Importance of higher order exchange terms? The MP2 ground state energy of a molecule or solid can be written as

$$E^{(2)} = \frac{1}{2}\sum^{\text{occ.}}_{ij}\sum^{\text{virt.}}_{ab} \frac{\langle ij | ab \rangle [\langle ab|ij \rangle - \langle ab|ji\rangle]}{\varepsilon_i + \varepsilon_j - \varepsilon_a - \varepsilon_b}$$

where the first part of the numerator is commonly called "direct MP2" and the second part "exchange MP2", since i and j are exchanged. 

However, for energy differences, it is often sufficient to calculate only the direct MP2 energy, since the contribution of the exchange part is mostly negligible. For the same reason the random phase approximation, which consists of these "direct terms" up to infinite order, works so well for many materials. 

But are there good examples (materials or molecules) where the exchange term is important? I guess magnetic systems or situations where Pauli repulsion is important... but does someone know particular examples?
Exchange is important and is present in any system. To be more specific, in MP2 you already have exact exchange for the ground state from the zeroth-term(HF) while in the second-term (MP2) you have a correction to correlation plus some more exchange from the excited state.

MP2 allows the electrons to be partly in the unoccupied MO however, it is limited to double excitations. It turns out that you end up having a small correction to exchange and a big correction to correlation which is completely missing in HF. So the exchange correction will be small in every system due to the type of excitation treated in MP2.