# User:Ivan Shmakov/Potopt 2014

Potopt is a software package to perform (potential) w:energy minimization of a multiple-particle system.

## Morse potential

The potential at the πth particle ππ is ππ and is the sum of potentials πππ created by selected (π·eππββ β0) other individual particles. (Where π·eππ, πeππ, and πeππ are per-pair parameters.)

align} V _i &= \sum _{j \ne i} V _{ij}; \\ V _{ij} &= D _{\mathrm {e}ij} \left[1 - \exp (- a _{ij} (\rho _{ij} - \rho _{\mathrm {e}ij}))\right] ^2;\\ \rho _{ij} &= \sqrt {(\bar r _i - \bar r _j) ^2}.\\ \end {align

In order to apply the w:gradient descent method, we derive βπππ as follows.

align} \nabla V _{ijk} &= D _{\mathrm {e}ij} \frac {\partial \left[1 - \exp (- a _{ij} (\rho _{ij} - \rho _{\mathrm {e}ij}))\right] ^2} {\partial x _{ik}}\\ &= 2 a _{ij} D _{\mathrm {e}ij} \left[1 - \exp (- a _{ij} (\rho _{ij} - \rho _{\mathrm {e}ij}))\right] ^2 \exp (- a _{ij} (\rho _{ij} - \rho _{\mathrm {e}ij})) \frac {x _{ik} - x _{jk}} {\rho _{ij}}\\ &= 2 a _{ij} D _{\mathrm {e}ij} (1 - \kappa _{ij}) \kappa _{ij} \frac {x _{ik} - x _{jk}} {\rho _{ij}};\\ \nabla \bar V _{ij} &= 2 a _{ij} D _{\mathrm {e}ij} (1 - \kappa _{ij}) \kappa _{ij} \frac {\bar \rho _{ij}} {\rho _{ij}};\\ \bar \rho _{ij} &= \bar r _i - \bar r _j.\\ \end {align

### Parameters

The parameters for the CβC w:single bond are as follows.[1]

Bond π·e, 10β»ΒΉβΉ Nβββm πe, 10β»βΉ m πe, 10ΒΉβ° mβ»ΒΉ
CβC 6.03105 0.1421 2.625

## References

1. β Avinash Parashar; Pierre Mertiny (2012-10-26). "Multiscale model to investigate the effect of graphene on the fracture characteristics of graphene/polymer nanocomposites". Nanoscale Research Letters 7 (1): 595. doi:10.1186/1556-276X-7-595. Retrieved on 2014-02-08.