# Symmetry Adaptation of the Rotation-Vibration Theory for Linear Molecules

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## Abstract

**:**

**D**${}_{\infty \mathrm{h}}$ point group, is formulated in terms of lower-order symmetry groups

**D**${}_{n\mathrm{h}}$ with finite n. Character tables and irreducible representation transformation matrices are presented for

**D**${}_{n\mathrm{h}}$ groups with arbitrary n-values. These groups can subsequently be used in the construction of symmetry-adapted ro-vibrational basis functions for solving the Schrödinger equations of linear molecules. Their implementation into the symmetrisation procedure based on a set of “reduced” vibrational eigenvalue problems with simplified Hamiltonians is used as a practical example. It is shown how the solutions of these eigenvalue problems can also be extended to include the classification of basis-set functions using ℓ, the eigenvalue (in units of ℏ) of the vibrational angular momentum operator ${\widehat{L}}_{z}$. This facilitates the symmetry adaptation of the basis set functions in terms of the irreducible representations of

**D**${}_{n\mathrm{h}}$. ${}^{12}$C${}_{2}$H${}_{2}$ is used as an example of a linear molecule of

**D**${}_{\infty \mathrm{h}}$ point group symmetry to illustrate the symmetrisation procedure of the variational nuclear motion program Theoretical ROVibrational Energies (TROVE).

## 1. Introduction

**D**${}_{\infty \mathrm{h}}$ point group (see Table 1). While the molecular vibrational states (assuming a totally symmetric singlet electronic state) span the representations of this point group of infinite order, the symmetry properties of the combined rotation-vibration states must satisfy the nuclear-statistics requirements and transform according to the irreducible representations (irreps) of the finite molecular symmetry group

**D**${}_{\infty \mathrm{h}}$(M) are given in Table 2 (see also Table A-18 of Ref. [1]). Table 2 presents several alternative notations for the irreducible representations. These alternative notations and multiple names for the same concept are perhaps not quite in agreement with time-honored principles such as Occam’s Razor, but they represent a nice example of the development of spectroscopic notation. We note already here that

**D**${}_{\infty \mathrm{h}}$(M) is isomorphic (and, for a triatomic linear molecule A–B–A like CO${}_{2}$, identical) to the group customarily called

**C**${}_{2\mathrm{v}}$(M) (Table A-5 of Ref. [1]), the molecular symmetry group of, for example, the H${}_{2}$O molecule, whose equilibrium structure is bent. The molecular symmetry (MS) groups

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

**C**${}_{2\mathrm{v}}$(M) are determined by applying the principle of feasibility first introduced by Longuet-Higgins [2] (see also Ref. [1]), and one obviously obtains isomorphic MS groups for all chain molecules A–B–C–…–C–B–A, irrespective of these molecules having linear or bent equilibrium structures. Longuet-Higgins [2] (see also Ref. [1]) further showed that for a so-called rigid non-linear molecule (in this context, a rigid molecule is defined as one whose vibration can be described as oscillations around a single potential energy minimum), the MS group is isomorphic to the point group describing the geometrical symmetry at the equilibrium geometry. H${}_{2}$O is a rigid non-linear molecule, whose geometrical symmetry at equilibrium is described by the

**C**${}_{2\mathrm{v}}$ point group which is indeed isomorphic to the MS group

**C**${}_{2\mathrm{v}}$(M). For the rigid linear molecule CO${}_{2}$, however, as already mentioned, the geometrical symmetry at equilibrium is described by the infinite-order point group

**D**${}_{\infty \mathrm{h}}$ which obviously is not isomorphic to the MS group

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

**C**${}_{2\mathrm{v}}$(M) of order four.

**D**${}_{\infty \mathrm{h}}$ point group. We discuss the EMS group in more detail below.

**D**${}_{\infty \mathrm{h}}$ symmetry can be implemented into general nuclear-motion programs. As an example we use TROVE [5,6], a numerical variational method to solve for the ro-vibrational (rotational-vibrational) spectra of (small to medium) general polyatomic molecules, which has been used to simulate the hot spectra of various polyatomic molecules [7,8,9,10,11,12,13,14,15,16,17,18,19] as part of the ExoMol project [20,21], a database of ab initio spectra of molecules of astrophysical importance. Since calculations of this type are based on electronic properties of the molecule, primarily the potential energy and dipole moment surfaces, obtained by solving the electronic Schrödinger equation, one can claim that such calculations are also rovibronic. However, as the calculations are directly concerned with the rotation and vibration, we characterize them as ro-vibrational. The use of molecular symmetry has applications in diverse fields, including molecular spectroscopy and the construction of molecular wavefunctions, ligand-field theory, material science, and electronic structure calculations [1,22,23,24,25]. While the use of a symmetry-adapted basis set has been shown to make calculations of ro-vibrational energies far more efficient by reducing the size of the Hamiltonian matrix blocks to be diagonalized [22], it is not strictly necessary. This is in contrast to intensity calculations, which would hardly be practicable without knowledge about the symmetry of the ro-vibrational states, mainly due to the selection rules imposed by the nuclear spin statistics associated with different irreducible representations [17,22]. A symmetry-free intensity calculation would involve the actual numerical computation of a colossal number of vanishing intensities for transitions that do not satisfy the symmetry selection rules. In this context, it is also important that some energy levels have zero weights and do not exist in nature. Without knowledge of how the eigenvectors transform under the symmetry operations, it is therefore impossible—or at least immensely complicated—to describe the molecular spectrum correctly. We show how a symmetrization procedure similar to that of TROVE can be extended to enable symmetry classification, in particular of vibrational basis functions, in the

**D**${}_{\infty \mathrm{h}}$ point group and thereby introduce the possibility of labelling these basis functions by the value of the vibrational angular momentum quantum number ℓ (see, for example, Ref. [26]). In practice, it turns out that the infinitely many elements in

**D**${}_{\infty \mathrm{h}}$ represent a problem in numerical calculations; we circumvent this by employing, instead of

**D**${}_{\infty \mathrm{h}}$, one of its subgroups

**D**${}_{n\mathrm{h}}$ with a finite value of n. We discuss below how to choose an adequate n-value.

**D**${}_{n\mathrm{h}}$ suitable for symmetry classification in numerical calculations has an n-value determined by ${K}_{\mathrm{max}}$ = ${L}_{\mathrm{max}}$.

**D**${}_{n\mathrm{h}}$; this can be applied to general nuclear motion routines. The matrices are chosen to describe the transformation of vibrational basis functions that are eigenfunctions of the operator ${\widehat{L}}_{z}$ (with eigenvalues $\ell \hslash $) representing the vibrational angular momentum.

**D**${}_{\infty \mathrm{h}}$ have been reported in the literature although the corresponding character tables have been published many times (see, for example [29]). Hegelund et al. [30] have described the transformation properties of the customary rigid-rotor/harmonic-oscillator basis functions (see, for example, Refs. [1,26,31]) for

**D**${}_{n\mathrm{h}}$ point groups with arbitrary n ⩾ 3 (see also Section 12.4 of Ref. [1]). The basis functions span the irreducible representations of

**D**${}_{n\mathrm{h}}$ and the coefficients obtained, defining the transformation properties, can straightforwardly be organized as transformation matrices. The present paper aims at providing the missing information for

**D**${}_{\infty \mathrm{h}}$. As an illustration, we present how this symmetry information is implemented in TROVE as part of the automatic symmetry adaptation technique [22].

## 2. Rotational and Vibrational Symmetry

#### 2.1. The Groups **D**${}_{\infty \mathrm{h}}$(M), **D**${}_{\infty \mathrm{h}}$(EM), and **D**${}_{\infty \mathrm{h}}$

**D**${}_{\infty \mathrm{h}}$(M) of Equation (1) (Table 2), depending on whether the parity p is $+1$($-1$). The parity is the character under ${E}^{*}$: ${E}^{*}\phantom{\rule{0.166667em}{0ex}}{\mathsf{\Phi}}_{\mathrm{int}}$ = $p\phantom{\rule{0.166667em}{0ex}}{\mathsf{\Phi}}_{\mathrm{int}}$.

**D**${}_{\infty \mathrm{h}}$. It is also invariant under ${E}^{*}$ but it may have its sign changed by $\left(p\right)$. Thus, it can have ${{\mathsf{\Sigma}}_{\mathrm{g}}}^{+}$ or ${{\mathsf{\Sigma}}_{\mathrm{u}}}^{+}$ symmetry in

**D**${}_{\infty \mathrm{h}}$(M) (Table 2).

**D**${}_{\infty \mathrm{h}}$(M) [Equation (1)] is relevant for the discussion of Fermi-Dirac and Bose-Einstein statistics in connection with the complete internal wavefunction ${\mathsf{\Phi}}_{\mathrm{int}}$. ${E}^{*}$ ∈

**D**${}_{\infty \mathrm{h}}$(M) is also a “true” symmetry operation, but the operations in the point group

**D**${}_{\infty \mathrm{h}}$ do not occur naturally in this context. However, as mentioned above it is advantageous also to make use of the

**D**${}_{\infty \mathrm{h}}$ symmetry, and for this purpose Bunker and Papoušek [4] defined the EMS group

**D**${}_{\infty \mathrm{h}}$(EM) which is isomorphic to

**D**${}_{\infty \mathrm{h}}$. The operations in

**D**${}_{\infty \mathrm{h}}$(EM) can be written as (see also Section 17.4.2 of Ref. [1])

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

**D**${}_{\infty \mathrm{h}}$(EM) are listed in Table 1. Four of them are one-dimensional (1D): ${{\mathsf{\Sigma}}_{\mathrm{g}}}^{+}$, ${{\mathsf{\Sigma}}_{\mathrm{g}}}^{-}$, ${{\mathsf{\Sigma}}_{\mathrm{u}}}^{+}$, and ${{\mathsf{\Sigma}}_{\mathrm{u}}}^{-}$ and an infinite number are two-dimensional (2D): ${\mathsf{\Pi}}_{\mathrm{g}/\mathrm{u}}$, ${\Delta}_{\mathrm{g}/\mathrm{u}}$, ${\mathsf{\Phi}}_{\mathrm{g}/\mathrm{u}}$, ${\Gamma}_{\mathrm{g}/\mathrm{u}}$, ${H}_{\mathrm{g}/\mathrm{u}}$, ${I}_{\mathrm{g}/\mathrm{u}}$, …. As indicated in the table, an equivalent notation is ${A}_{1\mathrm{g}}$, ${A}_{2\mathrm{g}}$, ${A}_{2\mathrm{u}}$, ${A}_{1\mathrm{u}}$ for 1D irreps and ${E}_{n\mathrm{g}}$ and ${E}_{n\mathrm{u}}$, where $n=1,2,\dots ,\infty $ for 2D irreps, see Table 1. The rotational basis functions ${\mathsf{\Phi}}_{J,k}^{\mathrm{rot}}$ = $|J,k\rangle $ span the irreducible representations of

**D**${}_{\infty \mathrm{h}}$(EM) as given in Table 3.

**D**${}_{\infty \mathrm{h}}$(EM). This group is defined such that the effect of the operations on the vibronic coordinates are identical to those of the point group

**D**${}_{\infty \mathrm{h}}$. It follows from the discussion given above, however, that only the operations in the MS group

**D**${}_{\infty \mathrm{h}}$(M) (corresponding to $\epsilon $ = 0 for the operations in

**D**${}_{\infty \mathrm{h}}$(EM)) are relevant for determining the requirements of Fermi-Dirac and Bose-Einstein statistics.

