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The Mechanism of Covalent Bonding
W
George B. Bacskay and Jeffrey R. Reimers School of Chemistry, University of Sydney, Sydney, NSW 2006, Australia Sture Nordholm Department of Physical Chemistry, Göteborg University/CTH, S-412 96 Göteborg, Sweden
Covalent chemical bonding is one of the central concepts of modern chemistry. On the phenomenological level, most students of chemistry can readily appreciate the differences in the chemistry of compounds with different types of chemical bonds (covalent, ionic, dative, hydrogen). Yet, in our experience, few students clearly understand the quantum mechanical origin of covalent bonding—especially since many chemistry texts gloss over the basics, shy away from the quantum mechanics essential to appreciate the fundamentals of chemical bonding, or provide a glib and incorrect explanation. The picture that is often presented conveys the idea that covalently bonded atoms are held together by a kind of “electronic glue”—that electronic charge is accumulated between two nuclei in “bonding regions”, which results in an energy lowering because such “shared” charge is attracted to both nuclei. This kind of simplistic model is incorrect on at least two counts. It is an electrostatic description, which completely neglects the significance of electronic kinetic energy. Moreover, the accumulated charge in the bonding region is so small that any increased attraction between the nuclei and electrons as a result of such charge accumulation cannot overcome the nuclear repulsion; no strong bonding would be predicted. As will be discussed later, the mechanism that does decrease the total potential energy of a molecule as its constituent atoms move towards each other is orbital contraction, which increases the localization of the electrons around the nuclei and, incidentally, decreases the accumulated charge in the bonding region(s). The aim of this article and the computer visualization software that complements it is to show that the stabilization of a molecule is mainly due to the delocalization of the electronic motion over two or more atoms joined by chemical bonds. We shall also show that analysis of the energetics of bonding in terms of kinetic and potential energy indicates a delicate balance between the two that strongly depends on the orbital basis as well as on bond length. Thus the common interpretation of covalent bonding as an electrostatic potential effect, argued forcefully by Coulson (1) on the basis of the virial theorem (2), will be found to be misleading and lacking in generality. The problem with the traditional electrostatic interpretation was exposed in detail by Ruedenberg and coworkers (3–7) over 20 years ago, but their insights have not reached mainstream chemistry texts (presumably owing to the demands of familiarity with quantum chemistry that they placed on the reader). We are convinced that the perceived pedagogical difficulties associated with the basic mechanism of covalent bonding are not only surmountable but largely illusory, especially if one can draw upon modern teaching tools such as computer graphics–based techniques that allow ready visualization of basic concepts such as covalent bonding. We W This article appears with color images on JCE Online at http://jchemed.chem.wisc.edu/Journal/Issues/1997/Dec/ index.html .
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have produced computer software for the teaching of covalent bonding that is now part of the undergraduate chemistry curricula at our respective universities. In this article we present the essential theoretical concepts needed to teach covalent bonding. These concepts form the backbone of our computer software. We also briefly describe information generated when using the software. We hope that the combination of theoretical explanation and visualization we have developed will be useful to the chemical community, especially to teachers and students of chemical bonding. The figures in the article were generated by our visualization programs, which allow the arguments presented to be explored in detail. The General Principles of Covalent Bonding The starting point when discussing bonding of any sort is that the molecule that forms when two atoms (or ions, molecules, etc.) interact is more stable than its constituent separated fragments: a system of lower energy has been achieved. Conversely, energy is required to break a bond (dissociate the molecule). The intrinsic atomic or molecular energy is the Coulombic potential energy of interaction of the electrons and nuclei as well as their kinetic energies, provided we neglect relativistic effects—which are mostly very small. We concentrate first on the electronic energy of a molecule, ignoring the vibrational, rotational, and translational contributions, and study how the molecular (electronic) energy varies as a function of the nuclear geometry—for example, as two atoms come together forming a molecule. All the kinetic energy is therefore electronic in origin, but the potential energy at any geometry includes the nuclear repulsion term. Since energy is quantized, so we focus on the electronic ground-state energies of the atoms and molecules and ignore, for the time being, any electronic excited states. Other quantum mechanical concepts and tools required for an in-depth study of chemical bonding are atomic and molecular orbitals, linear superpositions (combinations) of basis states (such as atomic orbitals [AO], giving molecular orbitals [MO]), and the variation principle. An important aspect of quantum theory is the existence of zero-point energy; localized particles can never be completely stationary and always possess some kinetic energy. Neglect of the electrons’ kinetic energy contribution to covalent bonding is a fundamental flaw of the simplistic electrostatic picture of covalent bonding that students often acquire in first-year chemistry courses. Basic quantum theory tells us that the kinetic energy of an electron depends on its confinement by the potential energy well provided by the nuclei and other electrons—in crude terms, on the width of the potential well, as in the the particle-in-a-box problem. At this point we turn to the examples treated in detail by our program: the hydrogen atom, H2 +, and H2 molecules. They will be treated by the simplest forms of quantum chemistry, which nevertheless encapsulate pedagogically the physical mechanisms of covalent bonding.
Journal of Chemical Education • Vol. 74 No. 12 December 1997
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Figure 1. Coulomb potential well of hydrogen atom and energy levels.
Figure 2. The hydrogenic 4f z 3 atomic orbital.
