Yves Jean - Molecular Orbitals of Transition Metal Complexes (793957), страница 49
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We shall suppose that all theligands, and thus all the orbitals σi , are identical.23M416-23z6.6.1. Square-planar ML4 complexesC4,C2, S4⬘C 2bL3ML1L4⬙C 2aL2xy⬙C 2b⬘C 2a6-24Consider a complex in which the metallic atom is surrounded by fourligands that are placed at the corners of a square (6-23). The symmetry elements of this system are characteristic of the D4h point group.The axes are shown in 6-24. The planes of symmetry are xy (σh ), xz(σda ), and yz (σdb ), respectively, together with the planes that bisect xz−L bonds (σva and σvb , respectively).and yz and each contain two M−The inversion centre is of course at the origin, coincident with thecentral atom.Elements of group theory and applications6.6.1.1.
Reduction of the Ŵσ representationThe action of a symmetry operation on one of the orbitals σi can transform it only into itself or into an orbital σj on another ligand. Followingthe procedure that we have already established for bases ŴH constitutedby 1sH orbitals located on hydrogen atoms (§ 6.3.2), the character of therepresentation Ŵσ is obtained simply by counting the number of ligandswhose position stays unchanged. It is easy to see that the identity operation (E) and reflection in the molecular plane (σh ) leave the positions ofthe four ligands unchanged (χ = 4). Rotations around the C2′ axes andreflections in the σv planes maintain the positions of the two ligands situated on the symmetry element concerned (χ = 2).
The other symmetryoperations move all the ligands (χ = 0). The characters obtained forthe representation are listed in Table 6.18, together with the charactersof the irreducible representations of the D4h point group.The order of the group (h, the number of symmetry operations)is 16 (first line of Table 6.18). The reduction formula (6.5) enables usto decompose the four-dimensional representation Ŵσ into a sum ofirreducible representations, i ai Ŵi . The only non-zero values of ai are:1[(4 × 1 × 1) + (2 × 1 × 2) + (4 × 1 × 1) + (2 × 1 × 2)] = 1161aB1g = [(4 × 1 × 1) + (2 × 1 × 2) + (4 × 1 × 1) + (2 × 1 × 2)] = 1161aEu = [(4 × 2 × 1) + (2 × 0 × 2) + (4 × 2 × 1) + (2 × 0 × 2)] = 116aA1g =henceŴσ = A1g ⊕ B1g ⊕ Eu(6.14)Table 6.18. Character table for the D4h point group and characters of the reducible representation Ŵσ of asquare-planar complex ML4D4hE2C4C22C2′2C2′′i2S4σh2σv2σdA1gA2gB1gB2gEgA1uA2uB1uB2uEuŴσ1111211112411−1−1011−1−1001111−21111−201−11−101−11−1021−1−1101−1−110011112−1−1−1−1−2011−1−10−1−111001111−2−1−1−1−1241−11−10−11−11021−1−110−111−100x 2 + y2 , z2x 2 − y2xy(xz, yz)z(x, y)Symmetry-adapted orbitals in several MLn complexesThe representation Ŵσ is thus decomposed into two one-dimensionalrepresentations (A1g and B1g ) and one degenerate two-dimensionalrepresentation (Eu ).6.6.1.2.
Symmetry-adapted orbitalsApplication of the projection formula (6.10) requires that we know howone of the ligand orbitals (the generating function) is transformed byall the symmetry operations. We shall consider two generating functionsin turn, σ1 and σ2 , where the numbering of the orbitals is shown in6-23. The results of the action of the symmetry operations on these twoorbitals, Rk (σ1 ) and Rk (σ2 ), are given in Table 6.19, together with thecharacters of the irreducible representations A1g , B1g , and Eu .The symmetry-adapted functions are obtained by multiplying thefunction Rk (σ1 ) (or Rk (σ2 )) for each symmetry operation by the character of the irreducible representation considered, and adding the sumof these products for all the symmetry operations (formula (6.10)).By using σ1 as a generating function for the symmetries A1g and B1g ,then σ1 and σ2 successively for the symmetry Eu (a two-dimensional,degenerate representation), we obtain:φA1g = 4 × (σ1 + σ2 + σ3 + σ4 )φB1g = 4 × (σ1 − σ2 + σ3 − σ4 )φEu (1) = 4 × (σ1 − σ3 )φEu (2) = 4 × (σ2 − σ4 )Table 6.19.
