Range separated hybrid density functional with long-range Hartree ...

THE JOURNAL OF CHEMICAL PHYSICS 127, 054101 共2007兲

Range separated hybrid density functional with long-range Hartree-Fock exchange applied to solids Iann C. Gerbera兲 and János G. Ángyánb兲 Laboratoire de Cristallographie et Modélisation des Matériaux Minéraux et Biologiques, UMR 7036, CNRS, Faculté des Sciences, Nancy-Université, B.P. 239, F-54506 Vandœuvre-lès-Nancy, France

Martijn Marsman and Georg Kressec兲 Institut für Materialphysik, Universität Wien, A-1090 Wien, Austria and Center for Computational Material Science, Sensengasse 8, A-1090 Wien, Austria

共Received 14 May 2007; accepted 20 June 2007; published online 2 August 2007兲 We report a plane wave-projector augmented wave implementation of the recently proposed exchange-only range separated hybrid 共RSHX兲 density functional 关Gerber and Ángyán, Chem. Phys. Lett. 415, 100 共2005兲兴 and characterize its performance in the local density approximation 共RSHXLDA兲 for a set of archetypical solid state systems, as well as for some transition metal oxides. Lattice parameters, bulk moduli, band gaps, and magnetic moments of the transition metal oxides have been calculated at different values of the range separation parameter and compared with results obtained with standard local density approximation 共LDA兲, gradient corrected 共PBE兲, and hybrid 共HSE兲 functionals. The RSHX functional, which has the main feature of providing a correct asymptotic behavior of the exchange potential, has a tendency to improve the description of structural parameters with respect to local and generalized gradient approximations. The band gaps are too strongly opened by the presence of the long-range Hartree-Fock exchange in all but wide-gap systems. In the difficult case of transition metal oxides, the gap is overestimated, while magnetic moments and lattice constants are slightly underestimated. The optimal range separation parameter has been found around 0.4 a.u., slightly lower than the value of 0.5 a.u., recommended earlier for molecular systems. © 2007 American Institute of Physics. 关DOI: 10.1063/1.2759209兴 I. INTRODUCTION

In spite of the spectacular success of the density functional theory 共DFT兲 in the local density1 共LDA兲 and generalized gradient approximations 共GGA兲,2,3 local and semilocal density functionals show several major shortcomings, due to the so-called self-interaction error 共SIE兲 and due to the absence or incomplete description of certain type of long-range dynamic and nondynamic correlation effects. These shortcomings are manifested in various situations, such as in the spurious fractional charges on dissociated atoms,4–6 the strong underestimation of the barrier of atom transfer reactions,7–9 the incorrect band structure and magnetic properties of transition metal oxides,10 the wrong description of bond-breaking processes,11 or the failure to describe London dispersion forces.12 In a general sense, these shortcomings are rooted at the heart of the LDA and GGA, in the Ansatz of locality. Local and semilocal functionals attempt to describe electron exchange and correlation interactions on the basis of the properties of the homogeneous electron gas. While this hypothesis of transferability, assuming that the properties of an inhomogeneous system are locally well approximated by those of the electron gas of the same density, is quite plaua兲

Present address: Laboratory of Physics, Helsinki University of Technology, P.O. Box 1100, 02015 Helsinki, Finland. b兲 Electronic mail: [email protected] c兲 Electronic mail: [email protected] 0021-9606/2007/127共5兲/054101/9/$23.00

sible at short range, its validity becomes questionable as far as long-range electron-electron interactions are concerned. For instance, the incorrect long-range behavior of the exchange potential derived from the exchange energy of the electron gas leads to considerable errors, related to violation of exact conditions, such as the derivative discontinuity of the total energy. Another class of problems, not treated in the present work, is related to electron correlation. The lack of long-range dynamical correlation is responsible for the bad description of dispersion forces in van der Waals interactions, while the lack of nondynamic correlation, characteristic for system with quasidegenerate ground states, explains failure for the “strongly correlated” solids. By now, a well established strategy to improve upon the local and semilocal density approximations is the construction of hybrid functionals,13,14 that include a certain amount of 共nonlocal兲 Hartree-Fock exchange. Hybrid functionals, such as B3LYP,13 PBE0,15,16 and HSE,17,18 present a considerable improvement over pure density functionals, in the description of formation enthalpies, equilibrium geometries, and vibrational frequencies of molecular systems,19–21 and have been shown to bring similar advantages to the description of the lattice constants, bulk moduli, heats of formation, and band gaps of archetypical solid state systems.22–28 The success of these hybrid functionals can be attributed to a partial reduction of the SIE present in pure local or semilocal density functionals, through the introduction of a portion of Hartree-Fock exchange.

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Gerber et al.

