Symmetry properties orbitals

In this chapter the symmetry properties of atomie, hybrid, and moleeular orbitals are treated. It is important to keep in mind that both symmetry and eharaeteristies of orbital energetics and bonding "topology", as embodied in the orbital energies themselyes and the interaetions (i.e., hj yalues) among the orbitals, are inyolyed in determining the pattern of moleeular orbitals that arise in a partieular moleeule. 

Properties can be computed by finding the expectation value of the property operator with the natural orbitals weighted by the occupation number of each orbital. This is a much faster way to compute properties than trying to use the expectation value of a multiple-determinant wave function. Natural orbitals are not equivalent to HF or Kohn-Sham orbitals, although the same symmetry properties are present. 

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. 

Extended Hiickel gives a qualitative view of the valence orbitals. The formulation of extended Hiickel is such that it is only applicable to the valence orbitals. The method reproduces the correct symmetry properties for the valence orbitals. Energetics, such as band gaps, are sometimes reasonable and other times reproduce trends better than absolute values. Extended Hiickel tends to be more useful for examining orbital symmetry and energy than for predicting molecular geometries. It is the method of choice for many band structure calculations due to the very computation-intensive nature of those calculations. 

Let us now examine the Diels-Alder cycloaddition from a molecular orbital perspective Chemical experience such as the observation that the substituents that increase the reac tivity of a dienophile tend to be those that attract electrons suggests that electrons flow from the diene to the dienophile during the reaction Thus the orbitals to be considered are the HOMO of the diene and the LUMO of the dienophile As shown m Figure 10 11 for the case of ethylene and 1 3 butadiene the symmetry properties of the HOMO of the diene and the LUMO of the dienophile permit bond formation between the ends of the diene system and the two carbons of the dienophile double bond because the necessary orbitals overlap m phase with each other Cycloaddition of a diene and an alkene is said to be a symmetry allowed reaction... 

For the orbital parts of the electronic wave functions of two electronic states the selection rules depend entirely on symmetry properties. [In fact, the electronic selection rules can also be obtained, from symmetry arguments only, for diatomic molecules and atoms, using the (or and Kf point groups, respectively but it is more... 

The key to understanding the mechanism of the concerted pericyclic reactions was the recognition by Woodward and Hoffmann that the pathways of such reactions were determined by the symmetry properties of the orbitals that were directly involved. Their recognition that the symmetry of each participating orbital must be conserved during the... 

A complete mechanistic description of these reactions must explain not only their high degree of stereospecificity, but also why four-ir-electron systems undergo conrotatory reactions whereas six-Ji-electron systems undergo disrotatory reactions. Woodward and Hoifinann proposed that the stereochemistry of the reactions is controlled by the symmetry properties of the HOMO of the reacting system. The idea that the HOMO should control the course of the reaction is an example of frontier orbital theory, which holds that it is the electrons of highest energy, i.e., those in the HOMO, that are of prime importance. The symmetry characteristics of the occupied orbitals of 1,3-butadiene are shown in Fig. 11.1. 

Fig. 11.3. Symmetry properties of cyclobutene (top) and butadiene (bottom) orbitals.

Fig. 11.9. Symmetry properties of ethylene, butadiene, and cyclohexene orbitals with respect to cycloaddition.

In the ordinary Hartree-Fock scheme, the total wave function is approximated by a single Slater determinant and, if the system possesses certain symmetry properties, they may impose rather severe restrictions on the occupied spin orbitals see, e.g., Eq. 11.61. These restrictions may be removed and the total energy correspondingly decreased, if instead we approximate the total wave function by means of the first term in the symmetry adapted set, i.e., by the projection of a single determinant. Since in both cases,... 

Symmetry properties which have so far been successfully treated by the projection operator method, include translational symmetry in crystals, cyclic systems, spin, orbital and total angular momenta, and further applications are in progress. ... 