Action of the symmetry operations of the D4h pointgroup on the orbitals σ1 and σ2 (see 6-23 and 6-24 for thenumbering of the orbitals and the symmetry elements) and thecharacters of the irreducible representations A1g , B1g , and EuD4hERk (σ1 )Rk (σ2 )A1gB1gEuσ1σ2112D4hiRk (σ1 )Rk (σ2 )A1gB1gEuσ3σ411−2′C2a′C2bσ3σ411−2σ1σ4110σ3σ2110σ4σ31−10σ2σ11−10S43σhσvaσvbσdaσdbσ2σ31−10σ1σ2112σ1σ4110σ3σ2110C41C43C2σ4σ11−10σ2σ31−10S41σ4σ11−10′′C2aσ4σ31−10′′C2bσ2σ11−10Elements of group theory and applicationswhich gives the following normalized expressions for the orbitals:ABM1gEu(1)1gMM1φA1g = (σ1 + σ2 + σ3 + σ4 )21φB1g = (σ1 − σ2 + σ3 − σ4 )21φEu (1) = √ (σ1 − σ3 )21φEu (2) = √ (σ2 − σ4 )2MorEu(2)MMFigure 6.6. Symmetry-adapted σ orbitals for asquare-planar ML4 complex.
Two differentrepresentations are shown for the degenerateEu orbitals.(6.15a)(6.15b)(6.15c)(6.15d)These symmetry-adapted orbitals are shown in Figure 6.6. The solutionfound for the degenerate Eu orbitals is not unique, since any pair ofindependent linear combinations of these orbitals is also a basis for thisrepresentation. Another choice is often made, the (normalized) sumand difference of the two functions φEu (1) and φEu (2) (right-hand sideof Figure 6.6).6.6.2. Tetrahedral ML4 complexesConsider a complex ML4 in which the four ligands are situated atthe apices of a tetrahedron. Each ligand has a σ orbital which pointstowards the metallic centre (6-25).
The symmetry elements, which arecharacteristic of the Td point group, are:41M236-25C3dL1ML4L3L26-26zC2 and S4••••−L bonds;four C3 axes, each of which is co-linear with one of the M−−M−−L angles;three C2 axes that bisect the L−three S4 axes that are co-linear with the C2 axes;−L bonds.six σd planes, each of which contains two M−An example of each of these elements is shown in 6-26.6.6.2.1.
Reduction of the Ŵσ representationTo obtain the characters that are associated with the reducible representation Ŵσ , we just need to count the number of ligands that are leftunmoved by the various symmetry operations. For the identity operation, we clearly have χ = 4. Rotation around a C3 -axis leaves one ligandin its original position (χ = 1), the one located on that axis. Rotationaround the C2 and S4 axes moves all the ligands (χ = 0). Reflection in aσd plane leaves the two ligands that are in this plane in their original positions (χ = 2).
The characters that are obtained are given in Table 6.20,together with those for the irreducible representations of the Td pointgroup.The order of the group (the number of symmetry operations, h) is24 (first line of Table 6.20). Application of the reduction formula (6.5)Symmetry-adapted orbitals in several MLn complexesTable 6.20. Character table for the Td point group and the charactersof the reducible representation Ŵσ in a tetrahedral ML4 complexTdE8C33C26S46σdA1A2ET1T2Ŵσ11233411−1001112−1−101−101−101−10−112x 2 + y2 + z2(2z2 − x 2 − y2 , x 2 − y2 )(x, y, z)(xy, xz, yz)gives:aA1 =1[(4 × 1 × 1) + (1 × 1 × 8) + (0 × 1 × 3) + (0 × 1 × 6)24+ (2 × 1 × 6)] = 1a A21= [(4 × 1 × 1) + (1 × 1 × 8) + (0 × 1 × 3) − (0 × 1 × 6)24− (2 × 1 × 6)] = 0aE =1[(4 × 2 × 1) − (1 × 1 × 8) + (0 × 2 × 3) − (0 × 0 × 6)24+ (2 × 0 × 6)] = 0a T1 =1[(4 × 3 × 1) + (1 × 0 × 8) − (0 × 1 × 3) + (0 × 1 × 6)24− (2 × 1 × 6)] = 0a T2 =1[(4 × 3 × 1) + (1 × 0 × 8) − (0 × 1 × 3) − (0 × 1 × 6)24+ (2 × 1 × 6)] = 1henceŴσ = A1 ⊕ T2(6.16)The representation Ŵσ is thus decomposed into a one-dimensional representation (A1 , the totally symmetric representation) and a degeneratethree-dimensional representation (T2 ).6.6.2.2.
