Why the Generalized Gradient Approximation Works and How to Go Beyond It

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1 Why the Generalized Gradient Approximation Works ow to Go Beyond It KIERON BURKE, JON P. PERDEW, AND MATTIAS ERNZEROF Department of Physics Quantum Theory Group, Tulane University, New Orleans, Louisiana 708 Received September 5, 995; revised manuscript received December 26, 995; accepted December 28, 995 ABSTRACT The local spin density Ž LSD. approximation, while of only moderate accuracy, has proven extremely reliable over three decades of use. We argue that any gradient-corrected functional should preserve the correct features of LSD, even if the system under study contains no regions of small density gradient. The PerdewWang 99 Ž PW9. functional respects this condition, while, e.g., the LeeYangParr Ž LYP. correlation functional violates it. We extend this idea to the next generation of density functionals, those which incorporate exact exchange via the optimized effective potential Ž OEP., with a model in which the correlation hole is constructed from the exact exchange hole. The resulting exchange-correlation hole is deeper less diffuse than the exact exchange hole. We denote such a functional as locally correlated artreefock list a variety of conditions such a functional should satisfy. We demonstrate the promise of this approach with a crude simple model. 997 John Wiley & Sons, Inc. Introduction Summary of Conclusions F or many years, the local spin density Ž LSD. approximation has been the mainstay of electronic structure calculations in solid-state * To whom correspondence should be addressed. Permanent address after July, 996: Department of Chemistry, Rutgers University, Camden, NJ physics 2. This approximation may be written Ž. LSD 3 Exc n, n drnrxc n r,n r, Ž. where Ž n, n. xc is the exchange-correlation en- ergy per particle of a orm electron gas Ž jellium. 57. The LSD exchange-correlation energies are insufficiently negative Ž by about 0%. for almost all atoms, molecules, solids 2. The LSD is a reliable, moderate-accuracy approximation. ŽBy the ( ) International Journal of Quantum Chemistry, Vol. 6, John Wiley & Sons, Inc. CCC / 97 /

2 BURKE, PERDEW, AND ERNZEROF reliability of an approximation, we mean that errors are regular chemical trends are reproduced.. For many solid-state purposes, the LSD level of accuracy is sufficient. owever, LSD is not accurate enough for most chemical applications, which require the determination of energy differences with considerable precision. ence the disinterest of the quantum chemistry community toward density functional methods 8 until recently. New, gradient-corrected functionals, of the form Ž. GGA 3 E n, n drf n Ž. r,nr,n Ž.,n xc Ž. 2 have reduced LSD atomization energy errors by about a factor of 5 9. Unfortunately Ž in marked contrast to LSD. a plethora of functions f are in use in the literature 0, each yielding different energies for the same system. These functionals may be divided into two broad classes: locally based functionals, whose construction starts from the orm electron gas 6, semiempirical functionals 7, 8, which contain one or more parameters fitted to a particular finite system Ž or class of systems.. Such functionals are called generalized gradient approximations Ž GGAs. in the literature Žalthough the term originally 2 meant only the locally based functionals.. The culmination of many years of theoretical work has produced the most modern locally based functional, PerdewWang 99 Ž PW9., which contains no empirical input, while Becke exchange 7 LeeYangParr Ž LYP. correlation 8 Ž BLYP. are perhaps the most popular semiempirical formulas. ow should one choose between the various GGAs currently being used? The answer to this question lies in the unique choice of LSD which has been used for many years. The particular form of exchange-correlation