Density-functional theory of the electronic structure of molecules.

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Annu. Rev. Phys. Chern. 1995.46: 701-28 Copyright © 1995 by Annual Reviews Inc. All rights reserved

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DENSITY-FUNCTIONAL THEORY OF THE ELECTRONIC Annu. Rev. Phys. Chem. 1995.46:701-728. Downloaded from www.annualreviews.org by Cornell University on 05/26/12. For personal use only.

STRUCTURE OF MOLECULES' Robert C. Parr

Department of Chemistry, University of North Carolina, Chapel Hill, North Carolina 27599 Weitao Yang

Department of Chemistry, Duke University, Durham, North Carolina 27708-0346 KEY WORDS:

molecular modeling, computational chemistry, computational physics, many-body methods, electron correlation

ABSTRACT

Recent fundamental advances in the density-functional theory of elec tronic structure are summarized. Emphasis is given to four aspects of the subject: (a) tests of functionals, (b) new methods for determining accurate exchange-correlation functionals, (c) linear scaling methods, and (d) devel opments in the description of chemical reactivity.

PREFACE The density-functional theory (dft) of the electronic structure of molecules ( 1,2) has been evolving rapidly since it was reviewed in the Annual Review ofPhysical Chemistry in 1983 (3). There have been many exciting advances, and now there are many more workers in the field. As a result, thousands of papers have been published. In the present review only certain topics are covered. We have singled out for discussion contributions to the basic theory, contributions that enhance the applicability to very large molecules, and contributions reI-

70 1 0066-426X/95/ 1 10 1-070 1$05.00

Annu. Rev. Phys. Chem. 1995.46:701-728. Downloaded from www.annualreviews.org by Cornell University on 05/26/12. For personal use only.

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evant for the understanding of chemical reactivity. Limited space is devoted to comparisons between dft calculations and more conventional calculations for particular chemical systems. Given the availability since 1989 of a number of comprehensive books on dft (4--7), most pre-1989 references have been omitted. Theoretical niceties are sacrificed in favor of increased breadth. Although we do not cover topics such as Thomas-Fermi-type dft, excited-state dft, relativistic dft, momentum-space dft, time-dependent dft, semiempirical dft, sym metry problems, numerical methods, and routine applications, there are fortunately a number of edited books (8-17) and many good reviews (e.g. 18-20) available, in addition to the rich original literature. In fact the subject has begun to enter standard textbooks in quantum chemistry (21) . Standard methods of quantum-chemical calculation have been refined to a high degree. The current considerable interest in dft stems from the computational efficiency and accuracy of dft methods. Particularly attractive also is the potential in the methods for achieving even higher efficiency for very large systems and for offering higher accuracy by design ing better functionals. Dft has already become competitively acurate; it puts less computational demand t han standard quantum-chemical methods; and it appears to have unique advantages for qualitatively describing chemical behavior . Our apologies t o the authors of the many important dft papers that are not referenced in the present review.

DENSITY�FUNCTIONAL THEORY Density-functional theory is p rincipally a theory of an atomic or molecular The system of interest has N electrons and a fixed set of nuclear positions. The nuclei give rise to an external potential, v(r), in which the electrons move and repel each other. The time-independent ground-state electronic wave function may be obtained by solution of the Schrodinger equation. The wavefunction is determined by N and v , as is the electronic energy: E[N, v]. The electron density p(r) is N times the integral at the square of the wavefunction over all electronic space and spin coordinates except the space coordinates of one. The density deter mines v (r) and N uniquely; hence E[N, v] = E[p]. P rovided p is normalized to N, E[p] is a minimum when p is the correct ground-state density . The foregoing two sentences describe the foundation stones of dft . Their simplicity is exquisite, and their power is mighty. They justify the use of the density in place of the wavefunction as the basic descriptor for an electronic system. The total electronic energy is given by electronic g round state.

703

DENSITY·FUNCTIONAL THEORY

E[p]

=

f

F[p] + v(r)p(r)dr,

1.

where the functional F[p], a universal function of p, is the sum of the kinetic energy functional T[p] and the electron-electron repulsion func tional Vee[p). The variational principle determining the density is


c5{E[p]-,uN[p]}

=

2.

0,

where ,u is a Lagrange multiplier J p(r)dr = N[p] = N. Otherwise put, ,u

=

v Cr) + c5F[p]jc5p(r)

=

constant.

for

normalization

of

p:

3.

The quantity ,u is the chemical potential of the system of interest. Differentials for any change from one ground state to another are given by

f

4.

f

5.

dE = ,udN + p(r)dv(r)dr,

and d,u

=

YfdN +

f(r)dv(r)dr.

