Understanding the Jahn–Teller Effect in Octahedral Transition-Metal

Laboratory Experiment pubs.acs.org/jchemeduc

Understanding the Jahn−Teller Effect in Octahedral Transition-Metal Complexes: A Molecular Orbital View of the Mn(β-diketonato)3 Complex Roxanne Freitag and Jeanet Conradie* Department of Chemistry, P.O. Box 339, University of the Free State, 9300 Bloemfontein, Republic of South Africa S Supporting Information *

ABSTRACT: Density functional theory calculations are utilized to calculate and visualize the compression and elongation Jahn−Teller distortion in selected Mn(β-diketonato)3 complexes. Students often struggle to understand this effect, due to the lack of visualization of the repulsion effect between charges on the metal and the axial ligands. Here, a visualization of the molecular orbitals involved in the elongation and compression Jahn−Teller distortion provides a great understanding of the effect.

KEYWORDS: Graduate Education/Research, Upper-Division Undergraduate, Physical Chemistry, Laboratory Instruction, Computer-Based Learning, Textbooks/Reference Books, Computational Chemistry, Molecular Properties/Structure

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upon their orientation relative to the six ligand coordination positions. Two d orbitals, dx2−y2 and dz2 (called the eg set), which have lobes that point at the ligands, will ascend in energy, due to the repulsion between electrons in the orbitals and the negative charges of the ligands. To maintain the total energy, the three d orbitals, dxy, dxz, and dyz (named the t2g set), which have lobes that lie between ligands experiencing less repulsion, will descend in energy. The 5-fold degeneracy among the d orbitals is thus lifted. For a high-spin d4 metal complex, there is one vacancy in the eg orbital group, either in the dx2−y2 or in the dz2 orbital. The two possible ways to fill these orbitals, d1x2−y2d0z2 or d0x2−y2d1z2, are of equal energy (see Figure 1A,B). Similarly, for a d9 metal complex, the two d2x2−y2d1z2 or d1 x2−y2d2z2 occupations of the eg orbital group are of equal energy (see Figure 1C,D). According to Jahn and Teller, this electronic situation is not stable and the octahedron must distort (be reduced in symmetry) in such a way that the energy of the two electronic configurations as mentioned are not equal. When looking for example at the configuration of d0x2−y2d1z2 (d4) and d1x2−y2d2z2 (d9), the ligand(s) along the z axis are much more screened from the charge of the central metal ion than the other four ligands along the x and y axes,6 because most of the electron density will be concentrated in the dz2 orbital between the metal and the two ligands on the z axis. Consequently, there will be greater electrostatic repulsion between the electron(s) in the dz2 orbital and the ligands along the z axis than between the electrons in the t2g set of orbitals pointing between the axes and the ligands

he Jahn−Teller effect (JTE) is formally described as the geometrical distortion of molecules and ions that are associated with certain electron configurations. When a molecule exhibits a spatially degenerate electronic ground state, it will undergo a geometrical distortion that removes this degeneracy to lower the overall energy of the species.1 The Jahn−Teller effect is encountered both in organic compounds (e.g., cyclobutadiene2,3 and cyclooctatetraene4) and in transitionmetal complexes (e.g., the hexaaquacopper(II) complex ion, [Cu(OH2)6]2+5). This article focuses on the JTE observed in octahedral transition-metal complexes. This exercise can be utilized in the physical chemistry curriculum at the upper-level undergraduate or introductory graduate level as a computational exercise or a lecture to introduce the Jahn−Teller effect.

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BACKGROUND Jahn and Teller proved that no nonlinear molecule can be stable in a degenerate electronic state. Such a molecule has to be distorted to lower symmetry to break the degeneracy. In octahedral complexes with nine d electrons (d9), low-spin d7, and high-spin d4 metal electrons, this effect is more pronounced. The d electrons of an isolated metal ion are equienergetic. If this transition-metal ion is placed in a hypothetical spherical field equivalent to the sum of the charges on six ligands, the energies of all five d orbitals would rise together (degenerately) as a result of the repulsions between the negative charges of the electrons in the metal orbitals and the negative charges on the ligands. However, in a real octahedral (Oh) environment, the energy of the orbitals relative to the perturbed energy of the hypothetical spherical field depends © XXXX American Chemical Society and Division of Chemical Education, Inc.