The Hydrogen Atom
Because its kinetic energy is the same as in an isolated hydrogen atom, a stable molecule can form only if the attraction between the electron and the bare proton can overcome the nuclear repulsion. As will be demonstrated later, use of the simple localized atomic wave function described above results in no stabilization at all! Of course, the hydrogen atom would be polarized by the proton, producing an (induced) dipole–charge interaction. An adequate description of polarization requires a more flexible wave function than used here. The resulting stabilization, however, is about an order of magnitude smaller than the covalent bond energy. Delocalization can be introduced by first recognizing that the electron could be in either of the Coulomb wells of the nuclei, described by the localized wave functions φa (r) and φb(r), which become delocalized by superposition. The simplest delocalized wave function ψ for H2+ is thus the molecular orbital (MO) expressed as a linear combination of the two hydrogenic (ground state) 1s atomic orbitals:
The nuclear Coulomb potential and the resulting electronic energy levels are shown in Figure 1. The total energy (E) and its kinetic (T) and potential (V) components in a particular quantum state, labeled by the principal quantum number n (≥1), are given as
En = { 1 2 ; 2n
< T >n = 1 2 ; 2n
< V >n = { 12 n
(1)
where the Dirac brackets < > indicate that the property is known only as an average. We emphasize the shape of the Coulomb potential well, the quantization of energy, the degeneracy of the energy levels (which necessitates the introduction of orbital angular momentum and magnetic quantum numbers, l and m, respectively), and the inverse relationship between the apparent width of the potential and the spacing of the energy levels. It follows that if the Coulomb well were filled with an electron fluid, a drop in the fluid (Fermi) level would result in an increase in average kinetic energy and a corresponding decrease in average potential energy, since the occupied part of the Coulomb potential well becomes “deeper” as the well is emptied of electron fluid and the Fermi level falls. This Coulomb paradox was discussed in detail by Nordholm (8). The visualization program also allows students to study the shapes of the hydrogenic atomic orbitals, to investigate the dependence of kinetic, potential, and total energies on the orbital exponent, and to experiment with the Gaussian expansion of the orbitals. To illustrate, Figure 2 shows various 1- and 3-dimensional representations of a 4f orbital. The H2+ Molecule The simplest way to visualize H2 + is to think of it as a hydrogen atom and a proton separated by a distance R. The corresponding wave function is
ΨR r = ψa r = π{1/ 2exp { r – ra
(2)
which is just the wave function of a hydrogen atom in its ground state located at ra. The proton is at rb, a distance R away, but it is not allowed to affect the form of the electronic wave function localized around the other proton. The electron is thus confined to the atomic potential well around nucleus a.
ψ = c aφa + c bφb
(3)
where ca and cb are variational parameters determined by the application of the variation principle—that is, minimizing the total molecular energy
ψ * Hψ dτ E = E c a, c b =
2
ψ dτ
(4)
^ where H is the nonrelativistic electronic Hamiltonian of H 2+ at a specified internuclear distance R. The integration is with respect to the electron coordinates, collectively denoted by τ. It is important to note that the energy is a simple function of the variational parameters ca and cb . The simplest physical explanation of this process of superposition is that we allow the two atomic 1s waves to interfere. When the electron is in the vicinity of one of the nuclei the wave function ψ is approximately given by the appropriate atomic orbital φa or φb , while in the region of overlap, the two AO’s make comparable contributions, resulting in interference. A linear superposition of the AO’s will therefore yield molecular wave functions, which, owing to the equivalence of the protons, will be either symmetric or antisymmetric linear combinations, namely, MO’s
ψ± = c ± φa ± φ b
Vol. 74 No. 12 December 1997 • Journal of Chemical Education
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Information • Textbooks • Media • Resources where the normalization constants are given as
c ± = 2 1 ± S ab
{1/ 2
(6)
and Sab is the orbital overlap
S ab = φa*φbdτ
(7)
The corresponding energies, obtained by minimization of E(ca, cb ), are H ± H ab E ± = aa (8) 1 ± S ab where the matrix elements H aa and Hab are defined as
H ab =
φa*Hφbdτ
(9)
As equations 5 and 8 show, application of the variation principle yields, in addition to the ground state (bonding, energy and wave function, E + and ψ+), an antibonding excited state wave function ψ{ with energy E { . The energies of these two states and their kinetic and potential components, as a function of the internuclear separation R, are shown in Figure 3, along with the energies of the separated H + H+ system discussed above. Study of the energies and their distance dependence permits several observations about covalent bonding. Concentrating on the bonding state first, we note that as the two atoms approach each other there is a drop in the kinetic energy that more than compensates for the repulsive nuclear potential energy. The result is net binding—that is, a stable molecule with an equilibrium distance of 2.5 a0 and dissociation energy of 0.0648 Eh. The total potential energy is actually repulsive at all distances, indicating that the Coulomb interaction between the nuclei and the electrons is smaller than the nuclear repulsion. At smaller-than-equilibrium separations the rise in kinetic and potential energies results in a repulsive “wall”. The behavior of the kinetic energy is directly related to the degree of spatial freedom the electron has in the molecule, that is, delocalization. While such a process is easy to understand when studying the particlein-a-box problem, it does not at first seem quite so straightforward here, as the initial kinetic energy drop occurs as the molecule actually becomes “smaller”. The complicating factor is the Coulombic double well potential. At large separations the electron is actually trapped by one of the nuclei. Any delocalization must take place by tunneling through the barrier between the nuclei. Rigorous exploration of the process requires a more complex analysis than we wish to embark on here. The end result, however, is very simple: the rate of electron transfer from one nucleus to the other is inversely proportional to the energy difference between the bonding and antibonding states—which depends on the cou^ pling matrix element , whose magnitude varies inversely with distance. Thus we get increasing dynamical delocalization as the rate of interatomic electron transfer increases with decreasing distance. As the nuclei move closer to each other, complete delocalization is effectively achieved while the potential is gradually transformed into one that more and more resembles the single well potential of He+. This results in greater spatial confinement for the electron, and hence an increase in kinetic energy, in accord with the particle-in-a-box picture once again. Our visualization program can display the nuclear double well potential and the corresponding bonding and antibonding energy levels at any internuclear separation specified.
