Quantum Chemistry of Solids / 17-Kohn-Sham LCAO Method for Periodic Systems

7

Kohn–Sham LCAO Method for Periodic Systems

7.1 Foundations of the Density-functional Theory

7.1.1 The Basic Formulation of the Density-functional Theory

Density-functional theory has its conceptual roots in the Thomas–Fermi model of a uniform electron gas [325, 326] and the Slater local exchange approximation [327]. A formalistic proof for the correctness of the Thomas–Fermi model was provided by Hohenberg–Kohn theorems, [328]. DFT has been very popular for calculations in solidstate physics since the 1970s. In many cases DFT with the local-density approximation and plane waves as basis functions gives quite satisfactory results, for solid-state calculations, in comparison to experimental data at relatively low computational costs when compared to other ways of solving the quantum-mechanical many-body problem.

It took a long time for quantum chemists to recognize the possible contribution of DFT. A possible explanation of this is that the molecule is a very di erent object to the solid as the electron density in a molecule is very far from uniform [329]. DFT was not considered accurate enough for calculations in molecular quantum chemistry until the 1990s, when the approximations used in the theory were greatly refined to better model the exchange and correlation interactions. DFT is now a leading method for electronic-structure calculations in both fields. In quantum chemistry of solids DFT LCAO calculations now have become popular especially with the use of so-called hybrid functionals including both HF and DFT exchange.

Traditional quantum chemistry starts from the electronic Schr¨odinger equation (SE) and attempts to solve it using increasingly more accurate approaches (Hartree– Fock and di erent post-Hartree–Fock methods, see Chapters 4 and 5). These approaches are based on the complicated many-electron wavefunction (and are therefore called wavefunction-based methods) and in these ab-initio methods no semiempirical parameters arise. Such an approach can be summarized by the following se-

SE	Ψ |...|Ψ
quence [330]: V (r) −−→ Ψ (r1	, r2, . . . , rN ) −−−−−→ observables, i.e. one specifies the

system by choosing potential V (r), plugs it into the Schr¨odinger’s equation, solves that equation for the wavefunction Ψ (r1, r2, . . . , rN ), and then calculates observables by taking the expectation values of operators with this wavefunction. One among the observables that are calculated in this way is the one-electron density

232 7 Kohn–Sham LCAO Method for Periodic Systems

ρ(r) = N	d3r2 d3r3 . . .	d3rN Ψ (r, r2, . . . , rN )Ψ (r, r2, . . . , rN )	(7.1)

In DFT, the density (7.1) becomes the key variable: DFT can be summarized by the sequence

ρ(r) → Ψ (r1, r2, . . . , rN ) → V (r)

i.e. knowledge of ρ(r) implies knowledge of the wavefunction and the potential, and hence of all other observables. This also represents the fact that ultimately the electron density and not a wavefunction is the observable. Although this sequence describes the conceptual structure of DFT, it does not really represent what is done in actual applications of it and does not make explicit the use of many-body wavefunctions. Some chemists until now consider DFT as a containing “ semiempirism”(not ab-initio) method but recognize that the small number of semiempirical parameters are used in DFT and these parameters are “universal to the whole chemistry” [329].

Two core elements of DFT are the Hohenberg–Kohn (HK) theorems [328,331] and the Kohn–Sham equations [332]. The former is mainly conceptual, but via the second the most common implementations of DFT have been done.

Whereas the many-electron wavefunction is dependent on 3N variables, three spatial variables for each of the N electrons, the density is only a function of three variables and is a simpler quantity to deal with both conceptually and practically.

The literature on DFT and its applications is large. Some representative examples are the following: books [333–337], separate chapters of monographs [8, 10, 102] and review articles [330, 338–340].

