1.4 Pseudopotentials
1.4.4 The Projector Augmented Wave method
While the norm-conserving pseudo-potentials (in their various implementations: Hamann, Schlueter and Chiang [23], Bachelet, Hamann, Schlueter [34], Troullier-Martins [25], Rappe-Rabe-Kaxiras-Joannopoulos (RRKY) [26], Kleinman-Bylander [28]) have proved to be quite successful in treating various elements (particularly semiconductors and sim-ple metals), they have however their limitations [33] [35] [36] [37] [22]. Thus, the norm-conserving pseudo-potentials generated for the p states of the elements in the first row of the periodic table (e.g. oxygen 2p), or for the d states of the second row transition metals (e.g. copper 3d ) are quite hard, i.e. their representation in reciprocal space still requires a large number of plane waves, see Fig. 1.2. This is due to the fact that for these valence states there are no core states of the same angular momentum to which they can be orthogonal. Consequently, the all-electron 2p or 3d wave-functions are nodeless and quite compressed, compared with other valence states. A smooth pseudo-wavefunction or potential could be obtained for these states only if one uses a large cut-off radius. Within the norm-conserving pseudopotential, however, it is not really possible to choose an rcut
much larger than the outermost maximum of the all-electron wavefunction as this would spoil the norm-conservation rule. For this reason the norm-conserving pseudization does not improve too much the smoothness of these states.
Efforts directed towards the reduction of the plane-wave cut-off, used in the description of the systems containing highly localized valence orbitals, have been focused on relaxing the norm-conservation condition. One of the approaches in this direction is the Projected Augmented Wave (PAW) method by P. Bl¨ochl [35]. In this method the pseudo-wave-function ( ˜Ψ) is required to be equal to the all-electron wave-function (Ψ) outside rcut, as with norm-conserving pseudopotentials, but inside rcut, ˜Ψ can be as soft as possible, i.e.
the norm-conservation constraint is removed. This greatly reduces the planewave cut-off needed in the calculations, since quite large values for rcut can be used in this scheme.
All variational calculations are performed onto the smooth wavefunction ˜Ψ, which is represented in a plane-wave basis set. Quite importantly, Bl¨ochl has shown also that at the end of these calculations, one can obtain via a clever linear transformation T -from the smooth wavefunction ˜Ψ the all electron wave-function Ψ, with its correct nodal structure, i.e.
|Ψ >= T | ˜Ψ > (1.68)
As the pseudo-wave function is equal to the all-electron wave-function outside rcut, the transformation T modifies the smooth pseudo wavefunction only within the core region, and it reconstructs the full all-electron wavefunction Ψ with its correct nodal structure
Figure 1.2: (Up)All electron wave-functions for oxygen(left) and copper(right) atoms.
(Middle) Norm-conserving pseudo-wave-functions for oxygen and copper [32]. (Down) 2p oxygen wavefunction: solid line - the all-electron wave-function, dotted line - the norm-conserved wave-function, dashed line - the non norm-conserved pseudo-wave-function. Notice the difference in smoothing between the norm-conserved and the non norm-conserved wave-functions [33].
1.4 Pseudopotentials 31 near the nuclei. Therefore, one can write the transformation T as identity plus a sum of atomic contributions Ta
T = 1 +X
a
Ta (1.69)
where each Ta acts only within the rcut region around the atom a. For every atom a, Ta
adds to ˜Ψ the difference between the true and the pseudo-wave function within the rcut sphere. In order to define Ta Bl¨ochl introduced three types of functions:
(1) the all-electron partial wave-functions φi,a(r), which are nothing else than the so-lutions of the Schr¨odinger equation for an isolated atom, at a reference energy ǫn and for an angular momentum quantum number L=(l,m). The index i from φi,a(r) is a shorthand notation for the angular momentum quantum number L=(l,m), and the reference energy ǫn, i.e. i={l,m,n}. These wave-functions form the basis set in which the all-electron wave-function Ψ is expanded within the augmentation sphere (the rcut sphere):
|Ψ >=X
i
ci,a|φi,a>, within Ωa (1.70)
(2) for each of the all-electron partial waves φi,a(r), one constructs a pseudo-electron partial wave ˜φi,a(r) with the constraints that this has to be smooth inside the aug-mentation sphere and match identically φi,a(r) outside:
| ˜φi,a(r) >= |φi,a(r) >, outside Ωa (1.71) Due to the cancellation property (1.71), φi,a(r) and ˜φi,a(r) are never evaluated be-yond r > rcut, although they are continuous for all r.
If the pseudo-electron partial waves form a complete basis set, one can expand ˜Ψ into it within the rcut sphere:
| ˜Ψ >=X
i
ci,a| ˜φi,a>, within Ωa (1.72)
(3) besides ˜φi,a(r), for each all-electron partial wave Bl¨ochl also introduces a projector function |˜pi,a >. These functions are introduced in order to pick out the weight with which a chosen atomic state is contained in ˜Ψ. Thus, if one calculates the scalar product between a 1s-type projector function and a wave-function ˜Ψ, this will pick out the 1s-character of the wave-function ˜Ψ. If one calculates, however, the scalar product between a 2s-type projector function and the same wave-function, this will pick out the 2s-character of the wave-function ˜Ψ, etc. Similarly for the p- or the d - type projector functions. The projector functions vanish for r > rcut and satisfy the orthogonality property:
< ˜pi,a| ˜φj,a >= δij (1.73)
In terms of the functions φi,a, ˜φi,a and ˜pi,a the projector Ta can be written as:
Ta=X
i
|φi,a> −| ˜φi,a >
< ˜pi,a| (1.74) From (1.68), (1.69) and (1.74) we see that the all-electron wavefunction can be obtained from the pseudo-electron wavefunction by the following procedure:
|Ψ > = T | ˜Ψ >
term corrects | ˜Ψ > for the correct nodal behavior in the vicinity of each atom. The projectors < ˜pi,a| are chosen such thatX
i,a
| ˜φi,a >< ˜pi,a| ˜Ψ >
cancels | ˜Ψ > within the augmentation region at atom a, i.e. within Ωa one is left with
|Ψ >=X
a,i
|φi,a >< ˜pi,a| ˜Ψ >. Thus, while far from the atoms |Ψ >= | ˜Ψ > (as the partial waves are not considered beyond r = rcut), close to the atoms |Ψ > is generated from partial waves that contain the proper nodal structure, as the pseudo-wave function and its partial wave expansion cancel one another within rcut.
Due to (1.75), in the PAW formalism the true valence charge ρ(r) can be decomposed into three contributions:
ρ(r) = ˜ρ(r) +
ρΩa(r) − ˜ρΩa(r)
(1.76) where ˜ρ(r) represents the charge over the entire space due to the smooth wave-function
| ˜Ψ >, ρΩa(r) represents the ”true” charge within the augmentation sphere Ωa (and it is due to the all electron wave-function expansion within Ωa), and ˜ρΩa(r) is the ”smooth”
charge within the augmentation sphere due to the pseudo-wavefunction | ˜Ψ >. The charge difference
ρΩa(r) − ˜ρΩa(r)
is typically called compensation charge or augmentation charge. In principle both ρΩa(r) and ˜ρΩa(r) can contain contributions from the core states, but most often one uses a frozen core approximation and restricts the explicit calculation to a certain number of valence states.