Generalized Gradient Approximation (GGA)

In the LDA approximation, it is assumed that the effects of exchange-correlation are local and depend uniquely in the electron density at each point. The next step consists in introducing density gradients in the description of the exchange-correlation effects. In this way, these effects depend on the density value at a given point and in the form in which the density vary in the proximity of that point. Kohn and Sham (1965) proposed this form to improve the LDA functional, supposing that the LDA exchange-correlation energy expression was the zero-order term in the Taylor expansion of electron density and including an additional term which introduces non-local information in the charge density (gradient expansion approximation, GEA). In the GEA, the exchange-correlation energy is given by:

𝐸 [𝜌] = 𝜌(𝒓)𝜖 [𝜌(𝒓)]𝑑𝒓 + 𝐶 [𝜌(𝒓)] ∇𝜌(𝒓)

𝜌(𝒓) / 𝑑𝒓 (3.35)

Unfortunately, the GEA violates the rule of conservation of the sum of exchange-correlation hole: 𝜌 (𝒓, 𝒓 )𝑑𝒓 = 𝟏 ∀ 𝒓 (3.36) where the Exchange-correlation hole, 𝜌 (𝒓, 𝒓 ), is the probability of finding an electron in 𝒓 , if we know that the other electron is placed in 𝒓.

Based on these ideas, a functional was later constructed which satisfies this rule. This approach is known as the generalized gradient approximation (GGA), and the corresponding GGA functional may be written as:

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Perdew and Wang constructed a GGA functional which satisfies the rule of conservation of the sum of exchange-correlation hole, together with other exact conditions (Perdew, 1991; Perdew et al., 1992; Perdew et al. 1996b):

𝐸 [𝜌(𝒓)] = 𝐸 [𝜌(𝒓)] + 𝐸 𝜌↓_{(𝒓), 𝜌}↑_{(𝒓) (3.38. 𝑎)} 𝐸 [𝜌(𝒓)] = 𝜌(𝒓)𝜖 [𝜌(𝒓)]𝐹(𝑠)𝑑𝒓 (3.38. 𝑏) 𝐸 𝜌↓(𝒓), 𝜌↑_{(𝒓) = 𝜌(𝒓) 𝜖 [𝜌(𝒓), 𝜉(𝒓)] + 𝐻[𝑡, 𝜌(𝒓), 𝜉(𝒓)] 𝑑𝒓 (3.38. 𝑐)} with 𝑠 = |𝛻𝜌(𝒓)|/2𝑘 (3.39. 𝑎) 𝑘 = (3𝜋𝜌) / (3.39. 𝑏) and 𝑡 ≅ |𝛻𝜌(𝒓)|/2 𝑘 / (3.40. 𝑎) 𝜉(𝒓) = (𝜌↑(𝒓) − 𝜌↓(𝒓))/𝜌(𝒓) (3.40. 𝑏) The 𝐹 and 𝐻 functions are determined from the conditions imposed on the exchange-correlation hole. This functional, called GGA-PW91, gives better results than LDA in the study of the ground state of the system. In general, the GGA corrections improve the geometries, frequencies, and charge densities obtained with LDA. The average errors are of 6 kcal/mol in thermochemical tests and it works reasonably for systems with hydrogen bonds. It gives better equilibrium volumes and cohesion energies and also improves the results for the magnetic properties of the system. Nevertheless, as LDA, it fails in the description of complexes bonded by van der Waals forces. In order to describe correctly these interactions, it is possible, for example, to include the empirical dispersion of Grimme (2006) which is described in a later section.

3.2.2.3.3.1 Perdew-Burke-Ernzerhof Functional (PBE)

In the generalized gradient approximation (GGA) (Perdew et al., 1992), we may write the exchange-correlation energy functional in the general form:

