Teaching the Jahn–Teller Theorem: A Simple Exercise That Illustrates

Article pubs.acs.org/jchemeduc

Teaching the Jahn−Teller Theorem: A Simple Exercise That Illustrates How the Magnitude of Distortion Depends on the Number of Electrons and Their Occupation of the Degenerate Energy Level Adam Johannes Johansson* Applied Physical Chemistry, Royal Institute of Technology (KTH), Teknikringen 30, S-100 44, Stockholm, Sweden S Supporting Information *

ABSTRACT: Teaching the Jahn−Teller theorem offers several challenges. For many students, the first encounter comes in coordination chemistry, which can be difficult due to the already complicated nature of transition-metal complexes. Moreover, a deep understanding of the Jahn−Teller theorem requires that one is well acquainted with quantum mechanics and group theory. A comparatively simple way to illustrate the anatomy of the Jahn− Teller effect is presented here. Chemistry teachers are reminded of a sometimes forgotten aspect, namely, that the orbital degeneracy itself is not a sufficient criterion for the Jahn−Teller effect to appear, it is necessary that the occupation of the degenerate energy level is asymmetric. The article can serve as an introduction to the Jahn− Teller theorem, either as a lecture or as a computational exercise. KEYWORDS: Graduate Education/Research, Upper-Division Undergraduate, Physical Chemistry, Organic Chemistry, Textbooks/Reference Books, Computational Chemistry, Molecular Properties/Structure, Quantum Chemistry, Theoretical Chemistry

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give the students an understanding of the anatomy of the Jahn−Teller effect before he or she is confronted with it in advanced quantum chemistry or in coordination chemistry. In transition-metal complexes, for which the Jahn−Teller theorem is often of structural importance,4b−4d the situation is typically much more complex due to several factors such as the involvement of d orbitals, higher degree of degeneracy (e.g., t2g level in tetrahedral- and octahedral complexes), near degeneracy (especially first row transition metals), larger number of electrons, multiple modes of distortion, and lower symmetry (or higher complexity) in the molecular structure.5 It could thus be helpful for the student of coordination chemistry if he or she has been introduced to the Jahn−Teller effect in a comparatively simple system, such as cyclobutadiene (C4H4).6

hen lecturing on Hückel molecular orbital (HMO) theory, I, as many textbooks on the subject, introduced the method by using, among others, the molecule cyclobutadiene (C4H4).1 After the lecture, a group of alert students confronted me with the following reasoning: In your derivation of the π orbitals of cyclobutadiene today, you reached the conclusion that the highest occupied molecular orbital is degenerate. If we then apply the Aufbau principle and Hund’s rule, which you have taught us in a previous lecture, and the fact that cyclobutadiene has four π electrons, we reach the conclusion that it has two electrons unpaired in the ground-state. Is cyclobutadiene paramagnetic? This chain of arguments presented me with a problem, because the Jahn−Teller effect,2−4 responsible for the rectangular planar structure of cyclobutadiene, was not supposed to be taught in the introductory course on quantum chemistry. Instead of just referring to the Jahn−Teller theorem as an axiom or a quantum mechanical fact, I entered into discussion with the students concerning the electronic and molecular structure of cyclobutadiene and its monovalent and divalent anions and cations. The discussion gave me a most enlightening experience of how the students understood the mechanisms determining molecular structure, and the qualitative relationship between symmetries in electronic and molecular structures. The discussion made me realize that investigating cyclobutadiene and its ions with computational quantum chemical methods can serve as a simple but illustrative introduction to the Jahn−Teller theorem. This exercise could © 2012 American Chemical Society and Division of Chemical Education, Inc.

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DESCRIPTION OF THE EXERCISE It is appropriate to introduce the system with a short discussion of the potential energy surface (PES) for the isomerization of cyclobutadiene (Figure 1).7 Alternatively, the students can investigate the PES using density functional theory (DFT) and some standard quantum chemical software package.8 Geometry optimizations of C4H4 in the closed-shell singlet (Q = 0, S = 0, M = 1, where Q, S, and M, represents total charge, electronic spin angular momentum, and spin multiplicity (M = 2S + 1), respectively) and triplet (Q = 0, S = 1, M = 3) states reveals that the singlet state is significantly more stable (26 kJ mol−1) than Published: November 2, 2012 63

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Figure 2. π Orbital diagram of square planar (D4h) cyclobutadiene in the triplet state (Q = 0, S = 1, M = 3). The green and red arrows represents attractive and repulsive forces due to the occupation of each of the degenerate eg orbitals in the square planar configuration of the triplet state.

