The idea is to find a matrix which multiplied with the residual vector gives the
exact error in the wavefunction. Formally this matrix (the Greens function) can be written
down and is given by
where $ \epsilon_n$ is the exact eigenvalue for the band in interest.
Actually the evaluation of this matrix is not possible, recognizing that the
kinetic energy dominates the Hamiltonian for large -vectors
(i.e. ), it
is a good idea to approximate the matrix by a diagonal
function which converges to for large Failed to parse (syntax error): {\displaystyle \mathbf[G}}
vectors, and possess
a constant value for small vectors.
We actually use the preconditioning function proposed by Teter et. al[1]
with being the kinetic energy of the residual vector.
The preconditioned residual vector is then simply
References