Symmetry of Tersoff parameters


I am experimenting with the Tersoff potential in lammps in the case of 2-element systems. Considering only two atoms of different type, and only the repulsive term of the Tersoff potential, without even the distance dependence (\lambda_1=0), it seems to me that according to the formula given in the lammps manual (pair_style tersoff command — LAMMPS documentation), the total energy should be E=\frac{1}{2}\sum_i\sum_{j\neq i} V_{ij}=(A_{12}+A_{21})/2.

However, when I do this calculation with lammps (version 24 Dec 2020), I do obtain either A_{21} or A_{12} as the total energy of the 2-atom system, depending on whether the atom of type 1 or 2 is listed first, respectively.

I do understand that having pair parameters that are not symmetrical may not be reasonable. However, the manual seems to indicate otherwise, it states: “Note that the twobody parameters in entries such as SiCC and CSiSi are often the same, due to the common use of symmetric mixing rules, but this is not always the case.”

Here is the lammps script I am using:

units                   metal

region                  mybox block -10 10 -10 10 -10 10
create_box              2 mybox

create_atoms            1 single -0.5 0 0
create_atoms            2 single 0.5 0 0

mass                    1 1
mass                    2 1

pair_style              tersoff
pair_coeff              * * test.tersoff C Si

run                     0

and the test.tersoff potential file:

C C C    1 0 0 0 0 0 0 0 0 0 2 0.1 0 0
C C Si   1 0 0 0 0 0 0 0 0 0 2 0.1 0 0
C Si C   1 0 0 0 0 0 0 0 0 0 2 0.1 0 0
C Si Si  1 0 0 0 0 0 0 0 0 0 2 0.1 0 1
Si C C   1 0 0 0 0 0 0 0 0 0 2 0.1 0 0
Si C Si  1 0 0 0 0 0 0 0 0 0 2 0.1 0 0
Si Si C  1 0 0 0 0 0 0 0 0 0 2 0.1 0 0
Si Si Si 1 0 0 0 0 0 0 0 0 0 2 0.1 0 0

The relevant entries are “C Si Si” and “Si C C”, and in particular the last number which corresponds to the A prefactor, which is set here asymmetrically: A_{CSi}=1 and A_{SiC}=0.

If you run the script above, the total energy is 0, but when you create the atom of type 2 first, the total energy becomes 1.

Can someone please advise me on this subject? Is there a problem with my script, or the way I use the Tersoff formula? Or does the manual mean something different when mentioning SiCC and CSiSi pair parameters that are not necessarily the same? Sorry, I am rather new to this, and I may have missed something in the documentation.

I don’t understand why you are wondering that you get asymmetric results when you assign asymmetric parameters. LAMMPS will compute pairwise interactions only once per pair. The statement you are referring to does not imply that you get symmetric results from asymmetric parameters, and most certainly those parameters will not differ as much as in your parameter file.

Thank you very much for your reply.

I am surprised by the asymmetric results because when I do the calculation on paper, the resulting total energy is not asymmetric. The sums according to the formula as presented in the manual run over i and j\neq i, and not over i and j <i.

I do understand that in the implementation, pairwise interactions are computed only once per pair, but this would prohibit as I understand it any asymmetric pair parameters (as in this case, the total energy would depend on the order of the atoms) which is, however, explicitly allowed by the manual.

Let me repeat: the manual says that parameters may be asymmetric, but it does not say that you will get symmetric results in that case when only looking at the pair wise parameters.

Ok, thanks, just one last point to clarify: not getting symmetric results means that the total energy depends on the order of the atoms?

Why would one want to have the total energy depend on the order of atoms? This seems unphysical to me and I do not understand why the manual would say that asymmetric parameters are sometimes possible in this case.

When you are looking at an A-B-A and B-A-B triple, you are not looking at an exactly symmetric interaction. While A-B and B-A are the same if looked at on their own, they are under the influence of a different third atom.

Thank you very much