Baixe kohn - lecture e outras Notas de estudo em PDF para Ciência da Atividade Física e do Esporte, somente na Docsity! Walter Kohn 213 ELECTRONIC STRUCTURE OF MATTER — WAVE FUNCTIONS AND DENSITY FUNCTIONALS Nobel Lecture, January 28, 1999 by WaLTER ROHN Department of Physics, University of California, Santa Barbara, CA 93106- 9530, USA 1. INTRODUCTION The citation for my share of the 1998 Nobel Prize in chemistry refers to the “development of the density functional theory”. The initial work on Density Functional Theory (DFT) was reported in two publications: the first with Pierre Hohenberg in 1964!) and the next with Lu J. Sham!2! in 1965. This was almost 40 years after E. Schroedinger!?! published his first epoch-making paper marking the beginning of wave-mechanies. The Thomas-Fermi theory, the most rudimentary form of DFT, was put forward shortly afterwards!* 5) and received only modest attention. There is an oral tradition that, shortly after Schroedinger's equation for the electronic wave-function 'P had been put forward and spectacularly validated for simple small systems like He and A, P.M. Dirac declared that chemistry had come to an end — its content was entirely contained in that powerful equation. Too bad, he is said to have added, that in almost all cases, this equa- tion was far too complex to allow solution, In the intervening more than six decades cnormous progress has been made in finding approximate solutions of Schroedinger's wave equation for systems with several elecirons, decisively aided by modern electronic com- puters. The outstanding contributions of my Nobel Prize cowinner John Pople are in this area. The main objective of the present account is to ex- plicate DFT, which is an alternative approach to the theory of electronic structure, in which the electron density distribution n(?), rather than the many electron wavefunction plays a central role. I felt that it would be useful to do this in a comparative context; hence the wording “Wavefunctions and Density Functionals” in the title. In my view DFT makes two kinds of contribution to the science of multi- particle quantum systems, including problems of electronic structure of molecules and of condensed matter: The first is in the area of fundamental understanding. Theoretical chemists and physicists, following the path of the Schroedinger equation, have become accustomed to think in a truncated Hilbert space of single particle orbitais. The spectacular advances achieved in this way attest to che fruitfulness of this per- spective. However, when high accuracy is required, so many Slater deter- 214 Chemistry 1998 minants are required (in some calculations up to - 10º!) thac comprehension be- comes difficult. DET provides a complementary perspective. It focuses on quantities in the real, 3-dimensional coordinate space, principally on the electron density n(7) of the groundstate. Other quantities of great interest are: the exchange correlation hole density 7, (47) which describes how the presence of an electron at the point 1 depletes the total density of the other electrons at the point 7 and the linear response function, y(47;0), which de- scribes the change of total density at the point r due to a perturbing potential at the point 7, with frequency q. These quantities are physicai, independent of representation, and easily visualisable even for very large systems. Their un- derstanding provides transparent and complementary insight into the nature of multiparticle systems. The second contribution is practical. Traditional multiparticle wavefunc- tion methods when applied to systems of many particles encounter what 1 call an exponential wall when the number of atoms, N, exceeds a critical value which, for “chemical accuracy”, currently is in the neighborhood of N, = 10 (to within a factor of about 2) for a system without symmetries. A major im- provement along present lines in the analytical and/or computational aspects of these methods will lead to only modest increases in A, Consequently, problems requiring the simultaneous consideration of very many interacting atoms, N/N, > 1, such as large organic molecules, molecules in solution, drugs, DNA, etc. overtax these methods. On the other hand, in DFT, comput- ing time Yrises much more moderately with the number of atoms, currently as T- Nºwith a = 2-3, with ongoing progress in bringing a down towards a = 1 (so-called