# Artificial Symmetries for Calculating Vibrational Energies of Linear Molecules

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## Abstract

**:**

**C**${}_{2\mathrm{v}}$(M) molecular symmetry group. This, in turn, necessitates purpose-built codes that specifically deal with linear molecules. In the present work, we describe an alternative scheme and introduce an (artificial) group that ensures that the condition ${\ell}_{3}=k$ is automatically applied solely through symmetry group algebra. The advantage of such an approach is that the application of symmetry group algebra in ro-vibrational calculations is ubiquitous, and so this method can be used to enable ro-vibrational calculations of linear molecules in polyatomic codes with fairly minimal modifications. To this end, we construct a—formally infinite—artificial molecular symmetry group

**D**${}_{\infty \mathrm{h}}$(AEM), which consists of one-dimensional (non-degenerate) irreducible representations and use it to classify vibrational and rotational basis functions according to ℓ and k. This extension to non-rigorous, artificial symmetry groups is based on cyclic groups of prime-order. Opposite to the usual scenario, where the form of symmetry adapted basis sets is dictated by the symmetry group the molecule belongs to, here the symmetry group

**D**${}_{\infty \mathrm{h}}$(AEM) is built to satisfy properties for the convenience of the basis set construction and matrix elements calculations. We believe that the idea of purpose-built artificial symmetry groups can be useful in other applications.

## 1. Introduction

^{−1}[1]), a molecular kinetic energy operator (KEO) in terms of the 3 vibrational (e.g., two stretching and one bending) and three rotational coordinates (usually Euler angles) [2] is constructed. The three vibrational modes correspond to the standard bent-molecule methodology that is based on the $3N-6$ vibrational modes. We traditionally start from selecting a zero-order Hamiltonian ${\widehat{H}}_{\mathrm{bent}}^{\left(0\right)}$=${\widehat{H}}_{\mathrm{vib},\mathrm{bent}}^{\left(0\right)}$+${\widehat{H}}_{\mathrm{rot},\mathrm{bent}}^{\left(0\right)}$ appropriate for a simplified molecule with the rotation-vibration interaction neglected. ${\widehat{H}}_{\mathrm{vib},\mathrm{bent}}^{\left(0\right)}$ is the Hamiltonian for the vibrational motion, obtained by entirely neglecting rotation in the complete Hamiltonian, while ${\widehat{H}}_{\mathrm{rot},\mathrm{bent}}^{\left(0\right)}$ is the Hamiltonian for the molecule rotating rigidly in its equilibrium configuration. The eigenfunctions of ${\widehat{H}}_{\mathrm{bent}}^{\left(0\right)}$ are products ${\psi}_{\mathrm{vib},\mathrm{bent}}^{\left(0\right)}$×${\psi}_{\mathrm{rot},\mathrm{bent}}^{\left(0\right)}$ of an eigenfunction ${\psi}_{\mathrm{vib},\mathrm{bent}}^{\left(0\right)}$ of ${\widehat{H}}_{\mathrm{vib},\mathrm{bent}}^{\left(0\right)}$ and an eigenfunction ${\psi}_{\mathrm{rot},\mathrm{bent}}^{\left(0\right)}$ of ${\widehat{H}}_{\mathrm{rot},\mathrm{bent}}^{\left(0\right)}$, and these products are used as basis functions for solving the Schrödinger problem for the complete rotating and vibrating molecule, which is done either by perturbation theory or in a variational approach.

**C**${}_{\infty \mathrm{v}}$ or

**D**${}_{\infty \mathrm{h}}$, which are not only an appropriate choice for linear molecules, but also give access to powerful symmetry group (or representation-theory) methods. As far as the ro-vibrational calculations are concerned, the more appropriate symmetry groups for description of the zero-order solutions of ${\widehat{H}}_{\mathrm{vib},\mathrm{lin}}^{\left(0\right)}$ and ${\widehat{H}}_{\mathrm{rot},\mathrm{lin}}^{\left(0\right)}$, (i.e., for the classification of the physically correct nuclear-rotation-vibrational states) are the finite groups

**C**${}_{\infty \mathrm{v}}$(M) or

**D**${}_{\infty \mathrm{h}}$(M). Ref. [2] describes the extensions to the infinitely large extended molecular groups

**C**${}_{\infty \mathrm{v}}$ (EM) and

**D**${}_{\infty \mathrm{h}}$ (EM) [9] isomorphic to the point groups

**C**${}_{\infty \mathrm{v}}$ and

**D**${}_{\infty \mathrm{h}}$ [2], respectively, which we discuss further below. The symmetry group methods have proved to be important when solving the corresponding nuclear motion problems, see e.g., Chubb et al. [10].

**C**${}_{\infty \mathrm{v}}$ or

**D**${}_{\infty \mathrm{h}}$. Indeed, for a centrosymmetric molecule, such as CO${}_{2}$ they transform as

**D**${}_{\infty \mathrm{h}}$(M) (isomorphic to

**C**${}_{2\mathrm{v}}$) and only span four irreducible representations (irreps), ${\Sigma}_{g}^{+}$, ${\Sigma}_{g}^{-}$, ${\Sigma}_{u}^{+}$, and ${\Sigma}_{u}^{-}$. For a non-centrosymmetric linear molecule, such as HCN, they transform as

**C**${}_{\infty \mathrm{v}}$(M) (isomorphic to

**C**${}_{\mathrm{s}}$) and they span two irreducible representations, ${\Sigma}^{+}$ and ${\Sigma}^{-}$. The vibrational basis set is orthogonal in ℓ due to the constraint $k=\ell $.

**D**${}_{\infty \mathrm{h}}$ (EM) or

**C**${}_{\infty \mathrm{v}}$ (EM) [9] with their much more detailed options for classification of the rotational and vibrational wavefunctions. In these groups, the vibrational and rotational basis functions span irreducible representations of

**C**${}_{\infty \mathrm{v}}$ or

**D**${}_{\infty \mathrm{h}}$ and transform as $\Sigma $, $\Pi $, $\Delta $, $\Phi $, ... for ℓ= 0, 1, 2, 3, ... and k= 0, 1, 2, 3, respectively. Moreover, their intrinsic properties offer efficient tools of associated irreducible representations of molecular symmetry groups when constructing the Hamiltonian matrix elements or building symmetry adapted ro-vibrational basis functions, which are not available for the bent molecule approaches.

