# Understanding the physical meaning of the Gay-Berne potential

Hello,

I am not sure if this forum accepts general questions about the physics of simulations, but I did not know where else to ask this question, so here goes. Apologies if I should redirect this to some other forum.

I am following this document: https://docs.lammps.org/PDF/pair_gayberne_extra.pdf.

I am following these references to guide my understanding…
Original anisotropic potential paper: Cookie Absent

however, the above two papers and document do not seem to employ the same notation and is sending me for a loop.

I am trying to understand what the η_12 and χ_12 orientation term mean, physically. Let’s say I have two ellipsoidal particles in a box, [arbitrary positions, in 2D for ease of visualization].

As seen in the above image.

I understand that r in the above image is r_12. My question is, what exactly are A1 and A2 from the document written by Mike Brown, and what is their physical meaning? I understand the equation of a general ellipse is xTYx = 1, but how are these A matrices helping us out here. Do e1, e2, f1, f2 and/or their magnitudes have any influence on A_1 and A_2?

The shape matrix S: is this matrix simply a diagonal matrix of the radii of these elliptical particles? Is it required condition that a>b>c, or a<b<c? How do we know what the major/minor axes are?

In the document, the is another matrix G_12 that is necessary to obtain the eta-dependent energy. What is the make up of G? Why is G=(A_i)^T(S_i)^2A_1 a relevant quantity?

I would appreciate any insight you have on these problems with understanding I am having!

It is OK to ask these kind of things here. Worst case: if it is too generic, we can always move the discussion to the “science talk” category. But in my opinion this would still be on-topic since it discusses a feature in LAMMPS and refers to files bundled with LAMMPS.

Unfortunately, I suspect that there are only very few people in the extended LAMMPS community that could answer your questions and it is unlikely they hang out here.
Your best shot is probably to contact Mike Brown directly.

Hi there! You can have a look at this 1998 paper describing the generalized GB potential for biaxial ellipsoids. There, you have the dimensionless pre-exponential strength coefficient \varepsilon_{AB}(\omega_1,\omega_2), which depends on the dimensions of the two ellipsoids and their orientations, and the anisotropic distance function \sigma_{AB}(\omega_1,\omega_2,\vec{r}_{12}), both depending on the symmetric overlap matrix \mathbf{A}_{AB}(\omega_1,\omega_2).

Here is a graphical demonstration of the variation of the GB potential w.r.t. the orientation, courtesy of Matteo Ricci.

Thank you for your response @hothello. Sorry if this is a basic question, but how does one define a the symmetric overlap matrix A_{AB}(\omega _1, \omega _2)?

I am trying to construct a concrete example so the concept solidifies in my head… Let’s take the above diagram for example. How do I define \omega for the red and blue ellipses? In the simpler, 2D case, for example.

The matrix \mathbf{A}_{AB}(\omega_1,\omega_2) is defined in equation (5) of the 1998 paper. You need to build the diagonal ‘shape’ matrix \mathbf{S} with elements \sigma_x,\sigma_y,\sigma_z and the rotation matrix \mathbf{M} transforming from laboratory to molecular frame. In the 1998 paper this matrix is defined in terms of the Euler angles \omega_i=\{\alpha_i,\beta_i,\gamma_i\}, but in LAMMPS is implemented directly in terms of the quaternion \mathbf{q}_i of each ellipsoid.
Please refer to this link on how to convert a quaternion into a rotation matrix. Then it’s just linear algebra.

I hope this helps.
Otello