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

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over contracted Gaussian-type functions. For these calculations the algorithm of Gill and Pople [416] was modified [408] so that the evaluation of integrals (7.62) is only slightly more time consuming than the regular electron-repulsion integrals.

To calculate the screened Coulomb exchange PBE functional the exchange model hole JxP BE of the PBE functional constructed in a simple analytical form by Ernzerhof and Perdew [417] is used. The model hole reproduces the exchange energy density of the PBE approximation for exchange and accurately describes the change in the exchange hole upon the formation of single bonds. In the HSE functional this PBE exchange hole is screened by employing the short-range Coulomb potential from (7.59).

The PBE long-range exchange contribution is then defined as the di erence of the exchange-hole-based PBE and the SR PBE exchange energy densities.

In the so-called revised HSE03 hybrid functional [409] the improvement was introduced in the calculation in the integration procedure of the PBE exchange hole. This modification made the calculations numerically more stable and ensures that the HSE03 hybrid functional for ω = 0 is closer to the PBE0 hybrid. The HSE03 (denoted also as EωP BE ) functional was incorporated into the development version of the GAUSSIAN code [418]. It was demonstrated for molecular systems that the HSE03 hybrid functional delivers results (bond lengths, atomization energies, ionization potentials, electron a nities, enthalpies of formation, vibrational frequencies), that are comparable in accuracy to the nonempirical PBE0 hybrid functional [409, 411, 412].

The HSE03 hybrid functional was extended to periodic systems [409]. The calculations were made for 21 metallic, semiconducting and insulating solids. The examined properties included lattice constants, bulk moduli and bandgaps. The results obtained with HSE03 exhibit significantly smaller errors than pure DFT calculations.

The preliminary screening of the integrals is necessary to take advantage of the rapid decay of short-range exchange integrals for periodic systems [409]. Two di erent screening techniques are used. In the first technique (Schwarz screening) substituting the SR integrals in place of the 1/r integrals yields an upper bound of the form

4		4
|(µν|λσ)SR| ≤	(µν|µν)SR (λσ|λσ)SR			(7.63)

The (µν|λσ) integrals are then evaluated for each batch of integrals and only batches with nonnegligible contributions are used in calculating the HF exchange. The SR screening integrals are evaluated by the same procedure as the SR exchange integrals.

The second – distance-based multipole screening technique – uses multipole moments introducing the following screening criterion

				1
Tn =	Cµνmax						Cλσmax	(7.64)
		r	l+M	µν	+M	λσ
M µν M λσ

where Tn is an estimate for the contribution of a shell quartet and M is a multipole in the multipole expansion of a given shell pair. Mijlow are the lowest-order multipoles that can contribute to the integral and Cijmax are the maximum coe cients in each order of multipoles. Replacing the 1/r potential with the erf c(ωr)/r short-range potential yields

T SR =	Cmax erf (ωrl+Mµνlow			+Mλσlow )	(7.65)
n	µν	r	l+Mlow +M low
M µν M λσ			µν	λσ

262 7 Kohn–Sham LCAO Method for Periodic Systems

This provides a distance-based upper bound for the SR exchange contribution of a given shell quartet.

The implementation of periodic boundary conditions relies on evaluating all terms of the Hamiltonian in real space [379]. The HF exchange is evaluated using replicated density matrices. All interactions within a certain radius from a central reference cell are calculated and the rest are neglected. This so-called near-field exchange (NFX) method [413] allows calculation of the HF exchange in time, scaling linearly with system size. It works reasonably well for insulating solids since the corresponding density matrix elements decay rapidly. In systems with smaller bandgaps, however, the spatial extent of nonnegligible contributions to the exchange energy is large. This large extent results in a large number of significant interactions. To render the computation tractable, the truncation radius must be decreased. Thus, significant interactions are neglected that leads to errors in the total energy of the system and introduces instabilities into the self-consistent field procedure.

