Molecular symmetry. An introduction to symmetry analysis

Transcription

1 Molecular symmetry 6 Symmetry governs te bonding and ence te pysical and spectroscopic properties of molecules In tis capter we explore some of te consequences of molecular symmetry and introduce te systematic arguments of group teory We sall see tat symmetry considerations are essential for constructing molecular orbitals and analysing molecular vibrations Tey also enable us to extract information about molecular and electronic structure from spectroscopic data Te systematic treatment of symmetry makes use of a branc of matematics called group teory Group teory is a ric and powerful subject, but we sall confine our use of it at tis stage to te classification of molecules in terms of teir symmetry properties, te construction of molecular orbitals, and te analysis of molecular vibrations and te selection rules tat govern teir excitation We sall also see tat it is possible to draw some general conclusions about te properties of molecules witout doing any calculations at all An introduction to symmetry analysis Tat some molecules are more symmetrical tan oters is intuitively obvious ur aim toug, is to define te symmetries of individual molecules precisely, not just intuitively, and to provide a sceme for specifying and reporting tese symmetries It will become clear in later capters tat symmetry analysis is one of te most pervasive tecniques in inorganic cemistry 61 Symmetry operations, elements and point groups Key points: Symmetry operations are actions tat leave te molecule apparently uncanged; eac symmetry operation is associated wit a symmetry element Te point group of a molecule is identified by noting its symmetry elements and comparing tese elements wit te elements tat define eac group A fundamental concept of te cemical application of group teory is te symmetry operation, an action, suc as rotation troug a certain angle, tat leaves te molecule apparently uncanged An example is te rotation of an H 2 molecule by 180º around te bisector of te HH angle (Fig 61 Associated wit eac symmetry operation tere is a symmetry element, a point, line, or plane wit respect to wic te symmetry operation is performed Table 61 lists te most important symmetry operations and teir corresponding elements All tese operations leave at least one point uncanged (te centre of te molecule, and ence tey are referred to as te operations of point-group symmetry Te identity operation, E, consists of doing noting to te molecule Every molecule as at least tis operation and some ave only tis operation, so we need it if we are to classify all molecules according to teir symmetry Te rotation of an H 2 molecule by 180º around a line bisecting te HH angle (as in Fig 61 is a symmetry operation, denoted In general, an n-fold rotation is a symmetry operation if te molecule appears uncanged after rotation by 360º/n Te corresponding symmetry element is a line, an n-fold rotation axis, C n, about wic te rotation is performed Tere is only one rotation operation associated wit a axis (as in H 2 because clockwise and anticlockwise rotations by 180º are identical Te trigonal-pyramidal H 3 An introduction to symmetry analysis 61 Symmetry operations, elements and point groups 62 Caracter tables Applications of symmetry 63 Polar molecules 64 Ciral molecules 65 Molecular vibrations Te symmetries of molecular orbitals 66 Symmetry-adapted linear combinations 67 Te construction of molecular orbitals 68 Te vibrational analogy Representations 69 Te reduction of a representation 610 Projection operators FURTHER READIG EXERCISES PRBLEMS 180 Figure 61 An H 2 molecule may be rotated troug any angle about te bisector of te HH bond angle, but only a rotation of 180 (te operation leaves it apparently uncanged

2 180 6 Molecular symmetry 120 C 3 Table 61 Symmetry operations and symmetry elements Symmetry operation Symmetry element Symbol Identity wole of space E Rotation by 360 /n n-fold symmetry axis C n Reflection mirror plane Inversion centre of inversion i Rotation by 360 /n followed by n-fold axis of improper rotation S n reflection in a plane perpendicular to te rotation axis 120 C 3 C 3 2 Figure 62 A treefold rotation and te corresponding C 3 axis in H 3 Tere are two rotations associated wit tis axis, one troug 120 (C 3 and one troug 240 (C 32 C 4 v d ote te equivalences S 1 = and S 2 = i molecule as a treefold rotation axis, denoted C 3, but tere are now two operations associated wit tis axis, one a