energy per particle is always chosen as that of the orm electron gas. ŽVWN 5, PZ 6, PW 7 are almost identical parametrizations of this quantity.. If the form Eq. Ž. is applied to a set of atomic molecular energies, the function Ž n, n. xc fitted thereto, the result will differ from Ž n, n. xc. But, as explained in detail in the following section, LSD is reliable because it respects conditions on the exact universal density functional. Thus the fitted form of Ž n, n. xc, while more accurate for the specific systems it was fitted to Ž, possibly, for similar systems., may fail when applied to different systems, while Ž n, n. will Ž almost. xc always yield a moderately accurate energy. Thus the semiempirical approach yields essentially a limited-range interpolation scheme, while LSD produces a controlled justified extrapolation scheme, being expected to give moderately accurate results, systematic errors, for systems not yet even imagined. We can extend this logic to the choice of GGA. If a user is confident that her or his system is covered by the interpolation scheme of the semiempirical formulas, has no need to underst the origin of the error made by the functional, then such semiempirical functionals should prove useful. The hybrid functionals which mix GGA with artreefock are typically based on this philosophy 9. Otherwise, the locally based functionals, which incorporate those features which LSD gets right Ž see the third section., are best, with PW9 being the latest of these. Note that a given energy evaluation may have little or no contribution from regions where the gradient is small therefore be insensitive to how a functional behaves in this limit. ŽZupan co-workers are currently studying these kinds of questions with a new density gradient analysis 20.. On this basis, one might say that the behavior of a functional is not important in this limit. owever, a good locally based functional should incorporate all the correct features of LSD, including the correct energy per particle of the orm gas. Thus, by construction, a locally based functional reduces to LSD in the orm density limit, this is an identifying characteristic of such functionals. Thus LYP Ž see the third section. the crude version of locally correlated artree Fock Ž see the fourth section. do not fall in this category. These arguments can be extended to the next generation of functionals, those based on exact exchange, as incorporated in the optimized effective potential 2 Ž OEP.. Once OEP calculations become part of the stard repertoire of quantum chemical techniques, there will doubtless be an explosion of semiempirical formulas. owever, a functional which can combine the good features of both LSD GGA with those of OEP, such as the locally correlated artreefock Ž LCF. discussed in the fourth section, can be hoped to achieve another factor of 5 reduction in energy errors, so yield chemical accuracy, for all those systems for which LSD yields moderate accuracy! 288 VOL. 6, NO. 2

3 GENERALIZED GRADIENT APPROXIMATION A useful aid in the construction of such a functional would be if results were reported using all levels of functionals, especially LSD. Such results give guidance as to how well functionals have been constructed, what is still missing in them. Achievement of an accurate LCF functional might produce a stard technique which will dominate quantum chemistry calculations as thoroughly as LSD has dominated solid-state electronic structure calculations 22. WY LSD IS RELIABLE The exchange-correlation energy may be written in terms of the exchange-correlation hole 23 : nrn Ž. Žr,r. 3 3 xc, E ddrdr Ž 3. xc 2rr 0 where n Žr, r. xc, is the exchange-correlation hole at coupling strength, 2. The on-top value of the hole is especially well approximated in LSD 23, 25, 26 : Ž. Ž. xc, xc, n r,r n n r, n r ; rr 0, because the local approximation for the hole works best in the vicinity