Screened Coulomb hybrid functionals do not need to rely on the decay of the density matrix to allow calculations in extended systems [409]. The SR HF exchange interactions decay rapidly and without noticeable dependence on the bandgap of the system. The screening techniques do not rely on any truncation radius and provide much better control over the accuracy of a given calculation. In addition, the thresholds can be set very tightly, without resulting in extremely long alculations.

A series of benchmark calculations [409] on three-dimensional silicon (6-21G basis was used, see Chap. 8) demonstrates the e ectiveness of the screening techniques. The time per SCF cycle was studied given as a function of the distance up to which exchange interactions were included in the calculation. As this radius grows, the number of replicated cells grows as O(N 3). The PBE0 curve tracks this growth since regular HF exchange has a large spatial extent. The relatively small bandgap of silicon (1.9 eV in this calculation) is insu cient for density matrix elements to decay noticeably. HSE, on the other hand, only shows a modest increase in CPU time as the system becomes larger. Beyond 10 ˚A, the CPU time only increases due to the time spent on screening. The HSE calculation of the total energy converges very rapidly and only cells up to 10 ˚A from the reference cell contribute to the exchange energy. PBE0, by comparison, converges significantly slower. Thus, HSE not only reduces the CPU time drastically, but also decreases the memory requirements of a given calculation since fewer replicated density matrices need to be stored in memory. In practice, HSE calculations can be performed with the same amount of memory as pure DFT calculations, whereas traditional hybrid methods have larger memory demands. Given the fast decay of the SR HF exchange interactions and the high screening e ciency, the HSE hybrid functional was e ciently applied to a variety of three-dimensional solids. We mention only some of these applications: 1) 21 metallic, semiconducting and insulating solids [409]; 2) semiconductor set of 40 solids, containing 13 group IIA-VI systems, 6 group IIB-VI systems, 17 group III-V systems, and 4 group IV systems [419]; 3) the zinc-blende and rocksalt phases of platinum nitride [420] and platinum monoxide [421];

4)lattice defects and magnetic ordering in plutonium oxides PuO2 and Pu2O3 [422];

5)carbon nanotubes [423]; 6) actinide oxides UO2, PuO2 and Pu2O3 [424]. It was demonstrated that the screened Coulomb hybrid density-functional HSE not only reduces the amounts of memory and CPU time needed, when compared to its parent functional PBE0, but it is also at least as accurate as the latter for structural, optical

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and magnetic properties of solids. The PBE0 functional and other hybrid functionals are transferred from molecules to solids. The possibility of such a transfer we discuss in the next section.

7.2.4 Are Molecular Exchange-correlation Functionals Transferable to Crystals?

The question in the title of this subsection arises due to the well-known fact that the properties of crystals are quite often di erent from those of molecular systems. The di erence is particularly evident for ionic and semi-ionic solids, in which the longrange electrostatic forces provide a strong localizing field for the electronic states that is not present in molecules. In this situation the shortcomings of standard LDA and GGA functionals, linked to the missing electronic self-interaction, are likely to be most severe [371]. This explains why the electronic-structure calculations in molecular quantum chemistry and solid-state physics were for a long time developing along two independent lines. While in the molecular quantum chemistry the wavefunction-based approaches (HF and post-HF) in the LCAO approximation were mainly used, in solidstate physics DFT-based methods with plane wave (PW) basis were popular. These two standard approaches were for a long time poorly transferable between the two fields [371]: early DFT functionals underperform post-HF techniques in reproducing the known properties of small molecules, while the extension of accurate post-HF methods to solid-state systems is di cult or has prohibitive computational expense.

The formulation of hybrid HF/DFT exchange-correlation functionals and their extension to periodic systems changed the situation; the combined HF-DFT approach has adequate accuracy for most needs in both quantum chemistry of molecules and solids, retaining a tractable computational cost and even allowing the system-size linear-scaling. One appealing feature of this approach is that it can readily exploit the progress and tools available to the quantum chemists for calculating the HF exchange, as it was demonstrated in Sections 7.2.2 and 7.2.3.

