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Commit 63203651 authored by peastman's avatar peastman
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Merge pull request #568 from peastman/docs 

Fixes to documentation
parents 05d64aea 1f54f70c
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Showing with 179 additions and 156 deletions
docs-source/CMakeLists.txt
View file @ 63203651
......@@ -84,15 +84,19 @@ ENDIF(DOXYGEN_EXECUTABLE)
SET(SPHINX_BUILD_DIR "${CMAKE_BINARY_DIR}/sphinx-docs/")
FILE(MAKE_DIRECTORY "${SPHINX_BUILD_DIR}")
FILE(GLOB_RECURSE USER_GUIDE_FILES ${CMAKE_CURRENT_SOURCE_DIR}/usersguide/*)
FILE(GLOB_RECURSE DEVELOPER_GUIDE_FILES ${CMAKE_CURRENT_SOURCE_DIR}/developerguide/*)
ADD_CUSTOM_COMMAND(
OUTPUT "${SPHINX_BUILD_DIR}/userguide/latex/OpenMMUsersGuide.pdf"
COMMAND "${CMAKE_MAKE_PROGRAM}" BUILDDIR="${SPHINX_BUILD_DIR}/userguide" OPENMM_VERSION="${OPENMM_MAJOR_VERSION}.${OPENMM_MINOR_VERSION}" latexpdf
DEPENDS ${USER_GUIDE_FILES}
WORKING_DIRECTORY "${CMAKE_CURRENT_SOURCE_DIR}/usersguide"
COMMENT "Generating PDF user guide"
)
ADD_CUSTOM_COMMAND(
OUTPUT "${SPHINX_BUILD_DIR}/developerguide/latex/OpenMMDeveloperGuide.pdf"
COMMAND "${CMAKE_MAKE_PROGRAM}" BUILDDIR="${SPHINX_BUILD_DIR}/developerguide" OPENMM_VERSION="${OPENMM_MAJOR_VERSION}.${OPENMM_MINOR_VERSION}" latexpdf
DEPENDS ${DEVELOPER_GUIDE_FILES}
WORKING_DIRECTORY "${CMAKE_CURRENT_SOURCE_DIR}/developerguide"
COMMENT "Generating PDF developer guide"
)
......@@ -103,14 +107,16 @@ ADD_CUSTOM_TARGET(sphinxpdf
ADD_CUSTOM_COMMAND(
OUTPUT "${SPHINX_BUILD_DIR}/userguide/html/index.html"
COMMAND "${CMAKE_MAKE_PROGRAM}" BUILDDIR="${SPHINX_BUILD_DIR}/userguide" OPENMM_VERSION="${OPENMM_MAJOR_VERSION}.${OPENMM_MINOR_VERSION}" html
DEPENDS ${USER_GUIDE_FILES}
WORKING_DIRECTORY "${CMAKE_CURRENT_SOURCE_DIR}/usersguide"
COMMENT "Generating PDF user guide"
COMMENT "Generating HTML user guide"
)
ADD_CUSTOM_COMMAND(
OUTPUT "${SPHINX_BUILD_DIR}/developerguide/html/index.html"
COMMAND "${CMAKE_MAKE_PROGRAM}" BUILDDIR="${SPHINX_BUILD_DIR}/developerguide" OPENMM_VERSION="${OPENMM_MAJOR_VERSION}.${OPENMM_MINOR_VERSION}" html
DEPENDS ${DEVELOPER_GUIDE_FILES}
WORKING_DIRECTORY "${CMAKE_CURRENT_SOURCE_DIR}/developerguide"
COMMENT "Generating PDF developer guide"
COMMENT "Generating HTML developer guide"
)
ADD_CUSTOM_TARGET(sphinxhtml
DEPENDS "${SPHINX_BUILD_DIR}/userguide/html/index.html" "${SPHINX_BUILD_DIR}/developerguide/html/index.html"
......
docs-source/sphinx/autonumber.py
View file @ 63203651
......@@ -43,6 +43,7 @@ def doctree_resolved(app, doctree, docname):
chapter = sectionNumbers[chapter.attributes['ids'][0]][0]
if chapter != lastChapter:
index = {}
nextNumber = 1
newNode = Text('%s %d-%d' % (category, chapter, nextNumber))
lastChapter = chapter
else:
......
docs-source/usersguide/application.rst
View file @ 63203651
......@@ -58,7 +58,7 @@ and tell it to install the command line tools. With Xcode 4.2 and earlier, the
command line tools are automatically installed when you install Xcode.)
3. (Optional) If you have an Nvidia GPU and want to use the CUDA platform,
download CUDA 5.5 from https://developer.nvidia.com/cuda-downloads. Be sure to
download CUDA 6.0 from https://developer.nvidia.com/cuda-downloads. Be sure to
install both the drivers and toolkit.
4. (Optional) If you plan to use the CPU platform, it is recommended that you
......@@ -100,7 +100,7 @@ location, you can set OPENMM_CUDA_COMPILER to tell OpenMM where to find it. For
example,
::
export OPENMM_CUDA_COMPILER=/opt/CUDA/cuda-5.5/bin/nvcc
export OPENMM_CUDA_COMPILER=/opt/CUDA/cuda-6.0/bin/nvcc
7. Verify your installation by running the “testInstallation.py” script found in
the “examples” folder of your OpenMM installation. To run it, cd to the
......@@ -132,7 +132,7 @@ into a console window.
3. (Optional) If you want to run OpenMM on a GPU, install CUDA and/or OpenCL.
* If you have an Nvidia GPU, download CUDA 5.5 from
* If you have an Nvidia GPU, download CUDA 6.0 from
https://developer.nvidia.com/cuda-downloads. Be sure to install both the
drivers and toolkit. OpenCL is included with the CUDA drivers.
* If you have an AMD GPU, download the latest version of the Catalyst driver
......@@ -179,7 +179,7 @@ location, you can set OPENMM_CUDA_COMPILER to tell OpenMM where to find it. For
example,
::
export OPENMM_CUDA_COMPILER=/opt/CUDA/cuda-5.5/bin/nvcc
export OPENMM_CUDA_COMPILER=/opt/CUDA/cuda-6.0/bin/nvcc
7. Verify your installation by running the “testInstallation.py” script found in
the “examples” folder of your OpenMM installation. To run it, cd to the
......@@ -214,7 +214,7 @@ and ignore it.)
4. (Optional) If you want to run OpenMM on a GPU, install CUDA and/or OpenCL.