**D**${}_{\infty \mathrm{h}}$(EM) operations with $\epsilon $ > 0 are artificial (in the sense that the complete ro-vibrational Hamiltonian does not commute with them [1]) and therefore the basis function ${\mathsf{\Phi}}_{J,k,v,l}$ from Equation (2) must be invariant to them—we can view this as a “reality check” of ${\mathsf{\Phi}}_{J,k,v,l}$, which turns out to be invariant to the artificial operations for k = ℓ. It is seen from Table 1 that consequently, ${\mathsf{\Phi}}_{J,k,v,l}$ can only span one of the four irreducible representations ${{\mathsf{\Sigma}}_{\mathrm{g}}}^{+}$, ${{\mathsf{\Sigma}}_{\mathrm{g}}}^{-}$, ${{\mathsf{\Sigma}}_{\mathrm{u}}}^{+}$, and ${{\mathsf{\Sigma}}_{\mathrm{u}}}^{-}$ of the EMS group

**D**${}_{\infty \mathrm{h}}$(EM). In the Introduction, we already gave examples of the weird and wonderful universe of spectroscopic notation. We now extend this universe by pointing out that according to the labelling scheme of Ref. [32], the four irreps ${{\mathsf{\Sigma}}_{\mathrm{g}}}^{+}$, ${{\mathsf{\Sigma}}_{\mathrm{g}}}^{-}$, ${{\mathsf{\Sigma}}_{\mathrm{u}}}^{+}$, and ${{\mathsf{\Sigma}}_{\mathrm{u}}}^{-}$ are also denoted e ortho, e para, f ortho and f para. The correspondence depends on whether J is even or odd and is given in Table 4.

**D**${}_{\infty \mathrm{h}}$(EM) and there are no restrictions as to their possible symmetries. However, the fact that the product function ${\mathsf{\Phi}}_{J,k,v,l}$ must transform according to a 1D irrep introduces restrictions on the possible combinations of ${\mathsf{\Phi}}_{J,k}^{\mathrm{rot}}$ and ${\mathsf{\Phi}}_{v,\ell}^{\mathrm{vib}}$; these restrictions limit the physically useful combinations to those with k = ℓ. For example, the vibrational state ${\nu}_{5}$ (with vibrational basis functions ${\mathsf{\Phi}}_{{v}_{5}=1,\ell =\pm 1}^{\mathrm{vib}}$ of ${\mathsf{\Pi}}_{\mathrm{u}}$ symmetry in

**D**${}_{\infty \mathrm{h}}$(EM)) of acetylene C${}_{2}$H${}_{2}$ can be combined with the ${\mathsf{\Phi}}_{J,k}^{\mathrm{rot}}$ rotational wavefunctions having $(J,k)$ = $(1,\pm 1)$ (and ${\mathsf{\Pi}}_{\mathrm{g}}$ symmetry) to produce three ro-vibrational combinations with symmetries ${{\mathsf{\Sigma}}_{\mathrm{u}}}^{+}$, ${{\mathsf{\Sigma}}_{\mathrm{u}}}^{-}$ and ${\mathsf{\Pi}}_{\mathrm{u}}$ in

**D**${}_{\infty \mathrm{h}}$(EM). However, only the ${{\mathsf{\Sigma}}_{\mathrm{u}}}^{+}$ and ${{\mathsf{\Sigma}}_{\mathrm{u}}}^{-}$ states can be used in practice and the ${\mathsf{\Pi}}_{\mathrm{u}}$ state must be discarded.

#### 2.2. The Point Groups **D**${}_{n\mathrm{h}}$ and Their Correlation with **D**${}_{\infty \mathrm{h}}$

**D**${}_{\infty \mathrm{h}}$ is the geometrical symmetry group of a (horizontal, say) circular disc whose upper and lower surfaces are equivalent so that one can turn the disc upside-down without any observable change resulting from this. Similarly,

**D**${}_{n\mathrm{h}}$ is the geometrical symmetry group of a (horizontal, say) regular polygon with n vertices (i.e., a regular n-gon) whose upper and lower surfaces are equivalent. That is, we can think of

**D**${}_{\infty \mathrm{h}}$ as the limiting case of a progression of

**D**${}_{n\mathrm{h}}$ groups:

**D**${}_{\infty \mathrm{h}}$ = ${\mathrm{lim}}_{n\to \infty}$

**D**${}_{n\mathrm{h}}$. As mentioned above, we aim at implementing

**D**${}_{\infty \mathrm{h}}$ symmetry for the ro-vibrational basis functions employed in variational calculations. However, owing to the infinitely many operations of

**D**${}_{\infty \mathrm{h}}$ and the corresponding infinitely many irreps, this is impracticable. Consequently, we resort to the strategy often used in numerical calculations and approximate ∞ by a large, finite number n or, in other words, we approximate

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

**D**${}_{n\mathrm{h}}$ with a suitably large n-value. In order to do this, we must discuss the correlation between

**D**${}_{n\mathrm{h}}$ and

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

**D**${}_{n\mathrm{h}}$. It is seen that we must distinguish between n even and odd. The difference in group structure—and an accompanying difference in the labelling of the irreps—are caused by the fact that for n even, the point group inversion i (as explained in connection with Equations (4)–(7) of Ref. [1], the point group inversion operation i is different from the spatial inversion operation ${E}^{*}$ and should be careful to distinguish between the two) is present in

**D**${}_{n\mathrm{h}}$, whereas for n odd it is not. Since i ∈

**D**${}_{\infty \mathrm{h}}$, in some sense an even-n

**D**${}_{n\mathrm{h}}$ is more similar to

**D**${}_{\infty \mathrm{h}}$ than an odd-n

**D**${}_{n\mathrm{h}}$. It could be argued that only even-n

**D**${}_{n\mathrm{h}}$ groups should be considered in the limit of $n\to \infty $; this is the approach we have taken in the example calculations of Section 4. However, for completeness we also discuss odd-n

**D**${}_{n\mathrm{h}}$ groups here. The corresponding information is potentially useful for the treatment of polyatomic molecules with an MS group isomorphic to an odd-n

**D**${}_{n\mathrm{h}}$ group.

**D**${}_{n\mathrm{h}}$ group used in a TROVE calculation to approximate

**D**${}_{\infty \mathrm{h}}$ depends on the maximum value ${L}_{\mathrm{max}}$ of the vibrational angular momentum number ℓ required for a given calculation. We have ${L}_{\mathrm{max}}$ = ${K}_{\mathrm{max}}$, the maximum value on the z-axis projection of the rotational angular momentum. In practical calculations we are usually limited by ${L}_{\mathrm{max}}$, as determined by the maximum total value of vibrational bending quanta, rather than by ${K}_{\mathrm{max}}$, as determined by the maximum quanta of rotational excitation.

**D**${}_{n\mathrm{h}}$ for arbitrary n is outlined in Section 3 below.

## 3. General Formulation of the Character Tables and the Irreducible Representation Transformation Matrices of the D${}_{\mathit{n}\mathbf{h}}$ Groups

#### 3.1. General Structure

**C**${}_{3\mathrm{v}}$. It contains the six operations

**C**${}_{3\mathrm{v}}$ have been expressed as products of the two operations ${C}_{3}$ and ${\sigma}^{\left(xz\right)}$. These operations are called the generating operations for

**C**${}_{3\mathrm{v}}$. It is clear that in order to symmetry classify an operator (or a function) in

**C**${}_{3\mathrm{v}}$, it is sufficient to know how the operator (or function) transforms under the generating operations ${C}_{3}$ and ${\sigma}^{\left(xz\right)}$. With this knowledge, Equations (5)–(8) can be used to construct the transformation properties under all other operations. All point groups can be defined in terms of generating operations. Hegelund et al. [30] have shown that for a general group

**C**${}_{n\mathrm{v}}$ the generating operations can be chosen as ${C}_{n}$ and ${\sigma}^{\left(xz\right)}$ by analogy with the choice for

**C**${}_{3\mathrm{v}}$.

**C**${}_{\mathrm{s}}$ and

**C**${}_{\mathrm{i}}$, can now be introduced:

**D**${}_{n\mathrm{h}}$ groups can be written as direct products of these simple groups:

**D**${}_{n\mathrm{h}}$ contains all elements R ∈

**C**${}_{n\mathrm{v}}$ together with all elements that can be written as $R\phantom{\rule{0.166667em}{0ex}}{\sigma}_{\mathrm{h}}$, and an even-n

**D**${}_{n\mathrm{h}}$ contains all elements R ∈

**C**${}_{n\mathrm{v}}$ together with all elements that can be written as $R\phantom{\rule{0.166667em}{0ex}}i$.

**D**${}_{n\mathrm{h}}$ group can be obtained as products involving three generating operations which are typically denoted by ${R}_{+}$, ${R}_{+}^{\prime}$, and ${R}_{-}$. The generating operations for the

**D**${}_{n\mathrm{h}}$ groups are summarized in Table 6. ${R}_{+}$ is chosen as ${C}_{n}$ for all n, but for n odd, $({R}_{+}^{\prime},{R}_{-})$ = $({\sigma}_{\mathrm{h}},{C}_{2}^{\left(x\right)})$, whereas for n even, $({R}_{+}^{\prime},{R}_{-})$ = $(i,{C}_{2}^{\left(x\right)})$, where ${C}_{2}^{\left(x\right)}$ is a rotation by $\pi $ about the molecule-fixed x axis.

