# Arguments for the powers used in the Lennard Jones Potential

#### grep6

Can anyone offer an explanation for the powers (6 and 12) used in the Lennard Jones potential? (other than simply saying weak attractive and strong repulsion forces)

I am especially looking for short derivation that can be used to argue that the attractive potential falls off with the sixth power of interatomic distance.

Also, I understand that the repulsive part is used as a convenient way to mimic an exponential. How do we go from the Pauli exclusion principle to an exponential barrier?

Thanks

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#### TeethWhitener

Gold Member
The 12th power repulsion is indeed a convenient approximation and has no physical basis. The 6th power attraction is based on induced dipole/induced dipole interactions. Dipole-dipole interactions have a potential energy of the form $V\propto 1/r^3$ (this can be found in most classical E&M books). To get the van der Waals potential, this interaction is treated quantum mechanically as a small perturbation. The first order perturbation is zero, so the largest term in the perturbation series is the second order term, meaning that the van der Waals potential is of the form $V_{pert}\propto 1/r^6$.

The repulsion term is a bit more complicated (and more difficult to find good references on). For a simple example of how this is derived, consider two closed shell atoms (like helium). We assume that the total wavefunction is a linear combination of the atomic wavefunctions:
$$|\Psi\rangle = a|\psi_a\rangle + b|\psi_b\rangle$$
Now we find the expectation value of the energy (and assuming that the atomic wavefunctions are normalized--but not necessarily orthogonal):
$$\langle \Psi|H|\Psi\rangle = |a|^2E_a+|b|^2E_b+a^*bE_b\langle \psi_a|\psi_b\rangle + b^*aE_a\langle \psi_b|\psi_a\rangle$$
It's clear that the energy dependence on distance will be contained in the overlap terms $\langle \psi_a|\psi_b\rangle$ and $\langle \psi_b|\psi_a\rangle$. These terms are quite difficult to calculate for multielectron systems, but the upshot is that an exponential decay $V_{rep}\propto \exp{(-r)}$ turns out to be a decent approximation for them, particularly in simple cases where the atoms are closed-shell and there is no bond formation.

"Arguments for the powers used in the Lennard Jones Potential"

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