Figure 1. Potential energy surface for the isomerization of cyclobutadiene in the singlet (S = 0) and triplet (S = 1) states (kJ mol−1).

the triplet state. In other words, the cyclobutadiene molecule has a singlet ground state, which makes it diamagnetic (Figure 1). Inspection of the optimized geometries shows that the closed-shell singlet state is planar and rectangular in the carbon skeleton, whereas the triplet state is perfectly square planar (Figure 1). In the following discussion, the group-theoretical notations D2h and D4h are used to denote rectangular and square planar geometries, respectively. For the purpose of the exercise, it is not necessary that the students are more familiar with group theory than the meaning of these notations. It is, however, important to realize that the square has a higher degree of symmetry than a rectangle, as the terms higher and lower symmetry will be used in the discussion. Geometrical considerations suggest that the transition state for the isomerization of cyclobutadiene should be square planar (D4h). The isomerization of cyclobutadiene is known to be rapid, but the energy of the closed-shell singlet state in the square planar geometry of the triplet state implies an isomerization barrier of 111 kJ mol−1, which corresponds to a rate constant (k) in the order of 10−7 s−1 at room temperature (Eyring equation), in other words, a far from rapid isomerization. However, using the unrestricted formalism and an initial-guess wave function with two unpaired electrons in antiparallel spin configuration gives a perfectly square planar (D4h) open-shell singlet state (a so-called broken symmetry state). A harmonic frequency analysis confirms that this D4h open-shell singlet is a transition state (i.e., it has one and only one imaginary vibrational frequency) for the isomerization of C4H4 in the singlet state. This transition state is only 11 kJ mol−1 higher than the rectangular (D2h) equilibrium geometry (Figure 1), implying a very rapid isomerization (k ∼ 1011 s−1). Knowing that the triplet state of cyclobutadiene is perfectly square planar (D4h), but that the rectangular (D2h) configuration of the singlet state is much more stable (26 kJ mol−1), the following questions should be addressed: What is it that makes the rectangular (less symmetric) configuration more stable than the square configuration (of higher symmetry), and what happens with the structure (and symmetry) if we add or remove electrons to or from the system? At this point it is appropriate to analyze the π orbitals of square planar cyclobutadiene (Figure 2). Either the π orbital diagram can be derived within HMO theory,1b−e or alternatively, a

visualization software can be used to inspect the molecular orbitals produced in a DFT calculation on the square planar triplet state.9 Analysis of the orbital nodes perpendicular to the molecular plane reveals that the a2u orbital is bonding with respect to all four C−C bonds, while the b2u orbital is totally antibonding. One of the two degenerate eg orbitals, here denoted eg1, is bonding with respect to C1−C2 and C3−C4, but antibonding in C1−C3 and C2−C4. For symmetry reasons, the other of the degenerate orbitals, denoted eg2, is bonding in C1−C3 and C2−C4, while antibonding in C1−C2 and C3−C4. According to the Aufbau principle and Hund’s rule of maximum spin multiplicity in degenerate states, square planar cyclobutadiene should be a diradical with a triplet electron configuration. In the triplet state, there is one electron occupying each of the eg orbitals, meaning that the opposite forces exerted by the occupation of the orbitals eg1 and eg2 are perfectly balanced.10 Thus, there is no net force acting on the atoms in the system (Figure 2), and therefore, cyclobutadiene remains square planar in the triplet state. What happens with these forces (and consequently the structure) if the electrons are paired and occupy the same orbital? If, for example, the spin of the electron in eg2 is inverted from spin-up to spin-down, and paired with the spinup electron in eg1, then two electrons occupy orbital eg1, while eg2 becomes unoccupied. Therefore, in the square planar equilibrium configuration of the triplet, this singlet electron configuration has a net attractive force in the C1−C2 (C3−C4) direction, while there is a net repulsive force in the C1−C3 (C2−C4) direction. This is the physical reason for the distortion that leads from the square planar (D4h) structure of triplet cyclobutadiene, to the rectangular (D2h) shape of cyclobutadiene in its singlet ground state when the electrons are paired. The analysis made herein requires that the student have a basic understanding of the mechanism of chemical bonding, that is, the internuclear accumulation of electron density due to constructive interference of bonding molecular orbitals. Even if this mechanism has been understood for the simple case of σ bonding in diatomic molecules, it could be useful to contemplate over the electronic probability density contours plotted in Figure 3. In the triplet state (3ψ), the single 64

