Density Functional Theory

Ann. Rev. Phys. Cher. 1983.34: 631-56 Copyri!ht © 1983 by Annual Reviews Inc. All rights reserved DENSITY FUNCTIONAL THEORY Robert G. Parr Department of Chemistry, University of North Carolina, Chapel Hill, North Carolina 27514 INTRODUCTION In writing this review I hope to alert the physical chemical community to the promise and the charm of the density functional theory of electronic structure of atoms and molecules. Density functional theory was born in an extraordinary 1964 paper by Hohenberg & Kohn ( 1), and its chief method of implementation in a 1965 paper �y Kohn & Sham (2). And yet each of these papers has been referenced only once in the Annual Review of Physical Chemistry, and then only in passing (3). I believe that these papers have transcendent importance for chemistry. See also the 1966 sequel (4). To be sure, the so-called XI method (5) has received much attention, and it has been twice reviewed in these pages (6, 7). However, it generally has been presented as an independent, self-contained method, and only the review elsewhere by Connolly (8) revealed it for what it is, an approximation to exact density functional theory. As for reviews of density functional theory itself, I should mention the one by Kohn & Vashishta (9, 10) and the one by Rajagopal (11). In the present review I shall survey the density functional theory as it applies to systems with few�as opposed to infnitely many-electrons. Fundamental aspects are emphasized, with specifc applications only touched upon. First I sketch the original theory. Then I consider refnements and extensions, problems within the theory, applications to the phenomenological description of chemical phenomena, and future pros­ pects. I also discuss connections with statistical mechanics and the intriguing conclusion that the distinction between microscopic and macro­ scopic systems is more blurred than one had thought. Uncertainties abound both in density functional theory and in its applications, and the literature of the subject is not free of errors. 631 0066-426X/83/1101-0631$02.00 A n n u . R e v . P h y s . C h e m . 1 9 8 3 . 3 4 : 6 3 1 - 6 5 6 . D o w n l o a d e d f r o m w w w . a n n u a l r e v i e w s . o r g b y S o u t h e r n I l l i n o i s U n i v e r s i t y - C a r b o n d a l e o n 0 3 / 2 3 / 1 2 . F o r p e r s o n a l u s e o n l y . Quick links to online content Further ANNUAL REVIEWS 632 PARR EXACT FORMULATIONS Density Theory There was an old model, the Thomas-Fermi model (12-15), from which originated the idea of an "electron gas." In this model all properties of a system turn out to be expressible in terms of the electron density p, the number of electrons per unit volume, as it varies through space. This fails to give an accurate description of electronic systems of chemical interest; for example, it cannot account for chemical binding (16). But it is now known to be but a frst approximation to an exact description of the ground state of any system in terms of its density-the density functional description. The density can be obtained by quadrature from the exact N-particle wave function 1(12·· N): p(1) " N JII1 2 d'2 d'3 .. d ON dsl . 1. 1/1 itself comes from solution of the Schrodinger equation, which can be expressed as a stationary principle, ({<.I�I.>-E[<.I.>-I]} 0 , where the energy E is a Lagrange multiplier and � is the Hamiltonian, � Li(f) + LIV(f)+ L"<v(1lr,v)· Since p integrates to N, N = N[p] ¯ J p(l) d Vl = J p d r , 2. 3. 4. and the operators t and l/r Iv are universal, this procedure is a way to begin with N and v and determine all properties, including p. The quantity p is of course of much interest, being directly accessible experimentally and readily visualizable-just the classical density of the electronic system. Four recent reviews of the properties of p should be mentioned (17-20), as should be the extraordinary work of Bader (21) and others (22), following Collard & Hall (23), in which the whole of structural chemistry is addressed by applying modern catastrophe theory to p and V p. But our concern here is rather the remarkable facts proved by Hohenberg & Kohn (1), that p determines v and hence everything, and a wrong p gives an energy above the true energy. Functionals T[p], v.[p], E[p], and so on therefore exist, and there is a stationary principle analogous to Eq. 2, c{E[p]-J[N[p]-N]} ¯ O. 5. Here J is the so-called chemical potential; T[p] and Ve[P] are functionals that give for any ground-state density p the kinetic energy of the electrons A n n u . R e v . P h y s . C h e m . 1 9 8 3 . 3 4 : 6 3 1 - 6 5 6 . D o w n l o a d e d f r o m w w w . a n n u a l r e v i e w s . o r g b y S o u t h e r n I l l i n o i s U n i v e r s i t y - C a r b o n d a l e o n 0 3 / 2 3 / 1 2 . F o r p e r s o n a l u s e o n l y . DENSITY FUNCTIONAL THEORY 633 and the electron-electron repulsion energy. The Euler equation cor­ responding to Eq. 5 is f ¯ o E/op = v + oF/o p 6. where P[ p ] " T[ p ] + v"e [P] . The right side of Eq. 6 varies from point to point through the system unless the density is the true density for the system, in which case the value is the same everywhere-the chemical potential. For a system of a given number of electrons (an important restriction), Levy (24) has shown how to determine F [p ] : F [ p ] ¨ min < tIT+ Velt). 7. !=p Compute for all antisymmetric t which give p the expectation value of T + fe; the minimum of all these is F[ p ] . This is not to say that the determination of F [ p ] is easy; therein indeed lies much of the challenge, and the charm, of density functional theory. For a change from one ground state to another, one fnds (25) the fundamental equation for the change in E [ N, v ] , oE ¯ foN + f p( l)ov ( l) dVl' Thus one has f (aE/aN)" and p(1) ¯ [ oE / ov ( l)h· 8. 9. 10. The analogy with macroscopic thermodynamics is clear, about which more later. That the chemical potential is no more no less than the electronegativity concept of structural chemistry follows from Eq. 9 and the famous electronegativity formula of Mulliken (26): I+A f = (aE/oN)" � - - 2 -= X M, 11. where I and A are the ionization potential and electron afnity of the species in question. Many properties of electronegativity follow (25, 27, 28). The electronic energy functional may be resolved into components as follows: E [p] = T[p ] + v., e [P] + v"e [ P] ¯ v.e[P] + F[p ], 1 2 . where v.e[ P ] ¯ fv(l)p(l) dVl and v"e[P] J[ p ]-K[ p ]; J[ p ] = ( 1 /2 ) ffp( l )p(2 ) (I/rd d Vl d V2' 13. A n n u . R e v . P h y s . C h e m . 1 9 8 3 . 3 4 : 6 3 1 - 6 5 6 . D o w n l o a d e d f r o m w w w . a n n u a l r e v i e w s . o r g b y S o u t h e r n I l l i n o i s U n i v e r s i t y - C a r b o n d a l e o n 0 3 / 2 3 / 1 2 . F o r p e r s o n a l u s e o n l y . 634 PARR For a molecule, the nuclear-nuclear repulsion must be added to give the total energy. K here is defned as J minus Ye; it includes correlation as well as exchange efects. Density Matrix Theory If the density determines all properties, so does the frst-order density matrix, 14. A whole formal theory can be constructed, therefore, using functionals of Y 1 rather than of p = Yl( l , 1), and that has been done (25, 29-32). [Note that this is not wave function theory restated as a problem in density matrices, which is a whole big, separate story.] The most general such theory uses an arbitrary representation OfYl and has many interesting aspects (29-31). Here we concentrate on the more compact theory (25) in which Yl is represented in terms of natural spin orbitals and their occupation numbers (33): Y I (1 ' , 1) Lk n kl:( l ')lk(I). 15. One expects (34, 35), and this point is vital, 0< nk < 1, 16. for a complete set of the I k ' or at worst almost a complete set. Functionals of p become functionals of the Ik and the nk. The kinetic energy as a functional ofYl is known explicitly, 17. while the electron-electron repulsion energy Ye[Yl] is not; the latter, however, txists by the Hohenberg-Kohn theorem. Minimization of the energy B[Yl] with respect to the nk and tk' subject to orthonormalization conditions, then gives (25, 31) 18. where J is the chemical potential of Eqs. 5 and 6. Also, the tk satisfy equations of the form F tk = Gktk, 19. where the ek are eigenvalues, and also ) = eJnk' for all k. 20. While these equations are not so easy to implement in practice (see below), the formal results are highly interesting. They depend critically on A n n u . R e v . P h y s . C h e m . 1 9 8 3 . 3 4 : 6 3 1 - 6 5 6 . D o w n l o a d e d f r o m w w w . a n n u a l r e v i e w s . o r g b y S o u t h e r n I l l i n o i s U n i v e r s i t y - C a r b o n d a l e o n 0 3 / 2 3 / 1 2 . F o r p e r s o n a l u s e o n l y . DENSITY FUNCTIONAL THEORY 635 Eq. 16. The verbal statement is that all natural spin orbitals have the same chemical potential or electronegativity. Equation 19 is not the Lowdin equation ( 33) for natural spin orbitals, but a density functional analogue thereof ( 32). The L6wdin equation in fact is not diagonal in the natural spin orbitals, whereas Eq. 19 is. Kohn-Sham Theory Kohn & Sham ( 2) introduced a quite diferent method for evaluating T[p ] : separate it into a major part that is known and a minor correction. Specifcally, a noninteracting (Ye = 0) system with density p would have a kinetic energy N T[ p ] L < ¢ kl- (1/2) V 21 ¢ k> 21 . k and one may then write, in place of Eq. 12, E[ p ] ¯ '[ p ] + Ye[P] +J[ p ] + Exc[ p ] 22. where Exc[p] = T[p] - T[ p ] -K[p] . 23. Also, 24. The "Kohn-Sham orbitals" ¢ k have densities that sum to the exact density. As far as the density is concerned, it is as if the wave function were a single Slater determinant built from the ¢ k . The exact wave function is not this determinant, however; the entire physical meaning of the ¢k is contained in Eq. 24. Minimization of the energy now leads to the Kohn-Sham equations for determining the orbitals ¢k , namely N one-electron equations of the form F KS ¢ k B � S ¢ k· Here FKS ¯ -(1/2)V2+ v + ¢+(JExc /J p) 25. 26. where ¢ is the classical electrostatic potential due to the whole electron distribution. These equations must be solved self-consistently, since F KS depends on p. The method is in principle exact, though in practice one must resort to approximations for Exc . None of the eigenvalues e � s is expected to have defnite physical signifcance except the highest, and even its meaning is tricky to establish (9, 36). A n n u . R e v . P h y s . C h e m . 1 9 8 3 . 3 4 : 6 3 1 - 6 5 6 . D o w n l o a d e d f r o m w w w . a n n u a l r e v i e w s . o r g b y S o u t h e r n I l l i n o i s U n i v e r s i t y - C a r b o n d a l e o n 0 3 / 2 3 / 1 2 . F o r p e r s o n a l u s e o n l y . 636 PARR Extensions SPIN DENSITY THEORY The density of Eq. 1 is spin free, but is of course a sum of spin-I density and spin-{ density. For states other than 1 S states, and indeed for practical purposes often even for 1 S states, it is necessary or convenient to use a generalization of the original Hohenberg-Kohn-Sham theory that invokes similar theorems for the separate density components (37, 38); we have, for example, E = E[Pa , Ppl Corresponding "spin­ polarized" versions have been developed of Kohn-Sham and XI methods (2, 37-40). Of particular importance is the possibility within such methods of incorporating self-energy corrections, which have been developed and exploited recently (41-47). EXCITED STATES Higher-energy solutions of Eq. 2 are automatically orthogonal to the ground-state solution. No such wave function ortho­ gonalities exist between higher-energy solutions of Eq. 5 to the ground­ state solution (except where there are symmetry diferences). For this reason, density functional theory for excited states (48) is more difcult than wave function theory for them; there is no such simple theorem that P for an excited state determines all of its properties. As has been said, "Density Functional theory is primarily a ground state theory" (49). The ground­ state density determines all excited state properties, however, and excited states have in fact been variously computed by density functional methods. Rigorous discussions of diferent possible procedures include those of Theophilou (50), Valone & Capitani (51), and Katriel (52). More work is needed in this area. TIME-DEPENDENT PROCESSES Using an energy-minimization principle based on the hydrodynamic formulation of quantum mechanics, Bartolotti (53) recently has derived a variational density functional theory for a time­ dependent ground state, and then (54) a corresponding Kohn-Sham procedure. A comprehensive discussion of much the same ground has been given by Deb & Ghosh (55, 56), who also give special attention to the problem of calculation of the dynamic polarizability (56). See also the review by Lundqvist (57). Such results will be essential as one sets out to describe systematically chemical processes using density functional concepts. MOMENTUM SPACE Electron density for a ground state, p ( r ) , and momen­ tum density for a ground state, p(p), are not simple Fourier transformations of each other, and so there is not a trivial transcription of Hohenberg-Kohn A n n u . R e v . P h y s . C h e m . 1 9 8 3 . 3 4 : 6 3 1 - 6 5 6 . D o w n l o a d e d f r o m w w w . a n n u a l r e v i e w s . o r g b y S o u t h e r n I l l i n o i s U n i v e r s i t y - C a r b o n d a l e o n 0 3 / 2 3 / 1 2 . F o r p e r s o n a l u s e o n l y . DENSITY FUNCTIONAL THEORY 637 theory into momentum space. However, there is a corresponding theory in momentum space, which has been given by Henderson (58). NON-BORN-OPPENHEIMER THEORY Among the extensions of Hohenberg­ Kohn theory to muiticomponent systems (9) is the extension to the molecule considered as a system of several components: the electrons plus nuclei of each different type of nucleus (e.g. H20 is a three-component system). This theory has been written out (59). Each component has its own chemical potential. Extensions to a phenomenological description of chemical reactions should be forthcoming. The Hartree-Fock Case The Hartree-Fock energy functional is a functional of the frst-order density matrix, and so the Hartree-Fock model can be viewed as an approximation to the density matrix theory described above (31, 32). However, there are two respects in which the usual Hartree-Fock theory falls short of being a full implementati�n of density functional ideas. First, Eq. 16 is not satisfed, and so Eq. 18 does not hold. Second, N is not generally treated as a variable, certainly not as a continuous variable. Nevertheless, Hartree-Fock theory as a density functional theory is of interest. The Hartree-Fock density (known to be very close to the true density) determines the Hartree-Fock potential and all properties for the ground state (including the true density) (60, 61). Following up on this result, Payne (62, 63) has introduced difference coordinltes in the Hartree-Fock density matrix, and produced an interest­ ing Hartree-Fock calculational scheme focusing on the density. In an important paper, Aashamar et al (64) have determined the exact Kohn-Sham orbitals and efective potential for Hartree-Fock atoms having Z : 54. The orbitals are very close to Hartree-Fock orbitals. The Exact Thomas-Fermi Limit A fundamental advance was made when Lieb & Simon (65) established that in the limit of high Nand Z, atoms conform to the original Thomas-Fermi model, in which 27. and K[p] is negligible. This is important because it establishes all of the properties of the exact Hohenberg-Kohn functional in one important limiting situation. However, one should note that N and Z "high" in this context means on the order of thousands, which rules out the possibility that Thomas-Fermi theory will govern elements actually in nature. The "hydrogenic" limit is much closer in reality¯the ve = 0 model (66, 67). A n n u . R e v . P h y s . C h e m . 1 9 8 3 . 3 4 : 6 3 1 - 6 5 6 . D o w n l o a d e d f r o m w w w . a n n u a l r e v i e w s . o r g b y S o u t h e r n I l l i n o i s U n i v e r s i t y - C a r b o n d a l e o n 0 3 / 2 3 / 1 2 . F o r p e r s o n a l u s e o n l y . 638 PARR LOCAL APPROXIMATIONS The Essence of Localization We now come to a very important point about this subject, the understand­ ing of which is essential for the appreciation of why formulas originally derived for the uniform electron gas work so well for such nonuniform systems as atoms and molecules. Consider an arbitrary given functional of the density, Q[p]. It is a local density functional if its functional derivative with respect to the density, JQ/Jp, at a point, is a function only of the density at that point (and not its derivatives or integrals) (68). Thus Jp dr is local; the integral J V p' V p dr is nonlocal. The exact Hohenberg-Kohn functionals T[p] and F[p] are nonlocal. Even though a given functional Q[p] is nonlocal, a good approximation to it often may be provided by a local functional: while a local formula Q[p] " J q( p ) dr may be incorrect, a formula Q[p] ¯ J q( p ) dr + corrections 28. may be extremely useful. If, further, the particular Q is known to have dimensions of kinetic energy or potential energy, the dependence of q( p ) on p is forced (69). Specifcally, we may write, without any error whatsoever, T[p] = Cdp5/3 dr + corrections 29. and K[p] ¯ Cxafp4/3 dr + corrections. \ 30. Note, however, that the Kahn-Sham "exchange-correlation" functional has potential and kinetic parts and cannot be written (except roughly) in the form ofEq. 30. Constants CF and CXa in these expressions can depend on N. For a uniform electron gas, the "corrections" in the above equations are zero and the constants are those of Thomas-Fermi-Dirac theory. For a nonuniform gas, there is no reason not to hope to use these same equations, with corrections as appropriate, and with N dependence in the coeffcients as appropriate. What is needed only is that the functionals in question are to a certain degree local, and that does not at all require t hat they be "statistical." Small N or large N, it does not matter! Gradient Expansions The idea of a gradient expansion of a functional Q[p] is to write, exactly, Q[p] " f q [p] dr, and to expand q [p] into a local part plus parts that depend successively on V p, V 2 p, etc, q [p] q o ( p ) + q l(P)V p + q 2a( p )V2 p + q2b(P)V p ' V P + . . . 