clockwise rotation by 120º and te oter an anticlockwise rotation by 120º (Fig 62 Te two operations are denoted C 3 and C 3 2 (because two successive clockwise rotations by 120º are equivalent to an anticlockwise rotation by 120º, respectively Te square-planar molecule XeF 4 as a fourfold C 4 axis, but in addition it also as two pairs of twofold rotation axes tat are perpendicular to te C 4 axis: one pair ( passes troug eac trans-fxef unit and te oter pair ( passes troug te bisectors of te FXeF angles (Fig 63 By convention, te igest order rotational axis, wic is called te principal axis, defines te z-axis (and is typically drawn vertically Te reflection of an H 2 molecule in eiter of te two planes sown in Fig 64 is a symmetry operation; te corresponding symmetry element, te plane of te mirror, is a mirror plane, Te H 2 molecule as two mirror planes tat intersect at te bisector of te HH angle Because te planes are vertical, in te sense of containing te rotational (z axis of te molecule, tey are labelled wit a subscript v, as in v and v Te XeF 4 molecule in Fig 63 as a mirror plane in te plane of te molecule Te subscript signifies tat te plane is orizontal in te sense tat te vertical principal rotational axis of te molecule is perpendicular to it Tis molecule also as two more sets of two mirror planes tat intersect te fourfold axis Te symmetry elements (and te associated operations are denoted v for te planes tat pass troug te F atoms and d for te planes tat bisect te angle between te F atoms Te d denotes diedral and signifies tat te plane bisects te angle between two axes (te FXeF axes To understand te inversion operation, i, we need to imagine tat eac atom is projected in a straigt line troug a single point located at te centre of te molecule and ten out to an equal distance on te oter side (Fig 65 In an octaedral molecule suc as SF 6, wit te point at te centre of te molecule, diametrically opposite pairs of atoms at te corners of te octaedron are intercanged Te symmetry element, te point troug Figure 63 Some of te symmetry elements of a square-planar molecule suc as XeF 4 v 5 1 i 2 3 v' Figure 64 Te two vertical mirror planes v and v in H 2 and te corresponding operations Bot planes cut troug te axis 3 2 Figure 65 Te inversion operation and te centre of inversion i in SF 6 1 5

3 An introduction to symmetry analysis 181 wic te projections are made, is called te centre of inversion, i For SF 6, te centre of inversion lies at te nucleus of te S atom Likewise, te molecule C 2 as an inversion centre at te C nucleus However, tere need not be an atom at te centre of inversion: an 2 molecule as a centre of inversion midway between te two nitrogen nuclei An H 2 molecule does not possess a centre of inversion o tetraedral molecule as a centre of inversion Altoug an inversion and a twofold rotation may sometimes acieve te same effect, tat is not te case in general and te two operations must be distinguised (Fig 66 An improper rotation consists of a rotation of te molecule troug a certain angle around an axis followed by a reflection in te plane perpendicular to tat axis (Fig 67 Te illustration sows a fourfold improper rotation of a CH 4 molecule In tis case, te operation consists of a 90º (tat is, 360 /4 rotation about an axis bisecting two HCH bond angles, followed by a reflection troug a plane perpendicular to te rotation axis eiter te 90º (C 4 operation nor te reflection alone is a symmetry operation for CH 4 but teir overall effect is a symmetry operation A fourfold improper rotation is denoted S 4 Te symmetry element, te improper-rotation axis, S n (S 4 in te example, is te corresponding combination of an n-fold rotational axis and a perpendicular mirror plane An S 1 axis, a rotation troug 360º followed by a reflection in te perpendicular plane, is equivalent to a reflection alone, so S 1 and are te same; te symbol is generally used rater tan S 1 Similarly, an S 2 axis, a rotation troug 180º followed by a reflection in te perpendicular plane, is equivalent to an inversion, i (Fig 68; te symbol i is employed rater tan S 2 EXAMPLE 61 Identifying symmetry elements Identify te symmetry elements in te eclipsed and staggered conformations of an etane molecule Answer We need to identify te rotations, reflections, and inversions tat leave te molecule apparently uncanged Don t forget tat te identity is a symmetry operation By inspection of te molecular models, we see tat te eclipsed conformation of a CH 3 CH 3 molecule (1 as te