* If you have an Nvidia GPU, download CUDA 5.5 from
* If you have an Nvidia GPU, download CUDA 6.0 from
https://developer.nvidia.com/cuda-downloads. Be sure to install both the
drivers and toolkit. For 64-bit machines, you should install the 64-bit driver,
but download the 32-bit version of the toolkit since the OpenMM binary is
......@@ -651,20 +651,20 @@ For the main force field, OpenMM provides the following options:
.. tabularcolumns:: |l|L|
===================== ================================================================================
============================= ================================================================================
File Force Field
===================== ================================================================================
amber96.xml AMBER96\ :cite:`Kollman1997`
amber99sb.xml AMBER99\ :cite:`Wang2000` with modified backbone torsions\ :cite:`Hornak2006`
amber99sbildn.xml AMBER99SB plus improved side chain torsions\ :cite:`Lindorff-Larsen2010`
amber99sbnmr.xml AMBER99SB with modifications to fit NMR data\ :cite:`Li2010`
amber03.xml AMBER03\ :cite:`Duan2003`
amber10.xml AMBER10
amoeba2009.xml AMOEBA 2009\ :cite:`Ren2002`. This force field is deprecated. It is
============================= ================================================================================
:code:`amber96.xml` AMBER96\ :cite:`Kollman1997`
:code:`amber99sb.xml` AMBER99\ :cite:`Wang2000` with modified backbone torsions\ :cite:`Hornak2006`
:code:`amber99sbildn.xml` AMBER99SB plus improved side chain torsions\ :cite:`Lindorff-Larsen2010`
:code:`amber99sbnmr.xml` AMBER99SB with modifications to fit NMR data\ :cite:`Li2010`
:code:`amber03.xml` AMBER03\ :cite:`Duan2003`
:code:`amber10.xml` AMBER10
:code:`amoeba2009.xml` AMOEBA 2009\ :cite:`Ren2002`. This force field is deprecated. It is
recommended to use AMOEBA 2013 instead.
amoeba2013.xml AMOEBA 2013\ :cite:`Shi2013`
charmm_polar_2013.xml CHARMM 2013 polarizable force field\ :cite:`Lopes2013`
===================== ================================================================================
:code:`amoeba2013.xml` AMOEBA 2013\ :cite:`Shi2013`
:code:`charmm_polar_2013.xml` CHARMM 2013 polarizable force field\ :cite:`Lopes2013`
============================= ================================================================================
The AMBER files do not include parameters for water molecules. This allows you
......@@ -674,17 +674,17 @@ files:
.. tabularcolumns:: |l|L|
=========== ============================================
=================== ============================================
File Water Model
=========== ============================================
tip3p.xml TIP3P water model\ :cite:`Jorgensen1983`
tip3pfb.xml TIP3P-FB water model\ :cite:`Wang2014`
tip4pew.xml TIP4P-Ew water model\ :cite:`Horn2004`
tip4pfb.xml TIP4P-FB water model\ :cite:`Wang2014`
tip5p.xml TIP5P water model\ :cite:`Mahoney2000`
spce.xml SPC/E water model\ :cite:`Berendsen1987`
swm4ndp.xml SWM4-NDP water model\ :cite:`Lamoureux2006`
=========== ============================================
=================== ============================================
:code:`tip3p.xml` TIP3P water model\ :cite:`Jorgensen1983`
:code:`tip3pfb.xml` TIP3P-FB water model\ :cite:`Wang2014`
:code:`tip4pew.xml` TIP4P-Ew water model\ :cite:`Horn2004`
:code:`tip4pfb.xml` TIP4P-FB water model\ :cite:`Wang2014`
:code:`tip5p.xml` TIP5P water model\ :cite:`Mahoney2000`
:code:`spce.xml` SPC/E water model\ :cite:`Berendsen1987`
:code:`swm4ndp.xml` SWM4-NDP water model\ :cite:`Lamoureux2006`
=================== ============================================
For the polarizable force fields (AMOEBA and CHARMM), only one explicit water model
......@@ -697,16 +697,16 @@ the following files:
.. tabularcolumns:: |l|L|
================= =================================================================================================
========================= =================================================================================================
File Implicit Solvation Model
================= =================================================================================================
amber96_obc.xml GBSA-OBC solvation model\ :cite:`Onufriev2004` for use with AMBER96 force field
amber99_obc.xml GBSA-OBC solvation model for use with AMBER99 force fields
amber03_obc.xml GBSA-OBC solvation model for use with AMBER03 force field
amber10_obc.xml GBSA-OBC solvation model for use with AMBER10 force field
amoeba2009_gk.xml Generalized Kirkwood solvation model\ :cite:`Schnieders2007` for use with AMOEBA 2009 force field
amoeba2013_gk.xml Generalized Kirkwood solvation model for use with AMOEBA 2013 force field
================= =================================================================================================
========================= =================================================================================================
:code:`amber96_obc.xml` GBSA-OBC solvation model\ :cite:`Onufriev2004` for use with AMBER96 force field
:code:`amber99_obc.xml` GBSA-OBC solvation model for use with AMBER99 force fields
:code:`amber03_obc.xml` GBSA-OBC solvation model for use with AMBER03 force field
:code:`amber10_obc.xml` GBSA-OBC solvation model for use with AMBER10 force field
:code:`amoeba2009_gk.xml` Generalized Kirkwood solvation model\ :cite:`Schnieders2007` for use with AMOEBA 2009 force field
:code:`amoeba2013_gk.xml` Generalized Kirkwood solvation model for use with AMOEBA 2013 force field
========================= =================================================================================================
For example, to use the GBSA-OBC solvation model with the Amber99SB force field,
......@@ -723,6 +723,12 @@ produce an error since no water parameters are defined:
forcefield = ForceField('amber10.xml')
Be aware that some force fields and water models include "extra particles", such
as lone pairs or Drude particles. Examples include the CHARMM polarizable force
field and all of the 4 and 5 site water models. To use these force fields, you
must first add the extra particles to the Topology. See section
:ref:`adding-or-removing-extra-particles` for details.
AMBER Implicit Solvent
======================
......@@ -739,17 +745,17 @@ allowed values for :code:`implicitSolvent`\ :
.. tabularcolumns:: |l|L|
===== ==================================================================================================================================
============= ==================================================================================================================================
Value Meaning
===== ==================================================================================================================================
None No implicit solvent is used.
