Symmetry-adapted group orbitals

Finally, it must be mentioned that localized orbitals are not always simply related to symmetry. There are cases where the localized orbitals form neither a set of symmetry adapted orbitals, belonging to irreducible representations, nor a set of equivalent orbitals, permuting under symmetry operations, but a set of orbitals with little or no apparent relationship to the molecular symmetry group. This can occur, for example, when the symmetry is such that sev-... 

For the point group D4h, the atomic d-orbital functions belong to four different group-theoretical representations. When the same representation occurs more than once, as it does, for example, in trigonal point groups, M-1 will contain cross terms between orbitals of the same symmetry, as shown in Table 10.3(a). In this case, we are interested in the population of the symmetry-adapted orbitals ysk, such as defined for the trigonal case by expression (10.2). The symmetry-adapted orbitals are linear combinations of the original functions, that is,... 

The z axis is the internuclear axis, and the z axes on atoms a and b point toward each other. Without using the full group-theory formalism, we can write down by inspection the following symmetry-adapted orbitals, which... 

Just as group theory enables one to find symmetry-adapted orbitals, which simplify the solution of the MO secular equation, group theory enables one to find symmetry-adapted displacement coordinates, which simplify the solution of the vibrational secular equation. We first show that the matrices describing the transformation properties of any set of degenerate normal coordinates form an irreducible representation of the molecular point group. The proof is based on the potential-energy expression for vibration, (6.23) and (6.33) ... 

Thus, it follows easily from symmetry, that the out-of-plane atomic vectorial contributions in the face-to-face conformation of trimethylene do not cancel but rather add In the point group, the symmetry-adapted orbitals A + B and A - B belong to a and bg, respectively, the space part of Tj therefore belongs to 02, and the overall symmetry of the three components and T belong to... 

This complex has the same symmetry as the preceding one (the D41, point group). Since the carbonyl groups are jr-acceptors, we consider the two empty jt and n orbitals on each, and these are combined in pairs to form the symmetry-adapted orbitals for the complex (3-23). The bonding combinations, and which are antisymmetric... 

Solve Exercise 3.1 (question 2) by making use of the results of Exercise 6.13 (Chapter 6) on the symmetry-adapted orbitals for ML4 complexes with a tt system on the ligands, and the character table for the D4J, point group (Table 6.18). 

Figure 6.2. Symmetry-adapted orbitals for H2O (C2v point group).

Figure 6.3. Symmetry-adapted orbitals for NH3 (Csv point group) and orbitals of the same symmetry on the central atom.

To construct the MO of the H2O molecule, we shall allow the atomic orbitals of the central oxygen atom to interact with the symmetry-adapted orbitals on the hydrogen atoms (the fragment method). We have already established the symmetry properties of the various orbitals Civ point group), and they are repeated in Figure 6.4. It is easy to check visually that the orbitals with the same symmetry on two different fragments have a non-zero overlap, whereas those with different symmetries have zero overlap ( by symmetry ). 

Characterize the E symmetry-adapted orbitals, using Ui followed by (ct2 — CT3) as the generating functions. The projection formula (6.10) and the character table for the 03 point group (Table 6.21) should be helpful. 

FIGURE 4.3. Assembly of the molecular orbital diagram for water C2y point group) using symmetry-adapted orbitals on the ligands and the central atom/ orbitals. 

To explain the properties of cyclopropane, of the cyclopropyl group and of other three-membered rings in terms of their electronic structure, two models were proposed in 1947. One model, due to Walshpostulates for the three-membered ring of carbon atoms a set of three bonding, symmetry-adapted orbitals (scheme 74) based on tangential 2p AOs and radial sp AOs p = 1,2,3) centred at each of the three carbon atoms. The other model, proposed by Coulson and Moffit, describes the electronic structure of cyclopropane in terms of three bent banana EBOs (scheme 75), in analogy to an earlier proposal by Forster... 

In the point groups of diatomic molecules, the classification of symmetry-adapted orbitals is according to the effect of rotations about the molecular axis. Notice that since diatomic molecules are cylindrical, rotation by any angle about the molecular axis is a symmetry operation. Orbital functions may possess nodal surfaces (surfaces where the wavefunction is zero), and these are planes that include the molecular axis as in the combination of 2p orbitals shown in Figure 10.7. Consider an end-on view of... 

Point-group symmetry-adapted rotations Since the exact wave function transforms as one of the irreducible representations of the Hamiltonian point group, it is convenient to work in a representation of symmetry-adapted orbitals. In the course of the optimization, we need then consider only totally symmetric rotations since these preserve the symmetry of the wave function. If symmetry-breaking geometrical distortions are applied, nontotally symmetric rotations must be considered as well. 

Within the Bom-Oppenheimer approximation, the exact stationary states form a basis for an irreducible representation of the molecular point group. We may enforce the same spatial symmetry on the approximate state by expanding the wave function in determinants constmcted from a set of symmetry-adapted orbitals. For atoms, in particular, the use of point-group... 

It is recommended that the reader become familiar with the point-group symmetry tools developed in Appendix E before proceeding with this section. In particular, it is important to know how to label atomic orbitals as well as the various hybrids that can be formed from them according to the irreducible representations of the molecule s point group and how to construct symmetry adapted combinations of atomic, hybrid, and molecular orbitals using projection operator methods. If additional material on group theory is needed. Cotton s book on this subject is very good and provides many excellent chemical applications. 

Because symmetry operators eommute with the eleetronie Hamiltonian, the wavefunetions that are eigenstates of H ean be labeled by the symmetry of the point group of the moleeule (i.e., those operators that leave H invariant). It is for this reason that one eonstruets symmetry-adapted atomie basis orbitals to use in forming moleeular orbitals. 

For example, the three NH bonding and three NH antibonding orbitals in NH3, when symmetry adapted within the C3V point group, cluster into ai and e mos as shown in the Figure below. The N-atom localized non-bonding lone pair orbital and the N-atom Is core orbital also belong to ai symmetry. 

To illustrate sueh symmetry adaptation, eonsider symmetry adapting the 2s orbital of N and the three Is orbitals of H. We begin by determining how these orbitals transform under the symmetry operations of the C3V point group. The aet of eaeh of the six symmetry operations on the four atomie orbitals ean be denoted as follows ... 

We now return to the symmetry analysis of orbital produets. Sueh knowledge is important beeause one is routinely faeed with eonstrueting symmetry-adapted N-eleetron eonfigurations that eonsist of produets of N individual orbitals. A point-group symmetry operator S, when aeting on sueh a produet of orbitals, gives the produet of S aeting on eaeh of the individual orbitals... 

Most ah initio calculations use symmetry-adapted molecular orbitals. Under this scheme, the Hamiltonian matrix is block diagonal. This means that every molecular orbital will have the symmetry properties of one of the irreducible representations of the point group. No orbitals will be described by mixing dilferent irreducible representations. 

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