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Figure 3. Energy curves of H + H+ and the bonding and antibonding MO’s of H2+ , calculated with a fixed orbital exponent α = 1. Arrow indicates equilibrium bond length in the bonding (ground) state.
Figure 4. Electron density and density difference maps of bond+ ing MO of H2 (orbital exponent α = 1 at R = 2a0).
Figure 5. Electron density and density difference maps of antibonding MO of H2+ (orbital exponent α = 1 at R = 2a0).
Journal of Chemical Education • Vol. 74 No. 12 December 1997
Information • Textbooks • Media • Resources Thus the key to the fundamental mechanism of covalent bonding is the process of delocalization that is accompanied by a faster internal electron transfer in the H 2+ molecule, and thereby a very much lower ground state energy, E+, than predicted by the localized model with the electron confined to one proton, as in an isolated H atom. Comparing the bonding and antibonding states, given by equation 5, we note that the most significant difference in the wave functions is the presence of a node in the antibonding MO. From the particle-in-a-box problem we know that kinetic energy rises with the number of nodes in the wave function, provided the spatial confinement by the potential (size of the effective “box”) changes little. This is so for H 2+, and the larger kinetic energy in the antibonding state more than compensates for the lower potential energy (see Fig. 3). The difference between the bonding and antibonding states, ψ± , is of course manifested in the corresponding electron densities, r± , which are 2
ρ± = ψ± = c ±
2
2
2
φa + φb ± 2 φaφb
(10)
The main difference is in the sign of the interference term 2|φaφb | that results in a buildup or depletion of charge in the region of high overlap, the “bonding” region, when compared with the sum of the atomic densities 2 ρatom = 1 φa + φb 2
2
(11)
Furthermore, since
c+ ≤ 1 ≤ c{ 2 2
2
(12)
it is easy to see that, to maintain normalization, the interference term is compensated by corresponding changes in the atomic terms |φa|2 and |φb |2. Contour maps of the densities ρ± are shown in Figures 4 and 5, along with density difference maps (ρ ± – ρatoms) that illustrate these points. Although the treatment of H2+ outlined above describes the bonding qualitatively correctly and quantitatively accounts for a considerable fraction of the binding energy as well, it has been criticized on the theoretical ground that it fails to satisfy the virial theorem (2), which requires that
Figure 6. Energy curves of H + H+ and bonding and antibonding MO’s of H2+ (orbital exponent α optimized for each wave function at all distances). Arrow indicates equilibrium bond length in the bonding (ground) state.
at equilibrium separation, as well as at infinity, the ratio of kinetic to potential energies be given as
T V
={1 2
(13)
Consequently, the kinetic, ∆〈T〉, and potential, ∆〈V〉, components of the binding energy ∆E (at equilibrium) should also be related as in equation 13, suggesting that binding occurs because of a drop in potential energy—that is, ∆〈V〉 < 0— despite a rise in the kinetic energy. On the basis of such behavior of the energy the conclusion has often been incorrectly reached that the potential energy drop is associated with the pileup of charge in the bonding (interference) region. The virial theorem is satisfied by the exact wave function as well as approximate (variational) wave functions that are fully optimized with respect to all variational parameters. In the case of the simple MO wave function for H2+ constructed from the AO’s φa and φb, this can be achieved by optimizing the AO’s themselves, by varying the orbital exponent α of the 1s AO’s, given as, for example
φa =
α3 / 2 exp { α r a π1/ 2
(14)
where ra is the distance of the electron from nucleus a. The behavior of the resulting energy and its components is shown in Figure 6 for both the bonding and antibonding states and the separated H + H+. Comparing the energies with those obtained using the fixed exponent of 1 shows that, while the behavior of the kinetic and potential energies is now very different, the total energy curves are qualitatively the same. More details on the bonding state, such as the distance dependence of the orbital exponent α are presented in Figure 7. The equilibrium distance of 2.0 a0 is in good agreement with experiment, and the dissociation energy of 0.0865 E h is 33% higher and thus considerably “better” than what had been obtained before (0.0648 Eh). The orbital exponent varies between 1 (at infinite separation) and 2 (in the united atoms limit), showing that as the molecule forms, an orbital contraction occurs. The effect of such a contraction is to decrease the atomic contribution to the potential energy while increasing the kinetic energy. We note that in
Figure 7. Detailed study of energetics of H2+ in its ground state (orbital exponent α optimized at all distances). Arrow indicates equilibrium bond length.
