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 (rnrn). Thus, V_n can diagonalize (rnrn), the (lambda_n)^2 is a diagonal matrix consisting of eigenvalues of (rnrn).
> 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 roundoff error becomes serious in the evaluation of offdiagonal Green's functions. One remedy for the serious problem is to evaluate the offdiagonal elements in a descending order instead of the ascending order. In this case, you will obtain a contitued fraction form to evaluate the offdiagonal elements similar to the diagonal term. (although I have not published anything about this issue.)
Best regards,
TO

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

