diff --git a/doc/src/Eqs/improper_class2.jpg b/doc/src/Eqs/improper_class2.jpg deleted file mode 100644 index ca0604f8d7..0000000000 Binary files a/doc/src/Eqs/improper_class2.jpg and /dev/null differ diff --git a/doc/src/Eqs/improper_class2.tex b/doc/src/Eqs/improper_class2.tex deleted file mode 100644 index 2c41df1a29..0000000000 --- a/doc/src/Eqs/improper_class2.tex +++ /dev/null @@ -1,13 +0,0 @@ -\documentclass[12pt]{article} - -\begin{document} - -\begin{eqnarray*} - E & = & E_i + E_{aa} \\ - E_i & = & K [ \frac{\chi_{ijkl} + \chi_{kjli} + \chi_{ljik}}{3} - \chi_0 ]^2 \\ - E_{aa} & = & M_1 (\theta_{ijk} - \theta_1) (\theta_{kjl} - \theta_3) + \\ - & & M_2 (\theta_{ijk} - \theta_1) (\theta_{ijl} - \theta_2) + \\ - & & M_3 (\theta_{ijl} - \theta_2) (\theta_{kjl} - \theta_3) -\end{eqnarray*} - -\end{document} \ No newline at end of file diff --git a/doc/src/Eqs/improper_cossq.jpg b/doc/src/Eqs/improper_cossq.jpg deleted file mode 100644 index bddcb3d867..0000000000 Binary files a/doc/src/Eqs/improper_cossq.jpg and /dev/null differ diff --git a/doc/src/Eqs/improper_cossq.tex b/doc/src/Eqs/improper_cossq.tex deleted file mode 100644 index 27e79c7b16..0000000000 --- a/doc/src/Eqs/improper_cossq.tex +++ /dev/null @@ -1,9 +0,0 @@ -\documentclass[12pt]{article} - -\begin{document} - -$$ - E = \frac{1}{2} K \cos^2{\left(\chi - \chi_0\right)} -$$ - -\end{document} diff --git a/doc/src/Eqs/improper_cvff.jpg b/doc/src/Eqs/improper_cvff.jpg deleted file mode 100644 index 14be3ffa03..0000000000 Binary files a/doc/src/Eqs/improper_cvff.jpg and /dev/null differ diff --git a/doc/src/Eqs/improper_cvff.tex b/doc/src/Eqs/improper_cvff.tex deleted file mode 100644 index 3a038b6b8c..0000000000 --- a/doc/src/Eqs/improper_cvff.tex +++ /dev/null @@ -1,9 +0,0 @@ -\documentclass[12pt]{article} - -\begin{document} - -$$ - E = K [1 + d \cos (n \phi) ] -$$ - -\end{document} \ No newline at end of file diff --git a/doc/src/Eqs/improper_dist-1.jpg b/doc/src/Eqs/improper_dist-1.jpg deleted file mode 100644 index aeb761bfbb..0000000000 Binary files a/doc/src/Eqs/improper_dist-1.jpg and /dev/null differ diff --git a/doc/src/Eqs/improper_dist-1.tex b/doc/src/Eqs/improper_dist-1.tex deleted file mode 100644 index 8ef5b5f4c3..0000000000 --- a/doc/src/Eqs/improper_dist-1.tex +++ /dev/null @@ -1,9 +0,0 @@ -\documentclass[12pt]{article} - -\begin{document} - -$$ - E = K_2 d^2 + K_4 d^4 -$$ - -\end{document} diff --git a/doc/src/Eqs/improper_distharm.jpg b/doc/src/Eqs/improper_distharm.jpg deleted file mode 100644 index e55036b5bf..0000000000 Binary files a/doc/src/Eqs/improper_distharm.jpg and /dev/null differ diff --git a/doc/src/Eqs/improper_distharm.tex b/doc/src/Eqs/improper_distharm.tex deleted file mode 100644 index 218de6a0dd..0000000000 --- a/doc/src/Eqs/improper_distharm.tex +++ /dev/null @@ -1,9 +0,0 @@ -\documentclass[12pt]{article} - -\begin{document} -\thispagestyle{empty} -$$ - E = K (d - d_0)^2 -$$ - -\end{document} diff --git a/doc/src/Eqs/improper_fourier.jpg b/doc/src/Eqs/improper_fourier.jpg deleted file mode 100644 index 0a1b13baf8..0000000000 Binary files a/doc/src/Eqs/improper_fourier.jpg and /dev/null differ diff --git a/doc/src/Eqs/improper_fourier.tex b/doc/src/Eqs/improper_fourier.tex deleted file mode 100644 index 47ac58ef01..0000000000 --- a/doc/src/Eqs/improper_fourier.tex +++ /dev/null @@ -1,9 +0,0 @@ -\documentclass[12pt]{article} - -\begin{document} - -$$ - E = K [C_0 + C_1 \cos ( \omega) + C_2 \cos( 2 \omega) ] -$$ - -\end{document} diff --git a/doc/src/Eqs/improper_harmonic.jpg b/doc/src/Eqs/improper_harmonic.jpg deleted file mode 100644 index c2c1eb4663..0000000000 Binary files a/doc/src/Eqs/improper_harmonic.jpg