Applications of Symmetry to Molecular Orbitals

Our treatment of polyatomic molecules thus far has not exploited the symmetry properties of the molecules. We have largely restricted our descriptions of chemical bonding to orbitals made from no more than two atomic orbitals, and have included hybrid orbitals in our basis functions to achieve this goal. The approximate LCAO molecular orbitals that we have created are not necessarily eigenfunctions of any symmetry operators belonging to the molecule. 

We call these new basis functions symmetry orbitals or symmetry-adapted basis functions. The labels on these linear combinations are explained in Appendix 1. The new basis set consists of the oxygen orbitals r/ris, fis, ipy, fip, and the symmetry orbitals and. The canonical orbitals will be simpler linear combinations of these basis orbitals than if we used the original basis set. 

Find the eigenvalue of each basis orbital for each of the operators that belong to the H2O molecule. 

Only basis orbitals of the same symmetry can be included in any one LCAO molecular orbital if it is to be an eigenfunction of the symmetry operators. The a basis function can combine with the Is, 2s, and 2p functions on the oxygen. The 2 basis function can combine with the 2py function on the oxygen, and the,2px function on the oxygen cannot combine with any of the other basis functions. Table 21.2 contains the values of the coefficients determined by the Hartree-Fock-Roothaan method for the seven canonical molecular orbitals, using Slater-type orbitals (STOs) as basis functions, with 

Sketch an approximate orbital region for the b2 LCAOMO in Table 21.2. Is this a bonding, a nonbonding, or an antibonding orbital  

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The complete set of symmetry operations for the molecule together form a mathematical entity known as a group. The properties of such groups are interesting and will form the basis of the application of symmetry to molecular orbital problems. There are four rules that must be satisfied ... 

These simplified VSEPR rules may seem a far cry from the more elegant application of symmetry and molecular orbitals to the beryllium hydride molecule and the nitrite ion (Chapter 5). or the BH2 molecule (Problem 6.27). Although the molecular orbital approach can rationalize these structures, the direct application of the VSEPR rules is by far the easier way to approach a new structure. 

We will consider the application of the Hiickel molecular orbital method to the benzene molecule and we will first see what happens when we do not make use of symmetry. The benzene molecule has a framework of six carbon atoms at the comers of a hexagon and each carbon atom contributes one 7r-electron. The tt-electron MOs will be constructed from six 2pc atomic orbitals, each located at one of the carbon atoms, thus, c... 

Covalent bonds can be described with a variety of models, virtually all of which involve symmetry considerations. As a means of illustrating the role of symmetry in bonding theory and laying some foundation for discussions to follow, this section will show the application of symmetry principles in the construction of hybrid orhitals. Since you will have encountered hybridization before now, hut perhaps not in a symmetry context, this provides a ladle introduction to the application of symmetry. You should remember that the basic procedure outlined here (combining appropriate atomic orbitals to make new orbitals) is applicable also to the derivation of molecular orbitals and ligand group orhitals, both of which will be encountered in subsequent chapters. 

By far, the theoretical approaches that experimental inorganic chemists are most familiar with and in fact nse to solve questions qnickly and qnalitatively are the simple Huckel method and Hoffinann s extended Hiickel theory. These approaches are nsed in concert with the application of symmetry principles in the bnUding of syimnetry adapted linear combinations (SALCs) or gronp orbitals. The ab initio and other SCF procednres ontlined above prodnce MOs that are treated by gronp theory as well, bnt that type of rigor is not usually necessary to achieve good qnahtative pictures of the character aud relative orderiugs of the molecular orbitals. 

More advanced applications of symmetry (not discussed here) involve the behaviour of molecular wavefunctions under symmetry operations. For example in a molecule with a centre of inversion (such as a homonuclear diatomic, see Topic C4). molecular orbitals are classified as u or g (from the German, ungerade and gerade) according to whether or not they change sign under inversion. In... 

Figure 6. Schematic of the state correlation diagram for the concerted addition of molecular oxygen to a diene. The electronic configurations for states of the (O2 + diene) complex are indicated on the left (state of O2 in the complex given in brackets). States of the cyclic peroxides with their appropriate electronic configurations are on the right in order of increasing energy. Dashed curves indicate the primitive correlations obtained by straightforward application of symmetry and spin restrictions. Solid curves indicate final correlations obtained by using the additional information contained in the orbital correlation...

The labels for the d y, and dy orbitals and e for the d and d zp orbitals come from the application of a branch of mathematics called group theory to crystal-field theory. Group theory can be used to analyze the effects of symmetry on molecular properties. 

Such comparisons apply in quantum mechanics, too. Recognizing the symmetry of an atomic or molecular system allows one to simplify the quantum mechanics, sometimes dramatically. We have already seen some aspects of symmetry odd and even functions, the spherical nature of the hydrogen atom s Is orbital, the cylindrical shape of H2 and H2. All these are applications of symmetry. In this chapter, we will develop a general understanding of symmetry using a mathematical tool called group theory. Then, we can see how symmetry applies to some aspects of quantum mechanics. 

Having discussed the symmetry of the spinors and products of spinors, we are now in a position to discuss the symmetries of the one- and two-electron integrals. For computational applications, the integrals over molecular orbitals are often divided into symmetry classes for convenient handling of symmetry. In addition, the consideration of symmetry may produce some simplification in the expressions for the many-electron Hamiltonian. In the relativistic case we must use both point-group and time-reversal symmetry. 

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. 

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