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import polymorphic pair style update from Xiaowang Zhou

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Axel Kohlmeyer 2020-04-15 11:31:30 -04:00
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doc/src/pair_polymorphic.rst
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@ -1,4 +1,5 @@
.. index:: pair_style polymorphic
<HTML><META HTTP-EQUIV="content-type" CONTENT="text/html;charset=utf-8">
<PRE>.. index:: pair_style polymorphic
pair_style polymorphic command
==============================
@ -18,22 +19,27 @@ Examples
.. code-block:: LAMMPS
pair_style polymorphic
pair_coeff * * TlBr_msw.polymorphic Tl Br
pair_coeff * * AlCu_eam.polymorphic Al Cu
pair_coeff * * GaN_tersoff.polymorphic Ga N
pair_coeff * * GaN_sw.polymorphic GaN
pair_coeff * * FeCH_BOPI.poly Fe C H
pair_coeff * * TlBr_msw.poly Tl Br
pair_coeff * * CuTa_eam.poly Cu Ta
pair_coeff * * GaN_tersoff.poly Ga N
pair_coeff * * GaN_sw.poly GaN
Description
"""""""""""
The *polymorphic* pair style computes a 3-body free-form potential
(:ref:`Zhou <Zhou3>`) for the energy E of a system of atoms as
(:ref:`Zhou &lt;Zhou3&gt;`) for the energy E of a system of atoms as
.. math::
E & = \frac{1}{2}\sum_{i=1}^{i=N}\sum_{j=1}^{j=N}\left[\left(1-\delta_{ij}\right)\cdot U_{IJ}\left(r_{ij}\right)-\left(1-\eta_{ij}\right)\cdot F_{IJ}\left(r_{ij}\right)\cdot V_{IJ}\left(r_{ij}\right)\right] \\
X_{ij} & = \sum_{k=i_1,k\neq i,j}^{i_N}W_{IK}\left(r_{ik}\right)\cdot G_{JIK}\left(\theta_{jik}\right)\cdot P_{IK}\left(\Delta r_{jik}\right) \\
\Delta r_{jik} & = r_{ij}-\xi_{IJ}\cdot r_{ik}
\begin{eqnarray}\nonumber
\left\{\begin{array}{l}
E = \frac{1}{2}\sum_{i=1}^{i=N}\sum_{j=1}^{j=N}\left[\left(1-\delta_{ij}\right)\cdot U_{IJ}\left(r_{ij}\right)-\left(1-\eta_{ij}\right)\cdot F_{IJ}\left(X_{ij}\right)\cdot V_{IJ}\left(r_{ij}\right)\right] \\
X_{ij} = \sum_{k=i_1,k\neq j}^{i_N}W_{IK}\left(r_{ik}\right)\cdot G_{JIK}\left(\theta_{jik}\right)\cdot P_{JIK}\left(\Delta r_{jik}\right) \\
\Delta r_{jik} = r_{ij}-\xi_{IJ}\cdot r_{ik}
\end{array}\right.
\end{eqnarray}
where I, J, K represent species of atoms i, j, and k, :math:`i_1, ...,
i_N` represents a list of *i*\ 's neighbors, :math:`\delta_{ij}` is a
@ -42,111 +48,157 @@ Dirac constant (i.e., :math:`\delta_{ij} = 1` when :math:`i = j`, and
constant that can be set either to :math:`\eta_{ij} = \delta_{ij}` or
:math:`\eta_{ij} = 1 - \delta_{ij}` depending on the potential type,
:math:`U_{IJ}(r_{ij})`, :math:`V_{IJ}(r_{ij})`, :math:`W_{IK}(r_{ik})`
are pair functions, :math:`G_{JIK}(\cos(\theta))` is an angular
function, :math:`P_{IK}(\Delta r_{jik})` is a function of atomic spacing
are pair functions, :math:`G_{JIK}(\cos\theta_{jik})` is an angular
function, :math:`P_{JIK}(\Delta r_{jik})` is a function of atomic spacing
differential :math:`\Delta r_{jik} = r_{ij} - \xi_{IJ} \cdot r_{ik}`
with :math:`\xi_{IJ}` being a pair-dependent parameter, and
:math:`F_{IJ}(X_{ij})` is a function of the local environment variable
:math:`X_{ij}`. This generic potential is fully defined once the
constants :math:`\eta_{ij}` and :math:`\xi_{IJ}`, and the six functions
:math:`U_{IJ}(r_{ij})`, :math:`V_{IJ}(r_{ij})`, :math:`W_{IK}(r_{ik})`,
:math:`G_{JIK}(\cos(\theta))`, :math:`P_{IK}(\Delta r_{jik})`, and
:math:`F_{IJ}(X_{ij})` are given. Note that these six functions are all
one dimensional, and hence can be provided in an analytic or tabular
:math:`G_{JIK}(\cos\theta_{jik})`, :math:`P_{JIK}(\Delta r_{jik})`, and
:math:`F_{IJ}(X_{ij})` are given. Here LAMMPS uses a global
parameter :math:`\eta` to represent :math:`\eta_{ij}`. When
:math:`\eta = 1`, :math:`\eta_{ij} = 1 - \delta_{ij}`, otherwise
:math:`\eta_{ij} = \delta_{ij}`. Additionally, :math:`\eta = 3`
indicates that the function :math:`P_{JIK}(\Delta r)` depends on
species I, J and K, otherwise :math:`P_{JIK}(\Delta r) = P_{IK}(\Delta r)`
only depends on species I and K. Note that these six functions are all
one dimensional, and hence can be provided in a tabular
