<<

Research: Science and Education

Chemical Bonding as a Superposition Phenomenon

Frank Weinhold Theoretical Institute and Department of Chemistry, University of Wisconsin, Madison, WI 53706; [email protected]

Chemistry and Functions where this complication can be ignored.) Whereas |φ(x)|2 is intrinsically a nonnegative quantity, as befits a probability, Perhaps the most difficult conceptual jump in all of the orbital φ(x) itself typically oscillates between positive and science is that from classical to (“wave”) . negative values in “wavy” fashion, giving rise to the character- us to surrender some of the most istic interference phenomena that underlie chemical bonding. self-evident tenets of our classical (pre-1925) picture of physical Each orbital φ of the full set of solutions of Schrödinger’s and causality. Such “quantum weirdness” deeply disturbed equation has potential chemical significance as a possible state Einstein, Schrödinger, de Broglie, and other pioneers of of the , whether actually occupied or not. (For example, (1). In recent years, increasingly stringent the transfer of the electron from one orbital state to another experimental tests (all passed with flying colors by quantum corresponds to a possible spectral excitation.) The Pauli prin- mechanics) have further underscored the remarkable di- ciple imposes the important quantal restriction that no more chotomy between common sense concepts and the observed than one electron (of given ) may occupy a given orbital. behavior of matter in the quantum domain (2). Thus the manifold of occupied orbitals constitutes the , above all, cannot escape the conceptual di- “” of an , while the manifold of lemma posed by quantum mechanics, for chemistry is in the low-lying unoccupied orbitals represents its capacity for elec- deepest sense a quantum phenomenon. Put another way, tronic give-and-take in chemical interactions with other . elementary particles that follow the classical (rather than The Schrödinger equation quantum) laws of cannot “do” chemistry. It therefore Ᏼ^φ φ follows that the fundamental concepts of chemical bonding n(x) = En n(x) (1) must be sought in the strange mathematical and logical rules of φ allows us to associate each orbital n with a definite the quantum domain, rather than in the comforting images Ᏼ^ En in a system described by quantum . More of 19th-century physics or their extensions in Bohr’s “old generally, we can consider that quantum mechanics provides quantum theory” of atomic structure. a functional Ᏹ[φ] (a “ of a function”) that associates These days, beginning chemistry students are quickly a definite energy Ᏹ with each chosen orbital φ. Specifically, plunged into the arcane conceptual constructs of quantum Ᏹ[φ] can be written as the Rayleigh–Ritz functional mechanics—“wave functions”, “orbital interactions”, “Pauli ∞ exclusion principle”, “hybridization”, “”, and the like φ* x Ᏼφ x dx (if not into mathematical details of the Schrödinger equation ᎑∞ Ᏹ φ itself). It is increasingly clear that such quantal concepts = ∞ (2) 2 provide the unified framework for satisfactory elucidation of φ(x) dx all chemical phenomena. Today, few would dispute ᎑∞ ’s prophetic 1929 statement, “The underlying physical laws [for] the whole of chemistry are thus completely under- whose stationary values are uniquely determined by eq 1. stood” (3), or Eyring’s similar 1944 conclusion, “In so far as However, we need not be concerned about details of how Ᏹ φ quantum mechanics is correct chemical equations are problems the [ ] functional actually works in order to recognize some in applied mathematics” (4 ). of its startling consequences. In the following, we cite some properties of the functional Ᏹ[φ] whose chemical conse- Some Basic Wave Mechanics quences can be deduced from simple algebra. The focal point of elementary bonding theory is the Superposition and “Spooky Action at a Distance” orbital. This can be defined as a 1-electron , that is, a solution of Schrödinger’s equation that describes a The most profound paradoxes of wave mechanics involve φ φ possible state of a single electron in a specified potential field superposition, the mixing of orbitals with plus sign ( 1 + 2, φ φ due to the nuclei and other of the system.1 More “in-”) or minus sign ( 1 – 2, “out-of-phase”). Energetic picturesquely, an orbital can be described as a “fuzzy orbit” consequences of follow from certain Ᏹ φ or “square root of the ”. Essentially, each mathematical properties of the energy functional [ ] that orbital φ(x) is a mathematical function of spatial position that may be quoted without proof: associates a wave φ with each point x, such that Invariance to Overall Sign Change the density ρ(x) of the electronic distribution (the relative Ᏹ φ Ᏹ ᎑ φ probability of finding the electron at point x) is given by [ ] = [ ] (3a) ρ(x) = |φ(x)|2 This property shows that the orbital energy does not depend on the overall sign of φ(x) (though it is sensitive to relative (More precisely, each [spin-] orbital is reserved for electrons sign changes within the orbital), consistent with the remark of particular spin [α = “up” or β = “down”], but we shall ini- that only the square of the orbital is related to experimen- tially restrict attention to the idealized single-electron case tally measurable probability density.

