# On the Physical Reasons for the Extension of Symmetry Groups in Molecular Spectroscopy

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Double Degenerate Vibrational Modes

_{a}and Q

_{b}. The respective harmonic Hamiltonians, in terms of normal coordinates and associated momenta, are

**Q**and

**P**represent the operators corresponding to a normal vibrational coordinate and its conjugate momentum.

**H**=

**H**

_{a}+

**H**

_{b}

**L**=

**Q**

_{a}

**P**

_{b}+

**Q**

_{b}

**P**

_{a}

**H**,

**H**

_{a}and

**H**

_{b}commute with each other, but

**L**commutes only with the total Hamiltonian

**H**. The figure also shows the quantum numbers associated with the eigenvalue of each operator. Thus, representing a general vibrational state as $|{\mathrm{v}}_{\mathrm{a}},{\mathrm{v}}_{\mathrm{b}},\mathrm{v},l\rangle .$, with v=v

_{a}+v

_{b}, the states of a degenerate fundamental pair can be taken either as $|1,0,1,-\rangle $ and $|0,1,1,-\rangle $ (for short $|{1}_{\mathrm{a}}\rangle $ and $|{1}_{\mathrm{b}}\rangle $ ) or as $|-,-,1,1\rangle $ and $|-,-,1,-1\rangle $ (for short $|{1}^{1}\rangle $ and $|{1}^{-1}\rangle $ ).

_{n}, C

_{nh}, S

_{n}), each of two $|{1}^{\pm 1}\rangle $ functions would constitute the basis of a one-dimensional irreducible representation. In spite of this group theoretical conclusion, the physical intuition and the equivalence of the directions normal to z in the case of high rotational symmetry about the z-axis still suggest that the a and b vibrational modes should occur in degenerate pairs. This is even more evident for the rotational states, because the moments of inertia about x and y would be equal anyway. It is for this reason that the one-dimensional symmetry species spanned by the components of a $|{1}^{\pm 1}\rangle $ pair in the mentioned class of molecules are called by Herzberg “separably degenerate irreducible representations” [2]. This odd situation is clarified if the time reversal operation is applied, and this can be understood without entering any mathematical treatment. In fact, the components of a $|{1}^{\pm 1}\rangle $ pair differ from each other by the sign of the vibrationally generated angular momentum, then they are interchanged if the direction of motion is inverted, and must form the basis of a two-dimensional corepresentation, in a group which includes the time reversal operation.

_{3}through direct multiplication by (

**E**,

**Θ**).

_{3}, and the behavior of the components of typical polar vectors (linear displacement T) and axial vectors (angular displacement R). Polar and axial vectors describing motion (linear momentum P and angular momentum J) also behave as T and R, respectively, since the group contains only geometrical symmetry operations. Note that the “separably degenerate” species E

_{+}and E

_{-}are two mono-dimensional species, and that circular vector components of the form x+iy and x-iy are basis of these two separate representations. On the contrary, their components along x and y are basis of a two-dimensional reducible representation, whose characters are the sum of those of E

_{+}and E

_{-}, that is 2,-1,-1, and which can be completely reduced to E

_{+}+E

_{-}by a transformation to circular components, such as that in equation (4).

_{3}are multiplied by (E, Θ ), one obtains the six operations shown on the top of Table 2. They form a group if Θ

^{2}= E, that is for systems with integral spin. Anyway, here we are dealing with vibration-rotation energy states and the resolution of spin structures is not considered. Also (E, Θ ) is a group if Θ

^{2}= E.

**Θ**(e

^{iφ}q) = e

^{-2iφ}(e

^{i}

^{φ}q) and

**Θ**(e

^{i}

^{φ}p) = - e

^{-2i}

^{φ}(e

^{i}

^{φ}p), for coordinates and momenta.

_{a}and Q

_{b}of perpendicular vibrational modes of C

_{3}-molecules can be chosen to transform as T

_{x}and T

_{y}, therefore their circular combinations, with defined angular momentum, span the non-degenerate species E

_{+}and E

_{-}of this point group, see Table 1. However, they transform according to the E-corepresentation of C

_{3}×(E, Θ ), therefore they are actually degenerate.

