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1854 J . Am. Chem. SOC. 991, 113, 1854-1855groups at the ambien
t side of the monolayer. A pictorial structureof an isolated
adsorbate molecule that incorporates featuresconsistent with ou r
data is given in Figure 3. Recent measure-ments on a monolayer
consisting of a diacid w ith a (CD2)6 segmentat the cen ter of the
chain demo nstrate that the fold must consistof at least six
methylene groups.28 These experiments, additionalones involving
more complex folded molecular structures, andfurther
characterization of the ordering in such structu res willbe
reported in detail in future publications.
Acknowledgment. Support from the N ational Science Foun-dation [
D M R 900-1270 (D.L.A.) and D M R 87-01586 ( R . G . S . ) ]and the
Natioinal Institutes of Health [ G M 27690 ( R . G . S . )
]sgratefully acknowledged.Registry No. Ag, 7440-22-4;
H02C(CH2)&02H, 14604-28-5.
Principle of Maximum HardnessRobert G . Parr* and Pratim K.
Chattaraj'
Department of Chemistry, University of North CarolinaChapel
Hill, North Carolina 27599Received Sep tember I I , I990
From his considerable experience with the concepts of
chemicalhardness and softness, in 1987 Ralph Pearson concluded
that"there seems to be a rule of nature that molecules
arrangethemselves so as to be as hard as possible." Subsequ ent
studiesof particular problems support this principle and imply that
itsvalidity may require conditions of constant temperature
andchemical potentiaL2 Following is a formal proof of the
principle,as so modified.Absolute hardness 7 nd absolute softness
S,of the equilibriumstate of an electronic system at temperature T
, are defined by
where p is the electronic chemical potential (constant through
the~ y s t e m ) , ~ is the number of electrons, and v ( r ) is the
potentialacting on an electron at r due to the nuclear attraction
plus suchother extern al forces as may be present. These
definitions arethe finit e-temp erature extensions of the ground
-state definitionsof these The chemical potential is the Lag rang
emultiplier for the norm alization constraint in the
finite-temperaturedensity-functional It also is the negative of the
absoluteele~tronegat ivi ty.~n terms of the Helmholtz free energy,
p =( C ~ A / B N ) ~ , ~ ) , , .ote that u ( r ) constant implies
total volumeconstant.Imagine a grand canonical ensemble consisting
of a largenumber of perfect replicas of a particular electronic
system ofinterest, i n equilibrium with a bath a t temperature Ta
nd chemicalpotential k , with the b ath also controlling v ( r ) .
The members ofthe ensemble may exchange energy and electrons with
each other.Equilibrium averages being denoted with brackets, N will
fluctuateabout ( N ) . The equilibrium softness is given by5
*Permanent address: Department of Chemistry, Indian Institute
ofTechnology, Kharag pur, 721 302, India.( I ) Pearson, R. G . J .
Chem. Educ. 1987, 64, 561-567.(2) (a) Zhou, 2.; Parr , R. G. ;
Garst , J . F. Teirahedron Leti. 1988, 29 ,4843-4846. (b ) Zhou,
Z.; Parr , R. G. J . Am. Chem. Soc . 1989, I l l ,7371-7379. (c)
Zhou, Z.; Parr , R. G. J . Am. Chem. Sor . 1990, 112,5720-5724.( 3
) Parr, R . G.;Donnelly, R. A .; Levy, M .; Palke, W . E. J . Chem.
Phys.(4) Parr, R. .; Pearson, R . G. J . Am. Chem. Soc. 1983,
105,7512-7516.( 5 ) Yang, W.; arr , R. G. Proc. Natl . Acad. Sci .
U.S.A. 1985, 82 ,
1978, 68, 3801-3807.6 7 2 3 4 7 2 6 ._._.(6) Parr , R . G.;
Yang, W . Density-Functional Theory of Atoms and
(7 ) Mermin, N . D. Phys. Reo. 1965, 137, A1441-Al443.Molecules;
Oxford: New York, 1989.