**D**${}_{\infty \mathrm{h}}$(M) to an infinite group with the group structure that is similar to

**D**${}_{\infty \mathrm{h}}$ (or, analogously, extend

**C**${}_{\infty \mathrm{v}}$(M) to an infinite group with a group structure that is similar to

**C**${}_{\infty \mathrm{v}}$ for a non-centro-symmetric linear molecule). The corresponding irreducible representations $\Pi $, $\Delta $, $\Phi $, ... (ℓ > 0, or k > 0, respectively) are doubly degenerate and, therefore, are not directly suitable for this purpose. Therefore, the aim of these extensions is to obtain a more flexible classification of the rotational and vibrational wavefunctions than that provided by

**D**${}_{\infty \mathrm{h}}$(M) and

**C**${}_{\infty \mathrm{v}}$(M), allowing for non-degenerate irreps, as required for bent molecules. The corresponding extended symmetries do not necessarily have to have any physical meaning or to be connected with energy conservation, as in the case of traditional "true” molecular symmetry groups. Instead, a group is constructed in such a way, so that the irreps have properties that are formalised from the outset, rather than the behaviour of the irreps being a result of the symmetry of the molecule itself. Thus, any "artificial” group, constructed in this way, which satisfies the standard group theorems and fulfills the purpose of being directly associated with the $\ell =k$ classification will do. The approach described here, then, is, in effect, a reversal of the standard procedure where the properties of the basis set are based on the symmetry group of the molecule. Here, we design the symmetry group to fulfill the required structure of the ro-vibrational basis functions and the matrix elements of the Hamiltonian.

**D**${}_{\infty \mathrm{h}}$(M) [

**C**${}_{\infty \mathrm{v}}$(M)] with some artificial irreps with the four [two] ‘physical’ irreps ${\Sigma}_{g}^{+}$, ${\Sigma}_{g}^{-}$, ${\Sigma}_{u}^{+}$ and ${\Sigma}_{u}^{-}$ [${\Sigma}^{+}$ and ${\Sigma}^{-}$] retained. The group is referred to as

**D**${}_{\infty \mathrm{h}}$(AEM) [

**C**${}_{\infty \mathrm{v}}$(AEM)], where AEM stands for ‘artificial extended molecular’ (group). We will also introduce a finite analogy

**D**${}_{n\mathrm{h}}$(AEM) [

**C**${}_{n\mathrm{v}}$(AEM)], where n is depends on the value of ${\ell}_{\mathrm{max}}$, and use it to classify the vibrational and rotational basis functions of the bent ($3N-6$ approach) similar to the classification used for linear,

**D**${}_{\infty \mathrm{h}}$[

**C**${}_{\infty \mathrm{v}}$] (EM)-based $3N-5$ systems.

**D**${}_{n\mathrm{h}}$(AEM) [

**C**${}_{n\mathrm{v}}$(AEM)] has been implemented in the variational program TROVE (Theoretical ROVibrational Energies) [20,21], with the aim to facilitate the computation of vibrational energies and wavefunctions for linear molecules. TROVE is a general, efficient computer program for simulating, by variational methods, hot rotation-vibration spectra of small- to medium-sized polyatomic molecules of arbitrary structure. It has been applied to a large number of polyatomic species [21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41] with various symmetry types [2], such as

**C**${}_{2\mathrm{v}}$,

**C**${}_{2\mathrm{h}}$,

**C**${}_{3\mathrm{v}}$(M),

**D**${}_{n\mathrm{h}}$(M),

**T**${}_{\mathrm{d}}$(M),

**G**${}_{36}$ etc. TROVE is one of the main tools of the ExoMol project [42,43]. Updates of TROVE have been recently reported in Refs. [19,43,44], with the most recent extension [19] being the implementation in TROVE of the $3N-6$ methodology with a singularity resolution based on the inclusion of interaction between the bending motions and the rotation about the a axis. Additionally, the adaptation of linear-molecule symmetry that was described in Ref. [10] was implemented. The theory and programming work reported in Ref. [19] and in the present work have already been employed for generating hot molecular line lists for SiO${}_{2}$ [45] and CO${}_{2}$ [46] as well as to produce a quadrupole spectrum of H${}_{2}$O [47]. TROVE uses an automatic approach for constructing a symmetry-adapted basis set to be used in setting up a matrix representation of the molecular rotation-vibration Hamiltonian [48], in the present case for

**D**${}_{n\mathrm{h}}$(AEM) [

**C**${}_{n\mathrm{v}}$(AEM)].

#### 1.1. Symmetry Properties of the Linear Molecule ($3N-5$) Basis Functions: **D**${}_{\infty \mathrm{h}}$(EM)

**D**${}_{\infty \mathrm{h}}$ (EM) [9] (see Table 1), with ${\ell}_{j}$ identifying the corresponding irrep $\Sigma $, $\Pi $, $\Delta $, …, for ${\ell}_{j}=0$, 1, 2, …, respectively, and ${\ell}_{j}$ spanning ${v}_{j},{v}_{j}-2,{v}_{j}-4,\dots ,-{v}_{j}$. When combined with the rotational, rigid-rotor, basis function $\mid J,k,m\rangle $, the constraint $k=\ell ={\sum}_{j}{\ell}_{j}$ is used, where ℓ is the total vibrational angular momentum (in units of ℏ) of the product-type vibrational basis. Here, J is the total angular momentum, $k\phantom{\rule{0.166667em}{0ex}}\hslash $ is the projection of the angular momentum on the molecular z (= a) axis and $m\phantom{\rule{0.166667em}{0ex}}\hslash $ is the projection on the laboratory-fixed Z axis. The rotational basis functions $\mid J,k,m\rangle $ transform according to the operations of

**D**${}_{\infty \mathrm{h}}$ (EM), with the similar mapping between the irreps and the rotational quantum number k, as found for ℓ: $\Sigma $ ($k=0$), $\Pi $ ($k=1$), $\Delta $ ($k=2$), etc. The correlation between the

**D**${}_{\infty \mathrm{h}}$ (EM) [9] irreps and the quantum numbers ℓ, k is very useful, as it enables the use of powerful group symmetry tools for classifying and even constructing the ro-vibrational (bending) basis functions, with the vibrational basis function in Equation (1) now given by:

**D**${}_{\infty \mathrm{h}}$ (EM), and they are fully defined by the corresponding values of $\ell \equiv {\ell}_{3}$ and k shown in Table 1. The ℓ-dependent vibrational basis functions span the g-type and u-type representations, while the k-dependent rotational functions can only be of g (i.e., do not change upon the ${\sigma}_{h}$ transformation). These functions are linked via the condition $\ell =k$. The ro-vibrational eigenfunctions of a linear XY${}_{2}$ type molecule can only transform as one of the four irreps of

**D**${}_{\infty \mathrm{h}}$(M) that are listed in Table 2 (in fact, due to the zero-nuclear spins of C and O, the ro-vibrational wavefunctions of CO${}_{2}$ can only span two representations, ${\Sigma}_{g}^{+}$ and ${\Sigma}_{u}^{-}$) . Therefore, physically meaningful combinations of $\mid {v}_{1},{v}_{2},{v}_{3}^{{\ell}_{3}}\rangle $ and rotational $\mid J,k,m\rangle $ from their direct products can only span these four irreps. These properties can be efficiently explored by the efficient symmetry group algebra via projection and symmetry reduction techniques, as was shown, e.g., in Chubb et al. [10].

