pearsonthe principle of maximum hardness

7/27/2019 PearsonThe Principle of Maximum Hardness

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250 Acc. Chem. Res. 1993,26, 2 5 e 2 5 5

The Principle of Maximum HardnessRALPHG . PEARSON

Chemistry Department, University of California, San ta Barbara, C alifornia 93106Received August IO, 1992

Density functional theory (DFT) is a branch ofquan tum mechanics in which the em phasis is on theone-electron density function, p , rather than on themore usual wave function.' It goes back to t he workof Hohenberg and Kohn, who proved t ha t th e ground-stat e energy of a chemical system is a functional of ponly. A chemical system is a collection of nuclei andelectrons. It may be an atom, an ion, a radical, amolecule, or several molecules, in a sta te of interaction.A functional is a recipe for turning a function into anumber, just as a function is a recipe for turning avariable into a number.There are many im portant applications of D FT tochem istry. One is in the calculation of prope rties ofatoms and molecules. These are much easier tha n andof sim ilar accuracy t o calcula tions using very good wavefunctions. For a system containing N electrons, thewave function depen ds on 4N space and spin coordi-nates. T he electron density depends on only three spacecoordinates, and N enters as a simple multiplicativefactor. Hence computation s are very much simpler inDF T, especially for large systems.Another important use for DFT is in elucidatingfamiliar chemical concepts. Since 1975,R. G. Parr andhis co-workers have been particularly ac tive in this area.Th e present paper will deal with two new properties ofa chemical system, arising from this work. The y arethe electronic chemical potential, p,2and the hardness,71.3 These ar e of great impo rtance in determining thebehavior of the system, and they lead to broadlyapplicable and useful principles.The definitions of these quantities are

p = (waw,; 271 = (apian3,= 2 1 ~ (1)where E is the energy, N is the number of electrons,an d v the potential due to he fixed nuclei. Th e softness,a = 1/71, is also a useful quan tity . The name electronicchemical potential comes from the thermodynamicequation

Here pois the ordinary chemical potential, an d N is thenumber of molecules. Th e two, p and PO,are alike intha t, a t equilibrium, they must be con stant everywherein the system.Classical thermod ynam ics also gives us(ap/ahT)V,T -V(ap/av) /M= V / M K (3)

where K is the compressibilityand N is again the numberof molecules. Comparison with eq 1 shows that theRalph 0 .Pearson is Professor Emeritus at the University of Californ ia, SantaBarbara. His undergraduate degree Is from the Lewis Instltute In Chicago. H isPh.D. was obtalned at Northwestern University in 1943. and he taught thereuntil 1976. Present research interests are chemical bonding, reactionmechanisms, and the application of density functional theory to chemistry.

0001-4842/93/0126-0250$04.00/0

chemical softness, u , is analogous to the mechanicalsoftness, K.In s pite of thes e similarities, there is no real rela-tionship between the two sets of quantities. Theelectronic chemical poten tial is not the electronic par tof the thermody nam ic chemical potential. Th e chem-ical hardness is not resistance to mechanical deforma-tion, but resistance to change in the num ber of electronsin the system, or to changes in the one-electron densityfunction, p .Operation al, and approxim ate, definitions of p an d71 are

(4)where I s the ionization potential an d A is the electronaffinity of th e system. x is the absolute electronega-tivity. It is similar to, bu t n ot equal, to the Mullikenelectronegativity.If we draw a plot of E vs N for any system, then pis simply the instantaneous slope of such a curve.Experim entally we only know po ints on the curve forintegral values of N , from data such as ionizationpotentials an d electron affinities. We do not know theins tan tan eou s slope of the curve. However, if we pickthe neutral species (or any other) as our starting point,we do know th e mean slope for the change from ( N -1)to N electrons. It is simply equ al to -I. Also themean slope from N to ( N + 1) electrons is simply -A,Using the m etho d of fin ite differences, we can approx-imate the slope at N as -(I+ A)/2.In th e sam e way, 271 is the curvature of a plo t of E vsN . Using finite differences again gives 7 = ( I - A)/2.Th e factor of 2 was added arbitrarily to make ~1 and 7symmetrical.According to Koopmans's theorem, the ionizationpote ntial is simply the orbita l energy of th e HOMO,with change in sign. For spin-pa ired molecules, theelectron affinity is the negative of the orbital energy ofth e LUM O; therefore, on an orbital basis, we can write

