====== Analytic potential functions ======
----
The following analytic functions are currently implemented in //potfit//.

For details see the ''functions.h'' and ''functions.c'' files.

If you want to add other analytic potentials see [[models:adding_analytic_functions|this guide]].

Each function is given in the following form:
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^ identifier | # of parameters | order of parameters | reference |
|  functional form  ||||

===== General potentials =====

Most of these potentials do not have any special properties. They may be used as regular pair
potentials as well as for advanced potentials like EAM, ADP or MEAM.

== Basic potentials ==

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^ ''const'' | 1 | $c$ | [none] |
| $$V(r)=c$$ ||||

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^ ''lj'' | 2 |$\varepsilon, \sigma$ | [[http://dx.doi.org/doi:10.1098/rspa.1924.0082|Link]] |
| $$V(r)=4\varepsilon\left[\left(\frac{\sigma}{r}\right)^{12}-\left(\frac{\sigma}{r}\right)^{6}\right]$$ ||||

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^ ''morse'' | 3 |$D_e, a, r_e$ | [[http://dx.doi.org/doi:10.1103PhysRev.34.57|Link]] |
| $$V(r)=D_e \left( \left[ 1-\exp\left(-a(r-r_e)\right) \right]^2 - 1 \right)$$ ||||

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^ ''power'' | 2 | $\alpha,\,\beta$ | [none] |
| $$V(r)=\alpha r^\beta$$ ||||

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^ ''softshell'' | 2 | $\alpha, \beta$ | [none] |
| $$V(r)=\left(\frac{\alpha}{r}\right)^\beta$$ ||||

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^ ''power_decay'' | 2 | $\alpha,\,\beta$ | [none] |
| $$V(r)=\alpha\left(\frac{1}{r}\right)^\beta$$ ||||

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^ ''exp_decay'' | 2 | $\alpha,\,\beta$ | [none] |
| $$V(r)=\alpha \exp\left(-\beta r\right)$$ ||||

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^ ''mexp_decay'' | 3 | $\alpha,\,\beta,\,r_0$ | [none] |
| $$V(r)=\alpha\exp\left(-\beta(r-r_0)\right)$$ ||||

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^ ''exp_plus'' | 3 | $\alpha,\,\beta,\,c$ | [none] |
| $$V(r)=\alpha\exp\left(-\beta r\right)+c$$ ||||

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^ ''sqrt'' | 2 | $\alpha,\,\beta$ | [[http://dx.doi.org/doi:10.1080/01418618408244210|Link]] |
| $$V(r)=\alpha\sqrt{r/\beta}$$ ||||

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^ ''born'' | 5 | $\alpha,\beta,\gamma,\delta,r_0$ | [[https://doi.org/10.1016/S0081-1947(08)60515-9|Link]] |
| $$V(r)=\alpha\exp(\frac{r_0 - r}{\beta})-\frac{\gamma}{r^6}+\frac{\delta}{r^8}$$ ||||

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^ ''harmonic'' | 2 | $\alpha,r_0$ | [none] |
| $$V(r)=\alpha(r-r_0)^2 $$ ||||

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^ ''acosharmonic'' | 2 | $\alpha,r_0$ | [none] |
| $$V(r)=\alpha(\arccos(r)-r_0)^2$$ ||||

== Advanced potentials ==

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^ ''eopp'' | 6 | $C_1,\,\eta_1,\,C_2,\,\eta_2,k,\,\varphi$ | [[http://arxiv.org/abs/0802.2926v2|Link]] |
| $$V(r)=\frac{C_1}{r^{\eta_1}}+\frac{C_2}{r^{\eta_2}}\cos\left(kr+\varphi\right)$$ ||||

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^ ''meopp'' | 7 | $C_1,\,\eta_1,\,C_2,\,\eta_2,\,k,\,\varphi,\,r_0$ | [[http://arxiv.org/abs/0802.2926v2|Link]] |
| $$V(r)=\frac{C_1}{\left(r-r_0\right)^{\eta_1}}+\frac{C_2}{r^{\eta_2}}\cos\left(kr+\varphi\right)$$ ||||

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^ ''eopp_exp'' | 6 | $C_1,\,\eta_1,\,C_2,\,\eta_2,\,k,\,\varphi$ | [[http://arxiv.org/abs/0802.2926v2|Link]] |
| $$V(r)=C_1\exp\left(-\eta_1r\right)+\frac{C_2}{r^{\eta_2}}\cos\left(kr+\varphi\right)$$ ||||

