} & = (S_{i}S_{j})^{1/2}
+
 
 Tersoff\_2 parameters R and S must be converted to the LAMMPS
 parameters R and D (R is different in both forms), using the following

doc/src/pair_tersoff_mod.rst

@@ -49,21 +49,40 @@ potential :ref:`(Tersoff\_1) <Tersoff_12>`, :ref:`(Tersoff\_2) <Tersoff_22>` wit
 modified cutoff function and angular-dependent term, giving the energy
 E of a system of atoms as
 
-.. image:: Eqs/pair_tersoff_mod.jpg
+.. math::
-   :align: center
 
-where f\_R is a two-body term and f\_A includes three-body interactions.
+  E & = \frac{1}{2} \sum_i \sum_{j \neq i} V_{ij} \\
+  V_{ij} & = f_C(r_{ij}) \left[ f_R(r_{ij}) + b_{ij} f_A(r_{ij}) \right] \\
+  f_C(r) & = \left\{ \begin{array} {r@{\quad:\quad}l}
+    1 & r < R - D \\
+    \frac{1}{2} - \frac{9}{16} \sin \left( \frac{\pi}{2} \frac{r-R}{D} \right) - \frac{1}{16} \sin \left( \frac{3\pi}{2} \frac{r-R}{D} \right) &
+      R-D < r < R + D \\
+    0 & r > R + D
+    \end{array} \right. \\
+  f_R(r) & = A \exp (-\lambda_1 r) \\
+  f_A(r) & = -B \exp (-\lambda_2 r) \\
+  b_{ij} & = \left( 1 + {\zeta_{ij}}^\eta \right)^{-\frac{1}{2n}} \\
+  \zeta_{ij} & = \sum_{k \neq i,j} f_C(r_{ik}) g(\theta_{ijk})
+                   \exp \left[ \alpha (r_{ij} - r_{ik})^\beta \right] \\
+  g(\theta) & = c_1 + g_o(\theta) g_a(\theta) \\
+  g_o(\theta) & = \frac{c_2 (h - \cos \theta)^2}{c_3 + (h - \cos \theta)^2} \\
+  g_a(\theta) & = 1 + c_4 \exp \left[ -c_5 (h - \cos \theta)^2 \right] \\
+
+
+where :math:`f_R` is a two-body term and :math:`f_A` includes three-body interactions.
 The summations in the formula are over all neighbors J and K of atom I
 within a cutoff distance = R + D.
 The *tersoff/mod/c* style differs from *tersoff/mod* only in the
 formulation of the V\_ij term, where it contains an additional c0 term.
 
-.. image:: Eqs/pair_tersoff_mod_c.jpg
+.. math::
-   :align: center
 
-The modified cutoff function f\_C proposed by :ref:`(Murty) <Murty>` and
+  V_{ij}  & = f_C(r_{ij}) \left[ f_R(r_{ij}) + b_{ij} f_A(r_{ij}) + c_0 \right]
+
+
+The modified cutoff function :math:`f_C` proposed by :ref:`(Murty) <Murty>` and
 having a continuous second-order differential is employed. The
-angular-dependent term g(theta) was modified to increase the
+angular-dependent term :math:`g(\theta)` was modified to increase the
 flexibility of the potential.
 
 The *tersoff/mod* potential is fitted to both the elastic constants
@@ -105,30 +124,30 @@ not blank or comments (starting with #) define parameters for a triplet
 of elements.  The parameters in a single entry correspond to
 coefficients in the formulae above:
 
-element 1 (the center atom in a 3-body interaction)
+* element 1 (the center atom in a 3-body interaction)
-element 2 (the atom bonded to the center atom)
+* element 2 (the atom bonded to the center atom)
-element 3 (the atom influencing the 1-2 bond in a bond-order sense)
+* element 3 (the atom influencing the 1-2 bond in a bond-order sense)
-beta
+* :math:`\beta`
-alpha
+* :math:`\alpha`
-h
+* h
-eta
+* :math:`\eta`
-beta\_ters = 1 (dummy parameter)
+* :math:`\beta_{ters}` = 1 (dummy parameter)
-lambda2 (1/distance units)
+* :math:`\lambda_2` (1/distance units)
-B (energy units)
+* B (energy units)
-R (distance units)
+* R (distance units)
-D (distance units)
+* D (distance units)
-lambda1 (1/distance units)
+* :math:`\lambda_1` (1/distance units)
-A (energy units)
+* A (energy units)
-n
+* n
-c1
+* c1
-c2
+* c2
-c3
+* c3
-c4
+* c4
-c5
+* c5
-c0 (energy units, tersoff/mod/c only):ul
+* c0 (energy units, tersoff/mod/c only):ul
 