We give here basic formulation of DFT and the Kohn–Sham method in the spirit of a “bird’s-eye view of DFT ” [330], referring the reader for the mathematical details to the original papers. The original Hohenberg–Kohn (HK) theorems [328] held only for the ground state in the absence of a magnetic field, although they have since been generalized. The theorems can be extended to the time-dependent domain DFT (TDDFT), which can also be used to determine excited states [341]. Nevertheless, the inability of the DFT method to describe for molecular systems the spin and space degenerate states [342], as the diagonal element of the full density matrix is invariant with respect to all operations of the symmetry group, was proven.

The first HK theorem demonstrates the existence of a one-to-one mapping between the ground-state electron density (7.1) and the ground-state wavefunction of a manyparticle system. The first Hohenberg–Kohn theorem is only an existence theorem, stating that the mapping exists, but does not provide any such exact mapping. It is in these mappings that approximations are made. Let us rewrite (4.7) for the manyelectron function of the system in the form

	1		N	N
			i
HΨ = [T + V + U ] Ψ = −		2	=1	∆i + V (ri) +	i<j	U (ri, rj ) Ψ = EΨ (7.2)
				i=1

where H is the electronic Hamiltonian, N is the number of electrons and U is the electron–electron interaction. The operators T and U are so-called universal operators as they are the same for any system, while V is system dependent (nonuniversal). The potential V is called the external potential. This may be not only that of nuclei but also for cases when the system is exposed to an external electrostatic or magnetic field.

7.1 Foundations of the Density-functional Theory

233

N

We can write it as a sum of one-electron potentials V (ri) but we do not know this

i=1

potential in advance. The actual di erence between a single-electron problem and the much more complicated many-electron problem just arises from the electron–electron interaction term U .

Hohenberg and Kohn proved [328] that the relation (7.1) can be reversed, i.e. to a given ground-state density ρ(r) it is in principle possible to calculate the corresponding ground-state wavefunction Ψ0(r1, r2, . . . , rN ). This means that for a given ground-state density for some system we cannot have two di erent external potentials V . This means that the electron density ρ(r) defines all terms in the Hamiltonian (7.2) and therefore we can, in principle, determine the complete N electron wavefunction for the ground-state by only knowing the electron density. The HK theorem shows only that it is possible to calculate any ground-state property when the electron density is known but does not give the means to do it. Since the wavefunction is determined by the density, we can write it as Ψ0 = Ψ0[ρ], which indicates that Ψ0 is a function of its N spatial variables, but a functional of ρ(r). More generally, a functional F [n] can be defined as a rule for going from a function to a number, just as a function y = f (x) is a rule (f ) for going from a number (x) to a number (y). A simple example of a

functional is the total number of electrons in a system N

N = d3rρ(r) = N [ρ(r)]

(7.3)

which is a rule for obtaining the number N , given the function ρ(r). Note that the name given to the argument of ρ is completely irrelevant, since the functional depends on the function itself, not on its variable. Hence we do not need to distinguish F [ρ(r)] from, e.g., F [ρ(r )]. Another important case is that in which the functional depends on a parameter, such as in

VH [ρ(r)] =	d3r |r − r |		(7.4)
		ρ(r )

that is a rule that for any value of the parameter r associates a value VH [ρ(r)] with the function ρ(r ). This term is the so-called Hartree potential, introduced in (4.16) and is a potential of the Coulomb field created by all electrons of the system, with the electron in question included.

DFT explicitly recognizes that nonrelativistic Coulomb systems di er only by their external potential V (r), and supplies a prescription for dealing with the universal operators T and U once and for all. This is done by promoting the electron density ρ(r) from just one of many observables to the status of the key variable, on which the calculation of all other observables can be based. In other words, Ψ0 is a unique functional of ρ, i.e. Ψ0[ρ] and consequently all other ground-state observables O are also functionals of ρ

ˆ	[ρ]	(7.5)
O [ρ] = Ψ0[ρ]|O|Ψ0

From this it follows, in particular, that also the ground-state energy is a functional of

ρ

E0 = E[ρ] = Ψ0[ρ]|T + V + U |Ψ0[ρ]

(7.6)

where the contribution of the external potential can be written explicitly in terms of the density

