# What are B and n in this expression for charge repulsion?

In a study of the creation of ionic bonds of in this case $\ce{Na}^+$ and $\ce{Cl}^-$ into $\ce{NaCl}$, I have come across this equation for the overall energy of the system:

$$E=-\frac{e^2}{4\pi\epsilon_0 R}+\frac{B}{R^n}+1.4 \,\mathrm{eV}$$

$e$: electron charge. $\epsilon_0$: vacuum permittivity. $R$: the distance between them.

• The first term is the attraction energy between the two equal but oppositely charged ions $\ce{Na}^+$ and $\ce{Cl}^-$.
• The second term is the repulsion between the like charges within the shells. When the distance $R$ is very small, this term has large (opposite) effect.
• The last term is the difference in free creation energy for the two free reactions ($\ce{Na \rightarrow Na^+ + e^-}$ requires $+5.1\,\mathrm{eV}$ and $\ce{Cl + e^- \rightarrow Cl^-}$ gives $-3.7\,\mathrm{eV}$)

My question is two-fold and about the repulsion energy term: What is $B$? And what is $n$? I have been told that the exponent is around $n \approx 10$, but is this a general constant or material specific?

• B is the magnetic field. See en.wikipedia.org/wiki/Magnetic_flux. BTW, B was used by J. C. Maxwell in the electromagnetic equations; he simply used Latin alphabet letters A through H. – DrMoishe Pippik May 29 '15 at 21:14
• No, B can't be a magnetic field here. The units wouldn't even work out. – Maria May 29 '15 at 22:01

There isn't an exact form for the Pauli Repulsion term, but it is often approximated as a $\frac{1}{r^n}$ term for simplicity. For example, in the Lennard-Jones expression for Van Der Waals interactions, $$E(r) = \epsilon \times[(\frac{\sigma}{r})^6-(\frac{\sigma}{r})^{12}]$$, the $\frac{1}{r^6}$ term is the London force term (which in reality, does go as $\frac{1}{r^6}$), and the $\frac{1}{r^{12}}$ is the Pauli repulsion term (just an approximation, but in this approximation, $n=12$).
In the OPLS force field (which uses a Lennard-Jones expression for Van Der Waals interactions), the parameters for $\ce{Na+}$ and $\ce{Cl-}$ are:

$\ce{Na+}$: $\sigma = 3.500 A$, $\epsilon = 0.0660\ \ce{kcal/mol}$

$\ce{Cl-}$: $\sigma = 4.180 A$, $\epsilon = 0.11779\ \ce{kcal/mol}$.

OPLS is parameterized for a geometric mixing rule, meaning that for an interaction between the two, the $\epsilon$ and $\sigma$ parameters are equal to the geometric mean of the parameters for each atom. Thus, for an $\ce{Na+ - Cl-}$ interaction:

$\sigma = 3.8249 A, \epsilon = 0.08817\ \ce{kcal/mol}$

Now, the Pauli part of the Lennard Jones expression is $\epsilon \times (\frac {\sigma}{r})^{12}$, so $B$ in this case is $\epsilon \times \sigma^{12} = 864554\ \ce{kcal A^12 /mol}$

However, there are other force fields parameterized differently, and some of these approximate the Pauli repulsion using a different value of $n$ (and some like $MM3$ don't even assume a $\frac{1}{r^n}$ dependency). So since $n$ is assumed (and since different force fields are parameterized differently), there are other possible values for $B$ and $n$ to use in the approximation.

• Thank you for a good answer. What is the "OPLS force field"? – Steeven May 30 '15 at 8:01
• en.wikipedia.org/wiki/OPLS – Mithoron May 30 '15 at 12:29
• OPLS (Optimized Potential for Liquid Simulations) is a commonly used force field. A force field is simply a specification for the various types of interactions between atoms (e.g. Van Der Waals interactions, bond stretching behavior, torsional energies, etc...), usually used for computational purposes such as simulations, structure prediction, potential energy scans, etc. – Maria May 30 '15 at 14:06