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

[math]\displaystyle{ \frac{1}{{\bf H} - \epsilon_n}, }[/math]

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

[math]\displaystyle{ \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{und} \quad x = \frac{\hbar^2}{2m} \frac{G^2} {1.5 E^{\rm kin}( \mathbf{R}) }, }[/math]

with [math]\displaystyle{ E^{\rm kin}(\bold{R}) }[/math] being the kinetic energy of the residual vector. The preconditioned residual vector is then simply

[math]\displaystyle{ | p_n \rangle = {\bf K} | R_n \rangle. }[/math]

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