Yves Jean - Molecular Orbitals of Transition Metal Complexes (793957), страница 45
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Although a rotation of 3×2π/3 (=360◦ ) around the z-axis doesleave the positions of the five fluorine atoms unchanged, an odd numberof reflections (three) in the xy plane interchanges the positions of theatoms F4 and F5 (σh3 = σh ). We therefore obtain S33 = σh . To obtain theidentity operation, the operation S36 must be performed (a rotation oftwice 360◦ and an even number (six) of reflections in the plane). Of thesix S3m (m = 1–6) that seem possible, only the S31 and S35 operation needto be considered, as the others are equivalent to symmetry operationsassociated with the C3 -axis the σh plane or the identity operation in thecase of S36 .zF4F1PF4F2 120°F3xy F3F5PF5F5F1F2refl.F3PF1F2F46-136.2.
Symmetry groups6.2.1. DefinitionsThe set of symmetry operations associated with a molecule makes up agroup, in the mathematical sense of the term. One can indeed verify (i) IfA, B, and C are three symmetry operations, the relation A(BC) = (AB)Cholds, that is, the combination of symmetry operations is associative,(ii) there is a null element (the identity operation E): AE = EA = A,(iii) for each symmetry operation A, there is an inverse operation A−1such that A−1 A = AA−1 = E (see Exercise 6.5).Symmetry groupsMoreover, all the symmetry elements intersect at a single point(e.g. the oxygen atom in dimethylether (6-1), the nitrogen atom inammonia (6-7), the middle of the carbon–carbon bond in ethylene(6-5)).
The expression point-group symmetry is therefore used. Foreach molecule, there is a corresponding point group that completelycharacterizes its symmetry properties.The different symmetry operations can be grouped into classes. Froma mathematical point of view, two symmetry operations A and B belongto the same class if there is a symmetry operation X in the group, suchthat B = X −1 AX.
Verification of this relationship can be tedious, and itwill be sufficient for us to note that a class gathers together symmetryoperations of the same ‘nature’, even if this term is rather vague: asexamples, we may give the rotations C31 and C32 associated with a C3 -axis,the rotations C41 and C43 associated with a C4 -axis, and the reflections intwo σv planes or in two σd planes in [PtCl4 ]2− (see the comment in §6.1.3). As far as notation is concerned, one often groups together theoperations in a given class, for example, replacing C31 and C32 by 2C3 , C41and C43 by 2C4 , etc.6.2.2. Determination of the symmetry point groupA set of symmetry elements (or of symmetry operations) enables usto define a symmetry point group which is represented by a symbol.There is a simple method to determine the point group, and it is notnecessary to establish a complete list of all the symmetry operations inthe molecule under consideration.1.
Initially, check whether the molecule adopts one of the four mosteasily identifiable geometries: octahedral (e.g. [FeCl6 ]4− (6-4)), tetrahedral (e.g. CH4 ), linear with an inversion centre (e.g. CO2 ), orlinear without an inversion centre (e.g. HCN). If this is the case,the problem is solved; the symbols associated with these four pointgroups are Oh , Td , D∞h , and C∞v respectively.2. If none of these groups is appropriate, and if there is no rotationaxis, there are three possibilities: the molecule (i) does not possessand symmetry element (the point group C1 , in which the identityoperation is the only symmetry operation); (ii) possesses a plane ofsymmetry (the point group Cs ); (iii) possesses an inversion centre(the point group Ci ).3.
If the molecule possesses a single rotation axis (of order n), there arefour possibilities: (i) there is no other symmetry element besides thisaxis, with the possible exception of a co-linear axis of lower order.The point-group symbol is then Cn ; (ii) the existence of a reflectionplan σh (perpendicular to the Cn -axis) indicates that the point groupis of the type Cnh ; (iii) if there are nσv planes (that contain the CnElements of group theory and applicationsaxis), the point-group symbol is Cnv ; (iv) the presence of an improperaxis S2n , co-linear with the Cn -axis, indicates that the point group isof the type S2n .4. If the molecule possesses several rotation axes (the principal axisbeing of order n), there are three possibilities: (i) there are no othersymmetry elements besides these axes (Dn point-groups); (ii) thereare nσd planes that bisect the C2 axes (Dnd point groups); (iii) there isalso a σh plane, perpendicular to the principal axis (Dnh point groups).6.2.3.
Basis of an irreducible representationThe application of group theory to the construction of molecularorbitals leads us to study the way in which atomic orbitals (or linearcombinations of these orbitals) transform when the operations of thepoint group are applied. Several definitions, which are also appropriate for other types of function besides orbitals, will be presented andillustrated by some simple examples.6.2.3.1. Basis for a representationIf a set of functions f = {f1 , f2 , .
. . , fi , . . . , fn } is such that any symmetryoperation, Rk , of the group G transform one of the functions, fi , intoa linear combination of the various functions of the set f , the set issaid to be globally stable and to constitute a basis for the representation ofthe group G. As the symmetry operations maintain the positions of theatoms or interchange the positions of equivalent atoms, it can be shownthat the set of atomic orbitals (AO) of a molecule constitute a basis for therepresentation of the point-group symmetry of the molecule. In what follows,we shall adopt the usual notation in group theory, and indicate a basisfor a representation by Ŵ.6.2.3.2. Basis for an irreducible representationSuppose that a basis Ŵ, of dimension n, can be decomposed into severalbases Ŵi , whose dimensions are smaller (ni ), each of which is globallystable with respect to all the symmetry operations of the group.
