Research: Science and Education
Advanced Chemistry Classroom and Laboratory
edited by
Joseph J. BelBruno Dartmouth College Hanover, NH 03755
When One Configuration Is Not Enough David R. McMillin Department of Chemistry, Purdue University, West Lafayette, IN 47907-2084;
[email protected] Understanding the electron distribution in atoms and molecules is obviously one of the principal challenges of chemistry. In theory, the Schrödinger wave equation can provide the answers to all questions, but exact solutions are often elusive and model systems present limitations. The hydrogen atom is an exception to the rule in that exact wave functions are available in the form of the familiar atomic orbitals: 1s, 2s, 2p, and so forth. Corresponding analytical solutions are lacking for the helium atom owing to a major complication, the repulsive interaction between electrons. By ignoring the repulsion term, one can model the ground-state wave function of helium as a product of two functions 1s(1) and 1s(2), where the numbers in parentheses designate the coordinates of electrons 1 and 2, respectively. The 1s2 configuration serves as shorthand notation for the ground-state wave function. Even though both electrons reside in the 1s shell, the model wave function is compatible with the Pauli exclusion principle so long as the electrons have different ms spin quantum numbers. In an attempt to improve the model (also known as the zero-order approximation), Parr and Taylor introduced an admixture of the 1s12s1 configuration into the ground-state wave function of helium (1). The expectation was that allowing each electron to spend part time in the 2s orbital would increase the average distance between the electrons and reduce the calculated energy. However, the average separation from the nucleus also increases, and in the end, Parr and Taylor found that the augmented wave function provided little improvement in the calculated energy. The shortcoming of this attempt is worth pursuing a bit further. Subsequent analysis has revealed that the distance between the electrons of helium is not as important as the angular relationship. Wave functions that allow the electrons to tend toward opposite sides of the helium nucleus actually predict a lower total energy and therefore provide a more accurate picture (2, 3). The angular correlation is simply not very strong in the 1s12s1 configuration. In semi-classical terms, the synchronization between the motion of the 1s and 2s electrons is poor because the average kinetic energy is so different in the two shells. The lesson is that, to be effective, the corrective action needs to address a critical shortcoming of the model. In molecules as well, one usually describes an electronic state in terms of a configuration-state function or an electronic configuration for short. The lowest energy configuration is the ground state, and it involves pairing up electrons in the lowest energy orbitals available according to the Aufbau principle. In obtaining the molecular orbitals, the simplest approach is to regard the electrons as independent particles that do not interact with each other. More sophisticated treatments utilize Hartree– Fock self-consistent field (SCF) theory (3). The latter approach assumes that each electron resides in the time-averaged field of the others, even though that requires having orbitals available
at the outset. The computations are nevertheless feasible if one inputs initial guesses and refines them by the method of successive approximations. Even SCF calculations do not tell the whole story, however, because electrons actually interact with each other in real time as opposed to a time-averaged fashion. The correlation energy is the difference between the calculated SCF energy and the true energy of the system. In a molecule finding a way to distribute the electrons properly among atoms is often the key to recovering the correlation energy. Introducing an admixture of another configuration can help improve the picture, but perturbation theory reveals that the mixing coefficient varies inversely with the energy gap when two states interact. For that reason configuration interaction often has the most dramatic consequences in the excited-state manifold, where the density of states is high. However, in the examples presented below configuration interaction is important even in the ground electronic state. Anti-Ferromagnetically Coupled Copper Centers As the first illustration, consider oxyhemocyanin, the dioxygen-containing form of hemocyanin. The copper-containing protein hemocyanin occurs in molluscs and is responsible for reversible binding of dioxygen (4). Dioxygen uptake by the protein entails a formal reduction of the substrate to peroxide and concomitant oxidation of two copper(I) centers functioning together in the active site. Each of the resulting copper(II) centers has an odd number of electrons, but the peroxide adduct is diamagnetic due to an anti-ferromagnetic