Molecular symmetry: Why permutation-inversion (PI) groups don’t render the point groups obsolete

Accepted Manuscript Molecular symmetry: Why permutation-inversion (PI) groups don’t render the point groups obsolete Peter Groner PII: DOI: Reference:

S0022-2852(17)30236-9 http://dx.doi.org/10.1016/j.jms.2017.07.011 YJMSP 10936

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Journal of Molecular Spectroscopy

Received Date: Revised Date: Accepted Date:

15 May 2017 14 July 2017 17 July 2017

Please cite this article as: P. Groner, Molecular symmetry: Why permutation-inversion (PI) groups don’t render the point groups obsolete, Journal of Molecular Spectroscopy (2017), doi: http://dx.doi.org/10.1016/j.jms.2017.07.011

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Molecular symmetry: Why permutation-inversion (PI) groups don’t render the point groups obsolete

Peter Groner

Department of Chemistry, University of Missouri - Kansas City, MO 64113, USA

Abstract

The analysis of spectra of molecules with internal large-amplitude motions (LAMs) requires molecular symmetry (MS) groups that are larger than and significantly different from the more familiar point groups. MS groups are described often by the permutation- inversion (PI) group method. It is shown that point groups still can and should play a significant role together with the PI groups for a class of molecules with internal rotors. In molecules of this class, several simple internal rotors are attached to a rigid molecular frame. The PI groups for this class are semidirect products like H ^ F, where the invariant subgroup H is a direct product of cyclic groups and F is a point group. This result is used to derive meaningful labels for MS groups, and to derive correlation tables between MS groups and point groups. MS groups of this class have many parallels to space groups of crystalline solids.

1

1

Introduction

The permutation- inversion (PI) group method [1], [2] to analyze the molecular symmetry of molecules with internal large-amplitude motions (LAMs) seems to have made applications of regular point groups in this context obsolete. Of course, it is impossible to treat the full symmetry of molecules with LAMs with point groups alone. On the other hand, a good number of molecules with LAMs are remarkably symmetric. For instance, in the absence of vibrational motions, ethane is always a prolate symmetric rotor, and the internal rotation in ethane changes the point group symmetry periodically from D3d to D3 to D3h to D3 back to D3d . Moreover, the symmetry of the equilibrium structures of molecules with LAMs can (almost) always be described by a specific point group. After the advent of the PI group method, there have been several attempts, among them by Altmann [3], [4], Woodman [5], and Günthard and coworkers [6] - [8], to generate molecular symmetry groups from smaller groups, including point groups. Since the point group of ethane in the model just described is always at least D3 , there should be a way to express that in the construction and notation of the molecular symmetry group. All attempts at a generally applicable decomposition of molecular symmetry groups provided valuable insights into many aspects of molecular symmetry, but some of them failed to reach the ultimate goal. The most directly useful result has been obtained by Woodman [5], who showed that for a class of molecules with LAMs, that the PI group G can be represented as a semidirect product G=H^F. The invariant “torsional subgroup” H is a pure permutation group without permutation- inversion (or “starred”) operators, and the “frame subgroup” F is isomorphic to a point group. Such a decomposition is possible for molecules with (periodic) internal rotations and it is non-trivial when at least one internal rotor has a period of 2π/n with n > 1. For molecules with internal large-amplitude motions (LAMs), without the definition of a molecule-fixed axes system, e.g. with PI operators but without equivalent rotations, one can “only” characterize the symmetry group, specify the selection rules in the space- fixed axes system, and determine the species of the nuclear spin functions and the spin statistical weights of the total wavefunction. Only after the definition of a molecule- fixed axes system is it possible to associate equivalent rotations with PI operators, and to investigate the symmetry properties of 2

overall-rotation wavefunctions. From this point on, as will be shown in this paper, point groups can really be useful in different ways. This paper will use the fact that F actually is a point group even if sometimes it is only C1 . It will be shown how derive F from the equivalent rotations, how it has been used to provide more meaningful labels for molecular symmetry groups than, e.g. G72 , and that transformation properties of the rotational wavefunctions and the components of the dipole moment and polarizability tensor can be obtained quickly from the standard character table of F. Because F is a subgroup of the molecular symmetry group G, the correlation of irreducible representations between groups and subgroups can be used to determine qualitative splitting patterns in cases where the interactions caused by the LAMs are not prohibitively large. The next two sections contain a historical perspective of the early days of molecular symmetry for molecules with LAMs, and a critical view on nomenclature (when are two molecular symmetry groups are the identical or “the same”) and stress the advantages of an alternate notation that involve point groups. These advantages are used in subsequent sections to discuss and compare the properties of different molecular systems with molecular symmetry groups of order 18 and separately groups isomorphic to the group of acetone, to predict qualitative splitting patterns, and to point out relations between molecular symmetry groups and space groups. The last section contains discussion and conclusions.

2

Historical pe rspective

In 1959, Lin & Swalen [9] published a review of the microwave spectroscopy of molecules with internal rotation and other large-amplitude internal motions (LAMs). Their list of molecules for which effects of one LAM have been observed and studied in microwave or infrared spectra includes NH3 [10], CH3 CH3 [11], [12], H2 O 2 [13], phenol, and many molecules with one methyl group like in CH3 OH [14], but also CH3 BF2 [15] and CH3 NO 2 [16]. In these examples, molecular symmetry had been treated in an “ad hoc” fashion based on an assumed Hamiltonian by using D3 (or C3v ) for methanol and D6 (or C6v ) for CH3 BF2 and CH3 NO2 . After the publication of this review, the first experimental results for molecules with two or more LAMs were published for (CH3 )2 CO [17], (CH3 )2 O [18], (CH3 )2 SiH2 [19], 2,3-epoxy-2-butene 3

