 Re: Some questions about the "Note on Recursion Methods" ( No.1 ) |
- Date: 2007/09/25 16:53
- Name: T.Ozaki
- Hi,
> 1. What is V_n in Eq. (1.42) at page 10 of your notes and how to compute V_n?
V_n is a matrix consisting of eigenvectors for (rn|rn).
Thus, V_n can diagonalize (rn|rn), the (lambda_n)^2 is
a diagonal matrix consisting of eigenvalues of (rn|rn).
> Similarly, how to compute B_{n+1} and C_{n+1} in Eq. (2.38) at page 40
> of your notes. Is there an equation missing which will describe
> the relationship between V_n, B_n and C_n?
Any B_{n+1} and C_{n+1} can be possible if they satisfy Eq.(2.38).
In practice, they can be calculated from the LU decomposition,
the singular value decomposition (SVD), or the diagonalization of
(\tilde{rn}|rn).
> This is a question about your paper, Phys. Rev. B 64, 195126 (2001).
> In Eq. (3) of your paper, to compute the dual basis |\tilde{i \alpha}>,
> we need compute > S^{-1}_{j \beta, i \alpha} first. To save the
> computational cost, for a give atom i, how many j atoms should you compute?
> Since the more j atoms you compute, the higher computational cost will
> need to compute \tilde{U}_n in Eq. (6).
> For the silicon case, do you think S^{-1}_{j \beta, i \alpha}=0 when
> the distance between the atoms i and j is larger than some cutoff distance R_c ?
> If so, then what is the value of R_c for silicon, 0.5nm, 1nm or 10nm?
The proper size should depend on the system and basis functions used.
In my case, I used the same truncation range as for the operation of S^{-1}H.
It would be better to check the convergence in a systematic way.
I would mention one thing. The recursion process may suffer from numerical
instability which is mainly attributed to Eq.(9) in Phys. Rev. B 64, 195126 (2001).
As the recursion process exceeds 15 iterations, the round-off error becomes serious
in the evaluation of off-diagonal Green's functions.
One remedy for the serious problem is to evaluate the off-diagonal elements
in a descending order instead of the ascending order. In this case, you will
obtain a contitued fraction form to evaluate the off-diagonal elements similar
to the diagonal term. (although I have not published anything about this issue.)
Best regards,
TO
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| One more question about the "Note on Recursion Methods" ( No.2 ) |
- Date: 2007/10/07 15:42
- Name: Yang Xu <yangxu@uiuc.edu>
- Dear Prof. Ozaki,
Thank you very much. I really appreciate your answer.
I just have one more question. It is really important for us.
I am wondering if there is a mistake in Eq. (2.37) on page 40 of your notes, should the \hat{H} be \hat{H'} where H'=S^{-1}H ?
To compute the |r_n) in Eq.(2.37), we need first explicitly compute H' by using H'=S^{-1}*H, right?
Your timely reply will be highly appreciated!!
Best regards,
Yang Xu
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Some questions about the "Note on Recursion Methods"