Projection operators 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. 

More generally, it is possible to combine sets of Cartesian displacement coordinates qk into so-called symmetry adapted coordinates Qrj, where the index F labels the irreducible representation and j labels the particular combination of that symmetry. These symmetry adapted coordinates can be formed by applying the point group projection operators to the individual Cartesian displacement coordinates. 

In the case of a symmetry-broken solution, these weights can not only diagnose the problem, but also quantify its extent. As for the projection operator considered in Section 3.2, this analysis is used primarily with Abelian point groups. 

The most important and frequent use for projection operators is to determine the proper way to combine atomic wave functions on individual atoms in a molecule into MOs that correspond to the molecular symmetry. As pointed out in Chapter 5, it is essential that valid MOs form bases for irreducible representations of the molecular point group, we encounter the problem of writing SALCs when we deal with molecules having sets of symmetry-equiv-... 

Thus, the problem of enumeration and construction of projective fullerenes reduces simply to that for centrally symmetric conventional spherical fullerenes. The point symmetry groups that contain the inversion operation are Q, C, h, (m even), Dmh (m even), Dmd (m odd), 7, Oh, and 7. A spherical fullerene may belong to one of 28 point groups ([FoMa95]) of which eight appear in the previous list C,-, C2h, Dm, Da, D3d, Du, 7, and /. Clearly, a fullerene with v vertices can be centrally symmetric only if v is divisible by four as p6 must be even. After the minimal case v = 20, the first centrally symmetric fullerenes are at v = 32 (Dm) and v = 36 (Dm). 

The tensor components that form bases for the IRs of the point groups are given in character tables, usually for 71(1), 7 (l)ax, and T 2) only. In all other cases, one may use the projection operator P1 in... 

Usually, an individual VB structure assembled from the localized bonding components does not share the point group symmetry of the molecule anymore. However, the overall VB wavefunction, PVB, should retain the same symmetry properties as the MO wavefunction (in the sense of full Cl, they are in fact identical). Therefore, TVs can be classified by an irreducible representation associated with a given point group. In order to sort vFra by symmetry, a project operator can be introduced as follows ... 

Applying the projection operators associated with C2v point group on these BTs results in Dewar benzene... 

For the D ,h point group, we can use the projection operators from the subgroup D4h. Applying the project operators Peg and PEu to the above 4 primary BTs, we obtain... 

It is often convenient to use the symmetry coordinates that form the irreducible basis of the molecular symmetry group. This is because the potential-energy surface, being a consequence of the Born-Oppenheimer approximation and as such independent of the atomic masses, must be invariant with respect to the interchange of equivalent atoms inside the molecule. For example, the application of the projection operators for the irreducible representations of the symmetry point group D3h (whose subgroup... 

Consider now the construction of the A i symmetry group orbital of the hydrogen s atomic orbitals in ammonia as an example of the application of the projection operator. (The various kinds of orbitals will be discussed in detail in Chapter 6.) The projection operator for the A irreducible representation in the C3v point group is... 

Symmetry coordinates can be generated from the internal coordinates by the use of the projection operator introduced in Chapter 4. Both the symmetry coordinates and the normal modes of vibration belong to an irreducible representation of the point group of the molecule. A symmetry coordinate is always associated with one or another type of internal coordinate—that is pure stretch, pure bend, etc.—whereas a normal mode can be a mixture of different internal coordinate changes of the same symmetry. In some cases, as in H20, the symmetry coordinates are good representations of the normal vibrations. In other cases they are not. An example for the latter is Au2C16 where the pure symmetry coordinate vibrations would be close in energy, so the real normal vibrations are mixtures of the different vibrations of the same symmetry type [7], The relationship between the symmetry coordinates and the normal vibrations can be... 

The numbers in the table, the characters, detail the effect of the symmetry operation at the top of the colurrm on each representation labelled at the front of the row. The mirror plane that contains the H2O molecule, a (xz), leaves an orbital of bi symmetry unchanged while a Ci operation on the same basis changes the sign of the wavefimction (orbital representations are always written in the lower case). An orbital is said to span an irreducible representation when its response upon operation by each symmetry element reproduces the same characters in the row for that irreducible representation. For atoms that fall on the central point of the point group, the character table lists the atomic orbital subscripts (e.g. x, y, z as p , Pj, p ) at the end of the row of the irreducible representation that the orbital spans. A central s orbital always spans the totally synunetric representation (aU characters = 1). For the central oxygen atom in H2O, the 2s orbital spans ai and the 2px, 2py, and 2p span the bi, b2, and ai representations, respectively (see (25)). If two or more atoms are synunetry equivalent such as the H atoms in H2O, the orbitals must be combined to form symmetry adapted hnear combinations (SALCs) before mixing with fimctions from other atoms. A handy mathematical tool, the projection operator, derives the functions that form the SALCs for the hydrogen atoms. 

Here a given function involves an n-fold product of MOs, to which is applied some projection operator or operators 0. As electrons are fermions, the solutions to Eq. (2) will be antisymmetric to particle interchange, and it is usually convenient to incorporate this into the n-particle basis, in which case the will be Slater determinants. The Hamiltonian given in Eq. (1) is also spin-independent and commutes with all operations in the molecular point group, so that projection operators for particular spin and spatial symmetries could also appear in 0. The O obtained in this way are generally referred to as configuration state functions (CSF s). 

Here Dr is the dimension of the representation T, g the order of the point group, Pr-y(S) is an operator corresponding to the symmetry-operation S of the point group that projects out the representation from the operand [Qj(k)]. 

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