coupling interaction between copper centers. Analogous coupling occurs in many other dimeric systems, and theoretical treatments are available (5, 6). Here the focus is on the electronic spectrum of oxyhemocyanin. In accordance with the treatment of Solomon and co-workers (7, 8), one can regard the dioxygen adduct as an idealized D2h structure (Figure 1). This model assumes the system is perfectly planar and ignores local differences in the copper environments. The structure reveals that the peroxide ligand bridges between copper centers and that the protein supplies the other in-plane ligands. Figure 1 also includes a truncated molecular orbital (MO) diagram that reveals the pathways of electron delocalization. In the model, the formally half-occupied dxy orbitals of CuA and CuB span the b2u and b1g representations of the D2h point group. The scheme also includes three peroxide-based orbitals. They are the in-plane π* orbital that has b1g symmetry, the out-of-plane counterpart that has b3g symmetry, and the σ* orbital that has b2u symmetry. Linear combinations of the component orbitals give rise to the molecular orbitals of the adduct. In theory, one arrives at the electronic configuration via orderly addition of electrons to the resulting energy-level diagram. To wit, the
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L
O Cu
L
L Cu
O
L b2u*
xy, xy
y
b1g*
x
T*(OO)
b2u Q*(OO) b3g b1g
b1g*
b2u
Figure 1. (top) Idealized D2h structure and coordinate system for oxyhemocyanin. (middle) Partial MO diagram constructed from the dxy orbitals of the two copper centers, as well as the π* and σ* orbitals of the peroxide moiety. The vertical arrows illustrate the two charge-transfer excitations observed in the absorption spectrum. (bottom) Schematic contours of the MO’s for the Cu2O2 fragment.
Aufbau principle predicts the adduct will contain two electron pairs in the relatively low energy b1g and b3g orbitals, both of which mainly localize on the peroxide moiety. The energy spacing between the next lowest molecular orbitals, b2u and b1g*, is a critical factor in determining the magnetism. The simple view is that, when the splitting is small in comparison to the pairing energy, one electron goes into each orbital, and the ground state has two unpaired electrons. Except in oxyhemocyanin the spacing between levels is large enough for the ground state to be diamagnetic. However, filling the b2u orbital is inconsistent with the electronic spectrum because Solomon and co-workers have assigned two low-lying ligand-to-metal charge-transfer (CT) transitions as b2u←b3g and b2u←b1g excitations (Figure 1). They based their assignments on the observation that each absorption has a molar extinction coefficient of the order of a thousand and therefore must have symmetry-allowed, gerade-to-ungerade character (7). The question, then, is how can the system be spin paired and still have excitations that terminate in the b2u molecular orbital. As the discussion to follow shows, the answer is that one has to describe the ground state as an admixture of the b1g2b3g2b2u2 and b1g2b3g2b1g*2 configurations. 860
To understand why one configuration does not suffice, consider the b2u orbital. As there is little or no direct overlap between the dxy orbitals, they interact relatively weakly via common overlap with the high energy σ* orbital of the peroxide bridge. The resulting MO is rich in metal character, and an electron in the bonding b2u orbital therefore ends up spending much of its time at one or the other copper(II) center. The predicted electron distribution is well-suited for a system with one electron, but the b2u2 configuration is clearly a high-energy arrangement for two electrons. The reason is the electrons repel each other, so having them spend significant quantities of time on one or the other copper atom and removed from the other nuclei is destabilizing. The electron density may accrue that way in an excited state, but the ground state is always the lowest possible energy state. A better approximation to the ground state would avoid the charge accumulation at the metal centers. Introduction of an admixture of another configuration proves to be an effective, though not immediately obvious, remedy. To see how the fix comes about, consider the neighboring b1g* orbital. It is a formally anti-bonding MO composed of the dxy orbitals and a π* orbital of peroxide. Filling this level gives rise to the b1g*2 state that is only slightly higher in energy than the b2u2 state owing to the small gap separating the b2u and b1g* levels. The b1g*2 and b2u2 states also have two other traits in common. Each has a spin of zero (S = 0), and each is totally symmetric by virtue of being a closed-shell configuration. Accordingly, the conditions are ideal for state mixing, and the Appendix (see the online supplement) shows in a bit more detail how a mixed configuration such as eq 1 can provide a lower-energy ground-state wave function.