[20], and NH2 NH2 (3 LAMs) [21], and more. Molecules with more than one LAM have more challenging symmetry groups as shown by Myers & Wilson [22] and others [21]. The first systematic study of symmetry groups for molecules with two CH3 internal rotors and frames with C2v , C2 , Cs, C's, or C1 point group symmetry was carried out by Dreizler [23]. In 1963, Longuet-Higgins’ paper on the symmetry groups of non-rigid molecules [1] introduced the Molecular Symmetry (MS) group as a group of permutation and permutationinversion (PI) operations of all particles and spins, subject to the condition of “feasibility”. The concept of the MS (or PI) group was illustrated using the examples of CH3 BF2 , B(CH3 )3 , CH3 CH3 and NH2 NH2 . Longuet-Higgins [1] cited Howard’s [11] and Wilson’s [12] work on CH3 CH3 , Wilson et al.’s paper on CH3 BF2 [24], as well as Kasuya's preliminary work on NH2 NH2 (see also [21]), but none of the other papers mentioned in the previous paragraph. Watson [25] first made a connection between the PI group approach and earlier work on acetone and its symmetry [17], [22]. The next decade brought forth not only a series of applications of the PI group approach to many systems starting with coaxial threefold rotors on a linear frame [26], [27], but also the first extended PI (EMS) group [26], the symmetry group of molecules of the type of XC(CH3 )3 derived by the “ad hoc” method [28], and also attempts to describe molecular symmetry groups in terms of smaller groups as direct or semidirect products. Altmann [3] sought to show that the Schrödinger supergroup S (essentially the PI group) is a semidirect product of an invariant isodynamic group I (consisting of isodynamic operations) and a Schrödinger group G, which is the point group for a particular molecular structure. Watson [29] provided a counter-example to show that this approach is not gene rally correct, and he stressed Altmann’s observation that S depends on the choice of G. In a follow-up paper, Altmann [4] conceded that it is necessary to consider the symmetry group of the full rovibronic Hamiltonian to achieve equivalence of the supergroup S with the corresponding PI group. Altmann’s examples were restricted to those mentioned by Longuet-Higgins [1] and Watson [29]. Woodman’s work [5] about the semidirect product form of many, but not all, MS groups has already been mentioned in Section 1. He derived PI groups of many molecules with internal rotors, including those mentioned by Longuet-Higgins [1], but none of them, except for ethane, had been studied by spectroscopists before the advent of the PI group method. 4

The isometric group concept for semi-rigid molecules (where LAMs move two or more rigid parts of a molecule against each other) by Günthard and coworkers [6], [7] was based on the premise that a symmetry operation must preserve the sets of internuclear distances between any two types of nuclei (e.g. distances between all 1 H nuclei or between all 12 C and 1 H nuclei). Such sets are obviously preserved in the PI group method [1]. Instead of the concept of feasibility, the allowed range of each LAM coordinate was defined. The isometric operations were defined as substitutions of LAM variables in explicit mathematical equations for the nuclear positions in a molecule- fixed axis system. (Hougen [26] also began the investigation of the symmetry properties of dimethyl acetylene with a definition of the LAM and displacement coordinates.) In this method, the equivalent rotations emerge naturally from the position vectors as functions of the LAM coordinates. It was shown eventually [8] that the full isometric group (which includes the infinitesimal vibrational coordinates) is isomorphic to the PI group or the extended PI (double) group. This method was used to derive the molecular symmetry groups for molecules with two methyl rotors attached to molecular frames with point group symmetry D 2h or any of its subgroups [30]. A review of applications to torsional infrared and Raman spectra has been published [31]. An approach unifying the approaches by Altmann and Günthard was formulated by Ezra [32]. Since about 1974, new exciting spectroscopic results prompted the development of different and much larger molecular symmetry groups. The symmetry groups of ethylene glycol [33] and the dimers of water [34] and methanol [35] were isomorphic to groups already known, but the dimer of ammonia [36] required a new group. Groups for molecules with three methyl rotors are necessary for dimethyl methylphosphonate [37], N,N-dimethyl acetamide [38] and (CH3 )3 XCl [39], [40]. The new experimental results also spawned a series of systematic explorations of larger molecular symmetry groups [41] - [44].

3

Nomenclature and notations

3.a

Isomorphism What is the exact meaning of statement “Two molecular symmetry groups are the same”?

Probably most of the time, we use that expression when two groups are isomorphic. This term 5

means that there is a one-to-one mapping of the elements of one group onto the elements of another group such that the multiplication laws of both groups are not violated. This results in identical character tables in the sense that the rows and/or columns of the character table of one group can be permuted so that they match exactly the characters in the table of the second group even though the labels of the rows/columns are generally different. For example, the point groups D6 , C6v, D3h and D3d are isomorphic to each other. This brings us to next question: Why do we not use the same symbol, D6 (or even G12 ) for example, for these four groups and their irreducible representations? Because, of course, the operators of these groups affect the functions of the translational and rotational coordinates differently even if the isomorphism maps the groups onto each other. And because there are conventions for the systematic notation for point groups (and space groups). Even Bunker and Jensen [2] use different notations for the groups C6v(M), D3d (M) and D3h (M). 3.b

When are two molecular symmetry groups the same? The PI groups of propane and acetone (Fig. 1) which both have two methyl internal rotors

on a C2v frame are both isomorphic to the abstract group G36 = D 3  D3 . (Group symbols in bold font refer to abstract groups.) But are they “the same”? Not really, because in propane more nuclei are involved in the feasible permutations. To demonstrate this, and considering only the permutation of the H atoms, the nuclei are labeled as follows: In methyl group a, the labels are 1, 2, and 3 for the H atoms. In methyl group b, the labels are 4, 5 and 6. The H atoms in the methylene group in propane are 7 and 8. The generating operators of the symmetry group are called C, D, T, and S are defined in Table I; they generate the group G36 . It is clear that the PI operator T is not identical for acetone and propane; the same is true for the operator S. Therefore, the groups are not identical, only isomorphic. Similarly, one could argue that the C2v point groups of the equilibrium structures of these two molecules were not identical because the generating PI operators T (or S, see Table I) of their C2v(M) groups are not identical. However, such a narrow definition of the PI operators for point groups does not conform with the established use of point groups (and symmetry groups of molecules with LAMs). To call two molecular symmetry groups identical - if we must - demands different criteria.

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1) The two groups should induce the same transformation properties of the functions of the Eulerian angles, have the same transformation properties of the dipole operator in space- fixed axes, and have the same transformation properties of the components of the dipole moment in the frame- fixed axes (like for point groups). 2) The two groups should involve the same number of periodic LAMs, and their respective periodicities should be identical. Accordingly, both propane and acetone have two equivalent periodic LAMs with period 2pi/3, the same species for the dipole moment operator in space-fixed axes, the same species for the components of the overall angular momentum and of the dipole moment in molecule fixed axes. A different example is encountered in the symmetry groups of cis- and trans-2,3-dimethyl oxirane (DMO) (Fig. 2). The labels of the methyl groups atoms are the same as for acetone. The H atoms on the oxirane ring are 7 and 8. For both molecules, the symmetry groups are of order 18. Besides the generators C and D, the operator U is feasible for the cis form whereas T is the feasible exchange operator for the trans form (see Table I). Given the commutation properties CD = DC, CT = TD, CU = UD-1 , the two groups are isomorphic but not the identical, because T is a pure permutation whereas U is a starred operator. In the group for trans-dimethyl oxirane, the dipole transition operator with respect to space-fixed axes belongs to the totally symmetric representation: this is not the case for cis-dimethyl oxirane! Therefore, the two groups are not identical. The comparison of the molecular symmetry groups of the two isomers of 2-butene (Fig. 3) provides another argument against a claim that two isomorphic symmetry groups are “the same”. The groups formed by the generating PI operators (123), (456), (23)(56)* = S and (14)(25)(36)(78) = T are identical. However, their sets of equivalent rotations are not. We can choose the axes so that the equivalent rotation of T is Rzπ for both molecules. For cis-2-butene, the equivalent rotation for S can be either Rxπ or Ryπ . However, for trans-2-butene, the equivalent rotation for S is Rzπ again. Does it make sense to claim that these two symmetry groups are the same?

