How to explain Born repulsion between ions in gas phase?

Question

In simple terms, the collision of two atoms $\ce{A}$ and $\ce{B}$ will result in ions $\ce{A^+}$ and $\ce{B^-}$ if $$I_a(\ce{A})+E_a(\ce{A})<I_a(\ce{B})+E_a(\ce{B})$$ where $I_a$ and $E_a$ are the ionisation energies and electron affinities, respectively. Let us view $\ce{A^+B^-}$ as a molecule of sorts, having ions of rigid, non-polarisable spheres. Say $r$ is the sum of the radii of such spheres. The potential energy $E_p$ of such a system would then be $$E_p=I_a(\ce{A})-E_a(\ce{B})-\frac{e^2}{r}+\frac{b}{r^n}.$$

It is intuitive that

• the term $I_a(\ce{A})-E_a(\ce{B})$ characterises the energy required to form isolated ions $\ce{A^+}$ and $\ce{B^-}$ in the gas phase;
• $-e^2/r$ takes into account culonic attraction between two ions.

It is not intuitive for me why ${b}/{r^n}$ is how it is.

The final term, introduced by Max Born, encompasses the repulsion generated by shells of electrons at either ion.

Of course, such effects cannot be ignored. But I am interested in why is it $b/r^n$. What is the justification? Feel free to provide the rigorous approach, even though it will probably$^{[1]}$ be above my abilities of comprehension.

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Browsing on Wikipedia, the information might be given in the book Advanced Inorganic Chemistry by F. Albert Cotton, Geoffrey Wilkinson, Carlos A. Murillo, and Manfred Bochmann.

The pertinent section “1-6. Energetics of Ionic Crystals” starts at page 18$^{[2]}$, however I cannot find the derivation. There is only that the value of $n$ relies on a compressibility measurement, given as fractional change in volume per unit change in pressure, or

$$\frac{\Delta V}{V\Delta P}.$$

Also, this is seems somewhat contradictory to the earlier assumption of rigid spheres as written in another book$^{[3]}$. So, if anything, the linked book confused me further.

$^{[1]}$ Most certainly.

$^{[2]}$ Page 28 in the PDF.

$^{[3]}$ U. Palm, V. Past. Physical Chemistry. (1974) [To my knowledge, not available in English.]

Answer 1

The term $+b/r^n$ is a power law repulsion term between the ions, where $b$ is the bare ion radius (twice the van der Waals radius) and $r$ the separation.

The hard sphere model considers the ions as 'billiard balls' which means that $n=\infty$ since $+b/r^n$ is effectively zero when $r \gt b$ and infinite when $r \lt b$.

Two other types of potentials are used (a) the power law potential when $n$ is some integer value, usually between 9 and 16 and (b) the exponential potential $+a_0exp(-r/a_1)$ where $a_0$ and $a_1$ are adjustable parameters with $a_1 $ of the order of 0.02 nm.
The justification for using these potentials is weak (the power law particularly so) and they are used as they represent a better approximation than the hard sphere potential and are mathematically convenient. The exponential form is presumably justified since the combination of all orbitals decay exponentially with distance.

Answer 2

You may be aware of the general potential energy vs r curve? In which at r=0 energy is very repulsive, possibly infinitely so for some approximations, and then drops rapidly into a trough, the minimum of which is the 'equilibrium' distance r(eq) and as r increases rises back up? From my ed.3 Cotton&Wilkinson, pg 59.(Somewhat abbreviated) The reason the ions reside at r(eq) from each other and do not move closer is there are short-range repulsive forces due to overlapping electron clouds [so much for rigid spheres, huh?] Born proposed the simple assumption that the repulsive force between two ions could be represented as B'/rⁿ where B' and n are constants characteristic of the ion pair. We can therefor write E(repulsion) = B/rⁿ where B is related to B' by the crystal geometry. so, the reason has more to due with the shape of the P.E. curve (of solid state crystals, but perhaps you know a bit about how ions in the gas phase can be considered to occupy a lattice?). I have not researched this, but my w.a. guess is that it's a fudge factor, forcing r(eq) to the correct value. It may turn out, upon doing the quantum mechanics, that there's a deep reason that the form of this term works out so well. My guess is that Born just wanted something easy to differentiate or integrate - why make it more difficult than needs be? - and no one has seen a need to improve upon it, given it's empirical derivation. It turns out that n can be determined based on compressibility (C&W are talking about solid crystals, so...) as well as theoretically and B = {-A*(Z-)(Z+) e²*[r(eq)^(n-1)]}/n from the lattice energy. C&W go on to state that corrections to their equation are sometimes used (keep in mind Edition 3 is ancient! 1972 !) and specifically that this repulsion term is "not strictly correct from quantum-mechanical considerations. More refined expressions do not greatly change the results, however." where A is the Madelung constant, Zs are the ions charges.