31. A n n u . R e v . P h y s . C h e m . 1 9 8 3 . 3 4 : 6 3 1 - 6 5 6 . D o w n l o a d e d f r o m w w w . a n n u a l r e v i e w s . o r g b y S o u t h e r n I l l i n o i s U n i v e r s i t y - C a r b o n d a l e o n 0 3 / 2 3 / 1 2 . F o r p e r s o n a l u s e o n l y . DENSITY FUNCTIONAL THEORY 639 The coefcient functions qo(p), ql(P), + . . , may depend on the choice of a reference situation and may or may not be easy to determine. The resultant series for Q[pJ may or may not converge; it may be useful even if it is divergent; a truncated form may give good numerical values of Q but poor values of oQ/op. KINETIC ENERGY Using Szasz' results (69), a gradient expansion of the kinetic energy functional must take the form fvp•vp T[pJ = COf p5/3 dr+C 2 p dr + C 4af(V2p)2p -5 /3 d r + C 4bf(V2p)(V p. V p)p -8/3 dT +C4cf(V p·Vp)2p-1l/3 dT + ... 32. where the coefcients Co, C 2 , etc, could in principle depend on N. The natural zero-order term is the Thomas-Fermi result 33. Perhaps the most natural second-order term is the correction suggested long ago by von Weizsacker (70)_ . Tw[pJ = � f V p � vp dr. 34. Note that this gives the whole correct T for a one-electron system or a two­ electron Hartree-Fock system. Note also that if included in a functional, this confers (if not interfered with by other terms) on oT/op the correct behavior near an atomic nucleus (cusp condition) and far from all atomic nuclei (long-range behavior) (71-73). Macke (74) in fact derived 35. but neglected a term of the same importance as Tw (12). This formula gives too large values of T . Systematic empirical studies on both atoms and molecules (75-78) give rather 36. where f � 0.20, a value which has very recently been theoretically justifed (79). This equation gives oT/ip = (iTo/ip)+f(iTw/ip); it is reasonable though generally impractical to treat f as a quantity that depends on position (71, 80). Using the Thomas-Fermi problem as reference, a gradient expansion of the form of Eq. 32 has been given by Hodges (81) up to fourth order; the sixth-order terms also have been obtained (82)= The second-order term A n n u . R e v . P h y s . C h e m . 1 9 8 3 . 3 4 : 6 3 1 - 6 5 6 . D o w n l o a d e d f r o m w w w . a n n u a l r e v i e w s . o r g b y S o u t h e r n I l l i n o i s U n i v e r s i t y - C a r b o n d a l e o n 0 3 / 2 3 / 1 2 . F o r p e r s o n a l u s e o n l y . 640 PARR turns out to be (1/9)Tw• This does violence to JT/Jp for small and large distances from a nucleus. Nevertheless, calculations of the total T for atoms gives some truly remarkable results (83-86). If one inserts Hartree-Fock densities, To is low for neutral atoms but no more than 10%, To + T2 is still low but less than 1 %, and To + T2 + T4 is even better (85). T6 is infnite for an atom (82). Recently it has been argued from detailed ftting of considerable data on atoms (73), that for atoms a better form than Eq. 36 is 37. with C = 1.332 � 21/3. This formula also has been theoretically derived (87-89). This accords with an analysis by Schwinger (90), to the efect that when Tw is added to describe correctly inner shells, a piece of To must be subtracted that incorrectly describes those shells. The derivative JT/Jp obtained from Eq. 37 still is incapable of correctly giving shell structure for many-electron atoms, however (91). THE COULOMB ENRGY The direct classical Coulomb energy J[p] ofEq. 13 is not a local functional, since its functional derivative, the potential 4 due to the electron distribution, is an integral over the entire electron distribution. In fact it has been shown (68) that an attractive and compact "completely local" model alternative to the Thomas-Fermi model exists, in which J[p] is replaced by a local approximation to it, h[p] = BN2/3Jp4/3 dr. 38. The value of B that best fts Hartree-Fock J values for atoms and ions is 0.9299 (68, 92), and this value can be rationalized theoretically (68, 93). An interesting corresponding theory in momentum space has also been given (94). A gradient correction for J also has been derived and discussed (92, 95). There results the nonlocal formula 39. with B " 1.2263 and C " 0.0027. Again the numerical values of the coefcients can be rationalized. EXCHANGE ENERGY AND OTHR ENERGY COMPONENTS In considering gradient expansions of exchange and correlation components, it is important to distinguish expansions ofthe quantity K[p] ofEq. 13, which is pure potential energy, from expansions of the quantity E x c [p] of Eq. 23, which comprises both potential energy and kinetic energy. There are many papers on these expansions. The classical formula for K[p] is the local functional of Thomas-Fermi- A n n u . R e v . P h y s . C h e m . 1 9 8 3 . 3 4 : 6 3 1 - 6 5 6 . D o w n l o a d e d f r o m w w w . a n n u a l r e v i e w s . o r g b y S o u t h e r n I l l i n o i s U n i v e r s i t y - C a r b o n d a l e o n 0 3 / 2 3 / 1 2 . F o r p e r s o n a l u s e o n l y . Dirac theory, DENSITY FNCTIONAL THEORY 641 Ko[p] = CxJp4/3 d., Cx = (3/4)(3/n)1/3. 40. If one lets the coefcient depend on N, one gets the exchange energy of XI theory (5), 41. where I = I(N). The best values of I for atoms probably are those of Schwartz (96, 97). From scaling properties, the frst gradient correction to K[p] is expected to be an integral Jp -4/3V p' V p d •. When this is given an arbitrary coefcient and added to Kx,[p], this gives what is called the X,p method (98). For atoms, however, Bartolotti (99) has gven persuasive theoretical and calculational reasons to prefer a correction in which the integrand depends on r as well as p. His formula is 42. wherey = d In p/d In r,C(N) = Cx +C 2 N-2/\andD(N) = D 2 N-2/3, with C2 and D2 chosen to give correct results for a one-electron atom. Combining a gradient expansion for K[p] with one for T[p] gives a total energy functional that reproduces atomic Hartree-Fock energies very well on input of Hartree-Fock densities (100). Excellent results have been obtained similarly in calculations of the surface energy of metals (101). Successive terms in gradient expansions unfortunately tend to be propor­ tional to one another; the gradient expansion for the Hartree-Fock K[p] in fact diverges. Inequalities among successive terms have also been established (102). When it comes to Exc[p], matters are much more complicated. Just to use the local formula 43. clearly is an oversimplifcation; the scaling is not right. This is what is being done in any simple XC calculation purporting to include correlation efects. Contemporary calculations with the Kohn-Sham method use instead a formula 44. where exc(p} is the correlation energy per particle for a uniform electron gas of density p. exc being a function of p, this equation defnes what is conventionally called the local density approximation (LDA). There an: various ways to handle exc (2, 4, 9, 103, 104); one generally employs complicated, often numerical, nearly exact expressions for it (37). Gradient A n n u . R e v . P h y s . C h e m . 1 9 8 3 . 3 4 : 6 3 1 - 6 5 6 . D o w n l o a d e d f r o m w w w . a n n u a l r e v i e w s . o r g b y S o u t h e r n I l l i n o i s U n i v e r s i t y - C a r b o n d a l e o n 0 3 / 2 3 / 1 2 . F o r p e r s o n a l u s e o n l y . 642 PARR corrections can be added, but successes with this have been limited (9, 105). For spin-density gradient expansions of impressive accuracy, see the recent work of Oliver & Perdew (106). ISSUES WITHIN THE THEORY The Problems of N Representability and v Representabilit y The Hohenberg-Kohn functional F[p] " T[p] + Ye[P] will be diferent for diferent statistics. Left implicit in the original statement of the theory (1), this point has caused confusion (107) but has since been explicitly stated (108). There is a real related problem, however, called the N represent ability problem. Consider a system of fermions. As we vary from one p to the next, in Eq. 5, for a given N, may we assume that every trial p is acceptable in the sense that it comes from an antisymmetric wave function? The answer in fact is yes, as was frst proved by Gilbert (29). He showed by explicit construction that for any p, there exists a single determinantal wave function which gives that p. Another construction is due to Harriman (109). Much more subtle is the problem of v representability, which is the question of whether trial p's always may be assumed to be associated with eigenfunctions of a Hamiltonian operator of the form of Eq. 3, with an appropriate v. This was assumed by Hohenberg & Kohn, who verifed its correctness for one limiting case (1, footnote on page B865). In general, however, the answer is now known to be negative. Early attempts to solve the problem were not conclusive (110), but recently the answer has been obtained. In a paper with the apt title "Electron Densities in Search of a Hamiltonian," Levy (111) gives a counterexample, as does also Lieb (112, 113). Fortunately, this result does not ruin the theory. Inserting even non-v­ representable p's in the Levy prescription of Eq. 7 does not hurt, because this functional achieves its absolute lower bound for a given system with the true v-representable p. As a constraint, v represent ability may be regarded as unnecessary (111). The Problem of N as a Continuous Parameter If the reason that local functionals of p for few-electron systems are useful is accounted for by the Szasz argument outlined above (69), there still can be raised another argument against use of density-functional theory for few­ electron systems. Density functional equations generally treat the para­ meter N as continuous, whereas actual atomic or molecular systems always possess an integral number of electrons. Does this invalidate use of density functional theory for such systems? A n n u . R e v . P h y s . C h e m . 1 9 8 3 . 3 4 : 6 3 1 - 6 5 6 . D o w n l o a d e d f r o m w w w . a n n u a l r e v i e w s . o r g b y S o u t h e r n I l l i n o i s U n i v e r s i t y - C a r b o n d a l e o n 0 3 / 2 3 / 1 2 . F o r p e r s o n a l u s e o n l y . DENSITY FNCTIONAL THEORY 643 No, for reasons that have been recently discussed in some detail (36, 114- 116). Data on properties as functions of N are generally smooth, which itself justifes interpolation procedures and demands development of interpolat­ ing theories. The earliest such interpolating theories were those of Thomas, Fermi & Dirac (12-14). The extensions of Kohn-Sham procedures due to Janak and others (117, 118) also qualify. Perhaps the most attractive argument is provided by the statistical mechanical view frst suggested by Gyftopoulos & Hatsopoulos (119). The average situation in a grand canonical ensemble can easily conform to a nonintegral number of electrons. This agrees with Valone's proposal that in any case the original Kohn-Hohenberg conception should be regarded from an ensemble viewpoint (120). Perdew et al (36) have proved a remarkable result for an isolated atom or molecule at 0 K: the energy E(N) as a function of the number of electrons is a broken straight line. A similar result has been obtained by Phillips & Davidson (121). More specifcally and completely, if k is an integer and N ¯ k+ w , (k ) E(k +l)-E(k-l) ' f lE(k + l)-E(k) if J + w = 2 1 E(k)-E(k-l) if In more familiar terms, we will have {-A J = -!(I +A) -I for a slightly negative species J for the neutral species for a slightly positive species 45. 46. Note that Mulliken's formula of Eq. 11 is verifed. Another interesting result is that for a combined species A • • B where A and B are noninteractirig, 47. where the min and max are of the values for the separate species A and B. Under other conditions, such as for T > 0 or such as for atoms in a molecule, diferent results will be obtained, but always = A<J< ¬ 1 + 48. There can be cases, in fact, in which the value of the chemical potential is imposed "from outside," as the case of a molecule physisorbed on a surface. Limits are imposed by Eq. 48, but otherwise there could be quite a change from one situation to the next. This could have bearing on heterogeneous A n n u . R e v . P h y s . C h e m . 1 9 8 3 . 3 4 : 6 3 1 - 6 5 6 . D o w n l o a d e d f r o m w w w . a n n u a l r e v i e w s . o r g b y S o u t h e r n I l l i n o i s U n i v e r s i t y - C a r b o n d a l e o n 0 3 / 2 3 / 1 2 . F o r p e r s o n a l u s e o n l y . 644 PARR catalysis (R. G. Parr, unpublished; Princeton University Seminar, December 13, 1982). A device helpful for separating the efect of change in N from change in density at constant N is to write (116) p = Na, Sa dv = 1. 49. The quantity a has been called a "shape factor." When this separation is elucidated, it becomes clear that in general there is an essential diference between functional derivatives of the type oQ/op for constant Nand oQ/op unrestricted. In particular, in Eq. 6 it is not (bF /bp)N but bF /op which enters-a matter that has caused confusion (122). The exact electron density for the ground state of a system of interest is not sufcient information to determine its chemical potential (116). The Problem of Well-Behavedness of Functionals Even if one accepts the existence and uniqueness of the Hohenberg-Kohn functional F[p] and other functionals we have been dealing with [and I believe one should (107, 108)], many questions remain that must be answered. Are they diferentiable with respect to p? With respect to parameters in v? Are they invertible? Do they need to be and are they convex? And so on. Early studies of highly rigorous character include those of Lieb & Simon (15, 65) on the Thomas-Fermi theory and Percus (123) on the general N­ fermion problem [anticipating in part the later work of Levy (24)]. But the defnitive study on mathematical aspects of Hohenberg-Kohn theory for Coulomb systems (which include atoms and molecules) is the new work by Lieb (112, 113). Lieb comes to constructive conclusions on many points. Support is provided for favoring Levy's principle of Eq. 7, over other formally equivalent ways to state the central stationary principle of the theory. One may remain cautiously optimistic about these mathematical matters. THREE COURSES OF THE THEORY The Molecule as a Single Thermodynamic System In this "nonhomogeneous electron gas" description of the electrons in a molecule that we have been discussing, we have what well may be thought of as a "thermodynamic" description; there are correct and useful analogies with a classical thermodynamics of such a system. The most obvious thermodynamic aspect is the chemical potential, which, as is stated above, measures the escaping tendency of electrons in just the way the chemical potential of Gibbs would do, for a macroscopic system with varying density in an external feld. A n n u . R e v . P h y s . C h e m . 1 9 8 3 . 3 4 : 6 3 1 - 6 5 6 . D o w n l o a d e d f r o m w w w . a n n u a l r e v i e w s . o r g b y S o u t h e r n I l l i n o i s U n i v e r s i t y - C a r b o n d a l e o n 0 3 / 2 3 / 1 2 . F o r p e r s o n a l u s e o n l y . DENSITY FUNCTIONAL THEORY 645 An important diference with classical thermodynamics is that the energy functional E[p] is not homogeneous of degree one in the number of .particles. If it were we would have E " Nf, which in general is not correct. There is a local pressure P that can be defned, however, as the negative one third of the trace of a classical stress tensor for a system (124). In terms of the Classical electrostatic potential due to electrons, ¢(r), and a new defned functional X[p] ¯ [Jp(tT/tp) dr -(5/3)T]-[ J p(tK/tp) dr -(4/3)K], 50. one then fnds, amohg other results, E [p] ¯ JPfo dr+ J p v dr+ (1/2) J p¢ dr - JP dr -X[p], where 10 is the chemical potential in the absence of felds. Also, 3 JP dr = 2T[p]-K[p]. 51. 