form. This allows users to design different potentials solely based on a
manipulation of these functions. For instance, the potential reduces to
Stillinger-Weber potential (:ref:`SW <SW>`) if we set
manipulation of these functions. For instance, the potential reduces a
Stillinger-Weber potential (:ref:`SW &lt;SW&gt;`) if we set
.. math::
\begin{eqnarray}\nonumber
\left\{\begin{array}{l}
\eta_{ij} = \delta_{ij},\xi_{IJ}=0 \\
U_{IJ}\left(r\right)=A_{IJ}\cdot\epsilon_{IJ}\cdot \left(\frac{\sigma_{IJ}}{r}\right)^q\cdot \left[B_{IJ}\cdot \left(\frac{\sigma_{IJ}}{r}\right)^{p-q}-1\right]\cdot exp\left(\frac{\sigma_{IJ}}{r-a_{IJ}\cdot \sigma_{IJ}}\right) \\
V_{IJ}\left(r\right)=\sqrt{\lambda_{IJ}\cdot \epsilon_{IJ}}\cdot exp\left(\frac{\gamma_{IJ}\cdot \sigma_{IJ}}{r-a_{IJ}\cdot \sigma_{IJ}}\right) \\
F_{IJ}\left(X\right)=-X \\
P_{IJ}\left(\Delta r\right)=1 \\
W_{IJ}\left(r\right)=\sqrt{\lambda_{IJ}\cdot \epsilon_{IJ}}\cdot exp\left(\frac{\gamma_{IJ}\cdot \sigma_{IJ}}{r-a_{IJ}\cdot \sigma_{IJ}}\right) \\
G_{JIK}\left(\theta\right)=\left(cos\theta+\frac{1}{3}\right)^2
\eta_{ij} = \delta_{ij} (\eta = 2~or~\eta = 0),\xi_{IJ}=0 \\
U_{IJ}\left(r\right) = A_{IJ}\cdot\epsilon_{IJ}\cdot \left(\frac{\sigma_{IJ}}{r}\right)^q\cdot \left[B_{IJ}\cdot \left(\frac{\sigma_{IJ}}{r}\right)^{p-q}-1\right]\cdot exp\left(\frac{\sigma_{IJ}}{r-a_{IJ}\cdot \sigma_{IJ}}\right) \\
V_{IJ}\left(r\right) = \sqrt{\lambda_{IJ}\cdot \epsilon_{IJ}}\cdot exp\left(\frac{\gamma_{IJ}\cdot \sigma_{IJ}}{r-a_{IJ}\cdot \sigma_{IJ}}\right) \\
F_{IJ}\left(X\right) = -X \\
P_{JIK}\left(\Delta r\right) = P_{IK}\left(\Delta r\right) = 1 \\
W_{IJ}\left(r\right) = \sqrt{\lambda_{IJ}\cdot \epsilon_{IJ}}\cdot exp\left(\frac{\gamma_{IJ}\cdot \sigma_{IJ}}{r-a_{IJ}\cdot \sigma_{IJ}}\right) \\
G_{JIK}\left(\theta\right) = \left(cos\theta+\frac{1}{3}\right)^2
\end{array}\right.
\end{eqnarray}
The potential reduces to Tersoff types of potential
(:ref:`Tersoff <Tersoff>` or :ref:`Albe <poly-Albe>`) if we set
The potential reduces to a Tersoff potential (:ref:`Tersoff &lt;Tersoff&gt;
` or :ref:`Albe &lt;poly-Albe&gt;`) if we set
.. math::
\begin{eqnarray}\nonumber
\left\{\begin{array}{l}
\eta_{ij}=\delta_{ij},\xi_{IJ}=1 \\
U_{IJ}\left(r\right)=\frac{D_{e,IJ}}{S_{IJ}-1}\cdot exp\left[-\beta_{IJ}\sqrt{2S_{IJ}\left(r-r_{e,IJ}\right)}\right]\cdot f_{c,IJ}\left(r\right) \\
V_{IJ}\left(r\right)=\frac{S_{IJ}\cdot D_{e,IJ}}{S_{IJ}-1}\cdot exp\left[-\beta_{IJ}\sqrt{\frac{2}{S_{IJ}}\left(r-r_{e,IJ}\right)}\right]\cdot f_{c,IJ}\left(r\right) \\
F_{IJ}\left(X\right)=\left(1+X\right)^{-\frac{1}{2}} \\
P_{IJ}\left(\Delta r\right)=exp\left(2\mu_{IK}\cdot \Delta r\right) \\
W_{IJ}\left(r\right)=f_{c,IK}\left(r\right) \\
G_{JIK}\left(\theta\right)=\gamma_{IK}\left[1+\frac{c_{IK}^2}{d_{IK}^2}-\frac{c_{IK}^2}{d_{IK}^2+\left(h_{IK}+cos\theta\right)^2}\right]
\eta_{ij} = \delta_{ij} (\eta = 2~or~\eta = 0),\xi_{IJ}=1 \\
U_{IJ}\left(r\right) = \frac{D_{e,IJ}}{S_{IJ}-1}\cdot exp\left[-\beta_{IJ}\sqrt{2S_{IJ}}\left(r-r_{e,IJ}\right)\right]\cdot f_{c,IJ}\left(r\right) \\
V_{IJ}\left(r\right) = \frac{S_{IJ}\cdot D_{e,IJ}}{S_{IJ}-1}\cdot exp\left[-\beta_{IJ}\sqrt{\frac{2}{S_{IJ}}}\left(r-r_{e,IJ}\right)\right]\cdot f_{c,IJ}\left(r\right) \\
F_{IJ}\left(X\right) = \left(1+X\right)^{-\frac{1}{2}} \\
P_{JIK}\left(\Delta r\right) = P_{IK}\left(\Delta r\right) = exp\left(2\mu_{IK}\cdot \Delta r\right) \\
W_{IJ}\left(r\right) = f_{c,IJ}\left(r\right) \\
G_{JIK}\left(\theta\right) = \gamma_{IK}\left[1+\frac{c_{IK}^2}{d_{IK}^2}-\frac{c_{IK}^2}{d_{IK}^2+\left(h_{IK}+cos\theta\right)^2}\right]
\end{array}\right.
\end{eqnarray}
where
.. math::
f_{c,IJ}=\left\{\begin{array}{lr}
1, & r\leq r_{s,IJ} \\
\frac{1}{2}+\frac{1}{2} cos \left[\frac{\pi \left(r-r_{s,IJ}\right)}{r_{c,IJ}-r_{s,IJ}}\right], & r_{s,IJ}<r<r_{c,IJ} \\
0, & r \geq r_{c,IJ} \\
\begin{eqnarray}\nonumber
f_{c,IJ}\left(r\right)=\left\{\begin{array}{l}
1, r\leq R_{IJ}-D_{IJ} \\
\frac{1}{2}+\frac{1}{2}cos\left[\frac{\pi\left(r+D_{IJ}-R_{IJ}\right)}{2D_{IJ}}\right], R_{IJ}-D_{IJ} < r < R_{IJ}+D_{IJ} \\
0, r \geq R_{IJ}+D_{IJ}
\end{array}\right.