-The n, eta, lambda2, B, lambda1, and A parameters are only used for
+The n, :math:`\eta`, :math:`\lambda_2`, B, :math:`\lambda_1`, and A parameters are only used for
-two-body interactions.  The beta, alpha, c1, c2, c3, c4, c5, h
+two-body interactions.  The :math:`\beta`, :math:`\alpha`, c1, c2, c3, c4, c5, h
 parameters are only used for three-body interactions. The R and D
 parameters are used for both two-body and three-body interactions.
 The c0 term applies to *tersoff/mod/c* only. The non-annotated

doc/src/pair_tersoff_zbl.rst

@@ -38,26 +38,53 @@ based on a Coulomb potential and the Ziegler-Biersack-Littmark
 universal screening function :ref:`(ZBL) <zbl-ZBL>`, giving the energy E of a
 system of atoms as
 
-.. image:: Eqs/pair_tersoff_zbl.jpg
+.. math::
-   :align: center
 
-The f\_F term is a fermi-like function used to smoothly connect the ZBL
+   E & = \frac{1}{2} \sum_i \sum_{j \neq i} V_{ij} \\
+   V_{ij} & =  (1 - f_F(r_{ij})) V^{ZBL}_{ij} + f_F(r_{ij}) V^{Tersoff}_{ij} \\
+   f_F(r_{ij}) & =  \frac{1}{1 + e^{-A_F(r_{ij} - r_C)}}\\
+   \\
+   \\
+   V^{ZBL}_{ij} & = \frac{1}{4\pi\epsilon_0} \frac{Z_1 Z_2 \,e^2}{r_{ij}} \phi(r_{ij}/a) \\
+  a & = \frac{0.8854\,a_0}{Z_{1}^{0.23} + Z_{2}^{0.23}}\\
+  \phi(x) & =  0.1818e^{-3.2x} + 0.5099e^{-0.9423x} + 0.2802e^{-0.4029x} + 0.02817e^{-0.2016x}\\
+  \\
+  \\
+  V^{Tersoff}_{ij} & = f_C(r_{ij}) \left[ f_R(r_{ij}) + b_{ij} f_A(r_{ij}) \right] \\
+  f_C(r) & = \left\{ \begin{array} {r@{\quad:\quad}l}
+    1 & r < R - D \\
+    \frac{1}{2} - \frac{1}{2} \sin \left( \frac{\pi}{2} \frac{r-R}{D} \right) &
+      R-D < r < R + D \\
+    0 & r > R + D
+    \end{array} \right. \\
+  f_R(r) & = A \exp (-\lambda_1 r) \\
+  f_A(r) & = -B \exp (-\lambda_2 r) \\
+  b_{ij} & = \left( 1 + \beta^n {\zeta_{ij}}^n \right)^{-\frac{1}{2n}} \\
+  \zeta_{ij} & = \sum_{k \neq i,j} f_C(r_{ik}) g(\theta_{ijk})
+                   \exp \left[ {\lambda_3}^m (r_{ij} - r_{ik})^m \right] \\
+  g(\theta) & =  \gamma_{ijk} \left( 1 + \frac{c^2}{d^2} - 
+                  \frac{c^2}{\left[ d^2 + (\cos \theta - \cos \theta_0)^2\right]} \right)
+
+
+The :math:`f_F` term is a fermi-like function used to smoothly connect the ZBL
 repulsive potential with the Tersoff potential.  There are 2
-parameters used to adjust it: A\_F and r\_C.  A\_F controls how "sharp"
+parameters used to adjust it: :math:`A_F` and :math:`r_C`.  :math:`A_F`
-the transition is between the two, and r\_C is essentially the cutoff
+controls how "sharp"
+the transition is between the two, and :math:`r_C` is essentially the cutoff
 for the ZBL potential.
 
