Generating tight-binding Hamiltonians with finite-difference methods

A method is presented for deriving a nearest-neighbor tight-binding Hamiltonian for electrons in solid, starting from a finite-difference Hamiltonian with atomic spheres embedded in it. The space is divided into cells surrounding the atoms. The basis states of the tight-binding Hamiltonian are the eigenstates of the finite-difference Hamiltonian in these cells with zero derivative boundary conditions at the cell boundaries. To calculate the matrix elements of the full Hamiltonian, the couplings over the links crossing the cell boundaries need to be restored which reads to a coupling between states in neighboring cells. The resulting tight-binding Hamiltonian is energy independent. Typically about 100 states per cell are needed to achieve reasonable accuracy.