4 Force Fields
4.1 Force Fields
A force field (or interatomic potential) is an analytical expression for the potential energy surface \(U(\mathbf{r}_1, \ldots, \mathbf{r}_N)\) that appears in Newton’s equations of motion (Eq. 3.1, Chapter 2).
4.2 Functional Forms
A typical classical force field decomposes \(U\) into bonded and non-bonded contributions:
\[ U = U_{\text{bond}} + U_{\text{angle}} + U_{\text{dihedral}} + U_{\text{vdW}} + U_{\text{elec}} \tag{4.1}\]
4.2.1 Pair Potentials
The simplest models use pair potentials — \(U\) depends only on interatomic distances \(r_{ij} = |\mathbf{r}_i - \mathbf{r}_j|\):
\[ U = \sum_{i < j} \phi(r_{ij}) \tag{4.2}\]
Lennard-Jones Potential
The Lennard-Jones (LJ) potential models van der Waals interactions:
\[ \phi_{\text{LJ}}(r) = 4\varepsilon \left[\left(\frac{\sigma}{r}\right)^{12} - \left(\frac{\sigma}{r}\right)^{6}\right] \tag{4.3}\]
where \(\varepsilon\) is the well depth and \(\sigma\) is the collision diameter (the distance at which \(\phi = 0\)). The minimum occurs at \(r_{\min} = 2^{1/6}\sigma\).
Embedded Atom Method (EAM)
For metals, the Embedded Atom Method (EAM) is more accurate than pair potentials. The energy is:
\[ U = \sum_i F_i(\bar{\rho}_i) + \frac{1}{2}\sum_{i \neq j} \phi(r_{ij}) \tag{4.4}\]
where \(F_i\) is the embedding energy, \(\bar{\rho}_i = \sum_{j\neq i} \rho(r_{ij})\) is the local electron density, and \(\phi(r_{ij})\) is a short-range pair repulsion.
4.3 Cutoffs and Periodic Boundary Conditions
This section is under construction.
Exercises
Plot Eq. 4.3 for argon (\(\varepsilon = 0.0104\) eV, \(\sigma = 3.40\) Å). At what distance is the force zero? At what distance is the potential minimum?
Compute the force \(F = -d\phi_{\text{LJ}}/dr\) and show that it is repulsive for \(r < r_{\min}\) and attractive for \(r > r_{\min}\).