Unfortunately, the aforementioned hybrid functionals do not correct all unfavorable features of the local and semilocal density functionals, and in some situations they still lead to qualitatively wrong results. For instance, the long-range asymptotic behavior of the exchange potential still remains incorrect, and the issue of long-range dynamic correlation, responsible for London dispersion forces, is not addressed at all. Building on the work of Savin and co-workers,29–31 Iikura et al. constructed a range separated hybrid functional that recovers the correct −1 / r long-range asymptotic form of the exchange potential.32 This long-range corrected DFT 共LC-DFT兲 hybrid functional provides a seamless match between a Hartree-Fock treatment of the exchange interactions at long range, and an LDA or GGA density functional that describes the complementary short-ranged part of the exchange. In a series of papers, Hirao and co-workers have demonstrated the importance of a correct description of the asymptotic behavior of exchange interactions, showing how the LC-DFT hybrid functional improves the description of 4s − 3d interconfigurational energies of first-row transition metals,32 polarizabilities of large molecules,32,33 Rydbergand charge-transfer excitation energies,34 and excited state geometries.35 Similar long-range exchange corrected hybrid functionals, e.g., CAM-B3LYP, RSHXLDA, and RSHXPBE, were shown to yield quite reasonable results for the thermochemical properties of Pople’s G2-1 set of small molecules, and an improved description of properties such as chargetransfer excitations,36,37 dissociation curves of two-center three-electron bonds of symmetric radical cations 共A+2 兲,38 and other features.39,40 In addition to the aforementioned benefits of a correct description of long-range exchange interactions per se, range separated hybrid functionals provide a starting point for methods that aim for an accurate description of long-range electronic correlations by means of many-body techniques. This, in fact, goes back to the original work of Leininger et al.,31 who used a seamless combination of an explicit wave function treatment of long-range exchange and correlation 关long-range Hartree-Fock exchange+ long-range configuration interaction 共CI兲兴, and a complementary local density functional description of short-range exchange and correlation interactions, to calculate the total energy of several closed shell atoms and the dissociation energies of first-row dimers. Treating the short-range correlations that make up the correlation cusp by means of a density functional greatly reduces the basis set requirements of the CI expansion. Similar combinations of short-range exchange-correlation density functionals with long-range Hartree-Fock exchange and an explicit many-body, MP2 or CCSD共T兲, treatment of longrange correlation were successfully applied to describe the van der Waals bonding in rare-gas12,41,42 and alkaline-earth dimers.43,44 So far, range separated hybrid functionals with longrange exchange 共RSHX兲 have been exclusively applied to atomic and molecular systems. In the case of solid state systems, RSHX functionals hold the promise of similar advantages. Foremost in our mind here is a computational strategy that consists in a combination of a range separated hybrid

effective one-electron treatment followed by explicit longrange correlation calculations. In some special cases, such as the rare gas solids, one can envisage the use of a second order MP2 treatment of the long-range correlation.45 Explicit long-range correlation effects can also be handled by means of many-body techniques such as GW 共Ref. 46兲 or the adiabatic-connection fluctuation-dissipation theorem 共ACFDT兲.47–50 The present work should be seen as a first step in that direction, by showing that the self-consistent inclusion of long-range Hartree-Fock exchange does not deteriorate the quality of the “parent” functional 共LDA in the present case兲, and on the contrary, as far as structural features are concerned, it leads to a net improvement. In view of an assessment of the general performance of the RSHXLDA for solid state systems, we have calculated the lattice parameters, bulk moduli, and band gaps for a selection of archetypical metallic, semiconducting, and ionic solid state systems. In addition, we study a set of simple cubic transition metal oxides. Our results will be analyzed in comparison to common density functionals 共LDA and PBE兲, as well as the HSE hybrid functional. The remainder of this paper is organized as follows: Section II recapitulates the construction of the RSHXLDA functional. The computational setup and the results of RSHXLDA, LDA, PBE, and HSE calculations of lattice parameters, bulk moduli, and band gaps, are presented in Secs. III and IV. Conclusions are drawn in Sec. V.

II. THEORY

The starting point of the construction of range separated hybrid functionals consists in an exact decomposition of the electron-electron interaction as a sum of short- and longrange component, erfc共␮r兲 erf共␮r兲 1 sr,␮ lr,␮ = wee + , 共r兲 + wee 共r兲 = r r r

共1兲

where the parameter ␮ defines the range separation, i.e., the reach 共spatial extent兲 of the short-range contribution, which is roughly proportional to 1 / ␮. Once this separation is defined, the short- and long-ranged parts will be handled in a different manner: the short-range component will be described by a density functional, while the long-range one by explicit wave function techniques. From a formal point of view, it means that the universal ˆ 兩⌿典, where Tˆ is the kidensity functional,51 F关n兴 = 具⌿兩Tˆ + W ee ˆ is the e-e interaction operator, netic energy operator, and W ee is decomposed as sr,␮ 关n兴, F关n兴 = Flr,␮关n兴 + EHxc

共2兲

ˆ lr,␮兩⌿典 is the long-range universal where Flr,␮关n兴 = 具⌿兩Tˆ + W ee sr,␮ functional and EHxc关n兴 is a complementary, short-range Hartree-exchange-correlation functional. Thus the exact ground-state energy of an N-electron system in an external nuclei-electron potential vne共r兲, where the search is over all N-representable densities, is expressed as


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Range separated hybrid density functional

⌿→N

再

+

drvne共r兲n⌿共r兲 .