## 3. Spin Double Groups and Kramers Degeneracy

^{(j)}. In particular, if the orientation of the components of a given Γ

^{(j)}set is chosen in such a way that they are simultaneous eigenfunctions of

**j**

^{2}and

**j**

_{z}, with good quantum numbers j and k, they transform under a rotation by φ about z as:

**R**

_{z}(φ) |j,±k〉 = exp(±ikφ) |j,±k〉

**R**, and to extend the group by direct multiplication by (

**E**,

**R**) [4,8]. The species symmetric under

**R**occur in integral spin states and are called single valued, those anti-symmetric under

**R**occur with half integral spin and are called double valued. It can be shown that states corresponding to double valued representations of the extended group occur always at the least in degenerate pairs (Kramers degenerate doublets). In low-symmetry groups, some or all spin functions may span non-degenerate species, but in this case, for half integral spin, each non-degenerate species occurs at least twice, corresponding to Kramers doublets of the same energy. However, Kramers degeneracy can be lifted by the presence of a magnetic field.

**R**is a new operation, distinct from any operation

**R**(2π) of the extended full rotation group (see table 55 of Ref. [8], where the operations

**R**(2π) are included under the symbol ${C}_{\infty}^{\phi}$, with φ=2π, and the new operation

**R**occurs as well). Another point is that we cannot explain why a magnetic field could lift the Kramers degeneracy. Both points are clarified if the fictitious operation

**R**is replaced by the square of time reversal operator,

**Θ**

^{2}[4]. This will also make the understanding of Kramers degeneracy in the absence of fields easier. One has to remember that all components of the spin operators, as for angular momenta, change sign under time reversal.

**Θ**|S,0〉 = (-1)

^{S}|S,0〉 for integral S

**Θ**|S,+1/2〉 = (-1)

^{S-1/2}|S,+1/2〉 for half integral S.

**S**

**are real and positive. With these conventions, building up the spin function $|S,-1/2\rangle $ by applying**

_{±}**S**

_{-}to $|S,+1/2\rangle $ and operating with time reversal, we also find from (7):

**Θ**on both sides of equation (9), one finds eventually:

**Θ**

^{2}, but those with half integral spin change sign, just as for the operation

**R**.

**Θ**transforms the above spin functions into different functions, orthogonal to them. Now if the Hamiltonian is invariant under time reversal, the eigenstates ψ and

**Θ**ψ (inclusive of spin) have the same energy (in fact, in this case the equation

**Θ**

**H**ψ =

**Θ**Eψ becomes

**HΘ**ψ = E

**Θ**ψ), therefore they must be degenerate if they are different functions, that is if S is half integral. On the other hand, in the presence of a magnetic field the Hamiltonian is not invariant under time reversal, and

**Θ**ψ and ψ are no longer bound to have the same energy. This explain why a magnetic field can lift the Kramers degeneracy.

^{2},Θ

^{3}), see Ref. [4].

## 4. The Extended Symmetry Group (E,${\mathcal{R}}^{\mathbf{\prime}}$)×(E,Θ), With Θ^{2} = E

**E**,${\mathcal{R}}^{\mathbf{\prime}}$). The ensemble of operations (

**E**,${\mathcal{R}}^{\mathbf{\prime}}$)

**×(E**,

**Θ**

**)**is also a group, as

**(E**,

**Θ**

**)**does, if Θ

^{2}

**=**E. This group can be effectively employed to classify vibration-rotation coordinates, operators and wavefunctions, including their real or imaginary nature [9]. Spin structure is not considered and one can assume that Θ

^{2}

**=**E. It is a subgroup of the extended group resulting by the multiplication of the point group or molecular group by

**(E**,

**Θ**

**)**. Table 3 shows the characters of the corepresentations of this group, which apply to bases of Cartesian components of vectors, with either real or imaginary coefficients, see also the upper part of Table 2.

