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

Question

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?

Answer 1

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.