52. Except for the term -X, Eq. 51 is completely classical; X measures the deviations from complete localizability, in the sense of Eqs. 29 and 30, of the functionals T[p] and K[p]. These results are for zero degrees K. An explicitly thermodynamic theory results if one considers what happens for a typical system in a grand canonical ensemble of identical systems, in equilibrium at some T > O. A remarkable result follows: the density again determines all properties, with a corresponding stationary principle for the free energy. This was frst shown by Mermin (125), using a basic inequality of Gibbs, and has since been generalized and used elsewhere in macroscopic physics, especially in the theory of surface phases (126-132). Even though the derivation is easy once Gibb's result is known, it apparently escaped discovery until the Hohenberg-Kohn-Mermin work (perhaps because of the lack of interest in nonhomogeneous systems). A summary is that there exists an equation of state even for a nonhomo­ geneous system: T and the local p everywhere determine all properties. There is a wide range of interesting and sometimes important identities and relations among partial derivatives (Maxwell relations), mandatory signs of derivatives (stability conditions), and alternative choices of independent variables (Legendre transformations) (133). These have been enumerated and discussed, both for Born-Oppenheimer and non-Born­ Oppenheimer systems (134-136). Many of the formulas are useful for describing actual physical phenomena. They also are important for assessing which of various possible stationary principles equivalent to Eq. 5 will be mathematically well behaved and computationally most feasible ( 112,113). The Single Hypersurface that is Chemistry Chemistry is what happens to atoms and molecules in the presence of external felds X, as a function of t, and so chemistry can be thought of as the A n n u . R e v . P h y s . C h e m . 1 9 8 3 . 3 4 : 6 3 1 - 6 5 6 . D o w n l o a d e d f r o m w w w . a n n u a l r e v i e w s . o r g b y S o u t h e r n I l l i n o i s U n i v e r s i t y - C a r b o n d a l e o n 0 3 / 2 3 / 1 2 . F o r p e r s o n a l u s e o n l y . 646 PARR st4dy of functionals of the form P[Z., Zp, ... ;N., Np, ... ;N;X; f], where P is some property of interest, Z., Z p, . . . , are atomic numbers (of all nuclei in the periodic table!); N�, N p, ... , are the numbers of the d i ferent nuclei present; and N is the number of electrons present. For example, one property of great interest is the total energy for all tire-independent ground states in the absence of felds; this is the hyper surface E [Z.,Zp,. ' .. ;N",Np, ... ;Nl This single function contains all of thermo- chemis t ry (at 0 K). " There is good reason to imagine all of the variables in E to be continuous if needs be, and in any case to try to codify as much information as possible by systematic exploitation of the calculus, or the fnite diference calculus, of this fu' nction. Many interesting recent studies can be viewed in this light. Particularly exciting is the success of Tal & Levy (137-141) in obtain i ng to an amazing accuracy the energies of all of the neutral atoms in the period + ¢ table by a one-parameter recursion procedure into which is input onlyihe Bohr formula for the energy of a hydrogen-like atom. Furthermore the parameter has a value that is theoretically justifable. Specifcally, what they have done is rationalize the formula EK ¯ EK-1 +so(K)-2/[so(K)EK_1]1/2, 53. where c o CK) is the noninteracting energy for the atom of atomic number K and P = 0.959906. The average percentage of error relative to Hartree­ Fock values is only 0.21. Other very recent studies in this general spirit include, for example, the works of Essen (142), Laurenzi (143), Mezey (144, 145), and Tal, Bartolotti & Bader (146-148). All studies of (l/Z) or (1/Nl/3) e xpansions are of this kind (28, 149-152); they ofen emphasize the noninteracting (v e = 0) limit, which ' in future may well replace the N = · Z = C limit as the prime reference point for the calculation of properties of real atoms (79, 153). The type of formula that is useful in such c o nnections is exemplifed by the formula for an atom (27) (iE/i Z )N/Z = (iEji N }z + (iE/i Z )N = J+(Vne/ Z ). 54. Note the appearance of the che�ical potential and the use of the Hellmann­ Feynman theorem. Connecting Wave Function Theory with Density Functional Theory Hohenberg-Kohn theory can be no more than a transcription of the usual Schrodinger theory, a renormalization of it so to speak (154). It therefore is pertinent to try to fnd the functionals F[p], Ve[P], T[p] using wave function calculations as appropriate, and to put to work on this problem A n n u . R e v . P h y s . C h e m . 1 9 8 3 . 3 4 : 6 3 1 - 6 5 6 . D o w n l o a d e d f r o m w w w . a n n u a l r e v i e w s . o r g b y S o u t h e r n I l l i n o i s U n i v e r s i t y - C a r b o n d a l e o n 0 3 / 2 3 / 1 2 . F o r p e r s o n a l u s e o n l y . DENSITY FUNCTIONAL THEORY 647 the tremendous know-how built up by the quantum chemists on how to compute wave functions. In the eforts so far on this problem many mistakes have been made, in particular in confusing constant-N calculations with non-constant-N calculations (see above). The prescription or ' Eq. 7 is a solution in principle for a given N (24). A specifc procedure for implementing this has been given by Freed & Levy (155). An early attempt to get a density functional procedure from wave functions was Macke's attempt (74) to take the high-N limit of Hartree­ Fock theory; this limit should be looked at again using the circulant transformation recently discovered (156). A program for determination, from wave functions for test cases, of the functional derivative �Ex c/ � p of Kohn-Sham theory has recently been initiated (157, 158 ) . One could hopeI to fnd a diferential equation for the Hartree-Fock density at fnite N (159, 160 ) . A procedure very recently proposed (161 ) can be considered to have achieved this. The result obtained is a set of coupled diferential equations, one of which is for the density, but which can be solved for the density starting only from a guessed density. For the (1sf(2s)2 ground-state of Be, replace the usual Is and 2s orbitals with circulant transformations of them (156), ¢ I = ( p/ 4) 1/ 2 exp( iO) , ¢2 = ( p/ 4) 1 1 2 exp { -iO). The Hartree-Fock energy functional then takes the perspicuous fon n E[p,O] (1/8) <Vp· Vp/p) + (1 /2) <pVO· V8) -Z<p/r) + (3 /4)J[p] - (1/ 2) JJp ( l ) p{ 2) cos[2 0(1)-20(2)] d 'l d'2/rI2' and the equations for determining p and 0 are the coupled equations � Vp · Vp � V 2 p � + �4(r)+ �VO· V8 8 p 2 4 P r 4 2 -} f p (2) cos [ 20(1)-28(2)] dr 2/ r 1 2 = fN+ f c cos 2 0 55. 56. 57. 1 John Platt once wrote (159): "We must . . . fnd an equation for . . . total electron density without going through the individual orbitals . . . . My guess is that there is a 'collective wave- density equation' in which the number of particles is only a parameter . . . A n n u . R e v . P h y s . C h e m . 1 9 8 3 . 3 4 : 6 3 1 - 6 5 6 . D o w n l o a d e d f r o m w w w . a n n u a l r e v i e w s . o r g b y S o u t h e r n I l l i n o i s U n i v e r s i t y - C a r b o n d a l e o n 0 3 / 2 3 / 1 2 . F o r p e r s o n a l u s e o n l y . 648 PARR and !V 2 e + !(Vp/p)Ve-! J p(2) sin[2e(I)-2e(2)J dr/r1 2 ¯ Jc sin 2e, 58. where JN and Jc are Lagrange multipliers. Guessing p, one may solve Eq. 58 for e, put the result in Eq. 57, and then determine a new p. Iterating, this is a pure density functional procedure. This result admits of generalization to arbitrary N (161). An enticing additional possibility exists, to fnd the best single determin­ autal wave function for a system such that all orbitals have the same electron density. That such orbitals should exist follows from some work of Harriman (109). They are not in general equivalent to Hartree-Fock orbitals, but the known properties of the circulant transformations of Hartree-Fock orbitals (156) suggest that they may be surprisingly close to Hartree-Fock orbitals. MISCELLANEOUS STUDIES Without pretending to be comprehensive, I now list some recent basic studies on various topics of interest. The pioneers in the application of density functionals ideas and methods to chemistry, exclusive of the XO practitioners, were Nikulin in Russia and Gordon & Kim in the US. In a series of papers beginning in 1970, these authors considered rare gas interactions, ion-ion interactions, and cor­ responding three-body efects (162-169). The essential idea of this work (which has since proved valuable elsewhere) is to take the electron density as a superposition of Hartree-Fock atomic densities, and to include in the total energy a contribution to the total energy of electron correlation, computed from formulas for the correlation energy for a uniform electron gas (as a function of density). Subsequent work has included calculation of crystal binding energies (170-172) and various static and collisional polarizabilities (173-181). Intermolecular forces also have been treated (182, 183). Inclusion of gradient corrections has been considered (184), and various diffculties with the theory (185-187). Wood & Pyper (185) conclude that the most reliable version of the method is that of Lloyd & Pugh (188, 189). Beginning with a calculation on H2 in 1976 (40), there have been a series of impressive Kohn-Sham-type density functional calculations on mole­ cules by Gunnarsson, Harris & Jones (190-197). True, arbitrary decisions have had to be made on how to determine E x c ' but as the description of E x c has improved, so have the results. Note in particular the improvement obtained (198) when the very recent Ceperley-Alder (199) accurate corre- A n n u . R e v . P h y s . C h e m . 1 9 8 3 . 3 4 : 6 3 1 - 6 5 6 . D o w n l o a d e d f r o m w w w . a n n u a l r e v i e w s . o r g b y S o u t h e r n I l l i n o i s U n i v e r s i t y - C a r b o n d a l e o n 0 3 / 2 3 / 1 2 . F o r p e r s o n a l u s e o n l y . DENSITY FUNCTIONAL THEORY 649 lation calculation for the uniform electron gas is incorporated. There seems to be no reason such methods should not be regarded as becoming competitive with traditional wave function methods for determining ground-state molecular properties. For calculations by the Xe method, the reader may refer to last year's comprehensive review in these pages (7). The electron afnities of the elements have recently been very successfully calculated (200) using the self­ interaction correction to the local spin density approximation for exchange and correlation. A number of alternative variational principles featuring the density have been given (201); these all require comparing the system of interest with another system (as does Kohn-Sham theory). Hierarchical principles involving density matrices have been derived, which ultimately should help connect density functional theory with density matrix theories (202-205). Also important for the future will be understanding of the geometry of density matrices (33, 206-209). There is need to fnd density matrix functional calculational methods that conform with Eqs. 16 and 18. The frst attempt at this failed to achieve the desired result (210). Given that the pure correlation energy is a functional of the electron density, plus the considerable, still advancing expertise of the quantum chemists in computing the density and the correlation energy, one would hope and expect that there would be much efort directed at fnding E corr[P] for atoms and molecules. So far there have been some, but not many, such studies (211-219), and the problem has not been solved. It has been shown that if electron densities constrained to be piece-wise (shell-wise) exponential are input into extremum principles of Thomas­ Fermi-Dirac-Weizsacker type for atoms, remarkably good electron den­ sities and energies can be obtained (220, 221), and other atomic properties as well. That atomic densities are indeed close to piecewise exponential (220) leads one to speculate that there is an information-theoretic approach to densities; a study aimed in that direction has appeared (222). There is no question more central to chemistry than the question of what is an atom in a molecule, and so addressing this question from density functional theory is important (25). If in molecule AB atoms A and B are in states A * and B* (in general distorted from A and B in their ground states, both with respect to energy and shape), we may require, and we should obviously require, 59. and JAB J1 J�. 60. A n n u . R e v . P h y s . C h e m . 1 9 8 3 . 3 4 : 6 3 1 - 6 5 6 . D o w n l o a d e d f r o m w w w . a n n u a l r e v i e w s . o r g b y S o u t h e r n I l l i n o i s U n i v e r s i t y - C a r b o n d a l e o n 0 3 / 2 3 / 1 2 . F o r p e r s o n a l u s e o n l y . 650 PARR Another condition is needed to fx A * and B* uniquely; we may take it to be a minimum promotional energy condition (25). The frst calculation with such a scheme has been carried out by Palke (223), the next is the elegant, similar work by Guse (224). Also of much interest in chemistry is the concept of electro negativity, about which there is a rich literature. Among recent density-functional studies of this quantity are studies of electro negativity equalization [a valid principle (25)] (225, 226), systematic calculations of electro negativity by the XI transition-state method (227-229) and by other density functional methods (230-232), and development of a useful electrostatic model (233). In one of these works (226), it was suggested that for an atom near neutrality, an exponentially decaying form be used for the electronegativity, X (N) " x(Z) exp [ -yeN -Z)]. 61. The value of y is surprisingly universal: y � 2.3; to the extent that it is universal a geometric mean principle of electronegativity equalization follows. In the local v e theory discussed above (68), the electron density turns out to be a simple function of the bare-nuclear potential, p(v). This roughly characterizes true molecular electron densities (234), and deserves intensive further study. Payne (235) has outlined a density functional theory of nonbonded interactions in molecules. Among recent modifcations of the original Thomas-Fermi ideas, one may mention the coreless Thomas-Fermi model of Ny den & Acharya (236), related works of Schwinger (90, 237-239), a "radially restricted" model (240), and an "orthogonality constrained" scheme of Nyden (241). The Thomas-Fermi atom has been solved in n dimensions (242). Insofar as the relativistic kinetic energy of a free electron gas can be expressed in terms of the density p, one can formulate a relativistic Thomas­ Fermi theory, and gradient-expansion improvements thereof. A key early paper is the work of Tomishima (243). Relativistic versions of general density functional theory and Kohn-Sham theory also exist, for which see the review of Rajagopal (11). There are many studies concerning the fact that energy components T, Y e, ve almost always are close to universal constant multiples of each other. Thomas-Fermi theory gives 7E = 3Vne " -21 ve. That the same ratios hold more generally was frst observed by Fraga (244). The cause is that the homogeneties in p represented in Eqs. 33 and 40 are generally good approximations (25, 245, 246). There are corresponding interesting impli­ cations for the electrostatic potentials at nuclei (247-250) and for the sums of orbital energies �)i (251). And there is a related homogeneity charac­ teristic of molecular energies with respect to nuclear charges (252, 253). A n n u . R e v . P h y s . C h e m . 1 9 8 3 . 3 4 : 6 3 1 - 6 5 6 . D o w n l o a d e d f r o m w w w . a n n u a l r e v i e w s . o r g b y S o u t h e r n I l l i n o i s U n i v e r s i t y - C a r b o n d a l e o n 0 3 / 2 3 / 1 2 . F o r p e r s o n a l u s e o n l y . DENSITY FUNCTIONAL THEORY 651 There are other related contributions by March and collaborators (254, 255). Of more purely theoretical interest, there has been the proof of a Hohenberg-Kohn theorem for a subspace (256, 257) and the discovery of a useful complementary variational principle for density functional theories (258). CONCLUSION It is commonplace to suppose that the theory of the electronic structure of molecules is fast becoming a closed subject, because of the magnifcent successes of the quantum chemists in solving the Schrodinger equation quantitatively (259, 260). The Hartree-Fock method sufces for most purposes and can be improved upon (261), one says, and the necessary computer programs are canned. Elsewhere I have given some of the reasons that in my opinion this attitude slights much of the subject (262, 263). The present review provides more. There is an attractive whole alternative theory, the density functional theory. And there are a whole set of alternative calculational schemes, of Xo or Kohn-Sham type. Such schemes, at least potentially, ofer hope of the same quantitative accuracy as the conventional schemes. And they promise much more as well: a perspicuity, an economy of description, a simplicity of interpretation, and a closeness to the classical ideas of structural chemistry. Much research is needed to bring density functional theory to maturity, but there will result a substantially broadened and more useful quantum chemistry. ACKNOWLEDGMENTS I have profted from prolonged discussion on density functional theory with my co-workers at the University of North Carolina, especially Dr. Libero Bartolotti, and many others. I thank Professor Mel Levy of Tulane University for helping me plan this review. Our research on this subject, and the preparation of this review, have been aided by grants from the National Institutes of Health and the National Science Foundation. The manuscript was completed while I was in residence at the Institute for Theoretical Physics, University of California at Santa Barbara. Literature Cited 1. Hohenberg, P., Kohn, W. 1 964. Phys. Rev. B 136: 86471 2. Kohn, W., Sham, L. 1. 