Bond Order potentials - ProkopHapala/FireCore GitHub Wiki

ReaxFF

Simplet Bond-Order potential

$$ O_i = \sum_j {\hat O}(r_{ij}) $$

$$ E_i = K (O_i - O_i^0)^2 $$

$$ F_{ij}(x_i) = \frac{ \partial E_i(r_{ij}) }{ \partial x_i } $$

$$ F_{ij}(x_{ij}) = \frac{ \partial E_i(r_{ij}) }{\partial O_i } \frac{\partial O_i(r_{ij}) }{\partial r_{ij}} \frac{\partial r_{ij} }{\partial x_{ij}} $$

$$ F_{ij}({\vec r}{ij}) = 2 K (O_i - O_i^0) {\hat r}{ij} \frac{ \partial O_i(r_{ij}) }{\partial r_{ij} } $$

Example O(r) of spline for Carbon-Carbon bond:

r      O(r)
-----------
1.10   4.0
1.20   3.0
1.34   2.0
1.53   1.0
1.8    0.4
2.5    0.05
3.8    0.0

visualize with cubic-spline-interpolation web

  • After summing bond order over all atoms we find 4 atoms with highest bond order and calculate angular forces
  • to guarantee that transition is smooth we calculate angles only on those who has bond-order >0.1
  • Bond order operatpor O(r) has cutoff at the same distance as non-bonding potential (Lennard-Jones or Morse) becomes positive

Non-bonded interactions instead of angular forces

  • Consider two carbon atoms (A,B) bonded to central carbon atom C.
  • Consider bond length r_B =1.5
  • Consider L-J interaction distance r_LJ = 2*1.9 = 3.8
  • For angle 120 deg between A-C-B we have r_AB = 2 * sin(60 deg) * r_B = 1.73 * 1.5 = 2.6
  • For atoms which are seriously bonded (e.i. O(r) >= ~0.8) we can sub-tract non-covalent forces paralel to the bond direction. i.e.

$${\vec F}{A} += {\vec F}{AB} - O(r_{AC}) \langle {\vec F}{AB} | {\hat r}{AC} \rangle {\hat r}_{AC}$$