Lennard-Jones potential for 3 atoms

In summary, the conversation discusses the calculation of total acceleration for a particle in a 3 atom lattice, taking into account the Lennard-Jones interatomic potential. The derived equation for the acceleration is a sum of two components, one for the interaction with atom 1 and one for the interaction with atom 3. This equation is then confirmed by a sanity check, showing that the sign of the acceleration is correct.
  • #1
barnflakes
156
4
If I have a 3 atom lattice laid out at positions ##x_1 = 1, x_2 = 2, x_3 = 3##

##x_1## and ##x_3## are stationary, but ##x_2## interacts with ##x_1## and ##x_3## via the Lennard-Jones interatomic potential.

I therefore worked out that the total acceleration for ##x_2## for ##m = \sigma = \epsilon = 1## is:[tex]a_2 = 24(2r_{21}^{-13} - r_{21}^{-7}) + 24(-2r_{32}^{-13} + r_{32}^{-7});[/tex]

where ##r_{21} = |x_2 - x_1| , r_{32} = |x_3 - x_2|##

Is this correct?
 
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  • #2
First, I assume that the in this particular case the Lennard-Jones potential is written as
$$
V_\mathrm{LJ}(r) = 4 \epsilon \left[ \left( \frac{\sigma}{r} \right)^{12} - \left( \frac{\sigma}{r} \right)^{6}
\right]
$$

barnflakes said:
Is this correct?
Yes, since
$$
a = \frac{F}{m} = \frac{1}{m} (- \nabla V)
$$

It is always good to make a sanity check. If ##r_{21} \rightarrow 0##, then atoms 1 and 2 are getting very close and the repulsive force dominates, so that should lead to an positive acceleration (i.e., to the right), and conversely for ##r_{32} \rightarrow 0##. The sign does appear to be correct.
 

1. What is the Lennard-Jones potential for 3 atoms?

The Lennard-Jones potential is an empirically derived mathematical model used to describe the interaction between atoms or molecules in a system. It combines both repulsive and attractive forces and is often used to study the behavior of atoms in a gas, liquid, or solid state.

2. How is the Lennard-Jones potential calculated?

The Lennard-Jones potential is calculated using the following equation:
V(r) = 4ε[(σ/r)^12 - (σ/r)^6], where r is the distance between two atoms, ε is the depth of the potential well, and σ is the distance at which the potential is zero.

3. What is the significance of the parameters ε and σ in the Lennard-Jones potential?

The parameters ε and σ in the Lennard-Jones potential represent the strength of the attractive and repulsive forces between atoms, respectively. They are often determined experimentally for a specific system and can be used to predict the behavior of the atoms in that system.

4. What is the physical interpretation of the Lennard-Jones potential?

The Lennard-Jones potential is a simple model that can be used to understand the behavior of atoms in a system. The attractive term in the equation represents the van der Waals forces between atoms, while the repulsive term represents the overlapping of electron clouds, resulting in a repulsive force.

5. What are the limitations of the Lennard-Jones potential for 3 atoms?

The Lennard-Jones potential is a simplified model and does not take into account all of the complexities of atomic interactions. It is only accurate for non-polar, spherical atoms and cannot be applied to systems with polar or charged atoms. Additionally, it does not consider quantum effects and cannot be used to accurately describe the behavior of atoms at very high energies.

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