J. Phys. Chem. 1994, 98, 1989-1992
1989
Principle of Maximum Physical Hardness Ralph C. Pearson Chemistry Department, University of California, Santa Barbara, California 93106 Received: October 14, 1993"
The proof of Parr and Chattaraj for the principle of maximum chemical hardness is analyzed in some detail. It is shown that it is very general and can be applied to many more observables. The case of the physical, or mechanical, hardness of a solid is taken as an example. It is shown that this also should be a maximum in an equilibrium system. Assuming the validity of the argument leads to new information about the compressibility of solids. A simple expression is also given for the Griineisen constant.
Recently Parr and Chattaraj (PC) gave a proof for a principle of maximum hardness.Is2 In this case hardness, 9 , referred to absolute, or chemical, hardness.3 The definition is 2?7 = ( 6 p / 6 N ) , = 21.
(1)
where p is the electronic chemical potential, N is the number of electrons in a chemical system, and v is the potential due to the nuclei. The softness, u, is the reciprocal of the hardness. The purpose of the present paper is to elucidate the proof of PC, to show that it is general and can be applied to many other observables, and to give examples of such applications. In particular, it is applied to the physical, or mechanical, hardness of solids. The very satisfying result is that the physical hardness, at equilibrium, is also a maximum. The proof of PC depended on the properties of grand canonical ensembles. The softness can be written as the fluctuation in N from the average value (N).4
= (6",,T
= P ( W - (W2)
(2)
where B = l/kT, as usual. Soft systems have large fluctuations. We next compare the fluctuations of an equilibrium ensemble with those of nonequilibrium ensembles with the same value of p as the equilibrium system, and where the departure from equilibrium is small. Since equilibrium in a grand canonical ensemble is determined by the grand potential, fl = E - Np, the nonequilibrium ensembles have a more positive energy. As time goes on,the nonequilibrium ensembles will relax toward equilibrium. The excess energy will be degraded to heat, by a mechanism of molecular collision. This means that the fluctuation-dissipation theorem will apply.6 This theorem dates back to early work by Nyquist' and Onsager6 and has been generalized by Callen and Weltonag In essence, it says that small deviations from equilibrium have the same relaxation times, whether they are spontaneous or induced. This follows because the mechanism is the same. The theorem applies to any observable, A, which relaxes from the nonequilibrium value 2,averaged over the ensemble, to the equilibrium value ( A ) . The relaxation of a spontaneous fluctuation in an equilibrium system is given by the time correlation function, C ( t ) . C ( t ) = ((40)- ( A ) ) ( A ( t )- ( A ) ) )=
(40)4 0 ) - ( A ) 2 (3) The correlation function decreases from a maximum value at t = 0 to zero at times long compared to the relaxation time, 7 . The fluctuation-dissipation theorem can then be written as Abstract published in Aduance ACS Abstracts, February 1, 1994.
(4)
This establishes that @ ( t ) - ( A ) )is directly proportional to C(t), but it does not give the sign of the constant of proportionality. Some observables in a nonequilibrium system are less than the equilibrium value, and some are greater. For example, consider concentrations of reactants and products in a chemical equilibrium. What Parr and Chattaraj did was to show that the constant of proportionality was simply equal to ( A ) - l , the average equilibrium value of A . The PC proof involves manipulation of probability distribution functions and follows an example given by Chandler.lO Accordingly, we can write
Now take A to be the observable that is the softness A = u. From eq 2, it is seen that A is always positive. Therefore in eq 5 all quantities on the right hand side are positive, and it must be true that
(A(0)- ( A ) )= (a - (a)) 1 0
(6)