JChemEd.chem.wisc.edu • Vol. 76 No. 8 August 1999 • Journal of Chemical Education 1141 Research: Science and Education

Invariance to Scale Change However, we can formulate additional consequences of eqs 3a–f whose implications for chemical bonding and electron Ᏹ[λφ] = Ᏹ[φ] (3b) delocalization are immediate and obvious. λ ≠ for any scale factor 0. This property shows that the orbital energy is unaltered by an overall nonzero multiplicative factor The Donor–Acceptor Paradigm λ ᎑ (such as = 1, the special case of eq 3a). Only relative ampli- φ φ Let us now specialize to the case in which 1, 2 are, tude changes within the orbital have physical significance, φ φ respectively, an occupied ( o) and a vacant ( v) orbital on two and we can therefore ignore overall normalization factors that φ nuclear centers. (For example, o might represent the occu- are sometimes included for mathematical convenience. φ pied 2s spin-orbital of a lithium atom and v the unoccu- Nondistributive Character pied 3p spin-orbital of a chlorine atom.) At large separation, Ᏹ φ φ ≠ Ᏹ φ Ᏹ φ this system is adequately described by an orbital of the form [ 1 + 2] [ 1] + [ 2] (3c) eq 3e, with corresponding energy Ᏹ from solution of eq 3f. Solv- ⑀ Ᏹ φ ⑀ In words, this inequality states that if 1 = [ 1] and 2 = ing the quadratic equation eq 3f in the usual manner, we ob- Ᏹ φ φ φ [ 2] are the orbital of 1 and 2, then the energy tain the two possible solutions, labeled Ᏹ±, Ᏹ φ φ φ φ [ 1 + 2] of their in-phase superposition 1 + 2 is not simply 1/2 ⑀ ⑀ Ᏹ φ φ ≠ ⑀ ⑀ 2 1 + 2. (Similarly, [ 1 – 2] 1 – 2 for an out-of-phase ⑀ + ⑀ ⑀ – ⑀ Ᏹ v o ± v o 2 superposition, and more generally, for any linear combination ± = + hov (4) of the orbitals.) As a useful general measure of this effect, we 2 2 can define the quantity h12 1⁄ Ᏹ φ φ Ᏹ φ φ ≠ We assume that the vacant orbital lies higher in energy than h12 = 2{ [ 1 + 2] – [ 1 – 2]} 0 (3d) ⑀ ⑀ the filled orbital: v > o. For sufficiently large separation which corresponds to the energy “splitting” separating the (small |hov|), the square root in eq 4 can be expanded as a ⑀ ⑀ 2 orbital energies of in-phase and out-of -phase superpositions Taylor series in powers of |2hov/( v – o)| to give the expression from their hypothetical average value. ⑀ ⑀ ⑀ ⑀ 2 Optimal Superposition of Two Orbitals v + o v – o hov Ᏹ± = ± + + … (5) ⑀ Ᏹ φ ⑀ Ᏹ φ ⑀ ⑀ If 1 = [ 1] and 2 = [ 2] are the orbital energies of 2 2 v – o φ φ 1, 2, and if h12 is as defined in eq 3d, then one can generally Ᏹ find an improved (lower energy) optimal superposition orbital To leading order, the low-energy solution – therefore becomes Φ opt of the form 2 Φ φ φ h opt = c1 1 + c2 2 (3e) Ᏹ = ⑀ – ov – o ⑀ ⑀ (6) Ᏹ Ᏹ Φ × v – o whose energy = [ opt] is found as a solution of the 2 2 determinantal equation ∆Ᏹ The energy lowering DA resulting from this interaction of φ φ ⑀ Ᏹ filled (“donor”) orbital o with vacant (“acceptor”) orbital v 1 – h12 ⑀ Ᏹ =0 (3f) can therefore be expressed as h12 2 – 2 h ∆Ᏹ = ov The numerical values of the coefficients c1,c2 (dependent on DA ⑀ ⑀ (7) ⑀ ⑀ v – o 1, 2, h12) will be discussed below. Note that the form of the superposed orbital Φ in eq 3e opt Equation 7 can also be recognized as the familiar 2nd-order can persist even if the component orbitals φ ,φ of the superpo- 1 2 perturbation theory expression for the stabilizing effect of a sition are separated by great distances. In fact, the Pauli occu- 1-electron donor-acceptor interaction. pancy restrictions and other quantal superposition constraints Note that although the specific approximate form of are maintained even if the distance between orbital centroids ∆Ᏹ , eq 7, is valid only under the assumptions of the deri- could not be traversed at the speed of on the time scale of DA vation (particularly, ⑀ ≠ ⑀ , |h | << |⑀ – ⑀ |), it can be shown measurements on the separated distributions! The superposed v o ov v o Φ more generally that the solutions are of the form orbital opt gives rise to probability distributions measurably φ φ Ᏹ ⑀ ∆Ᏹ distinct from those of 1, 2 or their uncorrelated average (e.g., – = o – DA (8a) with respect to the probability of finding the electron spins of a Ᏹ = ⑀ + ∆Ᏹ (8b) dissociated diatomic “up” or “down” along chosen + v DA ∆Ᏹ axis directions). Superposition therefore leads to persistent where DA depends only on the magnitude of hov (for ex- ∆Ᏹ ⑀ ⑀ correlations between arbitrarily separated probability distri- ample, DA = |hov| when v = o). Whether we employ the ∆Ᏹ butions that could not be caused by any physical signal approximate form in eq 7 or the exact expression for DA traveling at less than the (“spooky action at a will not materially alter the following discussion. distance” [5]), and are thus disturbingly inconsistent with Figure 1 schematically depicts the donor–acceptor inter- ⑀ ⑀ classical intuition. action, showing how the initial orbital energies o, v are split The paradoxes associated with quantum superposition to give net energy lowering of the occupied level (hence, are usually discussed in terms of two-slit experiments (6 ), mutual attraction between the systems). The occupied orbital Φ involving no specific assumptions about the nature of the – of the interaction system is now of delocalized superposi- particles (charge, mass, etc.) or obvious relevance to chemistry. tion form (cf. eq 3e),

