Subject: Generalizations of Lennard-Jones potential?

From: [email-masked]
Date: Sun, 3 Jan 1999 16:08:41 -0500 (EST)

I'm programming a simple model that requires being able to represent a
high degree of chemical heterogeneity among interacting particles.  As
a first pass, I would like to model point particles interacting via
spherically symmetrical potentials, with varying interaction
strengths, depending on the "chemical identity" of the particles.

I originally thought of using a Lennard-Jones 6-12 (LJ) potential,
suitably modified to represent various strengths of interaction, but
this has turned out to be a much harder problem than I expected (it
may not be obvious why this would be the case, but it has to do with
certain of the model's requirements.)

Does anyone know of a published study on generalizing the LJ
potential roughly along these lines?

Thanks in advance,

K. Jones

P.S. For the pathologically patient, here's a more detailed account of
the problem.

Fix a characteristic interparticle distance r0.  Let LJ(r) stand for a
standard Lennard-Jones 6-12 potential, with minimum -1 at r = r0, and
let LJR(r) stand for the repulsive part (~ (r0/r)^12) of LJ(r).  Let
U(m, r) stand for the desired generalization of this potential, where
the parameter m is defined as m := U(m, r0).  Ideally, we would like
U(m, r) to satisfy the following requirements:

1. U(m, r) must be continuous in both m and r, and have continuous
derivatives with respect to r of up to no less than order 2.

2. U(-1., r) = LJ(r) (i.e., we must be able to recover LJ(r) as a
special case of U(m, r)).

3. If m < 0 (the partially attractive case), U(m, r) must have exactly
one extremum, a minimum, at r0.

4. If m < 0, we want U(m, r) to "quickly" approximate |m|*LJ(r), as
r->oo.

5. For all m, we want U(m, r) to "quickly" approximate LJ(r) as r->0.

6. U(m, r) must be "not too exorbitant" computationally .

I could make the terms in quotes more precise, but this seems
premature at this stage.  Also, I realize that, most likely, the best
candidate will fit this set of requirements very imperfectly.

The first solution I tried was to use U(m, r) := |m|*LJ(r) for the
partially attractive case (m < 0), and U(m, r) := m*LJR(r) for the
purely repulsive case (m > 0).  This scheme has several serious
drawbacks, notably that it is not continuous at m = 0, and, for small
|m|, does not approach LJR(r) quickly enough as r->0, which translates
to a repulsive core that is too small.

Another partial answer, based on an idea originally proposed by Weeks
et al., (see p. 11 of Allen and Tildesley's "Computer Simulation of
Liquids") is:

For (m <= 0):

               LJ(r) + 1 + m    (r < r0)

  U(m, r) := 

               |m|*LJ(r)        (r >= r0)

The most serious objections to this idea is that it doesn't
readily generalize to the case m > 0.

---

From: "Peter Shenkin" <[email-masked]>
Date: Tue, 5 Jan 1999 19:06:10 -0500

On Jan 3,  4:08pm, [email-masked] wrote:
> Subject: CCL:Generalizations of Lennard-Jones potential?

> potential roughly along these lines?
 ...

You say *what* you want to do, but not *why* you want to do it.

Are you looking for greater accuracy vis-a-vis high-quality experimental
or q-mechanical results?  Are you trying to circumvent overflows in 
monte-carlo simulations when the particles get too close?  Or what?

Don't forget that your accuracy with respect to the Word of God
(quantum mechanics :-) ) is always limited by the neglect of 
3-and-more-body effects.

Still, Tom Halgren, in one of his MMFF papers (J. Comput. Chem., but
I don't have the ref. handy) did a study that led him to the use
of a 7-14 potential with rather complicated combining rules.  Whether or
not you like his solution, he makes careful comparisons between simple
molecular-mechanics nonbonded potentials and highly accurate pairwise-
central potentials that have been derives for specific simple systems.

Also, Tom and others do "buffering" of one kind or anotheer.  The simplest
such scheme is to replace r in the LJ equation by r+a, where a is a small
constant whose value is chosen to have little effect at "typical" distances,
but to avoid computational pathologies if/when r should become close
to zero.  In our era of IEEE arithmetic this particular motivation is
no longer of such great concern, but of course the buffering also changes
the functional form and can be adjusted to achieve a better fit to accurate
data.

Hope this helps,
-P.

-- 
*** "Freedom's just another word for nothing left to lose." (B. Yeltsin)***
*Peter Shenkin; Chemistry, Columbia U.; [email-masked] (212)854-5143*
* MacroModel URL: http://www.cc.columbia.edu/cu/chemistry/mmod/mmod.html  *