Diploma - Max Planck Institute for Solid State Research

4 2 Theoretical foundation
fore the electronic (represented by ψ(r, R)) and the nuclear part (represented by ψ(R))
decouple using a product ansatz Ψ (r, R) = φ (R) ψ (r, R):
H s e
{ }} {
(Te
s + Vee s + Vne) s ψ (r, R) = E e (R) · ψ (r, R) (2.2)
(T s
n + W s (R)) φ (R) = E ne · φ (R) (2.3)
with
W s (R) = E e (R) + V nn (R)
This procedure is called Born-Oppenheimer approximation (BO) [12]. Equation 2.2 is
referred to as electronic Schrödinger equation (SE) because the coordinates R of the
nuclei can be regarded as parameters since the spatial representation (of the operators)
does not contain a differential operator with respect to them. Eq. 2.2 can only be solved
approximately (e.g. for small molecules / compounds: Configuration Interaction; for
molecules as well as solids: Hartree-Fock, Thomas Fermi theory, density functional
theory (DFT); all mentioned algorithms are solved iteratively) despite the analytical
results for H and H + 2 . The solution of eq. 2.3 depends on the knowledge of the highlydimensional
Born-Oppenheimer surface and can just be solved for rather small or vastlysymmetric
molecules (cf. C 60 ).
Assuming that there is no interest in vibrational or rotational modes of the nuclei (i.e.
phonons in crystals), their positions can be locked and eq. 2.3 could be neglected. In
principle the statement is even stronger since the time frame for the motion of electrons
is considerably smaller than the one for the nuclei.
The solution of eq. 2.2 allows to deduce fundamental electronic and optical properties
of crystalline solids (e.g. if the solid is an insulator, a semiconductor or a metal)
by analyzing the dispersion of the eigenvalues (the so-called band structure) in the
reciprocal space (also called momentum space, k-space). The Wigner-Seitz cell (smallest
unit cell of the compound) transformed into the momentum space is called first Brillouin
Zone (BZ). The periodicity (originated in translational invariance of real space) is an
intrisic property of the momentum space introducing two different models: the reduced
and the extended BZ scheme. The latter is the reduction to the first BZ disregarding
the absolute value of the momentum (being defined as recurrent with period 2π a , a being
the lattice constant) which is – in most cases – sufficient. The former scheme is more
suitable for evaluating photoemission spectra because in this case one needs momentum
conservation.
Besides the afore mentioned iterative solutions (like DFT), there exists a second class
of methods for calculating the band structure of solids employing either pertubation
theory or the superposition of atomic solutions (e.g. k · p perturbation theory or tight
binding).