Yves Jean - Molecular Orbitals of Transition Metal Complexes (793957), страница 50
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Symmetry-adapted σ orbitals for atrigonal-planar ML3 complex.ME⬘(2)1These three orbitals are shown in Figure 6.8.6.6.4. Trigonal-bipyramidal ML5 complexes431M56-292In a trigonal-bipyramidal ML5 complex, three of the ligands are foundin equatorial positions (L1 , L2 , and L3 ), whereas the other two (L4 andL5 ) are in axial sites (6-29). The apices of the bipyramid are defined bythe axial ligands, its triangular base by the positions of the equatorialligands.
The angles between the equatorial bonds are 120◦ and the axialbonds are perpendicular to the equatorial plane. Like the trigonal-planarcomplex ML3 , a trigonal-bipyramidal molecule belongs to the D3h pointgroup (see 6-28 for the symmetry elements).6.6.4.1. Reduction of the representation ŴσThe C3 -axis maintains the positions of the two axial ligands (χ = 2),whereas the S3 -axis, which interchanges them, changes the positions ofall the ligands (χ = 0). A C2 -axis moves every ligand except the oneplaced on that axis (χ = 1).
The σh and σv planes do not move thethree ligands found in these planes (χ = 3). The characters of the Ŵσrepresentation are given in Table 6.22 (second line).The reduction formula (6.5), in combination with the charactersof the irreducible representations of the D3h point group found inTable 6.21, leads to:aA1′ =aA2′ =1[(5 × 1 × 1) + (2 × 1 × 2) + (1 × 1 × 3) + (3 × 1 × 1)12+ (3 × 1 × 3)] = 21[(5 × 1 × 1) + (2 × 1 × 2) − (1 × 1 × 3) + (3 × 1 × 1)12− (3 × 1 × 3)] = 01[(5 × 2 × 1) − (2 × 1 × 2) + (3 × 2 × 1)] = 1121= [(5 × 1 × 1) + (2 × 1 × 2) + (1 × 1 × 3) − (3 × 1 × 1)12− (3 × 1 × 3)] = 0aE ′ =aA1′′aA2′′ =aE′′ =1[(5 × 1 × 1) + (2 × 1 × 2) − (1 × 1 × 3) − (3 × 1 × 1)12+ (3 × 1 × 3)] = 11[(5 × 2 × 1) − (2 × 1 × 2) − (3 × 2 × 1)] = 012Symmetry-adapted orbitals in several MLn complexesTable 6.22.
Characters of the reduciblerepresentations Ŵσ , Ŵσ (eq) and Ŵσ (ax) ofan ML5 complex with a TBP geometryD3hE2C33C2σh2S33σvŴσŴσ (eq)Ŵσ (ax)532202110330000312henceŴσ = 2A1′ ⊕ A2′′ ⊕ E′(6.20)The Ŵσ representation is thus decomposed into three one-dimensionalrepresentations (2A1′ and A2′′ ) and one doubly degenerate representation(E′ ).It is important to note that no symmetry operation exchanges anaxial ligands with an equatorial one. As these two types of ligandsare therefore non-equivalent, both ‘chemically’ and according to grouptheory, they can be considered separately.
The characters of the representations Ŵσ (eq) and Ŵσ (ax) are given in Table 6.22. From the reductionformula (6.5), we find:Ŵσ (eq) = A1 ′ ⊕ E′Ŵσ (ax) = A1 ′ ⊕ A2 ′′(6.21)(6.22)It is easy to check that Ŵσ = Ŵσ (eq) ⊕ Ŵσ (ax).6.6.4.2. Symmetry-adapted orbitalsThe separation of Ŵσ into Ŵσ (eq) and Ŵσ (ax) leads to a considerablesimplification of the determination of the symmetry-adapted orbitals.For the orbitals on the equatorial ligands, they are identical to those wehave already determined for a trigonal-planar ML3 complex (§ 6.6.3.2,6.23a,b and c and Figure 6.9).1(6.23a)(φA1′ )eq = √ (σ1 + σ2 + σ3 )31(φE′ )eq (1) = √ (σ2 − σ3 )(6.23b)21(6.23c)(φE′ )eq (2) = √ (2σ1 − σ2 − σ3 )61(6.23d)(φA1′ )ax = √ (σ4 + σ5 )21(6.23e)(φA2′′ )ax = √ (σ4 − σ5 )2Elements of group theory and applications(A⬘) eq1(E⬘) eq (1)(E⬘) eq (2)Figure 6.9.
