The orbitals

Key points: Molecular orbitals are classified as π, Ϭ, or according to their rotational symmetry about the internuclear axis, and (in centrosymmetric species) as g or u according to their symmetry with respect to inversion. Our task is to see how MO theory can account for the features revealed by photoelectron spectroscopy and the other techniques, principally absorption spectroscopy, that are used to study diatomic molecules. We are concerned predominantly with outer-shell valence or bitals, rather than core orbitals. As with H₂ , the starting point in the theoretical discussion is the minimal basis set, the smallest set of atomic orbitals from which useful molecular orbitals can be built. In Period 2 diatomic molecules, the minimal basis set consists of the one valence s orbital and three valence p orbitals on each atom, giving eight atomic orbit als in all. We shall now see how the minimal basis set of eight valence shell atomic orbitals (four from each atom, one s and three p) is used to construct eight molecular orbitals. Then we shall use the Pauli principle to predict the ground-state electron configurations of the molecules. The energies of the atomic orbitals that form the basis set are shown on either side of the molecular orbital diagram in Fig. 2.12. We form orbitals by allowing overlap be tween atomic orbitals that have cylindrical symmetry around the internuclear axis, which (as remarked earlier) is conventionally labelled z. The notation signifies that the orbital has cylindrical symmetry; atomic orbitals that can form orbitals include the 2s and 2_pz orbitals on the two atoms (Fig. 2.13). From these four orbitals (the 2_s and the 2_pz orbitals on atom A and the corresponding orbitals on atom B) with cylindrical symmetry we can construct four Ϭ    molecular orbitals, two of which arise predominantly from interaction of the 2s orbitals, and two from interaction of the 2pz orbitals. These molecular orbitals are labelled 1Ϭg , 1Ϭu , 2Ϭg , and 2Ϭu , respectively. Their energies resemble those shown in Fig. 2.12 but it is difficult to predict the precise locations of the central two orbitals. Interaction between a 2s on one atom and a 2pz orbital on the other atom is possible if their relative energies are similar. The remaining two 2p orbitals on each atom, which have a nodal plane containing the z-axis, overlap to give π orbitals (Fig. 2.14). Bonding and antibonding π orbitals can be formed from the mutual overlap of the two 2_px orbitals, and also from the mutual overlap of the two 2_py orbitals. This pattern of overlap gives rise to the two pairs of doubly degenerate energy levels (two energy levels of the same energy) shown in Fig. 2.12 and labelled 1πu and 1πg.

For homonuclear diatomics, it is sometimes convenient (particularly for spectroscopic discussions) to signify the symmetry of the molecular orbitals with respect to their be haviour under inversion through the centre of the molecule. The operation of inversion consists of starting at an arbitrary point in the molecule, travelling in a straight line to the centre of the molecule, and then continuing an equal distance out on the other side of the centre. This procedure is indicated by the arrows in Figs 2.15 and 2.16. The orbital is designated g (for gerade, even) if it is identical under inversion, and u (for ungerade, odd) if it changes sign. Thus, a bonding orbital is g and an antibonding orbital is u (Fig. 2.15). On the other hand, a bonding π orbital is u and an antibonding π orbital is g (Fig. 2.16). Note that the Ϭ_g orbitals are numbered separately from the u orbitals, and similarly for the Ϭ_u orbitals, and similarly for the π orbitals.

The procedure can be summarized as follows:

1. From a basis set of four atomic orbitals on each atom, eight molecular orbitals are constructed.

 2. Four of these eight molecular orbitals are orbitals and four are orbitals.

3. The four orbitals span a range of energies, one being strongly bonding and another strongly antibonding; the remaining two lie between these extremes.

4. The four orbitals form one doubly degenerate pair of bonding orbitals and one doubly degenerate pair of antibonding orbitals.

To establish the actual location of the energy levels, it is necessary to use electronic absorption spectroscopy, photoelectron spectroscopy, or detailed computation. Photoelectron spectroscopy and detailed computation (the numerical solution of the Schrödinger equation for the molecules) enable us to build the orbital energy schemes shown in Fig. 2.17. As we see there, from Li2 to N2 the arrangement of orbitals is that shown in Fig. 2.18, whereas for O2 and F2 the order of the and orbitals is reversed and the array is that shown in Fig. 2.12. The reversal of order can be traced to the increasing separation of the 2s and 2p orbitals that occurs on going to the right across Period 2. A general principle of quantum mechanics is that the mixing of wavefunctions is strongest if their energies are similar; mixing is not important if their energies differ by more than about 1 eV. When the s,p energy separation is small, each molecular orbital is a mixture of s and p character on each atom. As the s and p energy separation increases, the molecular orbitals become more purely s-like and p-like. When considering species containing two neighbouring d-block atoms, as in Hg2 2 and [Cl₄ ReReCl₄]^-2, we should also allow for the possibility of forming bonds from d orbitals. A dz2 orbital has cylindrical symmetry with respect to the internuclear (z) axis, and hence can contribute to the Ϭ orbitals that are formed from s and pz orbitals. The dyz and dzx orbitals both look like p orbitals when viewed along the internuclear axis, and hence can contribute to the orbitals formed from px and _py . The new feature is the role of dx2 y2 and d_xy , which have no counterpart in the orbitals discussed up to now. These two orbitals can overlap with matching orbitals on the other atom to give rise to doubly degenerate pairs of bonding and antibonding orbitals (Fig. 2.19). orbitals are important for the discussion of bonds between d-metal atoms, in d-metal complexes, and in organometallic compounds.

Fig. 2.12 The molecular orbital energy level diagram for the later Period 2 homonuclear diatomic molecules. This diagram should be used for O₂ and F₂ .

Fig. 2.13 A orbital can be formed in several ways, including s,s overlap, s,p overlap, and p,p overlap, with the p orbitals directed along the internuclear axis.

Fig. 2.14 Two p orbitals can overlap to form a π orbital. The orbital has a nodal plane passing through the internuclear axis, shown here from the side

Fig 2.15 (a) Bonding and (b) antibonding Ϭ interactions with the arrow indicating the inversion.

Fig 2.16 (a) Bonding and (b) antibonding π interactions with the arrow indicating the inversions.

Fig. 2.17 The variation of orbital energies for Period 2 homonuclear diatomic molecules from Li₂ to F₂ .

Fig. 2.18 The molecular orbital energy level diagram for Period 2 homonuclear diatomic molecules from Li₂ to N₂ .

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المزيد

مواضيع ذات صلة

Polytypism

The close packing of spheres

Lattices and unit cells

Bonding and antibonding orbitals

Hybridization

Hypervalence

Promotion

Polyatomic molecules

Homonuclear diatomic molecules

The hydrogen molecule

The basic shapes

Resonance