,u, 1'/, and fer) are each of considerable chemical importance: J1 is the negative of the absolute electronegativity (22), Yf is the absolute chemical hardness (23), and fer) is the Fukui function or reactivity index (24). A factor of two included in the original definition of 1'/ is omitted here (as is now recommended). Kohn-Sham Method

The final fundamental idea in dft, the idea that made dft calculations actually feasible, is that an electron density can ordinarily be represented as the sum of squares of N orbital densities, with the orbitals themselves (more accurately, spin orbitals) defining a single Slater determinant. This idea gives rise to the Kohn-Sham method (2). The problem with Equation 3 as written is that F[p] contains two unknown pieces, T[p] and the nonclassical part of Vee[p]. There is a long history of approximating these pieces separately. Kohn & Sham proceeded differently. They introduced a non interacting reference system cor responding to each real system, a system with no electron-electron repul sion terms in its Hamiltonian but having the same ground-state electron density as the real system (but different external potential). Its kinetic

704

PARR & YANG

energy, written Ts[p], differs from T[p] by less than the Hartree-Fock ( H F) correlation energy (25). On incorporating this difference into a newly defined universal functional Exc[p], Equation 3 becomes J1 = v(r) + vjr) + vxcCr) + b Ts[p(r)]/bp(r),

6.

where vjr) is the classical potential due to per) and


vxc(r)

=

c5Exc[p(r)]/c5p(r),

7.

where Exc[p]

=

{ Vee[p]-J[p]} + { T[p]- Ts[p]}.

8.

The potential Vxc is the exchange-correlation potential; the energy Exc is the exchange-correlation energy. Equation 8 is the sum of potential-energy and kinetic-energy parts that are not easily related. An equivalent but more illuminating and useful expression for Exc is the adiabatic connection formula (26-29): 9.

where til is the many-electron wavefunction that gives the density p and minimizes the expectation value of T + A V ee, T is the kinetic-energy oper ator, Vee is the electron-electron repUlsion operator, and A is a coupling strength parameter. Equation 8 eliminates the need of dealing directly with the kinetic energy contribution to Exc and facilitates approximation. One may also write 10. where Exe> a function of the density, is the exchange correlation energy density. Equation 6 defines a separable N-fermion problem with external potential 1 1. The solution is a single determinant of N orthonormal spin orbitals, with spatial parts that are solutions of the Kohn-Sham equations [- �A 2 + VKS]o/i

=

6io/i.

12.

Also

l:;!cpl(r)1

=

per),

13.

DENSITY-FUNCTIONAL THEORY

705

where the sum is over the occupied orbitals. Self-consistent solution of Equations 12 and 13 gives orbitals and orbital energies. The highest occu pied orbital, the HOMO, has orbital energy /1, and /1 is the negative of the exact ionization potential (30, 3 1). The lowest unfilled orbital is the LUMO. The total electronic energy may be obtained from the formula E = Ts+J+ Exc.

14.


or the formula E

=

f

L/'-i-J+Exc- pvxcdr.

15.

This is the Kohn-Sham method. Various basis sets and various integration methods have been developed and tested for the solution of the Kohn-Sham equations, but these aspects are not emphasized in the present review. One may note, however, that in contrast to the nonlocal HF exchange operator, the dft operator Vxc is local and permits more convenient numerical treatment.

FOUR THEMES IN CONTEMPORARY RESEARCH Development and Test of Functionals

The realization that dft provides a powerful, competitive method for determining molecular properties resulted from two developments. First, exchange-correlation functionals have been devised that are good enough to produce accurate results. Second, much comprehensive testing has been done by computational chemists and physicists. Scores of papers in the last decade have dealt with improvement of functionals, the testing of functionals, and the specific chemical applications of functionals. Here the current state of affairs is only surveyed. Little is said about the numerical aspects; the emphasis is instead on functional development and com parisons among methods. Fewer than one hundred papers are explicitly considered, and these are discussed only briefly. DEVELOPMENT OF FUNCTIONALS The extant dft schemes can be char acterized by how they deal with the exchange-correlation energy. When the electron density is uniform, Exc[pJ is known accurately (32). This valuable information has been used in the local density approximation ( LDA) for the Exc functional (2). For a general inhomogeneous system with a density per) at r in the physical space, the LDA approximates the exchange-correlation energy density cxc[p(r)], defined in Equation 9, by the corresponding value of a homogeneous electron gas with density value equal to per). Slater's classic XC( approximation (33) takes Exc to be a