Symmetry-adapted σ orbitals for aTBP ML5 complex.L2L3ML66-30C4 and C2dhM6-31(A⬘) axM(A⬙) ax1M1M6.6.5. Octahedral ML6 complexesOf the numerous symmetry elements in an octahedral complex ML6(6-30), the only ones that leave the positions of certain ligands unchangedare shown in 6-31. The three C4 axes and their co-linear C2 axes bothconserve the positions of the two ligands situated on them (χ = 2).Reflection in the three σh planes, which are perpendicular to the C4axes, does not move the four ligands in these planes (χ = 4).
Each ofthe six σd planes contains only two ligands, so reflection therein gives acharacter of 2.6.6.5.1. Reduction of the Ŵσ representationThe characters of the resulting Ŵσ representation are given in Table 6.23(Ŵσ ), together with those of the irreducible representations of the Ohpoint group.The order of this group (h) is 48. Use of the reduction formula (6.5)shows that the only non-zero contributions are:aA1g =1[(6 × 1 × 1) + (2 × 1 × 6) + (2 × 1 × 3) + (2 × 1 × 6)48+ (4 × 1 × 3)] = 11[(6 × 2 × 1) + (2 × 2 × 3) + (4 × 2 × 3)] = 1481= [(6 × 3 × 1) + (2 × 1 × 6) − (2 × 1 × 3) + (2 × 1 × 6)48a Eg =aT1u+ (4 × 1 × 3)] = 1MFor the axial ligands, we only need to make normalized sums (A1′symmetry) and differences (A2′′ symmetry) (6.23d, 6.23e and Figure 6.9).L5L1L4MSymmetry-adapted orbitals in several MLn complexesTable 6.23.
Character table for the Oh point group and the characters of the reducible representationŴσ for an octahedral ML6 complexOhE8C36C2′6C43C2i8S66σd6S43σhA1gA2gEgT1gT2gA1uA2uEuT1uT2uŴσ1123311233611−10011−10001−10−111−10−1101−101−11−101−12112−1−1112−1−1211233−1−1−2−3−3011−100−1−110001−10−11−1101−121−101−1−110−1101−12−1−1−1−1−2114x 2 + y2 + z2(z2 , x 2 − y2 )(xy, xz, yz)(x, y, z)henceŴσ = A1g ⊕ Eg ⊕ T1u(6.24)The Ŵσ representation is therefore decomposed into a one-dimensionalrepresentation (A1g ), a doubly degenerate representation (Eg ), and atriply degenerate representation (T1u ).6.6.5.2. Symmetry-adapted orbitalsIn view of the large number of symmetry operations, we shall limitourselves here to giving the result of the application of the projectionformula (6.10).
The following orbitals are obtained; they are shown inFigure 6.10:1φA1g = √ (σ1 + σ2 + σ3 + σ4 + σ5 + σ6 )61φEg (1) = (σ1 − σ2 + σ3 − σ4 )21φEg (2) = √ (−σ1 − σ2 − σ3 − σ4 + 2σ5 + 2σ6 )121φT1u (1) = √ (σ1 − σ3 )21φT1u (2) = √ (σ2 − σ4 )21φT1u (3) = √ (σ5 − σ6 )2(6.25a)(6.25b)(6.25c)(6.25d)(6.25e)(6.25f )Elements of group theory and applicationsMMAE (1)1gE (2)gMFigure 6.10. Symmetry-adapted σ orbitals inan octahedral ML6 complex.MT (1)1uMT (2)1ugMT (3)1uThe degeneracy of the T1u orbitals is obvious from the coefficientsof the σi orbitals (6.25d–f ), but the same is not true of the Eg orbitals,whose coefficients are very different (6.25b, c).6.6.6. Trigonal-planar ML3 complexes with a ‘π system’on the ligandsp//1Mp//3p//26-32p1p2Mp36-33In this last example, we shall analyse a trigonal-planar ML3 complex inwhich the ligands are considered to have two p orbitals perpendicular−L bond as well as the σ orbital that points towards the metalto the M−(§ 6.6.3.).
These p orbitals are written p (6-32) and p⊥ (6-33), dependingon whether they are in the plane of the complex or perpendicular to it.By convention, each orbital pi is oriented in a clockwise sense (−picorresponds to the opposite orientation). This system can act as a modelfor a complex in which the three ligands are double-face π donors or πacceptors (Chapter 3).6.6.6.1.