For molecules, the first hybrid functionals have been coded into GAUSSIAN94 in the MO LCAO approximation traditional for molecules. The DFT PW computer codes, existing for solids, can be modified to include HF method and hybrid exchangecorrelation functionals [425]. However, the practical realization of HF method was made in LCAO approximation. The application of hybrid functionals to crystalline compounds (described under periodic boundary conditions (PBC)) was not possible until the late 1990s, when they were coded in CRYSTAL98 [426]. Numerous solid-state studies have been performed with molecular hybrid functionals, providing valuable experience on their accuracy and applicability. In the review article [371] the results are summarized of publications in which hybrid exchange functionals have been applied under PBC to represent crystalline solids. The list of later publications can be found on the site http://www.crystal.unito.it. The next step in the HF/DFT extension to solids was made by Scuseria and coworkers who coded the HF/DFT method with PBC in GAUSSIAN03 code [107] and included a linear-scaling hybrid exchange-correlation HSE functional.

The extension of HF-DFT methods to the crystalline solids encounters di culties [371]. In molecular chemistry, very accurate thermochemical and structural data exist for a selection of simple species. These data have been e ectively used to define

264 7 Kohn–Sham LCAO Method for Periodic Systems

representative sets against which to measure the results of calculations. The availability of these representative sets has enabled the empirical derivation of hybrid DFT functionals for molecules (for example, B3LYP and B3PW functionals). Comparison to experiment in the solid state is often indirect: most reliable experimental results on crystalline solids are not directly related to the equilibrium structure, energy and electronic density (as for molecules), but to the response of the solid to a perturbation, for instance to elastic distortions or to external electromagnetic fields. As an additional complexity, some fundamental electronic properties of solids cannot be measured directly; bandgaps, for instance, can be readily calculated as edge-to-edge di erences between specific filled and unfilled bands, but they cannot be measured directly. What experiments furnish are absorption edges and activation energies corresponding to processes that are often speculative. As another example, magnetic coupling constants can be neither measured nor calculated directly. Their extraction from calculations requires a mapping onto a phenomenological model of some sort, which by its very nature is inexact, while comparisons are made with indirect quantities such as disorder-transition temperatures, which in turn have to be expressed in terms of the phenomenological model.

The second di culty is the lack for solids of a reference theoretical method. PostHF techniques in molecular quantum chemistry can yield results with a controlled degree of accuracy. In the absence of experimental data, the results obtained with di erent DFT functionals could be compared against those calculated with the reference computational technique. Recent developments in wavefunction methods for solid-state systems (Chap. 5), GW perturbation theory [427], and quantum Monte Carlo (QMC) [428] are promising for future work, but at present they still su er from a limited applicability.

In most solids, HF/DFT calculations with the CRYSTAL code [23] the B3LYP hybrid functional was used with the standard molecular formulation (with the mixing parameter amix=0.2) and its results graded against a set of other Hamiltonians (HF, LDA, PWGGA, PBE). In some calculations of oxides the di erent mixing parameter values were used [429–432]. These studies point to the need specific formulations of the hybrid functionals for solids, in particular systematic investigations on the e ect of changing the mixing parameter. Such a systematic investigation on the e ect of changing the mixing parameter from 0% to 100% was performed [371] for perovskitestructured transition-metal oxides AMO3 (including ferroelectric materials BaTiO3 and KNbO3), rocksalt-structured oxides and fluorides and open-shell defects in these matrices. All these systems share the fact that standard DFT calculations, in either LDA or GGA formulations, fail to reproduce with su cient accuracy the correct electronic ground-state and/or the structural and electronic properties of the material. Several observables have been examined (structural parameters, elastic constants and bulk moduli, phonon spectra, bandgaps, thermochemical data, point defects and surfaces, magnetic ordering). We refer the reader to the review article [371] for comments on each observable separately. The details of the LCAO basis and an auxiliary Gaussians for exchange-correlation calculation choice are given in [371]. The truncation threshholds and BZ summation parameters are also given, ensuring convergence of the total energies to within 0.1 meV. As an example of observables calculations we show in Table 7.2 some results obtained in [371] for structural properties (lattice constants and bulk moduli) of cubic perovskites. The first line of the Table gives the