1142 Journal of Chemical Education • Vol. 76 No. 8 August 1999 • JChemEd.chem.wisc.edu Research: Science and Education

Φ φ φ – = co o + cv v (9a) Chemistry of Donor–Acceptor Interactions φ containing a partial admixture of the high-energy orbital v . Let us examine a few of the chemical phenomena that Three aspects of such donor-acceptor interactions are can be associated with donor–acceptor interactions. To make particularly noteworthy. the connection between the form of the superposition wave First, the corresponding interactions of two filled (spin-) function and the energetic quantities (eqs 6–8) more explicit, orbitals would not be similarly stabilizing. As eqs 8a,b and we can express eq 9a in the alternative form Figure 1 show, the two energy levels of the noninteracting Φ = cos(θ)φ + sin(θ)φ (9b) system are split by essentially equal and opposite amounts, so – o v the net stabilization of the interacting system would be nearly where the mixing angle θ that determines the extent of super- zero if both levels were occupied. In fact, consideration of position is given by θ ᎑∆Ᏹ higher-order terms shows that such filled–filled orbital inter- tan( ) = DA/hov (9c) actions generally lead to net energy increase, corresponding ∆Ᏹ Note that eq 9c remains correct when the exact value of DA to repulsions between the systems. Thus, the availability of is used in place of eq 7, and reduces to the proper symmetric an unoccupied orbital with which the filled orbital can in- θ superposition, tan( ) = 1, that is, co = cv, in the limiting case teract (i.e., into which the electron can partially delocalize) ⑀ ⑀ ∆Ᏹ v = o where DA = |hov|. is essential to the general attractive nature of superposition. The specific example of the previous section alludes to Second, the donor–acceptor energy lowering is generic, φ φ the o → v interaction of two nonidentical neutral atoms associated only with the general superposition expressions (e.g., Li + Cl). But we might equally well apply Figure 1 to (eqs 3d–f ), and virtually independent of the detailed form of Ᏼ^ ∆Ᏹ the case of a atom interacting with a bare proton, . In particular, the energy lowering DA depends only on Ᏼ^ + → + the magnitude of hov, not its sign, so any alteration of that H + H H2 (10) increases this magnitude (e.g., by increasing or decreasing 3 φ the prototype 1-electron . In this case, o and either kinetic or potential energy terms) would have an φ v are hydrogenic 1s atomic orbitals of the separated atom equivalent stabilizing effect. Of course, the details of the + Ᏼ^ and . At large separations of H and H , where |hov| is small kinetic and potential energy terms in (and their mutual and the superposition (eqs 9) is just beginning to develop, the relationship required by the virial theorem [7 ]) are impor- ⑀ ⑀ electron remains primarily localized on one center (co >> cv). tant for determining specific values of o, v, hov, and the However, at smaller separations where the interactions |h | quantitative energy lowering, but it is useful to recognize that ov becomes sufficiently large, the symmetric (co = cv) super- the qualitative chemical stabilizing effects of donor–acceptor position becomes the optimal solution. This corresponds to superposition are largely independent of these details. In the usual LCAO–MO form of the bonding σ orbital in the particular, it is fundamentally misleading to attempt to ration- + near-equilibrium region of H2 , alize the attractive chemical forces (e.g., in terms of classical ᎑ 2 σ 1/2 images of individual kinetic or potential energy terms ) AB = 2 {1sA + 1sB} (11) without reference to superposition. symmetrically delocalized between the nuclei A, B. Third, it is remarkable (and entirely characteristic) that The corresponding 2-electron covalent interaction the energy lowering ∆Ᏹ is achieved by mixing the low- DA ↑ ↓ → energy orbital with a higher-energy unoccupied orbital. Only H + H H2 (12) in quantum mechanics can one hope to lower the energy of can be considered as the composite of two such 1-electron an object by “blending in” some object of higher energy! This interactions, one for α and one for β spin. If we assume that counter-intuitive aspect of delocalizing interactions represents the α electron is initially on nucleus A and β on nucleus B, one of the deepest conceptual paradoxes of quantum theory, the two 1-electron interactions can be denoted (with “bar” yet such interactions pervasively underlie practically all chemi- denoting β spin) cal phenomena. A B B A φ →φ φ →φ o v , o v (covalent) (13) representing complementary delocalizations in opposite direc- tions (favored). However, the same final H–H orbital might arise from the coordinate covalent (“dative”) φ interaction of negative and positive hydrogen v ᎑ + → H + H H2 (14) expressed in terms of the two 1-electron interactions

φ A B A B o φ →φ φ →φ ∆ o v , o v (dative) (15) EDA delocalizing in the same direction (anion → cation). Such Φ φ φ – = co o + cv v dative-type delocalizations (“2-electron stabilizing interac- φ A 2 → φ B 2 tions” [8]), denoted ( o ) ( v ) , can also occur between Figure 1. Schematic perturbation diagram for 1-electron donor– neutral species, as in the prototype Lewis base–Lewis acid φ φ acceptor interaction between occupied ( o) and vacant ( v) spin- reaction ∆Ᏹ → orbitals, leading to energy lowering DA; cf. text eqs 8a,b. NH3 + BF3 H3N:BF3 (16)

JChemEd.chem.wisc.edu • Vol. 76 No. 8 August 1999 • Journal of Chemical Education 1143 Research: Science and Education