ˆ lr,␮兩⌿典 + Esr,␮关n 兴 E = min 具⌿兩Tˆ + W Hxc ⌿ ee

冕

冎

共3兲

The search is, in principle, carried out over all N-electron normalized 共multideterminantal兲 wave functions ⌿. In Eq. 共3兲, n⌿共r兲 is the density coming from the wave function ⌿, i.e., n⌿共r兲 = 具⌿兩nˆ共r兲兩⌿典 where nˆ共r兲 is the density operator. The minimizing wave function ⌿␮ in Eq. 共3兲 is given by the corresponding Euler-Lagrange equation, with the help of sr,␮ sr,␮ 关n⌿␮兴, defined as the functional derivative of EHxc 关n⌿␮兴 VˆHxc with respect to the density: ˆ lr,␮ + Vˆ + Vˆsr,␮关n ␮兴兲兩⌿␮典 = E␮兩⌿␮典. 共Tˆ + W ne Hxc ⌿ ee

共4兲

In the limiting case of ␮ → ⬁, the long-range electronˆ lr,␮=⬁ = 0, and one has the nonrepulsion operator vanishes, W ee interacting Kohn-Sham reference system. In this latter special case, the corresponding effective Schrödinger equation can be solved exactly in a monodeterminantal form. For intermediate values of the range separation parameter, 0 ⬍ ␮ ⬍ ⬁, the long-range exchange and correlation must be treated explicitly by appropriate wave function and many-body techniques. Although the above scheme is exact for any value of ␮, provided that the exact short-range exchange-correlation functional is known, for approximate functionals one has to find an optimum for the range separation parameter. In the context of the exchange-only range separated hybrid method a further approximation is introduced for the treatment of the correlation. It will be supposed that the standard 共e.g., LDA兲 correlation energy functional is valid for all the interaction range, which allows us to work with an effective one-electron equation, where only the long-range exchange is treated explicitly. This provides the final form of the RSHX exchange-correlation energy expression, sr,␮ lr,␮ ␮ = Ex,DFA + Ex,HF + Ec,DFA , Exc,RSHX

共5兲

where the abbreviation “DFA” stands for an appropriate density functional approximation, of LDA or GGA type; for instance, in the RSHXLDA scheme DFA= LDA. sr,␮ , can The short-range LDA exchange functional, Ex,LDA be derived from the exchange-hole of the homogeneous electron gas,52 interacting with a short-range e-e potential, leading to the exchange energy density sr,␮ ⑀x,LDA =−

冉冊冋冉冑冉冊冊册 18 ␲2

1/3

1 3 −A rs 8

2

␲ erf

1 2A

+ 共2A − 4A3兲e−1/共4A 兲 − 3A + 4A3

,

共6兲

where kF = 共3␲2n兲1/3, rs = 共共9␲兲 / 4kF3 兲1/3, and A = ␮ / 共2kF兲. At short range the local density exchange approximation is known to be exact,53 while at long-range the use of the Hartree-Fock exchange expression with a 1 / r asymptotic behavior of the exchange potential ensures a self-interactionfree description of long-range electron interactions. In other words, the self-interaction components of the long-range

SI,lr,␮ SI,lr,␮ Hartree 共EH 兲 and exchange 共Ex,HF 兲 energies cancel each SI,lr,␮ SI,lr,␮ other, EH + Ex,HF = 0. The long-range Hartree-Fock exchange energy is calculated in the plane wave projector augmented wave 共PAW兲 scheme54,55 according to the algorithm outlined in Ref. 56, by replacing the full Coulomb interaction by the long-range interaction, erf共␮兩r − r⬘兩兲 / 兩r − r⬘兩. While the handling of the plane wave contribution is straightforward, further considerations are needed in the case of the purely one-center and mixed terms. In particular, it will be supposed that the standard construction of the compensation charges ensures that the long-range potential originating from the difference of the one-center density and the sum of the pseudocharge and compensation-charge densities is vanishing.56 The range separated exchange integrals of the RSHXLDA functional were calculated on the radial augmentation grids, according to the usual PAW algorithm, using the recently derived spherical harmonic expression of the range separated interaction kernel.57 As mentioned above, in the present work the LDA correlation energy Ec,LDA is chosen to cover the full range of interaction and is given by the standard Vosko-Wilk-Nusair parametrization. It is to be noted that other possibilities exist. One of them consists in simply dropping the long-range correlation. Such an approach has been suggested on the basis of screening arguments,58 but the error compensations justifying this scheme are probably not operational in a sufficiently wide range of systems to consider it as a generally applicable method. On the contrary, dropping long-range correlation effects from the outset might be the recommended first step for explicit 共e.g., many-body兲 long-range correlation calculations.

III. COMPUTATIONAL SETUP

The calculations presented in this work were performed with an 共as yet兲 unreleased version of the Vienna ab initio simulation package 共VASP兲,59 which includes an implementation of the Hartree-Fock exchange operator56 and the closely related HSE 共Ref. 24兲共with ␻ = 0.207 Å−1兲 and RSHXLDA hybrid functionals. To describe the electron-ion interactions, 54,55 VASP employs the full-potential PAW method, adapted for range separated hybrid calculations. To facilitate a direct comparison of the results, obtained with the RSHXLDA hybrid functional, to those obtained with 共semi兲 local density functionals 共LDA and PBE兲 and the HSE hybrid functional, we have chosen the same set of bulk systems as used in Refs. 22 and 24. This set of solid state systems, originally proposed by Staroverov et al.,60 consists of several metals 共Li, Na, Al, Cu, Pd, and Ag兲, semiconductors 共cubic diamond, Si, BN, BP, SiC, GaAs, GaP, and GaN兲, and ionic compounds 共MgO, LiF, LiCl, NaF, and NaCl兲, all with only a single independent lattice parameter. The theoretical lattice constants and bulk moduli were determined by fitting the volume dependence of the static lattice energy 关weighted seven point fit; covering the interval 共0.9, 1.1兲⍀exp, where ⍀exp is the experimental equilibrium volume兴 to a Murnaghan equation of state.61 All Brillouin-zone integrations were performed