Thus the equilibrium ensemble has the minimum softness, or the maximum hardness. The beauty of the above proof is that the specific example of softness is not introduced until the last step. The proof, then, is valid for many other observables, provided certain restrictions are obeyed. One is that ( A ) must be positive; another is that the fluctuation-dissipation theorem must be applicable, and finally, certain variables must be held constant. These are the temperature, electronic chemical potential, and nuclear positions in the case of chemical softness.' They will be temperature and other variables in other cases. It seems likely that the restriction to constant p can be relaxed to some degree. The essential requirement is that the energy, E , be greater in the nonequilibrium ensembles. Evidence is accumulating on this issue.2Jl Equation 2 suggests that fluctuations in equilibrium systems will offer many examples of the minimum (maximum) principle. These are usually presented in terms of the variance, or second central moment, since the first moment, or average, is zero. The variance is always positive, as required, and its magnitude is usually taken as an inverse measure of goodness. That is, we want the variance of a measured, or calculated, variable to be as small as possible. It seems entirely reasonable that the equilibrium system would have the smallest variance. For example, the magnetic susceptibility x of a sample is given by the fluctuations of the local magnetization from its average value. x measures how easy it is to change the average
0022-365419412098- 1989%04.50/0 0 1994 American Chemical Society
1990 The Journal of Physical Chemistry, Vol. 98, No. 7, 1994
magnetization in the presenceof a magnetic field. Theequilibrium state, being the most stable, resists changes the most and has a minimum value of x. New information can be obtained by applying the principle of minimum variance. The fluctuations of the energy are given by1*
If T is constant, we conclude that the heat capacity of the equilibrium state is a minimum, compared to nearby nonequilibrium states. Clearly in the latter states, if there is excess energy, there will be a greater spread of energy values. A larger value of C, automatically follows form (7). The Principle of Maximum Physical Hardness Equations such as (1) and (2) have their origin in classical thermodynamics, where N is the number of molecules, rather than electrons, and p is the ordinary chemical potential. The analogous equations to (1) and ( 2 ) are
(aN/&),T = KN’/V= P ( ( N - ( N ) ) ’ )
(8)
where K is the compressibility. We see that the compressibility, which is certainly a measure of the physical, or mechanical, softness, is analogous to the chemical softness, u. In fact, Parr has shown that K - ~ Vwhere O, VOis the molecular volume, for various crystalline solids, is proportional to the crystal hardness as measured by the Mohs scale, and other such ~ca1es.l~ Thus, it seems reasonable to call K-~VO, which has the units of energy, the physical hardness. Arguing as before, we can then say that the equilibrium state of a condensed system, such as solids and liquids, will have a maximum physical hardness. The softness, K / V Owill , be a minimum. This is avery reasonable result. Chemical hardness isa measure of resistance to change of the electron cloud of a system. Physical hardness measures the resistance to change of the nuclear positions in a system. An equilibrium state should have the greatest resistance to change for both properties. It is important to bear in mind that the maxima are local. It is quite possible that a more stable state exists in which both the nuclear positions and the electron cloud have changed markedly. This would be the case in an isomerization, for example. Fortunately, it is possible to test the predictions with a variety of experimental evidence. This has already been done for chemical hardness.2 For physical hardness, we examine K-~VOto see if it is a maximum with respect to small changes in conditions. It is convenient to replace K - ~by its equal, B, the bulk modulus. The required conditionis that G(BV0)= 0. At this point, therestriction will be made to consider only cubic crystals, and only those of a single element, or binary AB compounds. In those cases, we can write VO= cRO3,where c is an irrelevant constant and Ro is the equilibrium distance between nearest neighbors at room temperature. Since the hardness is resistance to change of the nuclear positions, an important test is to see if BVo has a maximum value at Ro. We have the relationships
The potential energy, U, is expanded in a power series of x = ( R - Ro), where x is small.