234	7 Kohn–Sham LCAO Method for Periodic Systems
	V [ρ] = V (r)ρ(r)d3r	(7.7)

The functionals T [ρ] and U [ρ] are called universal functionals, while V [ρ] is obviously nonuniversal, as it depends on the system under study. Having specified a system, i.e. V is known, one then has to minimize the functional

E[ρ] = T [ρ] + U [ρ] + V (r)ρ(r)d3r

(7.8)

with respect to ρ(r), assuming one has got reliable expressions for T [ρ] and U [ρ]. The second HK theorem proves that the ground-state density minimizes the total electronic energy of the system. It states that once the functional that relates the electron density with the total electronic energy is known, one may calculate it approximately by inserting approximate densities ρ . Furthermore, just as for the variational method for wavefunctions, one may improve any actual calculation by minimizing the energy functional E[ρ ]. A successful minimization of the energy functional will yield the ground-state density ρ0 and thus all other ground-state observables. A practical scheme for calculating ground-state properties from electron density was provided by the approach of Kohn and Sham [332] considered in the next section.

7.1.2 The Kohn–Sham Single-particle Equations

Within the framework of Kohn–Sham (KS) DFT, the intractable many-body problem of interacting electrons in a static external potential is reduced to a tractable problem of noninteracting electrons moving in an e ective potential. The functional in (7.8) is written as a fictitious density functional of a noninteracting system

Eef f [ρ] = Ψef f [ρ]|Tef f + Vef f |Ψef f [ρ]

(7.9)

where Tef f denotes the noninteracting electrons’ kinetic energy and Vef f is an external e ective potential in which the electrons are moving. It is assumed that the fictitious (model) system has the same energy as the real system. Obviously, ρef f (r) = ρ(r) if Vef f is chosen to be

Vef f = V + U + (T − Tef f )

(7.10)

Thus, one can solve the so-called Kohn–Sham equations of this auxiliary noninteracting system with the e ective Hamiltonian

N		N
	1
Hef f = i=1 −	2	∆i + Vef f (ri) = i=1 hef f (ri)	(7.11)

that yields the orbitals ϕi(r) that reproduce the density ρ(r) of the original manyelectron system ρ(r) = ρef f (r) = Ni=1|ϕ(r)|2. The e ective single-electron potential Vef f (r) can be written as

Vef f (r) = V +	ρef f (r )
	|r − r | d3r + VXC [ρef f (r)]	(7.12)

where the second term denotes the so-called Hartree term describing the electron– electron Coulomb repulsion, while the last term VXC is called the exchange correlation

7.1 Foundations of the Density-functional Theory

235

potential and includes all the many-electron interactions. Since the Hartree term and VXC depend on ρ(r) that depends on the ϕi, which in turn depend on Vef f , the problem of solving the Kohn–Sham equation has to be done in a self-consistent (i.e. iterative) way. Usually, one starts with an initial guess for ρ(r), then calculates the corresponding Vef f and solves the Kohn–Sham equations for the ϕi. From these one calculates a new density and starts again. This procedure is then repeated until convergence is reached.

There are only single-particle operators in (7.11). Therefore, the solution to the Schr¨odinger equation for a model system of noninteracting electrons can be written exactly as a single Slater determinant Ψ = |ϕ1, ϕ2, . . . , ϕN |, where the single-particle orbitals ϕi are determined as solutions of the single-particle equation

hef f ϕi = εiϕi

(7.13)

Furthermore, ρ(r) = Ni=1|ϕi(r)|2, where the summation runs over the N orbitals with the lowest eigenvalues εi.

The main problem is that we do not know the exact form of the e ective potential Vef f , i.e. the so-called exchange-correlation energy and potential. There exist approximations to these (see Sections 7.1.3 and 7.1.4) that are good approximations in very many cases. On the other hand, by introducing an approximation in the Schr¨odinger equation for the model noninteracting particles, we do not know whether improved calculations (e.g., using larger basis sets) also lead to improved results (compared with, e.g., experiment or other calculations). The success of the Kohn–Sham approach is based on the assumption that it is possible to construct the model system of noninteracting particles moving in an e ective external potential. Thus, it is indirectly assumed that for any ground-state density there exists an e ective potential (the corresponding density matrix is called V representative). There exist (specifically constructed) examples where this is not the case, but in most practical applications, this represents no problem [8].