1965. Phys. Rev. A 140: 1133-38 3. Freed, K. 1971. Ann. Rev. Phys. Cher. 2: 313-46 4. Sham, L. 1., Kohn, W. 1966. Phys. Rev. 145:561-67 A n n u . R e v . P h y s . C h e m . 1 9 8 3 . 3 4 : 6 3 1 - 6 5 6 . D o w n l o a d e d f r o m w w w . a n n u a l r e v i e w s . o r g b y S o u t h e r n I l l i n o i s U n i v e r s i t y - C a r b o n d a l e o n 0 3 / 2 3 / 1 2 . F o r p e r s o n a l u s e o n l y . 652 PARR 5. Slater, J. C. 1974. Quantum Theory of Molecules and Solids, Vol. 4. New York : McGraw-Hili. 573 pp. 6. Johnson, K. H. 1975. Ann. Rev. Phys. Cher. 26 : 39-57 7. Case, D. 1982. Ann. Rev. Phys. Cher. 33 : 1 51-71 8. Connolly, J. W. D. 1977. Serierpirical Methods of Electronic Structure Calculation, Part A , Techniques, ed. G. A. Segal, pp. 105-32. New York : Plenum. 274 pp. 9. Kohn, W., Vashishta, P. 1983. See Ref lO, pp. 79-147 10. Lundqvist, S., March, N. H., eds. 1983. Theory ofthe Inhomogeneous Electron Gas. New York : Plenum. 410 pp. 1 1. Rajagopal, A. K. 1980. Adv. Cher. Phys. 41 : 59-193 12. March, N. H. 1957. Adv. Phys. 6 : 21- 101 13. March, N. H. 1983. See Ref. 10, pp. 1-77 14. March, N. H. 1975. Self-Consistent Fields in Atoms. London : Pergamon. 233 pp. 15. Lieb, E. H. 1981. Rev. Mod. Phys. 53 : 603-41 16. Balazs, N. L. 1967. Phys. Rev. 1 56: 42- 47 1 7. Smith, V. H. Jr., Absar, L 1977. Israel J. Cher. 1 6: 87-102 18. Barzai, A. S., Deb, B. M. 1981. Rev. Mod. Phys. 53 : 95126 19. Deb, B. M. 1981. The Force Concept in Chemistry. New York : Van Nostrand Reinhold. 502 pp. 20. Coppens, P., Hall, M. B., eds. 1982. Electron Distributions and the Chemical Bond. New York : Plenum. 479 pp. 21. Bader, R F. W., Tal, Y., Anderson, S. G., Nguyen-Dang, T. T. 1980. Israel J. Cher. 19 : 8-29 22. Melnick, S. J., Whitehead, M. A. 1982. J. Cher. Phys. 76: 1433-38 23. Collard, K., Hall, G. G. 1977. Int. J. Quantum Cher. 1 2 : 623-37 24. Levy, M. 1979. Proc. Nat/. Acad. Sci. USA 76 : 6062-65 25. Parr, R. G., Donnelly, R. A., Levy, M., Palke, W. E. 1978. J. Cher. Phys. 68 : 3801-7 26. Mulliken, R S. 1934. J. Cher. Phys. 2: 782-93 27. Ray, N. K., Parr, R. G. 1980. J. Cher. Phys. 73 : 1334-39 28. March, N. H., Parr, R. G. 1980. Proc. N atl. Acad. Sci. USA 77: 6285-88 29. Gilbert, T. L. 1975. Phys. Rev. B 1 2 : 21 1 1-20 30. Berrondo, M., Goscinski, O. 1975. Int. J. Quantum Cher. Syrp. 9 : 67-73 31. Donnelly, R A., Parr, R. G. 1978. J. Cher. Phys. 69 : 4431-39 32. Donnelly, R. A. 1979. J. Cher. Phys. 71 : 2874-79 33. Lowdin, P.-L. 1955. Phys. Rev. 97 : 1474-89 34. Morrell, M. M., Parr, R. G., Levy, M. 1975. J. Cher. Phys. 62 : 549-54; Levy, M., Parr, R. G. 1976. J. Cher. Phys. 64: 2707-8 35. Davidson, E. R. 1976. Reduced Density Matrices in Quantum Chemistry. New York : Academic. 1 35 pp. 36. Perdew, J. P., Parr, R. G., Levy, M., Balduz, J. L. Jr. 1982. Phys. Rev. Lett. 49 : 1691-94 37. Van Barth, V., Hedin, L. 1972. J. Phys. C Solid State Phys. 5 : 1629-42 38. Rajagopal, A. K., Callaway, J. 1973. Phys. Rev. B 7 : 1912-19 39. Stoddart, c., March, N. H. 1971. Ann. Phys. 64 : 174-210 4. Gunnarsson, 0., Lundqvist, B. L 1978. Phys. Rev. B 1 3 : 4274-98 41. Perdew, J. P. 1979. Cher. Phys. Lett. 64: 1 27-30 42. Perdew, J. P., Zunger, A. 1981. Phys. Rev. B 23 : 5048-59 43. Gunnarsson, 0., Jones, R. O. 1981. Solid State Corrun. 37 : 249-52 4. Tseng, T. J., Hong, S. H., Whitehead, M. A. 1980. J. Phys. B At. Mol. Phys. 1 3 : 4101-9 45. Manoli, S., Whitehead, M. A. 1981. Phys. Rev. A 23 : 215(55 46. Tseng, T. J., Whitehead, M. A. 1981 . Phys. Rev. A 24: 1 6-20 47. Tseng, T. 1., Whitehead, M. A. 1981. Phys. Rev. A 24: 21-28 48. Kohn, W., Vashishta, P. 1983. See Ref 10, pp. 13(1 49. Kohn, W. , Vashishta, P. 1983. See Ref. 10, p. 101 50. Theophilou, A. K. 1979. J. Phys. C 1 2 : 5419-30 51. Valone, S. M., Capitani, J. F. 1981. Phys. Rev. A 23 : 2127-33 52. Katriel, J. 1980. J. Phys. C 13 : L375- 76 53. Bartolotti, L. J. 1981. Phys. Rev. A 24 : 1682-88 54. Bartolotti, L. J. 1982. Phys. Rev. A 26: 2243-4 55. Deb, B. M., Ghosh, S. K. 1982. J. Cher. Phys. 77 : 342-48 56. Ghosh, S. K., Deb, B. M. 1979. Cher. Phys. 71 : 295-306 57. Lundqvist, S. 1983. See Ref 10, pp. 149- 88 58. Henderson, G. A. 1981. Phys. Rev. A 23 : 19-20 59. Capitani, J. F., Nalewajski, R. F., Parr, R. G. 1982. J. Cher. Phys. 76: 568-73 60. Meron, E., Katriel,J. 1977. Phys. Lett. A 61 : 19-21 A n n u . R e v . P h y s . C h e m . 1 9 8 3 . 3 4 : 6 3 1 - 6 5 6 . D o w n l o a d e d f r o m w w w . a n n u a l r e v i e w s . o r g b y S o u t h e r n I l l i n o i s U n i v e r s i t y - C a r b o n d a l e o n 0 3 / 2 3 / 1 2 . F o r p e r s o n a l u s e o n l y . 61. Payne, P. W. 1979. J. Cher. Phys. 71 : 490-96 62. Payne, P. W. 1980. Proc. Nat!. Acad. Sci. USA 77 : 6293-97 63. Payne, P. W. 1982. Proc. Natl. Acad. Sci. USA 79 : 6391-95 64. Aashamar, K., Luke, T. M., Talman, J. D. 1980. Phys. Rev. A 1 9: 616 65. Lieb, E., Simon, B. 1977. Adv. Math. 23 : 22-1 1 6 66. Tal, Y., Levy, M. 1981. Phys. Rev. A 23 : 408-1 5 67. Katriel, J., Nyden, M. R. 1 981 . J. Cher. Phys. 74: 1221-24 68. Parr, R. G., Gadre, S. R., Bartolotti, L. J. 1 979. Proc. Nat!. Acad. Sci. USA 76 : 2522-26 69. Szasz, L., Berrios-Pagan, I., McGinn, G. 1975. Z. Naturforsch. Teil A 30: 1 51 6-34 70. Von Weizsacker, C. F. 1935. Z. Phys. 96 : 431-58 71. Goodisman, J. 1970. Phys. Rev. A I : 1 574-76 72. Tal, Y., Bader, R. F. W. 1978. Int. J. Quantum Cher. Symp. 1 2 : 1 53-68 73. Acharya, P. K., Bartolotti, L. J., Sears, S. B., Parr, R. G. 1980. Proc. Natl. Acad. Sci. USA 77 : 6978-82 74. Macke, W. 1955. Ann. Phys. 1 7 : 1-9 75. Yonei, K., Tomishima, Y. 1965. J. Phys. Soc. Jpn. 20: 1051-57 76. Yonei, K. 1971. J. Phys. Soc. Jpn. 3 1 : 882-94 77. Gross, E. K. U., Dreizler, R. M. 1979. Phys. Rev. A 20: 1798-1807 78. Toepfer, A., Gross, E. K. U., Dreizler, R. M. 1979. Phys. Rev. A 20: 1 808-15 79. Lieb, E. H., Liberman, D. A. 1982. Numerical calculation of the Thoras­ F erri-von Weizsacker function for an infnite ator without electron repulsion, Los Alaros Nat!. Lab. Rep. LA-91 86- MS, Los Alamos. 22 pp. 80. Murphy, D. R., Parr, R. G. 1 979. Cher. Phys. Lett. 60: 377-79 81 . Hodges, C. H. 1973. Can. J. Phys. 5 1 : 1428-37 82. Murphy, D. R. 1981. Phys. Rev. A 24: 1 682-88 83. Wang, W.-P., Parr, R. G., Murphy, D. R., Henderson, G. 1976. Cher. Phys. Lett. 43 : 409-12 84. Shih, C. C. 1976. Phys. Rev. A 1 4: 919- 21 85. Murphy, D. R., Wang, W.-P. 1980. J. Cher. Phys. 72 : 429-33 86. Wang, W.-P. 1980. J. Cher. Phys. 73 : 416-18 87. Gazquez, J. L., Robles, J. 1982. J. Cher. Phys. 76 : 1467-72 88. Ludena, E. V. 1982. J. Cher. Phys. 76 : 31 57-60 DENSITY FCTIONAL THEORY 653 89. Acharya, P. K. 1983. J. Cher. Phys. 78 : 2101-2 90. Schwinger, J. 1980. Phys. Rev. A 22 : 1 827-32 91. Bartolotti, L. J., Acharya, P. K. 1982. J. Cher. Phys. 77 : 4576-85 92. Bartolotti, L. J., Parr, R. G. 1981. J. Cher. Phys. 75 : 4553-55 93. Gadre, S. R., Bartolotti, L. J., Handy, N. C. 1980. J. Cher. Phys. 72 : 1034-38 94. Pathak, R. K., Panat, P. V., Gadre, S. R. 1982. Phys. Rev. A 26 : 3073-77 95. Gazquez, J. L., Robles, J. 1 981 . J. Cher. Phys. 74: 5927-28 96. Schwarz, K. 1972. Phys. Rev. B 5 : 2466- 68 97. Schwarz, K. 1974. Theor. Chir. Acta 34: 225-31 98. Herman, F., Ortenberger, I. B., Van Dyke, J. P. 1970. Int. J. Quantur Cher. Syrp. 3 : 827-46 99. Bartolotti, L. J. 1982. J. Cher. Phys. 76 : 6057-59 100. Shih, C. c., Murphy, D. R., Wang, W.­ P. 1 980. J. Cher. Phys. 73 : 1340-43 101. Sahni, V., Gruenbaum, J. 1982. Phys. Rev. B 25 : 627 5-79 102. Pathak, R. K., Gadre, S. R. 1982. Phys. Rev. A 25 : 3426-28 103. Gunnarsson, 0., Jonson, M., Lundqvist, B. I. 1979. Phys. Rev. B 20 : 3136-64 104. Von Barth, U. 1 979. Phys. Rev. A 20: 1693-703 105. Geldart, D. J. W., Rasolt, M. 1 976. Phys. Rev. B 1 3 : 1477-88 106. Oliver, G. L.. Perdew, 1. P. 1979. Phys. Rev. A 20 : 397-403 1 07. Kryachko, E. S. 1980. Int. J. Quantur Cher. 18 : 1029-35 108. Levy, M., Perdew, 1. P. 1982. Int. J. Quantur Cher. 21 : 51 1-13 109. Harriman, J. E. 1981. Phys. Rev. A 24 : 680-82 1 10. Katriel, J., Appellof, C. J., Davidson, E. R. 1981. Int. J. Quantur Cher. 1 9: 293- 300 1 1 1. Levy, M. 1982. Phys. Rev. A 26 : 1 20- 8 1 12. Lieb, E. 1982. PhYSics as Natural Philosophy: Essays in Honor ofLaszlo Tisza on His 75th Birthday, ed. A. Shimony, H. Feshbach, pp. 1 1 1-9. Cambridge : MIT Press 1 13. Lieb, E. 1983. Int. J. Quantur Cher. In press 1 14. Katriel, J., Parr, R. G., Nyden, M. R. 198 1 . J. Cher. Phys. 74 : 2397-01 1 15. P arr, R. O. 1983. In Local Density Methods, ed. J. Avery, J. P. Dahl. New York : Plenum. In press 1 16. Parr, R. G., Bartolotti, L. J. 1983. J. Phys. Cher. In press (July) A n n u . R e v . P h y s . C h e m . 1 9 8 3 . 3 4 : 6 3 1 - 6 5 6 . D o w n l o a d e d f r o m w w w . a n n u a l r e v i e w s . o r g b y S o u t h e r n I l l i n o i s U n i v e r s i t y - C a r b o n d a l e o n 0 3 / 2 3 / 1 2 . F o r p e r s o n a l u s e o n l y . 654 PARR 1 17. Janak, J. F. 1978. Phys. Rev. B 1 8 : 7165- 68 1 18. Gopinathan, M. S., Whitehead, M. A. 1980. Israel J. Cher. 19 : 209-14 1 19. Gyftopoulos, E. P., Hatsopoulos, G. N. 1968. Proc. Natl. Acad. Sci. USA 60: 786-93 120. Valone, S. M. 1 980. J. Cher. Phys. 73 : 13349 1 21 . Phillips, P., Davidson, E. R. 1983. Int. J. Quantur Cher. 23 : 185-94 122. Nguyen-Dang, T. T., Bader, R. F. W., Essen, H. 1982. Int. J. Quantur Cher. 22 : 1049-58 123. Percus, 1. K. 1978. Int. J. Quantur Cher. 1 3 : 89-124 124. Bartolotti, L. J., Parr, R. G. 1980. J. Cher. Phys. 72 : 1 593-96 125. Mermin, N. D. 1965. Phys. Rev. A 137 : 1441-43 126. Ebner, C., Saam, W. F., Stroud, D. 1976. Phys. Rev. A 1 4: 2264- 73 . 127. Saam, W. F., Ebner, C. 1977. Phys. Rev. A 1 5 : 2566-68 128. Ebner, C., Punyanita, C. 1979. Phys. Rev. A 19 : 856-65 129. Evans, R. 1979. Adv. Phys. 28 : 143¯20 130. Evans, R., Slukin, J. J. 1980. Mol. Phys. 40: 41 3-35 1 31 . Robert, M. 1982. J. Cher. Phys. 77 : 2670-72 1 32. Percus, J. K. 1981. J. Cher. Phys. 75 : 1 31 6-19 133. Parr, R. G. 1980. Horizons ofQuant ur Cheristry, ed. K. Fukui, B. Pullman, pp. 5-15. Dordrecht/BostonfLondon : Reidel. 292 pp. . 1 34. Nalewajski, R. F., Parr, R. G. 1982. J. Cher. Phys. 77 : 399-407 1 35. Nalewajski, R. F., Capitani, J. F. 1982. J. Cher. Phys. 77 : 2514-26 136. Nalewajski, R. F. 1983. J. Cher. Phys. 78 : 61 12-20 1 37. Levy, M., Tal, Y 1 980. J. Cher. Phys. 72 : 3416-17 1 38. Politzer, P., Truhlar, D. G., eds. 1981. Cherical Applications of Atoric and Molecular Electrostatic Potential. New York/London : Plenum. 472 pp. 1 39. Levy, M., Clement, S. c., Tal, Y. 1981. See Ref. 138, pp. 29-50 140. Levy, M., Tal, Y., Clement, S. C. 1982. J. Cher. Phys. 77 : 3140-7 141. Tal, Y., Levy, M. 1982. Phys. Rev. A 25 : 1 838-44 142. Essen, H. 1982. Int. J. Quantur Cher. 21 : 717¯26 143. Laurenz, B. 1981. J. Cher. Phys. 74 : 1 840-51 144. Mezey, P. G. 1982. Int. J. Quantur Cher. 22 : 101-14 145. Mezey, P. G. 1982. Theoret. Chir. Acta 62 : 133-61 146. Tal, Y, Bartolotti, L. J., Bader, R. F. W. 1982. J. Cher« Phys. 76: 463-67 147. Tal, Y., Bartolotti, L. J. 1982. J. Cher. Phys. 76 : 2558-64 . 148. Tal" Y., Bartolotti, L. J. 1982. J. Cher. Phys. 76 : 4056-62 . 149. Castro, E. A., Fernandez, F. M. 1982. J. Cher. Phys. 76 : 1 177-78 1 50. March, N. H. 1982. J. Cher. Phys. 76 : 1 869-71 1 51 . Goruganthu, R. R., Bonham, R. A. 1982. Phys. Rev. A 26 : 1¯1 1 152. Aguilera-Navarro, V. c., de Llano, M., Rojo, 0., Verardi, S. L. L. 1982. J. Cher. Phys. 77 : 61 31-33 , 1 53. Lieb, E. H. 1983. Corrun. Math. Phys. 85 : 15-25 1 54. Primas, H. 1967. Int. J. Quantur Cher. 1 : 493-51 9 1 55. Freed, K. F., Levy, M. 1982. J. Cher. Phys. 77 : 396-98 1 56. Parr, R. G., Chen, M.-B. 1981. Proc. Natl. Acad. Sci. USA 78: 1 323-26 157. Smith, D. W., Jagannathan, S., Handler, G. S. 1979. Int. J. Quantur Cher. Syrp. 13 : 103-10 1 58. Handler, G. S., Smith, D. W. 1979. Int. J. Quantur Cher. Syrp. 13 : 1 1 1-14 1 59. Platt, J. R. 1968. Letter to R. G. Parr dated October 23 (unpublished) 160. Lawes, G. P., March, N. H. 1980. Phys. Sci. 21 : 402-8 161. Nyden, M. R., Parr, R. G. 1983. J. Cher. Phys. 78 : 4044-47 1 62. Gaydaenko, V. I. , Nikulin, V. K. 1970. Cher. Phys. Lett. 7 : 36062 163. Nikulin, V. K. 1971 . Zh. Tekhn. Fiz. 41 : 41-47 (Trans\. Sov. Phys. Tech. Phys. 1 6: 28-33) 164. Nikulin, V. K., Tsarev, Y. N. 1975. Cher. Phys. 10 : 433-37 165. Gordon, R. G., Kim, Y. S. 1972. J. Cher. Phys. 56 : 3122-33 1 66. Kim, Y. S., Gordon, R. G. 1974. J. Cher. Phys. 60 : 4332-44 1 67. Kim, Y. S., Gordon, R. G. 1974. J. Cher. Phys. 61 : 1-1 6 168. Kim, Y S. 1975. Phys. Rev. A 1 1 : 796- 804 169. Kim, Y. S. 1975. Phys. Rev. A 1 1 : 80- 13 1 70. Oxtoby, D. W. , Gelbart, W. M. 1975. Phys. Rev. A 12 : 2227-29 171. LeSar, R., Gordon, R. G. 1982. Phys. Rev. B 25 : 7221-37 1 72. LeSar, R., Gordon, R. G. 1982. J. Cher. Phys. 77 : 3682-92 173. Harris, R. A., Heller, D. F., Gelbart, W. M. 1974. J. Cher. Phys. pl : 3854-55 174. Heller, D. F., Harris, R. A., Geltart, W. M. 1975. J. Cher. Phys. 62 : 1947¯53 175. Harris, R. A., Heller, D. F. 1975. J. Cher. Phys. 62 : 3601-4 A n n u . R e v . P h y s . C h e m . 1 9 8 3 . 3 4 : 6 3 1 - 6 5 6 . D o w n l o a d e d f r o m w w w . a n n u a l r e v i e w s . o r g b y S o u t h e r n I l l i n o i s U n i v e r s i t y - C a r b o n d a l e o n 0 3 / 2 3 / 1 2 . F o r p e r s o n a l u s e o n l y . 176. Harris, R. A. 1975. Cher. Phys. Lett. 33 : 495-98 177. Dxtoby, D. W., Gelbart, W. M. 1975. Mol. Phys. 29 : 1 569-76 178. Oxtoby, D. W., Gelbart, W. M. 1975. Mol. Phys. 30 : 535-7 1 79. Mahan, G. D. 1982. J. Cher. Phys. 76 : 493-97 180. Stott, M. J., Zaremba, E. 1980. Phys. Rev. A 21 : 12-23 1 81 . Stott, M. J., Zaremba, E., Zobin, D. 1982. Can. J. Phys. 60 : 210-21 1 82. Ree, F. H., Winter, N. W. 1980. J. Cher. Phys. 73 : 322-36 1 83. Van Hemert, M. c., Berns, R. M. 1982. J. Cher. Phys. 76 : 354-61 184. Shih, C. C. 1979. Mol. Phys. 38 : 1225- 32 185. Wood, C. P ., P yper, N. C. 1981. Mol. Phys. 43 : 1371-83 1 86. Brual, G. B. Jr., Rothstein, S. M. 1978. J. Cher. Phys. 69 : 1 177-83 1 87. Brual, G. B. Jr., Rothstein, S. M. 1979. Cher. Phys. Ltt. 61 : 167-70 188. Rae, A. I. M. 1973. Cher. Phys. Lett. 1 8 : 574-77 189. Lloyd, J., Pugh, D. 1977. J. Cher. Soc. Faraday Trans. 2 73 : 234-44 190. Gunnarsson, D., Harris, J., Jones, R. D. 1977. J. Cher. Phys. 67 : 3970-79 191. Harris, 1., Jones, R. D. 1978. J. Cher. Phys. 68 : 1 1 90-93 192. Harris, J., Jones, R. D. 1979. J. Cher. Phys. 70 : 830-41 193. Harris, 1., Jones, R. D. 1978. Phys. Rev. A 1 8 : 2159-66 194. Jones, R. O. 1979. J. Cher. Phys. 71 : 1 300-18 195. Jones, R. D. 1980. J. Cher. Phys. 72 : 3197-200 196. Harris, J., Jones, R. D. 1979. Phys. Rev. A 1 9: 1 81 3-18 197. Gunnarsson, D. , Jones, R. D. 1980. Phys. Scr. 21 : 394-401 198. Jones, R. O. 1982. J. Cher. Phys. 76 : 3098-3101 199. Ceperley, D. M. , Alder, B. J. 1980. Phys. Rev. Lett. 45 : 566-69 200. Cole, L. A., Perdew, J. P. 1982. Phys. Rev. A 25 : 1 265-70 201. Nakatsuji, H., Parr, R. G. 1975. J. Cher. Phys. 63 : 1 1 12-17 202. Nakatsuji, H. 1976. Phys. Rev. A 1 4: 41-50 203. Cohen, L., Frishberg, C. 1976. Phys. Rev. A 1 3 : 927-30 204. Cohen, L., Frishberg, C. 1976. J. Cher. Phys. 65 : 4234-38 205. Harrimang J. E. 1979. Phys. Rev. A 19 : 1 893-95 206. Harriman, J. E. 1978. Phys. Rev. A 1 7 : 1249-56 207. Harriman, 1. E. 1978. Phys. Rev. A 1 7: DENSITY FUCTIONAL THEORY 655 1257-68 208. Harriman, J. E. 1979. Int. J. Quantur Chem. 1 5 : 61 1-43 209. Valone, S. M. 1980. J. Cher. Phys. 73 : 4653-55 210. Donnelly, R. A. 1981. Int. J. Quantur Cher. Syrp. 1 5 : 409-41 7 21 1 . Lie, G. C., Clementi, E. 1974. J. Cher. Phys. 60: 1 275-87 212. Lie, G. c., Clementi, E. 1974. J. Cher. Phys. 60: 1288-96 213. Colle, R., Salvetti, D. 1975. Theor. Chir. Acta 37 : 329-34 214. McWeeny, R. 1976. New World of Quantum Cheristry, ed. B. Pullman, R. G. Parr, pp. 3-31 . Dordrecht/Boston/ London : Reidel. 356 pp. 21 5. McKelvey, J. M., Streitwieser, A. Jr. 1977. J. Ar. Cher. Soc. 99 : 7121-25 216. Stoll, H., Paylidou, C. M. E., Preuss, H. 1978. Theor. Chir. Acta 49 : 143-49 217. Stoll, H., Golka, Preuss, H. 1980. Theor. Chir. Acta 55 : 29-41 218. Cohen, L., Santhanam, P., Frishberg, C. 1980. Int. J. Quantur Cher. Syrp. 1 4: 143-54 219. Perdew; 1. P., McMullen, E. R. 1981. Phys. Rev. A 23 : 2785-89 220. Wang, W.-P., Parr, R. G. 1 977. Phys. Rev. A 1 6: 891-902 221. Wang, W.-P. 1982. Phys. Rev. A 25 : 2901-12 222. Sears, S. B., Parr, R. G., Dinur, U. 1980. Israel J. Cher. 19 : 165-73 223. Palke, W. E. 1980. 1. Cher. Phys. 72 : 25 1 1-14 224. Guse, M. P. 1981. J. Cher. Phys. 75 : 828-33 225. Ray, N. K., Samuels, L., Parr, R. G. 1979. J. Cher. Phys. 70: 3680-84 226. Parr, R. G., Bartolotti, L. J. 1982. J. Ar. Cher. Soc. 104 : 3801-3 227. Bartolotti, L. J., Gadre, S. R., Parr, R. G. 1980. J. Ar. Cher. Soc. 1 02 : 2945- 48 228. Sen, K. D., Schmidt, P. c., Weiss, A. 1 980. Theor. Chir. Acta 58 : 69-71 229. Sen, K. D., Schmidt, . P. c., Weiss, A: 1981. J. Cher. Phys. 75 : 1037-38 230. Alonso, 1. A., Girifalco, L. A. 1980. J. Cher. Phys. 73 : 1 31 3-19 231. Balbas, L. c., Alonso, J. A., Iniguez, M. P. 1981 . Z. Phys. A At. Nuclei 302 : 307- 310 232. BalMs, L. c., Las Heras, E., Alonso, J. A. 1982. Z. Phys. A At. Nuclei 305 : 31- 37 233. Boyd, R. J., Markus, G. E. 1981. J. Cher. Phys. 75 : 5385-88 234. Parr, R. G., Berk, A. 1981. See Ref 138, pp. 51-62 235. Payne, P. W. 1978. J. Cher. Phys. 68 : 1242-47 A n n u . R e v . P h y s . C h e m . 