Abstract
In quantum mechanics the observable phenomena are interactions, expressed as matrix elements of operators in a function space. These spaces and operators are like communicating vessels, reality is neither the operation nor the representation, but the interaction. The evaluation of the corresponding matrix elements requires the coupling of representations, and can be factorized into an intrinsic scalar quantity that contains the physics of the interaction, and a tensorial coupling coefficient that contains its symmetry. This factorization is first illustrated for the case of overlap integrals, where the operator is just the unit operator, and then extended to the case of non-trivial operators, such as the Hamiltonian, and electric and magnetic dipole operators. The Wigner–Eckart theorem is introduced, together with the symmetry selection rules, both at the level of representations and subrepresentations. The results are applied to chemical reaction theory, and to the theory of the Jahn–Teller effect. Selection rules are illustrated for linear and circular dichroism. Finally, the polyhedral Euler theorem is introduced and applied to valence-bond theory for clusters.
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Notes
- 1.
The general case with complex irreps is exemplified for the coupling of spin representations in Sect. 7.4.
- 2.
Any permutation can be expressed as a sequence of transpositions of two elements. If the total number of transpositions is even, sgn(σ)=+1; if it is odd, sgn(σ)=−1. See also Sect. 3.3.
- 3.
For complex variables, variable and derivative have complex-conjugate transformation properties.
- 4.
The general time-reversal selection rules are discussed in Sect. 7.6.
- 5.
Such combinations can be cast in a higher-order symbol, known as 6Γ symbol, by analogy with the 6j coupling coefficients in atomic spectroscopy.
- 6.
Based on [13].
- 7.
In tris-chelate complexes Δ refers to a right-handed (dextro) helix. A left-handed helix (lævo) is denoted as Λ.
- 8.
The excitation creates an electron-hole pair, which can move from one ligand to another. This is called an exciton.
- 9.
In geometry a vertex is a point were two or more lines meet.
- 10.
This flow description provides a simple pictorial illustration of the abstract homology theory. The standard reference is: [23].
References
Wigner, E.P.: Group Theory. Academic Press, New York (1959)
Griffith, J.S.: The Irreducible Tensor Method for Molecular Symmetry Groups. Prentice Hall, Englewood-Cliffs (1962)
Butler, P.H.: Point Group Symmetry Applications, Methods and Tables. Plenum Press, New York (1981)
Ceulemans, A., Beyens, D.: Monomial representations of point-group symmetries. Phys. Rev. A 27, 621 (1983)
Lijnen, E., Ceulemans, A.: The permutational symmetry of the icosahedral orbital quintuplet and its implication for vibronic interactions. Europhys. Lett. 80, 67006 (2007)
Jahn, H.A., Teller, E.: Stability of polyatomic molecules in degenerate electronic states. I. Orbital degeneracy. Proc. R. Soc. A 161, 220 (1937)
Bersuker, I.B.: The Jahn–Teller Effect. Cambridge University Press, Cambridge (2006)
Domcke, W., Yarkony, D.R., Köppel, H.: Conical Intersections, Electronic Structure, Dynamics and Spectroscopy. Advanced Series in Physical Chemistry, vol. 15. World Scientific, Singapore (2004)
Shapere, A., Wilczek, F.: Geometric Phases in Physics. Advanced Series in Mathematical Physics, vol. 5. World Scientific, Singapore (1989)
Judd, B.R.: The theory of the Jahn–Teller effect. In: Flint, C.D. (ed.) Vibronic Processes in Inorganic Chemistry. NATO Advanced Study Institutes Series, vol. C288, p. 79. Kluwer, Dordrecht (1988)
Bersuker, I.B., Balabanov, N.B., Pekker, D., Boggs, J.E.: Pseudo Jahn–Teller origin of instability of molecular high-symmetry configurations: novel numerical method and results. J. Chem. Phys. 117, 10478 (2002)
Hoffmann, R., Woodward, R.B.: Orbital symmetry control of chemical reactions. Science 167, 825 (1970)
Halevi, E.A.: Orbital symmetry and reaction mechanism, the OCAMS view. Springer, Berlin (1992)
Rodger, A., Nordén, B.: Circular Dichroism and Linear Dichroism. Oxford Chemistry Masters. Oxford University Press, Oxford (1997)
Orgel, L.E.: Double bonding in chelated metal complexes. J. Chem. Soc. 3683 (1961). doi:10.1039/JR9610003683
Ceulemans, A., Vanquickenborne, L.G.: On the charge-transfer spectra of iron(II)- and ruthenium(II)-tris(2,2′-bipyridyl) complexes. J. Am. Chem. Soc. 103, 2238 (1981)
Day, P., Sanders, N.: Spectra of complexes of conjugated ligands. 2. Charge-transfer in substituted phenantroline complexes: intensities. J. Chem. Soc. A 10, 1536 (1967)
Yersin, H., Braun, D.: Localization in excited states of molecules. Application to \(\mbox{Ru(bipy)}_{3}^{2+}\). Coord. Chem. Rev. 111, 39 (1991)
Melvin, M.A.: Simplification in finding symmetry-adapted eigenfunctions. Rev. Mod. Phys. 28, 18 (1956)
Ceulemans, A.: The construction of symmetric orbitals for molecular clusters. Mol. Phys. 54, 161 (1985)
Biel, J., Chatterjee, R., Lulek, T.: Fiber structure of space of molecular-orbitals within LCAO method. J. Chem. Phys. 86, 4531 (1987)
Ceulemans, A., Fowler, P.W.: Extension of Euler theorem to the symmetry properties of polyhedra. Nature 353, 52 (1991)
Hilton, P.J., Wylie, S.: Homology Theory. Cambridge University Press, Cambridge (1967)
Hoffmann, R.: Building bridges between inorganic and organic chemistry (Nobel lecture). Angew. Chem., Int. Ed. Engl. 21, 711 (1982)
Mingos, D.M.P., Wales, D.J.: Introduction to Cluster Chemistry. Prentice Hall, Englewood-Cliffs (1990)
Fowler, P.W., Manolopoulos, D.E.: An Atlas of Fullerenes. Clarendon, Oxford (1995)
Fowler, P.W., Ceulemans, A.: Electron deficiency of the fullerenes. J. Phys. Chem. A 99, 508 (1995)
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Ceulemans, A.J. (2013). Interactions. In: Group Theory Applied to Chemistry. Theoretical Chemistry and Computational Modelling. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-6863-5_6
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