71 = (ELIJMO - EHOM0)/2 (5)If we make th e usual diagram of the MOs of a m oleculeas func tion of their energie s,p is avertic al line halfwaybetween the HOM O and the LUMO. Th e hardness isjust half the energy gap between theTh us a hard molecule has a large energy gap, and asoft molecule has a small gap. In MO theory, thefrontier o rbitals also define the lowest excited state of

-p = (I+ A)/2 = X ; 7= (I- A)/2

CL = (EHOMO + LUMO)/%

(1)For a general introduction, see: Parr, R. G.; Yang, W. DensityFunctional Theory for A t o m an d Molecules; Oxford Press: New York,1989.(2 ) Parr,R. G.; Donnelly,R. A.; Levy, M.; Palke,W . E. J.Chem.Phys .1978,68, 3801-3807.(3) Parr,R.G.;earson, R. G. J . A m . Chem. SOC.983,105,7512-7516.(4 ) Pearson, R.G. Proc. N a t l . Acad. Sci. U.S.A. 1986,83,8440-8441.0 1993 American Chem ical Society


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Principle of Maximum Hardness Acc. Chem. Res., Vol. 26, No. 5, 1993 251LUMO '\

\$

-UM O '\,-LUMO

\L -UMO HOMO \ \I 21 i 2 1II -------Al

- c + - - - -a ) ( b )

Figure 1. Orbita l energies for NH3: (a) pyramidal form; (b)planar form.the m olecule. A large energy gap means a large energydifference between the groun d sta te and the manifoldof excited states of the sam e multiplicity.This is of great importance, since the quantummechanical theory of response to a perturbation in achemical system depend s on this difference.5 Changesin electron density occur by mixing excited-state wavefunctions with the ground-state wave function. Themixing coefficient is inversely proportional tothe energydifference between th e states. There fore soft moleculesundergo changes in electron density, p , more readilythan hard molecules.Circumstantial Evidence Concerning Hardness

The foregoing suggests that it is a good thing for achemical system if i t can arran ge itself to be as hard aspossiblea6Since chem ical reactions obviously requirechanges in the electron density distribution patter n,we expe ct soft molecules to be more reactive than hardmolecules, in general. Th is is indeed the case.*J Forexample, HzS,with q =6.2 eV , is certain ly more reactiveand less stable than HzO, with q = 9.5 eV. HzSe andHzT e are softer and more reactive still.A classic example is provided by the benzenoidhydrocarbom8 A plot of the experimental values of I-A against the resonance energy per electron (RE PE )is a straight line. Th e REP E is a good measure ofaromatic character, that is, reactivity and stability.Benzene has th e largest value of I -A and is the moststable and least reactive. Pentacene has the sm allestvalue of I - A and is the most reactive.Theo reticians who do MO calcu lations of m olecularstructures and energies always seem to find th at themost stable structure has the largest HOMO-LUMOenergy g a ~ . ~ , ~igure 1 llustrates th is by showing anMO energy diagram for NH3 in its stable pyramidalform, and in th e unstable planar form.The smaller gap in th e latte r case arises because thepLorbital of nitrogen is removed from bo nding whilestill occupied. Thu s it becomes the HOM O. A linearcom bination of hydrogen 1s orbitals is removed froman an tibonding MO, is lowered in energy, and becom esthe LUMO.Table I shows the results of som e calculations by th eextend ed Huckel m ethod for the closo-borane anions.1( 5 ) This is often discussed under the heading of the second-orderJahn-Teller effect. See: Pearson, R. G. Symmetry Rule s fo r ChemicalReactions; Wiley-Interscience: New York, 1976.(6 ) Pearson, R. G. J. Chem. Educ. 1987,64,561-567.(7 ) Pearson, R. G. J. Am. Chem. SOC.1988,11 0, 2092-2097.(8) Zhou, Z.; Parr, R. G. J. Am. Chem. SOC.989,111,7371-7379.(9) (a) Bartell, L. S. J. Chem. Educ. 1968,45 , 754-758. (b)Burdett,J. K.; Coddens,B. K.; ulkarni,G. V. Inorg. Chem. 1988,27,3259-3261.

E t \wHOMO( 0 ) ( b )

Figure 2. Energy changes during the approach of two sub-systems: (a) HOMO-LUMO interaction; (b) HOMO-secondfilled orbital interaction.Table I. Results of Molecular Orbital Calculations forthe clom-Boranesaborane idealized AE gap, borane idealized AE ap,anion geometry eV anion geometry eVBsHs2- o h 6.6 B7H7% Cgu 0BsHs2- D3h 1.1 B8Hs2- DZ d 2.5B7W- Dsh 5.9 B a s 2 - D u 3.2B7H7'- C Z ~ 1.2 BsHs2- Czu 2.2

From ref 10.Th e HOMO-LUM O gap is given fo r B6Hs2- and B7HT2-,for both the stable structure and unstable ones. Inboth cases the energy gap is much larger for the stablestructure. For B8Hs2- the gap is calculated for threepossible structures. No strong preference for any ofthese is found. In agreement with this, the ion hasbeen found to be fluxional.An examination of orbital interactions when twopartial systems pproa ch each other can be informative.Assume that the HOMO and the LUMO play thedom inant role in these interactions. Figure 2a showsthe intera ction of th e frontier orbitals. Th e lower energyorbital, he H OMO, goes down in energy, and t he LUM Ogoes up. Th ere is a ne t energy lowering, and th e gapincreases. Figure 2b shows the case where the H OMOis interacting m ainly with anothe r filled orbital. Nowthe HOMO goes up in energy. Th e LUMO will mixprimarily with other empty orbitals, which will lowerits energy, as shown. The n et effect is th at th e energyis raised and the HOMO-LUMO gap is diminished.This is the result for orbital interactions, or covalentbonding. What about ionic bonding? Consider ananion and a cation approaching each other, with a netdecrease in energy. Th e HOMO will be an atom icorbital on the anion, and th e LUMO will be an orbitalof th e cation, in the usu al case. As he ions approach,th e pote ntial of th e cation will lower the orbital energyof the HOM O, and the p oten tial of the anion will raisethe orbital energy of the LUMO. The HOMO an dLUMO will move apa rt, just as in Figu re 2a.Th us covalent an d ionic bonding give similar results.However, it must be remembered th at these remarksapply only to the orbital energies. The total energymust also include the nuclear-nuclear repulsion.The preceding examples refer to hardness as afunction of nuclear positions. Equa lly imp ortan t isth erelationshipof hardnesstothe electron density function,p . Will the hardness increase to a maximum value 88

(10) Muetterties, E. L.; Beier, B. F. Bull . SOC. him. Belg. 1979,84,(11)See: Zhou, Z.; Parr, R. G. J. Am. Chem. SOC. 990, 112, 5720-397-401.5724.


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262 Acc. Chem. Res., Vol. 26, No. 5, 1993a set of trial wave functions becomes progressivelybetter, approaching the tr ue wave function?Some conclusions can be drawn, if we restrictourselves to LCAO-MO theory. Th e eigenfunctions,&,, are given by

Pearsonconclusive. Fortunately Pa rr and C hattar aj haverecently given a rigorous and general proof, using acombination of statistical mechanics and density func-tional theory.12One starts with a grand canonical ensemble ofidentical systems in equilibrium a t tempe rature T ndelectronic chemical potential p. Th e softness, u, canthe n be def ine 4 in terms of fluctuations in N from th eaverage value N.13

(9)where f l = l / k T , as usual. Sof t systems have largefluctuations.Th e softness, which is an average value, can also bewritten as

Q = (dN/dp), = f l ( (N-m2)where the $s are atomic orbitals. Th e orbital energiesare found from the determ inant