of the electron. In fact, Eq. Ž. is exact for the exchange hole Ž 0. when the KohnSham wave function is a single determinant; even at, Eq. Ž. is accurate over most of space, including both valence intershell regions 26. Furthermore, since both the LSD the exact angle-averaged exchange-correlation holes, d u n Ž r, u. n Ž r,ru., Ž 5. xc, xc, satisfy the cusp condition 27, 28 at u 0, Ž. averages 30, 3 of the hole appear in Eq. 3, i.e., ² Ž.: 3 n u drnrn Ž. Ž r,u,. Ž 8. xc, xc, N E N d du 2 u² n Ž u.:. Ž 9. xc 0 0 xc, The system-averaging unweights regions of small r Ž near the nucleus. in an atom, where LSD does less well, due to large higher-order reduced density gradients 26. In Figure, we plot this average hole 32 at for ooke s atom 33, a model system consisting of two electrons in an external oscillator potential, with spring constant k, both exactly in LSD. This analytically solved model system has electron density values typical of those for real valence electrons, so provides a fairly realistic test of density functional approximations. We see that the above restrictions ensure that the area under the curve, E xc,, also referred to as the potential contribution to Exc 3, will almost always be moderately well approximated in LSD. ŽIn fact, LSD does better at than at 0 or for the coupling-constant average 26.. The reason that LSD satisfies all these constraints is because it approximates the hole by that of another physical system, the orm electron gas. ŽAnother choice, the hydrogen atom hole, has been successfully invoked as a model for the spherically averaged hole by Becke Roussel 35, 36.. n Ž r, u0. n Ž r, u0. nž r., Ž 6. xc, xc, where n Ž r, u 0. n Ž r, u. u xc, xc, u0, this accuracy is extended 23 to the neighborhood r close to r. The hole also satisfies the charge conservation rule 29 : 3 drn Žr,r., Ž 7. xc, both exactly in LSD. Thus LSD describes both the size the shape of this hole quite well. Finally, we note that only the system angle FIGURE. Full-coupling strength system-averaged exchange-correlation hole, weighted by 2 u, for the k = / ooke s atom, both exactly within LSD. The area under the curve is E ( see text ). xc, = INTERNATIONAL JOURNAL OF QUANTUM CEMISTRY 289

4 BURKE, PERDEW, AND ERNZEROF ow a GGA Should Work From the preceding section, it should be clear that the reliability of LSD depends on more than just the particular choice of local energy per parti cle, Ž n, n. xc. Any GGA that hopes to emulate this reliability should retain the good features of LSD. The simplest form, already suggested by Kohn Sham, is to exp the energy in a gradient expansion include the n 2 terms. owever, this gradient expansion approximation Ž GEA. often does worse than LSD 2 because the associated hole is not the hole of any physical system, so violates Eq. Ž. 7 other exact conditions. Perdew 2 devised a simple procedure to overcome this difficulty, by removing all obviously unphysical contributions to the GEA hole, thereby restoring Eq. Ž. 7. This defines a numerical GGA which retains the good features of LSD, while improving the description of the average hole Ž therefore the energy. by using the gradient. PW9 is an analytic parameterization of this numerical functional, which incorporates several further exact conditions. Thus PW9 is expected to improve LSD accuracy while preserving LSD reliability. Clearly, functionals that have not been constructed in this way should not be expected to be as reliable. owever, at the exchange Ž or 0. level, the Becke 88 functional 7 is quantitatively very similar to numerical GGA, so should be reasonably reliable. Such agreement can be ascribed to the universal nature of the functional, to Becke s use of the correct Ž n, n. x, to the relatively simple nature of exchange. owever, LYP 8 underestimates the correlation energy of the orm electron gas by about a factor of 2 37, so cannot be Ž is not. close