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singlet ground state (Table 1). The reason for this destabilization is that the eg2 orbital is bonding and antibonding in directions perpendicular to those of the eg1 orbital (Figure 2). The destabilization of the eg2 orbital has no effect on the total energy of the molecule since it is unoccupied in the singlet state having two electrons in eg1. If instead both electrons would occupy the eg2 orbital, the rectangular shape would be inverted (Figure 1), the eg2 orbital would be stabilized and the empty eg1 orbital would be destabilized. Singlet C4H4 distorts into a rectangular structure in which two C−C bonds are contracted from 1.44 to 1.33 Å, while the other two C−C bonds are elongated from 1.44 to 1.58 Å. The ratio of the short and long C−C distances (denoted S/L) is 0.84, which can serve as a measure of rectangularity (or the degree of distortion from the quadratic structure of the carbon skeleton in the triplet state). To quantify the distortion energetically, it is useful to compare the energy of the geometrically relaxed singlet state (Eopt) with the single-point energy (Esp) of the singlet state in the square planar (D4h) geometry of the triplet state. The geometrical relaxation energy, defined as ΔEgeo = Eopt − Esp, is −111 kJ mol−1 for the singlet state of cyclobutadiene (Figure 1, Table 1). Although the energetic driving force for the distortion is determined by changes in the total energy of the system, we know from Walsh’s rules and Fukui’s frontier molecular orbital (FMO) theory that the change in the total energy is correlated with the change in the energy of the highest occupied molecular orbital (HOMO).11,12 It is thus of interest to analyze the energetic splitting of the eg level, and to compare the energy of the electrons occupying the eg level in the square planar and relaxed geometries.13 For the singlet state, in which two electrons occupy the eg1 orbital, this energy difference is

Figure 3. Contour plots of the electronic probability density in the triplet (3ψ) and singlet (1ψ) states of cyclobutadiene. Note the elliptic contours in the C1−C2 and C3−C4 bonds of the singlet (1ψ) state, which indicate enhanced probability density in these regions.

occupation of each of the eg orbitals (Figure 2) makes the electron density evenly distributed among the four C−C bonds, leading to a perfectly square planar molecule (Figure 3). In the singlet state (1ψ), the eg1 orbital is doubly occupied, which causes a concentration of the electron density to the C1−C2 and C3−C4 bonds and leads to a rectangular molecule (Figure 3). The distortion resulting from the double occupation of the eg1 orbital is accomplished by a stabilization of the eg1 orbital, from −0.1858 au (1 au = 2626 kJ mol−1) in the square planar triplet state to −0.1967 au in the rectangular singlet ground state (Table 1). The quantum mechanical reasons for this stabilization of the eg1 orbital are two: (i) the increased bonding overlap in eg1 as the molecule distorts with a contraction in the C1−C2 (C3−C4) direction and (ii) the decreased antibonding overlap associated with the elongation in the C1−C3 (C2−C4) direction. The same distortion that stabilizes eg1 has a destabilizing effect on eg2, from −0.1858 au in the square planar triplet state, to −0.0626 au in the rectangular planar

Table 1. Computational Results: Orbital (eg1, eg2), Absolute Energies (Esp, Eopt), Relative Energies (ΔEeg, ΔEgeo), and Bond Distances for C4H4 Ions 0 0

+1 1/2

−1 1/2

−2 0

−154.7057 α = 0.0701 β = 0.0998 α = 0.0775

−154.4677 −0.3187

−154.4713 −0.3136

−0.6378 1.44 1.44 1.00

−154.7157 α = 0.0566 β = 0.0789 α = 0.0894 1.39 1.51 0.92

−2 -

−26 −59

−9 -

Total charge Spin angular momentum

0 1

+2 0

Esp/au eg1/au

-

eg2/au

-

−0.1070

Eopt/au eg1/au

−154.7197 −0.1858

Optimized Geometry −154.7296 −154.4490 −0.1967 −0.4630

−153.8982 −0.6378

eg2/au C1−C2/Å C1−C3/Å S/L

−0.1858 1.44 1.44 1.00

−0.0626 1.33 1.58 0.84

ΔEgeo/(kJ mol−1) ΔEeg/(kJ mol−1)