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^ ''ms'' | 3 | $D_e,\,a,\,r_0$ | [[http://dx.doi.org/doi:10.1063/1.1513312|Link]] |
| $$V(r)=D_e\left[\exp\left(a\left(1-\frac{r}{r_0}\right)\right)-2\exp\left(\frac{a}{2}\left(1-\frac{r}{r_0}\right)\right)\right]$$ <wrap round important 440px center>
When using electrostatic interactions this function is defined differently!
</wrap>See section 2. Shifting in [[interactions:coulomb|Coulomb Interactions]] for details. This definition is available under the new name ''ms_non_es''. ||||

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^ ''strmm'' | 5 | $\alpha,\,\beta,\,\gamma,\,\delta,\,r_0$ | [[http://dx.doi.org/10.1103/PhysRevB.50.11996|Link]] |
| $$V(r)=2\alpha\exp\left(-\beta(r-r_0)/2\right) - \gamma\left[1+\delta(r-r_0)\exp\left(-\delta(r-r_0)\right)\right]$$ ||||

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^ ''double_morse'' | 7 | $E_1,\alpha_1,r_0^{(1)},E_2,\alpha_2,r_0^{(2)},\delta$ | [[http://dx.doi.org/doi:10.1557/proc-538-535|Link]] |
|  $$V(r)=E_1M(r,r_0^{(1)},\alpha_1)+E_2M(r,r_0^{(2)},\alpha_2)+\delta $$ $$ M(r,r_0,\alpha) = \exp\left(-2\alpha(r-r_0)\right)-2\exp\left(-\alpha(r-r_0)\right)$$ ||||

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^ ''double_exp'' | 5 | $a,\beta_1,r_0^{(1)},\beta_2,r_0^{(2)}$ | [[http://dx.doi.org/doi:10.1557/proc-538-535|Link]] |
| $$V(r)=a\exp\left(-\beta_1(r-r_0^{(1)})^2\right)+\exp\left(-\beta_2(r-r_0^{(2)})\right)$$ ||||

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^ ''poly_5'' | 5 | $F_0,\,F_2,\,q_1,\,q_2,\,q_3$ | [[http://dx.doi.org/doi:10.1557/proc-538-535|Link]] |
| $$V(r)=F_0+\frac{1}{2}F_2(r-1)^2+\sum_{n=1}^3q_n(r-1)^{n+2}$$ ||||

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^ ''buck'' | 3 | $\alpha, \beta, \gamma$ | [[http://dx.doi.org/doi:10.1098/rspa.1977.0049|Link]] |
| $$V(r)=\alpha\exp\left(-\frac{r}{\beta}\right)-\gamma\left(\frac{\beta}{r}\right)^6$$ <wrap round important 440px center>
When using electrostatic interactions this function is defined differently!
</wrap>See section 2. Shifting in [[interactions:coulomb|Coulomb Interactions]] for details. This definition is available under the new name ''buck_non_es''. ||||

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^ ''kawamura'' | 9 | $z_1,z_2,f_0,a_1,a_2,b_1,b_2,$ $c_1,c_2$ | [[http://dx.doi.org/10.1016/S0925-8388(00)00806-9|Link]] |
| $$V(r)=\frac{z_1z_2}{r}+f_0(b_1+b_2)\exp\left(\frac{a_1+a_2-r}{b_1+b_2}\right)-\frac{c_1c_2}{r^6}$$ ||||

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^ ''kawamura_mix'' | 12 | $z_1,z_2,f_0,a_1,a_2,b_1,b_2,$ $c_1,c_2,D,\beta,r_0$ | [[http://dx.doi.org/10.1016/S0925-8388(00)00806-9|Link]] |
| $$V(r)=\frac{z_1z_2}{r}+f_0(b_1+b_2)\exp\left(\frac{a_1+a_2-r}{b_1+b_2}\right)-\frac{c_1c_2}{r^6}$$ $$+ f_0 D \left[\exp\{-2\beta(r-r_0)\}-2\exp\{-\beta(r-r_0)\}\right]$$ ||||

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^ ''mishin'' | 6 | $A_0,B_0,C_0,r_0,y,\gamma$ | [[http://dx.doi.org/doi:10.1016/j.actamat.2005.05.001|Link]] |
| $$V(r)=A_0(r-r_0)^y\exp\left(-\gamma(r-r_0)\right)\left[1+B_0\exp\left(-\gamma(r-r_0)\right)\right]+C_0$$ ||||

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^ ''gen_lj'' | 5 | $V_0,b_1,b_2,r_1,\delta$ | [[http://dx.doi.org/doi:10.1016/j.actamat.2005.05.001|Link]] |
| $$V(r)=\frac{V_0}{b_2-b_1}\left(\frac{b_2}{(r/r_1)^{b_1}}-\frac{b_1}{(r/r_1)^{b_2}}\right)+\delta$$ ||||

