The problem of the occupied bandwidth of sodium has been discussed for decades. Sodium is a simple metal and the effect of the lattice potential on the valence bandstructure is very small. The valence electrons of sodium should therefore behave like a homogeneous electron gas at a density of r_s=4. However, angular resolved photoemission measurements of the occupied bandwidth of sodium have shown a narrowing of the occupied valence band by 0.7 eV compared with the free electron bandwidth.

Such effects can be investigated using a self-energy approach. The GW self-energy approximation was proposed by Hedin in 1965 and is based on the self-consistent Hedin equations. In this approximation vertex corrections in the expansion of the self-energy are neglected. The self-energy operator is then evaluated as the product of the interacting Green function G and the effective interaction W. This self-consistent set of equations relates the self-energy, the screened interaction W, the irreducible polarisation propagator P and the one-electron Green function G. The initial Green function is usually constructed from LDA or Hartree-Fock single-particle orbitals. Normally the calculation is stopped after one iteration, which gives a band narrowing for the homogeneous electron gas at rs=4 of 0.3 eV, while a self-consistent GW calculation gives a broadening of 0.5 eV.

The nature of the self-energy corrections to unoccupied states is also of interest. Experiments using Bremsstrahlung isochromat spectroscopy (BIS) on the nearly free-electron like metal aluminium have found very good agreement with the theoretical predictions of the non-self-consistent GW approximation. Recently, however, the validity of the GW for higher energies has also been challenged.

The accuracy of the GW approximation for the self-energy of interacting electron systems remains uncertain. In principle one can calculate the bandwidth of the interacting uniform electron gas directly using quantum Monte Carlo methods. However, our first calculations have shown a broadening of the bandwidth which is believed to originate from the poor quality of the trial wave function. The development of suitable trial wave functions is the subject of current research.

Furthermore, expressions for the self-energy corrections of the homogeneous electron gas both for high-energy excitations have been developed in our group. Both operators are expressed in terms of ground-state observables of the homogeneous electron gas which can be calculated with quantum Monte Carlo.

In this project we hope to develop a full picture of the energetic structure of the homogeneous electron gas over the full energy range. This could improve our understanding of the validity of the GW approximation and other methods for the calculation of excited states.