linear scaling). Phe current state of the art of applied DFT can handle systems with up to Ng = O(102)-O(10º) atoms. The following figures and legends illustrate what can currently be achiev- ed. In these examples the number of atoms is O(10º) and the number of elec- trons several times larger. In Section 1, I shall talk about traditional wavefunction methods and con- trast their great success for few-atom systems with their fundamental limita- tions in dealing with very-many-atom systems. Section 2 deals with DFT against the backdrop of wavefunction methods. The basic theory is summarized: First the original Hohenberg-Kohn (HK) variational principle, where n(?) is the variational variable, is described. This is followed by the Kohn-Sham (KS) sel£-consistent single-particle equations which involve the well-defined exchange - correlation functional, £, [n(1)]. In principle, when used with the exact £, , these single particle equations in- corporate al! many-body effects." Next the physics of E, [n(7) ] is discussed in terms of the concept of the ex- change correlation hole n, (31). T have found the concept of “nearsighted- ness” useful which, in the present context, says that the exchange correlation hole n,,(%7) for an electron at the point ris largely determined by 4 — v, 7), where | is the chemical potential and v,(7) is the effective single particle tItis however known that for some density distributons E [n(7)] cannot be defined. Walter Kohn 217 Figure 8. Fully Hydroxylated Aluminum (0001) Surface. (Red-O; blue-interior AI; grey-H-atoms; the green lines are H-bonds). Each surface Alatom in AGO, has been replaced by 3 Hratoms, The cs simulation at regu- dynamics of water ad- figure represents a superposition of configurations in a molecular dyna lar intervals of 1 ps. These calculations help to understand the comple sorption on aluminum (K.C. Haas et al, Science 289, 265 (1998)) V=W(rira, The Pauli principle requires that Pjy=-V, (2.8) where P,, permutes the space and spin coordinates of electrons j and j'. Al physical Pr operties of the electrons depend parametrically on the R, in par- ty n(9) and total energy E which play key roles in his paper: n(r) = n(r; By, ..Ry) (2.4) 'N). (2.9) E = (RR), (2.5) where N is the number of nuclei. A. Few Electron Systems-the H, Molecule The first demonstrations of the power of the Schroedinger equation in ch istry were calculations of the properties of the simplest multielectron mole- cule, Hy: lts experimental binding energy” and internuclear separation are “This is the observed dissociation energy plus the zero point energy of 0.27 ev. 218 Chemistry 1998 Eu: D=4756V, R=0.740 À. (2.6) Thc carliest quantum theoretical estimate was made by Heitler and London in 192718], who used the Ansatz Vaz = Alonlr — Rope(ra — Ra) + pulr— Ropulro — Rúlxo, (2.7) where g;(n, — Rj is the orbital wavefunction of electron 1 in its atomic groundstate around a proton located at R,, etc; 4, denotes the spin singlet function; and A is the normalization. The components of this wavefunction describe two hydrogen atoms, at &, and R,, with spins pointing in opposite di- rections. The combination satisfics the reflection symmetry of the molecule and the Pauli principle. The expectation value of the Hamiltonian as a fune- ton of R= |R, - R, | was calculated. Its minimum was found to occur at R= 0.87Ã, and the calculated dissociation energy was 3.14eV, in semiquantitative agreement with experiment. However the errors were far too great for the typical chemical requirements of |5 8 | 0.01 and |5D | < 0.1. An alternative Ansatz, analogous to that adopted by Bloch for crystal elec- trons, was made by Mullikan in 1928/71: Tom = Pmo(ri)Pmol(T2) - Xo + (2.8) where Pmolri) = A(pu(m — Ra) + qulr — Ro), (2.9) and A' is the appropriate normalization constant. In this function both elec- trons occupy the same molecular orbital Q,,(1). The spin function xo is again the antisymmetric singlet function. The results obtained with this function were R= 0.76 À, and D= 2.65eV, again in semiquantitative agreement with ex- periment. The Mullikan Ansatz can be regarded as the simplest version of a more general, so-called Hartree-Fock Ansatz, the Slater determinant 1 Var = pa Det | Pm(rio(Lom(r2)B(2) |, (2.10) where (p,(7) is à general molecular orbital and a and 8 denote up and down spin functions. For given R= IR, - R, |, minimization with