indeed, they are often considered as hydrophobic interactions (short-range interactions), despite the fact that in many occasions the formation of induced dipoles between electron clouds is present. As a guideline, consider polar interactions the ones that include polar atoms as oxygens and nitrogens, often in hydrogen bonds (with hydrogen bound to O and N) but also when a charged group is present (salt bridges, or dipoles).
In the concept of the potential of mean force, for example, when two methane molecules interact in water, the interaction of water-water, methane-water, methane-methane is taken into account. Methane-methane can be considered a van der Waals interaction. If we consider the interaction of two molecules of benzene in water, then it is necessary to take into account the pi-pi interaction of benzene-benzene, benzene-water, water-water. Strange as it may seem, the "pure" hydrophobic interaction in this water-water sum is strange.
The minimum W (r) is 2.1 kJ / mol at r = 0.5 nm at which the benzene molecules are oriented perpendicular to each other. P.Linse J.Am.Chem. Soc. 1993. 115. 8793.
I was NOT writing about neat benzene molecules, but about aromatic moieties with attached groups capable of forming H-bonds. in the case of such systems e.,g., diamines - two possible forms of molecular packing are possible - 1st - due to H-bonds, the 2nd one due to pi-pi interactions between oriented plane-over-plane pile of aromatic rings. In such situation the difference in formation of the above described 2 posssible structures is negligibly small.
The problem with grading of pi-interactions as hydrophobic is that these terms are mostly non-comparable. One can name as hydrophobic any molecular interactions that are weaker than those between water molecules in liquid water so that aggregation of solute is possible in aqueous solution. The terms “pi-interactions”, “CH-pi-interactions”, “halogen bonds”, etc. are from crystallography, which tries to describe relationships between structure of organic molecules and their crystal packing or melting parameters as if molecules in crystals have the same aggregation and orientation equilibrium as in liquids. This is not the case. The main difference between molecular interactions in liquids and in crystals is that they have different dominant geometric factors. For liquids, the surface of interacting molecules is important, while for crystals the molecular shape is relevant. Hence, an extrapolation of any relationships and estimations based on weak interactions in solids to liquids and vice versa may be compromised.