𝐸 𝜌↓, 𝜌↑ = 𝑓 𝜌↓, 𝜌↑, ∇𝜌↓, ∇𝜌↑ (3.41) The PW91 functional (Perdew, 1991) described in the previous subsection is an analytic fit to a numerical GGA, designed to satisfy several further exact conditions (Perdew et al., 1996b). PW91 incorporates inhomogeneity effects while retaining many of the best features of LSD, but has its own problems: (1) Its derivation is complex and long; (2) The analytic function f, fitted to the numerical results, is complicated and non-transparent; (3) f is over-parametrized; (4) The parameters are not smoothly joined, leading to spurious wiggles in the exchange-correlation potential for small and large density gradients, which can give problems in the construction of GGA-based electron-ion pseudopotentials; (5) Although the numerical GGA correlation energy functional behaves properly (Perdew et al., 1996b) under uniform scaling to the high-density limit, its analytic parametrization (PW91) does not; (6) Because PW91 reduces to the second- order gradient expansion for density variations that are either slowly varying or small, it describes the linear response of the density of a uniform electron gas less satisfactorily than does LSD. This last problem illustrates the fact that the above form of 𝐸 is too restrictive to reproduce all the known behaviours of the exact functional (Perdew et al., 1996b). In contrast to the construction of the PW91 functional, which was designed to satisfy as many exact conditions as possible, the GGA presented by Perdew, Burke and Ernzerhof (Perdew et al., 1996a) satisfies only those which are energetically significant. The six problems above were solved with a

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derivation of a simple new GGA functional in which all parameters selected were fundamental constants. Improvements over PW91 include an accurate description of the linear response of the uniform electron gas, correct behaviour under uniform scaling, and a smoother potential function. The two pieces of the PBE functional, 𝐸 and 𝐸 , may be written in the same way as for PW1 functional (see previous subsection):

𝐸 [𝜌] = 𝜌(𝒓)𝜖 [𝜌(𝒓)]𝐹(𝑠)𝑑𝒓 (3.42. 𝑎) 𝐸 𝜌↓, 𝜌↑ = 𝜌(𝒓) 𝜖 [𝜌(𝒓), 𝜉(𝒓)] + 𝐻[𝑡, 𝜌(𝒓), 𝜉(𝒓)] 𝑑𝒓 (3.42. 𝑏) However, the 𝐹 and 𝐻 are written differently. The 𝐻 function can be expressed as:

𝐻 = 𝑒 𝑎 𝛾𝜙 ×𝑙 1 + 𝛽 𝛾𝑡 1 + 𝐴𝑡 1 + 𝐴𝑡 + 𝐴 𝑡 (3.43. 𝑎) where, 𝐴 =𝛽 𝛾[𝑒𝑥𝑝{−𝜖 /(𝛾𝜙 𝑒 /𝑎 )} − 1] (3.43. 𝑏) and 𝜙(𝜉) = (1 + 𝜉) + (1 − 𝜉) /2 (3.43. 𝑐) And 𝛾 and 𝛽 are parameters, which assume the values (deduced from the asymptotic limits, 𝑡 → 0 and 𝑡 → ∞),

𝛾 = 0.031091 (3.44. 𝑎) 𝛽 = 0.066725 (3.44. 𝑏) Likewise, the 𝐹 function is written as,

𝐹(𝑠) = 1 + 𝜅 − 𝜅(1 + 𝜇𝑠 /𝜅) (3.45) where 𝜇 and 𝜅 are parameters which assume the values (deduced from the asymptotic limit (𝑠 → 0) and to satisfy the Lieb-Oxford bound):

𝜇 = 0.066725 (3.46. 𝑎) 𝜅 = 0.804 (3.46. 𝑏) Becke (1986) proposed this form, but with empirical coefficients (𝜅 = 0.967; 𝜇 = 0.235). The PBE functional retains correct features of LSD, and combines them with the most energetically important features of gradient-corrected nonlocality. The correct but less important features of PW91 functional which have been sacrificed are: (1) correct second-order gradient coefficients for 𝐸 and 𝐸 in the slowly varying limit (𝑡 → ∞), and (2) correct non-uniform scaling of 𝐸 in the limit where the reduced gradient s tends to ∞. Calculations of atomization energies for small molecules also show that this functional yield essentially the same results as the PW91. Except for the improvements in the description of the linear response of the uniform electron gas, the correct behaviour under uniform scaling, and the fact that the potential function is smoother, the PBE functional is close to PW91. However, its simpler form and derivation make it easier to understand, apply, and improve.

3.2.2.3.3.2 Perdew-Burke-Ernzerhof Functional for Solids (PBEsol)

While PBE represented a high point of non-empirical functional development, much has been learned about its limitations since its development. PBE reduces the chronic overbinding of the

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local spin density approximation (LSDA) but, while LSDA often slightly underestimates equilibrium lattice constants by about 1%, PBE usually overestimates them by about the same amount. Other equilibrium properties, such as bulk moduli, phonon frequencies, magnetism, and ferroelectricity, are sensitive to the lattice constant and, therefore, are also overcorrected by PBE (Wu and Cohen, 2006). Surface energies are too low in LSDA, but are made lower still by PBE (Constantin et al, 2006).