_{k}relative to the shift of k, since in the absence of external fields the Hamiltonian is diagonal in J and M. For non-degenerate oscillators, the values of the phase angles δ

_{v}(s) and δ

_{v}(a), relative to the shift in the vibrational quanta of modes symmetric or anti-symmetric under ${\mathcal{R}}^{\mathbf{\prime}}$, have to be defined. For two-dimensional oscillators, using circular components, the value of δ

_{v}

_{t}±δ

_{l}

_{t}, where δ

_{l}

_{t}is the phase angle relative to the shift of l

_{t}, the quantum number related to the angular momentum generated by the degenerate vibrational modes, has to be defined. For three-dimensional oscillators, using spherical components, one has to define the phase angles δ

_{v}

_{t}(s) ±δ

_{l}

_{t}(s), δ

_{m}

_{t}(s), δ

_{v}

_{t}(a) ±δ

_{l}

_{t}(a) and δ

_{m}

_{t}(a), where l and m apply to the angular momentum generated by the three-dimensional mode and its z-component, and (s) and (a) apply to three-dimensional modes with character 1 or -1 under ${\mathcal{R}}^{\mathbf{\prime}}$. Classifying the vibration-rotation operators according to the symmetry species of the group in Table 3, and exploiting the property that the rotational and vibrational factors in the Hamiltonian must have the same symmetry, we have shown that it is possible to determine values of δ

_{k}and of all the vibrational phase angles in such a way that all vibration-rotation matrix elements are real [9,10].

_{M}, relative to the shifts of J and M [9,10,11]. We have also shown that with the phase conventions leading to real vibration-rotation matrix elements, and with ${\eta}_{J}^{\lambda}$ either 0 or π (modulo 2π) the matrix elements of vibration-rotation transition moment operators are all real if ${\mathcal{R}}^{\mathbf{\prime}}$ is a reflection and all imaginary if ${\mathcal{R}}^{\mathbf{\prime}}$ is a rotation [9,10].

## 5. Symmetry Groups for Floppy Molecules

_{a}, x

_{pam}and z

_{pam}are principal inertia axes, z

_{ρ}is the axis about which the whole molecule must rotate to compensate for the angular momentum caused by the internal rotation of the methyl group (top) with respect to the aldehyde group H-C-O. We also shall make use of a coordinate system x

_{a}, y

_{a}, z

_{a}, fixed to the internally rotating top (methyl group), with x

_{a}and y

_{a}parallel to x and y in the reference conformation with the internal rotation angle τ equal to zero.

_{6}, consists of the identity, the two circular permutations (123) and (132), and the three permutation-inversion operations (12)*, (13)* and (23)*. The molecular geometries generated by the circular permutations can be brought back to the initial one again by an internal rotation, those generated by the permutation-inversion operations also require a rotation by π about an axis normal to the HCO plane (y

_{t}in Figure 2). The operations of the CNPI group containing permutations of a methyl hydrogen with the hydrogen of the HCO group, the permutation of the two carbon atoms, the permutations (12), (13) and (23), the permutation-inversion operations (123)* and (132)*, and the inversion through the mass center E* have been disregarded as “non feasible”, leading to geometries that the molecule cannot actually attain. The character table of the G

_{6}molecular symmetry group is reported in Table 4.

_{s}, with (23)* corresponding to the reflection through the symmetry plane. In fact, the energy levels of molecules with only small amplitude vibrational modes can be classified under the appropriate point group.

## 6. G_{6} and G_{12} Molecules with Internal Rotation

_{6}molecular symmetry group in Table 4. If the methyl group rotates from 0 to 2π radians about the C-C bond, with respect to the HCO plane, the molecule will assume six times a geometry of C

_{s}symmetry, where HCO is a symmetry plane. The H atom of the aldehyde group is staggered with respect to the hydrogens of the methyl group in three of the C

_{s}geometries, and eclipsed to one of them in the other three, see Figure 3.

_{s}point group geometry, is meaningless. Therefore molecular symmetry groups are fully adequate to classify the energy levels of molecules with free internal rotation, even if they contain no elements that could make the above mentioned correlations feasible. When the internal rotation is not free, the vibrational deformations of head and tail interact with each other in a manner which depends on the torsional angle τ, and the molecular motion behaves as if it had somehow “memory" of how it would be in the limit semi-rigid geometries. In this case it is instructive, and in general helpful, to establish a correlation of the vibrational symmetry species in the molecular symmetry group with those of point groups corresponding to characteristic symmetries attained at particular values of τ (e.g., the staggered and eclipsed conformations of acetaldehyde in Figure 3). Thus it may be convenient to extend the molecular symmetry group, if it contains no elements that could make the desired correlations feasible.