and /dev/null differ diff --git a/doc/src/Eqs/improper_harmonic.tex b/doc/src/Eqs/improper_harmonic.tex deleted file mode 100644 index c47020cf9c..0000000000 --- a/doc/src/Eqs/improper_harmonic.tex +++ /dev/null @@ -1,9 +0,0 @@ -\documentclass[12pt]{article} - -\begin{document} - -$$ - E = K (\chi - \chi_0)^2 -$$ - -\end{document} \ No newline at end of file diff --git a/doc/src/Eqs/improper_inversion_harmonic.jpg b/doc/src/Eqs/improper_inversion_harmonic.jpg deleted file mode 100644 index 793311e8ad..0000000000 Binary files a/doc/src/Eqs/improper_inversion_harmonic.jpg and /dev/null differ diff --git a/doc/src/Eqs/improper_inversion_harmonic.tex b/doc/src/Eqs/improper_inversion_harmonic.tex deleted file mode 100644 index a1607a1149..0000000000 --- a/doc/src/Eqs/improper_inversion_harmonic.tex +++ /dev/null @@ -1,15 +0,0 @@ -\documentclass[12pt]{article} - -\pagestyle{empty} -\begin{document} - -$$ - E = K \left(\theta - \theta_0\right)^2 -$$ - -\end{document} - -%%% Local Variables: -%%% mode: latex -%%% TeX-master: t -%%% End: diff --git a/doc/src/Eqs/improper_ring.jpg b/doc/src/Eqs/improper_ring.jpg deleted file mode 100644 index 81a79f7ce5..0000000000 Binary files a/doc/src/Eqs/improper_ring.jpg and /dev/null differ diff --git a/doc/src/Eqs/improper_ring.tex b/doc/src/Eqs/improper_ring.tex deleted file mode 100644 index b7bd553c37..0000000000 --- a/doc/src/Eqs/improper_ring.tex +++ /dev/null @@ -1,12 +0,0 @@ -\documentclass[12pt]{article} - -\begin{document} - -\begin{eqnarray*} - E & = &\frac{1}{6} K \left(\Delta_{ijl} + \Delta_{ijk} + \Delta_{kjl} \right)^6 \\ - \Delta_{ijl} & = & \cos{\theta_{ijl} - \cos{\theta_0}} \\ - \Delta_{ijk} & = & \cos{\theta_{ijk} - \cos{\theta_0}} \\ - \Delta_{kjl} & = & \cos{\theta_{kjl} - \cos{\theta_0}} -\end{eqnarray*} - -\end{document} diff --git a/doc/src/Eqs/improper_sqdistharm.jpg b/doc/src/Eqs/improper_sqdistharm.jpg deleted file mode 100644 index b1a84e995e..0000000000 Binary files a/doc/src/Eqs/improper_sqdistharm.jpg and /dev/null differ diff --git a/doc/src/Eqs/improper_sqdistharm.tex b/doc/src/Eqs/improper_sqdistharm.tex deleted file mode 100644 index 1b50a309a0..0000000000 --- a/doc/src/Eqs/improper_sqdistharm.tex +++ /dev/null @@ -1,9 +0,0 @@ -\documentclass[12pt]{article} - -\begin{document} -\thispagestyle{empty} -$$ - E = K (d^2 - d_0^2)^2 -$$ - -\end{document} diff --git a/doc/src/Eqs/improper_umbrella.jpg b/doc/src/Eqs/improper_umbrella.jpg deleted file mode 100644 index 8d6cbf81e7..0000000000 Binary files a/doc/src/Eqs/improper_umbrella.jpg and /dev/null differ diff --git a/doc/src/Eqs/improper_umbrella.tex b/doc/src/Eqs/improper_umbrella.tex deleted file mode 100644 index 53dfbf31d5..0000000000 --- a/doc/src/Eqs/improper_umbrella.tex +++ /dev/null @@ -1,13 +0,0 @@ -\documentclass[12pt]{article} - -\begin{document} - -$$ -E=\frac{1}{2}K\left( \frac{1}{\sin\omega_0}\right) ^2 \left( \cos\omega - \cos\omega_0\right) ^2 \qquad \omega_0 \neq 0^o -$$ - -$$ -E=K\left( 1-cos\omega\right) \qquad \omega_0 = 0^o -$$ - -\end{document} diff --git a/doc/src/dihedral_class2.rst b/doc/src/dihedral_class2.rst index 427dc299a3..b6afa130f0 100644 --- a/doc/src/dihedral_class2.rst +++ b/doc/src/dihedral_class2.rst @@ -38,15 +38,15 @@ The *class2* dihedral style uses the potential .. math:: - E & = & E_d + E_{mbt} + E_{ebt} + E_{at} + E_{aat} + E_{bb13} \\ - E_d & = & \sum_{n=1}^{3} K_n [ 1 - \cos (n \phi - \phi_n) ] \\ - E_{mbt} & = & (r_{jk} - r_2) [ A_1 \cos (\phi) + A_2 \cos (2\phi) + A_3 \cos (3\phi) ] \\ - E_{ebt} & = & (r_{ij} - r_1) [ B_1 \cos (\phi) + B_2 \cos (2\phi) + B_3 \cos (3\phi) ] + \\ - & & (r_{kl} - r_3) [ C_1 \cos (\phi) + C_2 \cos (2\phi) + C_3 \cos (3\phi) ] \\ - E_{at} & = & (\theta_{ijk} - \theta_1) [ D_1 \cos (\phi) + D_2 \cos (2\phi) + D_3 \cos (3\phi) ] + \\ - & & (\theta_{jkl} - \theta_2) [ E_1 \cos (\phi) + E_2 \cos (2\phi) + E_3 \cos (3\phi) ] \\ - E_{aat} & = & M (\theta_{ijk} - \theta_1) (\theta_{jkl} - \theta_2) \cos (\phi) \\ - E_{bb13} & = & N (r_{ij} - r_1) (r_{kl} - r_3) + E = & E_d + E_{mbt} + E_{ebt} + E_{at} + E_{aat} + E_{bb13} \\ + E_d = & \sum_{n=1}^{3} K_n [ 1 - \cos (n \phi - \phi_n) ] \\ + E_{mbt} = & (r_{jk} - r_2) [ A_1 \cos (\phi) + A_2 \cos (2\phi) + A_3 \cos (3\phi) ] \\ + E_{ebt} = & (r_{ij} - r_1) [ B_1 \cos (\phi) + B_2 \cos (2\phi) + B_3 \cos (3\phi) ] + \\ + & (r_{kl} - r_3) [ C_1 \cos (\phi) + C_2 \cos (2\phi) + C_3 \cos (3\phi) ] \\ + E_{at} = & (\theta_{ijk} - \theta_1) [ D_1 \cos (\phi) + D_2 \cos (2\phi) + D_3 \cos (3\phi) ] + \\ + & (\theta_{jkl} - \theta_2) [ E_1 \cos (\phi) + E_2 \cos (2\phi) + E_3 \cos (3\phi) ] \\ + E_{aat} = & M (\theta_{ijk} - \theta_1) (\theta_{jkl} - \theta_2) \cos (\phi) \\ + E_{bb13} = & N (r_{ij} - r_1) (r_{kl} - r_3) where :math:`E_d` is the dihedral term, :math:`E_{mbt}` is a middle-bond-torsion term, diff --git a/doc/src/dihedral_opls.rst b/doc/src/dihedral_opls.rst index 50ad0dcc5a..ef431b92d2 100644 --- a/doc/src/dihedral_opls.rst +++ b/doc/src/dihedral_opls.rst @@ -38,8 +38,8 @@ The *opls* dihedral style uses the potential .. math:: - E = \frac{1}{2} K_1 [1 + \cos(\phi)] + \frac{1}{2} K_2 [1 - \cos(2 \phi)] + - \frac{1}{2} K_3 [1 + \cos(3 \phi)] + \frac{1}{2} K_4 [1 - \cos(4 \phi)] + E = & \frac{1}{2} K_1 [1 + \cos(\phi)] + \frac{1}{2} K_2 [1 - \cos(2 \phi)] + \\ + & \frac{1}{2} K_3 [1 + \cos(3 \phi)] + \frac{1}{2} K_4 [1 - \cos(4 \phi)] Note that the usual 1/2 factor is not included in the K values. diff --git a/doc/src/dihedral_spherical.rst b/doc/src/dihedral_spherical.rst index 03dfcecb38..795bd5ad4d 100644 --- a/doc/src/dihedral_spherical.rst +++ b/doc/src/dihedral_spherical.rst @@ -32,10 +32,10 @@ The *spherical* dihedral style uses the potential: .. math:: - E(\phi,\theta_1,\theta_2) & = & \sum_{i=1}^N\nolimits\ C_i\ \Phi_i(\phi)\ \Theta_{1i}(\theta_1)\ \Theta_{2i}(\theta_2) \\ - \Phi_{i}(\phi) & = & u_i - \mathrm{cos}((\phi - a_i)K_i) \\ - \Theta_{1i}(\theta_1) & = & v_i - \mathrm{cos}((\theta_1-b_i)L_i) \\ - \Theta_{2i}(\theta_2) & = & w_i - \mathrm{cos}((\theta_2-c_i)M_i) + E(\phi,\theta_1,\theta_2) & = \sum_{i=1}^N\nolimits\ C_i\ \Phi_i(\phi)\ \Theta_{1i}(\theta_1)\ \Theta_{2i}(\theta_2) \\ + \Phi_{i}(\phi) & = u_i - \mathrm{cos}((\phi - a_i)K_i) \\ + \Theta_{1i}(\theta_1) & = v_i - \mathrm{cos}((\theta_1-b_i)L_i) \\ + \Theta_{2i}(\theta_2) & = w_i - \mathrm{cos}((\theta_2-c_i)M_i) For this dihedral style, the energy can be any function that combines the diff --git a/doc/src/dihedral_table_cut.rst b/doc/src/dihedral_table_cut.rst index a369d29302..529a911e0b 100644 --- a/doc/src/dihedral_table_cut.rst +++ b/doc/src/dihedral_table_cut.rst @@ -68,8 +68,8 @@ cutoff function: .. math:: - f(\theta) & = & K \qquad\qquad\qquad\qquad\qquad\qquad \theta < \theta_1 \\ - f(\theta) & = & K \left(1-\frac{(\theta - \theta_1)^2}{(\theta_2 - \theta_1)^2}\right) \qquad \theta_1 < \theta < \theta_2 + f(\theta) & = K \qquad\qquad\qquad\qquad\qquad\qquad \theta < \theta_1 \\ + f(\theta) & = K \left(1-\frac{(\theta - \theta_1)^2}{(\theta_2 - \theta_1)^2}\right) \qquad \theta_1 < \theta < \theta_2 The cutoff specifies an prefactor to the cutoff function. While this value diff --git a/doc/src/improper_class2.rst b/doc/src/improper_class2.rst index 9bb5c88618..260348a668 100644 --- a/doc/src/improper_class2.rst +++ b/doc/src/improper_class2.rst @@ -32,24 +32,33 @@ Description The *class2* improper style uses the potential -.. image:: Eqs/improper_class2.jpg - :align: center +.. math:: -where Ei is the improper term and Eaa is an angle-angle term. The 3 X -terms in Ei are an average over 3 out-of-plane angles. + E = & E_i + E_{aa} \\ + E_i = & K [ \frac{\chi_{ijkl} + \chi_{kjli} + \chi_{ljik}}{3} - \chi_0 ]^2 \\ + E_{aa} = & M_1 (\theta_{ijk} - \theta_1) (\theta_{kjl} - \theta_3) + \\ + & M_2 (\theta_{ijk} - \theta_1) (\theta_{ijl} - \theta_2) + \\ + & M_3 (\theta_{ijl} - \theta_2) (\theta_{kjl} - \theta_3) + + +where :math:`E_i` is the improper term and :math:`E_{aa}` is an +angle-angle term. The 3 :math:`\chi` terms in :math:`E_i` are an +average over 3 out-of-plane angles. The 4 atoms in an improper quadruplet (listed in the data file read by -the :doc:`read_data ` command) are ordered I,J,K,L. X\_IJKL -refers to the angle between the plane of I,J,K and the plane of J,K,L, -and the bond JK lies in both planes. Similarly for X\_KJLI and X\_LJIK. +the :doc:`read_data ` command) are ordered I,J,K,L. +:math:`\chi_{ijkl}` refers to the angle between the plane of I,J,K and +the plane of J,K,L, and the bond JK lies in both planes. Similarly for +:math:`\chi_{kjli}` and :math:`\chi_{ljik}`. Note that atom J appears in the common bonds (JI, JK, JL) of all 3 X terms. Thus J (the 2nd atom in the quadruplet) is the atom of -symmetry in the 3 X angles. +symmetry in the 3 :math:`\chi` angles. -The subscripts on the various theta's refer to different combinations -of 3 atoms (I,J,K,L) used to form a particular angle. E.g. Theta\_IJL -is the angle formed by atoms I,J,L with J in the middle. Theta1, -theta2, theta3 are the equilibrium positions of those angles. Again, +The subscripts on the various :math:`\theta`\ s refer to different +combinations of 3 atoms (I,J,K,L) used to form a particular angle. +E.g. :math:`\theta_{ijl}` is the angle formed by atoms I,J,L with J +in the middle. :math:`\theta_1`, :math:`\theta_2`, :math:`\theta_3` +are the equilibrium positions of those angles. Again, atom J (the 2nd atom in the quadruplet) is the atom of symmetry in the theta angles, since it is always the center atom. @@ -59,34 +68,35 @@ this is not required. See :ref:`(Sun) ` for a description of the COMPASS class2 force field. -Coefficients for the Ei and Eaa formulas must be defined for each +Coefficients for the :math:`E_i` and :math:`E_{aa}` formulas must be +defined for each improper type via the :doc:`improper_coeff ` command as in the example above, or in the data file or restart files read by the :doc:`read_data ` or :doc:`read_restart ` commands. -These are the 2 coefficients for the Ei formula: +These are the 2 coefficients for the :math:`E_i` formula: -* K (energy/radian\^2) -* X0 (degrees) +* :math:`K` (energy/radian\^2) +* :math:`\chi_0` (degrees) -X0 is specified in degrees, but LAMMPS converts it to radians +:math:`\chi_0` is specified in degrees, but LAMMPS converts it to radians internally; hence the units of K are in energy/radian\^2. -For the Eaa formula, each line in a +For the :math:`E_{aa}` formula, each line in a :doc:`improper_coeff ` command in the input script lists -7 coefficients, the first of which is "aa" to indicate they are +7 coefficients, the first of which is *aa* to indicate they are AngleAngle coefficients. In a data file, these coefficients should be -listed under a "AngleAngle Coeffs" heading and you must leave out the -"aa", i.e. only list 6 coefficients after the improper type. +listed under a *AngleAngle Coeffs* heading and you must leave out the +*aa*, i.e. only list 6 coefficients after the improper type. -* aa -* M1 (energy/distance) -* M2 (energy/distance) -* M3 (energy/distance) -* theta1 (degrees) -* theta2 (degrees) -* theta3 (degrees) +* *aa* +* :math:`M_1` (energy/distance) +* :math:`M_2` (energy/distance) +* :math:`M_3` (energy/distance) +* :math:`\theta_1` (degrees) +* :math:`\theta_2` (degrees) +* :math:`\theta_3` (degrees) The theta values are specified in degrees, but LAMMPS converts them to radians internally; hence the units of M are in energy/radian\^2. diff --git a/doc/src/improper_cossq.rst b/doc/src/improper_cossq.rst index 7428a7f02e..e8f2032d80 100644 --- a/doc/src/improper_cossq.rst +++ b/doc/src/improper_cossq.rst @@ -28,18 +28,20 @@ Description The *cossq* improper style uses the potential -.. image:: Eqs/improper_cossq.jpg - :align: center +.. math:: -where x is the improper angle, x0 is its equilibrium value, and K is a -prefactor. + E = \frac{1}{2} K \cos^2{\left(\chi - \chi_0\right)} + + +where :math:`\chi` is the improper angle, :math:`\chi_0` is its +equilibrium value, and :math:`K` is a prefactor. If the 4 atoms in an improper quadruplet (listed in the data file read -by the :doc:`read_data ` command) are ordered I,J,K,L then X -is the angle between the plane of I,J,K and the plane of J,K,L. +by the :doc:`read_data ` command) are ordered I,J,K,L then +:math:`\chi` is the angle between the plane of I,J,K and the plane of J,K,L. Alternatively, you can think of atoms J,K,L as being in a plane, and -atom I above the plane, and X as a measure of how far out-of-plane I -is with respect to the other 3 atoms. +atom I above the plane, and :math:`\chi` as a measure of how far +out-of-plane I is with respect to the other 3 atoms. Note that defining 4 atoms to interact in this way, does not mean that bonds necessarily exist between I-J, J-K, or K-L, as they would in a @@ -52,8 +54,8 @@ above, or in the data file or restart files read by the :doc:`read_data ` or :doc:`read_restart ` commands: -* K (energy) -* X0 (degrees) +* :math:`K` (energy) +* :math:`\chi_0` (degrees) ---------- diff --git a/doc/src/improper_cvff.rst b/doc/src/improper_cvff.rst index a6938c7a12..c1086797e2 100644 --- a/doc/src/improper_cvff.rst +++ b/doc/src/improper_cvff.rst @@ -31,8 +31,10 @@ Description The *cvff* improper style uses the potential -.. image:: Eqs/improper_cvff.jpg - :align: center +.. math:: + + E = K [1 + d \cos (n \phi) ] + where phi is the improper dihedral angle. @@ -54,9 +56,9 @@ above, or in the data file or restart files read by the :doc:`read_data ` or :doc:`read_restart ` commands: -* K (energy) -* d (+1 or -1) -* n (0,1,2,3,4,6) +* :math:`K` (energy) +* :math:`d` (+1 or -1) +* :math:`n` (0,1,2,3,4,6) ---------- diff --git a/doc/src/improper_distance.rst b/doc/src/improper_distance.rst index c170be50f2..c29386e010 100644 --- a/doc/src/improper_distance.rst +++ b/doc/src/improper_distance.rst @@ -22,8 +22,10 @@ Description The *distance* improper style uses the potential -.. image:: Eqs/improper_dist-1.jpg - :align: center +.. math:: + + E = K_2 d^2 + K_4 d^4 + where d is the distance between the central atom and the plane formed by the other three atoms. If the 4 atoms in an improper quadruplet @@ -43,8 +45,8 @@ The following coefficients must be defined for each improper type via the improper\_coeff command as in the example above, or in the data file or restart files read by the read\_data or read\_restart commands: -* K\_2 (energy/distance\^2) -* K\_4 (energy/distance\^4) +* :math:`K_2` (energy/distance\^2) +* :math:`K_4` (energy/distance\^4) ---------- diff --git a/doc/src/improper_distharm.rst b/doc/src/improper_distharm.rst index 09f835f575..585285b88e 100644 --- a/doc/src/improper_distharm.rst +++ b/doc/src/improper_distharm.rst @@ -22,23 +22,25 @@ Description The *distharm* improper style uses the potential -.. image:: Eqs/improper_distharm.jpg - :align: center +.. math:: + + E = K (d - d_0)^2 + where d is the oriented distance between the central atom and the plane formed by the other three atoms. If the 4 atoms in an improper quadruplet (listed in the data file read by the :doc:`read_data ` command) are ordered I,J,K,L then the L-atom is assumed to be the central atom. Note that this is different from the convention used -in the improper\_style distance. The distance d is oriented and can take -on negative values. This may lead to unwanted behavior if d0 is not equal to zero. +in the improper\_style distance. The distance :math:`d` is oriented and can take +on negative values. This may lead to unwanted behavior if :math:`d_0` is not equal to zero. The following coefficients must be defined for each improper type via the improper\_coeff command as in the example above, or in the data file or restart files read by the read\_data or read\_restart commands: -* K (energy/distance\^2) -* d0 (distance) +* :math:`K` (energy/distance\^2) +* :math:`d_0` (distance) ---------- diff --git a/doc/src/improper_fourier.rst b/doc/src/improper_fourier.rst index 1cc72c7319..734aa75eb9 100644 --- a/doc/src/improper_fourier.rst +++ b/doc/src/improper_fourier.rst @@ -28,8 +28,10 @@ Description The *fourier* improper style uses the following potential: -.. image:: Eqs/improper_fourier.jpg - :align: center +.. math:: + + E = K [C_0 + C_1 \cos ( \omega) + C_2 \cos( 2 \omega) ] + where K is the force constant, C0, C1, C2 are dimensionless coefficients, and omega is the angle between the IL axis and the IJK plane: @@ -45,10 +47,10 @@ above, or in the data file or restart files read by the :doc:`read_data ` or :doc:`read_restart ` commands: -* K (energy) -* C0 (unitless) -* C1 (unitless) -* C2 (unitless) +* :math:`K` (energy) +* :math:`C_0` (unitless) +* :math:`C_1` (unitless) +* :math:`C_2` (unitless) * all (0 or 1, optional) diff --git a/doc/src/improper_harmonic.rst b/doc/src/improper_harmonic.rst index 6c9496dc42..666bd425e8 100644 --- a/doc/src/improper_harmonic.rst +++ b/doc/src/improper_harmonic.rst @@ -34,18 +34,22 @@ Description The *harmonic* improper style uses the potential -.. image:: Eqs/improper_harmonic.jpg - :align: center +.. math:: -where X is the improper angle, X0 is its equilibrium value, and K is a -prefactor. Note that the usual 1/2 factor is included in K. + E = K (\chi - \chi_0)^2 + + +where :math:`\chi` is the improper angle, :math:`\chi_0` is its equilibrium +value, and :math:`K` is a prefactor. Note that the usual 1/2 factor is +included in :math:`K`. If the 4 atoms in an improper quadruplet (listed in the data file read -by the :doc:`read_data ` command) are ordered I,J,K,L then X +by the :doc:`read_data ` command) are ordered I,J,K,L then +:math:`\chi` is the angle between the plane of I,J,K and the plane of J,K,L. Alternatively, you can think of atoms J,K,L as being in a plane, and -atom I above the plane, and X as a measure of how far out-of-plane I -is with respect to the other 3 atoms. +atom I above the plane, and :math:`\chi` as a measure of how far out-of-plane +I is with respect to the other 3 atoms. Note that defining 4 atoms to interact in this way, does not mean that bonds necessarily exist between I-J, J-K, or K-L, as they would in a @@ -58,10 +62,10 @@ above, or in the data file or restart files read by the :doc:`read_data ` or :doc:`read_restart ` commands: -* K (energy/radian\^2) -* X0 (degrees) +* :math:`K` (energy/radian\^2) +* :math:`\chi_0` (degrees) -X0 is specified