VOLUME 77, NUMBER 18 P H Y S I C A L R E V I E W L E T T E R S 28 OCTOBER 1996 Generalized Gradient Approximation Made Simple John P. Perdew, Kieron Burke,* Matthias Ernzerhof Department of Physics and Quantum Theory Group, Tulane University, New Orleans, Louisiana 70118 (Received 21 May 1996) Generalized gradient approximations (GGA���s) for the exchange-correlation energy improve upon the local spin density (LSD) description of atoms, molecules, and solids. We present a simple derivation of a simple GGA, in which all parameters (other than those in LSD) are fundamental constants. Only general features of the detailed construction underlying the Perdew-Wang 1991 (PW91) GGA are invoked. Improvements over PW91 include an accurate description of the linear response of the uniform electron gas, correct behavior under uniform scaling, and a smoother potential. [S0031-9007(96)01479-2] PACS numbers: 71.15.Mb, 71.45.Gm Kohn-Sham density functional theory [1,2] is widely used for self-consistent-field electronic structure calcula- tions of the ground-state properties of atoms, molecules, and solids. In this theory, only the exchange-correlation energy EXC ��� EX 1 EC as a functional of the electron spin densities n"srd and n#srd must be approximated. The most popular functionals have a form appropriate for slowly varying densities: the local spin density (LSD) ap- proximation EXC LSD fn", n#g ��� Z d3r neXC unifsn", n#d, (1) where n ��� n" 1 n#, and the generalized gradient approxi- mation (GGA) [3,4] EXC GGAfn", n#g ��� Z d3r fsn", n#, =n", =n#d. (2) In comparison with LSD, GGA���s tend to improve total energies [4], atomization energies [4���6], energy barriers and structural energy differences [7���9]. GGA���s expand and soften bonds [6], an effect that sometimes corrects [10] and sometimes overcorrects [11] the LSD prediction. Typically, GGA���s favor density inhomogeneity more than LSD does. To facilitate practical calculations, eXC unif and f must be parametrized analytic functions. The exchange- correlation energy per particle of a uniform electron gas, eXC unif sn", n#d, is well established [12], but the best choice for fsn", n#, =n", =n#d is still a matter of debate. Judging the derivations and formal properties of various GGA���s can guide a rational choice among them. Semiempirical GGA���s can be remarkably successful for small molecules, but fail for delocalized electrons in the uniform gas [when fsn", n#, 0, 0d fi neXC unif sn", n#d] and thus in simple metals. A first-principles numerical GGA can be constructed [13] by starting from the second-order density-gradient expansion for the exchange-correlation hole surrounding the electron in a system of slowly varying density, then cutting off its spurious long-range parts to satisfy sum rules on the exact hole. The Perdew-Wang 1991 (PW91) [14] functional is an analytic fit to this numerical GGA, designed to satisfy several further exact conditions [13]. PW91 incorporates some inhomogeneity effects while retaining many of the best features of LSD, but has its own problems: (1) The derivation is long, and depends on a mass of detail. (2) The analytic function f, fitted to the numerical results of the real-space cutoff, is complicated and nontransparent. (3) f is overparametrized. (4) The parameters are not seamlessly joined [15], leading to spurious wiggles in the exchange-correlation potential dEXCydnssrd for small [16] and large [16,17] dimension- less density gradients, which can bedevil the construction of GGA-based electron-ion pseudopotentials [18���20]. (5) Although the numerical GGA correlation energy func- tional behaves properly [13] under Levy���s uniform scaling to the high-density limit [21], its analytic parametrization (PW91) does not [22]. (6) Because PW91 reduces to the second-order gradient expansion for density variations that are either slowly varying or small, it descibes the linear response of the density of a uniform electron gas less satisfactorily than does LSD [20,23]. This last problem illustrates a fact which is often over- looked: The semilocal form of Eq. (2) is too restrictive to reproduce all the known behaviors of the exact func- tional [13]. In contrast to the construction of the PW91 functional, which was designed to satisfy as many exact conditions as possible, the GGA