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PARR & YANG

constant times the integral of the four-thirds power of p. Despite the fact that the density in an atom or molecule is not homogeneous, LDA has been remarkably successful in structure prediction, in particular for tran sition metal complexes ( 19, 20) and for solid-state systems ( 18). LOA tends to give excessive binding energies for most types of systems. Beyond LOA, one has nonlocal density functionals. The commonly used nonlocal functionals are such that Bxc[p(r)] of Equation lOis a function of per) and its gradients (and Laplacian). Thus they are not nonlocal functionals in the most general mathematical sense. The generalized gradi ent approximation (GGA) is of this type (34, 35). The weighted density approximation (WOA) offers an example of a genuinely nonlocal func tional (36a,b). Gradient approximations are usually, but not always, derived by expansion methods starting from the LOA. Binding energies are significantly improved. The exchange-correlation energy often is separated into exchange and correlation parts (though it is important to note that this is not demanded by the theory and may in fact be counterproductive). For use in molecules, neither the exchange nor the correlation functional has to be modeled by making strong reference to uniform electron gas behavior, although exact universal functionals should satisfy the uniform-gas limit. Providing justification from the adiabatic connection formula of Equa tion 9. Becke (37) has shown that inclusion of a small percentage of the H F exchange in Exe improves energetics for many molecules. The resulting equations for the orbitals can be solved self-consistently, but the resultant approximate Vxc is not local. From Equation 8, one sees that to model Exe correctly requires modeling T - Ts as well as the nonclassical part of Vee' Scaling relations based on adiabatic connection, Equation 9, allow one of these quantities to be obtained from the other (38), and it has been shown that such use of scaling holds promise (33-37). Much else has been learned about exact properties of Vxe, e.g. long range behavior and integral properties (44-49). Thus there are rigorous implications on Vxc of the cusp conditions on density and density matrices. For example, an important ingredient of the correlation functional L YP (50), developed from the Colle-Salvetti treatment in wavefunction theory (51), is the imposition on the second-order density matrix of the proper fl2 cusp condition. When combined with an exchange formula by Becke (34), this produces the BL YP exchange-correlation functional, a functional that has been popular recently. Similarly, one has the BWP exchange correlation approximations, with the WP correlation part developed by Perdew & Wang using sophisticated GA ideas (52). Another well-founded correlation functional is by Wilson & Levy (53).



707

TESTS OF FUNCTIONALS Quantum chemistry has been and remains a prag matic subject: If a theoretical scheme works, fine; if it gives poor accuracy, discard it. So it is of the greatest importance to test, by actual computations on real systems, whether a proposed dft method (defined by the approxi mations characterizing it) is accurate enough to permit useful application. Here the standard must be, ultimately, the kilocalorie per mole accuracy required to correctly predict the small energy differences that chemistry is all about. To summarize, the many tests that have been performed lead to four conclusions: 1. Density-functional methods are competitive in accuracy with the more conventional quantum-chemical methods, that is, they approach a few kilocalories per mole accuracy. 2. To achieve the accuracy necessary for molecules, one must go to nonlocal dft methods of some kind. 3. There is as yet no clear choice as to what is the best dft calculational scheme for molecules. 4. For very large molecules dft methods have unique computational advantages. Conclusions 1 and 3 appear to conflict, but they do not. Density functional theory, as currently practiced, lacks any algorithm for sys tematic successive improvement; one knows how to take a calculation at one level to a higher level with certainty that the energy will get closer to the exact value. LDA is an internally consistent, well-defined, and solidly based model (the electron gas), but it simply is not good enough. And although certain contemporary ways to include nonlocality are doing very well, there is nothing to indicate that any particular method is the best or final one. Accordingly, research on changing and improving the methods continues, with hope but not certainty that an elegant solution will emerge. What has been done in the testing of functionals cannot be adequately summarized; the following is arbitrary at best. We emphasize contributions in which comparisons are stressed; many of the papers cited also contain considerable theoretical matter. Hundreds of good papers are not listed, especially older papers and papers dealing with the solid state. Because there is only a small difference between the electron density obtained from the H F method and that from dft, the H F density (and orbitals) alone may be used as input to the dft energy functional. This hybrid HF-dft method avoids solving the Kohn-Sham equation and is convenient for appending dft capability to conventional HF computer programs. This idea is in fact quite old (54-56). This constitutes what can be termed hybrid dft with SCF H F density as input. Results are encour aging (57-61). A more stringent test is to iterate to self-consistency. This is the original Xoc method, which was first done with the Slater-Dirac exchange. If the exact LDA is used, this is SCF LDA dft. As the rigorously correct theory

708

PARR & YANG

for a uniform electron gas, this is a beautiful, unambiguous model. Unfor tunately the results, of which many are available (62-67), reveal that for molecular binding energies corrections for nonuniformity are mandatory. Studies (40, 68-70) indicate, however, that the last word may not have been said about local theories with other than LDA form for the functionals, in particular Wigner-inspired formulas for exchange and/or correlation. Full SCF gradient-corrected nonlocal dft, then, is what is demanded. Annu. Rev. Phys. Chem. 1995.46:701-728. Downloaded from www.annualreviews.org by Cornell University on 05/26/12. For personal use only.

Tests on molecular energies and properties are very encouraging indeed

(7l-86b). Particular classes of molecules may give problems, but this is expected in view of the lack of conclusiveness in all that has been tried in the way of gradient-corrected functionals. Accuracy appears to be at the HF + MP2 level or better. Comparable accuracy and decreased computing time result (87-90) when one applies the nonlocal corrections after the SCF process is finished without them; this is nonlocal dft with SCF local dft density. Of course properties other than the energy are of widespread interest. Assessing errors in computed properties can reveal much about func tiona Is, and there are many such studies (91-105). The comprehensive study on benzene force constants by Handy et al (74) is particularly instructive. Magnetic properties require special theoretical treatment (1061 1 1). van der Waals forces still fail of adequate treatment ( 112- 1 14). Calculations on hydrogen bonding are promising (115-119), as are those on clusters (e.g. 120). The tr adit iona l interplay of co mp utationa l trial with mathematical theory-"Will it work?" with "How can I justifiably modify the theory to make it work better?"-has been important as dft has progressed. The range of studies has been broad, from careful theoretical analysis to the purely empirical, tolerating introduction of empirical constants where there is a physical basis, but hoping, and sometimes succeeding, to find parameter-free representations that are of value. The bibliography pro vides a sampling of recent papers that illustrate the research that has been going On (121-146). Linear Scaling Methods