#### 1.2. Symmetry Properties of the Bent $3N-6$ Basis Functions

**D**${}_{\infty \mathrm{h}}$(M) with four irreps (see Table 2) in contrast to the irreps of the infinite group

**D**${}_{\infty \mathrm{h}}$ (EM). Moreover, the bending wavefunctions ${\phi}_{{v}_{3},k}\left(\rho \right)$ can only be of ${\Sigma}_{g}^{+}$ symmetry of

**D**${}_{\infty \mathrm{h}}$(M). Yet, the property of orthogonality between basis functions with different ℓ-values is very similar to the orthogonality that arises from different irreps being generated by the wavefunctions in Equation (2) in the $3N-5$ treatment.

**D**${}_{\infty \mathrm{h}}$(M) symmetry description of the basis functions in Equation (3), naturally arising for the $3N-6$ case, can be extended in such a manner that it becomes similar to the much more detailed description, in terms of

**D**${}_{\infty \mathrm{h}}$ (EM), obtained for the $3N-5$ case. In particular, we would like a description in which basis functions with different ℓ-values span different irreps so that we can take advantage of the specific properties of the linear-molecule basis set functions. For example, the vibrational Hamiltonian matrix, constructed in an extended-symmetry-adapted basis set, will automatically be block diagonal in ℓ and thus its construction and diagonalization can be optimally cost-effective. There exists no “true” molecular symmetry group, based on permutation-inversion symmetry, with ℓ-dependent irreps. Thus, our theory does not automatically yield such irreps. However, in the following, we show that is possible to introduce an “artificial” symmetry group that is based on the property of cyclic groups of prime-order that produces the desired ℓ-dependent irreps. By analogy to the extended group

**D**${}_{\infty \mathrm{h}}$ (EM), the new artificially extended group will be referenced as

**D**${}_{\infty \mathrm{h}}$(AEM), where AEM stands for ‘artificial extended molecular’.

#### 1.3. “Artificial” Molecular Group Symmetry **D**${}_{\infty \mathrm{h}}$(AEM) for Centrosymmetric Triatomic Molecules

**D**${}_{n\mathrm{h}}$(AEM), as follows

**Z**${}_{2}$ is the cyclic group of order 2 and, therefore, it consists of the set $\{0,1\}$ with addition modulo 2. The integer n depends on the value of ${\ell}_{\mathrm{max}}$. Taking the limit $n\to \infty $ defines the infinite

**D**${}_{\infty \mathrm{h}}$(AEM) group. Note that this is a countable infinity, as opposed to the group

**D**${}_{\infty \mathrm{h}}$ (EM). The vibrational $\mid {v}_{1},{v}_{2},{v}_{3}^{\ell}\rangle $ and rotational $\mid J,k,m\rangle $ basis set functions of differing ℓ (or k) values are assigned to different irreps of this group.

**D**${}_{n\mathrm{h}}$(AEM) must fulfil certain conditions necessary for our purposes. First, all irreps should be one-dimensional and, for simplicity, real. The former condition is so that each irrep of

**D**${}_{n\mathrm{h}}$(AEM) is correlated with one irrep of

**D**${}_{\infty \mathrm{h}}$(M). The irreps of

**D**${}_{n\mathrm{h}}$(AEM) will be labelled as ${\Gamma}_{3}=$

**C**${}_{2\mathrm{v}}$ irreps with an extra superscript (see Table 2), e.g., ${A}_{1}^{4}$. The bending function ${\psi}_{{\nu}_{3},\ell}\left(\rho \right)$ or rotational function $\mid J,k,m\rangle $, if they transform as irrep $\Gamma $ in

**C**${}_{2\mathrm{v}}$, would be assigned to ${\Gamma}^{\ell}$ or ${\Gamma}^{k}$, respectively. For example, a vibrational function with $\ell =4$ and transforming as ${A}_{1}$ in

**C**${}_{2\mathrm{v}}$ would be assigned the symmetry ${A}_{1}^{4}$ in the

**D**${}_{n\mathrm{h}}$(AEM).

**D**${}_{\infty \mathrm{h}}$(AEM) and

**D**${}_{n\mathrm{h}}$(AEM), we select four elements and match each with a

**C**${}_{2\mathrm{v}}$ element. Subsequently, the characters of those elements for each 0-superscripted irrep should be the same as the corresponding

**C**${}_{2\mathrm{v}}$ irrep. When combining a bending function that transforms as ${\Gamma}_{1}^{\ell}$ with a rotational function which transforms as ${\Gamma}_{2}^{k}$, their product should transform as ${\Gamma}_{1}^{\ell}\times {\Gamma}_{2}^{k}={({\Gamma}_{1}\times {\Gamma}_{2})}^{m}$ for some $m\ne 0$ if $\ell \ne k$. If $\ell =k$, then they should transform as ${\Gamma}_{1}^{\ell}\times {\Gamma}_{2}^{\ell}={({\Gamma}_{1}\times {\Gamma}_{2})}^{0}$. For example, ${A}_{2}^{4}\times {B}_{1}^{4}$ should be ${B}_{2}^{0}$.

**D**${}_{4\mathrm{h}}$(AEM) in Table 3 for the characters and Table 4 for the multiplication table of the irreps.

**D**${}_{n\mathrm{h}}$(AEM) of the infinite group

**D**${}_{\infty \mathrm{h}}$, see Chubb et al. [10]. Accordingly, the practical procedure introduced and developed here involves

**D**${}_{n\mathrm{h}}$(AEM), rather than

**D**${}_{\infty \mathrm{h}}$(AEM). Here n is some integer value defined to cover the highest excitations of $k=\ell $ in the bending ${\phi}_{{v}_{3},\ell}\left(\rho \right)$ and rotational $\mid J,k,m\rangle $ functions.

