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Improve Chebyshev expansion for exponential

Sebastian Ohlmannrequested to merge
improve_chebyshev into main

Description

Use the normal recurrence instead of the Clenshaw algorithm. This allows to truncate the expansion with a certain tolerance once the coefficients (using Bessel functions) are small enough. This improves the accuracy and is much better than a fixed number of iterations to be specified.

Also, change the estimate for the maximum energy of the discrete Hamiltonian: it needs to be a sum over all dimensions. Otherwise it is too small by a factor of three, and the expansion does not converge for larger timesteps as a part of the spectrum is capped.

In general, this method typically needs more applications of the Hamiltonian as the Lanczos method, but it has two advantages: it needs less memory (only 3 additional batches) and it avoids orthogonalization, which can be costly.

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Improve Chebyshev expansion for exponential

Checklist

  • I have checked that my code follows the Octopus coding standards
  • I have added tests for all the new features added in this request.

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