**D**${}_{n\mathrm{h}}$ groups (Equations (11) and (12)) it would in fact have been more logical to choose ${\sigma}^{\left(xz\right)}$ as a generating operation for

**D**${}_{n\mathrm{h}}$ instead of ${C}_{2}^{\left(x\right)}$. However, this does not seem to be the customary choice (see, for example, Hegelund et al. [30]) and we attempt here to following accepted practice as much as possible. With the relations

**D**${}_{n\mathrm{h}}$ in terms of the chosen generating operations. Here, ${C}_{n}^{n/2}$ is a rotation by $\pi $ about the z axis.

**D**${}_{n\mathrm{h}}$ point group can be unambiguously constructed.

#### 3.2. Irreducible Representations

**D**${}_{n\mathrm{h}}$ point groups alternates for even and odd n-values. Consequently, so do the transformation matrices generated by the rotation-vibration basis functions (see Refs. [29,30] and Section 5.1.2 of Jensen and Hegelund [35]). The irreducible representations of

**D**${}_{n\mathrm{h}}$ point groups are easily constructed for arbitrary n as described in Section 5.8.2 of Ref. [1], as listed in Table 7. The irreps are expressed in terms of the characters under the generating operations ${R}_{+}$, ${R}_{+}^{\prime}$, and ${R}_{-}$ which are also given in Table 7 (The labelling of the irreducible representations by A, B, and E is recommended by the International Union of Pure and Applied Chemistry; see Section 2.3.3 of Ref. [36]. Why these three labels were chosen remains one of the mysteries in the weird and wonderful world of spectroscopic notation).

**D**${}_{n\mathrm{h}}$ group has four 1D irreps called ${A}_{1\mathrm{g}}$, ${A}_{2\mathrm{g}}$, ${A}_{1\mathrm{u}}$, and ${A}_{2\mathrm{u}}$ and $(n-2)$ 2D irreps, of which half are called ${E}_{r\mathrm{g}}$ and the other half ${E}_{r\mathrm{u}}$ (r = 1,2, …, $n/2-1$). All of these irreps correlate with irreps of

**D**${}_{\infty \mathrm{h}}$ denoted by the same names in Table 1. In addition, the even-n

**D**${}_{n\mathrm{h}}$ group has another four 1D irreps called ${B}_{1\mathrm{g}}$, ${B}_{1\mathrm{u}}$, ${B}_{2\mathrm{g}}$, ${B}_{2\mathrm{u}}$ associated with a sign change of the generating function under the ${C}_{n}$ rotation (Table 7). These B-type irreps have no counterparts in

**D**${}_{\infty \mathrm{h}}$ and so basis functions of these symmetries are useless, if not nonphysical, in the context of approximating

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

**D**${}_{n\mathrm{h}}$. We noted above that the point group inversion operation i is contained in

**D**${}_{\infty \mathrm{h}}$ and in even-n

**D**${}_{n\mathrm{h}}$, but not in odd-n

**D**${}_{n\mathrm{h}}$. Therefore the labelling of the irreps of odd-n

**D**${}_{n\mathrm{h}}$ differs from that used for

**D**${}_{\infty \mathrm{h}}$ and in even-n

**D**${}_{n\mathrm{h}}$. However, Table 8 gives the correspondence between the irreps of odd-n

**D**${}_{n\mathrm{h}}$ and those of even-n

**D**${}_{n\mathrm{h}}$ and

**D**${}_{\infty \mathrm{h}}$, and so we have established the correlation between the

**D**${}_{n\mathrm{h}}$ and the

**D**${}_{\infty \mathrm{h}}$ irreps for all n-values.

#### 3.3. Transformation Matrices

**D**${}_{n\mathrm{h}}$ group with an arbitrary finite n-value. For the 1D irreps (of type A and B, in the notation of Table 1 and Table 7) the $1\times 1$ transformation matrix is simply equal to the character in Table 7. For the 2D irreps (of type E, in the notation of Table 1) we require $2\times 2$ matrices whose traces are the characters in Table 7. Once a set of irreducible-representation matrices are known, symmetrized basis functions (with transformation properties defined by the irreducible-representation matrices) can in principle be determined by the projection-operator technique described in Section 6.3 of Ref. [1].

**D**${}_{n\mathrm{h}}$ group, for any $2\times 2$ matrix $\mathbf{V}$ with a non-vanishing determinant we can construct an equivalent representation consisting of the matrices $\mathbf{V}\phantom{\rule{0.166667em}{0ex}}{\mathbf{M}}_{R}\phantom{\rule{0.166667em}{0ex}}{\mathbf{V}}^{-1}$ as explained in Section 5.4.1 of Ref. [1]. We normally consider representation matrices effecting the transformation under the group operations of particular wavefunctions, coordinates or operators.

**D**${}_{n\mathrm{h}}$ group, spanned by $\left(\right|J,K,+\rangle ,|J,K,-\rangle )$ for K > 0. Towards this end, we use the fact that any element R ∈

**D**${}_{n\mathrm{h}}$ can be expressed as a product of the generating operations ${R}_{+}$, ${R}_{+}^{\prime}$, and ${R}_{-}$, and that the transformation matrix ${\mathbf{M}}_{R}^{\prime}$ generated by R can be expressed as the analogous matrix product of the representation matrices ${\mathbf{M}}_{{R}_{+}}^{\prime}$ ${\mathbf{M}}_{{R}_{+}^{\prime}}^{\prime}$, and ${\mathbf{M}}_{{R}_{-}}^{\prime}$ in Table 9. In forming the matrix products, one can make use of the fact that all ${\mathbf{M}}_{R}^{\prime}$ = $\mathbf{V}\phantom{\rule{0.166667em}{0ex}}{\mathbf{M}}_{R}\phantom{\rule{0.166667em}{0ex}}{\mathbf{V}}^{-1}$, where the ${\mathbf{M}}_{R}$ matrices are generated by $\left(\right|J,K\rangle ,|J,-K\rangle )$ (Table 9) for K > 0 and the matrix $\mathbf{V}$ is defined in Equation (15). For example, all

**D**${}_{n\mathrm{h}}$ groups contain the operations ${C}_{n}^{r}$, where r = 1, 2, …, $n-1$. The operation ${C}_{n}^{r}$ = ${R}_{+}^{r}$ thus generates the transformation matrix

**C**${}_{n\mathrm{v}}$, r = 1, 2, 3, …, $n-1$. In general,

**C**${}_{n\mathrm{v}}$ further contains n reflections in planes that contain the ${C}_{n}$ axis, customarily chosen as the z axis of the molecule-fixed axis system. As discussed for

**C**${}_{3\mathrm{v}}$ in connection with Equations (5)–(8), we can start with one such reflection, ${\sigma}^{\left(xz\right)}$ say, and then obtain the other $n-1$ reflections as ${C}_{n}^{r}\phantom{\rule{0.166667em}{0ex}}{\sigma}^{\left(xz\right)}$, r = 1, 2, 3, …, $n-1$. However, ${\sigma}^{\left(xz\right)}$ is not chosen as a generating operation for

**D**${}_{n\mathrm{h}}$ (see Table 7), but we can use Equations (13) and (14) to express the n reflections as

**C**${}_{n\mathrm{v}}$, and we straightforwardly augment this group by $2n$ elements; $R\phantom{\rule{0.166667em}{0ex}}i$ for n even and $R\phantom{\rule{0.166667em}{0ex}}{\sigma}_{\mathrm{h}}$ for n odd, with R ∈

**C**${}_{n\mathrm{v}}$ (Equations (11) and (12)). The $n-1$ operations ${C}_{n}^{r}\phantom{\rule{0.166667em}{0ex}}i$(${C}_{n}^{r}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}{\sigma}_{\mathrm{h}}$), r = 1, 2, 3,…, $n-1$ are improper rotations ${S}_{n}^{\left(r\right)}$ for n even(odd). We see from Equations (18)–(19) that the remaining operations of type ${\sigma}^{\left(r\right)}\phantom{\rule{0.166667em}{0ex}}i$ (n even) and ${\sigma}^{\left(r\right)}\phantom{\rule{0.166667em}{0ex}}{\sigma}_{\mathrm{h}}$ (n odd) can be written as

**D**${}_{n\mathrm{h}}$ can be expressed as a product of the generating operations given in Table 7. To generate a corresponding set of representation/transformation matrices, we must derive the analogous matrix products of the representation matrices in Table 9. The resulting representation matrices are given in Table 10 for n even and in Table 11 for n odd.

**D**${}_{n\mathrm{h}}$ only for n even. This is because, as explained in Section 4.5 of Ref. [1], the point group inversion i and its MS-group counterpart ${\widehat{O}}_{i}$ do not change the Euler angles, i.e., the rotational coordinates. Consequently, the rotational functions are invariant to these operations. We have added to Table 10 transformation matrices also for u-type (odd) irreps. These matrices can be thought of as generated by functions $\left(\right|J,K,\pm \rangle \phantom{\rule{0.166667em}{0ex}}|{v}_{\mathrm{u}}=1\rangle $, where $|{v}_{\mathrm{u}}=1\rangle $ is the vibrational wavefunction for the fundamental level of a (probably hypothetical) vibrational mode ${\nu}_{\mathrm{u}}$ of ${A}_{1\mathrm{u}}$ symmetry.

