3.2 Density Functional Theory calculations
3.2.2 Plane wave basis and pseudopotentials
To solve the KS equations (2.64) numerically in an actual calculations, the KS orbitalsϕi⟩have to be expanded into a suitable basis set χq
⟩ : ϕi⟩ =
∑
q χqiχq ⟩ , χqi =⟨χqϕi⟩. (3.1) Possible basis functions are Slater type orbitals (STO’s) [94], which correctly describe the wave function’s cusp at the nucleus; Gaussian type orbitals (GTO’s) [95], which allow for efficient calculation of overlap and exchange integrals like (2.21); finite elements or a real space grid [96, 97], which al- low high accuracy and good parallel scaling by domain decomposition; and plane waves. The latter are especially suited for periodic boundary condition calculations, both from a physical point of view, with electronic eigenfunc- tions (Bloch states) spread out over the entire system, and from a technical point of view, since the kinetic energy operator is diagonal, and symmetry in the periodic space simplifies the calculations’ complexity.Every plane wave q⟩ is characterized by its wave vector q, defined by ⟨
r q⟩ = eı˙q·r. In a periodic system, the wave vector q can be split into
q =k+G, wherekis in the first Brillouin zone, andGis a reciprocal lattice vector. That way, every electronic state can be expressed as
ϕnk⟩ =
∑
G χnk(G)kG ⟩ , χnk(G) = ⟨ kGϕnk ⟩ . (3.2) kG⟩denote the plane wave basis vectors,⟨rkG⟩ = exp ˙ı(k+G)·r. The sum over G has to be truncated in an actual calculation. A single parame- ter Gmax = max|G|, the maximum length of all included reciprocal lattice
vectors, controls the size of the plane wave basis. IncreasingGmax or, equiv-
alently, the kinetic energy cutoff Ec = G2max/2, increases the basis set size
monotonously. All basis functions are orthogonal, thus no basis set superpo- sition error occurs.
The number of basis functions scales like Ec3/2, and a typical DFT calcu-
the basis set expansion as much as possible. The inter-atomic part of the KS wave functions is generally well described by relatively few plane waves: over distance, the wave functions do not vary much. Near the nuclei, how- ever, the KS wave functions will exhibit nodes and strong fluctuations, which contain high spatial frequencies and thus require a large cutoff energy Ec
for a quantitatively good description. Much of the interesting physics and chemistry happens in the overlap regions of the valence orbitals; the core re- gions, and especially the core electrons, do not contribute significantly to a number of material properties such as equilibrium lattice constants, phonon frequencies, and optical properties. This argument justifies the “frozen core approximation”, where the core electron wave functions are not determined self-consistently, but rather kept frozen from an all-electron atomic calcula- tion. The KS equations (2.64) are then solved for the valence electrons only, however, with a modified one-electron potentialVenc :
HKSϕi⟩ = [ −1 2∇ 2+Vc en+VH[nv] +Vxc[nv] ] ϕi ⟩ =εiϕi⟩, (3.3) Venc =Ven+VH[nc] +Vxc[nc+nv]−Vxc[nv]. (3.4)
Here, nc (nv) denotes the core (valence) electron density. Due to the non-
linearity of Vxc in n(r), the core-valence exchange-correlation potential can
be written exactly only as the differenceVxc[nc+nv]−Vxc[nv]. It can be esti-
mated, however, as
Vxc[nc+nv]−Vxc[nv]≈
∑
sVxc[ncAE,s +nAEv,s]−Vxc[nAEv,s], (3.5)
where s runs over all atomic centres in the unit cell, and nAE denotes an electron density from an all-electron atomic calculation.
Applying the frozen-core approximation reduces the number of electronic degrees of freedom and can speed up calculations drastically. However, this is paid for by a certain unphysical description of the regions near the atomic nuclei. Going further, one can postulate that not only the description of the core but also of the valence electrons near the nuclei should have small in- fluence on a large number of physical and chemical properties. Thus, em- ploying apseudopotential, that deviates from the actual potential close to the
nuclei and leads to a numerically convenient description of the valence elec- tron wave functions, should be of much use while introducing only minor unphysical effects. The concept of pseudopotentials has received much at- tention since its introduction in atomic [98] and nuclear physics [99] and is a research field of ongoing activity. Whether or not the use of pseudopoten- tials will be rendered unnecessary by ever increasing computational power is difficult to say.
