User:Ivan Shmakov/Potopt 2014

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

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Morse potential
The potential at the 𝑖th particle 𝑟𝑖 is 𝑉𝑖 and is the sum of potentials 𝑉𝑖𝑗 created by selected (𝐷e𝑖𝑗 ≠ 0) other individual particles. (Where 𝐷e𝑖𝑗, 𝜌e𝑖𝑗, and 𝑎e𝑖𝑗 are per-pair parameters.)

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

In order to apply the method, we derive ∇𝑉𝑖𝑗 as follows.

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

Parameters
The parameters for the C─C are as follows.