Yves Jean - Molecular Orbitals of Transition Metal Complexes (793957), страница 7
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At a given distance r from the nucleus, the amplitudeof the dxz orbital is directly proportional to the product of the x andz coordinates for that point (formula (1.8)). The same applies for thedyz , dxy , and dx 2 −y2 orbitals. But the dz2 orbital is a special case. Itsname suggests that it is concentrated wholly along the z-axis. But infact, this orbital also has a small amplitude, of opposite sign, in the xyplane, so according to formula (1.11), it would be more logical to callit d2z2 −(x 2 +y2 ) .It is important to define carefully the graphical representations thatwe shall use for these orbitals throughout this book.
They show theorbitals’ symmetry properties, the regions of space where their amplitudes are largest and where they are zero (nodal surfaces), all importantaspects for our subsequent analysis of interactions between the d orbitalsSetting the scenezdyzyx1-27zdxyyx1-28zdxzyand orbitals on the ligands. Consider the dyz orbital as an example.
Itsanalytical expression (formula (1.9) shows that its amplitude is zero ify = 0 (i.e. at all points in the xz plane) and if z = 0 (xy plane): xz andxy are therefore two nodal planes for the dyz orbital. In contrast, theamplitude is greatest along the bisectors of the y- and z-axes. Finally, it ispositive where y and z have the same sign, but negative otherwise. All ofthese properties are clearly shown by the graphical representation 1-27.The dxy and dxz orbitals (formulae (1.7) and (1.8)) may be obtainedfrom the dyz orbital by a rotation of 90◦ around the y- and z-axes, respectively. They have analogous symmetry properties, with two nodal planes(xz and yz for dxy , xy and yz for dxz ), a maximum amplitude along thebisectors of the (x, y) or (x, z) axes and alternating signs for the lobes.Their graphical representation poses the same problem as that alreadymet for the px orbital (1-26), since the plane of the page is one of thenodal planes.
In the same way as before, we represent the intersection ofthe lobes with planes parallel to the plane of the page (yz), placed eitherin front or behind, with the back part of the orbital being partly hiddenby the front part (1-28 and 1-29).The dx 2 −y2 orbital (formula ((1.10)) has its maximum amplitudealong the x- and y-axes, and it also possesses two nodal planes which arethe planes bisecting the x- and y-axes (1-30a). An alternative representation of this orbital is given in 1-30b, where the x-axis is perpendicularto the plane of the page.
The lobes directed along this axis are nowrepresented by two offset circles.dx2–y2yx1-29znodal conex = 109.5°1-31oryx1-30adz 2zdx2–y2nodal planex1-30bThe shape of the dz2 orbital is very different from those we havealready seen. Its analytical expression (formula (1.11)) shows that itsmaximal amplitude lies along the z-axis, and that it is positive for bothpositive and negative z. But it is negative in the xy plane (z = 0), and thischange in sign implies the existence of a nodal surface. The equationof this surface, z2 = (x 2 + y2 )/2 from formula (1.11), defines a conewhose apex angle, θ, is equal to 109.5◦ (the tetrahedral angle).
Allof these properties are reproduced by the conventional representationgiven in 1-31.We close this section by noting that the sign in the analyticalexpressions of the orbitals is arbitrary. The same remark applies forMetal orbitalsTable 1.4. Energies (in eV) of the s and d orbitals for d-block transition elements obtained fromspectroscopic data.1st seriesε3dε4s2nd seriesε4dε5s3rd seriesε5dε6sScTiVCrMnFeCoNiCuZn−7.92−6.60−9.22−7.11−10.11−7.32−10.74−7.45−11.14−7.83−11.65−7.90−12.12−8.09−12.92−8.22−13.46−8.42−17.29−9.39YZrNbMoTcRuRhPdAgCd−6.48−6.70−8.30−7.31−8.85−7.22−9.14−7.24−9.25−7.21−9.31−7.12−9.45−7.28−9.58−7.43−12.77−7.57−17.85−8.99LuHfTaWReOsIrPtAuHg−5.28−7.04−6.13−7.52−7.58−8.45−8.76−8.51−9.70−8.76−10.00−8.81−10.21−8.83−10.37−8.75−11.85−9.22−15.58−10.43their graphical representations.
