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Next: The Lanczos recursion scheme Up: The Chebyshev method Previous: Example 1: Propagation operator.


Example 2: Relaxation method.

The operator $ f\left( \widehat{H'}\right) $ to approximate is $ \widehat{U}=e^{-\widehat{H'}\tau } $ with the maximal and minimal eigenvectors $ \lambda _{min}=E_{min} $, $ \lambda _{max}=E_{max} $. This is often referred to as propagation in imaginary time as it results from the above propagation operator by setting

$\displaystyle t=-i\hbar \tau .$

(Note that the propagation is actually in negative imaginary time.)

If the operator is applied to a mixed state and the result renormalised, the state with the lowest energy in the mixed state is produced and all other states are filtered out at the rate $ e^{-(E_{1}-E_{0})\tau } $, where $ E_{0} $ and $ E_{1} $ are the lowest eigenvalues. If a higher state is sought, the ground state has to be first produced and in the second run of the method projected out before every renormalization step. There are operators $ e^{-(\widehat{H'}-E)^{m}\tau },\, m $ an even number, which produce the eigenstate closest to the energy $ E $ directly without the need of producing the lower states first but I shall not go into any details about these [8].

The factor $ \Phi =e^{\left( \Delta E/2+E_{min}\right) \tau } $ for the relaxation operator $ \widehat{U} $ is not a phase factor anymore, $ f(z')=e^{-z'} $ and hence

$\displaystyle b_{n}=(-1)^{n}2I_{n}\left( \alpha \right) ,\, \alpha =\frac{\Delta E\tau }{2\hbar }$

with $ I_{n} $ the modified Bessel function of first kind of order n.

This time, the coefficients go to zero exponentially as $ n $ becomes greater than $ \sqrt{\alpha } $ so the method is very fast except for very short times where this exponentially decaying tail becomes dominant in the calculation time.

The same result is produced by

$\displaystyle f(z')=\frac{a_{0}}{2}+\sum ^{\infty }_{n=1}a_{n}\chi _{n}(z')$

with the recursion
$\displaystyle \chi _{0}$ $\displaystyle =$ $\displaystyle \psi ,$  
$\displaystyle \chi _{1}$ $\displaystyle =$ $\displaystyle -\widehat{H'}\psi ,$  
$\displaystyle \chi _{n+1}$ $\displaystyle =$ $\displaystyle -2\widehat{H'}\chi _{n}-\chi _{n-1}$  

(note the changed sign of $ \chi _{n-1} $ compared to the second algorithm in Ex. 1) and the coefficients

$\displaystyle a_{n}=2I_{n}\left( \alpha \right) .$

This can be shown by proving inductively that $ \chi _{n}=(-1)^{n}\phi _{n} $, as obviously $ b_{n}=(-1)^{n}a_{n} $:
$\displaystyle \chi _{n+1}$ $\displaystyle =$ $\displaystyle -2\widehat{H'}\chi _{n}-\chi _{n-1}$  
  $\displaystyle =$ $\displaystyle (-1)^{n+1}2\widehat{H'}\phi _{n}-(-1)^{n-1}\phi _{n-1}$  
  $\displaystyle =$ $\displaystyle (-1)^{n+1}\left( 2\widehat{H'}\phi _{n}-\phi _{n-1}\right)$  
  $\displaystyle =$ $\displaystyle (-1)^{n+1}\phi _{n+1}.$  


next up previous
Next: The Lanczos recursion scheme Up: The Chebyshev method Previous: Example 1: Propagation operator.
Andreas Markmann 2003-10-22