Supposealso that it is not possible to decompose any of the representations Ŵiinto representations whose dimensions are smaller than ni . The reduciblerepresentation Ŵ is said to have been decomposed into a sum of irreduciblerepresentations Ŵi , which is written:Ŵ = a1 Ŵ1 ⊕ a2 Ŵ2 ⊕ · · · ⊕ am Ŵm(6.1)Symmetry groups6.2.3.3. H2 O as an exampleOH2 z6-14yH1To illustrate this point, we shall consider the set of valence AO of theatoms in the water molecule: 1sH1 and 1sH2 on the hydrogen atoms and2s, 2px , 2py , and 2pz on the oxygen atom. The water molecule possessesa two-fold rotation axis (z), and two planes of symmetry, xz and yz (6-14);its point group is therefore C2v . The set of atomic orbitals constitutes abasis (Ŵ) for the representation of this point group.We apply one of the symmetry operations of the group (E, C2 z , σxz ,and σyz ) to these orbitals.
A rotation by 180◦ around the z-axis (C2 z )interchanges the orbitals 1sH1 and 1sH2 (6-15 and 6-16). On the oxygenatom, the 2s and 2pz are transformed into themselves (the axis of revolution for 2pz is identical to the rotation axis) (6-17 and 6-18), whereas 2pxand 2py are transformed into their opposites (6-19 and 6-20).C2zOzO1s H21s H16-15C2zO1s H2Oz1s H16-162sHz2sC2zHHH6-172pzzC2HH2pzHHz6-18If the same exercise is undertaken for the three other symmetryoperations, we obtain the results that are presented in Table 6.1.Elements of group theory and applicationsz2 pxH–2pxC2zHHH6-192pyz–2pyC2H zHHH6-20Table 6.1. Transformation of the atomic orbitalsof the hydrogen and oxygen atoms of the H2 Omolecule by the action of the symmetryoperations of the C2v point groupC2vE1sH11sH22s2px2py2pz1sH11sH22s2px2py2pzC2zσxzσyz1sH21sH12s−2px−2py2pz1sH21sH12s2px−2py2pz1sH11sH22s−2px2py2pzNo matter which symmetry operation is applied, it is clear that the2s orbital on the oxygen atom is transformed into itself.
It thereforeconstitutes by itself a set that is globally stable, and so it is a basis forthe representation of the point group. As the dimension of this basisis 1, it is impossible to reduce it further: the 2s orbital is a basis of aone-dimensional irreducible representation in the C2v point group. Thesame applies for the other AO of the oxygen atom (2px , 2py , and 2pz ):the action of the symmetry operations transforms each of them eitherinto itself, or into its opposite (Table 6.1), so each of them constitutes aglobally stable set.We now examine the action of the symmetry operations on the 1sH1and 1sH2 orbitals on the hydrogen atoms (the representation ŴH ).
It isimmediately clear that neither 1sH1 nor 1sH2 constitutes by itself a set thatis globally stable, since some symmetry operations transform 1sH1 into1sH2 , and vice-versa (Table 6.1). However, the set {1sH1 , 1sH2 } is stable,and so it is a basis for a representation of the group. Moreover, if thesymmetry operations are applied to the two linear combinations (1sH1 +1sH2 ) and (1sH1 − 1sH2 ), we see that they are globally stable towards theaction of the symmetry operations, since each is transformed either intoitself or into its opposite (Table 6.2).
Both these linear combinations areSymmetry groupsTable 6.2. Transformation of the linear combinations (1sH1 + 1sH2 ) and(1sH1 − 1sH2 ) of the atomic orbitals of the hydrogen atoms in the H2 Omolecule by the action of the symmetry operations of the C2v point groupC2vEC2zσxzσyz1sH1 + 1sH2 1sH1 + 1sH2 1sH2 + 1sH11sH2 + 1sH11sH1 + 1sH21sH1 − 1sH2 1sH1 − 1sH2 −(1sH1 − 1sH2 ) −(1sH1 − 1sH2 ) 1sH1 − 1sH21These two functions are also called thesymmetry-adapted linear combinations of theatomic orbitals 1sH1 and 1sH2 (we shall returnto this point in greater detail in § 6.4).therefore bases for an irreducible representation whose dimension is 1in the C2v point group.16.2.4. Characters6.2.4.1. Representation of the groupAs we have already remarked, the action of a symmetry operation Rkon a basis for a representation f = {f1 , f2 , .
. . , fi , . . . , fn } of the grouptransforms each of the functions fi into a linear combination fi′ of thedifferent functions of the set f . This action can be represented by a matrixMk such that:Mk fi = fi′(6.2)This is an (n × n) matrix, where n is the dimension of the basis. Thecollection of matrices associated with the different symmetry operationsconstitutes a representation of the point group.6.2.4.2. Characters: the C3v point groupv (3)v (1)NH3H1H2zv (2)6-21The following symmetry elements are present in the NH3 molecule:−Hi bonds anda C3 (z) axis and three planes σv (i) that contain the N−−N−−H angle (6-21). These symmetry elementsbisect the opposite H−are characteristic of the C3v point group.Consider the orbitals 1sH1 , 1sH2 , and 1sH3 which constitute a basiswritten ŴH .
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