7230
J. Am. Chem. SOC.1992, 114, 723C-7244
(in >1,4 relationships) is given by E,,, where t * = with representing the "hardness" of atoms i and k (kcal-mol-'), r* = ri i- rk, the sum of the van der Waals radii of atoms i and k (A), and r is the distance between the two atoms (A). When r*/r > 3.31 1, eq A5 reduces to E5 = t*(336.176)r*/r to prevent two atoms fusing when they come very close together. E6 is the dipole interaction energy, where w iand wj are the bond moments (D) of two bonds close in space, x is the angle between the dipoles (deg), R is the line between midpoints of the bonds, aiand ajare the angles (deg) between the dipole axes and the lines along which R is measured, 14.39418 converts ergs.molecule-I to kcal-mol-],
and D is the dielectric constant (default value = 1.5).
ti and tk
Registry No. Ni(porph), 15200-33-6; Fe(porph), 15213-42-0; Zn-
(porph),
137626-05-2; Pb(porph), 30993-25-0; [P(TPP)(OH),]+, 87374-07-0; Ni(OEP), 24803-99-4; Fe(TPP), 16591-56-3; Zn(TPP), 14074-80-7; Pb(TPrP), 73395-80-9.
Supplementary Material Available: Listing of selected observed and MM-calculated bond lengths, bond angles, and torsional angles and a listing of modifications made to the MM2(87) program (42 pages). Ordering information is given on any current masthead page.
Theoretical Increments and Indices for Reactivity, Acidity, and Basicity within Solid-state Materials Richard Dronskowskit Contribution from the Department of Chemistry and Materials Science Center, Cornell University, Ithaca, New York 14853- 1301. Received November 27, 1991
Abstract: With the aim of aiding the design of solid-state chemical syntheses, we construct some quantum mechanical indicators of reactivity, acidity, and basicity in crystal chemistry. These definitions are based on three-dimensional electronic structure calculations. After referring back to the concept of density-functional theory of Kohn and Sham and, in particular, the definition of absolute hardness due to Parr and Pearson, we first give a short overview of what is known in the field of molecular quantum chemistry. We then proceed to derive local reactivity, electrophilicity (acidity), and nucleophilicity (basicity) increments and indices both for atoms and bonds in any possible crystal structure. Our definitions are formulated in terms of a one-electron picture, and the first concrete calculations are performed within the framework of the semiempiricalextended Huckel tight-binding method. However, the definitions are not restricted to the latter method and are quite easily generalized for ab initio numerical techniques to solve the complex eigenvalue problem in k-space. As an illustrative application, we investigate the acid-base solid-state reaction from K2Ti409to K2Ti8OI7.A comparison of our approach, which is suited (but not restricted) to the solid state, with another scheme from molecular orbital calculations is attempted. In detail, we (i) determine the compound's resistance to electronic attack for different electron counts, (ii) analyze all atoms with respect to their electronic reactivity, acidity, and basicity, and (iii) clearly identify the chemically most basic oxygen atom by its outstanding atomic nucleophilicity (basicity) index. In fact, the changing connectivity of this single 0 atom governs the structural change from K2Ti409to K2Ti80,,. We perform a numerical investigation of Rouxel's hypothesis that the basicity of this particular 0 atom could be decreased by replacing one Ti atom by a Nb atom, and finally we elucidate the resulting changes in the electronic structure in detail.
1. Introduction Even though the comparatively young discipline of solid-state inorganic chemistry has made great progress during the last three decades, there still seems to be a strange discrepancy in its methodology. On one hand, modern solid-state chemistry is unthinkable without X-ray crystal structure analysis, which allows a detailed and unambiguous description of the geometry and stoichiometry of the often complex chemical structures. Therefore, one might well take the view that these huge molecules synthesized by solid-state chemists are at least as well characterized as the "typical" small molecules of solution chemistry, whose structures are often determined by various spectroscopic techniques, as well as by crystallography in the solid state. On the other hand, a solid-state chemist engaged in generating three-dimensional periodic structures can plan the individual synthetical steps only in a very rudimentary way and must often rely on the classical "shake and bake" techniques.' While organic chemistry faces up each day to incredibly complicated organic (natural or unnatural) molecules whose syntheses are planned step by step, an even approximately similar strategy to retro~ynthesis,~,~ for example, seems to be out of the question for inorganic solidstate chemistry. As a matter of fact, there is no simple Ansatz Present address: Max-Planck-Institut fur Festkorperforschung, Heisenbergstr. 1, 7000 Stuttgart 80, Germany.