_{2}will become clear later, with reference to the expanded molecular group G

_{6}(EM). In the presence of interaction between top and frame, the coordinates ${\mathrm{S}}_{\mathrm{x}}^{\mathrm{a}}$(A

_{1}) and ${\mathrm{S}}_{\mathrm{y}}^{\mathrm{a}}$(A

_{2}), spanning A

_{1}and A

_{2}symmetry species under the molecular symmetry group G

_{6}, respectively, are no longer degenerate and can mix with deformations of the frame of the same symmetries. As their orientation is fixed in an axis system fixed at the HCO plane, A

_{1}coordinates are always symmetric (A′ under C

_{s}) and A

_{2}coordinates are always anti-symmetric (A″ under C

_{s}) when the molecule passes through the conformations shown in Figure 3, where the HCO plane becomes a symmetry plane. Other typical coordinates obtained by combining ${\mathrm{S}}_{\mathrm{x}\mathrm{a}}^{\mathrm{a}}$ and ${\mathrm{S}}_{\mathrm{y}\mathrm{a}}^{\mathrm{a}}$ with τ-dependent coefficients are the following:

_{1}(B

_{2}) coordinates are A′(A″) in the staggered conformations and A″(A′) in the eclipsed conformations.

_{1}and B

_{2}coordinates rotate by 180°, that is they change sign, if the torsional angle changes by an odd multiple of 120°, therefore they change sign under the operation

**E′**, consisting of a complete internal rotation by 360° (double valued coordinates). Therefore if B

_{1}and B

_{2}coordinates occur in a given treatment,

**E′**can no longer be considered equal to the identity E, and the G

_{6}molecular symmetry group has to be multiplied by (

**E**,

**E′**) yielding the extended molecular symmetry group G

_{6}(EM), see Table 5.

^{i(M}

^{a}

^{-}

^{ρ}

^{k}

^{ρ}

^{)}

^{τ}, where M

_{a}is the quantum number relative to the angular momentum of the top about the internal rotation axis, k

_{ρ}is the quantum number of the angular momentum of the whole molecule about the z

_{ρ}(the axis about which the whole molecule must rotate in order to cancel the angular momentum generated by the rotation of the top about the internal rotation axis z, see Figure 2), and ρ is the ratio of the moment of inertia of the top about z and that of the whole molecule about z

_{ρ}[15,16]. Thus from equations (16) and (19) one finds that, for given k

_{ρ}:

_{a}(E

_{2})

_{±}= M

_{a}(E)

_{±}∓ 1

_{a}(E

_{1})

_{±}= M

_{a}(E)

_{±}± 1/2

_{a}(E

_{2})

_{±}= M

_{a}(E

_{1})

_{±}∓ 3/2

_{a}changes with the adopted vibrational basis, because different amounts of the torsional angular momentum are incorporated in the vibrational function, for different vibrational basis sets.

_{12}, which applies to molecules like nitromethane or toluene, contains already four non-degenerate species: A

_{1}′ and A

_{2}′, symmetric under the permutation of the two oxygen atoms, and A

_{1}″ and A

_{2}″, anti-symmetric under the mentioned permutation. It can be shown that the A

_{1}′ and A

_{2}′ coordinates are single valued and are given by the equations (14) and (15), whereas the A

_{1}″ and A

_{2}″ coordinates are double valued and are given by the equations (17) and (18). However this molecular symmetry group does not need to be extended, because it can be shown that a torsion by 2π, applied to a nitromethane molecule distorted by a mode that would be double valued under G

_{6}(EM), would transform the molecule into a geometry which could be brought back to the initial one by the permutation of the two oxygen atoms in the frame [13]. Thus Table 3 also applies to G

_{12}molecules if the operation

**E′**is replaced by the permutation of the two oxygen atoms, and the species A

_{1}, A

_{2}, B

_{1}, B

_{2}, E

_{2}and E

_{1}are replaced by A

_{1}′, A

_{2}′, A

_{1}′, A

_{2}′, E′ and E′, respectively. Similar operations replacing

**E′**can be identified in toluene (permutations of all carbons and hydrogens in the right half of the benzene ring with those in the left half), and other G

_{12}molecules.