respect to (p,(9) of the expectation value of H leads to the nondocal Hartree Fock equations!8! for the molecular orbital 4,(7), whose solution gives the following results: R=0.74 À, D=3.630V. The most complete early study of H, was undertaken by James and Coolidge in 1933/, They made the very gencral variational Ansatz Vyc = U(rs,ra)xo, (2.11) where 'P(n,1,) is a general, normalized function of 1 and 1,, symmetric under interchange of +, and x, and respecting the spatial symmetries of the mole- cule. The trial function 'W was written as depending on a number of parame- Walter Kohn 219 ters, ffi>-- ny SO that for given |R, — R, |, the expectation value of the Schroedinger Hamiltonian in “FP, an upper bound to the true groundstate energy, became a function of the parameters p,, E = Ely...) The calcula- tions were made with M up to 13. Minimization of H(p,....)y) with respect 1o the p resulted in R= 0.740 Áand D= 4.70 eV, in very good agreement with experiment. More recent variational calculations of the same gencral charac- ter give theoretical results whose errors are estimated to be much smaller than experimental uncertainties, and other theoretical corrections. Before leaving the variational calculation for H,, we want to make a very rough “guesstimate” of the number of parameters M needed for a satisfactory resul. The number of continuous variables of W(r,,t;) is 6 by 1 reflecting axial symmetry. Let us call the number of parameter per vari- able needed for the desired accuracy p. Since a fractional accuracy of O(10) is needed for the energy, implying a fractional aceuracy of O(10"!) in 'P, we guess that 3 « p « 10. Hence M= pº = 3º — 105 = 102 105. By using symmetries and chemical and mathematical insights, this number can be significantly reduced. Such relatively modest numbers are very manag- able on today's (and even yesterday's) computers. It is thus not surprising that for sufficiently smail molecules, wavefunction methods give excellent resulis. = 5, the reduction B. Many Electrons - Encountering an Exponential Wall In the same spirit as our last “guesstimates” for A, let us now consider a ge- neral molecule consisting of N atoms with a total of N'interacting electrons, where N> 10 say. We ignore symmetries and spin, which will not affect our general conelusions. Reasoning as betore, we see that the number M of para- meter required is M=pN,3<p<h. (2.12) “The energy needs to be minimized in the space of these M parameters. Call M he maximum valuc feasible with the hest available computer software and hardware; and N the corresponding maximum number of electrons. Then, trom Eq. (2.12) we find 1 logM N= 2.13 3 Togp (2.18) Let us optimistically take Mx 10º and p= 3. This gives the shocking result 19 N=0—=6. . 3 pag 60) 214) Im practice, by being “clever”, one can do better than this, perhaps by one half order of magnitude, up to say N = 20. But the exponential in Eg. (2.12) represents a “wall” severely limiting N. Let us turn this question around and ask what is the needed Mfor N = 100. By Eq. (2.12) and taking p= 3 we find 222 Chemistry 1998 non-integral atomic number Z = xZ, + (1-2) Z2(Z, = 29, Z, = 30). This model nicely explained the linear dependence of the electronic specific heat on x. On the other hand the low temperature resistance is roughly proportional to x(1-2), reflecting the degree of disorder among the two constituents. While isolated Cu and Zw atoms are, of course, neutral, in a Cu-Zn alloy there is transfer of charge between Cu and Zn unit cells on account of their chemical differences, The electrostatic interaction energy of these charges is an im- portant part of the total energy. Thus in considering the energetics of this system there was a natural emphasis on the electron density distribution n(7). Now a very crude theory of electronic energy in terms of the electron den- sity distribution, (7), the Thomas-Fermi (TF) theory, had existed since the 1920s/4) [5], Tt was quite useful for describing some qualitative trends, e.g. for total encrgies of atoms, but for questions of chemistry and materials science, which involve valence electrons, it was of almost no use; for example it did not lead to any chemical binding. However the theory had one feature which interested me: It considered interacting electrons moving in an external po- tential v(7), and provided a highly over-simplified one-to-one implicit relation