0002-786319211514-7230$03.00/0
in sight to predict nontrivial reaction paths toward imagined structures. Besides the greater elemental variety of solid-state chemistry compared to organic chemistry, there is another fundamental reason for this finding. The majority of solid-state inorganic compounds is, in fact, only stable in the solid phase. This might sound trivial, but in reality it represents a singularity compared to the behavior of molecules that arise from solution chemistry. A typical giant solid-state molecule showing fascinating structural details in the crystal decomposes at the melting point because its chemical bonding is intimately connected to the ordered crystalline state. Thus, the confinement to a single small unit (the molecule), which is so successful in organic synthesis, does not seem to work. In addition, there is obviously no easy way to see or define a "functional group" in a solid-state compound, although this concept is so extremely important in organic synthesis. Even worse, one has to face the sad fact that solid-state counterparts to "functional group interconversions" or "synthons" are extremely rare. What is retained in inorganic as well as in organic chemistry, in the solid state or in solution, is the time-honored and useful concept of acidity and basicity, except that this idea, and its ~
~~
(1) Rouxel, J. Giuing the Baker lectures at Cornell Uniuersity, 1991. ( 2 ) Corey, E. J. Angew. Chem. 1991, 103, 469-479; Angew. Chem., Int. Ed. Engl. 1991, 30, 455-465. (3) Warren, S. Organic Synthesis: The Disconnection Approach; John Wiley: Chichester, New York, 1982.
0 1992 American Chemical Society
Reactivity, Acidity, and Basicity in Solid-state Materials associated notions of electrophilicity and nucleophilicity, is not well defined in the solid state. This paper will face up to the demand for a solid-state synthetic language including descriptors for reactivity, acidity, and basicity.
2. Motivation and Starting Point Fortunately, especially in the last decade, some solid-state syntheses of new structures give rise to the hope that the construction of a synthetic language including reactivity or acidity and basicity for the solid state should be worth trying, in principle. In the solid-state chemistry of low-dimensional solids, Rouxel and others4v5have brought “soft chemistry” to work, incorporating (i) redox chemistry involving intercalation/deintercalationprocesses, (ii) acid-base reactions followed by structural recondensations, and (iii) grafting reactions using van der Waals gaps separating internal surfaces of solids. Another example is the beautiful solid-state liquid-state donoracceptor chemistry that has allowed infinite lattices to be ”cut” into smaller pieces. As Tarascon and DiSalvo have shown, some representatives of the AMo3X3structure type with A = Li, Na and X = Se, Te can be dissolved when treated with polar solvents, finally giving purely inorganic transition-metal polymer solutions.6 In general it should be very helpful to detect the intrinsic reactive or sensitive atomic parts of a given structure in the solid state. In other words, which sites of a structure will be most influenced by a change in the overall electronic conditions? A natural way of intellectually dividing a structure into smaller units could thereby be achieved, without referring back to the language of, for example, ionic or covalent bonding. There is nothing wrong with the latter, except for the tendency of each to overestimate the specific subparts they want to see (coordination polyhedra, clusters, metal-metal or metal-nonmetal bonds, and so on). Our approach will not investigate the strength of chemical bonds in given crystal structures. Besides the fact that there are already empirical bond length-bond strength formulas a ~ a i l a b l e , ~one -I~ might well imagine a bonding situation where a strong chemical bond is likely to break because of a stronger one formed, while a weak bond remains inert because no alternative stronger bond can be generated, for steric reasons, for instance. Again, it seems more reasonable to seek the reactive parts of a structure, concentrating on them while planning a synthesis. Therefore, we will try to detect and quantify, by means of electronic structure calculations, the reactive sites in a given structure. As will be seen, we believe that structures do naturally “fall apart” into sublattices of donors and acceptors. In this paper we will set up different reactivity descriptorsfor atoms and bonds within solid-state compounds. These indicators are generated via quantum mechanical calculations of the electronic structure in the solid. The underlying concept is densityfunctional theory including the concept of absolute hardness, while the first calculations are performed within the framework of extended Huckel (EH) theory in its tight-binding approach. 