# Notes¶

The charge-optimized multibody (COMB) potential is an extended version of the Tersoff potential (cf. TersoffSingleTypePotential), which additionally allows to take interatomic charge transfer into account [Yas96], [YSP07].

The total energy is composed of the following contributions.

$E = \sum_i \phi_i + \frac{1}{2} \sum_{j \neq i} \left [ U_{ij}^\mathrm{rep} + U_{ij}^\mathrm{sht} + U_{ij}^\mathrm{ion} + U_{ij}^\mathrm{corr} \right ] + \frac{1}{2} \sum_{j \neq i} \sum_{k\neq i\atop k\neq j} U^\mathrm{triple}_{ijk}$

Using the smoothed cutoff function $$f_C$$ from TersoffSingleTypePotential we can write the potential components as follows:

$\phi_i = J1_i(q_i - q_i^0) + \frac{1}{2} J2_i(q_i - q_i^0)^2 + J3_i (q_i - q_i^0)^3 + J4_i (q_i - q_i^0)^4 \, ,$
$U^\mathrm{rep}_{ij} = f^S_{ij} A_{ij} \exp(-\lambda_{ij} r_{ij}) \left(1 + K_{ij}(1 - r_{ij} / r_{ij}^0)^2\right) \, ,$
$U^\mathrm{sht}_{ij} = -f^S_{ij} b_{ij} B_{ij} \exp(-\mu_{ij} r_{ij}) \, ,$
$U^\mathrm{ion}_{ij} = \frac{f^L_{ij} \eta_i\eta_j q_i q_j}{4\pi\varepsilon_0 r_{ij}} \, ,$
$f^S_{ij} = f_C(r_{ij}, R_{ij}, S_{ij}) \, ,$
$A^S_i = A_i \exp(\lambda_i D_i) \, ,$
$B^S_i = B_i \exp(\mu_i D_i) \left(a^B_i - |b^B_i(q_i - Q^O_i)|^{n^B_i} \right) \, ,$
$D_i = D^U_i + |b^D_i(Q^U_i - q_i)|^{n^D_i} \, ,$
$b^D_i = \frac{(D^L_i - D^U_i)^{1 / n^D_i}}{Q^U_i - Q^L_i} \, ,$
$n^D_i = \frac{\ln(-D^U_i / (D^U_i - D^L_i))}{\ln(Q^U_i / (Q^U_i - Q^L_i)} \, ,$
$b_{ij} = \left( 1 + \left(\beta_i \sum_{k\neq i\atop k\neq j} \zeta_{ijk}\right)^{n_i} \right)^{-1/(2n_i)} \, ,$
$\zeta_{ijk} = f^S_{ik} \exp(\mu_{ij}^{m_i} (r_{ij} - r_{ik})^{m_i}) \left(1 + \frac{c_i^2}{d_i^2} - \frac{c_i^2}{d_i^2 + (h_i - \cos \theta_{i,j,k})^2} \right) \, ,$
$\Delta Q_i = \frac{1}{2} (Q^U_i - Q^L_i) \, ,$
$Q^O_i = \frac{1}{2} (Q^U_i + Q^L_i) \, ,$
$a^B_i = \frac{1}{1 - |Q^O_i / \Delta Q_i|^{n^B_i}} \, ,$
$b^B_i = \frac{|a^B_i|^{1/n^B_i}}{\Delta Q_i} \, .$

These parameters are specified in COMBParticle, or, for interactions between different particle types, in COMBPairPotential.

The charges $$q_i$$ are not static. Instead, they are recalculated in each simulation step by minimizing $$E$$ with respect to the charges.

There are several additions to the basic COMB potential. One can for example add additional bond bending terms (cf. COMBTriplePotential):

$U^\mathrm{triple}_{ijk} = f^S_{ij} f^S_{ik} \sum_{d = 0}^6 K_{ijk}^d L^d\left(\cos \theta_{ijk} - \cos \theta_{ijk}^0\right) ,$

where $$L^d$$ denotes the Legendre polynomial of degree $$d$$.

In [DSC+11], a self-energy correction term was introduced (cf. COMBSelfEnergyCorrection):

$U^\mathrm{corr}_{ij} = \frac{q_i^2}{4\pi\varepsilon_0} \left( \frac{a_i q_j}{r_{ij}^3} + \frac{b_i q_j^2}{r_{ij}^5}\right) .$

Alternatively, the modification proposed in [SDK+10],

$U^\mathrm{corr}_{ij} = \frac{1}{4\pi\varepsilon_0} \left( \frac{a_i q_j}{r_{ij}^5} + \frac{b_i q_j^2}{r_{ij}^5}\right) ,$

can be used by specifying COMBSelfEnergyCorrectionShan.

To add electrostatic interactions, you either use damped point-charge interactions,

$U^\mathrm{ion}_{ij} = \frac{f^S_{ij} \eta_{ij} q_i q_j}{4\pi\varepsilon_0 r_{ij}} \, ,$

with parameters specified in COMBPointWiseCoulomb, or resort to the Streitz-Mintmire model (cf. COMBSMCoulomb),

$U^\mathrm{ion}_{ij} = \frac{1}{4\pi\varepsilon_0 r_{ij}} J_{ij} q_i q_j \, ,$

where $$J_{ij}$$ is calculated by integration over the charge densities $$\rho_i$$ [DSC+11],

$\rho_i(x) = \frac{\xi^3}{\pi} \exp(-2\xi_i |x - r_i|) .$

Instead of specifying all interaction parameters between different particle types manually via COMBPairPotential, these are often determined by employing the following combination rules:

$A_{ij} = \sqrt{A_i A_j} \, ,$
$B_{ij} = \sqrt{B_i B_j} \, ,$
$\lambda_{ij} = 0.5(\lambda_i + \lambda_j) \, ,$
$\mu_{ij} = 0.5(\mu_i + \mu_j) \, .$

These combination rules are enabled by using the COMBMixitPotential for the desired particle type combination.

 [DSC+11] (1, 2) B. Devine, T.-R. Shan, Y.-T. Cheng, A. J. H. McGaughey, M. Lee, S. R. Phillpot, and S. B. Sinnott. Atomistic simulations of copper oxidation and cu/cu2o interfaces using charge-optimized many-body potentials. Phys. Rev. B, 84:125308, Sep 2011. doi:10.1103/PhysRevB.84.125308.
 [SDK+10] T.-R. Shan, B. D. Devine, T. W. Kemper, S. B. Sinnott, and S. R. Phillpot. Charge-optimized many-body potential for the hafnium/hafnium oxide system. Phys. Rev. B, 81:125328, Mar 2010. doi:10.1103/PhysRevB.81.125328.
 [Yas96] Akio Yasukawa. Using An Extended Tersoff Interatomic Potential to Analyze The Static-Fatigue Strength of SiO2 under Atmospheric Influence. JSME international journal. Ser. A, Mechanics and material engineering, 39(3):313–320, jul 1996. URL: http://ci.nii.ac.jp/naid/110002964467/en/.
 [YSP07] J. Yu, S. B. Sinnott, and S. R. Phillpot. Charge optimized many-body potential for the si∕sio2 system. Phys. Rev. B, 75:085311, Feb 2007. doi:10.1103/PhysRevB.75.085311.