## 4. Symmetrization Using the TROVE Approach

**D**${}_{n\mathrm{h}}$ representations. TROVE uses a general numerical symmetrization approach to build a symmetry-adapted ro-vibrational basis set [22]. The procedure will be outlined here and extended to include classification based on the vibrational angular momentum quantum number, ℓ, as necessary for dealing with linear molecules of

**D**${}_{\infty \mathrm{h}}$ point group symmetry, using ${}^{12}$C${}_{2}$H${}_{2}$ as an example. This classification is general and can be implemented into other similarly constructed variational routines.

**D**${}_{n\mathrm{h}}$ that the basis functions, ${\mathsf{\Psi}}_{\mu}^{J,{\Gamma}_{s},\alpha}$ and ${\mathsf{\Psi}}_{{\mu}^{\prime}}^{J,{\Gamma}_{t},{\alpha}^{\prime}}$, transform according to, and $\alpha $ and ${\alpha}^{\prime}$ represent their degenerate components (if present). The block diagonal structure of a Hamiltonian matrix in the

**D**${}_{n\mathrm{h}}$ irreducible representation is given in Figure 1; the symmetry blocks of non-vanishing matrix elements can be diagonalized separately.

#### Symmetrization of the Basis Set for ${}^{12}$C${}_{2}$H${}_{2}$ Using the $(3N-5)$ Coordinate TROVE Implementation

**D**${}_{n\mathrm{h}}$ group being used in place of

**D**${}_{\infty \mathrm{h}}$, we show an example of the construction of the vibrational basis set in case of the linear molecule ${}^{12}$C${}_{2}$H${}_{2}$. We use the recent implementation of the $(3N-5)$ coordinates approach in TROVE (see Ref. [18]) and select a set of seven vibrational coordinates used for ${}^{12}$C${}_{2}$H${}_{2}$: $\Delta R$, $\Delta {r}_{1}$, $\Delta {r}_{2}$, $\Delta {x}_{1}$, $\Delta {y}_{1}$, $\Delta {x}_{2}$, $\Delta {y}_{2}$, as illustrated in Figure 2. The transformation matrices defining their symmetry properties are listed in Table 13 (with even n used in this example). These relate to the symmetry operations of Table 5, and the general irrep transformation matrices for

**D**${}_{n\mathrm{h}}$ of even n given in Table 10.

**D**${}_{n\mathrm{h}}$ (even n), the eigenfunctions of ${\widehat{H}}^{\left(1D\right)}$ span the ${A}_{1\mathrm{g}}$ irrep, while the eigenfunctions of ${\widehat{H}}^{\left(2D\right)}$ span the ${A}_{1\mathrm{g}}$ and ${A}_{2\mathrm{u}}$ irreps.

**D**${}_{n\mathrm{h}}$, and $\alpha $ indicates a degenerate component in the case of 2D irreps.

**D**${}_{\infty \mathrm{h}}$ (see Section 3) and the $K=L$ condition for linear molecules in the $(3N-5)$-approach [18] was applied. Note that the symmetrized basis functions use K and L instead of k and ℓ in Equation (23).

## 5. Numerical Example

#### 5.1. Symmetrization

#### 5.2. Even vs. Odd **D**${}_{\infty n}$ Symmetries

**D**${}_{n\mathrm{h}}$ that are outlined below the primitive and contracted basis sets were controlled by the polyad number as given by Equation (34), with ${P}_{\mathrm{max}}$ = 8 for the primitive basis set and reduced to 6 after contraction (see Refs. [5,18] for more details).

**D**${}_{\infty \mathrm{h}}$, we use a finite group

**D**${}_{n\mathrm{h}}$, with a value of n large enough to cover all required excitations of the vibrational angular momentum $L=\left|\ell \right|$ up to up ${L}_{\mathrm{max}}$ and of the rotational quantum number K up to ${K}_{\mathrm{max}}$ (with the constraint ${L}_{\mathrm{max}}={K}_{\mathrm{max}}$) such that $n=2{L}_{\mathrm{max}}+1$ or $n=2{L}_{\mathrm{max}}+2$ (depending on whether n is odd or even, respectively). For example, in order to be able to cover the rotational excitation up to $K=10$ (${E}_{10g}$ and ${E}_{10u}$), it is necessary to use at least the

**D**${}_{21\mathrm{h}}$ symmetry.

**D**${}_{n\mathrm{h}}$ in the symmetrization approach described in Section 4.

**D**${}_{n\mathrm{h}}$ to use. However, it is also dependent on ${K}_{\mathrm{max}}$, the maximum value required for the z-projection of rotational angular momentum quantum number J, which would ideally be limited by ${J}_{\mathrm{max}}$ (working under the assumption that $K=L$ for the $(3N-5)$-approach to dealing with linear molecules; see [18] and, for example, [27,28]).

## 6. Conclusions

**D**${}_{\infty \mathrm{h}}$ point group using finite

**D**${}_{n\mathrm{h}}$ symmetry (with arbitrary user-defined n) in numerical calculations, and given the implementation in nuclear motion routine TROVE as an example application. We have shown how a symmetrization scheme such as that used in TROVE can be extended by including the vibrational angular momentum operator ${\widehat{L}}_{z}$ into the set of commuting operators, allowing the classification of basis sets based on vibrational angular momentum quantum number, L. Character tables and irreducible representation transformation matrices for

**D**${}_{n\mathrm{h}}$ of general integer odd or even n have been presented, along with some numerical examples for ${}^{12}$C${}_{2}$H${}_{2}$.

## Author Contributions

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**The block diagonal structure of a Hamiltonian matrix in the

**D**${}_{n\mathrm{h}}$ irreducible representation. The empty (white) cells indicate blocks of vanishing matrix elements. It should be noted that, although B-symmetries will be present for even values of n, they are not physical and do not appear as a block of matrix elements to be diagonalized.

**Figure 2.**HCCH as described using the $(3N-5)$ coordinates employed in TROVE. $\overrightarrow{R}$ is the vector (of length R) pointing from the first to the second carbon atom, C${}_{1}$ to C${}_{2}$, while ${\overrightarrow{r}}_{i}$ are the two CH${}_{i}$ bond vectors (of lengths ${r}_{i}$). The $\Delta {x}_{1}$, $\Delta {x}_{2}$, $\Delta {y}_{1}$ and $\Delta {y}_{2}$ notation of this diagram is to reflect the Cartesian projections of the CH${}_{i}$ bond vectors.

**Table 1.**Common character table for the point group

**D**${}_{\infty \mathrm{h}}$ and the Extended Molecular Symmetry (EMS) group

**D**${}_{\infty \mathrm{h}}$(EM) ${}^{a}$.

D${}_{\mathbf{\infty}\mathbf{h}}$(EM): | ${\mathit{E}}_{\mathbf{0}}$ | ${\mathit{E}}_{\mathit{\epsilon}}$ | ⋯ | $\mathbf{\infty}{\mathit{E}}_{\mathit{\epsilon}}{}^{\mathbf{*}}$ | ${\mathbf{\left(}\mathbf{12}\mathbf{\right)}}_{\mathit{\pi}}{}^{\mathbf{*}}$ | ${\mathbf{\left(}\mathbf{12}\mathbf{\right)}}_{\mathit{\pi}\mathbf{+}\mathit{\epsilon}}^{\mathbf{*}}$ | ⋯ | $\mathbf{\infty}{\mathbf{\left(}\mathbf{12}\mathbf{\right)}}_{\mathit{\epsilon}}$ |
---|---|---|---|---|---|---|---|---|

1 | 2 | ⋯ | ∞ | 1 | 2 | ⋯ | ∞ | |

D${}_{\mathbf{\infty}\mathbf{h}}$: | E | $\mathbf{2}{\mathit{C}}_{\mathbf{\infty}}{}^{\mathit{\epsilon}}$ | ⋯ | $\mathbf{\infty}{\mathit{\sigma}}_{\mathbf{v}}^{\mathbf{(}\mathit{\epsilon}\mathbf{/}\mathbf{2}\mathbf{)}}$ | i | $\mathbf{2}{\mathit{S}}_{\mathbf{\infty}}^{\mathit{\pi}\mathbf{+}\mathit{\epsilon}}$ | ⋯ | $\mathbf{\infty}{\mathit{C}}_{\mathbf{2}}^{\mathbf{(}\mathit{\epsilon}\mathbf{/}\mathbf{2}\mathbf{)}}$ |

${\mathsf{\Sigma}}_{\mathrm{g}}{}^{+}$, ${A}_{1\mathrm{g}}$: | 1 | 1 | ⋯ | 1 | 1 | 1 | ⋯ | 1 |

${\mathsf{\Sigma}}_{\mathrm{u}}{}^{+}$, ${A}_{2\mathrm{u}}$: | 1 | 1 | ⋯ | 1 | $-1$ | $-1$ | ⋯ | $-1$ |

${\mathsf{\Sigma}}_{\mathrm{g}}{}^{-}$, ${A}_{2\mathrm{g}}$: | 1 | 1 | ⋯ | $-1$ | 1 | 1 | ⋯ | $-1$ |

${\mathsf{\Sigma}}_{\mathrm{u}}{}^{-}$, ${A}_{1\mathrm{u}}$: | 1 | 1 | ⋯ | $-1$ | $-1$ | $-1$ | ⋯ | 1 |

${\mathsf{\Pi}}_{\mathrm{g}}$, ${E}_{1\mathrm{g}}$: | 2 | $2cos\epsilon \phantom{2}$ | ⋯ | 0 | 2 | $2cos\epsilon \phantom{2}$ | ⋯ | 0 |

${\mathsf{\Pi}}_{\mathrm{u}}$, ${E}_{1\mathrm{u}}$: | 2 | $2cos\epsilon \phantom{2}$ | ⋯ | 0 | $-2$ | $-2cos\epsilon \phantom{2}$ | ⋯ | 0 |

${\Delta}_{\mathrm{g}}$, ${E}_{2\mathrm{g}}$: | 2 | $2cos2\epsilon $ | ⋯ | 0 | 2 | $2cos2\epsilon $ | ⋯ | 0 |

${\Delta}_{\mathrm{u}}$, ${E}_{2\mathrm{u}}$: | 2 | $2cos2\epsilon $ | ⋯ | 0 | $-2$ | $-2cos2\epsilon $ | ⋯ | 0 |