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on ⌫-centered, symmetry-reduced 共16⫻ 16⫻ 16兲 Monkhorst-Pack62 k-point meshes, using the tetrahedron method with Blöchl corrections.63 Other details of the computational setup, such as the choice of the PAW data sets and the plane wave cutoff energies, have been chosen to be the same as in Ref. 24. To investigate the dependence of the RSHXLDA results on the range separation parameter ␮, the lattice constants and bulk moduli of the aforementioned set of solid state systems were calculated for ␮ = 0共⬅LDA兲, 0.5, 0.75, and 1.0 Å−1 共0.26, 0.40, 0.53 bohr−1兲. As will be shown in Sec. IV, a range separation parameter of ␮ = 0.75 Å−1 共0.40 a.u.兲 seems to be a reasonable choice for solid state systems. In addition to the above described set of solids, we present calculations of the lattice constants, magnetic moments, and band gaps of the transition-metal monoxides MnO, FeO, CoO, and NiO. These calculations were performed for ␮ = 0.5 Å−1, using 共6 ⫻ 6 ⫻ 6兲 ⌫-centered Monkhorst-Pack k-point meshes, and a plane wave basis set energy cutoff of 400 eV.

IV. RESULTS

For the main part this work focuses on the structural properties, i.e., equilibrium lattice constants and bulk moduli 共Sec. IV A兲 of the systems in the set of Staroverov et al. Band gaps for GaAs, Si, cubic diamond, MgO, NaCl, and Ar, obtained using the RSHXLDA functional, are discussed in Sec. IV B, and compared to LDA, PBE, and HSE results. Section IV C presents a comparison between the LDA, B3LYP, and RSHXLDA description of the structural and electronic properties of the antiferromagnetic 共AFMII兲 transition-metal monoxides MnO, FeO, CoO, and NiO.

A. Lattice parameters and bulk moduli

Table I lists the equilibrium lattice constants a0 and bulk moduli B0 for the considered test set of metallic, semiconducting, and ionic systems, obtained from LDA, PBE, HSE 共␻ = 0.207 Å−1兲, and RSHXLDA calculations using four different values of the range separation parameter, ␮ = 0.5, 0.75, and 1.0 Å−1 共␮ = 0.265, 0.397, and 0.529 a.u.兲. Three statistical measures, the mean absolute 共unsigned兲 relative error 共MARE兲 and the mean 共signed兲 relative error 共MRE兲, as well as the standard deviation 共STD兲 characterize the quality of the results in comparison to experiment. As can be seen from the MAREs 共all solids兲 in Table I, the usual global trends are recovered for the pure density functionals 共LDA and PBE兲. The LDA systematically underestimates the lattice parameters 共MARE= 1.6%, MRE = −1.6%兲, while the PBE functional has an overall tendency to overestimate them 共MARE= 1.0%, MRE= 0.8%兲. Since the calculated bulk moduli are quite sensitive to the equilibrium volume at which they are evaluated, an error in the theoretical lattice constant with respect to experiment translates into a comparatively large discrepancy in the bulk moduli B0. Generally speaking the underestimation of lattice constants is in a one-to-one correspondence with the overes-

