PAW Transformed Operators

Considering the all-electron operator ˆOand its matrix element with respect to the all-electron states|Ψniand|Ψmi, we insert Equation (3.4) to find its matrix element evaluated in terms of smooth wave functions|Ψ˜niand|Ψ˜mi. Note that we use the

|i-notation here for all kinds of wave functions even though these are in general

3.2. PAW Transformed Operators 29

3.2.1. Local and Semi-local Operators

The above expressions hold for any general operator ˆO. However, the density op-erator and thus also the overlap opop-erator, the effective potential and the kinetic en-ergy operator are local and semi-local operators, respectively. For these operators the expression can be simplified considerably. The correction functionshr|φ^b−φ˜^bi are localized within the sphere S^b. Thus all mixed-spheres expectation values hφ^a−φ˜^a|Oˆ_loc|φ^b−φ˜^bi vanish for different spheres a 6= b if the requirement of disjoint spheres (Equation 3.7) is fulfilled. This implies for the mixed expressions, second line in Equation (3.15), that we can use the expansion of |Ψ˜ni and hΨ˜m| inside the spheres. Inserting the completeness relation from Equation (3.11) we obtain

which yield (ex-pectation value of a mixed pair of smooth and true partial waves) cancel out exactly.

Also, the expressions hφ˜^a_i|Oˆ_loc|φ˜^b_jifrom the second and third line cancel. Thus a (semi-)local operator matrix elements in the PAW formalism consists of three con-tributions,

hΨm|Oˆ_loc|Ψni=hΨ˜m|Oˆ_loc|Ψ˜ni+X

aij

c^a_mj^∗ hφ^a_j|Oˆ_loc|φ^a_ii−hφ˜^a_j|Oˆ_loc|φ˜^a_ii

c^a_ni. (3.17) Since we did not impose any restrictions onto the states Ψn and Ψm, Equation (3.17) allows us read off a representation of any (semi-)local operator with respect to smooth states ˜Ψ Therefore, we can apply Equation (3.17) to find the density of a Kohn-Sham state

|Ψi

hΨ|rihr|Ψi=hΨ|r˜ ihr|Ψ˜i+X

aij

c^a_i^∗ hφ^a_i|rihr|φ^a_ji−hφ˜^a_i|rihr|φ˜^a_ji

c^a_j. (3.19) The transformed valence density operator then reads

ˆ

ρ(r) =ρ(r) +ˆ˜ X

aij

|p˜^a_ii hφ^a_i|rihr|φ^a_ji−hφ˜^a_i|rihr|φ˜^a_ji

hp˜^a_j|. (3.20) In various situations, the smooth and true densities in the atomic sphere need to be treated. Therefore, an atomic density matrixD^ais defined by

D^a_ijσ=X

nk

f_nσkc^a_inσk^∗ c^a_jnσk (3.21)

3.2. PAW Transformed Operators 31 With this, the representation of the valence density reads

n^σ_v(r) = n˜^σ_v(r) +X

a

D^a_ijσ φ^a_i(r)φ^a_j(r) −φ˜^a_i(r)φ˜^a_j(r)

(3.22) where the smooth valence density ˜n^σ_v(r)is constructed from the smooth KS wave functions

˜

n^σ_v(r) = X

nk

f_nσk

Ψ˜_nσk(r)

2. (3.23)

3.2.3. Overlap Matrix

The PAW transformation can be —but does not necessarily have to be —norm-conserving. However, the true wave functions must be normalizable in order to find the correct number of particles in the system. The requirementhΨ|Ψi=1 can be translated into the space of smooth wave functions by writing the norm as the expectation value of the unity operator which is clearly local. According to Equa-tion (3.17), we find

1=hΨ|1|Ψi=hΨ|˜ Ψ˜i+X

aij

c^a_i^∗ hφ^a_i|φ^a_ji−hφ˜^a_i|φ˜^a_ji

c^a_j. (3.24) We have therefore introduced an overlap operator ˆSthat is a metric to normalize the smooth wave functionshΨ|˜ S|ˆ Ψ˜i=1, where

Tˆ^†Tˆ =Sˆ =1+X

aij

|p˜^a_ii∆q^a_ijhp˜^a_j| (3.25) with the norm deficit matrix∆q^a_ij = hφ^a_i|φ^a_ji−hφ˜^a_i|φ˜^a_ji. For the special case of a norm-conserving construction of the smooth partial waves, the entire norm deficit matrix vanishes and an implementation of the overlap operator becomes redun-dant; see also Section A.0.2. The norm deficit matrix is often called charge deficit matrix because the elementary chargee=1 in (Hartree) atomic units and the same matrix is applied in the context of the electrostatic monopole deficit. This ensures that the normalization conditionhΨ|˜ S|ˆΨ˜i = 1 also leads to a normalized general-ized density.

3.2.4. Kinetic Energy Operator

The kinetic energy operator relies on spatial derivatives and is thus not a local but a semi-local operator since we can compute derivatives of a function at a given point holding information about this point’s surrounding. This is done on real-space grids with the finite difference approximation that leads to a non-local but

localized operator. The PAW transformation for the kinetic energy operator reads and the transformed operator with respect to the smooth wave functions is given as Tˆ^†TˆTˆ = −1 The matrix elementsE^a,kin_ij of the true and smooth partial waves with kinetic energy operator are evaluated on the radial grid. Furthermore, we can exploit that the true partial waves have been found as solutions of a local potential to a given energy eigenvalue, such that ˆT|φ_ℓni= [ǫ_ℓn−V_ref^a(r)]|φ_ℓniwhich helps to evaluateE^a,kin_ij .

As shown in the following chapter of this thesis the derivatives ∆_r are approxi-mated by finite-differences on the real-space grid. This procedure involves a lim-ited number if neighboring grid points such that the kinetic energy operator is non-local with a range finite Nfh. However, in the limit of small grid spacings, h → 0, this range vanishes and the PAW transformation for local operators may still be applied.

3.2.5. Potential Operator

In the previous cases we used the PAW transformation in the sphere merely to replace the true wave functions by an expansion in smooth partial waves. The potential operator is treated differently in the sense that also the potential shape is replaced by a much smoother one. As mentioned in the beginning of the chapter, the singular Coulomb potential of the nucleus cannot be represented in a basis that captures smooth quantities only. We therefore augment the singularity and replace it with an —in principle arbitrary, but preferably —smooth potential ˜Vinside the sphereS^a

hΨ|V|Ψi=hΨ|˜ V|˜ Ψ˜i−X

aij

c^a_i^∗ hφ^a_i|V|φ^a_ji−hφ˜^a_i|V|˜ φ˜^a_ji

c^a_j. (3.28) Thus, the potential operator with respect to the smooth wave functions reads

Tˆ^†VTˆ =V˜ −X

aij

|˜p^a_ii hφ^a_i|V|φ^a_ji−hφ˜^a_i|V|˜ φ˜^a_ji

hp˜^a_j|. (3.29) The freedom in the shape of ˜V is usually restricted by the constraints that the local potential on the grid ˜V(r)is as smooth as possible. The construction of very smooth potentials can be achieved by a potential shape correction term, see Section A.0.2.