\end{eqnarray}
The potential reduces to Rockett-Tersoff (:ref:`Wang <Wang3>`) type if we set
The potential reduces to a modified Stillinger-Weber potential (:ref:`Zhou &lt;Zhou3&gt;`) if we set
.. math::
\begin{eqnarray}\nonumber
\left\{\begin{array}{l}
\eta_{ij}=\delta_{ij},\xi_{IJ}=1 \\
U_{IJ}\left(r\right)=\left\{\begin{array}{lr}
A_{IJ}\cdot exp\left(-\lambda_{1,IJ}\cdot r\right)\cdot f_{c,IJ}\left(r\right), & r\leq r_{s,1,IJ} \\
A_{IJ}\cdot exp\left(-\lambda_{1,IJ}\cdot r\right)\cdot f_{c,IJ}\left(r\right)\cdot f_{c,1,IJ}\left(r\right), & r_{s,1,IJ}<r<r_{c,1,IJ} \\
0, & r\ge r_{c,1,IJ}
\end{array}\right. \\
V_{IJ}\left(r\right)=\left\{\begin{array}{lr}
B_{IJ} \cdot exp\left(-\lambda_{2,IJ}\cdot r\right)\cdot f_{c,IJ}\left(r\right), & r\le r_{s,1,IJ} \\
B_{IJ} \cdot exp\left(-\lambda_{2,IJ}\cdot r\right)\cdot f_{c,IJ}\left(r\right)+A_{IJ}\cdot exp\left(-\lambda_{1,IJ}\cdot r\right)\cdot & \\ ~~~~~~ f_{c,IJ}\left(r\right)\cdot \left[1-f_{c,1,IJ}\left(r\right)\right], & r_{s,1,IJ}<r<r_{c,1,IJ} \\
B_{IJ} \cdot exp\left(-\lambda_{2,IJ}\cdot r\right)\cdot f_{c,IJ}\left(r\right)+A_{IJ}\cdot exp\left(-\lambda_{1,IJ}\cdot r\right)\cdot & \\ ~~~~~~ f_{c,IJ}\left(r\right) & r \ge r_{c,1,IJ}
\end{array}\right. \\
F_{IJ}\left(X\right)=\left[1+\left(\beta_{IJ}\cdot X\right)^{n_{IJ}}\right]^{-\frac{1}{2n_{IJ}}} \\
P_{IJ}\left(\Delta r\right)=exp\left(\lambda_{3,IK}\cdot \Delta r^3\right) \\
W_{IJ}\left(r\right)=f_{c,IK}\left(r\right) \\
G_{JIK}\left(\theta\right)=1+\frac{c_{IK}^2}{d_{IK}^2}-\frac{c_{IK}^2}{d_{IK}^2+\left(h_{IK}+cos\theta\right)^2}
\eta_{ij} = \delta_{ij} (\eta = 2~or~\eta = 0),\xi_{IJ}=0 \\
U_{IJ}\left(r\right) = \varphi_{R,IJ}\left(r\right)-\varphi_{A,IJ}\left(r\right) \\
V_{IJ}\left(r\right) = u_{IJ}\left(r\right) \\
F_{IJ}\left(X\right) = -X \\
P_{JIK}\left(\Delta r\right) = P_{IK}\left(\Delta r\right) = 1 \\
W_{IJ}\left(r\right) = u_{IJ}\left(r\right) \\
G_{JIK}\left(\theta\right) = g_{JIK}\left(cos\theta\right)
\end{array}\right.
\end{eqnarray}
The potential reduces to a Rockett-Tersoff potential (:ref:`Wang &lt;Wang3&gt;`) if we set
.. math::
f_{c,IJ}=\left\{\begin{array}{lr}
1, & r\leq r_{s,IJ} \\
\frac{1}{2}+\frac{1}{2} cos \left[\frac{\pi \left(r-r_{s,IJ}\right)}{r_{c,IJ}-r_{s,IJ}}\right], & r_{s,IJ}<r<r_{c,IJ} \\
0, & r \geq r_{c,IJ} \\
\begin{eqnarray}\nonumber
\left\{ \begin{array}{l}
\eta_{ij} = \delta_{ij} (\eta = 2~or~\eta = 0),\xi_{IJ}=1 \\
U_{IJ}\left(r\right) = A_{IJ}exp\left(-\lambda_{1,IJ}\cdot r\right)f_{c,IJ}\left(r\right)f_{ca,IJ}\left(r\right) \\
V_{IJ}\left(r\right) = \left\{\begin{array}{l}B_{IJ}exp\left(-\lambda_{2,IJ}\cdot r\right)f_{c,IJ}\left(r\right)+ \\ A_{IJ}exp\left(-\lambda_{1,IJ}\cdot r\right)f_{c,IJ}\left(r\right) \left[1-f_{ca,IJ}\left(r\right)\right]\end{array} \right\} \\
F_{IJ}\left(X\right) = \left[1+\left(\beta_{IJ}X\right)^{n_{IJ}}\right]^{-\frac{1}{2n_{IJ}}} \\
P_{JIK}\left(\Delta r\right) = P_{IK}\left(\Delta r\right) = exp\left(\lambda_{3,IK}\cdot \Delta r^3\right) \\
W_{IJ}\left(r\right) = f_{c,IJ}\left(r\right) \\
G_{JIK}\left(\theta\right) = 1+\frac{c_{IK}^2}{d_{IK}^2}-\frac{c_{IK}^2}{d_{IK}^2+\left(h_{IK}+cos\theta\right)^2}
\end{array}\right.
\end{eqnarray}
where :math:`f_{ca,IJ}(r)` is similar to the :math:`f_{c,IJ}(r)` defined above:
.. math::
f_{c,1,IJ}=\left\{\begin{array}{lr}
1, & r\leq r_{s,1,IJ} \\
\frac{1}{2}+\frac{1}{2} cos \left[\frac{\pi \left(r-r_{s,1,IJ}\right)}{r_{c,1,IJ}-r_{s,1,IJ}}\right], & r_{s,1,IJ}<r<r_{c,1,IJ} \\
0, & r \geq r_{c,1,IJ} \\
\begin{eqnarray}\nonumber
f_{ca,IJ}\left(r\right)=\left\{\begin{array}{l}
1, r\leq R_{a,IJ}-D_{a,IJ} \\
\frac{1}{2}+\frac{1}{2}cos\left[\frac{\pi\left(r+D_{a,IJ}-R_{a,IJ}\right)}{2D_{a,IJ}}\right], R_{a,IJ}-D_{a,IJ} < r < R_{a,IJ}+D_{a,IJ} \\
0, r \geq R_{a,IJ}+D_{a,IJ}
\end{array}\right.