1 9 8 3 . 3 4 : 6 3 1 - 6 5 6 . D o w n l o a d e d f r o m w w w . a n n u a l r e v i e w s . o r g b y S o u t h e r n I l l i n o i s U n i v e r s i t y - C a r b o n d a l e o n 0 3 / 2 3 / 1 2 . F o r p e r s o n a l u s e o n l y . 656 PARR 236. Nyden, M. R., Acharya, P. K. 1981. J. Cher. Phys. 75 : 4567-71 237. Schwinger, J. 1 981 . Phys. Rev. A 24: 2353-61 238. De Raad, L. L. Jr., Schwinger, J. 1982. Phys. Rev. A 25 : 2399-2401 239. Aguilera-Navarro, v. c., De Llano, M., Rojo, 0. 1982. J. Cher. Phys. 77 : 61 31- 33 240. Kemister, G., Nordholm, S. 1982. J. Cher. Phys. 76 : 5043-50 241 . Nyden, M. R. 1983. J. Cher. Phys. 78 : 4048-51 242. Kventsel, G. F., Katriel, J. 1981 . Phys. Rev. A 24: 2299-2301 243. Tomishima, Y. 1969. Prog. Theor. Phys. 42 : 437-47 24. Fraga, S. 1 964. Theor. Chir. Acta 2 : 406-10 245. March, N. H. 1977. J. Cher. Phys. 67 : 4618-19 246. Mucci, J. F., March, N. H. 1979. J. Cher. Phys. 71 : 5270- 75 247. Politzer, P., Parr, R. G. 1974. J. Cher. Phys. 61 : 4258-62 248. Politzer, P. 1976. J. Cher. Phys. 64: 4239-40 249. Politzer, P. 1980. Israel, J. Cher. 19 : 224-32 250. Politzer, P. 1981. See Ref. 138, pp. 7-25 251. Ruedenberg, K. 1977. J. Cher. Phys. 66 : 375-76 252. Parr, R. G., Gadre, S. R. 1980. J. Cher. Phys. 72 : 3669-73 253. Fliszar, S., Foucrault, M., Beraldin, M.­ T., Bridet, J. 1981. Can. J. Cher. 59 : 1074-82 254. Mucci, J. F., March, N. H. 1979. J. Cher. Phys. 71 : 1495-97 255. Pucci, R., March, N. H. 1981 . J. Cher. Phys. 74 : 2936-37 256. Srebrenik, S. 1975. J. Cher. Phys. 65 : 2478-79 257. Riess, J., Munch, W. 1981. Theor. Chir. Acta 58 : 295-300 258. Berk, A. 1983. Phys. Rev. A 27 : 1-1 1 259. Schaefer, H. F. III. 1976. Ann. Rev. Phys. Cher. 27 : 261-90 260. Goddard, W. A. III, Harding, L. B. 1978. Ann. Rev. Phys. Cher. 29 : 363-96 261 . Bartlett, R. J. 1981 . Ann. Rev. Phys. Cher. 32 : 359-401 262. Parr, R. G. 1975. Proc. Nat!. Acad. Sci. USA 72 : 763-71 263. Parr, R. G. 1982. See Ref. 20, pp. 95-100 A n n u . R e v . P h y s . C h e m . 1 9 8 3 . 3 4 : 6 3 1 - 6 5 6 . D o w n l o a d e d f r o m w w w . a n n u a l r e v i e w s . o r g b y S o u t h e r n I l l i n o i s U n i v e r s i t y - C a r b o n d a l e o n 0 3 / 2 3 / 1 2 . F o r p e r s o n a l u s e o n l y . 632 PARR EXACT FORMULATIONS Density Theory Annu. Rev. Phys. Chem. 1983.34:631-656. Downloaded from www.annualreviews.org by Southern Illinois University - Carbondale on 03/23/12. For personal use only. There was an old model, the Thomas-Fermi model (12-15), from which originated the idea of an "electron gas." In this model all properties of a system turn out to be expressible in terms of the electron density p, the number of electrons per unit volume, as it varies through space. This fails to give an accurate description of electronic systems of chemical interest; for example, it cannot account for chemical binding (16). But it is now known to be but a first approximation to an exact description of the ground state of any system in terms of its density-the density functional description. The density can be obtained by quadrature from the exact N-particle wave function 1/1(12·· N): p(1) = NJII/I12 d'2 d'3 .. dON dsl . 1. 1/1 itself comes from solution of the Schrodinger equation, which can be expressed as a stationary principle, (){<.pI�I.p>-E[<.pI.p>-I]} � = 0, 2. 3. where the energy E is a Lagrange multiplier and � is the Hamiltonian, = Li(ft)+ LI'V(ft)+ L"<v(1lr,,v)· = Since p integrates to N, N = N[p] Jp(l) dVl = Jp dr, 4. and the operators t and l/rI'v are universal, this procedure is a way to begin with N and v and determine all properties, including p. The quantity p is of course of much interest, being directly accessible experimentally and readily visualizable-just the classical density of the electronic system. Four recent reviews of the properties of p should be mentioned (17-20), as should be the extraordinary work of Bader (21) and others (22), following Collard & Hall (23), in which the whole of structural chemistry is addressed by applying modern catastrophe theory to p and Vp. But our concern here is rather the remarkable facts proved by Hohenberg & Kohn (1), that p determines v and hence everything, and a wrong p gives an energy above the true energy. Functionals T[p], v..[p], E[p], and so on therefore exist, and there is a stationary principle analogous to Eq. 2, c5{E[p]-Jt[N[p]-N]} = O. 5. Here Jt is the so-called chemical potential; T[p] and V.e[P] are functionals that give for any ground-state density p the kinetic energy of the electrons For a change from one ground state to another.34:631-656.e[P]+F[p].elt/l).DENSITY FUNCTIONAL THEORY 633 and the electron-electron repulsion energy. 11.Carbondale on 03/23/12. one finds (25) the fundamental equation for the change in E[N. 1983. Annu. where I and A are the ionization potential and electron affinity of the species in question. Levy (24) has shown how to determine F[p] : = F[p] = min <t/lIT+ V. and the charm. in which case the value is the same everywhere-the chemical potential. oE fJ. where P[p] T[p] + v"e[P]. 27. For personal use only. 28). 6 varies from point to point through the system unless the density is the true density for the system. The right side of Eq.e[P] + v"e[P] = = v. Chem. Many properties of electronegativity follow (25. !/t=p 7.oN + fp(l)ov(l) dVl' 8. v]. 5 is fJ. the minimum of all these is F[ p].. of density functional theory. The Euler equation cor­ responding to Eq. Compute for all antisymmetric t/I which give p the expectation value of T + f:oe.. therein indeed lies much of the challenge. 9. about which more later. . Phys. 9 and the famous electronegativity formula of Mulliken (26): fJ. The electronic energy functional may be resolved into components as follows: E[p] = T[p]+ v. and p(1) = = fJ. Downloaded from www.e[P] v"e[P] = fv(l)p(l) dVl and = J[p]-K[p]. For a system of a given number of electrons (an important restriction). 10. This is not to say that the determination of F[p] is easy. Thus one has = (aE/aN)" [oE/ov (l)h· The analogy with macroscopic thermodynamics is clear. = oE/op = v+ oF/op 6.org by Southern Illinois University . where v.. Rev. 12. J[p] (1/2)ffp(l)p(2)(I/rd dVl dV2' 13. = (aE/oN)" � - -2 I+A == - XM.annualreviews. That the chemical potential is no more no less than the electronegativity concept of structural chemistry follows from Eq. = YI(1'. therefore.634 PARR For a molecule. While these equations are not so easy to implement in practice (see below). Here we concentrate on the more compact theory (25) in which Yl is represented in terms of natural spin orbitals and their occupation numbers (33): Lknkl/t:(l')l/tk(I). Also. 1983. where Jl. for a complete set of the I/t k' or at worst almost a complete set. while the electron-electron repulsion energy Y. 17. and this point is vital. K here is defined as J minus Y. 35). 1).e[Yl] is not. Density Matrix Theory If the density determines all properties.annualreviews.Carbondale on 03/23/12. where the ek are eigenvalues. 20. The kinetic energy as a functional ofYl is known explicitly. the formal results are highly interesting. 3 1) 18. = eJnk' for all k. )J.e. so does the first-order density matrix. < 1. and that has been done (25. [Note that this is not wave function theory restated as a problem in density matrices. 16. using functionals ofY1 rather than of p =Yl(l. Annu. For personal use only. separate story. Chem. Functionals of p become functionals of the I/tk and the nk. They depend critically on . A whole formal theory can be constructed.org by Southern Illinois University . the nuclear-nuclear repulsion must be added to give the total energy. 14. then gives (25. the latter. is the chemical potential of Eqs. however. 5 and 6. and also Gk Ft/lk = 19.34:631-656.] The most general such theory uses an arbitrary representation OfYl and has many interesting aspects (29-31). the t/lk satisfy equations of the form t/l k. One expects (34. Rev. Phys. it includes correlation as well as exchange effects. which is a whole big. 29-32). Downloaded from www. 1) 0< nk 15. Minimization of the energy B[Yl] with respect to the nk and t/lk' subject to orthonormalization conditions. t!xists by the Hohenberg-Kohn theorem. where ¢ is the classical electrostatic potential due to the whole electron distribution. a noninteracting (Yee = 0) system with density p would have a kinetic energy T.[p] = L <¢kl. the entire physical meaning of the ¢k is contained in Eq. 36). FKS¢k B� ¢k· = = 22. 19 is.T. and even its meaning is tricky to establish (9. None of the eigenvalues e�s is expected to have definite physical significance except the highest. but a density functional analogue thereof (32). As far as the density is concerned. N 2 1.DENSITY FUNCTIONAL THEORY 635 Eq. Kohn & Sham (2) introduced a quite different method for evaluating T[p] : separate it into a major part that is known and a minor correction. since FKS depends on p. namely N one-electron equations of the form S 25. Rev. 24.(1/2)V21¢k> k +J[p]+ Exc[p] 'Tg[p]+ Yne[P] T[p] . . The "Kohn-Sham orbitals" ¢k have densities that sum to the exact density. however.[p] -K[p]. though in practice one must resort to approximations for Exc. 1983. The method is in principle exact. in place of Eq. 24. Kohn-Sham Theory Annu. Phys. The verbal statement is that all natural spin orbitals have the same chemical potential or electronegativity. and one may then write. The exact wave function is not this determinant. Chem.Carbondale on 03/23/12. Downloaded from www. it is as if the wave function were a single Slater determinant built from the ¢k. == Here FKS = -(1/2)V2+ v+¢+(JExc/Jp ) 26.org by Southern Illinois University . Equation 19 is not the Lowdin equation (33) for natural spin orbitals. E[p] where Exc[p] Also. These equations must be solved self-consistently. The L6wdin equation in fact is not diagonal in the natural spin orbitals. Minimization of the energy now leads to the Kohn-Sham equations for determining the orbitals ¢k.34:631-656. 16. For personal use only. whereas Eq. Specifically. 23. 12.annualreviews. 38). and Katriel (52). 56). "Density Functional theory is primarily a ground state theory" (49). SPIN DENSITY THEORY STATES Higher-energy solutions of Eq. Rigorous discussions of different possible procedures include those of Theophilou (50). and indeed for practical purposes often even for 1S states. however. For personal use only. and momen­ tum density for a ground state. Bartolotti (53) recently has derived a variational density functional theory for a time­ dependent ground state. Downloaded from www. for example. 37-40). 1 is spin free. Rev. TIME-DEPENDENT Electron density for a ground state. For states other than 1S states. but is of course a sum of spin-IX density and spin-{3 density. who also give special attention to the problem of calculation of the dynamic polarizability (56). Of particular importance is the possibility within such methods of incorporating self-energy corrections. Valone & Capitani (51). Ppl Corresponding "spin­ polarized" versions have been developed of Kohn-Sham and XIX methods (2. More work is needed in this area. and so there is not a trivial transcription of Hohenberg-Kohn MOMENTUM SPACE . there is no such simple theorem that P for an excited state determines all of its properties. E = E[Pa. Chem.636 PARR Extensions Annu. The density of Eq. The ground­ state density determines all excited state properties.annualreviews. As has been said. p(r). A comprehensive discussion of much the same ground has been given by Deb & Ghosh (55. we have. it is necessary or convenient to use a generalization of the original Hohenberg-Kohn-Sham theory that invokes similar theorems for the separate density components (37. No such wave function ortho­ gonalities exist between higher-energy solutions of Eq. are not simple Fourier transformations of each other. p(p). Phys. Such results will be essential as one sets out to describe systematically chemical processes using density functional concepts. which have been developed and exploited recently (4 1-47).org by Southern Illinois University . and then (54) a corresponding Kohn-Sham procedure. 2 are automatically orthogonal to the ground-state solution.Carbondale on 03/23/12. 1983. and excited states have in fact been variously computed by density functional methods. EXCITED PROCESSES Using an energy-minimization principle based on the hydrodynamic formulation of quantum mechanics. 5 to the ground­ state solution (except where there are symmetry differences).34:631-656. For this reason. See also the review by Lundqvist (57). density functional theory for excited states (48) is more difficult than wave function theory for them. 61). 32). 18 does not hold. However. For personal use only.org by Southern Illinois University . Hartree-Fock theory as a density functional theory is of interest. H20 is a three-component system). The Hartree-Fock Case The Hartree-Fock energy functional is a functional of the first-order density matrix. Rev. Payne (62. and so Eq. Annu.annualreviews. which rules out the possibility that Thomas-Fermi theory will govern elements actually in nature. Extensions to a phenomenological description of chemical reactions should be forthcoming. However.Carbondale on 03/23/12. Aashamar et al (64) have determined the exact Kohn-Sham orbitals and effective potential for Hartree-Fock atoms having Z :s. NON-BORN-OPPENHEIMER THEORY Among the extensions of Hohenberg­ Kohn theory to muiticomponent systems (9) is the extension to the molecule considered as a system of several components: the electrons plus nuclei of each different type of nucleus (e. there are two respects in which the usual Hartree-Fock theory falls short of being a full implementati�n of density functional ideas. 54. and produced an interest­ ing Hartree-Fock calculational scheme focusing on the density.g. This theory has been written out (59). The Exact Thomas-Fermi Limit A fundamental advance was made when Lieb & Simon (65) established that in the limit of high N and Z. Nevertheless. 16 is not satisfied. 1983. 67). N is not generally treated as a variable.34:631-656. The orbitals are very close to Hartree-Fock orbitals. The "hydrogenic" limit is much clo ser in reality the v'e == 0 model (66. Following up on this result. Phys. The Hartree-Fock density (known to be very close to the true density) determines the Hartree-Fock potential and all properties for the ground state (including the true density) (60. and so the Hartree-Fock model can be viewed as an approximation to the density matrix theory described above (31. which has been given by Henderson (58). certainly not as a continuous variable. one should note that N and Z "high" in this context means on the order of thousands. there is a corresponding theory in momentum space. Each component has its own chemical potential. In an important paper. This is important because it establishes all of the properties of the exact Hohenberg-Kohn functional in one important limiting situation. Second. Chem. Downloaded from www. 63) has introduced difference coordinl)tes in the Hartree-Fock density matrix. atoms conform to the original Thomas-Fermi model. in which 27. - . First.DENSITY FUNCTIONAL THEORY 637 theory into momentum space. and K[p] is negligible. However. Eq. Q[p] fq[p] dr.org by Southern Illinois University . V2p. Rev.Carbondale on 03/23/12. exactly. Q[p]. T[p] and K[p] = = Cdp5/3 dr + corrections Cxafp4/3 dr + corrections. It is a local density functional if its functional derivative with respect to the density. Consider an arbitrary given functional of the density. \ 29. the dependence of q(p) on p is forced (69). Chem. 31. however. We now come to a very important point about this subject. it does not matter ! Gradient Expansions The idea of a gradient expansion of a functional Q[p] is to write. Note. Specifically. at a point. that the Kahn-Sham "exchange-correlation" functional has potential and kinetic parts and cannot be written (except roughly) in the form ofEq. The exact Hohenberg-Kohn functionals T[p] and F[p] are nonlocal. the "corrections" in the above equations are zero and the constants are those of Thomas-Fermi-Dirac theory." Small N or large N.34:631-656. we may write. the integral JVp' Vp dr is nonlocal. Thus Jp dr is local. 30. For a nonuniform gas. Downloaded from www. a good approximation to it often may be provided by a local functional: while a local formula Q[p] Jq(p) dr may be incorrect. may be extremely useful. q[p] qo(p) +ql(P)Vp + q2a(p)V2p +q2b(P)Vp' VP + .annualreviews. etc. For a uniform electron gas. without any error whatsoever. 