=0.2,
mix
(a
parameter
mix
andbulkmodulusB(GPa)forcubicperovskites,asafunctionofa functional([371])
˚	DFTandhybridB3LYP
latticeparametera(A)
Table7.2.Equilibrium	0.4,0.6,0.8,1.0)inthe

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265

Expt	3.84		3.90	179	4.01	195	4.02		4.15		4.19		3.94		4.01
HF	3.870	218.0	3.937	212.3	4.020	203.9	4.150	181.4	4.197	176.1	4.250	176.5	3.962	226.7	4.013	221.5	3.941	189.8
1.0	3.802	248.5	3.867	245.6	3.947	235.0	4.081	205.0	4.124	201.1	4.175	201.7	3.905	235.1	3.949	251.0	3.881	208.5
0.8	3.820	236.5	3.887	232.0	3.966	222.9	4.098	196.7	4.143	191.9	4.193	193.5	3.927	238.7	3.972	235.6	3.900	187.7
0.6	3.840	223.2	3.909	217.2	3.987	209.7	4.117	188.1	4.164	182.0	4.212	184.8	3.951	222.9	3.998	218.8	3.923	176.0
0.4	3.863	208.9	3.935	201.1	4.011	195.4	4.137	178.9	4.187	171.7	4.232	175.4	3.980	205.4	4.028	200.5	3.951	164.8
0.2	3.890	193.0	3.965	183.2	4.039	179.8	4.160	168.9	4.214	160.1	4.255	165.5	4.013	186.6	4.063	181.0	3.982	156.8
B3LYP	3.885	194.6	3.959	185.0	4.033	181.6	4.155	170.1	4.208	161.4	4.249	166.9	4.008	188.2	4.057	182.5	3.978	156.6
BLYP	3.921	175.8	4.001	163.3	4.072	162.5	4.186	157.9	4.244	147.5	4.282	154.6	4.051	168.3	4.103	158.9	4.018	155.4
PW	3.884	187.2	3.960	176.8	4.024	177.8	4.149	167.4	4.203	157.3	4.235	166.0	4.018	178.0	4.063	172.1	3.992	146.4
LDA	3.817	213.8	3.888	207.7	3.953	204.7	4.084	187.8	4.133	179.4	4.167	189.3	3.960	203.7	3.996	197.4	3.938	144.7
	a	B	a	B	a	B	a	B	a	B	a	B	a	B	a	B	a	B
	3		3		3		3		3		3		3		3		3
Solid	CaTiO		SrTiO		BaTiO		CaZrO		SrZrO		BaZrO		NaNbO		KNbO		KTaO

266 7 Kohn–Sham LCAO Method for Periodic Systems

exchange-correlation functional type. For the HF-DFT hybrid functional both the standard molecular B3LYP functional and the BLYP exchange-correlation functional with a fraction amix of HF exchange (the numerical values of amix are given in the first line of Table 7.2) were used. For comparison, the available experimental data are given in the last column of this table. The following trends can be observed from the results given in Table 7.2. The lattice constant a increases systematically on increasing the mixing parameter amix and has a variation of approximately 3% in the series 0 < amix < 1. The bulk modulus also increases with increasing amix, with a change for most materials of about 80% between amix = 0 and amix = 1. Increasing the percentage of the HF exchange in the functional, therefore, makes the structure more compact and harder, and one may expect this feature to influence the behavior of the material towards structural distortions.