\end{eqnarray}
The potential becomes embedded atom method (:ref:`Daw <poly-Daw>`) if we set
The potential becomes embedded atom method (:ref:`Daw &lt;poly-Daw&gt;`) if we set
.. math::
\begin{eqnarray}\nonumber
\left\{\begin{array}{l}
\eta_{ij}=1-\delta_{ij},\xi_{IJ}=0 \\
U_{IJ}\left(r\right)=\phi_{IJ}\left(r\right) \\
V_{IJ}\left(r\right)=1 \\
F_{II}\left(X\right)=-2F_I\left(X\right) \\
P_{IJ}\left(\Delta r\right)=1 \\
W_{IJ}\left(r\right)=f_{K}\left(r\right) \\
G_{JIK}\left(\theta\right)=1
\eta_{ij} = 1-\delta_{ij} (\eta = 1),\xi_{IJ}=0 \\
U_{IJ}\left(r\right) = \phi_{IJ}\left(r\right) \\
V_{IJ}\left(r\right) = 1 \\
F_{II}\left(X\right) = -2F_I\left(X\right) \\
P_{JIK}\left(\Delta r\right) = P_{IK}\left(\Delta r\right) = 1 \\
W_{IJ}\left(r\right) = f_{J}\left(r\right) \\
G_{JIK}\left(\theta\right) = 1
\end{array}\right.
\end{eqnarray}
In the embedded atom method case, :math:`\phi_{IJ}(r_{ij})` is the pair
In the embedded atom method case, :math:`\phi_{IJ}(r)` is the pair
energy, :math:`F_I(X)` is the embedding energy, *X* is the local
electron density, and :math:`f_K(r)` is the atomic electron density function.
electron density, and :math:`f_J(r)` is the atomic electron density function.
The potential reduces to another type of Tersoff potential
(:ref:`Zhou &lt;Zhou4&gt;`) if we set
.. math::
\begin{eqnarray}\nonumber
\left\{\begin{array}{l}
\eta_{ij} = \delta_{ij} (\eta = 3),\xi_{IJ}=1 \\
U_{IJ}\left(r\right) = \frac{D_{e,IJ}}{S_{IJ}-1}\cdot exp\left[-\beta_{IJ}\sqrt{2S_{IJ}}\left(r-r_{e,IJ}\right)\right]\cdot f_{c,IJ}\left(r\right) \cdot T_{IJ}\left(r\right)+V_{ZBL,IJ}\left(r\right)\left[1-T_{IJ}\left(r\right)\right] \\
V_{IJ}\left(r\right) = \frac{S_{IJ}\cdot D_{e,IJ}}{S_{IJ}-1}\cdot exp\left[-\beta_{IJ}\sqrt{\frac{2}{S_{IJ}}}\left(r-r_{e,IJ}\right)\right]\cdot f_{c,IJ}\left(r\right) \cdot T_{IJ}\left(r\right) \\
F_{IJ}\left(X\right) = \left(1+X\right)^{-\frac{1}{2}} \\
P_{JIK}\left(\Delta r\right) = \omega_{JIK} \cdot exp\left(\alpha_{JIK}\cdot \Delta r\right) \\
W_{IJ}\left(r\right) = f_{c,IJ}\left(r\right) \\
G_{JIK}\left(\theta\right) = \gamma_{JIK}\left[1+\frac{c_{JIK}^2}{d_{JIK}^2}-\frac{c_{JIK}^2}{d_{JIK}^2+\left(h_{JIK}+cos\theta\right)^2}\right] \\
T_{IJ}\left(r\right) = \frac{1}{1+exp\left[-b_{f,IJ}\left(r-r_{f,IJ}\right)\right]} \\
V_{ZBL,IJ}\left(r\right) = 14.4 \cdot \frac{Z_I \cdot Z_J}{r}\sum_{k=1}^{4}\mu_k \cdot exp\left[-\nu_k \left(Z_I^{0.23}+Z_J^{0.23}\right) r\right]
\end{array}\right.
\end{eqnarray}
where :math:`f_{c,IJ}(r)` is the as defined above. This Tersoff potential
differs from the one above because the :math:`\P_{JIK}(\Delta r)` function
is now dependent on all three species I, J, and K.
If the tabulated functions are created using the parameters of sw,
tersoff, and eam potentials, the polymorphic pair style will produce
@ -157,37 +209,36 @@ corresponding tersoff and eam pair styles. However, due to a different
partition of global properties to atom properties, the polymorphic
pair style will produce different atom properties (energies and
stresses) as the sw pair style. This does not mean that polymorphic
pair style is different from the sw pair style in this case. It just
means that the definitions of the atom energies and atom stresses are
different.
pair style is different from the sw pair style. It just means that the
definitions of the atom energies and atom stresses are different.
Only a single pair_coeff command is used with the polymorphic style
which specifies an potential file for all needed elements. These are
mapped to LAMMPS atom types by specifying N additional arguments after
the filename in the pair_coeff command, where N is the number of
LAMMPS atom types:
Only a single pair_coeff command is used with the polymorphic pair
style which specifies an potential file for all needed elements.
These are mapped to LAMMPS atom types by specifying N additional
arguments after the filename in the pair_coeff command, where N
is the number of LAMMPS atom types:
* filename
* N element names = mapping of Tersoff elements to atom types
See the pair_coeff doc page for alternate ways to specify the path for
the potential file. Several files for polymorphic potentials are
included in the potentials directory of the LAMMPS distribution. They
the potential file. Several files for polymorphic potentials are
included in the potentials directory of the LAMMPS distribution. They
have a "poly" suffix.
