Density functional theory (DFT) approximate functionals.

In this section I shall expose some of the theory behind the most popular approximate DFT methods. The performance of these methods is illustrated by the comparison between different methods and experiment, demonstrated in Tables 5-1 to 5-5.

Local density approximation (LDA). LDA is the simplest approximation to -^xc[p]> although the local expression (5.3.3.2) for the kinetic energy is very poor for atoms, local exchange-correlation is rather accurate. In LDA, the kinetic energy is calculated from orbitals using (5.3.3.4), rather that from the electron-gas formula (5.3.3.2).

The exchange-correlation energy is calculated as:

^xc[p]= |d r p ( r ) e j p ( r ) ) , (5 3 .4 .1 )

where the exchange energy density of a homogeneous electron gas is known exactly from (5.3.3.3):

3_

4n 471 (5.3.4.2)

and the correlation energy density is know n very accurately from q uantum M onte Carlo sim ulations o f C eperley and A lder (1980) and O rtiz and B allone (1994). A n analytical representation satisfying the exactly know n high- and low -density analytical lim its and fitting the num erical quantum M onte C arlo results for the spin- u npolarised case, is (Perdew & Z unger, 1981):

g ,(p ) = 0.031 lln r, - 0.048 + 0.0020r,lnr, - 0 .0 1 16r, ( r ,< 1),

e,(p) = -0.1423/(1 + 1.0529\A^ + 0 .3 3 3 4 rJ ( r ,> 1) (5.3.4.3) Fig. 5-4 show s different contributions to the total energy o f the h o m o geneous electron gas and clearly indicates the im portance o f the correlation contribution.

0.2 0.1 kinetic s . H F p 0.0 oa CD

I

c z CD total -0.1 V correlation ex c h a n g e -0.2 0.0 2.0 4.0 6.0 r s ( B o h r ) 8.0 10.0

Fig. 5-4. Energy contributions for the homogeneous electron gas (per 1 electron). Average r, values o f the valence electrons o f several metals are shown. After Pisani (1996).

Perdew and W ang (1992) expressed the correlation energy o f the hom o g en eo u s electron gas m ore accurately by a single analytical form ula, w hich also satisfies the high- and low -density lim its and quantum M onte Carlo data:

1

«c(p) = -2co(l+ai0 1n[l+

131

where Co=0.031091 and c,=0.046644, P ,= -^ e x p (--^ )= 7 .5 9 5 7 , a , =0.21370, 2co 2co

p2=2cop' = 3.5876, p3=1.6382, and p^=0.49294 for the spin-unpolarised (Ç=0) electron gas. For a fully spin-polarised case (^=1) Cq=0.0 15545 and c,=0.025599, a , =0.20548, pg=3.3662, and P4=0.62517.

The LDA generalised for spin-polarised systems is called LSDA (local spin density approximation). For the exchange energy we have a simple exact spin scaling relation (5.3.3.35), which implies:

e.(Pr,Pi) = e M ^ ^ (5.3.4.5)

The most popular spin-interpolation formula used for the correlation energy is that due to Vosko et al. (1980):

e.CPt.P*) = e.(p) + a , ( p ) | ^ ( l - 4 ‘) + k (P .O )-e,(p)]f(4)Ç‘ , (53.4.6)

. 4/ 3 . / I c \ 4 / 3

(\ -f c5V + n — E) — 2

where % )= -— —— — —--- , and -a^p ) can be accurately represented by a

2 — 2

function of the type (5.3.4.4) with Cq=0.016887, c,=0.035475, a,=0.11125,

P)=0.88026, p4=0.49671.

LDA leads to significant errors in exchange (about -5%) and correlation (about +100%), but these largely cancel each other, explaining the successes o f the LDA. This approximation does not give accurate exchange-correlation holes for atoms, but gives reasonable spherical averages for these holes - luckily, it is only the spherically-symmetric part of the multipole expansion that is important. In addition, and perhaps most importantly, LDA exactly satisfies all the sum rules (5.3.3.17) and (5.3.3.19) that are true for the exact functional. The self-interaction error is largely cancelled by the exchange-correlation potential; e.g., in the H atom -95% of this error is cancelled at the LSDA level of theory (Thijssen, 1999). However, this error becomes important for highly localised electronic states (such as d-states in transition metal atoms). Perdew and Zunger (1981) devised a simple method to incorporate the self-interaction correction in the LDA approach (LDA+SIC). LDA+SIC offers a significant improvement for atomic energies, but is difficult to apply to crystals,

where the electronic orbitals are usually represented by delocalised Bloch functions, whose self-interaction is zero’.