J. Chem. Phys. 127, 054101 共2007兲

timation of bulk moduli, and vice versa, as witnessed by the mean relative errors for LDA 共MRE= 6.9% 兲 and for PBE 共MRE= −8.8% 兲. The HSE results in Table I illustrate the performance of state-of-the-art hybrid functionals 共e.g., a0: MARE= 0.5%, MRE= 0.2%兲, that present a general improvement over the 共semi兲local density functional results for practically all categories of solid state systems considered, perhaps in a slightly lesser extent for metals than for insulating systems. When analyzing RSHXLDA calculations, one has to consider the dependence of the results on the range separation parameter ␮ as well. To this end the RSHXLDA results in Table I are listed in order of an increasing range separation parameter, ␮ = 0.0共⬅LDA兲, 0.5, 0.75, and 1.0 Å−1. As far as the lattice constants are concerned, the overall MARE in the RSHXLDA results 共all solids兲 is quite insensitive to the choice of ␮ 共MARE⬇ 1.0%, for 0.5 Å−1 艋 ␮ 艋 1.0 Å−1兲 and presents an improvement upon the LDA. As shown in Table I, this mainly stems from an improved description of the lattice constants of metallic and large gap systems. The favorable performance of the RSHXLDA functional for metallic systems is somewhat surprising since an unscreened Hartree-Fock description of long-range exchange interactions is usually thought to be inconsistent with the metallic behavior. This question will be further discussed in the context of the band structure, in Sec. IV B. Equally noteworthy is the fact that the RSHXLDA does not present a significant improvement over the LDA description of the lattice constants of small/medium gap systems, for which PBE-based hybrids, such as PBE0 and HSE, were shown to be especially efficient.22–24 The RSHXLDA bulk moduli show a considerably stronger dependence on the range separation parameter ␮ than the lattice constants. Judging from the overall MARE 共all solids兲 of the RSHXLDA bulk moduli alone, one might choose an optimal value for the range separation parameter as ␮ = 0.5 Å−1 共MARE= 4.7%, MRE= 2.7%兲. At this particular value of the ␮ the RSHXLDA consistently improves the description of the bulk moduli with respect to the LDA for all categories of systems, although not by much for the small/ medium gap systems, and the performance of the RSHXLDA 共␮ = 0.75 and 0.5兲 seems to be at least equal and even slightly better, than that of the HSE hybrid functional. However, one should be cautious by drawing general conclusions on the basis of bulk moduli that are prone to relatively large experimental errors. Therefore, we are on a safer ground to propose an optimal range separation value for this family of simple solids by relying mainly on the behavior of the lattice constants. The optimum for all solids is at 0.75 Å−1, which seems to be slightly better than the performance of PBE, and has a tendency to underestimate lattice constants, such as LDA. A further advantage of choosing ␮ = 0.75 Å−1 = 0.4 a.u. is that it is close to value of ␮ = 0.5 a.u., widely accepted for molecular systems.38–40,64 The relative errors in the LDA, PBE, HSE, and RSHXLDA 共␮ = 0.75 Å−1兲 lattice parameters and bulk moduli, with respect to experiment, are depicted in Figs. 1 and 2, respectively. The characteristic behavior of metals, semiconductors, and large gap ionic systems can be easily appreciated on


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TABLE I. Lattice constants a0 共Å兲 and bulk moduli B0 共GPa兲, obtained from LDA, PBE, HSE, and RSHXLDA 共for ␮ = 0.50, 0.75, and 1.00 Å−1兲 calculations. Experimental results were taken from Ref. 22, PBE and HSE 共␻ = 0.207 Å−1兲 results from Ref. 24. Mean absolute 共unsigned兲 relative error 共MARE兲 and mean 共signed兲 relative error 共MRE兲, as well as standard deviations 共STD兲 with respect to experiment are given in percent. a0 共Å兲

B0 共GPa兲

Solid

LDA

0.50

0.75

1.00

PBE

HSE

Expt.

LDA

0.50

0.75

1.00

PBE

HSE

Expt.

Na Li Al Cu Pd Ag Si GaAs BP GaP SiC ␤-GaN C BN MgO NaCl LiCl NaF LiF

4.048 3.361 3.988 3.521 3.840 4.006 5.407 5.613 4.492 5.395 4.330 4.460 3.538 3.584 4.154 5.464 4.964 4.505 3.914

4.100 3.411 3.950 3.532 3.840 4.037 5.405 5.593 4.499 5.391 4.326 4.455 3.540 3.589 4.162 5.596 5.088 4.582 3.980

4.165 3.458 3.935 3.561 3.856 4.081 5.401 5.590 4.496 5.389 4.315 4.448 3.533 3.584 4.163 5.654 5.144 4.620 4.013

4.211 3.495 3.951 3.609 3.888 4.138 5.396 5.593 4.490 5.390 4.304 4.440 3.524 3.576 4.160 5.677 5.165 4.635 4.025

4.200 3.438 4.040 3.635 3.943 4.147 5.469 5.752 4.547 5.506 4.380 4.546 3.574 3.626 4.258 5.698 5.150 4.707 4.068

4.225 3.460 4.022 3.638 3.921 4.142 5.435 5.687 4.521 5.462 4.348 4.494 3.549 3.600 4.210 5.659 5.115 4.650 4.018

4.225 3.477 4.032 3.603 3.881 4.069 5.430 5.648 4.538 5.451 4.358 4.520 3.567 3.616 4.207 5.595 5.106 4.609 4.010

9.1 14.9 79.3 174 211 129 92.7 71.5 170 85.1 222 194 453 386 170 30.9 39.8 59.8 84.3

8.3 13.8 82.9 158 197 111 101 81.1 174 92.9 231 206 460 390 170 25.1 32.7 50.8 74.3

7.7 13.0 82.5 137 174 93.4 106 85.7 179 97.5 242 217 475 403 173 24.0 31.2 48.0 72.0

7.4 12.5 91.8 112 144 77.3 110 88.5 184 101 251 223 491 416 177 23.9 31.3 47.8 72.9

7.8 13.7 76.6 136 166 89.1 87.8 59.9 160 75.3 210 169 431 370 149 23.4 31.5 44.6 67.3

8.0 13.6 82.0 126 161 85.9 97.7 70.9 173 86.6 230 196 467 402 169 24.5 33.1 49.3 72.7

7.5 13.0 79.4 142 195 109 99.2 75.6 165 88.7 225 210 443 400 165 26.6 35.4 51.4 69.8

