M. Hargittai, I. Hargittai - Symmetry through the Eyes of a Chemist (793765), страница 40

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Molecules253sum of products of the one-electron molecular orbitals. Thus, the finalscheme is as follows:One-electron atomic orbitals (AOs)||LCAO↓One-electron molecular orbitals (MOs)||multiplication|(and summation)↓Total molecular wave functionAlthough both atomic orbitals and molecular orbitals are oneelectron wave functions, the shape and symmetry of the molecularorbitals are different from those of the atomic orbitals of the isolatedatom. The molecular orbitals extend over the entire molecule, andtheir spatial symmetry must conform to that of the molecular framework. Of course, the electron distribution is not uniform throughoutthe molecular orbital.

In depicting these orbitals, usually only theportions with substantial electron density are emphasized.When constructing molecular orbitals from atomic orbitals, theremay be a large number of possible linear combinations of the atomicorbitals. Many of these linear combinations, however, are unnecessary. Symmetry is instrumental as a criterion in choosing among them.The following statement is attributed to Michelangelo: “The sculptureis already there in the raw stone; the task of a good sculptor is merelyto eliminate the unnecessary parts of the stone.” In the LCAO procedure, the knowledge of symmetry eliminates the unnecessary linearcombinations.

All those linear combinations must be eliminated thatdo not belong to any irreducible representation of the molecular pointgroup. The reverse of this statement constitutes the fundamental principle of forming molecular orbitals: Each possible molecular orbitalmust belong to an irreducible representation of the molecular pointgroup. Another equally important rule for the construction of molecular orbitals is that only those atomic orbitals can form a molecular orbital that belong to the same irreducible representation of themolecular point group.

This rule follows from the general theorem(see p. 210) about the value of an energy integral. This theorem can2546 Electronic Structure of Atoms and Moleculesbe restated for the special case of MO construction as follows: Anenergy integral will be nonzero only if the atomic orbitals used forthe construction of molecular orbitals belong to the same irreduciblerepresentation of the molecular point group.The atomic orbitals in an isolated atom possess spherical symmetry.When they are used for MO construction, however, their symmetrymust be considered in the symmetry group of the particular molecule.When two atomic orbitals of the same symmetry form a molecularorbital, the symmetry of the molecular orbital will be the same as thatof the component atomic orbitals.In addition to complying with the symmetry rules, successful MOconstruction requires certain energy conditions. In order for twoorbitals to interact appreciably, their energies cannot be too different.The so-called overlap integral Sij is a useful guide in constructingmolecular orbitals.

It is symbolized asSij = ␺i ␺ j dτ(6-5)where ␺i and ␺j are the two participating atomic orbitals. The physicalmeaning of Sij is related to the measure of volume in which there iselectron density contributed by both atoms i and j. The knowledgeof the sign and magnitude of Sij is especially instructive; they can bearrived at via the following considerations.Positive overlap results from the combination of adjacent lobesthat have the same “sign.” The electron density originating from bothatoms will increase and concentrate in the region between the twonuclei. The resulting MO is a bonding orbital.

Some typical bondingatomic orbital combinations are presented in Figure 6-9. Two kinds ofmolecular orbitals are shown in this figure. A ␴ orbital is concentratedprimarily along the internuclear axis. On the other hand, a ␲ orbitalhas a nodal plane going through this axis, and its electron density ishighest on either side of this nodal plane. The ␴ orbitals are nondegenerate, while the ␲ orbitals are always doubly degenerate.Negative overlap results from the combination of adjacent lobesthat have opposite “sign.” In such an instance, there will be nocommon electron density in the region between the two nuclei;instead, electron density will concentrate in the outside regions. Suchan MO is an antibonding orbital and is illustrated in Figure 6-10.6.3.

Molecules255(a)(b)Figure 6-9. Illustration of positive overlap between atomic orbitals. The result is abonding orbital: (a) ␴ orbitals; (b) ␲ orbitals.(a)(b)Figure 6-10. Formation of antibonding orbitals by the combination of differentlobes of atomic orbitals: (a) ␴ antibonding orbitals; (b) ␲ antibonding orbitals.2566 Electronic Structure of Atoms and MoleculesFigure 6-11.

