104 Density functional theory
the energy spectrum, according to the time–energy uncertainty relation. Therefore,
we can no longer speak of a discrete energy level, but we can still find a fingerprint
of the spectrum in quantities such as the macroscopic dielectric function, which is
the long-wavelength limit of the microscopic functionε(r,r,ω). This will exhibit
peaks as a function ofωwhose centres can be viewed as energy levels, and the
widths as lifetimes.
Experimentally, excited states are studied by using spectroscopy techniques. In
directphoto-emission, an incident photon excites an electron to sufficiently high
energy that it can leave the system (ionisation). Ininversephoto-emission we send
an electron into the material to occupy an unoccupied state, causing emission of a
photon whose energy is detected (electron addition). In absorption spectroscopy,
the electron or photon that is sent into the system is also detected when it leaves
the system. It may meanwhile have changed its energy by interaction with another
electron in the material which is excited to a higher state.
The first two processes, ionisation and electron addition, are calledone-particle
processes; the third is atwo-particleprocess. The two-particle character arises
because, when an electron is excited in a system, it leaves a (positive) hole behind. If
the electron remains in the system, it interacts with the hole, and in particular it may
form anexciton: a bound state of the particle–hole system. We shall briefly describe
the analysis for one-particle processes, and then review two-particle methods.
In the previous subsection, the problem of finding the ionisation energy was
addressed. In general, when performing spectroscopy experiments, levels other
than the highest one may be excited. Of course, one could try to use a generalised
delta-SCF procedure for these, but this is difficult because for a band structure
calculation, we would need many calculations as eachk-vector in the Brillouin
zone has its own particular excitation. Another problem is that for a band state,
DFT differs essentially from Hartree–Fock, which allows for calculating excited
states: the HF orbital energies can be interpreted as excitation energies accord-
ing to Koopman’s theorem (which only holds for the highest band in DFT, see
above). This theorem is based on the assumption that the orbitals do not relax
when the configuration changes by emptying full, or filling empty levels. This
approximation fails miserably in solids, for example in diamond, where the band
gap is in HF predicted to be 15 eV, more than twice the experimental gap of
about 7 eV.
What is missing from the description of a ground state system, is the fact that an
electron added to the system does not feel the pure Coulomb interaction from the
ground state charge distribution: the resident electrons will re-order in the presence
of the visiting electron, and tend toscreenthe effect of the Coulomb interaction. A
many-body theory for band structure takes these effects into account; HF and DFT
do not.