${\mathsf{\Phi}}_{\mathrm{g}}$, ${E}_{3\mathrm{g}}$: | 2 | $2cos3\epsilon $ | ⋯ | 0 | 2 | $2cos3\epsilon $ | ⋯ | 0 |

${\mathsf{\Phi}}_{\mathrm{u}}$, ${E}_{3\mathrm{u}}$: | 2 | $2cos3\epsilon $ | ⋯ | 0 | $-2$ | $-2cos3\epsilon $ | ⋯ | 0 |

⋮ | ⋮ | ⋮ | ⋯ | ⋮ | ⋮ | ⋮ | ⋯ | ⋮ |

**D**${}_{\infty \mathrm{h}}$ are defined as follows: ${C}_{\infty}{}^{\epsilon}$ is a rotation by $\epsilon $ about the molecular axis, ${\sigma}_{\mathrm{v}}^{(\epsilon /2)}$ is a reflection in a plane containing the molecular axis, i is the point group inversion operation, ${S}_{\infty}^{\pi +\epsilon}$ is an improper rotation by $\pi +\epsilon $ about the molecular axis, and ${C}_{2}^{(\epsilon /2)}$ is a rotation by $\pi $ about an axis perpendicular to the molecular axis (see also Ref. [1]). Here, $\epsilon $ = 0$\cdots 2\pi $. See the text for the definitions of the

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

**Table 2.**Character table for the

**D**${}_{\infty \mathrm{h}}$(M) Molecular Symmetry (MS) group ${}^{a}$.

${\mathbf{\Gamma}}_{\mathbf{1}}$ | ${\mathbf{\Gamma}}_{\mathbf{2}}$ | ${\mathbf{\Gamma}}_{\mathbf{3}}$ | ${\mathbf{\Gamma}}_{\mathbf{4}}$ | ${\mathbf{\Gamma}}_{\mathbf{5}}$ | ${\mathbf{\Gamma}}_{\mathbf{6}}$ | E | $\mathbf{\left(}\mathit{p}\mathbf{\right)}$ | ${\mathit{E}}^{\mathbf{*}}$ | ${\mathbf{\left(}\mathit{p}\mathbf{\right)}}^{\mathbf{*}}$ |
---|---|---|---|---|---|---|---|---|---|

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

${\mathsf{\Sigma}}_{\mathrm{u}}{}^{+}$ | $+a$ | ${B}_{2}$ | ${B}^{+}$ | ${B}_{\mathrm{u}}$ | ${A}_{2\mathrm{u}}$ | 1 | $-1$ | 1 | $-1$ |

${\mathsf{\Sigma}}_{\mathrm{g}}{}^{-}$ | $-a$ | ${B}_{1}$ | ${B}^{-}$ | ${B}_{\mathrm{g}}$ | ${A}_{2\mathrm{g}}$ | 1 | $-1$ | $-1$ | 1 |

${\mathsf{\Sigma}}_{\mathrm{u}}{}^{-}$ | $-s$ | ${A}_{2}$ | ${A}^{-}$ | ${A}_{\mathrm{u}}$ | ${A}_{1\mathrm{u}}$ | 1 | 1 | $-1$ | $-1$ |

**D**${}_{\infty \mathrm{h}}$(M). ${\Gamma}_{3}$ is customarily used for

**C**${}_{2\mathrm{v}}$(M) and ${\Gamma}_{5}$ is for

**C**${}_{2\mathrm{h}}$(M) (Table A-8 of Ref. [1]). g and u stand for the German gerade (even) and ungerade (odd), related to the permutation-inversion operation ${\left(p\right)}^{*}$.

**Table 3.**The irreducible representation $\Gamma $ of

**D**${}_{\infty \mathrm{h}}$(EM) spanned by the rotational wavefunction $|J,k\rangle $ of a linear molecule in the absence of external electric and magnetic fields. The irrep depends on k, the z-axis projection in units of ℏ, of the rotational angular momentum.

k | $\mathbf{\Gamma}$ | |
---|---|---|

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

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

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

±2 | ${\Delta}_{\mathrm{g}}$ | |

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

⋮ | ⋮ |

**Table 4.**Symmetry labels for the ro-vibrational states of a linear molecule such as ${}^{12}$C${}_{2}$H${}_{2}$. The $e/f$ labels are defined in Ref. [33] and $ortho/para$ define the nuclear-spin state [32,34]. ${\Gamma}_{a}$, ${\Gamma}_{b}$ and ${\Gamma}_{c}$ are alternative notations for the irreducible representations of

**D**${}_{\infty \mathrm{h}}$(M) (see Table 2 for an expanded list).

${\mathbf{\Gamma}}_{\mathit{a}}$ | ${\mathbf{\Gamma}}_{\mathit{b}}$ | ${\mathbf{\Gamma}}_{\mathit{c}}$ | $\mathit{e}\mathbf{/}\mathit{f}$ | $\mathit{Ortho}\mathbf{/}\mathit{Para}$ | |
---|---|---|---|---|---|

J odd: | ${\mathsf{\Sigma}}_{\mathrm{g}}{}^{+}$ | ${A}_{1\mathrm{g}}$ | $+s$ | f | $para$ |

${\mathsf{\Sigma}}_{\mathrm{u}}{}^{-}$ | ${A}_{1\mathrm{u}}$ | $-s$ | e | $para$ | |

${\mathsf{\Sigma}}_{\mathrm{g}}{}^{-}$ | ${A}_{2\mathrm{g}}$ | $-a$ | e | $ortho$ | |

${\mathsf{\Sigma}}_{\mathrm{u}}{}^{+}$ | ${A}_{2\mathrm{u}}$ | $+a$ | f | $ortho$ | |

J even: | ${\mathsf{\Sigma}}_{\mathrm{g}}{}^{+}$ | ${A}_{1\mathrm{g}}$ | $+s$ | e | $para$ |

${\mathsf{\Sigma}}_{\mathrm{u}}{}^{-}$ | ${A}_{1\mathrm{u}}$ | $-s$ | f | $para$ | |

${\mathsf{\Sigma}}_{\mathrm{g}}{}^{-}$ | ${A}_{2\mathrm{g}}$ | $-a$ | f | $ortho$ | |

${\mathsf{\Sigma}}_{\mathrm{u}}{}^{+}$ | ${A}_{2\mathrm{u}}$ | $+a$ | e | $ortho$ |

**Table 5.**Symmetry operations of the

**D**${}_{n\mathrm{h}}$ groups, for even and odd n. ${\sigma}_{\mathrm{h}}$, ${\sigma}_{\mathrm{v}}$ and ${\sigma}_{\mathrm{d}}$ represent reflections in planes perpendicular to the molecular axis, containing the molecular axis, and bisecting the angle between a pair of ${C}_{2}$ axes, respectively. An improper rotation ${S}_{n}^{r}$ is a rotation by $r\left(\frac{2\pi}{n}\right)$ ($r=1\cdots n-2$) followed by a reflection in the plane perpendicular to the molecular axis and containing the nuclear center-of-mass. ${C}_{n}^{r}$ represents rotations by $r\left(\frac{2\pi}{n}\right)$, where $r=1\cdots n-1$. See Ref. [1] for further details on these symmetry operations.

Symmetry Operation | Number of Operations | Description |
---|---|---|

Even n: | ||

E | 1 | Identity |

${C}_{n}^{r}$ | $n-1$ | Rotations about the n-fold molecular axis |

${C}_{2}^{\prime}/{C}_{2}^{\u2033}$ | n | n rotations by $\pi $ about axes |

perpendicular to the molecular axis | ||

i | 1 | Point group inversion |

${S}_{n}^{r}$ | $n-2$ | Improper rotation (see caption) |

${\sigma}_{\mathrm{h}}$ | 1 | Horizontal reflection (see caption) |

${\sigma}_{\mathrm{v}}$ | $n/2$ | Vertical reflection (see caption) |

${\sigma}_{\mathrm{d}}$ | $n/2$ | Diagonal reflection (see caption) |

Total: | $4n$ | |

Odd n: | ||

E | 1 | Identity |

${C}_{n}^{r}$ | $n-1$ | Rotations about the n-fold molecular axis |

${C}_{2}^{\prime}$ | n | n rotations by $\pi $ about axes |

perpendicular to the molecular axis | ||

${S}_{n}^{r}$ | $n-1$ | Improper rotation (see caption) |

${\sigma}_{\mathrm{h}}$ | 1 | Horizontal reflection (see caption) |

${\sigma}_{\mathrm{v}}$ | n | Vertical reflection (see caption) |

Total: | $4n$ |

Point Group | ${\mathit{R}}_{\mathbf{+}}$ | ${\mathit{R}}_{\mathbf{+}}^{\mathbf{\prime}}$ | ${\mathit{R}}_{\mathbf{-}}$ |
---|---|---|---|

D${}_{n\mathrm{h}}$, n odd | ${C}_{n}$ | ${\sigma}_{\mathrm{h}}$ | ${C}_{2}^{\left(x\right)}$ |

D${}_{n\mathrm{h}}$, n even | ${C}_{n}$ | i | ${C}_{2}^{\left(x\right)}$ |

**Table 7.**Irreducible representations for the

**D**${}_{n\mathrm{h}}$ groups and their characters under the generating operations ${R}_{+}$, ${R}_{+}^{\prime}$ and ${R}_{-}$.

