Maximum Chemical and Physical Hardness

Research: Science & Education

Maximum Chemical and Physical Hardness Ralph G. Pearson Chemistry Department, University of California, Santa Barbara, CA 93106

The purpose of this article is to bring the reader up to date on the subject of chemical hardness and its relationship to the more common physical, or mechanical, hardness. Recent advances particularly include the postulates that, at equilibrium, these properties have maximum values with respect to small departures from equilibrium. These developments on hardness come from the application of density functional theory (DFT). Density functional theory (DFT) is a form of quantum mechanics that uses the one-electron density function, ρ, instead of the more usual wave function, ψ , to describe a chemical system (1). Such a system is any collection of nuclei and electrons. It may be an atom, a molecule, an ion, a radical or several molecules in a state of interaction. Hohenberg and Kohn proved in 1964 that the groundstate energy of a chemical system is a functional of ρ only (2). A functional is a recipe for turning a function into a number, just as a function is a recipe for turning a variable into a number. For example, the energy is also a functional of the wave function. The variational method is one recipe for turning ψ into a number, E. The density, ρ, can be obtained by squaring ψ (which is assumed to be antisymmetric) and integrating over the coordinates of all the electrons but one. This is then multiplied by N, the total number of electrons, to get the number of electrons per unit volume, ρ, which is a function of the three space coordinates only. It is a quantity easily visualized and experimentally measurable by X-ray diffraction, though the accuracy is not adequate for chemical purposes. There are many important applications of DFT to chemistry. One is in the calculation of properties of atoms and molecules. These are much easier than calculations using very good wave functions, and of similar accuracy. For a system containing N electrons, the wave function depends on 4N space and spin coordinates. The electron density depends on only three space coordinates, and N enters as a simple multiplicative factor. Hence computations are very much simpler in DFT, especially for large systems. Another important use for DFT is in elucidating familiar chemical concepts. Since 1975, R. G. Parr and his coworkers have been particularly active in this area. Two new properties of a chemical system arise from this work. They are the electronic chemical potential (3), µ, and the hardness (4), η. These are of great importance in determining the behavior of the system, and they lead to broadly applicable and useful principles. The definitions of these quantities are µ = (∂ E/∂N )v ;

2η = (∂µ/∂N )v = 2/σ

(1)

where E is the energy, N is the number of electrons, and v the potential due to the fixed nuclei. The softness, σ = 1/η, is also a useful quantity. The name electronic chemical potential comes from the thermodynamic equation

µ0 = (∂E/∂N)S,V

(2)

Here µ0 is the ordinary chemical potential and N is the number of molecules. The two, µ and µ0, are alike in that, at equilibrium, they must be constant everywhere in the system. Classical thermodynamics also gives us (∂µ0/∂N )V,T = ᎑V 2( ∂P/∂V )/N 2 = V/N 2 κ

(3)

where κ is the compressibility and N is again the number of molecules. Comparison with eq 1 shows that the chemical softness is analogous to the mechanical softness, or compressibility. The correspondence between µ and µ0 arises from an old practice in solid state physics and electrochemistry. In both of these fields, it is often useful to think of electrons as independent components of a system, and not just appendages of the atoms. The novelty introduced by Parr is to take N as the number of electrons in a single molecule, and not the total number of electrons in a macroscopic sample. In place of the definition in eq 1, we also have alternative definitions which come from the calculus of variations (1). µ = [δE/δρ]v ;

2η = [δµ/δρ] v

(4)

The use of δ in eq 4 is read as “the functional dependence of E and µ on ρ”. This serves to remind us that the chemical hardness measures the resistance to changes in ρ. But ρ can be changed in two ways: by changing the number of electrons, N, or by changing the one-electron density function, ρ/N, keeping N constant. Operational, and approximate, definitions of µ and η are ᎑ µ = (I + A)/2 = χ ;

η = (I – A)/2

(5)

where I is the ionization potential and A is the electron affinity of the system; χ is the absolute electronegativity, similar, but not equal, to the Mulliken electronegativity. If we draw a plot of E vs N for any system, then µ is simply the instantaneous slope of such a curve. There is no assurance that this curve will be continuous. Experimentally we only know points on the curve for integral values of N, from data such as ionization potentials and electron affinities. We do not know the instantaneous slope of the curve. However, if we pick the neutral species (or any other) as our starting point, we do know the mean slope for the change from (N – 1) to N electrons. It is simply equal to ᎑I. Also, the mean slope from N to (N + 1) electrons is simply ᎑A. Using the method of finite differences, we can approximate the slope at N as ᎑(I + A)/2. In the same way, 2η is the curvature of a plot of E vs N. Using finite differences again gives η = (I – A)/2. The factor of 2 was added arbitrarily to make µ and η symmetrical. For a bound system, µ will always be negative. But η will be positive at all times. According to Koopmans’ theorem, the ionization

JChemEd.chem.wisc.edu • Vol. 76 No. 2 February 1999 • Journal of Chemical Education

267

Research: Science and Education

potential is simply the orbital energy of the HOMO, with change in sign. For spin-paired molecules, the electron affinity is the negative of the orbital energy of the LUMO; therefore, on an orbital basis, we can write µ = (εHOMO + εLUMO)/2; η = (εLUMO – εHOMO)/2

(6)

