# Arguments for the powers used in the Lennard Jones Potential

• grep6
In summary, the powers 6 and 12 in the Lennard Jones potential can be explained by induced dipole/induced dipole interactions and the Pauli exclusion principle. The attractive potential falls off with the sixth power of interatomic distance due to a quantum mechanical perturbation series, while the repulsion term is derived from the overlap of atomic wavefunctions and is approximated by an exponential decay.
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

Last edited:
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.

## What is the Lennard Jones potential?

The Lennard Jones potential is a mathematical model used to describe the interaction between two neutral atoms or molecules. It takes into account both attractive and repulsive forces between the particles.

## Why is the Lennard Jones potential important?

The Lennard Jones potential is important because it is widely used in molecular simulations and studies of intermolecular interactions. It allows for the prediction of properties such as equilibrium distances, binding energies, and phase transitions.

## What are the parameters used in the Lennard Jones potential?

The Lennard Jones potential has two main parameters: the depth of the potential well (ε) and the distance at which the potential is zero (σ). These parameters can vary depending on the specific atoms or molecules being studied.

## How is the Lennard Jones potential derived?

The Lennard Jones potential is derived from the interatomic or intermolecular forces, which can be described using the Lennard Jones 6-12 potential function. This function is based on the assumption that the forces between particles can be approximated by a combination of short-range repulsive and long-range attractive forces.

## What are some limitations of the Lennard Jones potential?

The Lennard Jones potential is a simplified model and does not take into account more complex interactions such as dipole-dipole or quadrupole-quadrupole interactions. It also does not account for the effects of temperature or pressure. Additionally, it is limited in its applicability to only neutral particles and cannot accurately describe interactions between charged particles.

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