I am trying to understand and implement the Gay-Berne potential, but I do not quite understand how one goes about constructing them. I am following this paper to parse through the idea.

I understand how Lennard-Jones/Coulombic potentials work in MD engines: every particle has some $\epsilon, \sigma, q$ associated with it, and these potentials are purely a function of the distance $r=|\mathbf{r}_{ij}|$ between two particles. It does not depend upon the orientation of the two particles. $$U = U(r)$$

In the above paper, a particle $i$ at position $\mathbf{r}_i$ and orientation $\omega$ with respect to the laboratory frame is represented by a Gaussian function $G_i (\omega _i, \mathbf{r})$ which is $$ G_i(\omega _i, \mathbf{r}) = \exp [-\frac{1}{2}(\mathbf{r}-\mathbf{r}_i)^T M_i ^T S_i ^{-2} M_i (\mathbf{r}-\mathbf{r}_i)]$$ where S is a diagonal 'shape' matrix with elements $\sigma _x, \sigma _y, \sigma _z$, the axes of the ellipsoid representing the molecule measures with respect to a unit length $\sigma _0$, and $M_i$ is the rotation matrix transforming from laboratory to a molecular frame.

What I don't understand is, why do we need a Gaussian function to denote a particle? A 3-dimensional ellipse can be represented with the shape matrix S, and its orientation which can be given by the matrix M. Where does $\omega$ figure into the above definition? If anybody has any recommendations on how to go about understanding the implementation of this idea, I would greatly appreciate it.


1 Answer 1


I don't have a ton of time to answer and this is somewhat outside my area, but I think I can at least address some parts of your question.

To start, they mention just after equation 6 of your linked paper that $\omega=(\alpha, \beta,\gamma)$, the Euler angles that define the orientation. This set of Euler angles is used to define the rotation matrix $M$ in a succinct way.

My understanding of why a Gaussian is used is to simply model the interaction between particles. Since the exponent measures distance from the ellipse, the Gaussian will be largest at the border of the ellipse and will slowly decay as you move further away from it. If you wanted to just model hard ellipses, you could just consider when the ellipses of each particle made contact to model collisions, but with a Gaussian you include interactions from when the particles are slightly further away.


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