Atomic orbitals, cylindrical symmetry

Both MOs described in Fig. 1.10 have cylindrical symmetry about the internuclear axis. MOs of this type are called sigma (a) MOs. Since the inter-nuclear axis is usually defined as the z axis atomic orbital has a a-symmetry about this axis and thus it can be combined with other orbitals of the same symmetry (per orbitals), p and Py have however other symmetry. Rotation about the internuclear axis is not symmetric and there is a nodal plane containing this axis. These kind of orbitals are said to have 7r-symmetry. The linear combination of n atomic orbitals leads to bonding (ti ) and antibonding (tc ) MOs. Analogously atomic orbitals with symmetry 5, i.e. two nodal planes containing the internuclear axis, may be combined to give MOs with the same symmetry. That will be discussed separately in Sect. 1.3. The formation of dinuclear molecular orbitals with two different classes of symmetry is illustrated in Fig. 1.14. 

Double and triple bonds, particularly those in carbon molecules, are often described in MO terms. Thus a double bond is described as consisting of a a bond formed by the overlap of an sp hybrid orbital on each carbon atom and a tt bond formed by the sideways overlap of either the 2p or 2p orbitals (Figure 3.18). A cr orbital has cylindrical symmetry like an atomic s orbital whereas a ir orbital, like an atomic p orbital, has a planar node passing through the nucleus of each of the bonded atoms. A triple bond is similarly described as consisting of a cr orbital and two ir orbitals formed from both the 2p and 2pv orbitals on... 

A sigma bond is a molecular orbital that looks like an s-type atomic orbital when viewed down its axis and has cylindrical symmetry. A tt bond looks like a p-type of atomic orbital from the same... 

The cylindrical symmetry about the inter-nuclear axis leads to the solutions of the molecular Schrodinger equation, eqn (3.3), having either a or character. Taking the z axis along the axis of the molecule, the a eigenfunctions will comprise linear combinations of the , , and atomic orbitals so that we can write the molecular orbital as... 

First, we set up molecular orbitals from the available atomic orbitals in the complex, just as we would for a molecule. Consider an octahedral complex of a d-metal in Period 4, such as iron, cobalt, or copper. We need to consider the 4s-, 4p-, and 3J-orbitals of the central metal ion, because all these orbitals have similar energies. To simplify the discussion, we use only one atomic orbital on each of the ligands. For instance, for a Cl- ligand, we use the C13p-orbital directed toward the metal atom for an NH3 ligand, we use the sp3 lone-pair orbital. These six ligand orbitals are represented by tear-shaped objects in Fig. 16.42. The orbitals have cylindrical symmetry around the metal-ligand axis, so they are ready to form cr-orbitals. 

There are totally six molecular orbitals (cts, a, az, a, tt", and tt") formed by the six atomic orbitals (2s and 2p orbitals on Be and 1 s orbitals on the hydrogens). Note that the a molecular orbitals have cylindrical symmetry around the molecular axis, while the nonbonding n orbitals do not. Another important characteristic of these orbitals is that they are delocalized" in nature. For example, an electron occupying the ers orbital has its density spread over all three atoms. Table 3.4.1 summarizes the way the molecular orbitals of BeH2 are formed by the atomic orbitals on Be and H, where the linear combinations of H orbitals are normalized. 

Pi orbitals, on the other hand, require the presence of two atomic p orbitals on adjacent atoms. Most important, the charge density in the n orbital is concentrated above and below the molecular plane it is almost zero along the line-of-centers between the two atoms.lt is this perpendicular orientation with respect to the molecular plane (and the consequent lack of cylindrical symmetry) that defines the n orbital. The combination of a a bond and a % bond extending between the same pair of atoms constitutes the double bond in molecules such as ethylene. 

Figure 5. Shapes of the XeF ions based on steric activity of the nonbonding xenon valence-electron pairs. (Arrows indicate directions of maximum polarizing effect.) [These models represent the nonbonding xenon electrons in a formalistic way. In the Xe-F case the model cannot be realistic since such a cation has cylindrical symmetry. The postulated axial polarizing behavior can also be seen to be a consequence of Xe-F bond formation. Thus we can synthesize XeF by bringing F ( D) up to the spherical Xe atom. If we use a p-orbital pair of electrons of the Xe atom to form the Xe-F bond, the electron density will be diminished trans to the bond.]...

Because of the cylindrical symmetry of cr bonds (Section 3.8), orbital overlap in the C-C single bond of ethane is exactly the same regardless of the geometric relationships among other atoms attached to the carbons (Figure 4.1). The different arrangements of atoms that result from rotation about a single bond are called conformations, and a specific conformation is called a conformer (conformational isomer). Unlike constitutional isomers, which have different connection. of atoms, different conformers have... 

The delocalized picture of m.o.s for polyatomic molecules requires some adaptation of the distinction between a and tt m.o.s, particularly for nonlinear molecules. If tt m.o.s continue to be defined as linear combinations of p orbitals having parallel axes, a m.o.s are to be regarded as the remaining m.o.s, even though, in general, they no longer have internuclear lines as axes of cylindrical symmetry. In Chapter 8, we will see how to reconcile delocalized m.o.s with the classical structural formulae involving bonds between adjacent atoms. 

It has already been mentioned that hybrid orbitals can be chosen so as to minimize the residual interactions between localized molecular orbitals . It is not uncommon for the hybrids chosen in this way to have axes that do not conform to the molecular geometry. For example, when it is said that the Cl atoms of chloroform increase the s-character of the C hybrid involved in the CH bond, an angle between the axes of the other three hybrid orbitals smaller than 109.5° is implied. In fact, the actual Cl-C-Cl bond angles are larger. Again, bent quasi-localized molecular orbitals occur independently of any deviations of the electron density in each CCl sector with respect to cylindrical symmetry around the C-Cl nuclear axis. 

Some of the possible combinations of atomic orbitals are shown in Fig. 5.11. Those orbitals which are cylindrically symmetrical about the internuclear axis are called cr orbitals, analogous to an s orbital, the atomic orbital of highest symmetry. If the internuclear axis lies in a nodal plane, a n bond results. In S bonds (Chapter 16) the internuclear axis lies in two mutually perpendicular nodal planes. All antibonding orbitals (identified with an ) possess an additional nodal plane perpendicular to the internuclear axis and lying between the nuclei. In addition, the molecular orbitals may or may not have a center of symmetry. Of particular interest in this regard are orbitals, which are ungerade, and tt orbitals, which are gerade. 

If the molecular orbital has cylindrical symmetry about the axis connecting the atoms that are bonded together, i.e. there is not a nodal plane that passes through both of the bonded nuclei, then it is called a a bond. An example is a bond formed between an sp3 hybridised atomic orbital and an s atomic orbital,... 

An idealized single bond is a sigma bond—one that has cylindrical symmetry. In contrast, a p-orbital or pi-bond orbital has pi symmetry—one that is antisymmetric with respect to reflection in a plane passing through the atomic centers with which it is associated. In ethene, the pi-bonding orbital is symmetric with respect to reflection in a plane perpendicular to and bisecting the C-C bond, whereas the pi-star-anti bonding orbital is antisymmetric with respect to this operation. 

So far we have considered only cases in which the only important MO s are formed by overlap of an s orbital on each of the two atoms. MO s of this type are called a sigma) MO s and the property that so classifies them is their cylindrical symmetry about the internuclear axis. That such symmetry must exist should be clear from the fact that each of the two s orbitals composing them is symmetrical about this axis. 

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