Vol. 74 No. 12 December 1997 • Journal of Chemical Education
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Information • Textbooks • Media • Resources the H atom itself the kinetic and potential energies of the electron are (in Eh)
T = 1 α2 ; 2
V = {α
(15)
However, as seen in Figure 7, the net increase in kinetic energy is actually due to an increase in the bond-perpendicular (x,y) components. The bond-parallel component remains attractive over a much greater range. The importance of such an analysis of the kinetic energy has been emphasized by Ruedenberg and coworkers (2–6). The density maps in Figure 8 show the combined effects of delocalization and orbital contraction for the bonding state. The virial theorem, as stated in equation 13, applies to a system of particles where the potentials are Coulombic. Other potentials, such as the harmonic one, give rise to a different partitioning of kinetic and potential energies. The somewhat peculiar nature of Coulomb potentials and their effect on the kinetic and potential energies of a delocalized electron fluid were discussed by Nordholm (7). The delocalization of the fluid implies that in each Coulomb well the Fermi level is lower than in a single well containing the same quantity of fluid. A drop in the Fermi level in a Coulomb well, however, increases the average kinetic energy while decreasing the average potential energy, in agreement with the virial theorem. In summary, our study of bonding in H2+ shows, first, that electron delocalization over the two nuclei is the key mechanism. Second, it shows that the stabilization due to delocalization may appear as a drop in kinetic energy for a minimal AO basis set with fixed exponent or as a drop in potential energy in accord with the virial theorem if the AO’s in the minimal basis are optimized with respect to the exponent α at all geometries. If a larger, more complete AO basis is used to construct the MO’s, namely,
ψ=
Σ c iφi i =1 n
(16)
the orbital contraction in the MO’s is resolved by the linear variational parameters {ci}. This accounts for the virial theorem and the great stability of bond energies and bond lengths with respect to choice of basis set as long as delocalization is properly accounted for. It is a paraphrasing of Ruedenberg’s picture of orbital contraction followed by kinetic energy lowering, simplified so as to allow its presentation in elementary chemistry texts. The studies of Hurley (8) highlight another interesting aspect of covalent bonding and the roles of kinetic and potential energies. Using the virial and electrostatic theorems, Hurley showed that the behavior of the kinetic energy with distance provides a “touchstone” for bonding. This follows from the integrated kinetic energy theorem, according to which
∆E = { 1 Re
∞
T(∞) – T(R) dR Re
(17)
Thus the kinetic energy T (R) must be smaller than that of the separated atoms T(∞) over a large enough range to ensure that the total interaction energy ∆E is negative (attractive). These findings neatly summarize an aspect of bond formation, the “competition” between delocalization and orbital contraction. Initially, as the atoms move toward each other, the lowering of energy is almost entirely due to delocalization (kinetic in origin). Orbital contraction takes place considerably later, when the molecule is near its equilibrium configuration. The ideas developed in this section and applied to H 2+ are generalizable to many-electron molecules. For example,
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Figure 8. Electron density and density difference maps of bonding MO of H2+ (orbital exponent α optimized at R = 2a0).
the popular self-consistent field (SCF) MO wave functions are constructed in terms of delocalized MO’s, which in turn could be formulated in terms of valence AO’s (which are contracted in comparison with the AO’s of the free atoms). However, this is rarely done because interest is usually in the properties of the molecules themselves rather than in the way they are formed from their constituent atoms. However, many-electron systems have their own subtleties that will be explored in the next section. The H2 Molecule The hydrogen molecule is the simplest molecule that is stable under “normal” terrestrial conditions, and its electronic structure has been the subject of extensive experimental and theoretical study. It provides the simplest and most popular textbook example of covalent bonding at all levels of theory ranging from Lewis’s electron pair and octet theory to the rigorous quantum mechanical treatments of Kolos and coworkers (9, 10). In this section we discuss various approaches to the construction of wave functions for H2, using just the atomic 1s orbitals as the basis set and keeping the formalism as simple as possible. The issues that we address include the structure of two-electron wave functions, electron indistinguishability and permutation symmetry, importance of electron spin, limitations of single configuration MO wave functions, excited electronic states, electron correlation, and the correct description of (two-electron) covalent bonds. The wave functions are presented in unnormalized form; normalization is explicitly considered only when necessary. Starting with two hydrogen atoms labeled a and b, the simplest H2 wave function is the Hartree product of the 1s AO’s φa and φb, where the two electrons, 1 and 2, are arbitrarily assigned to the atoms:
ΨHP = φa(1) φb(2)
(18)
The molecular wave function ΨHP, as all the H2 wave functions that will be discussed, is a two-electron wave function: that is, it is a function of both electrons’ coordinates r1 , r2, which are simply abbreviated as 1 and 2. This wave function is clearly appropriate at large internuclear separation where the approximation accurately represents the wave function as that of two independent hydrogen atoms. The simple Hartree formalism, however, fails to describe the formation of a strong covalent bond in H2 . It predicts an equilibrium