 For the ZBL portion, there are two terms. The first is the Coulomb
 repulsive term, with Z1, Z2 as the number of protons in each nucleus,
-e as the electron charge (1 for metal and real units) and epsilon0 as
+e as the electron charge (1 for metal and real units) and :math:`\epsilon_0`
-the permittivity of vacuum.  The second part is the ZBL universal
+as the permittivity of vacuum.  The second part is the ZBL universal
 screening function, with a0 being the Bohr radius (typically 0.529
 Angstroms), and the remainder of the coefficients provided by the
 original paper.  This screening function should be applicable to most
 systems.  However, it is only accurate for small separations
 (i.e. less than 1 Angstrom).
 
-For the Tersoff portion, f\_R is a two-body term and f\_A includes
+For the Tersoff portion, :math:`f_R` is a two-body term and :math:`f_A`
+includes
 three-body interactions. The summations in the formula are over all
 neighbors J and K of atom I within a cutoff distance = R + D.
 
@@ -102,29 +129,32 @@ in the formula above:
 * element 2 (the atom bonded to the center atom)
 * element 3 (the atom influencing the 1-2 bond in a bond-order sense)
 * m
-* gamma
+* :math:`\gamma`
-* lambda3 (1/distance units)
+* :math:`\lambda_3` (1/distance units)
 * c
 * d
-* costheta0 (can be a value < -1 or > 1)
+* :math:`\cos\theta_0` (can be a value < -1 or > 1)
 * n
-* beta
+* :math:`\beta`
-* lambda2 (1/distance units)
+* :math:`\lambda_2` (1/distance units)
 * B (energy units)
 * R (distance units)
 * D (distance units)
-* lambda1 (1/distance units)
+* :math:`\lambda_1` (1/distance units)
 * A (energy units)
-* Z\_i
+* :math:`Z_i`
-* Z\_j
+* :math:`Z_j`
 * ZBLcut (distance units)
 * ZBLexpscale (1/distance units)
 
-The n, beta, lambda2, B, lambda1, and A parameters are only used for
+The n, :math:`\beta`, :math:`\lambda_2`, B, :math:`\lambda_1`, and A
-two-body interactions.  The m, gamma, lambda3, c, d, and costheta0
+parameters are only used for
+two-body interactions.  The m, :math:`\gamma`, :math:`\lambda_3`, c, d,
+and :math:`\cos\theta_0`
 parameters are only used for three-body interactions. The R and D
 parameters are used for both two-body and three-body interactions. The
-Z\_i,Z\_j, ZBLcut, ZBLexpscale parameters are used in the ZBL repulsive
+:math:`Z_i`, :math:`Z_j`, ZBLcut, ZBLexpscale parameters are used in the
+ZBL repulsive
 portion of the potential and in the Fermi-like function.  The
 non-annotated parameters are unitless.  The value of m must be 3 or 1.
 
@@ -153,7 +183,8 @@ SiCC entry.
 The parameters used for a particular
 three-body interaction come from the entry with the corresponding
 three elements.  The parameters used only for two-body interactions
-(n, beta, lambda2, B, lambda1, and A) in entries whose 2nd and 3rd
+(n, :math:`\beta`, :math:`\lambda_2`, B, :math:`\lambda_1`, and A)
+in entries whose 2nd and 3rd
 element are different (e.g. SiCSi) are not used for anything and can
 be set to 0.0 if desired.
 
@@ -172,12 +203,19 @@ different but equivalent form for alloys, which we will refer to as
 Tersoff\_2 potential :ref:`(Tersoff\_2) <zbl-Tersoff_2>`.
 
 LAMMPS parameter values for Tersoff\_2 can be obtained as follows:
-gamma = omega\_ijk, lambda3 = 0 and the value of
+:math:`\gamma = \omega_{ijk}`, :math:`\lambda_3 = 0` and the value of
 m has no effect.  The parameters for species i and j can be calculated
 using the Tersoff\_2 mixing rules:
 
-.. image:: Eqs/pair_tersoff_2.jpg
+.. math::
-   :align: center
+
+   \lambda_1^{i,j} & = \frac{1}{2}(\lambda_1^i + \lambda_1^j)\\
+   \lambda_2^{i,j} & = \frac{1}{2}(\lambda_2^i + \lambda_2^j)\\
+   A_{i,j} & = (A_{i}A_{j})^{1/2}\\
+   B_{i,j} & = \chi_{ij}(B_{i}B_{j})^{1/2}\\
+   R_{i,j} & = (R_{i}R_{j})^{1/2}\\
+   S_{i,j} & = (S_{i}S_{j})^{1/2}\\
+
 
 Tersoff\_2 parameters R and S must be converted to the LAMMPS
 parameters R and D (R is different in both forms), using the following