Much progress has been made over the years in solving the Kohn-Sham equations (e.g. 12) with any given exchange-correlation functional. Many res earc h groups have developed the capability for dft on molecules, and it has become a routine task to perform df t calculations, local or nonlocal, for small and mid-size molecules (up to a few dozen atoms). However, realistic modeling of large systems of molecules in the gas phase or in the condensed phase is still out of reach. This is because conventional quantum-mechanical methods are severely limited by the


DENSITY·FUNCTIONAL THEORY

709

size of systems. For a system with N electrons, the computational costs of the H F method increase as N4 or N3; those of the Kohn-Sham method, as N3• In both these electronic structure approaches, it is necessary to cal culate a set of single-electron orbitals, each orbital accommodating one or two electrons. Such a set of orbitals is global, i.e. the electron density at any given point in the space is represented by the same set of orbital functions. The global orbital description requires computational efforts that scale at least as the cube of the number of electrons, as manifested in the matrix diagonalization of the LCAP approach or in the orbital orthogonalization of the plane-wave iterative approach. With current com puter technologies, we are approaching the point where this N3 scaling is the time-limiting consideration. Note that scaling is much worse than N3 when correlation is included in the classical orbital methods. Within dft, it is possible in principle, to eliminate the N3 scaling associ ated with the use of global molecular orbitals by the description of elec tronic structure solely in terms of the electron density. The Thomas-Fermi theory does that with an approximate kinetic energy functional, but the Thomas-Fermi kinetic energy functional is not accurate enough to make it useful for electronic structure calculations. Other methods must be sought. The possibility of accurate first-principle electronic structure cal culations that scale linearly with the size of the system was demonstrated in 1991 in the divide-and-conquer approach ( 147, 148). Subsequently, there has been a surge of interest in linear scaling methods (149-174). The key to achieving linear scaling (order-N) is the localization of the electronic degrees of freedom; namely, it is necessary to limit to a local region of space the physical span of electronic degrees of freedom, whether it is the electron density, density matrix, or local orbitals. This is the crucial e lement in all the order-N approaches. Methods for localization developed so far can be grouped into two categories. The first category is based on the electron density or the first order reduced density matrix. Yang's method is described below. Cortona designed a scheme for crystalline solids whereby the electron densities of symmetry-nonequivalent atoms are represented by solutions to a spheri cally-averaged Schrodinger-like equation ( 150, 151). Tests of Cortona's method have been successful but are limited to alkali halides where the valence electrons are particularly localized and the densities around each nucleus are nearly spherical. Baroni & Giannozzi suggested a linear scaling method based on a finite-difference representation of the Hamiltonian, and a recursive Green's function approach to obtain the real space point wise approximation of the electron density (152). Many steps in the recur sion and a very fine spatial resolution in the finite-difference scheme are


710

PARR & YANG

needed for satisfactory accuracy. Linear scaling methods based on direct minimization with the density matrix have been proposed by Li et al (156), Nunes & Vanderbilt (170), Daw (157), Drabold & Sankey (158), and Godecker & Colombo ( 17 1). Linear scaling approaches mainly based on the Lanczos recursion for the density matrix have been formulated by Zhong et al (160), Gibson et al (161), and Aoki (162). Most of these methods have only been used for tight-binding cases and have not been tested in a self-consistent approach. The second category of methods is based on local molecular orbitals. Galli & Parrinello proposed a localized orbital formulation within the pseudopotential plane-wave approach (153). Later, unconstrained mini mum principles for the simultaneous minimization of the Kohn-Sham energy functional and approximate orthogonalization of orbitals were proposed by Mauri et al (154) and by Ozdejon et al (155, 166). These new variational principles appear to perform well in finding the ground-state energy in conjunction with a tight-binding approach. Kohn has given a prescription for determining generalized Wannier orbitals with linear scal ing ( 159). Recently, Stechel et al (168) proposed another linear s caling method, which explores the localization of electrons and replaces the diagonalization of the Hamiltonian matrix with block diagonalization into occupied, partially occupied and unoccupied subspace. These several schemes are still under development, and it is too early to make a critical comparison. Here the focus is just On the density-based divide-and-conquer method (147). This method uses the electron density as the basic computational variable, in place of the molecular orbitals that are the usual theoretical building blocks in the other first-principle computational approaches. It adopts a divide-and-conquer strategy: Divide a large molecule into many subsystems, then determine the electron density of each individual subsystem, and finally add the contributions from all the subsystems to obtain the total molecular electron density (and energy). The procedure and effort for determining the density for a subsystem are about the same as those in the calculation of a small molecule of the size of the subsystem. In the divide-and-conquer method, the electron density is the sum of contributions from subsystems,

per) 2IP·(r)Ifp(eF-e�)I"'�(r)12, =

"

16.

where IX is the index for subsystems, P"(r) is the partition function for subsystem IX, hex) is the Fermi function with inverse temperature {3, and "'� and e� are the subsystem local eigenfunctions and eigenvalues. The

DENSITY·fUNCTIONAL THEORY

711

Fermi energy eF is set by the normalization of the electron densities and is required to be the same for all subsystems (a key point): N

=

f

p(r)dr

=

2�

�fp(eF-e�),

27.

i

and

f

Q[p] = p[

-

(r)/2

-

v xc (r)Jdr + Exc [ pJ

·

28.