2.1. The Hard/Soft Acid-Base Principle. There is no doubt that the formal Lewis reaction Lewis acid + Lewis base = Lewis acid-base complex is one of the most universal chemical reactions one can think of. The Lewis acid represents the electron acceptor, whereas the Lewis base is the electron donor. The infinite inorganic solid, of course, may be seen as the most striking example of a strong acid-base complex. Almost 30 years ago, the Lewis theory of acids and bases was greatly augmented by the qualitative description of “hard” and (4) Rouxel, J . Chem. Scr. 1988, 28, 33-40. (5) Rouxel. J. 2Olst National Meeting of the American Chemical Society, Atlanta, GA, Spring 1991. (6) Tarascon, J . M.; DiSalvo, F. J.; Chen, C. H.; Carroll, P. J.; Walsh, M.; Rupp, L. J. Solid Sfare Chem. 1985, 58, 290-300. (7) Pauling, L. The Nature of the Chemical Bond, 3rd ed.; Cornell University Press: Ithaca, N Y , 1960. (8) Brown, I. D.; Wu, K. K. Acta Crysr. 1976, B32, 1957-1959. (9) Brown, I . D. In OKeeffe, M., Navrotsky, A,, Eds.; Structure and Bonding in Crystals; Academic Press: New York, 1981; Vol. 11. (IO) Brown, 1. D.; Altermatt, D. Acta Crysr. 1985, 8 4 1 , 244-247.
J . Am. Chem. Soc., Vol. 114, No. 18, 1992 7231 “soft” acids and bases, as done by Pearson.”-I4 According to Pearson’s early thesis, a “soft” acid is a big species (cation), not too highly charged, whose valence shell is easily polarizable, in contrast to a “hard” acid, which is well represented by a small and highly charged species (cation), difficult to polarize. On the other hand, a “soft” base is an easily polarizable, easily oxidizable species (anion) with a small electronegativity and low-lying orbitals. Consequently, a “hard” base is generated by a species (anion) which is difficult to polarize or oxidize, with a high electronegativity and high-lying empty orbitals. Pearson’s rules that “hard” prefers “hard” and “soft” prefers “soft” have already become textbook material. It cannot be overemphasized that, in contrast to Pauling’s electronegativity rule,’ Pearson’s concept manages to explain why the solid-state reaction LiI + CsF = LiF + CsI is highly exothermic (AH -130 kJ/mol) on going to the right side which is the “softsoft“ and “hard-hard” one. As convincing as the latter example is, it is surprising that Pearson’s concept has not become more popular in solid-state chemistry. It was Klopman who set up a quantum-chemical treatment of chemical reactivity15J6 for molecules using polyelectronic perturbation theory. His ideas could at least rationalize the initial guess of hard-hard combinations being bonded mainly by ionic forces and softsoft combinations by mainly covalent forces. While referring to the position of the H O M O and LUMO, Klopman introduced the terms “charge-controlled” and “orbital-controlled” for hard-hard and softsoft combinations, respectively. Moreover, he introduced reactivity scales and managed to connect them to measurable quantities such as formation enthalpies. However, although Pearson’s concept proved to be a useful one in estimating specific stabilities while combining different ions or molecules, it more or less remained an empirical criterion for 20 years. The quantification of “hard” and “soft” behavior was only set up within empirical formula schemes such as those of Drago, Wayland, and Edwards, for example. These typically include empirical “softness” and “strength” values for different molecular species, somehow linked to stability c o n ~ t a n t s . l ~ - ~ ~ A definite breakthrough came with the paper of Parr and Pearson in 1983 in which they simultaneously introduced absolute electronegativity x and absolute hardness v.21 Both were defined in terms of the ionization potential Z and the electron affinity A of a system having total energy E and total electron number N according to x = -cc (1)
= Y2(1 + A )
(3)
where p is the chemical potential and M is the number of electrons within the neutral molecular species, and t = Y2(a2E/dW~
l/z(I
(4)
-A) L 0
( 1 1 ) Pearson, R. G. J. A m . Chem. SOC.1963, 85, 3533-3539. (12) Pearson, R. G. Science 1966, 151, 172-177. (13) Pearson, R. G.; Songstad, J. J. Am. Chem. Soc. 1967,89, 1827-1836. (14) Pearson, R. G. J. Chem. Educ. 1968, 4 5 , 581-587, 643-648. ( 1 5 ) Klopman, G.;Hudson, R. F. Theor. Chim. Acta 1967,8, 165-174. (16) Klopman, G.J. A m . Chem. SOC.1968, 90, 223-234. (17) Ho, Tse-Lok Hard and Soft Acids and Bases Principle in Organic Chemistry; Academic Press: New York, 1977. (18) Gutmann, V . The Donor-Accepror Approach to Molecular Interactions; Plenum Press: New York, 1978. (19) Jensen, W. B. The Lewis Acid-Base Concept; John Wiley: New York, 1980. (20) Finston, H. L.; Rychtman, A. C. A New View of Currenr Acid-Base Theories; John Wiley: New York, 1982. (21) Parr, R. G.;Pearson, R. G. J. Am. Chem. SOC.1983,105,7512-7516.