between v(1) and the density distribution n(7): L,2m n(r) = (ue — vers) PP (5 = amp, (8.1) n(r” , , vess(r) = v(r) + [Ear , (3.2) where | is the r-independent chemical potential; Eq. (3.1) is based on the ex- pression n=Yu-— vu? (3.8) for the density of'a uniform, non-interacting, degenerate electron gas in a con- stant external potential v; and the second term in (3.2) is just the classically computed electrostatic potential times (-1), generated by the electron densi- ty distribution 2(?). Since Eq. (3.1) ignores gradients of v, (9) it was clear that the theory would apply best for systems of slowly varying density. In subsequent years various refinements (gradient-, exchange-and correla- tion corrections) were introduced, but the theory did not become signifi- cantly more useful for applications to the electronic structure of matter. It was clear that TF theory was a rough representation of the exact solution of the many-electron Schroedinger equation, but since TF theory was expressed in terms of n(7) and Schroedinger theory in term of 'P(r,,...,1,), it was not clear how to establish a strict connection between them. This raised a general question in my mind: Is a complete, exast description of groundstate electronic structure in terms of n(7) possible in principle. A key question was whether the density n(7) completely characterized the system. It was true in TF theory, where n(?), substituted in Eg. (3.1) yields, (v, 49) — u and, by (3.2), (v(r) — 4). In addition, n(r) also yields the total number of elec- Walter Kohn 223 trons by integration. Thus the physical system is completely specified by n(?). It was also simple to check that the same was true for any 1-particle system, as well as for a weakly perturbed, interacting, uniform electron gas vlr)=v tur) (A<1), (3.4) n(r) = no + Am(r) +... (3.5) for which w,(7) can be explicitly calculated in term of m (1) by means of the wave-number-dependent susceptibility of the uniform gas. This suggested the hypothesis that « knowledge of the groundstate density of n(r) for any electronic system, (with or without interactions) uniquely determines the system. This hypothesis became the starting point of modem DFT. IV. THE HONENBERG-KOHN FORMULATION OF DENSITY FUNCTIONAL THEORY A, The Density n(r) as the Basic Variable The Basic Lemma of HK. The groundstate density n(1) of à bound system of interacting electrons in some external potential v(7) determines this potenti- al uniquelyl!, Remarks: (1) The term “uniquely” means here up to an uninteresting additive con- stant. (2) In the case of a degenerate groundstate, the lemma refers to any ground- state density n(7). (3) This lemma is mathematically rigorous. The proof is very simple. We present it for a non-degenerate groundstate. Let n(7) be the non-degenerate groundstate density of N electrons in the potential v,(7), corresponding to the groundstate 'Y,, and the energy E. Then, Ei= (Bo $i) = fontr)n(r)dr + (Wi (T+ UNO, (4a) where H is the total Hamiltonian corresponding to vw, and Tand Uare the kinetic and interaction energy operators. Now assume that there exists a second potential w(7), not equal to vw (1) + constant, with groundstate “Po, necessarily + e'º*P,, which gives rise to the same (1). Then E= foten(rdár+ [Go (PAU). qu) Since 'W, is assumed to be non-degenerate, the Rayleigh-Ritz minimal prin- ciple for 'Y, gives the inequality, 224 Chemistry 1998 Ei < (Pa, Ho) = f o(rIn(e)dr + (Do (T+ UNE) =E+ [(ntr)-ur)n(ndr as) Similarly E, < (HU) = E + [ln(r) —olrnrdr , (44) where we use < since the non-degeneracy of Y, was not assumed. Adding (4.3) and (4.4) leads to the contadiction E +E<B +. (4.5) We conclude by reductivo ad absurdum that the assumption of the existence of a second potential v,(1), which is unequal to v, (1) + constant and gives the same n(7), must be wrong. The lemma is thus proved for a non-degenerate groundstate. Since n(?) determines both Nand «(») (ignoring an irrelevant additive con- stant) it gives us the full 4 and N for the electronic system. Hence (7) determines implicitly al! properties derivable from H through the solution of the time-independent or time-dependent Schrocdinger equation (even in the presence of additional perturbations like electromagnetic fields), such as: the many-body cigenstates FO (rr), POC), -. the 2eparticle Green's function G(r,4, 14), the frequency dependent electric polarizability a(o), and so on. We repeat that all this information is implicit in 7(7), the groundstate density. Remarks: 1. The requirement of non-degeneracy can easily be lifted!2!, 2. Of course the lemma remains valid for the special case of non-interacting electrons. 