-

−111 −202

Single-Point SCF Calculation in Triplet Geometry, [C1−C2] = [C1−C3] = 1.44 −154.6874 −154.4370 −153.8975 −0.1583 −0.4400 −0.6396 −0.3793

−0.3542 1.37 1.50 0.91 Relative Energies −32 −60 65

−0.6396

−0.3187

−0.3136 1.47 1.47 1.00

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Figure 4. Flow scheme illustrating how the Jahn−Teller effect is explored in cyclobutadiene, C4H4, and the ions: C4H42+, C4H4+, C4H4−, and C4H42‑.

which should be reflected by the magnitude of distortion upon relaxation (as well as in the quantities ΔEgeo and ΔEeg, defined above). As predicted, the distortion in the C4H4+ ion is smaller than for singlet C4H4.14 In the relaxed geometry of C4H4+, the short C−C bonds are 1.37 Å, whereas the long bonds are 1.50 Å. The S/L ratio for C4H4+ is thus 0.91, which is significantly larger than in C4H4 (Table 1). As discussed above, the distortion is accomplished by an energetic splitting in the eg level. The stabilization of the electron in the singly occupied eg1 orbital of C4H4+ is

ΔEeg = 2eg1(D2h) − 2eg1(D4h) = 2( −0.1967 − (− 0.1583)) au = −202 kJ mol−1

Here, 2eg1(D2h) means the energy of two electrons occupying the eg1 orbital in the relaxed rectangular (D2h) geometry, whereas 2eg1(D4h) is the energy of the two electrons occupying the eg1 orbital in the square planar (D4h) geometry of the triplet state, which is used as a reference geometry. What was just described is well-known; the Jahn−Teller effect causes cyclobutadiene (in its singlet ground state) to be rectangular (D2h), not square planar (D4h).2 However, although it is seldom discussed in textbooks,1b,c,e,12b it is apparently so that the orbital degeneracy itself is not sufficient for the distortion to appear. It is necessary that the electrons are unevenly or asymmetrically distributed among the degenerate orbitals, otherwise there is no energy to gain in the distortion from the square planar geometry.1d,4c−4d To generalize this analysis of the dependence on electronic configuration, it is useful to consider the ionic states C4H42+, C4H4+, C4H4−, and C4H42‑, in which electrons have been removed from or added to the system. To most clearly illustrate the Jahn−Teller effect, it is particularly useful to add or remove electrons to or from the square planar triplet state of cyclobutadiene and to analyze the effects of geometrical relaxation in terms of distortion from the square planar geometry and energetic changes. The analysis is illustrated by the flow scheme in Figure 4.

ΔEeg = eg1(D2h) − eg1(D4h) = ( −0.4630 − (− 0.4400)) au = −60 kJ mol−1

This value is only about 30% of ΔEeg in C4H4 (Table 1) and reflects the smaller distortion in C4H4+ (and the fact that the eg1 orbital is singly occupied in the C4H4+ ion). Similarly, the geometrical relaxation energy (ΔEgeo) of C4H4+ is −32 kJ mol−1 (Table 1), which is 29% of ΔEgeo in C4H4. C4H4−

We will now analyze the effect of adding an electron to the square planar triplet state of C4H4. The addition of a spin-down electron results in a square planar C4H4− ion in a doublet electron configuration (Q = −1, S = 1/2, M = 2). Since the degenerate eg level is unevenly occupied, the C4H4− ion is expected to distort from the square planar structure when the geometry is relaxed and, as discussed above for singlet C4H4 and doublet C4H4+, the distortion from the square planar (D4h) to the rectangular (D2h) geometry is accomplished by the removal of the degeneracy of the eg level. For the C4H4− ion, this means that if the eg1 orbital is doubly occupied and stabilized by the distortion (i.e., contraction of C1−C2 and C3−C4, elongation of C1−C3 and C2−C4), the eg2 orbital must be singly occupied and destabilized. There is thus a qualitative difference between the C4H4− ion on one side, and the C4H4+ ion and the (singlet) C4H4 molecule on the other side, since the destabilized eg2 orbital is unoccupied in the latter two. What does the occupation of the destabilized eg2 orbital mean for the magnitude of the distortion in the C4H4− ion? Simple reasoning suggests that since the distortion lowers the energy of two electrons in eg1, but raises the energy of one electron in eg2, the net effect on the energy should be similar to the effect in

