COMERCIO GRUPO 2.

In order to make DFT implementation practical an approximation to the XC functional needs to be made. The quality of the approximation will depend on the amount of many-body effects that are included in the functional. Unfortunately, the quality is paid in price of the complexity hence, the most sophisticated approximations are rarely used in practice. In fact, the success of the DFT is mostly due to the simplest possible ap- proximation, namely the local density approximation (LDA) and its generalization, the generalized gradient approximation (GGA). Both approximations are based on the exact results obtained for the uniform electron gas (UEG). The central idea is to split the XC energy density into the exchange,X, and the correlation contribution,C. The former

is obtained using the HF approximation for the UEG [52, Ch. 5], where an analytical solution is known, and the latter is fitted from the results of MC calculations, e.g. [70], which are used as a reference. For spin polarized systems the LDA XC functional is given by:

E_XC(LDA)[n↑, n↓] =

Z

dr n(r) [X(n↑(r), n↓(r)) +C(n↑(r), n↓(r))] . (1.39)

The non-polarized case is obtained by setting n↑ = n↓. The functional is a sim-

ple integral over all points in space, where the contribution at each point depends only on the density at that point. The LDA XC functional is thus said to be local. Similarly, the GGA is obtained by rescaling the XC with a dimensionless function

FXC(n↑(r), n↓(r),∇n↑(r),∇n↓(r)), where the density gradient appears through a di-

mensionless parameter,s= _2k1

F

|∇n|

n , known as the generalized gradient [71]. The Fermi

wavevector,kF = (3π2n)1/3, represents the inverse of the characteristic screening length

of the system. The main improvement of the GGA over the LDA is to reduce the bind- ing energy and thus to improve the bond lengths and the crystal lattice parameters in

molecules and solids, respectively. Strictly speaking the GGA XC functional is semi- local, in the sense that it depends on the gradient of the density. However, this is not a severe limitation and an efficient numerical implementation of the method is possible. Beyond the LDA and the GGA, which are by far the two most commonly used ap- proximations, there exists an entire hierarchy of the XC functionals [72]. By continuing with the gradient expansion we arrive at the meta-GGA functionals [73], which include explicitly the dependence on ∇2_{n and on the KS kinetic energy density,τ}

σ(r), where τσ(r) = occup X i 1 2|∇φiσ(r)| 2 _.

Next the orbital dependent functionals follow, where the XC energy is expressed as a functional of orbitals, instead of the density. Despite the increased computational requirements, orbital dependent functionals are of particular interest, since they natu- rally include the derivative discontinuity, with respect to the particle number, and thus yield better bandgaps. A different type of orbital dependent approaches are the self- interaction corrected (SIC) and the DFT+U. Both approaches are aimed at systems where the e-e interaction is strong, due to electron confinement. The former tries to improve the functional by explicitly removing the self-interaction error. The approach is straightforward to implement in finite systems, however, in solids the definition of the SIC becomes somewhat arbitrary, due to de-localized states (see Ch. 8 in Ref. [52]). The self-interaction error becomes smaller in the latter case. The LDA+U approach attempts to address the strong on-site correlations in localized orbitals, which are rele- vant in certain classes of materials, e.g. transition metal-oxides. In this case the on-site interaction, which is described at a “mean-field” level in DFT, is replaced by an effective Hubbard model for a given band, usually the transition-metal d band (see Ch. A8 in Ref. [51]). In order to make the approach trulyab initio, the parameters for the model need to be obtained from a constrained DFT calculation. Finally, we have the hybrid functionals [74, 75]. They are based on an exact expression for the XC energy, in terms of the adiabatic connection formula [76]. The adiabatic connection allows one to express the XC energy using an integral over the e-e coupling-strength constant,λ, as

EXC = 1 2 Z dr dr0 n(r)n(r 0₎ |r−r0_| Z 1 0 dλ gλ(r,r0)−1 , (1.40)

wheregλ(r,r0) is theλ-dependent pair distribution function. The functional dependence

on the coupling constant, which is used to scale the e-e interaction, is then approximated in between the end points of the integral, for which the exact result is known. The most widely used functional of this family is the HSE06 [75, 77]. The hybrid functionals provide the most accurate energies and systematically improve the description of the

bandgap [78]. However, in metals they tend to overestimate the exchange contribution, which in turn results in too narrow bandwidth [79].

1.1.2.5 Pseudopotentials

The independent-particle approaches rely on single-particle orbitals to form a complete basis set in which the many-body wavefunction can be represented. The unfavorable scaling of the problem with the number of electrons then leads to numerical difficulties for large systems. Therefore, an important aspect of practical electronic structure cal- culations is to reduce the number of electrons to be explicitly described by the EOM. Conceptually this is not a problem. Since all material properties are determined by the valence electrons it would be enough to consider a subspace of the Hamiltonian involving only these states. The remaining electrons, occupying energy levels below the valence band, are called “core” electrons. The total electron density, n, can thus be split into the valence, nval, and the core contribution, ncore,

n(r) =nval(r) +ncore(r) (1.41)

where the latter is assumed to remain fixed when the atom forms chemical bonds, and onlynval needs to be determined. In order to have a complete description of the system,

an appropriate, reduced, Hamiltonian needs to be constructed. The latter also needs to include the effect of the core states on the valence band. For a single atom, stripped of the valence electrons, this results in an effective ionic potential, which is known as the pseudopotential (PP). To appreciate the simplification, one can consider the case of Pb, which has 82 electrons in total, of which only 3 are in the valence band.