HK theorems and KS equations can be extended to the spin-polarized systems where the electrondensity components ρα(r), ρβ (r) for spin-up and spin-down orbitals di er i.e. the spin-density ρs(r) = ρα(r) − ρβ (r) is nonzero.

The result is that the total energy and any other ground-state properties become a functional not only of ρ(r) but also of spin-density ρs(r), i.e. Eef f = E[ρ(r), ρs(r)]. Spin-density-functional theory (SDFT) is a widely implemented and applied formalism of DFT.

Alternative DFT formulations that use other variables in addition to (or instead of) the spin densities are also useful [343].

The single-particle orbitals and complete N -electron wavefunctions in KS DFT belong to irreducible representations of the group for the symmetry operations of the system of interest (point-symmetry group for molecules and space-symmetry group for crystals). This means that we can consider each irreducible representation separately. For each of these, the variational principle will apply (since any state of a given irreducible representation will only contain contributions from exactly that irreducible representation). We can use this in generalizing the HK theorems that then apply for the energetically lowest state of each irreducible representation. This means that if we have the representations Dα, Dβ for a given system symmetry group then we can apply DFT in studying the energetically lowest state for the Dα representation, that

236 7 Kohn–Sham LCAO Method for Periodic Systems

for the Dβ representation and so on, but not energetically higher ones of any of the representations.

Since 1990 there has been an enormous amount of comparison for molecules between DFT KS and HF (MO) theory. Such a comparison is easily extended to crystals when we consider CO instead of MO. The discussion of advantages and disadvantages of DFT compared to MO theory can be found for example, in [102] and is briefly reproduced here. The most fundamental di erence between DFT and MO theory is the following: DFT optimizes an electron density, while MO theory optimizes a wavefunction. So, to determine a particular molecular property using DFT, we need to know how that property depends on the density, while to determine the same property using a wavefunction, we need to know the correct quantummechanical operator. As a simple example, consider the total energy of interelectronic repulsion. Even if we had the exact density for some system, we do not know the exact exchange-correlation energy functional, and thus we cannot compute the exact interelectronic repulsion. However, with the exact wavefunction it is a simple matter of evaluating the expectation value for the interelectronic repulsion operator to determine this energy. Thus, it is easy to become confused about whether there exists a KS “wavefunction”. Formally, the KS orbitals are pure mathematical constructs useful only in construction of the density. In practice, however, the shapes of KS orbitals tend to be remarkably similar to canonical HF MOs and they can be quite useful in qualitative analysis of chemical properties. If we think of the procedure by which they are generated, there are indeed a number of reasons to prefer KS orbitals to HF orbitals. For instance, all KS orbitals, occupied and virtual, are subject to the same external potential. HF orbitals, on the other hand, experience varying potentials, and, in particular, HF virtual orbitals experience the potential that would be felt by an extra electron being added to the molecule. As a result, HF virtual orbitals tend to be too high in energy and anomalously di use compared to KS virtual orbitals. This fact is especially important for crystalline solids and explains why HF bandgaps are overestimated compared to those in DFT and experiment. Unfortunately, for some choices of exchange-correlation potential DFT bandgaps are too small in comparison with the experimental data. In exact DFT, it can also be shown that the eigenvalue of the highest KS MO is the exact first ionization potential, i.e. there is a direct analogy to Koopmans’ theorem for this orbital – in practice, however, approximate functionals are quite poor at predicting IPs in this fashion without applying some sort of correction scheme, e.g. an empirical linear scaling of the eigenvalues.