C4H4+

Let us continue the investigation by removing one electron from the triplet state of C4H4, and compare the effects with those of electron spin pairing discussed above. Initially, the removal of one electron from the eg2 orbital creates a square planar C4H4+ ion (vertical ionization). Computationally, this is modeled by a single-point self consistent field (SCF) calculation of the cation C4H4+ (Q = +1, S = 1/2, M = 2) in the square planar (D4h) geometry of the triplet state of C4H4 (Figure 4). Just as for the singlet state of C4H4, only the eg1 orbital is occupied. However, while the eg1 orbital in singlet C4H4 is doubly occupied, the eg1 orbital of the C4H4+ ion is singly occupied. This difference in orbital occupation has the consequence that the accumulation of electron density between the C1−C2 and C3−C4 atoms is smaller in the C4H4+ ion than in the C4H4 molecule. Thus, the distorting forces in square planar C4H4+ are smaller than in square planar singlet C4H4, 66

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the C4H4+ ion, for which the energy of the single electron occupying the eg1 orbital is lowered by the geometrical distortion. As we will see below, simple reasoning should not be underestimated. The relaxed geometry of C4H4− is almost identical to that of C4H4+ (Table 1).15 The contracted C−C bonds are 1.39 Å, the elongated C−C bonds are 1.51 Å, and the S/L ratio becomes 0.92. For C4H4−, the change in the energy of the three electrons occupying the eg level becomes slightly more complicated due to the splitting of the alpha (spin-up) and beta (spin-down) orbitals in the unrestricted formalism applied:9

the ions of cyclobutadiene are quadratic or rectangular in the carbon skeleton. Distortion from the highly symmetric square planar (D4h) structure, to the rectangular (D2h) structure of lower symmetry, occurs only in those states in which the electrons occupy the degenerate eg level in an asymmetric way, that is, C4H4+, C4H4−, and singlet C4H4. It is even possible to make qualitative predictions about the relative magnitudes of the energetic and geometric effects and the predictions are easily verified and quantified by DFT calculations. Hopefully, this exercise gives a practical understanding of the anatomy of the Jahn−Teller effect. It is appropriate to conclude the exercise with a suitable formulation of the general theorem, originally postulated by Hermann A. Jahn and Edward Teller (with some help from Lev D. Landau) in 1937:3a Geometries where the electrons occupy the components of a degenerate HOMO in an asymmetric way are always unstable with respect to some lower symmetry arrangement.12b

ΔEeg = (α + β)[eg1(D2h) − eg1(D4h)] + α[eg2(D2h) − eg2(D4h)] = (0.0566 + 0.0789 − 0.0701 − 0.0998 + 0.0894 − 0.0775) au = −59 kJ mol−1

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This value is almost identical to the corresponding value for C4H4+ (ΔEeg = −60 kJ mol−1). Moreover, the geometrical relaxation energy (ΔEgeo = Eopt - Esp) for C4H4− is −26 kJ mol−1 (Table 1), which is also very close to the corresponding value for C4H4+ (ΔEgeo = −32 kJ mol−1).

ASSOCIATED CONTENT

S Supporting Information *

An appropriate continuation of the exercise presented herein could be to investigate the structure of benzene and its ions; C6H6−, C6H62−, C6H63−, C6H64−, C6H6+, C6H62+, C6H63+, and C6H64+ in various spin configurations. An introduction and summary of geometrical results is given. This material is available via the Internet at http://pubs.acs.org.

C4H42+

The removal of two electrons from the square planar triplet state of C4H4 initially results in the square planar dication C4H42+ in a closed-shell singlet state (Q = +2, S = 0, M = 1). As there are no electrons occupying the degenerate eg orbitals, we should not expect any distortion from the square planar geometry (D4h) upon relaxation of the geometry. Just as expected, the structure of the carbon skeleton remains unaffected by relaxation. All four C−C bonds remain at 1.44 Å meaning that the S/L ratio is equal to one and that the C4H42+ ion is quadratic in the carbon skeleton (Figure 4). The small destabilization of the eg level (Table 1) is unimportant since the degeneracy remains and the orbitals are unoccupied. The meager relaxation energy, ΔEgeo = −2 kJ mol−1, reflects a minor elongation of the C−H bonds from 1.08 to 1.09 Å.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The author would like to express his gratitude to the students taking the course, Molecular structure, given at the Royal Institute of Technology (KTH), Stockholm, during the winter semester of 2011. The author is further thankful to Lars Kloo, Ulf Henriksson, and the Journal’s reviewers for useful discussions and constructive critisism.