If we make the usual diagram of the MOs of a molecule as a function of their energies, µ is a vertical line halfway between the HOMO and the LUMO. The hardness is just half the energy gap between the two (5). It is useful to relate µ (or ᎑ χ) and η to the usual MO energy diagrams. Figure 1 shows such a diagram for a molecule where I = 10 eV and A = ᎑2 eV. Then χ = 4 eV and η = 6 eV. The figure also shows the case of a free radical where the frontier orbital (SOMO) is half-filled, and where I = 10 eV and A = 2 eV. Then χ = 6 eV and η = 4 eV, as shown. The unknown energy of the LUMO plays no role. (I – A) = 2η is just the mean energy of repulsion between the two electrons in the SOMO (6 ). For the usual case of doubly occupied orbitals, we see that a hard molecule has a large energy gap and a soft molecule has a small energy gap. In MO theory, the energy gap between the HOMO and the LUMO also defines the energy difference between the ground state and the manifold of excited states of the same multiplicity, singlet in this case. This energy gap is of great importance, since the quantum mechanical theory of response to a perturbation in a chemical system depends on this difference. Changes in electron density occur by mixing excited-state wave functions with the groundstate wave function. The mixing coefficient is inversely proportional to the energy difference between the states. Therefore soft molecules undergo changes in electron density, ρ, more readily than hard molecules. There are also symmetry restrictions on the mixing of wave functions, but these need not concern us here. The foregoing suggests that it is a good thing for a chemical system if it can arrange itself to be as hard as possible (7 ). Since chemical reactions obviously require changes in the electron density distribution pattern, we expect soft molecules to be more reactive than hard molecules, in general. This is indeed the case. For example, H2S, with η = 6.2 eV, is certainly more reactive and less stable than H2O,

Figure 1. Orbital energy diagram for a molecule (left) and a radical (right). The hardness, η, and µ, the electronic chemical potential, are shown. SOMO is singly occupied MO.

268

with η = 9.5 eV. H2Se and H2Te are softer and more reactive still. Many more examples of the relationship between reactivity and the size of the HOMO–LUMO energy gap may be cited (8, 9). Theoreticians who do calculations on molecular structures always seem to find that the thermodynamically most stable structure is the one with the largest gap between the occupied and the empty orbitals (7, 9). For a given basis set of atomic orbitals, the solution which gives the lowest energy also gives the largest value of η (9). As atoms or radicals combine to give stable molecules, the hardness always seems to increase as the energy decreases (9, 10). What happens when two systems approach each other to form a single more stable system is that the HOMO goes down in energy and the LUMO goes up. Figure 2 illustrates this for the case of covalent bonding, using H2+ as a simple example. The HOMO is the bonding 1σg orbital, and the LUMO is the antibonding 1σ u orbital. As the two nuclei approach each other, the separation between the orbital energies increases until R = 0. At zero internucleus separation, the 1σg orbital becomes the 1s orbital of He+ with an energy of ᎑2 au. The 1σu orbital becomes the 2pz orbital of He+, with an energy of ᎑1/2 au. Pure ionic bonding shows a similar effect. In simple terms, the HOMO will be an atomic orbital on the anion and the LUMO will be an atomic orbital on the cation. As the ions approach each other, the orbital energy of the anion is decreased because of the potential of the cation, whereas the orbital energy of the cation is increased by the potential of the anion. In both covalent and ionic bonding the gap keeps increasing as we go to R = 0. But the molecule has its lowest total energy at some value of R greater than zero. Orbital energies are part of the electronic energy only. To get the total energy, we must add in the nuclear–nuclear repulsion energy

Figure 2. Electronic energy curves for H2+, showing the HOMO (1σg) and LUMO (1σu) energies as a function of internuclear separation.

Journal of Chemical Education • Vol. 76 No. 2 February 1999 • JChemEd.chem.wisc.edu


E tot = E el + Vnn

(7)

Note that eqs 1 and 2 require that Vnn is to be held constant. Therefore the E involved can be either the total energy or the electronic energy. The equilibrium value of R occurs when the derivative of the total energy is equal to zero. Maximum Hardness Principle From much circumstantial evidence of the kind given in the previous section, it was concluded that “There seems to be a rule of nature that molecules arrange themselves to be as hard as possible” (7). But none of the evidence is rigorous enough or general enough to be conclusive. Fortunately Parr and Chattaraj have given a rigorous and widely general proof of the “Principle of Maximum Hardness”, or PMH (11).

Statistical Mechanics Their proof is based on a combination of statistical mechanics and the fluctuation–dissipation theorem. Since it is not easy to follow, some preliminary discussion is needed. The statistical mechanics part depends on the properties of a grand canonical ensemble. Such an ensemble consists of a large number of systems, each identical with the system of interest. These can exchange energy and particles with each other. Therefore the various members of the ensemble can have different numbers of particles and different values of the total energy. These quantities will fluctuate. The Hohenberg and Kohn theorem applies to ground states at the absolute zero of temperature. Fortunately there is a finite temperature version of DFT, first proved by Mermin (12). The equilibrium properties of a grand canonical ensemble are determined by the grand potential, Ω, which is defined as follows: Ω = E – µN – TS

