Charge Density Considerations. The major benefit of AHS pseudopotentials for developing quantum-based interatomic potentials, such as the GPT, is that there exists a unique transformation between the pseudowavefunctions {ϕk} and the ‘true’ wavefunctions {ψk} [93] |ψk⟩ = Ck (1 − Pc) |ϕk⟩ (2.64) This is effectively a Gram-Xxxxxxx orthogonalisation procedure multiplied by a nor- malisation constant Ck. A pseudopotential for which Ck = 1 is said to be norm- conserving [187]. A norm-conserving pseudopotential produces pseudowavefunctions with the property that the integrated charge, inside the core radius Rc, matches that of the ‘true’ wavefunction Qk = ∫0Rc ∫ r2drdΩ |ϕk (r)|2 = ∫0Rc ∫ r2drdΩ |ψk (r)|2 . (2.65) where Ω is the solid angle. Whilst norm-conservation is a useful property for DFT practitioners, where norm-conservation means that the diagonalisation of H is an ordinary eigenvalue problem, it is non-essential for our purposes. Norm-conserving pseudopotentials tend to be MPs which means that there is no longer an exact trans- val formation from the pseudowavefunctions to the ‘true’ wavefunctions. This will mean that the valence electron density nval will have a missing term called the orthogonali- sation hole density noh. To see this explicitly, we write the valence pseudodensity nps as n ps val (r) = 2 fkϕk∗ (r) ϕk (r) . (2.66) Σ k val If the pseudopotential is norm-conserving, the valence pseudodensity nps will be pre- val cisely the same as the ‘true’ valence electron density nval outside of the inner core region. However, inside the inner core region, there is no way of determining the ‘true’ valence electron nval density from the valence pseudodensity nps . For an AHS potential, we can use Eq.(2.64) to find val nval (r) = nps (r) + δnoh (r) (2.67) where δnoh is the orthogonalisation hole correction. In second-order pseudopoten- tial perturbation theory, the correction has the form [99] δnc (r) = . Z∗ − 1Σ nunif + Σ noh (r − R ) (2.68) oh Z i where Z∗ is an effective valence occupation (Z∗ ≥ Z) and noh is a localized hole density centred on a particular atom. For a simple metal, noh is confined to the inner core region of the atom at site i. Both Z∗ and noh depend on the properties of the pseudopotential. For an non-local, energy-dependent AHS pseudopotential, we have Z∗ = Z + ∫ dk fk⟨k|Pc|k⟩ (2.69) and (2π) c x x
