Approximations for the Exchange-Correlation Functional

In the following sections, we will discuss approximations for the exchange-correlation functional (E_XC) Def.2.2.1necessary in any real application since the true functional is not available. The Kohn-Sham approach maps the many-body problem exactly onto a one-electron problem. It is possible to solve the problem if approximations forE_XC can be found. Exact solutions are available for very few systems, but good approximations can be applied to a large class of problems. The search for better functionals is an active field of research.

2.3.1. Local-Density Approximation (LDA)

In the pioneering paper byKohn and Shamin 1965 an approximation for the exchange-correlation functional (E_XC) was proposed - the local-density approximation (LDA). Kohn and Sham’s idea was to approximate the ex-change-correlation functional (EXC) of an heterogenous electronic system as locally homogenousand useE_XC corresponding to the homogenous electron gas at pointrin space. Assuming a slowly varying electron density (n(r)), they expressed

EXC[n(r)] = Z

d³r n(r)e^xc-LDA[n(r)], (2.12)

wheree^xc-LDA[n(r)]denotes the exchange-correlation energy per electron of the homogenous electron gas.e^xc-LDA[n(r)]can be rewritten as a sum of the exchange and correlation contributions

e^xc-LDA[n(r))] =e^x-LDA[n(r)] +e^c-LDA[n(r)] (2.13) The exchange parte^x-LDA[n(r)]of the homogeneous electron gas was evalu-ated analytically by Dirac [73].

wherers is the mean inter-electronic distance expressed in atomic units [73].

For the correlation part the low- and high-density limits are known, but an analytic expression for the range between the limits is not known. However, for the homogeneous electron gas, excellent parametrisations of accurate quantum Monte Carlo (QMC) calculations exist [52]. TheLDAappears to be oversimplified, in particular for strongly varying densities, as one might expect in real materials. Nonetheless,LDAforms the basis for most of the widely used approximations toE_XC.

2.3.2. Generalised Gradient Approximation (GGA)

The LDAonly takes the density at a given point r into account. One of the first extension to theLDAwas the generalised gradient approximation (GGA). In theGGA, the treatment of inhomogeneities in the electron density is refined by a first order Taylor series with respect to density gradients

E_XC ≈E^XC-GGA[n(r)] =

whereF^XCis a dimensionless enhancement factor. As before, the exchange-correlation energy per electronε^xc-GGAis split into an exchange and correl-ation contribution. The exchange contribution is the same as in theLDA (Eq.2.14). Unlike theLDA, the functional form ofGGAs and thus F^XCis not unique. As a result, a large number ofGGAs have been proposed. A comprehensive comparison of differentGGAs have been given for example by Filippiet al.[90, chap 8] or Korth and Grimme [172].

TheGGA suggested by Perdewet al. is used throughout this work. The aim of the Perdew-Burke-Ernzerhof generalised gradient approximation

[246] (PBE) is to satisfies as many formal constraints and known limits as possible, sacrificing only those being energetically less important [246].

PBEimproves the description of bulk properties like cohesive energies and lattice constants, as compared toLDA[311,312,328].

2.3.3. Hybrid Functionals

In the approximations for the exchange-correlation functional (E_XC) dis-cussed so far, a spurious self-interaction error remains. The self-interaction error, as the name suggests, is the spurious interaction of the electron dens-ity with itself. For example, the Hamiltonian of an one-electron system depends only on the kinetic energy and the potential due to the nuclei.

In the Kohn-Sham framework the energy of such a system is given by Eq.2.6and contains two additional terms, the Hartree term (Eq.2.3) and the exchange-correlation functional (EXC) (Def.2.2.1). The exact exchange-correlation functional (E_XC) would cancel the self-interaction introduced by the Hartree term. However, inDFTmost approximations to the exchange-correlation functional (EXC) leave a spurious self-interaction error. This is different to the Hartree-Fock approximation, where the self-interaction error is cancelled by the exchange term

E^HF_x =−¹

However, Hartree-Fock theory neglects all correlation except those required by the Pauli exclusion principle. This leads to sizeable errors in the de-scription of chemical bonding. CombiningDFTwith Hartree-Fock exact-exchange is a pragmatic approach to deal with the problem. These so-called hybrid functionals were first introduced byBecke[16]. His main idea was to replace a fraction of theDFTexchangeE^DFT_X energy by the exact exchange E^HF_X energy. In this work, we use the Heyd-Scuseria-Ernzerhof hybrid func-tional [138,175] (HSE), therefore the discussion focuses onHSE. For hybrid functionals,EXC is reformulated as

E_XC^hyb=αE^HF_X + (1−α)E^DFT_X +E_C^DFT (2.17) whereαspecifies the fraction of exact exchangeE_X^HF. This approach works well for a varity of systems reaching from semiconductors to molecules and mitigates the effects of the self-interaction error reasonably well. Heyd et al.introduced a screening parameterωto separate the Coulomb operator

1

That way only the short-range part of the Hartree-Fock exchange is included [138,175]. Combining Eq.2.17and Eq.2.18

E_XC^HSE(α,ω) = αE_X^HF,SR(ω) + (1−α)E^PBE,SR_X (ω) +E^PBE,LR_X (ω) +E^PBE_C (2.19) gives the so-calledHSEfunctional. Equation2.19contains two parameters αandω; one controls the amount of exact-exchangeαand the second separ-ates the short- and long-range regimeω. In the case ofα =0 andω =0 we fully recoverPBE,α =0.25 andω =0 gives the PBE0 exchange-correlation functional [248]. In the2006version ofHSEα is set to 0.25 and the range-separation parameterωto 0.11 bohr⁻¹[175], so-calledHSE06. The authors chose the value ofωby testing the performance of the functional for differ-ent test sets of atoms, molecules and solids covering insulators, semicon-ductors and metals. For bulk systems, they showed that band gaps are very sensitive to variations of the screening parameterω, while bulk properties like the bulk modulus or lattice parameters are less affected. In addition to an improved description of the underlying physics, the range-separation reduces the computational effort if compared to hybrid functionals without range-separation, mainly because the Hartree-Fock exchange decays slowly with distance (1/r-decay).