to the results of numerical GGA. Any successes of this functional for specific systems should not be expected to transfer to other systems. The ground-state structure of C may be a case in point Locally Correlated artreefock Approach It is now possible to incorporate exact exchange via construction of the optimized effective potential Ž OEP. 2 for an orbital-dependent functional. ŽIn practice, one often uses the KriegerLeeIafrate Ž KLI. approximation 39, which introduces negligible errors.. owever, exact exchange is too nonlocal to be combined with any of the locally based correlation functionals discussed above. The exact exchange hole in a molecule is often much more diffuse than the exchange-correlation hole, as discussed in Ref. 0. To overcome this difficulty, we suggest modeling the angle-averaged exchangecorrelation hole of Eq. Ž. 5 by Ž. Ž. Ž. Ž. n r, u n r, u K r, u, 0 xc, x xc, where n n, K Ž r, u. x xc, 0 xc, is a local cor- relating factor, to be modeled using the restrictions already discussed. By including exact exchange, such a model retains the benefits of OEP : It is self-interaction free, the exchange-correlation potential has the correct asymptotic behavior Ž r. far from infinite systems, it contains the derivative discontinuity with respect to particle number, etc. It has the further advantage of confining the correlation hole to regions of the system where the density is significant, a property not shared by LSD. This approach is similar in spirit to the ColleSalvetti approximation for the wave function ŽEqs. Ž. Ž 2. of Ref. 2.. The conditions given in the second section may be written as conditions on the correlating factor: Eq. Ž. implies Ž. xc, xc, K r,u0 K n r, n r ; u0, Ž. Ž. Ž Eq. 6 implies when the KohnSham wave function is a single determinant. 2 K Ž r, u0. K Ž r, u0. xc, xc, 2, Ž. r Ž 2. where Ž. r n Ž. r n Ž. r nž. r is the local relative spin polarization, the sum rule of Eq. Ž. 7 requires 3 Ž. Ž. Ž. dun r,u K r, u. 3 x xc, The first two of these conditions are simple enough to fulfill, but the last may require solving an integral equation at every point in the system. Finding a useful approximation for K xc, is a challenge for the next generation of density functionals. To demonstrate the power of this approach, we make a crude first gaff at the correlating factor, 290 VOL. 6, NO. 2

5 GENERALIZED GRADIENT APPROXIMATION with the form Ž. Ž. K r, u K r, u c, xc, u džr. Kc, n r, n r ; u0 e Ž Ž. Ž. C ruu d r. ere d Ž. r is chosen to satisfy the cusp condition of Eq. Ž. 6, n n Ž. r, n Ž. c, r ; u0 dž. r, Ž 5. n n Ž. r, n Ž. r ; u0 xc, C Ž. r is chosen to satisfy the sum rule, Eq. Ž. 7. Thus this crude interpolation consists of a shortranged contribution taken from LSD a longranged contribution which restores the sum rule. Because of the linear sum of long- short-ranged contributions, C Ž. r may be found simply by evaluating two integrals over the spherically averaged exchange hole at each point in the system: Ž. c, C r K n r, n r ; u0 2 Ž. Ž. 0 du u nx r, u exp ud r. Ž. 2 2 du un r,uu dž r. 0 x Ž. 6 We are not suggesting this approximation for use in real electronic structure calculations, but merely as a pedagogical demonstration of the potential of this approach. In fact, we have shown elsewhere 3 that K Ž r, u. xc, a finite value as u, just as in Eq. Ž.. owever, since it is unclear if this limit is achieved at any energetically meaningful value of u, we do not use it to determine C in Eq. Ž.. In Figure 2 we plot the spherically averaged correlation hole at full coupling strength for ooke s atom with spring constant k, for several values of r, both exactly within the approximation of Eq. Ž.. Clearly this approximation is not too bad. We can define a correlation energy density via Ž. r 2 du un Ž r, u., Ž 7. c, c, 0 which is plotted in Figure 3 within LSD, from Eq. Ž., exactly. We see that this function is very well