The Slater determinant formed from the KS orbitals is the exact wavefunction for the fictitious noninteracting system having the same density as the real system. This KS Slater determinant has certain interesting properties by comparison to its HF analogs. It is an empirical fact that DFT is generally much more robust in dealing with open-shell systems where UHF methods show high spin contamination, i.e. incorporates some higher spin states (doublets are contaminated by quartets and sextets, while triplets are contaminated by pentets and heptets). The degree of spin contamination can be estimated by inspection of S2 , which should be 0.0 for a singlet, 0.75 for a doublet, 2.00 for a triplet, etc. Note, incidentally, that the expectation values of S2 are sensitive to the amount of HF exchange in the functional. A “pure” DFT functional nearly always shows very small spin contamination, and each added per cent of HF exchange tends to result in a corresponding percentage of the spin con-

7.1 Foundations of the Density-functional Theory

237

tamination exhibited by the HF wavefunction. This behavior can make the hybrid HF-DFT functionals useful for open-shell systems (see Sect. 4.1.2).

The formal scaling behavior of DFT is, in principle, no worse than N 3, where N is the number of basis functions used to represent the KS orbitals. This is better than HF by a factor of N , and very substantially better than other methods that include electron correlation (see Chap. 5).

The most common methods for solving KS equations proceed by expanding the Kohn–Sham orbitals in a basis set. DFT has a clear advantage over HF in its ability to use basis functions that are not necessarily contracted Gaussians. The motivation for using contract GTOs is that arbitrary four-center two-electron integrals can be solved analytically. In the electronic-structure programs where DFT was added as a new feature to an existing HF code the representation of the density in the classical electron-repulsion operator is carried out using the KS orbital basis functions. Thus, the net e ect is to create a four-index integral and these codes inevitably continue to use contracted GTOs as basis functions. In particular, such a scheme is used in the CRYSTAL code [23]. However, if the density is represented using an auxiliary basis set, or even represented numerically, other options are available for the KS orbital basis set, including Slater-type functions. The SIESTA density-functional code [344] for crystalline solids uses numerical AO basis instead of GTO. Slater-type orbitals (STO) enjoy the advantage that fewer of them are required (since they have the correct cusp behavior at the nuclei) and certain advantages associated with symmetry can more readily be taken, so they speed up calculations considerably. The Amsterdam density-functional code and its BAND version for solids [345] makes use of STO basis functions covering atomic numbers 1 to 118. Some information about the three mentioned computer LCAO codes for solids is given in Appendix C.

Another interesting possibility is the use of plane waves as basis sets in periodic infinite systems (crystalline solids) represented using periodic boundary conditions. While it takes an enormous number of plane waves to properly represent the decidedly aperiodic densities that are possible within the unit cells of interesting chcmical systems, the necessary integrals are particularly simple to solve, and thus this approach has found wide use in solid-state physics.

Meanwhile, plane-wave basis use excludes the possibility of calculations with hybrid DFT-HF functionals.

7.1.3 Exchange and Correlation Functionals

in the Local Density Approximation

The e ective potential (7.10) includes the external potential and the e ects of the Coulomb interactions between the electrons, e.g. the exchange and correlation interactions. In principle, it also includes the di erence in kinetic energy between the fictitious noninteracting system and the real system. In practice, however, this difference is ignored in many modern functionals as the empirical parameters appear, which necessarily introduce some kinetic-energy correction if they are based on experiment [102].

Modeling the exchange and correlation interactions becomes di cult within KS DFT as the exact functionals for exchange and correlation are not known except for the homogeneous (uniform) electron gas. However, approximations exist that permit the calculation of real systems.

238 7 Kohn–Sham LCAO Method for Periodic Systems

The simplest approximation is the local-density approximation (LDA), based upon the exact exchange energy for a uniform electron gas, which can be obtained from the Thomas–Fermi (TF) model, and from fits to the correlation energy for a uniform electron gas.