1232 J . Am. Chem. SOC.,Vol. 114, No. 18, 1992
Dronskowski
The very last line is only valid for stable systems, stating that the first ionization potential is larger than the corresponding electron affinity. In fact, for atoms and molecules there are no counterexamples to eq 6 although a mathematical proof has not been given so far. For example, if eq 6 is not true for a system S, then 2 s is unstable with respect to S+ and S-. Such systems would be described with a negative Hubbard U parameterz2 (attractiue electron correlation). The above statements can be exactly expressed in terms of quantum-mechanical quantities with the help of density-functional theory, Le., as different derivatives of E versus N (see above). Thereby the ‘soft” and “hard” terms are given a physical basis (similar to other vivid expressions in the language of modern science, as, for example, the terms of “color”, “flavor”, or ‘charm” which are widely used in elementary particle physics). 2.2. Total Hardness and Density-FunctionalTheory. In contrast to traditional quantum chemical methods, density-functional (DF) theory is based on the electron density rather than on the electronic wave function +.23-25 As introduced by Hohenberg, Kohn, and Sham,26*z7 DF theory explicitly includes many-particle effects that are essential for chemical bonding, and it has been proven to give an accurate description of a system’s electronic ground state by Levy.z* Within D F theory the ground state (GS) is a functional of the one-particle density n(i), i.e.
(7) The GS energy Eo is given as the minimum of the functional E(n(i))which can be separated into contributions of the kinetic energy of noninteracting electrons, classical Coulomb energy of the charge density and a functional for exchange and correlation
Number of electrons N Figure 1. Total energy E of an atomic or molecular system as a function of the electron number N according to density-functional theorySz4
scheme for exchange and correlation. On the other hand, schemes well beyond the local-density approximation are already available.29 In short, a t the present time experience suggests D F calculations to be practical and accurate schemes for treating complex systems. From a chemical viewpoint, it is most interesting to investigate how the total energy E of the ground state might change with the particle (electron) number N . If one expresses E as a power series of N while keeping the potential u constant, Le., the positions of nuclei and the atomic numbers remain fixed,
E=Eo+(g)fldN+i($)fldp+... = Eo where ucx‘(i) is the Coulomb potential of the fixed nuclei. introducing the local-density approximation
ExC(n( i)}
1
din( 7) P ( n (i))
(9)
which is exact in the limit of slowly varying densities, the solution of the many-particle problem is reduced to the self-consistent solution of the Kohn-Sham equations
with an effectiue one-particle potential of the form u(i) = u y i )
+
s
2n(i3
-di’ + uxc(n(i)) (i- i l
(1 1 )
which consists of an external potential, a Hartree potential, and a potential for exchange and correlation. Neglecting its comparatively fast computation time, by solving Hartree-like equations in a self-consistent manner, D F theory is truly size-consistent and remains an orbital theory, thereby offering advantages in interpretation compared to configuration interaction (CI) methods. Unfortunately, the electronic ‘interplay” between different configurations, which is so important in chemical reactions (transition states), is lost, and moreover, an in principle exact DF calculation is always only as reliable as the incorporated parametrization (22) Hubbard, J . Proc. R . SOC.London 1963, A276, 238-257. (23) Jones, R. 0.;Gunnarsson, 0. Reu. Mod. Phys. 1989, 61, 689-746. (24) Parr, R. G.; Yang, W. Density-Functional Theory of Atoms and Molecules; Oxford University Press: New York, 1989. (25) Dreizler, R. M.; Gross, E. K. U. Density Functional Theory: An Approach fo rhe Quantum Many-Body Problem; Springer: Berlin, Heidelberg, New York, 1990. (26) Hohenberg, P.; Kohn, W. Phys. Reu. 1964, 136, B864-B871. (27) Kohn, W.; Sham, L. J. Phys. Rev. 1965, 140, A1 133-A1 138. (28) Levy, M. Proc. Nafl. Acad. Sci. U.S.A. 1979, 76, 6062-6065.