The lattice constant and bulk modulus are the ground-state properties defined by the total energy and its derivatives. The measured bandgap is defined by the excitation energies and its calculation includes the eigenvalues corresponding to the bottom of the conduction band. Nevertheless, recent calculations with hybrid functionals show that the observed bandgaps are reliably reproduced in a wide variety of materials. As an example, we give in Table 7.3 the results of B3LYP LCAO bandgap calculations [433] in comparison with the experimental data.

To establish the reliability of the hybrid functional the di erent types of materials have been studied: semiconductors (diamond and GaAs), semi-ionic oxides (ZnO, Al2O3, TiO2), sulfides (FeS2, ZnS), an ionic oxide MgO and the transition-metal oxides (MnO, NiO). For each system the AO basis sets were developed and full structural optimization of the cell and internal coordinates was performed [433]. Typically, the optimized structural parameters are within ±2% of the experimental values. The bandgaps were determined from the band structure of the optimized system as the di erence in the converged eigenvalues. As is seen from Table 7.3 even the molecular version of the B3LYP exchange correlation functional is capable of reproducing bandgaps for a wide variety of materials in good agreement with the experimental

Table 7.3. A comparison of observed bandgaps with those calculated using the B3LYP functional for a wide range of materials (all values are given in eV) [433]

Material	Expt	B3LYP
Diamond	5.5	5.8
GaAs	1.4	1.5
ZnO	3.4	3.2
Al2O3	9.0	8.5
Cr2O3	3.3	3.4
MgO	7.8	7.3
MnO	3.6	3.8
NiO	3.8	3.9
TiO2	3.0	3.4
FeS2	1.0	2.0
ZnS	3.7	3.5

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data (the latter even measured under well-controlled conditions are reliable to about 5%, [433]). This agreement is at least as good as that obtained with more sophisticated correlated calculations or perturbation theories.

The latter studies confirmed the e ciency of HF/DFT approach for solids and also use of the HSE hybrid functional. In [419] a set of 40 semiconductors was chosen based on the following criteria: all considered systems were both closed shell and of simple or binary composition; the majority of them have simple zincblende or rocksalt structures but also several systems with wurtzite structure were included. In addition, the availability of experimental data for lattice constants (and to a lesser extent bandgaps) was an important factor (the references to the experimental data can be found in [419]). These criteria led to the semiconductor/40 (SC/40) set of 40 solids containing 13 group IIA-VI systems, 6 group IIB-VI systems, 17 group III-V systems, and 4 group IV systems.

Table 7.4 contains a full list of all compounds and the calculated lattice constants and bandgaps. Based on the results of previous calculation with the HSE screened hybrid functional [409] the short-range – long-range splitting parameter was chosen as system independent and equal to 0.15, the PBE0 functional mixing parameter was taken to be 0.25. The results in Table 7.4 are given for the LSDA, PBE, meta-GGA TPSS (Tao–Perdew–Staroverov–Scuseria, [360]) and HSE hybrid functionals. All calculations employed the PBC option of the GAUSSIAN03 code, total energies are accurate to 10−6 a.u. All basis sets used are modified molecular Gaussian basis sets (all di use basis functions with exponents below 0.12 were removed). The modification of the atomic basis sets for solids is discussed in Chap. 8 in more detail. For 31 of 40 systems the inclusion of relativistic e ects in the pseudopotential was examined. In Chap. 8 we discuss relativistic pseudopotentials in more detail. Table 7.4 gives in comparison with the experimental data the calculated lattice constants and bandgaps for 40 semiconductors (with optical bandgaps ranging from 0.2 to 7.2 eV). The comparison of results is given for three nonhybrid (LSDA, PBE and TPSS) functionals and hybrid HSE functional. Table 7.5 contains the error statistics for predicting lattice constants a (in ˚A) and bandgaps (in eV).

