# Potential shapes¶

Setting the Verbosity to Verbose the engine prints the potential formula and the parameters used.

The ForceField engine has a couple of potentials defined.

• Stretch: harmonic
$V^\text{stretch/harm} = \frac{1}{2} f_c (r-r_0)^2$
• Angle: harmonic and cyclic
\begin{align}\begin{aligned}V^\text{bend/harm} = \frac{1}{2} f_c (\phi-\phi_0)^2\\V^\text{bend/cycl} = f_c \sum_{m=0}^n c_m \cos(m \phi)\end{aligned}\end{align}
• Torsions: cyclic, possibly linearly combined. The same torsion occurs more than once in the printed table, and the energies are added.
\begin{align}\begin{aligned}V^\text{torsion/harm} = \frac{1}{2} f_c (\phi-\phi_0)^2\\V^\text{torsion/cycl} = f_c \sum_{m=0}^n c_m \cos(m \phi)\end{aligned}\end{align}
• Inversions: either angle or distance based. The angle based one depends on the order of the three atoms connected to the central atom. UFF averages over the three permutations.
\begin{align}\begin{aligned}V^\text{inversion/harm} = \frac{1}{2} f_c (\phi-\phi_0)^2\\V^\text{inversion/cycl} = f_c \sum_{m=0}^n c_m \cos(m \phi)\\V^\text{inversion/amber} = f_c (1 + \cos(2 \phi- \phi_0))\\V^\text{inversion/dist} = f_c \; d^2\end{aligned}\end{align}
• Dispersion: Lennard-Jones. Neglect up to second neighbors, possibly scale contribution from third neighbors.
$V^\text{dispersion/LJ} = d \; ( (x/r)^{12} -2 * (x/r)^6 )$
• Coulomb: Neglect up to second neighbors, possibly scale contributions from third neighbors.

In general which formula is used depends on the parameter files. Note that the scaling of the third neighbors contributions is only possible when using .ff parameter files.

# APPLE&P potential shapes¶

In general, APPLE&P uses similar expressions for the potentials, with some differences. For completeness’ sake we list all APPLE&P potentials below.

• Bond: the same as the stretch potential above.
$V^\text{bond} = \frac{1}{2} f_c (r-r_0)^2$
• Bend: the same as the harmonic angle potential above.
$V^\text{bend} = \frac{1}{2} f_c (\phi-\phi_0)^2$
• Torsion: cyclic.
$V^\text{torsion} = - \sum_{m=1}^n c_m \cos(m \phi)$
• Out-of-plane angle: sum of three harmonic terms, each corresponding to an angle between the Rij bond and the (jkl) plane, where j is the central atom and i, k, l are permutations of the other three atoms.
$V^\text{oop} = \frac{1}{2} f_c (\phi_1^2 + \phi_2^2 + \phi_3^2)$
• Dispersion: mix of the Buckingham and Lennard-Jones potentials.
$V^\text{dispersion} = A e^{-Br} - \frac{C}{r^6} + \frac{D}{r^{12}}$
• Electrostatic potential: charge-charge, charge-dipole and dipole-dipole. Interaction are excluded for the 1-2 and 1-3 neighbors and can be scaled for the 1-4 ones. For each atom, the self-consistent induced dipole moment is computed from its polarizability and the electric field due to other charges and dipoles. The latter includes the Thole damping.
$V^\text{elstat} = \sum_{i>j} {} \frac{q_i q_j}{4 \pi \epsilon_0 r_{ij}} + \sum_{i} {} \vec{\mu_i} \cdot \vec{E_i}$