Ghosts

GHOST STATES, RESONANCES, and DETECTION

Bound states of the radial Schroedinger equation with a local potential are strictly ordered in energy by their number of nodes. This is not necessarily true for separable non-local pseudopotentials. Spurious bound states with more nodes can arise below or between the zero- and one-node states which the projectors are designed to reproduce. While in the case of single Kleinman- Bylander projectors, a simple test can predict their presence from the bound states of the local potential and the non-local projector coefficient, (Ref. 25 of the paper), no such test exists for multiple-projector potentials.

New Approach

A new "ghost detector" has been introduced in 3.3.0. This will detect deeply bound ghosts unlikely to found in the log-derivaive plots because they are below the limits for any useful scan of the valence region. The routine solves the problem of the pseudo-atom surrounded by a spherical hard wall by a basis function method, mimicking how ghosts would show up in a plane-wave calculation. For each angular momentum, the routine computes a set of radial basis functions that allow efficient calculation of a Hamiltonian matrix of the walled pseudopotential, and diagonalizes it. Eigenvalues lying below the wanted bound states (of the un-walled atom) are tagged as ghosts. "Proper" eigenvalues fall above these states because of the wall, although very close for deeply-bound states. These are simply compared to the unwalled radial Schroeinger equation eigenvalues in the new "Testing for bound ghosts" section of the output file. The effective basis cutoff energies of the basis functions are also listed. Ghosts are reliably found even if they are too localized to be well converged.

Note that the "old" log-derivative plots of the valence region remain the only way to assess the general quality of the potential and identify shallow bound ghosts of ghost resonances above 0.

Our old 14_Si_GHOST.dat illustration proves to be an embarrassment. While one bound ghost (l=1, -3.03 Ha) and two ghost resonances are found in the log plots, the new routine finds 4 more deeply-bound ghosts. The choice of local potential here was deliberately very non-realistic. A new example is 60_Nd_GHOST, first found by a user with strange results in a solid-state calculation. Here, there is one extremely deep f ghost (-208 Ha), whose basis-set-derived wave function is plotted in Nd_ghost.png in this directory. This ghost goes away if lpopt is set to 5 instead of 3.

There is no real theory for getting rid of ghosts, but adjusting the local potential to be closer to the semi-local potential of the ghost l often works.

This test should be considered only "beta" until users have more experience with it. It has not yet (as of 3.3.0) been implemented for relativistic calculations, but a scalar-relativistic calculation with the same input should be a reliable test.

Old Approach

Our approach is to compute phi(E) = arctan(R * d psi(r)/dr |_R) for some R greater than the core radius, where psi is the solution of the non-local radial equation regular at the origin (ie., the outward-integrated solution). Scanning downward in energy, usually from a positive energy of a few Ha, phi(E) for the corresponding all-electron potential takes smooth positive steps of pi, with the steps becoming sharper and corresponding to increasingly localized bound states at increasingly negative energies. Since it is only possible to compute phi on [-pi, pi], steps of 2 pi must be added for continuity as E decreases. For a well-designed pseudopotential, the corresponding phi(E) will closely track that of the all-electron potential over a wide range of energies from well-below to well-above the valence and semi-core states of interest. The steps of pi indicate localized pseudo wave functions. Spurious steps of pi indicate "ghost" states, which are localized states than on investigation turn out to have more nodes than appropriate for their energies.

Ghosts are illustrated by the badly-designed 12_Si_GHOST psp generated by the corresponding data file. (To get something so bad an absurdly attractive and long-ranged local potential was introduced.) Examining the phi plots, the s channel has a rather sharp "ghost resonance" at +1.8 Ha. The p channel has an extremely sharp ghost at -3 Ha, about 0.5 Ha above the core state seen as a similar jump in the all-electron phi. This psp isn't supposed to produce core states. Finally, the d channel has an extremely sharp scattering resonance at +1.3 Ha. When this resonant scattering state is normalized on [0, 6 a_B], it has three peaks and two nodes inside 2 a_B, and the slowly-periodic tail from 2 to 6 is nearly zero. One wonders what sort of a spurious conduction band this would yield!

The routine vkbphsft implements the phi calculations aligning the full- potential and pseudopotential plots at the energy of the first projector using an appropriate multiple of pi. There is, however, a difficulty which can result in a spurious indication of a ghost. Due to the structure of the separable non-local radial equation, the signs of psi and d psi/dr can change abruptly at a discrete energy, giving rise to a spurious jump of pi in phi which can in fact be either positive or negative. This does not correspond to a spurious localized state -- except for the arbitrary sign, psi changes smoothly with no localized features around this energy. Having identified this problem, in release 2.0.1 we have implemented an algorithm which has so far proven accurate is distinguishing rapid but continuous pi steps from such "sign-change" abrupt jumps. For the abrupt jumps, pi is added or subtracted to cancel them out. Sharp but continuous jumps are let stand, be they desired semi-core states or ghosts to be identified as such.

As the search algorithm implemented in the last section of vkbphsft was developed and tested, steps corresponding to deep but correct semi-cores were sometimes canceled. This was corrected by driving the search to smaller energy intervals and loosening the criterion for "less than a change of pi," but the absence of an expected semi-core step might still occur. If the diagnostic tests in the output file show that the psp has an accurate bound semi-core it does, and its absence in the plot can be ignored. It is also possible that a perversely sharp ghost step could be mistakenly canceled. However, our two-projector potentials are remarkably free of real ghosts, while sign-changes have been found often enough to warrant the new procedure.