Yves Jean - Molecular Orbitals of Transition Metal Complexes (793957), страница 11
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The three bonding orbitals differ from each other only by theirorientations: they have the same energy, and are therefore degenerate.The same is clearly true for the antibonding orbitals.We now turn to the interactions that involve the d orbitals of themetal centre. We consider first the three orbitals xy, xz, and yz that eachcontain two nodal planes, (xz, yz), (xy, yz), and (xy, xz), respectively.The six ligands, placed on the x-, y-, or z-axes (2-2), are all located inone of the two nodal planes of the d orbitals, and sometimes even at theintersection of these two planes.
The overlap between any one of theσi orbitals and the yz (2-15), xy (2-16) or xz orbitals (2-17) is thereforeequal to zero.6yz2-13(bonding)2-14(antibonding)6112-15 (S = 0)61xy2-16 (S = 0)xz2-17 (S = 0)Therefore, there cannot be any interaction between the three xy,xz, and yz orbitals and the σi orbitals on the ligands. In an octahedralcomplex, these three d orbitals therefore form a degenerate nonbondingset, located only on the metal (2-18 to 2-20).Principal ligand fields: σ interactionsyz2-18 (nonbonding)61x2–y22-21 (S = 0)2-23(bonding)62-25 (S = 0)2-22 (S = 0)2-24(antibonding)1z22-27(bonding)x2–y2z22-26 (S = 0)2-28(antibonding)xy2-19 (nonbonding)xz2-20 (nonbonding)The situation is different for the two remaining d orbitals.
The x 2−y2orbital points towards the ligands located on the x- and y-axes (L1 –L4 ).The overlaps with the orbitals on these four ligands (σ1 in 2-21) aretherefore non-zero and all equal (in absolute value), since the x 2−y2orbital has the same amplitude (in absolute value), at a given distancefrom the metal, in the direction of each of the ligands.
But the ligands L5and L6 are located on the z-axis, that is, the intersection of the two nodalplanes of the x 2−y2 orbital (the planes which bisect the x- and y-axes).No overlap is therefore possible between x 2−y2 and the orbitals on theselast two ligands (σ6 in 2-22).The interaction of the x 2−y2 orbital with the ligand orbitals thereforeleads to a bonding MO, mainly concentrated on the ligands (2-23), andan antibonding MO, essentially the metal orbital (2-24).
In each of theseMO, the coefficients on the ligands L1 –L4 are equal in absolute value,since the overlaps concerned are identical, but the coefficients on L5 andL6 are zero.The last orbital to be considered is the z2 orbital. It is mainly orientedalong the z axis, but also, to a lesser degree, in the xy plane; none ofthe ligands is located in its nodal cone (Chapter 1, Scheme 1-31). It cantherefore interact with the orbitals of all six ligands, though the overlapwith a ligand placed on the z-axis (σ6 in 2-25) is larger than that with aligand in the xy plane (σ1 in 2-26).The bonding (2-27) and antibonding (2-28) MO formed from thez2 orbital therefore contain contributions from the six ligands, with thecoefficients for L5 and L6 , placed on the z-axis, being larger than thosefor L1 –L4 , in the xy plane (Chapter 6, § 6.6.5).This initial analysis of the orbital structure of octahedral complexeshas relied simply on the existence, or the absence, of an overlap betweenthe orbitals of the metal and those on the ligands, and on the difference in electronegativity between the metal and the ligands.
It enablesus to obtain several important results: (i) there are six bonding MO,mainly concentrated on the ligands, and six antibonding MO, mostlyon the metal; (ii) there are three remaining orbitals located on themetal which take no part in the metal–ligand bonds (nonbonding orbitals). This description is consistent with the simplified scheme givenin Chapter 1 (§ 1.6.1, Figure 1.8) which predicted the formation of ℓbonding MO, ℓ antibonding MO, and (9−ℓ) nonbonding MO in MLℓcomplexes.
It also provides a qualitative description of the shape ofOctahedral ML6 complexesthese MO, and shows that the three nonbonding orbitals are, in theparticular case of an octahedral complex, pure d orbitals on the metalliccentre.2.1.2. Complete interaction diagramThe analysis presented above allows us to establish the main features ofthe orbital structure of octahedral complexes. It also shows that certainmolecular orbitals are degenerate. This is the case, for example, for thethree bonding MO constructed from the metal p orbitals (2-9, 2-11, and2-13), or for the three nonbonding orbitals (2-18 to 2-20). However,other degeneracies exist which cannot readily be shown by the analysisabove.