_{6}(EM), either single valued (A) or double valued (B), in an actual problem is a matter of convenience, and the preferred basis is not necessarily the one which is closer to the eigenstates. For instance, it has been found that the perpendicular vibrations of the methyl group of methanol are closer to the B-limit, see Ref. [12,13,17] and references therein, but it may be easier to work with A-vibrational bases in the numerical treatment of an actual problem. One point for this choice may be the handling of selection rules, which are simpler with vibrational coordinates with fixed orientation in the x,y,z-system.

## 7. G_{36}(EM) Molecules

_{s}after the group extension discussed hereafter). The two components in the a-unit, in circular coordinates, are called ${\mathrm{S}}_{\mathrm{a}\pm}^{\mathrm{a}}$, and generate overall and torsional angular momenta with the same sign. The corresponding circular coordinates ${\mathrm{S}}_{\mathrm{b}\pm}^{\mathrm{b}}$ in the b-unit generate overall and torsional angular momenta with opposite signs.

_{a}and z

_{b}), and with x and y bisecting the angles between x

_{a}and x

_{b}and y

_{a}and y

_{b}, respectively. The torsional angle τ is set to zero when the three systems of axes are coincident, therefore a torsion by τ implies that the two moieties a and b rotate about z by τ/2 and –τ/2, respectively. Often the angle γ=τ/2 is used to define the torsion. The interaction of a and b causes the mixing of their deformations, and the components of a G

_{s}-mode in a and b mix together generating double degenerate modes. As before, typical limit situations, with characteristic forms of degenerate modes, can be defined.

_{s}-mode on the x and y axes, we obtain vibrational coordinates with fixed orientations in the x,y,z-system [18,19]. Their circular components are

^{±i}

^{π}= -1 as τ changes by 2π, therefore they are double valued and require the extension of the molecular symmetry group to G

_{36}(EM) = G

_{36}×(E,E′), just as for G

_{6}molecules. Other typical coupled coordinates, single valued, are obtained by projecting the displacements in the a-unit of a G

_{s}-mode on an axis system which rotates about z at the angular velocity $\frac{3}{2}\dot{\mathsf{\tau}}$, and the displacements in the b-units on an axis system rotating at the angular velocity -$\frac{3}{2}\dot{\mathsf{\tau}}$. Thus their directions in a and b rotate by 90° and -90° if τ changes by 60°, showing that the directions of the deformations in a and b of modes described by these coordinates change from cis(trans) to trans(cis) at any conversion between eclipsed and staggered conformations.

_{1d}and E

_{2d}modes or of E

_{1s}and E

_{2s}modes can generate torsional angular momentum.

**Combinations and overtones in G**

_{36}(EM) molecules_{1d}and E

_{2d}basis vibrational functions for the degenerate fundamentals, owing to the properties of vibrational modes with fixed orientations in the chosen x,y,z molecular axis system. However, a complication arises in the presence of combinations or overtones with the excitation of an even number of degenerate vibrational quanta. In fact, from group theory it follows that the degenerate components of these combinations and overtones should be E

_{2s}or E

_{1s}[23], see also Ref. [24]. This is only a formal complication, because the total vibration-rotation-torsion functions are always single valued, and the problem only concerns the choice about their partition into the partial factors. We do not investigate this point in detail here, but we note that since the above partition is arbitrary, it is possible to refer to E

_{1d}or E

_{2d}basis functions also for overtones and combinations with an even number of quanta of double valued degenerate vibrational modes.