3. Lastly we come to the question whether any well-behaved positive function n(?), which integrates to a positive integer N, is a possible groundstate den- sity corresponding to some v(?). Such a density is called «representable (VR). On the positive side it is easy to verify that, in powers of À, any near- ly uniform, real density of the form n(7) = n,+ AZ n(g)é% is VR, and that for a single particle any normalized density n() = ly (1) Bis also VR. On the other hand Levy!!! and Lieb DM have shown by an example which in- volves degenerate groundstates, that there do exist well-behaved densities which are not VR. The topology of the regions of vrepresentability in the abstract space of all (7) continues to be studied. But this issue has so far not appeared as a timitation in practical applications of DFT. B. The Hohenberg-Kohn Variational Principle The most important property of an electronic groundstate is its energy E. By wavefunction methods £ could be calculated either by direct approximate Walter Kohn 227 sel£consistent Harwee equations. One may start from a first approximation for n(1), (e.g. from TF theory), construet v, (7), solve (4.13) for the j; and recalcutate 2(*) from Eq. (4.14), which should he the same as the initial (7). Hit is not one iterates appropriately until it is. In the winter of 1964, I returned from France to San Diego, where 1 found my new post-doctoral fellow, Lu Sham. I knew that the Hartree equations de- seribed atomic groundstates much better than TF theory. The difference be- tween them lay in the different treatments of the kinctic energy T (See Egs. (4.10) and (4.13). [ set ourselves the task of extracting the Hartree equations from the HK variational principle for the energy, Egs. (4.9), (4.7), (4.8), which 1 knew to be formally exact and which therefore had to have the Hartree equations and improvements “in them”. In fact it promised a Hartree-like formulation, which - like the HK minimal principle — would be formally exact. The Hartree difterential equation (4.13) had the form of the Schroedinger equation for non-interacting electrons moving in the external potential v,,. Could we learn something useful from a DFT for noninteracting electrons moving in a given external potential v(r)?. For such a system, the HK vari- ational principle takes the form fotmitriar + Tolfi(r)] (4.15) DE, (4.16) Eumlh where (assuming that à(7) is VRfor non-interacting electrons), Tjlã(v)) = kinetic energy of the groundstate of nor-interacting (4.17) electrons with density distribution à(m). The Euler-Lagrange equations, embodying the fact that the expression (4.14) is stationary with respect to variations of R(1) which leave the total number of electrons unchanged, is SEsfilr)] = f óRi(r) (ol) + quit lion —ejdr = 0, (4.18) where à(v) is the exact groundstate density for v(7). Here sis a Lagrange mul- tiplyer to assure particle conservation. Now in this soluble, nor-interacting case we know that the groundstate energy and density can be obtained by cal- culating the eigenfunctions P(7) and eigenvalues é, of nominteracting, single-particle equations 1 (5 VE +o(r) —eitr) =0, (4.19) yielding N N E=-Sssd)=5 ler P. (4.20) ja j=a 228 Chemistry 1998 (Here j labels both orbital quantum numbers and spin indices, + 1). Returning now to Lhe problem of interacting electrons, which had previous- ly been addressed approximately by the single-particledike Hartree equations, we deliberately wrote the functional F[R(r)) of Eq. (4.8) in the form riatol= Dino 5 [EE qdrae! + Bi) (42 where T[5(7)] is the kinetic energy functional for non-interacting electrons, Eq. (4.15). The last term, E, [4(9)], the so<alled exchange-correlation enes- gy functional is then defined by Eq. (4.21). The HK variational principle for in- teracting electrons now takes the form, Esi(r)]= f ulritrar + Tito) + 5 :/ es [9 irdr' + Esc(fi(r)] (4.22) ZE. The corresponding Euler-Lagrange equations, for a given total number of electrons has the form sBfnto] = [ 586)ven(r) + Fr Inca —eJdr = 0, (423) where vess(r) = o(r) + f ] no. [+ tao) (4.24) and 8 Ver) = say Pesltr)] liitente): (4.25) Now the form of Eg. (4.23) is identical to that of Eq. (4.18) for non-interacting particles moving in an effective external potential v,, instead of (7), and so we conclude that the minimizing density 2(7) is given by solving the single- particle equation 1 (59 +uestr) — e9) lr) =0, (4.26) with N am) =5 er) É, (427) vess(r) = v(r ni o + Vrelr), (4.28) where v,.