Reduction of the representations Ŵp and Ŵp⊥To obtain the characters of the representations Ŵp and Ŵp⊥ , it is nownot sufficient just to count the number of ligands whose position isunchanged by the action of the symmetry operations of the (D3h )point group. Although that procedure was acceptable in the previousexamples, the situation is more complicated now, since an orbital can betransformed into itself (χ = 1) or into its opposite (χ = −1).We consider first the Ŵp representation. Action of the C3 and S3axes changes all three orbitals (χ = 0), a C2 -axis transforms the orbitalof the atom on that axis into its opposite and also interchanges the twoothers while changing their sign (6-34 for the axis that passes throughSymmetry-adapted orbitals in several MLn complexesthe ligand L1 ).
The associated character is therefore −1. The molecularplane σh maintains the three orbitals (χ = 3) in their original positions,whereas reflection in a σv plane has the same consequences as rotationaround a C2 -axis (6-35, χ = −1).–p//1p//1C2p//3p//2–p//2–p//36-34v–p//1p//1vp//2p//3–p//3–p//26-35We now turn to the Ŵp⊥ representation. The character is still zerofor the C3 and S3 axes, and still −1 for a C2 -axis (6-36). Reflection in theσh plane changes each orbital into its opposite (6-37, χ = −3), whereasa σv plane maintains one orbital and interchanges the two others (6-38,χ = 1).p⊥1–p⊥1C2p⊥ 3p⊥ 2–p⊥ 3–p⊥ 26-36The characters associated with the Ŵp and Ŵp⊥ representationsare listed in Table 6.24, together with the characters of the irreduciblerepresentations of the D3h point group.Elements of group theory and applicationshp⊥1–p⊥1hp⊥ 3p⊥ 2–p⊥ 3–p⊥ 26-37vp⊥1p⊥1vp⊥ 3p⊥ 3p⊥ 2p⊥ 26-38Table 6.24. Character table for the D3h point group and thecharacters of the reducible representations Ŵp and Ŵp⊥ for atrigonal-planar ML3 complexD3hE2C33C2σh2S33σvA1′A2′E′A1′′A2′′E′′ŴpŴp⊥1121123311−111−1001−101−10−1−1112−1−1−23−311−1−1−11001−10−110−11x 2 + y2 , z2(x, y)(x 2 − y2 , xy)z(xz, yz)For the Ŵp representation, application of the reduction formula(6.5) gives:aA1′ =aA2′ =aE ′ =aA1′′ =aA2′′ =aE′′ =1[(3 × 1 × 1) − (1 × 1 × 3) + (3 × 1 × 1) − (1 × 1 × 3)] = 0121[(3 × 1 × 1) + (1 × 1 × 3) + (3 × 1 × 1) + (1 × 1 × 3)] = 1121[(3 × 2 × 1) + (3 × 2 × 1)] = 1121[(3 × 1 × 1) − (1 × 1 × 3) − (3 × 1 × 1) + (1 × 1 × 3)] = 0121[(3 × 1 × 1) + (1 × 1 × 3) − (3 × 1 × 1) − (1 × 1 × 3)] = 0121[(3 × 2 × 1) − (3 × 2 × 1)] = 012Symmetry-adapted orbitals in several MLn complexeshenceŴp = A2′ ⊕ E′(6.26)For the Ŵp⊥ representation, we obtain:aA1′ =aA2′ =aE ′ =aA1′′ =aA2′′ =aE′′ =1[(3 × 1 × 1) − (1 × 1 × 3) − (3 × 1 × 1) + (1 × 1 × 3)] = 0121[(3 × 1 × 1) + (1 × 1 × 3) − (3 × 1 × 1) − (1 × 1 × 3)] = 0121[(3 × 2 × 1) − (3 × 2 × 1)] = 0121[(3 × 1 × 1) − (1 × 1 × 3) + (3 × 1 × 1) − (1 × 1 × 3)] = 0121[(3 × 1 × 1) + (1 × 1 × 3) + (3 × 1 × 1) + (1 × 1 × 3)] = 1121[(3 × 2 × 1) + (3 × 2 × 1)] = 112henceŴp⊥ = A2′′ ⊕ E′′(6.27)6.6.6.2.
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