_{1s}or E

_{2s}function of circular type is a combination of E

_{1d}terms multiplied by factors containing $\mathrm{cos}\frac{3}{2}\mathsf{\tau}$ or $\mathrm{i}\mathrm{sin}\frac{3}{2}\mathsf{\tau}$ and E

_{2d}terms multiplied by factors containing $\mathrm{i}\mathrm{sin}\frac{3}{2}\mathsf{\tau}$ or $cos\frac{3}{2}\mathsf{\tau}$ [20], consistently with the symmetries of $\mathrm{cos}\frac{3}{2}\mathsf{\tau}$ (A

_{1d}) and $\mathrm{i}\mathrm{sin}\frac{3}{2}\mathsf{\tau}$ (A

_{3d}). The presence of τ-dependent coefficients in these transformations gives evidence to the fact that the torsional bases associated to the single valued and double valued vibrational bases are different.

## References

- Wilson, E.B.; Decius, J.C.; Cross, P.C. Molecular Vibrations; Mc-Graw-Hill: New York, NY, USA, 1955. [Google Scholar]
- Herzberg, G. Molecular Spectra and Molecular Structure II. Infrared and Raman Spectra of Polyatomic Molecules; D. Van Nostrand: Princeton, NJ, USA, 1964. [Google Scholar]
- Califano, S. Vibrational States; J. Wiley: London, UK, 1976. [Google Scholar]
- Bunker, P.R.; Jensen, P. Molecular Symmetry and Spectroscopy; NRC Research Press: Ottawa, Canada, 1998. [Google Scholar]
- Wigner, E.P. Group Theory; Academic Press: New York, NY, USA, 1959. [Google Scholar]
- Landau, L.; Lifchitz, E. Mécanique Quantique; Mir: Moscow, Russia, 1974. [Google Scholar]
- Henry, L.; Amat, G. Sur les coefficients d’intéraction entre la vibration et la rotation dans les molécules polyatomiques. II. Cahiers de Physique
**1960**, 118, 230–256. [Google Scholar] - Herzberg, G. Molecular Spectra and Molecular Structure III. Electronic Spectra of Polyatomic Molecules; D. Van Nostrand: Princeton, NJ, USA, 1966. [Google Scholar]
- di Lauro, C.; Lattanzi, F.; Graner, G. Phase conventions that render all matrix elements of the vibration-rotation Hamiltonian real. Mol. Phys.
**1990**, 6, 1285–1302. [Google Scholar] [CrossRef] - di Lauro, C.; Lattanzi, F. Phase Angles in the Matrix Elements of Molecular Spectroscopy, in Vibration-Rotational Spectroscopy and Molecular Dynamics; World Scientific: Singapore, 1997; Volume 9. [Google Scholar]
- di Lauro, C.; Lattanzi, F. Angles de phases dans les éléments matriciels en spectroscopie ro-vibronique. J. Physique
**1981**, 42, 693–703. [Google Scholar] [CrossRef] - Wang, X.; Perry, D.S. An internal coordinate model of coupling between the torsion and C-H vibrations in methanol. J. Chem. Phys.
**1998**, 109, 10795–10805. [Google Scholar] [CrossRef] - Lattanzi, F.; di Lauro, C. Vibrational symmetry classification and torsional tunneling splitting patterns in G
_{6}(EM), G_{12}and G_{36}(EM) molecules. Mol. Phys.**2005**, 103, 697–708. [Google Scholar] [CrossRef] - Hougen, J.T. Coordinates, hamiltonian, and symmetry operations for the small amplitude vibrational problem in methyl-top internal-rotor molecules like CH
_{3}CHO. J. Mol. Spectrosc.**1997**, 181, 287–296. [Google Scholar] [CrossRef] - Lin, C.C.; Swalen, J.D. Internal rotation and microwave spectroscopy. Rev. Mod. Phys.
**1959**, 31, 841–892. [Google Scholar] [CrossRef] - Hougen, J.T.; Kleiner, I.; Godefroid, M. Selection rules and intensity calculations for a C
_{s}asymmetric top molecule containing a methyl group internal rotor. J. Mol. Spectrosc.**1994**, 163, 559–586. [Google Scholar] [CrossRef] - Fehrensen, B.; Luckhaus, D.; Quack, M.; Willeke, M.; Rizzo, T.R. Ab initio calculations of mode selective tunneling dynamics in
^{12}CH_{3}OH and^{13}CH_{3}OH. J. Chem. Phys.**2003**, 119, 5534–5544. [Google Scholar] [CrossRef] - Bunker, P.R. Dimethylacetylene: An analysis of the theory required to interpret its vibrational spectrum. J. Chem. Phys.
**1967**, 47, 718–738. [Google Scholar] [CrossRef] - Bunker, P.R.; di Lauro, C. Dimethylacetylene: The theory required to interpret the infrared and Raman perpendicular bands. Chem. Phys.
**1995**, 190, 159–169. [Google Scholar] [CrossRef] - Lattanzi, F.; di Lauro, C. Head and tail deformations, torsional Coriolis coupling, and E
_{1d}-E_{2d}vibrational mixing in ethane-like molecules. J. Mol. Spectrosc.**1999**, 198, 304–314. [Google Scholar] [CrossRef] [PubMed] - di Lauro, C.; Lattanzi, F. A GF-matrix approach to the end-to-end coupling in ethane-like molecules. J. Mol. Spectrosc.
**1993**, 162, 375–396. [Google Scholar] [CrossRef] - di Lauro, C.; Lattanzi, F.; Avellino, R. Vibration-torsion dynamics of ethane-like molecules in degenerate vibrational states. J. Mol. Spectrosc.
**1994**, 167, 450–463. [Google Scholar] [CrossRef] - Hougen, J.T. Perturbations in the vibration-rotation-torsion energy levels of an ethane molecule exhibiting internal rotation splitting. J. Mol. Spectrosc.
**1980**, 82, 92–116. [Google Scholar] [CrossRef] - Lattanzi, F.; di Lauro, C. Symmetries and torsional splitting patterns in the overtone and combination vibrational states of molecules like ethane and methanol. J. Mol. Struct.
**2006**, 795, 105–113. [Google Scholar] [CrossRef]