**D**${}_{n\mathrm{h}}$(AEM) has the required properties, and it is effectively the only group that does, first note that, since the irreps are one-dimensional, the group is abelian, and, thus, by the fundamental theorem of finite abelian groups [52], can be expressed as a direct product of cyclic groups of prime order, i.e.,

**D**${}_{n\mathrm{h}}$(AEM) has ${2}^{n}$ representations, hence

**Z**${}_{2}$. It has character table shown in Table 5.

**Z**${}_{2}$ is written

**C**${}_{2\mathrm{v}}$ =

**Z**${}_{2}$⊗

**Z**${}_{2}$. Hence,

**D**${}_{2\mathrm{h}}$(AEM) works for ${k}_{\mathrm{max}}=0$, where we label the 0-superscripted irreps of

**D**${}_{2\mathrm{h}}$(AEM) in the same order as Table 3. We call the

**D**${}_{2\mathrm{h}}$(AEM) character table matrix G, i.e.,

**D**${}_{2\mathrm{h}}$(AEM), the result of ${\Gamma}_{1}\times {\Gamma}_{2}$ is another irrep in

**D**${}_{2\mathrm{h}}$(AEM). If we now consider the case ${\ell}_{\mathrm{max}}=4$, we have the character matrix

**D**${}_{4\mathrm{h}}$(AEM), if they have the same superscript, ${\Gamma}_{1}\times {\Gamma}_{2}$ will cancel the −s of the Gs and the result will be an irrep of the 0-superscripted block. Moreover, this will be the same irrep one would obtain had one used the

**C**${}_{2\mathrm{v}}$ group. If the irreps do not have the same superscript, then we will have $-G\times G=-G$ type multiplication whose result will be an irrep not in the 0-superscripted block. We can see that this easily generalises to any ${\ell}_{\mathrm{max}}$. This concludes the definition of

**D**${}_{n\mathrm{h}}$(AEM). Table 6 describes how the characters for

**D**${}_{n\mathrm{h}}$(AEM) are defined for an arbitrary n, although, in practice, one would use the outer product formulation to build the character table.

**G**(EM), which are used in the ro-vibrational problems of non-rigid molecules, such as hydrogen peroxide (H

_{2}O

_{2}) [34] and ethane (C

_{2}H

_{6}) [53].

**D**${}_{n\mathrm{h}}$(AEM) is given by

**G**:

**G**${}_{36}$(EM), the MS group of ethane C${}_{2}$H${}_{6}$, can transform as odd or even under ${E}^{\prime}$, labelled as s and d, respectively. Likewise, the full rotation-vibrational wavefunction can only be of s type and the artificial d types must be eliminated [53].

## 2. Example of **D**${}_{n\mathrm{h}}$(AEM) in Ro-Vibrational Calculations of CO${}_{2}$ Energies with TROVE

**D**${}_{5\mathrm{h}}$(AEM), but our basis functions can be only one of (for ${k}_{\mathrm{max}}=4$) $4(4+1)=20$ irreps, and the remainder is not utilised. However, they are still necessary when combining basis functions, as was illustrated in Table 4.

**D**${}_{5\mathrm{h}}$(AEM)) eigenfunctions for both the bending and stretching Hamiltonians, the standard TROVE symmetrisation procedure is applied, as described in [48]. The resulting irreps are the 0-superscripted ones of

**D**${}_{5\mathrm{h}}$(AEM) and they correspond to the

**C**${}_{2\mathrm{v}}$(M) ones. To correctly symmetrise this way, the operation ${O}^{m}$, using the notation of Table 3, has the same behaviour on the coordinates as operation ${O}^{0}$.

**D**${}_{5\mathrm{h}}$(AEM). The irreps that are generated by the bending basis functions can only be ${A}_{1}$, and they are assigned to ${A}_{1}^{\ell}$ for a given ℓ. In this case, $\ell \le 4$. The stretching functions are all assigned as a 0-superscripted irrep. This ensures that, when they are combined with the bending function, the “base” irrep obtained (i.e., the letter and subscript of the irrep label) would have been the same if we had used the

**C**${}_{2v}$ group. The only 0-superscripted irrep is the one where $\ell =k$, as expected, and only this would be retained from this set.

## 3. Conclusions

**D**${}_{\infty \mathrm{h}}$(AEM) introduced here was designed to take advantage of the detailed symmetry information that is available in the linear-molecule $3N-5$ treatment, even when the bent-molecule $3N-6$ approach is used. The construction of

**D**${}_{\infty \mathrm{h}}$(AEM) is based on the cyclic group of order 2.

**D**${}_{\infty \mathrm{h}}$(AEM) is shown to have all the properties required: basis functions with different ℓ or k functions (vibrational or rotational, respectively) generate different irreducible representations of

**D**${}_{\infty \mathrm{h}}$(AEM). This is a natural property for the linear molecule $3N-5$ treatment and we can now also employ it for quasi-linear or bent molecules treated in the $3N-6$ approach, but using non-degenerate irreps that are required for bent systems.

**C**${}_{\infty \mathrm{v}}$ (EMS) and

**D**${}_{\infty \mathrm{h}}$(EMS) of Chapter 17 of Ref. [2]. As discussed above, these groups are obtained by, among other changes, extending the MS groups

**C**${}_{\mathrm{s}}$(M) and

**C**${}_{2\mathrm{v}}$(M) [2] by artificial operations, such as ${C}_{\infty}$${}^{\u03f5}$, a rotation of an arbitrary angle $\u03f5$ about the a axis. In the resulting EMS groups

**C**${}_{\infty \mathrm{v}}$(EM) and

**D**${}_{\infty \mathrm{h}}$(EM) the vibrational basis function $|{v}_{1},{v}_{2},\dots ,{v}_{N-1};{v}_{N}^{{\ell}_{N}},{v}_{N+1}^{{\ell}_{N+1}},{v}_{N+2}^{{\ell}_{N+2}},\dots ,{v}_{2N-3}^{{\ell}_{2N-3}}\rangle $ generates a $\Sigma $, $\Pi $, $\Delta $, $\Phi $ .... Similarly, $|J,k,m\rangle $ generates a $\Sigma $, $\Pi $, $\Delta $, $\Phi $ … symmetry for k= 0, 1, 2, 3, … See Tables 17-1 and 17-2 of Ref. [2]. The introduction of

**C**${}_{\infty \mathrm{v}}$(EM) and

**D**${}_{\infty \mathrm{h}}$ (EM) was, in principle, motivated by the same ideas that inspired the present work. These groups are more closely related to actual symmetries than the results presented here, since they utilise the near-symmetry of arbitrary rotations about the a axis; the ideas of the present work are more abstract.

**C**${}_{\infty \mathrm{v}}$(EM) and

**D**${}_{\infty \mathrm{h}}$(EM). Moreover, although more physically relevant, the

**C**${}_{\infty \mathrm{v}}$(EM) and

**D**${}_{\infty \mathrm{h}}$(EM) have the disadvantage that they involve the degenerate irreps $\Pi $, $\Delta $, $\Phi $ .... The double degeneracy is a simple consequence of the fact that any rotation about the a axis exists in two equivalent forms, which we can call clockwise and anti-clockwise, respectively. The 2 × 2 transformation matrices of these irreps are not uniquely determined and more complicated to apply than the characters of non-degenerate representations.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. TROVE Calculation Details Used in the CO 2 Example

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**Table 1.**The irreducible representation $\Gamma $ of

**D**${}_{\infty \mathrm{h}}$ (EM) spanned by the rotational $\mid J,k,m\rangle $ or vibrational $\mid {v}_{3},{\ell}_{3}\rangle $ wavefunction of a linear molecule in the absence of external electric and magnetic fields. The irrep depends on k or ℓ.