described by Eq. Ž.. In order to get the coupling-constant-averaged quantities, which yield the usual definition of the correlation energy, one could imagine applying Eq. FIGURE 2. Angle-averaged full-coupling strength correlation hole in k = / ooke s atom, weighted by 2 u. The numbers indicate the values of r. The solid curves are exact, while the circles are locally-correlated artreefock, approximated in Eq. ( ). Ž. for each value of, integrating the results over. owever, a much simpler approach, probably no less accurate, is to do only one calculation, in which all the quantities have already been averaged over, i.e., K Ž r, u. K Ž r, u. c xc u džr. Kc n r, n r ; u0 e Ž Ž. Ž. C r u u d r, 8 where the lack of explicit -dependence implies the averaged quantities. In this case, we have no FIGURE 3. Full-coupling strength correlation energy per electron, as defined in Eq. ( 3 ), for the k =/ ooke s atom; exactly, within LSD, within Eq. ( ). INTERNATIONAL JOURNAL OF QUANTUM CEMISTRY 29

6 BURKE, PERDEW, AND ERNZEROF TABLE I Energies in artrees for the k = within several approximations. ooke s atom ( ) Energy LSD PW9 Eq. Exact E c, = E c Exc Exc, = exact calculations of holes to compare with, but we can compare the energies. In Table I, we compare several results for the k ooke s atom. The crude model of Eq. Ž. does better than even PW9 for both the full-coupling strength correlation energy for the coupling-constant average. Even when exchange correlation are added together, a fairer test against functionals which incorporate some cancellation of errors, Eq. Ž. does best. Only when the full-coupling strength exchange-correlation energy is used, where the cancellation of errors in the LSD PW9 functionals is greatest 26,, do they do better than the crude model. We make a similar comparison in Table II for e, finding here that the crude model is comparable to PW9. Note that the crude model of Eq. Ž. has not been designed to recover the orm gas limit, a crucial component of a locally based functional. So we can test its accuracy on the orm gas itself. In Figure, we compare the coupling-constantaveraged correlation energy in the unpolarized orm electron gas from Eq. Ž 8. with the exact value 5. Clearly this crude approximation does not do well here, being particularly poor in the high-density Ž r 0. s regime. In fact, it does even worse than LYP so should not be considered as accurate enough for state-of-the-art calculations. To underst the origin of this failure, we plot in Figure 5 the Ž coupling-constant-averaged. correlation factor of Eq. Ž 8. for both a high-density Ž r 0.5. a low-density case Ž r 5., as a s s FIGURE. Correlation energy in the spin-unpolarized [ ] ( ) orm electron gas both exactly 7 within Eq. 8. function of the dimensionless separation, y kfu. The correlation hole itself becomes negligible at about y 8. We see that for rs 5, the qualitative shape of the approximate hole is somewhat close to that of the exact value Žwith cancellation of errors over the integral yielding the energy., primarily due to the fact that C 0.6, which is reasonably close to its exact value of, due to the cancellation of the exchange correlation holes at large separations in the orm gas 5. But for rs 0.5, there is a big difference, producing the large energy difference apparent in Figure. Satisfaction of the cusp condition has caused the ap- TABLE II Energies in artrees for the e atom within several approximations. ( ) Energy LSD PW9 Eq. Exact E c, = E c Exc Exc, = FIGURE 5. Correlation factor in the spin-unpolarized orm electron gas both exactly [ 5] ( solid lines) within Eq. ( 8) ( circles) for a high density ( r = 0.5) s a low density ( r = 5. ) s 292 VOL. 6, NO. 2

