Preconditioning
The idea is to find a matrix that 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_{\mathbf{G},\mathbf{G'}} \to \delta_{\mathbf{G},\mathbf{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[1]
- [math]\displaystyle{ \langle \mathbf{G} | {\bf K} | \mathbf{G'}\rangle = \delta_{\mathbf{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}) }, }[/math]
with [math]\displaystyle{ E^{\rm kin}(\mathbf{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]