in degrees, but LAMMPS converts it to radians +:math:`\chi_0` is specified in degrees, but LAMMPS converts it to radians internally; hence the units of K are in energy/radian\^2. diff --git a/doc/src/improper_hybrid.rst b/doc/src/improper_hybrid.rst index 75bc592c68..aac53bbb66 100644 --- a/doc/src/improper_hybrid.rst +++ b/doc/src/improper_hybrid.rst @@ -38,9 +38,9 @@ In the improper\_coeff command, the first coefficient sets the improper style and the remaining coefficients are those appropriate to that style. In the example above, the 2 improper\_coeff commands would set impropers of improper type 1 to be computed with a *harmonic* -potential with coefficients 120.0, 30 for K, X0. Improper type 2 -would be computed with a *cvff* potential with coefficients 20.0, -1, -2 for K, d, n. +potential with coefficients 120.0, 30 for :math:`K`, :math:`\chi_0`. +Improper type 2 would be computed with a *cvff* potential with coefficients +20.0, -1, 2 for K, d, and n, respectively. If the improper *class2* potential is one of the hybrid styles, it requires additional AngleAngle coefficients be specified in the data @@ -67,7 +67,8 @@ MOLECULE package. See the :doc:`Build package ` doc page for more info. Unlike other improper styles, the hybrid improper style does not store -improper coefficient info for individual sub-styles in a :doc:`binary restart files `. Thus when restarting a simulation from a +improper coefficient info for individual sub-styles in a :doc:`binary restart files `. +Thus when restarting a simulation from a restart file, you need to re-specify improper\_coeff commands. Related commands diff --git a/doc/src/improper_inversion_harmonic.rst b/doc/src/improper_inversion_harmonic.rst index 735b46a1f6..86cdbc7c8f 100644 --- a/doc/src/improper_inversion_harmonic.rst +++ b/doc/src/improper_inversion_harmonic.rst @@ -26,19 +26,21 @@ Description The *inversion/harmonic* improper style follows the Wilson-Decius out-of-plane angle definition and uses an harmonic potential: -.. image:: Eqs/improper_inversion_harmonic.jpg - :align: center +.. math:: -where K is the force constant and omega is the angle evaluated for -all three axis-plane combinations centered around the atom I. For -the IL axis and the IJK plane omega looks as follows: + E = K \left(\omega - \omega_0\right)^2 + + +where :math:`K` is the force constant and :math:`\omega` is the angle +evaluated for all three axis-plane combinations centered around the atom I. +For the IL axis and the IJK plane :math:`\omega` looks as follows: .. image:: JPG/umbrella.jpg :align: center Note that the *inversion/harmonic* angle term evaluation differs to the :doc:`improper_umbrella ` due to the cyclic -evaluation of all possible angles omega. +evaluation of all possible angles :math:`\omega`. The following coefficients must be defined for each improper type via the :doc:`improper_coeff ` command as in the example @@ -46,12 +48,12 @@ above, or in the data file or restart files read by the :doc:`read_data ` or :doc:`read_restart ` commands: -* K (energy) -* omega0 (degrees) +* :math:`K` (energy) +* :math:`\omega_0` (degrees) -If omega0 = 0 the potential term has a minimum for the planar -structure. Otherwise it has two minima at +/- omega0, with a barrier -in between. +If :math:`\omega_0 = 0` the potential term has a single minimum for +the planar structure. Otherwise it has two minima at +/- :math:`\omega_0`, +with a barrier in between. ---------- diff --git a/doc/src/improper_ring.rst b/doc/src/improper_ring.rst index 3f6bec718b..1554439380 100644 --- a/doc/src/improper_ring.rst +++ b/doc/src/improper_ring.rst @@ -28,11 +28,17 @@ Description The *ring* improper