3.c

Labels for groups The practice of calling a group simply G g where g is the order of the group is not really

informative. The label G72 for a molecular symmetry group with 72 elements does only tell us 7

that it has 72 elements. Symbols like C6  C6  C2 or (C6  C3 ) ^ (C2  C2 ) would provide information about the mathematical structures of groups. The structures of most molecules are 3dimensional objects in space and most have several LAMs. Information about the number and nature of those LAMs would be much more useful than the number of elements or the structure of abstract groups. Since our interest is on symmetry groups of molecules with internal LAMs, a notation conveying information about the LAMs of the molecule type in question and even some information about the structure would be preferable over e.g. G72 . In many simple cases with several internal rotors attached to a rigid frame, Woodman’s invariant torsional subgroup H [5] is a direct product of cyclic groups, H = Cn  Cn'  Cn"  .... Each of these cyclic groups Cn consists of the cyclic permutations generated by one internal rotation LAM coordinate and whose equivalent rotations are all equal to the identity rotation R0 (see CH3 BF2 example below). For these cases, the notation G = [nn'n"...]F has been proposed [44] to label the group where the bracketed part stands for the direct product H of the cyclic groups and the point group F is Woodman’s frame subgroup. Subgroup F can be obtained either by the method used by Groner [44] which is based on the isometric group concept [6] - [8] or by derivation from the equivalent rotations introduced by Longuet-Higgins [1]. His Figure 1 for H2 O (without LAMs) clearly shows that the combination of a PI operation with its equivalent rotation has the same effect as a point group operation. Specifically, for H2 O, E R0 = E, (12) Ryπ = C2 , E* Rzπ = σxy, and (12)* Rzπ = σyz. It is obvious that the equivalent rotations together form a point group that contains only proper rotations, e.g. either Cn , Dn , T, O, or I. However, if the group G contains a starred PI operation, its equivalent rotation can be combined with the inversion operation implied by the asterisk to obtain an improper rotation (a point group operation for a symmetry plane, a center of inversion, or a rotation-reflection). Combining these operations from the starred PI operators with the equivalent rotations of the unstarred PI operators leads to a regular point group which is Woodman’s frame subgroup F. Thus, in Longuet-Higgins’ example CH3 BF2 , [1] the invariant subgroup H is formed by E, (123) and (132) which are the elements of the cyclic group C3 . The equivalent rotation of the (23)* operator which is C2 (z) becomes the σxy operator and C2 (y) from (23)(45)* becomes σxy; together with C2 (x) from the (45) operator and the identity, they form C2v. Therefore, the molecular symmetry group for CH3 BF2 is G12 = [3]C2v.

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This alternate notation for molecular symmetry groups provides the following at just a glance: (i) The order of the group = product of numbers within the square brackets multiplied by the order of the frame point group; (ii) the number of periodic LAMs (internal rotors) and (iii) their periodicities in 2π. Unfortunately, it does not convey information about LAMs that are not simple internal rotors. The point group symbol conveys quickly whether the molecule is (or could be) a spherical or symmetric rotor and what the symmetry species of the rotational and translational degrees of freedom are (through the symbols Tα, Rα or Jα, α = x, y, z) on the right side of the conventional character tables of the point group. The symmetry species of Γ* is the product of the species Tα and Rα. This alternate notation for molecular symmetry groups will be used throughout this paper as additional information for all molecular symmetry groups that can be represented this way. If a group G is a semidirect product like Woodman’s decomposition of the molecular symmetry group, its irreducible representations (IRs) can be obtained from the irreducible representations of the invariant subgroup H by induction as described by Altmann [45] and Ezra [46]. During the induction, the IRs of H are grouped into orbits by the actions of the elements of the subgroup F. Each orbit forms the basis for one or more of the IRs of G. In fact, each IR of G can be labeled {abc}RF where {abc} is the label of an orbit, and RF is the label of an IR of the little co- group of that orbit [45] where the little co-group is a subgroup of the frame subgroup F. Such labels for the symmetry group of acetone- like molecules, derived by Ezra [46], have recently been used by Van et al. [47]. The symbol for an orbit is essentially a string of the symmetry numbers σ1 , σ2 , .. etc. like the one proposed as a “more rational” notation for the IRs of simple molecules with internal rotors by Meier et al. [48]. The symmetry number σ is a parameter in the definition of the basis functions commonly used for internal rotation problems. If the internal rotation period is 2π/n, the basis functions of a single internal rotor are usually defined as

where j is integer, τ is the internal rotation coordinate, and 0 ≤ σ < n is the integer symmetry number. Each IR arising from the orbit {000 ..} correlates one-to-one with an IR of the frame subgroup F.

9

4

Molecular symmetry groups of order 18 and groups isomorphic to G36 The use of the same symbol for different symmetry groups of molecules with quite

