content_image/373660.jpg content_image/373661.jpg content_image/373662.jpg content_image/373663.jpg
so efficient matrix square roots algorithms that avoid diagonalization are computationally advantageous [21]. 

In terms of convergence, Eq. (38) is superior to the standard integral form of Eqs. (19) or (23). There is only a single convergence parameter which is the size of the basis set of transitions {| t 〉} in the calculation. For the integral forms, one must converge additionally the continuous ω integral by dealing with issues of grid spacings, truncation at large ω , etc. In Section 11, we will highlight the superior convergence properties of Eq. (38) numerically for atoms. 

## 8. UNBOUNDEDNESS OF Φ GW xc AND LACK OF EXTREMUM OF F 

In this section, we prove that Φ GW xc [ G$_{0}$ ] is in fact unbounded from below: its minimum value is negative infinity. Furthermore, the total energy functional F can not have an ex- tremum when optimized freely over non-interacting Green’s functions G$_{0}$ that are generated by non-local potentials U$_{0}$ . We then discuss the meaning and consequences of these facts. 

Beginning with the total energy functional of Eq. (14), we see that the only term depend- ing on the ϵ$_{n}$ is the exchange-correlation part Φ$_{xc}$[ G$_{0}$ ]. For the GW -RPA approximation, we have the exact expression of Eq. (38) for Φ GW xc . This functional is unbounded from below if varied freely over ϵ$_{n}$ . Consider sending all the eigenvalues towards the Fermi en- ergy, ϵ$_{n}$ → µ . This makes all transition energies tend to zero from above, Δ$_{t}$ → 0 $^{+}$. The RPA/Casida Eq. (36) immediately shows that the plasmon energies must also tend to zero ω$_{p}$ → 0 + as well. Thus Φ GW xc will turn into 

Writing this out explicitly in coordinate space 

<img src='content_image/373660.jpg'>

<img src='content_image/373661.jpg'>

Using completeness 

where v ′ labels occupied states, we find 

<img src='content_image/373662.jpg'>

<img src='content_image/373663.jpg'>

Since V ( x, x $^{′}$) = δ$_{σ,σ}$ ′ / | r − r $^{′}$| diverges to positive infinity for x = x $^{′}$, we conclude that Φ GW xc diverges to negative infinity when all eigenvalues approach the Fermi energy. This particular choice of variation of the ϵ$_{n}$ is perfectly permissible if one allows for arbitrary variations of the non-local potential U$_{0}$ . We give numerical evidence of this behavior for atoms in Section 11. 

