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The Pseudopotential Approximation

It has been shown by the use of Bloch's theorem, that a plane wave energy cut-off in the Fourier expansion of the wavefunction and careful k-point sampling (see Section 3.5 and 3.6 respectively) that the solution to the Kohn-Sham equations for infinite crystalline systems is now tractable. Unfortunately a plane wave basis set is usually very poorly suited to expanding the electronic wavefunctions because a very large number are required to accurately describe the rapidly oscillating wavefunctions of electrons in the core region.

It is well known that most physical properties of solids are dependent on the valence electrons to a much greater degree than that of the tightly bound core electrons. It is for this reason that the pseudopotential approximation is introduced. This approximation uses this fact to remove the core electrons and the strong nuclear potential and replace them with a weaker pseudopotential which acts on a set of pseudo wavefunctions rather than the true valence wavefunctions. In fact, the pseudopotential can be optimised so that, in practice, it is even weaker than the frozen core potential[43].

The schematic diagram in Figure 3.4 shows these quantities. The valence wavefunctions oscillate rapidly in the region occupied by the core electrons because of the strong ionic potential. These oscillations maintain the orthogonality between the core and valence electrons. The pseudopotential is constructed in such a way that there are no radial nodes in the pseudo wavefunction in the core region and that the pseudo wavefunctions and pseudopotential are identical to the all electron wavefunction and potential outside a radius cut-off tex2html_wrap_inline5224 . This condition has to be carefully checked for as it is possible for the pseudopotential to introduce new non-physical states (so called ghost states) into the calculation.

  figure605
Figure 3.4: An illustration of the full all-electronic (AE) wavefunction and electronic potential (solid lines) plotted against distance, r, from the atomic nucleus. The corresponding pseudo wavefunction and potential is plotted (dashed lines). Outside a given radius, tex2html_wrap_inline5224 , the all electron and pseudo electron values are identical. 

The pseudopotential is also constructed such that the scattering properties of the pseudo wavefunctions are identical to the scattering properties of the ion and core electrons. In general, this will be different for each angular momentum component of the valence wavefunction, therefore the pseudopotential will be angular momentum dependent. Pseudopotentials with an angular momentum dependance are called non-local pseudopotentials.

The usual methods of pseudopotential generation firstly determine the all electron eigenvalues of an atom using the Schrödinger equation

  eqnarray611

where tex2html_wrap_inline5688 is the wavefunction for the all electron (AE) atomic system with angular momentum component l. The resulting valence eigenvalues are substituted back into the Schrödinger equation but with a parameterised pseudo wavefunction function of the form[43]

  eqnarray620

Here, tex2html_wrap_inline5692 are spherical Bessel functions. The coefficients, tex2html_wrap_inline5694 , are the parameters fitted to the conditions listed below. In general the pseudo wavefunction is expanded in three or four spherical Bessel functions.

The pseudopotential is then constructed by directly inverting the Kohn-Sham equation with the pseudo wavefunction, tex2html_wrap_inline5696 .

A pseudopotential is not unique, therefore several methods of generation also exist. However they must obey several criteria. These are:

  1. The core charge produced by the pseudo wavefunctions must be the same as that produced by the atomic wavefunctions. This ensures that the pseudo atom produces the same scattering properties as the ionic core. This will be discussed further in Section 3.7.2.
  2. Pseudo-electron eigenvalues must be the same as the valence eigenvalues obtained from the atomic wavefunctions.
  3. Pseudo wavefunctions must be continuous at the core radius as well as its first and second derivative and also be non-oscillatory.
  4. On inversion of the all electron Schrödinger equation for the atom, excited states may also be included in the calculation (if appropriate for a given condensed matter problem), for example, generating a d component for a non-local pseudopotential when the ground state of an atom does not contain these angular momentum components.
There are several ways in which these conditions can be satisfied leading to the non-uniqueness of a pseudopotential. This can be traced back to the expansion of the wavefunction in terms of a plane wave basis set. The set of plane waves, tex2html_wrap_inline5700 , forms a complete basis set (assuming a high enough cutoff) and the additional core states to which they are orthogonal results in a linearly dependent spanning set, that is, an over complete basis set. This linear dependance leads to non-unique pseudopotentials.


next up previous
Next: Norm Conserving Pseudopotentials Up: Ion-Electron Interactions Previous: Ion-Electron Interactions

Stewart Clark
Thu Oct 31 19:32:00 GMT 1996