elements E, C 3,,, and S 3 Te staggered conformation (2 as te elements E, C 3, d, i, and S 6 Self-test 61 Sketc te S 4 axis of an H 4 ion How many of tese axes does te ion possess (a i i (b Figure 66 Care must be taken not to confuse (a an inversion operation wit (b a twofold rotation Altoug te two operations may sometimes appear to ave te same effect, tat is not te case in general H C C 3 1 A C 3 axis (1 Rotate S 1 C 4 (2 Reflect H σ (a C (1 Rotate S 2 S 6 (2 Reflect i 2 An S 6 axis (b Figure 67 A fourfold axis of improper rotation S 4 in te CH 4 molecule Figure 68 (a An S 1 axis is equivalent to a mirror plane and (b an S 2 axis is equivalent to a centre of inversion

4 182 6 Molecular symmetry Te assignment of a molecule to its point group consists of two steps: 1 Identify te symmetry elements of te molecule 2 Refer to Table 62 Table 62 Te composition of some common groups Point group Symmetry elements Sape Examples C 1 E SiHClBrF E, H 2 2 C s E, HF 2 v E, S 2 Cl 2, H 2 C 3v E, 2C 3, 3 v H 3, PCl 3, PCl 3 C v E,, 2C, v CS, C, HCl D 2 E, 3, i, 3 2 4, B 2 H 6 D 3 E, 2C 3, 3,, 2S 3, 3 v BF 3, PCl 5 D 4 XeF E, 2C4,, 2, 2 4,, i, 2S 4,, 2 v, 2 d trans-[ma 4 B 2 ] D E,, 2C, i, v, 2S C 2, H 2, H 2 T d E, 8C 3, 3, 6S 4, 6 d CH 4, SiCl 4 E, 8C 3, 6, 6C 4, 3, i, 6S 4, 8S 6, 3, 6 d SF 6 In practice, te sapes in te table give a very good clue to te identity of te group to wic te molecule belongs, at least in simple cases Te decision tree in Fig 69 can also be used to assign most common point groups systematically by answering te questions at eac decision point Te name of te point group is normally its Scoenflies symbol, suc as v for a water molecule EXAMPLE 62 Identifying te point group of a molecule To wat point groups do H 2 and XeF 4 belong Answer We need to work troug Fig 69 (a Te symmetry elements of H 2 are sown in Fig 610 H 2 possesses te identity (E, a twofold rotation axis (, and two vertical mirror planes ( v and v Te set

5 An introduction to symmetry analysis 183 of elements (E, corresponds to te group v (b Te symmetry elements of XeF 4 are sown in Fig 63 XeF 4 possesses te identity (E, a fourfold axis (C 4, two pairs of twofold rotation axes tat are perpendicular to te principal C 4 axis, a orizontal reflection plane in te plane of te paper, and two sets of two vertical reflection planes and d Tis set of elements identifies te point group as D 4 Self-test 62 Identify te point groups of (a BF 3, a trigonal-planar molecule, and (b te tetraedral S 4 2 ion v' D i C v Linear groups Molecule Linear i Two or more C n, n > 2 σ C n Select C n wit igest n; ten is n C n σ nσ d nσ v S 2n σ i v' Figure 610 Te symmetry elements of H 2 Te diagram on te rigt is te view from above and summarizes te diagram on te left v v C 5 D n D nd D n C n C nv C s C i C 1 C S 2n C n I T d Cubic groups 3 C 2 (D Figure 69 Te decision tree for identifying a molecular point group Te symbols of eac point refer to te symmetry elements C S It is very useful to be able to recognize immediately te point groups of some common molecules Linear molecules wit a centre of symmetry, suc as H 2, C 2 (3, and HC CH belong to D A molecule tat is linear but as no centre of symmetry, suc as HCl or CS (4 belongs to C v Tetraedral (T d and octaedral ( molecules ave more tan one principal axis of symmetry (Fig 611: a tetraedral CH 4 molecule, for instance, as four C 3 axes, one along eac CH bond Te and T d point groups are known as cubic groups because tey are closely related to te symmetry of a cube A closely related group, te icosaedral group, I, caracteristic of te icosaedron, as 12 fivefold axes (Fig 612 Te icosaedral group is important for boron compounds (Section 1311 and te C 60 fullerene molecule (Section 146 Te distribution of molecules among te various point groups is very uneven Some of te most common groups for molecules are te low-symmetry groups C 1 and C s Tere are many examples of polar molecules in groups v (suc as S 2 and C 3v (suc as H 3 Tere are many linear molecules, wic belong to te groups C v (HCl, CS and D (Cl 2 and C 2, and a number of planar-trigonal molecules, D 3 (suc as BF 3, 5, trigonal-bipyramidal molecules (suc as PCl 5, 6, wic are D 3, and square-planar molecules, D 4 (7 So-called octaedral molecules wit two identical substituents opposite eac oter, as in (8, are also D 4 Te last example sows tat te point-group classification of a molecule is more precise tan te casual use of te terms octaedral or tetraedral tat indicate molecular geometry