The shortcomings of the LDA include the following: 1 .Underestimation of the energies of isolated atoms - LDA treats well the valence electrons, but significantly underbinds the core electrons, 2. Overbinding (by -20% ) in molecules and crystals, 3. Bond lengths are usually ~1% too short, 4. Reaction barriers are usually too low compared to experiment, 5. Large errors for weakly bonded systems (e.g., hydrogen bonds), 6. van der Waals bonding cannot be treated correctly within the LDA, 7. Sometimes the energy differences between polymorphs are inaccurate (e.g., for quartz-stishovite the huge energy difference of -0.5 eV is not reproduced: even the sign is wrong!), 8. Often a wrong ground state is predicted (e.g., paramagnetic fee instead of ferromagnetic bcc for Fe), 9. Serious errors for many Mott insulators (often magnetic moments are underestimated or low-spin states are predicted to be more stable, LDA fails to properly split the d-levels of transition metal ions and predicts no band gap®), 10. In many cases, the dissociation products are not neutral atoms, but atoms with non-physical fractional charges (due to the self-interaction error - see a very interesting paper by Becke (2000)).

These serious shortcomings inspire further developments of functionals. The impressive successes of the LDA suggest it as a reference point in these developments.

Table 5-1. Total energies of atoms (in a.u.): comparison of experim ent with several approxim ate methods. Data were taken from Perdew & Zunger (1981), Parr & Y ang

Atom HF LSDA LSDA+SIC GGA

(PW91) Exp. H -0.500 -0.479 -0.500 -0.500 -0.500 He -2.86 -2.835 -2.918 -2.900 -2.904 Ne -128.55 -128.228 -129.268 -128.947 -128.937 A r -526.82 -525.938 -528.289 -527.539 -527.60

’’ In such cases, it is necessary to select and localise som e orbitals (e.g., d- and f-orbitals o f transition metal ions) - see Szotek & Temmerman (1993) for details and application to transition metal oxides. * This is artificially corrected by the LD A +U method, w hich enforces splitting o f the d-levels.

133

Generalised gradient approximation (GGA). The most obvious way to construct an improved functional is to expand the exchange-correlation energy in powers o f the density gradients. This approximation, called GEA (Gradient Expansion Approximation), exact for an electron gas with a slowly varying density, turns out to be worse than the LDA for atoms. The main reason of this is that GEA does not satisfy the sum rules (5.3.3.17) and (5.3.3.19). The short-range part of the exchange- correlation hole is improved over the LDA, but the long-range part is worsened and has spurious undamped oscillations, due to which GEA does not satisfy the sum rules and exchange hole sometimes becomes positive. It is possible to construct a very accurate functional by setting the exchange-correlation hole to zero everywhere the GEA exchange hole is positive and everywhere beyond a certain cut-off radius, chosen so as to enforce the sum rules.

Table 5-2. Exchange-correlation energies of atoms (in a.u.): comparison of LDA, GGA (PW91), and exact values. Data were taken from Parr & Yang (1989) and Perdew & Kurth (1998). In this table, GGA is represented by the PBE functional, numerically very similar to the PW91 functional, used in Table 5-1.

Atom HF LSDA GGA

(PBE) Exact H -0.31 -0.29 -0.31 -0.31 He -1.03 -1.00 -1.06 -1.09 Li -1.78 -1.69 -1.81 -1.83 Ne -12.11 -11.78 -12.42 -12.50

GGA is constructed in such a way as to preserve all the correct features of the LDA and add some more. There are several popular GGA functionals, the best of which seem to be the PW91 functional (Wang & Perdew, 1991) and a very similar functional PBE (Perdew et al., 1996) - below I will mainly describe their features. Good GGA exchange-correlation functionals satisfy several conditions:

Reduce to GEA for slowly varying densities. Reduce to LDA for the homogeneous electron gas. Satisfy sum rules (5.3.3.17) and (5.3.3.19).

Satisfy conditions (5.3.3.15,18,20). Have proper scaling (5.3.3.31,32). Have proper spin scaling (5.3.3.35).

• Obey the Lieb-Oxford bound (5.3.3.34). GGA exchange is defined as:

4 [p .V p ]= Jd rF ,(5)p(r)e^(p(r)) , (5.3.4.7)

where e^CpCr)) is the exchange energy of the homogeneous electron gas per 1 electron (given by (5.3.5.1)), and the exchange enhancement factor FJs) is a function o f the reduced density gradient s:

Set in this form, GGA exchange automatically has the correct scaling behaviour (5.3.3.31). For the total exchange-correlation functional the definition is:

4 c[p.Vp] = Jd rF ’,,(p,5)p(r)e,,(p(r)) ^ , (5.3.4.9) where the enhancement factor 7\c(p,s) now depends on both the density and its reduced gradient. No GGA functional is expected to be accurate for s » l ; happily, only the range 0<y<3 is important in atoms, molecules, and solids. Values o f 5>3

correspond to atomic tails and are unimportant (Perdew & Kurth, 1998). The Lieb-Oxford bound (5.3.4.34) can be rewritten in terms of

F Jp,5)< 2.273 (5.3.4.10)

A summary of several functionals in terms of their enhancement factors is given in Fig. 5-5.