C4H42−

Let us finally take a look at the anion C4H42‑, created by adding two spin-down electrons to the triplet state of C4H4. In the closed-shell singlet state of the C4H42− ion (Q = −2, S = 0, M = 1), both orbitals of the degenerate eg level are doubly occupied. Therefore, just as for the triplet state of C4H4, the competing forces exerted by the occupation of the eg orbitals are perfectly balanced, and there is no resultant force that causes any distortion from the square planar (D4h) geometry and the quadratic carbon skeleton.16 Nevertheless, relaxation of the geometry is followed by elongation of all four C−C bonds from 1.44 to 1.47 Å, and the C−H bonds from 1.08 to 1.10 Å. The corresponding relaxation energy is ΔEgeo = −9 kJ mol−1 (Table 1).

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REFERENCES

(1) (a) Murrell, J. N., Kettle, S. F. A., Tedder, J. M. Valence Theory; John Wiley & Sons Ltd.; London; 1966. (b) Cotton, F. A. Chemical Applications of Group Theory; John Wiley & Sons, Inc.; New York; 1971. (c) Zimmerman, H. E. Quantum Mechanics for Organic Chemists; Academic Press, Inc.; New York; 1975. (d) Coulson, C. A., O’Leary, B., Mallion, R. B. Hückel Theory for Organic Chemists; Academic Press, Inc.; London; 1978. (e) March, N. H., Mucci, J. F. Chemical Physics of Free Molecules; Plenum Press: New York; 1993. (2) Just as the use of a square planar structure for cyclobutadiene in teaching HMO theory has its drawback by misleading the students to conclude a triplet ground state, there are admittedly some flaws also in the exercise here proposed. The distortion in singlet cyclobutadiene (1Ag) is formally a result of the second order or pseudo-Jahn−Teller effect, caused by the vibronic coupling between the ground state and some exited state. The pseudo-Jahn−Teller effect, together with the Jahn−Teller and Renner−Teller effects, are jointly referred to as vibronic coupling effects. The formal distinction between these phenomena is beyond the scope of this exercise, yet it may be useful for the instructor to be aware of its existance. For thorough discussions

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CONCLUSION In summary, we have learned that through inspection of the degenerate highest occupied molecular orbital (HOMO) and simple reasoning, it is possible to predict whether there will be a distortion from the square planar (D4h) structure when electrons are spin-paired, added to, or removed from the triplet state of cyclobutadiene. It is thus possible to predict whether 67