**Figure 1.**Commutativity of energy and momentum operators for an isotropic two-dimensional harmonic oscillator.

**Figure 3.**The H-C-O and methyl groups of acetaldehyde, projected on a plane normal to the internal rotation axis C-C, in six different torsional C

_{s}conformations. The orientations of the components normal to the C-C axis of vibrational modes of A and B symmetries are shown at the bottom of the figure.

**Figure 4.**Top-fixed axes (x

_{a},y

_{a}) and (x′,y′)-axes, rotating at angular velocities $\dot{\mathsf{\tau}}$ and 3$\dot{\mathsf{\tau}}$/2 with respect to the molecular axis system (x,y) fixed to the frame. The directions of vibrational coordinates of species A

_{1}, A

_{2}, B

_{1}and B

_{2}under G

_{6}(EM) are also shown.

**Table 1.**Character table and symmetry species of the point group C

_{3}. Although E

_{+}and E

_{-}are uni-dimensional representations under C

_{3}, the vibrational modes of these symmetries occur in pairs of the same energy, behaving as components of a two-dimensional corepresentation E.

C_{3} | E | ${\mathit{C}}_{3}^{1}$ | ${\mathit{C}}_{3}^{2}$ | ||
---|---|---|---|---|---|

A | 1 | 1 | 1 | T_{z} | R_{z} |

1 | exp(2πi/3) | exp(-2πi/3) | T_{x}+iT_{y} | R_{x}+iR_{y} | |

1 | xp(-2πi/3) | exp(2πi/3) | T_{x}-iT_{y} | R_{x}-iR_{y} |

**Table 2.**Characters and symmetry species of the extended group C

_{3}×(E, Θ), with Θ

^{2}

**=**E. The behavior of vector components, also multiplied by the imaginary unit i, is shown in the upper part. The lower part applies when x and y components are combined in the form x±iy. See text.

