Yves Jean - Molecular Orbitals of Transition Metal Complexes (793957), страница 47
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Characters of a reducible representationzyxpz1pz2We need intially to establish the characters of a reducible representationŴ. For a given symmetry operation R, only the diagonal terms of thematrix associated with this operation contribute to the character χR (§6.2.4.2). If the symmetry operation transforms the orbital under consideration into itself, the contribution to the character is +1.
If, however,it transforms the orbital into its opposite, the contribution is −1. If theorbital is transformed into another one, the contribution is zero. Wehave already given an example of the calculation of these characters forthe 1sH orbitals on the hydrogen atoms in NH3 (Table 6.4). We shallnow consider tow other examples, for the molecules H2 O and C2 H4 .Consider the basis ŴH constituted by the two orbitals 1sH1 and 1sH2on the hydrogen atoms in the water molecule. From Table 6.1, we noticethat the two orbitals are transformed into themselves by the operations Eand σyz (characters equal to 1+1 = 2), whereas they are interchanged bythe operations C2z and σxz (characters equal to 0+0 = 0).
The charactersassociated with the basis ŴH , listed in Table 6.7, do not correspond toany of the irreducible representations of the C2v point group, which areall one-dimensional (Table 6.5). This is therefore a basis for a reduciblerepresentation.The examples of H2 O and NH3 (Table 6.4) illustrate a characteristicproperty of the ŴH representations constituted by the set of 1sH1 orbitalson the hydrogen atoms of a molecule. An orbital of this type can only betransformed either into itself (contribution to the character equal to +1)or into an orbital located on another hydrogen atom (contribution to thecharacter of 0). To determine the character associated with a particularsymmetry operation, it is therefore sufficient to count the number ofhydrogen atoms that are left unchanged by this operation. This commentis also applicable to other orbitals that possess this property (e.g. thes orbitals on heavy atoms that are equivalent by symmetry).In other cases, it is necessary to examine carefully the ways in whichthe orbitals are transformed by the symmetry operations.
Consider, forexample, the ethylene molecule (6-22). It belongs to the D2h point groupyand contains the following symmetry elements: the C2x , C2 , and C2z axes,6-22Table 6.7. Characters associated withthe representation ŴH constituted bythe 1sH1 and 1sH2 orbitals of thehydrogen atoms in the H2 O molecule(C2v point group)C2vEC2zσxzσyzŴH2002Elements of group theory and applicationsan inversion center i located in the middle of the carbon–carbon bondand the three planes σxy , σxz , and σyz (§ 6.2.2).The results of applying the symmetry operations to the pz1 and pz2orbitals are given in Table 6.8. It is clear that an orbital can be transformedinto its opposite even if the symmetry operations does not move the atom onwhich it is found (e.g. the action of C2x transform pz1 into −pz1 , eventhough the position of the C1 atom is unaltered).
The characters associated with the basis Ŵpz cannot therefore be obtained simply by countingthe number of carbon atoms whose position is left unchanged by eachsymmetry operation, in contrast to those of the ŴH representation(Table 6.8).Table 6.8. Transformation of the atomic orbitals pz1 and pz2 on the carbonatoms of the ethylene molecule (6-22) by the action of the symmetryoperations of the D2h point group, and the characters of the Ŵpz and ŴHrepresentationsyD2hEC2zC2C2xiσxyσxzσyzpz1pz2ŴpzŴHpz1pz224pz2pz100−pz2−pz100−pz1−pz2−20−pz2−pz100−pz1−pz2−24pz1pz220pz2pz1006.3.3. Applications6.3.3.1. H2 O as an exampleWe shall now apply the reduction formula (6.5) to some of the reduciblerepresentations that we have already studied.
Table 6.9 contains thecharacter table for the C2v point group and the characters that we haveobtained for the ŴH basis that is formed by the two 1sH1 and 1sH2 orbitalsin the H2 O molecule (Table 6.7).Table 6.9. Characters of the irreduciblerepresentations of the C2v point group andof the representation ŴH in the H2 OmoleculeC2vEC2zσxzσyzA1A2B1B2ŴH1111211−1−101−11−101−1−112The reduction formulaAs the order of the group (h, the number of symmetry operations)is 4, we obtain:1aA1 = [(2 × 1 × 1) + (0 × 1 × 1) + (0 × 1 × 1) + (2 × 1 × 1)] = 141aA2 = [(2 × 1 × 1) + (0 × 1 × 1) − (0 × 1 × 1) − (2 × 1 × 1)] = 041aB1 = [(2 × 1 × 1) + (0 × 1 × 1) + (0 × 1 × 1) − (2 × 1 × 1)] = 041aB2 = [(2 × 1 × 1) − (0 × 1 × 1) + (0 × 1 × 1) + (2 × 1 × 1)] = 14which leads to the expressionŴH = A1 ⊕ B2(6.6)6.3.3.2.