Symmetry-adapted orbitalsThe linear combinations of the σi orbitals that are bases for the A1 andT2 irreducible representations can be determined from the projectionformula (6.10). Since there are many symmetry operations in this pointgroup (24), the exercise is rather tedious, and we shall be content hereElements of group theory and applicationsjust to give the result. The symmetry-adapted orbitals are:A1φA1 = (σ1 + σ2 + σ3 + σ4 )21φT2 (1) = √ (σ1 − σ2 )21φT2 (2) = √ (σ3 − σ4 )21φT2 (3) = (σ1 + σ2 − σ3 − σ4 )2M1T2(1)MT2(2)Mφ T 2(3)MMorMMFigure 6.7.
Symmetry-adapted σ orbitals in atetrahedral ML4 complex. Two differentrepresentations are given for the degenerateT2 orbitals.3126-27hL1zMC3 and S3L3 yL26-28(6.17b)(6.17c)(6.17d)These orbitals are shown in Figure 6.7.The expression of the T2 orbitals given in (6.17b–d) is not unique;as we have already seen for square-planar complexes in § 6.6.1.2, anyindependent linear combination of these functions is acceptable. Onesuch, which ‘favours’ a vertical C3 -axis, is shown on the right-hand sideof Figure 6.7 (see Exercise 6.10).6.6.3. Trigonal-planar ML3 complexesA trigonal-planar ML3 complex (6-27) belongs to the D3h point groupand contains the following symmetry elements: the molecular plane−L(σh ), three perpendicular planes (σv ), each of which contains one M−bond, a C3 -axis and an S3 -axis which are co-linear and perpendicular tothe σh plane, and three C2 axes, each of which is co-linear with one ofthe bonds.
Each type of symmetry element is illustrated in 6-28.6.6.3.1. Reduction of the representation ŴσMv(6.17a)C2Rotations around the C3 and S3 axes change the positions of all theligands (χ = 0), but a rotation around a C2 -axis leaves the ligand onthat axis unmoved (χ = 1). Reflections in the σh and σv planes leaveall three ligands in their original positions (χ = 3). The charactersobtained from these considerations are given in Table 6.21 (Ŵσ ), togetherTable 6.21. Character table for the D3h point group and the charactersof the reducible representation Ŵσ for a trigonal-planar ML3 complexD3hE2C33C2σh2S33σvA1′A2′E′A1′′A2′′E′′Ŵσ112112311−111−101−101−101112−1−1−2311−1−1−1101−10−1101x 2 + y2 , z2(x, y)(x 2 − y2 , xy)z(xz, yz)Symmetry-adapted orbitals in several MLn complexeswith those of the different irreducible representations of the D3h pointgroup.As the order of the group (h) is 12, application of the reductionformula (6.5) leads to (products where at least one factor is zero havebeen omitted):1aA1′ = [(3 × 1 × 1) + (1 × 1 × 3) + (3 × 1 × 1)12+ (1 × 1 × 3)] = 1aA2′ =1[(3 × 1 × 1) − (1 × 1 × 3) + (3 × 1 × 1)12− (1 × 1 × 3)] = 0aA1′′1[(3 × 2 × 1) + (3 × 2 × 1)] = 1121= [(3 × 1 × 1) + (1 × 1 × 3) − (3 × 1 × 1)12aA2′′1= [(3 × 1 × 1) − (1 × 1 × 3) − (3 × 1 × 1)12aE ′ =− (1 × 1 × 3)] = 0+ (3 × 1 × 3)] = 0aE′′ =1[(3 × 2 × 1) − (3 × 2 × 1)] = 012henceŴσ = A1′ ⊕ E′(6.18)The representation Ŵσ is thus decomposed into a one-dimensional representation (A1′ , the totally symmetric representation) and a degeneratetwo-dimensional representation (E′ ).6.6.3.2.
Symmetry-adapted orbitalsAs in the previous examples, the orbital with A1′ symmetry (the totallysymmetric representation) is a linear combination of all the σi orbitalswith coefficients that are equal in sign and magnitude. The orbitalswith E′ symmetry are obtained by applying the projection formula tothe functions σ1 and (σ2 − σ3 ), in turn (see Exercise 6.11), as for NH3(§ 6.4.2.2.). The results are:1φA1′ = √ (σ1 + σ2 + σ3 )31φE′ (1) = √ (2σ1 − σ2 − σ3 )61φE′ (2) = √ (σ2 − σ3 )2(6.19a)(6.19b)(6.19c)Elements of group theory and applicationsE⬘(1)1A⬘1MMFigure 6.8.
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