Preconditioning: Difference between revisions

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The idea is to find a matrix which multiplied with the residual vector gives the
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
exact error in the wavefunction. Formally this matrix (the Greens function) can be written
down and is given by
down and is given by


<math>
::<math>
  \frac{1}{{\bf H} - \epsilon_n},
  \frac{1}{{\bf H} - \epsilon_n},
</math>
</math>
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Actually the evaluation of this matrix is not possible, recognizing that the
Actually the evaluation of this matrix is not possible, recognizing that the
kinetic energy dominates the Hamiltonian for large <math>\mathbf{G}</math>-vectors
kinetic energy dominates the Hamiltonian for large <math>\mathbf{G}</math>-vectors
(i.e. <math>H_{G,G'} \to \delta_{G,G'} \frac{\hbar^2}{2m} \mathbf{G}^2</math>), it
(i.e. <math>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
is  a good idea to approximate the matrix by a diagonal
function which converges to  <math>\frac{2m}{\hbar^2 \mathbf{G}^2}</math> for large <math>\mathbf{G}</math> vectors, and possess
function which converges to  <math>\frac{2m}{\hbar^2 \mathbf{G}^2}</math> for large <math>\mathbf{G}</math> vectors, and possess
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We actually use the preconditioning function proposed by Teter et. al{{cite|teter:prb:1989}}
We actually use the preconditioning function proposed by Teter et. al{{cite|teter:prb:1989}}


<math>
::<math>
  \langle \mathbf{G} | {\bf K} | \mathbf{G'}\rangle = \delta_{\bold{G} \mathbf{G'}} \frac{ 27 + 18 x+12 x^2 + 8x^3}
  \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
       {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}) },
   x = \frac{\hbar^2}{2m} \frac{G^2} {1.5 E^{\rm kin}( \mathbf{R}) },
</math>
</math>
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The preconditioned residual vector is then simply
The preconditioned residual vector is then simply


<math>
::<math>
| p_n \rangle = {\bf K} |  R_n \rangle.
| p_n \rangle = {\bf K} |  R_n \rangle.
</math>
</math>
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<references/>
<references/>
----
----
[[Category:Electronic Minimization]][[Category:Electronic Minimization Methods]][[Category:Theory]]
[[Category:Electronic minimization]][[Category:Theory]]

Latest revision as of 15:44, 6 April 2022

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

where 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 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