# 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?

• Complete guess, with a rather indirect supporting argument: $\ce{F2}$. – hBy2Py Apr 29 '18 at 1:06
• @hBy2Py It would be more conclusive to look at an energy difference, say its IP. Do you what kind of bonding behavior the cation displays? – pentavalentcarbon Apr 29 '18 at 13:30
• IPs are a very good example, see e.g. here journals.aps.org/prl/abstract/10.1103/PhysRevLett.112.096401. But this I already knew ;-) – thyme Apr 29 '18 at 13:39
• I actually don't think this paper presents enough information to be conclusive. Would you say the differences between the GW0, TC-TC, and gamma methods are significant for carbon, MgO, ZnO, and CdO? It's a difference of 0.5 eV at most. – pentavalentcarbon Apr 29 '18 at 13:53

## 1 Answer

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.

• Thank you for the answer. I know all that, but I believe that there are systems where the MP2 exchange is not just tiny and negligible. You wrote "exchange is important in any system" but you also wrote "the exchange correction will be small". Yes, mostly, for energy differences, you can disregard it. And I am wondering if there are systems where you can not disregard it, without introducing a significant error. Please note, that I only talk about the MP2 exchange (not exact exchange from Hartree-Fock). – thyme Apr 29 '18 at 7:16
• Look at the S66 collection (cdn-pubs.acs.org/doi/10.1021/ct2002946) that analyzes weak, non-covalent interactions. Exchange is extremely important in many of these. – LordStryker Jun 28 '18 at 11:43