(7)Hij- SijEl= 0The conditions for the solutions are th at

(dE,/dC,,) = 0; (dE/dC,,) = 0 (8)All the roots are maxima or minima on the energy-coefficient hypersurface. Th e owest root isthe absoluteminimum, th e highest root is the absolute maximum,and the remaining roots a re local maxima a nd minima.Th e occupied orbitals, in v irtually all cases, will havenegative curvatures, (&/dC,12), corresponding o bond-ing MOs, or zero curvatu re, nonbonding orbitals. Th eemp ty orbitals will have po sitive curvatu res, antibond -ing, or zero curvature. This applies to the HOMO a ndLUMO, in particular.The best values of the Cmls define the best wavefunction, and the best value of p , tha t can be obtainedfrom the selected basis set of AOs. Any change fromthe best values will cause the HOMO t o rise in energy,or be unchanged, an d th e LUM O will fall in energy, orbe unchanged. Th us the energy gap between them isa maximum for the b est values of th e coefficients, orthe bestelectron density function. Usually, of course,this will not be t he tr ue density function.The conclusion that the hardness has a maximumvalue for the lowest energy solution can be readilyverified in simple cases, such a s the Huckel MO theory .Inclusion or om ission of overlap integrals ha s no effect.Examples where all atoms are th e sam e and cases wherethe atoms and t he orbitals are different give the sameresult.Provided th e SC F condition is met, calculations a tthe H artree-Fock level also obey the m athem atics ofeq s 4 and 5. Therefore th e HOMO-LUMO gap shouldalso be amaximum in these cases. Because the solutionsare normalized an d orthogonal, an d because the atom icorbitals are conserved, the coefficients for different MOsare not independ ent. There fore wrong coefficients inone orb ital will usually lead to w rong coefficients in allorbitals.We can also change the trial wave function for asystem by enlarging th e basis se t of ato mic orbitals. Ifthis is done, what usually happens is tha t the HOMOis changed only slightly, usually to a more negativeenergy. Bu t the LUM O is decreased in energy muchmore, so tha t the HOMO-LUMO gap is smaller, eventhough the total energy isdecreased. A t the same time,the value of p becomes more negative, because of thedecrease in ELUMO. This turns out to be important, aswill be seen in the next section.The Principle of Maximum Hardness

T he evidence of t he previous section is suggestive ofsome principle of maximum hardness (PM H). Bu t noneof it is rigorous enough or general enough to be

where the PONSare the equilibrium probabilities.pN,i = exP[-fl(E~,i NP)l (11)

F s the grand partition function.A nonequilibrium ensemble would be characterizedby a set of probabilities, P N , ~ ,ifferent from theequilibrium values. We can write

Now the fluctuation-dissipation theorem of statisticalmechanics14may be used to show tha t d u > 0. Thusthe equilibrium state has the minimum softness, ormaximum hardne ss, of all the possible states.The proof depends on p, v, and T being constant.The potential p appears in eq 11because it is also theLagrange multiplier for the normalization constraint.While the details are not simple, they are standardstatistical mechanics. N is the number of electrons,not the number of molecules, as is more usual.The requirement that p and v remain c ons tant is veryrestrictive. However, there are examples where it canbe met. Th e pote ntial of the nuclei,v, s constan t if th egeometry is fixed. For two-orbital, two-electron cases,the HOMO and the LUMO approach each othersymm etrically, when overlap is ignored. The ir mid-point, which is p, s unchanged. Alterna nt hydrocarbonsbehave in the same way.This agrees with the observations of the previoussection, tha t the HOMO-LUMO gap is a maximum forthe b est values of the C,! coefficients. In other cases,while p is not co nstan t, its change may be very small.T o what extent this is true for Hartree-Fock calcula-tions is a question still to be examined.Fortunately, there are also ways to examine theprinciple of maximum hardness, even for changes inthe nuclear positions.15 Start with a molecule in itsequilibrium geometry an d calculate the orbital energiesat , or near, the H F level. Now distort the molecule asma ll distance along directions given by the vibration alsymmetry coordinates, and recalculate the orbitalenergies. By using th e comp lete se t of symm etry(12) arr, R.G.; hattaraj, P. K. J.Am. Chem. Soc. 1991,113,1854-(13)Yang, W.; arr, R. G. Proc. Natl . Acad. Sci. U.S.A. 1985, 2,(14)Chandler, D. ntroduction to Modern Statistical Mechanics;(15) earson, R.G.; alke,W. E. J.Phys. Chem.1992,96,3283-3285.

1856.6723-6726.Oxford Press: New York, 1987;Chapter 8.