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of vibronic coupling effects see for instance: Burdett, J. K. Chemical Bonds: A Dialogue; John Wiley & Sons Ltd.: Chichester, 1997 or Bersuker, I. B. Modern aspects of the Jahn−Teller effect theory and applications to molecular problems. Chem. Rev. 2001, 101, 1067− 1114. (3) The original papers are (a) Jahn, H. A.; Teller, E. Stability of polyatomic molecules in degenerate electronic states. I. Orbital degeneracy. Proc. R. Soc. London, Ser. A 1937, 220−235. (b) Jahn, H. A. Stability of polyatomic molecules in degenerate electronic states. II. Spin degeneracy. Proc. R. Soc. London, Ser. A 1938, 117−131. (4) Notable papers and textbooks on the Jahn−Teller theorem: (a) Landau, L. D., Lifshitz, E. M. Course of Theoretical Physics; Vol. 3; Quantum Mechanics (Non-Relativistic theory); Pergamon Press Ltd.: Oxford, 1965. (b) Cotton, F. A., Wilkinson, G. Advanced Inorganic Chemistry; John Wiley & Sons: New York, 1972. (c) Kettle, S. F. A. Studies in Modern Chemistry. Coordination Compounds; Thomas Nelson and Sons Ltd.: Ontario, Canada, 1977. (d) Burdett, J. K. Use of the Jahn−Teller theorem in inorganic chemistry. Inorg. Chem. 1981, 20, 1959−1962. (e) Anslyn, E. V., Dougherty, D. A. Modern Physical Organic Chemistry; University Science Books: Sausalito, CA, 2005. (5) (a) Roos, K.; Siegbahn, P. E. M. Density functional theory study of the manganese-containing ribonucleotide reductase from Chlamydia trachomatis: Why manganese is needed in the active complex. Biochemistry 2009, 48, 1878−1887. (b) Roos, K.; Siegbahn, P. E. M. Oxygen cleavage with manganese and iron in ribonucleotide reductase from Chlamydia trachomatis. J. Biol. Inorg. Chem. 2011, 16, 553−565. (6) Educational papers at various levels concerning different aspects of the Jahn−Teller theorem have been published in this Journal: (a) Heilbronner, E. Why do some molecules have symmetry different from that expected? J. Chem. Educ. 1989, 66, 471−478. (b) Senn, P. A simple quantum mechanical model that illustrates the Jahn−Teller effect. J. Chem. Educ. 1992, 69, 819−821. (c) Bacci, M. A simple approach to the Jahn−Teller effect. J. Chem. Educ. 1982, 59, 816−818. (d) Boulil, B.; Henri-Rousseau, O.; Deumié, M. Born-Oppenheimer and pseudo-Jahn-Teller effects as considered in the framework of the time-dependent adiabatic approximation. J. Chem. Educ. 1988, 65, 395−399. (7) Eckert-Maksic, M.; Vazdar, M.; Barbatti, M.; Lischka, H.; Maksic, Z. B. Automerization reaction of cyclobutadiene and its barrier hight: an ab initio benchmark multireference average-quadratic coupled cluster study. J. Chem. Phys. 2006, 125, 064310. (8) All values presented herein were calculated at the UB3LYP/ccpvtz(-f) level of theory using Schrödingers Jaguar 7.7 (Jaguar, version 7.7; Schrödinger LCC: New York, NY, 2010). The same trends were obtained using the smaller 6-31G basis set. The vertical or nonadiabatic singlet−triplet splitting of C4H4 is 16 kJ mol−1 at the UB3LYP/cc-pvtz(-f) level (Figure 1), which can be compared to ∼24 kJ mol−1, as predicted by highly correlated wave functions (see reference 7). The isomerization barrier at the UB3LYP/cc-pvtz(-f) level is only 11 kJ mol−1 (Figure 1), which can be compared with experimental predictions of 7−22 kJ mol−1 ( Withman, D. W.; Carpenter, B. K. Limits on the activation parameters for automerisation of cyclobutadiene-1,2-d2. J. Am. Chem. Soc. 1982, 104, 6473− 6474. Carpenter, B. K. Heavy-atom tunneling as the dominant pathway in a solution-phase reaction? Bond-shift in antiaromatic annulenes. J. Am. Chem. Soc. 1983, 105, 1700−1701. Maier, G.; Wolf, R. and Kalinowski. How high is the barrier for valence isomerization of cyclobutadiene. Angew. Chem., Int. Ed. Engl. 1992, 31, 738−740 ), and multireference coupled cluster calculations giving 26 kJ mol−1 (see ref 7). The accuracy of the theoretical level here applied can also be evaluated for the ionization energy of C4H4, which is 176 kJ mol−1 at the UB3LYP/cc-pvtz(-f) level of theory, while 185 kJ mol −1 experimentally ( Zhang, M.-Y.; Westdimiotis, C.; Marchetti, M.; Danis, P. O.; Ray, J. C., Jr.; Carpenter, B. K.; McLafferty, F. W. Characterization of four C4H4 molecules and cations by neutralizationreionization mass spectroscopy. J. Am. Chem. Soc. 1989, 111, 8341− 8346 ). It can thus be concluded that the B3LYP functional underestimates the singlet−triplet splitting as well as the activation energy for isomerization of C4H4. Nevertheless, the magnitudes of the