7 GENERALIZED GRADIENT APPROXIMATION proximate correlation factor to drop very quickly at small y, with a value of C 0.05, producing far too little correlation energy. Thus, much work remains to be done to construct a reliable locally correlated artreefock functional. ACKNOWLEDGMENTS This work has been supported by National Science Foundation grant DMR by the Deutsche Forschungsgemeinschaft. After this work was performed, we learned that van Leewen von Barth were also pursuing a similar line of reasoning 6. References. W. Kohn L. J. Sham, Phys. Rev. A 0, 33 Ž R. O. Jones O. Gunnarsson, Rev. Mod. Phys. 6, 689 Ž R. G. Parr W. Yang, Density Functional Theory of Atoms Molecules Ž Oxford, New York, R. M. Dreizler E. K. U. Gross, Density Functional Theory Ž Springer-Verlag, Berlin, S.. Vosko, L. Wilk, M. Nusair, Can. J. Phys. 58, 200 Ž J. P. Perdew A. Zunger, Phys. Rev. B 23, 508 Ž J. P. Perdew Y. Wang, Phys. Rev. B 5, 32 Ž P. Fulde, Electron Correlations in Molecules Solids Ž Springer-Verlag, Berlin, K. Burke, J. P. Perdew, M. Levy, in Modern Density Functional Theory: A Tool for Chemistry, J.M.Seminario P. Politzer, Eds. Ž Elsevier, Amsterdam, J. P. Perdew K. Burke, in Proceedings of the 8th International Congress of Quantum Chemistry, 92 June, 99, Prague; Int. J. Quantum Chem. to appear.. D. C. Langreth M. J. Mehl, Phys. Rev. B 28, 809 Ž J. P. Perdew, Phys. Rev. B 33, 8822 Ž 986.; 3, 706 Ž 986. Ž E.. 3. J. P. Perdew Y. Wang, Phys. Rev. B 33, 8800 Ž 986.; 0, 3399 Ž 989. Ž E... J. P. Perdew, in Electronic Structure of Solids 9, P. Ziesche. Eschrig, Eds. Ž Akademie Verlag, Berlin, J. P. Perdew, J. A. Chevary, S.. Vosko, K. A. Jackson, M. R. Pederson, D. J. Singh, C. Fiolhais, Phys. Rev. B 6, 667 Ž 992.; Ž 993. Ž E.. 6. J. P. Perdew, K. Burke, Y. Wang, unpublished. 7. A. D. Becke, Phys. Rev. A 38, 3098 Ž C. Lee, W. Yang, R. G. Parr, Phys. Rev. B 37, 785 Ž A. D. Becke, J. Chem. Phys. 98, 372 Ž A. Zupan, J. P. Perdew, K. Burke, M. Causa, Int. J. Quantum Chem., to appear. 2. T. Grabo E. K. U. Gross, Chem. Phys. Lett. 20, Ž 995., references therein. 22. M. L. Cohen, Solid State Commun. 92, 5 Ž K. Burke J. P. Perdew, Int. J. Quantum Chem. 56, 99 Ž D. C. Langreth J. P. Perdew, Solid State Commun. 7, 25 Ž K. Burke, J. P. Perdew, D. C. Langreth, Phys. Rev. Lett. 73, 283 Ž K. Burke, M. Ernzerhof, J. P. Perdew, unpublished. 27. J. C. Kimball, Phys. Rev. A 7, 68 Ž E. R. Davidson, Reduced Density Matrices in Quantum Chemistry Ž Academic Press, New York, O. Gunnarsson B. I. Lundqvist, Phys. Rev. B 3, 27 Ž O. Gunnarsson, M. Jonson, B. I. Lundqvist, Solid State Commun. 2, 765 Ž O. Gunnarsson, M. Jonson, B. I. Lundqvist, Phys. Rev. B 20, 336 Ž K. Burke, J. P. Perdew, M. Ernzerhof, unpublished. 33. S. Kais, D. R. erschbach, N. C. y, C. W. Murray, G. J. Laming, J. Chem. Phys. 99, 7 Ž M. Levy, in Density Functional Theory, R. Dreizler E. K. U. Gross, Eds., NATO ASI Series ŽPlenum, New York, A. D. Becke M. R. Roussel, Phys. Rev. A 39, 376 Ž R. Neumann, R.. Nobes, N. C. y, Mol. Phys., to appear. 37. M. Ernzerhof, J. P. Perdew, K. Burke, in Density Functional Theory, R. Nalewajski, Ed. ŽSpringer-Verlag, Berlin, J. C. Grossman, L. Mitas, K. Raghavachari, Phys. Rev. Lett. 75, 3870 Ž 995.; 76, 006 Ž 996. Ž E J. B. Krieger, Y. Li, G. J. Iafrate, Phys. Rev. A 5, 0 Ž V. Tschinke T. Ziegler, J. Chem. Phys. 93, 805 Ž J. B. Krieger, Y. Li, G. J. Iafrate, in Density Functional Theory, R. Dreizler E. K. U. Gross, Eds., NATO ASI Series Ž Plenum, New York, R. Colle O. Salvetti, Theoret. Chim. Acta 37, 329 Ž M. Ernzerhof, K. Burke, J. P. Perdew, unpublished.. K. Burke, J. P. Perdew, M. Levy, Phys. Rev. A, to appear. 5. J. P. Perdew Y. Wang, Phys. Rev. B 6, 297 Ž R. van Leeuven U. von Barth, private communication. INTERNATIONAL JOURNAL OF QUANTUM CEMISTRY 293

Virial Exchange Correlation Energy Density in Hooke s Atom

Virial Exchange Correlation Energy Density in Hooke s Atom Virial Exchangeorrelation Energy Density in ooke s Atom KIN-UNG LAM, FEDERIO G. RUZ, KIERON BURKE Department of hemistry, Rutgers University, 35 Penn Street, amden, New Jersey 080 Received 8 August 997;