As an example, imagine the SiC_tersoff.poly file has tabulated
functions for Si-C tersoff potential. If your LAMMPS simulation has 4
atoms types and you want the 1st 3 to be Si, and the 4th to be C, you
As an example, imagine the GaN_tersoff.poly file has tabulated
functions for Ga-N tersoff potential. If your LAMMPS simulation has 4
atoms types and you want the 1st 3 to be Ga, and the 4th to be N, you
would use the following pair_coeff command:
.. code-block:: LAMMPS
pair_coeff * * SiC_tersoff.poly Si Si Si C
pair_coeff * * GaN_tersoff.poly Ga Ga Ga N
The 1st 2 arguments must be \* \* so as to span all LAMMPS atom
types. The first three Si arguments map LAMMPS atom types 1,2,3 to the
Si element in the polymorphic file. The final C argument maps LAMMPS
atom type 4 to the C element in the polymorphic file. If a mapping
types. The first three Ga arguments map LAMMPS atom types 1,2,3 to the
Ga element in the polymorphic file. The final N argument maps LAMMPS
atom type 4 to the N element in the polymorphic file. If a mapping
value is specified as NULL, the mapping is not performed. This can be
used when an polymorphic potential is used as part of the hybrid pair
style. The NULL values are placeholders for atom types that will be
@ -203,67 +254,79 @@ and are ignored by LAMMPS. The next line lists two numbers:
ntypes :math:`\eta`
Here ntypes represent total number of species defined in the potential
file, and :math:`\eta = 0` or 1. The number ntypes must equal the total
number of different species defined in the pair_coeff command. When
:math:`\eta = 1`, :math:\eta_{ij}` defined in the potential functions
above is set to :math:`1 - \delta_{ij}`, otherwise :math:`\eta_{ij}` is
set to :math:`\delta_{ij}`. The next ntypes lines each lists two numbers
and a character string representing atomic number, atomic mass, and name
of the species of the ntypes elements:
file, :math:`\eta = 1` reduces to embedded atom method, :math:`\eta = 3`
assumes three spcies dependent :math:`P_{JIK}(\Delta r)` function, and
all other :math:`\eta` assumes two species dependent
:math:`P_{JK}(\Delta r)` function. The number ntypes must equal the total
number of different species defined in the pair_coeff command. The next
ntypes lines each lists two numbers and a character string representing
atomic number, atomic mass, and name of the species of the ntypes elements:
.. parsed-literal::
atomic_number atomic-mass element (1)
atomic_number atomic-mass element (2)
atomic-number atomic-mass element-name(1)
atomic-number atomic-mass element-name(2)
...
atomic_number atomic-mass element (ntypes)
atomic-number atomic-mass element-name(ntypes)
The next line contains four numbers:
.. parsed-literal::
nr ntheta nx xmax
Here nr is total number of tabular points for radial functions U, V, W, P,
ntheta is total number of tabular points for the angular function G, nx is
total number of tabular points for the function F, xmax is a maximum
value of the argument of function F. Note that the pair functions
:math:`U_{IJ}(r)`, :math:`V_{IJ}(r)`, :math:`W_{IJ}(r)` are uniformly
tabulated between 0 and cutoff distance of the IJ pair,
:math:`G_{JIK}(\theta)` is uniformly tabulated between -1 and 1,
:math:`P_{JIK}(\Delta r)` is uniformly tabulated between -rcmax
and rcmax where rcmax is the maximum cutoff distance of all pairs, and
:math:`F_{IJ}(X)` is uniformly tabulated between 0 and xmax. Linear
extrapolation is assumed if actual simulations exceed these ranges.
The next ntypes\*(ntypes+1)/2 lines contain two numbers:
.. parsed-literal::
cut :math:`xi` (1)
cut :math:`xi` (2)
cut :math:`xi`(1)
cut :math:`xi`(2)
...
cut :math:`xi` (ntypes\*(ntypes+1)/2)
cut :math:`xi`(ntypes\*(ntypes+1)/2)
Here cut means the cutoff distance of the pair functions, :math:`\xi` is
the same as defined in the potential functions above. The
ntypes\*(ntypes+1)/2 lines are related to the pairs according to the
sequence of first ii (self) pairs, i = 1, 2, ..., ntypes, and then then
sequence of first ii (self) pairs, i = 1, 2, ..., ntypes, and then
ij (cross) pairs, i = 1, 2, ..., ntypes-1, and j = i+1, i+2, ..., ntypes
(i.e., the sequence of the ij pairs follows 11, 22, ..., 12, 13, 14,
..., 23, 24, ...).
The final blocks of the potential file are the U, V, W, P, G, and F
In the final blocks of the potential file, U, V, W, P, G, and F
functions are listed sequentially. First, U functions are given for
each of the ntypes\*(ntypes+1)/2 pairs according to the sequence
described above. For each of the pairs, nr values are listed. Next,
similar arrays are given for V, W, and P functions. Then G functions
are given for all the ntypes\*ntypes\*ntypes ijk triplets in a natural
sequence i from 1 to ntypes, j from 1 to ntypes, and k from 1 to
ntypes (i.e., ijk = 111, 112, 113, ..., 121, 122, 123 ..., 211, 212,
...). Each of the ijk functions contains ng values. Finally, the F
functions are listed for all ntypes\*(ntypes+1)/2 pairs, each
containing nx values. Either analytic or tabulated functions can be
specified. Currently, constant, exponential, sine and cosine analytic
functions are available which are specified with: constant c1 , where
f(x) = c1 exponential c1 c2 , where f(x) = c1 exp(c2\*x) sine c1 c2 ,
where f(x) = c1 sin(c2\*x) cos c1 c2 , where f(x) = c1 cos(c2\*x)
Tabulated functions are specified by spline n x1 x2, where n=number of
point, (x1,x2)=range and then followed by n values evaluated uniformly
over these argument ranges. The valid argument ranges of the
functions are between 0 <= r <= cut for the U(r), V(r), W(r)
functions, -cutmax <= delta_r <= cutmax for the P(delta_r) functions,
-1 <= :math:`\cos\theta` <= 1 for the G(:math:`\cos\theta`) functions,
and 0 <= X <= maxX for the F(X) functions.
similar arrays are given for V and W functions. If P functions
depend only on pair species, i.e., :math:`\eta \neq 3`, then P
functions are also listed the same way the next. If P functions
depend on three species, i.e., :math:`\eta = 3`, then P functions
are listed for all the ntypes*ntypes*ntypes IJK triplets in a
natural sequence I from 1 to ntypes, J from 1 to ntypes, and K from
1 to ntypes (i.e., IJK = 111, 112, 113, ..., 121, 122, 123 ..., 211,
212, ...). Next, G functions are listed for all the ntypes*ntypes*ntypes
IJK triplets similarly. For each of the G functions, ntheta values
are listed. Finally, F functions are listed for all the
ntypes*(ntypes+1)/2 pairs in the same sequence as described above.
For each of the F functions, nx values are listed.
**Mixing, shift, table tail correction, restart**\ :
This pair styles does not support the :doc:`pair_modify <pair_modify>`
This pair styles does not support the :doc:`pair_modify &lt;pair_modify&gt;`
shift, table, and tail options.