presented here satisfies only those which are energetically significant. For exa- mple, in the pseudopotential theory of simple metals, the linear-response limit is physically important. On the other hand, recovery of the exact second-order gradient expan- sion in the slowly varying limit makes little difference to the energies of real systems. We solve the 6 prob- lems above with a simple new derivation of a simple new GGA functional in which all parameters [other than those in eXC unif sn", n#d] are fundamental constants. Although the derivation depends only on the most general features of the real-space construction [13] behind PW91, the result- ing functional is close to numerical GGA. We begin with the GGA for correlation in the form EC GGAfn", n#g ��� Z d3r nfeC unifsrs, z d 1 Hsrs, z , tdg, (3) 0031-9007y96y77(18)y3865(4)$10.00 �� 1996 The American Physical Society 3865

VOLUME 77, NUMBER 18 P H Y S I C A L R E V I E W L E T T E R S 28 OCTOBER 1996 where rs is the local Seitz radius (n ��� 3y4prs 3 ��� kF 3 y 3p 2), z ��� sn" 2 n#dyn is the relative spin polarization, and t ��� j=njy2fksn is a dimensionless density gradi- ent [13,14]. Here fsz d ��� fs1 1 z d2y3p1 s1 2 z d2y3gy 2 is a spin-scaling factor [24], and ks ��� 4kF ypa0 is the Thomas-Fermi screening wave number (a0 ��� ��2yme2). h =z corrections to Eq. (3), which are small for most pur- poses, will be derived in later work. We construct the gradient contribution H from three conditions: (a) In the slowly varying limit (t ! 0), H is given by its second-order gradient expansion [24] H ! se2ya0dbf3 t2, (4) where b . 0.066 725. This is the high-density (rs ! 0) limit [25] of the weakly rs-dependent gradient coefficient [26] for the correlation energy [with a Yukawa interaction se2yud exps2kud in the limit k ! 0], and also the coef- ficient which emerges naturally from the numerical GGA [13] discussed earlier. (b) In the rapidly varying limit t ! `, H ! 2eC unif, (5) making correlation vanish. As t ! ` in the numerical GGA, the sum rule R d3u nCsr, r 1 ud ��� 0 on the corre- lation hole density nC is only satisfied by nC ��� 0. For example, in the tail of the electron density of a finite sys- tem, the exchange energy density and potential dominate their correlation counterparts in reality, but not in LSD. (c) Under uniform scaling to the high-density limit [nsrd ! l3nslrd and l ! `, whence rs ! 0 as l21 and t ! ` as l1y2], the correlation energy must scale to a constant [21]. Thus [27] H must cancel the logarithmic singularity of eC unif [28] in this limit: eC unifsrs, z d ! se2yao df3fg lnsrsya0d 2 vg, where g and v are weak functions [12] of z which we shall replace by their z ��� 0 values, g ��� s1 2 ln 2dyp 2 . 0.031 091 and v . 0.046 644, so H ! se2ya0dgf3 ln t2. (6) Conditions (a), (b), and (c) are satisfied by the simple ansatz H ��� se2ya0dgf3 3 ln �� 1 1 b g t2 ��� 1 1 At2 1 1 At2 1 A2t4 ���ae , (7) where A ��� b g fexph2eC unifysgf3e2ya0dj 2 1g21. (8) The function H starts out from t ��� 0 like Eq. (4), and grows monotonically to the limit of Eq. (5) as t ! ` thus EC GGA # 0. (H appears as one of two terms in the PW91 correlation energy, but with g ��� 0.025.) Under uniform scaling to the high density limit, EC GGA tends to 2 e2 a0 Z d3r ngf3 3 ln ��� 1 1 1 xs2yf2 1 sxs2yf2d2 ��� , (9) where s ��� j=njy2kFn ��� srs ya0d1y2ftyc is another dimensionless density gradient, c ��� s3p 2y16d1y3 . 1.2277, and x ��� sbygdc2 exps2vygd . 0.721 61. The correlation energy for a two-electron ion of nuclear charge Z ! ` is 2` by LSD, 1` by PW91, 20.0482 Hartree by Eq. (9), and 20.0467 exactly [29]. For a finite system, s cannot vanish identically, except on sets of measure zero, so Eq. (9) is finite for an infinite jellium, s vanishes everywhere, and Eq. (9) reduces to 2` as GGA reduces to LSD. The GGA for the exchange energy will be constructed from four further conditions: (d) Under the uniform density scaling described along with condition (c) above, EX must scale [30] like l. Thus, for z ��� 0 everywhere, we must have EX GGA ��� Z d3r neX unifsndFXssd, (10) where eX unif ��� 23e2kFy4p. To recover the correct uni- form gas limit, FXs0d ��� 1. FIG. 1. Enhancement factors of Eq. (15) showing GGA non- locality. Solid curves denote the present GGA, while open circles denote the PW91 of Refs. [4,13,14]. 3866