The linear scaling of the method can be derived as follows: If the molecule has M atoms and each su bsy stem has m atoms, then there are cM/m subsystems, where c > 1. For the calculations of the density of one par ticular subsystem, the computational cost scales as m\ independent of the size of the large system. The calculation of long-range electrostatic poten tial from the total density can be formulated to scale as M, or M In M. The overall scaling should be Mn2, linear in the system size. The use of the local basis set for the calculation of a subsystem density, instead of the entire set of atomic orbitals as in the LCAO Kohn-Sham method, apparently introduces a truncation error. Buffer atoms are intro duced to create a buffer zone for the better representation of the density of the subsystem and hence to lessen the truncation error. The increasing use of buffer atoms according to their distance from a subsystem, therefore, constitutes a systematic procedure for improving the accuracy of the divide-and-conquer approach. The convergence of the divide-and-conquer total energies towards those of the Kohn-Sham method was first demonstrated with the linear tetra glycine molecule by using the non-self-consistent Harris functional (148, 174, 175a,b). The good accuracy of the divide-and-conquer method has been further evidenced by its capacity to describe the electronic energy changes in molecular internal rotation around a single or a partial double bond ( 149). Such energy changes do not involve alteration in the length of any chemical bond and are hence small in magnitude. Thus molecular internal rotation presents a stringent test. It has been found that atoms up to third nearest-neighbor are needed as buffer atoms to describe molecular internal rotation. This is crucial to the applications of the method. Other


713


tests performed include the energy and structure of the benzene molecule and the total electronic density of states for the tetraglycine molecule_ In all the tests, the divide-and-conquer method accurately reproduces the Kohn-Sham results (148, 164, 173) . The SCF energy gradient calculation of the method has also been developed (176). There are already initial applications of these order N methods (164, 165, 169). One expects that such methods will become a very important tool for molecular modeling in the near future, possibly sooner than the four or five years predicted by Teter (163) . Exact Exchange-Correlation Potentials

As stated above, a major center of attention in dft research is the exchange correlation potential vxc[p(r)] of Equation 7, and to establish better approximations to it always has been an active area of research within dft. There is a unique approach to the problem that has been evolving only very recently: Find exact (very accurate) exchange-correlation potentials v xc(r) for specific systems, then learn therefrom. Because of the availability of accurate electron densities from con ventional quantum chemistry, it is possible, if v(r) and an accurate per) are known, to find an accurate vxcCr). This has been known for some time (177-179). There are several methods to accomplish this, all of which amount to self-consistent solution of Equation 12 for the given p held fixed. Results of this kind of investigation are of much interest because they provide (a) data that approximate functionals must be able to repro duce, (b) hints for how to improve approximation schemes, and (most importantly if perchance this pans out in the end) (c) a possible path to the ultimate exact treatment of the exchange-correlation. Equation 12 may be thought of as the variational equation for deter mining the Kohn-Sham Ts[p]. Among all possible Slater determinants D that give p, the ground-state D is the one that minimizes the expectation value of the kinetic energy operator (180, 181). If one imposes the density constraint with a local Lagrange multiplier A(r), one finds from the unique ness of the local vKS(r) that, up to a constant, A.(r) = vKS(r). The elusive Vxc then follows from Equation 7. The Kohn-Sham orbital energies follow as well, up to a constant. Usually, the constant is determined by setting the HOMO orbital energy equal to the negative of the exact first ionization potential. A number of studies are now available that reveal a lot about Vxc and generally agree very well with each other. The atom Be has been exhaustively studied (25, 182-185), but atoms through Ar also have been treated (186-193). Two-electron systems are straightforward to deal with (25, 189). The ongoing careful studies of Baerends and coworkers are of

7 14

PARR & YANG


special interest (49, 191-194); these workers already have been examining accurate Vxc for molecules (49). To simply summarize what has been found to date, none of the contemporary approximations to Vxc accurately repro duces the exact Vxc for the cases studied. There is a way to proceed that uses a single global Lagrange mUltiplier for the density constraint (25, 186). Let the ground-state density for a system with external potential v b e Po; an approximation to it, Then an N and S condition for to be identical to Po is

p C[p] � ff[p(rl)-po(rl)][p(r2)-po(r2)](l/r12)drldrz =

p.

=

O.

29.

Attaching a global Lagrange multiplier A to this, we find the orbital equations

[ iA+Av�]t/J t/J j

-

=

30.

E:j j,

or, with t he additional terms adding ease of calculation,

[-�L\+v+v/(I-I/N)+Ave.l]t/J1 E:1t/J1. =

3 1.

