Yves Jean - Molecular Orbitals of Transition Metal Complexes (793957), страница 12
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It is important to realizethat the two Eg orbitals that are degenerate by symmetry necessarily have thesame energy (Chapter 6). In contrast to the case of the three orbitals ofT1u symmetry, this result is not at all obvious just from the coefficientsfor the ligands in the two orbitals. This additional information, obtainedafter proper consideration of the octahedral symmetry of the complex,enables us to produce a complete description of their orbital structure.CommentIn what follows, the symmetry of atomic or molecular orbitals will berepresented with lower-case letters (a1g , t1u , eg , t2g , etc.), while uppercase letters will be reserved for the different irreducible representations incharacter tables, for the symmetry of electronic states and for spectroscopicterms.2a1g2t1u2.1.2.3.
Interaction diagramp t1us a1g2egt2ged tg2gegt1ua1g1eg1t1u1a1gLML MLLLLLLLLLLFigure 2.2. Interaction diagram (σ only) foran octahedral complex ML6 : the metalorbitals are on the left and thesymmetry-adapted ligand orbitals on the right.We can now construct all the MO of an octahedral complex, by allowing the metal orbitals (Figure 2.2, left-hand side) to interact withthe symmetry-adapted combinations of orbitals on the six ligands(Figure 2.2, right-hand side).The energetic ordering adopted for the metal orbitals follows theusual rule for the d-block transition metals: ε(d) < ε(s) < ε(p)(Chapter 1, § 1.4.2).
The six ligand-orbital combinations are not allat the same energy. To understand the energetic ordering in Figure 2.2,one must analyse the bonding or antibonding character of each forthe ligand–ligand interactions (Figure 2.1). The a1g orbital is the moststable, since all the interactions are bonding. In the t1u orbitals, thereis an antibonding interaction between two ligands that are trans toeach other. Due to the large distance between these ligands, the orbitals may be considered to be practically nonbonding.
For the two egorbitals (which necessarily have the same energy), it is easier to consider the first: the four interactions between the closer ligands (cis)are antibonding, but two weakly bonding interactions occur betweenthe trans ligands. Overall, the eg orbitals are therefore antibonding,leading to the energetic ordering: ε(a1g ) (bonding) < ε(t1u ) (nonbonding) < ε(eg ) (antibonding).
One must remember that the bonding orPrincipal ligand fields: σ interactionsantibonding interactions just described never involve atoms that aredirectly bonded to each other. The overlaps involved are thereforeweak, due to the distance between the centres concerned (e.g. if themetal–ligand bond distances are 2.2 Å, the corresponding cis-ligandseparation is 3.11 Å, and as much as 4.40 Å for the trans ligands).This explains why the energy separations between the different ligandorbitals are small (Figure 2.2, right-hand side). We conclude by notingthat the ligand orbitals are placed lower in energy than those on themetal, since the ligands are in general more electronegative than themetal.The molecular orbitals are obtained by allowing the orbitals of theM and L6 fragments to interact (Figure 2.2).
We note features that wereestablished in § 2.1.1: the formation of six bonding MO (1a1g , 1t1u , and1eg ), by bonding interactions involving the fragment orbitals whosesymmetries are a1g , t1u , and eg , and the formation of six correspondingantibonding MO (2eg , 2a1g , and 2t1u ). The new feature concerns the degeneracy of the eg orbitals (bonding or antibonding): it was not possible to establishthis just from the shapes of the orbitals that were already determined (2-23 and2-27 for the bonding MO, 2-24 and 2-28 for the antibonding MO).
Threenonbonding MO of t2g symmetry are found at an intermediate energylevel. They are pure metal d orbitals (xy, xz, and yz), which remainnonbonding as there are no orbitals of the same symmetry on the L6fragment.This general scheme, with six bonding MO, three nonbondingMO, and six antibonding MO is appropriate for all octahedral complexes, since it is based on the symmetry properties of the fragmentorbitals.
Moreover, the lowest antibonding MO (situated above thenonbonding t2g orbitals) are always the two degenerate 2eg orbitals,since they involve the metal d orbitals, which are lower in energythan the s and p orbitals which contribute to the antibonding MO2a1g and 2t1u , respectively.
However, the energetic ordering given inFigure 2.2 for both the bonding and antibonding MO can depend onthe nature of the metal and the ligands. For example, the bondinglevel 1eg can be found between the bonding 1a1g and 1t1u levels, orthe antibonding 2a1g may be placed lower than the antibonding 2t1ulevel.CommentThere are six valence atomic orbitals on the metal which contribute to theformation of the six metal–ligand bonds: the s orbital, the three p orbitals,and two of the d orbitals. This result may be compared with that givenby hybridization theory, where one forms six equivalent hybrids, directedtowards the six vertices of an octahedron.
One does indeed talk of d2 sp3hybridisation, each hybrid orbital being a suitable linear combination oftwo d orbitals, the s orbital, and the three p orbitals.Octahedral ML6 complexes2.1.2.4. Molecular orbitals and the definition of the d block2a1g2t 1ud-block2e gz2x2 –y 2t 2gxyxzyz1eg1t 1u1a1gFigure 2.3.