HCT Hawkins-Cramer-Truhlar GBSA model\ :cite:`Hawkins1995` (corresponds to igb=1 in AMBER)
OBC1 Onufriev-Bashford-Case GBSA model\ :cite:`Onufriev2004` using the GB\ :sup:`OBC`\ I parameters (corresponds to igb=2 in AMBER).
OBC2 Onufriev-Bashford-Case GBSA model\ :cite:`Onufriev2004` using the GB\ :sup:`OBC`\ II parameters (corresponds to igb=5 in AMBER).
============= ==================================================================================================================================
:code:`None` No implicit solvent is used.
:code:`HCT` Hawkins-Cramer-Truhlar GBSA model\ :cite:`Hawkins1995` (corresponds to igb=1 in AMBER)
:code:`OBC1` Onufriev-Bashford-Case GBSA model\ :cite:`Onufriev2004` using the GB\ :sup:`OBC`\ I parameters (corresponds to igb=2 in AMBER).
:code:`OBC2` Onufriev-Bashford-Case GBSA model\ :cite:`Onufriev2004` using the GB\ :sup:`OBC`\ II parameters (corresponds to igb=5 in AMBER).
This is the same model used by the GBSA-OBC files described in section :ref:`force-fields`.
GBn GBn solvation model\ :cite:`Mongan2007` (corresponds to igb=7 in AMBER).
GBn2 GBn2 solvation model\ :cite:`Nguyen2013` (corresponds to igb=8 in AMBER).
===== ==================================================================================================================================
:code:`GBn` GBn solvation model\ :cite:`Mongan2007` (corresponds to igb=7 in AMBER).
:code:`GBn2` GBn2 solvation model\ :cite:`Nguyen2013` (corresponds to igb=8 in AMBER).
============= ==================================================================================================================================
You can further control the solvation model in a few ways. First, you can
......@@ -785,15 +791,15 @@ The :code:`nonbondedMethod` parameter can have any of the following values:
.. tabularcolumns:: |l|L|
================= ===========================================================================================================================================================================================================================================
========================= ===========================================================================================================================================================================================================================================
Value Meaning
================= ===========================================================================================================================================================================================================================================
NoCutoff No cutoff is applied.
CutoffNonPeriodic The reaction field method is used to eliminate all interactions beyond a cutoff distance. Not valid for AMOEBA.
CutoffPeriodic The reaction field method is used to eliminate all interactions beyond a cutoff distance. Periodic boundary conditions are applied, so each atom interacts only with the nearest periodic copy of every other atom. Not valid for AMOEBA.
Ewald Periodic boundary conditions are applied. Ewald summation is used to compute long range interactions. (This option is rarely used, since PME is much faster for all but the smallest systems.) Not valid for AMOEBA.
PME Periodic boundary conditions are applied. The Particle Mesh Ewald method is used to compute long range interactions.
================= ===========================================================================================================================================================================================================================================
========================= ===========================================================================================================================================================================================================================================
:code:`NoCutoff` No cutoff is applied.
:code:`CutoffNonPeriodic` The reaction field method is used to eliminate all interactions beyond a cutoff distance. Not valid for AMOEBA.
:code:`CutoffPeriodic` The reaction field method is used to eliminate all interactions beyond a cutoff distance. Periodic boundary conditions are applied, so each atom interacts only with the nearest periodic copy of every other atom. Not valid for AMOEBA.
:code:`Ewald` Periodic boundary conditions are applied. Ewald summation is used to compute long range interactions. (This option is rarely used, since PME is much faster for all but the smallest systems.) Not valid for AMOEBA.
:code:`PME` Periodic boundary conditions are applied. The Particle Mesh Ewald method is used to compute long range interactions.
========================= ===========================================================================================================================================================================================================================================
When using any method other than :code:`NoCutoff`\ , you should also specify a
......@@ -902,14 +908,14 @@ The :code:`constraints` parameter can have any of the following values:
.. tabularcolumns:: |l|L|
======== =============================================================================================================================================
================ =============================================================================================================================================
Value Meaning
======== =============================================================================================================================================
None No constraints are applied. This is the default value.
HBonds The lengths of all bonds that involve a hydrogen atom are constrained.
AllBonds The lengths of all bonds are constrained.
HAngles The lengths of all bonds are constrained. In addition, all angles of the form H-X-H or H-O-X (where X is an arbitrary atom) are constrained.
======== =============================================================================================================================================
================ =============================================================================================================================================
:code:`None` No constraints are applied. This is the default value.
:code:`HBonds` The lengths of all bonds that involve a hydrogen atom are constrained.
:code:`AllBonds` The lengths of all bonds are constrained.
:code:`HAngles` The lengths of all bonds are constrained. In addition, all angles of the form H-X-H or H-O-X (where X is an arbitrary atom) are constrained.
================ =============================================================================================================================================
The main reason to use constraints is that it allows one to use a larger
......@@ -1331,6 +1337,8 @@ Allowed values for :code:`positiveIon` are 'Cs+', 'K+', 'Li+', 'Na+', and
some force fields do not include parameters for all of these ion types, so you
need to use types that are supported by your chosen force field.
.. _adding-or-removing-extra-particles:
Adding or Removing Extra Particles
**********************************
......@@ -2426,9 +2434,10 @@ with the following signature to parse the XML tag and create the generator:
:code:`element` is the XML tag (an xml.etree.ElementTree.Element object) and
:code:`forcefield` is the ForceField being created. This method should
create a generator and add it to the ForceField:
::
generator = MyForceGenerator()
forcefield._forces.append(generator)
generator = MyForceGenerator()
forcefield._forces.append(generator)
It then should parse the information contained in the XML tag and configure the
generator based on it.
......
docs-source/usersguide/library.rst
View file @ 63203651
......@@ -20,11 +20,11 @@ high level interface for running simulations. This layer is targeted at
computational biologists or other people who want to run simulations, and who
may or may not be programmers.
Part I of this guide focused on the application layer and described how to run
The first part of this guide focused on the application layer and described how to run
simulations with it. We now turn to the lower level libraries. We will assume
you are a programmer, that you are writing your own applications, and that you
want to add simulation features to those applications. Part II of this guide
describes how to do that with OpenMM.
want to add simulation features to those applications. The following chapters
describe how to do that with OpenMM.
How to get started
==================
......@@ -42,7 +42,8 @@ License
========
Two different licenses are used for different parts of OpenMM. The public API,
the low level API, and the reference platform are all distributed under the MIT
the low level API, the reference platform, the CPU platform, and the application
layer are all distributed under the MIT
license. This is a very permissive license which allows them to be used in
almost any way, requiring only that you retain the copyright notice and
disclaimer when distributing them.