For instance, a molecule may be called octaedral (tat is, it as octaedral geometry even if it as six different groups attaced to te central atom However, te octaedral molecule belongs to te octaedral point group only if all six groups and te lengts of teir bonds to te central atom are identical and all angles are 90º (a 4 CS (C v 62 Caracter tables Key point: Te systematic analysis of te symmetry properties of molecules is carried out using caracter tables We ave seen ow te symmetry properties of a molecule define its point group and ow tat point group is labelled by its Scoenflies symbol Associated wit eac point group is (b Figure 611 Sapes aving cubic symmetry (a Te tetraedron, point group T d (b Te octaedron, point group

6 184 6 Molecular symmetry C 5 F Cl B P Figure 612 Te regular icosaedron, point group I, and its relation to a cube 5 BF 3 (D 3 6 PCl 5 (D 3 Cl Pt 7 [PtCl 4 ] 2 (D 4 2 a caracter table A caracter table displays all te symmetry elements of te point group togeter wit a description, as we explain below, of ow various objects or matematical functions transform under te corresponding symmetry operations A caracter table is complete: every possible object or matematical function relating to te molecule belonging to a particular point group must transform like one of te rows in te caracter table of tat point group Te structure of a typical caracter table is sown in Table 63 Te entries in te main part of te table are called caracters, (ci Eac caracter sows ow an object or matematical function, suc as an atomic orbital, is affected by te corresponding symmetry operation of te group Tus: Caracter Significance 1 te orbital is uncanged 1 te orbital canges sign 0 te orbital undergoes a more complicated cange X M 8 trans-[mx 4 2 ] (D 4 For instance, te rotation of a p z orbital about te z axis leaves it apparently uncanged (ence its caracter is 1; a reflection of a p z orbital in te xy plane canges its sign (caracter 1 In some caracter tables, numbers suc as 2 and 3 appear as caracters: tis feature is explained later Te class of an operation is a specific grouping of symmetry operations of te same geometrical type: te two (clockwise and anticlockwise treefold rotations about an axis form one class, reflections in a mirror plane form anoter, and so on Te number of members of eac class is sown in te eading of eac column of te table, as in 2C 3, denoting tat tere are two members of te class of treefold rotations All operations of te same class ave te same caracter Eac row of caracters corresponds to a particular irreducible representation of te group An irreducible representation as a tecnical meaning in group teory but, broadly speaking, it is a fundamental type of symmetry in te group (like te symmetries represented Table 63 Te components of a caracter table ame of Symmetry Functions Furter rder of point operations R functions group, group arranged by class (E, C n, etc Symmetry Caracters ( Translations and Quadratic functions species ( components of suc as z 2, xy, etc, dipole moments (x, y, z, of relevance to Raman of relevance to IR activity activity; rotations Scoenflies symbol

7 An introduction to symmetry analysis 185 by and π orbitals for linear molecules Te label in te first column is te symmetry species (essentially, a label, like and π of tat irreducible representation Te two columns on te rigt contain examples of functions tat exibit te caracteristics of eac symmetry species ne column contains functions defined by a single axis, suc as translations or p orbitals (x,y,z or rotations (R x,r y,r z, and te oter column contains quadratic functions suc as d orbitals (xy, etc Caracter tables for a selection of common point groups are given in Resource section 4 EXAMPLE 63 Identifying te symmetry species of orbitals Identify te symmetry species of te oxygen valence-sell atomic orbitals in an H 2 molecule, wic as v symmetry Answer Te symmetry elements of te H 2 molecule are sown in Fig 610 and te caracter table for v is given in Table 64 We need to see ow te orbitals beave under tese symmetry operations An s orbital on te atom is uncanged by all four operations, so its caracters are (1,1,1,1 and tus it as symmetry species Likewise, te 2p z orbital on te atom is uncanged by all operations of te point group and is tus totally symmetric under v : it terefore as symmetry species Te caracter of te 2p x orbital under is 1, wic means simply tat it canges sign under a twofold rotation A p x orbital also canges sign (and