The residual minimization method

Algorithm

  1. Initial guess for wave functions (\(\tilde{\psi}_n\)).
  2. Orthogonalize wavefunctions (make sure \(\langle \tilde{\psi}_n | \hat{S} | \tilde{\psi}_m \rangle = \delta_{nm}\)).
  3. Calculate density (\(\tilde{n}\), \(D_{ij}^a\)).
  4. Calculate potential (\(\tilde{v}\), \(\Delta H_{ij}^a\)).
  5. Apply hamiltonian (\(\hat{H}\tilde{\psi}_n\)).
  6. Subspace diagonalization (rotate \(\tilde{\psi}_n\) so that \(\langle \tilde{\psi}_n | \hat{H} | \tilde{\psi}_m \rangle = \delta_{nm} \epsilon_n\)).
  7. Calculate residuals (\(R_n = \hat{H}\tilde{\psi}_n - \epsilon_n \hat{S}\tilde{\psi}_n\)).
  8. Improve wave functions using the RMM-DIIS algorithm (see below).
  9. Back to (2).

RMM-DIIS step

For each wave function we calculate the residual:

\[R_n = (\hat{H} - \epsilon_n \hat{S}) \tilde{\psi}_n\]

New improved wave function: \(\tilde{\psi}_n' = \tilde{\psi}_n + \lambda \hat{P} R_n\), where \(\hat{P}\) is a preconditioner. Find step length \(\lambda\) by minimizing the norm of:

\[R_n' = (\hat{H} - \epsilon_n \hat{S}) \tilde{\psi}_n'\]

Since we already have \(R_n'\), we might as well use it to take an extra step (with the same step length as for the first step):

\[\tilde{\psi}_n \leftarrow \tilde{\psi}_n' + \lambda \hat{P} R_n' = \tilde{\psi}_n + \lambda \hat{P} R_n + \lambda \hat{P} R_n'\]

See [Kresse96] for details.

Preconditioning

The ideal preconditioner would be:

\[\hat{P} = -(\hat{H} - \epsilon_n \hat{S})^{-1}.\]

For the short wavelength parts of the residuals, \(\hat{H} - \epsilon_n \hat{S}\) will be dominated by the kinetic energy operator, so we have approximately \(\hat{P} \simeq -\hat{T}^{-1}\).

We calculate preconditioned residuals (\(\tilde{R}_n = \hat{P} R_n\)) by solving \(\hat{T} \tilde{R}_n = -R_n\) or equivalently

\[\frac{1}{2} \nabla^2 \tilde{R}_n = R_n\]

approximately using multigrid techniques as described in [Briggs95].

References

[Kresse96]G. Kresse, J. Furthmüller: Phys. Rev. B 54, 11169 - 11186 (1996) “Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set”
[Briggs95]E. L. Briggs, D. J. Sullivan and J. Bernholc: Phys. Rev. B 52, R5471 (1995), “Large Scale Electronic Structure Calculations with Multigrid Acceleration”