This pair style does not write their information to :doc:`binary restart files <restart>`, since it is stored in potential files. Thus, you
This pair style does not write their information to :doc:`binary restart files &lt;restart&gt;`, since it is stored in potential files. Thus, you
need to re-specify the pair_style and pair_coeff commands in an input
script that reads a restart file.
@ -277,31 +340,34 @@ input script. If using read_data, atomic masses must be defined in the
atomic structure data file.
This pair style is part of the MANYBODY package. It is only enabled if
LAMMPS was built with that package. See the :doc:`Build package <Build_package>` doc page for more info.
LAMMPS was built with that package. See the :doc:`Build package &lt;Build_package&gt;` doc page for more info.
This pair potential requires the :doc:`newtion <newton>` setting to be
This pair potential requires the :doc:`newtion &lt;newton&gt;` setting to be
"on" for pair interactions.
The potential files provided with LAMMPS (see the potentials
directory) are parameterized for metal :doc:`units <units>`. You can use
directory) are parameterized for metal :doc:`units &lt;units&gt;`. You can use
any LAMMPS units, but you would need to create your own potential
files.
Related commands
""""""""""""""""
:doc:`pair_coeff <pair_coeff>`
:doc:`pair_coeff &lt;pair_coeff&gt;`
----------
.. _Zhou3:
**(Zhou)** X. W. Zhou, M. E. Foster, R. E. Jones, P. Yang, H. Fan, and
F. P. Doty, J. Mater. Sci. Res., 4, 15 (2015).
**(Zhou)** X. W. Zhou, M. E. Foster, R. E. Jones, P. Yang, H. Fan, and F. P. Doty, J. Mater. Sci. Res., 4, 15 (2015).
.. _Zhou4:
**(Zhou)** X. W. Zhou, M. E. Foster, J. A. Ronevich, and C. W. San Marchi, J. Comp. Chem., 41, 1299 (2020).
.. _SW:
**(SW)** F. H. Stillinger-Weber, and T. A. Weber, Phys. Rev. B, 31, 5262 (1985).
**(SW)** F. H. Stillinger, and T. A. Weber, Phys. Rev. B, 31, 5262 (1985).
.. _Tersoff:
@ -309,8 +375,7 @@ F. P. Doty, J. Mater. Sci. Res., 4, 15 (2015).
.. _poly-Albe:
**(Albe)** K. Albe, K. Nordlund, J. Nord, and A. Kuronen, Phys. Rev. B,
66, 035205 (2002).
**(Albe)** K. Albe, K. Nordlund, J. Nord, and A. Kuronen, Phys. Rev. B, 66, 035205 (2002).
.. _Wang3:
@ -319,3 +384,4 @@ F. P. Doty, J. Mater. Sci. Res., 4, 15 (2015).
.. _poly-Daw:
**(Daw)** M. S. Daw, and M. I. Baskes, Phys. Rev. B, 29, 6443 (1984).
</PRE>

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127
src/MANYBODY/pair_polymorphic.cpp
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@ -46,6 +46,7 @@ PairPolymorphic::PairPolymorphic(LAMMPS *lmp) : Pair(lmp)
nelements = 0;
elements = NULL;
match = NULL;
pairParameters = NULL;
tripletParameters = NULL;
elem2param = NULL;
@ -168,7 +169,7 @@ void PairPolymorphic::compute(int eflag, int vflag)
firstneighW1 = new int[neighsize];
}
if (eta) {
if (eta == 1) {
iparam_ii = elem2param[itype][itype];
PairParameters & p = pairParameters[iparam_ii];
emb = (p.F)->get_vmax();
@ -193,7 +194,7 @@ void PairPolymorphic::compute(int eflag, int vflag)
PairParameters & p = pairParameters[iparam_ij];
// do not include the neighbor if get_vmax() <= epsilon because the function is near zero
if (eta) {
if (eta == 1) {
if (emb > epsilon) {
iparam_jj = elem2param[jtype][jtype];
PairParameters & q = pairParameters[iparam_jj];
@ -255,7 +256,7 @@ void PairPolymorphic::compute(int eflag, int vflag)
evdwl,0.0,fpair,delx,dely,delz);
}
if (eta) {
if (eta == 1) {
if (emb > epsilon) {
@ -356,7 +357,7 @@ void PairPolymorphic::compute(int eflag, int vflag)
PairParameters & q = pairParameters[iparam_ik];
(q.W)->value(r2,wfac,1,fpair,0);
(q.P)->value(r1-(p.xi)*r2,pfac,1,fpair,0);
(trip.P)->value(r1-(p.xi)*r2,pfac,1,fpair,0);
(trip.G)->value(costheta,gfac,1,fpair,0);
zeta_ij += wfac*pfac*gfac;
@ -397,7 +398,7 @@ void PairPolymorphic::compute(int eflag, int vflag)
iparam_ik = elem2param[itype][ktype];
PairParameters & q = pairParameters[iparam_ik];
attractive(&q,&trip,prefactor,r1,r2,delr1,delr2,fi,fj,fk);
attractive(&p,&q,&trip,prefactor,r1,r2,delr1,delr2,fi,fj,fk);
f[i][0] += fi[0];
f[i][1] += fi[1];
@ -586,7 +587,7 @@ void PairPolymorphic::read_file(char *file)
error->all(FLERR,"Incorrect number of elements in potential file");
match = new int[nelements];
ptr = strtok(NULL," \t\n\r\f"); // 1st line, 2nd token
eta = (atoi(ptr)>0) ? true:false;
eta = atoi(ptr);
// map the elements in the potential file to LAMMPS atom types
for (int i = 0; i < nelements; i++) {
@ -654,17 +655,9 @@ void PairPolymorphic::read_file(char *file)
p.cut = atof(ptr);
p.cutsq = p.cut*p.cut;
ptr = strtok(NULL," \t\n\r\f"); // 2nd token
p.xi = (atoi(ptr)>0) ? true:false;
p.xi = atof(ptr);
}
// set cutmax to max of all params
cutmax = 0.0;
for (int i = 0; i < npair; i++) {
PairParameters & p = pairParameters[i];
if (p.cut > cutmax) cutmax = p.cut;
}
cutmaxsq = cutmax*cutmax;