U = Uo+ f2x2 + gx3+ hx4
(10)
Pearson TABLE 1: Pressure Derivative of the Bulk Modulus for Single Crystals at Room Temperature substance dBldP substance dBldP A1 4.4 Mi30 4.4 BaF2 5.1 KBz 5.4 CaF2 5.5 KCl 5.4 CsBr 5.1 KF 5.0 CSI 5.4 RbBr 5.36 cu 5.5 RbCl 5.5 GaSb 4.1 RbI 5.3 4.5
GaAs
Si
4.2c
Au 5.2 Ag 5.1 Fe 5.3 NaBr 5.0 Pb 5.5 NaCl 5.1 LiF 4.8 NaF 5.2 Data from ref 16; cubic crystals only. At 220 A. Static method. Ignoring h when x is small, (9) and (10) are compatible only if g = -f/Ro. The principleof maximum physical hardness has given a definite relationship between the harmonic force constant, f, and the anharmonic constant, g. We know at once that the negative sign for g is right and that the magnitude of g is right, since the anharmonic term in (10) determines the coefficient of thermal expansion. The variation of the compressibility with pressure also depends on g. For the latter case, the volume as a function of pressure can be written as
+ a2P (dK/dP) = -26 + a2 K
= a - 2bP
(13)
The last two equations are valid at moderate pressures. From the definition of the bulk modulus, we then find
(dB/dP) = (2b/a2)- 1
(14)
Pauling and Waser have shown that14 2g = (1 - b / a 2 )f / R o
Setting g = -f/Ro, we find b/a2 = 3, and (dB/dP) = 5 . Therefore, for cubic solids, the pressure derivative of the modulus is equal to 5.00, a constant and dimensionless number.15 The value of a, the compressibility a t very low pressure, is well-known for many solids. However, b is difficult to determine. Older, static methods are not reliable. Better results are obtained by ultrasonic pulse methods on single crystals, but even here the results of different investigators can differ by 10-20%. Table 1 shows experimental values of (dB/dP) at room temperature.16 They have been averaged when several values are available, and in one or two cases, values markedly different from the mean have been dropped. The closeness of these numbers to the predicted value of 5.00 is indeed remarkable. It may well be that the theoretical value is more reliable than many of the experimental values. However, there are approximations in the theory. For example, writing the pressure as (SU/SV), ignores the contributions of the lattice vibrations to the pressure. This term should be small, but it will contribute varying amounts for different substances.
The Griineisen Constant The thermal expansion of a solid depends on the anharmonic term in (10). Because of this term, the average value of ( R - & ) is different from zero and increases with temperature. A simple treatment, partly classical and partly quantum mechanical, gives17
The Principle of Maximum Physical Hardness Cl$
f f = - - - -
The Journal of Physical Chemistry, Vol. 98, No. 7, 1994 1991
cu
TABLE 2: Criineisen Constants for Selected Covalent Solids at Room Temperature
PRO - fR; where CY is thelinear expansion coefficient. C, is the heat capacity per mol of atoms. The value of the force constant, f, can be found from the compressibility. The procedure is to equate the pressure-volume work to the energy needed to compress the chemical bonds between nearest neighbor atoms. The result for a cubic crystal is given by Waser and P a ~ 1 i n g . l ~
Here n is the number of nearest neighbors, or coordination number, CN, divided by 2, or the number of bonds per atom. VOis the volume per mole of atoms. Equations 16 and 17 can be combined to calculate a. However, it is more interesting to calculate the Gruneisen constant, y. Operationally, this is defined as
Theoretically, y was first defined in terms of the variation of the eigenfrequencies of the solid as the volume changed.’* While (18) can be derived from this definition, using the model of an elastic continuum, the value of y is not calculated. It has values ranging from 0.40 to 2.60 for different substances. Using(18),(17),and (16), thevalueofycannowbecalculated. The result is a simple and novel one, y = 4 3 . Thus y is 0.67 for crystals with the diamond, or zinc blend, structure, y = 2.00 for cubic close packing, and so on. Table 2 contains experimental values of the Gruneisen constant for a number of covalent solids. The number can be compared with the predicted n/3. While the agreement is not quantitative, the trend with changing coordination number is unmistakable. For example, the bcc metals have an average value of y = 1.67, consistent with n = 4. The fcc metals have an average value of y = 2.17, consistent with n = 6. Equation 17 was derived on a model of covalent bonds between nearest neighbors. It is not strictly applicable to ionic solids.19 The repulsive part of the potential energy must be similar for ionic and covalent cases, but the attractive part for ionic solids must also include the sum of the Coulombic interactions with the remainder of the lattice. To see the magnitude of their effects, compare (b2U/8R2)h for a NaCl molecule and for solid NaC1, using the Born-Mayer potential function in both cases.
