1854 J. Am. Chem. Soc. 1991, 113,
1854-1855
groups
at the ambient side of the
monolayer.
A
pictorial
structure
of an isolated adsorbate molecule that
incorporates
features
consistent with our data is
given
in
Figure
3. Recent measure-
ments on a
monolayer consisting
of a diacid with a
(CD2)6 segment
at the center of the chain demonstrate that the fold must consist
of at least six
methylene groups.28
These
experiments,
additional
ones
involving
more
complex
folded molecular
structures,
and
further characterization of the
ordering
in such structures will
be
reported
in detail in future
publications.
Acknowledgment. Support
from the National Science Foun-
dation
[DMR
900-1270
(D.L.A.)
and DMR 87-01586
(R.G.S.)]
and the Natioinal Institutes of Health
[GM
27690
(R.G.S.)]
is
gratefully acknowledged.
Registry
No.
Ag, 7440-22-4; HO2C(CH2)30CO2H,
14604-28-5.
Principle
of Maximum Hardness
Robert G. Parr* and Pratim K.
Chattaraj*
Department of Chemistry, University of
North Carolina
Chapel Hill, North Carolina 27599
Received
September 11, 1990
From his considerable
experience
with the
concepts
of chemical
hardness and softness,
in 1987
Ralph
Pearson concluded that
“there seems to be a rule of nature that molecules
arrange
themselves so as to be as hard as
possible.”*
1
Subsequent
studies
of
particular problems support
this
principle
and
imply
that its
validity may require
conditions of constant
temperature
and
chemical
potential.2 Following
is a formal
proof
of the
principle,
as so modified.
Absolute hardness and absolute softness
S,
of the
equilibrium
state of an electronic
system
at
temperature T,
are defined
by
where
µ
is the electronic chemical
potential (constant through
the
system),3
N is the number of electrons,
and
v(r)
is the
potential
acting
on an electron at r due to the nuclear attraction
plus
such
other external forces as
may
be
present.
These definitions are
the
finite-temperature
extensions of the
ground-state
definitions
of these
quantities.4"6
The chemical
potential
is the
Lagrange
multiplier
for the normalization constraint in the
finite-temperature
density-functional theory.6,7
It also is the
negative
of the absolute
electronegativity.3
In terms of the Helmholtz free
energy, µ
=
{dA/dN)^,)j.
Note that
u(r)
constant
implies
total volume
constant.
Imagine
a
grand
canonical ensemble
consisting
of a
large
number of
perfect replicas
of a
particular
electronic
system
of
interest,
in
equilibrium
with a bath at
temperature
T and chemical
potential µ,
with the bath also
controlling v(i).
The members of
the ensemble
may exchange energy
and electrons with each other.
Equilibrium averages being
denoted with
brackets,
N will fluctuate
about
(N).
The
equilibrium
softness is
given by5
Permanent address:
Department
of
Chemistry,
Indian Institute of
Technology, Kharagpur,
721 302, India.
(1) Pearson, R. G. J. Chem. Educ. 1987, 64, 561-567.
(2) (a) Zhou, Z.; Parr, R. G.; Garst, J. F. Tetrahedron Lett. 1988, 29,
4843-4846.
(b) Zhou, Z.; Parr, R. G. J. Am. Chem. Soc. 1989, 111,
7371-7379. (c) Zhou, Z.; Parr, R. G. J. Am. Chem. Soc. 1990, 112,
5720-5724.
(3) Parr, R. G.; Donnelly,
R. A.; Levy, M.; Palke, W. E. J. Chem.
Phys.
1978, 68, 3801-3807.
(4) Parr, R, G.; Pearson, R. G. J. Am. Chem. Soc. 1983, 105, 7512-7516.
(5) Yang, W.; Parr, R. G. Proc. Natl. Acad. Sci. U.S.A. 1985, 82,
6723-6726.
(6) Parr, R. G.; Yang,
W.
Density-Functional Theory of
Atoms and
Molecules; Oxford: New York, 1989.
(7) Mermin, N. D.
Phys.
Rev. 1965, 137, A1441-A1443.
[d(N)
1
-3—
=ß(( - </V»2> (2)
Jvtrl.r
where
ß
=
1
/kT.
Or,
<5>
=
ß °»
-
<7V»2 (3)
N,i
where
n,,
=
(1/2) exp[-/?(£M·
-
TVM)] (4)
and
2
=
LexpHtFw
-
µ)] (5)
N,i
a is the
grand partition
function. The
probabilities F%j
define
the
equilibrium
distribution.
They
are functions of the
parameters
that characterize the ensemble:
ß, µ,
and
y(r); dependence
on
u(r)
comes
through
the fact that the
Em depend
on
u(r).
Now consider the countless
other, nonequilibrium
ensembles
of the same
system
of
interest,
at
temperature
T but characterized
by probabilities PNi
different from the canonical
probabilities P°Nj.
An
average
in
any
such ensemble
may
be denoted with
overbars,
as for
example
S
=
ß ,/ ,/ 7- ( 7))2.
Consider
only
those
of these
nonequilibrium
ensembles that can be
generated
as
equilibrium
ensembles
by changing
the bath
parameters µ
and
y(r), by
small amounts. For
any
one of
these,
it will be shown
that
5
-
(5)
=
ß (
-
(N)HPn¡í
-
P%,,)
>
0
(6)
N,i
Consequently, among
all these
states,
the
equilibrium
state
may
be characterized as
having
minimum softness.
Proof of
eq
6 follows from the
fluctuation-dissipation
theorem
of statistical mechanics.8 For
simplicity employing
classical
statistical mechanics,
let the
equilibrium probability
distribution
function for the
system
of
interest, with
grand potential ^, 7'7)
=
H(rN,pN)
-
µ ,
be
/(r7V,p7V) (corresponding
to
P°NJ)
and let the
corresponding arbitrary nearby
distribution be
F(rN,pN) (corre-
sponding
to
PNf).
Then the
physical perturbation ( 7 , )
generating
F at time t
=
0 must
satisfy
F(r7V,p7V)
(-ß )
( (-0 )>
/( p") (7)
=
(A)-1 A(rfp")/(rfp") (8)
where
CA(rN,pN)
=
(-/3 )
and
=
In
[C4(r",p")] (9)
f(rN,pN)
and
(A)
are
independent
of time, and C is a
positive
constant
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 Exercise
8.3, p 242,
of ref 8.
Accordingly,
it follows that
(A)[A(t)
-
(A)]
=
<(A(0)
-
(A))(A(t)
-
(A))) (10)
and
A(0)-(A)
=
( )- ( {0)
-
(A))2) (11)
Here
A{t)
is the
average
of
A(t)
for the
nonequilibrium
distribution
F. Now take A to be the observable that is the softness,
A
=
ß(
-
(TV))2 (12)
Since this
{A)
is
positive, eq
11
implies
5(0)
-
(5)
>0
(13)
which is the
inequality
of
eq
6.
(8) Chandler, D. Introduction to Modern Statistical Mechanics; Oxford:
New York, 1987; Chapter
8.
0002-7863/91 /1513-1854S02.50/0
© 1991 American Chemical
Society
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