\(\renewcommand{\AA}{\text{Å}}\)

# angle_style amoeba command

## Syntax

    angle_style amoeba

## Examples

    angle_style amoeba
    angle_coeff * 75.0 -25.0 1.0 0.3 0.02 0.003
    angle_coeff * ba 3.6551 24.895 1.0119 1.5228
    angle_coeff * ub -7.6 1.5537

## Description

The amoeba angle style uses the potential

\[\begin{split}E & = E_a + E_{ba} + E_{ub} \\ E_a & = K_2\left(\theta - \theta_0\right)^2 + K_3\left(\theta - \theta_0\right)^3 + K_4\left(\theta - \theta_0\right)^4 + K_5\left(\theta - \theta_0\right)^5 + K_6\left(\theta - \theta_0\right)^6 \\ E_{ba} & = N_1 (r_{ij} - r_1) (\theta - \theta_0) + N_2(r_{jk} - r_2)(\theta - \theta_0) \\ E_{UB} & = K_{ub} (r_{ik} - r_{ub})^2\end{split}\]

where \(E_a\) is the angle term, \(E_{ba}\) is a bond-angle term, \(E_{UB}\) is a Urey-Bradley bond term, \(\theta_0\) is the equilibrium angle, \(r_1\) and \(r_2\) are the equilibrium bond lengths, and \(r_{ub}\) is the equilibrium Urey-Bradley bond length.

These formulas match how the Tinker MD code performs its angle calculations for the AMOEBA and HIPPO force fields. See the Howto amoeba page for more information about the implementation of AMOEBA and HIPPO in LAMMPS.

Note that the \(E_a\) and \(E_{ba}\) formulas are identical to those used for the angle_style class2/p6 command, however there is no bond-bond cross term formula for \(E_{bb}\). Additionally, there is a \(E_{UB}\) term for a Urey-Bradley bond. It is effectively a harmonic bond between the I and K atoms of angle IJK, even though that bond is not enumerated in the “Bonds” section of the data file.

There are also two ways that Tinker computes the angle \(\theta\) in the \(E_a\) formula. The first is the standard way of treating IJK as an “in-plane” angle. The second is an “out-of-plane” method which Tinker may use if the center atom J in the angle is bonded to one additional atom in addition to I and K. In this case, all 4 atoms are used to compute the \(E_a\) formula, resulting in forces on all 4 atoms. In the Tinker PRM file, these 2 options are denoted by angle versus anglep entries in the “Angle Bending Parameters” section of the PRM force field file. The pflag coefficient described below selects between the 2 options.

Coefficients for the \(E_a\), \(E_{bb}\), and \(E_{ub}\) formulas must be defined for each angle type via the angle_coeff command as in the example above, or in the data file or restart files read by the read_data or read_restart commands.

These are the 8 coefficients for the \(E_a\) formula:

- pflag = 0 or 1

- ubflag = 0 or 1

- \(\theta_0\) (degrees)

- \(K_2\) (energy)

- \(K_3\) (energy)

- \(K_4\) (energy)

- \(K_5\) (energy)

- \(K_6\) (energy)

A pflag value of 0 vs 1 selects between the “in-plane” and “out-of-plane” options described above. Ubflag is 1 if there is a Urey-Bradley term associated with this angle type, else it is 0. \(\theta_0\) is specified in degrees, but LAMMPS converts it to radians internally; hence the various \(K\) values are effectively energy per radian^2 or radian^3 or radian^4 or radian^5 or radian^6.

For the \(E_{ba}\) formula, each line in a angle_coeff command in the input script lists 5 coefficients, the first of which is “ba” to indicate they are BondAngle coefficients. In a data file, these coefficients should be listed under a “BondAngle Coeffs” heading and you must leave out the “ba”, i.e. only list 4 coefficients after the angle type.

- ba

- \(N_1\) (energy/distance^2)

- \(N_2\) (energy/distance^2)

- \(r_1\) (distance)

- \(r_2\) (distance)

The \(\theta_0\) value in the \(E_{ba}\) formula is not specified, since it is the same value from the \(E_a\) formula.

For the \(E_{ub}\) formula, each line in a angle_coeff command in the input script lists 3 coefficients, the first of which is “ub” to indicate they are UreyBradley coefficients. In a data file, these coefficients should be listed under a “UreyBradley Coeffs” heading and you must leave out the “ub”, i.e. only list 2 coefficients after the angle type.

- ub

- \(K_{ub}\) (energy/distance^2)

- \(r_{ub}\) (distance)

## Restrictions

This angle style can only be used if LAMMPS was built with the AMOEBA package. See the Build package doc page for more info.

## Related commands

angle_coeff

## Default

none