Short-Ranged Intramolecular Potentials

2.5. Short-Ranged Intramolecular Potentials#

2.5.1. Features of Interaction#

  • Applies between all particles in the system, pairwise and additive.

  • Some force fields exclude interactions between certain bonded particles:

    • 1–2 (bonded): ignored

    • 1–3 (two bonds apart): ignored

    • 1–4 (three bonds apart): rescaled (e.g., by 50%)

  • Interactions are “short-ranged” and can be truncated at a cutoff \(r_c\):

    • \(\Delta u = 2\pi \rho \int_{r_c}^{\infty} \underbrace{r^2 \underbrace{u(r)}_{\sim 1/r^6}}_{\sim 1/r^4} dr \) converges

    • In MD, truncate forces: \( F(r > r_c) = 0 \)

2.5.2. Nonbonded Potentials#

2.5.2.1. Lennard-Jones Potential#

\( u(r) = 4\varepsilon \left[ \left( \frac{\sigma}{r} \right)^{12} - \left( \frac{\sigma}{r} \right)^6 \right] \)

The most wildely used intramolecular potential. We will focus is on the Lennard-Jones (LJ) potential due to its common use.

Tip

Once of the reasons why the LJ potential is so popular, is that one can re-use the \( \left( \frac{\sigma}{r} \right)^6\) term by squaring it - thus reducing computational cost. This is much less important with improved hardware.

2.5.2.2. Mie Potential#

\( u(r) = C \varepsilon \left[ \left( \frac{\sigma}{r} \right)^n - \left( \frac{\sigma}{r} \right)^m \right] \) with \( C = \frac{nm}{n - m} \left( \frac{n}{m} \right)^{\frac{m}{n - m}} \)

This is a generaized LJ potential.

2.5.2.3. Exponential-6 Potential#

\( u(r) = -A e^{-Br} + \frac{C}{r^6} \)

2.5.3. Force Calculation#

\[\begin{align*} \vec{F}_{ij} &= -\nabla_i u(r_{ij}) \\ &= \underbrace{-\frac{du}{dr}}_{=F(r)} \nabla_i {r}_{ij} \\ & = - F(r)\hat{r}_{ij} \end{align*}\]

Example: Lennard-Jones Force

\[\begin{align*} u(r) = 4\varepsilon \left[ \left( \frac{\sigma}{r} \right)^{12} - \left( \frac{\sigma}{r} \right)^6 \right] \end{align*}\]
\[\begin{align*} F(r) &= -\frac{du}{dr} &= -4\varepsilon \left[ -12\frac{\sigma^{12}}{r^{13}} + 6 \frac{\sigma^{6}}{r^{7}}\right] & = \frac{24\varepsilon}{r} [2 \left( \frac{\sigma}{r} \right)^{12} - \left( \frac{\sigma}{r} \right)^{6} ] \end{align*}\]

When written in this form, the \(\left( \frac{\sigma}{r} \right)^6\)-term can be reused from the \(u(r)\) calculation.

Tip

Write \(F_i= -F(r) \hat{r}_{ij} = - \frac{F(r_{ij})}{r_{ij}} \vec{r}_{ij}\). Then one can write the LJ force in such a way, that only ever \(r^2_{ij}\) is needed, not \(r_{ij}\), thus avoiding the need for a expensive sqrt operation.

2.5.4. Truncating and Shifting#

Truncate:

\[\begin{align*} u(r) = u_o(r)H(r_c-r) =\begin{cases} U_o(r)\quad r\leq r_c \\ 0 \quad r\ge r_c \end{cases} \end{align*}\]

Note that this means that the force has a “jump” at the cutoff, because \(\frac{\partial u}{\partial r}\) now has a \(\delta(r_c-r)\) term from the Heaviside function, if \(U_o(r_c) \neq 0\).

Truncate and shift:

\[\begin{align*} u(r) = \left[u_o(r)-u_o(r_c)\right]H(r_c-r) =\begin{cases} U_o(r)-u_o(r_c)\quad r\leq r_c \\ 0 \quad r\ge r_c \end{cases} \end{align*}\]

Here \(\frac{\partial u}{\partial r}\) is finite but might be discontinous. One can apply smoothing functions, e.g. polynominals (xplor).

Example: Weeks-Chandler-Anderson Potential

Use truncate and shift scheme at \(r_c=2^{1/6}\sigma\) for LJ potential. The result is a purely repulsive potential, the WCA potential, often used to model purely excluded volume interactions.

For truncating, truncating and shifting, and or smoothing, thermodynamic properties will be influenced by the exact scheme. Those effects can be corrected with long-range integral approximations/corrections.

2.5.5. Mixing Rules#

For unlike atom types \(i\) and \(j\), use mixing rules on LJ parameters. There are many different ones, falling into two categories, arithmetic (\(x_{ij}=(x_j+x_i)/2\)) or geometric (\(x_{ij}=\sqrt{x_ix_j}\)).

Commonly used mixing rules are:

  • Lorentz-Berthelot:
    \(\begin{align*} \sigma_{ij} &= \frac{\sigma_{ii}+\sigma_{jj}}{2}\\ \epsilon_{ij} &= \sqrt{\epsilon_{ii}\epsilon_{jj}} \end{align*}\)

  • Kong:
    \(\begin{align*} \epsilon_{ij}\sigma_{ij}^{6}&=\left(\epsilon _{ii}\sigma_{ii}^{6}\epsilon_{jj}\sigma_{jj}^{6}\right)^{1/2}\\ \epsilon_{ij}\sigma_{ij}^{12}&=\left[{\frac {(\epsilon_{ii}\sigma_{ii}^{12})^{1/13}+(\epsilon_{jj}\sigma_{jj}^{12})^{1/13}}{2}}\right]^{13} \end{align*}\)

2.5.6. Additional Resorces#

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