The first added term gives the correct cusp, while the second added term gives the so-called exchange-correlation hole the correct normalization and correct long-range behavior (49) and so puts less of a burden on the term Superscripts A are used here on the orbitals and other quantities because self-consistent solutions for these will depend on the value of A. The sum of the squares of the occupied orbitals is equal to t he approximate density The potential Vc in Equation 31 is the potential due to the error in t he density, that is, due to the density /' Comparing Equation 31 with Equations 10 and 1 1, one sees that Equation 3 1 is an approximate Kohn Sham equation with a defined exchange-correlation potential.

AVe. p".

V�c

=

-(lIN)v�+AV�.

-Po.

32.

Now what is the value of A for w hich C = O? It must be infinity because both C and Vc are zero at the solution poin t . So one must solve self consistently for a series of A values and then extrapolate to A equals infinity! T his works marvelously well (187-189). Furthermore, no arbitrary constant appears in the limiting Vxc, which then is the accurate (exact) Vxc for the system of interest. All quantities in the Kohn-Sham equations are determined. Equation 3 provides an extraordinary representation of the exact Vxc: It is the limit of a sum of classical electrostatic potentials [a result that reminds of some important work b y Harbola & Sahni (126)].



715

This procedure solves the problem of determining the orbitals (here the Kohn-Sham orbitals) from the electron density, and as such has many possible interesting applications (187, 189)_ The Kohn-Sham orbital ener gies also can be accurately studied now. Also, and this could be a fruitful line for further research, one can now address the problem of going from the electron density directly to the total energy. At first this goal seems already reached, because the accurate Vxc is in hand. However, full knowledge of Vxe is not, unfortunately, the same as full knowledge of Exe, and it is Exc that is required to implement the energy formulas of Equation 14 or 15. A new paper ( 189) examines how far one can go. As van Leeuwen & Baerends (194) have emphasized, one can get one Exe from another by functional integration of Vxe; it is a matter of finding the right reference state and performing the integration. Furthermore, Equation 15 shows that what one needs is not Exe but the difference between it and the density-weighted average Dxc (187, 189). In this difference, any portion of Exe[P] that is homogeneous of first order in p cancels. If one were to assume that Exe indeed had this homogeneity, Equation 15 would become

This formula gives energies with errors on the order of magnitude of correlation energies (189). This is remarkable and indicative that further studies along these lines are in order. A side result of this work is the numerical demonstration (187) that the exact Exe for atoms cannot be local, in spite of other results (40,43) demonstrating that a local functional is surprisingly accurate. The exchange-correlation potential for the argon atom determined by the method just described (188) is shown in Figure 1. The shape is charac teristic. Note how completely the component -(1/N)vJ takes care of the correct long-range tail of Vxc• One could of course separate out the exact Kohn-Sham exchange poten tial or treat the exchange-only Kohn-Sham theory. In these and related connections, recent work of Krieger and coworkers is pertinent (195-198). Chemical Reactivity DFT AS A TOOL FOR DESCRIBING CHEMICAL REACTIVITY Because dft for ground states is equivalent to wavefunction theory for ground states, the two must in effect be identical. One expects hardly anything new if one is just computing. One hopes rather to obtain new insights or improved formulations of old insights. Time-dependent rate theory is not discussed here; instead we discuss the simple signals about chemical reactivity pro-

716

PARR & YANG

5 .. . .. ....�;:;�: .:;! /�:;::

.

0

: .


u

)( >

.·o·J!: �.:; :.::; :.::;: .::; ::':!!�';: �'o'�! ��.�! .'o·�' ,.. ....... .

.

". ,

-5 ...... ... ......

..

-10 -15 -20

o

2 r

Figure J Vc.

Exchange-correlation potential for argon.

(dots) radial distribution.

(Solid) V.c' (dotdash)

-(l/N)vj,

(dashes)

vided by the ground-state description itself, through the first- and second order responses of a system to perturbations it may undergo. W hat one finds in dft is a lucid description of these properties (199). Describing reactivity is mainly a matter of interpreting and expounding upon Equations 4 and 5. Equation 4 gives an energy change to first order. The second term on the right is a familiar first-order perturbation result: The first-order response of energy to a change in external potential is determined by the density. The first term on the right obviously is very important for chemistry. Almost more than anything, chemical processes involve redistributions of electrons in a system or parts of a system. The chemical potential determines how much the energy changes to first order when the electron number changes: 34. This clinches the identification of J1 with negative electronegativity in the sense of Mulliken (22,200). Because J1 is a constant through the system, it also verifies Sanderson's principle of electronegativity equalization (22). Approximately, Jl =

-(l/2)(I+A),

35.


717

where I and A are the ionization potential and electron affinity of the system of interest. The big payoff comes with the second-order effects, however. It long had been known that the response of the density at a point to a change of potential at another point was important for determining many properties, for example polarizability:


[op(r)/ov(r')]N = [op(r')/ov(r)]N'

36.

The symmetry here has clear (perhaps untapped) physical implications for reactivity; it is the generalization of the symmetry of the atom-atom polarizability tensor of classical Huckel theory . What are very new and different are the second-order effects associated with the electron number. From Equations 4 and 5 we see that these are the quantities

37. and

fer)

=

[bJi/bv(r)]N

=

[bp(r)/bN]".