Zero overlap between atomic orbitals. There is no net interaction.Zero overlap means that there is no net interaction between the twoatomic orbitals. They have both positive and negative overlaps thatcancel each other. Some examples are shown in Figure 6-11.The energy changes in the formation of homonuclear and heteronuclear diatomic molecules are illustrated in Figure 6-12. The energy ofthe bonding MO is smaller (larger negative value) than is the energyof the interacting atomic orbitals.

On the other hand, the energy of theantibonding MO is larger than is the energy of the interacting atomicorbitals. The largest energy changes occur when the two participatingatomic orbitals have equal energies. As the energy difference betweenthe participating atomic orbitals increases, the stabilization of the(a)(b)Figure 6-12. Energy changes during MO formation: (a) Homonuclear molecules;(b) Heteronuclear molecules.6.3. Molecules257bonding MO decreases. Molecular orbitals are not formed when theparticipating atomic orbitals possess very different energies.Thus, both symmetry and energy requirements must be fulfilled inorder to form molecular orbitals.

Energetically, the 2s and 2p atomicorbitals are suffı̀ciently similar to form molecular orbitals with eachother. For symmetry reasons, however, the px and py orbitals of oneatom of a homonuclear diatomic molecule cannot combine with the2s orbital of the other atom because they belong to different irreducible representations (see Figure 6-13a). On the other hand, forexample, the radial extension of the 4f orbitals of the lanthanideelements is very small and they are well separated from the valenceregion, therefore they cannot form molecular orbitals with manyligand orbitals for energetic reasons despite their matching symmetries (see Figure 6-13b) [21].(a)(b)Figure 6-13. (a) Combination of the 2s and 2px (or 2py ) atomic orbitals does notresult in a molecular orbital because their symmetries do not match; (b) Combination of the 4f orbital of dysprosium and the 2p orbitals of chlorine does not result ina molecular orbital because their energies (radial extention) are too different [22].2586 Electronic Structure of Atoms and MoleculesKnowledge of the symmetry of the MOs is important for practical reasons.

The energy of the orbitals can be calculated by costlyquantum chemical calculations. The symmetry of the molecularorbitals, on the other hand, can be deduced from the molecular pointgroup and with the use of character tables, a process that requiresmerely paper and pencil. Then, when all possible solutions that arenot allowed by symmetry have been excluded, only the energies ofthe remaining orbitals need to be calculated.We are, of course, concerned with the symmetry aspects of theMOs and their construction. As was discussed before, the degeneracy of atomic orbitals is determined by ml . Thus, all p orbitalsare threefold degenerate, and all d orbitals are fivefold degenerate.The spherical symmetry of the atomic subshells, however, necessarilychanges when the atoms enter the molecule, since the symmetry ofmolecules is nonspherical. The degeneracy of atomic orbitals will,accordingly, decrease; the extent of decrease will depend upon molecular symmetry.Various methods (described in Chapter 4) can be used to determinethe symmetry of atomic orbitals in the point group of a molecule,i.e., to determine the irreducible representation of the molecular pointgroup to which the atomic orbitals belong.

There are two possibilitiesdepending on the position of the atoms in the molecule. For a centralatom (like O in H2 O or N in NH3 ), the coordinate system can alwaysbe chosen in such a way that the central atom lies at the intersectionof all symmetry elements of the group. Consequently, each atomicorbital of this central atom will transform as one or another irreducible representation of the symmetry group. These atomic orbitalswill have the same symmetry properties as those basis functions inthe third and fourth areas of the character table which are indicatedin their subscripts.

For all other atoms, so-called “group orbitals” or“symmetry-adapted linear combinations” (SALCs) must be formedfrom like orbitals. Several examples below will illustrate how this isdone.First, however, consider the symmetry properties of the centralatom orbitals. Take the C4v point group as an example. Its charactertable is presented in Table 6-1. The pz and dz2 atomic orbitals of thecentral atom belong to the totally symmetric irreducible representationA1 , the dx2 −y2 orbital belongs to B1 and dxy to B2 . The symmetry properties of the (px , py ) and (dxz , dyz ) orbitals present a good opportunity6.3.

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