-( 5 -2)
a2u- e2
diatomic
aR2 Ri P
;a2u ;;;1=Ae2 %(: R
- 2)
(19)
solid
The latter is greater by a factor of A , the Madelung constant. Then ( A - 1) must be the effect of the remainder of the lattice. This suggests using the “corrected” result y = nA/3 for ionic solids. In effect, the number of bonds is increased because of bonding to the remainder of the lattice. Table 3 shows the “corrected” results for a number of ionic solids. The assumption is made that solids with coordination number 6 (rocksalt structure) and coordination number 8 (CsC1 structure) are sufficiently ionic that the “corrected” values are needed.20 This expectation is clearly met. The larger values of y for CN8 compared to CN6 are found, as expected. Only B e 0 has CN4, but still has a value of y that shows ionic bonding. This agrees with recent calculations of the cohesive
substance CC Si Ge
Li Na K Rb Ca
Y
0.86 0.45b 0.72b 1.17 1.25 1.34 1.48 1.29 1.62 1.57 1.57 1.75 2.17 1.87
1113 0.67 0.67 0.67 0.67 1.33 1.33 1.33 1.33 1.33 1.33 1.33 1.33 2.00 2.00
substance Ni
cu
Pd Ag Pt
Y
nl3
1.88 1.86 2.23 2.40 2.54 2.40 2.23 0.49 0.84 0.72 0.54 0.5@ 0.66 0.76
2.00 2.00 2.00 2.00 2.00 2.00 2.00 0.67 0.67 0.67 0.67 0.67 0.67 0.67
Au Pb CdS W CUCl Zn GaAs Mo InAs Ta InSb A1 ZnO co ZnS Data for metals from ref 22. Remainder calculated for this work, except as indicated. Data for calculation from refs 13, 16, and 23. Gibbons, D. F. Phys. Rev. 1958, 112, 136. Diamofid. TABLE 3 Griineisen Constants for Selected Ionic Solids at Room Temperature substance y An13 substance Y Ant3 BeOb 1.54 1.10 AgCl 1.90b 1.75 LiF 1.58 1.75 AgBr 2.056 1.75 LiCl 1.54 1.75 Mi30 1.59c 1.75 NaF 1.51 1.75 CaO 1.5lC 1.75 NaCl 1.57 1.75 SrO 1.52c 1.75 NaBr 1.57 1.75 CsF 1.49 1.75 NaI 1.71 1.75 CsBr 1.93 2.34 KF 1.48 1.75 CSCl 1.97 2.34 KCI 1.45 1.75 CSI 2.00 2.34 KBr 1.43 1.75 TlCl 2.30b 2.34 KI 1.47 1.75 TlBr 2.19b 2.34 RbI 1.50 1.75 Data for the alkali halides from refs 18 and 24. Calculated for this work. Ruppin, R. Solid State Commun. 1972, 10, 1053. energies of this solid, and similar solids.20 The CN4 is forced on B e 0 because the small size of BeZ+will not allow six oxide ions around it (the radius-ratio effect). In contrast, the y of ZnO shows that the bonding is largely covalent and that a highly ionic model is inappropriate. The theory so far has only shown that BVOhas an extremium value, not that it is a maximum. The answer to that question lies in the value of h in (10). This number must be negative, or have a small positive value, in order for BVo to be a maximum. There is very little evidence concerning the magnitude, or even the sign, of the quartic term. It is usually considered to be a “softening” term (h negative).” This is supported by some heat capacity data at high temperatures.21 Only a local maximum is predicted by the theory. Indeed, it must be true that BVbecomes very much larger when R is much less than Ro. At high pressures, solids essentially become incompressible. While the local maximum may be quite shallow, and apparently of little consequence, Tables 1-3 show that important new results can be found nevertheless. Of course, the tables show that y = n/3, or An/3, is only a first approximation. For a given n, the values of y show considerable variation. This is not surprising, since both (16) and (17) are simple approximations for more complex phenomena. The original interpretation of y was that it measured the effect of volume changes on the lattice vibrations. This view, based on a continuum model, is not incompatible with the view based on bonded atoms. A larger number of nearest neighbors could lead to a greater change in frequency for a given change in volume. For an isolated crystal at constant V and T, the chemical potential is given by p = (SA/SN),where A is the Helmholtz free energy. At the equilibrium value of R, p is a minimum. Changes in R will lead to increases in p. However, it appears that this is
1992 The Journal of Physical Chemistry, Vol. 98, No. 7, 1994
not a problem since the energy also increases. As already mentioned, this is the necessary requirement. Summary This work began with an analysis of the proof of Parr and Chattaraj for the principle of maximum chemical hardness. The proof seems to be valid, and the restriction to constant electronic chemical potential can be relaxed to some extent. It is pointed out that the proof is very general and can be applied to many other observables. A requirement is that theobservable must always havea positive sign. This means that variances can often be shown to have a minimum value in an equilibrium system. This seems to be a very reasonable result. A fertile field for application of the PC proof is that of fluctuations, which are usually given as thevariance of the property. An important example of fluctuations is that of the particle number, N , in theseparate systems of a grand canonical ensemble. Minimizing this quantity leads to a conclusion that K/Vo is a minimum, or BVOis a maximum, at equilibrium. This in turn leads to new information about experimental numbers, much as the pressure derivative of the bulk modulus, and Griineisen’s constant . While these results are interesting, they are not sufficiently quantitative to be considered definite confirmation of the PC proof. Further limitationson thearguments of Parr and Chattaraj may still exist. For this reason, it cannot be concluded that all fluctuations and variances are a minimum at equilibrium. Acknowledgment. The author wishes to thank W. E. Palke and R. G. Parr for helpful discussions. References and Notes (1) Parr, R. G.; Chattaraj, P. K. J . Am. Chem. SOC.1991, 113, 1854.
