Band parallelization

The orthogonalization can be paralleized over k-points, spins, domains (see Orthogonalizing the wave functions), and bands, described below.

Let’s say we split the bands in five groups and give each group of wave functions to one of five processes:

The overlap matrix contains 5x5 blocks. These are the steps:

rank:      1           2           3           4           5

       A . . . .   . B . . .   . . C . .   . . . . .   . . . . .
       . . . . .   . A . . .   . . B . .   . . . C .   . . . . .
S:     . . . . .   . . . . .   . . A . .   . . . B .   . . . . C
       C . . . .   . . . . .   . . . . .   . . . A .   . . . . B
       B . . . .   . C . . .   . . . . .   . . . . .   . . . . A
  1. Each process calculates its block in the diagonal and sends a copy of its wave functions to the right (rank 5 sends to rank 1).
  2. Rank 1 now has the wave functions from rank 5, so it can do the row 5, column 1 block of \(\mathbf{S}\). Rank 2 can do the row 1, column 2 block and so on. Shift wave functions to the right.
  3. Rank 1 now has the wave functions from rank 4, so it can do the row 4, column 1 block of \(\mathbf{S}\) and so on.

Since \(\mathbf{S}\) is symmetric, we have all we need:

A B C . .
. A B C .
. . A B C
C . . A B
B C . . A

With \(B\) blocks, we need \((B - 1) / 2\) shifts.

Now we can calculate \(\mathbf{L}^{-1}\) and do the matrix product \(\tilde{\mathbf{\Psi}}_0 \mathbf{L}^{-1}\) which requires \(B - 1\) shifts of the wave functions.