Journal of Chemical Education • Vol. 74 No. 12 December 1997
Information • Textbooks • Media • Resources bond length of 1.9 a0 and a dissociation energy of just 0.0196 Eh, results that are in very poor agreement with the experimental values of 1.4 a0 and 0.1745 Eh. The Hartree predictions are in fact characteristic of weak van der Waals–type interactions rather than of a covalent bond. So what is wrong with the Hartree approach? The answer is simple: it does not allow for electron delocalization. The wave function is such that electron 1 is always associated with nucleus a and electron 2 is associated with nucleus b. A related problem is that electrons, being identical particles, ought to be indistinguishable. The Hartree wave function does not account for this. Having worked through the H2 + problem the solution to the first of the above problems is obvious: allow the electrons to occupy delocalized, H2 +-like MO’s, ψ± , as defined in equation 4. At this point we need to appeal to the Pauli exclusion principle and consider electron spin. To form the simplest single configuration ground-state MO wave function, both electrons are assigned to the bonding MO ψ+ with opposite spins. The spatial wave function is therefore just ψ+(1) ψ+(2). This satisfies the indistinguishability requirement as well, since it is symmetric with respect to the permutation (interchange) of the electrons. The resulting equilibrium distance of 1.60 a0 and dissociation energy of 0.0991 Eh, calculated with a fixed orbital exponent of 1, are in much better agreement with experiment than the HP predictions, indicating that the MO model is valid—at least near the equilibrium geometry. As for H2 +, the orbital exponent can be optimized at each separation, yielding better estimates of the bond length (1.39 a0) and dissociation energy (0.128 Eh) and satisfying also the virial theorem. The arguments concerning delocalization, its effect on the kinetic and potential energy contributions to the dissociation energy, and the role of orbital contraction are exactly the same as for H 2+. Distributing the two electrons differently from the ground state configuration results in excited state wave functions— for example, where one electron is promoted from the bonding MO ψ+ to the antibonding MO ψ{ (singly excited configuration). Now, however, the Pauli principle allows the two electrons to be of the same spin as well as opposite. Thus the singly excited state can be either triplet or singlet, with somewhat different energies. Given that we are working within the confines of a minimal basis of two AO’s and hence two MO’s, the only other possible excited-state configuration is one where both electrons occupy ψ{: a doubly excited configuration that is also a singlet state with the spins antiparallel. Electron Spin and the Structure of Many-Electron Wave Functions We now discuss the structure of many-electron wave functions, concentrating on the two-electron case. The full wave function consists of space and spin components. The latter consist of one of two possible spin functions representing the two components (ms = ± 1 /2) of each electron’s spin (s = 1/2), denoted α and β, respectively, where s and ms are the quantum numbers associated with the electron’s spin angular momentum and its z-component. The possible two-electron spin functions θSM, where S and MS are quantum numbers that pertain to the total two-electron spin angular momentum, are 1θ a = α (1) β (2) – 1 3θ s = α (1) α (2) , 1 3θ s = α (1) β (2) + 2 3θ s = β (1) β (2) , 3
β (1) α (2) , β (1) α (2) ,
S = 0, M S = 0 S = 1, M S = 1 S = 1, M S = 0 S = 1, M S = {1
(19)
that is, a singlet (S = 0, MS= 0) that is antisymmetric (a) with respect to the interchange of the electrons and three triplet states (S = 1, MS = 1,0,{1), which are symmetric (s) with respect to electron interchange.
Now, the general formulation of the Pauli exclusion principle is that the total wave function must be antisymmetric with respect to the permutation of electrons. Therefore the spin functions need to be combined with spatial wave functions having the correct permutational symmetry. In the case of two MO’s, the possible symmetric and antisymmetric spatial wave functions are Φ1s = ψ+(1) ψ+(2) Φ2s = ψ+(1) ψ{(2) + ψ{(1) ψ+(2) Φ3s = ψ{(1) ψ{(2) Φ1a = ψ+(1) ψ{(2) – ψ{(1) ψ+(2)
(20)
Thus, the ground state (singlet) wave function is a product of Φ1s and 1 θ1a, while the lowest excited triplet state can be represented by the product of Φ1a and one of the triplet spin functions. The obvious difference between the spatial wave functions of the excited triplet (Φ1a, antisymmetric) and the lowest excited singlet (Φ2s, symmetric) states is the origin of singlet/triplet splitting. (The usual nonrelativistic molecular Hamiltonian is spin-independent. Thus the spin function does not directly affect the energy of a given state.) The MO, VB, and CI Wave Functions Returning to the MO description of the ground state of H2, we reiterate that the single-configuration MO model adequately describes bonding around the equilibrium geometry. However, it fails as the molecule dissociates. The energy of the molecule does not tend to the correct limit—the energy of two noninteracting hydrogen atoms—as the bond length becomes infinite, but rather tends very slowly to some higher energy. A simple analysis of the wave function indicates what the problem is. Let us express the spatial part of the MO wave function in terms of its constituent AO’s: ΨMO = ψ+ (1) ψ +(2) = [φa(1) + φb(1)][φ a(2) + φb(2)] = φa(1) φa(2) + φb(1) φb(2) + φ a(1) φ b(2) + φb(1) φa(2)
(21)