(8)

where S is the entropy. Equilibrium corresponds to a minimum value of Ω. Equation 8 applies either to the case of ordinary thermodynamics, where E, N, and µ have their usual meaning, or to the case where E is the electronic energy, N the number of electrons, and µ the electronic chemical potential. In either case, Mermin showed that the grand potential is a unique functional of the density for a system at finite temperature. Also, the correct density for the system will give a minimum value of Ω. Thus we have a DFT for finite temperature by taking a grand canonical ensemble of the system of interest and calculating its properties. For example, for ordinary thermodynamics it is found that ( ∂ N/ ∂µ0) V,T = β 冬(N – 冓N 冔) 2冭/冓N 冔

(9)

where β = 1/kT, 冓N 冔 is the average value of N over the ensemble, which is constant, and N is the value for each member of the ensemble that can vary. For an ensemble of molecular systems we would have (13) ( ∂冓N 冔/ ∂µ ) v,T = β 冬(N – 冓N 冔) 冭/冓N 冔 = 2冓σ 冔 2

(10)

N is the number of electrons in each molecule, which can vary by transfer between molecules. However in the cases of interest, the average value, 冓N 冔, will be constant. The restriction of constant v, the nuclear potential, is equivalent to the

constant volume, V, of eq 9. The equilibrium softness of the ensemble is 冓σ 冔. Note that this is not the same as the molecular softness, σ0, but it is related. In either case, the softness depends on the fluctuations of N. Soft systems have large fluctuations. A large value of σ0 means I is small and A is large. Both gain and loss of electrons are easier in such molecules. In statistical mechanics we do not measure individual values directly. Instead we observe an average over all possible values. The averaging is done by means of a probability distribution function, which in classical mechanics is averaged over all of phase space. Let us compare an equilibrium ensemble with grand potential Ω, and an arbitrary nearby ensemble prepared by a small perturbation, ∆Ω. Let the equilibrium probability distribution function be f, and that for the nearby ensemble be F. The latter is the equilibrium distribution function for the grand potential Ω + ∆Ω. F could also be written as f + ∆ f. At time t = 0 we imagine that ∆Ω is turned off in some way. Then F will relax to f, usually by a first-order process characterized by a relaxation time, τ. Now let A (not the electron affinity) be a dynamic variable whose observed average value will change as F changes to f. Then we will have F = 冓A 冔᎑1Af

(11)

where 冓A 冔 is the average value of the observable in the equilibrium ensemble. Both f and 冓A 冔 are constants, independent of time. But A and F will relax together for small perturba¯ tions. At t = 0 the value of A can be written as A(0), where the overbar indicates an average over the non-equilibrium ensemble. All averaging must be done with the µ, v, and τ of the equilibrium ensemble. The reason for µ constant is that it also is the Lagrange undetermined multiplier which ensures that 冓N 冔 remains constant. Looking at eq 8, we can see that if µ is constant, the free energy of the equilibrium ensemble will be lower than that of the nonequilibrium ones.

Fluctuation–Dissipation Theorem 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 (14 ). This theorem dates back to early work by Nyquist and Onsager, and has been generalized by Callen and Welton. In essence, it says that small deviations from equilibrium, whether spontaneous or induced, have the same relaxation times. This follows because the mechanism is the same. The theorem applies to any observable, A, which relaxes from the nonequilibrium value A,¯ averaged over the ensemble, to the equilibrium value 冓A 冔.The relaxation of a spontaneous fluctuation in an equilibrium system is given by the time correlation function, C(t). C(t) = 冬(A(0) – 冓A 冔)(A(t) – 冓A 冔) 冭 = 冓A(0)A(t)冔 – 冓A 冔2

(12)

The correlation function decreases from a maximum value at t = 0 to zero at times long compared to the relaxation time, τ. The fluctuation–dissipation theorem can then be written


269


A(t) – A C(t) ᎑t/ τ =e = C(0) A(0) – A

(13)

¯ – 冓A 冔) is directly proportional This establishes that (A(t) 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. To find the constant of proportionality, we follow Parr and Chattaraj (11), who followed Chandler (14a). The value ¯ is found from the distribution function, but instead of A(t) of F, we use its equal from eq 11. The result is ¯ = 冓A 冔᎑1A(0)A(t) A(t) Combining this with eq 12, we find that ¯ – 冓A 冔 ) C(t) = 冓A 冔( A(t)

(14)

(15)

The unknown constant of proportionality is just 冓A 冔. Now take the observable to be the softness σ, which is always positive. Then it must be true that ¯ – 冓A 冔 ) = (¯ ( A(0) σ – 冓 σ 冔) ≥ 0 (16) Thus the equilibrium ensemble has the minimum softness, or the maximum hardness. This result is valid for nearby nonequilibrium distributions that obey eq 11. This is a common assumption of linear response theory. The restriction to constant µ and v is severe, and greatly limits the usefulness of the proof. However it is likely that these restrictions can be relaxed somewhat. The most important feature is that the nonequilibrium system has a higher free energy than the equilibrium one, so that the fluctuation– dissipation theorem can be applied. The result of interest is that the constant of proportionality is a quantity that is always positive. Its exact value of 冓A 冔 is not actually used. An interesting feature of the Parr–Chattaraj proof of the PMH is that the specific example of chemical softness is not introduced until the last step. The proof should then be valid for many other observables, provided that certain restrictions are met. One requirement is that the observable always have a positive value (or in some cases always a negative one). Equation 9 suggests that the fluctuations in an equilibrium system will offer many examples of the minimum (maximum) principle. These fluctuations are usually presented as the variance, or second central moment, since the average, or first moment, is zero (15). 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. Indeed, it has been shown that the variance of many observables is a minimum for the equilibrium system (16, 17 ). There have been many tests made of the PMH (9, 17, 18). Usually these have been calculations of the energy and the HOMO –LUMO gap as a function of changing internuclear distances. The results have always agreed with the