MARE MRE STD

1.6 −1.6 1.0

1.1 −1.1 0.7

0.9 −0.7 0.5

1.0 −0.3 0.5

1.0 0.8 0.6

0.6 0.2 0.5

9.9 6.9 7.5

4.7 2.7 3.0

6.8 0.8 4.2

11.8 0.2 7.5

9.8 −8.8 5.9

6.3 −2.7 5.3

MARE MRE STD

2.2 −2.2 1.3

1.8 −1.8 0.8

1.1 −1.0 0.8

0.8 0.0 0.8

1.1 0.5 0.6

Metals 共Na–Ag兲 0.8 14.2 0.5 14.2 0.6 8.6

5.9 5.9 4.3

5.9 −3.7 5.5

16.2 −11.0 11.5

8.4 −5.2 6.5

10.8 −5.9 7.3

MARE MRE STD

1.4 −1.4 0.8

0.8 −0.8 0.4

0.9 −0.5 0.4

1.1 −0.5 0.4

1.0 1.0 0.7

No metals 共Si–LiF兲 0.4 7.9 0.1 3.5 0.3 6.4

4.2 1.3 2.2

7.2 2.9 3.6

9.7 5.3 3.7

10.5 −10.5 5.8

4.2 −1.2 2.3

MARE MRE STD

0.8 −0.8 0.3

0.9 −0.9 0.3

1.0 −1.0 0.3

1.1 −1.1 0.3

0.7 0.7 0.6

Small/medium gap 共Si–BN兲 0.4 4.2 −0.1 −2.9 0.2 2.2

3.8 2.7 1.9

7.2 7.2 3.9

10.7 10.7 4.1

10.8 −10.8 7.1

3.7 −0.5 2.3

MARE MRE STD

2.2 −2.2 0.6

0.6 −0.5 0.4

0.6 0.2 0.5

0.9 0.5 0.5

1.5 1.5 0.5

Large gap/ionic 共MgO–LiF兲 0.5 13.7 0.5 13.7 0.5 6.7

4.8 −1.0 2.6

7.3 −4.1 3.6

8.1 −3.4 2.8

9.9 −9.9 3.8

5.0 −2.4 2.2

All solids

these figures. The significant improvement of the lattice parameter for ionic systems is quite obvious, as well as the fact that the bulk moduli of semiconductors are systematically overestimated by the RSHXLDA functional, while LDA, PBE, and HSE have all a tendency of underestimating it. B. Band gaps

The electronic structure of solids is usually characterized by their band structure, which corresponds to electron addition and removal energies, measurable by direct and inverse angle-resolved photoemission experiments. The common

practice to interpret band structures in terms of Kohn-Sham eigenvalues obtained from approximate exchange-correlation functionals is based on an a priori unjustified interpretation of such eigenvalues as genuine one-particle energies. However, taking into account the complexity of all the effects that come into play for a complete description of these energy levels, the performance of simple functionals, such as LDA, GGA, and some hybrids, should be qualified as surprisingly good and presumably due mostly to an interplay of extensive error compensations.65 In the following, we analyze one particular aspect of the


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Gerber et al.

FIG. 1. 共Color online兲 Relative errors in the LDA 共black兲, PBE 共blue兲, HSE 共green兲, and RSHXLDA 共␮ = 0.75 Å−1兲共red兲 lattice parameters with respect to experiment.

FIG. 2. 共Color online兲 Relative errors in the LDA 共black兲, PBE 共blue兲, HSE 共green兲, and RSHXLDA 共␮ = 0.75 Å−1兲共red兲 bulk moduli with respect to experiment.

band structure as obtained in the RSHXLDA calculations, namely, the band gap for a series of selected systems, GaAs 共zinc blende兲, Si 共cubic diamond兲, C 共cubic diamond兲, MgO 共B1兲, NaCl 共B1兲, and Ar 共fcc兲. The RSHXLDA band gaps obtained for ␮ = 0.5 and 0.75 Å−1 are presented in Table II together with LDA, PBE, and HSE results, taken from Ref. 24 and compared with experimental data. All calculations were carried out at the experimental lattice constant. The LDA, PBE, and HSE results listed in Table II illustrate the typical behavior of these functionals: The 共semi兲local density functional calculations 共LDA and PBE兲 consistently underestimate the band gaps, whereas the HSE hybrid functional provides a fair description of the band gaps for small/medium gap systems but underestimates the band gaps for large gap materials.22–24 As clearly shown in Table II, the RSHXLDA functional strongly overestimates the band gaps in all systems, except for the extreme case of the solid Ar, which has the largest gap and the RSHXLDA band gap is in good agreement with the experiment.

To understand these general trends one should realize that the width of the band gap is generally governed by the long-range decay of the exchange potential. The exponential decay of the exchange potential derived from the 共semi兲local density functionals leads to an underestimation of the band gaps, whereas the long-range 1 / r decay of the bare 共unscreened兲 Hartree-Fock potential generally induces an overestimation. The long-range decay of the HSE exchange potential is a particular blend of the exponential decay of the PBE exchange potential and the erfc共␮r兲 / r decay of a screened Hartree-Fock exchange operator, that seems to be well balanced to describe small/medium gap systems. It is only in very large gap systems, i.e., systems with very weak dielectric screening, where the nonlocal exchange interactions are well described by the bare long-range Hartree-Fock exchange operator of the RSHXLDA functional. For small/ medium gap systems, where we are faced with a strong or intermediate dielectric screening, the bare long-range