1983. the particular Q is known to have dimensions of kinetic energy or potential energy. Constants CF and CXa in these expressions can depend on N. is a function only of the density at that point (and not its derivatives or integrals) (68). the understand­ ing of which is essential for the appreciation of why formulas originally derived for the uniform electron gas work so well for such nonuniform systems as atoms and molecules. Phys. a formula = Q[p] = Jq(p) dr + corrections 28. and with N dependence in the coefficients as appropriate. with corrections as appropriate. further. and to expand q[p] into a local part plus parts that depend successively on Vp. Even though a given functional Q[p] is nonlocal.638 PARR LOCAL APPROXIMATIONS The Essence of Localization Annu. = = . . JQ/Jp. For personal use only. 30. and that does not at all require that they be "statistical. there is no reason not to hope to use these same equations. What is needed only is that the functionals in question are to a certain degree local. If. . .annualreviews. Using the Thomas-Fermi problem as reference. The 2 natural zero-order term is the Thomas-Fermi result 33. Macke (74) in fact derived 35.Carbondale on 03/23/12. This formula gives too large values of T. a truncated form may give good numerical values of Q but poor values of oQ/op.org by Southern Illinois University . but neglected a term of the same importance as Tw (12). . ql(P). KINETIC ENERGY Using Szasz' results (69).34:631-656. the sixth-order terms also have been obtained (82) The second-order term . Note that this gives the whole correct T for a one-electron system or a two­ electron Hartree-Fock system. 1983. this confers (if not interfered with by other terms) on o T/op the correct behavior near an atomic nucleus (cusp condition) and far from all atomic nuclei (long-range behavior) (71-73).. 32. a value which has very recently been theoretically justified (79). Phys. etc. For personal use only. Systematic empirical studies on both atoms and molecules (75-78) give rather 36. 2 f vp•vp p dr +C4af(V2p)2p. Note also that if included in a functional. a gradient expansion of the kinetic energy functional must take the form T[pJ = COf p5/3 dr+C Annu. 32 has been given by Hodges (81) up to fourth order. it is reasonable though generally impractical to treat f3 as a quantity that depends on position (71. where f3 � 0. Rev. . Chem. where the coefficients Co. C . This equation gives o T/i5p = (i5To/i5p)+f3(i5Tw/i5p). could in principle depend on N. a gradient expansion of the form of Eq. . may depend on the choice of a reference situation and may or may not be easy to determine. it may be useful even if it is divergent. Downloaded from www.5/3 dr +C4bf(V2p)(Vp. . 34.20.DENSITY FUNCTIONAL THEORY 639 The coefficient functions qo(p). . 80). Tw[pJ = � f Vp�vp dr. . The resultant series for Q[pJ may or may not converge. Perhaps the most natural second-order term is the correction suggested long ago by von Weizsacker (70) . Vp)p -8/3 dT +C4cf(Vp·Vp)2p-1l/3 dT+ . and To + T2 + T4 is even better (85). Recently it has been argued from detailed fitting of considerable data on atoms (73). with C = 1. THE COULOMB ENERGY h[p] = BN2/3Jp4/3 dr. 13. = = ENERGY AND OTHER ENERGY COMPONENTS In considering gradient expansions of exchange and correlation components. Chem. If one inserts Hartree-Fock densities. For personal use only. 1983.0027. which comprises both potential energy and kinetic energy. To + T2 is still low but less than 1 %.640 PARR turns out to be (1/9)Tw• This does violence to JT/Jp for small and large distances from a nucleus. In fact it has been shown (68) that an attractive and compact "completely local" model alternative to the Thomas-Fermi model exists. 38. 95). There are many papers on these expansions. An interesting corresponding theory in momentum space has also been given (94). Again the numerical values of the coefficients can be rationalized. a piece of To must be subtracted that incorrectly describes those shells. to the effect that when Tw is added to describe correctly inner shells.Carbondale on 03/23/12. with B 1. A gradient correction for J also has been derived and discussed (92. it is important to distinguish expansions ofthe quantity K[p] ofEq. 36 is 37. This accords with an analysis by Schwinger (90). 13 is not a local functional. Rev. since its functional derivative. the potential 4J due to the electron distribution. To is low for neutral atoms but no more than 10%. Annu.2263 and C 0. Nevertheless. T6 is infinite for an atom (82). from expansions of the quantity Exc[p] of Eq. Phys. which is pure potential energy. The derivative JT/Jp obtained fromEq. The direct classical Coulomb energy J[p] ofEq. that for atoms a better form than Eq. The classical formula for K[p] is the local functional of Thomas-FermiEXCHANGE . in which J[p] is replaced by a local approximation to it.34:631-656.annualreviews. 37 still is incapable of correctly giving shell structure for many-electron atoms.9299 (68. Downloaded from www. 92). 93). however (91). calculations of the total T for atoms gives some truly remarkable results (83-86). This formula also has been theoretically derived (87-89). The value of B that best fits Hartree-Fock J values for atoms and ions is 0. and this value can be rationalized theoretically (68. There results the nonlocal formula 39.332 � 2 1 / 3 . is an integral over the entire electron distribution. 23.org by Southern Illinois University . this equation defines what is conventionally called the local density approximation (LDA). . Successive terms in gradient expansions unfortunately tend to be propor­ tional to one another. There an: various ways to handle exc (2. When this is given an arbitrary coefficient and added to Kx. Excellent results have been obtained similarly in calculations of the surface energy of metals (101). matters are much more complicated. the scaling is not right. 104). where exc(p} is the correlation energy per particle for a uniform electron gas of density p. From scaling properties. with C2 and D2 chosen to give correct results for a one-electron atom. Chem. Bartolotti (99) has given persuasive theoretical and calculational reasons to prefer a correction in which the integrand depends on r as well as p. 9. clearly is an oversimplification. When it comes to Exc[p].org by Southern Illinois University .34:631-656.. Gradient Annu.p method (98).annualreviews. For personal use only.. 103. 1983. exc being a function of p. This is what is being done in any simple XC( calculation purporting to include correlation effects.[p]. where IX = IX(N). nearly exact expressions for it (37). the first gradient correction to K[p] is expected to be an integral Jp -4/3Vp' Vp d•. Just to use the local formula 43. 41.DENSITY FUNCTIONAL THEORY 641 Dirac theory. Combining a gradient expansion for K[p] with one for T[p] gives a total energy functional that reproduces atomic Hartree-Fock energies very well on input of Hartree-Fock densities (100). 4. 97). If one lets the coefficient depend on N. His formula is 42. Cx = (3/4)(3/n)1/3. The best values of IX for atoms probably are those of Schwartz (96.C(N) = Cx + C2N-2/\andD(N) = D2N-2/3. Inequalities among successive terms have also been established (102). Contemporary calculations with the Kohn-Sham method use instead a formula 44. one generally employs complicated. wherey = d In p/d In r. the gradient expansion for the Hartree-Fock K[p] in fact diverges. Downloaded from www. one gets the exchange energy of XIX theory (5). For atoms. Rev. 40.. often numerical. Ko[p] = CxJp4/3 d.Carbondale on 03/23/12. Phys. this gives what is called the X. however. Chem. but successes with this have been limited (9.annualreviews. Density functional equations generally treat the para­ meter N as continuous. with an appropriate v. this point has caused confusion (107) but has since been explicitly stated (108). because this functional achieves its absolute lower bound for a given system with the true v-representable p. He showed by explicit construction that for any p." Levy (111) gives a counterexample. As a constraint. In general. however. Left implicit in the original statement of the theory (1). For personal use only. Much more subtle is the problem of v representability. Another construction is due to Harriman (109). but recently the answer has been obtained. may we assume that every trial p is acceptable in the sense that it comes from an antisymmetric wave function? The answer in fact is yes. Consider a system of fermions. there still can be raised another argument against use of density-functional theory for few­ electron systems. which is the question of whether trial p's always may be assumed to be associated with eigenfunctions of a Hamiltonian operator of the form of Eq. 7 does not hurt. In a paper with the apt title "Electron Densities in Search of a Hamiltonian. footnote on page B865). v representability may be regarded as unnecessary (111). This was assumed by Hohenberg & Kohn. The Problem of N as a Continuous Parameter If the reason that local functionals of p for few-electron systems are useful is accounted for by the Szasz argument outlined above (69). Phys. The Hohenberg-Kohn functional F[p] T[p] + Yee[P] will be different for different statistics. 5. Early attempts to solve the problem were not conclusive (110). as does also Lieb (112. Does this invalidate use of density functional theory for such systems? . 113). Fortunately.642 PARR corrections can be added. as was first proved by Gilbert (29). Downloaded from www. 3. however. ISSUES WITHIN THE THEORY The Problems of N Representability and = v Representability Annu. 105). the answer is now known to be negative. As we vary from one p to the next.org by Southern Illinois University . there exists a single determinantal wave function which gives that p. who verified its correctness for one limiting case (1. in Eq. For spin-density gradient expansions of impressive accuracy. for a given N. There is a real related problem.34:631-656. 1983.Carbondale on 03/23/12. Rev. this result does not ruin the theory. Inserting even non-v­ representable p's in the Levy prescription of Eq. see the recent work of Oliver & Perdew (106). called the N representability problem. whereas actual atomic or molecular systems always possess an integral number of electrons. 1983. More specifically and completely. The earliest such interpolating theories were those of Thomas. Another interesting result is that for a combined species A B where A and B are noninteractirig. 47. The average situation in a grand canonical ensemble can easily conform to a nonintegral number of electrons. There can be cases. This agrees with Valone's proposal that in any case the original Kohn-Hohenberg conception should be regarded from an ensemble viewpoint (120). • • J 46. in which the value of the chemical potential is imposed "from outside. we will have for a slightly negative species for the neutral species for a slightly positive species Jl = {-A -I l E(k + l)-E(k) E(k + l)-E(k-l) 2 E(k)-E(k-l) if 1 ' f 45. Perdew et al (36) have proved a remarkable result for an isolated atom or molecule at 0 K: the energy E(N) as a function of the number of electrons is a broken straight line. 48. Chem. which itself justifies interpolation procedures and demands development of interpolat­ ing theories. if k is an integer and N k+w. but otherwise there could be quite a change from one situation to the next. where the min and max are of the values for the separate species A and B.Carbondale on 03/23/12. 48. Rev." as the case of a molecule physisorbed on a surface. This could have bearing on heterogeneous .34:631-656. different results will be obtained. A similar result has been obtained by Phillips & Davidson (121). 11 is verified. in fact. if -!(I + A) Note that Mulliken's formula of Eq.org by Southern Illinois University . Data on properties as functions of N are generally smooth. = Jl(k + w) = In more familiar terms. such as for T > 0 or such as for atoms in a molecule. 114116). Limits are imposed by Eq. but always - A<Jl< - 1 . Fermi & Dirac (12-14). Downloaded from www. Phys. For personal use only. 118) also qualify. for reasons that have been recently discussed in some detail (36. Perhaps the most attractive argument is provided by the statistical mechanical view first suggested by Gyftopoulos & Hatsopoulos (119). No.DENSITY FUNCTIONAL THEORY 643 Annu. The extensions of Kohn-Sham procedures due to Janak and others (117.annualreviews. Under other conditions. G. 1983. 6 it is not (bF/bp)N but bF/op which enters-a matter that has caused confusion (122). THREE COURSES OF THE THEORY The Molecule as a Single Thermodynamic System In this "nonhomogeneous electron gas" description of the electrons in a molecule that we have been discussing. Are they differentiable with respect to p? With respect to parameters in v? Are they invertible? Do they need to be and are they convex? And so on. Princeton University Seminar. The exact electron density for the ground state of a system of interest is not sufficient information to determine its chemical potential (116). Annu. 7. Chem. But the definitive study on mathematical aspects of Hohenberg-Kohn theory for Coulomb systems (which include atoms and molecules) is the new work by Lieb (112.annualreviews. 108)]. 49. The most obvious thermodynamic aspect is the chemical potential. Sa dv = 1. Support is provided for favoring Levy's principle of Eq. Downloaded from www. 1982). In particular. many questions remain that must be answered. in Eq. Phys. as is stated above. there are correct and useful analogies with a classical thermodynamics of such a system. One may remain cautiously optimistic about these mathematical matters. The quantity a has been called a "shape factor. Early studies of highly rigorous character include those of Lieb & Simon (15. A device helpful for separating the effect of change in N from change in density at constant N is to write (116) p = Na.34:631-656. The Problem of Well-Behavedness of Functionals Even if one accepts the existence and uniqueness of the Hohenberg-Kohn functional F[p] and other functionals we have been dealing with [and I believe one should (107. . which. 65) on the Thomas-Fermi theory and Percus (123) on the general N­ fermion problem [anticipating in part the later work of Levy (24)]. Lieb comes to constructive conclusions on many points. it becomes clear that in general there is an essential difference between functional derivatives of the type oQ/op for constant Nand oQ/op unrestricted. unpublished. we have what well may be thought of as a "thermodynamic" description. over other formally equivalent ways to state the central stationary principle of the theory. measures the escaping tendency of electrons in just the way the chemical potential of Gibbs would do." When this separation is elucidated.org by Southern Illinois University . December 13. for a macroscopic system with varying density in an external field. For personal use only.Carbondale on 03/23/12.644 PARR catalysis (R. 113). Parr. Rev. There is a wide range of interesting and sometimes important identities and relations among partial derivatives (Maxwell relations). An explicitly thermodynamic theory results if one considers what happens for a typical system in a grand canonical ensemble of identical systems. 52. and a new defined functional = X[p] Annu. Downloaded from www. and so chemistry can be thought of as the . especially in the theory of surface phases (126-132). There is a local pressure P that can be defined.34:631-656. The Single Hypersurface that is Chemistry Chemistry is what happens to atoms and molecules in the presence of external fields X. For personal use only. in equilibrium at some T > O. 3JP dr = 2T[p]-K[p]. 51 is completely classical. both for Born-Oppenheimer and non-Born­ Oppenheimer systems (134-136). Phys. as the negative one third of the trace of a classical stress tensor for a system (124). 51. and alternative choices of independent variables (Legendre transformations) (133). in the sense of Eqs.113). E[p] = where 110 is the chemical potential in the absence of fields. In terms of the Classical electrostatic potential due to electrons.particles. These have been enumerated and discussed. = [Jp(t5T/t5p) dr-(5/3)T]-[Jp(t5K/t5p) dr-(4/3)K]. Except for the term -X. 1983. and has since been generalized and used elsewhere in macroscopic physics. These results are for zero degrees K. Chem. using a basic inequality of Gibbs. 5 will be mathematically well behaved and computationally most feasible ( 112. ¢(r).org by Southern Illinois University . Also. A summary is that there exists an equation of state even for a nonhomo­ geneous system: T and the local p everywhere determine all properties. as a function of t. They also are important for assessing which of various possible stationary principles equivalent to Eq. amohg other results.Carbondale on 03/23/12. This was first shown by Mermin (125). it apparently escaped discovery until the Hohenberg-Kohn-Mermin work (perhaps because of the lack of interest in nonhomogeneous systems). Rev. one then finds. Eq. with a corresponding stationary principle for the free energy. Many of the formulas are useful for describing actual physical phenomena. however.DENSITY FUNCTIONAL THEORY 645 An important difference with classical thermodynamics is that the energy functional E[p] is not homogeneous of degree one in the number of . 29 and 30. X measures the deviations from complete localizability. of the functionals T[p] and K[p]. JPflo dr+Jpv dr+(1/2)Jp¢ dr-JP dr-X[p]. 