Inconsistent index ordering in GW self-energies

[YannIntVeld]

This is a problem in the unstable version of TPRF.

The problem

The order of the orbital-indices of the rank-4 Coulomb tensor W in the gw_sigma and g0w_sigma routines is not consistent with those of other two-particle Greens functions used in TPRF. It is also not consistent with the definition in the 'theory and notation' section of the documentation.

For example, the convention used for the density-density terms in the current implementation is $V{abab}$, whereas density-density terms in other parts of TPRF would be $V{aabb}$.

Possible solution

Update the order of the orbital indices in the current GW implementation and implement a benchmark for the GW routines which depends on the order of indices. Some ideas for benchmarks are:

• Hubbard dimer
• Simple two-band semi-conductor

[HugoStrand]

I propose that we work on this in the pull request #40.