Group orbitals character

The matrix Rij,kl = Rik Rjl represents the effect of R on the orbital products in the same way Rjk represents the effect of R on the orbitals. One says that the orbital products also form a basis for a representation of the point group. The character (i.e., the trace) of the representation matrix Rij,kl appropriate to the orbital product basis is seen to equal the product of the characters of the matrix Rjk appropriate to the orbital basis %e2(R) = Xe(R)%e(R)i which is, of course, why the term "direct product" is used to describe this relationship. 

The linear water molecule belongs to the D point group. The classifications of atomic and group orbitals must be carried out using the character table. The molecular axis, C, is arranged to coincide with the z axis. 

Q Look up the transformation properties of the 2s and 2p orbitals of the nitrogen atom in the character tables of the C3v point group in Appendix 1 to confirm the content of Table 6.1. Carry out the procedure for classifying the Is orbitals of the three hydrogen atoms as group orbitals in the pyramidal molecule. 

This representation of the group orbitals may be converted to its constituent irreducible representations by considering that it is likely that they are identical with the representations to which the 2s and 2p atomic orbitals of the boron atom belong. The subtraction of the characters of the a, representation gives the result ... 

The axially symmetric metal carbonyl fragments M(CO)n (n = 1, 3, 4) have three outpointing hybrid orbitals with a high proportion of s and p orbital character, which are suitable for forming cluster skeletal molecular orbitals (77, 78, 238). The number and radial characteristics of these frontier molecular orbitals, which are illustrated schematically in Fig. 26a, are reminiscent of the frontier orbitals of a main group diatomic hydride fragment E—H, where E = C or B (Fig. 26b). To describe this similarity the term isolobal has been introduced (77). Molecular orbital... 

The simplest molecules are atoms, which belong to point group %h (often called the full rotation-reflection group). The character table (which we omit) contains irreducible representations of dimensions 1,3,5,... these representations correspond to energy levels with electronic orbital angular-momentum quantum number /=0,1,2,... we have the (2/+1)-fold degeneracy associated with different values of the quantum number... 

We may use the set of five d wave functions as a basis for a representation of the point group of a particular environment and thus determine the manner in which the set of d orbitals is split by this environment. Let us choose an octahedral environment for our first illustration. In order to determine the representation for which the set of d wave functions forms a basis, we must first find the elements of the matrices which express the effect upon the set of wave functions of each of the symmetry operations in the group the characters of these matrices will then be the characters of the representation we are seeking. 

Fig. 11.18 Identification of the symmetries of ligand group orbitals and metal orbitals involved in the o bonds (represented as vectors) of an octahedral MU. complex. The characters of the reducible representation, r, are derived by counting the number of vectors that remain unmoved under each symmetry operation of the Oi, point group The irreducible components of C, are obtained by application of Eq. 3.1-...

Fig. 11.24 Identification of the symmetries of ligand group orbitals and metal orbitals capaMf of participating in ir bonds (represented na vectors) in an octahedral Ml complex. The characters and irreducible components of the reducible representation, T,. were derived by application of the same methods used for the c-only system (Fig. 11.(8).

Various methods (described in Chapter 4) can be used to determine the symmetry of atomic orbitals in the point group of a molecule, i. e., to determine the irreducible representation of the molecular point group to which the atomic orbitals belong. There are two possibilities depending on the position of the atoms in the molecule. For a central atom (like O in H20 or N in NH3), the coordinate system can always be chosen in such a way that the central atom lies at the intersection of all symmetry elements of the group. Consequently, each atomic orbital of this central atom will transform as one or another irreducible representation of the symmetry group. These atomic orbitals will have the same symmetry properties as those basis functions in the third and fourth areas of the character table which are indicated in their subscripts. For all other atoms, so-called group orbitals or symmetry-adapted linear combinations (SALCs) must be formed from like orbitals. Several examples below will illustrate how this is done. 

Table 6-4. The C3v Character Table and the Reducible Representation of the Hydrogen Group Orbitals of Ammonia...

The next step is to determine how these group orbitals transform in the D(,h point group. The D6h character table is given in Table 6-7. Since most of the AOs in the suggested group orbitals are transformed into another AO by most of the symmetry operations, the representations will be quite simple, though still reducible ... 

We shall now imagine that the optimized function Vlo for the L group of electrons has been obtained, and look in more detail at the variational calculation specified by Eqs. (3-6 a 3-10) when the functions Wmot are expanded in terms of Slater determinants constructed from the M subset of the orthonormal spin-orbital basis f, the ten spin-orbitals of d-orbital character. 

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. 

In this cluster, which has C2, symmetry, the calculated molecular-orbital energy levels should be compcu-ed with those of the FeO cluster in Fig. 4.26(a). As for the latter cluster, the molecular orbitals can be grouped into sets in relation to their atomic-orbital character, with the dominantly Fe 3d (crystal-field-type) orbitals being at the top of the valence band. Now, however, as shown in Fig. 4.29, the orbitals below these in energy fall into four groups O 2p nonbonding orbitals, Fe-(9... 

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