Attempts to construct a better GGA functional encounters the following dilemma: Those with an enhanced gradient dependence improve atomization and total energies, but worsen bond lengths, while more recent suggestions of a GGA for solids (like the very first GGA of Langreth and Mehl (1983)) have a reduced gradient dependence and typically do improve lattice parameters and/or surface energies, but worsen atomization energies. Advanced functionals as the meta-GGAs, using also the orbital kinetic-energy densities, provide greater accuracy over a wider range of systems and properties (Tao et al., 2003). But current meta-GGAs do not improve lattice constants as dramatically as surface energies, and meta-GGAs are not yet available in all solid-state codes (or are included only with a reduced functionality). Besides, the use of meta-GGAs is much more time consuming.

In the work by Perdew et al. (2008) the origin of this dilemma is explained. They show that the GGA functionals cannot improve atomization and total energies and at the same time to improve bond lengths and lattice parameters. Accurate atomic exchange energies require violating the gradient expansion for slowly-varying densities, which is valid for solids and their surfaces. At the GGA level, one must choose. A pragmatic approach to lattice properties is therefore to use a modified functional especially for solids which recovers the gradient expansion for exchange over a wide range of density gradients, including small and moderate values as required to describe adequately these systems. Restoration of the gradient expansion for exchange requires a complementary alteration for correlation. By using the PBE form but simply altering two parameters, we retain all other exact conditions that make PBE so reliable.

Any GGA that recovers the uniform gas limit has an 𝐹(𝑠) function determining the GGA exchange functional (see previous section), of the form

𝐹(𝑠) = 1 + 𝜇𝑠 + ⋯ (𝑠 → ∞) (3.47) The gradient expansion that is accurate for slowly-varying electron gases (Antoniewicz and Kleinman, 1985) has

𝜇 = 𝜇 = 0.1235 (3.48) But any GGA that is accurate for the exchange energies of free neutral atoms must have 𝜇 ≈ 2 𝜇 (Perdew et al., 2008). PBE is accurate, although its value of 𝜇 = 0.21951 was found from a different non-empirical argument. The B88 functional is also accurate as it was fitted to the exchange energies of noble gas atoms (Becke, 1986). Thus, to attain accurate exchange energies of atoms any GGA must strongly violate the gradient expansion for slowly varying densities. However, the densities of real solids and their surfaces are often almost slowly varying over space. Restoring the gradient expansion should improve their description (but worsen atomization energies). The GGA is a limited form, and cannot satisfy both conditions. In PBEsol, 𝜇 ≈ 𝜇 is chosen.

Now, for a GGA correlation functional that recovers the uniform gas limit, the gradient expansion must behave as:

𝐸 𝜌↓, 𝜌↑ = 𝜌(𝒓)[𝜖 [𝜌(𝒓), 𝜉(𝒓)] + 𝛽𝑡 (𝑟) + ⋯ )]𝑑𝒓 (3.49. 𝑎) For slowly varying high densities 𝛽 = 𝛽 = 0.066725. Since we are interested in weakly-varying valence-electron densities in densely packed solids, we wish to retain the excellent behaviour of LSDA. If

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the LSDA response is recovered beyond LSDA. In PBE, the gradient expansion for correlation is respected, i.e., 𝜇 ≈ 2 𝜇 and 𝛽 = 𝛽 . But we have already argued that 𝜇 ≈ 2 𝜇 is harmful for many condensed matter applications. Exact satisfaction of the previous equation for 𝜇 would yield 𝛽 = 0.0375, but a compromise value will satisfy another, more relevant constraint for solid- state applications. In PBEsol, the value used for this parameter:

𝛽 = 0.046 (3.49. 𝑐) (together with 𝜇 ≈ 𝜇 ) provides the best fit to the results of the TPSS meta-GGA (Tao et al., 2003; Constantin et al., 2006) because it gives very good surface energies. Thus, the previous equation is violated in favour of good surface energies. But the value is considerably closer to that of the linear response requirement (0.0375) demanded by complete restoration of the gradient expansion.

PBEsol becomes exact for solids under intense compression, where real solids and their surfaces become truly slowly varying, and exchange dominates over correlation. PBEsol should improve most surface energies over LSDA, whereas PBE worsens them. To test our functional, a test set of 18 solids was used. The PBEsol greatly reduces the PBE overestimate of lattice constants by a factor of almost 4, except for semiconductors. PBEsol is much less accurate than PBE for atomization energies, but halves the error of LSDA.