As is seen from Table 7.2.4 LSDA underestimates lattice constants in nearly all cases, while PBE and TPSS always overestimate them. The screened hybrid functional HSE reduces the overestimation of PBE (on which it is based) drastically, leading to the best predictions overall. The 20% accuracy improvement of HSE over LSDA is noteworthy since LSDA is the most widely used method for lattice optimizations in solids. In addition, the overestimation of lattice constants by using the pesudopotentials (instead of all-electron calculations) is partly responsible for the observed errors with HSE. It is possible to expect that all-electron calculations yield even better results with HSE while the underestimation exhibited by LSDA would only be exaggerated. Bandgaps calculated with all three nonhybrid density-functionals are severely underestimated (mean absolute errors are about 1 eV, see Table 7.5), in extreme cases such as MgO, by as much as 2.88 eV. In addition, several small-bandgap systems (Ge, GaSb, InN, InAs, and InSb) are predicted to be quasimetallic. All bandgaps correspond to lattices optimized with the respective functional. The HSE hybrid functional yields a drastically reduced mean absolute error of only 0.26 eV and predicts even small-bandgap systems correctly. The HSE errors for bandgaps are comparable to those obtained with the GW approximation [434]. However, these calculations do not

268 7 Kohn–Sham LCAO Method for Periodic Systems

Table 7.4. Lattice constants (˚A) and bandgaps (eV) for the set of 40 simple and binary semiconductors ( [419]), a–sphalerite structure, c–wurtzite structure

			Lattice constants						Bandgaps
Solid		LSDA	PBE	TPSS	HSE	Expt. LSDA PBE TPSS HSE Expt.
C		3.537	3.579	3.579	3.553	3.567	4.23	4.17	4.21	5.49	5.48
Si		5.410	5.479	5.466	5.444	5.430	0.59	0.75	0.82	1.28	1.17
Ge		5.634	5.776	5.744	5.701	5.658	0.00	0.00	0.00	0.56	0.74
SiC		4.355	4.404	4.394	4.372	4.358	1.40	1.46	1.42	2.39	2.42
BN		3.584	3.629	3.629	3.603	3.616	4.45	4.51	4.52	5.98	6.22
BP		4.509	4.567	4.566	4.543	4.538	1.31	1.41	1.45	2.16	2.40
BAs		4.750	4.829	4.821	4.794	4.777	1.16	1.27	1.29	1.92	1.46
BSb		5.201	5.291	5.280	5.251	n/a	0.80	0.88	0.81	1.37	n/a
AlN	(a)	3.112	3.153	3.147	3.127	3.111	4.9	4.95	5.01	6.45	6.13
	(c)	4.974	5.045	5.028	5.000	4.981
AlP		5.436	5.508	5.497	5.472	5.463	1.60	1.83	1.90	2.52	2.51
AlAs		5.639	5.733	5.713	5.691	5.661	1.40	1.62	1.71	2.24	2.23
AlSb		6.079	6.188	6.172	6.146	6.136	1.29	1.40	1.63	1.99	1.68
GaN	(a)	3.167	3.233	3.224	3.198	3.189	2.09	1.70	1.73	3.21	3.50
	(c)	5.165	5.272	5.244	5.204	5.185
β − GaN		4.476	4.569	4.552	4.518	4.523	1.93	1.55	1.56	3.03	3.30
GaP		5.418	5.534	5.522	5.484	5.451	1.59	1.71	1.98	2.47	2.35
GaAs		5.626	5.771	5.745	5.705	5.648	0.43	0.19	0.52	1.21	1.52
GaSb		6.043	6.208	6.183	6.140	6.096	0.09	0.00	0.08	0.72	0.73
InN	(a)	3.523	3.599	3.589	3.555	3.537	0.02	0.01	0.00	0.71	0.69
	(c)	5.684	5.807	5.765	5.729	5.704
InP		5.839	5.970	5.961	5.909	5.869	0.83	0.68	0.90	1.64	1.42
InAs		6.038	6.195	6.170	6.120	6.058	0.00	0.00	0.00	0.39	0.41
InSb		6.430	6.608	6.585	6.535	6.479	0.00	0.00	0.00	0.29	0.23
ZnS		5.319	5.467	5.465	5.432	5.409	2.25	2.16	2.39	3.42	3.66
ZnSe		5.588	5.751	5.736	5.707	5.668	1.21	1.19	1.48	2.32	2.70
ZnTe		6.017	6.195	6.174	6.150	6.089	1.28	1.14	1.45	2.19	2.38
CdS		5.776	5.934	5.944	5.896	5.818	1.01	1.11	1.34	2.14	2.55
CdSe		6.025	6.210	6.195	6.152	6.052	0.34	0.48	0.73	1.39	1.90
CdTe		6.422	6.626	6.610	6.568	6.480	0.61	0.62	0.88	1.52	1.92
MgO		4.178	4.268	4.247	4.218	4.207	4.92	4.34	4.56	6.50	7.22
MgS		5.618	5.721	5.719	5.681	5.622	3.54	3.57	3.88	4.78	5.4
MgSe		5.417	5.532	5.520	5.499	5.40	1.70	1.76	2.07	2.62	2.47
MgTe		6.381	6.517	6.517	6.478	6.42	2.74	2.68	3.12	3.74	3.6
CaS		5.572	5.715	5.710	5.698	5.689	2.16	2.56	2.69	3.59	n/a
CaSe		5.799	5.962	5.955	5.946	5.916	1.74	2.11	2.23	3.02	n/a
CaTe		6.209	6.387	6.386	6.381	6.348	1.25	1.60	1.72	2.37	n/a
					to be continued