The wave function ΨMO is a linear combination of four configurations, two of which, φa(1)φa(2) and φb(1)φb (2), assign both electrons to the same atom, while the remaining two, φa(1)φb (2) and φb(1)φa (2), assign one electron to each atom. These configurations may be called ionic and atomic, as individually they represent H{/H+, H+/H{, and H/H ion and atom pairs. These configurations are present in the MO wave function in the same proportion at all H–H distances and imply that the molecule dissociates into species with both atomic and ionic character—or as usually said, into a mixture of atoms and ions. In reality, however, the molecule must dissociate into atoms, which, although reactive, are far more stable than a pair of ions. This deficiency of a many-electron single-configuration MO wave function is universal. The breaking of a covalent bond described by one or more doubly occupied MO’s implies dissociation into atoms and/or molecular radicals. Singleconfiguration MO wave functions, irrespective how complex and extensive the individual MO’s (such as SCF MO’s) are, cannot describe such processes correctly for the same reason as in the case of the H2 molecule. In principle the problem is easily remedied. The simplest solution is to omit the offending ionic terms—to expand the molecular wave function in terms of atomic configurations only. The resulting H2 wave function, known as the Heitler– London valence bond (VB) wave function, is ΨVB = φa(1) φ b(2) + φ b(1) φa(2)
(22)
This represents the first (approximate) solution of the Schrödinger equation for the ground state of the H2 molecule,
Vol. 74 No. 12 December 1997 • Journal of Chemical Education
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Information • Textbooks • Media • Resources proposed in 1927, just one year after the publication of Schrödinger’s wave equation. The full wave function is again the product of the symmetric spatial function (eq 22) and the singlet spin function. Since it describes covalent bonding and its formation in an essentially correct manner, the atomic configurations that comprise the VB wave function are usually referred to as covalent terms. Within the confines of VB theory we could also generate the lowest triplet excited state, by combining one of the triplet spin functions with the antisymmetric spatial VB wave function a ΨVB = φa(1) φb(2) – φb(1) φa(2)
(23)
An alternative approach to solving the problem of incorrect dissociation is to relax the restriction that the MO wave function be a single configuration. The technique of formulating and calculating multiconfiguration wave functions is generally referred to as configuration interaction (CI). For the H2 molecule, given the minimal MO basis {ψ+,ψ{}, we can construct four spatial configurations, as given by equation 20. Thus, the CI wave function for a singlet state is formulated as a linear superposition (combination) of the three symmetric configurations, each multiplied by the same antisymmetric spin function 1 u1a : Ψ CI = [c1Φ1s + c2 Φ2s + c 3Φ3s ]1θ1a
Figure 9. Ground state energy curves of H2 calculated using HP, MO, VB, and CI methods (orbital exponent α optimized for each wave function at all distances). Arrows indicate equilibrium bond lengths.
(24)
where the variational coefficients c1 , c2 , and c3 are determined by using the variation principle. The calculations yield three distinct energies and wave functions in the form of distinct sets of coefficients, which represent the ground state and two excited states. Since the configuration Φ2s has different spatial symmetry from the other two, the ground state CI wave function is in fact a linear superposition of just two configurations, so it can be written and expanded in terms of atomic and ionic configurations as ΨCI = c 1ψ + (1) ψ + (2) + c3 ψ { (1) ψ { (2) = (c1 – c3) [φa (1) φ b(2) + φ b(1) φ a(2)] + (c1 + c3) [φa (1) φ a(2) + φ b(1) φ b(2)]
(25)
We see that the CI wave function contains both the atomic and ionic configurations that are present in the single configuration MO wave function, but now their contributions are determined variationally, such that the CI energy is minimized. Thus, at large internuclear separations c1 ≈ { c3 and the ionic terms disappear, as they must to ensure correct dissociation. Where it is energetically advantageous, the ionic terms are present, to the optimal degree. Of course, it is perfectly valid to view the CI wave function (eq 25) as the generalization of the VB wave function (eq 22) so as to include ionic terms also. Therefore, despite the different starting points, the MO-based CI and AO-based VB methods are equivalent once all possible configurations are included in the wave function. The HP, MO, VB, and CI approaches to calculating the ground state wave function and energy of H2 are compared in Figure 9. The inadequacy of the HP approach is immediately apparent. In the equilibrium region the MO, VB, and CI methods provide comparable descriptions of the energetics, although at the quantitative level the superiority of CI is obvious. The incorrect asymptotic behavior of the MO wave function is also evident. The kinetic and potential components of the energy behave in a similar manner to their H2+ ground state counterparts, the exception being those from the MO calculation, which tend to the “wrong” dissociation limits. The ground and excited state CI energies and their distance dependence are shown in Figure 10. The triplet state is a repulsive (dissociative) state, whereas the singlet excited states are weakly bound states with equilibrium separa-
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Figure 10. Ground and excited state energy curves of H2 calculated by CI (orbital exponent α optimized for each wave function at all distances). Arrows indicate equilibrium bond lengths.
Figure 11. One-electron density maps of H2 in its ground and excited states calculated by CI (orbital exponent α optimized for each wave function at R = 1.4a 0).
