content_image/367936.jpg
the simplest COHSEX self-energy of Eq. (52) and COHSEX correlation energy of Eq. (51) contain the non-local density matrix ρ$_{0}$ that projects the action of the screened Coulomb interaction W onto the occupied states. If the occupied states are derived from localized states such as 3 d or 4 f orbitals of transition metals, then the projection is automatically onto this manifold. Namely, already at the COHSEX level, we expect to recover the benefits of an LDA+U type treatment. This is no surprise since a static and localized approximation to the GW self-energy yields the kernel of the LDA+U approach [7]. As an added benefit, a COHSEX type approach should automatically include a properly screened U parame- ter, as opposed to U being an externally chosen parameter in the usual LDA+U method. The dynamic formulae of Eqs. (44) and (45) extend these results to include the frequency dependence of the screening. What is missing from these approximate results are the con- tributions to the RPA correlation energy which are physically distinct from those stemming from screened interband transitions: these are the neglected residue contributions in Eq. (43) from the poles from the screening modes themselves. Unfortunately, at present we are un- able to provide simple physical examples of situations where these neglected contributions play the dominant role and our key approximation to fail for basic physical reasons. This question is a subject of present investigation. 

## 11. NUMERICAL TESTS: ATOMS 

In this section, we describe numerical results on atomic systems. The aim of this section is not to present an exhaustive and comprehensive treatment. That is the subject of a future investigation. Rather, the main aim is to demonstrate the numerical efficacy of the plasmon formula Eq. (36) for the GW -RPA correlation energy and to test the main approximation of high frequency screening used in the approximate results of Eqs. (44) and (45). 

Our atomic code uses a standard non-relativistic approach. The atomic eigenfunctions are assumed to take a spherical form: for an eigenstate with spin index σ = ± 1, the spatial part is R$_{nlσ}$ ( r ) Y$_{lm}$ ( θ, φ ). The radial part R$_{nl}$ ( r ) is represented numerically on a radial grid of exponentially-spaced points [63, 64]. The spherical harmonics Y$_{lm}$ are included up to a some maximum angular momentum l$_{max}$ , typically l$_{max}$ = 3 below. When building up the one-particle density matrix ρ$_{σ}$ for spin σ , 

<img src='content_image/367936.jpg'>

we allow for m -dependent state fillings f$_{nlmσ}$ ∈ { 0 , 1 } . The local spin-density approximation (LSDA) [62] or unrestricted Hartree-Fock (HF) [65] equations are solved by minimizing the appropriate total energy over arbitrary occupied orthonormal radial functions. We have tested our code with available high quality LSDA [64] and HF [66] data to ensure agreement to at least one part in 10 8 in total energies. When computing various correlation energies below, we make a spherical approximation which is consistent with assuming spherical eigen- functions. This means that when we compute the polarizability ( P or P$_{t}$ ) and the screened interaction ( W or W$_{t}$ ), we assume the fillings have no m -dependence ( i.e. , shell occupancies f$_{nlσ}$ instead of orbital occupancies f$_{nlmσ}$ ): different m -channels will not interact so that taking both l and m as good quantum numbers is a self-consistent assumption. However, 