Molecular orbitals for anoctahedral ML6 complex in the orientationgiven in Scheme 2-2.The shapes of the molecular orbitals may be obtained without difficultyfrom the shapes of the fragment orbitals (Figure 2.3). Given the greaterelectronegativity of the ligands, the bonding MO are more concentratedon them, whereas the antibonding MO are more concentrated on themetal. Among these MO, seven involve contributions from the metal dorbitals: the two bonding MO 1eg , the three nonbonding t2g MO and thetwo antibonding MO 2eg .
The term ‘d block’ of an octahedral complex(and also for other types of complex) is used to refer to the five molecularorbitals that are mainly concentrated on the d orbitals of the metal. The threenonbonding MO of t2g symmetry clearly belong to the d block, sincethey are formed from pure d orbitals (xy, xz, and yz). For the MO of egsymmetry, the bonding combinations (1eg ) are mainly concentrated onthe ligands while the largest coefficient for the antibonding combinations(2eg ) is on the metal. It is these last two orbitals which are considered tobelong to the d block. The d block of an octahedral complex is therefore madeup of three nonbonding degenerate (t2g ) MO and two antibonding MO whichare also degenerate (2eg ). These latter orbitals are often represented asx 2−y2 and z2 , with the antibonding contributions on the ligands beingimplicitly taken into account in this simplified notation.2.1.2.5.
Two other representations of the d blockThe orbitals of the d block, presented in the way we have just established,correspond to the arrangement of the ligands that is indicated in 2-2: twoligands placed on each of the x-, y-, and z-axes. The three nonbondingorbitals are then xy, xz, and yz, and the two antibonding orbitals arex 2−y2 and z2 . But other arrangements of the ligands with respect tothe axes are of course possible. Two in particular will be useful to usin what follows: (i) where four metal–ligand bonds bisect the x- andy-axes (2-29); (ii) where the z-axis is a threefold symmetry axis (C3 ) ofthe octahedron (2-30).
It is clear that a change in the axis system cannothave any consequence for the structure of the d block, which consists ofthree degenerate nonbonding orbitals and two antibonding orbitals thatare also degenerate. However, the expressions of some of these orbitalsin the new axes systems do change.L6L4ML1L3zyL3L2L52-29xL6L4ML2L5L12-30Principal ligand fields: σ interactionszyxz2xyyzxzx 2–y 2Figure 2.4.
Representation of the orbitals ofthe d block for an octahedral complex ML6 inthe orientation shown in Scheme 2-29.zL6yxL52-31We examine first the situation represented in 2-29. The six ligandsare still placed in one or other of the two nodal planes of the xz and yzorbitals. These two orbitals are therefore still nonbonding (Figure 2.4),as they were in the preceding case (Figure 2.3).
Nor is there any changefor the antibonding orbital z2 , since there are still two ligands on thez-axis and four in the xy plane. However, the xy orbital which wasnonbonding now points towards all four ligands placed on the bisectorsof the x- and y-axes: this orbital becomes one of the antibonding orbitalsof the d block. On the other hand, these four ligands are now situatedin the nodal planes of the x 2−y2 orbital, which becomes, with xz andyz, the third nonbonding orbital of the d block.
This new representationof the d block (Figure 2.4) may be distinguished from the preceding one(Figure 2.3) by ‘exchange’ of the xy and x 2−y2 orbitals.Analysis of the situation is more complex for the geometricalarrangement shown in 2-30. The only orbital for which the result isobtained easily is z2 . The angle between the M–L bonds and the z-axisas it is now defined is exactly equal to 109.5◦ , which is the value of theangle of the nodal plane of the z2 orbital (Chapter 1, Scheme 1-31).As the six ligands are located on the nodal surface of this orbital, nointeraction of σ type is possible, so z2 is one of the three nonbondingorbitals.
The difficulty in finding the two other nonbonding orbitalsarises because none of the four ligands L1 –L4 is now placed in one of thenodal planes of the xy, xz, yz, or x 2−y2 orbitals. None of these orbitalsis therefore nonbonding. Nor is any as antibonding as the 2eg orbitalswere in the preceding representations, since they do not point directlytowards the ligands. An example is given in 2-31 for the xz orbital. Theligands L5 and L6 are in the nodal plane yz (coefficients zero).
The orbitals on the ligands L1 –L4 interact with xz, though they are not placed onthe bisectors of the x- and z-axes where the amplitude of this orbital isgreatest.Octahedral ML6 complexeszyx3z2412Figure 2.5. Representation of the orbitals ofthe d block of the octahedron when the z-axiscoincides with a C3 axis (Scheme 2-30).In fact, rather than considering the metal d orbitals individually, wemust form new combinations that either ‘avoid’ the six ligands (nonbonding orbitals) or point towards them (antibonding orbitals). Theselinear combinations, which allow the original orbitals to be reoriented,are indicated below and shown graphically in Figure 2.5.φ1φ2φ3φ42 212(x − y ) −yz=3321xy −xz=331 222=(x − y ) +yz3312xy +xz=33(2.1)(2.2)(2.3)(2.4)The first two combinations (φ1 and φ2 ) correspond to two orbitalswhich, with z2 , form the t2g block.
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