Bibliography.

[KX] Σ. Κομηνέας, Ε. Χαρμανδάρης, “Μαθηματική Μοντελοποίηση”.

Research article: J.B. Adams, “Bonding energy models”, Encyclopedia of Materials: Science and Technology, 763767 (2001). (Sec. 2.1)

Lennard-Jones (LJ) potential

The chemical bond between two atoms results from the interaction between them. There are various forces that give such interactions. These can be

• electrical forces (protons and electrons in atoms have electric charges)
• quantum mechanical forces. (The Pauli exclusion principle tells us - in a very rough form - that two electrons cannot be very close to each other. More precisely, it tells us that electron clouds may only have small overlap).
• magnetic, etc.

The Lennard–Jones potential (LJ) is a model used to describe the interactions between atoms, characterized by a balance of weak van der Waals attractive forces (dipole–dipole interactions) and short-range repulsive forces (due to Pauli exclusion principle).

Let us consider two atoms at a distance $r$. An empirical potential that describes the interaction between them is the potential Lennard-Jones,

$$ V(r) = D\left(\frac{R}{r}\right)^{12} - 2 \left(\frac{R}{r}\right)^6 $$

where $D, R$ are constants.

[It was proposed in 1924 by the British mathematician Sir John Edward Lennard-Jones.]

The potential has two terms

• The attractive term (second term) is assumed to decay with distance as the inverse sixth power (dipole–dipole interactions). Remember that a Coulomb potential between two charges is $V\sim-1/r$. Since in this case we have interaction between dipoles, the potential should decay more rapidly at large distances.
• The repulsive term (first term) is assumed to decay more rapidly for large distances (inverse twelfth power) and be significant only at short distances.

Note that

$$ \begin{aligned} & V(r) \to \infty,\qquad \text{when}\;\; r\to 0 \\ & V(r) \to 0,\qquad\,\, \text{when}\;\; r\to \infty. \end{aligned} $$

• The potential energy is set to zero at infinity (when the atoms are far from each other).
• The atoms can never get on top of each other (at $r=0$).

Solid line: Lennard-Jones potential. (Values of parameters $D=1, R=5$.)

To start studying the functional form of the potential, we find its derivative,

$$ \frac{dV}{dr} = \frac{12}{r}\,\left[ \left(\frac{R}{r}\right)^6 - D \left(\frac{R}{r}\right)^{12} \right]. $$