// start reading tabular functions
double * singletable = new double[nr];
for (int i = 0; i < npair; i++) { // U
@ -694,14 +687,53 @@ void PairPolymorphic::read_file(char *file)
p.W = new tabularFunction(nr,0.0,p.cut);
(p.W)->set_values(nr,0.0,p.cut,singletable,epsilon);
}
for (int i = 0; i < npair; i++) { // P
cutmax = 0.0;
for (int i = 0; i < npair; i++) {
PairParameters & p = pairParameters[i];
if (comm->me == 0) {
grab(fp,nr,singletable);
if (p.cut > cutmax) cutmax = p.cut;
}
cutmaxsq = cutmax*cutmax;
if (eta != 3) {
for (int j = 0; j < nelements; j++) { // P
if (comm->me == 0) {
grab(fp,nr,singletable);
}
MPI_Bcast(singletable,nr,MPI_DOUBLE,0,world);
for (int i = 0; i < nelements; i++) {
TripletParameters & p = tripletParameters[i*nelements*nelements+j*nelements+j];
p.P = new tabularFunction(nr,-cutmax,cutmax);
(p.P)->set_values(nr,-cutmax,cutmax,singletable,epsilon);
}
}
for (int j = 0; j < nelements-1; j++) { // P
for (int k = j+1; k < nelements; k++) {
if (comm->me == 0) {
grab(fp,nr,singletable);
}
MPI_Bcast(singletable,nr,MPI_DOUBLE,0,world);
for (int i = 0; i < nelements; i++) {
TripletParameters & p = tripletParameters[i*nelements*nelements+j*nelements+k];
p.P = new tabularFunction(nr,-cutmax,cutmax);
(p.P)->set_values(nr,-cutmax,cutmax,singletable,epsilon);
TripletParameters & q = tripletParameters[i*nelements*nelements+k*nelements+j];
q.P = new tabularFunction(nr,-cutmax,cutmax);
(q.P)->set_values(nr,-cutmax,cutmax,singletable,epsilon);
}
}
}
}
if (eta == 3) {
for (int i = 0; i < ntriple; i++) { // P
TripletParameters & p = tripletParameters[i];
if (comm->me == 0) {
grab(fp,nr,singletable);
}
MPI_Bcast(singletable,nr,MPI_DOUBLE,0,world);
p.P = new tabularFunction(nr,-cutmax,cutmax);
(p.P)->set_values(nr,-cutmax,cutmax,singletable,epsilon);
}
MPI_Bcast(singletable,nr,MPI_DOUBLE,0,world);
p.P = new tabularFunction(nr,-cutmax,cutmax);
(p.P)->set_values(nr,-cutmax,cutmax,singletable,epsilon);
}
delete[] singletable;
singletable = new double[ng];
@ -730,6 +762,22 @@ void PairPolymorphic::read_file(char *file)
fclose(fp);
}
// recalculate cutoffs of all params
for (int i = 0; i < npair; i++) {
PairParameters & p = pairParameters[i];
p.cut = (p.U)->get_xmax();
if (p.cut < (p.V)->get_xmax()) p.cut = (p.V)->get_xmax();
if (p.cut < (p.W)->get_xmax()) p.cut = (p.W)->get_xmax();
p.cutsq = p.cut*p.cut;
}
// set cutmax to max of all params
cutmax = 0.0;
for (int i = 0; i < npair; i++) {
PairParameters & p = pairParameters[i];
if (cutmax < p.cut) cutmax = p.cut;
}
cutmaxsq = cutmax*cutmax;
}
/* ---------------------------------------------------------------------- */
@ -771,15 +819,16 @@ void PairPolymorphic::setup_params()
// for debugging, call write_tables() to check the tabular functions
// if (comm->me == 0) {
// write_tables(51);
// errorX->all(FLERR,"Test potential tables");
// }
// error->all(FLERR,"Test potential tables");
}
/* ----------------------------------------------------------------------
attractive term
------------------------------------------------------------------------- */
void PairPolymorphic::attractive(PairParameters *p, TripletParameters *trip,
void PairPolymorphic::attractive(PairParameters *p, PairParameters *q,
TripletParameters *trip,
double prefactor, double rij, double rik,
double *delrij, double *delrik,
double *fi, double *fj, double *fk)
@ -793,7 +842,7 @@ void PairPolymorphic::attractive(PairParameters *p, TripletParameters *trip,
rikinv = 1.0/rik;
vec3_scale(rikinv,delrik,rik_hat);
ters_zetaterm_d(prefactor,rij_hat,rij,rik_hat,rik,fi,fj,fk,p,trip);
ters_zetaterm_d(prefactor,rij_hat,rij,rik_hat,rik,fi,fj,fk,p,q,trip);
}
/* ---------------------------------------------------------------------- */
@ -802,46 +851,39 @@ void PairPolymorphic::ters_zetaterm_d(double prefactor,
double *rij_hat, double rij,
double *rik_hat, double rik,
double *dri, double *drj, double *drk,
PairParameters *p, TripletParameters *trip)
PairParameters *p, PairParameters *q,
TripletParameters *trip)
{
double gijk,gijk_d,ex_delr,ex_delr_d,fc,dfc,cos_theta;
double dcosdri[3],dcosdrj[3],dcosdrk[3];
cos_theta = vec3_dot(rij_hat,rik_hat);
(p->W)->value(rik,fc,1,dfc,1);
(p->P)->value(rij-(p->xi)*rik,ex_delr,1,ex_delr_d,1);
(q->W)->value(rik,fc,1,dfc,1);
(trip->P)->value(rij-(p->xi)*rik,ex_delr,1,ex_delr_d,1);
(trip->G)->value(cos_theta,gijk,1,gijk_d,1);
costheta_d(rij_hat,rij,rik_hat,rik,dcosdri,dcosdrj,dcosdrk);
// compute the derivative wrt Ri
// dri = -dfc*gijk*ex_delr*rik_hat;
// dri += fc*gijk_d*ex_delr*dcosdri;
// dri += fc*gijk*ex_delr_d*(rik_hat - rij_hat);
vec3_scale(-dfc*gijk*ex_delr,rik_hat,dri);
vec3_scaleadd(fc*gijk_d*ex_delr,dcosdri,dri,dri);
vec3_scaleadd(fc*gijk*ex_delr_d,rik_hat,dri,dri);
vec3_scaleadd(fc*gijk*ex_delr_d*(p->xi),rik_hat,dri,dri);
vec3_scaleadd(-fc*gijk*ex_delr_d,rij_hat,dri,dri);