50. mandatory signs of derivatives (stability conditions).annualreviews. Even though the derivation is easy once Gibb's result is known. which in general is not correct. If it were we would have E Nfl. A remarkable result follows: the density again determines all properties. N". Z. . Laurenzi (143). one property of great interest is the total energy for all tirqe-independent ground states in the absence of fields. where P is some property of interest.e = 0) limit.21.. f]. The average p'ercentage of error relative to Hartree­ Fock values is only 0. . Note the appearance of the che�ical potential and the use of the Hellmann­ Feynman theorem. . . For personal use only. are the numbers of the d ifferent nuclei present. For example. the works of Essen (142). Phys. There is good reason to imagine all of the variables in E to be continuous if needs be.N.. 149-152). 54. what they have done is rationalize the formula EK = EK-1 +so(K)-2/3[so(K)EK_1]1/2. T[p] using wave function calculations as appropriate. 145). The type of formula that is useful in such connections is exemplified by the formula for an atom (27) (iJE/iJZ)N/Z = (iJEjiJN}z +(iJE/iJZ)N = J. for example..X. . . Zp. . Rev.e[P]. of this fu'nction.l+(Vne/Z).Carbondale on 03/23/12. Np. where coCK) is the noninteracting energy for the atom of atomic number K and P = 0. Furthermore the parameter has a value that is theoretically justifiable. and to put to work on this problem .annualreviews.. are atomic numbers (of all nuclei in the periodic table!). and N is the number of electrons present. Chem. Many interesting recent studies can be viewed in this light.34:631-656. st4dy of functionals of the form P[Z. .. and in any case to try to codify as much information as possible by systematic exploitation of the calculus.. It therefore is pertinent to try to find the functionals F[p]. Mezey (144. All studies of (l/Z) or (1/Nl/3) expansions are of this kind (28. .. . N�. this is the hyper surface ' E[Z. ..Zp. 153).N. Zp. V.Nl This single function contains all of thermo" chemistry (at 0 K).org by Southern Illinois University . Downloaded from www... or the finite difference calculus. Specifically. and Tal. Other very recent studies in this general spirit include....646 PARR Annu.959906.Np. Bartolotti & Bader (146-148). Particularly exciting is the success of Tal & Levy (137-141) in obtaining to an amazing accuracy the energies of all of the neutral atoms in the period+¢ table by a one-parameter recursion procedure into which is input onlyihe Bohr formula for the energy of a hydrogen-like atom. a renormalization of it so to speak (154). 1983.. · ' which in future may well replace the N = Z = CfJ limit as the prime reference point for the calculation of properties of real atoms (79. they often emphasize the noninteracting (v. ... Np. 53.. Connecting Wave Function Theory with Density Functional Theory Hohenberg-Kohn theory can be no more than a transcription of the usual Schrodinger theory. f. but which can be solved for the density starting only from a guessed density. ' The prescription or Eq. An early attempt to get a density functional procedure from wave functions was Macke's attempt (74) to take the high-N limit of Hartree­ Fock theory.iO). Chem. . .Carbondale on 03/23/12. Downloaded from www. . For the (1sf(2s)2 ground-state of Be. ¢I = 2 (p/4)1/ exp ( iO). the tremendous know-how built up by the quantum chemists on how to compute wave functions. .1-N+ f. 1983. of the functional derivative �Exc/�p of Kohn-Sham theory has recently been initiated (157. total ele ctron density without going through the individual orbitals. in particular in confusing constant-N calculations with non-constant-N calculations (see above). replace the usual Is and 2s orbitals with circulant transformations of them (156). 7 is a solution in principle for a given N (24). A procedure very recently proposed (161) can be considered to have achieved this. . 55. A specific procedure for implementing this has been given by Freed & Levy (155). find an equation for . from wave functions for test cases. .1-c cos 20 John Platt once wrote ( 1 5 9): "We must . . One could hopeI to find a differential equation for the Hartree-Fock density at finite N (159. . The result obtained is a set of coupled differential equations. 158). 160). this limit should be looked at again using the circulant transformation recently discovered (156). Phys.annualreviews.O] = (1/8) <Vp· Vp/p) + (1/2) <pVO· V8) -Z<p/r) + (3/4)J[p] 56. My guess is that there is a 'collective wavedensity equation' in which the number of particles is only a parameter . A program for determination.DENSITY FUNCTIONAL THEORY 647 Annu. -(1/2)JJp(l)p{2) cos[20(1)-20(2)] d'l d'2/rI2' and the equations for determining p and 0 are the coupled equations � Vp 8 Vp 2 p · _ � V2 p 4 P _ � + �4>(r)+ �VO· V8 r 4 2 -} f = 1 p(2) cos [20(1)-28(2)] dr2/r12 57.34:631-656.org by Southern Illinois University . ." . Rev. ¢2 = 2 (p/4)11 exp{. In the efforts so far on this problem many mistakes have been made. one of which is for the density. The Hartree-Fock energy functional then takes the perspicuous fonn E[p. For personal use only. = 58. The essential idea of this work (which has since proved valuable elsewhere) is to take the electron density as a superposition of Hartree-Fock atomic densities. and then determine a new p. MISCELLANEOUS STUDIES Without pretending to be comprehensive. For personal use only. Inclusion of gradient corrections has been considered (184). these authors considered rare gas interactions. I now list some recent basic studies on various topics of interest. arbitrary decisions have had to be made on how to determine Exc' but as the description of Exc has improved. one may solve Eq. The pioneers in the application of density functionals ideas and methods to chemistry. but the known properties of the circulant transformations of Hartree-Fock orbitals (156) suggest that they may be surprisingly close to Hartree-Fock orbitals. 1 83). this is a pure density functional procedure. ion-ion interactions.org by Southern Illinois University . Note in particular the improvement obtained (198) when the very recent Ceperley-Alder (199) accurate corre- . exclusive of the XOC practitioners.648 PARR and !V2e+!(Vp/p)Ve-!Jp(2) sin[2e(I)-2e(2)J d-r/r12 J. That such orbitals should exist follows from some work of Harriman (109). to find the best single determin­ autal wave function for a system such that all orbitals have the same electron density. Phys. Rev.1c sin 2e. where J. Iterating. so have the results. were Nikulin in Russia and Gordon & Kim in the US. 58 for e.annualreviews. An enticing additional possibility exists. Annu.1N and J. put the result in Eq. computed from formulas for the correlation energy for a uniform electron gas (as a function of density). and to include in the total energy a contribution to the total energy of electron correlation. and cor­ responding three-body effects (1 62-169). 189). Intermolecular forces also have been treated (182. Chem. Beginning with a calculation on H2 in 1 976 (40).34:631-656.1c are Lagrange multipliers. Subsequent work has included calculation of crystal binding energies (170-172) and various static and collisional polarizabilities (173-181). and various difficulties with the theory (185-187). In a series of papers beginning in 1 970. Downloaded from www. Wood & Pyper (185) conclude that the most reliable version of the method is that of Lloyd & Pugh (188. there have been a series of impressive Kohn-Sham-type density functional calculations on mole­ cules by Gunnarsson. This result admits of generalization to arbitrary N (161). Harris & Jones (190-197). True. 1983. 57. They are not in general equivalent to Hartree-Fock orbitals.Carbondale on 03/23/12. Guessing p. A number of alternative variational principles featuring the density have been given (201). one would hope and expect that there would be much effort directed at finding Ecorr[P] for atoms and molecules. If in molecule AB atoms A and B are in states A* and B* (in general distorted from A and B in their ground states. both with respect to energy and shape). a study aimed in that direction has appeared (222). still advancing expertise of the quantum chemists in computing the density and the correlation energy. and the problem has not been solved. For personal use only.34:631-656. 16 and 18. these all require comparing the system of interest with another system (as does Kohn-Sham theory). There seems to be no reason such methods should not be regarded as becoming competitive with traditional wave function methods for determining ground-state molecular properties. 59. 221). we may require. 1983. plus the considerable. It has been shown that if electron densities constrained to be piece-wise (shell-wise) exponential are input into extremum principles of Thomas­ Fermi-Dirac-Weizsacker type for atoms. There is need to find density matrix functional calculational methods that conform with Eqs. The electron affinities of the elements have recently been very successfully calculated (200) using the self­ interaction correction to the local spin density approximation for exchange and correlation. . such studies (211-219). Jl1 = Jl�. and we should obviously require. Hierarchical principles involving density matrices have been derived. Downloaded from www. Phys. Rev.DENSITY FUNCTIONAL THEORY 649 lation calculation for the uniform electron gas is incorporated. but not many. 60. So far there have been some. and other atomic properties as well. 206-209). which ultimately should help connect density functional theory with density matrix theories (202-205). the reader may refer to last year's comprehensive review in these pages (7). For calculations by the Xex method. and so addressing this question from density functional theory is important (25).org by Southern Illinois University . There is no question more central to chemistry than the question of what is an atom in a molecule. Chem. remarkably good electron den­ sities and energies can be obtained (220.Carbondale on 03/23/12. Also important for the future will be understanding of the geometry of density matrices (33. Given that the pure correlation energy is a functional of the electron density. and JlAB = Annu. That atomic densities are indeed close to piecewise exponential (220) leads one to speculate that there is an information-theoretic approach to densities. The first attempt at this failed to achieve the desired result (210).annualreviews. This roughly characterizes true molecular electron densities (234). and gradient-expansion improvements thereof. The first calculation with such a scheme has been carried out by Palke (223). 237-239).annualreviews. the electron density turns out to be a simple function of the bare-nuclear potential. Yoe. That the same ratios hold more generally was first observed by Fraga (244). There are corresponding interesting impli­ cations for the electrostatic potentials at nuclei (247-250) and for the sums of orbital energies �)i (251). one may mention the coreless Thomas-Fermi model of Nyden & Acharya (236). we may take it to be a minimum promotional energy condition (25). p(v). 1983. a "radially restricted" model (240). it was suggested that for an atom near neutrality. In one of these works (226).3.e. Phys. Among recent modifications of the original Thomas-Fermi ideas. and deserves intensive further study. and development of a useful electrostatic model (233). 245.650 PARR Annu. The cause is that the homogeneties in p represented in Eqs. for which see the review of Rajagopal (11). 226). related works of Schwinger (90. X(N) = x(Z) exp [-yeN-Z)]. In the local v'e theory discussed above (68).org by Southern Illinois University . Downloaded from www. 253). A key early paper is the work of Tomishima (243). 246). Also of much interest in chemistry is the concept of electronegativity. v'e almost always are close to universal constant multiples of each other. 61. Chem. Payne (235) has outlined a density functional theory of nonbonded interactions in molecules. The Thomas-Fermi atom has been solved in n dimensions (242). to the extent that it is universal a geometric mean principle of electronegativity equalization follows. Relativistic versions of general density functional theory and Kohn-Sham theory also exist. one can formulate a relativistic Thomas­ Fermi theory. The value of y is surprisingly universal: y � 2. 33 and 40 are generally good approximations (25. And there is a related homogeneity charac­ teristic of molecular energies with respect to nuclear charges (252. There are many studies concerning the fact that energy components T. Among recent density-functional studies of this quantity are studies of electronegativity equalization [a valid principle (25)] (225. similar work by Guse (224). and an "orthogonality constrained" scheme of Nyden (241). the next is the elegant. = . Insofar as the relativistic kinetic energy of a free electron gas can be expressed in terms of the density p.Carbondale on 03/23/12.34:631-656. systematic calculations of electronegativity by the XIX transition-state method (227-229) and by other density functional methods (230-232). an exponentially decaying form be used for the electronegativity. Rev. For personal use only. about which there is a rich literature. Thomas-Fermi theory gives 7E = 3Vne -21 v. Another condition is needed to fix A* and B* uniquely. Elsewhere I have given some of the reasons that in my opinion this attitude slights much of the subject (262. A 140: 1133-38 1. Phys.34:631-656. but there will result a substantially broadened and more useful quantum chemistry. Downloaded from www. Much research is needed to bring density functional theory to maturity. have been aided by grants from the National Institutes of Health and the National Science Foundation. at least potentially. one says. 260). And they promise much more as well: a perspicuity. Sham. offer hope of the same quantitative accuracy as the conventional schemes. Chern. 2: 313-46 4.. It is commonplace to suppose that the theory of the electronic structure of molecules is fast becoming a closed subject. W. an economy of description. P. 255). 145:561-67 . because of the magnificent successes of the quantum chemists in solving the Schrodinger equation quantitatively (259. For personal use only.Carbondale on 03/23/12. Sham. and a closeness to the classical ideas of structural chemistry. 1 964. the density functional theory. Rev. Chem. The manuscript was completed while I was in residence at the Institute for Theoretical Physics. and the preparation of this review.. Of more purely theoretical interest. Literature Cited Rev. Rev. 1966. Kohn. University of California at Santa Barbara. Kohn. 197 1 . Kohn. and the necessary computer programs are canned. 1983. The present review provides more. 257) and the discovery of a useful complementary variational principle for density functional theories (258). 1965. K. ACKNOWLEDGMENTS I have profited from prolonged discussion on density functional theory with my co-workers at the University of North Carolina. W. Hohenberg.org by Southern Illinois University . Libero Bartolotti. Phys. There is an attractive whole alternative theory. 3. Phys. The Hartree-Fock method suffices for most purposes and can be improved upon (261). Phys. Rev.. of Xoc or Kohn-Sham type. And there are a whole set of alternative calculational schemes. Our research on this subject. Freed.DENSITY FUNCTIONAL THEORY 651 There are other related contributions by March and collaborators (254. 1. W. B 136: 864-71 2. especially Dr. L. I thank Professor Mel Levy of Tulane University for helping me plan this review. and many others. a simplicity of interpretation. Rev. Phys.annualreviews. Ann. 1. CONCLUSION Annu. 263). there has been the proof of a Hohenberg-Kohn theorem for a subspace (256. L. Such schemes. Lundqvist. 1 980. 1978. 1977. March... Phys.. R. Chern.. Phys. Phys.' Techniques. 198 1 . Meron. Rev. 9 : 67-73 3 1 . 274 pp. Stoddart. Chern. G. 1982. 573 pp. Parr. Serniernpirical Methods o f Electronic Structure Calculation.. 49 : 1 69 1-94 37. D. M.. Kohn. 410 pp. Chern.. Hall.. 68 : 3801-7 26. B. pp. J. 1983. G. B. A 23 : 19-20 59. 1 980. Slater. 77 : 342-48 56. N. R . Levy. G. Self -Consistent Fields in Atoms. Anderson. Bartolotti. Natl. Chern. Capitani.. J. T. J.... N. 1979. J. H . Phys. Phys. M. Sci. 1 1. 1980. R F. A. For personal use only. H. Phys. Rev. 1. A. H. B. Bartolotti. 1 56 : 4247 1 7. 1 3 : 41 0 1-9 45. W. Int. S.. p. Phys. Nalewajski. lO. Manoli. Adv. Segal. 1982. Rev. H. Whitehead. USA 76 : 6062-65 25. M. 1973. G. . The Force Concept in Chemistry. New York : Plenum. 1975. Phys. A 24 : 1 682-88 54. Rev. M. F.. Acad... 1977. 4. Phys. 6 : 21101 13. Phys. Kohn. J. J. Phys. Solid State Cornrnun. L. Henderson. R. P. 1 9 8 1 . B. 10. R. Ann. 1982. M. Rev. Phys. New York : Van Nostrand Rei nhold.. 233 pp. Balazs. W. J. N. A 6 1 : 19-21 Annu. 105-32. Vashishta. C 1 3 : L37576 53. 1972. 1980. Phys. R. March. Phys. Valone. J. 64 : 174-210 40. 1979. V. Perdew.. T. Parr. J. R. Phys. Int. G. Vol. USA 77: 6285-88 29. Gunnarsson.org by Southern Illinois University . Phys. 73 : 1 334-39 28. Perdew. Phys. Mol. B 1 3 : 4274-98 41. 101 50. Parr. Downloaded from www. 1983. G. J. C Solid State Phys. 1 2 : 623-37 24. 71 : 295-306 57. 1 980. J. Barnzai. pp. See Ref. J. 5 : 1629-42 38. K. M. Phys. A. Lett. G. A. Vashishta.. L. J. A 23 : 21 5(}-55 46. W. Chern.. A 23 : 2 1 27-33 52. J. Goscinski.. Phys. 36. Israel J. See Ref. 502 pp. Donnelly.Carbondale on 03/23/12. T. c. H. 19 : 8-29 22. Nat/. Whitehead. J. M. M. B. H. B. A. K. Quantum Chern. Ghosh.. Van Barth. O. Ann. S. L. J. B 1 2 : 2 1 1 1-20 30. N. Ray. P. Chern. A. P. Gilbert. M. Tseng.. F. Adv. Deb. Mod. Chern. Part A .. Melnick. 