In the TF model it is suggested that the number of electrons is so large that the system could be treated using quantum-statistical arguments. The approximation in the TF model concerns the kinetic energy. For the homogeneous interaction-free electron gas the density is constant and the average kinetic energy per particle is εhomt = Cn5/3. The kinetic energy per unit volume in this model is nεhomt . If the

electron density varies su ciently slowly in space T LDA =		d3rεthom(ρ(r)) may serve
as a workable approximation for the kinetic-energy	functional.
		/

If this is combined with the expression for the nuclei–electron attractive potential and the electron–electron Hartree repulsive potential we have the TF expression for the energy of a homogeneous gas of electrons in a given external potential:

ET F [ρ(r)] = CF

ρ5/3(r)d3r − j

Zj

ρ(r) +

1

ρ(r)ρ(r )

d3rd3r (7.14)

|

r

−

Rj

|

2

|

r

−

r

|

The importance of this equation is not so much how well it is able to really describe the energy even of an atom, but that the energy is given completely in terms of the electron density ρ(r).

This is an example of a density-functional for energy allowing us to map a density ρ(r) onto an energy E without any additional information required. Furthermore, the TF model employs the variational principle assuming that the ground state of the system is connected to the electron density for which the energy (7.14) is minimized under the constraint of N = / d3rρ(r).

Before the KH theorems were proved it was not known either whether expressing the energy as a density-functional is physically justified or whether employing the variational principle on the density is really allowed [337].

Slater’s approximation of HF exchange [346] is another example of exploiting the electron density as the central quantity. In this case, the nonlocal HF exchange energy

is approximated by α · C ·		ρ(r)4/3dr.
This approximate	expression depends only on the local values of the electron
		/

density ρ(r) representing a density-functional for the exchange energy (in the TF model exchange and correlation e ects are completely neglected). This formula was originally derived as an approximation to the HF exchange, without any reference to the density-functional theory but it is conceptually connected with this theory. The 4/3 power law for dependence of the exchange interaction on the electron density was also obtained from a completely di erent approach using the concept of the uniform electron gas [347]. Combined with the TF energy this approximation is known as the Thomas–Fermi–Dirac model having conceptual importance for DFT methods. In particular, it seems natural that the exchange-correlation energy EXC is approximated by sum of the exchange EX and correlation EC energies.

In LDA for the exchange energy calculation the Dirac–Slater exchange energy is

used
EX [ρ] = CX		ρ4/3(r)d3r	(7.15)

or the more complicated suggested by Barth and Hedin [348].

7.1 Foundations of the Density-functional Theory

239

For the correlation energy functional EC [ρ(r)] the situation is more complicated since even for a homogeneous electron gas it is not known exactly. Early approximate expressions for correlation in homogeneous systems were based on applying the perturbation theory and were suggested by Barth and Hedin [348], Gunnarsson and Lundqvist [349]. With the advent of highly precise calculations of correlation energy for the electron liquid by Ceperley and Alder (CA) [350] the approximations for the correlation energy in a homogeneous system are made by parametrization of CA data for a free-electron gas. There are known parametrizations of Vosko– Wilk–Nisair [351], Perdew–Zunger [352] and Perdew–Wang [353]. The three latter parametrizations of the LDA are implemented in most standard DFT program packages (both for molecules and solids) and in many cases give almost identical results. On the other hand, the earlier parametrizations of the LDA, based on perturbation theory can deviate substantially and are better avoided.

The functional dependence of EXC on the electron density is expressed as an interaction between the electron density and “an energy density” εXC that is dependent on the electron density

EXC [ρ(r)] = ρ(r)εXC [ρ(r)]d3r (7.16)

The energy density εXC is treated as a sum of individual exchange and correlation contributions. Two di erent kinds of densities are involved [102]: the electron density is a per unit volume density, while the energy density is a per particle density. The LDA for EXC formally consists in

EXCLDA[ρ(r)] =				d3rεXChom[ρ(r)]
	d3r εXhom[ρ(r)] + εChom[ρ(r)]	=			(7.17)

The energy densities εhomX , εhomC refer to a homogeneous system, i.e. the exchangecorrelation energy is simply an integral over all space with the exchange-correlation energy density at each point assumed to be the same as in a homogeneous electron gas with that density. Nevertheless, LDA has proved amazingly successful, even when applied to systems that are quite di erent from the electron liquid that forms the reference system for the LDA.