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orbitals of the metal change in energy. The octahedral symmetry Oh has no distortion. The t2g orbital, however, splits into an a and e component when going from Oh to D3 symmetry. When any of the degenerate e orbitals in D3 symmetry are partially occupied, the system is Jahn−Teller unstable and it will spontaneously distort to C2 symmetry. Under C2 symmetry, the degeneracy of the e orbitals is split into a and b components to stabilize the complex. A D3 distortion leads either to a flattening (z-in) of the octahedron or to increasing (z-out) its height.7 High-spin transition-metal complexes such as d2 and d4 species, as well as d9 species, would be Jahn−Teller unstable and would therefore spontaneously distort to C2 symmetry. Various low- and high-spin octahedral complexes can show the Jahn−Teller effect, but the distortion is more pronounced when the degeneracy occurs in the eg orbital group, as these orbitals point directly toward the ligands. Figures S1 and S2 (in the Supporting Information) show all the different electronic states of both low- and high-spin octahedral complexes, as well as indicating for which states the Jahn−Teller effect is possible (namely, uneven occupation of the eg orbitals). Students often struggle to fully grasp the Jahn−Teller effect, as they see it as a complex theorem. This exercise illustrates the JTE by the visualization of the density functional theory (DFT) calculated highest-occupied molecular orbital (HOMO) of three selected octahedral MnIII(β-diketonato)3 complexes, where the β-diketonato ligands are acetylacetonato (CH3COCHCOCH3)−, dibenzoylmethanato (C6H5COCHCOC6H5)− and hexafluoroacetylacetonato (CF3COCHCOCF3)−; see Figure 3 for the structure.

Figure 1. Presentation of the two possible ways of electron filling of the d orbitals of (A) and (B) high-spin d4 octahedral metal complexes and (C) and (D) d9 octahedral metal complexes.

on the x and y axes itself. The ligands on the z axis will thus move further away from the central metal ion and thus lower the symmetry (z-out or elongation Jahn−Teller distortion). This causes the dz2 orbital to be more stable (lower in energy) than the dx2−y2 orbital, as is shown in Figure 2. The opposite

Figure 3. The structure of MnIII(β-diketonato)3 complexes. The numbers refer to Mn−O bonds. The z axis is defined along the Mn−O bonds 5 and 6 as indicated. R = CH3, C6H5, or CF3.

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COMPUTATIONAL METHOD Three octahedral MnIII(β-diketonato)3 complexes, with the β-diketonato ligands being acetylacetonato (CH3COCHCOCH3)−, dibenzoylmethanato (C6H5COCHCOC6H5)−, and hexafluoroacetylacetonato (CF3COCHCOCF3)−, as well as MnII(CH3COCHCOCH3)3−, were chosen for this study. All structures were optimized with the ADF (Amsterdam Density Functional) 2012 program,8 using the GGA (Generalized Gradient Approximation) functional OLYP (Handy−Cohen and Lee−Yang−Parr),9,10 the TZP (Triple ζ polarized) basis set,11 a fine mesh for numerical integration, a-spinunrestricted formalism, applying tight convergence criteria, using C2 or C1 symmetry as indicated. The OLYP functional proved to be a good choice to correctly calculate the ground state of paramagnetic complexes.12

Figure 2. The effect of z elongation and compression Jahn−Teller distortion on the M−L bond lengths and energy of the eg and t2g set of orbitals.

distortion is expected from the d2x2−y2d1z2 (d9) and d1x2−y2d0z2 (d4) configurations (namely, z-in or compression Jahn−Teller distortion). The effect that the Jahn−Teller distortion has on the M−L bond lengths, as well as on the orbital energies, is illustrated in Figure 2. A z-out or elongation Jahn−Teller distortion leads to an elongation of the M−L bond lengths on the z axis or a shortening of the bond lengths on the x and y axes. The z-in or compression Jahn−Teller distortion causes just the opposite, the shortening of M−L bond lengths on the z axis or the elongation of the M−L bond lengths on the x and y axes. The Jahn−Teller distortion can also be explained by the use of point groups. If the symmetry of a complex is lowered, the complex will belong to a point group of lower symmetry. When there is a change in the point group of the complex, the d

