Plane Waves

The PW expansion of a wave function ψ_i in a periodic system can be written as:

φi,k(r) = e^ik·rfi(r) (3.42) where k is defined in the first Brillouin zone of the system and f_i takes the form

fi(r) =^X

G

ci,Ge^iG·r (3.43) with G being reciprocal lattice vectors. The basis set for a given wavevector k is therefore discrete but infinite. In order to use the basis in a computational implementation of the KS iteration, some approximations are needed. Firstly, a cutoff energy ^|k+G|₂ ² ≤ E_c is used to restrict to a sphere in reciprocal space the size of the sum over G in Eq. 3.43. In PW based DFT implementations, the cutoff energy is the only parameter that controls the accuracy of the basis set.

This represents a substantial advantage over other basis sets which often require many parameters to control the basis expansion and in some cases, no systematic scheme for convergence is available (i.e.

a variational principle is not available). The second approximation consists of removing the core region of the Coulomb potential which would require a prohibitively large number of plane waves to reach convergence. This is discussed in the next section.

Pseudopotentials

Pseudopotentials are always used in conjunction with the PW basis to remove the¹/rbehaviour of the Coulomb potential which otherwise would make the use of this basis functions prohibitive. The need for pseudopotentials is nicely shown by Fig. 3.2 where the rapid oscilla-tions which maintain the orthogonality between the core and valence electron states are depicted. The description of such rapid oscillations would require a large cutoff energy and is usually useless since in many cases one is interested in the description of atomic bonding which only requires an accurate description of the region where valence electrons overlap.

Ψ ψ˜

−^Zr

Vpseudo

rc

r

r

Figure 3.2.: Schematic illustration of the pseudopotential concept. For r > rc the pseudo wave function, ˜φ, and the local part of the pseudopotential, Vloc, are equal to the all electron functions.

The concept of pseudization was introduced to overcome all these problems. Moreover for heavier atoms in which relativistic effects are important, and so the Dirac equation is required, the valence electrons can be treated non-relativistically. Therefore removal of the core elec-trons also allows to maintain the non-relativistic approach.

Pseudopotentials remove localised core states by modifying the charge state of the nucleus. This will lead to a modified valence wave function ψ that replaces the true valence eigenfunction ψ and is characterized˜ by a smoothed form between the nucleus and some cut-off radius r_c. Beyond this radius, ˜ψ is identical to the full all-electron wave function ψ.

In what follow we will consider only ab initio pseudopotential which are a class of pseudopotentials that are obtained from the inverse

3.3. Basis sets solution of the KS Schrödinger equation. Consider the Schrödinger equation for the l component of the radial part of the atomic orbital φ_l(r)

"

−1 2

d²

dr² +l(l + 1) 2r² −Ze

r + V_H(r) + V_xc(r) − E_l

#

φ_l(r) = 0 (3.44)

where φ = rψ and V_H(r) and V_xc(r) were introduced in Eq. 3.21. To remove the core divergence we have to replace the −^Ze_r term with an attractive V_ps which provide a different eigenfunction but the same eigenvalue. The problem can be formally written as

"

−1 2

d²

dr² +l(l + 1)

2r² + V_ps+ V_H(r) + V_xc(r) − E_l

#

φ˜_l(r) = 0 (3.45)

Usually one defines the form of the pseudo wave function ˜ψ and the cutoff radius. Then V_ps, which is assumed to have the form

V_ps= V_loc+^X

lm

B_l|χ_lmihχ_lm| (3.46)

where orbitals χ_lm vanish outside the core region and are obtained from the solution of

χ_lm(r) = (

E_l−

"

−1 2

d²

dr² +l(l + 1)

2r² + V_loc+ V_H(r) + V_xc(r)

#) ψ˜ (3.47) where

B_l= 1

hχ_lm| ˜ψi (3.48)

It is clear from the above equation that we still have some freedom in the form of V_loc. This is used to fulfil two requirements that the pseu-dopotential should provide: softness and transferability. The former concept is a measure of the number of planewaves that are needed to describe the smooth wave function. The transferability is the ac-curacy that the pseudopotential provides for smooth wave functions different from the one for which it was generated. Accuracy and good

transferability are obtained, at the expense of softness, by introduc-ing norm conservintroduc-ing pseudopotentials [85]. These are defined by the condition:

Z rc

0

r²dr |ψ(r)|²= Z rc

0

r²dr | ˜ψ(r)|² (3.49) together with the requirement that, for a given atomic configuration, the pseudopotential provides the exact eigenvalues of the all electron potential. This constraints fix the form of V_loc and therefore also V_ps. However, norm-conservation requires exceedingly high cutoff ener-gies for the first row elements and for transition metals. To solve this issue, Vanderbilt introduced the concept of ultrasoft pseudopo-tentials. The key idea is to relieve the norm-conserving condition for the smooth wave functions but guarantee the correct evaluation of the core charge. This is achieved by introducing a generalised eigenvalue problem of the form

H| ˜ˆ ψli = ε_iS| ˜ˆψli (3.50) h ˜ψ_l| ˆS| ˜ψ_l⁰i = δ_ll⁰ (3.51) where the requirement for charge conservation is included in the operator ˆS. For more details on this method, the reader is referred to the original article Ref. [86].

Finally, shortly after the introduction of ultrasoft pseudopotentials, Blöchl proposed the Projector Augmented Wave (PAW) method [87].

Somehow resembling the APW method discussed in the next chapter, in the PAW method the core part of the smooth wave function is reconstructed by means of a linear transformation acting on ˜φ. This is obtained by defining a set of projector operators. The projection operators allow to describe the rapidly varying core wave function with linear combination of smooth wavefaunction like, for example, polynomials or Bessel functions.

We conclude this section by mentioning that the plane wave basis set offers a number of advantages, including the simplicity of the basis functions and, as a consequence, of matrix elements evaluation, the parallel efficiency and the easily tuning accuracy. Moreover, the adop-tion of plane-wave basis set provides simple expression for forces and

3.3. Basis sets stress tensor calculations and enables the full relaxation of the struc-ture to minimize the forces in the system efficiently and accurately.