different LAMs and the use of different symbols for the same the groups for the same type of molecules is rather confusing. Groups of order 18 have been reported for six different types of molecules (Table II). Types 1a and 1b refer to molecules with one internal rotor whereas types 2a-2d refer to two-rotor molecules. Historically the first one has been obtained by Wilson [12] in the study of the heat capacity of ethane subject to internal rotation of the methyl groups against each other. He reported the pure permutation group of the H atoms to have 18 elements and used it to determine the spin statistical weights of the energy levels and he showed that this group is isomorphic to the abstract group C3 D 3 . After the successful analysis of microwave spectra of the two-methyl-rotor molecule acetone [17] whose symmetry group of order 36 became known as C3v-C3v + [22] or G 36 [25], Dreizler [23] derived all subgroups of this group that are applicable to other types of molecules with two methyl internal rotors. Among them were three different groups of order 18 which he called “Gruppe 18” (“Group 18”) for 12 CH3 -S-13 CH3 (symmetry plane containing S and C atoms), C3 -  C3v+ for CH3 -S(O)-CH3 (with symmetry plane mapping the internal rotor axes onto each other), and C3v-  C3 + for CH3 SSCH3 (C2 axis mapping the internal rotor axes onto each other). The latter two groups are isomorphic. Bunker [27] determined the PI group of methyl silyl acetylene CH3 CCSiH3 . Groner and Durig [30] reported other symmetry groups of molecules with two methyl rotors, among them another group of order 18 for molecules where the internal rotors are attached to a molecular frame with Ci symmetry although no sample molecule was given. Table II also includes selected newer references to the same models. The column “Model” refers to the notation used in [30]. The PI groups for the molecules of type 2b and 2c cannot be identical since type 2c has starred PI operators whereas type 2b has only pure permutations. The alternate notation (Section 3.c) distinguishes clearly between the different types of molecules that have symmetry groups of order 18. The symbol G 36 is used often for molecular symmetry groups that are isomorphic to the direct product of two dihedral groups of order 6, e.g. D3  D 3 , (D3 is isomorphic to the point groups D3 and C3v). Such a group was used for the first time to explain the observed splittings in the microwave spectra of acetone [17] and dimethyl ether [18]. It was called C3v-  C3v+ by 10

Myers and Wilson [22] and by Dreizler [23]. PI groups isomorphic to G36 have been obtained by Longuet-Higgins for ethane [1], by Watson for acetone [25], by Koput for dimethyl peroxide [53], and by Ohashi and Hougen for the dimer of methanol [35]. The same PI group was used for the ammonia dimer [43], [54] although a group of order 144 has also been proposed [36]. Another group isomorphic to G36 has been derived for trans-2-butene (two methyl groups on a rigid frame with C2h point group symmetry) [30]. It is sort of surprising that so many different molecular models have molecular symmetry groups that are isomorphic to G36 . For instance, ethane and dimethyl acetylene have only one torsional coordinate, whereas acetone, dimethyl ether and trans-2-butene have two and dimethyl peroxide has three. The dimers of ammonia and methanol, however, have six and eight LAM coordinates, respectively. A similar diversity is exhibited by the observed or assumed equilibrium structures. Their point group symmetries are D3d for ethane, C2v for acetone and dimethyl ether, C2h for trans-2-butene, Cs for the ammonia dimer, C2 for dimethyl peroxide, and C1 for the methanol dimer. The point group formed by the equivalent rotations are D2 for acetone, C2 for trans-2-butene, the ammonia dimer and the methanol dimer. For the models chosen for dimethyl acetylene [26] and dimethyl peroxide [53] with separate coordinates for internal and overall rotation, it was necessary to use the extended PI groups G 36 † (= G72 ). The point group of the equivalent rotations are in these cases D6 and D2 , respectively. The observations described in the previous paragraph are summarized in Table III. Additional information in the table includes point groups of alternate equilibrium structures that are in the feasible ranges of the LAM coordinates, but are not subgroups of the equilibrium point groups. For instance, the point group of dimethyl peroxide is C2 as long as the methyl torsional coordinates are equal, but when the coordinate for the internal rotation about the O-O bond (γ in [53]) is 0 or π/2, the point group is C2v or C2h , respectively. The last column in Table III contains the alternate notation of the symmetry group (Section 3.c). The last two lines include the information for butane and for p-xylene. The model for dimethyl peroxide also applies directly to butane, CH3 -CH2-CH2-CH3 , with three coordinates of internal rotation; of course, their symmetry groups are isomorphic. In this case however, two different equilibrium structures have been determined: C2h for trans-(anti-) butane and C2 for gauche butane. The symmetry group of pxylene is of order 72 and isomorphic to the abstract group D 3  D3  C3 and to G 36 † (= G72 ), although it is NOT a double group [55]. According to work cited there, the equilibrium structure 11

in the electronic ground state S0 is C2h but C2v in the S1 excited state. The group G 72 is necessary to derive the correct selection rules for the analysis of the S1  S0 electronic transition. The alternate labels (Section 3.c) for these molecules are G36 = [33]C2v for acetone, G36 = [33]C2h for trans-2-butene and the dimers of ammonia and methanol, G72 = [33]D2h for p-xylene and G36 † = [3]D6h for ethane and dimethyl acetylene. For butane and dimethyl peroxide, one has the choice of two labels: [331]D2h or [33]D2h . The preferred first choice emphasizes that there is a third motion which is periodic in 2π. But deleting “1” in the label changes neither the group order nor anything else because the underlying transformation defined by Koput remain the same. The second choice however could lead to confusion with the symbol for the group of pxylene. For each molecule in the table, the point group formed by the equivalent rotations is a proper or improper subgroups of the point group in the alternate notation of the group. The same is true also for the point group(s) of the equilibrium structure (real or potential). A note here about Bunker and Jensen’s Table A-28 [2] containing the character table for G36 . All information in that table applies to acetone for which G36 = [33]C2v. It can also be used for cis-2-butene whose symmetry group is also G36 = [33]C2v (see also Section 3.b). For trans-2butene, the main part of the character table A-28 with the notations for the PI operators and the irreducible representation could be used as well. However, and it is important to stress that, the right- most column in the table with the information on the transformation properties of rotational wavefunctions and the components of dipole moment and polarizability tensor must not be used for trans-2-butene, because its symmetry group is G36 = [33]C2h with has a different set of equivalent rotations.

5

Prediction of s plitting patterns

In a recent paper on pinacolone, CH3 -CO-C(CH3 )3 , the splitting pattern observed in the microwave spectrum was rather unusual [56]. It was expected from observations of other ketones of the type CH3 -CO-R that the primary splitting would be caused by the acetyl methyl group CH3 -CO-, and that the barriers of the CH3 internal rotors of the -C(CH3 )3 group would be to high for observable tunneling splittings. The ab initio calculations for pinacolone predicted an 12