: M1 b1g2 b3g2 b2u2 M 2 b1g2 b3g2 b1g* 2
(1)
(In qualitative terms, the b1g*2 electron density distribution is like b2u2 to the extent that it requires both electrons to spend a fair quantity of time together at each copper center. If one imagines the molecule to be a neighborhood of atoms, then subtracting some measure of one electron distribution from the other and renormalizing the resultant has the effect of reducing double occupancy at the copper units.) Because the original b2u2 and b1g*2 states have such similar energies, the absolute values of the λ1 and λ2 coefficients are apt to be approximately equal in eq 1. The λ2 term is of pivotal importance for CT absorption because the presence of this term ensures that the occupancy of the b2u shell is not complete. In other words, this term provides for the possibility of b2u←b3g as well as b2u←b1g excitation. To be sure, oxyhemocyanin represents an extreme case because the zeroorder configurations are so close in energy. Nevertheless, similar effects occur in other systems as well, if to a lesser degree. The Delta Bond The octachlorodirhenate(III) system is a case in point that is interesting to compare. Here, too, combinations of the two dxy orbitals give rise to what are formally regarded as the HOMO (highest occupied MO) and LUMO (lowest unoccupied MO) levels. The anticipated energies of the HOMO, LUMO, and other orbitals relevant to metal–metal bonding appear in Figure 2. In the dirhenate complex, the two dxy orbitals directly overlap in a delta (face-to-face) fashion and form the fourth metal– metal bond (9). Pairing electrons in the bonding molecular
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Research: Science and Education y
orbitals predicts formation of a closed shell, totally symmetric, S = 0 ground state with 1A1g state symmetry. However, detailed calculations by Hay yield the following wave functions for the ground-state Ψ(1A1g ) and the first single excited-state Ψ(1A2u) (10):
: 1A 2 u 0.85 T 2 Q 4 E1 E *1
0.19 T Q E * Q * |
0.79 T Q E 0.46 T Q E *
0.24 T Q E Q * | 0.48 T 2 Q 3 E 2 Q *1 2
: 1A1g
2
3
2
2
2 2
2
4
Cl
dz 2
where nb denotes the number of electrons in the bonding orbital, and na denotes the number of electrons in the anti-bonding counterpart. The na correction to the bond order tends to be small when overlap is strong. Thus, the EBO is close to 1 when two hydrogen atoms interact at the equilibrium bond length but close to naught at the dissociation limit. When there are multiple bonds between atoms, the contribution to the EBO varies with the nature of the overlap. Consider the Cr2 molecule that forms when two Cr atoms with a 3d54s1 configuration come together. Six distinct bonding interactions are theoretically possible; however, Roos et al. calculate that the EBO is only about 3.52. The analysis shows that the d orbitals oriented in a δ, or face-to-face, fashion contribute almost nothing to the EBO. For those levels, the overlap is so weak that nb is only slightly larger than na (11).
E*
dxy
E
dz 2
dz 2 Q*
dxy
T
dxz dyz dxy dxy
Q
dxz dyz
(3)
The discussion ends with an application due to Roos and co-workers who have adopted a bonding perspective (11). They start with the idea that bond formation typically entails formation of bonding and anti-bonding molecular orbitals. In the development they are careful to work with the “natural orbitals” that result from specific theoretical calculations; nevertheless, some of the basic ideas discussed above carry forward. In particular, Roos et al. find that whenever there is population of the bonding orbital, there is also some population of the anti-bonding orbital, as one might expect from electron repulsion considerations. Since population of the anti-bonding level weakens bonding, they define an effective bond order (EBO): n na (4) EBO b 2
T*
dxz dyz
2
Effective Bond Order
Cl x
(2)
Here, the ground state has a preponderant configuration, namely σ2π4δ2, but the inclusion of other configurations is necessary to generate a realistic picture of the electron distribution. Analogous to the case of oxyhemocyanin, including an admixture of the σ2π4δ*2 configuration obviates some of the electron–electron repulsions endemic to the pure σ2π4δ2 configuration. The z component of the dipole moment operator is capable of inducing Ψ(1A2u)←Ψ(1A1g ) excitation and thereby producing the higher energy 1A2u state. In keeping with one’s intuition, the transition has considerable δ*←δ character, but inspection of eqs 2 and 3 reveals that the transition has π*←π character as well.