$\mathit{k}/\mathit{\ell}$ | ${\mathbf{\Gamma}}_{\mathbf{vib}}$ | ${\mathbf{\Gamma}}_{\mathbf{rot}}$ | |
---|---|---|---|

0 | ${\Sigma}_{\mathrm{g}}{}^{+}$ | ${\Sigma}_{\mathrm{g}}{}^{+}$ | (J even) |

0 | ${\Sigma}_{\mathrm{g}}{}^{-}$ | (J odd) | |

±1 | ${\Pi}_{\mathrm{u}}$ | ${\Pi}_{\mathrm{g}}$ | |

±2 | ${\Delta}_{\mathrm{g}}$ | ${\Delta}_{\mathrm{g}}$ | |

±3 | ${\Phi}_{\mathrm{u}}$ | ${\Phi}_{\mathrm{g}}$ | |

⋮ | ⋮ |

**Table 2.**Character table for the molecular symmetry (MS) group

**D**${}_{\infty \mathrm{h}}$(M) ${}^{a}$. The last four columns show the group operations, with two labels for each operation.

$\mathbf{\Gamma}$ | $\mathit{E}$ | $\left(\mathit{p}\right)$ | ${\left(\mathit{p}\right)}^{\ast}$ | ${\mathit{E}}^{\ast}$ |
---|---|---|---|---|

$\mathit{E}$ | ${\mathit{C}}_{\mathbf{2}}\left(\mathit{z}\right)$ | $\sigma \left(\mathit{xz}\right)$ | ${\sigma}_{\mathit{h}}\left(\mathit{xy}\right)$ | |

${\Sigma}_{\mathrm{g}}{}^{+}$ | 1 | 1 | 1 | 1 |

${\Sigma}_{\mathrm{g}}{}^{-}$ | 1 | $-1$ | 1 | $-1$ |

${\Sigma}_{\mathrm{u}}{}^{-}$ | 1 | 1 | $-1$ | $-1$ |

${\Sigma}_{\mathrm{u}}{}^{+}$ | 1 | $-1$ | $-1$ | 1 |

^{a}g and u stand for the German gerade (even) and ungerade (odd), related to the permutation-inversion operation (p)

^{∗}.

**Table 3.**Character table for the

**D**${}_{4\mathrm{h}}$(AEM) group. The operations of the group that are 0-superscripted correspond to the

**C**${}_{2\mathrm{v}}$ group. Note that the characters of the 0-superscripted irreps for these operations are the same as those of the corresponding irreps for the

**C**${}_{2\mathrm{v}}$ group.

D${}_{4\mathbf{h}}$(AEM) | ${\mathit{E}}^{0}$ | ${\mathit{C}}_{2}^{0}$ | ${\mathit{\sigma}}^{0}$ | ${\mathit{\sigma}}_{\mathit{v}}^{0}$ | ${\mathit{E}}^{1}$ | ${\mathit{C}}_{2}^{1}$ | ${\mathit{\sigma}}^{1}$ | ${\mathit{\sigma}}_{\mathit{v}}^{1}$ | ${\mathit{E}}^{2}$ | ${\mathit{C}}_{2}^{2}$ | ${\mathit{\sigma}}^{2}$ | ${\mathit{\sigma}}_{\mathit{v}}^{2}$ | ${\mathit{E}}^{3}$ | ${\mathit{C}}_{2}^{3}$ | ${\mathit{\sigma}}^{3}$ | ${\mathit{\sigma}}_{\mathit{v}}^{3}$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

${A}_{1}^{0}$ | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | $\phantom{-}1$ | 1 | 1 | 1 | 1 | 1 | 1 | 1 |

${B}_{1}^{0}$ | 1 | $-1$ | 1 | $-1$ | 1 | $-1$ | 1 | $-1$ | $\phantom{-}1$ | $-1$ | 1 | $-1$ | 1 | $-1$ | 1 | $-1$ |

${A}_{2}^{0}$ | 1 | 1 | $-1$ | $-1$ | 1 | 1 | $-1$ | $-1$ | $\phantom{-}1$ | 1 | $-1$ | $-1$ | 1 | 1 | $-1$ | $-1$ |

${B}_{2}^{0}$ | 1 | $-1$ | $-1$ | 1 | 1 | $-1$ | $-1$ | 1 | $\phantom{-}1$ | $-1$ | $-1$ | 1 | 1 | $-1$ | $-1$ | 1 |

${A}_{1}^{1}$ | 1 | 1 | 1 | 1 | $-1$ | $-1$ | $-1$ | $-1$ | $\phantom{-}1$ | 1 | 1 | 1 | $-1$ | $-1$ | $-1$ | $-1$ |

${B}_{1}^{1}$ | 1 | $-1$ | 1 | $-1$ | $-1$ | 1 | $-1$ | 1 | $\phantom{-}1$ | $-1$ | 1 | $-1$ | $-1$ | 1 | $-1$ | 1 |

${A}_{2}^{1}$ | 1 | 1 | $-1$ | $-1$ | $-1$ | $-1$ | 1 | 1 | $\phantom{-}1$ | 1 | $-1$ | $-1$ | $-1$ | $-1$ | $-1$ | 1 |

${B}_{2}^{1}$ | 1 | $-1$ | $-1$ | 1 | $-1$ | 1 | 1 | $-1$ | $\phantom{-}1$ | $-1$ | $-1$ | 1 | $-1$ | 1 | 1 | $-1$ |

${A}_{1}^{2}$ | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | $-1$ | $-1$ | $-1$ | $-1$ | $-1$ | $-1$ | $-1$ | $-1$ |

${B}_{1}^{2}$ | 1 | $-1$ | 1 | $-1$ | 1 | $-1$ | 1 | $-1$ | $-1$ | 1 | $-1$ | 1 | $-1$ | 1 | $-1$ | 1 |

${A}_{2}^{2}$ | 1 | 1 | $-1$ | $-1$ | 1 | 1 | $-1$ | $-1$ | $-1$ | $-1$ | 1 | 1 | $-1$ | $-1$ | 1 | 1 |

${B}_{2}^{2}$ | 1 | $-1$ | $-1$ | 1 | 1 | $-1$ | $-1$ | 1 | $-1$ | $-1$ | 1 | 1 | $-1$ | 1 | 1 | $-1$ |

${A}_{1}^{3}$ | 1 | 1 | 1 | 1 | $-1$ | $-1$ | $-1$ | $-1$ | $-1$ | $-1$ | $-1$ | $-1$ | 1 | 1 | 1 | 1 |

${B}_{1}^{3}$ | 1 | $-1$ | 1 | $-1$ | $-1$ | 1 | $-1$ | 1 | $-1$ | 1 | $-1$ | 1 | 1 | $-1$ | 1 | $-1$ |

${A}_{2}^{3}$ | 1 | 1 | $-1$ | $-1$ | $-1$ | $-1$ | 1 | 1 | $-1$ | $-1$ | 1 | 1 | 1 | 1 | $-1$ | $-1$ |

${B}_{2}^{3}$ | 1 | $-1$ | $-1$ | 1 | $-1$ | 1 | 1 | $-1$ | $-1$ | 1 | 1 | $-1$ | 1 | $-1$ | $-1$ | 1 |

**Table 4.**Multiplication table for the irreps of the

**D**${}_{4\mathrm{h}}$(AEM) group. The vertical and horizontal lines demarcate the blocks of different superscript values. Note that the diagonal blocks are all 0-superscripted, while off diagonal ones are non-0-superscripted.