errors are within the expected accuracy of B3LYP and of less importance for the purpose of the exercise. It is emphasized that pure Hartree−Fock (HF) should not be used, as it fails to predict the singlet−triplet splitting qualitatively correct, i.e., the triplet becomes the ground state at the HF level. (9) When inspecting the molecular orbitals, students might be more familiar with the results of a restricted open-shell (RODFT) calculation, as it gives a restricted MO diagram below the degenerate eg level. This is how most textbooks illustrate MO diagrams, with the same spatial orbital for different spins. For some of the relative energies (ΔEgeo and ΔEeg, defined in text), the author has compared the UDFT results in Table 1 with RODFT calculations, with essentially the same results. If the splitting of alpha- and beta orbitals in the unrestricted formalism is unfamiliar to the students, it is probably better to consequently use RODFT. It can be useful for the instructor to know that for some of the ions, the highest doubly occupied orbital is not the a2u π orbital, as derived by HMO theory, but an antibonding σ orbital that is always close to degenerate with the a2u π-orbital. This is unimportant for the exercise because we do not remove more than two electrons from C4H4. The MO surfaces in Figure 2 and the electron density contour plots in Figure 3 were produced with Maestro, version 9.2; Schrödinger, LLC: New York, NY, 2011. It is further useful for the instructor to know that there is an alternative representation of the degenerate HOMO in terms of "rhomboidal" orbitals having zero MO coefficients on two opposite carbon atoms. It has been shown that this representation gives a slightly higher energy than the "rectangular" HOMO used herein (Roeselova, M.; Bally, T.; Jungwirth, P.; Carsky, P. An ab initio study of the Jahn−Teller surface. Chem. Phys. Lett. 1995, 234, 395−404). (10) Borden, W. T.; Davidson, E. R.; Hart, P. The Potential Surfaces for the Lowest Singlet and Triplet States of Cyclobutadiene. J. Am. Chem. Soc. 1978, 100, 388−392. (11) Walsh, A. D. The electronic orbitals, shape, and spectra of polyatomic molecules. Part I. AH2 molecules. J. Chem. Soc. 1953, 2260−2266 This was the first in a series of 10 papers published by Walsh in 1953.. (12) (a) Fukui, K.; Yonezawa, T.; Shingu, H. A molecular orbital theory of reactivity in aromatic hydrocarbons. J. Chem. Phys. 1952, 20, 722−725. (b) Jean, Y., Volatron, F. An Introduction to Molecular Orbitals; Burdett, J., Ed.; Oxford University Press: New York, 1993. (13) The difference between ΔEgeo and ΔEeg should not be allowed to confuse the students, who must be informed that (i) the total energy of a quantum mechanical system is not just the sum of the occupied orbital energies and (ii) it may be so that some other occupied orbitals are destabilized by the Jahn−Teller distortion, which cancels part of the stabilization in the eg level. A more sophisticated analysis of the energetic components of the pseudo-Jahn−Teller effect in cyclobutadiene (C4H4) can be found in; Koseki, S.; Toyota, A. Energy component analysis of the pseudo-Jahn-Teller effect in the ground and electronically exited states of the cyclic conjugated hydrocarbons: cyclobutadiene, benzene, and cyclooctatetraene. J. Phys. Chem. A 1997, 101, 5712−5718. (14) Roeselova, M.; Bally, T.; Jungwirth, P.; Carsky, P. Cyclobutadiene radical cation. Ab initio study of the Jahn-Teller surface. Chem. Phys. Lett. 1995, 234, 395−404. (15) Although the planar structures of singlet and triplet C4H4, and C4H4+, are local minima on the PES, the planar structures of C4H4− and C4H42+ are formally transition states for the bending of the hydrogen atoms out of the plane. However, as long as the initial guess geometry is planar (or even slightly distorted), the geometry optimization will converge towards a planar structure. To localize any of the two minima, the hydrogen atoms must be bent out significantly from the plane of the carbon atoms. If the students would get the minimum for C4H4−, it is still not a severe problem for the analysis and discussion made herein, as it is only 6 kJ mol−1 more stable than the square planar transition state. In the case of C4H42−, the situation is more complicated, since planar C4H42− is a stationary point on the PES having four imaginary frequencies. Thus, it is not a 68

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transition state (first order saddle point on the PES) but a state of more complicated, but less chemical, meaning. (16) When the first crystal structure of a transition metal− cyclobutadiene complex was elucidated, it turned out that the C4H4 ligand was quadratic (D4h) in the carbon skeleton ( Harvey, P. D.; Schaefer, W. P.; Gray, H. B.; Gilson, D. F. R.; Butler, I. S. Structure of tricarbonyl(η4-cyclobutadienyl)iron(0) at −45 °C. Inorg. Chem. 1988, 57−59 ). Electronically, this is explained by the formal oxidation of iron(0) to iron(II), which results in the reduction of C4H4 to C4H42−..

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dx.doi.org/10.1021/ed300295r | J. Chem. Educ. 2013, 90, 63−69