D${}_{\mathit{n}\mathbf{h}}$(n even) | E | ${\mathit{R}}_{\mathbf{+}}$(=${\mathit{C}}_{\mathit{n}}$) | ${\mathit{R}}_{\mathbf{+}}^{\mathbf{\prime}}$(=i) | ${\mathit{R}}_{\mathbf{-}}$(=${\mathit{C}}_{\mathbf{2}}^{\mathbf{\left(}\mathit{x}\mathbf{\right)}}$) |

${A}_{1\mathrm{g}}$ | 1 | −1 | 1 | 1 |

${A}_{2\mathrm{g}}$ | 1 | −1 | 1 | −1 |

${B}_{1\mathrm{g}}$ | 1 | −1 | 1 | 1 |

${B}_{2\mathrm{g}}$ | 1 | −1 | 1 | −1 |

${E}_{r\mathrm{g}}$${}^{a}$ | 2 | $2cos\frac{2\pi r}{n}$ | 2 | 0 |

${A}_{1\mathrm{u}}$ | 1 | −1 | −1 | 1 |

${A}_{2\mathrm{u}}$ | 1 | −1 | −1 | −1 |

${B}_{1\mathrm{u}}$ | 1 | −1 | −1 | 1 |

${B}_{2\mathrm{u}}$ | 1 | −1 | −1 | −1 |

${E}_{r\mathrm{u}}$${}^{a}$ | 2 | $2cos\frac{2\pi r}{n}$ | −2 | 0 |

D${}_{\mathit{n}\mathbf{h}}$(n odd) | E | ${\mathit{R}}_{\mathbf{+}}$(=${\mathit{C}}_{\mathit{n}}$) | ${\mathit{R}}_{\mathbf{+}}^{\mathbf{\prime}}$(=${\mathbf{\sigma}}_{\mathbf{h}}$) | ${\mathit{R}}_{\mathbf{-}}$(=${\mathit{C}}_{\mathbf{2}}^{\mathbf{\left(}\mathit{x}\mathbf{\right)}}$) |

${A}_{1}^{\prime}$ | 1 | 1 | 1 | 1 |

${A}_{2}^{\prime}$ | 1 | 1 | 1 | −1 |

${E}_{r}^{\prime}$${}^{b}$ | 2 | $2cos\frac{2\pi r}{n}$ | 2 | 0 |

${A}_{1}^{\u2033}$ | 1 | 1 | −1 | 1 |

${A}_{2}^{\u2033}$ | 1 | 1 | −1 | −1 |

${E}_{r}^{\u2033}$${}^{b}$ | 2 | $2cos\frac{2\pi r}{n}$ | −2 | 0 |

**Table 8.**The correspondence between the $g/u$ (gerade/ungerade) notation of the irreps of

**D**${}_{n\mathrm{h}}$ (even n) and the ${}^{\prime}{/}^{\u2033}$ notation of the irreps of

**D**${}_{n\mathrm{h}}$ (odd n), based on K (the absolute value of the projection, in units of ℏ, onto the molecule-fixed z-axis of the rotational angular momentum).

K | $\mathbf{\Gamma}$ (even n) | $\mathbf{\Gamma}$ (odd n) | D${}_{\mathbf{\infty}\mathbf{h}}$(EM) |
---|---|---|---|

0 | ${A}_{1\mathrm{g}}$ | ${A}_{1}^{{}^{\prime}}$ | ${\mathsf{\Sigma}}_{\mathrm{g}}{}^{+}$ |

${A}_{1\mathrm{u}}$ | ${A}_{1}^{{}^{\u2033}}$ | ${\mathsf{\Sigma}}_{\mathrm{u}}{}^{+}$ | |

${A}_{2\mathrm{g}}$ | ${A}_{2}^{{}^{\prime}}$ | ${\mathsf{\Sigma}}_{\mathrm{g}}{}^{-}$ | |

${A}_{2\mathrm{u}}$ | ${A}_{2}^{{}^{\u2033}}$ | ${\mathsf{\Sigma}}_{\mathrm{u}}{}^{-}$ | |

>0, even | ${E}_{k\mathrm{g}}$ | ${E}_{k}^{{}^{\prime}}$ | ${\Delta}_{\mathrm{g}},{\Gamma}_{\mathrm{g}},{I}_{\mathrm{g}}\cdots $ |

${E}_{k\mathrm{u}}$ | ${E}_{k}^{{}^{\u2033}}$ | ${\Delta}_{\mathrm{u}},{\Gamma}_{\mathrm{u}},{I}_{\mathrm{u}}\cdots $ | |

>0, odd | ${E}_{k\mathrm{g}}$ | ${E}_{k}^{{}^{\u2033}}$ | ${\mathsf{\Pi}}_{\mathrm{g}},{\mathsf{\Phi}}_{\mathrm{g}},{H}_{\mathrm{g}}\cdots $ |

${E}_{k\mathrm{u}}$ | ${E}_{k}^{{}^{\prime}}$ | ${\mathsf{\Pi}}_{\mathrm{u}},{\mathsf{\Phi}}_{\mathrm{u}},{H}_{\mathrm{u}}\cdots $ |

**Table 9.**Transformation matrices for the

**D**${}_{n\mathrm{h}}$ groups generated by the rotational basis functions $|J,0,+\rangle $ = $|J,0\rangle $ for K = 0 and $\left(\right|J,K\rangle ,|J,-K\rangle )$ and $\left(\right|J,K,+\rangle ,|J,K,-\rangle )$ for K > 0, with $\epsilon $ = $2\pi /n$.

D${}_{\mathit{n}\mathbf{h}}$ (n even) | E | ${\mathit{R}}_{\mathbf{+}}\mathbf{=}{\mathit{C}}_{\mathit{n}}$ | ${\mathit{R}}_{\mathbf{+}}^{\mathbf{\prime}}\mathbf{=}\mathit{i}$ | ${\mathit{R}}_{\mathbf{-}}\mathbf{=}{\mathit{C}}_{\mathbf{2}}^{\mathbf{\left(}\mathit{x}\mathbf{\right)}}$ |

$|J,0,+\rangle $ | 1 | 1 | 1 | 1 |

$\left(\begin{array}{c}|J,K\rangle \hfill \\ |J,-K\rangle \hfill \end{array}\right)$ | $\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)$ | $\left(\begin{array}{cc}{\mathrm{e}}^{+\mathrm{i}K\epsilon}& 0\\ 0& {\mathrm{e}}^{-\mathrm{i}K\epsilon}\end{array}\right)$ | $\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)$ | $\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)$ |

$\left(\begin{array}{c}|J,K,+\rangle \hfill \\ |J,K,-\rangle \hfill \end{array}\right)$ | $\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)$ | $\left(\begin{array}{cc}cosK\epsilon & -sinK\epsilon \\ sinK\epsilon & \phantom{-}cosK\epsilon \end{array}\right)$ | $\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)$ | $\left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right)$ |

D${}_{\mathit{n}\mathbf{h}}$ ($\mathit{n}$ odd) | $\mathit{E}$ | ${\mathit{R}}_{\mathbf{+}}\mathbf{=}{\mathit{C}}_{\mathit{n}}$ | ${\mathit{R}}_{\mathbf{+}}^{\mathbf{\prime}}\mathbf{=}{\mathit{\sigma}}_{\mathbf{h}}$ | ${\mathit{R}}_{\mathbf{-}}\mathbf{=}{\mathit{C}}_{\mathbf{2}}^{\mathbf{\left(}\mathit{x}\mathbf{\right)}}$ |

$|J,0,+\rangle $ | 1 | 1 | 1 | ${(-1)}^{J}$ |

$\left(\begin{array}{c}|J,K\rangle \hfill \\ |J,-K\rangle \hfill \end{array}\right)$ | $\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)$ | $\left(\begin{array}{cc}{\mathrm{e}}^{+\mathrm{i}K\epsilon}& 0\\ 0& {\mathrm{e}}^{-\mathrm{i}K\epsilon}\end{array}\right)$ | $\left(\begin{array}{cc}{(-1)}^{K}& 0\\ 0& {(-1)}^{K}\end{array}\right)$ | $\left(\begin{array}{cc}0& {(-1)}^{J}\\ {(-1)}^{J}& 0\end{array}\right)$ |

$\left(\begin{array}{c}|J,K,+\rangle \hfill \\ |J,K,-\rangle \hfill \end{array}\right)$ | $\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)$ | $\left(\begin{array}{cc}cosK\epsilon & -sinK\epsilon \\ sinK\epsilon & \phantom{-}cosK\epsilon \end{array}\right)$ | $\left(\begin{array}{cc}{(-1)}^{K}& 0\\ 0& {(-1)}^{K}\end{array}\right)$ | $\left(\begin{array}{cc}{(-1)}^{J}& 0\\ 0& -{(-1)}^{J}\end{array}\right)$ |

**Table 10.**Irreducible-representation transformation matrices of the

**D**${}_{n\mathrm{h}}$ group for n even, generated by the rotational basis functions $\left(\right|J,K,+\rangle ,|J,K,-\rangle )$ for K> 0. $\epsilon $ = $\frac{2\pi}{n}$, r is an integer used to identify the group operations, and $\kappa $ = $|K+nt|$; the integer t is determined such that 1 ⩽ $\kappa $ ⩽ $n/2-1$.

${\mathit{\epsilon}}_{\mathit{r}}$ | r | ${\mathit{E}}_{\mathit{\kappa}\mathbf{g}}$ | ${\mathit{E}}_{\mathit{\kappa}\mathbf{u}}$ | |
---|---|---|---|---|

E | $\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)$ | $\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)$ | ||

${C}_{n}^{r}$ | $rK\epsilon $ | $1\dots n-1$ | $\left(\begin{array}{cc}cos{\epsilon}_{r}& -sin{\epsilon}_{r}\\ sin{\epsilon}_{r}& cos{\epsilon}_{r}\end{array}\right)$ | $\left(\begin{array}{cc}cos{\epsilon}_{r}& -sin{\epsilon}_{r}\\ sin{\epsilon}_{r}& cos{\epsilon}_{r}\end{array}\right)$ |

${C}_{2}^{\prime}$ | $2rK\epsilon $ | $0\dots \frac{n}{2}-1$ | ${\left(-1\right)}^{K}$$\left(\begin{array}{cc}cos{\epsilon}_{r}& \phantom{-}sin{\epsilon}_{r}\\ sin{\epsilon}_{r}& -cos{\epsilon}_{r}\end{array}\right)$ | ${\left(-1\right)}^{K}$$\left(\begin{array}{cc}cos{\epsilon}_{r}& \phantom{-}sin{\epsilon}_{r}\\ sin{\epsilon}_{r}& -cos{\epsilon}_{r}\end{array}\right)$ |

