.. _list-of-potential-forms: *********************** List of Potential Forms *********************** .. contents:: :local: .. rubric:: Key to features: The **Features** field in the following potential description may contain the following values: * **potential-form** - can be used in the ``[Pair]``\ , ``[EAM-Embed]`` and ``[EAM-Density]`` sections of a :ref:`potable ` input file. * **potential-function** - can also be used as a function in sections of input files that accept mathematical expressions. * **deriv** - potential form provides an analytical derivative with respect to separation. * **deriv2** - provides an analytical second derivative with respect to separation. .. _potform-bornmayer: Born-Mayer (bornmayer) ====================== .. math :: V(r_{ij}) = A \exp \left( \frac{-r_{ij}}{\rho} \right) :potable signature: `as.bornmayer` :math:`A` :math:`\rho` :Features: potential-form, potential-function, deriv, deriv2 .. seealso:: * :ref:`potform-buck` .. _potform-buck: Buckingham (buck) ================= Potential form due to R.A. Buckingham [Buckingham1938]_ .. math :: V(r_{ij}) = A \exp \left( - \frac{r_{ij}}{\rho} \right) - \frac{C}{r_{ij}^6} :potable signature: ``as.buck`` :math:`A` :math:`\rho` :math:`C` :Features: potential-form, potential-function, deriv, deriv2. .. seealso:: * :ref:`potform-bornmayer` * `Wikipedia - Buckingham potential `_ .. _potform-buck4: Buckingham-4 (buck4) ==================== Four-range Buckingham potential due to B. Vessal et al. [Vessal1989]_\, [Vessal1993]_\ . .. math:: V(r_{ij}) = \begin{cases} A \exp(-r_{ij}/\rho) , & 0 \leq r_{ij} \leq r_\text{detach}\\ a_0 + a_1 r_{ij} +a_2 r_{ij}^2+a_3 r_{ij}^3+a_4 r_{ij}^4+a_5 r_{ij}^5, & r_\text{detach} < r_{ij} < r_\text{min}\\ b_0 +b_1 r_{ij}+b_2 r_{ij}^2+b_3 r_{ij}^3 , & r_\text{min} \leq r_{ij} < r_\text{attach}\\ -\frac{C}{r_{ij}^6} , & r_{ij} \geq r_\text{attach}\\ \end{cases} In other words this is a Buckingham potential in which the :ref:`Born-Mayer ` component acts at small separations and the disprsion term acts at larger separation. These two parts are linked by a fifth then third order polynomial (with a minimum formed in the spline at :math:`r_\text{min}`). The spline parameters (:math:`a_{0\ldots 5}` and :math:`b_{0\ldots 3}`) are subject to the constraints that :math:`V(r_{ij})`, first and second derivatives must be equal at the boundary points and the function must have a stationary point at :math:`r_{min}`\ . The spline coefficients are automatically calculated by this potential-form. .. note:: Due to the complexity of calculating the spline-coefficients this potential form does not have an equivalent in the atsim.potentials.potentialfunctions module. :potable signature: ``as.buck4`` :math:`A` :math:`\rho` :math:`C` :math:`r_\text{detach}` :math:`r_\text{min}` :math:`r_\text{attach}` :Features: potential-form, deriv, deriv2 .. seealso:: * :ref:`spline-buck4` * :ref:`aspot-splining` .. _potform-constant: Constant (constant) =================== Potential form that always evaluates to a constant value. .. math:: V(r_{ij}) = C :potable signature: ``as.constant`` C :Features: potential-form, potential-function, deriv, deriv2 .. _potform-coul: Coulomb (coul) ============== Electrostatic interaction between two point charges. .. math :: V(r_{ij}) = \frac{ q_i q_j }{4 \pi \epsilon_0 r_{ij} } .. note:: Constant value appropriate for :math:`r_{ij}` in angstroms and energy in eV. :potable signature: ``as.coul`` :math:`q_i` :math:`q_j` :Features: potential-form, potential-function, deriv, deriv2 .. _potform-exponential: Exponential (exponential) ========================= General exponential form. .. math :: V(r_{ij}) = A r_{ij}^n :potable signature: ``as.exponential`` :math:`A` :math:`n` :Features: potential-form, potential-function, deriv, deriv2 .. _potform-exp_spline: Exponential Spline (exp_spline) =============================== Exponential spline function (as used in splining routines). .. math:: V(r_{ij}) = \exp \left( B_0 + B_1 r_{ij} + B_2 r_{ij}^2 + B_3 r_{ij}^3 + B_4 r_{ij}^4 + B_5 r_{ij}^5 \right) + C Where :math:`B_0`\ , :math:`B_1`\ , :math:`B_2`\ , :math:`B_3`\ , :math:`B_4`\ , :math:`B_5`\ , :math:`C` are spline coefficients. :potable signature: ``as.exp_spline`` :math:`B_0` :math:`B_1` :math:`B_2` :math:`B_3` :math:`B_4` :math:`B_5` :math:`C` :Features: potential-form, potential-function, deriv, deriv2 .. seealso:: * :ref:`spline-exp_spline` * :ref:`aspot-splining` .. _potform-hbnd: Hydrogen Bond 12-10 (hbnd) ========================== .. math:: V(r_{ij}) = \frac{A}{r_{ij}^{12}} - \frac{B}{r_{ij}^{10}} :portable signature: ``as.hbnd`` :math:`A` :math:`B` :Features: potential-form, potential-function, deriv, deriv2 .. _potform-lj: Lennard-Jones 12-6 (lj) ======================= Potential form first proposed by John Lennard-Jones in 1924 [Lennard-Jones1924]_\ . .. math :: V(r_{ij}) = 4 \epsilon \left( \frac{\sigma^{12}}{r_{ij}^{12}} - \frac{\sigma^6}{r_{ij}^6} \right) :math:`\epsilon` defines depth of potential well and :math:`\sigma` is the separation at which :math:`V(r_{ij})` is zero. :potable signature: ``as.lj`` :math:`\epsilon` :math:`\sigma` :Features: potential-form, potential-function, deriv, deriv2 .. seealso:: * `Wikipedia - Lennard-Jones potential `_ .. _potform-morse: Morse (morse) ============= .. math :: V(r_{ij}) = D \left[ \exp \left( -2 \gamma (r_{ij} - r_*) \right) - 2 \exp \left( -\gamma (r - r_*) \right) \right] :math:`-D` is the potential well depth at an equilibrium separation of :math:`r_*`\ . :potable signature: ``as.morse`` :math:`\gamma` :math:`r_*` :math:`D` :Features: potential-form, potential-function, deriv, deriv2 .. seealso:: * `Wikipedia - Morse potential `_ .. _potform-polynomial: Polynomial (polynomial) ======================= Polynomial of arbitrary order. .. math:: V(r_{ij}) = C_0 + C_1 r_{ij} + C_2 r_{ij}^2 + \dots + C_n r_{ij}^n This function accepts a variable number of arguments which are :math:`C_0, C_1, \dots, C_n` respectively. :potable signatures: ``as.polynomial`` :math:`C_0 ... C_n` :Features: potential-form, potential-function, deriv, deriv2 .. seealso:: * :ref:`potable-published-spline-coefficients` .. _potform-sqrt: Square Root (sqrt) ================== Potential function is: .. math:: U(r_{ij}) = G\sqrt{r_{ij}} :potable signature: ``as.sqrt`` :math:`G` :Features: potential-form, potential-function, deriv, deriv2 .. _potform-tang-toennies: Tang-Toennies (tang_toennies) ============================= This potential form was derived to describe the Van der Waal's interactions between the noble gases (He to Rn) by Tang and Toennies [Tang2003]_\ . This has the following form: .. math:: V(r) = A \exp(-br) - \sum_{n=3}^N f_{2N} (bR) \frac{C_{2N}}{R^{2N}} Where: .. math:: f_{2N}(x) = 1- \exp(-x) \sum_{k=0}^{2n} \frac{x^k}{k!} :potable signature: ``as.tang_toennies`` :math:`A` :math:`b` :math:`C_6` :math:`C_8` :math:`C_{10}` :Features: potential-form, potential-function, deriv, deriv2 .. _potform-zero: Zero (zero) =========== Potential form which returns zero for all separations. .. math:: V(r) = 0 :potable signature: ``as.zero`` :Features: potential-form, potential-function, deriv, deriv2 .. _potform-zbl: Ziegler-Biersack-Littmark (zbl) =============================== Ziegler-Biersack-Littmark screened nuclear repulsion for describing high energy interactions [Ziegler2015]_\ . .. math:: V(r) & = \; & \frac{1}{4\pi\epsilon_0} \frac{Z_1}{Z_2} \phi(r/a) + S(r) \\ a & = & \frac{0.46850}{Z_i^{0.23} + Z_j^{0.23}} \\ \phi(x) & = & 0.18175 \exp(-3.19980x) \\ & & \quad + 0.50986 \exp(-0.94229x) \\ & & \quad + 0.28022\exp(-0.40290x) \\ & & \quad + 0.02817\exp(-0.20162x) \\ Where :math:`Z_i` and :math:`Z_j` are the atomic numbers of two species. :potable signature: ``as.zbl`` :math:`Z_i` :math:`Z_j` :Features: potential-form, potential-function, deriv, deriv2