Cl Re
1
2
4 2
Cl
dxz dyz dz 2
Figure 2. (bottom) Molecular orbital scheme showing quadruple bond formation in the ground electronic state. (top) Side-on and end-on views of the D4h ion Re2Cl82− along with the coordinate system.
Conclusion According to the tenets of molecular orbital theory, the lowest energy electronic configuration is a prescription for the ground electronic state. This is usually a good approximation when orbital overlap is strong and electron delocalization is the dominant factor shaping the electron distribution. However, the results are less straightforward when orbital overlap is weak. The molecular orbital framework is still useful for describing the system, but a single electronic configuration no longer suffices to describe the ground state. Using more than one configuration may seem strange at first because it inevitably entails transferring electron density into an anti-bonding orbital (or orbitals). However, electron–electron interactions are inescapably a part of the picture, and the goal is minimization of the total energy. Points To Ponder
1. What contributions in eqs 2 and 3 ensure that the lowest energy transition of Re2Cl82– has π*←π as well as δ*←δ character?
2. In the H2 molecule both electrons are sometimes on the same atom, but the Ψ2 configuration overestimates this effect and requires the γ2 term in eq A7 as a correction (see the online supplement). Why does the γ2/γ1 ratio vary with the bond length and increase as the molecule approaches dissociation (12)?
3. In oxyhemocyanin, optical excitation from the singlet state (1B3u) associated with the b2u1b1g*1 configuration to either of the two 1Ag states formally derived from the b2u2 and b1g*2 configurations is symmetry allowed. Show that each transition has a polarization (involves a charge shift) along the copper– copper axis (the x direction in Figure 1). The transition moments associated with the zero-order states oscillate at similar frequencies and cannot avoid interacting with each other (8). That they couple to give in-phase and out-of-phase combinations is another manifestation of the inevitable mixing of the b2u2 and b1g*2 configurations.
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4. Reference 11 quotes an effective bond order of 5.19 for W2. Identify at least one reason for the impressive increase in the calculated bond order relative to diatomic chromium.
Acknowledgment The National Science Foundation helped fund this research through grant CHE-0550241. Literature Cited 1. Taylor, G. R.; Parr, R. G. Proc. Natl. Acad. Sci. U.S.A. 1952, 38, 154–160. 2. Olsson, L. F.; Kloo, L. J. Chem. Educ. 2004, 81, 138–141. 3. Fock, V. Z. Physik 1930, 61, 126–148. Pilar, F. L. Elementary Quantum Chemistry; McGraw-Hill: New York, 1968; Chapter 10. 4. Magnus, K. A.; Hazes, B.; Tonthat, H.; Bonaventura, C.; Bonaventura, J.; Hol, W. G. J. Proteins: Struct., Funct., Genet. 1994, 19, 302–309. 5. Hay, P. J.; Thibeault, J. C.; Hoffmann, R. J. Am. Chem. Soc. 1975, 97, 4884–4899. 6. Kahn, O. Molecular Magnetism; Wiley-VCH: New York, 1993.
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7. Solomon, E. I.; Baldwin, M. J.; Lowery, M. D. Chem. Rev. 1992, 92, 521–542. Tuczek, F.; Solomon, E. I. J. Am. Chem. Soc. 1994, 116, 6916–6924. 8. McMillin, D. R. Physical Methods in Bioinorganic Chemistry; Que, L. J., Ed.; University Science: Sausalito, CA, 2000; Chapter 1. 9. Cotton, F. A.; Murillo, C. A.; Walton, R. A. Multiple Bonds Between Metal Atoms, 3rd ed.; Cotton, F. A., Murillo, C. A., Walton, R. A., Eds.; Springer Science and Business Media, Inc.: New York, 2005; Chapter 1. 10. Hay, P. J. J. Am. Chem. Soc. 1982, 104, 7007–7017. 11. Roos, B. O.; Borin, A. C.; Gagliardi, L. Angew. Chem., Int. Ed. 2007, 46, 1469–1472. 12. Matito, E.; Duran, M.; Sola, M. J. Chem. Educ. 2006, 83, 1243– 1248.
Supporting JCE Online Material
http://www.jce.divched.org/Journal/Issues/2008/Jun/abs859.html Abstract and keywords Full text (PDF) Links to cited JCE articles Supplement Appendix containing details of a mixed configuration
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