⊗ | ${\mathit{A}}_{1}^{0}$ | ${\mathit{B}}_{1}^{0}$ | ${\mathit{A}}_{2}^{0}$ | ${\mathit{B}}_{2}^{0}$ | ${\mathit{A}}_{1}^{1}$ | ${\mathit{B}}_{1}^{1}$ | ${\mathit{A}}_{2}^{1}$ | ${\mathit{B}}_{2}^{1}$ | ${\mathit{A}}_{1}^{2}$ | ${\mathit{B}}_{1}^{2}$ | ${\mathit{A}}_{2}^{2}$ | ${\mathit{B}}_{2}^{2}$ | ${\mathit{A}}_{1}^{3}$ | ${\mathit{B}}_{1}^{3}$ | ${\mathit{A}}_{2}^{3}$ | ${\mathit{B}}_{2}^{3}$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

${A}_{1}^{0}$ | ${A}_{1}^{0}$ | ${B}_{1}^{0}$ | ${A}_{2}^{0}$ | ${B}_{2}^{0}$ | ${A}_{1}^{1}$ | ${B}_{1}^{1}$ | ${A}_{2}^{1}$ | ${B}_{2}^{1}$ | ${A}_{1}^{2}$ | ${B}_{1}^{2}$ | ${A}_{2}^{2}$ | ${B}_{2}^{2}$ | ${A}_{1}^{3}$ | ${B}_{1}^{3}$ | ${A}_{2}^{3}$ | ${B}_{2}^{3}$ |

${B}_{1}^{0}$ | ${B}_{1}^{0}$ | ${A}_{1}^{0}$ | ${B}_{2}^{0}$ | ${A}_{2}^{0}$ | ${B}_{1}^{1}$ | ${A}_{1}^{1}$ | ${B}_{2}^{1}$ | ${A}_{2}^{1}$ | ${B}_{1}^{2}$ | ${A}_{1}^{2}$ | ${B}_{2}^{2}$ | ${A}_{2}^{2}$ | ${B}_{1}^{3}$ | ${A}_{1}^{3}$ | ${B}_{2}^{3}$ | ${A}_{2}^{3}$ |

${A}_{2}^{0}$ | ${A}_{2}^{0}$ | ${B}_{2}^{0}$ | ${A}_{1}^{0}$ | ${B}_{1}^{0}$ | ${A}_{2}^{1}$ | ${B}_{2}^{1}$ | ${A}_{1}^{1}$ | ${B}_{1}^{1}$ | ${A}_{2}^{2}$ | ${B}_{2}^{2}$ | ${A}_{1}^{2}$ | ${B}_{1}^{2}$ | ${A}_{2}^{3}$ | ${B}_{2}^{3}$ | ${A}_{1}^{3}$ | ${B}_{1}^{3}$ |

${B}_{2}^{0}$ | ${B}_{2}^{0}$ | ${A}_{2}^{0}$ | ${B}_{1}^{0}$ | ${A}_{1}^{0}$ | ${B}_{2}^{1}$ | ${A}_{2}^{1}$ | ${B}_{1}^{1}$ | ${A}_{1}^{1}$ | ${B}_{2}^{2}$ | ${A}_{2}^{2}$ | ${B}_{1}^{2}$ | ${A}_{1}^{2}$ | ${B}_{2}^{3}$ | ${A}_{2}^{3}$ | ${B}_{1}^{3}$ | ${A}_{1}^{3}$ |

${A}_{1}^{1}$ | ${A}_{1}^{1}$ | ${B}_{1}^{1}$ | ${A}_{2}^{1}$ | ${B}_{2}^{1}$ | ${A}_{1}^{0}$ | ${B}_{1}^{0}$ | ${A}_{2}^{0}$ | ${B}_{2}^{0}$ | ${A}_{1}^{3}$ | ${B}_{1}^{3}$ | ${A}_{2}^{3}$ | ${B}_{2}^{3}$ | ${A}_{1}^{2}$ | ${B}_{1}^{2}$ | ${A}_{2}^{2}$ | ${B}_{2}^{2}$ |

${B}_{1}^{1}$ | ${B}_{1}^{1}$ | ${A}_{1}^{1}$ | ${B}_{2}^{1}$ | ${A}_{2}^{1}$ | ${B}_{1}^{0}$ | ${A}_{1}^{0}$ | ${B}_{2}^{0}$ | ${A}_{2}^{0}$ | ${B}_{1}^{3}$ | ${A}_{1}^{3}$ | ${B}_{2}^{3}$ | ${A}_{2}^{3}$ | ${B}_{1}^{2}$ | ${A}_{1}^{2}$ | ${B}_{2}^{2}$ | ${A}_{2}^{2}$ |

${A}_{2}^{1}$ | ${A}_{2}^{1}$ | ${B}_{2}^{1}$ | ${A}_{1}^{1}$ | ${B}_{1}^{1}$ | ${A}_{2}^{0}$ | ${B}_{2}^{0}$ | ${A}_{1}^{0}$ | ${B}_{1}^{0}$ | ${A}_{2}^{3}$ | ${B}_{2}^{3}$ | ${A}_{1}^{3}$ | ${B}_{1}^{3}$ | ${A}_{2}^{2}$ | ${B}_{2}^{2}$ | ${A}_{1}^{2}$ | ${B}_{1}^{2}$ |

${B}_{2}^{1}$ | ${B}_{2}^{1}$ | ${A}_{2}^{1}$ | ${B}_{1}^{1}$ | ${A}_{1}^{1}$ | ${B}_{2}^{0}$ | ${A}_{2}^{0}$ | ${B}_{1}^{0}$ | ${A}_{1}^{0}$ | ${B}_{2}^{3}$ | ${A}_{2}^{3}$ | ${B}_{1}^{3}$ | ${A}_{1}^{3}$ | ${B}_{2}^{2}$ | ${A}_{2}^{2}$ | ${B}_{1}^{2}$ | ${A}_{1}^{2}$ |

