Category:Electronic minimization: Difference between revisions

The algorithms VASP offers for electronic minimisation (i.e., determining the electronic ground state) can be divided in two categories:

• Iterative matrix diagonalisation + density mixing, aka the "Self-Consistency Cycle" (SCC).
• Direct optimisation of the orbitals.

The Self-Consistency Cycle

1. The SCC starts with an initial guess for the electronic density of the system under consideration: VASP uses the approximation of overlapping atomic charge densities. This density defines the initial Hamiltonian.
2. By means of iterative matrix diagonalisation techniques one obtains the N lowest lying eigenstates of the Hamiltonian (where N is of the order of the number of electrons in the unit cell). The iterative matrix diagonalisation algorithms implemented in VASP are the blocked-Davidson algorithm and the Residual Minimization Method with Direct Inversion in the Iterative Subspace (RMM-DIIS). Per default VASP uses the blocked-Davidson algorithm (ALGO = Normal).
3. After the eigenstates and eigenvalues have been the determined with sufficient accuracy, they are used to to compute the total energy of the system, and to construct a new electronic density.
4. In principle, this new density could be taken to define a new Hamiltonian. However, in order to arrive at a stable algorithm this new density is not used as is, but is mixed with the old density. By default VASP uses a Broyden mixer. The resulting density then defines the new Hamiltonian for the next round of iterative metrix diagonalisation (step 2).

Steps 2-4 are repeated until the change in the total energy from one cycle to the next drops below a specific threshold (EDIFF).

Note that when starting from scratch (ISTART = 0), the self-consistency cycle procedure of VASP always begins with several (NELMDL) cycles where the density is kept fixed at the initial approximation (overlapping atomic charge densities). This ensures that the wave functions that are initialised with random numbers have converged to a something sensible before they are used to construct a new charge density.

A (slightly) more detailed description of the SCC may be found here.

Direct optimisation

Similar to the SCC procedure described above, when starting from scratch (ISTART = 0), the direct optimisation procedures in VASP always begin with several (NELMDL) self-consistency cycles where the density is kept fixed at the initial approximation (overlapping atomic charge densities). This ensures that the wave functions that are initialised with random numbers have converged to reasonable starting point for the subsequent direct optimisation.

The direct optimisation of the orbitals uses the gradient of the total energy with respect to the orbitals to move towards the ground state of the system: the orbitals are changed such that the total energy is lowered, using, e.g. the Conjugate Gradient Approximation, or Damped Molecular Dynamics.

After every change of the orbitals, the total energy and electronic density are recomputed. Per default, the electronic density is constructed directly from the orbitals at each step along the way, without any density mixing. Optionally, though, density mixing may be used to stabilise these optimisation procedures when charge sloshing occurs.

Similar to the SCC described above, the direct optimisation of the orbitals stops when the change of the total energy drops below EDIFF.

• Electronic minimization: Algorithms used in VASP to calculate the electronic groundstate.
  • Preconditioning of residuals: Preconditioning.
  • Simple Davidson iteration scheme: Davidson iteration scheme.
  • Single band steepest descent scheme: Single band steepest descent scheme.
  • Efficient single band eigenvalue-minimization: Efficient single band eigenvalue-minimization.
  • Conjugate gradient optimization: Conjugate gradient optimization.
  • Residual minimization scheme, direct inversion in the iterative subspace (RMM-DIIS): RMM-DIIS.
• Wrap-around errors in FFT meshes - convolutions: Wrap-around errors.
• Integration of k points: k-point integration.
• Theory of density mixing

How to

• A description to obtain band decomposed charge densities is given here: Band decomposed charge densities.
• k-point projection scheme: LKPROJ.