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^ ''gljm'' | 12 | $V_0,b_1,b_2,r_1,\delta,m,A_0,$ $B_0,C_0,r_0,y,\gamma$ | [[http://dx.doi.org/doi:10.1016/j.actamat.2005.05.001|Link]] |
| $$V(r)=\frac{V_0}{b_2-b_1}\left(\frac{b_2}{(r/r_1)^{b_1}}-\frac{b_1}{(r/r_1)^{b_2}}\right)+\delta $$ $$ + m\left[A_0(r-r_0)^y\exp\left(-\gamma(r-r_0)\right)\left[1+B_0\exp\left(-\gamma(r-r_0)\right)\right]+C_0\right]$$ ||||

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^ ''vas'' | 2 | $\alpha, \beta$ | [[http://dx.doi.org/doi:10.1016/0022-3093(94)90351-4|Link]] |
| $$V(r)=\exp\left(\frac{\alpha}{r-\beta}\right)$$ ||||

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^ ''vpair'' | 7 | $\alpha, \beta, \gamma, \delta, a, b, c$ | [[http://dx.doi.org/doi:10.1016/0022-3093(94)90351-4|Link]] |
| $$V(r)=14.4\left[\frac{\alpha}{r^\beta}-\frac{a\delta^2+b\gamma^2}{r^4}\exp\left(-\frac{r}{c}\right)\right]$$ ||||

===== EAM embedding functions =====

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^ ''universal'' | 4 | $F_0,\,p,\,q,\,F_1$ | [[http://dx.doi.org/doi:10.1557/jmr.1989.1195|Link]] |
| $$F(n)=F_0\left[\frac{q}{q-p}n^p-\frac{p}{q-p}n^q\right]+F_1n$$ ||||

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^ ''bjs'' | 3 | $F_0,\,\gamma,\,F_1$ | [[http://dx.doi.org/doi:10.1103/PhysRevB.37.6632|Link]] |
| $$F(n)=F_0\left[1-\gamma\ln n\right]n^\gamma+F_1n$$ ||||

===== EAM transfer functions =====

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^ ''parabola'' | 3 | $a,\,b,\,c$ | [none] |
| $$\rho(r)=ar^2+br+c$$ ||||

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^ ''csw'' | 4 | $a_1,\,a_2,\,\alpha,\,\beta$ | [[http://dx.doi.org/doi:10.1103/PhysRevB.53.14080|Link]] |
| $$\rho(r)=\frac{1+a_1\cos\left(\alpha r\right)+a_2\sin\left(\alpha r\right)}{r^\beta}$$ ||||

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^ ''csw2'' | 4 | $a_1,\,\alpha,\,\varphi,\,\beta$ | [[http://dx.doi.org/doi:10.1103/PhysRevB.53.14080|Link]] |
| $$\rho(r)=\frac{1+a_1\cos\left(\alpha r+\varphi\right)}{r^\beta}$$ ||||

===== Tersoff functions =====

These functions do not correspond directly to the Tersoff potential. 
They only hold the parameters for the potentials.

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^ ''tersoff_pot'' | 11 | $A, B, \lambda, \mu, \gamma, n,c, d, h, S, R$ | [[http://dx.doi.org/doi:10.1103/PhysRevB.39.5566|Link]] |
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^ ''tersoff_mix'' | 2 | $\chi, \omega$ | [[http://dx.doi.org/doi:10.1103/PhysRevB.39.5566|Link]] |
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===== modified Tersoff function =====

This dummy function holds all 16 parameters required to defined a modified Tersoff potential.

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^ ''tersoff_mod_pot'' | 16 | $A, B, \lambda, \mu, \eta, \delta, \alpha, \beta, c_1, c_2, c_3, c_4, c_5, h, R_1, R_2$ | [[http://dx.doi.org/10.1016/j.commatsci.2006.07.013|Link]] |
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===== Stillinger-Weber functions =====

The ''stiweb_2'' functions is equvalent to the $V_2$ potential in the Stillinger-Weber potential model.
The ''stiweb_3'' function, however, only accounts for the exponential function in $V_3$.

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^ ''stiweb_2'' | 6 | $A, B, p, q, \delta, a$ | [[http://dx.doi.org/doi:10.1103/PhysRevB.31.5262|Link]] |
| $$V_2(r) = \left(\frac{A}{r^p}-\frac{B}{r^q}\right)\exp\left(\frac{\delta}{r-a}\right)$$ ||||

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^ ''stiweb_3'' | 2 | $\gamma, b$ | [[http://dx.doi.org/doi:10.1103/PhysRevB.31.5262|Link]] |
| $$V_3^\text{part} = \exp\left(\frac{\gamma}{r-b}\right)$$ ||||

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^ ''stiweb_lambda'' | $N^2(N+1)/2$ | $\lambda_{ijk}$ | [[http://dx.doi.org/doi:10.1103/PhysRevB.31.5262|Link]] |
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