38.

The first of these is the chemical hardness (23); the second is the Fukui function (24). These are of major importance for reactivity, as Equations 37 and 38 already imply. Softness S is the inverse of hardness. Local softness is the global S times the Fukui function. The hardness is from Pearson's HSAB Principle (20 I). It is the derivative of the chemical potential with respect to electron number or the second derivative of the energy (curvature of the energy plot as a function of N). It measures resistance to flow of electrons at constant external potential, as driven by chemical potential (electronegativity) differences. Approxi mately, 11 =

I-A.

39.

Hardness is an exact quantity for which the band gap is an approximation (202). For conjugated organic molecules, hardness is a good measure of what is conventionally called the aromaticity (203). Note that Equation 39 applies equally to inorganic molecules, organic molecules, solids, and clusters. The Fukui function is a quantity that for isolated systems has one value for adding electrons and another for subtracting them (24); these are exact generalizations of the familiar LUMO orbital density and HOMO orbital density of approximate orbital theories (204). Fukui proved in 1987 that the HOMO and LUMO densities were useful for discussing reactivity

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(205). Clearly, then, the Fukui function is a correct and natural reactivity index. In the last few years many papers have been written extending and applying these concepts (206-278). Utility is clearly established, but there remains much to do before one can really understand reaction processes from the dft point of view. More detailed theory certainly will be required, possibly, even probably, involving the three ingredients described below.

Due originally to Mermin (279), there is a finite-temperature version of dft that is potentially neces sary for understanding molecules: The density and temperature determine everything, even for nonhomogeneous systems. The grand canonical ensemble of statistical mechanics is the natural construct in this theory, and in this ensemble one has remarkable formulas relating softness and local softness to density and number fluctuations,


FINITE-TEMPERATURE DFT AND FLUCTUATIONS

s

=

f3[(N2)-(N>(N>],

4 0.

and s(r)

=

f3[(Np(r»-(N>(p(r»].

41.

The last formula (280) agrees with the Falicov-Somorjai claim (281) that in certain cases low-energy density fluctuations are determinative for catalysis. But there is more: Other ensembles, too, are necessary, e.g. the isomorphic ensemble, for which the fluctuation external potential itself (BG Baekelant, A Cedillo & RG Parr, Submitted). THE HARDNESS-SOFTNESS HIERARCHY AND THE PROBLEM OF LOCAL HARDNESS

There exists a whole set of hierarchical equations for hardness and softness quantities: Two variable kernels, l1(r,r') and s(r,r'), one-variable local quan tities l1(r) and s(r), and global quantities 1'/ and S (283, 284). Reciprocal relations are S=

1/1'/, Js(r)1'/(r)dr 1, JJs(r,r')1'/(r',r")dr' o(r-r"). =

=

42.

The original definitions are s(r)

=

J(r)S = [op(r)/oll)v.

1'/(r)

=

[OIl/Op(r)]N;

43.

and s(r,r')

=

-

bp(r)/bu(r'),

vCr)

1J(r,r')

=

bu(r)/bp(r')

=

b2F/bp(r)bp(r').

44.

where u(r) = -11 and p and u are universal functionals of each other. The reduction formulas for softness and hardness are


s

'1

=

=

f f

s (r) dr,

s(r)

=

f

719 45.

s(r,r')dr';

f

f(r)'1(r)dr, '1(r) = x(r)'1(r,r')dr';

46.

where x(r) x[p(r)] is any functional of p that integrates to unity. From the first formula for s(r) in Equation 43, s(r) is clearly vital. It integrates to S and when divided by S, gives fir). It therefore contains all of the information in both the Pearson hardness and the Fukui reactivity index. From the second formula for '1(r,r') in Equation 44, '1(r,r') also is centrally important. It is the second functional derivative of the universal Kohn-Honenburg functional F[p]. As such it must be an entity of the greatest fundamental importance. However, one should not be all that sanguine about this hierarchy as described. The softness kernel s(r,r') is difficult to visualize (although a sim ple formula relates it to the response function of Equation 36). And there is a peculiar problem about the local hardness: Equation 4 3 is unacceptable as a definition of '1(r) if '1(r) is to have physical meaning, because there is an essential ambiguity in the last derivative in Equation 43. Consequences are the arbitrariness of x(r) in Equation 46 and a need for additional considerations. One attractive idea is to take x f This yields I](r) = 1], which is a nice result (285), except that it appears to obliterate '1{r) as a significant local measure. This would be fine if one could either do without the term local hardness in the vocabulary or find another local hardness quantity that could better play the role. We should call attention to the landmark works by Nalewajski, Mortier, and collaborators modeling the hardness kernel (286-294).


