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

$\langle \mathbf{G} | {\bf K} | \mathbf{G'}\rangle = \delta_{\bold{G} \mathbf{G'}} \frac{ 27 + 18 x+12 x^2 + 8x^3} {27 + 18x + 12x^2+8x^3 +16x^4} \quad \mbox{and} \quad x = \frac{\hbar^2}{2m} \frac{G^2} {1.5 E^{\rm kin}( \mathbf{R}) },$

with $E^{\rm kin}(\bold{R})$ 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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