Journal of Chemical Education • Vol. 74 No. 12 December 1997
Information • Textbooks • Media • Resources tions substantially larger than that in the ground state. The nature of these states and the differences between them can be further studied through their respective one-electron density maps shown in Figure 11. Electron Correlation In addition to describing correctly the breaking of the covalent bond, the VB and CI wave functions allow for another very important phenomenon: electron correlation. Since electrons repel each other, their “motion” should be correlated such that they “keep out of each other’s way”. Given an arbitrary wave function, a straightforward way to demonstrate the presence (or absence) of correlation is to analyze the corresponding two-electron probability distribution (density), function ρ12. For a two-electron system such as H2 , ρ12 (r 1,r 2) = |Ψ (r 1,r2)|2
(26)
describes the probability of electrons 1 and 2 being simultaneously at the points r1 and r2 respectively. Now, the electrons are uncorrelated if ρ12 can be factorized as the product of the one-electron density functions ρ 1(r1) and ρ1(r2). Consider the single configuration MO wave function, for which ρ12 (r1,r 2) |ψ + (r 1)|2|ψ + (r 2)|2 = ρ1(r1) ρ1(r2)
(27)
This simple result, showing that the electrons are uncorrelated, was to be expected, since each electron had been assigned to the same spatial orbital ψ+. The situation is different for the VB wave function, where
1
ρ12 r1,r2 = 2
2 1 + S ab
[φ
2 2 2 2 a (r1) φb (r2) + φb (r1) φa (r2)
(28)
]
+ 2φa(r1) φb(r1) φa(r2) φb(r2)
and cannot be factorized into ρ 1(r1)ρ 1(r2), the one-electron density function being
1
ρ1 r =
φa2 (r) + φb2 (r) + 2S abφa(r) φb(r)
2
(29)
2 1 + S ab where Sab is the overlap integral between the atomic orbitals φa and φb, assumed real. (The normalized VB wave function was used to derive equations 28 and 29.) Thus in this case, as for the CI wave function, the two electrons are correlated— which on reflection may seem obvious, since the atomic configurations in the VB and CI wave functions assign the electrons to different orbitals. Therefore, if electron 1 is “around” atom a, electron 2 is most likely to be near atom b, and vice versa. This type of correlation in diatomic molecules is described as left–right correlation, distinguishing it from radial and angular correlation, whose description requires atomic configurations involving at least the 2s and 2p AO’s. An interesting and important aspect of electron correlation can be demonstrated by considering the triplet excited state of H2, for which the spatial component of the VB wave function is antisymmetric, as shown in equation 23. The two-electron density function corresponding to it is therefore
1
ρ12 r1,r2 = 2
2 1 – S ab
[φa2 (r1) φb2 (r2) + φb2 (r1) φa2 (r2)
– 2φa(r1) φb(r1) φa(r2) φb(r2)]
(30)
Figure 12. Two-electron wave function and density of H2 along its internuclear z axis in ground state calculated by CI (orbital exponent α optimized at R = 1.4a0). Upper sections show probability of finding electron 1 at location (0,0,z 1) given that electron 2 is at the point indicated by the arrow, (0,0,0.4).
which vanishes when r1 = r2 (when the two electrons are at the same point). A more detailed study of ρ12(r1,r2) would show that it gradually decreases as the two electrons move closer together. Each electron is thus said to be surrounded by a correlation hole, which cannot be penetrated by the other electron. In the ground state, where the spatial wave function is symmetric, the hole around each electron is much less pronounced. Indeed, as indicated by the appropriate two-electron density, the probability is nonzero even when r1 = r2. The high correlation between two electrons with parallel spins is a direct consequence of the Pauli exclusion principle and, since it applies to fermions such as electrons, the resulting correlation hole is called the Fermi hole. A simple way to visualize a two-electron wave function or density is to restrict the electron coordinates to lie on the internuclear axis (z) and to produce contour maps of Ψ(z1, z2) and ρ12(z1, z2). Examples of these for the ground state CI wave function are shown in Figure 12, demonstrating that ρ12(z1, z2) is high when the electrons are at opposite ends of the molecule and low when they are in the same region (z1 ≈ z2). This is well illustrated by the difference map ∆ρ12(z1 , z2 ) = ρ 12(z 1, z2) – ρ1(z1) ρ1 (z2 ). Alternatively, one-dimensional plots of ρ12(z1, z2) and ∆ρ12(z1, z2 ) can be displayed as functions of z1 at a fixed value of z2. Our visualization program allows the user to alter z2 by dragging an arrow on the screen and thus dynamically scan the two-electron density and difference maps, displaying the correlation hole. For a many-electron single configuration wave function, where each (molecular) orbital is doubly occupied with electrons of opposite spin, we could readily show what may seem obvious by now: electrons in the same orbital are not correlated, whereas electrons in different orbitals are, the degree of correlation depending on whether the two electrons have parallel or opposite spins and on the accuracy and sophistication of the variational wave function. It is the convention, however, to refer to single configuration wave functions such as SCF wave functions as uncorrelated, unless the double occupancy of the orbitals is relaxed. The degree of correlation in some arbitrary multiconfigurational (CI) wave function is monitored by the correlation energy, Ecorr, defined as Ecorr = ECI – ESCF
(31)
where E CI and E SCF are the energies of the CI and SCF wave functions.