......@@ -425,12 +426,18 @@ Get the OpenMM source code
You will also need the OpenMM source code before building OpenMM from source.
To download and unpack OpenMM source code:
#. Browse to https://simtk.org/home/openmm/.
#. Browse to https://simtk.org/home/openmm.
#. Click the "Downloads" link in the navigation bar on the left side.
#. Download OpenMM<Version>-Source.zip, choosing the latest version.
#. Unpack the zip file. Note the location where you unpacked the OpenMM source
code.
Alternatively, if you want the most recent development version of the code rather
than the version corresponding to a particular release, you can get it from
https://github.com/SimTk/openmm. Be aware that the development code is constantly
changing, may contain bugs, and should never be used for production work. If
you want a stable, well tested version of OpenMM, you should download the source
code for the latest release as described above.
Other Required Software
=======================
......@@ -540,10 +547,10 @@ There are several variables that can be adjusted in the CMake interface:
install OpenMM.
Configure (press “\ :code:`c`\ ”) again. Adjust any variables that cause an
Configure (press “c”) again. Adjust any variables that cause an
error.
Continue to configure (press “\ :code:`c`\ ”) until no starred/red CMake
Continue to configure (press “c”) until no starred/red CMake
variables are displayed. Congratulations, you have completed the configuration
step.
......@@ -566,7 +573,7 @@ Windows
Mac and Linux
=============
* Press :code:`g` to generate the Makefile.
* Press “g” to generate the Makefile.
* If CMake does not exit automatically, press “q” to exit.
......@@ -1321,8 +1328,10 @@ to change the build environment.
double stepSizeInFs,
std::string& platformName);
static void myStepWithOpenMM(MyOpenMMData*, int numSteps);
static void myGetOpenMMState(MyOpenMMData*, bool
wantEnergy,double& time, double& energy,
static void myGetOpenMMState(MyOpenMMData*,
bool wantEnergy,
double& time,
double& energy,
MyAtomInfo atoms[]);
static void myTerminateOpenMM(MyOpenMMData*);
......@@ -3544,7 +3553,7 @@ Hamiltonian is given by
.. math::
H=\sum _{k=1}^{n}\left(\frac{{p}_{{k}^{2}}}{2m}+E({q}_{k})+\frac{m({k}_{B}Tn)^{2}}{2h^{2}}({q}_{k}-{q}_{k-1})^{2}\right)
H=\sum _{k=1}^{n}\left(\frac{{p}_{{k}^{2}}}{2m}+E({q}_{k})+\frac{m({k}_{B}Tn)^{2}}{2\hbar^{2}}({q}_{k}-{q}_{k-1})^{2}\right)
This Hamiltonian resembles that of a system of classical ring polymers where
......@@ -3560,7 +3569,7 @@ needed to converge a calculation can be estimated using\ :cite:`Markland2008`\
.. math::
n>\frac{h\omega_{max}}{{k}_{B}T}
n>\frac{\hbar\omega_{max}}{{k}_{B}T}
where :math:`\omega_{max}` is the highest frequency in the problem. For example, for
......
docs-source/usersguide/theory.rst
View file @ 63203651
......@@ -109,10 +109,10 @@ Each harmonic angle is represented by an energy term of the form
.. math::
E=\frac{1}{2}k{\left(q-{q}_{0}\right)}^{2}
E=\frac{1}{2}k{\left(\theta-\theta_0\right)}^{2}
where :math:`\theta` is the angle formed by the three particles, :math:`\theta` :sub:`0` is
where :math:`\theta` is the angle formed by the three particles, :math:`\theta_0` is
the equilibrium angle, and *k* is the force constant.
As with HarmonicBondForce, be aware that some force fields define their harmonic
......@@ -127,7 +127,7 @@ Each torsion is represented by an energy term of the form
.. math::
E=k\left(1+\text{cos}\left(\mathit{nq}-{q}_{0}\right)\right)
E=k\left(1+\text{cos}\left(n\theta-\theta_0\right)\right)
where :math:`\theta` is the dihedral angle formed by the four particles, :math:`\theta_0`
......@@ -141,7 +141,7 @@ Each torsion is represented by an energy term of the form
.. math::
E=\sum _{i=0}^{5}{C}_{i}{\left(\text{cos}f\right)}^{i}
E=\sum _{i=0}^{5}{C}_{i}{\left(\text{cos}\phi\right)}^{i}
where :math:`\phi` is the dihedral angle formed by the four particles and
......@@ -160,7 +160,7 @@ Each torsion pair is represented by an energy term of the form
.. math::
E=f\left({q}_{1},{q}_{2}\right)
E=f\left(\theta_1,\theta_2\right)
where :math:`\theta_1` and :math:`\theta_2` are the two dihedral angles
......@@ -182,7 +182,7 @@ an energy term of the form
.. math::
E=4e\left({\left(\frac{s}{r}\right)}^{\text{12}}-{\left(\frac{s}{r}\right)}^{6}\right)
E=4\epsilon\left({\left(\frac{\sigma}{r}\right)}^{\text{12}}-{\left(\frac{\sigma}{r}\right)}^{6}\right)
where *r* is the distance between the two particles, :math:`\sigma` is the distance
......@@ -193,14 +193,14 @@ to zero. Because the interaction decreases very quickly with distance, the
cutoff usually has little effect on the accuracy of simulations.