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STUDENT EXERCISE This is a dry lab experiment. Students can work individually. A student handout is provided in the Supporting Information. A desktop computer with suitable DFT software13 is required. Three B

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hours in total should be budgeted to familiarize the students with the DFT software, to run three single-point calculations (of ca. 10 min each on a desktop computer) on the provided optimized coordinates, as well as for interpretation of the results. Take note that if nonoptimized coordinates are used, full geometry optimizations are required, which may run for 3−12 h each. First, the students have to optimize high-spin S = 5/2 (sextet) MnII(CH3COCHCOCH3)3− without any symmetry constraint (C1), to evaluate the optimized lowest-energy geometry of this complex. An example input file is provided in the Supporting Information. The input geometry was chosen specifically to have elongated Mn−O bonds along the z axis and to be without any symmetry. Second, the students have to optimize high-spin S = 2 (quintet) MnIII(CH3COCHCOCH3)3, to find the geometry and energy of the ground state of complex MnIII(CH3COCHCOCH3)3 in C2 symmetry. An example input file is provided in the Supporting Information. Once the optimized geometry is obtained, the electron occupation has to be determined from the output file (A 49//46 and B 45//44). Manganese(III) octahedral complexes are known to be high-spin complexes that undergo Jahn−Teller distortion7,14 because of the partial filling of the doubly degenerate eg orbital group. The electron filling of the d orbitals of Mn(βdiketonato)3 is thus expected to be either t32gd1z2 or t32gd1x2−y2. The highest-occupied molecular d orbital (HOMO) of Mn(βdiketonato)3 is therefore either dz2 or dx2−y2. Once the groundstate geometry is obtained, an alternative electron occupation can be specified to obtain the geometry of the alternative Jahn−Teller distortion. This can be achieved by moving the electron in the HOMO (A49) of the ground-state geometry, from an A to a B irreducible representation. The alternative occupation of the electrons (A 48//46 and B 46//44) should thus be included in the second geometry optimization; see the Supporting Information for an example input file. Ensure the input geometry in both cases to be C2. The Supporting Information also provides the optimized coordinates of the dz2 and dx2−y2 electronic states. Once the optimized geometries of both the ground state and the alternative electron occupation of MnIII(CH3COCHCOCH3)3 are obtained, the students have to evaluate the bond lengths and relative energies of the two electronic states and also have to create a file to visualize the HOMO of each electronic state. The electronic energy can be used as an approximation of free energy G, as the difference between electronic energy and Gibbs free energy is small. The electronic energy is printed at the end of the output file. Results on MnIII(C6H5COCHCOC6H5)3 and MnIII(CF3COCHCOCF3)3 are also provided and discussed, should additional exercise for the students be needed. Alternatively, three groups of students could each solve the two electronic states of one of the three complexes, comparing the results.

Figure 4. ADF/OLYP optimized geometry and the electronic configuration of the high-spin (S = 5/2) state of MnII(CH3COCHCOCH3)3−. Bond lengths shown in Å.

Mn(CH3COCHCOCH3)3, where only one of the eg orbitals has an electron, and two degenerate electronic states are possible, Figure 1A or 1B. During the optimization of the high-spin (S = 2) state of Mn(CH3COCHCOCH3)3, the geometry is distorted to break the degeneracy of the eg orbitals. The geometry and HOMO of the DFT optimized ground electronic state of the Mn(CH3COCHCOCH3)3 complex are presented in Figure 5A and a summary of the bond lengths (see Figure 3 for the numbering of the bonds) is tabulated in Table 1. The two bonds in the z direction are ca. 0.25 Å longer than the other four bonds. The HOMO of this electronic state is mainly of dz2 character, pointing directly in the directions of the ligands on the z axis. The antibonding character between the dz2 orbital of the manganese center and pz orbital of Oβ‑diketonato on the z axis is clear and leads to an elongation Jahn−Teller distortion and subsequent elongation of the Mn−O bond lengths along the z axis (bond lengths 5 and 6 in Table 1). The geometry and HOMO of Mn(CH3COCHCOCH3)3 with the alternative electron filling of the d orbitals, t32gd1x2−y2, is illustrated in Figure 5B. Here the two bonds along the z axis

