Paolo, is the LaTeX script below part of a greater treatise you've written? If so, where is it for downloading (pdf preferred, but script is OK too)?
-Paul Paul M. Grant, PhD Principal, W2AGZ Technologies Visiting Scholar, Applied Physics, Stanford University EPRI Science Fellow (Retired) IBM Research Staff Member Emeritus w2agz at pacbell.net http://www.w2agz.com ? ? -----Original Message----- From: pw_forum-bounces at pwscf.org [mailto:[email protected]] On Behalf Of Paolo Giannozzi Sent: Tuesday, November 06, 2007 5:54 AM To: PWSCF Forum Subject: Re: [Pw_forum] Davidson algorithm On Nov 2, 2007, at 21:08 , Nichols A. Romero wrote: > Can anyone recommend a basic reference (e.g. thesis, personal notes, > computional article) that explains the Davidson algorithm used in > PWSCF > in more detail? The basic reference is E. R. Davidson, "The iterative calculation of a few of the lowest eigenvalues and corresponding eigenvectors of large real-symmetric matrices," Comput. Phys. 17, 87-94 (1975). The only PWscf-specific stuff I have is the following --- \section{Iterative diagonalization} Iterative diagonalization can be used whenever i) the number of states to be calculated is much smaller than the dimension of the basis set, and ii) a reasonable and economical estimate of the inverse operator $H^{-1}$ is available. Both conditions are satisfied in practical calculation in a PW basis set: the number of PW's is usually much larger than the number of bands, and the Hamiltonian matrix is dominated by the kinetic energy at large $\G$ ( the Hamiltonian is {\em diagonally dominant}). Iterative methods are based on a repeated refinement of a trial solution, which is stopped when satisfactory convergence is achieved. The number of iterative steps cannot be predicted in advance. It depends heavily on the structure of the matrix, on the type of refinement used, and on the starting point. A well-known and widely used algorithm is due to Davidson. In this method, a set of correction vectors $|\delta\psi_i\rangle$ to the $M$ trial eigenvectors $|\psi_i\rangle$ are generated as follows: \begin{equation} |\delta\psi_i\rangle = {1 \over D-\epsilon_i} (H-\epsilon_i)|\psi_i\rangle \end{equation} where the $\epsilon_i=\langle\psi_i|H|\psi_i\rangle$ are the trial eigenvalues. The $|\delta\psi_i\rangle$'s are orthogonalized and the Hamiltonian is diagonalized (with conventional techniques) in the subspace spanned by the trial and correction vectors. A new set of trial eigenvectors is obtained and the procedure is iterated until convergence is achieved. A good set of starting trial vectors is supplied by the eigenvectors at the preceding iteration of the potential. An important point is the following. The Hamiltonian matrix is never explicitly required excepted for its diagonal part. Only $H \psi_i$ products are required, which can be calculated in a very convenient way by applying the {\em dual-space technique}. In fact the kinetic energy is diagonal in \G-space, whereas the local potential term is diagonal in real space. Using FFT's (see below) one can go quickly back and forth from real to reciprocal space and perform the products where it is more convenient. There is still a nonlocal term which appears to require the storage of the matrix. The trick is to write $V_{NL}$ in a {\em separable} form: \begin{equation} V_{NL}(\k+\G,\k+\G')= \sum_{\mu=1}^{N_{at}} \sum_{j=1}^n f^\mu_j(\k+\G) g^\mu_j(\k+\G'), \end{equation} where $n$ is a small number and $N_{at}$ is the number of atoms in the unit cell. This allows us to perform the products by storing only the $f$ and $g$ vectors. --- Paolo Giannozzi, Dept of Physics, University of Udine via delle Scienze 208, 33100 Udine, Italy Phone +39-0432-558216, fax +39-0432-558222 _______________________________________________ Pw_forum mailing list Pw_forum at pwscf.org http://www.democritos.it/mailman/listinfo/pw_forum
