Molecules symmetry orbitals

Atoms, linear molecules, and non-linear molecules have orbitals which can be labeled either according to the symmetry appropriate for that isolated species or for the species in an environment which produces lower symmetry. These orbitals should be viewed as regions of space in which electrons can move, with, of course, at most two electrons (of opposite spin) in each orbital. Specification of a particular occupancy of the set of orbitals available to the system gives an electronic configuration. For example,... 

Molecular orbitals are not unique. The same exact wave function could be expressed an infinite number of ways with different, but equivalent orbitals. Two commonly used sets of orbitals are localized orbitals and symmetry-adapted orbitals (also called canonical orbitals). Localized orbitals are sometimes used because they look very much like a chemist s qualitative models of molecular bonds, lone-pair electrons, core electrons, and the like. Symmetry-adapted orbitals are more commonly used because they allow the calculation to be executed much more quickly for high-symmetry molecules. Localized orbitals can give the fastest calculations for very large molecules without symmetry due to many long-distance interactions becoming negligible. 

The total electron density contributed by all the electrons in any molecule is a property that can be visualized and it is possible to imagine an experiment in which it could be observed. It is when we try to break down this electron density into a contribution from each electron that problems arise. The methods employing hybrid orbitals or equivalent orbitals are useful in certain circumsfances such as in rationalizing properties of a localized part of fhe molecule. Flowever, fhe promotion of an electron from one orbifal fo anofher, in an electronic transition, or the complete removal of it, in an ionization process, both obey symmetry selection mles. For this reason the orbitals used to describe the difference befween eifher fwo electronic states of the molecule or an electronic state of the molecule and an electronic state of the positive ion must be MOs which belong to symmetry species of the point group to which the molecule belongs. Such orbitals are called symmetry orbitals and are the only type we shall consider here. 

Atoms are special, because of their high symmetry. How do we proceed to molecules The orbital model dominates chemistry, and at the heart of the orbital model is the HF-LCAO procedure. The main problem is integral evaluation. Even in simple HF-LCAO calculations we have to evaluate a large number of integrals in order to construct the HF Hamiltonian matrix, especially the notorious two-electron integrals... 

In 1937 Jahn and Teller applied group-theoretical methods to derive a remarkable theorem nonlinear molecules in orbitally degenerate states are intrinsically unstable with respect to distortions that lower the symmetry and remove the orbital degeneracy.37 Although Jahn-Teller theory can predict neither the degree of distortion nor the final symmetry, it is widely applied in transition-metal chemistry to rationalize observed distortions from an expected high-symmetry structure.38 In this section we briefly illustrate the application of Jahn-Teller theory and describe how a localized-bond viewpoint can provide a complementary alternative picture of transition-metal coordination geometries. 

Usually the electronic structure of diatomic molecules is discussed in terms of the canonical molecular orbitals. In the case of homonuclear diatomics formed from atoms of the second period, these are the symmetry orbitals 1 og, 1 ou, 2ag,... 

Fig. 12 a—c. A schematic representation of the frontier % MO s of an S3N3X molecule, as derived from those of SsNf (a). The diagram illustrates the effects of (b) electronegative perturbation at the tricoordinate sulfur atom and (c) conjugation with the n symmetry orbitals of the substituent... 

Fig. 13 a—c. A correlation diagram for the Huckel a MO s of (a) S3N3 and (b) a hypothetical SjNj" ion. In (c), the ab initio HFS energy levels of a planar HjPS Nj molecule are shown. For purposes of clarity, only the n symmetry orbitals are labelled. Unoccupied orbitals are indicated by an asterisk... 

Symmetry Properties of the Hydrogen 1 s Orbitals in the Water Molecule Group Orbitals... 

Construct symmetry orbitals for each of the molecules in Problem 9.23. 

As an example, consider a tetrahedral molecule in T symmetry, with two singly-occupied t2 symmetry orbitals, say tfy1. The direct product T2 (8) T2 reduces to A E Ti T2, so we obtain singlet states Mi, 1E, 1Ti, and 1T2, and triplet states Mi, 3E, 37), and 3T2. A handy check on the correctness of this sort of analysis is to add up the toted spin and spatial degeneracies of all the states and verify that it equals the spin and spatial degeneracy of the original orbital product (36 in this case). 

Figure 11.25 The electronic ground state and a few excited states of a hypothetical molecule. Above, orbital energy-level diagrams specify symmetry properties and illustrate orbital occupancies for various states the molecular orbital set is the same for each state in this approximation. Below, the states are shown on an energy scale. Each state function is a product of orbital functions for the electrons, with state symmetry determined by multiplying the symmetries of the electrons (+1 for S, — 1 for A).

The 0 e Renner-Teller vibronic system describes an orbital doublet ( ) interacting with a two-dimensional vibrational mode of s symmetry. In Section 2 we determine the general formal structure of the electron-phonon interaction matrices with orbital electronic functions of different symmetry (p-like, d-like, /-like, etc.), exploiting their intuitive relation with the Slater-Koster matrices of the two-center integrals. A direct connection with the form obtained through the molecule symmetry is discussed in Section 3. 

Within the independent electron and single active electron approximations, the symmetries of the contributing photoelectron partial waves will be determined by the symmetry of the orbital(s) from which ionization occurs, and so the PAD will directly reflect the evolution of the molecular orbital configuration. Example calculations demonstrating this are shown in Fig. 3 for a model Civ molecule, where a clear difference in the PAD is observed according to whether ionization occurs from an a or an ai symmetry orbital [55] (discussed in more detail below). 

Figure 3. Calculated LF PADs for ionization of a model C v molecule. PADs are shown for ionization of a and aj symmetry orbitals for the same set of dynamical parameters. The molecular axis distribution in these calculations was described as a cos2 0 distribution, where 0 is the angle between the direction of linear polarization of the pump laser and the principal molecular axis. The linear probe polarization is along the z axis. Panel (a) shows PADs for parallel pump and probe polarizations, while panel (b) shows PADs for perpendicular pump and probe polarizations. See Ref. [55] for the dynamical parameters used in these calculations.

Figure 4. The MF PADs for single-photon ionization of ai and <22 symmetry orbitals of a model C3V molecule for light linearly polarized along different axes of the molecule (indicated in parentheses). Note that no photoionization can occur from the <22 orbital for light polarized along the z axis (molecular symmetry axis). The same dynamical parameters as for the calculations of the LF PADs shown in Fig. 3 were used. For further details see Ref. [55],...

---