This means that one can change allthe signs in the representation of an orbital, for example, representing the dx 2 −y2 orbital by negative (white) lobes along the x-axis andpositive (grey) lobes along the y-axis. All the orbital’s properties, suchas the regions of maximum amplitude, the changes in sign betweenneighbouring lobes or the nodal surfaces, are retained in this newrepresentation.NotationIn the interest of simplification, the five d orbitals are often written xy, xz,yz, x 2 − y2 , and z2 . This is the notation that we shall use henceforth.1.4.2.
Orbital energies10J. B. Mann, T. L. Meek, E. T. Knight,J. F. Capitani, L. C. Allen J. Amer. Chem. Soc.122, 5132 (2000). The calculations use theionization potential of the atom and theenergy of the atom in its ground state,as well as the energy of the cation formed byremoval of an electron from the s orbital orfrom a d orbital.It is possible to determine the energy of the nd and (n + 1)s orbitalsfor transition metals in their ground-state electronic configuration nda(n + 1)sb from spectroscopic data.10The values that are obtained are presented in Table 1.4. They inviteseveral comments which will be helpful when we come to the construction of diagrams for the interaction between metal and ligandorbitals. On moving from left to right in a given series, the energyof both the s and d orbitals decreases (becoming more negative).
Thisdecrease in orbital energy arises from the increase in nuclear chargewhich strengthens the interaction between the nucleus and the electrons. The variation is nonetheless less pronounced for the s orbitalsthan for the d orbitals, because the (n + 1)s electrons are strongly shiel∗ded by the nd electrons. As a consequence, the effective charge Z(n+1)sSetting the scene(the nuclear charge Z reduced by the screening σ(n+1)s ) experienced bythe s electrons varies little from one element to the next: the increase ofthe nuclear charge by one unit is largely cancelled by the presence of anadditional d electron with its screening effect. However, the d electrons∗are only weakly screened by the s electrons, so the effective charge Zndincreases by close to one unit from one element to the next, leading toa substantial stabilization of the energy of the d orbitals.
When we consider the variation within a group, the energetic ordering of the d orbitalsin the first four columns is ε3d < ε4d < ε5d , but there is an inversion ofthe 4d and 5d levels for the four following groups (ε3d < ε4d > ε5d ). Forall the elements except four (Y, Lu, Hf, and Ta), the nd orbital is lower inenergy than the (n + 1)s orbital. The (n + 1)p orbital is always higher inenergy than the (n + 1)s, as it is everywhere in the periodic table.
Forthe great majority of the d-block transition metals, the orbital energyordering is therefore, εnd < ε(n+1)s < ε(n+1)p .1.5. Ligand orbitalsIt is not possible to define a single set of orbitals that can be used todescribe the interactions with the metal for any type of ligand. Twoconditions must be met: the ligand orbitals must be close in energy tothose on the metal, and their overlap must also be substantial (§ 1.3.2).Depending on the nature of the ligand, one or several orbitals may satisfythese criteria.11A σ bond means, in this context, a bonddescribed by an MO that possesses cylindricalsymmetry about the M-ligand axis.
Thisnotation is widely used by chemists for singlebonds (see § 1.3.4). However, in group theory,the σ notation is reserved for linear molecules.M–H1sH1-321.5.1. A single ligand orbital: σ interactionsThe case where it is clearest that only one orbital need be consideredinvolves the ligand H, since it possesses only one valence orbital, 1sH .This orbital, which contains one electron (an X-type ligand), can be usedto form a σM−−H bond by combination with a metal orbital such as thez2 orbital (1-32).11For certain more complicated ligands, it is also possible, as a firstapproximation, to consider only a single orbital to describe the metal–ligand interaction. This is the case for ligands of the type AH3 (or moregenerally AR 3 ) whose orbital structure is summarized in Figure 1.4.Therefore, for an amine or phosphine (L-type ligands), it is in generalsufficient to consider the nonbonding orbital 2a1 (Figure 1.4) that characterizes the lone pair on the nitrogen or phosphorus atom (1-33a).Analogous remarks may be made for the methyl ligand, CH3 , or moregenerally for an alkyl radical CR 3 , the nonbonding orbital being onlysingly occupied in this case (an X-type ligand) (1-33b).
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