(7) is the local exchange-correlation potential, depending functio- nally on the entire density distribution (7), as given by Eq. (4.25). These self- consistant equations are now called the Rohn-Sham (KS) equations. Walter Kohn 229 The groundstate energy is given by E = 56, + Esdn(m)]— [ vmelr)n(r)do 5 If one neglects £, and v, altogether, the KS equations (4.26)-(4.29), reduce to the self-consistent Hartree equations. “The KS theory may be regarded as the formal exactification of Hartree theory. With the exact E, and 2, all many body effects are in principle includ- ed. Clearly this directs attention to the functional E, [a(r)]. The practical use- fulness of groundstate DFT depends entirely on whether approximations for the functional E, [1 (7)] could be found, which are at the same time suffici- ently simple and suíficiently accurate, The next section V briefly describes the development and current status of such approximations. Remarks: - The exact effective single particle potential v, 7) of KS theory, Eq. (4.28) can be regarded as that unique, fictitious external potential which leads, for non-interacting particles, to the same physical density n(7) as that of the interacting electrons in the physical externat potential v(r. Thus if the physical density 2(7) is independently known (from experiment or-for small systems-from accurate, wavefunction-based calculations) v, fo and hence also v, (7) can be directly obtained from the density n() sf! Because of their linkage to the exact physical density n(7), the KS single particle wavefunctions 9;(7) may be considered as “density-optimal”, while, of course, the Hartree-Fock HF wavefunctions er (7) are “toial energy-op- timal” im the sense that their normalized determinant leads to the lowest groundstate energy attainable with a single determinant. Since the advent of DFT the term “exchange energy” is often used for the exchange energy computed with the exact KS & (1), and not with the HF e! (For the uniform electron gas the two definitions agree; typically the differences Ed are very small). 3, Neither the exact KS wavefunctions nor energies E; have any known, directly observable, strict meaning, except for a) the connection (4.27) between the q, and the true, physical density n(1); and b) the fact that the magnitude of the highest occupied &, relative to the vacuum equals the ionization energy 18], In concluding this Section we remark that most practical application of DFT use the KS$ equations, rather than the generally less accurate HK for- mulation. V. APPROXIMATION FOR E, [n(7)]: FROM MATHEMATICS TO PHYSICAL SCIENCE So far DFT has been presented as a formal mathematical framework for viewing electronic structure from the perspective of the electron-density n(r). This mathematical framework has been motivated by physical considerations, but, 232 Chemistry 1998 tron interaction. Of course, like everything else, it is a functional of the den- sity distribution (7). To define the average xe hole one introduces a fictitious, à-depentent Hamiltonian, H, for the many body system, 0 <A < 1, which dif fers from the physical Hamiltonian, 77 . by the two replacements (5.7) vlr) — lr), (5.8) where the fictitious x, (3) is so chosen that for all À in the interval (0,1) the corresponding density equals the physical density, n(7): mal”) = meai(r) = n(r). (5.9) The procedure (5.2), (5.3) represents an interpolation between the KS system (A = 0) and the physical system (A = 1). The average xc hole density à(yr') is then defined as «1 Aselr,r) =[ EMgelr; TA). (5.10) Its importance stems from the exact result, proved independently in three im- Pp! F y portant publications!!9), that Em = (5.11) An equivalent expression is/2 Eu- 5 farmtmAZt al 812) where Role) = far pº 6 Rae(rs In) ente DD , (5.13) is the moment of degree (-1) of 4,,(7), Le, minus the inverse radius of the iraveraged xe hole. Comparison of Egs. (5.12) and (5.1) gives the very physi- cal, formally exact relation eselr; [n(P)]) = SRH lr; [n()D). (514) Walter Kohn 283 Gradient Expansion and Generalized Gradient Approximation