Optionally you can use a switching function to make the energy go smoothly to 0
at the cutoff distance. When :math:`r_{switch} < r < r_{cutoff}`\ , the energy is multiplied by
at the cutoff distance. When :math:`r_\mathit{switch} < r < r_\mathit{cutoff}`\ , the energy is multiplied by
.. math::
S=1-{6x}^{5}+15{x}^{4}-10{x}^{3}
where :math:`x = (r-r_{switch})/(r_{cutoff}-r_{switch})`. This function decreases smoothly from 1 at
:math:`r = r_{switch}` to 0 at :math:`r = r_{cutoff}`, and has continuous first and
where :math:`x = (r-r_\mathit{switch})/(r_\mathit{cutoff}-r_\mathit{switch})`. This function decreases smoothly from 1 at
:math:`r = r_\mathit{switch}` to 0 at :math:`r = r_\mathit{cutoff}`, and has continuous first and
second derivatives at both ends
When an exception has been added for a pair of particles, :math:`\sigma` and :math:`\epsilon`
......@@ -209,10 +209,10 @@ from the parameters of the individual particles using the Lorentz-Bertelot
combining rule:
.. math::
s=\frac{{s}_{1}+{s}_{2}}{2}
\sigma=\frac{\sigma_1+\sigma_2}{2}
.. math::
e=\sqrt{{e}_{1}{e}_{2}}
\epsilon=\sqrt{\epsilon_1 \epsilon_2}
When using periodic boundary conditions, NonbondedForce can optionally add a
term (known as a *long range dispersion correction*\ ) to the energy that
......@@ -220,11 +220,11 @@ approximately represents the contribution from all interactions beyond the
cutoff distance:\ :cite:`Shirts2007`\
.. math::
{E}_{\text{cor}}=\frac{{8pN}^{2}}{V}\left(\frac{\langle e_{ij}\sigma_{ij}^{12}\rangle}{{9r}_{{c}^{9}}}-\frac{\langle e_{ij}{s}_{ij}^{6}\rangle}{{3r}_{{c}^{3}}}\right)
{E}_{\text{cor}}=\frac{{8\pi N}^{2}}{V}\left(\frac{\langle \epsilon_{ij}\sigma_{ij}^{12}\rangle}{{9r_c}^9}-\frac{\langle \epsilon_{ij}\sigma_{ij}^{6}\rangle}{{3r_c}^3}\right)
where *N* is the number of particles in the system, *V* is the volume of
the periodic box, *r*\ *c* is the cutoff distance, :math:`\sigma`\ *ij* and
:math:`\epsilon`\ *ij* are the interaction parameters between particle *i* and
the periodic box, :math:`r_c` is the cutoff distance, :math:`\sigma_{ij}` and
:math:`\epsilon_{ij}` are the interaction parameters between particle *i* and
particle *j*\ , and :math:`\langle \text{...} \rangle` represents an average over all pairs of particles in
the system. When a switching function is in use, there is also a contribution
to the correction that depends on the integral of *E*\ ·(1-\ *S*\ ) over the
......@@ -236,22 +236,20 @@ The Lennard-Jones interaction is often parameterized in two other equivalent
ways. One is
.. math::
E=e\left({\left(\frac{{r}_{\text{min}}}{r}\right)}^{\text{12}}-2{\left(\frac{{r}_{\text{min}}}{r}\right)}^{6}\right)
E=\epsilon\left({\left(\frac{{r}_{\mathit{min}}}{r}\right)}^{\text{12}}-2{\left(\frac{{r}_{\mathit{min}}}{r}\right)}^{6}\right)
where :math:`r_{min}` (sometimes known as :math:`d_{min}`; this is not a
where :math:`r_\mathit{min}` (sometimes known as :math:`d_\mathit{min}`; this is not a
radius) is the center-to-center distance at which the energy is minimum. It is
related to :math:`\sigma` by
.. math::
s=\frac{{r}_{\text{min}}}{{2}^{1/6}}
\sigma=\frac{{r}_{\mathit{min}}}{{2}^{1/6}}
In turn, :math:`r_{min}` is related to the van der Waals radius by :math:`r_{min} = 2r_{vdw}`\ .
In turn, :math:`r_\mathit{min}` is related to the van der Waals radius by :math:`r_\mathit{min} = 2r_\mathit{vdw}`\ .
Another common form is
......@@ -266,12 +264,12 @@ The coefficients A and B are related to :math:`\sigma` and :math:`\epsilon` by
.. math::
s={\left(\frac{A}{B}\right)}^{1/6}
\sigma={\left(\frac{A}{B}\right)}^{1/6}
.. math::
e=\frac{{B}^{2}}{4A}
\epsilon=\frac{{B}^{2}}{4A}
Coulomb Interaction Without Cutoff
......@@ -297,19 +295,19 @@ distance is a solvent with a uniform dielectric constant.\ :cite:`Tironi1995`
.. math::
E=\frac{{q}_{1}{q}_{2}}{4{\text{pe}}_{0}}\left(\frac{1}{r}+{k}_{\text{rf}}{r}^{2}-{c}_{\text{rf}}\right)
E=\frac{{q}_{1}{q}_{2}}{4\pi\epsilon_0}\left(\frac{1}{r}+{k}_{\mathit{rf}}{r}^{2}-{c}_{\mathit{rf}}\right)
.. math::
{k}_{\text{rf}}=\left(\frac{1}{{r}_{{\text{cutoff}}^{3}}}\right)\left(\frac{{\epsilon}_{\text{solvent}}-1}{2{\epsilon}_{\text{solvent}}+1}\right)
{k}_{\mathit{rf}}=\left(\frac{1}{{r_\mathit{cutoff}}^3}\right)\left(\frac{{\epsilon}_{\mathit{solvent}}-1}{2{\epsilon}_{\mathit{solvent}}+1}\right)
.. math::
{c}_{\text{rf}}=\left(\frac{1}{{r}_{\text{cutoff}}}\right)\left(\frac{3{\epsilon}_{\text{solvent}}}{2{\epsilon}_{\text{solvent}}+1}\right)
{c}_{\mathit{rf}}=\left(\frac{1}{{r}_{\mathit{cutoff}}}\right)\left(\frac{3{\epsilon}_{\mathit{solvent}}}{2{\epsilon}_{\mathit{solvent}}+1}\right)
where *r*\ *cutoff* is the cutoff distance and :math:`\epsilon_{solvent}` is
the dielectric constant of the solvent. In the limit :math:`\epsilon_{solvent}` >> 1,
where :math:`r_\mathit{cutoff}` is the cutoff distance and :math:`\epsilon_\mathit{solvent}` is
the dielectric constant of the solvent. In the limit :math:`\epsilon_\mathit{solvent}` >> 1,
this causes the force to go to zero at the cutoff.
