1.. index:: pair_style cosine/squared 2 3pair_style cosine/squared command 4================================= 5 6Syntax 7"""""" 8 9.. code-block:: LAMMPS 10 11 pair_style cosine/squared cutoff 12 13* cutoff = global cutoff for cosine-squared interactions (distance units) 14 15.. code-block:: LAMMPS 16 17 pair_coeff i j eps sigma 18 pair_coeff i j eps sigma cutoff 19 pair_coeff i j eps sigma wca 20 pair_coeff i j eps sigma cutoff wca 21 22* i,j = a particle type 23* eps = interaction strength, i.e. the depth of the potential minimum (energy units) 24* sigma = distance of the potential minimum from 0 25* cutoff = the cutoff distance for this pair type, if different from global (distance units) 26* wca = if specified a Weeks-Chandler-Andersen potential (with eps strength and minimum at sigma) is added, otherwise not 27 28Examples 29"""""""" 30 31.. code-block:: LAMMPS 32 33 pair_style cosine/squared 3.0 34 pair_coeff * * 1.0 1.3 35 pair_coeff 1 3 1.0 1.3 2.0 36 pair_coeff 1 3 1.0 1.3 wca 37 pair_coeff 1 3 1.0 1.3 2.0 wca 38 39Description 40""""""""""" 41 42Style *cosine/squared* computes a potential of the form 43 44.. math:: 45 46 E = 47 \begin{cases} 48 -\epsilon& \quad r < \sigma \\ 49 -\epsilon\cos\left(\frac{\pi\left(r - \sigma\right)}{2\left(r_c - \sigma\right)}\right)^2&\quad \sigma \leq r < r_c \\ 50 0& \quad r \geq r_c 51 \end{cases} 52 53between two point particles, where (:math:`\sigma, -\epsilon`) is the 54location of the (rightmost) minimum of the potential, as explained in 55the syntax section above. 56 57This potential was first used in :ref:`(Cooke) <CKD>` for a coarse-grained lipid 58membrane model. It is generally very useful as a non-specific 59interaction potential because it is fully adjustable in depth and width 60while joining the minimum at (sigma, -epsilon) and zero at (cutoff, 0) 61smoothly, requiring no shifting and causing no related artifacts, tail 62energy calculations etc. This evidently requires *cutoff* to be larger 63than *sigma*\ . 64 65If the *wca* option is used then a Weeks-Chandler-Andersen potential 66:ref:`(Weeks) <WCA>` is added to the above specified cosine-squared potential, 67specifically the following: 68 69.. math:: 70 71 E = \epsilon \left[ \left(\frac{\sigma}{r}\right)^{12} - 72 2\left(\frac{\sigma}{r}\right)^6 + 1\right] 73 , \quad r < \sigma 74 75In this case, and this case only, the :math:`\sigma` parameter can be equal to 76*cutoff* (:math:`\sigma =` cutoff) which will result in ONLY the WCA potential 77being used (and print a warning), so the minimum will be attained at 78(sigma, 0). This is a convenience feature that enables a purely 79repulsive potential to be used without a need to define an additional 80pair style and use the hybrid styles. 81 82The energy and force of this pair style for parameters epsilon = 1.0, 83sigma = 1.0, cutoff = 2.5, with and without the WCA potential, are shown 84in the graphs below: 85 86.. image:: JPG/pair_cosine_squared_graphs.jpg 87 :align: center 88 89---------- 90 91Mixing, shift, table, tail correction, restart, rRESPA info 92""""""""""""""""""""""""""""""""""""""""""""""""""""""""""" 93 94Mixing is not supported for this style. 95 96The *shift*, *table* and *tail* options are not relevant for this style. 97 98This pair style writes its information to :doc:`binary restart files <restart>`, so pair_style and pair_coeff commands do not need 99to be specified in an input script that reads a restart file. 100 101These pair styles can only be used via the *pair* keyword of the 102:doc:`run_style respa <run_style>` command. They do not support the 103*inner*, *middle*, *outer* keywords. 104 105---------- 106 107Restrictions 108"""""""""""" 109 110The *cosine/squared* style is part of the EXTRA-PAIR package. It is only 111enabled if LAMMPS is build with that package. See the :doc:`Build package <Build_package>` page for more info. 112 113Related commands 114"""""""""""""""" 115 116:doc:`pair_coeff <pair_coeff>`, 117:doc:`pair_style lj/cut <pair_lj>` 118 119Default 120""""""" 121 122none 123 124.. _CKD: 125 126**(Cooke)** "Cooke, Kremer and Deserno, Phys. Rev. E, 72, 011506 (2005)" 127 128.. _WCA: 129 130**(Weeks)** "Weeks, Chandler and Andersen, J. Chem. Phys., 54, 5237 (1971)" 131