  These lecture notes introduce some simple effective Hamiltonians (also known
as semi-empirical models) that have widespread applications to solid state and
molecular systems. They are aimed as an introduction to a beginning graduate
student. I also hope that it may help to break down the divide between the
physics and chemistry literatures. After a brief introduction to second
quantisation notation, which is used extensively, I focus of the "four H's":
the Huckel (or tight binding), Hubbard, Heisenberg and Holstein models. Some
other related models, such as the Pariser-Parr-Pople model, the extended
Hubbard model, multi-orbital models and the ionic Hubbard model, are also
discussed. Finally, I discuss the epistemological basis of effective
Hamiltonians and compare and contrast this with that of ab initio methods as
well as discussing the problem of parametrising effective Hamiltonians.