In the local spin-density approximation (LSDA) the exchange-correlation energy can be written in terms of either of two spin densities ρα(r) and ρβ (r)

EXCLSDA[ρα(r), ρβ (r)] =

d3rρ(r)εXChom[ρα(r), ρβ (r)]

=

εXhom

+ εChom

d3rρ(r)

ρα(r), ρβ (r)

ρα(r), ρβ (r)

(7.18)

or

the total

density

ρ(r)

and

the fractional

spin polarization

ζ(r) = (ρ

α(r)

−

β

(r))/ρ(r).

−ρ

For many decades the LDA has been applied in, e.g., calculations of band structures and total energies in solid-state physics. The LDA provides surprisingly good results for metallic solids with delocalized electrons, i.e. those that most closely resemble the uniform electron gas (jellium). At the same time, there are well-known disadvantages of LDA for solids. LDA revealed systematic shortcomings in the description of systems with localized electrons and as a result the underestimation of

240 7 Kohn–Sham LCAO Method for Periodic Systems

bond distances and overestimation of binding energies. LDA calculations as a rule give calculated bandgaps that are too small. In quantum chemistry of molecules LDA is much less popular because the local formulation of the energy expression does not account for the electronic redistribution in bonds. For well-localized electrons the nonexact cancellation of the self-energy part (self-interaction) of the Hartree term in the LDA exchange functional is important (in HF energy the self-energy part in the Hartree term is cancelled by the corresponding part of the exchange term). LDA fails to provide results that are accurate enough to permit a quantitative discussion of the chemical bond in molecules (so-called “chemical accuracy” requires calculations with an error of not more than about 1 kcal/mol per particle). LDA exploits knowledge of the density at point ρ(r). The real systems, such as molecules and solids, are inhomogeneous (the electrons are exposed to spatially varying electric fields produced by the nuclei) and interacting (the electrons interact via the Coulomb interaction). The way density-functional theory, in the local-density approximation, deals with this inhomogeneous many-body problem is by decomposing it into two simpler (but still highly nontrivial) problems: the solution of a spatially uniform many-body problem (the homogeneous electron liquid) yields the uniform exchange-correlation energy, and the solution of a spatially inhomogeneous noninteracting problem (the inhomogeneous electron gas) yields the particle density. Both steps are connected by the local-density approximation, which shows how the exchange-correlation energy of the uniform interacting system enters the equations for the inhomogeneous noninteracting system.

We note that both the local density approximation and the local exchange approximation use only the diagonal part of the density matrix ρ(r, r ), i.e. ρ(r) = ρ(r, r). However these approximations are di erent by their nature. In LDA the local density is used to include both the exchange and correlation of electrons, in the local exchange approximation to HF exchange the electron correlation is not taken into account at all.

The particular way in which the inhomogeneous many-body problem is decomposed, and the various possible improvements on the LDA, are behind the success of DFT in practical calculations, in particular, materials. The most important improvement of LDA is connected with the attempt to introduce a spatially varying density and include information on the rate of this variation in the functional. The corresponding functionals, known as semilocal functionals [330], are considered in the next section.

7.1.4 Beyond the Local Density Approximation

The first successful extensions for the LDA were developed in the early 1980s when it was suggested to supplement the density ρ(r) at a particular point r with information about the gradient of the electron density at this point in order to account for the nonhomogeneity of the true electron density [337]. LDA was interpreted as the first term of a Taylor expansion of the uniform density, the form of the functional was termed the gradient expansion approximation (GEA). It was expected to obtain better approximations of the exchange-correlation functional by extending the series with the next lowest term. In practice, the inclusion of low-order gradient corrections almost never improves on the LDA, and often even worsens it. The reason for this failure is that for GEA the exchange-correlation hole has lost many of the properties that