Symmetry properties representation

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. 

As described above, the ground state vibrational wavefunction is totally symmetric for most common molecules. Therefore, the product, -(1)0 must at least contain a totally symmetric component. The direct product of two irreducible representations contains the totally symmetric representation only if the two irreducible representations are identical. Therefore, transitions can occur from a symmetrical initial state only to those states that have the same symmetry properties as the transition operator, 0. 

Spin Warns.—In Application to Point Groups, above, we considered the irreducible representations of magnetic point groups. These would be useful in obtaining the symmetry properties of localized states in magnetic crystals, as impurity or single ion states in the tight... 

It is important to distinguish between mmetiy properties of wave functions on one hand and those of density matrices and densities on the other. The symmetry properties of wave functions are derived from those of the Hamiltonian. The "normal" situation is that the Hamiltonian commutes with a set of symmetry operations which form a group. The eigenfunctions of that Hamiltonian must then transform according to the irreducible representations of the group. Approximate wave functions with the same symmetry properties can be constructed, and they make it possible to simplify the calculations. 

It is also of interest to study the "inverse" problem. If something is known about the symmetry properties of the density or the (first order) density matrix, what can be said about the symmetry properties of the corresponding wave functions In a one electron problem the effective Hamiltonian is constructed either from the density [in density functional theories] or from the full first order density matrix [in Hartree-Fock type theories]. If the density or density matrix is invariant under all the operations of a space CToup, the effective one electron Hamiltonian commutes with all those elements. Consequently the eigenfunctions of the Hamiltonian transform under these operations according to the irreducible representations of the space group. We have a scheme which is selfconsistent with respect to symmetty. 

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. 

It should be noticed that lower case Mulliken symbols are used to indicate the irreducible representations of orbitals. The upper case Mulliken symbols are reserved for the description of the symmetry properties of electronic states. 

The classification of the 2s and 2p atomic orbitals of the central oxygen atom. The 2s(0) orbital transforms as Gg, the 2px and 2p orbitals transform as the doubly degenerate nu representation, and the 2pz orbital transforms as om+. In some texts the + and superscripts are omitted, but throughout this one there is strict adherence to the use of the full symbols for all orbital symmetry representations. Unlike the 90° case, the 2s and 2p7 orbitals have different symmetry properties so there is no question of their mixing. 

First then, for methane, we must obtain I 71. To do this let us associate with each carbon hybrid orbital a vector pointing in the appropriate direction and let us label these vectors vv vs, v , v4 (see Fig. 11-3.1). All of the symmetry properties of the four hybrid orbitals will be identical to those of the four vectors. The reducible representation using these vectors (or hybrids) as a basis can be obtained from 4... 

For nonlinear molecules, each electronic wave function is classified according to the irreducible representation (symmetry species) to which it belongs the symmetry properties of i cl follow accordingly. For example, for the equilibrium nuclear configuration of benzene (symmetry ), the... 

Let us examine the direct product of a representation with itself. Let the two different sets of functions Fx,F2,...,Fm and Fl,F2,...,Fm each form a basis for the same representation TF of some point group. If we consider the symmetry properties of the m2 functions... 

It will be obvious from the content of Chapter 5 why such combinations are desired. First, only such functions can, in themselves, constitute acceptable solutions to the wave equation or be directly combined to form acceptable solutions, as shown in Section 5.1. Second, only when the symmetry properties of wave functions are defined explicitly, in the sense of their being bases for irreducible representations, can we employ the theorems of Section 5.2 in order to determine without numerical computations which integrals or matrix elements in the problem are identically zero. 

The procedure just used, although routine and reliable, is lengthy, particularly for the two-dimensional representation. The results could have been obtained with less labor by recognizing that the rotational symmetry, the behavior of the SALCs upon rotation about a principal axis, of order 3 or more, alone fixes their basic form. Their behavior under the other symmetry operations is a direct consequence of the inherent symmetry of an individual orbital toward these operations, ah or a C2 passing through it, being added to the symmetry properties under the pure rotations about the principal axis. Let... 

As pointed out in Section 6.3, for the (CH)3 case, all the essential symmetry properties of the LCAOs we seek are determined by the operations of the uniaxial rotational subgroup, Cft. When the set of six pn orbitals is used as the basis for a representation of the group C6, the following results are obtained ... 

We first note that all types of A orbitals (in D3h) have the same symmetry properties with respect to the rotations constituting the subgroup C3 also, both and " orbitals have the same properties with respect to these rotations. Thus we can use the group C3 to set up some linear combinations that will be correct to this extent. Since these rotations about the C3 axis do not interchange any of the orbitals 0, 02, 03 with those of the set 4, 05, 6, we can, temporarily, treat the two sets separately. We thus first write down linear combinations corresponding to the A and representations of C3. As shown in Section 7.3 for such cyclic systems, the characters are the correct coefficients, and we can thus write, by inspection of the character table for the group C3 ... 

Many of the symmetry properties of a point group, including its characteristic operations and irreducible representations, are conveniently displayed in an array known as a character table. The character table for C2o is17... 

In the columns on the right are some of the basis functions which have the symmetry properties of a given irreducible representation. R, Ry, and R. stand for rotations around the specified axes. The binary products on the far right indicate, for example, how the d atomic orbitals will behave ( transform ) under the operations of the group. 

The atomic orbitals suitable for combination into hybrid orbitals in a given molecule or ion will he those that meet certain symmetry criteria. The relevant symmetry properties of orbitals can be extracted from character tables by simple inspection. We have already pointed out (page 60) that the p, orbital transforms in a particular point group in the same manner as an x vector. In other words, a px orbital can serve as a basis function for any irreducible representation that has "x" listed among its basis functions in a character table. Likewise, the pr and p. orbitals transform as y and vectors. The d orbitals—d d dy, d >, t, and d ,—transform as the binary products xy, xz, yr, x2 — y2, and z2, respectively. Recall that degenerate groups of vectors, orbitals, etc, are denoted in character tables by inclusion within parentheses. 

Johnson and Rice used an LCAO continuum orbital constructed of atomic phase-shifted coulomb functions. Such an orbital displays all of the aforementioned properties, and has only one obvious deficiency— because of large interatomic overlap, the wavefunction does not vanish at each of the nuclei of the molecule. Use of the LCAO representation of the wavefunction is equivalent to picturing the molecule as composed of individual atoms which act as independent scattering centers. However, all the overall molecular symmetry properties are accounted for, and interference effects are explicitly treated. Correlation effects appear through an assigned effective nuclear charge and corresponding quantum defects of the atomic functions. 

---