Research: Science and Education Chemical Bonding as a Superposition Phenomenon Frank Weinhold Theoretical Chemistry Institute and Department of Chemistry, University of Wisconsin, Madison, WI 53706; [email protected] Chemistry and Wave 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 quantum (“wave”) mechanics. negative values in “wavy” fashion, giving rise to the character- Quantum mechanics forces 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 reality 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 electron, whether actually occupied or not. (For example, quantum theory (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 spin) may occupy a given orbital. behavior of matter in the quantum domain (2). Thus the manifold of occupied orbitals constitutes the Chemists, above all, cannot escape the conceptual di- “electron configuration” of an atom, 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 atoms. elementary particles that follow the classical (rather than The Schrödinger equation quantum) laws of physics cannot “do” chemistry. It therefore *^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 energy the quantum domain, rather than in the comforting images *^ En in a system described by quantum operator . 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 %[φ] (a “function of a function”) that associates These days, beginning chemistry students are quickly a definite energy % with each chosen orbital φ. Specifically, plunged into the arcane conceptual constructs of quantum %[φ] can be written as the Rayleigh–Ritz functional mechanics—“wave functions”, “orbital interactions”, “Pauli ∞ exclusion principle”, “hybridization”, “resonance”, and the like φ* x *φ x dx (if not into mathematical details of the Schrödinger equation {∞ % 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 scientists would dispute {∞ Dirac’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 % 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 %[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 wave function, 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 electrons of the system.1 More “in-phase”) or minus sign ( 1 – 2, “out-of-phase”). Energetic picturesquely, an orbital can be described as a “fuzzy orbit” consequences of quantum superposition follow from certain % φ or “square root of the electron density”. 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 amplitude φ with each point x, such that Invariance to Overall Sign Change the density ρ(x) of the electronic distribution (the relative % φ % { 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 %[λφ] = %[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, % φ φ ≠ % φ % φ this system is adequately described by an orbital of the form [ 1 + 2] [ 1] + [ 2] (3c) eq 3e, with corresponding energy % from solution of eq 3f. Solv- e % φ e In words, this inequality states that if 1 = [ 1] and 2 = ing the quadratic equation eq 3f in the usual manner, we ob- % φ φ φ [ 2] are the orbital energies of 1 and 2, then the energy tain the two possible solutions, labeled %±, % φ φ φ φ [ 1 + 2] of their in-phase superposition 1 + 2 is not simply 1/2 e e % φ φ ≠ e e 2 1 + 2. (Similarly, [ 1 – 2] 1 – 2 for an out-of-phase e + e e ± e % 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⁄ % φ φ % φ φ ≠ We assume that the vacant orbital lies higher in energy than h12 = 2{ [ 1 + 2] – [ 1 – 2]} 0 (3d) e e 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 e e 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 e e e e Optimal Superposition of Two Orbitals v + o v ± o hov %± = ± + + … (5) e % φ e % φ e e 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 % 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) % = e ± ov ± o e e (6) % % Φ × v ± o whose energy = [ opt] is found as a solution of the 2 2 determinantal equation ∆% The energy lowering DA resulting from this interaction of φ φ e % filled (“donor”) orbital o with vacant (“acceptor”) orbital v 1 ± h12 e % =0 (3f) can therefore be expressed as h12 2 ± 2 h ∆% = ov The numerical values of the coefficients c1,c2 (dependent on DA e e (7) e e 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 ∆% , eq 7, is valid only under the assumptions of the deri- could not be traversed at the speed of light on the time scale of DA vation (particularly, e ≠ e , |h | << |e – e |), 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 % e ∆% 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 % = e + ∆% (8b) dissociated diatomic molecule “up” or “down” along chosen + v DA ∆% axis directions).