Norm-conserving Pseudopotentials
Introduced by Kleinman and Phillips [100], and after further work by Hamannet al.[101, 102], an easy way to construct so called norm-conserving pseudopotentials was presented by Troullier and Martins [103]. Starting from a desired analytical fit of the wave function near the nucleus, the inverse radial Schrödinger equation is solved for each angular momentum compo- nent l for the pseudopotential. The pseudo wave function fulfills several physical and numerical conditions:
• Transferability : atomic all-electron and pseudo eigenvalues are equal,
ϵAE
l =ϵPPl , and wave functions are equal outside a cutoff radiusrl. • Norm conservation: ⟨ϕPPi ϕiPP⟩ =1.
• Softness: ϕiPP⟩ is nodeless, and has no cusp atr=0.
• Smoothness: atr=rl, pseudo and all-electron wave function are equal
up to fourth radial derivative.
Ultrasoft Pseudopotentials
Studying first-row elements with pseudopotentials of the Troullier-Martins type is numerically expensive, since the valence 2porbitals are localized close to the nuclei. The cutoff radius rp is thus rather small, and the smoothness
and norm conservation conditions lead to a pseudo wave function that is not much different from the all-electron wave function. Thus, high plane wave cutoff energies (around 35 a.u. for oxygen) are necessary. Vanderbilt introduced so-called ultrasoft pseudopotentials, based on the following gen- eralisations of the Troullier-Martins scheme [78]:
• Scattering properties of all-electron and pseudo state should not be equal at the eigenvalueϵAEbut at an arbitrarily chosen energy ˜ϵthat is closer to the chemically interesting energy range.
• Scattering properties should be equal atseveralarbitrarily chosen ener- gies ˜ϵi.
• Allow non-norm conserving pseudo wave functions.
The last condition transforms the KS problem into a generalized eigenvalue problem. The additional computational cost will in general be outweighed by the smaller plane wave basis set.
Central quantity is the overlap operatorS between the pseudo wave func- tions: S =1+
∑
i,j Qijβi ⟩⟨ βj, (3.6) Qij =⟨ϕiAEϕjAE⟩−⟨ϕiPPϕPPj ⟩ (3.7) βi⟩ =∑
j B−ij1χj⟩ (3.8) Bij = ⟨ ϕPP i χj ⟩ (3.9) χi⟩ = (ϵ˜i−Te−Vloc)ϕPPi ⟩ (3.10) It fulfills the orthonormalization condition⟨ϕiPPSϕjPP⟩ =δij. The KS equa- tions read [ −1 2∇ 2+Vc en+VH[nv] +Vxc[nv] +∑
ij Dijβi⟩⟨βj ] ϕi⟩ =ϵiSϕi⟩ (3.11) Dij = ∫ d3r (VH[nv] +Vxc[nv])Qij(r) (3.12)Relaxing the norm-conservation condition allows significant reduction of the plane wave cutoff. First-row elements can be described accurately using a cutoff energy as low as 15 a.u. in case of oxygen [85].
Projector Augmented Wave Method
The central quantity in the ultrasoft pseudopotential scheme is the overlap operator S. Recognizing that S = T†T is the square of a transformation operator T is the essential step to extend this scheme to the projector aug- mented wave method (PAW) [104]. UsingT instead ofS means constructing a frozen core approximation, combined with a dual basis set ansatz similar to LMTO or LAPW approaches. In the PAW method, radial projector functions transform between all-electron wave functions defined on a radial grid, and soft wave functions expanded into a plane wave basis set. This way, the all- electron wave functions near the nuclei are always available, unlike in the ultrasoft scheme, where they are lost. It also means that every observable will be evaluated on the plane wave grid and on the radial grid using the all-electron wave functions near the core. The PAW method is much better suited to compute e.g. optical properties, at a computational cost for the self- consistent step that is comparable to the ultrasoft pseudopotential scheme.