${A}_{1}^{2}$ | ${A}_{1}^{2}$ | ${B}_{1}^{2}$ | ${A}_{2}^{2}$ | ${B}_{2}^{2}$ | ${A}_{1}^{3}$ | ${B}_{1}^{3}$ | ${A}_{2}^{3}$ | ${B}_{2}^{3}$ | ${A}_{1}^{0}$ | ${B}_{1}^{0}$ | ${A}_{2}^{0}$ | ${B}_{2}^{0}$ | ${A}_{1}^{1}$ | ${B}_{1}^{1}$ | ${A}_{2}^{1}$ | ${B}_{2}^{1}$ |

${B}_{1}^{2}$ | ${B}_{1}^{2}$ | ${A}_{1}^{2}$ | ${B}_{2}^{2}$ | ${A}_{2}^{2}$ | ${B}_{1}^{3}$ | ${A}_{1}^{3}$ | ${B}_{2}^{3}$ | ${A}_{2}^{3}$ | ${B}_{1}^{0}$ | ${A}_{1}^{0}$ | ${B}_{2}^{0}$ | ${A}_{2}^{0}$ | ${B}_{1}^{1}$ | ${A}_{1}^{1}$ | ${B}_{2}^{1}$ | ${A}_{2}^{1}$ |

${A}_{2}^{2}$ | ${A}_{2}^{2}$ | ${B}_{2}^{2}$ | ${A}_{1}^{2}$ | ${B}_{1}^{2}$ | ${A}_{2}^{3}$ | ${B}_{2}^{3}$ | ${A}_{1}^{3}$ | ${B}_{1}^{3}$ | ${A}_{2}^{0}$ | ${B}_{2}^{0}$ | ${A}_{1}^{0}$ | ${B}_{1}^{0}$ | ${A}_{2}^{1}$ | ${B}_{2}^{1}$ | ${A}_{1}^{1}$ | ${B}_{1}^{1}$ |

${B}_{2}^{2}$ | ${B}_{2}^{2}$ | ${A}_{2}^{2}$ | ${B}_{1}^{2}$ | ${A}_{1}^{2}$ | ${B}_{2}^{3}$ | ${A}_{2}^{3}$ | ${B}_{1}^{3}$ | ${A}_{1}^{3}$ | ${B}_{2}^{0}$ | ${A}_{2}^{0}$ | ${B}_{1}^{0}$ | ${A}_{1}^{0}$ | ${B}_{2}^{1}$ | ${A}_{2}^{1}$ | ${B}_{1}^{1}$ | ${A}_{1}^{1}$ |

${A}_{1}^{3}$ | ${A}_{1}^{3}$ | ${B}_{1}^{3}$ | ${A}_{2}^{3}$ | ${B}_{2}^{3}$ | ${A}_{1}^{2}$ | ${B}_{1}^{2}$ | ${A}_{2}^{2}$ | ${B}_{2}^{2}$ | ${A}_{1}^{1}$ | ${B}_{1}^{1}$ | ${A}_{2}^{1}$ | ${B}_{2}^{1}$ | ${A}_{1}^{0}$ | ${B}_{1}^{0}$ | ${A}_{2}^{0}$ | ${B}_{2}^{0}$ |

${B}_{1}^{3}$ | ${B}_{1}^{3}$ | ${A}_{1}^{3}$ | ${B}_{2}^{3}$ | ${A}_{2}^{3}$ | ${B}_{1}^{2}$ | ${A}_{1}^{2}$ | ${B}_{2}^{2}$ | ${A}_{2}^{2}$ | ${B}_{1}^{1}$ | ${A}_{1}^{1}$ | ${B}_{2}^{1}$ | ${A}_{2}^{1}$ | ${B}_{1}^{0}$ | ${A}_{1}^{0}$ | ${B}_{2}^{0}$ | ${A}_{2}^{0}$ |

${A}_{2}^{3}$ | ${A}_{2}^{3}$ | ${B}_{2}^{3}$ | ${A}_{1}^{3}$ | ${B}_{1}^{3}$ | ${A}_{2}^{2}$ | ${B}_{2}^{2}$ | ${A}_{1}^{2}$ | ${B}_{1}^{2}$ | ${A}_{2}^{1}$ | ${B}_{2}^{1}$ | ${A}_{1}^{1}$ | ${B}_{1}^{1}$ | ${A}_{2}^{0}$ | ${B}_{2}^{0}$ | ${A}_{1}^{0}$ | ${B}_{1}^{0}$ |

${B}_{2}^{3}$ | ${B}_{2}^{3}$ | ${A}_{2}^{3}$ | ${B}_{1}^{3}$ | ${A}_{1}^{3}$ | ${B}_{2}^{2}$ | ${A}_{2}^{2}$ | ${B}_{1}^{2}$ | ${A}_{1}^{2}$ | ${B}_{2}^{1}$ | ${A}_{2}^{1}$ | ${B}_{1}^{1}$ | ${A}_{1}^{1}$ | ${B}_{2}^{0}$ | ${A}_{2}^{0}$ | ${B}_{1}^{0}$ | ${A}_{1}^{0}$ |

Z${}_{2}$ | 0 | 1 |
---|---|---|

A | 1 | 1 |

B | 1 | $-1$ |

**Table 6.**The character table for the

**D**${}_{n\mathrm{h}}$(AEM) group for some n. The character corresponding to the ith row and jth column is given by $f(i,j)=f(j,i)$. Here, k is given by $k={2}^{n-2}-1$. Starting the row and column number from zero, the output of the function f is as follows: first, i and j are converted into binary numbers and their bitwise sum is calculated. If the number of 1s in the result is odd, character is $-1$; if it is even, the character is 1. For example, 7 and 4 would be $111\&100=100$, so the number of 1s is 1 (odd) and, thus, the character is $-1$.

D${}_{\mathit{n}\mathbf{h}}$ (AEM) | ${\mathit{E}}^{0}$ | ${\mathit{C}}_{2}^{0}$ | ${\mathit{\sigma}}^{0}$ | ${\mathit{\sigma}}_{\mathit{v}}^{0}$ | … | ${\mathit{\sigma}}_{\mathit{v}}^{\mathit{k}}$ |
---|---|---|---|---|---|---|

${A}_{1}^{0}$ | 1 | 1 | 1 | 1 | $f(4k+3,0)$ | |

${B}_{1}^{0}$ | 1 | $-1$ | 1 | $-1$ | $f(4k+3,1)$ | |

${A}_{2}^{0}$ | 1 | 1 | $-1$ | $-1$ | $f(4k+3,2)$ | |

${B}_{2}^{0}$ | 1 | $-1$ | $-1$ | 1 | $f(4k+3,3)$ | |

⋮ | ||||||

${B}_{2}^{k}$ | $f(4k+3,4k+3)$ |

**Table 7.**The $J=0$ vibrational states, including the symmetry $\Gamma $ in

**D**${}_{\infty \mathrm{h}}$(AEM) of the full state. ${v}_{1}$, ${v}_{2}$, ℓ and ${v}_{3}$ are the linear molecule quantum numbers of CO${}_{2}$.