NH 3 as an exampleIn the same way, we can decompose the ŴH basis in the NH3 molecule(Table 6.10): its characters have already been obtained (Table 6.4):Table 6.10. Characters of the irreduciblerepresentations of the C3v point groupand of the ŴH representation in theNH3 moleculeC3vE2C33σvA1A2EŴH112311−101−101We obtain1aA1 = [(3 × 1 × 1) + (0 × 1 × 2) + (1 × 1 × 3)] = 161aA2 = [(3 × 1 × 1) + (0 × 1 × 2) + (1 × (−1) × 3)] = 061aE = [(3 × 2 × 1) + (0 × (−1) × 2) + (1 × 0 × 3)] = 16soŴH = A1 ⊕ E(6.7)The three-dimensional representation ŴH is therefore decomposedinto a one-dimensional (A1 ) and a two-dimensional (E) irreduciblerepresentation.Elements of group theory and applications6.3.4.
Direct productsIt is often useful to be able to determine the symmetry of a functionthat is a product of two (or more) functions whose symmetry is alreadyknown. This need arises, for example, when we consider a polyelectronicwave function that is written as a product of monoelectronic functions(atomic or molecular orbitals), or when we are interested in the overlapbetween two orbitals (see § 6.5.1).Consider two functions φ1 and φ2 , whose symmetries are B1 and B2 ,respectively, in the C2v point group. The characters associated with theproduct function (φ1 × φ2 ) are obtained, for each symmetry operationR, from the product of the characters χR (B1 ) and χR (B2 ) of the B1and B2 representations. Inspection of the character table for the C2vpoint group (Table 6.5) shows that the characters that are obtainedfor the product function are identical to those of the A2 irreduciblerepresentation (Table 6.11).Table 6.11.
Direct product of the B1 and B2irreducible representations in the C2vpoint groupC2vEC2zσxzσyzχR (B1 )χR (B2 )χR (B1 ) × χR (B2 )χR (A2 )1111−1−1111−1−1−1−11−1−1This result can be written as follows:B1 × B2 = A2(6.8)Note that if the two functions φ1 and φ2 have the same symmetryin a group whose irreducible representations are one-dimensional, theproduct of the characters is necessarily equal to 1 for all the symmetryoperations.
The product function is therefore a basis for the totallysymmetric representation (A1 in the case of the C2v point group). Butthe situation is more complicated when the point group contains atwo- or three-dimensional (degenerate) representation. Consider, forexample, the product E × E in the C3v point group (Table 6.12). Thecharacters that are obtained do not correspond to any of the irreduciblerepresentations of this group (see Table 6.6).
The representation E × Ein fact forms a basis for a reducible four-dimensional representation;when this is reduced, the totally symmetric representation (A1 ) of theC3v group appears as one component.E × E = A1 ⊕ A 2 ⊕ E(6.9)Symmetry-adapted orbitalsTable 6.12. The E × E directproduct in the C3v point groupC3vE2C33σvχR (E)χR (E) × χR (E)24−1100In general, the product of two functions of the same symmetry is eithera basis for the totally symmetric representation of the group, or is a basisfor a reducible representation that contains it.
In contrast, the product oftwo functions whose symmetries are different never contains the totallysymmetric representation.6.4. Symmetry-adapted orbitalsOnce the decomposition of a reducible representation into a sum ofirreducible representations has been achieved, the following step consistsof finding the linear combination of orbitals that are bases for theseirreducible representations. These are often referred to as symmetryadapted linear combinations of orbitals (SALCO).6.4.1. Projection operatorTo find these combinations, we use a ‘projection operator’ P, whoseaction on a function φ is defined as follows (without consideringnormalization):Pφ =2If the character is complex, the complexconjugate is taken.χi (Rk )Rk φk(6.10)where χi (Rk ) is the character associated with the operation Rk for theirreducible representation being considered, Ŵi ,2 Rk is the symmetryoperation whose character is χi (Rk ) and φ is one of the orbitals of thereducible representation or a linear combination of these orbitals.
It iscalled a generating function.Note that the summation (k) in this formula is carried out over allthe symmetry operations. If there are several operations in a given class ofthe group, we must take account of each of them.6.4.2. Application6.4.2.1. The ŴH basis in H2 OA simple example will show how this formula is used. Consider the ŴHbasis that is constituted by the two 1sH1 and 1sH2 orbitals in the H2 OElements of group theory and applicationsmolecule.
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