φ A φ B where o is a nonbonding nN orbital and v is a symmetric equilibrium structure. (In this respect, is 1 vacant 2pB orbital. As previously remarked, quanti- somewhat analogous to the intermolecular limiting case [eq tative ab initio descriptions of eqs 12, 14, and 16 involve 19].) The propensity toward bond-equalization and delocal- higher-level configuration interaction effects, but the physical ization in π-conjugated can therefore be seen as a gist of chemical bonding is well captured in all cases by the direct consequence of the strength of π → π* donor–acceptor simple LCAO–MO picture based on eqs 3d–f. superposition. Weaker forms of 2-electron donor-acceptor interactions Numerous other examples of donor–acceptor superpo- φ π σ occur when the acceptor orbital v is a vacant 2-center sition might be cited, including “hyperconjugative” → *, antibond orbital (rather than ), with additional σ → π*, and σ → σ* interactions of importance in spectros- nodal structure, higher energy, and reduced acceptor strength. copy (10), rotation barrier phenomena (11), hypervalency σ For example, the HF* antibond orbital of HF (the out-of- (12), and (13). However, the above examples may σ phase counterpart of the HF bonding orbital) can interact serve to illustrate the central role of the quantum mechanical with the ammonia lone pair, in a manner similar to eq 16, superposition principles (eqs 3) in a broad array of chemical φ A 2 φ B 2 to give the weaker form of ( o ) → ( v ) interaction that phenomena. leads to a , … Teaching the Donor–Acceptor Concept NH3 + HF → H3N HF (17) φ A φ B σ Although the donor–acceptor concepts sketched above with o = nN, v = HF*. Such an interaction corresponds, in resonance terms, to a partial admixture of the alternative are by now rather widely recognized in the chemistry research ion-pair resonance structure, community, their influence on freshman chemistry textbook writers and curriculum reformers is as yet virtually imper- … ↔ +… ᎑ H3N HF H3N–H F (18) ceptible. This is unfortunate, for these concepts are inherently 12 simpler and more highly visualizable (as well as substantially in which the roles of covalent bond and hydrogen bond are more accurate) than the familiar melange of freshman chemis- partially blurred. For this H-bonded complex, the weighting try “rationalizations”. We briefly sketch how these concepts might of the alternative ion-pair structure 2 is much smaller than be integrated into a reformed, unified framework that could truly serve as a “” introduction to advanced studies that of 1 (w1 >> w2), and the electrons remain well localized as pictured in structure 1. However, when the amine base in organic, inorganic, physical, and . ᎑ A donor–acceptor-based freshman chemistry curriculum NH3 is replaced by the stronger (anionic) donor F , the resulting interaction should generally emphasize quantal “orbital intuition”, includ- ing orbital visualization of interactions and charge distributions, ᎑… ↔ +… ᎑ F HF FH F (19) rather than electron-counting schemes and quasi-classical 3 4 particulate description of electron behavior. Topics to be φ φ σ promoted by this shift of emphasis include particle–wave (with o = nF, v = HF*) leads to the more strongly delocalized bifluoride ion FHF᎑ species, with symmetric equilibrium duality, fractional charge, polar covalency, bond and geometry and complete obliteration of the distinction be- antibond orbitals, and donor-acceptor mixing, as discussed tween covalent bond and H-bond (w3 = w4). Such a donor– below. acceptor picture allows one to account for many of the known Particle–Wave Duality regularities of H-bonding in terms of simple Lewis acid–base concepts (9). As stressed throughout this article, teachers should em- Still another well-known form of donor–acceptor inter- phasize the deeply counterintuitive implications of quantal action involves 2-center π orbitals as both donors and acceptors. particle/wave duality A simple example is butadiene (H2C=CH–CH=CH2), whose electron ↔ φ(x) (21) classical contains two filled π bond orbitals π π π rather than attempt to rationalize quantum behavior in terms ( A, B) and the corresponding unfilled antibond orbitals ( A*, π σ of comforting (but ultimately incorrect) classical models. This B*), held in proximity by the underlying bond network. Each π bond interacts with the adjacent π antibond would tend to encourage student interest in more rigorous and mathematical treatment of particle–wave duality (e.g., π 2 → π 2 π 2 → π 2 [( A) ( B*) , ( B) ( A*) ] in a future course), rather than compla- cency with misleading, infertile classical images. to give the strong conjugative delocalizations characteristic of extended polyene systems Fractional Charge C=C–C=C ↔ ؒC–C=C–Cؒ (20) The quantal association (eq 21) requires one to picture electronic charge in a distributed fashion (corresponding to 56 the spatial distribution of |φ|2). Students should be taught to In butadiene and other cyclic polyenes, the π → π* interactions think in terms of fractional charges (“partial charges”) and are not sufficient to entirely erase the bond alternation pattern molecular (or supramolecular) charge distributions, based on of the principal resonance structure (w5 > w6). However, in shapes of underlying orbitals. Less emphasis would thereby benzene and other cyclic aromatic systems, these delocalizing be placed on formal electron counting scheme (i.e., formal interactions are so strong as to obliterate the distinction charge, oxidation number, VSEPR) and idealized atomic or between single and double bonds (w5 = w6), leading to a ionic configurations of integer net charge.