				7.2 Density-functional LCAO Methods for Solids							269
Solid	LSDA	PBE	TPSS	HSE	Expt.	LSDA	PBE	TPSS	HSE	Expt.
SrS	5.926	6.066	6.052	6.037	5.99	2.27	2.68	2.79	3.59		n/a
SrSe	6.151	6.306	6.290	6.282	6.234	1.89	2.26	2.37	3.09		n/a
SrTe	6.543	6.714	6.703	6.701	6.64	1.51	1.89	2.00	2.57		n/a
BaS	6.303	6.436	6.433	6.413	6.389	2.05	2.44	2.61	3.28		3.88
BaSe	6.517	6.671	6.659	6.649	6.595	1.76	2.12	2.26	2.87		3.58
BaTe	6.897	7.062	7.062	7.051	7.007	1.49	1.87	2.01	2.50		3.08

Table 7.5. Lattice constant a ( in ˚A) and bandgap (in eV) error statistics for the set of 40 simple and binary semiconductors( [419])

		LSDA	PBE	TPSS	HSE
Mean error	a	–0.046	0.076	0.063	0.035
	bandgap	–1.14	–1.13	–0.98	–0.17
Mean absolute error	a	0.047	0.076	0.063	0.037
	bandgap	1.14	1.13	0.98	0.26
Root-mean-square error	a	0.058	0.084	0.071	0.044
	bandgap	1.24	1.25	1.12	0.34
Maximum positive deviation	a	0.017	0.158	0.143	0.100
	bandgap	...	...	...	0.32
Maximum negative deviation	a	0.139	...	...	–0.014
	bandgap	–2.30	–2.88	–2.66	–0.72

include any excitonic or quasiparticle e ects. While these e ects are small for some systems, they can be nonnegligible for others. Some systems in the SC/40 test set exhibit significant spin-orbit coupling e ects that can split the bandgap by 1 eV, for example ZnTe. For these systems, the comparison is made to the weighted average of the split experimental bandgap, but a better description of the systems is highly desirable. As follows from [419] the computational e ort involved in HSE calculations presents only a modest increase (a factor of less than 2) over pure DFT calculations giving a vast improvement in accuracy.