TABLE II. Direct and indirect gaps 共eV兲 for GaAs, Si, C, MgO, NaCl, and Ar for the ⌫ and X points using the LDA, PBE, HSE03, and RSHXLDA 共␮ = 0.5 and 0.75 Å−1兲 functionals. Experimental, PBE, and HSE results were taken from Ref. 24. LDA

PBE

HSE

⌫15v → ⌫1c ⌫15v → X1c

0.35 1.34

0.56 1.46

GaAs 1.30 1.88

⌫⬘25v → ⌫15c ⌫⬘25v → X1c

2.53 0.60

2.57 0.71

3.16 1.14

⌫⬘25v → ⌫15 ⌫⬘25v → X1c

5.54 4.70

5.59 4.76

6.74 5.68

⌫15 → ⌫1

4.67

4.75

6.24

⌫15 → ⌫1

4.79

5.20

6.31

⌫15 → ⌫1

8.19

8.68

10.07

␮ = 0.5

␮ = 0.75

Expt.

4.22 5.60

5.29 6.38

1.52 1.90

6.98 4.96

7.90 5.78

3.05, 3.34–3.36 1.13, 1.25

10.88 10.28

12.22 11.61

7.3

10.65

12.57

7.7

7.96

8.33

8.5

14.27

15.92

14.2

Si

C

MgO

NaCl

Ar


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son, the LDA DOS of bcc Na 共dashed line兲, obtained using the tetrahedron method with Blöchl’s corrections, is shown as well. It can be concluded that the RSHXLDA functional, although quite reliable for ground-state total energies, is unable to provide effective one-electron states that can be approximately assigned to quasiparticle states. This feature is manifested in a general overestimation of the band gap and it is particularly pronounced for metallic systems, where the singular behavior of the Hartree-Fock DOS is reproduced. As far as one is interested in a physically coherent description of the band structure, range separated hybrid calculations must be followed by the application of many-body techniques to include explicitly the necessary correlation effects to remove the divergence in metals and correct the band gap in general. FIG. 3. Density of states 共DOS兲 for bcc Na, obtained from LDA 共dashed line兲 and RSHXLDA 共full line兲 calculations. The zero of energy was placed at the Fermi level.

C. Transition metal monoxides

Hartree-Fock exchange in the RSHXLDA functional leads to an overestimation of the band gaps. The situation in metallic systems represents an extreme case. As is known from a thorough analysis of the HartreeFock exchange in metals,66,67 the combination of the longrange decay of the bare Hartree-Fock exchange interaction and the discontinuity in the population function, that describes a partially filled band at the Fermi level, leads to a vanishing density of states 共DOS兲 at the Fermi level. This singularity is present in the RSHXLDA calculations on metallic systems as well. Figure 3 shows the RSHXLDA DOS of bcc Na 共solid line兲, obtained with a dense 共48⫻ 48⫻ 48兲 ⌫-centered Monkhorst-Pack k-point mesh and a Gaussian smearing with a width of ␴ = 0.1 eV. To facilitate a compari-

Table III lists the results of our LDA and RSHXLDA 共␮ = 0.5 Å−1兲 calculations for the lattice constants a0, local spin magnetic moments M s, and band gaps ⌬ of the antiferromagnetic 共AFMII兲 transition metal monoxides 共TMO兲 MnO, FeO, CoO, and NiO, in comparison with experimental data, as well as with B3LYP hybrid functional results, obtained with the CRYSTAL code,26,68,69 and LDA and PBE0 data from full-potential 共linearized兲 augmented plane wave plus local orbital 关FP共L兲APW+ lo兴 calculations with 70 WIEN2K. In this latter case, the PBE0 hybrid functional was applied only within the augmentation sphere of transition metal atoms. The calculated magnetic moments reported in Table III do not contain the orbital contributions, which could be obtained by taking into account the spin-orbit coupling.

TABLE III. Lattice parameters a0 共Å兲, spin magnetic moments M s 共␮B兲, and band gaps ⌬ 共eV兲 for the transition metal monoxides MnO, FeO, CoO, and NiO, obtained from LDA and RSHXLDA 共␮ = 0.5 Å−1兲 calculations. B3LYP results were obtained by the CRYSTAL code; LDA and PBE0 values, as well as experimental reference data, were taken from Ref. 70 共and references therein兲. LDAa

PBE0a

a0 Ms ⌬

4.32 4.19 0.8

4.51 4.40 1.3

a0 Ms ⌬

4.18 3.35 0.0

a0 Ms ⌬

a0 Ms ⌬

B3LYP

LDAb

RSHXLDA

Expt.

MnO 共Ref. 68兲 4.50 4.31 4.73 4.14 3.92 0.4

4.36 4.39 7.1

4.45 4.58 3.9

4.40 3.55 1.2

FeO 共Ref. 26兲 4.37 4.17 3.26 3.70 0.0

4.24 3.48 6.2

4.33 3.32, 4.2 2.4

4.11 2.36 0.0

4.32 2.66 2.1

CoO 共Ref. 68兲 4.32 4.10 2.69 2.23 3.63 0.0

4.16 2.50 6.2

4.25 3.35, 4.0 2.5

4.07 1.21 0.4

4.24 1.78 2.9

NiO 共Ref. 69兲 4.23 4.06 1.67 1.06 4.1 0.4

4.09 1.47 7.0

4.17 1.64 4.0

a

Reference 70. Present work.

b


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Gerber et al.