+ F d N + q dN2 + ...
(12) (13)
to a reasonable accuracy, the change of the electronic ground state due to a change in the number of electrons is expressed by the chemical potential p which equals the slope of the function E versus N at N“. It measures the escaping tendency of a charged particle and, as a Lagrange multiplier, it plays a similar role within D F theory as the energy plays in the Ritz variational principle of wave-function theory. As shown in Figure 1 , which gives an atom’s or molecule’s E versus N function in a schematical way, the slope of this function ca? be easily approximated by taking the arithmetic average of the slope on the “left” side (equal to the negative ionization potential I) and the slope on the “right” side (equal to the negative electron affinity A ) as done in eq 3 . Upon comparing Parr and Pearson’s finite-difference approximation to with the formally identical Mulliken expression for electronegativity, it emerges that electronegativity is a concept surprisingly consistent and moreover justified within an accurate quantum-mechanical description of an electronic system. There is, however, a serious drawback as pointed out early by Perdew et al.,30investigations on fractional electron numbers as a time average led to the conclusion that, strictly speaking, the E versus N curve is a series of straight line segments with slope discontinuities at integral N. [Thisdoes not necessarily hold for an atom or functional group in a molecule, and not necessarily for a species of finite T.] Consequently, the chemical potential jumps because of irregularities in the exchange-correlation potential. Interestingly, this fact is known as the band gap problem in calculating semiconductor band structures using D F theory. We will face it again in section 3 . 3 . On the other hand, the second derivative q, measuring the curvature of the E versus N curve, is of comparable chemical interest as it measures the electronic tendency of a system to disproportionate and the sensitivity of the electronegativity to (29) Svane, A,; Gunnarsson, 0. Phys. Reo. B 1988, 37, 9919-9922. (30) Perdew, J. P.; Parr, R. G.; Levy, M.; Balduz, J. L., Jr. Phys. Rev. Letf. 1982, 49, 1691-1694.
Reactivity, Acidity, and Basicity in Solid-state Materials change in the number of electrons. If the second derivative is greater than zero, the ground-state E is concave upward with changing particle number, and the system is stable against falling into charged pieces. Again the meaning of this statement can easily be visualized by looking at Figure 1. From elementary differential analysis, the finite-difference approximation for the curvature 7 is 7)
= !/2(E+ + E-) -Eo
expressions, and so alternative methods have been chosen to investigate local properties in molecules. The molecular softness S is defined39 as half the inverse of the total hardness TJ
s=
and
introduced by Parr and Pearson2’ and substantially refined by Nalewajski (by incorporating electrostatic effects and resolving the “hard-hard paradox”34)allow the semiquantitative measure of electron transfer AN and energy transfer AE during a chemical reaction. To further characterize the local consequences of total hardness at an atom within a molecule, a new quantitative scheme was p r o p o ~ e d ,using ~ ~ . ~the ~ expressions
+ $n(i)dv(i)di
(17)
d p = 217 d N + $f(i)dv(i)di
(18)
and introducing the so-called Fukui functionfli) by the Maxwell relation
As is obvious from the last equation, the Fukui function is strongly related to the electron density of the frontier orbitals and permits one to draw contour maps for reactivity tendencies within a molecule. Moreover, Klopman’s initial treatment of chemical reactivity is reinstalled. It turned out to be difficult to calculate local h a r d n e ~ s e s because ~ ~ ~ ~ *of highly complicated computational Pariser, R. J . Chem. Phys. 1953, 21, 568-569. Pearson, R. G. Inorg. Chem. 1988, 27, 734-740. Pearson, R. G. Chem. Brit. 1991, 444-447. Nalewajski, R. F. J . A m . Chem. SOC.1984, 106, 944-945. Parr, R. G.; Yang, W. J . Am. Chem. SOC.1984, 106, 4049-4050. Yang, W.; Parr, R. G.; Pucci, R. J . Chem. Phys. 1984, 81, 2862-2863. (37) Berkowitz, M.; Ghosh, S. K.; Parr, R. G. J . Am. Chem. SOC.1985, 107, 6811-6814. (38) Ghosh, S. K. Chem. Phys. Lett. 1990, 172, 77-82.
