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arxiv: 1906.11734 · v1 · pith:I4KAKD5Knew · submitted 2019-06-27 · ⚛️ physics.comp-ph · quant-ph

Transformation Properties under the Operations of the Molecular Symmetry Groups G₃₆ and G₃₆(EM) of Ethane H₃CCH₃

Thomas M. Mellor , Sergey N. Yurchenko , Barry P. Mant , Per Jensen This is my paper

Pith reviewed 2026-05-25 13:49 UTC · model grok-4.3

classification ⚛️ physics.comp-ph quant-ph
keywords molecular symmetry groupethaneG36transformation matricesgeneratorsvariational calculationstorsional motionro-vibrational spectra
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0 comments X

The pith

Ethane's symmetry groups G36 and G36(EM) have explicit irreducible transformation matrices derived from four or five generators.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper sets out to supply the symmetry transformation rules needed to build block-diagonal Hamiltonians for variational ro-vibrational calculations on ethane, a molecule whose two methyl groups can tunnel through large-amplitude torsion. It works with the 36-element group G36 and its 72-element extension G36(EM), deriving the full set of irreducible matrices that show how every group operation acts on chosen basis functions. Algorithms are given that generate all matrices numerically once a small set of generators is fixed, and the same matrices are used to symmetrize both the potential-energy surface and the rotation-torsion-vibration basis. A reader would care because these matrices let the Hamiltonian be factored into smaller independent blocks, which is essential for feasible computations on non-rigid eight-atom molecules.

Core claim

We present the derivation of irreducible transformation matrices for all 36 (72) operations of G36 (G36(EM)) and also describe algorithms for a numerical construction of these matrices based on a set of four (five) generators. The derived transformation matrices associated with G36(EM) have been implemented in the variational nuclear motion program TROVE. The methodology is illustrated on the construction of the symmetry-adapted representations both of the potential energy function of ethane and of the rotation, torsion and vibration basis set functions.

What carries the argument

The set of four (five) generators that produce the full set of irreducible transformation matrices for the 36 (72) group operations acting on the ro-vibrational basis.

If this is right

  • The Hamiltonian matrix for ethane factors into independent blocks labeled by the irreducible representations of G36(EM).
  • Symmetry-adapted basis functions for rotation, torsion and vibration can be generated automatically for any size of basis.
  • The same generator procedure yields symmetry-adapted expansions of the potential energy surface.
  • The matrices are already coded inside TROVE, so the method is immediately available for numerical work on ethane and similar species.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The generator technique could be reused for other molecules that possess internal rotation and the same extended symmetry structure.
  • Numerical generation from generators removes the need to tabulate hundreds of matrices by hand and may reduce transcription errors.
  • Once the matrices exist, it becomes straightforward to project any operator or basis onto individual symmetry species without re-deriving the action each time.

Load-bearing premise

The selected generators produce every group operation on the chosen basis without extra phase or sign conventions that would have to be adjusted by hand when the basis grows or the molecule changes.

What would settle it

A direct multiplication check in which the matrix for the product of two generators differs from the product of their individual matrices when both are applied to the same basis function.

Figures

Figures reproduced from arXiv: 1906.11734 by Barry P. Mant, Per Jensen, Sergey N. Yurchenko, Thomas M. Mellor.

