Preconditioning

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

${\frac {1}{{\bf {H}}-\epsilon _{n}}},$

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 $G$-vectors (i.e. $H_{G,G'}\to \delta _{G,G'}{\frac {\hbar ^{2}}{2m}}G^{2}$), it is a good idea to approximate the matrix by a diagonal function which converges to ${\frac {2m}{\hbar ^{2}G^{2}}}$ for large $G$ vectors, and possess a constant value for small $G$ vectors. We actually use the preconditioning function proposed by Teter et. al

Failed to parse (unknown function "\tp"): {\displaystyle \langle \bold{G} | {\bf K} | \bold{G'}\rangle = \delta_{\bold{G} \bold{G'}} \frac{ 27 + 18 x+12 x^2 + 8x^3} {27 + 18x + 12x^2+8x^3 +16x^4} \quad \mbox{und} \quad x = \tp \frac{G^2} {1.5 E^{\rm kin}( \bR) }, }

with $E^{\rm {kin}}({\mathbf {R}})$ being the kinetic energy of the residual vector. The preconditioned residual vector is then simply

$|p_{n}\rangle ={\bf {K}}|R_{n}\rangle .$

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