GGA significantly improves the description of atomic core electrons and to some extent the valence electrons as well. Total energies are much better than in LSDA and even better than in HF. LSDA overbinding is corrected by the GGA. Energy differences and especially reaction barriers are often significantly improved, as well as the description of magnetic systems. Exchange energies of atoms are reproduced with a typical error of only 0.5%; the typical error for the correlaton energy is 5% (Perdew & Kurth, 1998). GGA exchange-correlation cannot be formally self­ interaction-free; however, for the H atom, the self-interaction error is practically completely cancelled within the GGA. GGA, like LSDA, cannot reproduce the derivative discontinuity (5.3.3.39) on passing an integer number of electrons and does not have the correct (5.3.3.40) long-range behaviour.

135 w 2.0 1 . 8 1. 6 fe' 1 .4 1 . 2 1.0 r , ” ~ r , - 1 8 - : S ' r . - e ‘ J r . - 2 . . r . r o , 1 . . . . 1 •n 0 1 2 s = |V n |/2 k p n ^^2 .0 1M o p 6 1 2 5 s = iVn|/2kpH

Fig. 5-5. Exchange-correlation enhancement factors. a)LDA, b)GEA, c-d)GGA - PW91 (circles) and PBE (solid lines). Figures a-b (after Perdew & Burke, 1996) refer to the spin-unpolarised cases; figures c and d (after Perdew et al., 1996) refer to the unpolarised and fully polarised cases, respectively.

Som e o f the failings o f the G G A include: 1. B ond lengths are usually overestim ated by ~1% , 2. Like in the LSD A , but less often, electronic ground states o f atom s, m olecules, and solids can som etim es be incorrect. 3. S om etim es the energy differences are incorrect. G G A is particularly inaccurate for the heavy tran sitio n m etals (e.g.. Au). For bond lengths the G G A ‘overcorrects’ the LD A.

It can be dem onstrated (Zupan et ah, 1998; P erdew & K urth, 1998) that gradient correction w ould favour a process, for w hich:

, (5.3.4.11) < J > < r. >

w here r, and 5 are the W igner-S eitz radius (see caption to Fig. 5-3 and eq. (5.3.4.2)) and the reduced density gradient (see eq. (5.3.4.8)), respectively.

In a process tow ards a less dense structure (e.g., atom isation, bond stretching) d<r^> > 0 and often (but not alw ays) d<y> > 0. This suggests that the total effect o f the gradient corrections should be sm all - hence m any o f the successes o f the LSD A . H ow ever, usually the left-hand side o f (5.3.4.11) is larger than the right, explaining

why gradient corrections usually lead to longer bond lengths and more open structures.

Table 5-3. Com parison of HF, LDA and GGA for solids. Data are com piled from Lee & Martin (1997), Lichanot (2000), D ovesi (1996), Zupan et al, (1998). HF+c m eans Hartree-Fock calculation with an a p o ste rio ri estimate o f the correlation energy by a density functional.

Property HF

(HF+c)

LSDA PW91

(PBE)

Experim ent

Periclase MgO [G(GA = PW86]

4.191 4.160 4.244 4.20

-7.32 (-9.69) - - -10.28

Æ ,GPa 186 198 157 167

Ferrom agnetic bcc-Fe

- 10.44 (11.34) 11.77 Æ ,GPa - 260 (200) 172 1C - 4.6 (4.5) 5.0 Diamond C ûo, 3.58 3.53 3.57 (3.57) 3.567 ^ .,e V -5.2 (-7.4) -8.87 -7.72 (-7.72) -7.55 Æ ,GPa 471 455 438 (439) 442

Table 5-4. Com parison between LSDA, LDA+U, GGA, and experim ent for antiferrom agnetic FeO. Data are taken from Isaak et al. (1993), Cohen et al. (1 9 9 7 ), Fang et al. (1 9 9 9 ), Gramsch et al. (2001).

LSDA LDA+U GGA Experim ent

Uo, À' 141.5 161.2 165.8,156.8, 157.1 162.82

A ,G P a 173 186 178, 180, 158 142-180

Æ' 4.3 3.43 4.2, 3.55, 3.05 4.9

3.4 3.4 -, 3.46, 3.5 4.2

N otes: 1. The ground state o f FeO at 0 K is rhombohedral antiferromagnetic; how ever, the N eél temperature o f FeO (198 K) is below room temperature and m ost experiments refer to the spin- disordered cubic phase. This is a different phase, w hose properties are, however, similar. 2. The L SD A entry for the volum e is ambiguous: Isaak et al. (1993) give only the lattice parameter, w hich they seem to define as the cubic root o f the volum e o f the rhombohedrally distorted cell. 3. Experimental m agnetic m oments o f Fe atoms have a large orbital com ponent, which is not included in usual calculations. For the spin m agnetic moments agreement between theory and experim ent is good.