270

predictions, provided attention is paid to the restriction. Small variations in µ and v are allowed, but not large ones. For example, the stable isomer of a chemical system need not have the largest gap, though it often does. Different isomers have quite different values of the nuclear–nuclear repulsion, which invalidates constant v. We can change the calculated electron density function of a test molecule by increasing the size of the basis set of the trial wave function. What usually happens is that the HOMO is changed to a slightly more negative energy. But the LUMO is decreased in energy much more, so that the HOMO– LUMO gap, or hardness, is smaller, even though the total energy is decreased. But µ is changed markedly at the same time, which invalidates any test of the PMH. As might be expected, if we calculate the hardness and the energy for a transition state in a chemical reaction, we find that the hardness is a local minimum, just as the energy is a local maximum (19). This is consistent with the great reactivity, or instability, of a transition state. A formal proof has been given (20). A test of the PMH that is quite different, and more general, has recently appeared (21). It follows earlier work by Gyftopoulos and Hatsopoulos, who used a grand canonical ensemble with a limited number of discrete energy levels, so that the distribution function was known (22). The latter authors calculated the electronic chemical potential, µ, which was found to be µ° = (I + A)/2. The ensemble was a collection of systems containing the three species M°, M+, M᎑ and with energy levels E °, (E ° + I ), and (E ° – A). The ensemble average electronic energy can then be calculated in the usual way, and the softness from eq 10:

E – E°=

2η° exp(᎑ βη°) 1 + 2exp(᎑ βη°)

σ exp(᎑ βη°) = β 1 + 2exp(᎑ βη°)

(17)

(18)

These are the equilibrium values. The softness 冓σ 冔 is not the same as σ ° = 1/ηo where ηo = (I – A)/2. It may be thought of as an additional softness due to the finite temperature, since at zero Kelvin, 冓σ 冔 = 0. The application of DFT to excited states is not trivial, since the Hohenberg–Kohn theorem is only for ground state electron densities (23). However, if we assume that eq 5 is a valid operational definition for excited states, both µ* and η* can be calculated for any excited state (17 ). It is found that µ* = µo and η* = ηo – E*, where E* is the excitation energy. Any kind of excess energy, translational, rotational, vibrational or electronic, will lower the hardness. This result is certainly consistent with the properties expected for a soft molecule. Energetic molecules will be more reactive than ground state molecules. They will also have a more diffuse electron cloud than a similar hard molecule. A more diffuse cloud can be distorted more easily than a compact cloud. The compactness of any particle can be evaluated by applying the uncertainty principle to its various degrees of freedom (17). Parr and Zhou have made another argument for the PMH (24 ). Shells and subshells and the special stability of



filled shells are properties of matter at all levels, from atomic nuclei to atoms to molecules to large clusters of metal atoms. Again, using eq 5 as a measure, it is found that filled shells are always associated with a maximum value of (I – A), compared to systems with unfilled or overfilled shells. For nuclei, eq 6 is used, where ε is a nuclear energy level. It would be very useful if a quantitative relationship could be found between hardness and the energy of chemical systems. Both µ and η are small numbers compared to the total energies of such systems. Quantities of chemical interest, such as bond energies and activation energies, are calculated as small differences between very large numbers. This greatly magnifies the errors of calculation. If instead we could calculate changes in µ or η and convert them to changes in energy, in principle greater accuracy could be achieved. A problem exists, however, in that we can calculate energies with considerable accuracy, but calculations of µ and η are much less accurate. Much information is available, for example, using eq 6 as a source of changes in µ and η as Eel changes. But the uncertainty in the value of εLUMO makes any correlation difficult to quantify. Another possible procedure is to use DFT to establish a purely theoretical relationship between E el, µ, and η. Using functional expansion methods, an equation has been derived of the form (25) Eel = Nµ – N 2 η + other terms

(19)

The other terms involve quantities that depend on the changes in ρ with changes in potential. They are not easy to deal with. In the case of H2+ and H2 (probably) it can be shown that the other terms in eq 19 are either zero or close to zero, respectively. That is Eel = Nµ – N 2η for the entire range of R, from separated atoms to united atoms. But for larger atoms and molecules, where N is larger, the other terms become increasingly important. For diatomic molecules, it is found that ∆Eel is a linear function of ∆η, with a negative slope, as eq 19 predicts, but the slope is not equal (26 ) to N 2. Physical Hardness The physical, or mechanical, hardness of a solid refers to its resistance to a change in its shape or volume. Changes in shape are easier than changes in volume and are more important from a practical point of view. They determine whether a solid can be used as a material of construction or manufacture. The term mechanical strength is often used to describe hardness. Oddly enough, there has been no exact definition of hardness. Its value for a given sample is usually determined by very empirical methods such as the scratch test, which gives the Moh scale of hardness, or the indentation caused by dropping a weight on the sample. Such numbers are very useful, but difficult to interpret in a fundamental way. Also, the results are very dependent on the past history of the sample and its purity. In the Vickers test for hardness, which is the most quantitative, the size of the indentation left by a diamond stylus under a fixed load is measured. The hardness number can be expressed in pressure units, usually kg/mm 2. This test and the scratch test are irreversible. That is, the sample does not