style uses the potential -.. image:: Eqs/improper_ring.jpg - :align: center +.. math:: -where K is a prefactor, theta is the angle formed by the atoms -specified by (i,j,k,l) indices and theta0 its equilibrium value. + E = &\frac{1}{6} K \left(\Delta_{ijl} + \Delta_{ijk} + \Delta_{kjl} \right)^6 \\ + \Delta_{ijl} = & \cos{\theta_{ijl} - \cos{\theta_0}} \\ + \Delta_{ijk} = & \cos{\theta_{ijk} - \cos{\theta_0}} \\ + \Delta_{kjl} = & \cos{\theta_{kjl} - \cos{\theta_0}} + + +where :math:`K` is a prefactor, :math:`\theta` is the angle formed by +the atoms specified by (i,j,k,l) indices and :math:`\theta_0` its +equilibrium value. If the 4 atoms in an improper quadruplet (listed in the data file read by the :doc:`read_data ` command) are ordered i,j,k,l then @@ -56,8 +62,8 @@ above, or in the data file or restart files read by the :doc:`read_data ` or :doc:`read_restart ` commands: -* K (energy) -* theta0 (degrees) +* :math:`K` (energy) +* :math:`\theta_0` (degrees) ---------- diff --git a/doc/src/improper_sqdistharm.rst b/doc/src/improper_sqdistharm.rst index ec1e68959a..b9bf53ba34 100644 --- a/doc/src/improper_sqdistharm.rst +++ b/doc/src/improper_sqdistharm.rst @@ -22,10 +22,12 @@ Description The *sqdistharm* improper style uses the potential -.. image:: Eqs/improper_sqdistharm.jpg - :align: center +.. math:: -where d is the distance between the central atom and the plane formed + E = K (d^2 - {d_0}^2)^2 + + +where :math:`d` is the distance between the central atom and the plane formed by the other three atoms. If the 4 atoms in an improper quadruplet (listed in the data file read by the :doc:`read_data ` command) are ordered I,J,K,L then the L-atom is assumed to be the @@ -36,10 +38,10 @@ The following coefficients must be defined for each improper type via the improper\_coeff command as in the example above, or in the data file or restart files read by the read\_data or read\_restart commands: -* K (energy/distance\^4) -* d0\^2 (distance\^2) +* :math:`K` (energy/distance\^4) +* :math:`{d_0}^2` (distance\^2) -Note that d0\^2 (in units distance\^2) has be provided and not d0. +Note that :math:`{d_0}^2` (in units distance\^2) has be provided and not :math:`d_0`. ---------- diff --git a/doc/src/improper_umbrella.rst b/doc/src/improper_umbrella.rst index 8515a94598..33f549ffc8 100644 --- a/doc/src/improper_umbrella.rst +++ b/doc/src/improper_umbrella.rst @@ -30,18 +30,21 @@ The *umbrella* improper style uses the following potential, which is commonly referred to as a classic inversion and used in the :doc:`DREIDING ` force field: -.. image:: Eqs/improper_umbrella.jpg - :align: center +.. math:: -where K is the force constant and omega is the angle between the IL + E = & \frac{1}{2}K\left( \frac{1}{\sin\omega_0}\right) ^2 \left( \cos\omega - \cos\omega_0\right) ^2 \qquad \omega_0 \neq 0^o \\ + E = & K\left( 1-cos\omega\right) \qquad \omega_0 = 0^o + + +where :math:`K` is the force constant and :math:`\omega` is the angle between the IL axis and the IJK plane: .. image:: JPG/umbrella.jpg :align: center -If omega0 = 0 the potential term has a minimum for the planar -structure. Otherwise it has two minima at +/- omega0, with a barrier -in between. +If :math:`\omega_0 = 0` the potential term has a minimum for the planar +structure. Otherwise it has two minima at :math:`\omega +/- \omega_0`, +with a barrier in between. See :ref:`(Mayo) ` for a description of the DREIDING force field. @@ -51,8 +54,8 @@ above, or in the data file or restart files read by the :doc:`read_data ` or :doc:`read_restart ` commands: -* K (energy) -* omega0 (degrees) +* :math:`K` (energy) +* :math:`\omega_0` (degrees) ----------