To obtain a more complete description of the orbital structure,it is necessary to exploit all the symmetry properties of the octahedronand not simply this or that symmetry element as we have done so far.There are four stages in the general procedure for constructing theMO of an MLℓ complex:2In other words, the linear combinationsof orbitals which are bases for an irreduciblerepresentation of the symmetry point group(see Chapter 6, § 6.4).1.
Find the appropriate point-group symmetry. In this case, it is theoctahedral group, Oh .2. Determine the symmetry properties of the orbitals on the centralmetal atom. They are given directly by the character table for thepoint group.3. Do not consider the ligand orbitals individually, but use linear combinations of these orbitals which are ‘adapted’ to the symmetry ofthe complex.24. Allow the metal and ligand orbitals to interact (fragment method).Only orbitals of the same symmetry can interact, since their overlapis non-zero.2.1.2.1. The symmetry of metal orbitals3The dimension of the representation isgiven by the value of the character associatedwith the identity operation E (3 in the case ofT1u ).Examination of the character table of the octahedral point group (Oh )gives us the symmetry of the metal orbitals directly, as this atom islocated at the origin of the x, y, z frame (Table 2.1).
The last two columnsindicate the symmetry of several functions of x, y, and z which have adirect link with the analytical expressions, in Cartesian coordinates, ofthese orbitals (Chapter 1, § 1.4.1).The s orbital, which has spherical symmetry, has the same symmetryproperties as the function x 2 +y2 +z2 . It is therefore a basis of the irreducible representation A1g (the totally symmetric representation of the Ohgroup), or more simply ‘it has A1g symmetry’.
The px , py , and pz orbitalstransform as x, y, and z, respectively. Their symmetry is therefore T1u(a three-dimensional, or triply degenerate, representation).3 Finally, thePrincipal ligand fields: σ interactionsTable 2.1. Character table of the point group OhOhE8C36C2′6C43C2i8S66σd6S43σhA1gA2gEgT1gT2gA1uA2uEuT1uT2u112331123311−10011−1001−10−111−10−111−101−11−101−1112−1−1112−1−111233−1−1−2−3−311−100−1−11001−10−11−1101−11−101−1−110−111−12−1−1−1−1−211x 2 + y2 + z2(z2 , x 2 − y2 )(xy, xz, yz)(x, y, z)d orbitals have Eg symmetry for the z2 and x 2−y2 pair, but T2g for thexy, xz, and yz set. It is thus important to realize that the five d orbitals,which clearly have the same energy for an isolated atom, are separated into two groups according to their symmetry properties: a doublydegenerate representation Eg and a triply degenerate set T2g .A1gs2.1.2.2.
Symmetry-adapted orbitals on the ligandspxT1upypzx2–y 2Egz2Figure 2.1. Symmetry-adapted σ orbitals foran octahedral complex ML6 , and orbitals ofthe same symmetry on the metal centre(consult Chapter 6, § 6.6.5 for the analyticalexpressions of these symmetry-adaptedorbitals and the method for finding them).None of the ligand orbitals σi , considered individually, is a basis forthe irreducible representations of the Oh group.
For the construction ofthe orbital interaction diagram, it is therefore necessary to use linearcombinations of these orbitals which are adapted to the symmetry of theoctahedron. These are presented in Figure 2.1, next to the metal orbitalswith the same symmetry.In the orbital of A1g symmetry, all the coefficients have the samevalue and sign.
It is clearly a totally symmetric orbital, just like the sorbital on the central atom. Out-of-phase combinations on two ligandsin trans positions appear in the three T1u orbitals, along the x, y, or z axes.It is clear by inspection of the shape of these orbitals that they differ onlyin their orientations and that they have the same symmetry propertiesas the px , py , and pz orbitals of the metal centre (Figure 2.1). Thesecombinations of ligand orbitals are therefore degenerate by symmetry,just like the metal p orbitals (T1u representation).In the case of the first orbital of Eg symmetry, the coefficients forthe four ligands in the xy plane are all of equal value, but of oppositesign, depending whether they are on the x- or the y-axis.
The symmetryOctahedral ML6 complexesproperties of this orbital are therefore just the same as those of themetal x 2 − y2 orbital, which also has Eg symmetry. The second Egorbital contains contributions from all six ligands. The coefficients forthe ligands located on the z-axis are twice as large as those for theligands in the xy plane, and of opposite sign. The z2 orbital, which alsohas Eg symmetry, shares these properties; its amplitude is proportionalto (2z2 − x 2 − y2 ) (Chapter 1, formula (1.11)).
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