Figure 1
Figure 1. Figure 1: The structure of ethane in the staggered configuration. 2. The Structure of the G36 Group Longuet-Higgins [38] has shown that the group G36 can be written as a direct product of two smaller groups C (−) 3v and C (+) 3v G36 = C (−) 3v × C (+) 3v ; (1) both of these groups are of order 6 and isomorphic to the C3v point group. The top row and leftmost column of [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The labelling of the ethane nuclei. obtained as C (−) i × C(+) j , that is, a class of G36 contains all elements RS where R ∈ C(−) i and S ∈ C(+) j . In the top row and leftmost column of [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Representative members of three of the vibrational coordinate subsets. Here R is the C – C bond length, r is one of the six C–Hk bond lengths rk , and α is one of the six ∠(Hk -C-C) bond angles αk . The last vibrational subset is obtained form six dihedral angles θ12, θ23, θ31, θ45, θ56, θ64, one of which is labelled θ in [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: A Newman projection of ethane, with the CH3 group containing protons 1, 2, and 3, indicated by solid C – H bonds, being closest to the viewer. The x and y components of the coordinate axes attached to each CH3 group is shown, the subscript a signifying that the coordinate axes are for the CaH3 group. To ensure the coordinate system is right handed, the z axis (the same for both groups) points from Cb to Ca… view at source ↗
Figure 5
Figure 5. Figure 5: A Newman projection of ethane, with the CH3 group containing protons 1, 2, 3, indicated by solid C – H bonds, being closest to the viewer. One of the dihedral angles used in the vibrational subsets is labelled by θ and the torsional angle is labelled by τ and is measured in the counterclockwise direction. The x axis halves the dihedral angle between the H1 – C – C and H4 – C – C planes. a 1 3 2 b 4 5 6 [P… view at source ↗
Figure 6
Figure 6. Figure 6: Ethane in the eclipsed configuration. In conclusion, the coordinate subsets, for which we initially diagonalize reduced Hamiltonians, are 1. the C – C bond length R, 2. six C – H bond lengths rk , k = 1, 2, . . . , 6, 3. six bond angles ∠(Hk -C-C) = αk , k = 1, 2, . . . , 6, 4. four dihedral-angle coordinates γ1, γ2, δ1, and δ2, 5. the torsional angle τ, and 6. the three rotational angles (θ, φ, χ) [PITH_… view at source ↗
Figure 7
Figure 7. Figure 7: Newman projections of ethane showing the effects of the G36(EM) generators (and their G36 partners; see [PITH_FULL_IMAGE:figures/full_fig_p016_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: The torsional potential energy as a function of the torsion angle τ. The allowed energy values are marked by blue horizontal lines. Each energy may correspond to more than one eigenfunction of a given irreducible representation. Dashed lines indicate states of d-type symmetry. 0.8 0.9 1 1.1 1.2 1.3 145.220 145.225 145.230 145.235 145.240 A1s E3d E3s A1d τ (rad) Energy/hc (cm −1 ) [PITH_FULL_IMAGE:figures/… view at source ↗
Figure 9
Figure 9. Figure 9: A enlarged detail of [PITH_FULL_IMAGE:figures/full_fig_p021_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: The ground state torsional wavefunction. 0 π 2 π 3π 2 2π 5π 2 3π 7π 2 4π −0.5 0 0.5 τ (rad) Amplitude 0 π 2 π 3π 2 2π 5π 2 3π 7π 2 4π −0.5 0 0.5 τ (rad) Amplitude [PITH_FULL_IMAGE:figures/full_fig_p022_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: The degenerate first excited state torsional wavefunctions. Left display: The two wavefunctions which generate the E3d irrep, but which do not transform as given by the transformation matrices of Section B.1, displayed as a red and a blue curve, respectively. Right display: The corresponding two E3d wavefunctions obtained as transforming according to the transformation matrices of Section B.1 and displaye… view at source ↗
Figure 12
Figure 12. Figure 12: Four nuclei 1, 4, a, and b. The dihedral angle θ is the angle between the 1–a–b and 4–a–b planes. We define the positive direction of the angle by the right hand rule with the thumb pointing in the rab direction. 1 4 θ [PITH_FULL_IMAGE:figures/full_fig_p032_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Top down view of [PITH_FULL_IMAGE:figures/full_fig_p032_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: The y axis. We define the dihedral angles θij and τij in terms of Eq. (C.5) and, by means of this equation, we can determine their transformation properties under the generating operations of G36. We note that generally E ∗arctan2(y, x) = arctan2(−y, x) = 2π − arctan2(y, x). The dihedral angle θ23, for example, is defined by θ23 = arctan2[ez · (hez × r3ai × hez × r2ai), (ez × hez × r3ai) · (ez × hez × r2a… view at source ↗
read the original abstract

In the present work, we report a detailed description of the symmetry properties of the eight-atomic molecule ethane, with the aim of facilitating the variational calculations of rotation-vibration spectra of ethane and related molecules. Ethane consists of two methyl groups $\text{CH}_3$ where the internal rotation (torsion) of one $\text{CH}_3$ group relative to the other is of large amplitude and involves tunneling between multiple minima of the potential energy function. The molecular symmetry group of ethane is the 36-element group $G_{36}$ but the construction of symmetrized basis functions is most conveniently done in terms of the 72-element extended molecular symmetry group $G_{36}\text{(EM)}$. This group can subsequently be used in the construction of block-diagonal matrix representations of the ro-vibrational Hamiltonian for ethane. The derived transformation matrices associated with $G_{36}\text{(EM)}$ have been implemented in the variational nuclear motion program TROVE (Theoretical ROVibrational Energies). TROVE variational calculations will be used as a practical example of a $G_{36}\text{(EM)}$ symmetry adaptation for large systems with a non-rigid, torsional degree of freedom. We present the derivation of irreducible transformation matrices for all 36 (72) operations of $G_{36}\text{(M)}$ ($G_{36}\text{(EM)}$) and also describe algorithms for a numerical construction of these matrices based on a set of four (five) generators. The methodology presented is illustrated on the construction of the symmetry-adapted representations both of the potential energy function of ethane and of the rotation, torsion and vibration basis set functions.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The paper derives irreducible transformation matrices for all 36 operations of the molecular symmetry group G36 and all 72 operations of the extended group G36(EM) for ethane, presents algorithms for their numerical construction from a fixed set of four (five) generators, and describes their implementation in the TROVE variational program for symmetry-adapted ro-vibrational basis functions and potential energy surfaces.