return to its original state. The deformation is said to be plastic, rather than elastic. The study of mechanical strength falls under the heading of elasticity, the science of the response of a solid sample to applied forces. The forces are described by tensors called stresses, which give the direction of the force and the plane to which it is applied. The responses, called strains, are also given by tensors, which give the relative changes in dimensions or shape. The ratio of a stress to its corresponding strain is called an elastic modulus. For small stresses the modulus is a constant and the material behaves elastically. It returns to its original condition when the stress is removed. For larger stresses, the elastic limit may be exceeded, and the sample undergoes a permanent, or plastic, deformation. Important stresses are compressional, in which the force acts in one dimension only; hydrostatic, in which the force acts equally in all directions; and shearing. In the last, forces act to move parallel planes of the sample past each other. At the microscopic level, shear stresses cause the gliding of planes of atoms over each other. This is the commonest and easiest way for a solid to change its shape. The hardness, or the force needed, is very dependent on the presence of crystal defects. Even a pure crystal in the process of being formed will necessarily contain many defects. The important defects for gliding motion are the edge and screw dislocations. A shearing stress will cause the dislocations to move. Minute amounts of added material can change the strength greatly. For example, added carbon atoms in iron can act as dislocation traps and halt the gliding motion. Work hardening of metals is a process whereby many of the dislocations pile up and become immobilized. The movement of dislocations can be studied at various temperatures and the activation energy can be found. At 1500 °C plastic flow can be seen, even in diamond. The activation energy for flow is a function of the band gap, or HOMO–LUMO gap, for solids (27). It is large for insulators such as diamond, and small for metals. Each of the elastic constants is a measure of hardness for a particular deformation. There are two fundamental elastic moduli for isotropic materials: the bulk modulus, which measures volumetric stiffness, and the shear modulus, which measures shape stiffness. The bulk modulus requires changes in bond distances only; the shear modulus depends mainly on changes in bond angles. The energies needed depend on the force constants for those changes (28). As expected, it is easier to change bond angles than bond distances. To an engineer or materials scientist, it is clear that the word hardness means resistance to change in shape. Its magnitude for a given sample depends on its purity and previous history, which are extrinsic properties. The intrinsic property most closely related is the shear modulus (29). This hardness is adequately measured by scratch and indentation tests. It is difficult to relate such a property to more fundamental characteristics of the solid. The bulk modulus, on the other hand, is an intrinsic property easy to comprehend. It depends only on the potential energy–interatomic distance relationship, at least for isotropic solids. It will be closely related to other properties that depend on the energy–distance curve, such as cohesive energies. This is the energy of atomization of a solid:


271


∆ Ecoh

AB(s) = A(g) + B(g)

(20)

Since the cohesive energy determines the stability, reactivity, and structure of a solid, it is the property of most interest to chemists. There is another reason to select the bulk modulus, B, as the hardness factor of interest to chemists. It is defined as

B = ᎑V ∂P ∂V

T

1 =κ

(21)

The reciprocal of the bulk modulus is the compressibility, κ . For an isotropic solid, a force exerted equally in three dimensions will produce a change in volume by a uniform decrease of distances between nearest neighbors. Bond angles will not be changed. The inverse relationship between B and κ resembles the relationship between hardness and softness. Certainly one expects a solid that is physically soft to be compressible, and one that is hard to resist compression. Equation 3 can be expressed as

∂µ ∂N

2

= V2 = BV 0 T,V N κ

V = cR 3; dV/V = 3dR/R; P = –(∂U/∂V ) T

(23)

where c is an inconsequential constant. For cubic crystals we can relate changes in the volume to changes in the nearestneighbor distance, R, in a simple way. We then get 2

= T

V 2 2 ∂ U/∂V V

where R0 is the equilibrium value. 272

T

=

2 R0

9V 0

BV 0 =

(22)

V0 is the molar volume and N is the number of molecules. The factor of N 2 is needed for dimensional purposes. B has the units of pressure and BV0 has the units of energy. If V0 and µ are calculated per mol of atoms, then N 2 may be ignored. Because of eq 22, Yang, Parr, and Uytterhoeven have proposed that BV0 be called the physical hardness, H (30). It has the units of energy, the same as chemical hardness. Its reciprocal is the softness, proportional to κ, as desired. Their proposal was reinforced by showing that BV0 for a number of substances followed much the same ordering as the Moh hardness of those substances. Table 1 contains experimental values of H = BV0 for a number of solids of simple type with cubic structures. For comparison, the values of the cohesive energy, ∆E coh, are also included. In the same units, kcal/mol, the hardness numbers are of the same magnitude as the energies of atomization. There is also a definite trend for H to increase as ∆E coh increases. However it is not very regular. For ionic solids, H is always less than the cohesive energy. For covalent solids, including metals, H is always much greater than ∆E coh, except for Li. We also see that W, Pt, and a few other noble metals have H greater than carbon. On this scale, they are harder than diamond, which means that H is no longer matching the scratch test for hardness. This does not invalidate BV0 as a legitimate scale for hardness. We can obtain a new equation for B from its definition in eq 21 and the relations