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RESULTS AND DISCUSSION The DFT optimized geometry of the high-spin (S = 5/2) state of the MnII(CH3COCHCOCH3)3− complex is presented in Figure 4. It is observed that, although the bonds along the z axis in the starting geometry were longer than the other four Mn−O bonds, the optimized geometry gives an almost perfect octahedral structure, with equal Mn−O bonds of 2.248 Å up to three decimal values. For this complex, the eg orbitals have one electron each, that is, having even eg occupation. This high-spin d5 configuration (see the inset in Figure 4) is thus nondegenerate, symmetric, and allowed. Therefore, the distorted symmetry in the input file is restored to an octahedron in the optimized geometry. However, this is not the case with the high-spin (S = 2) state of

Figure 5. ADF/OLYP geometry and HOMO orbital of the optimized Mn(CH3COCHCOCH3)3 complex: (A) the minimum energy electronic ground state showing mainly dz2 character and (B) the alternative electronic occupation showing mainly dx2−y2 character. Bond lengths shown in Å. C

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Table 1. Mn−O Bond Lengths of the Indicated Optimized Mn(β-diketonato)3 Complexes Bond Lengtha (Mn−O)/Å Complex

1

Mn(CH3COCHCOCH3)3 compression JT 2.118b elongation JT 1.974 Mn(C6H5COCHCOC6H5)3 compression JT 2.100 elongation JT 1.965 Mn(CF3COCHCOCF3)3 compression JT 2.117 elongation JT 1.998 a

2

3

4

5

6

Energy/(kJ mol−1)

Population (%)

2.118 1.974

2.121 1.993

2.121 1.993

1.944 2.245

1.944 2.245

3.6 0

21 79

2.100 1.965

2.112 1.974

2.112 1.974

1.934 2.217

1.934 2.217

5.6 0

9 91

2.117 1.998

2.129 1.970

2.129 1.970

1.949 2.240

1.949 2.240

3.5 0

19 81

Refer to Figure 3 for Mn−O numbering. bElongated bond lengths are indicated in bold format.

are ca. 0.17 Å shorter than the other four bonds. It is clear that the HOMO of the alternative electron occupation shows mainly dx2−y2 character. The repulsion along the x and y axes between the dx2−y2 HOMO of Mn and the px and py orbitals of Oβ‑diketonato on the x and y axes, respectively, as can be seen in the antibonding character of the HOMO in in Figure 5B, leads to the elongation of the Mn−O bond lengths along the x and y axes; see bond lengths 1−4 in Table 1. This result is known as a compression Jahn−Teller distortion (elongation of the Mn−O bond lengths along the x and y axes, or shortening of the Mn−O bond lengths along the z axis). Careful evaluation of the bonds in Table 1 show that for the elongation Jahn−Teller distortion we have two long Mn−O bonds along the z axis and two medium to short bonds, as well as two short bonds in the xy plane. This asymmetric kind of distortion, especially when more pronounced, is called orthorhombic distortion and is consistent with Jahn−Teller behavior.14 The Jahn−Teller theorem only predicts that distortion must occur for degenerate states; it does not give any indication of what kind of geometrical distortion will occur, or how large this distortion will be. We also observe from the bonds tabulated in Table 1 that the Jahn−Teller distortion along the z axis is more pronounced than in the xy plane. This is understandable, since the dz2 orbital protrudes further along the z axis than the dx2−y2 orbital along the x and y axes. Consequently, there is a stronger repulsion between the dz2 orbital of Mn, and the pz orbital of Oβ‑diketonato along the z axis than the repulsion between the dx2−y2 orbital of Mn and the px and py orbitals of Oβ‑diketonato along the x and y axes. The DFT calculated energy of the electronic state with alternative electron filling is 3.6 kJ mol−1 higher than the lowest-energy state. Using the Boltzmann equation (eq 1), we find that the population of Mn(CH3COCHCOCH3)3 that will occur with an elongation Jahn−Teller distortion is 79%, relative to the 21% population found to have compression Jahn−Teller distortion. ln

nj ni

=−

Figure 6. ADF/OLYP molecular orbital energy-level diagram of the optimized structures of Mn(CH3COCHCOCH3)3, showing elongation (left) and compression (right) Jahn−Teller distortion. The dbased orbitals of the Mn complex are indicated in red. Only the top frontier MOs are shown. Arrows pointing up and downward represent α- and β-electrons, respectively.