equilibrium structure with C1 symmetry with the methyl C atom not in the plane formed by the O=C-C atoms. Only about half of the measured transitions could be assigned using combination difference loops. The assigned transitions could not be fit satisfactorily using the programs XIAM, BELGI-C1, and ERHAM for an assumed C1 equilibrium structure; nor could XIAM, BELGI-Cs, RAM36 and ERHAM do the job for a Cs equilibrium structure with the C-CO-C frame in the plane of symmetry. The authors concluded that another tunneling motion, most likely a tunneling motion of the whole CH3 group through O=C-C plane, perhaps even internal rotation of the complete -C(CH3 )3 group. Watson [25] has shown how the correlation between the irreducible representations of a group and its subgroups can be used to determine qualitative splitting patterns of transitions for high and intermediate barriers. The molecular symmetry group for pinacolone with only the CH3 group rotation is G6 = [3]Cs, isomorphic to the symmetry group for methanol. If the C-C(CH3 )3 internal rotation is also considered to be a feasible motion, the symmetry group is G18 = [33]C's. In what follows, the correlation method is used determine the splitting pattern expected for pinacolone with Cs or C1 equilibrium structures. The character tables for G18 = [33]C's and G6 = [3]Cs with their correlations to Cs and C1 are shown in Tables IV and V. The representations A1 and A2 in both tables become A' and A", respectively, in Cs, and A in C1 . All E representations split into A' and A" in Cs, and 2A in C1 . According to Watson, the correlation table can be read backwards to obtain the splitting patterns as follows as shown in Table VI. In G6 = [3]Cs, for equilibrium symmetry Cs, there are two components, either A1 + E for A' or A2 + E for A", whereas for C1 equilibrium symmetry, there are four components: A1 + A2 + 2 E or (A1 + E) + (A2 + 2 E). This may well explain why only about 50% of the observed transition frequencies could be assigned for pinacolone [56]. Similarly, in G18 = [33]C's and Cs equilibrium symmetry. there are five components: A1 + E1 + E2 + E3 + E4 for A' or A2 + E1 + E2 + E3 + E4 for A". In G18 = [33]C's for C1 equilibrium, there are 10 components: A1 + A2 +2 E1 + 2 E2 + 2 E3 + 2 E4 or (A1 + E1 + E2 + E3 + E4 ) + (A2 + E1 + E2 + E3 + E4 ). Zhao et al. [56] also calculated the pure torsional energy levels (J = 0) for pinacolone from a two-dimensional potential function predicted by ab initio calculations. The lowest energy levels were predicted as shown in Table VII. According to the correlation method, any energy level of A symmetry in C1 should split into four components in group G6 . That means that the vibrational quantum number of the lowest four levels should be 0 because it the vibrational and torsional quantum numbers 13

are zero in the ground state. In the revised assignment in Table VII, the sign in parentheses refers to the sign in the linear combination of the two tunneling components with respect to the hypothetical symmetry plane. The calculations also showed that the A and E levels shown in Table VII split into 2 and 3 components, respectively. But those additional splittings are too small to be experimentally resolvable for the foreseeable future. That means that for both v = 0 and v =1 (revised assignment), there are 10 components each, in agreement with the correlationbased prediction for Group G18 = [33]C's if the equilibrium symmetry is C1 . The splitting patterns were derived by the correlation method for molecules with two methyl rotors attached to molecular frames with C1 , C's (nonequivalent rotors), Cs (equivalent rotors), C2 , C2v, C2h (for all possible equilibrium symmetries) and D2h (for C2h and C2v symmetries. (Tables VIII - X). Observable splittings of energy levels and transitions due to LAMs occur only if potential barriers are not too high. But the splitting patterns described in Section 5 are useful only for states sufficiently below any barriers to other minima in the potential surface belonging to an indistinguishable or a different conformer.

6

Relation between molecular symmetry groups with space groups of crystalline solids

A similar kind of splittings as the one described above is observable in vibrational spectra of crystalline solids where it is called Davydoff or factor group splitting. Crystals are thought to be made up of an infinite number of 3-dimensional identical space- filling unit cells. Their elements of symmetry are operators from the 32 crystallographic point groups but may also include screw-axes or glide-planes. Together with translation operators (motion from one point in a unit cell to the equivalent point in another cell), they form one of the 230 space groups [57]. The 73 “symmorphic” space groups which lack screw-axes or glide-planes are semidirect products of an invariant translation group and a point group where the translation group is a direct product of three cyclic groups (similar to H in case of several internal rotors attached to frame, Section 3.c). Cn  Cn'  Cn" , where n, n', n" ,→ ∞

14

The Schönflies notation for a space groups consists of the symbol of a point group with a superscripted number like in

. A point on a symmetry plane or a Cn rotation axis is said to

have a site symmetry Cs or Cn , respectively. Of course, at intersections of two or more symmetry elements, the site symmetry is higher. Points not on elements of symmetry have the trivial site symmetry C1 . The possible site symmetries are all proper or improper subgroups of the point group used in the symbol for the space group, but not all subgroups may be site groups for a specific space group [58]. The unit cell of

has sites with Cs symmetry; to conform with all

symmetry elements, the unit cell therefore must have two identical molecules in equivalent Cs sites. The energy levels of the identical molecules would be degenerate but they split because of the interactions between them; this splitting is called factor group splitting. The “unit cell” in the molecular symmetry problem for a molecule with two 3- fold internal rotors is a square with side length 2π/3. If there is only one molecule within each square, the only splitting is due to interaction within the 9 identical squares that cover the whole range of the torsional coordinates (2π for each). Within such a unit cell, there are sites where the instantaneous geometry has non-trivial point group symmetry. They correspond to fixed points mentioned by Günthard and coworkers [6] - [8]. In contrast to molecules with multiple periodic internal rotors which have symmetry groups of a finite order, the order of the space groups is infinity because there are in infinite numbers of unit cells. Therefore, each energy level splits into an infinite number of levels which are usually illustrated by graphs of phonon energy versus the length |k| of k-vectors into the same direction. For space groups, |k| is continuous from 0 to |k|max . But the intensity for absorption or emission between different phonons are forbidden unless |k| = 0. In our molecular symmetry groups, the equivalents of the k-vectors are the sets of the symmetry numbers σ1 , σ2 , .. or orbits (Section 3.c). For symmetry groups of finite orders, there are only a finite number of those [59].

7

Discussion and conclusions

Some examples in Section 3.b demonstrating that two different groups are not identical or “the same” may seem to be just semantics. Clearly, it depends on what we think about an operator like (23)(56)*. If it just means a permutation multiplied by the E* operator, there is no 15

