Notes on Tersoff-Brenner Potential¶
Several modifications of the original Tersoff potential (see Notes on Tersoff Potentials) have been proposed. One of them was introduced in [1] and is a combination of the Tersoff, Brenner and Tanaka potentials. For the sake of brevity we call it the Tersoff-Brenner potential.
The potential energy of the Tersoff-Brenner potential is given as the sum of attractive and repulsive potentials:
The repulsive interactions are the same as in the Tersoff potential
but a different tapering function \(f^{TB}_{ij}\) is used:
The attractive potential \(U^{att}_{ij}\) is given by the term
The function \(g(\theta_{ijk})\), where \(\theta_{ijk}\) is the angle between the particles i-j and i-k, is either
or
The potential is activated in several steps.
To begin with, the pair-dependent parameters must be set by adding a TersoffBrennerPairPotential for each particle pair.
The parameters in the constructor of TersoffBrennerPairPotential map the the potential parameters as follows:
a |
The parameter \(A_{ij}\) |
b |
The parameter \(B_{ij}\) |
lambda |
The parameter \(\lambda_{ij}\) |
mu |
The parameter \(\mu_{ij}\) |
re |
The parameter \(R^{(e)}_{ij}\) that is only used if additional bond-order terms are activated |
r1 |
The parameter \(R_{ij}\) in the taper function |
r2 |
The parameter \(S_{ij}\) in the taper function |
This sets all pairwise parameters except \(\eta_{ij}\) and \(\delta_{ij}\) which are set to 0, thereby deactivating all three-body interactions.
These parameters can be set by the TersoffBrennerBOPairPotential. Not that these parameters are used in a non-symmetric way, which means a TersoffBrennerBOPairPotential should be specified for each ij and ji.
The three-body parameters are set in a similar way. To use \(g(\theta)\)
from equation () you need to set up a
TersoffBrennerTriplePotential. For the alternative form
() the TersoffBrennerTriplePotential2 should be used.
Note that particle_type1
denotes the type of the central particle during
the angle calculation. The remaining parameters are the following:
alpha |
The parameter \(\alpha_{ijk}\) |
beta |
The parameter \(\beta_{ijk}\) |
g_a |
The parameter \(a_{ijk}\) (TersoffBrennerTriplePotential2 only) |
g_c |
The parameter \(c_{ijk}\) |
g_d |
The parameter \(d_{ijk}\) |
g_h |
The parameter \(h_{ij}\) |
Another correction modifies the term \(b_{ij}\) in equation () as follows:
\(H_{ij}\) is an arbitrary 2D-function that is given as a bicubic spline. \(N^{(1)}_{ij}\) and \(N^{(2)}_{ij}\) are given by
\(P^{(1)}_{ij}\) and \(P^{(2)}_{ij}\) are two lists of particle types that specify which interactions are taken into account for the calculation of \(N^{(1)}_{ij}\) and \(N^{(2)}_{ij}\). To enable this correction, a TersoffBrennerSplinePotential can be used where the parameters are defined as follows.
particleType1 |
The particle type referred to as i |
particleType2 |
The particle type referred to as j |
activeTypes1 |
The type list \(P^{(1)}_{ij}\) |
activeTypes2 |
The type list \(P^{(2)}_{ij}\) |
x |
The x-coordinates of the grid of the spline function |
y |
The y-coordinates of the grid of the spline function |
f |
The spline values on the x-y-grid |
Note that this potential acts in a non-symmetric way, meaning that it will only act on ij type pairs, but not on ji type pairs.
Finally, the term \(\bar{b }_{ij}\) in equation () can be modified in the following way:
The function \(F_{corr}\) is an arbitrary 3D-function that is given as a tricubic spline. \(N^{(t)}_{ij}\) is the coordination number of particle \(i\), excluding \(j\), i.e.
\(N^{(conj)}_{ij}\) is defined as follows:
\(T_{ij}\) is a new type of tapering function given by
\(P^{(conj)}_{ij}\) is a list of particle types that specify which interactions are taken into account for the calculation of \(N^{(conj)}_{ij}\). For this correction the TersoffBrennerCorrectionPotential can be used.
particleType1 |
The particle type referred to as i |
particleType2 |
The particle type referred to as j |
activeTypes |
The type list among which neighbors of atoms i and j should be searched for. |
L |
The lower cutoff in \(T_{ij}\) |
U |
The upper cutoff in \(T_{ij}\) |
x |
The x-coordinates of the grid of the spline function. The same values are used for the y-coordinates |
z |
The z-coordinates of the grid of the spline function |
f |
The spline values on the x-y-z-grid |
In contrast to the TersoffBrennerSplinePotential, this correction acts in a symmetric way between the types ij.