B = ᎑V ∂P/∂V

The curvature (∂2U/∂R 2)T may be identified with the force constant, f, when U is the potential energy function for a diatomic molecule. It has also been identified as an important property determining the scratch hardness of a solid (31). The relationship between the hardness, H, and the cohesive energy now seems to be the same as that between the force constant of a diatomic molecule and its dissociation energy, D0. When one increases, the other usually does also. But there is no simple relation between them (32). Force constants depend more strongly on the repulsive part of the potential energy function than bond energies do. Models for covalent and ionic bonding give quite different relations between BV0 and ∆E coh (33). Equation 24 suggests that we calculate BV0 from the point of view of force constants for the bonds undergoing compression. This has already been done for cubic crystals by Pauling and Waser (34 ). Their procedure was to equate the pressure–volume work done to the energy of compressing the bonds. The result was

∂2U/∂R

2

(24) T

nf R 0 9

(25)

where n = m/2, the number of bonds per atom. The coordination number m is divided by two because each bond is shared by two atoms. Equation 25 is completely consistent with eq 24, since the total change in energy depends on the number of bonds, Table 1. Hardness Numbers, BV0, for Simple Solids Solid

H/ ∆ Ecoh/ kcal mol ᎑1 kcal mol ᎑1

Solid

H/ ∆ Ecoh/ kcal mol ᎑1 kcal mol ᎑1

Elements Li

38

38

Zn

128

31

Na

39

26

Ag

246

68

Cs

35

19

Cd

137

27

Mg

113

35

In

147

58

Al

173

78

Pb

180

47

Ca

107

43

W

790

202

Sr

100

39

Pt

651

135

Cr

332

95

C

478

171

Mn

250

67

Si

282

109

Fe

280

99

Ge

270

90

Cu

235

81

Sn(grey)

285

73

Compounds a LiF

78

204

BaS

165

218

NaCl

79

153

ZnO

249

174

KCl

81

155

ZnS

217

147

KI

53

124

ZnTe

186

107

RbF

48

171

CdTe

208

97

CuBr

119

133

InSb

218

128

AgCl

136

127

InP

264

155

MgO

207

239

GaAs

246

156

CaO

229

254

GaSb

230

139

SrO

217

240

SiC

367

269

MnO

240

219

TiC

349

328

N OTE: Data are from ref 31. a∆E coh for compounds should be divided by 2 to compare with elements.



as well as their force constants. V0 is the volume per mol of atoms. The force constants of a number of solids have been calculated from eq 25 (33). They are less than for diatomic molecules, but this can be rationalized. The possibility of calculating force constants for bonds in solids has many potential applications. The physical properties of solids are very dependent on the vibrational motions of the particles. These must depend greatly on the force constants, as well as on the masses of the particles. We can expect that there will be simple relationships between BV0 and such properties as the velocity of longitudinal sound waves and the Debye temperature.

The PMPH gives us a definite relationship between the force constant and the anharmonicity constant. We know that, since f is always positive, the negative sign for g is correct, from experimental evidence to be considered next. There are two properties of solids that depend on the anharmonicity term most directly. One is the coefficient of thermal expansion, and the other is the variation of the bulk modulus with pressure, (∂B/∂P)T . The experimental results in the latter case are given either by a power series equation V = V0(1 – aP + bP 2) or by the Murnaghan equation

The Principle of Maximum Physical Hardness, PMPH A great advantage of using the thermodynamic definition of hardness is that we can find other thermodynamic equations involving H. For example, taking a grand canonical ensemble, we have not only eq 22, but also eq 9. Since κ/V 0 = 1/H, we can say that crystals that are physically soft have large number fluctuations. In this case the systems of the ensemble are crystals of identical volume, but with varying numbers of component molecules. Using the proof of Chattaraj and Parr we can conclude that the mechanical softness is a minimum, and the hardness is a maximum, for the equilibrium state (35). This would apply to any condensed system, solid or liquid. This is a very reasonable result. Chemical hardness is the resistance to change in the electron distribution, and physical hardness measures the resistance to change of the nuclear positions. An equilibrium system should have the greatest resistance to change for both of these properties. To test this prediction for solids, we must show that BV0 is a maximum for the equilibrium state of the crystal. That is, δ (BV0) = 0, subject to certain restrictions, such as constant volume and temperature. The restriction is also made to consider only cubic crystals containing a single element, or the binary AB compounds. This is so we can use the information in eqs 11 and 12. We wish to prove that the equilibrium crystal with V = V0, the experimental volume per mol of atoms, is harder than crystals with larger or smaller volumes. Differentiate BV with respect to V, using the definition of B in eq 21 and the definition of P in eq 23: B + V (∂B/∂V )T,V 0 = 0 ; ᎑2(∂ 2U /∂V 2) = V0(∂3U/∂V 3)

(26a)

With the relation between V and R in eq 11, this can be written as 2

᎑ 2R 0 ∂2U 2 9 ∂R

T, R 0

R 0 ∂3U 27 ∂R 3

B P B 1 ln V = ᎑ ln 1 + 1 V0 B0

(29)

Expanding the logs in eq 29 for moderate pressures, we obtain eq 28 again, with a = 1/B0 and b = (1 + B1)/2B02. B0 is the modulus at zero pressure and B1 is