contest in calling them to be the same in two different systems. But as soon as we have a picture with a molecule- fixed axes system in mind, things change. Then it matters whether the operator permutes nuclei across the x-z or x-y plane because the equivalent rotations and the transformation properties of the rotational basis functions depend on choice of that plane. Another example not presented in that section is the difference between the symmetry groups for CH3 -CD3 and CH3 -O-CD3 . Looking only at the PI operators (without worrying about axes systems or bent vs. linear molecular frames), the PI groups are identical. But if the C-O-C angle in the second molecule never comes close to π, we need two LAM coordinates - for CH3 -O-CD3 , whereas there is only a single one for ethane-d3 . In that sense, these two groups should not be called “the same” even though they are isomorphic. In the proposed alternate notation, the groups for ethane-d3 and dimethyl ether-d3 are G 18 = [3]C3v and G 18 = [33]C's, respectively. The alternate label for molecular symmetry groups (Section 3.c) distinguishes clearly between different numbers of internal rotors and their periodicity, and provides information about spherical, symmetric or asymmetric rotor systems through the point group symbol. A look at the character table of the point group informs the reader quickly about the transformation properties of the components of the dipole moment and polarizability tensor. The possible point groups of equilibrium structures, saddle point structures etc. do have structures conforming to the subgroups of the point group. And that point group for the label is easy to generate from the equivalent rotations as shown in Section 3.c. A label for a molecular symmetry group of the type “G36 = [33]C2h ” is clearly the best to specify the group. They also facilitate the correlation between point groups and molecular symmetry groups for (many) molecules with LAMs and make they derivation of the qualitative splitting patterns by Watson’s method [25] much easier. The string of the symmetry numbers (or the orbit) as part of the label of IRs associates a representation with the properties of the internal rotor basis functions. In the notation used by Van et al. [47], e.g. (00)·B2 or (12)·A", the orbit is in parentheses; the second part of that label identifies the label of an irreducible species of a point group, called the little co- group. Unfortunately, the derivation of this group, a subgroup of the point group F in the alternate group label, generally requires significant work. The notation used in [31] in groups of many two-rotor molecules uses plus and minus signs to distinguish the irreducible representations belonging to a given orbit: in the labels 00st, 11s and 12t, s and t are the signs of the characters of the representation under the operator T = F4 = (14)(25)(36) and U = F3 F4 = (14)(26)(35)*, 16

respectively. The character tables in [31] also illustrate the correlation of IRs of the orbit {000 ..} to the IRs of the point group symbol in the group label. Using point groups (and space groups) in many different ways to describe crystals has a long tradition. Sections 6 illustrates a number of parallels (and the big difference) between space groups and symmetry groups of molecules with several internal rotors on a rigid frame. After all, crystals and these molecules are in some ways geometrical objects. Point groups are useful for crystals. As shown in this paper, they are also useful for some multi- rotor molecules. Further exploration of similarities and differences between space groups and molecular symmetry groups may be worthwhile.

Acknowledgement The author thanks the unknown referees for their careful and diligent reviews of the original submission, for spotting typos and other mistakes and above all for their constructive questions and suggestions. This research did not receive any specific grant from funding agencies in the public, commercial, or not- for-profit sectors.

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HC Longuet-Higgins, Mol. Phys., 6 (1963) 445-460

[2]

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[3]

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[9]

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[10] CH Townes, AL Schawlow, Microwave Spectroscopy (1955, McGraw-Hill, New York) [11] JB Howard, J. Chem. Phys. 5 (1937) 442-450 [12] EB Wilson, J. Chem. Phys. 6 (1938) 740-745 [13] JT Massey, DR Bianco, J. Chem. Phys. 22 (1954) 442-448 [14] JS Koehler, DM Dennison, Phys. Rev. 57 (1940) 1006-1021 [15] RE Naylor, EB Wilson, J. Chem. Phys. 26 (1957) 1057-1060 [16] E Tannenbaum, RJ Myers, WD Gwinn, J. Chem. Phys. 25 (1956) 42-47 [17] JD Swalen, CC Costain, J. Chem. Phys. 31 (1959) 1562-1574 [18] PH Kasai, RJ Myers, J. Chem. Phys. 30 (1959) 1096-1097 [19] L Pierce, J. Chem. Phys. 34 (1961) 498-506 [20] ML Sage, J. Chem. Phys. 35 (1961) 142-148 [21] T Kasuya, T Kojima, J. Phys. Soc. Japan 18 (1963) 364-368 [22] RJ Myers, EB Wilson, J. Chem. Phys. 33 (1960) 186-191 [23] H Dreizler, Z. Naturforsch. A, 16 (1961) 1354-1367 [24] EB Wilson, CC Lin, DR Lide, J. Chem. Phys. 23 (1955) 136-142 [25] JKG Watson, Can. J. Phys. 43 (1965) 1996-2007 [26] JT Hougen, Can. J. Phys. 42 (1964) 1920-1937 [27] PR Bunker, Mol. Phys. 9 (1965) 257-264 [28] KD Möller, HG Andresen, J. Chem. Phys. 39 (1963) 17-22 [29] JKG Watson, Mol. Phys. 21 (1971) 577-585 [30] P Groner, JR Durig, J. Chem. Phys. 66 (1977) 1856-1874 [31] P Groner, JF Sullivan, JR Durig, in: JR Durig (ed.), Vibrational Spectra and Structure (1981, Elsevier, Amsterdam, Vol. 9) pp 405-496 [32] GS Ezra, Mol. Phys. 38 (1979) 863-875 [33] T-K Ha, H Frei, R Meyer, HH Günthard, Theor. Chim. Acta 34 (1974) 277-292 [34] TR Dyke, J. Chem. Phys. 66 (1977) 492-497 [35] N Ohashi, JT Hougen, J. Mol. Spectrosc. 163 (1994) 86-107 [36] JA Odutola, TR Dyke, BJ Howard, JS Muenter, J. Chem. Phys. 70 (1979) 4884-4886 [37] N Ohashi, JT Hougen, J. Mol. Spectrosc. 211 (2002) 119-126 [38] M Fujitake, Y Kubota, N Ohashi, J. Mol. Spectrosc. 236 (2006) 97-109 [39] M Schnell, J-U Grabow, Angew. Chem. Int. Ed. 45 (2006) 3465 –3470 18