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Principle of Maximum Hardness Acc. Chem. Res., Vol. 26 , No. 5, 1993 253Table 11. Distortions of Carbon Dioxide from the Eauilibrium Geometry

Ri R2 0 -HOMO CLUMO -2pb 2Vb1.1352 1.1352" 180 " 0.5423 0.2307 0.3116 0.77301.1052 1.1652 180' 0.5405 0.2313 0.3092 0.77181.1652 1.1052 180' 0.5405 0.2313 0.3092 0.77181.1052 1.1052 180 0.5438 0.2391 0.3047 0.78291.1652 1.1652 180' 0.5407 0.2066 0.3341 0.74731.1352 1.1352 175' 0.5419 0.2098 0.3321 0.75171.1352 1.1352 185' 0.5419 1.2098 0.3321 0.7517

a Equilibrium values. From ref 16. In atomic units.

Figure 3. Vibrational modes of COz in D-h point group.coordinates, he h ardness can be probed for all possiblechanges in t he equilibrium geom etry.Table 11shows th e resu lts of su ch calculations for th eCOZ molecule.16 In order to und ersta nd th e results,Figure 3 shows the nor ma l modes of vibr ation of COZ.The asymmetric modes, nu nd Z,, differ from thesymm etric mode, Z,, n severa l respec ts. Fi rs t of all,they des troy some elements of sym metry , changing thepoint group to Czuand C respectively.Secondly, positive deviations from e quilibrium pro-duce a configuration which gives the same averagenuclear potential as negative deviations. Therefo re,CHOMO, CLUMO,1,and q mu st be the same for both. If wele t Q epresent a symm etry coordinate, the n both ( d p /dQ) and (dq/dQ) must be equal to 0 at the equilibrium

Also, if we expan d th e ene rgy as a power series in AQ ,the linear term mu st vanish, and the qua dratic term isthe first nonvanishing one. Symm etry arguments canbe used toshow tha t (dvJdQ) an d (dvJdQ) are separatelyequal to 0 when averaged." He re un and u e refer to th epotentials of the nuclear repulsion and the nuclear-electron attraction, respectively.Hence, for the no n-totally symm etric distortions, wehave met the restrictions of Parr and Chattaraj.Accordingly, the h ardness shou ld be a maximum a t80.Table I shows tha t this is the case. Note tha t p caneither decrease or increase upon d istortio n from equi-librium.Th e totally symm etric mode gives different results.The hardness and p both increase steadily as the nucleiapproach each other. If the nuclei coalesced, p wouldbe a maximum. This does not happen because a t Qowe have the condition

((dv,/dQ) + (dv,/dQ) ) = 0 (13)Thu s the equilibrium value of Q s determined by th eHellman-Feynman theorem of balanced forces, and no t

80-

(16) Palke, W. E. Unpublished calculations.(17) Pearson, R. G . Acc. Chem. Res. 1971,4, 152-160.

Table 111. Changes in q for Bond-FormingReactionsreaction V R , ~ V V P , ~ V

N a + C1= NaClLi + F = Li FLi + H = Li HH + C1= HClC + O = C OH + OH = H 20CH3 + F = CH3FCH3 + C1= CH3ClCH3 + Br = CH3BrCH3+ I = CH31N i + CO = NiCOCr + 6CO = Cr(CO)GFe + 2CsHs = Fe(C5Hs)zCr !8&H6 = Cr(C6H&

0.81.02.34.74.95.73.23.13.23.43.33.12.83.1

4.85.43.88.07.99.59.47.55.84.73.64.53.83.3

a Hardness of reactants. Hardness of products. From ref 18.by the m aximum value of p . Th is is not a violation ofthe P M H since neither u nor p remains constant.Th ese results for COZ are typical. Th e same featureshave been found for H20,l6 NH3,15 and Th elatter molecule is a good test, since there are threesymm etric modes and nine asymm etric ones. Whilethe equilibrium distances and angles in a molecule aredetermined by eq 13, the existence of symmetry in amolecule is determined by t he hardness.If the hardness decreases upon an y distortion th atdestroys an element of symm etry, th at element is stable.If 7 ncreases, the molecule will spontaneo usly distortan d the eleme nt will vanish. An example would beNH3 in a planar form.15 These results are strikinglysimilar to those d educed from th e second-order Jahn-T elle r e f f e ~ t . ~Ot he r Obse rva t i ons

