I can not agree with david. The cause of a flat band structure is not the electronic correlation. In fact, it is the atomic structure of the material that generates flat bands, if one uses the band theory to solve it. After getting the flat bands, we immediately know that the band theory is not...
Once we have solved the electronic structure problem, we have the Bloch waves, then we are able to Fourier transform the Bloch to Wannier. But how can we solve the electronic structure problem? Using atomic orbitals as basis set for expansion of the Bloch waves is one of the solution.
given that you have a perfect boundary condition: U=0 at r=1. The Sturm-Lioiville eigenproblem is designed to deal with such proper boundary condition. The boundary condition combined with the spherical coordinate gives orthorgonal basis set, each function of the basis set is product of...
Looking at the form of the equation, it involves time t, so Laplace transformation with respect to time t would be helpful. Looking at the form of the solution, it is the Laplace transformation of a convolution. So Laplace transformation is the solution.
For a finite system, computing Green's function is easy: to compute (zI-H)^-1. If you are only interested in a subsystem of a finite system, the concept of self-energy can be introduced. The self-energy is more helpful when you considering an infinite system. I suppose you want the Green's...