Direct optimization of the orbitals

With "direct optimisation of the orbitals" we denote a category of electronic minimisation algorithms that use the gradient of the free 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.

This gradient of the free energy with respect to an orbital $\psi_n$ is given by:

$| g_n \rangle = f_n \Big(1-\sum^N_{m=1} \vert \psi_m \rangle \langle \psi_m \vert\Big) \hat{H} \vert \psi_n \rangle + \sum^N_{m=1} \frac{1}{2} {\bf H}_{nm} (f_n - f_m) \vert \psi_m \rangle$

where $\{ f_i | i=1,..,N \}$ are the partial occupancies, and

${\bf H}_{nm}=\langle \psi_m \vert \hat{H} \vert \psi_n \rangle$

is the Hamiltonian expressed within the subspace spanned by the current orbitals $\{ \psi_i | i=1,..,N \}$.

The structure of the gradient may be understood as follows: the first part on the right-hand side describes the change of the free energy with respect to changes in the orbitals that are outside (orthogonal) the subspace spanned by the current set of orbitals, whereas the second part describes the changes of the free energy due to a unitary transformation between the orbitals within this subspace.

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.

Note that, 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), using the blocked-Davidson algorithm to optimise the orbitals. This ensures that the orbitals, that are initialised with random numbers, have converged to reasonable starting point for the subsequent direct optimisation.