 BornOppenheimer approximation 
Definition:  The Born–Oppenheimer
approximation is one
of the most
effective
simplifications in
quantum mechanics.
By decoupling the
motions of the
electrons and the
nuclei, it
introduces the
concept of
electronic structure
and allows to solve
Schrödinger’s
equation for
molecules and
solidstate
materials. 
Explanation:  The fundamentals of
quantum mechanics
ascertain that the
energy eigenvalue,
E,
of a system of
nuclei and electrons
can be calculated by
applying the
Hamiltonian
operator,
Ĥ,
onto the wave
function, Ψ,
as stated in the
famous
timeindependent
Schrödinger
equation: ĤΨ=EΨ.
Note that the
Hamiltonian Ĥ
contains two
kinetic energy
operators: T̂_{e}
of the electrons and
T̂_{N}
of the nuclei, as
well as
three
potential energy
operators:
nuclei–electron
attraction (V̂_{Ne}),
electron–electron
(V̂_{ee})
as well as
nuclei–nuclei
repulsion (V̂_{NN}).
Because of a large
mass difference of
nuclei and
electrons (note that
the nucleus is about
10000 times heavier
than the electron in
a Fe case),
the latter move more
rapidly and,
therefore,
instantaneously
adjust to the
nucleus movement.
Hence, the moving
electrons may be
treated separately
from the slowly
moving
nuclei, and the
Hamiltonian is
simplified
into:
Ĥ=T̂_{e}+V̂_{Ne}+V̂_{ee}.
In other words, the
kinetic energy of
the nuclei is set to
zero assuming that
the nuclei are
stationary; then,
the repulsion
between nuclei
becomes a constant
value which may be
added once the
electronic structure
is solved.
In practice, a
ground state of a
system can be
found iteratively by
alternatively
calculating the
electronic wave
function in a set of
fixed nucleus,
followed by another
optimization of the
nucleus
configuration by
calculating the
Hellmann–Feynman
forces acting on the
nucleus from the
wave
function. 
Picture / Figure / Diagram: 

Max Born (1882‒1970, left) and Julius Robert Oppenheimer (1904‒1967, right).


SFBLink:  Real solid state materials can be treated in quantummechanical terms only by allowing approximations such as Born–Oppenheimer among others. It is used in any ab initio method used in SFB 761. 
References:  Born, Max; Oppenheimer, J. Robert (1927). "Zur Quantentheorie der Molekeln", Annalen der Physik 389 (20): 457–484. 

