Subject: Energy conservation of molecular dynamics in the presence of electrostatic forces

From: Charles Xie <[email-masked]>
Date: Thu, 06 Mar 2003 09:20:27 -0500

Dear CCLers,

I have a general question about totoal energy conservation of classical
molecular dynamics (MD) in the presence of electrostatic forces.

Total energy is perfectly conserved when we do a typical MD with
only the wan der Waals (Lennard-Jones) potentials, using a time
step of one femtosecond (e.g. for the Argon gas as a classical example).

Problems, however, arise when molecules are charged. Because
electrostatic forces are much more long-ranged (1/r dependence) and
typically stronger, the magnitude of electrostatic forces on atoms can
be several hundred (or even thousand) times greater than that of van
der Waals forces. As a result, the time step has to be accordingly
reduced for two or three orders of magnitude, in order for the total
energy to conserve. This means a time step of 0.001-0.01 femtosecond
has to be applied.

(EXPLAIN: Most standard algorithms, such as the Verlet method, the Gear
predictor-corrector method and the Runge-Cutta method, depend on using
the term a*dt^2 to compute the solution stepwisely, where a is the 
acceleration,
dt is the time step. If this term is too large, the numerical error will 
rapidly
propogate, and the solution will diverge. Compared with pure Lennard-Jones
simulations, adding the electrostatic forces increase accelerations. 
Therefore,
dt must be decreased in order to keep a*dt^2 down.)

I can see only two ways to solve this problem, both of which turn out to be
impractical, as you will see later.

One is to accept the reality, use a smaller time step. But, reducing the time
step to 0.001 femtosecond will slow down the simulation 1000 times (in
comparison to pure Lennard-Jones simulations). You will probably never see
any emerging behavior of the model because losing your temper.

The other way around is to use smaller charges, as most molecular mechanics
force fields do. For example, instead of using 1 for a free radical's charge,
one can use 0.01 (and self-explain that this is an effective-field 
approximation).
While this would maintain energy conservation, you might not see the
emerging behaviors due to charges either --- because the electrostatic forces
are so weaker than they ought to be, they might not be able to produce
results that only strong interactions can exhibit, such as self-organization.

Energy conservation is generally not an concern for most classic molecular
dynamics simulations that assume a heat bath (for controlling the temperature
to a desired value, thus remove the numerical errors resulted from large 
a*dt^2).
However, getting the energetics right is of paramount importance to modeling
exoergicity/endoergicity of chemical reactions. Unfortunately, I am not aware
of anyone mentioning this before. All the molecular simulation books I have
(Allen and Tildesley, Leach, Rapaport and so on) do not even mention it at all.
My prior experience with CHARMm v27, if I was doing it right, is that it 
doesn't
seem to do energy conservation, even though the NVE protocol is literally 
specified.

I would greatly appreciate any advice and opinion.

Thank you,

Charles Xie

International Center
The Concord Consortium
www.concord.org