1/27
(20)
whereas the local softness is given by the product of the Fukui function and molecular softness according to
(14)
This means that the system is more likely to stay at l?‘ instead of breaking up into two pieces having energies of E+ and E , indicated by a positive value of 7. Therefore, it is obvious to think of the curvature value 7 as a resistance indicator of the system against any electronic (chemical) attack. Most importantly, using the finite-difference approximation which connects 7 to the HOMO-LUMO gap of molecules, there are no difficulties in definitions arising from those discontinuities in the E versus N curve by the tXCjump. Probably the first one to recognize the importance of this finite-difference approximation was Pariser who connected it to the self-repulsion integral.31 There are already available several tables of atoms’ and small molecules’ absolute electronegativity and absolute hardness values,24J2s33calculated using numerical estimates of electron affinities and ionization potentials from gas-phase measurements. In general, these data are in fascinating agreement with chemical knowledge. The approximate formulas
dE = pdN
J . Am. Chem. SOC.,Vol. 114, No. 18, 1992 7233
43 = Mi)
(21)
It is important to note that local softness and hardness values are not simply reciprocals of each other. The usefulness of the last definition has been shown in the course of electronegativity equalization calculation^^*^^ in order to compute effective electronegativities as well as atomic charges. Typically, for the investigation of small molecules, various local softnesses are obtained with the help of so-called condensed Fukui functions,43based on gross Mulliken charges q such as
fi+
= qi(N
+ 1) - qi(N)
(22)
This allows the estimation of reactivities for nucleophilic, electrophilic, and radical attacks. The language of this kind of theoretical chemistry is quite m a t ~ r e ,and ~ ~ all . ~important ~ relations between global and local hardnesses and softnesses as well as more elaborate definitions of so-called hardness and softness kernels q(i,i’),s(i,i’), true reciprocals of each other, have already been developed. Quite recently, even higher-order derivatives were presented.45 Applications for small molecules have been publ i ~ h e d ~analyzing ~,~’ possible reaction paths by looking at the condensed Fukui functions. An overview of the possible calculational routes can be found in ref 48. A slightly different approach was introduced by Nalewajski et al. who pragmatically set up an atom-in-a-molecule hardness matrix 8 whose diagonal elements are given by the formula of Parr and Pearson (eq 6), whereas the nondiagonal entries are approximated semiempirically, finally giving access to normal displacement modes in electron populations. Moreover, reactive tendencies within molecules have been analyzed, especially those incorporating donor-acceptor contribution^.^^^^^ Keeping all this in mind, the theoretical notation of absolute hardness and its consequences for crystal (solid-state) chemistry compared to molecular chemistry seems to have been neglected. With the exception of an early paper about structure diagrams that involves a Philips-Van Vechten-like descriptions1of the solid state, on the one hand,52and the theoretical analysis of a bulk metal’s Fukui function that comes out to be the normalized local on the other, a hardness or density-of-states at the Fermi (39) Yang, W.; Parr, R. G. Proc. Narl. Acad. Sci. U.S.A. 1985, 82, 6723-6726. (40) Yang, W.; Lee, C.; Ghosh, S. K. J . Phys. Chem. 1985,89,5412-5414. (41) Mortier, W.; Ghosh, S. K.; Shankar, S. J . Am. Chem. Soc. 1986,108, 43 15-4320. (42) Van Genechten, K. A.; Mortier, W. J.: Geerlings, P. J . Chem. Phys. 1987, 86, 5063-507 1. (43) Yang, W.; Mortier, W. J. J . Am. Chem. SOC.1986,108,5708-571 1. (44) Berkowitz, M.; Parr, R. G.J . Chem. Phys. 1988. 88, 2554-2557. (45) Fuentealba, P.; Parr, R. G. J . Chem. Phys. 1991, 94, 5559-5564. (46) Lee, C.; Yang, W.; Parr, R. G. J . Mol. Srrucf. (THEOCHEM)1988, 163, 305-3 13. (47) Langenaeker, W.; De Decker, M.; Geerlings, P.; Raeymaekers, P. J . Mol. Srruct. (THEOCHEM)1990, 207, 115-130. (48) Glquez, J. L.; Vela, A.; Galvln, M. Srrucf. Bonding 1987, 66,79-97. (49) Nalewaiski, R. F.; Korchowiec, J.; Zhou, Z. Int. J . Quantum Chem. 1988, 22, 349-j66. (50) Nalewajski, R. F. Znt. J . Quantum Chem. 1991, 40, 265-285. (51) Phillips, J. C.; Van Vechten, J . A. Phys. Reu. B 1970, 2 , 2147-2160. (52) Shankar. S.; Parr, R. G. Proc. Natl. Acad. Sci. U.S.A. 1985, 82, 264-266.
