On mathematics behind molecular orbital energies When explaining the mathematics behind calculating molecular orbital energy , namely HOMO and LUMO energies, using quantum chemistry, Chainer Chemistry documentation states that 


  From mathematical viewpoint it requires a solution of an internal
  eigenvalue problem for a Hamiltonian matrix.



I have never heard of internal eigenvalue problem in mathematics. Is this a term specific to quantum chemistry and what it means?
Is the Hamiltonian matrix same as Kohn-Sham Hamiltonian?
An internal or interior eigenvalue problem seeks a solution somewhere in the middle of the spectrum of eigenvalues as opposed to trying to find the smallest or largest eigenvalue. Its generally trickier to solve for a particular eigenvalue than it is to solve for the smallest/largest one. The HOMO and LUMO are necessarily somewhere in the middle of the eigenspectrum.
The Kohn-Sham Hamiltonian is the orbital-DFT approximation of the molecular Hamiltonian. The Hamiltonian describes the interactions within the molecule and allows one to solve for the molecular wavefunction through the Schrödinger equation. Most of electronic structure is focused on approximations of the Hamiltonian or the form of the wave-function that make solving this problem tractable. There is a lot more that can be said about Hamiltonians than would ever fit in an answer here, so you should try to find good textbooks and papers that discuss them.