[40] M Schnell, JT Hougen, J-U Grabow, J. Mol. Spectrosc. 251 (2008) 38-55 [41] K Balasubramanian, J. Chem. Phys. 72 (1980) 665-677 [42] JA Odutola, DL Alvis, CW Curtis, TR Dyke, Mol. Phys. 42 (1981) 267-282 [43] L Coudert, JT Hougen, J. Mol. Spectrosc. 149 (1991) 73-98 [44] P Groner, Spectrochim. Acta A 49 (1993) 1935-1946 [45] SL Altmann, Induced Representations in Crystals and Molecules. Point, space and nonrigid molecule groups (1977. Academic Press, London, New York, San Francisco) [46] GS Ezra, Symmetry Properties of Molecules, Lecture Notes in Chemistry 28 (1982, Springer-Verlag, Berlin, Heidelberg, New York) [47] V Van, W Stahl, HVL Nguyen, Phys. Chem. Chem. Phys. 17 (2015) 32111-32114 [48] J Meier, A Bauder, HH Günthard, J. Chem. Phys. 57 (1972) 1219-1236 [49] N Ohashi, JT Hougen, RD Suenram, FJ Lovas, Y Kawashima, M Fujitake, J Pyka, J. Mol. Spectrosc. 227 (2004) 28-42 [50] M Meyer, J. Mol. Spectrosc. 148 (1991) 310-323 [51] D Sutter, H Dreizler, H-D Rudolph, Z. Naturforsch. A 20 (1965) 1676-1681 [52] H Hartwig, H Dreizler, Z. Naturforsch A 51 (1996) 923-932 [53] J Koput, J. Mol. Spectrosc. 141 (1990) 118-133 [54] DD Nelson, W Klemperer, J. Chem. Phys. 87 (1987) 139-149 [55] AM Gardner, WD Tuttle, P Groner, TG Wright, J. Chem. Phys. 146 (2017) 124308 [56] Y Zhao, HVL Nguyen, W Stahl, JT Hougen, J. Mol. Spectrosc. 318 (2015) 91-100 [57] CJ Bradley, AP Cracknell, Representation theory for point groups and space groups (1972, Clarendon Press, Oxford) [58] WG Fateley, FR Dollish, NT McDevitt, FF Bentley, Infrared and Raman selection rules for molecular and lattice vibrations: The correlation method (1972, Wiley-Interscience, New York) [59] P. Groner, J. Mol. Struct. 659 (2003) 1-11

19

Table I Definition of operator symbols in terms of PI operators (U = TS = ST)

Acetone Propane cis-2,3-DMO trans-2,3-DMO cis-2-butene trans-2-butene

C (123) (123) (123) (123) (123) (123)

D (456) (456) (456) (456) (456) (456)

T (14)(25)(36) (14)(25)(36)(78)

S (23)(56)* (23)(56)(78)*

U (14)(26)(35)* (14)(26)(35)* (14)(26)(35)(78)*

(14)(25)(36)(78) (14)(25)(36)(78) (14)(25)(36)(78)

(23)(56)* (23)(56)*

(14)(26)(35)* (14)(26)(35)*

20

Table II Symmetry groups of order 18 for molecules with internal rotors Year

Ref.

Molecule

1938 1965 1961 1972 2002 2004 1961 1961 1991 2002 1961 1965 1996 1977

[12] [27] [23] [48] [37] [49] [23] [20] [50] [37] [23] [51] [52] [30]

CH3 CH3 CH3 CCSiH3 12 CH3 S13CH3 CH3 N=CHCH3 CH3 PO(OCH3 )2 CH3 CONHCH3 CH3 SOCH3 cis-2,3-DMO CH3 CHFCH3 CH3 PO(OCH3 )2 CH3 SSCH3 CH3 SSCH3 trans-2,3-DMO

Type Group label b 1a 1b G 18 PI 2a Gruppe 18 2a G 18 2a G'18 PI c 2a G 18 PI 2b C-3  C+3v 2b 2b C-3  C+3v 2b G 18 PI c 2c C-3v  C+3 2c C-3v  C+3 2c C-3v  C+3 2d

Model [44]

Cs(n) Cs(n) Cs(n) Cs(n) Cs(e) Cs(e) Cs(e) Cs(e) C2 (e) C2 (e) C2 (e) Ci(e)

AN a [3]D3 [3]C3v [33]C's [33]C's [33]C's [33]C's [33]Cs [33]Cs [33]Cs [33]Cs [33]C2 [33]C2 [33]C2 [33]Ci

Abstract group C3  D3 D3,3 d D3,3 D3,3 D3,3 D3,3 C3  D3 C3  D3 C3  D3 C3  D3 C3  D3 C3  D3 C3  D3 C3  D3

a

Alternate notation (section 3.c) Pure permutation group c The PI group of CH3 PO(OCH3 )2 is G 54 = C3  G'18 , and G 18 = C3v  C3 is an approximate PI group of G54 d D3,3 : A generalized dihedral group with generating elements C, D, and A, and C3 = D3 = A2 , CD = DC, ACjDk = CjDkA. b

21

Table III Symmetry groups isomorphic to G36 Molecule Ref. CH3 -CO-CH3 [17], [22] CH3 -CH3 [1] CH3 -CC-CH3 [26] CH3 -CO-CH3 [25] t-CH3 -CHCH-CH3 [30] (NH3 )2 [54], [43] CH3 -O-O-CH3 [53] (CH3 OH)2 [35] CH3 -CH2 -CH2 -CH3 this work p-CH3-C6 H4-CH3 [55] h a b c d e f g h

LAMs a 2 1 1 2 6 3 8 3 2

b

G 36 G 36 † G 36 G 36 G 36 † G 36

PG Eq c C2v D3d ? C2v C2h Cs C2 C1 C2h , C2 C2h , C2v

PG alt Eq D3h D3d , D3h

C2h C2h , C2v C2h C2v

d

PG ER D2 D6 g D6 g D2 n C2 C2 D2 g C2 D2 g D2

e

AN f [33]C2v [3]D6h g [3]D6h g [33]C2v [33]C2h [33]C2h [331]D2h g [33]C2h [331]D2h g [33]D2h

Number of LAMs PI group symbol used in reference PG Eq: Point group of equilibrium structure PG alt Eq: Point group of alternate equilibrium structures PG ER: Point group of equivalent rotations Alternate notation (section 3.c) Extended PI group G 36 † whose order is 72 Group of order 72 is NOT a double group! [55]

22

Table IV Character table of G18 = [33]C's and correlation to the point groups Cs and C1 G18 [49] A1 A2 E1 E2 E3 E4

[33]C's [31] 00+ 0010 01 11 12

E 1 1 1 2 2 2 2

(456) 2 1 1 2 -1 -1 -1

(123) 2 1 1 -1 2 -1 -1

(123)(456) (123)(465) (23)(56)* 2 2 9 1 1 1 1 1 -1 -1 -1 0 -1 -1 0 2 -1 0 -1 2 0

Cs A' 1 0 1 1 1 1

A" 0 1 1 1 1 1

C1 A 1 1 2 2 2 2

23

Table V Character table of G6 = [3]Cs and correlation to the point groups Cs and C1 G6

[3]Cs

A1 A2 E

0+ 01

E 1 1 1 2

(123) 2 1 1 -1

(23)* 3 1 -1 0

Cs A' 1 0 1

A" 0 1 1

C1 A 1 1 2

24

Table VI Splitting of rovibronic energy levels for equilibrium symmetry Cs or C1 in the molecular symmetry groups G18 = [33]C's and G 6 = [3]Cs Equilibrium symmetry Cs A' Cs A" C1 A