Standard density functionals, e.g., the LDA, underestimate the TMO lattice parameters and the local magnetic moments on the transition metal ions and severely underestimate the TMO bands gaps, such that FeO and CoO are even predicted to have a metallic character. These deficiencies are commonly attributed to the SIE in the density functional description of the strongly localized transition metal 3d states. The partial reduction of the SIE, through the admixture of a certain amount of Hartree-Fock exchange in hybrid functionals generally leads to a better description of these TMO systems,71 as illustrated by the PBE0 and B3LYP results listed in Table III. The general trends of the RSHXLDA results are in broad agreement with the conclusions drawn from the test set of simple materials 共see Secs. IV A and IV B兲. The inclusion of long-range Hartree-Fock exchange leads to a slight increase of the LDA lattice parameters, although they remain underestimated with respect to the experiment. As expected, the band gaps are largely opened and they are significantly overestimated, due to the unscreened 1 / r decay of the HartreeFock exchange, as discussed in Sec. IV B. It is interesting to note that with respect to full Hartree-Fock results72 the RSHXLDA band gaps are smaller by a factor of about 0.5. The RSHXLDA local spin magnetic moments increase in comparison to the LDA calculations, although they remain smaller than the experiment. The comparison of calculated magnetic moments among themselves and with respect to the experiment is not simple because their values depend on the definitions adopted in various computational schemes, such as the size of the sphere in which the magnetic moment is calculated. This effect can be roughly estimated by comparing the LDA results obtained by VASP and WIEN2K, reported in Table III. The RSHXLDA magnetic moments, readjusted by the difference of the two LDA calculations, 4.44␮B, 3.57␮B, −2.63␮B, and 1.68␮B, are close to the experimental values corrected for the orbital moments of 0.9␮B, and 1.0␮B for the FeO and CoO, respectively. In comparison with augmentation-sphere-only PBE0 spin magnetic moments obtained by WIEN2K and with spin moments obtained by CRYSTAL using the B3LYP functional, the agreement is quite good. V. CONCLUSIONS

In view of constructing a hybrid functional with a correct behavior of the long-range exchange potential, the RSHXLDA scheme seems to be the simplest possibility, made of a combination of short-range LDA exchangecorrelation functional with long-range Hartree-Fock exchange and long-range LDA correlation. The use of the range separated hybrid functional admittedly increases computational costs for plane wave 共and also for local, atomic orbital兲 basis sets, due to the slow convergence of the exchange energy with respect to the number of k points and/or long interaction range of small gap and metallic systems in real space. The extra computational effort may vary between a factor of 10–200 per self-consistency cycle, depending on the type of the system and on the Brillouin-zone sampling.

J. Chem. Phys. 127, 054101 共2007兲

The performance of the RSHXLDA range separated exchange functional has been assessed against the properties of a set of simple cubic systems. By considering structural properties, we have shown that RSHXLDA provides reliable lattice parameters, with an accuracy at least as good as that of the PBE calculations. The lattice parameters of this set of systems are quite insensitive with respect to the range separation parameter. Moreover, by setting the ␮ parameter of 0.75 Å−1共⬇0.4a−1 0 兲 we can obtain a reasonable compromise to describe both bulk and molecular properties. The RSHX gap is strongly overestimated in some cases, as a direct consequence of the Hartree-Fock treatment of the long-range exchange. This is the price to pay for having ensured a correct asymptotic behavior of the exchange potential and removing long-range self-interaction error. Therefore, in contrast to conventional hybrid functionals, which have the flavor of an effective one-electron theory, the range separated hybrid method is supposed to be followed by an explicit many-body treatment of long-range correlation effects, at least for the correct evaluation of the band structure. We have also considered the difficult case of TMOs and pointed out the limits of a simple local description of shortrange exchange and correlation effects. A better description might be obtained with a combination of GGA-type shortrange functionals, such as RSHXPBE, called also as LC-␻PBE.40 To conclude, we believe that the main interest of this study is by no means to suggest a method that is able to compete with existing GGA or hybrid functionals, rather to show that the range separated hybrid approach offers a very good starting point for further improvements. In some cases, where the major shortcoming of conventional functionals is limited to the SIE, the RSHXLDA approach in itself can be used to achieve significant improvements. For instance, according to some ongoing work on the difficult problem of charge transfer in donor-acceptor materials,73,74 where standard functionals systematically fail in the description of both the neutral and ionic phases,75,76 mainly due to the problem related to the SIE,77 RSHXLDA seems to provide a satisfactory account of both phases, and predicts charge transfers78 in agreement with recent experiments.74 Finally, it is clear that the transfer of the short-range LDA functionals from the electron gas to a nonhomogeneous electronic system opens the way for a physically wellfounded description of the electron-electron cusp region, without running into the problems of slow convergence with respect to the one-electron basis. A safer, but obviously much more demanding approach could consist in calculating longrange correlation effects explicitly, by appropriate manybody techniques. Band structures could be improved in the future by long-range-only GW corrections and total energies could be treated by some variants of the ACFDT methods, applied again to long-range electron-electron interactions, while leaving the task of handling short-range correlation by well transferable LDA 共or eventually GGA兲 functionals. Works in these directions are in progress.


054101-9

ACKNOWLEDGMENTS

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