vec3_scale(prefactor,dri,dri);
// compute the derivative wrt Rj
// drj = fc*gijk_d*ex_delr*dcosdrj;
// drj += fc*gijk*ex_delr_d*rij_hat;
vec3_scale(fc*gijk_d*ex_delr,dcosdrj,drj);
vec3_scaleadd(fc*gijk*ex_delr_d,rij_hat,drj,drj);
vec3_scale(prefactor,drj,drj);
// compute the derivative wrt Rk
// drk = dfc*gijk*ex_delr*rik_hat;
// drk += fc*gijk_d*ex_delr*dcosdrk;
// drk += -fc*gijk*ex_delr_d*rik_hat;
vec3_scale(dfc*gijk*ex_delr,rik_hat,drk);
vec3_scaleadd(fc*gijk_d*ex_delr,dcosdrk,drk,drk);
vec3_scaleadd(-fc*gijk*ex_delr_d,rik_hat,drk,drk);
vec3_scaleadd(-fc*gijk*ex_delr_d*(p->xi),rik_hat,drk,drk);
vec3_scale(prefactor,drk,drk);
}
@ -919,26 +961,29 @@ void PairPolymorphic::write_tables(int npts)
}
for (int i = 0; i < nelements; i++) {
for (int j = 0; j < nelements; j++) {
for (int k = 0; k < nelements; k++) {
strcpy(line,elements[i]);
strcat(line,elements[j]);
strcat(line,elements[k]);
strcat(line,"_P");
strcat(line,tag);
fp = fopen(line, "w");
int iparam_ij = elem2param[i][j];
PairParameters & pair = pairParameters[iparam_ij];
int iparam_ij = elem3param[i][j][k];
TripletParameters & pair = tripletParameters[iparam_ij];
xmin = (pair.P)->get_xmin();
xmax = (pair.P)->get_xmax();
double xl = xmax - xmin;
xmin = xmin - 0.5*xl;
xmax = xmax + 0.5*xl;
for (int k = 0; k < npts; k++) {
x = xmin + (xmax-xmin) * k / (npts-1);
for (int n = 0; n < npts; n++) {
x = xmin + (xmax-xmin) * n / (npts-1);
(pair.P)->value(x, pf, 1, pfp, 1);
fprintf(fp,"%12.4f %12.4f %12.4f \n",x,pf,pfp);
}
fclose(fp);
}
}
}
for (int i = 0; i < nelements; i++) {
for (int j = 0; j < nelements; j++) {
for (int k = 0; k < nelements; k++) {

87
src/MANYBODY/pair_polymorphic.h
View File

@ -104,32 +104,59 @@ class PairPolymorphic : public Pair {
}
void set_values(int n, double x1, double x2, double * values, double epsilon)
{
int i0;
i0 = n-1;
// shrink (remove near zero points) reduces cutoff radius, and therefore computational cost
// do not shrink when x2 < 1.1 (angular function) or x2 > 20.0 (non-radial function)
if (x2 >= 1.1 && x2 <= 20.0) {
for (int i = n-1; i >= 0; i--) {
if (fabs(values[i]) > epsilon) {
i0 = i;
break;
}
int shrink = 1;
int ilo,ihi;
double vlo,vhi;
ilo = 0;
ihi = n-1;
for (int i = 0; i < n; i++) {
if (fabs(values[i]) <= epsilon) {
ilo = i;
} else {
break;
}
}
// do not shrink when when list is abnormally small
if (i0 < 10/n) {
i0 = n-1;
} else if (i0 < n-1) {
values[i0] = 0.0;
i0 = i0 + 1;
values[i0] = 0.0;
for (int i = n-1; i >= 0; i--) {
if (fabs(values[i]) <= epsilon) {
ihi = i;
} else {
break;
}
}
xmin = x1;
xmax = x1 + (x2-x1)/(n -1)*i0;
if (ihi < ilo) ihi = ilo;
vlo = values[ilo];
vhi = values[ilo];
for (int i = ilo; i <= ihi; i++) {
if (vlo > values[i]) vlo = values[i];
if (vhi < values[i]) vhi = values[i];
}
// shrink (remove near zero points) reduces cutoff radius, and therefore computational cost
// do not shrink when x2 < 1.1 (angular function) or x2 > 20.0 (non-radial function)
if (x2 < 1.1 || x2 > 20.0) {
shrink = 0;
}
// do not shrink when when list is abnormally small
if (ihi - ilo < 50) {
shrink = 0;
}
// shrink if it is a constant
if (vhi - vlo <= epsilon) {
// shrink = 1;
}
if (shrink == 0) {
ilo = 0;
ihi = n-1;
}
xmin = x1 + (x2-x1)/(n -1)*ilo;
xmax = xmin + (x2-x1)/(n -1)*(ihi-ilo);
xmaxsq = xmax*xmax;
n = i0+1;
n = ihi - ilo + 1;
resize(n);
memcpy(ys,values,n*sizeof(double));
for (int i = ilo; i <= ihi; i++) {
ys[i-ilo] = values[i];
}
initialize();
}
void value(double x, double &y, int ny, double &y1, int ny1)
@ -228,32 +255,32 @@ class PairPolymorphic : public Pair {
struct PairParameters {
double cut;
double cutsq;
bool xi; // "indicator"
double xi;
class tabularFunction * U;
class tabularFunction * V;
class tabularFunction * W;
class tabularFunction * P;
class tabularFunction * F;
PairParameters() {
cut = 0.0;
cutsq = 0.0;
xi = true;
xi = 1.0;
U = NULL;
V = NULL;
W = NULL;
P = NULL;
F = NULL;
};
};
struct TripletParameters {
class tabularFunction * P;
class tabularFunction * G;
TripletParameters() {
P = NULL;
G = NULL;
};
};
double epsilon;
bool eta; // global indicator
int eta;
int nx,nr,ng; // table sizes
double maxX;
@ -283,11 +310,13 @@ class PairPolymorphic : public Pair {
void setup_params();
void write_tables(int);
void attractive(PairParameters *, TripletParameters *, double, double,
double, double *, double *, double *, double *, double *);
void attractive(PairParameters *, PairParameters *, TripletParameters *,
double, double, double, double *, double *, double *,
double *, double *);
void ters_zetaterm_d(double, double *, double, double *, double, double *,
double *, double *, PairParameters *, TripletParameters *);
double *, double *, PairParameters *, PairParameters *,
TripletParameters *);
void costheta_d(double *, double, double *, double,
double *, double *, double *);