Lots of ways to solve Schroedinger's equation, divided broadly into
- so-called ``abinitio'' methods where the Schroedinger equation is solved
with some ``good'' approximations, [ABINIT]
- ``tightbinding'' methods where the Schroedinger equation is solved in a
tightbinding approximation, [OXON][PLATO]
- classical potential methods, where interatomic potentials are deduced from
experiment and ``abinitio'' results, [Technion codes]. Many people (usually engineers or experimental physicists) claim that
``Classical potentials are a waste of time''. NOT TRUE, the
only real problem with their use is that you can't get band gaps
out of them.
To include dynamics must also solve
equations of motion. This can be done in combination with any method of solving
the Schroedinger equation, with ``abinitio'' its called Car-Parinello Molecular
Dynamics. Also use static algorithms like Metropolis/Simulated Annealing,
Simulated Tempering and Conjugate Gradient.