User:Ivan Shmakov/Potopt 2014

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


Morse potential[edit]

The potential at the 𝑖th particle π‘Ÿπ‘– is 𝑉𝑖 and is the sum of potentials 𝑉𝑖𝑗 created by selected (𝐷eπ‘–π‘—βŸβ‰ βŸ0) other individual particles. (Where 𝐷e𝑖𝑗, 𝜌e𝑖𝑗, and π‘Že𝑖𝑗 are per-pair parameters.)

Failed to parse (unknown function "\begin"): \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 w:gradient descent method, we derive βˆ‡π‘‰π‘–π‘— as follows.

Failed to parse (unknown function "\begin"): \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}


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

Bond 𝐷e, 10⁻¹⁹ NβŸβ‹…βŸm 𝜌e, 10⁻⁹ m π‘Že, 10¹⁰ m⁻¹
C─C 6.03105 0.1421 2.625