Limitations of pairwise additive forcefields

Question

What are some of the limitations on the use of a pairwise additive forcefields in classical molecular dynamics simulations?

Answer 1

The Axilrod-Teller force is not a "pairwise" force but a "3-body" force whose potential has the formula:

$$\tag{1} V_{ijk}= E_{0} \left( \frac{1 + 3 \cos\gamma_{i} \cos\gamma_{j} \cos\gamma_{k}}{\left( r_{ij} r_{jk} r_{ik} \right)^3} \right), $$

as opposed to a pairwise force like the Van der Waals force which is commonly used in molecular dynamics because of its presence in the Lennard-Jones potential:

$$ V_{ij}= E_{0} \left( \frac{1}{r_{ij}^6} \right). $$

A pairwise additive forcefield would probably not do well in cases where 3-body interactions like the Axilrod-Teller force are important (e.g. at very low temperatures and when very high accuracy is needed).

Answer 2

Pairwise force-fields neglects 3 body and higher interactions (many body). As Nike pointed out, there are force-fields that can take into account higher order interactions and still be somewhat pairwise, but parameterizing them is hard(er). It is easier to parameterize a simple model, and often, more accurate. We have put alot of work into simple pairwise FF's. and are pretty good at fudging them, I mean, parameterizing them.

An important piece of the many body problem is polarization. Classical pairwise FF's are fixed charge. They are impervious to their environment which is an obvious weakness, since the electrons in a molecule would shift around depending on distance, orientation and type of molecules surrounding them.

Generally pairwise FF's use quadratic potentials (Hooke's law) for describing intramolecular bonding. This means it is impossible to break or form a bond. This is a classic example of a weakness, but this is only really a weakness if you are interested in kinetics. If you are after thermodynamic properties, for instance, chemical reaction equilibria, you do not need to bother with manually breaking and forming bonds. You can calculate the chemical potentials of the molecules of interest(alchemical free energy calculations are one way) and use thermodynamics to figure out what and how far the reactions will progress. You could also use Reaction Ensemble Monte Carlo, which is again calculates chemical equilibria without actually breaking or forming bonds. The take-home from this paragraph is that when people tell you you can't do something, just be clever and figure out a way.

There are two strengths to pairwise FF's, one of which is also their downfall. Pairwise FF's are very fast to use, so time or ensembles can be thoroughly sampled. However, this speed is due to their reliance on parameters which essentially fudge the fact that they ignore quite a bit of physics, such as many body interactions. Parameters are fit so that the FF will replicate some kind of experimental(or computational) data. The FF is constrained to systems generally similar to that of the data to which it was fit. Some typical examples of this being an issue is using a FF for the same molecule in different phases. Of this, the solid phase is the most troubling. Most FF's are fit for gas or condensed phase simulations, and are poor at solid phases, which is a problem considering how important the solid phase is to the booming biotech/pharma industry.

The nice thing though about pairwise FF is that it is not all that hard to fit them to specific systems. This is happening alot, particularly in the crystal structure prediction community (they often use multipole FF's as well though). Hundreds of thousands or even millions of possible crystal structures for pure or multi-component crystals are generated, pairwise or multipole Force-Fields are fit to many of the crystals structures and then the crystals are evaluated using the custom FF, and the majority is thrown out. More sophisticated methods, like plane-wave DFT are then used to figure out the best structure. This is because, even when custom fit, a pairwise (or even multipole expansion) FF is missing essential physics and not quite accurate enough.

That said, OpenEye is doing some interesting things with their multipole FF that is performing as well as plane-wave DFT for crystal structure prediction, but is an order of magnitude faster, which just goes to show, people give up on simple models way too fast a.k.a. it is easier to publish research if you claim something new is needed, but alas, here we are still using pairwise FF's. I have a feeling one could probably perform as well as the multipole models by simply adding off-center charges and doing a good parameterization ;)

Answer 3

There are many deficiencies in pairwise force fields that are balancing to perform well at some specific task. Pairwise interactive force fields are almost always parameterized to reproduce some kind of experimental data. That experimental data is gathered at some finite temperature and pressure and most likely does not appear in the actual potential. That is, many potentials strive to reproduce some radial distribution function, but the RDF is not actually part of the interaction potential, so the connection between parameters and experiment is somewhat fabricated. This means that most pairwise potentials are not even representations of the Born-Oppenheimer potential energy surface. Hence, the dynamics generated with them are occurring on some kind of pseudo free-energy surface. This is a drawback if you are interested in phenomena happening over a large range of temperatures, particularly in the low temperature limit. When going to low temperature, nuclear quantum effects will probably become relevant, but doing a path integral simulation with pairwise potentials would not be physically reliable since quantum effects are already parameterized in to some extent.

Additionally, pairwise potentials totally fail to simultaneously capture both energetic and structural components of systems that have large many-body effects. Water is the most important example in this case. Nearly all pairwise additive water models exhibit the wrong trend in structures going from the gas-phase to the condensed phase. This is because in the condensed phase, the average dipole moment on each water increases due to polarization from the surrounding environment. Without polarization (i.e. with fixed charges), all you can do to increase the dipole is narrow the bond angle (bring charges more in line), when experimentally the bond angle increases going from the gas phase to clusters to liquid to ice.

The most accurate water models either explicitly fit electronic energies for the short-range part of the potential (MB-Pol, HBB2, etc.) or use induced dipoles to add in many-body effects. These potentials actually live on the Born-Oppenheimer surface, so it is actually more appropriate to carry out nuclear quantum simulations with them than ordinary molecular dynamics. Even though these potentials tend to be much slower to evaluate, they give the right answers for the right reasons, and hence are transferable across the entire phase diagram. They also manage to simultaneously describe molecular structure and energetics correctly while capturing appropriate dynamics.

There is a certainly a place for both types of potentials, but whenever pairwise potentials give you the right answer, it's because they've been manufactured to do so for the system you're interested in. You shouldn't expect it to happen again in a different situation. Many-body models are much more reliable. They are, however, much harder to make and can't access the same timescales as pairwise additive models.