1144 Journal of Chemical Education • Vol. 76 No. 8 August 1999 • JChemEd.chem.wisc.edu Research: Science and Education

Polar Covalency benefits of such an approach might be realized with relatively Corresponding to eq 3e or eq 11, the electrons of a general subtle shifts of emphasis in concepts listed above. A–B chemical bond should be pictured in terms of a general bond orbital function (in-phase superposition) of the form Conclusion φ The quantum leads to a virtually AB = cAhA + cBhB (22a) certain mechanism for rewarding interactions of a chemical where hA, hB are atomic hybrid orbitals and cA, cB are general nature. Given that nature favors evolution of matter toward coefficients (satisfying the normalization condition 2 2 states of lower energy, it is virtually certain that two electronic cA + cB = 1) that vary smoothly between 0 and 1 according systems, each allowed to probe the unoccupied orbitals of to the relative electronegativities of A and B. The simple the other, will find at least one of the two possible superpo- relationships between hybrid character and bond polarity φ φ ′ φ φ ′ φ ′ λφ sitions ( o + v or o – v , for some scaled v = v) to be expressed by Bent’s rule (14 ) should be strongly emphasized. energetically favorable. Hence, nature tends to favor bringing As Sanderson (15) has pointed out, a polar covalent descrip- electronic systems into closer spatial proximity (“bonding”) tion properly expresses the continuous variation between because this leads to nonvanishing interactions hov (of either covalent (cA = cB) and ionic (cA >> cB) limits, correcting the φ → φ sign) that will allow favorable o v delocalizations to lower unfortunate impression of extreme bonding “types” (each the energy.4 with atoms of integer electron and charge assignments) that The examples of this paper show that many combinations still pervade many elementary discussions of bonding theory. of occupied and vacant orbitals (1-center or bond type, Valence Bond and Antibond Orbitals intramolecular or intermolecular, σ or π, etc.) can lead to Emphasis should be placed on recognizing the charac- donor–acceptor superposition and stabilization. The favorable superposition patterns can often be qualitatively predicted teristic shapes of bond orbitals (eq 22a) corresponding to Ᏼ^ electron pairs of the localized Lewis structure diagram, as well with only minimal knowledge of the actual form of . This as to the corresponding antibonding orbitals justifies the conviction of many chemists that qualitative awareness of orbital shapes and mixing patterns (17) (rather φ AB* = cBhA – cAhB (22b) than, for example, details of individual kinetic and potential which represent residual unfilled (acceptor) valence-shell energy contributions) provides the most important insight capacity. Just as effort is expended to learn the shapes and in understanding and predicting chemical phenomena. energies of filled and unfilled atomic orbitals (to comprehend Owing to its pervasive dependence on the superposition hybrid and bond formation), so should the shapes and energies phenomenon, chemistry truly deserves to be called “the of leading bond and antibond orbitals (eqs 22a, b) be mastered quantum science”. to explain noncovalent interactions and electron delocalization Notes (“resonance”) effects. This represents only a slight gener- alization of what is commonly taught in the standard 1. While more accurate treatment of Schrödinger’s equation in- “homonuclear diatomic molecules” module, but would troduces further complexities, this simple “ picture” considerably extend the pedagogical use of these orbitals (e.g., is known to capture the essential physical gist of chemical bonding in beyond the paramagnetism of O ). Examples of the type of all systems considered herein. 2 2. For example, freshman chemistry textbooks often attempt to bond–antibond orbital diagrams that could enhance freshman “explain” chemical bonding in terms of “expanded box length” (to lower chemistry books are shown in the Encyclopedia of Computa- the kinetic energy) or “attraction of the electron to both nuclei” (to tional Chemistry (16 ). lower the potential energy). Each of these expresses only a partial truth. 3. In this case the formal -adapted eigenfunctions of Ᏼ^ Donor–Acceptor Mixing contains equal contributions from two degenerate, localized, long-lived The same type of reasoning and visualization that leads solutions of the time-dependent Schrödinger equation, only one of which is physically relevant. However, one can use a formal perturbative to formation of covalent bonds from overlapping hybrids hA, h should be employed to describe the interactions of over- treatment, based on an initial neutral H atom perturbed by a distant B proton, to obtain the unsymmetric solution that is appropriate in the lapping bond–antibond orbitals in forming delocalized present case. molecular (or supramolecular) orbitals, using the mnemonic 4. Of course, general aggregation of matter is opposed by both perturbation diagram (Fig. 1) or eqs 3–8 as a unifying frame- steric forces (filled-filled orbital interactions) and the randomizing work. As described in the section on chemistry of donor– effects of ambient thermal energies. acceptor interactions, this leads naturally to unified treatment of a variety of conjugative, hyperconjugative, and noncovalent Literature Cited phenomena that are usually considered “beyond” freshman 1. Einstein, A.; Podolsky, B.; Rosen, N. Phys. Rev. 1935, 47, 777– chemistry. 780. Jammer, M. The Philosophy of Quantum Mechanics; Wiley: In summary, a donor–acceptor-based approach should New York, 1974; Chapter 2. Yam, P. Sci. Am. 1997, 276(6), 124. aim to instill in the freshman chemistry student (i) increased Marshall, I.; Taylor, D. Who’s Afraid of Schrödinger’s Cat; Morrow: awareness that many fundamental chemical phenomena can New York, 1997. be traced back to basic wave-mixing principles embodied in 2. Bell, J. S. Physics 1964, 1, 195–200. D’Espagnat, B. Sci. Am. 1979, 241(5), 158. Watson, A. Science 1997, 277, 481. Figure 1 and (ii) increased familiarity with the shapes and 3. Dirac, P. A. M. Proc. R. Soc. London 1929, 123, 714. energies of leading atomic and bond–antibond orbitals 4. Eyring, H.; Walter, J.; Kimball, G. E. ; Wiley: needed to treat the phenomena of bonding “resonance” de- New York, 1944; p iii. localization, , and hydrogen bonding. The 5. This phrase, from Einstein’s remarks at the 5th Solvay Conference