Coulomb Interaction With Ewald Summation
......@@ -321,19 +319,19 @@ space sum*\ , the *reciprocal space sum*\ , and the *self-energy term*\ .\
.. math::
E=E_{\text{dir}}+{E}_{\text{rec}}+{E}_{\text{self}}
E=E_{\mathit{dir}}+{E}_{\mathit{rec}}+{E}_{\mathit{self}}
.. math::
E_{\text{dir}}=\frac{1}{2}\sum _{i,j}\sum _{n}{q}_{i}{q}_{j}\frac{\text{erfc}\left({\mathit{\alpha r}}_{ij,n}\right)}{{r}_{ij,n}}
E_{\mathit{dir}}=\frac{1}{2}\sum _{i,j}\sum_\mathbf{n}{q}_{i}{q}_{j}\frac{\text{erfc}\left({\mathit{\alpha r}}_{ij,\mathbf{n}}\right)}{r_{ij,\mathbf{n}}}
.. math::
E_{\text{rec}}=\frac{1}{2{\pi}V}\sum _{i,j}q_i q_j\sum _{\mathbf{k}{\neq}0}\frac{\text{exp}(-(\pi \mathbf{k}/\alpha)^2+2\pi i \mathbf{k} \cdot (\mathbf{r}_{i}-\mathbf{r}_{j}))}{\mathbf{m}^2}
E_{\mathit{rec}}=\frac{1}{2{\pi}V}\sum _{i,j}q_i q_j\sum _{\mathbf{k}{\neq}0}\frac{\text{exp}(-(\pi \mathbf{k}/\alpha)^2+2\pi i \mathbf{k} \cdot (\mathbf{r}_{i}-\mathbf{r}_{j}))}{\mathbf{m}^2}
.. math::
E_{\text{self}}=-\frac{\alpha}{\sqrt{p}}\sum _{i}{q}_{{i}^{2}}
E_{\mathit{self}}=-\frac{\alpha}{\sqrt{\pi}}\sum _{i}{q}_{{i}^{2}}
In the above expressions, the indices *i* and *j* run over all
......@@ -351,7 +349,7 @@ width of the periodic cell, the number of terms in this sum is never greater
than the square of the number of particles.
The error made by applying the direct space cutoff depends on the magnitude of
:math:`\text{erfc}({\alpha}r_{cutoff})`\ . Similarly, the error made in the reciprocal space
:math:`\text{erfc}({\alpha}r_\mathit{cutoff})`\ . Similarly, the error made in the reciprocal space
sum by ignoring wave numbers beyond k\ :sub:`max` depends on the magnitude
of :math:`\text{exp}(-({\pi}k_{max}/{\alpha})^2`\ ). By changing :math:`\alpha`, one can decrease the
error in either term while increasing the error in the other one.
......@@ -361,14 +359,14 @@ instead asks the user to choose an error tolerance :math:`\delta`. It then calc
.. math::
\alpha =\sqrt{-\text{log}\left(2{\delta}\right)}/{r}_{\text{cutoff}}
\alpha =\sqrt{-\text{log}\left(2{\delta}\right)}/{r}_{\mathit{cutoff}}
Finally, it estimates the error in the reciprocal space sum as
.. math::
\text{error}=\frac{k_{\text{max}}\sqrt{d\alpha}}{20}\text{exp}(-(\pi k_\text{max}/d\alpha)^2)
\mathit{error}=\frac{k_{\mathit{max}}\sqrt{d\alpha}}{20}\text{exp}(-(\pi k_\mathit{max}/d\alpha)^2)
where *d* is the width of the periodic box, and selects the smallest value
......@@ -376,7 +374,7 @@ for k\ :sub:`max` which gives *error* < :math:`\delta`\ . (If the box is not sq
k\ :sub:`max` will have a different value along each axis.)
This means that the accuracy of the calculation is determined by :math:`\delta`\ .
:math:`r_{cutoff}` does not affect the accuracy of the result, but does affect the speed
:math:`r_\mathit{cutoff}` does not affect the accuracy of the result, but does affect the speed
of the calculation by changing the relative costs of the direct space and
reciprocal space sums. You therefore should test different cutoffs to find the
value that gives best performance; this will in general vary both with the size
......@@ -402,27 +400,27 @@ computed very quickly, giving performance that scales as O(N log N) in the
number of particles (assuming the volume of the periodic box is proportional to
the number of particles).
As with Ewald summation, the user specifies the direct space cutoff :math:`r_{cutoff}`
As with Ewald summation, the user specifies the direct space cutoff :math:`r_\mathit{cutoff}`
and error tolerance :math:`\delta`\ . NonbondedForce then selects :math:`\alpha` as
.. math::
\alpha =\sqrt{-\text{log}\left(2d\right)}/{r}_{cutoff}
\alpha =\sqrt{-\text{log}\left(2\delta\right)}/{r}_\mathit{cutoff}
and the number of nodes in the mesh along each dimension as
.. math::
{n}_{\text{mesh}}=\frac{2\alpha d}{{3d}^{1/5}}
n_\mathit{mesh}=\frac{2\alpha d}{{3d}^{1/5}}
where *d* is the width of the periodic box along that dimension. Alternatively,
the user may choose to explicitly set values for these parameters. (Note that
some Platforms may choose to use a larger value of :math:`n_\text{mesh}` than that
some Platforms may choose to use a larger value of :math:`n_\mathit{mesh}` than that
given by this equation. For example, some FFT implementations require the mesh
size to be a multiple of certain small prime numbers, so a Platform might round
it up to the nearest permitted value. It is guaranteed that :math:`n_\text{mesh}`
it up to the nearest permitted value. It is guaranteed that :math:`n_\mathit{mesh}`
will never be smaller than the value given above.)