terefore as caracter 1 wen reflected in te yz-plane ( v, but is uncanged (caracter 1 wen reflected in te xz-plane ( v It follows tat te caracters of an 2p x orbital are (1, 1,1, 1 and terefore tat its symmetry species is B 1 Te caracter of te 2p y orbital under is 1, as it is wen reflected in te xz-plane ( v Te 2p y is uncanged (caracter 1 wen reflected in te yz-plane ( v It follows tat te caracters of an 2p y orbital are (1, 1, 1,1 and terefore tat its symmetry species is B 2 Self-test 63 Identify te symmetry species of all five d orbitals of te central Xe atom in XeF 4 (D 4, Fig 63 Table 64 Te v caracter table v E v v = z x 2, y 2, z 2 A R z B x, R y xy B y, R x zx, yz Te letter A used to label a symmetry species in te group v means tat te function to wic it refers is symmetric wit respect to rotation about te twofold axis (tat is, its caracter is 1 Te label B indicates tat te function canges sign under tat rotation (te caracter is 1 Te subscript 1 on means tat te function to wic it refers is also symmetric wit respect to reflection in te principal vertical plane (for H 2 tis is te plane tat contains all tree atoms A subscript 2 is used to denote tat te function canges sign under tis reflection ow consider te sligtly more complex example of H 3, wic belongs to te point group C 3v (Table 65 An H 3 molecule as iger symmetry tan H 2 Tis iger symmetry is apparent by noting te order,, of te group, te total number of symmetry operations tat can be carried out For H 2, = 4 and for H 3, = 6 For igly symmetric molecules, is large; for example = 48 for te point group Inspection of te H 3 molecule (Fig 613 sows tat wereas te 2p z orbital is unique (it as symmetry, te 2p x and 2p y orbitals bot belong to te symmetry representation E In oter words, te 2p x and 2p y orbitals ave te same symmetry caracteristics, are degenerate, and must be treated togeter Te caracters in te column eaded by te identity operation E give te degeneracy of te orbitals: Symmetry label Degeneracy A, B 1 E 2 T 3 Table 65 Te C 3v caracter table C 3v E 2C 3 3 v = z z 2 A R z E (x, y (R x, R y (zx, yz (x 2 y 2, xy

8 186 6 Molecular symmetry Figure 613 Te nitrogen 2p z orbital in ammonia is symmetric under all operations of te C 3v point group and terefore as symmetry Te 2p x and 2p y orbitals beave identically under all operations (tey cannot be distinguised and are given te symmetry label E + p z p x p y + + E Be careful to distinguis te italic E for te operation and te roman E for te label: all operations are italic and all labels are roman Degenerate irreducible representations also contain zero values for some operations because te caracter is te sum of te caracters for te two or more orbitals of te set, and if one orbital canges sign but te oter does not, ten te total caracter is 0 For example, te reflection troug te vertical mirror plane containing te y-axis in H 3 results in no cange of te p y orbital, but an inversion of te p x orbital EXAMPLE 64 Determining degeneracy Can tere be triply degenerate orbitals in BF 3 Answer To decide if tere can be triply degenerate orbitals in BF 3 we note tat te point group of te molecule is D 3 Reference to te caracter table for tis group (Resource section 4 sows tat, because no caracter exceeds 2 in te column eaded E, te maximum degeneracy is 2 Terefore, none of its orbitals can be triply degenerate Self-test 64 Te SF 6 molecule is octaedral Wat is te maximum possible degree of degeneracy of its orbitals Applications of symmetry Important applications of symmetry in inorganic cemistry include te construction and labelling of molecular orbitals and te interpretation of spectroscopic data to determine structure However, tere are several simpler applications, one being to use group teory to decide weter a molecule is polar or ciral In many cases te answer may be obvious and we do not need to use group teory However, tat is not always te case and te following examples illustrate te approac tat can be adopted wen te result is not obvious Tere are two aspects of symmetry Some properties require a knowledge only of te point group to wic a molecule belongs Tese properties include its polarity and cirality ter properties require us to know te detailed structure of te caracter table Tese properties include te classification of molecular vibrations and te identification of teir IR and Raman activity We illustrate bot types of application in tis section 63 Polar molecules Key point: A molecule cannot be polar if it belongs