Concludig the discussion of hybrid functionals we stress that the HSE functional is universally applicable and does not contain any system-dependent parameter. It yields excellent results, in molecules and solids, for many di erent properties. In contrast to other methods, HSE can be employed for both structural and electronic properties; HSE provides a unique and powerful alternative for the study of large complex systems, such as chemisorption at surfaces and three-dimensional impurities in semiconductors. These fields of HSE functional applications await future study.

In the next subsection we consider two orbital-dependent exchange-correlation functionals used for solids with the strong electron correlation. These functionals are used mainly with non-LCAO (PW, LMTO) basis and from this point of view are not quantum-chemical approaches. Therefore, we discuss them only briefly.

270 7 Kohn–Sham LCAO Method for Periodic Systems

7.2.5 Density-functional Methods for Strongly Correlated Systems: SIC DFT and DFT+U Approaches

DFT (LDA, GGA approximations) have a great predictive power for solids only as long as one is treating electronic states extended over the whole system, so that electrons can be considered completely delocalized [151]. In the case of 3d-states in transitionmetal atoms and 4f -states in rare-earth atoms and their compounds the metal-atom electrons partially preserve their atomic-like (localized) nature. The Coulomb correlations between localized electrons are strong, so that DFT results in many cases are in disagreement with experiment. A well-known example of such a disagreement is represented by some insulating transition-metal oxides, which LDA predicts to be metallic. The DFT methods are in serious trouble for the strongly correlated electrons as these methods describe noninteracting electrons moving in an e ective self-consistent mean field. For the strongly correlated molecular systems post-HF methods in the LCAO approximation are traditionally applied(see Chap. 5). In solids with strong correlations approaches based on LDA (GGA) approximations are applied as a starting point and additional terms intended to treat strong Coulomb correlations between electrons are introduced. As a rule, the LCAO approximation is not used in these approaches. More traditional for solid-state physics localized mu n tin orbitals (LMTO) or augmented plane waves (APW) are popular in the calculations of strongly correlated systems. Various methods have been developed to extend the DFT approach to the strongly correlated systems. The best suited for strongly correlated systems is the so-called GW approximation, which is formally the first term in the perturbation expansion of the self-energy operator in powers of the screened Coulomb interaction (we return to the Coulomb-screening problem later). The reader can find the detailed description of GW method in the review article by Aryasetiawan [435] and other publications. The GW approximation is computationally very heavy and some simpler methods were proposed. Two of these methods (self-interaction corrected (SIC) DFT and DFT +

U) are briefly discussed in this subsection.										ˆ
/	The HF operator for a system of N electrons includes Coulomb Jj ψi(x1) =
	i			/	ψ	ˆ				ˆ
	|ψi (x2)|	2	ˆ			(x	)ψ	(x	)
					j	2	i	2
	|r2−r1|		dx2ψi(x1) and exchange Kj ψi(x1) =			|r2−r1|				dx2ψj (x1) operators

(here, ψ (x) are spin-orbitals and x = r, σ – the space r and spin σ coordinates). From the definitions of these operators it follows that Jiψi = Kiψi. Without this

ˆ ˆ N ˆ − ˆ identity the HF operator would have been orbital dependent F = H + Jj Kj ,

j =i

i.e. we would have di erent HF operators for di erent orbitals. By including the i = j term we obtain the same HF operator for all orbitals and allow the electron to interact with itself, in the Coulomb part and also in the exchange part. But these two self-interaction terms cancel. The HF approximation uses the exact expression of the exchange energy, but omits the correlation energy, and the resulting e ective potential is nonlocal (it depends on the electron-density matrix ρ(r, r )). The DFT (LDA, GGA approaches) uses an approximate local form of the exchange-correlation energy and the Kohn–Sham e ective potential is local (depends on the electron density ρ(r, r) = ρ(r)). In particular, the LDA is exact for a completely uniform system of noninteracting particles, and thus is self-interaction-free in this limit. In other versions of DFT the approximate exchange-correlation energy functionals are used so that the self-interaction terms may not cancel. The self-interaction-corrected (SIC)