∂B = 2b – 1 = B 1 ∂P a2

(30)

Experiments show that the modulus is indeed a linear function of the pressure, over the range of pressures available. The slope is given by eq 30. The intercept is B0. Pauling and Waser interpreted both B0 and B1 in terms of force constants (34 ). In eq 25 it is B0 that appears and gives a value for f. B1 then gives the value of g , again equating pressure–volume work to the energy of compressing the bonds. They found that g = (1 – b/a 2)f /2R0

(31)

Using the result from the PMPH, f = ᎑gR 0, we find (b/a 2 ) = 3 and B1 = 5.0. Therefore, for cubic solids, the pressure derivative of the modulus is equal to a constant dimensionless number, 5.0. Fairly good data for (∂B/∂P) at room temperature are available for a large number of cubic crystals. They range in value from 4.4 to 5.5. On average, the results are remarkably close to the predicted value of 5.0. Variations in the experimental results are 10–20%, by different investigators. If the potential energy function of a solid were simply harmonic, there would be no thermal expansion.

α = 1 ∂V 3V ∂T

P

= 1 ∂x = 0 R 0 ∂T

(32)

The average displacement,

3

=

(28)

(26b)

x = R – Ro

T, R 0

U is expanded as a power series in x = (R –R0), where x is small. 4 U = U0 + f x2 + g x3 + h x + … 2

(27)

The parameter f is the force constant and g is the anharmonicity constant. We have no term linear in x at equilibrium. That is, (∂U/∂R ) = 0, (∂2U/∂R 2) = f, and (∂3U/∂R 3) = 6g, at R = R0. Putting this information into eq 26 gives us f = ᎑gR 0.

would be zero. But the term in x3 (eq 17) makes a positive value of x more probable than a negative value. This is based on the Boltzmann distribution law and a negative value for g. A calculation gives (3)

x=

Cv T g f

2

;

α=

Cv g 2

f R0

=

Cv 2

f R0


(33)

273


In the last equality, we have used the maximum hardness result, ᎑f = gR 0 . We can use eq 25 to eliminate f R02 and relate α , the coefficient of thermal expansion, to κ, the compressibility. The value of α could then be calculated. But experimentally it is better to use the ratio of α to κ by introducing the Gruneisen constant, γ (36 ).

γ=

3αV 0 Cv κ

(34)

The Gruneisen constant occurs in the model of a solid as an elastic continuum. Its value experimentally ranges from 0.4 to 2.60 for different substances. The reason for this variation was not well understood. However by combining eqs 25, 33, and 34, we can find a theoretical value for γ . The result is a simple and novel one, γ = n/3 or m/6. This prediction has been tested for a large number of solids (35, 37 ). To make the test, we must distinguish between covalent and ionic bonding. Equation 25 is derived on the basis of covalent bonding, or bonds that are not highly ionic. But for solids such as NaCl, we know that the bonding comes not only from the interaction of each ion with its nearest neighbors, but also with the ions of the rest of the crystal. This leads to an increased number of bonds by a factor of M, the Madelung constant (35, 38). Metals certainly represent covalent bonding but binary compounds, AB, can be covalent or ionic. A variety of evidence suggests that solids with a coordination number (CN) of four are highly covalent, whereas those with CN = 6 or 8 are highly ionic (39, 40). A recent paper shows that the cohesive energies of CN 4 solids are well represented by a completely covalent bonding model, and those of CN 6 or 8 by a completely ionic model (38). Accordingly, the value of Mm/6 = γ is taken for AB compounds with CN 6 or 8, and m/6 = γ for metals and CN 4 AB compounds. Table 2 shows the results for the average experimental values compared with the theoretical ones for a large number of solids. The agreement is very good, and it appears that γ can be used as a criterion for distinguishing between ionic and covalent bonding. For example, BeO has γ = 1.54 and ZnO has γ = 0.66. This agrees with the observation that the ionic model fits the cohesive energy of BeO and the covalent model fits for ZnO, though both have a coordination number of four (38). The BeO structure is a result of the radius ratio rule, which usually is unreliable. The theory so far has only shown that BV0 has an extremum value, not that it is a maximum. The answer to that question lies in the value of h in eq 27. This number must be negative, or have a small positive value, in order for BV0 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). The actual value of BV becomes larger as V becomes smaller because the compressibility goes to zero. This is not a contradiction of the prediction that BV0 is a maximum at V = V0 for the ensemble average. The ensemble members are crystals of the same volume, but containing different numbers of particles. Since the total number of particles is constant, some crystals will have more than the average number, but others will have 274

less in the non-equilibrium ensemble. The average of these will give a BV less than the equilibrium case. Summary

Table 2. Gruneisen Constants for Solids as a Function of Coordination Number γ, exp a γ, theory CN