where p = I / k T . Or,(S ) = PCpoN,i(N- ( N ) ) 2
p0N.i = ( l / z ) exp[-P(ENi - NP)I
Z = CeXp[-P(ENi - N p ) ]
( 3 )N,iwhere
(4 )
( 5 )an d
N,iis the gran d partition function. Th e probabilities & i
definethe equilibrium distribution. They ar e functions of the
parametersthat characterize the ensemble: P, p , an d u ( r ) ;
dependence onv ( r ) comes through the fact that the EN i depend on
v ( r ) .Now consider the countless other, nonequilibrium
ensemblesof the same system of interest, at temperature T but
characterizedby probabilities PN,idifferent from the canonical
probabilities PoN,[.An average in an y such ensemble may be denoted
with overbars,as for example = @CN,iPN,i(N-IV))~.onsider only
thoseof these nonequilibrium ensembles that can be generated
asequilibrium ensembles by changing the bath parame ters /I andu(r
) ,by small amou nts. For any one of these, it will be shown
that(6)
Consequently, among all these states, the equilibrium state
maybe characterized as having minimum softness.Proof of eq 6
follows from the fluctuation-dissipation theoremof statistical
mechanics.* For simplicity employing classicalstatistical
mechanics, let the equilibrium probability distributionfunction for
the system of interest, with grand potential Q(r",f)= H(rN,pN) p N
, be f ( r N , p N )corresponding to P",,,)and let thecorresponding
arbitrary nearby distribution be F(rN,pN)corre-sponding to PN,i).
Then the physical perturbation A Q ( r N , p N )generating F at t
ime t = 0 must satisfy
S - ( S ) = P C ( N - ( N ) ) 2 ( P N , j Po$,[) 0N, i
( 7 )= ( A ) - ' A ( r N ,p N )( r N , p N ) (8 )
whereC A ( r N , p N ) exp(-pAQ) and AQ .
f ( r N , p N ) nd ( A ) are independent of time, and C s a
positiveconstant serving the purpose of a field component of the
per-turbation which couples with A ; other quantities depend on
time.Equation 8 shows that the conditions are satisfied for
Exercise8.3, p 242, of ref 8. Accordingly, it follows that( A ) [ A
( t ) ( A ) ]= ( ( A ( 0 )- ( A ) ) ( A ( t ) ( A ) ) ) (10)
A ( 0 ) - ( A ) = ( A ) - ' ( ( A ( O ) ( A ) ) * ) ( 1 1 )an
d
Here A( t ) s the average of A ( t ) or the nonequilibrium
distributionF. Now take A to be the observable that is the
softness,A = @ (N ( N ) ) 2S(0)- (S ) L 0
(12)
(13)Since this ( A ) s positive, eq I 1 implies
which is the inequality of eq 6.(8) C handler, D. Introduciion
to Modern Statistical Mechanics;Oxford:New York, 1987; Chapter
8.
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Society
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J . Am. Chem. Sot .The derivation is finished if the class of
neighboring statesdefined by eqs 9 and 12 , tha t is, by
C @ ( N- ( N ) ) = exp(-PAO) (14)can be shown to include all
possible physically acceptable nearbynonequilibrium distributions
th at can evolve into the given originalequilibrium distribution.
Presumably it does, as eq 14 is a minimalrequirement to produce ( A
) = (S), nd such assumptions arestanda rd in traditional linear
response theory. Note th at constantT and p are required because
the averaging after the perturbationis removed is done with the
pN,ir f ( r N , p N ) .Is there certainly a rule of nature that
chemical systems atconstant temperature and chemical potential
evolve towardminimum softness or maximum hard ness? A major concern
withthe proof just given should be whether statistical mechanics
indeedapplies to electrons in individual molecules in th e sense
employed.Evidence is accumulating tha t it does, a t least to some
reasonableaccuracy,6 but one should not, a t this time, imply
certainty.Nevertheless, with some confidence one may assert, with
Pearson,the maxim um hardness principle. If one also asserts, with
San -derson? the electronegativity (chemical
p~tential)~qualizationprinciple, then one has reached two basic,
broadly applicableelectronic-structure principles.
Acknowledgment. Comments of Zhongxiang Zhou, WeitaoYang , and
Man oj Harbola have been helpful. Financial supportfrom the
National Science Foundation is gratefully acknowledged.(9)
Sanderson, R. T. Science 1981, 114 , 670-672.
HSAB PrinciplePratim K. Chat taraj ,+Hsing Lee, and Robert G.
Parr*
Department of Chemistry, University of North CarolinaChapel
Hill, North Carolina 27599Received September 11 , I990
It is generally accepted th at a variety of acid-base reaction
scan be described by the HS AB principle: Ha rd likes hard an dsoft
likes soft. It is not, however, easy to theoretically establishthis
principle. He re two proofs are offered of this principle witha
restriction a dded : Amo ng potential partn ers of a given
elec-tronegativity, hard likes hard and soft likes soft.For the
ground state of an N-electron system at 0 K, or for theequilibrium
state a t the temperature T , according to density-functional theo
ry2 hardn ess and softness are given by3*4
where p an d v ( r ) are the chemical and external potentials,
re-spectively. When two species comb ine to give a third,
theirchemical potentials are equalized. Chem ical potential is
thenegative of electronegativity. Th e maximum hardness principleor
minimum softness principle states that, at constant p , u(r) ,andT
, systems evolve toward minim um to tal softness.5s6First Proof.
