Yves Jean - Molecular Orbitals of Transition Metal Complexes (793957), страница 6
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Jean and F.Volatron, Volume 2, Chapter 13, Dunod,Paris (2003).two quantities are proportional to the overlap S between the interacting orbitals.6 Therefore, as this overlap increases, the stabilization ofthe bonding MO and the destabilization of the antibonding MO bothbecome larger.We shall often be interested later in this book by interactionsinvolving either two or four electrons.
In the first case (1-20), after theinteraction the two electrons both occupy the bonding MO, producinga stabilization of the electronic energy equal to 2E + .7 We deduce thatthe stabilization associated with a two-electron interaction between orbitals ofthe same energy is proportional to the overlap S.1-201-21In the case of a four-electron interaction (1-21), both the bondingand antibonding orbitals are doubly occupied. Since E − is larger thanE + , the four-electron interaction is destabilizing, and it can be shownthat the destabilization is proportional to the square of the overlap, S2 .1.3.2. Interaction between two orbitals withdifferent energiesWe now consider the more general case, where the two orbitals χ1 andχ2 , have different energies (ε1 < ε2 , Figure 1.2).
Their interaction leadsto the formation of a bonding orbital (φ+ ), lower in energy than thelowest orbital (χ1 ), and an antibonding orbital (φ− ), higher in energythan the highest orbital (χ2 ). As in the preceding case, the stabilization (E + ) of the bonding orbital, compared to the energy of χ1 , issmaller than the destabilization (E − ) of the antibonding orbital compared to the energy of χ2 (Figure 1.2).
It can be shown that these twoquantities are both proportional to the square of the overlap betweenthe orbitals and inversely proportional to their energy difference (ε),that is, proportional to S2 /ε. A strong interaction therefore requiresboth a good overlap between the orbitals and a small energy differencebetween them.CommentThis formula is approximate and cannot be used when the two orbitals aretoo close in energy. It is clear that the expression tends to infinity as ε tendsSetting the scenetowards zero (orbitals of the same energy).
For orbitals whose energies areonly slightly different, it is safer to use the result from the preceding paragraph(proportional to S). An example will be discussed in Chapter 4, § 4.1.3.–∆E –21∆E ++Figure 1.2. Interaction diagram for twoorbitals with different energies.1-22As far as the coefficients are concerned, the bonding orbital (φ+ )is concentrated on the centre (or the fragment) that has the lowestenergy orbital (χ1 here), whereas the opposite polarization is foundfor the antibonding orbital (φ− ), where the coefficient is larger for χ2(Figure 1.2).
From a chemical viewpoint, this means that the bondingMO is mainly based on the more electronegative centre (or fragment),but the antibonding MO on the less electronegative centre (or fragment).If we consider a two-electron interaction between doubly occupiedχ1 and empty χ2 (1-22), the two electrons are stabilized by 2E + . Thestabilization associated with a two-electron interaction between two orbitalsof different energy is therefore proportional to the square of the overlap andinversely proportional to the energy difference between the two orbitals, that is,proportional to S2 /ε.
However, a four-electron interaction is destabilizing, since E − is larger than E + (1-23). It can be shown that thisfour-electron destabilization is proportional to the square of the overlap, S2 .The two-orbital interaction diagrams (1-20 to 1-23) enable us toestablish a link between the idea of a bonding pair in the Lewis sense andthe MO description.
The bonding pair corresponds to double occupationof the bonding MO with the antibonding MO empty. There is thusa single bond in H2 (identical orbitals, 1-20) and in HeH+ (differentorbitals, 1-22). However, if four electrons are involved, the antibondingorbital is doubly occupied and no chemical bond exists between the twoatoms. This is the situation in He2 , for example (identical orbitals, 1-21),and HeH− (different orbitals, 1-23), species where the two atoms remainseparate.1.3.3.
The role of symmetry1-23The interaction between two orbitals χ1 and χ2 leads to a stabilization(destabilization) of the bonding (antibonding) MO, proportional to theoverlap if the orbitals have the same energy but to S2 /ε if their energiesare different. In both cases, there is clearly no interaction if the overlapis zero. Now S is equal to the integral over all space of the product ofthe functions χ1∗ and χ2 .
In order for this integral to be non-zero, thesetwo functions must be bases for the same irreducible representation ofthe molecular symmetry group, or, in simpler terms, they must have thesame symmetry (Chapter 6, § 6.5.1). If they have different symmetries,the integral is exactly equal to zero, and one says that the overlap is zeroby symmetry.Metal orbitalsIn the general case, where two fragments each with several orbitalsinteract, this comment allows us to simplify the interaction diagramsvery considerably: only orbitals of the same symmetry interact.1.3.4.
σ and π interactions1-24 (σ )P1-25 (π)*CC*CCnpnpCCnnTwo types of interactions are often distinguished: σ interactions,which concern an axial orbital overlap, and π interactions, wherethe orbital overlap occurs laterally, or ‘sideways’. These two types ofoverlap are illustrated in 1-24 and 1-25, respectively, for two p orbitalswhose axes of revolution are either co-linear (axial overlap) or parallel(sideways overlap). Notice that another way to characterize π interactions is to observe that the orbitals involved share a common nodalplane (P, 1-25).In general, σ interactions are stronger than π interactions, since axialoverlap is more efficient than sideways overlap.