Pearson (2) See: Pearson, R. G. Acc. Chem. Res. 1993,26, 250. Parr, R. G.; Zhou, Z. Ibid., 225. (3) Parr, R. G.; Pearson, R. G. J . Am. Chem. Soc. 1983,106,7512. (4) Yang, W.; Parr, R. G. Proc. Natl. Acad. Sci. U.S.A. 1985.82,6723. (5) Parr, R. G.; Yang, W. Density Functional Theory for Aroms and Molecules; Oxford Press: New York, 1989. (6) For general discussions, see: Forster, D. In Hydrodynamic Nuctuations, Broken Symmetry and Correlation Functions; Benjamin, W.A., Ed.; New York 1975. Keizer, J. Staristical Thermodynamics of Nonequilibrium Processes; Springer-Verlag: New York, 1987. (7) Nyquist, H. Phys. Rev. 1928, 32, 110. (8) Onsager, L.Phys. Rev. 1931, 37, 4053;38,2265. (9) Callen, H. B.; Welton, T. A. Phys. Rev. 1951,83, 34. (10) Chandler, D.Introduction to ModernStatistical Mechanics;Oxford Press: New York, 1987. (11) See: Gasquez, J. L.;Martinez, A. L.; Mendez, F. J. Phys. Chem. 1993,97,4055. (12) For an elementary discussion of fluctuations, see: McQuarrie, D. A. Statistical Mechanics; Harper and Row: New York, 1973;Chapter 3. (13) Yang, W.; Parr, R. G.; Uytterhoeven, L.Phys. Chem. Miner. 1987, 15, 191. (14) Waser, J.; Pauling, L. J . Chem. Phys. 1950,18, 747. (15) In the notation of Waser and Pauling, f = k and -6g = k’. (16) Simmons, G.; Wang, H. Single Crystal Elastic Constants; MIT Press: Cambridge, MA, 1971. (17) Kittel, C. Introduction to Solid Srate Physics, 2nd 4.; John Wiley and Sons: New York, 1956;p 152ff. Vonsovsky, S. V.; Katsnelson, M. I. Quantum Solid Stare Physics; Springer Verlag: New York, 1989;p 121. (18) For example, see: Ashcroft, N. W.; Mermin, N. D. Solid State Physics; Saundern: New York, 1976;p 492ff. (19) Pearson, R. G. J. Am. Chem. Soc. 1977,99,4969. (20) Pearson, R. G. J. Mol. Struct. Theochem. 1992,260,11. (21) quations, such as the Born-Mayer equation, for the potential energy are not reliable enough to calculate the cubic terms, much less the quartic terms. (22) Zwikker, C. Physical Properties of Solid Materials; Interscience: New York, 1958;p 156. (23) Krishnan, R. S. Thermal Expansion of Crystals; Pergamon Press: New York, 1979. (24) Born, M.;Huang, K. Dynamical Theory of Crystal Lattices; Clarendon Press: Oxford, 1954; p 52.