=

=

In 1987 Pearson put forward a new prin ciple, that molecules tend to rearrange themselves so as to achieve maximum hardness (266), and he has since extensively discussed and given many examples of this general behavior (268, 269, 295-299). Others also have labored to delimit this principle, which is clearly important to do (300-319). Constraints must be expected, just as they always occur in the extreme principles of classical thermodynamics. A first case is the decay to electronic equilibrium at some constant temperature of a molecule at a fixed nuclear configuration. A somewhat formal statistical mechanical proof has been given (300), that the hardness (band gap) almost always (309, 3 10) increases. The idea in the proof is that a nonequilibrium state near an equilibrium state can be conceived MAXIMUM HARDNESS PRINCIPLE


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& YANG

as an equilibrium state for changed external parameters. The decay to equilibrium is instigated by reverting the parameters to their equilibrium values. In wave-mechanical terms, the nonequilibrium wavefunction is a combination of the ground and various excited states of the equilibrium system. The excited states are softer; as they are purged during the return to equilibrium, the system becomes harder. Alternatively, one can think of the situation as follows. Increasing a band gap tends to push occupied levels down and thus to stabilize the system. This is most likely to be true when the change involved leaves the Fermi energy (average of HOMO and LUMO energies-the chemical potential) more or less unchanged. So we have constant chemical potential as a likely condition to accompany a valid maximum hardness principle. Matters are more complicated than that, however, because sometimes one is interested in equilibrium states changing to other equilibrium states, whereas other times one is interested in the effect of a molecular vibration or internal rotation on the hardness of the electronic cloud, in comparing a molecule with a minor perturbation of itself, in comparing clusters of d ifferent sizes, or in reactions of certain molecules to give others. One must keep tabs on what constraints are operative in a given case and on what questions should be asked. The result is that the preferred direction is almost always toward the greater hardness. Among the more unique results obtained to date are the facts that (almost always) (a) closed shells in the periodic table come at atomic numbers for which the hardness is a maximum (314); (b) along a reaction coordinate the hardness is minimum at the transition state (301); (c) the favored side in a chemical reaction has the greater hardness (302); (d) a symmetrical normal molecular vibration away from equilibrium involves decrease of hardness (299); and (e) the most stable internal rotation isomer is the most hard (211). These are hut examples. The maximum hardness principle is the second of Pearson's wonderful principles, the first being the HSAB principle: Hard likes hard, and soft likes soft (201). One could hope to prove one from the other, and that was the structure of the paired proofs in 1991 (300, 315). The HSAB proof given then (315) assumed the correctness of the maximum hardness principle as proved then (300). In the second paper (315), a "principle of equal satisfaction", was proposed, which presaged the later appearance as a major player on the scene of the system's grand potential [see the recent discussion of the so-called hardness functional (317)]. A final definitive statement of the maximum hardness principle still needs to be laid down. Ultimately of course, nothing will be found that does not follow from the known laws of quantum and statistical mechanics as applied to inhomogeneous systems. This does not imply, though, that


721


there can never be new valid quantitative and useful principles that apply to important chemical circumstances . We again recall classical thermo dynamics, where the great energy and entropy laws magically transform into what is most important for constant T and p: the minimum principle for the Gibbs free energy. The discussion in this section has largely been about constant nuclear configuration. Clearly the role of nuclear motion must be expounded in any complete description of molecules in course of chemical reaction. Strides in this direction recently have been taken by Cohen et al (320, 321).

CONCLUSIONS In a very recent paper on df t calculation of one-electron properties, Duffy et al (322) remark that "Density functional theory is a field enjoying a tremendous recent surge in popularity among theoretical and practical chemists alike." This is true. Not only are dft methods feasible and highly useful, but also dft possesses attributes that make research in the subject exciting: subtle mathematical difficulties, great opportunities for improv ing approximations, and manifold challenges to shape the language of chemical thought. Calculational (quantitative) dft now has become competitive with con ventional methods for computing ground states, for all but tiny molecules. It rivals in accuracy large-basis-set MP2 level conventional calculations, including correlation effects. For large molecules, the computational requirements of dft are much less than those of conventional methods. Linear-in-N methods makes this even more so. The excited-state advan tages of dft are less clear. With the probable (but not certain) continued failure of attempts to find the exact energy functional (the holy grail!), we must aggressively pursue more accurate exchange-correlation functionals. Conceptual (qualitative) dft holds much untapped promise. The begin nings have been laid down: the constructs of electronegativity, hardness, Fukui function, global and local hardness and softness, hardness and softness kernels, and so on. Completion of the language awaits com prehensive expositions of chemical bond formation and Pearson's two principles. It awaits other advances as well. Nevertheless, there appear to be sound grounds for an optimistic outlook concerning the ultimate value of conceptual df t. As the new and next-generation dft computational methods develop, together with the qualitative, conceptual language that goes with the density-functional formulas, we believe that there will be a more unified understanding of molecular science.

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YANG


ACKNOWLEDGMENTS

We have benefited from discussions about this subject with many people . Each of us has received research support from the National Science Foun dation and the North Carolina Supercomputer Center . RGP also has been supported by the Petroleum Research Fund of the American Chemical Society and the Exxon Education Foundation; WT by Cray Research, the U S Environmental Protection Agency, and the Sloan Foundation. For special help we thank EJ Baerends, LJ Bartoiotti, PK C hattaraj, S Liu, and P Politzer. Any Annual Review chapter, as well as any article cited in an Annual Review chapter, may be purchased from the Annual Reviews Preprints and Reprints service. 1-800-347-8007; 415-259-5017; email: [email protected]

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