Significance. If the matrices and generator-driven algorithm are correct and close under the group, the work supplies a practical, reproducible route to block-diagonal Hamiltonian matrices for molecules with large-amplitude torsion; this directly supports high-accuracy variational spectra calculations for ethane and related systems. The explicit implementation in TROVE and the generator-based numerical method are concrete strengths that lower the barrier for symmetry adaptation in large non-rigid molecules.

major comments (1)
  1. [Abstract / numerical construction section] Abstract and the section on numerical construction: the claim that the chosen set of four (five) generators produces all irreducible matrices without additional phase or sign conventions for the torsional factors e^{i m τ} is load-bearing for the 'automatic' algorithm. No explicit verification is shown that the generated matrices satisfy the full G36(EM) multiplication table (including operations such as (123)(456) and extended inversion) for an enlarged vibrational basis; a single counter-example would require case-by-case adjustments and undermine the generator-only procedure.
minor comments (2)
  1. The notation for the basis functions (rotation, torsion, vibration) should be defined once with explicit phase conventions before the generator action is applied.
  2. A short table listing the four (five) generators and their explicit action on each coordinate would improve readability of the algorithm description.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful review and for highlighting a key aspect of the numerical construction. We respond to the major comment below and will revise the manuscript to incorporate the requested verification.

read point-by-point responses

  1. Referee: [Abstract / numerical construction section] Abstract and the section on numerical construction: the claim that the chosen set of four (five) generators produces all irreducible matrices without additional phase or sign conventions for the torsional factors e^{i m τ} is load-bearing for the 'automatic' algorithm. No explicit verification is shown that the generated matrices satisfy the full G36(EM) multiplication table (including operations such as (123)(456) and extended inversion) for an enlarged vibrational basis; a single counter-example would require case-by-case adjustments and undermine the generator-only procedure.

    Authors: We agree that an explicit verification of closure under the full multiplication table would strengthen the presentation. The four (five) generators were selected to obey the defining relations of G_{36}(EM), so that the generated matrices form a faithful representation by construction and require no additional phase or sign conventions for the torsional factors. During implementation we confirmed that composite operations, including (123)(456) and the extended inversion, are reproduced correctly by products of the generated matrices for the vibrational basis employed in TROVE. To make this transparent we will add a short verification subsection (or appendix) in the revised manuscript that tabulates selected products for both the original and an enlarged vibrational basis, confirming that the generator-only procedure holds without case-by-case adjustments. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation rests on standard group-theory generators and definitions

full rationale

The paper derives irreducible transformation matrices for G36 and G36(EM) via explicit construction from a fixed set of four (five) generators, then implements them in TROVE. No equations reduce a claimed prediction or matrix element to a fitted parameter or self-citation chain inside the paper; the construction follows ordinary group-theory multiplication rules applied to the chosen basis (rotation, torsion, vibration functions). The central claim is therefore self-contained against external group-theory benchmarks and does not collapse by definition to its inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on the standard definition of the molecular symmetry group G36 for ethane and the extension to G36(EM); no new free parameters, ad-hoc axioms, or invented entities are introduced.

axioms (2)
  • domain assumption The 36-element group G36 is the correct molecular symmetry group for ethane with feasible internal rotation.
    Invoked in the opening paragraph to justify the choice of symmetry group.
  • domain assumption The 72-element extension G36(EM) is the appropriate group for constructing symmetrized basis functions before projection onto G36.
    Stated as the reason for working with the larger group.

pith-pipeline@v0.9.0 · 5872 in / 1406 out tokens · 30194 ms · 2026-05-25T13:49:17.245629+00:00 · methodology

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Reference graph

Works this paper leans on

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