G18 = [33]C's A1 + E1 + E2 + E3 + E4 A2 + E1 + E2 + E3 + E4 A1 + A2 +2 E1 + 2 E2 + 2 E3 + 2 E4

G 6 = [3]Cs A1 + E A2 + E A1 + A2 + 2 E

25

Table VII Torsional energy levels for pinacolone according to [56] Assignment 0A 0E 1E 1A 2A 2E 3E 3A a b

Energy (GHz) 2798.7 2816.8 2868.5 2880.2 4019.6 4075.2 4238.1 4317.5

Degeneracy 2+1=3 2+2+2=6 2+2+2=6 2+1=3 2+1=3=6 2+2+2=6 2+2+2=6 2+1=3

a

b

0A1 (+) 0E(+) 0E(-) 0A2 (-) 1A1 (+) 1E(+) 1E(-) 1A2 (-)

1A1 (-), 1A2 (+), 1A2 (-) 1E(-), 1E(+) 1A2 (+), 1A1 (-), 1A1 (+)

Proposed revised assignment (see text) Other possibilities for revised assignment

26

Correlation derived splitting patterns (Part 1) a

Table VIII Table VIII-1 [33]C1 G9 dim C1 Table VIII-2 [33]C's G18 dim Cs C1 Table VIII-3 [33]Cs G18 dim Cs C1 Table VIII-4 [33]C2 dim C2 C1 a

b c

Splitting patterns from G9 = [33]C1 [31] [49] A

[31] [49] A' A" A

[31] [37] A' A" A

00 11 b 12 b 01 b 10 b AA EE EE* AE EA 1 1 1 1 1 1 1 1 1 1 Splitting patterns from G18 = [33]C's

NC c

00+ A1 1 1

NC c

00A2 1

11 12 01 10 E3 E4 E2 E1 2 2 2 2 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 Splitting patterns from G18 = [33]Cs

00+ A1 1 1

00A2 1 1 1

1

11 E 2 1 1 2

12+ b A11 1 1

12- b A21 1 1

01 b E2 1 1 2

5

5 5 10

NC c

4 4 8

Splitting patterns from G18 = [33]C2 [31] A B A

00+ 1 1 1

001 1 1

11+ b 11- b 1 1 1 1 1 1

12 2 1 1 2

01 b 2 1 1 2

NC c 4 4 8

In Tables VIII and IX, the operators F in Ref. [31] correspond to the PI operators of the methyl group H atoms as follows: F1 ↔ (123); F2 ↔ (456); F3 , F'3 , F 3 ↔ (23)(56)*; F4 ↔ (14)(25)(36); F3 F4 , F'3 F4 , F 3 F4 ↔ (14)(26)(35)* Separately degenerate pairs Number of components

27

Table IX Table IX-1 [33]C2v G36 C2v

C2 Cs(e) Cs(n)

Correlation derived splitting patterns (Part 2) a Splitting patterns from G36 = [33]C2v [31] [2] dim A1 A2 B1 B2 A B A' A" A' A" 00

00++ A1 1 1

00+A3 1

00-A4 1

11+ E3 2 1 1

1

11E4 2

1 1

1 1 1 1

12+ E1 2 1

1 1

1 1 1

1 1 2

2 1 1 1 1 2

1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 Splitting patterns from G36 = [33]C2h 00+- 00-+ A3 A2 1 1

00-A4 1

1

11+ E3 2 1 1

1 2 1 1 1 1 1

1 1 1

1 1

1

1 1 1 1 2

12+ E1 2 1

2 1 1 2

12E2 2 1

1 1

1 1

11E4 2

2 1 1 1 1 2

1 1 1 2 1 1 2

NC c

01 G 4 1 1 1 1 2 2 2 2 2 2 4

1

2 1 1

12E2 2 1

1

C1 1 Table IX-2 [33]C2h [31] 00++ G36 [2] A1 dim 1 C2h Ag 1 Au Bg Bu C2 A 1 B Ci(e) Ag 1 Au Cs(n) A' 1 A" C1 A 1 a, c

00-+ A2 1

1 1 1 2 1 1 2

4 4 4 4 8 8 8 8 8 8 16

01 G 4 1 1 1 1 2 2 2 2 2 2 4

NC c

4 4 4 4 8 8 8 8 8 8 16

See footnotes to Table VIII

28

Table X Correlation derived splitting patterns from G72 = [33]D2h [33]D2h G72 D2h (M)

C2v(x)

C2v(y)

C2h (x)

C2h (y)

a

[55] [55] dim Ag Au B1g B1u B2u B2g B3u B3g A1 A2 B1 B2 A1 A2 B1 B2 Ag Au Bg Bu Ag Au Bg Bu

00g++ 00g+A1 ' A3 ' 1 1 1 1

00g-+ A2 ' 1

00g-A4 ' 1

1 1

11g+ 11gE3 ' E4 ' 2 2 1 1 1 1

12g+ 12gE1 ' E2 ' 2 2 1 1 1 1

01g G' 4 1 1 1 1

00u++ 00u+A1" A3" 1 1

00u-+ A2" 1

00u-A4 " 1

1

11u+ E3" 2

1 1

1 1 1 1

1 1

1 1 1

1 1

1

1 1

1 1 1

1

1 1

1 1 1 1 1

1 1

1 1

1

1 1

1 1

1

1 1

1 1

1

1

1 1 1 1

1 1

1 1

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 1

1 1

1 1

1

1 1

1 1

1

1 1

1 1

1

1 1

1 1

1

1 1

1 1

1

1 1

1 1

1

12uE2" 2

1

1 1

1

12u+ E1" 2

1 1 1

1 1

11uE4" 2

1 1 1 1

1 1

1 1

01u NC a G" 4 4 4 4 4 1 4 1 4 1 4 1 4 1 8 1 8 1 8 1 8 1 8 1 8 1 8 1 8 1 8 1 8 1 8 1 8 1 8 1 8 1 8 1 8

Number of components 29

Figure captions Fig. 1.

Structures and H-atom labels for acetone (left) and propane (right)

Fig. 2.

Structures and H-atom labels for 2,3-dimethyl oxirane: trans (left) and cis (right)

Fig. 3.

Structures and H-atom labels for 2-butene: trans (left) and cis (right)

30

Fig. 1

31

Fig. 2

32

Fig. 3

33

Molecular symmetry: Why permutation-inversion (PI) groups don’t render the point groups obsolete Peter Groner

Graphical abstract

34

Molecular symmetry: Why permutation-inversion (PI) groups don’t render the point groups obsolete Peter Groner

Highlights Molecular symmetry group for LAMs in historical context Nomenclature and notation Alternate label for molecular symmetry group involving point group Tunneling splitting pattern by correlation to point group Parallels between molecular symmetry groups and space groups

35