Meta-GGA. The next step after GGA is to include the Laplacian of the electron density V^p. The inclusion of this variable is very promising, since the Laplacian of

137

the density proved to be very important in the Bader analysis of chemical bonding (Bader, 1990; Coppens, 1997; Tsirelson, 1993). Quantum Monte Carlo simulations (Nekovee et ah, 2001) demonstrated that often GGA would worsen the LDA exchange-correlation hole. They also showed that inclusion of the Laplacian o f the density is essential in reasonable modelling o f the exact hole. There have been several attempts to construct meta-GGA functionals. E.g., Perdew et al. (1999) have constructed a meta-GGA functional including the Laplacian of the density and the kinetic energy density, which is defined as:

T(r) = ‘/2SV^(|.i(r) (5.3.4.12)

Table 5-5. Atomisation energies (in eV) of several molecules: experim ent versus theory. Experimental values include a correction for zero-point vibrational energy. A ll calculations are spin-polarised. (Taken from Perdew et al., 1996; m eta-GGA results: from Perdew et al., 1995). M eta-GGA calculations used GGA electron densities.

Molecule HF LSDA GGA

(PBE)

GGA (PW91)

Meta-GGA Experim ent

Hz 3.64 4.90 4.55 4.55 4.97 4.73 OH 2.95 5.38 4.77 4.77 4.67 4.64 HjO 6.72 11.58 10.15 10.19 9.98 10.06 HF 4.21 7.03 6.16 6.20 6.01 6.11 Liz 0.09 1.00 0.82 0.87 0.98 1.04 Nz 4.99 11.58 10.54 10.49 9.94 9.93 Oz 1.43 7.59 6.24 6.20 5.70 . 5.25 Fz -1.60 3.34 2.30 2.34 1.87 1.69 CH4 14.22 20.03 18.21 18.26 18.26 18.17 CzH4 18.56 27.45 24.76 24.85 24.35 24.41

Becke (2000) showed that x(r) is a useful indicator o f delocalisation of the exchange hole. Conventional functionals fail for systems with significantly delocalised exchange holes (such as stretched HjQ because of the large self­ interaction error - see Zhang & Yang (1998). Including x(r) it is possible to accurately model the exchange energy of systems with highly delocalised exchange holes - e.g., Becke (2000). x(r) can even be used to recognise the regions of such delocalisation (as regions where x ( r ) » V,o(3ti^)^V^^^)- Becke (2000) has constructed a meta-GGA exchange functional, which very accurately simulated the exact exchange. Perdew et al. (1999) have constructed a meta-GGA functional, which has

two fitted parameters. Among its good properties are self-interaction-free correlation and reduction to LDA for the uniform electron gas. The performance of this meta- GGA for atomisation energies is significantly better than GGA, but bond lengths (and lattice parameters) seem to be slightly worse. There are other meta-GGA functionals and much ongoing work on the derivation and testing o f meta-GGAs.

Hybrid functionals. Adding local (LDA or GGA) correlation to the exact (HF) exchange turns out to give worse results than pure GGA exchange-correlation. Nevertheless, as emphasised in A.D. Becke's seminal paper (Becke, 1993), exact exchange is important in constructing accurate functionals. Becke proposed a hybrid functional, whose exchange part is a mixture of local and exact exchange and whose correlation part is a local functional, schematically:

E,, = E j - ' + a„,,(E “ “ -E,'“ ^) , (5.3.4.13)

where a^^-^ is a fitted parameter equal to 0.20. Hybrid functionals generally are highly accurate in predicting structures and atomisation energies, as well as total energies of atoms. There are a host of hybrid functionals, the most popular of which is called B3LYP.

Van der Waals bonding One of the most difficult problems for quantum- mechanical simulations has been to reproduce van der Waals bonding. Originating from dynamical correlation, van der Waals forces are by definition absent in the Hartree-Fock theory. Even at the DFT level, there are problems as these long-range forces cannot be adequately reproduced by any o f the local functionals. LDA is too crude for correlation generally; GGA (PBE functional) can give some dispersion forces and even give reasonable bond lengths for noble-gas dimers. However, due to its local nature, GGA gives an exponential, rather than dependence for the van der Waals energy. Kohn et al. (1998) have proposed a practical recipe in order to get accurate dispersion forces and their long-range behaviour within the DFT framework.

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