# Gay-Berne Potential (option uniax)

The Gay-Berne potential models the pair interaction of uniaxial molecules. It depends not only on the scalar distance of their centres of mass but also on their mutual orientation. Due to this complicated dependence the Gay-Berne interaction cannot be tabulated. In IMD it is calculated from its analytical expression. Forces and torques are also evaluated analytically as derivatives of the potential.

## Gay-Berne model

The Gay-Berne potential for uniaxial molecules (UNIAX) was implemented by Joachim Stelzer. It is a model for the pair interaction of two axially symmetric ellipsoidal molecules. The original model was invented by J. G. Gay and B. J. Berne J. Chem. Phys. 74, 3316 (1981).

## Molecular Properties

In conventional IMD an atom is completely described by its mass m and the vectors [rx ry rz] and [vx vy vz] which denote its position and orientation, respectively. For uniaxial molecules the data structure is enlarged. Each translational quantity is supplemented by a rotational one.

• The molecule possesses both mass m and moment of inertia I.
• Its centre of mass is located at [rx ry rz] and the orientation of its main axis is given by a unit vector [ux, uy, uz], with ux^2 + uy^2+ uz^2 = 1.
• Its velocity is [vx vy vz] and its angular velocity is [wx wy wz].
• The total force exerted on the molecule under consideration is [fx fy fz]. In addition the rotational state of the molecule is changed due to the torque [tx ty tz].
• The shape of the uniaxial molecule is given by the lengths of its principal axes [sx sy sz]. Thereby, sx must be equal to sy. The original version of the Gay-Berne model deals with ellipsoids that possess a shape of sx : sy : sz = 1 : 1 : 3. However, the current implementation in IMD allows for arbitrary values of sx and sz.
• The anisotropy of the interaction energy is given by the well depths of the potential [ex ey ez]. They measure the interaction of a pair of molecules whose main axes are mutually parallel along the z direction. The three values then correspond to the intermolecular separation vector pointing along the x y z direction, respectively. For uniaxial molecules ex must be equal to ey. The original version of the Gay-Berne model deals with an energy anisotropy of ex : ey : ez = 1 : 1 : 1/5. However, the current implementation in IMD allows for arbitrary values of ex and ez.

The moment of inertia, particle shape, and potential depths, which are common to all particles, are specified in the parameter file, and the other properties in the configuration file.

Limitation: The current implementation is limited to a single type of Gay-Berne particle.

## Gay-Berne Potential

For fixed shape and energy anisotropy of the unaxial ellipsoidal molecules the Gay-Berne pair potential is depending on four quantities: the scalar separation rij of the centres of mass, the two angles thetai and thetaj between the molecular separation vector and the main axis of molecule i and j, respectively, and, finally, the angle phiij for a twist of the molecular axes out of plane.

Due to this complicated dependence the Gay-Berne potential cannot be tabulated. Instead it is calculated from its analytical expressions. Likewise, the forces and torques are evaluated analytically as derivatives of the potential.

## Integrators for Uniaxial Molecules

Newton's equations of motion govern the time development of the translational degrees of freedom of the molecules. Uniaxial molecules in addition possess rotational degrees of freedom (main axis, angular velocity). These quantities change according to a simplified version of Euler's gyroid equations. Therefore the integration algorithms implemented in IMD must be supplemented. This has been performed for the ensembles with the option NVE, NVT, and NPT_ISO.

Due to the constraint of the main axis [ux uy uz] being a unit vector, the integration of the gyroid equations is more complicated than only a one-to-one mapping of the corresponding way for solving Newton's equations. In particular, the angular velocity has always to be perpendicular to the main axis, because a uniaxial molecule cannot rotate around its main axis. The number of its rotational degree of freedoms is only two.

The Nosé-Hoover thermostat has been supplemented by an analogous method that keeps the total rotational energy on average at the value of 2/2 * kT, according to the equipartition theorem.

## Parameters

No potential table is necessary for the Gay-Berne interaction, because the potential, forces and torques are calculated analytically. However, the potential and other parameters must be specified in the parameter file.

Parameter  Description
uniax_inert moment of inertia perpendicular to the molecule's axis
uniax_sig shape of the molecule, given by the three half axes; the first two must be equal (uniaxial molecule)
uniax_eps potential depths in the three main axis directions; the first two values must be equal (uniaxial molecule)
rcut cutoff radius for Gay-Berne interaction. rcut = 4.0 for original parametrization of the model.
tau_r time constant for coupling to the heat bath of Nosé-Hoover thermostat for rotational motion. Can be set equal to tau_eta.

## Compiling IMD for UNIAX

When compiling IMD for UNIAX, the option uniax has to be specified, which is implemented only in three dimensions. Only the NVE, NVT, and NPT_ISO ensembles are available, both in the serial and the parallel version. Examples:

```   gmake imd_nve_uniax
gmake imd_mpi_nvt_uniax
```

## Generating Initial Configurations

A separate code (in c) is provided in the IMD user directory: /imd/imd-samples/uniax/init_conf_uniax.c. It should be run in interactive mode. Input quantities are requested from the keyboard. Its output is a IMD configuration file for uniaxial molecules.

A sample both for a configuration file config.inp and for a parametre file param.inp is available in the same directory.