Since Rjtr) is a functional of n(7), expected ta be (predominantty) short- sighted, we can formally expand x(7) around the point rwhich we take to be the origin: 1 as =n + mi ts ngfis+ o (5.15) where n= n(0), x,= V, (9 | o etc, and then consider R,(?) as a function of the coefficients 1,n,%; .. Ordering in powers of the differential operators and respecting the ccalar nature of Rj gives Rzo(r) = Fo(r(r)) + Fos(n(r)) 7? nr) + Fro(n(r)) x 5Ventr) (Valn(r)) +. When this is substituted into Eq. (5.19) for E, it leads (after an integration by parts) to the gradient expansion (5.16) Es = ELPA + [a (n)( (onár + float Ven +. Jdr+.., (617) where G;(n) is a universal functional of n!2), In application to real systems this expansion has generally bcen disappointing, indeed often worsencd the results of the LDA. The series (5.15) can however be formally resummed to result in the following sequence E, = f enlr)nlrdr (LDA), (618) E = [ 1Mntr),| nto) Dlrjar (GG), (as) EP = [19 vn) | vino, 620) ES is the (LDA), requiring the independently caleulated function of one variable, = n. E(Dthe so called generalized gradient approximation (GGA) requires the independently calculated function of two variables, x = 1, = Iva l ete. Thanks to much thoughtful work important progress has been made in deriving successful GGA's of the form (5.19). Their contruction has made use of sum rules, general scaling properties, asymptotic behavior of effective potentials and densities in the tail regions of atoms and their aggregates. In addition, A. Becke in his work on GGAs, introduced some numerical fitting parameters which he determined by optimizing the accuracy of atomization 234 Chemistry 1998 energies of standard sets of molecules. This subject was recently reviewed!!! We mention here some of the leading contributors: AD. Becke, D.C. Langreth, M. Levy, R.G. Parr, | P. Perdew, C. Lee, W. Yang. In another approach A. Becke introduced a successful hybrid method Et = BS + (1 EÇS*, (5.21) where E is the exchange energy calculated with the exact KS wavefunctions, is an appropriate GGA, and a is a fitting parameter!22), “The form of this linear interpolation can be rationalized by the Mintegration in Eg. (5.10), with the lower limit corresponding to pure exchange. Usc of GGAs and hybrid approximations instead of the LDA has reduced errors of atomization energies of standard sets of small molecules, consisting of light atoms, by factors of typically 35. The remaining errors are wypically + (2-3) kg moles per atom, about twice as high as for the hest current wave- function methods. This improved accuracy, the ease of calculation, together with the previously emphasized capability of DFT to deal with systems of very many atoms, has, over a period of relatively few years beginning about 1990, made DFT a significant component of quantum chemistry. For other kinds of improvements of the LDA, including the weighted den- sity approximation (WDA) and selfinteraction corrections (SIC) we refer the reader to the literature, e.g. Before closing this section | remark that the treatments of xc-effects in the LDA and all of ils improvements, mentioned above, is completely inappro- priate for all those systems or subsystems for which the starting point of an electron gas of slowly varying density n(7) is fundamentally incorrect. Examples are a) the electronic Wigner crystal; b) Van der Wuals (or polariza- tion) energies benween non-overlapping subsystems; c) the electronic tails evanescing into the vacuum near the surfaces of bounded electronic systems, However this does not preclude that DFT with appropriate, different approx- imations could successfully deal with such problems (See Sec. VTN). VI. GENERALIZATIONS AND QUANTITATIVE APPLICATIONS While DFT for non-degenerate, non-mugnetic systems has continued to progress over the last several decades, the DFT paradigm was also greatly ex- tended and generalized in several directions. The purpose of this section is to give the briefest mention of these developments. For further details we refer 16 two monographs!21 [24 and a recent set of lecture notes!?!] A. Generalizations a. Spin DFT for spin polarized systems: v(9). BD; 2(0, (4 (D- 24 (0). b. Degenerate groundstates: v(1); nm v= 1,.M; Ep e. Multicomponent systems (electron hole droplets, nuclei): v, (1); n,(7: Ho: 4. Ensemble DFT for M degenerate groundstates: u(9); n(1) (= M! (Tr RÃ): Ep