There are still further reasons to believe that in-creasing hardness accompanies the approach of achemical system to equilibrium. Consider th e overallprocessN(g) + 3H(g) = NH,(g) (14)

which is very favorable energetically. We can calculateth e overall changes in p and in 7,using experimentalvalues of I and A and eq 4. We find tha t p increasesfrom -7.2 to -2.6 eV, an d q increases from 6.4 to 8.2 eV.Exam ination of a large numbe r of reactions where afew atoms or radicals form a molecule always showsth at the hardnes s increases.18 Th e electronic chemicalpote ntial may increase or decrease. Tab le I11 gives anumber of examples of the changes in 7.l9These results are con sistent with Figure 1. As longas the energy is decreasing, th e HOMO-LUMO gap is(18) Pearson, R. G. Inorg. Chim. Acta 1992, 198-200, 781-786.(19) T o find I and A for a mixed system , take the smallest I and themost positive value of A . Perdew,J.P.; Parr, R. G .;Balduz,J. L. Phys .Reu. Le t t . 1982, 49, 1691-1694.


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254 Acc. Chem. Res., Vol. 26, N o. 5, 1993 Pearsonnumbers the hardness shows a local maximum. In-creased stability is accompanied by increasinghardness.As N becomes very large, we approach metalliclithium. I becomes equal to A , and both are equal tothe work function for the m etal. Th e reason for this,of course, is th e band s truc tur e of the solid. In a metal,the valence band is only partly filled, so the highestoccupied level has virtually the same energy as thelowest empty one.

The process Li(g) = Li(s) (16)is downhill in energy, bu t th e hardness decreases from2.4 to 0.0 eV. Th e potential p remains surprisinglyconstant, being -3.0 eV for lithium atoms and -3.1 eVfor lithium m etal.22 Since the nuclear potential chan gesgreatly in eq 16 , there is no violation of the PMH.For in sulating solids an d semicond uctors there is alsoan overall decrease in q in going from the gas phase tothe s01id.l~However, jus t as for single molecules, thespace group symmetry of solids appears to be deter-mined by the energy gap. Th e most stable structureis the one with the largest gap.g

Jus t as for the aromatic hydrocarbons, the stabilityof the fullerenes, C,, dep end s on the HOMO-LUMOgapaZ 3 n ad dition, there is a geometric factor in th atthere must be the right number of carbon atoms toform a suitable cage. Calculations, using densityfunctional theory, show that C ~ Z ,60, CTO, W, nd CIOOhave a large ga p and should be stable; bu t Clz and Cz4have a zero gap and should not exist.Concluding Remarks

The examples of the preceding reactions are allconsistent with the statement th at at equilibrium,chemicalsystems are ashard aspossible. Th e hardnessis a m aximum for changes th at do not also change p orv, as predicted by Par r and Cha ttaraj. Some relaxationof these restrictions seems to be allowable, and it ispossible th at a more gen eral, and less restrictive, rulemay be found.A potential use for the P M H is in finding the bestappro ximate wave function for a chemicalsystem. Sincechanging the basis set usually changes 1.1 as well as 7,this is a case where the more general rule is needed. Inthe case of isom ers where a difference in the bondingbetween atom s occurs, the PM H does not predict tha tthe m ost stable isomer has he largest HOMO-LUMOgap. Such isomers will necessarily have quite differen tvalues of v, though p can be fairly constan t. Anexamination of a number of isomers show that thenuclear-nuclear repulsions most often dominate therelative stabilities.Usually each isomer has a local energy minimum inth e energy-nuclear position hypersurface. Th e samerules asoutlined for COZ,NH3, CZH6, an d H ZO ill ap plyto each isomer. Th e hardness should be a maximumwith respect to antisymmetric distortions from thegeometry of the local energy minimum.Th e factors tha t change the hardness are fairly wellunders tood, though a calculation from first principles

(22) The values of p are remarkably similar for all metal atoms and(23) Kurita, M. ; Kobayoshi, K.; Kum ahora, H.; Tago, K.; Ozawa, K.their solid metal forms. Pearson, R. G. Chem. Br. 1991, 444-447.Chem. Phys . L e t t . 1991,188, 181-185.