The energy separationbetween the resulting orbitals is therefore larger for σ (bonding) and σ ∗(antibonding) MO than for the π and π ∗ MO.The ethylene molecule provides a typical example. The constructionof the σCC and πCC MO from nonbonding orbitals (represented by nσand np ) of the CH2 fragments is presented in Figure 1.3: the order ofthe four resulting MO, in terms of increasing energy, is σCC < πCC <∗ < σ∗ .πCCCCCCFigure 1.3. Construction of the σCC and πCCMO in ethylene from nσ and np orbitals oneach CH2 fragment.1.4. Metal orbitalsIn the case of monometallic transition metal complexes, it seems quitenatural to construct the MO by allowing the orbitals on the metal centreto interact with those on the ligands.
We are now going to examine justwhich orbitals one must consider on the metal (§ 1.4) and on the ligands(§ 1.5) so as to obtain, after interaction, a satisfactory description of theorbital structure of the complex.For the metal centre, the atomic orbitals (AO) describing the coreelectrons will not be considered for the construction of the complex’sMO. This approximation can be justified by noting that the amplitudeof these orbitals is significant only close to the nucleus, so they cantherefore play only a negligible role in bond formation. One must,however, consider the valence AO that are occupied in the groundstate of the isolated atom (nd and (n + 1)s), see Table 1.1), togetherwith the (n + 1)p orbitals, which, even though they are empty in theisolated atom, do contribute to bond formation in the complexes oftransition metals.
There are, therefore, nine atomic orbitals in all whichparticipate on the metal, five d-type orbitals, one s-type, and three p-typeorbitals.Setting the scene1.4.1. Description of the valence orbitals8Y. Jean and F. Volatron in An Introductionto Molecular Orbitals, Oxford University Press,NY.
(1993). Chapter 2.For the s and p orbitals, we shall use the usual conventionalrepresentation8 (1-26) which takes their essential features into account:1. The spherical symmetry of the s orbital.2. The existence of an axis of revolution for the px , py , and pz orbitals(the Ox, Oy, or Oz directions, respectively) and of a nodal planeperpendicular to this axis (the yOz, xOz, or xOy planes, respectively),that is, a plane in which the orbital amplitude is zero. The p orbitalschange sign on crossing the nodal plane, which is why they arerepresented by two ‘lobes’, one grey (positive amplitude), the otherwhite (negative amplitude).zzzyxsyxzyxpxpyyxpz1-269In these expressions, the angular part isexpressed using the ratios x/r, y/r, and z/rrather than the spherical coordinates (r, θ , andφ).
This transformation enables us to makethe link between the analytical expression ofthe orbital and the name which is attributedto it.It should be noted that the representation of the orbital whoseaxis of revolution is perpendicular to the plane of the page (px , 1-26)poses a special problem. This function is exactly zero in this plane(the nodal plane yOz), the positive lobe being directed towards thereader and the negative lobe away from him or her. When one takesaccount of the symmetry of revolution around the Ox axis, the intersections of these lobes with planes parallel to the nodal plane, either‘above’ or ‘below’ it, are circles. The conventional representation ofthis orbital shows two offset circles, which represent each lobe seen inperspective.The presence of five valence d-type orbitals is, of course, the chiefcharacteristic of the metals of the first three transition series.
Forhydrogenoïd atoms (those with only one electron but a nuclear chargeequal to +Z), exact analytical solutions to the Schrödinger equationcan be obtained (which is not the case for polyelectronic atoms ingeneral). The expressions for the 3d orbitals are given below (formulae (1.7)–(1.12)), where both the radial (R3,2 (r)) and angular parts arenormalized.9 Analogous expressions are obtained for the 4d and 5d orbitals of the hydrogenoïd atoms, only the radial part of the functions (R4,2Metal orbitalsand R5,2 ) being modified.60 xy16π r 2(1.7)60 xz16π r 2(1.8)60 yz16π r 2(1.9)3dx 2 −y2 = R3,2 (r)15 x 2 − y216π r 2(1.10)3dz2 = R3,2 (r)5 2z2 − x 2 − y2,16πr2(1.11)Z3a033dxy = R3,2 (r)3dxz = R3,2 (r)3dyz = R3,2 (r)whereR3,2 (r) =4√81 30Zra0Zr 2.exp −3a0(1.12)In this last expression (1.12), a0 is the Bohr radius, equal to 0.529 Å,and Z is the nuclear charge.
To what extent are these hydrogenoïdorbitals suitable to describe the d orbitals of transition metals? Inpolyelectronic atoms, it is only the radial part of the orbitals that isdifferent from hydrogenoïd orbitals; it is modified to take account of thecharge on the nucleus and the screening effect created by the other electrons. Since the angular part of the orbitals is conserved, the expressionsthat are obtained for the 3d orbitals of hydrogenoïd atoms enable us toanalyse the symmetry properties of the d orbitals of all the transitionmetals.We note first that the names given to these orbitals(dxy , dxz , dyz , dx 2 −y2 , dz2 ) are directly related to the formulae of theirangular parts.
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