The objective when constructing a good PP is the following: the number of valence electrons should be as small as possible and the resulting effective potential should be as simple as possible, while accurate and transferable. The latter condition guarantees that the PP will remain accurate when used in real systems, e.g. molecules and solids. However, there is no way to enforce the transferability, except by careful testing. With respect to the PP complexity, one can talk about hard and soft PPs, where the latter have a more compact representation in Fourier space. The “hardness” of a PP is measured by the number of Fourier components that are needed to represent the potential. The idea that a soft PP can be employed instead of the true hard potential comes from scattering theory. It was shown that, far from the nucleus, the true potential can be replaced by a simpler effective potential (see Ch. 11 in Ref. [52]). The phase of the scattered wave, however, remains the same. This means that the radial part of a valence orbital can be split into two parts, the core and the tail, where only the latter

is relevant for the formation of the chemical bond. In addition, the atomic potential is spherically symmetric and hence, it is sufficient to consider the PP for the radial part of the wavefunction for a given angular momentum,l. For a smooth pseudo-wavefunction (PWF), ˜ψ(r), the radial Schr¨odinger equation,

−1 2 d2 dr2 + l(l+ 1) 2r2 + ˜Vl(r) rψ˜l(r) =rψ˜l(r) , (1.42)

can be inverted to obtain the l-dependent PP, ˜Vl(r), which satisfies Eq. (1.42) for a

corresponding all-electron (AE) eigenvalue,. The tail of the PWF, ˜ψ, is always chosen to match the AE result, while the core part is replaced by a smooth function, which matches the AE part at the core boundary, rcore. In general, the PP is constructed to

reproduce the PWF for a given range of energies. However, if the chemical environment causes a large enough change in the true wavefunction, the PWF produced by the PP may become a bad approximation.

There are three different types of PPs, which are commonly used today. These are the norm-conserving (NCPP), the ultra-soft (USPP), and the projector augmented wave (PAW) PP [80, 81]. The difference between these PPs is in the way they achieve accuracy and transferability. The NCPP defines a set of PWF for which the appropriate potential is constructed so that a set of predefined rules is satisfied. One of these is that the norm of the PWF within the core, i.e. the core charge, Qcore, has to be conserved in the

construction of the PP.Qcore is defined as

Qcore=

Z

|r|≤rcore

dr |ψ(˜ r)|2 . (1.43)

The conservation implies that the change of the PWF at the core boundary will be the same as the change of the AE wavefunction, up to linear order in energy. The expansion is valid only close to the atomic eigenvalue corresponding to the wavefunction. In other words, the matching condition at the boundary is preserved for small perturbations. This ensures the transferability of the PP. The norm conservation imposes a limitation on the shape of the PWF in the core region, which is a strong constraint for the already nodeless wavefunctions, e.g. the 3dorbitals. The resulting shape of the PWF has to be similar to the AE wavefunction, which precludes the construction of a potential that is both smooth and accurate (see Ch. 11 in Ref. [52]). This constraint is loosened by USPPs, leading to a smoother, but more complex PP. The PWF are no longer orthogonal, which in turn leads to a generalized eigenvalue problem. The accuracy is simultaneously improved by allowing the PWF to reproduce the AE results for an arbitrary energy range. The PAW approach shares some similarities with the USPP, however, the starting point for the construction of the potential is different. In the PAW approach [81] the AE wavefunction,

|ψi, is replaced by a smooth wavefunction, |ψ˜i, using a linear transformation,T, as

|ψi=T|ψ˜i. (1.44) The transformation leaves the wavefunction identical to the AE solution outside the core region, while in the core a set of support, i.e. augmentation, wavefunctions is used to represent the fast oscillations. The advantage is that in this approach all electrons are kept in the PP and the AE Hamiltonian, H, is related to the pseudo-Hamiltonian, ˜H, via the same linear transformation, T, that acts on the wavefunctions, namely

H =THT˜ † . (1.45)

The transformation then leads to the generalized eigenvalue problem. The accuracy is, in principle, limited only by the completeness of the projectors, which are needed to rep- resent the AE wavefunction on the augmentation basis set. In practice, the core electron levels are not allowed to change, which is known as the “frozen core” approximation. The PAW PP are nowadays the method of choice, being both soft and accurate. The results that can be obtained using these PPs are comparable to those of AE calculations [82]. However, the computational cost is significantly lower in the former case.