${C}_{2}^{\u2033}$ | $(2r+1)K\epsilon $ | $0\dots \frac{n}{2}-1$ | ${\left(-1\right)}^{K}$$\left(\begin{array}{cc}cos{\epsilon}_{r}& sin{\epsilon}_{r}\\ sin{\epsilon}_{r}& -cos{\epsilon}_{r}\end{array}\right)$ | ${\left(-1\right)}^{K}$$\left(\begin{array}{cc}cos{\epsilon}_{r}& sin{\epsilon}_{r}\\ sin{\epsilon}_{r}& -cos{\epsilon}_{r}\end{array}\right)$ |

i | $\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)$ | $\left(\begin{array}{cc}-1& \phantom{-}0\\ \phantom{-}0& -1\end{array}\right)$ | ||

${\sigma}_{\mathrm{h}}$ | ${(-1)}^{K}$$\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)$ | $-{(-1)}^{K}$$\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)$ | ||

${\sigma}_{\mathrm{v}}$ | $2rK\epsilon $ | $0\dots \frac{n}{2}-1$ | ${\left(-1\right)}^{K}$$\left(\begin{array}{cc}cos{\epsilon}_{r}& sin{\epsilon}_{r}\\ sin{\epsilon}_{r}& -cos{\epsilon}_{r}\end{array}\right)$ | $-{\left(-1\right)}^{K}$$\left(\begin{array}{cc}cos{\epsilon}_{r}& sin{\epsilon}_{r}\\ sin{\epsilon}_{r}& -cos{\epsilon}_{r}\end{array}\right)$ |

${\sigma}_{\mathrm{d}}$ | $(2r+1)K\epsilon $ | $0\mathrm{t}\dots \frac{n}{2}-1$ | $-{\left(-1\right)}^{K}$$\left(\begin{array}{cc}cos{\epsilon}_{r}& sin{\epsilon}_{r}\\ sin{\epsilon}_{r}& -cos{\epsilon}_{r}\end{array}\right)$ | |

${S}_{n}^{\left(r\right)}$ | $rK\epsilon $ | 1, 2, 3, …, $n/2-1$, $n/2+1$, …, $n-1$ ${}^{a}$ | ${\left(-1\right)}^{K}$$\left(\begin{array}{cc}cos{\epsilon}_{r}& -sin{\epsilon}_{r}\\ sin{\epsilon}_{r}& \phantom{-}cos{\epsilon}_{r}\end{array}\right)$ | $-{\left(-1\right)}^{K}$$\left(\begin{array}{cc}cos{\epsilon}_{r}& -sin{\epsilon}_{r}\\ sin{\epsilon}_{r}& \phantom{-}cos{\epsilon}_{r}\end{array}\right)$ |

**Table 11.**Irreducible-representation transformation matrices of the

**D**${}_{n\mathrm{h}}$ group for n odd, generated by the rotational basis functions $\left(\right|J,K,+\rangle ,|J,K,-\rangle )$ for K > 0. ${}^{a}$ $\epsilon $ = $\frac{2\pi}{n}$, r is an integer used to identify the group operations, and $\kappa $ = $|K+nt|$; the integer t is determined such that 1 ⩽ $\kappa $ ⩽ $(n-1)/2$.

${\mathit{\epsilon}}_{\mathit{r}}$ | r | ${\mathit{E}}_{\mathit{\kappa}}^{\mathbf{\prime}}$ | ${\mathit{E}}_{\mathit{\kappa}}^{\mathbf{\u2033}}$ | |
---|---|---|---|---|

E | $\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)$ | $\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)$ | ||

${C}_{n}^{r}$ | $rK\epsilon $ | $1\dots n-1$ | $\left(\begin{array}{cc}cos{\epsilon}_{r}& -sin{\epsilon}_{r}\\ sin{\epsilon}_{r}& cos{\epsilon}_{r}\end{array}\right)$ | |

${C}_{2}^{\prime}$ | $rK\epsilon $ | $0\dots n-1$ | ${\left(-1\right)}^{J}$$\left(\begin{array}{cc}cos{\epsilon}_{r}& sin{\epsilon}_{r}\\ sin{\epsilon}_{r}& -cos{\epsilon}_{r}\end{array}\right)$ | ${\left(-1\right)}^{J}$$\left(\begin{array}{cc}cos{\epsilon}_{r}& sin{\epsilon}_{r}\\ sin{\epsilon}_{r}& -cos{\epsilon}_{r}\end{array}\right)$ |

${\sigma}_{\mathrm{h}}$ | $\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)$ | $\left(\begin{array}{cc}-1& 0\\ 0& -1\end{array}\right)$ | ||

${\sigma}_{\mathrm{v}}$ | $rK\epsilon $ | $0\dots n-1$ | ${\left(-1\right)}^{J}$$\left(\begin{array}{cc}cos{\epsilon}_{r}& \phantom{-}sin{\epsilon}_{r}\\ sin{\epsilon}_{r}& -cos{\epsilon}_{r}\end{array}\right)$ | ${\left(-1\right)}^{J}$$\left(\begin{array}{cc}cos{\epsilon}_{r}& \phantom{-}sin{\epsilon}_{r}\\ sin{\epsilon}_{r}& -cos{\epsilon}_{r}\end{array}\right)$ |

${S}_{n}^{\left(r\right)}$ | $rK\epsilon $ | $1\dots n-1$ |

**Table 12.**Irreducible representations of the

**D**${}_{n\mathrm{h}}$ groups generated by the rotational basis functions $|J,0\rangle $ for K = 0 and $\left(\right|J,K,+\rangle ,|J,K,-\rangle )$ for K > 0. The integer t is determined such that 0 ⩽ $\kappa $ ⩽ $n/2$ for n even and 0 ⩽ $\kappa $ ⩽ $(n-1)/2$ for n odd, with $\kappa $ = $|K+nt|$.

K | $\mathit{\kappa}$ | D${}_{\mathit{n}\mathbf{h}}$(n even) | D${}_{\mathit{\infty}\mathbf{h}}$(EM) | |

=0 | 0 | ${A}_{1\mathrm{g}}$ | ${\mathsf{\Sigma}}_{\mathrm{g}}{}^{+}$ | |

>0 | 0 | ${A}_{1\mathrm{g}}$⊕${A}_{2\mathrm{g}}$ | ${\mathsf{\Sigma}}_{\mathrm{g}}{}^{+}$⊕${\mathsf{\Sigma}}_{\mathrm{g}}{}^{+}$ | |

>0 | $n/2$ | ${B}_{1\mathrm{g}}$⊕${B}_{2\mathrm{g}}$ | ${\mathsf{\Sigma}}_{\mathrm{g}}{}^{+}$⊕${\mathsf{\Sigma}}_{\mathrm{g}}{}^{+}$ | |

>0 | $\kappa $ = 1, 2, …, $n/2-1$ | ${E}_{\kappa \mathrm{g}}$ | ${\mathsf{\Pi}}_{\mathrm{g}}$, ${\Delta}_{\mathrm{g}}$, ${\mathsf{\Phi}}_{\mathrm{g}}$, ${\Gamma}_{\mathrm{g}}$, … | |

$\mathit{K}$ | $\mathit{\kappa}$ | D${}_{\mathit{n}\mathbf{h}}$($\mathit{n}$odd) | D${}_{\mathbf{\infty}\mathbf{h}}$(EM) | |

=0 | 0 | J even | ${A}_{1}^{\prime}$ | ${\mathsf{\Sigma}}_{\mathrm{g}}{}^{+}$ |

J odd | ${A}_{2}^{\prime}$ | ${\mathsf{\Sigma}}_{\mathrm{g}}{}^{-}$ | ||

>0, odd | 0 | ${A}_{1}^{\prime}$⊕${A}_{2}^{\prime}$ | ${\mathsf{\Sigma}}_{\mathrm{u}}{}^{+}$⊕${\mathsf{\Sigma}}_{\mathrm{u}}{}^{-}$ | |

>0, even | 0 | ${A}_{1}^{\prime}$⊕${A}_{2}^{\prime}$ | ${\mathsf{\Sigma}}_{\mathrm{g}}{}^{+}$⊕${\mathsf{\Sigma}}_{\mathrm{g}}{}^{-}$ | |

>0, odd | 1, 2, …, $(n-1)/2$ | ${E}_{\kappa}^{\u2033}$ | ${\mathsf{\Pi}}_{\mathrm{g}},{\mathsf{\Phi}}_{\mathrm{g}},{H}_{\mathrm{g}}\cdots $ | |

>0, even | 1, 2, …, $(n-1)/2$ | ${E}_{\kappa}^{\prime}$ | ${\Delta}_{\mathrm{u}},{\Gamma}_{\mathrm{u}},{I}_{\mathrm{u}}\cdots $ |

**Table 13.**Transformation properties based on those of Table 10 for the

**D**${}_{n\mathrm{h}}$ group (relating to the symmetry operations of Table 5), where n is even, for transforming the set of 7 vibrational coordinates ($\Delta R$, $\Delta {r}_{1}$, $\Delta {r}_{2}$, $\Delta {x}_{1}$, $\Delta {y}_{1}$, $\Delta {x}_{2}$, $\Delta {y}_{2}$) used in the calculations of Ref. [18] for linear molecule ${}^{12}$C${}_{2}$H${}_{2}$, as illustrated in Figure 2. The two-component vectors ${\overrightarrow{\rho}}_{1}={(\Delta {x}_{1},\Delta {y}_{1})}^{T}$ and ${\overrightarrow{\rho}}_{2}={(\Delta {x}_{2},\Delta {y}_{2})}^{T}$ transform as ${E}_{1u}$, with the transformation matrices ${\mathbf{M}}_{R}^{{E}_{1u}}$ from Table 10. m is an integer for the bounds given for each operation, used to form ${\epsilon}_{m}$, where $\epsilon =\frac{2\pi}{n}$ in all cases.