$\mathbf{\Gamma}$ | E (cm${}^{-1}$) | ${\mathit{v}}_{1}$ | ${\mathit{v}}_{2}$ | ℓ | ${\mathit{v}}_{3}$ |
---|---|---|---|---|---|

${A}_{1}^{0}$ | 0.00 | 0 | 0 | 0 | 0 |

${A}_{1}^{1}$ | 667.75 | 0 | 1 | 1 | 0 |

${A}_{1}^{0}$ | 1285.40 | 0 | 2 | 0 | 0 |

${A}_{1}^{2}$ | 1336.67 | 0 | 2 | 2 | 0 |

${A}_{1}^{0}$ | 1388.21 | 1 | 0 | 0 | 0 |

${A}_{1}^{1}$ | 1932.82 | 0 | 3 | 1 | 0 |

${A}_{1}^{3}$ | 2006.73 | 0 | 3 | 3 | 0 |

${A}_{1}^{1}$ | 2077.23 | 1 | 1 | 1 | 0 |

${B}_{2}^{0}$ | 2349.17 | 0 | 0 | 0 | 1 |

${A}_{1}^{0}$ | 2548.34 | 1 | 2 | 0 | 0 |

${A}_{1}^{2}$ | 2586.55 | 0 | 4 | 2 | 0 |

${A}_{1}^{0}$ | 2671.14 | 2 | 0 | 0 | 0 |

${A}_{1}^{4}$ | 2677.94 | 0 | 4 | 4 | 0 |

${A}_{1}^{2}$ | 2762.27 | 1 | 2 | 2 | 0 |

${A}_{1}^{0}$ | 2797.16 | 1 | 2 | 0 | 0 |

${B}_{2}^{1}$ | 3004.45 | 0 | 1 | 1 | 1 |

${A}_{1}^{1}$ | 3181.79 | 1 | 3 | 1 | 0 |

**Table 8.**The $J=2$ ro-vibrational states including the symmetry of the full state and the symmetry of the rotational, stretching, and bending parts. The assignment of ${v}_{1},{v}_{2},\ell ,{v}_{3}$ and K is approximate and based on the largest contribution to the eigenfunction. Only states with non-zero nuclear statistical weights are shown.

${\mathbf{\Gamma}}_{\mathbf{tot}}$ | E (cm${}^{-1}$) | ${\mathbf{\Gamma}}^{\mathbf{rot}}$ | K | ${\mathbf{\Gamma}}^{\mathbf{stretch}}$ | ${\mathbf{\Gamma}}^{\mathbf{bend}}$ | ${\mathit{v}}_{1}$ | ${\mathit{v}}_{2}$ | ℓ | ${\mathit{v}}_{3}$ |
---|---|---|---|---|---|---|---|---|---|

${A}_{1}^{0}$ | 2.34 | ${A}_{1}^{0}$ | 0 | ${A}_{1}^{0}$ | ${A}_{1}^{0}$ | 0 | 0 | 0 | 0 |

${A}_{2}^{0}$ | 669.71 | ${A}_{2}^{1}$ | 1 | ${A}_{1}^{0}$ | ${A}_{1}^{1}$ | 0 | 1 | 1 | 0 |

${A}_{1}^{0}$ | 1287.75 | ${A}_{1}^{0}$ | 0 | ${A}_{1}^{0}$ | ${A}_{1}^{0}$ | 0 | 2 | 0 | 0 |

${A}_{1}^{0}$ | 1337.46 | ${A}_{1}^{2}$ | 2 | ${A}_{1}^{0}$ | ${A}_{1}^{2}$ | 0 | 2 | 2 | 0 |

${A}_{1}^{0}$ | 1390.55 | ${A}_{1}^{0}$ | 0 | ${A}_{1}^{0}$ | ${A}_{1}^{0}$ | 1 | 0 | 0 | 0 |

${A}_{2}^{0}$ | 1934.78 | ${A}_{2}^{1}$ | 1 | ${A}_{1}^{0}$ | ${A}_{1}^{1}$ | 0 | 3 | 1 | 0 |

${A}_{2}^{0}$ | 2079.19 | ${A}_{2}^{1}$ | 1 | ${A}_{1}^{0}$ | ${A}_{1}^{1}$ | 1 | 1 | 1 | 0 |

${A}_{1}^{0}$ | 2550.69 | ${A}_{1}^{0}$ | 0 | ${A}_{1}^{0}$ | ${A}_{1}^{0}$ | 1 | 2 | 0 | 0 |

${A}_{1}^{0}$ | 2587.33 | ${A}_{1}^{2}$ | 2 | ${A}_{1}^{0}$ | ${A}_{1}^{2}$ | 0 | 4 | 2 | 0 |

${A}_{1}^{0}$ | 2673.48 | ${A}_{1}^{0}$ | 0 | ${A}_{1}^{0}$ | ${A}_{1}^{0}$ | 2 | 0 | 0 | 0 |

${A}_{1}^{0}$ | 2763.06 | ${A}_{1}^{2}$ | 2 | ${A}_{1}^{0}$ | ${A}_{1}^{2}$ | 1 | 2 | 2 | 0 |

${A}_{1}^{0}$ | 2799.50 | ${A}_{1}^{0}$ | 0 | ${A}_{1}^{0}$ | ${A}_{1}^{0}$ | 1 | 2 | 0 | 0 |

${A}_{1}^{0}$ | 3006.39 | ${B}_{2}^{1}$ | 1 | ${B}_{2}^{0}$ | ${A}_{1}^{1}$ | 0 | 1 | 1 | 1 |

${A}_{2}^{0}$ | 3183.76 | ${A}_{2}^{1}$ | 1 | ${A}_{1}^{0}$ | ${A}_{1}^{1}$ | 1 | 3 | 1 | 0 |

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## Share and Cite

**MDPI and ACS Style**

Mellor, T.M.; Yurchenko, S.N.; Jensen, P.
Artificial Symmetries for Calculating Vibrational Energies of Linear Molecules. *Symmetry* **2021**, *13*, 548.
https://doi.org/10.3390/sym13040548

**AMA Style**

Mellor TM, Yurchenko SN, Jensen P.
Artificial Symmetries for Calculating Vibrational Energies of Linear Molecules. *Symmetry*. 2021; 13(4):548.
https://doi.org/10.3390/sym13040548

**Chicago/Turabian Style**

Mellor, Thomas M., Sergei N. Yurchenko, and Per Jensen.
2021. "Artificial Symmetries for Calculating Vibrational Energies of Linear Molecules" *Symmetry* 13, no. 4: 548.
https://doi.org/10.3390/sym13040548