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2017 | Supplement | Buchkapitel

7. Group Theory and Molecular Symmetry

verfasst von : Fabien Gatti, Benjamin Lasorne, Hans-Dieter Meyer, André Nauts

Erschienen in: Applications of Quantum Dynamics in Chemistry

Verlag: Springer International Publishing

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Abstract

In various chapters of this book, we have mentioned how a group-theoretical approach could be applied to molecular symmetry and help in the context of vibrational and vibronic problems.

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Vorheriges Kapitel The Kinetic Energy Operator in Curvilinear Coordinates

Nächstes Kapitel Introduction to Numerical Methods

Charge conjugation (C), parity transformation (P), and time reversal (T) form together the famous CPT symmetry under which all physics laws are assumed to be invariant.

These aspects are sometimes explained using the concept of version. The various versions of a structural formula are produced upon permuting the labels of identical nuclei in all possible ways. Versions are considered distinct if they cannot be interconverted under a mere rigid-body rotation. In some cases, the set of permutational versions may get doubled upon further considering the image of each under space inversion (two enantiomers of the same isomer are considered as two distinct versions). As a result, a degenerate set of distinct versions corresponds to equivalent but distinct multiple wells in the energy landscape.

From a quantum-mechanical perspective, an unfeasible transformation means that tunnelling is too weak to be detected experimentally. When so, the unsuperable barrier between two equivalent wells can be considered as infinite and this structural degenerescence can be omitted from the description.

In this chapter, we will limit our discussion to non-degenerate cases, whereby body-fixed axes are turned into themselves or their negatives. However, degenerate cases may involve transformations that mix pairs of equivalent axes together.

Note that alternative conventions are sometimes found in the literature. The x- and y-axes are often swapped, for example. Or, in Ref. [1], the x-axis is defined as the rotation axis while the y-axis is perpendicular to the molecular plane.

Pseudovectors (or axial vectors), as opposed to true vectors (or polar vectors) actually are antisymmetric second-order tensors for which the three independent components are gathered as vectors. They usually are the results of vector products (cross products) of two true vectors. If an operation turns true vectors into their negatives, it leaves pseudovectors unchanged: \({\mathbf u} \times {\mathbf v} \rightarrow (-{\mathbf u}) \times (-{\mathbf v}) = {\mathbf u} \times {\mathbf v}\). Angular momenta are typical examples of pseudovectors.

In fact, an infinitesimal translation expressed in terms of body-fixed coordinates reflects the constraint that the body-fixed frame keeps being centred at the centre of mass when the nuclei move. It thus corresponds to displacements multiplied by masses rather than a collective translation that would shift the centre of mass with respect to its original position in the laboratory-fixed frame.

The treatment of large-amplitude deformations (contortions) able to connect two equivalent isomers (for example in the case of a symmetric double well) will be treated in the last section of this chapter.

Following Ref. [1], a conceptual distinction must be made between the molecular point group of a rigid non-linear molecule and the point group of a solid object of the same shape. Using the properties of the point group in place of the molecular point group (what we have called the standard approach above) is less rigorous but works in practice because both yield the same results as long as vibronic coordinates are concerned. The molecular-symmetry group is made of operations composed of operations belonging to: (i) the molecular point group; (ii) the molecular rotation group, \(\mathcal {K}(\text {mol})\); and (iii) the nuclear-spin-permutation group. For the water molecule, \(\mathcal {K}(\text {mol})\) is made of \(R_0\), \(R_{\pi }(x)\), \(R_{\pi }(y)\), and \(R_{\pi }(z)\) whereby the three axes are distinguishable, as expected for an antisymmetric top molecule (it thus identifies to the point group \(\mathcal {D}_2\)). Because the molecular point group concerns vibronic coordinates in the body-fixed frame, it is sometimes called molecular vibronic group, although the latter concept is more general and also applies to the case of non-rigid molecules (see last section).

Such monomials appear in the expressions of the real and imaginary parts of the spherical harmonics. When considered as origin-centred (or centred on a nucleus that does not move), s-orbitals behave as pure numbers and thus belong to the totally-symmetric irreducible representation, \((p_x,p_y,p_z)\)-orbitals behave as (x, y, z), and d-orbitals behave as linear combinations of second-order monomials of x, y, and z (for example, \(d_{x^2-y^2}\)-orbitals behave as \(x^2-y^2\)). For f-orbitals and beyond, the behaviours of higher-order monomials are obtained upon further considering the direct product table to generate them (see Table 7.3).

