Criteria for Choosing a Potential

What science are you wanting to address? What does it require, as far as accuracy of forces or energetics, and so on?

The three main criteria for choosing a potential are:

Accuracy
Reproduce properties of interest as closely as possible.
Transferability
Applicable to more situations for which the potential was NOT fit.
Computational Speed
Force calculations are the most time consuming part of simulation, e.g., so they should be as fast as possible (meaning potentials should be as simple as possible). A 2-body potential V(r_ij) depends only on distance between atoms, i.e., |r_i-r_j|; whereas a 3-body potential will depend on orientation which will require more computation.

Typical emphasis for various discplines include:

CHEMISTRY: Accuracy
Rate constants, for example, require very accurate reaction barriers.
STATISTICAL MECHANICS: Computational Speed
Complexity can result from simple potentials (e.g., the near-neighbor Ising Model).
MATERIALS SCIENCE:    Computational Speed   (due to complexity; in some case all 3)
    Calculation of equilibrium positions of atoms around defects, etc., for comparison to
    EXAFS or LEED signals, or High-resolution TEM.
BIOCHEMISTRY: Perhaps all 3
Protein structures with different conformations using approximate forces.

Finding the actual parameterizations for the analytic functions one has chosen is an important but perhaps difficult task. It can be technically elaborate and an "art form" in itself.

Many applications use no more than 3-body terms, which implicitly assumes higher-order terms are negligible to the energy and forces. This is equivalent to assuming a rapidly convergent interaction series. Basically, a general 3-body equation is not known and forms are chosen for particular interest, e.g. Si (3-body with angular bonding information required) vs. Au (metallic 2-body that is density dependent). Notably, convergence of many-body interactions really may not converge very rapidly (if at all).

Therefore, what do you want to model, how accurate must you be, and how long/much are you willing to spend calculating?

Commonly Used Potentials

Here are some common approaches, with various levels of sophistication, to obtaining V(r) which in turn determines the forces:

Analytic potential based on functional forms, built-in physics, and assumptions.

Potentials derived from concepts from quantum-mechanical bonding arguments, e.g., EAM, EMT, Bond-order, Finnis-Sinclair.

Forces obtained directly via electronic-structure (quantum-mechanical-based) calculations, e.g., Carr-Parrinello Phys. Rev. Lett. 55, 2471 (1985).

Force-Field Methods (used widely in chemistry) are tremendously valuable, but never transferable. Usually, these are much simpler in form and can be parameterized to achieve very good molecular structures.

For example, to model Polyethylene, one requires a potential to deal with bond-stretching (2-body), angular bending (3-body), stretch-stretch and stretch-bend (3-body + higher), bend-bend (4-body + higher). Such potentials may be composed of pieces: (1) intramolecular (Valence Force Field with bonded atoms in molecule) and (2) intermolecular (L-J + Coulomb interaction for non-bonding interaction).

Combination of approaches 2 and 3 are being used in order to increase system sizes yet provide more realistic representations of the systems. (See, for example, multiscale scale materials science modeling.)

Approaches 1 and 2 fit parameters from chosen functional form via some restricted data base of physical properties of the system (or sub-systems) of interest. For molecules, e.g., bond length, cohesive energies, vibrational energies, etc., could be used. For solids, lattice constants, elastic moduli, configurational or defect energies could be used, which may be also modified to include molecular data, or lower-symmetry data of interested in surfaces. Based on electronic-structure methods, approach 3 requires a large computational effort and restricts system sizes, so while important it has limited application to large simulations. (See Review, A.E. Carlsson, Solid State Physics 43, 1 (1990).)

For some brief remarks about other simplified Quantum-based methods, see comments on Tight-binding related methods.

Calendar

Aug 1998, Sept. 4, 1999, Sept. 2001 by D.D. Johnson