The reader will be aware at this point (for example, by comparing the H2S orbitals of Figures 3 and 4) that the final result is not unique, this being of course consistent with the qualitative character of these methods. The symmetry properties are, however, preserved,... 

The subsets of d orbitals in Fig. 3-4 may also be labelled according to their symmetry properties. The d ildxi y2 pair are labelled and the d yldxMyz trio as t2g. These are group-theoretical symbols describing how these functions transform under various symmetry operations. For our purposes, it is sufficient merely to recognize that the letters a ox b describe orbitally i.e. spatially) singly degenerate species, e refers to an orbital doublet and t to an orbital triplet. Lower case letters are used for one-electron wavefunctions (i.e. orbitals). The g subscript refers to the behaviour of... 

The left superscript indicates that the arrangements are all spin triplets. The letter T refers to the three-fold degeneracy just discussed and it is in upper case because the symbol pertains to a many-electron (here two) wavefunction (we use lower-case letters for one-electron wavefunctions or orbitals, remember). The subscript g means the wavefunctions are even under inversion through the centre of symmetry possessed by the octahedron (since each d orbital is of g symmetry, so also is any product of them), and the right subscript 1 describes other symmetry properties we need not discuss here. More will be said about such term symbols in the next two sections. 

The names 2 g ond Sg are derived from symmetry properties of the orbitals that are not important for general chemistry. 

Although the band model explains well various electronic properties of metal oxides, there are also systems where it fails, presumably because of neglecting electronic correlations within the solid. Therefore, J. B. Good-enough presented alternative criteria derived from the crystal structure, symmetry of orbitals and type of chemical bonding between metal and oxygen. This semiempirical model elucidates and predicts electrical properties of simple oxides and also of more complicated oxidic materials, such as bronzes, spinels, perowskites, etc. 

The LCAO-MO expressions corresponding to the HMO orbital energies (3.141) for the pi MOs symmetry properties of the cyclic C topology.66 For benzene, for example, the results are (renumbered in... 

The conclusion above that optimisation of the non-linear parameters in the AO basis leads to a basis with correct spatial symmetry properties cannot be true for all intemuclear separations. At R = 0 the orbital basis must pass over into the double-zeta basis for helium i.e. two different 1 s orbital exponents. It would be astonishing if this transition were discontinuous at R = 0. While considering the variation of basis with intemuclear distance it is worth remembering that the closed-shell spin-eigenfunction MO method does not describe the molecule at all well for large values of R the spin-eigenfunction constraint of two electrons per spatial orbital is completely unrealistic at large intemuclear separation. With these facts in mind we have therefore computed the optimum orbital exponents as a function of R for three wave functions ... 

We therefore conclude that, for a combination of model, numerical and conceptual reasons the OHAO basis is well-adapted to a theory of valence. The hybrid orbital basis (for simple molecules) has a distinctive symmetry property it carries a permutation representation of the molecular symmetry group the equivalent orbitals are always sent into each other, never into linear combinations of each other. This simple fact enables the hybrid orbital basis to be studied in a way which is physically more transparent than the conventional AO basis. 

The symmetry properties of sigma orbital of a C-C-covalent bond is having a mirror plane symmetry and because a rotation of 180° through its mind point regenerates the same o orbital, it is also having C2 symmetry. The o orbital would be antisymmetric with respect to both m and C2 shown as follows ... 

The symmetry properties of molecular orbitals of cyclohexadiene are given in the following... 

These indices depend on MO symmetry properties which are well reproduced by any type of calculation. On the other hand, orbital energies and total energies do depend crucially on the type of calculation employed. For example, the three occupied pi MO s of cis 1,2-difluoroethylene are calculated as... 

In the first term, Uc, usually called the Coulombic term, the initially excited electron on D returns to the ground state orbital while an electron on A is simultaneously promoted to the excited state. In the second term, called the exchange term, Liex, there is an exchange of two electrons on D and A. The exchange interaction is a quantum-mechanical effect arising from the symmetry properties of the wavefunctions with respect to exchange of spin and space coordinates of two electrons. 

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