Vol. 74 No. 12 December 1997 • Journal of Chemical Education
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Information • Textbooks • Media • Resources If we were to apply the above measure of correlation to the H2 molecule we would find from the data in Figure 9 that in the neighborhood of the equilibrium geometry the correlation energy is fairly constant, but increases as the atoms move further apart. Asymptotically it reaches a value of about 0.3 Eh. The rise in the correlation energy reflects the falling energy difference between the ground state configuration ψ+(1) ψ+(2) and the excited singlet state ψ{(1) ψ{(2), which allows the electrons to stay out of each other’s way at decreasing cost in kinetic energy. Thus, according to the above definition of correlation energy, the incorrect asymptotic behavior of the MO energy is due to the neglect of correlation. On this point, however, there are conflicting interpretations. One school of thought accepts the validity of the above; another argues that we should differentiate between dynamical and nondynamical correlation. The former describes the situation where electrons are dynamically avoiding each other in a repulsive scattering event, so as to reduce the interelectron repulsion. Nondynamical correlation occurs when electrons are correlated at a distance (i.e., they rarely come close enough to scatter off each other). Thus as a molecule dissociates, the nuclei drag their appropriate share of electrons with them. As the nuclei move further apart the repulsion between electrons on different atoms becomes progressively smaller, resulting in less and less dynamical correlation. Ensuring that the wave function does allow the electrons to become atomic in character—that it has the flexibility to make ionic terms vanish—is nondynamical correlation. Except in limiting cases, there is no rigorous way to separate and quantify the two types of correlation. Nevertheless, it is important to understand what electron correlation is and the role it plays in covalent bonding.
since, by definition, no delocalization is present. Delocalization can be introduced by symmetrizing the Hartree product, resulting in the VB model for the ground state. The VB form of the wave function exaggerates electron correlation around the equilibrium geometry but still gives a very good description of the covalent bond in H2. It becomes exact in the asymptotic limit. The simplest way to introduce the covalent bonding mechanism is to allow delocalization at the one-electron level, as in H2+. This leads to the MO (SCF) wave function, which, despite lack of correlation, gives a good description of H2 around the equilibrium geometry. Correlation is, however, needed for large separations to eliminate ionic character and provide a valid picture of the dissociation. We conclude that delocalization is the key mechanism of covalent bonding. It can be introduced at the two-electron level if the orbital basis is localized, as in the VB method, or at the one-electron level, by delocalizing the orbitals, giving rise to MO-based methods. Conclusion We believe that the teaching of the mechanism of covalent bonding is in need of revision. We hope to aid that process with the publication of our ideas on the subject, which have developed over years of undergraduate teaching, and with the distribution of our computer visualization software. Although we dealt with the simplest of molecules, the basic principles developed are equally applicable to more complex molecules where an even greater degree of delocalization, orbital contraction, and electron correlation is generally possible. Note Added in Proof
Covalent Bonding in H2 In the description of covalent bonding in the ground state of H2 , with the exception of the Hartree model, all the methods discussed allow for electron delocalization and are able to provide a qualitatively correct description of the bond length and energy. If the orbital exponents of the AO’s are held fixed, the behavior of the kinetic and potential energies with interatomic distance is the same as in H2 +—the decrease in kinetic energy is responsible for bond formation. Optimizing the orbital exponent at each distance results in orbital contraction; thus at equilibrium the virial theorem is satisfied and, in comparison with the free atoms, there are a net increase in kinetic energy and the appropriate decrease in potential energy. Following Ruedenberg (2–6), we also look at the behavior of the bond-parallel and bondperpendicular components of the kinetic energy. Despite the orbital contraction, the former still behaves as expected with distance, decreasing with decreasing distance until the equilibrium value, while the net increase in kinetic energy is entirely due to the perpendicular components. We can summarize our results for H2 as follows. The use of localized atomic orbitals, as in the Hartree product wave function, fails to represent the covalent bonding mechanism,
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In addition to the papers of Ruedenberg and co-workers (3–7), we wish to draw attention to Kutzelnigg’s excellent article (10) entitled “The Physical Origin of the Chemical Bond”, which presents an extensive and thorough discussion of bonding in the spirit of Ruedenberg’s work. Literature Cited 1. Coulson, C. A. Valence, 2nd ed.; Oxford University: London, 1961. 2. Slater, J. C. Quantum Theory of Matter, 2nd ed.; McGraw-Hill: New York, 1968; p 400. 3. Ruedenberg, K. Rev. Mod. Phys. 1962, 34, 326. 4. Feinberg, M. J.; Rudenberg, K.; Mehler, E. L. In Advances in Quantum Chemistry, Vol. 5; Löwdin, P. O., Ed.; Academic: New York, 1970; p 28. 5. Feinberg, M. J.; Rudenberg, K. J. Chem. Phys. 1971, 54, 1495. 6. Feinberg, M. J.; Rudenberg, K. J. Chem. Phys. 1971, 55, 5804. 7. Ruedenberg, K. In Localization and Delocalization in Quantum Chemistry, Vol. 1; Chalvet, O.; Daudel, R.; Diner, S.; Malrieu, J. P., Eds.; Reidel: Dordrecht, 1975; p 223. 8. Nordholm, S. J. Chem. Educ. 1988, 65, 581. 9. Hurley, A. C. Int. J. Quantum Chem. 1982, 22, 241. 10. Kutzelnigg, W. In Theoretical Models of Chemical Bonding; Maksic, Z. B., Ed.; Springer: Berlin, 1990; p 1.
Journal of Chemical Education • Vol. 74 No. 12 December 1997