7234 J . Am. Chem. SOC.,Vol. 114, No. 18, 1992
Dronskowski
reactivity theory for solid-state compounds (infinite molecules) is unknown. This is very astonishing as there are already collections of empirical facts concerning the catalytic properties of solid surfaces, for example, which are surely a strong sign of acidic and basic material proper tie^.^^ The definition of the condensed Fukui function as a working tool is based on definitions of Mulliken charges, which themselves are calculated via overlap populations. In "conservative" molecular quantum mechanics, one typically deals with a well-known and still improving set of Gaussian- or Slater-type functions where the success and failure of the Mulliken overlap population scheme is well investigated. In contrast, in solid-state quantum mechanics, because of the quite different computational problems that had to be solved, a large set of most different methods was invented, typically working with sets of various basis functions such as plane waves, Gaussians, analytical atomic partial wave expansions, spherical waves, Hankel- and Bessel-type functions (often energyand potential-dependent), and so on. To formulate a reactivity and acid-base theory that is as general as possible, we will circumvent any difficulties due to different degrees of basis set orthogonality and therefore avoid the use of overlap populations and Fukui functions. Moreover, using such an Ansatz one immediately loses one of the Fukui function's limitations, namely, it being an intrinsically relative measure of reactivity. As already said above, local hardness h ( i ) and local softness s ( i ) are not true reciprocals of each other. The latter number is calculated indirectly by multiplication of S (global value) withfli) (local value). In addition, the Fukui function Ai) only measures differences in electron occupations at the frontier orbitals without telling anything about their bonding or antibonding character. This can, of course, be done "by hand" while investigating the transparent M O order of a small molecule. In the solid state, however, the delocalization of all levels and the energy spectrum as a consequence make this a quite difficult undertaking. From a heuristic standpoint, therefore, we will break down the solid-state ensemble's hardness directly into atomic (or bonding) contributions. As atoms condense to a solid with its given crystal structure, all absolute electronegativities equilibrate because of the governing consequences of density-functional theory. As has been pointed out earlier,54the electronegativity is the negative of the chemical potential, and it is the driving force of chemistry to equilibrate chemical potentials and to transfer electrons. At this point, chemical intuition asks for different chemical reactivities within the solid. Although all chemical potentials have been equilibrated, the atoms will still retain different characteristics while being chemically attacked. In other words, their resistance toward a global change in the electronic system will changefrom atom to atom, and that is exactly what we are going to look for.