the ground state of both Mn(C6H5COCHCOC6H5)3 and Mn(CF3COCHCOCF3)3 also have an elongation Jahn−Teller distortion, with two bonds over 0.2 Å longer than the other four bonds. The population of the elongation Jahn−Teller distortion of Mn(C6H5COCHCOC6H5)3 is 91%, explaining why the two known crystal structures of Mn(C6H5COCHCOC6H5)3 both show an elongation Jahn−Teller distortion.15,16 The only crystal structure of MnIII(CF3COCHCOCF3)3 published to date, according to the author’s knowledge,17,18 also showed an elongation Jahn−Teller distortion, as was expected from the DFT calculations in this study, which predicted a 81% population of the elongation Jahn−Teller distortion. Structures of MnII(CF3COCHCOCF3)3− are octahedral.19,20 This also was expected, because MnII(CF3COCHCOCF3)3− is a high-spin d5 complex with five unpaired electrons: one electron in each orbital of the eg orbital group, and one electron in each orbital of the t2g orbital group. Lastly, it is important to evaluate whether the metal orbitals of the optimized structures of Mn(CH3COCHCOCH3)3 indeed have the order as predicted in Figure 2. Figure 6 gives a molecular energy level diagram of the relative energies of the molecular orbitals of the two optimized structures of

(Ej −Ei) kT

(1)

nj = the number of atoms with energy Ej relative to the number of atoms ni with energy Ei at temperature T, k = Boltzmann constant. The ADF/OLYP results of this study thus show that Mn(CH3COCHCOCH3)3 exists mainly with an elongation Jahn− Teller distortion, although compression Jahn−Teller distortion is also possible, as indeed has been observed experimentally.14 The bond lengths of the lowest energy DFT optimized structure of Mn(C6H5COCHCOC6H5)3 and Mn(CF3COCHCOCF3)3 are given in Table 1. As has been found for Mn(CH3COCHCOCH3)3, D

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(8) te Velde, G.; Bickelhaupt, F. M.; Baerends, E. J.; Fonseca Guerra, C.; van Gisbergen, S. J. A.; Snijders, J. G.; Ziegler, T. Chemistry with ADF. J. Comput. Chem. 2001, 22, 931−967. (9) Handy, N. C.; Cohen, A. J. Left-Right Correlation Energy. Mol. Phys. 2001, 99, 403−412. (10) (a) Lee, C.; Yang, W.; Parr, R. G. Development of the ColleSalvetti Correlation-Energy Formula into a Functional of the Electron Density. Phys. Rev. B. 1988, 37, 785−789. (b) Johnson, B. G.; Gill, P. M. W.; Pople, J. A. The Performance of a Family of Density Functional Methods. J. Chem. Phys. 1993, 98, 5612−5626. (c) Russo, T. V.; Martin, R. L.; Hay, P. J. Density Functional Calculations on First Row, Metals. J. Chem. Phys. 1994, 101, 7729−7737. (11) van Lenthe, E.; Baerends, E. J. Optimized Slater-Type Basis Sets for the Elements 1−118. J. Comput. Chem. 2003, 24, 1142−1156. (12) Conradie, J.; Wondimagegn, T.; Ghosh, A. Spin States at a Tipping Point: What Determines the dz2 Ground State of Nickel(III) Tetra(tbutyl)porphyrin Dicyanide? J. Phys. Chem. B. 2008, 112, 1053− 1056. (13) ADF Modeling Suite; see https://www.scm.com/trial for a one month free trail license (accessed Oct 2013). (14) Geremia, S.; Demitri, N. Crystallographic Study of Manganese(III) Acetylacetonate: An Advanced Undergraduate Project with Unexpected Challenges. J. Chem. Educ. 2005, 82, 460−465. (15) Zaitseva, E. G.; Baidina, I. A.; Stabnikov, P. A.; Borisov, S. V.; Igumenov, I. K. Crystal and Molecular Structure of Tris(Dibenzoylmethanato)manganese(III). Zh. Strukt. Khim. 1990, 31, 184−189. (16) Barra, A.-L.; Gatteschi, D.; Sessoli, R.; Abbati, G. L.; Cornia, A.; Fabretti, A. C.; Uytterhoeven, M. G. Electronic Structure of Manganese(III) Compounds from High-Frequency EPR Spectra. Angew. Chem., Int. Ed. Engl. 1997, 36, 2329−2331. (17) Cambridge Structural Database (CSD), Version 5.34, Feb 2013 update. (18) Bouwman, E.; Caulton, K. G.; Christou, G.; Folting, K.; Gasser, C.; Hendrickson, D. N.; Huffman, J. C.; Lobkovsky, E. B.; Martin, J. D.; Michel, P.; Tsai, H.; Xue, Z. Doubly-Hydrated Hexafluoroacetylacetone as a Tetradentate Ligand: Synthesis, Magnetochemistry, and Thermal Transformations of a MnIII2 Complex. Inorg. Chem. 1993, 32, 3463−3470. (19) Bryant, J. R.; Taves, J. E.; Mayer, J. M. Oxidations of Hydrocarbons by Manganese(III) tris(hexafluoroacetylacetonate). Inorg. Chem. 2002, 41, 2769−2776. (20) Villamena, F. A.; Dickman, M. H.; Crist, D. R. Nitrones as Ligands in Complexes of Cu(II), Mn(II), Co(II), Ni(II), Fe(II), and Fe(III) with N-tert-Butyl-α-(2-pyridyl)nitrone and 2,5,5-Trimethyl-1pyrroline-N-oxide. Inorg. Chem. 1998, 37, 1446−1453.

