I was wondering: in common DFT (or Hartree-Fock) algorithms/codes, and in particular in solving the Roothan equation, what is the most computationally expensive part:
1- evaluate all the terms of the Hamitonian matrix (a.k.a. Fock matrix)
or
2- solve the eigenvalue problem once the H matrix had been calculated
?
And by how much?
Many Thanks to whomever will give useful answers. References (articles or reviews) on that very topic will also be much appreciated.
In general, when evaluating a n by n hermitian matrix you need to compute (1/2)n(n+1) distinct matrix elements but a single iteration; diagonalization of a matrix by whichever algorithm you use is in general a multi iteration process where each such iteration will involve recomputation of these (1/2)n(n+1) matrix elements, in my opinion diagonalization might be more computationally expensive as far as no. of iterative steps are involved in convergence to threshold level of tolerances.
The detailed answer to your problem can be found in standard texts on numerical linear algebra, I would recommend the following books in this regard:
1. Numerical Linear Algebra and its applications
Author: B.N.Dutta
2. Matrix Computations by Golub and Van loan.
3. Matrix Analysis by Horn and Johnson.
In particular look into the QR algorithm, that is one of the most efficient algorithms to compute eigenvalues of a matrix simultaneously.
Thank You Debopam Ghosh . What you wrote totally makes sense. I guess it boils down to how expensive is to compute the single matrix element in DFT (during diagonalization one doesn't recompute the elements from scratch, I believe it is only a matter of additions and subtractions among matrix elements).
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