Plane wave basis and Pseudopotentials

To solve the Kohn-Sham equation (2.2.24) in practice, the problem is written in a single-particle basis to get a matrix representation of HKS. Then, diagonalizing this

Due to the translational invariance in solids, plane waves χk(r) are often used as

a possible basis set. The plane wave expansion of the eigenstates yields ψik(r) = X m ci,k+Gmχk+Gm(r) = 1 √ Ω X m ci,k+Gme i(k+Gm)·r _(2.2.30)

with the reciprocal lattices vectors Gm, the crystal volume Ω and the expansion

coefficients ci,k+Gm of the wave function. With that the Kohn-Sham equation (2.2.24)

becomes a matrix equation X

m0

Hm,m0(k)c_i,m0(k) = ε_i(k)c_i,m(k) (2.2.31)

with the Hamilton matrix Hm,m0(k) =

~2

2me|k + Gm| 2_δ

m,m0+ v_KS(G_m− G_m0). (2.2.32)

In theory the sum over m has to be infinite, but is in practice truncated by an energy cutoff

~2

2me |k + Gm| 2

< Ecut. (2.2.33)

The computational effort is thus directly linked to the number of involved basis functions. The basis set should on the one hand be able to describe the wave functions near the nuclei, where they show strong oscillations and on the other hand describe the rather smooth area in the bonding regions between the atoms. However, for the description of the strong oscillations near the nuclei a larger number of basis functions is needed for the plane wave expansion resulting in a large computational effort. To reduce the needed computational power mainly two strategies are used: augmented plane waves (APW) or pseudopotentials (PP).

In the augmented wave methods, which have been introduced 1937 by Slater [63], the space is divided into atom-centered spheres (augmentation spheres) and intersti- tial regions between the atoms (comparable to a muffin-tin). In the region between the atoms, the smoothly varying parts of the wave functions are represented by so called envelope functions, which are plane waves or other smooth functions. Within the augmentation sphere atom-like functions, e.g. spherical harmonics times radial functions, are utilized as basis functions. The partial solutions of both regions are then matched at the interface to compose the complete basis functions.

The method of pseudopotentials is based on the orthogonalized plane wave method by Herring [64] and was further developed by Philipps [65] and Antonˇc´ık [66]. The main idea is to remove the oscillating structure of the wave function around the core. Therefore, the strong Coulomb potential of the nucleus and the effects of the bound

core electrons are replaced by an effective potential which results in nodeless wave functions around the nuclei while the wave functions outside this region are com- pletely reproduced. With that the valence electrons can be treated with a reasonable amount of plane waves. One further advantage is, that core electrons can also be included in the PP. As a result the number of treated electrons in DFT can be even more reduced. This is possible, when the material properties depend only slightly on the core electrons and are primarily described by the valence electrons which is the case for a wide range of solids.

The PP can be constructed with an all-electron DFT calculation for a spherical atom yielding atomic potentials and wave functions Φl(r). To describe the valence

electrons, so called pseudo wave functions ˜Φl(r) are generated, which are nodeless

within the core region and the same as Φl(r) anywhere else. To be transferable, i.e.

the PP constructed for single atoms can also describe molecules and solids, ˜Φl(r)

should have the same norm as Φl(r), thus they should be norm- conserving [24,

67, 68]. From the pseudo wave functions ˜Φl(r) the effective pseudopotential can be

derived [39]. There are many PP constructed and studied in detail by for example Haman, Schl¨uter and Chiang [67], Kerker [69] or Troullier and Martins [70] to name just a few.

In order to be numerically reasonable, PP should not only be transferable but also smooth in the sense, that the number of expansion components needed is minimal. Transferability usually needs a small radius of the core region while smooth functions need a larger radius. Norm-conserving PP have quite accurate results but often have to sacrifice the smoothness of the function. Especially when valence states at the beginning of an atomic shell (1s, 2p, 3d, etc.) are treated, norm-conserving PP are difficult to find.

A different approach to overcome this problem was introduced by Vanderbilt [71, 72] with the so called ultrasoft PP where he generalized the PP method to non norm- conserving pseudopotentials leading to accurate and smooth pseudo wave functions. Another way to deal with the oscillations in the core region introduced by Bl¨ochl [73] is the projector augmented waves (PAW) method which combines ideas from the augmented wave method and pseudopotential approach. Using local projectors within an augmentation sphere, the full all-electron wave function is mapped onto smooth, numerically convenient pseudo wave functions. Outside of the sphere, the pseudo wave function coincides with the true wave function. Thus, all involved evaluations of integrals can be carried out as a combination of integrals of smooth functions and localized contributions within an augmentation sphere. If a suitable projection is found, the pseudo wave functions can be expanded in a basis like plane waves. All physical properties can be evaluated after the reconstruction of the true wave functions.

In this work the DFT calculations were performed using the PAW formalism as it was implemented in VASP [47].