C_{3} ×(E, Θ) | E | ${\mathit{C}}_{3}^{1}$ | ${\mathit{C}}_{3}^{2}$ | Θ | ${\mathit{C}}_{3}^{1}$ | ${\mathit{C}}_{3}^{2}$ | Bases |
---|---|---|---|---|---|---|---|

A_{1} | 1 | 1 | 1 | 1 | 1 | 1 | T_{z}, R_{z} or iP_{z}, iJ_{z} |

A_{2} | 1 | 1 | 1 | -1 | -1 | -1 | P_{z}, J_{z} or iT_{z}, iR_{z} |

E_{1} | 2 | -1 | -1 | 2 | -1 | -1 | (T_{x}, T_{y}), (R_{x}, R_{y}) or (iP_{x,}, iP_{y}), (iJ_{x,}, iJ_{y} ) |

E_{2} | 2 | -1 | -1 | -2 | 1 | 1 | (P_{x,}, P_{y}), (J_{x,}, J_{y} ) or (iT_{x}, iT_{y}), (iR_{x}, iR_{y}) |

A_{1} | 1 | 1 | 1 | 1 | 1 | 1 | T_{z}, R_{z} or iP_{z}, iJ_{z} |

A_{2} | 1 | 1 | 1 | -1 | -1 | -1 | T_{z}, R_{z} or iP_{z}, iJ_{z} |

E | 2 | -1 | -1 | 0 | 0 | 0 | T_{x}±iT_{y}; R_{x}±iR_{y}; P_{x}±iP_{y}; J_{x}±iJ_{y} |

**Table 3.**Character table and symmetry species of the group (E,${\mathcal{R}}^{\prime}$)×(E,Θ ), with Θ

^{2}

**=**E. See text.

(E,${\mathcal{R}}^{\mathbf{\prime}}$)×(E,Θ ) | E | ${\mathcal{R}}^{\mathbf{\prime}}$ | Θ | Θ${\mathcal{R}}^{\mathbf{\prime}}$ |
---|---|---|---|---|

${\mathsf{\Gamma}}_{1}^{1}$ | 1 | 1 | 1 | 1 |

${\mathsf{\Gamma}}_{-1}^{1}$ | 1 | -1 | 1 | -1 |

${\mathsf{\Gamma}}_{1}^{-1}$ | 1 | 1 | -1 | -1 |

${\mathsf{\Gamma}}_{-1}^{-1}$ | 1 | -1 | -1 | 1 |

(123) | (12)* | |||

E | (132) | (13)* | ||

G_{6} | (23)* | |||

A_{1} | 1 | 1 | 1 | T_{z,} T_{x}, J_{y} |

A_{2} | 1 | 1 | -1 | T_{y}, J_{x}, J_{z}, J_{tors} |

E | 2 | -1 | 0 |

(123) | (12)* | E′(123) | E′(12)* | ||||

E | (132) | (13)* | E′ | E′(132) | E′(13)* | ||

G_{6}(EM) | (23)* | E′(23)* | |||||

A_{1} | 1 | 1 | 1 | 1 | 1 | 1 | T_{z,} T_{x}, J_{y} |

A_{2} | 1 | 1 | -1 | 1 | 1 | -1 | T_{y}, J_{x}, J_{z}, J_{tors} |

E_{2} | 2 | -1 | 0 | 2 | -1 | 0 | |

B_{1} | 1 | 1 | 1 | -1 | -1 | -1 | |

B_{2} | 1 | 1 | -1 | -1 | -1 | 1 | |

E_{1} | 2 | -1 | 0 | -2 | 1 | 0 |

© 2010 by the authors; licensee Molecular Diversity Preservation International, Basel, Switzerland. This article is an open-access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

## Share and Cite

**MDPI and ACS Style**

Lattanzi, F.; Di Lauro, C.
On the Physical Reasons for the Extension of Symmetry Groups in Molecular Spectroscopy. *Symmetry* **2010**, *2*, 213-229.
https://doi.org/10.3390/sym2010213

**AMA Style**

Lattanzi F, Di Lauro C.
On the Physical Reasons for the Extension of Symmetry Groups in Molecular Spectroscopy. *Symmetry*. 2010; 2(1):213-229.
https://doi.org/10.3390/sym2010213

**Chicago/Turabian Style**

Lattanzi, Franca, and Carlo Di Lauro.
2010. "On the Physical Reasons for the Extension of Symmetry Groups in Molecular Spectroscopy" *Symmetry* 2, no. 1: 213-229.
https://doi.org/10.3390/sym2010213