# Can Newton's Equation of Motion be Integrated Analytically for Certain Chemical Systems?

In molecular dynamics, one typically integrates Newton's equation of motion, for example using a leap-frog algorithm. Are there chemical systems, for which an analytic integration is a viable alternative? Are there even simulation programs doing this?

• I am not sure about the existence of chemical systems that would need analytic integration, but when you ask if there are simulation programs doing analytical integration, do you mean performing integration using symbols through a computer algebra system? Because other than this, any "program" performing integration is doing numerical integration. May 13, 2020 at 13:30
• @Mythreyi: I was thinking of having the benefits (accurracy, speed of reevaluation, ...) for a chemical system rather then an actual "need", since I suppose the analytic solution is traditionally more difficult and therefore no system "needs" it. And for your question, "doing analytical integration, do you mean performing integration using symbols through a computer algebra system?": yes, exactly, just like e.g. Mathematica does it. May 13, 2020 at 18:21

## 1 Answer

Analytic integration is only possible for the simplest potentials (a lot of them are covered in the introductory quantum mechanics course). Thus, this is only possible if the model for the process is simple enough that the motion decouples into a series of uncoupled harmonic oscillators, rigid rotors, spins and central potentials. As you can imagine, this is not at all a typical situation.

If this happens, however, no simulation program is needed as the dynamics is analytic.

As an aside: it is fairly common to decompose the dynamics into parts, that can be solved analytically and then obtain numerically exact answers to parts of your dynamics (e.g. the leap-frog algorithm you cited is an example of that with the free particle sub-step being obtained exactly - of course the error is then buried in the Trotter splitting).

The examples that come to mind from my personal experience are the ring-polymer molecular dynamics (RPMD) simulations, where the free ring-polymer evolution sub-step is a collection of uncoupled harmonic oscillators, that can be integrated analytically. The algorithm is well-described in section IIB of this paper.

Shameless self-advertising: It turns out that this is a good, but not the best way to do it. Instead, one should integrate the approximation of the free ring-polymer dynamics exactly.