Irrep | ${\mathit{\epsilon}}_{\mathit{m}}$ | m | Transformation |
---|---|---|---|

E | $\Delta R$ | ||

$\Delta {r}_{1}$ | |||

$\Delta {r}_{2}$ | |||

${\overrightarrow{\rho}}_{1}$ | |||

${\overrightarrow{\rho}}_{2}$ | |||

${C}_{n}^{m}$ | $m\epsilon $ | $1\cdots n-1$ | $\Delta R$ |

$\Delta {r}_{1}$ | |||

$\Delta {r}_{2}$ | |||

${\mathbf{M}}_{{C}_{n}^{r}}^{{E}_{1u}}\left({\epsilon}_{m}\right)\xb7{\overrightarrow{\rho}}_{1}$ | |||

${\mathbf{M}}_{{C}_{n}^{r}}^{{E}_{1u}}\left({\epsilon}_{m}\right)\xb7{\overrightarrow{\rho}}_{2}$ | |||

${C}_{2}^{{}^{\prime}}$ | $2m\epsilon $ | $0\cdots \frac{n}{2}-1$ | $\Delta R$ |

${C}_{2}^{{}^{\u2033}}$ | $\epsilon (2m+1)$ | $0\cdots \frac{n}{2}-1$ | $\Delta {r}_{2}$ |

$\Delta {r}_{1}$ | |||

${\mathbf{M}}_{{C}_{2}^{{}^{\prime}{/}^{\u2033}}}^{{E}_{1u}}\left({\epsilon}_{m}\right)\xb7{\overrightarrow{\rho}}_{2}$ | |||

${\mathbf{M}}_{{C}_{2}^{{}^{\prime}{/}^{\u2033}}}^{{E}_{1u}}\left({\epsilon}_{m}\right)\xb7{\overrightarrow{\rho}}_{1}$ | |||

i | $\Delta R$ | ||

$\Delta {r}_{2}$ | |||

$\Delta {r}_{1}$ | |||

$-{\overrightarrow{\rho}}_{2}$ | |||

$-{\overrightarrow{\rho}}_{1}$ | |||

${\sigma}_{\mathrm{h}}$ | $\Delta R$ | ||

$\Delta {r}_{2}$ | |||

$\Delta {r}_{1}$ | |||

${\overrightarrow{\rho}}_{2}$ | |||

${\overrightarrow{\rho}}_{1}$ | |||

${\sigma}_{\mathrm{d}}$ | $\epsilon (2m+1)$ | $0\cdots \frac{n}{2}-1$ | $\Delta R$ |

${\sigma}_{v}$ | $2m\epsilon $ | $0\cdots \frac{n}{2}-1$ | $\Delta {r}_{1}$ |

$\Delta {r}_{2}$ | |||

${\mathbf{M}}_{{\sigma}_{\mathrm{d}/\mathrm{v}}}^{{E}_{1u}}\left({\epsilon}_{m}\right)\xb7{\overrightarrow{\rho}}_{1}$ | |||

${\mathbf{M}}_{{\sigma}_{\mathrm{d}/\mathrm{v}}}^{{E}_{1u}}\left({\epsilon}_{m}\right)\xb7{\overrightarrow{\rho}}_{2}$ | |||

${S}_{n}^{m}$ | $m\epsilon $ | $1\cdots n-2$ | $\Delta R$ |

$\Delta {r}_{2}$ | |||

$\Delta {r}_{1}$ | |||

${\mathbf{M}}_{{S}_{n}^{r}}^{{E}_{1u}}\left({\epsilon}_{m}\right)\xb7{\overrightarrow{\rho}}_{2}$ | |||

${\mathbf{M}}_{{S}_{n}^{r}}^{{E}_{1u}}\left({\epsilon}_{m}\right)\xb7{\overrightarrow{\rho}}_{1}$ |

**Table 14.**Symmetries of the symmetrized rotational basis set used by TROVE, Equations (31) and (32) for different combinations of J, K and $\tau $ (where $\tau $ $(=0,1)$ and $K=\left|k\right|$); each 2D representation ${E}_{Kg}$ state has an a and b component, represented by the different values of $\tau $. See Table 8 for an explanation of the differing notation of ${\Gamma}_{rot}$ for even and odd values of n.

K | $\mathit{\tau}$ | ${\mathbf{\Gamma}}_{\mathit{rot}}$ | |
---|---|---|---|

Even n | Odd n | ||

0 | 0 | ${A}_{1\mathrm{g}}$ | ${A}_{1}^{{}^{\prime}}$ |

1 | ${A}_{2\mathrm{g}}$ | ${A}_{2}^{{}^{\prime}}$ | |

>0, odd | 0 | ${E}_{k\mathrm{g}b}$ | ${E}_{kb}^{{}^{\u2033}}$ |

1 | ${E}_{k\mathrm{g}a}$ | ${E}_{ka}^{{}^{\u2033}}$ | |

>0, even | 0 | ${E}_{k\mathrm{g}a}$ | ${E}_{ka}^{{}^{\prime}}$ |

1 | ${E}_{k\mathrm{g}b}$ | ${E}_{kb}^{{}^{\prime}}$ |

**Table 15.**An example of some rotational, vibrational and ro-vibrational assignments (see Section 4 for the meaning of the rotational assignments and e.g., [32] for the vibrational assignments) with associated symmetries (${\Gamma}_{r}$, ${\Gamma}_{v}$ and ${\Gamma}_{r-v}$, respectively) from ro-vibrational calculations using TROVE of ${}^{12}$C${}_{2}$H${}_{2}$ using different (even/odd) values of n for

**D**${}_{n\mathrm{h}}$. In each case ${L}_{\mathrm{max}}$ = 4. The energies are identical for symmetries of higher n than those shown here, but converge towards the experimental values as the polyad number Equation (34) is increased; a low value is used here for demonstration purposes. The symmetry assignment will remain unchanged for more accurate calculations; these will be published elsewhere.

Exp. Energy (cm${}^{\mathbf{-}\mathbf{1}}$) [32] | Energy (cm${}^{\mathbf{-}\mathbf{1}}$) | J | K | $\mathit{\tau}$ | ${\mathit{\nu}}_{\mathbf{1}}{\mathit{\nu}}_{\mathbf{2}}{\mathit{\nu}}_{\mathbf{3}}{\mathit{\nu}}_{\mathbf{4}}^{{\mathit{l}}_{\mathbf{4}}}{\mathit{\nu}}_{\mathbf{5}}^{{\mathit{l}}_{\mathbf{5}}}$ | D${}_{\mathbf{12}\mathbf{h}}$ | D${}_{\mathbf{13}\mathbf{h}}$ | ||||
---|---|---|---|---|---|---|---|---|---|---|---|

This Work | ${\mathbf{\Gamma}}_{\mathit{r}\mathbf{-}\mathit{v}}$ | ${\mathbf{\Gamma}}_{\mathit{r}}$ | ${\mathbf{\Gamma}}_{\mathit{v}}$ | ${\mathbf{\Gamma}}_{\mathit{r}\mathbf{-}\mathit{v}}$ | ${\mathbf{\Gamma}}_{\mathit{r}}$ | ${\mathbf{\Gamma}}_{\mathit{v}}$ | |||||

2.353286 | 2.356491 | 1 | 0 | 1 | ${0000}^{0}{0}^{0}$ | ${A}_{2\mathrm{g}}$ | ${A}_{2\mathrm{g}}$ | ${A}_{1\mathrm{g}}$ | ${A}_{2}^{{}^{\prime}}$ | ${A}_{2}^{{}^{\prime}}$ | ${A}_{1}^{{}^{\prime}}$ |

614.044355 | 625.810547 | 1 | 1 | 1 | ${0001}^{1}{0}^{0}$ | ${A}_{2\mathrm{g}}$ | ${E}_{1\mathrm{g}}$ | ${E}_{1\mathrm{g}}$ | ${A}_{2}^{{}^{\prime}}$ | ${E}_{1}^{{}^{\u2033}}$ | ${E}_{1}^{{}^{\u2033}}$ |

1232.749162 | 1283.603736 | 1 | 0 | 1 | ${0002}^{0}{0}^{0}$ | ${A}_{2\mathrm{g}}$ | ${A}_{2\mathrm{g}}$ | ${A}_{1\mathrm{g}}$ | ${A}_{2}^{{}^{\prime}}$ | ${A}_{2}^{{}^{\prime}}$ | ${A}_{1}^{{}^{\prime}}$ |

7.059822 | 7.069433 | 2 | 0 | 0 | ${0000}^{0}{0}^{0}$ | ${A}_{1\mathrm{g}}$ | ${A}_{1\mathrm{g}}$ | ${A}_{1\mathrm{g}}$ | ${A}_{1}^{{}^{\prime}}$ | ${A}_{1}^{{}^{\prime}}$ | ${A}_{1}^{{}^{\prime}}$ |

618.745653 | 630.518518 | 2 | 1 | 1 | ${0001}^{1}{0}^{0}$ | ${A}_{1\mathrm{g}}$ | ${E}_{1\mathrm{g}}$ | ${E}_{1\mathrm{g}}$ | ${A}_{1}^{{}^{\prime}}$ | ${E}_{1}^{{}^{\u2033}}$ | ${E}_{1}^{{}^{\u2033}}$ |

1235.874392 | 1276.518756 | 2 | 2 | 0 | ${0002}^{2}{0}^{0}$ | ${A}_{1\mathrm{g}}$ | ${E}_{2\mathrm{g}}$ | ${E}_{2\mathrm{g}}$ | ${A}_{1}^{{}^{\prime}}$ | ${E}_{2}^{{}^{\prime}}$ | ${E}_{2}^{{}^{\prime}}$ |

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Chubb, K.L.; Jensen, P.; Yurchenko, S.N.
Symmetry Adaptation of the Rotation-Vibration Theory for Linear Molecules. *Symmetry* **2018**, *10*, 137.
https://doi.org/10.3390/sym10050137

**AMA Style**

Chubb KL, Jensen P, Yurchenko SN.
Symmetry Adaptation of the Rotation-Vibration Theory for Linear Molecules. *Symmetry*. 2018; 10(5):137.
https://doi.org/10.3390/sym10050137

**Chicago/Turabian Style**

Chubb, Katy L., Per Jensen, and Sergei N. Yurchenko.
2018. "Symmetry Adaptation of the Rotation-Vibration Theory for Linear Molecules" *Symmetry* 10, no. 5: 137.
https://doi.org/10.3390/sym10050137