The comments in the previous section regarding the interpretation of :math:`\delta` for Ewald
......@@ -450,11 +448,11 @@ molecule. The Generalized Born energy is given by\ :cite:`Onufriev2004`
.. math::
E\text{=-}\frac{1}{2}\left(\frac{1}{\epsilon_{\text{solute}}}-\frac{1}{\epsilon_{\text{solvent}}}\right)\sum _{i,j}\frac{{q}_{i}{q}_{j}}{{f}_{\text{GB}}\left({d}_{ij},{R}_{i},{R}_{j}\right)}
E\text{=-}\frac{1}{2}\left(\frac{1}{\epsilon_{\mathit{solute}}}-\frac{1}{\epsilon_{\mathit{solvent}}}\right)\sum _{i,j}\frac{{q}_{i}{q}_{j}}{{f}_{\text{GB}}\left({d}_{ij},{R}_{i},{R}_{j}\right)}
where the indices *i* and *j* run over all particles, :math:`\epsilon_\text{solute}`
and :math:`\epsilon_\text{solvent}` are the dielectric constants of the solute and solvent
where the indices *i* and *j* run over all particles, :math:`\epsilon_\mathit{solute}`
and :math:`\epsilon_\mathit{solvent}` are the dielectric constants of the solute and solvent
respectively, :math:`q_i` is the charge of particle *i*\ , and :math:`d_{ij}` is the distance
between particles *i* and *j*\ . :math:`f_\text{GB}(d_{ij}, R_i, R_j)` is defined as
......@@ -467,7 +465,7 @@ between particles *i* and *j*\ . :math:`f_\text{GB}(d_{ij}, R_i, R_j)` is defin
.. math::
{R}_{i}=\frac{1}{{\rho}_{{i}^{-1}}-{\rho}_{{i}^{-1}}\text{tanh}\left(\alpha \Psi_{i}-{\beta \Psi}_{{i}^{2}}+{\gamma \Psi}_{{i}^{3}}\right)}
R_i=\frac{1}{\rho_i^{-1}-r_i^{-1}\text{tanh}\left(\alpha \Psi_{i}-{\beta \Psi}_{{i}^{2}}+{\gamma \Psi}_{{i}^{3}}\right)}
where :math:`\alpha`, :math:`\beta`, and :math:`\gamma` are the GB\ :sup:`OBC`\ II parameters :math:`\alpha` = 1, :math:`\beta` = 0.8, and :math:`\gamma` =
......@@ -478,7 +476,7 @@ spheres of all particles outside particle *i*\ :
.. math::
\Psi_{i}=\frac{{\rho }_{i}}{4p}{\int }_{\text{VDW}}q\left(\mid r\mid -{\rho }_{i}\right)\frac{1}{{\mid r\mid }^{4}}{d}^{3}r
\Psi_i=\frac{\rho_i}{4\pi}\int_{\text{VDW}}\theta\left(|\mathbf{r}|-{\rho }_{i}\right)\frac{1}{{|\mathbf{r}|}^{4}}{d}^{3}\mathbf{r}
where :math:`\theta`\ (\ *r*\ ) is a step function that excludes the interior of particle
......@@ -491,11 +489,11 @@ The surface area term is given by\ :cite:`Schaefer1998`\ :cite:`Ponder`
.. math::
E=4\pi \cdot 2\text{.}\text{26}\sum _{i}{\left({r}_{i}+{r}_{\text{solvent}}\right)}^{2}{\left(\frac{{r}_{i}}{{R}_{i}}\right)}^{6}
E=4\pi \cdot 2\text{.}\text{26}\sum _{i}{\left({r}_{i}+{r}_{\mathit{solvent}}\right)}^{2}{\left(\frac{{r}_{i}}{{R}_{i}}\right)}^{6}
where :math:`r_i` is the atomic radius of particle *i*\ , :math:`r_i` is
its Born radius, and :math:`r_\text{solvent}` is the solvent radius, which is taken
its Born radius, and :math:`r_\mathit{solvent}` is the solvent radius, which is taken
to be 0.14 nm.
......@@ -510,11 +508,11 @@ energy is given by Equation 2 of the referenced paper:
.. math::
E=-\frac{1}{2}\left(\frac{1}{{\epsilon }_{\text{solute}}}-\frac{1}{{\epsilon }_{\text{solvent}}}\right)\sum _{i,j}\frac{{q}_{i}{q}_{j}}{{f}_{\text{GB}}\left({d}_{ij},{R}_{i},{R}_{j}\right)}+\sum _{i}^{n}{\gamma }_{i}{\left(\frac{{r}_{i}}{{R}_{i}}\right)}^{3}
E=-\frac{1}{2}\left(\frac{1}{{\epsilon }_{\mathit{solute}}}-\frac{1}{{\epsilon }_{\mathit{solvent}}}\right)\sum _{i,j}\frac{{q}_{i}{q}_{j}}{{f}_{\text{GB}}\left({d}_{ij},{R}_{i},{R}_{j}\right)}+\sum _{i}^{n}{\gamma }_{i}{\left(\frac{{r}_{i}}{{R}_{i}}\right)}^{3}
where the indices *i* and *j* run over all n particles, :math:`\epsilon_\text{solute}`
and :math:`\epsilon_\text{solvent}` are the dielectric constants of the solute
where the indices *i* and *j* run over all n particles, :math:`\epsilon_\mathit{solute}`
and :math:`\epsilon_\mathit{solvent}` are the dielectric constants of the solute
and solvent respectively, :math:`q_i` is the charge of particle *i*\ ,
:math:`d_{ij}` is the distance between particles *i* and *j*\ , :math:`r_i`
are the input particle radii, and the :math:`\gamma_i` are adjustable
......@@ -530,7 +528,7 @@ where V(d,r,S) is given by
.. math::
V\left(d,r,S\right)=\left\{\begin{array}{ccc}L\left(d,x,S\right){\mid }_{x=\text{max}\left(r,d-S\right)}^{x=d+S}& \mid r-S\mid <d& \\ 0& 0\le d\le r-S& \\ L\left(d,x,S\right){\mid }_{x=d-S}^{x=d+S}& 0\le d\le S-r& \end{array}\right\}
V\left(d,r,S\right)=\left\{\begin{array}{ccc}L\left(d,x,S\right){\mid }_{x=\mathrm{max}\left(r,d-S\right)}^{x=d+S}& \mid r-S\mid <d& \\ 0& 0\le d\le r-S& \\ L\left(d,x,S\right){\mid }_{x=d-S}^{x=d+S}& 0\le d\le S-r& \end{array}\right\}
and
......@@ -544,7 +542,7 @@ The S\ :sub:`i` are derived from the covalent topology of the solute:
.. math::
{S}_{i}=0\text{.}\text{95}\cdot\text{max}\left\{0,{\nu }_{i}^{}\right\}
{S}_{i}=0\text{.}\text{95}\cdot\mathrm{max}\left(0,\nu_i^{1/3}\right)
......@@ -727,10 +725,10 @@ user. That is, the interaction energy of each angle is given by
.. math::
E=f\left(q\right)
E=f\left(\theta\right)
where *f*\ (\ :math:`\theta`\ ) is a user defined mathematical expression.
where :math:`f(\theta)` is a user defined mathematical expression.
In addition to depending on the angle :math:`\theta`\ , the energy may also depend on an
arbitrary set of user defined parameters. Parameters may be specified in two
......@@ -750,10 +748,10 @@ by the user. That is, the interaction energy of each angle is given by
.. math::
E=f\left(q\right)
E=f(\theta)
where *f*\ (\ :math:`\theta`\ ) is a user defined mathematical expression. The angle
where :math:`f(\theta)` is a user defined mathematical expression. The angle
:math:`\theta` is guaranteed to be in the range [-π, π]. Like PeriodicTorsionForce, it
is defined to be zero when the first and last particles are on the same side of
the bond formed by the middle two particles (the *cis* configuration).