When acid A and base B interact to give AB,two things happen, which
can be taken as happening in succession.First there is a charge
transfer producing a common chemicalpotential, and then there is a
reshuffling of the ch arge distributions.In the first step there is
an energy gain proportional to the squarePermanent address:
Department of Chemistry, Indian Institute of( I ) Pearson, R. G. .
Am. Chem. Soc . 1963, 85 , 3533-3539.(2) Parr, R. G.;Yang, W .
Density-Functional Theory of Atoms and(3) Parr, R. G.; earson, R.
G. . Am . Che m .Soc .1983, 105, 7512-7516.(4) Yang, W.; Parr, R.
G. Proc. Narl. Acad. Sci. U.S.A. 1985, 82,(5) Pearson, R.G . .
Chem.Educ. 1987, 6 4 , 561-567.(6) Parr, R . G.; hat taraj , P . K
. J . A m . Chem. SOC. , receding commu-
Technology, Kharagpur 721 302, India.Molecules; Oxford: New
York, 1989.6723-6726.nication i n this issue.
1991, 13 , 1855-1856 1855of the original chemical potential
difference and inversely pro-portional to the resistance to charge
transfer which is the hardnesssum. The speci fi c o ld fo r m ~ l a
, ~
need not be precisely valid; the essential point is the
indicateddependence on softnesses. Th e greater S A S B / ( S A +
SB)s, themore stabilizing is the charg e transfer. For a given SA,
he largerSB s, the better.In the n ext, reshuffling step, which is
at cons tant p an d T , heminimum-softness principle applies. Th e
total softness is, at leastroughly, SA+ SB y application of eq 1 to
nonoverlapping A plusB with a total number of electrons NA + NE.
The preferencein this step is for SAan d SB o be as small as
possible. So, fora given SA, he smaller SB s, the better. Thus
there are twoopposing tendencies, and the optimum situation will be
a com-promise.Suppose SA s fixed, let SB= asA, nd consider th e
questionof what will be the best value of SB r a. The supremum
valueof a/ ( + a ) , 1 , corresponds to maximum initial energy gain
andmaximum softness, while the infimum value, 0, has these
rolesreversed. If nothing is known about the relative importan ce
ofthe two conditions, a most natural compromise would be to takethe
average value, viz., a / ( l + a ) = This is the HSABprinciple: a =
I . In this analysis SAan d SB re softnesses either
before or after the chemical potential equalization step;
fortunatelythese quantities are known to be insensitive to the
number ofelectrons.* Th e final chemical bond formation may be
assumednot to much affect the partitioning SA+ SB nd not to much
affectthe components SA an d SB .To quantify the argument: For a
given Ap an d SA, he problemis to find the most favorable SB , r
the most favorable a . Fromeq 2 one would want a / ( + a ) o be as
large as possible, whichfavors a large a value. But to minimize S =
SA+ SB= SA(+ a) ne would want a to be as small as possible. Simult
aneou ssatisfaction of the two conditions being impossible, what on
e mustassess is the relative impor tance of the two conditions. Let
X bethe weight of the first relative to the second. Then a is
determinedfromd[(l a ) A (& ) ] = 0d a (3 )
This gives a = vX - 1. For a fixed value of X ( X may be hopedto
be a more or less universal constant), the result is that,
fromamong a series of Bs with the same chemical potential
(elec-tronegativity), a given A will prefer one, and similarly with
otherAs. Th e HS AB principle follows precisely if one takes X =
4,for then a = 1 an d SB SA . An argument that X should be closeto
4 s not easy to construct.Note that AB + AB - B + AB is predicted
to be ahardness-raising or softness-lowering process if SA= SB,
A,=S B r . The maximum hardness (minimum softness) principle
isdemanded by the HSAB principle.Second Proof. To obta in a second
proof, rewrite eq 2 in theformA E = ARA + ARB (4)where
( p B - f i A ) ?AA R A = -(? A + ?E)
Assume that for a given pB- p A an d vB, AQ A is minimized
withrespect to q,,. There follows(7) Nalewajski, R. F.; Korchowiec,
J . ; Zhou, Z . In ? . J . Quantum Chem.(8) Fuentealba, P.; Parr,
R. G. . Chem. Phys. I n press.1988, 22, 349-366.
0002-7863/91/1513-1855$02.50/0 0 1991 American Chemical
Society