41 : 59-193 12. 1975. 76 : 1 433-38 23. 76 : 568-73 60.. 1976. R. 1981. L 1978. S. M. Rev. 1981 . A. J. pp. 7 1 : 2874-79 33. K. Chern. Syrnp. 62 : 549-54 . Zunger. D. 1983.. eds. Quantum Chern. M. Connolly. L.. A. See Ref. J. Gunnarsson. M. S. New York : McGraw-Hili. 53 : 9 5. Phys. 13(}-41 49. Morrell. J. Theory o the Inhomogeneous Electron f Gas. London : Pergamon. J. Hall. Levy. March. New York : Plenum. Chern. J. R. 33 : 1 5 1-71 8. T. W. P.. G. 198 1 . Chern. N. K. Hong. A.34:631-656. 1 979. 64 : 1 27-30 42. 1955. 10.. 1979. J.. Phys. 69 : 4431-39 32. H. J. Rajagopal. 1983. K. R. Absar. S. Chern. P. Rev. H. Phys. Phys. V. 21. P. 1982. 1975. Tseng. B At. J. 1 98 1 . 1981. Jr. 64 : 2707-8 35. Proc. Parr. 1982. Katriel. Phys. B 7 : 1 9 1 2-19 39. Phys. Israel J. Phys. Berrondo. A. Katriel. B 23 : 5048-59 43. J. Chern. R A. S. G. Lundqvist. Parr. Chern. T. L 1977. M. 10. 1967. A 26 : 2243-44 55. Theophilou. K. Rev. E. Chem. Phys. A. 1983. Donnelly. Kohn.. Ghosh. P. Phys. J. R S. E. Smith. 1976. Ann. L. A. Rev. A. Lieb. See Ref. Deb. L. Rev. Phys. Tal. 37 : 249-52 44. 9. K. Coppens. 79-147 10... 26 : 39-57 7. Jones. 1957. 0. 198 1 . 479 pp. G. Lett. Bader. W. T. 1981. C. Sci. 1 5 . 1934. E. Levy. Phys. M. Lett. Rev. See Ref. Deb. Lundqvist. N. pp. Chern. Phys. P. Levy. New York : Academic. 0. 1977. Rev. Rev. S. G. 2 : 782-93 27. Parr. Callaway. Electron Distributions and the Chemical Bond. M. S. New York : Plenum. A 24 : 1 6-20 47.. Palke. Jr. M. Rev. March. Levy.. E. 1 980. 1 6 : 87-102 18.. 1975. 20. Whitehead. O. Johnson. 1981. Hedin. Nguyen-Dang. Y. M... Parr. Chern. N. 1983. eds. Balduz. C 1 2 : 5419-30 5 1 . 10. Chern. S. Davidson. J. Reduced Density Matrices in Quantum Chemistry. Proc. 1 35 pp. Acad. H. Lowdin. R. S. Donnelly. 97 : 1 474-89 34. A. R. Rev. Rev. Mod. 1983. 1971. Quantum Theory o f Molecules and Solids. 1974. A 24 : 21-28 48.annualreviews. March. G. ed. J. Mulliken . J. N. K. 1982. Tseng. Phys. H. 1979. Phys. Whitehead. Vashishta. 14988 58. Phys. J. Chern.1 26 19.652 PARR 5. 53 : 603-41 16. W. Phys. Parr. M. Phys.. 1978. Whitehead. Case. F. Rev. R. Phys. 1975.. 1-77 14. 1 98 1 .J. J.. Deb.-L. March. Rev. Collard. Phys. Phys. Phys. Capitani. 1982. Rev. Rajagopal... 6. pp. G. M.. Perdew. R. Pathak. Jpn. Chern. Gruenbaum. Gunnarsson. Y.. Rev. 1 9 8 1 . Oliver. Int. Lab.1 5 67. Lett. D. R. Gadre. R.. Murphy.. G. A 20 : 1 808-15 79. C. Geldart. R. Murphy. 1 979. Lett. Gazquez.. 1 980. Quanturn Chern. 1979. Bartolotti. J. . 1981. Rev. J. Proc. 72 : 1034-38 94. Wang. R.DENSITY FUNCTIONAL THEORY 6 1 . W. Phys. Gadre. T. R. 73 : 1340-43 1 0 1 . Natl. Sci. L. K. D. M. Bartolotti. Int. Rev.. Katriel. Bader. Rev. Natur orsch. J.. J. F.1 1 6 66. Sahni. 80. J. Chern. 1 7 : 1-9 75. Syrnp. Chern.. Ann. M. 1 9 8 1 . A 24 : 1 682-88 83. Bartolotti. Ludena. R. Chern. R. Herman. K. 1980. 1 9 8 1 . A 22 : 1 827-32 91. Int. P.. In Local Density Methods.. Wang. PhYSics as Natural Philosophy : Essays in Honor o Laszlo f Tisza on His 75th Birthday. Phys. Avery. Parr. R.. Los Alamos. Hodges. Quanturn Chern. 0. K. U. Nat!. USA 79 : 6391-95 64.. Gazquez. 1982. P. 21 : 5 1 1-1 3 109.. Phys. Schwarz. Phys. W. W. Appellof. Phys. B 25 : 6275-79 102. 1980. J. K. J. 7 1 : 490-96 62. Parr. Parr. L. D. Proc. Int. 1982. Tal. Gadre. L. 1 2 : 1 53-68 73. Chirn. A 20 : 397--403 1 07. J. Phys. Shimony. C. A 1 9 : 6-16 65. W. Lieb. Z. J. E. M . W. In press (July) Annu. Jpn. R. G. M.. P. K. Perdew. Bartolotti. Von Weizsacker. Wang. H. Phys. E. G. J. 73 : 416-18 87. J. J. P. 60 : 377-79 8 1 . Lieb. Phys. Berrios-Pagan. J. Tomishima. 1 1 1--49. A I : 1 574-76 72. Nyden. J. W. E. Bartolotti. J. Parr. G. J . Acad. Rev. U. 1980. Phys. 1980. R. J. 653 8 9 . J.. 3 : 827--46 99. G. Chern.. R. Rev. In press 1 1 6.. Lieb. B. McGinn. Natl. Acharya.annualreviews. Goodisman. E. J. Robles. Phys. Acta 34 : 225-3 1 98. J. M . Gross. L. Int. D. M. J. Acharya. Chern. K. 1975. V. W.. K. 1982. A 24 : 680-82 1 10.. Parr.-P..org by Southern Illinois University . S. Rev. 1983. 1983... 43 : 409-12 84. Rev. L. Rev. Phys. Symp. J. K.. G. J. S. K. R. B 1 3 : 1477-88 106. A 23 : 408. 1981. USA 77 : 6293-97 63. A 26 : 1 2008 1 12. Phys. L. E. S. Soc.. 1982. J. I. Parr.. N. B 5 : 246668 97. Liberman. 1982. R. 1. R. Ortenberger.. A 20 : 1 693-703 105. Lundqvist. Z. B. Schwarz. Phys. Phys. 1979. Tal. J. L.. Handy. R . 1965. D. Rev. Luke. Panat. J. Y. B 20 : 3 1 36-64 104. 1979. D. K. 1 980. L. 74 : 2397--401 1 1 5. 72 : 429-33 86. 76 : 3 1 57-60 . Phys. Quanturn Chern. 1 974... L. S. Chern. Soc. 1 98 1 .Carbondale on 03/23/12. F. 1982. Wang. E . New York : Plenum. R. J. A. Rev. E. A 20 : 1 798-1807 78.. Math. Bartolotti. D. A 1 4 : 9 1 921 8 5 . Chem. J . Payne. Yonei. 1982. 1 976. Bartolotti. 1970.. Rev. Phys. Parr. U. 22 pp. Phys. 1979. Phys. Chern. Phys.. Payne. Pathak. Schwinger. G. 76 : 1467-72 88. A. P. 1979. V. Levy. Robles.-P. Chern. P. M . Phys. Rep. S. Parr. Van Dyke. K. Phys. Chern. Gross. 1982. Cambridge : MIT Press 1 13. 1 8 : 1029-35 108. J. LA-9 1 86MS. Teil A 30 : f 1 5 1 6-34 70. 1977. J. 3 1 : 882-94 77. 197 1 . Dreizler. Macke. R. J. Dreizler. 76 : 6057-59 1 00. Phys. Rev. C. 1 979.. Von Barth. Sci. Phys. Phys. Quanturn Chern. Murphy. Sears.. Rev. H. J . R. c. 1982. Acharya. Los Alarnos Nat!.-P.. Acad. USA 77 : 6978-82 74. Sci. M. Acad. Phys. Int. W. A 25 : 3426-28 103. M. Proc. Phys. 5 1 : 1428-37 82. 1972. J. Phys. Yonei. 1980. S. Sci. 96 : 431-58 71. J. 1. 1 98 1 . Chern. Phys. Y. Payne. Phys. R. H. Katriel. J. Henderson. G. 1976. K. J.. M. Chern. Chern. For personal use only. 1982. R. P.. L. ed.. Aashamar. O. Rev. Levy. Phys. 1 9 : 293300 1 1 1 . Chern. Jonson. Phys. Murphy. Phys. In press 1 14. F. 77 : 4576-85 92. D. Chern. J. Theor. J. J. 75 : 4553-55 93. R. USA 76 : 2522-26 69. Can.. Feshbach. Davidson. Nyden. Gadre. Phys. A 26 : 3073-77 95. 198 1 . 1983. R.. 1982. Katriel. 23 : 22.. J. Murphy. Acad. C. 1983. Phys. Phys. J. 1973. Chern. Rev. Simon. 1970. Perdew. P. B. B. J. G. Numerical calculation o the Thornas­ f Ferrni-von Weizsacker f unction f or an in finite atorn without electron repulsion.... 1978. W. Downloaded from www. J. G. 1 980. Phys. Dahl. Chern. Shih. Nat!. Rev. Szasz.. ed. 78 : 2 1 01-2 90. Proc. R. R. E. E. Phys. I. Phys. Phys. 1955. 1976.­ P. Toepfer A. 1980. 1 9 8 3 . C. Harriman. J. L. 74 : 122 1-24 68.34:631-656. Adv. 1935. Phys. R. Rev. J. V.. 74 : 5927-28 96. 1982. Rasolt.. C. P. J. Phys. Kryachko. I. Lieb. Talman. W. J. Levy. P. 20 : 1051-57 76. Quanturn Chern. P. 1 979. E. pp. J. Shih. Rev.. Quantum Chern. C. Harris. R. Kim. 198 3 . Chern. R. J. 1975. Phys. H. Phys. S. Phys. G. J. Ebner.5 1 144. 1 979. 1983. R. Geltiart. M. 1982. D. 1980. 77 : 61 3 1-33 . Chern. Chern. Math.. D. Nalewajski. E. 1982. Quanturn Chern. Ebner. 1975. M. 1979. Phys. 1976.. Chern. 1982. P. Phys. A 1 1 : 804-13 1 70. 7 6 : 463-67 147. G. Jagannathan. LeSar. J. M. P. P. 56 : 3122-33 1 66. 28 : 143 -2 00 130. A. G. Gordon. Aguilera-Navarro. 1974. 1975.. 1982. 1 55. V. Quanturn Chern. F.. M. Tsarev.. Smith. R. C. R. Int. Fernandez. . Castro. M. Mermin. J. G. Smith. Gordon.annualreviews. 1980. Handler.. 1982. 22 : 101-14 145. F. B 1 8 : 7 16568 1 1 8. eds.. Phys. 1983. Gordon. N. Acad.org by Southern Illinois University . 1982. N. Phys. 19 74. R. Phys. 1 53. Phys. Proc. D. pp. 1982. Rev. S. A 19 : 856-65 129. 10 : 433-37 1 65. Bartolotti. 77 : 3140-47 141. 72 : 3416-17 Sci. W. Rev. Rev. 1975. H. T. J. Clement. Whitehead. Chern. pp. 1981. de Llano.. J. R. B. 1982. J. 1983. R. 472 pp.. J. Phys. Evans. Adv. W.34:631-656.. Oxtoby. J. A 1 4 : 2264--7 3 . R. 1982. 19 81 .. J. Chern. R. R. L. Rev. L. A 26 : 1. C. Y. Y. Gyftopoulos. S. S. C. Chern. 1972. 21 : 7 1 7-26 143.-B.. Phys. M..654 PARR 146. Y S.. D. 76 : 4056-62 . G. D. Rev. Lett. L. Int. Gaydaenko.. Chen. G. S.. J. Quanturn Chern. Saam. Rev. Tal. J.. Israel J. . J. Ebner. R. R. G. 1982. Chern. R. Chern. F. Theoret. 23 : 1 8 5-94 122. F. Chern. J. Mezey. Gelbart. R. A. V.. Rev. Platt. W. F. M. E. K. R. J. Phys. J. Quanturn Chern. Phys. Lieb. C. Nguyen-Dang. Percus. 1975. Kim. Proc. Clement. G. G. Tal. Tal. G. Phys. USA 60 : 786-93 1 20. Parr. C. J . Percus. Punyanita. Pullman. 62 : 3601-4 1 7 1 . Phys. Kim. Nikulin. Nalewajski. 1 3 : 89-124 1 24. Chern. Mol. J. A 1 1 : 796804 169. F. Nyden. 4 1 : 41-47 (Trans\. pl : 3854-55 1 74. K. Phys. 1982. 77 : 396-98 1 56.. Tal. F. G. G. Kim. G. Slukin. 1982. J.. Harris. Phys. A. N. N. Freed. c. Int. J. J. Phys. S. R.. K. Bader. J. Chern.. Int. Rev. 127. 1980. Davidson. Nikulin. Phys. J. D. M. 74 : 25 : 1 838-44 1 38. Syrnp. Gopinathan. 60 : 4332-44 1 67. 1974. 1978. Fukui. Phys.. Natl. Chern. E. Cornrnun. Phys. S. A. Phys.. G. Kim. Phys. 6 1 : 1-1 6 1 68. S.Carbondale on 03/23/12. Quanturn Chern. 1 9 8 1 . K.. Phillips. H. Chern. Y. P. J. W. Chern. Rojo. F. M. F. Int. Quanturn Chern. Parr. Primas. J. 1982. Goruganthu. F. Int. Phys. Y. M. Heller. For personal use only. Sci. Y. 1 9 : 209-14 1 19. Horizons o Quanturn f 136. Saam. V. 1982. 1975. 1983.. R. Acad. R. 133. 77 : 2514--26 Chernical Applications o Atornic and f Molecular Electrostatic Potential. 292 pp. M. Chern. L. Phys. S. Rev. Parr 1 : 493-5 1 9 Phys. Downloaded from www. 2670-72 1 32. Gordon. R. Y. Chern. B 25 : 7221-37 1 72. W . Parr. J. Chern. Quanturn Chern.. R.. 197 1 . 1. Phys. Lawes. G. J. 1979. J. 7 : 360-62 1 63. Sci. Bader. 40 : 41 3-35 1 3 1 . Phys. Heller. 1979. Acta 62 : 1 3 3-61 Rev. 1 3 : 1 1 1-14 1 59. Gelbart. Y. Valone. J. Chern. R. W. Bartolotti. 1 39. P. Janak. Tekhn. Evans. 21 : 402-8 1 6 1 . H. Chirn. 1967. R. Sov. P.. A. R. USA 7 8 : 1 323-26 157.. c. J. 76 : 2558-64 1 5 1 . F. 85 : 15-25 1 54. Chem. Phys. J. Chern. M. 77 : Phys.1 1 152. Phys. E. 75 : 1 3 1 6-19 1 1 7. March. Nikulin. J. Quanturn Chern. Capitani.. 1970. Tech. G. Letter to R. J. Heller. Bartolotti. Int. J. Rev. 1982. W. 78 : 4044-47 1 62. Phys.. Phys. Harris. Essen. L. Verardi. S. Stroud. H. Chernistry. Hatsopoulos. 149. ed. March. Phys.. 76 : 1 177-78 1 50. Politzer. Phys. P hys. D. 76 : 1 869-71 Annu. 138. See Ref. J. J. Chern. Zh. F.. Essen. J. I. Handler. Phys. Phys. Chern. W. Parr. J. Parr. 1982. 1968. V . B. M. Chern. 1 34. M. 1965.. Nalewajski. Chern. J. dated October 23 (unpublished) 1 840.. Rev.. Robert. R. R. 1977. Laurenzi. Truhlar. 1980. Y. K. K. 78 : 61 12-20 1 37. Phys. Chern. 1 6 : 28-33) 164. 1968. 1982. 73 : 1334-49 1 2 1 . J. L. Levy. H. Phys. 1982. V. 0. D. Chern. Phys. 72 : 1 593-96 125. Phys. 77 : 3682-92 173. Tal. J. G. N.. M. T. G.. Dordrecht/BostonfLondon : Reidel. R. Y. F. Levy. 1982. A 1 5 : 2566-68 128. 148. 1981. R. A. Syrnp. Fiz. J. 1980. 5-15. K. 62 : 1 947-5 3 175. LeSar. New York/London : Plenum. Phys. Phys.. Tal" Y. 29-50 1 40. Mezey. S. Chern.. Y 1 980. Int. S. Bartolotti. F. W. 1 60. Levy. A 137 : 1441-43 126... Y. Levy. A 12 : 2227-29 .... J. Tal. Levy. Natl. 1 9 8 1 . A 142. 1 3 : 103-10 1 58. 22 : 1049-58 123. 1 980. Chern... Phys. 1978. 77 : 399-407 1 35. M. W. D. Gordon. Bonham.. 1983. Soc. Phys. W. Scr. Acta 58 : 69-71 229. Cohen. Zaremba. 72 : 25 1 1-14 224. J. Harris. 1978. 1 4 : 143-54 2 1 9. Chern. Brual.. Phys. Chirn. Faraday Trans. Mol.. J. G. 72 : 3 1 97-200 196. M . 1980. A 23 : 2785-89 220. J. R. Phys. Downloaded from www. 1978.. E.. P. Chern. Phys. S. W. M. Mol. 1 974. J. M. 1 02 : 294548 228. Rev. R. Harriman. A 19 : 1 893-95 206. G . 1980. C. 104 : 3801-3 227. 45 : 566-69 200. 1975. Theor.. Chern.. 1979. U. Pyper. Phys. J. Shih. Gunnarsson. G. Parr. 138. Rev. c. Phys. J. Phys. Brual.-P. Rev. Rae. 1 9 8 1 . Weiss. B. R. Jr. 1976. Chern. E. 1982. P. H. Salvetti. Phys. Zaremba. Acta 37 : 329-34 2 1 4. Nakatsuji. Phys. 1976. c. Phys. Harriman. 67 : 3970-79 1 9 1 . Phys. 1978.. Harris. Chern. C. D . Phys. D. J. Palke. pp. Phys. Streitwieser. Chern. 1982. J.. Lett. D. A 25 : 2901-12 222. R. Rev. E. J. McWeeny. G.. Gadre. Chern. 1976. Parr. 68 : 1 1 90-93 192. Berns. Alonso. Dxtoby. 1979. 70 : 830-41 193. E. Phys. c. A . R. J. A.. D. Rev. Wang. Samuels. Colle. M. 1979.. Chern. Phys. Acta 49 : 143-49 2 1 7. M cKelvey. J. M. Cohen. H. K. F. L... E . A.. M. J. Phys. G. 1980. 73 : 4653-55 210.. 1. A 1 6 : 89 1-902 22 1 . A 1 3 : 927-30 204. 1980. J. L. Frishberg. Chern.. E. G. W. Chern. D. Phys. J. Chern. Nakatsuji. 70 : 3680-84 226. Syrnp. 1980. Phys. L. N.. Perdew. Phys. Chem. J. c. Lett. Chern. P. Ceperley. D. J. Stott. Phys. Payne. 1 9 8 1 . Phys. D. J . A At. W.. J. 1979. Phys. 1979. 1. J.Carbondale on 03/23/12. D. R. 30 : 535-47 1 79. Jones. Chern. Cohen. J. M. 21 : 394-401 198. Preuss. Lett. 76 : 354-61 184. 1 980. J.. P. 1980. J. 1982. 75 : 828-33 225. E. Z.. 1979. Markus. R. McMullen. Quanturn Chern. Jones.. Rev.. C. Alonso. A 25 : 1 265-70 201. 1980. Rev. J. Golka.-P. 1981. D. Wang. R. Phys. Preuss. D. J. Van Hemert. E. R. A. Chern. Schmidt. Rev. 1 8 : 574-77 189. N. 1979. I. Chirn. Chern. Gelbart. Phys.. Donnelly. B.. R. . Harris. 198 1 . M. 1978.34:631-656... O. M. Stoll. Sen. R. 43 : 1371-83 1 86.. 7 1 : 1 300-18 195. D. 1 98 1 . Phys. Chern. Lloyd.. Wood. M. N. R. R. D. J . Alder. 68 : 1242-47 Annu. 63 : 1 1 12-17 202. R. Soc. Pugh. Chern. Jr. Perdew. G. E. J. Chern. Stott. J. R. Rev. Phys. Chern. Can. See Ref. M. G. Quanturn Chem. J. L.. Parr. D. S. pp. D. Phys.. J. Phys. B. M. Chern. Quanturn Chern. Rothstein. C. C. 2 73 : 234-44 190. Int. E. A. H. J. J. Phys. Stoll. Alonso. R. D. Ray. A 1 4 : 41-50 203. J. New World o f Quantum Chernistry. Chern. J. R. Cole. P. 1975. For personal use only. Int. 1982. Frishberg. Arn.. J. L . 1. 1977. Sen. Jones. 76 : 3098-3 1 0 1 1 9 9 . O. Chern. Harriman. Parr. Chern. R. C. Rothstein. R. R. L. J. 1980. Paylidou. Phys. Phys... 1980. R. 1982. . Phys. D. G. Chern. 1. Balbas. 2 1 5. A 1 9 : 1 8 1 3-18 197. Chirn. G.. Soc. C. A. Rev. Parr.. 1 9 8 1 ... 33 : 495-98 177. Phys. Arn. D. W.annualreviews. Schmidt. L. Valone. Rev. A 1 8 : 2 1 59-66 194. G. 65 : 4234-38 205. R. Phys. Jones. Nuclei 302 : 307310 232. K. Rev. M. 1973. S. Phys. 29 : 1 569-76 178. H. Soc. Frishberg. Santhanam. 1 980. Weiss. G. Phys.. Parr. Phys. Phys. W. 1980.. A. Acta 55 : 29-4 1 2 1 8. Theor. 60 : 1 288-96 213. 75 : 5385-88 234. Mahan. 1978. Iniguez. 69 : 1 1 77-83 1 87. 1975. J. Sears. E. K. Berk... Int. Nuclei 305 : 3 137 233. BalMs. G. 1975. Girifalco. A At.DENSITY FUNCTIONAL THEORY 176. Phys. 1 5 : 6 1 1-43 209. 1975. P. Phys. J. Dordrecht/Boston/ London : Reidel. 1982. B. Zobin. 75 : 1037-38 230. Ree. 1979. . W. Dinur.. Lie. H.. Mol. Jones. Phys. J. M. B. 1978. Phys. J. Jones. L. A 2 1 : 1 2-23 1 8 1 . Jones. 19 : 165-73 223. Parr. P. A 1 7 : 655 1257-68 208. 1980. 99 : 7 1 21-25 2 1 6. Chern. Jr. J. S. Israel J. J. H arriman. L. R. J. Chern. 1 5 : 409-41 7 21 1 . J.. 73 : 322-36 1 83. Lett. Harris. J. Phys.. Theor. J. 1977.. 76 : 493-97 180. Las Heras. 1978.. S. Chirn. Parr. Phys. Chern. A: 1 9 8 1 . Phys. Clementi. Rev. H.. Winter.. Lie. P. C. A. L. 1. 60 : 2 1 0-21 1 82. Pullman. Chern. W. 1982. 1982. 1. R. A. Harris. Boyd. 61 : 167-70 188. c. E. Bartolotti. Guse. Gunnarsson. J.. A. E. Theor. 1976. A 1 7 : 1 249-56 207. Chern. A. W. 1 974. c. Jones. R. D.org by Southern Illinois University . Phys. 60 : 1 275-87 212. 1981. Bartolotti. Mol. Harris. Jones. H. Oxtoby. M. 1977. Syrnp. C. Z. Phys. Chern. Rev. 51-62 235. 38 : 122532 185.. P. Arn. 1 977. 356 pp. ed. G. 3-3 1 . Gelb art. 73 : 1 3 1 3-19 231. Clementi. Israel. M. J. J. Phys. Rojo. Chern. Katriel. Tomishima. Parr. March. J. 1976. J. Chern. For personal use only. Theor. Ruedenberg. J.. R. Chern. 0. Acad. Phys. M. P. J. Phys. Rev. R. Chirn. Phys. March. N. c. Phys. 1982... Riess. 198 1 . Pucci.. Schaefer. 72 : 3669-73 253. 76 : 5043-50 241 . Phys. N. 1 964. Downloaded from www. A 25 : 2399-2401 239. Nordholm. Kventsel.Carbondale on 03/23/12. 59 : 1 074-82 254. 27 : 261-90 260. 1983.. Phys. 1976. Goddard. 138. P. Mucci. M. A 24 : 2353-61 238. Bartlett. 61 : 4258-62 248. 74 : 2936-37 256. Rev. Chern. A. 198 1 . Chem. 1983. Chern. Y. M. Phys.. Nyden. 66 : 375-76 252. Harding. Aguilera-Navarro. R. 1983. R. 75 : 4567-7 1 237. Chern. Politzer. J.34:631-656.. Chern.annualreviews. 32 : 359-401 262. Gadre. K. Phys. Can.. P. III. Phys. Phys.. J. USA 72 : 763-71 263. Phys. Rev. N. R. Ann. III. Phys. 1969. P. 1981. F. Rev. Sci. Fraga. 1979. Acta 2 : 406-. J. J. Phys. Kemister. Ann.org by Southern Illinois University . Phys. 7 1 : 5270--7 5 247. 78 : 4048-5 1 242. Politzer. P. Acta 58 : 295-300 258. 67 : 4618-19 246. J. A 27 : 1-1 1 259. 1975. Fliszar. Munch.. 1 98 1 . Nat!. Chirn. G. G. Chern. Mucci. J. 198 1 . Chern. J. De Raad. W. Phys. 20. Chern. 29 : 363-96 261 . 1977. H.. v. J. Chern. 1974.656 PARR 224-32 250. Chern. Parr. Berk. Chern. G. March. See Ref. K.. S. 198 1 . J. Foucrault. 1979. Phys. 7-25 251. 1978. G. Chern. S. J. F. Rev. S. A 24 : 2299-2301 243. Chern. Ann. N. Parr. B. Prog. 1 98 1 . L. 1980. M.­ T. 7 1 : 1495-97 255. 1982. Phys. J. 198 1 . 1982. G. J.. Chern. Nyden. pp. Phys. R. G. F. Politzer. R. 1977. Phys. R. H. F.. Schwinger. Rev. March. 95-100 Annu. Phys. Beraldin. 42 : 437-47 244. Bridet. J. S. 65 : 2478-79 257. L. Rev. H. See Ref. Phys.. W. S. H. Theor. J. J. 1975. De Llano. Jr. J. A. L. Proc. Srebrenik. Rev. Acharya. H. 1 980. pp. 64 : 4239-40 249. Schwinger. Politzer. 19 : . Chern. Chern. 77 : 6 1 3 133 240. 236. 1982.1 0 245. J. R. Parr. Theor.