Mn(CH3COCHCOCH3)3, showing both elongation and compression Jahn−Teller distortion. We observe that the order of the energy levels is indeed as expected. First, the three d orbitals of the t2g set, the dxy, dxz, and dyz orbitals, are more stable (lower energy) than the dz2 or dx2−y2 orbitals, as would be expected for a distorted octahedral ligand field. Second, the filled dz2 orbital (HOMO) is lower in energy than the empty dx2−y2 orbital (LUMO) in the case of the elongation Jahn−Teller distortion, and vice versa for the compression Jahn−Teller distortion.

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CONCLUSION Summarizing, we have illustrated that DFT computational chemical methods can serve as a simple but illustrative representation, aiding the understanding of the Jahn−Teller theorem. DFT provides a convenient way to predict the relative geometric and energetic effect due to the JTE. DFT also provides the method to calculate the opposite Jahn−Teller distortion and to predict the population of each. The calculated ratio of 79:21 for the elongation/compression Jahn−Teller distortion for Mn(CH3COCHCOCH3)3 complexes explains why both the compression and elongation Jahn−Teller distortion is observed in solid state crystal structures.

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ASSOCIATED CONTENT

S Supporting Information *

[Optional, except for laboratories.] Student handouts, optimized coordinates; example input files of the DFT calculations. This material is available via the Internet at http://pubs.acs.org.

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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 This work has received support from the South African National Research Foundation and the Central Research Fund of the University of the Free State, Bloemfontein.

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REFERENCES

(1) Jahn, H. A.; Teller, E. Stability of Polyatomic Molecules in Degenerate Electronic States. I. Orbital Degeneracy. Proc. R. Soc. London, Ser. A 1937, 161, 220−235. (2) Senn, P. A Simple Quantum Mechanical Model That Illustrates the Jahn−Teller effect. J. Chem. Educ. 1992, 69, 819−821. (3) Johansson, A. J. 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. J. Chem. Educ. 2013, 90, 63−69. (4) Klärner, F.-G. About the Anti-aromaticity of Planar Cyclooctatetraene. Angew. Chem., Int. Ed. 2001, 40, 3977−3981. (5) Janes, R.; Moore, E. A. Metal-Ligand Bonding; Royal Society of Chemistry: Cambridge, 2004. (6) Cotton, F. A.; Wilkinson, G.; Gaus, P. L. Basic Inorganic Chemistry, 3rd ed.; John Wiley & Sons Ltd: New York, 1995; pp 539. (7) Diaz-Acosta, I.; Baker, J.; Hinton, J. F.; Puley, P. Calculated and Experimental Geometries and Infrared Spectra of Metal TrisAcetylacetonates: Vibrational Spectroscopy As a Probe of Molecular Structure for Ionic Complexes. Part II. Spectrochim. Acta, Part A 2003, 59, 363−377. E

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