2.0 I

-0.54 10 20 30 40 50 s o 70N

Figure 4. Second energy differences, A2, or lithium clustersversus the number of atom s,N . Reprinted with permission fromref 21. Copyright 1992 Nationa l Academy of Scien ces.

0 10 20 30 4 0 50 60 70NFigure 5. Chemical hardnes s, - A , versus the number of atom s,N , for lithium clusters. Reprinted with permission from ref 21.

Copyright 1992 Nation al Academy of Sciences.increasing. A maximum in q is not reached when th etota l energy is a minimum, because Figure 1 does notinclude the nuclear repulsion term. Th e PM H is notviolated because v and p are not constant.The mass spectra of clusters, both metallic andnonm etallic, show enhanced intensity for certain num -bers of atom s, called magic numbers. For cluste rs ofalkali metal atoms, the magic numbers are N = 2, 8,(18), 20, 34, 40, and 58.20 Figures 4 and 5 show th eresults of some theoretical calculations on lithiumclusters using the spherical ellium background model.21Th e cluster stability is shownby plotting the secondenergy difference.

Az=E(N+l)- E(N-1) -Z(N) (15)Figure 4 shows tha t A2 is close to 0, except for N =2 , 8 , 18 , 20 , 30 , 40 ,and 58. Th e peaks a t these numbersshow tha t th e m odel is working well.Figure 5 shows a plot of q as a function of N, usingeq s 4 and 5. Th ere is a general decrease of (I -A ) asN increases. However, the re are pronounced peak sagain a t N = 2 , 8 , 1 8 , 2 0 , 3 4 , 4 0 ,and 58 . A t the m agic(20) de Heer, W. A.; Knight, W. D.; Chou, M. Y.; Cohen, M. L. Solid(21) Harbola,M. K. Proc. N a t l . Acad.Sci. U S A . 992,89,1036-1039.Sta te Physics 1987,40,93-181.


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Principle of Maximum Hardnessis n ot s t r a i g h t f o r ~ a r d . ~ ~he factors are the changesin the kinetic energy and th e interelectronic repulsionas p is changed. Changes in p are due to changes in N ,or to changes in the shape factor of p at constant N.When a subshell is being filled, the changes ininterelectronic repulsion are dom inant. When a shellis filled, there is a big change in kinetic energy for th enext electron added. This causes a large change in q S z 5The re are a numbe r of im portan t consequences of this.These will be discussed in th e following Account in th isissue, by Zhou and P arr.A corollary of th e P M H has to do with the propertiesof the transition state. Long ago Bader pointed outthat an activated complex must have very low lyingexcited states.26 We can now speculate that it m usthave a minimum energy gap between the HOMO andthe LUMO. Certainly this should be so for a reactioncoordinate (for decomposition) th at is not totally

(24) Berkowitz, M.; Ghosh,S.K.; Parr, R. G. J . Am. Chem. SOC. 985,(25) Peareon, R. G. J.Am. Chem. SOC. 986,107,6801-6806.(26) Bader, R. F. W. Can.J.Chem. 1962,40, 1164-1168.(2 7 )Datta, D. J . Phys. Chem. 1992,96,2409-2410.

107,6811-6815.

Acc. Chem. Res., Vol. 26, No. 5,1993 255symmetric. The re is some evidence th at th e gap is aminimum, even when the transition state is of low

It appears that th e PM H is a fundamental, broadlyapplicable electronic-structure ule. In this it resemblesthe other general rule derived from density functionaltheory: the electron ic chemical potential mu st becons tant everywhere in a chem ical system a t equilib-rium!2 This is the famous electronegativityequaliza-tion rule , if eq 4 is used to define the electron egativity.To appreciate the broad applicability of the P MH ,one need only look a t the proof as given by eqs 9-12.These use only the laws of prob ability applied to largesystems. Yet the conclusion can be applied at once toproperties of the frontier orbitals of single molecules.It is noteworthy that this application seems to work sowell, even though eq 5 is a fairly severe approxim ation.Also all levels of MO theory seem to be covered equallywell.Z am indebted to R . G . Parr and W. . Palke for manydiscussions.

pearsonthe principle of maximum hardness

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