......@@ -778,7 +776,7 @@ interaction energy between each pair of particles is given by
.. math::
E=f\left(r\right)
E=f(r)
where *f*\ (\ *r*\ ) is a user defined mathematical expression.
......@@ -797,7 +795,7 @@ API documentation for details.
When using a cutoff, a switching function can optionally be applied to make the
energy go smoothly to 0 at the cutoff distance. When
:math:`r_{switch} < r < r_{cutoff}`\ , the energy is multiplied by
:math:`r_\mathit{switch} < r < r_\mathit{cutoff}`\ , the energy is multiplied by
......@@ -805,9 +803,9 @@ energy go smoothly to 0 at the cutoff distance. When
S=1-{6x}^{5}+15{x}^{4}-10{x}^{3}
where :math:`x=(r-r_{switch})/(r_{cutoff}-r_{switch})`\ .
This function decreases smoothly from 1 at :math:`r=r_{switch}`
to 0 at :math:`r=r_{cutoff}`\ , and has continuous first and
where :math:`x=(r-r_\mathit{switch})/(r_\mathit{cutoff}-r_\mathit{switch})`\ .
This function decreases smoothly from 1 at :math:`r=r_\mathit{switch}`
to 0 at :math:`r=r_\mathit{cutoff}`\ , and has continuous first and
second derivatives at both ends.
When using periodic boundary conditions, CustomNonbondedForce can optionally add
......@@ -817,7 +815,7 @@ cutoff distance:\ :cite:`Shirts2007`
.. math::
{E}_{cor}=\frac{2\pi N^2}{V}\langle\underset{{r}_{cutoff}}{\overset{\infty}{\int }}E\left(r\right)r^{2}dr\rangle
{E}_{cor}=\frac{2\pi N^2}{V}\left\langle\underset{{r}_\mathit{cutoff}}{\overset{\infty}{\int}}E(r)r^{2}dr\right\rangle
where *N* is the number of particles in the system, *V* is the volume of
......@@ -827,7 +825,7 @@ contribution to the correction given by
.. math::
E_{cor}^\prime=\frac{2\pi N^2}{V}\langle\underset{{r}_{switch}}{\overset{{r}_{cutoff}}{\int }}E\left(r\right)\left(1-S\left(r\right)\right)r^{2}dr\rangle
E_{cor}^\prime=\frac{2\pi N^2}{V}\left\langle\underset{{r}_\mathit{switch}}{\overset{{r}_\mathit{cutoff}}{\int }}E(r)(1-S(r))r^{2}dr\right\rangle
The long range dispersion correction is primarily useful when running
......@@ -843,7 +841,7 @@ is given by
.. math::
E=f\left(x,y,z\right)
E=f(x,y,z)
where *f*\ (\ *x*\ , *y*\ , *z*\ ) is a user defined mathematical
......@@ -870,7 +868,7 @@ of four particles. That is, the interaction energy of each bond is given by
.. math::
E=f\left(\left\{{x}_{i}\right\},\left\{{r}_{i}\right\},\left\{{q}_{i}\right\},\left\{{f}_{i}\right\}\right)
E=f(\{x_i\},\{r_i\},\{\theta_i\},\{\phi_i\})
where *f*\ (\ *...*\ ) is a user defined mathematical expression. It may
......@@ -985,7 +983,7 @@ group is given by
.. math::
E=f\left(\left\{{r}_{i}\right\},\left\{{q}_{i}\right\},\left\{{f}_{i}\right\}\right)
E=f(\{r_i\},\{\theta_i\},\{\phi_i\})
where *f*\ (\ *...*\ ) is a user defined mathematical expression. It may
......@@ -1021,7 +1019,7 @@ expressions. These expressions are specified as character strings, and may
involve a variety of standard operators and mathematical functions.
The following operators are supported: + (add), - (subtract), * (multiply), /
(divide), and ^ (power). Parentheses “(“and “)” may be used for grouping.
(divide), and ^ (power). Parentheses “(“ and “)” may be used for grouping.
The following standard functions are supported: sqrt, exp, log, sin, cos, sec,
csc, tan, cot, asin, acos, atan, sinh, cosh, tanh, erf, erfc, min, max, abs,
......@@ -1145,7 +1143,7 @@ of the integration error:
.. math::
error={\left(\Delta t\right)}^{2}\sum _{i}\frac{\mid \mathbf{f}_{i}\mid}{m_i}
error={\left(\Delta t\right)}^{2}\sum _{i}\frac{|\mathbf{f}_{i}|}{m_i}
where :math:`\mathbf{f}_i` is the force acting on particle *i* and :math:`m_i`
......@@ -1159,7 +1157,7 @@ specified error tolerance:
.. math::
\Delta t=\sqrt{\frac{\delta}{\sum _{i}\frac{\mid \mathbf{f}_i\mid}{m_i}}}
\Delta t=\sqrt{\frac{\delta}{\sum _{i}\frac{|\mathbf{f}_i|}{m_i}}}
where :math:`\delta` is the error tolerance. This is the largest step that may be
......@@ -1320,14 +1318,14 @@ specific types of rules. They are:
average of the positions of three particles:
.. math::
\mathbf{r}={w}_{1}\mathbf{r}_{1}+{w}_{2}\mathbf{r}_{{2}_{1}}+{w}_{3}\mathbf{r}_{3}
\mathbf{r}={w}_{1}\mathbf{r}_{1}+{w}_{2}\mathbf{r}_{2}+{w}_{3}\mathbf{r}_{3}
* OutOfPlaneSite: The virtual site location is computed as a weighted average
of the positions of three particles and the cross product of their relative
displacements:
.. math::
\mathbf{r}={r}_{1}+{w}_{12}\mathbf{r}_{12}+{w}_{13}\mathbf{r}_{13}+{w}_{cross}\left(\mathbf{r}_{12}\times \mathbf{r}_{13}\right)
\mathbf{r}=\mathbf{r}_{1}+{w}_{12}\mathbf{r}_{12}+{w}_{13}\mathbf{r}_{13}+{w}_\mathit{cross}\left(\mathbf{r}_{12}\times \mathbf{r}_{13}\right)
..
where :math:`\mathbf{r}_{12} = \mathbf{r}_{2}-\mathbf{r}_{1}` and
......
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