# Van der Waals forces [duplicate]

What exactly are Van der Waals forces? How do they arise and how can an instantaneous dipole cause another dipole? Shouldn't this make a chain reaction that causes all matter to eventually become polar?

• – Tan Yong Boon Oct 29 '19 at 6:42

The post that was linked to you in comments contains a nice, short description of the 'traditional' categories of van der Waals interactions. I suggest you take a look at that, it might just be enough to clear things up. The reason I am writing this post is that the categories introduced there are defined somewhat arbitrary, so I'll try to introduce some terminology that (as of today) is considered correct in the field.

So given a molecular system, it is quite easy to factorize the forces into intermolecular and intramolecular forces. Intramolecular forces are what hold the molecules together. These are (almost completely) covalent bonds, and these interactions would be same for one single molecule and the bulk phase of the matter. But we know, from the real life, that matter does not only exist as an ideal gas: we got liquids, solids, and even the gases are non-ideal. So evidently, there has to be forces between molecules. These forces are collectively called van der Waals forces for historical reasons.

Now there are three different types of interaction between molecules. There are electrostatic, induction and dispersion interactions.

Electrostatic effects are simple classical interaction between charged particles. Thesy simply come from the Coulomb law, can be either attractive or repulsive, and they are strictly additive (because Coulomb law is additive)

It is tricky to properly understand induction and dispersion. These two effect come from quantum mechanical perturbation theory. Without getting into details, perturbation theory means that the wavefunction of two molecules together is not simply the sum (or average) of the wavefunctions of the separate system, but there is an interaction between them, but this interaction can be expressed in terms of the interactionless wavefunction. The expression for the second-order energy correction is given by the Rayleigh-Schrodinger picture as follows (this is called the polarization approximation):

$$W^{(2)} = - \sum_{mn}' \frac{ <00|H'|mn> }{W_{mn}-W_{00}}$$

In this expression, $$|m>$$ represents the mth excited state of the original wavefunction of molecule A and $$|n>$$ represents the nth excited state of the original wavefunction of molecule B. This expression could be read: the energy difference between the two separate systems and the interacting system depends on the excitations in the separate systems through an interaction operator $$H'$$.

The sum can be factorized into three components:

$$W^{(2)} = - \sum_{m\neq0}' \frac{ <00|H'|m0> }{W_{m0}-W_{00}} - \sum_{m\neq0}' \frac{ <00|H'|0n><0n|H'00> }{W_{0n}-W_{00}} - \sum_{m\neq0,n\neq0}' \frac{ <00|H'|mn> }{W_{mn}-W_{00}}$$

Here, the first term contains the energy contributions of the interactions of the ground state of A with the excited states of B. This is called the induction energy of B. The second term is the induction energy of $$A$$, which again, expresses the total response of the molecule A to the ground state of B. The third term is the so-called dispersion interaction, which is then a mixing of the excited states of both molecules.

So these expressions are the defining expressions for induction and dispersion. One should know that both of these interactions are always attractive , they are non-additive, and they are always present, between charged and neutral particles too. Also don't get confused that the energies are expressed in terms of excited states: there is no excitation going on in the system, it's just the mathematical tool that we used as definitions.

The total intermolecular force, or van der Waals force is then usually the sum of electrostatic, induction and dispersion terms. For the sake of completeness we might include resonant energy couplings and magnetostatic interactions too, but these are either rare or so small that we can neglect them.