to any group tat includes a centre of inversion, any of te groups D and teir derivatives, te cubic groups (T,, te icosaedral group (I, and teir modifications A polar molecule is a molecule tat as a permanent electric dipole moment A molecule cannot be polar if it as a centre of inversion Inversion implies tat a molecule as matcing carge distributions at all diametrically opposite points about a centre, wic rules out a dipole moment For te same reason, a dipole moment cannot lie perpendicular to any mirror plane or axis of rotation tat te molecule may possess For example, a mirror plane demands identical atoms on eiter side of te plane, so tere can be no dipole moment across te plane Similarly, a symmetry axis implies te presence of identical atoms at points related by te corresponding rotation, wic rules out a dipole moment perpendicular to te axis

9 Applications of symmetry 187 In summary: 1 A molecule cannot be polar if it as a centre of inversion 2 A molecule cannot ave an electric dipole moment perpendicular to any mirror plane 3 A molecule cannot ave an electric dipole moment perpendicular to any axis of rotation EXAMPLE 65 Judging weter or not a molecule can be polar Te rutenocene molecule (9 is a pentagonal prism wit te Ru atom sandwiced between two C 5 H 5 rings Can it be polar Answer We sould decide weter te point group is D or cubic because in neiter case can it ave a permanent electric dipole Reference to Fig 69 sows tat a pentagonal prism belongs to te point group D 5 Terefore, te molecule must be nonpolar Self-test 65 A conformation of te ferrocene molecule tat lies 4 kj mol 1 above te lowest energy configuration is a pentagonal antiprism (10 Is it polar Ru 9 Fe Ciral molecules Key point: A molecule cannot be ciral if it possesses an improper rotation axis (S n A ciral molecule (from te Greek word for and is a molecule tat cannot be superimposed on its own mirror image An actual and is ciral in te sense tat te mirror image of a left and is a rigt and, and te two ands cannot be superimposed A ciral molecule and its mirror image partner are called enantiomers (from te Greek word for bot parts Ciral molecules tat do not interconvert rapidly between enantiomeric forms are optically active in te sense tat tey can rotate te plane of polarized ligt Enantiomeric pairs of molecules rotate te plane of polarization of ligt by equal amounts in opposite directions A molecule wit an improper rotation axis, S n, cannot be ciral A mirror plane is an S 1 axis of improper rotation and a centre of inversion is equivalent to an S 2 axis; terefore, molecules wit eiter a mirror plane or a centre of inversion ave axes of improper rotation and cannot be ciral Groups in wic S n is present include D n, D nd, and some of te cubic groups (specifically, T d and Terefore, molecules suc as CH 4 and i(c 4 tat belong to te group T d are not ciral Tat a tetraedral carbon atom leads to optical activity (as in CHClFBr sould serve as anoter reminder tat group teory is stricter in its terminology tan casual conversation Tus CHClFBr (11 belongs to te group C 1, not to te group T d ; it as tetraedral geometry but not tetraedral symmetry Wen judging cirality, it is important to be alert for axes of improper rotation tat migt not be immediately apparent Molecules wit neiter a centre of inversion nor a mirror plane (and ence wit no S 1 or S 2 axes are usually ciral, but it is important to verify tat a iger-order improper-rotation axis is not also present For instance, te quaternary ammonium ion (12 as neiter a mirror plane (S 1 nor an inversion centre (S 2, but it does ave an S 4 axis and so it is not ciral H F Br Cl 11 CHClFBr (C 1 CH 3 H CH [(CH 2 CH(CH 3 CH(CH 3 CH 2 2 ] + EXAMPLE 66 Judging weter or not a molecule is ciral Te complex [Mn(acac 3 ], were acac denotes te acetylacetonato ligand (CH 3 CCHCCH 3, as te structure sown as (13 Is it ciral acac Answer We begin by identifying te point group in order to judge weter it contains an improper-rotation axis eiter explicitly or in a disguised form Te cart in Fig 69 sows tat te ion belongs to te point group D 3, wic consists of te elements (E, C 3, 3 and ence does not contain an S n axis eiter explicitly or in a disguised form Te complex ion is ciral and ence, because it is long-lived, optically active Self-test 66 Is te conformation of H 2 2 sown in (14 ciral Te molecule can usually rotate freely about te bond: comment on te possibility of observing optically active H 2 2 Mn 13 [Mn(acac 3 ] (D 3d