Metals 8

1.46

1.33

There seems to be a law of 12 2.17 2.00 nature that in an equilibrium Compounds system, the chemical hardness 4 0.64 0.67 and the physical hardness have 6 1.58 1.75 b maximum values, compared to 8 2.08 2.34 b nearby non-equilibrium states. NOTE : Data are from ref 37. However, it must not be inferred aAverage value for all cases. that these maximum principles bMadelung correction applied. are being proposed to take the place of established criteria for equilibrium. Instead, they are necessary consequences of these fundamental laws. It is very clear that the principle of maximum hardness for electrons is a result of the quantum mechanical criterion of minimum energy. Similarly, Sanchez has recently derived the relationship (∂B/∂P) = 5 by a straightforward manipulation of the thermodynamic equation of state (41). The PMPH is a result of the laws of thermodynamics. It is, of course, very reasonable that the equilibrium state has the greatest resistance to change, both of electron distribution and of nuclear positions. After all, the non-equilibrium states must all change these properties to reach equilibrium. It should be emphasized that the maximum physical hardness principle requires the definition of a scale of hardness that is not the one envisioned by engineers and materials scientists. They are interested in the resistance of a solid to changes in shape. The shear modulus of a solid substance is the major intrinsic factor. It is likely that a principle of maximum hardness also applies to shear moduli. Literature Cited 1. Parr, R. G.; Yang, W. Density Functional Theory of Atoms and Molecules; Oxford University Press: New York, 1989. For short reviews see Ziegler, T. Chem. Rev. 1991, 91, 651 and Kohn, W.; Becke, A. D.; Parr, R.G. J. Phys. Chem. 1996, 100, 12974. 2. Hohenberg, P.; Kohn, W. Phys. Rev. 1964, 136, 13864. 3. Parr, R. G.; Donnelly, R. A.; Levy, M.; Palke, W. E. J. Chem. Phys. 1978, 68, 3801. 4. Parr, R. G.; Pearson, R. G. J. Am. Chem. Soc. 1983, 105, 7512. 5. Pearson, R. G. Proc. Natl. Acad. Sci. U.S.A. 1986, 83, 8440. 6. Pariser, R. J. Chem. Phys. 1953, 21, 568. 7. Pearson, R. G. J. Chem. Educ. 1987, 64, 561. 8. Pearson, R. G. J. Am. Chem. Soc. 1988, 110, 2092. 9. Pearson, R. G. Acc. Chem. Res. 1993, 26, 250. 10. Pearson, R. G. Inorg. Chim. Acta 1992, 200, 781. 11. Parr, R. G.; Chattaraj, P. K. J. Am. Chem. Soc. 1991, 113, 1854. 12. Mermin, N. D. Phys. Rev. 1965, 137, A1441. 13. Yang, W.; Parr, R.G. Proc. Natl. Acad. Sci. U.S.A. 1985, 82, 6723. 14. (a) Chandler, D. Introduction to Modern Statistical Mechanics; Oxford University Press: New York, 1987. (b) Keizer, J. Statistical Thermodynamics of Nonequilibrium Processes; Springer: New York, 1987. 15. McQuarrie, D. A. Statistical Mechanics; Harper and Row: New York, 1973; Chapter 5. 16. Pearson, R. G. J. Phys. Chem. 1994, 98, 1989. 17. Pearson, R. G. Chemical Hardness; VCH: Weinheim, Germany; 1997; Chapter 4.


Research: Science & Education 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36.

Pearson, R. G.; Palke, W. E. J. Phys. Chem. 1992, 96, 3283. Datta, D. J. Phys. Chem. 1992, 96, 2409. Parr, R. G.; Gázquez, J. L. J. Phys. Chem. 1995, 99, 5325. Chattaraj, P. K.; Liu, G.H.; Parr, R. G. Chem. Phys. Lett. 1995, 237, 171. Gyftopoulos, E F. P.; Hatsopoulos, G. N. Proc. Natl. Acad. Sci. U.S.A. 1965, 60, 786. Parr, R. G.; Yang, W. Density Functional Theory of Atoms and Molecules; Oxford University Press: New York, 1989; pp 204–208. Parr, R. G.; Zhou, Z. Acc. Chem. Res. 1993, 26, 256. Liu, S.; Parr, R. G. J. Chem. Phys. 1997, 106, 5578. Gázquez, J. L.; Martinez, A.; Méndez, F. J. Phys. Chem. 1989, 93, 3068. Gilman, J. J. Science 1993, 261, 1436. Harrison, W. A. Electronic Structure and Properties of Solids; Freeman: San Francisco, 1980; pp 193ff. Gilman, J. J. Materials Research Innovations, in press. Yang, W.; Parr, R. G.; Uytterhoeven, T. Phys. Chem. Min. 1987, 15, 191. Goble, R. Y.; Scott, S. D. Can. Mineral. 1985, 23, 273. Pearson, R. G. J. Mol. Struct. (Theochem) 1993, 300, 519. Pearson, R. G. Chemical Hardness; VCH: Weinheim, Germany; 1997; Chapter 6. Pauling, L.; Waser, J. J. Chem. Phys. 1950, 18, 747. Pearson, R. G. J. Phys. Chem. 1994, 98, 1989. Kittel, C. F. Introduction to Solid State Physics, 3rd ed.; Wiley: New York, 1967; pp 183 ff.

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Pearson, R. G. Int. J. Quantum Chem. 1995, 56, 211. Pearson, R. G. J. Mol. Struct. (Theochem) 1992, 260, 11. Pearson, W. B. J. Phys. Chem. Solids 1962, 23, 103. Phillips, J. C. Revs. Mod. Phys. 1971, 42, 317. Sanchez, I. A. J. Phys. Chem. 1993, 97, 6120.


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Maximum Chemical and Physical Hardness

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