Thus, in this section \({\varvec{\mu }}({{\varvec{q}}})\) corresponds to \({\varvec{\mu }}_{0 0}({{\varvec{q}}})\) in Sect. 3.4 of Chap. 3.

The terms normal mode and normal coordinate are often used in place of one another in the literature. Strictly speaking, a normal mode is the set of rectilinear directions followed collectively by the nuclei for a given vibration and the normal coordinate is the amplitude of the corresponding displacement. For a one-dimensional motion along an axis, the mode would be the direction of the axis and the coordinate would be the position along this axis.

As in the case of \(\mathcal {C}_{2\text {v}}\), an alternative convention is sometimes found in the literature, whereby x and y are swapped. In fact, there are six possible conventions here because there are three axes of same degree.

Dimensionless normal coordinates \(q_j\) are scaled (frequency-mass-weighted) upon multiplying lengths by the factor \(\sqrt{\frac{\mu _j \omega _j}{\hbar }}\), where \(\mu _j\) is the reduced mass of mode \(\nu _j\) and \(\hbar \omega _j\) is the corresponding quantum of vibrational energy.

As already pointed out, we have limited our discussion to non-degenerate cases, whereby body-fixed axes are turned into themselves or their negatives and the same for orbitals. Degenerate cases may involve transformations that mix pairs of equivalent orbitals together.

The molecular-symmetry group is valid for any relative arrangement of the nuclei, as opposed to the point group that characterises a specific arrangement. This is why the former can be considered as “containing” all possible subgroups that can be connected dynamically during the course of a reactive process. Molecular-symmetry groups are sometimes called dynamical groups for this reason.

Bunker PR, Jensen P (1998) Molecular symmetry and spectroscopy, 2nd edn. NRC Research Press, Ottawa

Weyl H (1950) The theory of groups and quantum mechanics. Dover Publications, New York

Herzberg G (1992) Molecular spectra and molecular structure. Krieger Pub Co

Mulliken RS (1955) Report on notation for the spectra of polyatomic molecules. J Chem Phys 23:1997

Mulliken RS (1956) Erratum: report on notation for the spectra of polyatomic molecules.J Chem Phys 24:1118

Wilson E, Decius J, Cross P (1955) Molecular vibrations. McGraw-Hill, New York

Cotton FA (1990) Chemical applications of group theory, 3rd edn. Wiley, New York

Wales DJ (2003) Energy landscapes—applications to clusters. Biomolecules and glasses. Cambridge University Press, Cambridge, UK

Raab A, Worth G, Meyer H-D, Cederbaum LS (1999) Molecular dynamics of pyrazine after excitation to the S\(_2\) electronic state using a realistic 24-mode model Hamiltonian. J Chem Phys 110:936

10.

Cattarius C, Worth GA, Meyer H-D, Cederbaum LS (2001) All mode dynamics at the conical intersection of an octa-atomic molecule: multi-configuration time-dependent Hartree (MCTDH) investigation on the butatriene cation. J Chem Phys 115:2088

11.

Marquardt R, Sanrey M, Gatti F, Quere FL (2010) Full-dimensional quantum dynamics of vibrationally highly excited NHD\(_2\). J Chem Phys 133:174302CrossRef

12.

Sala M, Guérin S, Gatti F, Marquardt R, Meyer H-D (2012) Laser induced enhancement of tunneling in NHD\(_2\). J Chem Phys 136:194308CrossRef

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Snels M, Hollenstein H, Quack M (2003) The NH and ND stretching fundamentals of \(^{14}\)ND\(_2\)H. J Chem Phys 119:7893CrossRef

Titel: Group Theory and Molecular Symmetry
verfasst von: Fabien Gatti
Benjamin Lasorne
Hans-Dieter Meyer
André Nauts
Verlag: Springer International Publishing
Buch: Applications of Quantum Dynamics in Chemistry
Print ISBN: 978-3-319-53921-8

Electronic ISBN: 978-3-319-53923-2

Copyright-Jahr: 2017
DOI: https://doi.org/10.1007/978-3-319-53923-2_7

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