JChemEd.chem.wisc.edu • Vol. 76 No. 8 August 1999 • Journal of Chemical Education 1145 Research: Science and Education

(Jammer, ref 1, Sec. 5), was often echoed (in various languages) 12. Reed, A. E.; Schleyer, P. v. R. J. Am. Chem. Soc. 1990, 112, 1434. by pioneer quantum theorists. See, e.g., Ehrenfest, P. Z. Phys. Cleveland, T.; Landis, C. R. J. Am. Chem. Soc. 1996, 118, 6020. 1932, 78, 555–559. Pauli, W. Z. Phys. 1933, 80, 573–586. Landis, C. R. Adv. Mol. Struct. Res. 1996, 2, 129–161. 6. Feynman, R. P.; Hibbs, A. R. Quantum Mechanics and Path 13. Deslongchamps, P. Stereoelectronic Effects in ; Integrals; McGraw-Hill: New York, 1965; Chapter 1. Pergamon: New York, 1983. Fleming, I. Frontier Orbitals and 7. Karplus, M.; Porter, R. N. Atoms and Molecules: An Introduction Organic Chemical Reactions; op. cit. for Students of Physical Chemistry; Benjamin/Cummings: Menlo 14. Bent, H. A. Chem. Rev. 1961, 61, 275. Park, CA, 1970; p 21. 15. Sanderson, R. T. Polar Covalence; Academic: New York, 1983. 8. Fleming, I. Frontier Orbitals and Organic Chemical Reactions; 16. Weinhold, F. Natural Bond Orbital Methods; In Encyclopedia of Wiley: New York, 1976; pp 24–27. ; Schleyer, P. v. R.; Allinger, N. L.; Clark, 9. Reed, A. E.; Curtiss, L. A.; Weinhold, F. Chem. Rev. 1988, 88, 899. T.; Gasteiger, J.; Kollman, P. A., Eds.; Wiley: Chichester, UK, 10. Mulliken, R. S. J. Chem. Phys. 1939, 7, 339. Crawford, V. A. Q. 1998; Vol. 3, pp 1792–1811. Rev. Chem. Soc. 1949, 3, 226. 17. For example, see Woodward, R. B.; Hoffmann, R. Conservation 11. Reed, A. E.; Weinhold, F. Isr. J. Chem. 1993, 31, 277–285. of Orbital Symmetry; Verlag Chemie: Weinheim, Germany, 1970.

1146 Journal of Chemical Education • Vol. 76 No. 8 August 1999 • JChemEd.chem.wisc.edu