wherc 2 runs over all atomic positions in the primitive unit cell and T over all its translations. Consequently, the important difference between the molecular and the solid-state case is that we have a spectrum of different states in energy which vary with k, in contrast to the discrete energy levels of the molecule. Therefore, a well-suited description is an energy-resolved k-dependent (spectral) density-of-states matrix, here restricted to the non-spin-polarized case,
P J ~ , R ) = CJ n , c ~ j ( i ) c v j ( Z )-~t(j )t
(27)
The total electronic energy (equivalent to the total energy within E H t h e ~ r y up ~ ~to~the ~ ~Fermi ) energy is given by
(28) 73 characterizes the real parts of the complex off-diagonal entries. To simplify the notation, we average the k-dependent densityof-states matrix over the whole Brillouin zone and get
Thus, the total electronic energy is rewritten as
assuming the exchange integrals h to be potentially energy-dependent. As in a molecular case, the first part of the formula is centered at the atoms R while the m n d part represents the bonds within the crystal. By using the three-point finite-difference approximation (eq 14) the absolute hardness of a crystal is written as
3. Reactivity, Acidity, and Basicity A short theoretical outline of the concept for an isolated molecule may be taken from the Appendix. However, the following description for the solid state should be self-explanatory. 3.1. Reactivity within the Solid Stgte. According to Bloch's theorem there is a reciprocal vector k for a wave function that fulfills Schrodinger's equation such that a translation by a real lattice vector T i s equivalent to the multiplication with a phase factor
dji(il,r +
5 = eiiTdj(i,q
(25)
j is the 'band index". If the Bloch function is constructed as a
(26)
where the integration in energy is to be performed for the N 1 case (up to E:), for the N case (up to e:), and for the N 1 case (up to t i ) . It is clear that the crystal's hardness is given as a summ of what is to be defined as an atomic reactivity increment f R
(53) Tanabe, K.; Misono, M.; Ono, Y . ;Hattori, H. New Solid Acids and Bases; Elsevier Science Publishers: Amsterdam, 1989. (54) Donnelly, R. A.; Parr, R . G.J. Chem. Phys. 1978, 69, 4431-4439. (55) Hall, G. G.Proc. R. SOC.London 1952, A213, 113-123. (56) Manning, P. P. Proc. R. SOC.London 1955, A230, 424-428. (57) Ruedenberg, K. Rev. Mod. Phys. 1962, 34, 326-376.
(58) Koopmans, T. Physica 1933, I , 104-113. (59) Hoffmann, R.; Lipscomb, W. N. J . Chem. Phys. 1962, 36, 2179-2189. (60) Hoffmann, R. J . Chem. Phys. 1963, 39, 1397-1412. (61) Baird, N. C. Tetrahedron 1970, 26, 2185-2190.
linear combination of atomic centered orbitals, we can write c,,,(i)pr(7- r? -
$j(&7') = x e i i T x T
R
0
r E R
+
Reactivity, Acidity, and Basicity in Solid-state Materials t
= CfR R
(34)
where the definition of the absolute atomic reactivity increment, which is an energy value measured in eV, is formulated as
JfFh;,R[P,,(e)]ds - 2 f ' P h ~ v R [ P , , v ( ede) )]
1
J . Am. Chem. SOC.,Vol. 114, No. 18, 1992 7235 to be upper case Greek letters instead of lower case. Because of the normalization, they are not transferable between different compounds. To simplify the calculations we may assume the exchange integrals to be explicitly energy-indepenzent. Thus the simplified absolute atomic reactivity increment tRwill read
(35)
The reader might recognize that this definition of the tRincorporates sums and differences of so-called atomic Hamilton populations. Compared to the common overlap population S,,,P,, which adds up to the total number of electrons if integrated over all occupied electronic states, the Hamilton population h,?, adds up to the total electronic energy, as already mentioned before. Only within E H theory is there a linear relationship between them, based on the approximation of Wolfsberg and Helmholz62
Introducing a "frozen band approximation" which is the solid-state analogue of Koopmans' theorem,j8 we can further simplify the integrals' differences to
~ : P , , , ( E )de - 2f"P,,((c) dc - Z l p , , , ( s ) de)
+
where K is a proportionality constant of the order 1.75. Moreover, in eq 35 the Hamilton populations both combine atomic (terms 1 to 3) and bonding parts (terms 4 to 6 ) . W e could call this a gross increment. However, if we artificially "cut" all bonds from an atom, its atom-centered Hamilton populations will then form a net atomic reactivity increment: fR,ne, P
cfc
f t ' h ~ , ~ , , ( c dc ) + f'*hh;,P,,(c) de -
d
R
2f"h&P,,(O
de) (37)
If we only focus on a bond's reactivity, the investigation of the absolute bond reactivity increment of the bond between the atoms R and R' would be of interest, and would read fRR:bond
C f ( S f f h ~ v R [ ~ p v ( de e )+ l r 8 R v
