Td character table

Reference to the Td character table shows that this representation can be reduced in the following way ... 

This means that SALCs of , Tu and T2 symmetries can be constructed from the eight n orbitals of the B atoms. But, n MOs can be formed only if there are appropriate AOs on atom A with which they can interact. (Strictly speaking, a SALC on the B atoms alone is also a MO, but it is a nonbonding one.) Inspection of the Td character table shows that the following AOs are available. 

This result implies that, among the four required atomic orbitals (on the central atom), one must have A i symmetry and the other three must form a 7 2 set. From Areas III and IV of the Td character table, we know that the. v orbital has A symmetry, while the three p orbitals, or the d, dv-, and dxz orbitals, collectively form a T2 set. In other words, the hybridization scheme can be either the well-known sp3 or the less familiar sd3, or a combination of these two schemes. 

The first two of the shapes are extremely common in chemistry, while the third shape is important in boron chemistry and many other cluster molecules (a cluster is defined as a molecule in which three or more identical atoms are bonded to each other) and ions. The three special shapes are associated with point groups and their character tables and are labelled, Td, Oh and Ih, respectively. The point group to which a molecule belongs may be decided by the answers to four main questions ... 

As a first illustration let us consider the optical transitions in a tetrahedral complex of Co(II). The ground state belongs to the A2 representation of the tetrahedral point group Td, and there are two excited states of F, symmetry and one of T2 symmetry. The character table for Td tells us that the coordinates x, y, and z form a basis for the T2 representation. For the A2 F, transitions we then see that the intensity integral will span the representations in the direct product of A2 x F, x F2, and this reduces as follows ... 

Determine correlation relations between the IRs of (a) Td and C3v, and (b) Oh and D3d. [Hints. Use character tables from Appendix A3. For (a), choose the C3 axis along [111] and select the three dihedral planes in Td that are vertical planes in C3v. For (b), choose one of the C3 axes (for example, that along [11 1]) and identify the three C2 axes normal to the C3 axis.]... 

Using the character table of Td in Appendix A3 we find for A that... 

The splitting pattern of the 4f orbitals in a tetrahedral crystal field can be deduced in a similar manner. If we adopt the coordinate system shown in Fig. 8.11.4, we can obtain the splitting pattern shown in the same figure. As expected the 1 3 3 pattern for the tetrahedral crystal field is just the reverse of the 3 3 1 pattern of the octahedral field. Also, the symmetry classification of the orbitals in a tetrahedral complex can be readily obtained from the character table of the Td group. 

As we proceed to molecules of higher symmetry the vibrational selection rules become more restrictive. A glance at the character table for the Td point group (Table A.41 in Appendix A) together with Equation (6.56) shows that, for regular tetrahedral molecules such as CH4, the only type of allowed infrared vibrational transition is... 

Construct all the MO of a tetrahedral complex, making use of symmetry-adapted combinations of ligand orbitals and the data in the character table for the Td group (Chapter 6, 6.6.2), following the procedure adopted in Figure 2.2 for an octahedral complex. 

This group is isomorphic with Td so the character tables differ only in the nature of the group elements and the column. 

Unambiguous assignments of the band maxima of the allowed electronic transitions are only possible for the longest-wavelength bands of ions with Td and C3p symmetry. However, all bands of Table 1 of Chapter 16.1 can be assigned qualitatively to metal reduction transitions (from MOs with predominantly ligand character to ones essentially localized on the central atom). Empirically, this... 

Example 6.1-1 This example describes a bonding in tetrahedral AB4 molecules. The numbering of the B atoms is shown in Figure 6.1. Denote by ar a unit vector oriented from A along the bond between A and Br. With ((TX matrix representatives (MRs) T(T) from T(rr (a V(T) since we only need the character system -/T of the representation To-. Every ar that transforms into itself under a symmetry operator T contributes +1 to the character of that MR T(T), while every oy that transforms into point group. The values of, t( T) for the point group Td are given in Table 6.1. This is a reducible representation, and to reduce it we use the prescription... 

Table 6.1. The character system Xafar the representation Va in the point group Td.

These waters are derived from both pristine and polluted carbonate systems. The analyses are ordered according to their increasing TDS contents. The freshest waters have a prevalent chemical character (PCC) of the calcium-bicarbonate type. As TDS values increase, the waters become relatively more enriched in Na, Cl, and/or SO4. The high Na and Cl content of analysis number 9 in Table 6.7 (TDS = 1269 mg/L) is derived from its pollution by sewage. The even higher Na and Cl concentrations of analysis number 10 reflect the fact that waters in the X-Can well, which is in coastal Yucatan, have been mixed with seawater. 

The tetrahedral sulfate ion (Td) exhibits two unoccupied antibonding molecular orbitals ai and h of 3s and 3p character, respectively (Sekiyama et al. 1986 Tyson et al. 1989 Hawthorne et al., 2000 Myneni 2000). The 15— ai transitions are dipole-forbidden (because of the 5-character of a orbitals), and the intense features in the NEXAFS spectra of sulfate correspond to the 15—orbitals (Fig. 27). The high-energy features above these intense bound state transitions correspond to the Rydberg-type transitions of 3d character or continuum state transitions (Table 4—in Appendix). 

Early MO calculations provided the composition of the frontier MOs and the character of the electronic transitions in different metallonitrosyls. Briefly, the HOMO has been described as metal-centered for SNP ( 65-70%) (67) and for many other systems (45). HoAvever, in the series of tw 5-[Ru(NH3)4(NO) (L)] complexes, the HOMO is still metal centered for L = NH3, H2O, Cl, and OH, though it is mainly py- or pz-centered for the N-heterocyclic ligands (68). For the [Ru(Me3[9]aneN3)(bpy)(NO)] ion (Table 1), a combined experimental UV—vis/(TD)DFT approach allowed assigning the detailed ordering of frontier orbitals (15), with the HOMO defined as a t(bpy) orbital. On the other hand, the LUMO and LUMO +1 are mostly located on the nitrosyl moiety with 27% contribution firom metal center orbitals, reflecting a substantial Jt-backbonding, comparable to the one calculated in [Ru(tpm)(bpy)(NO)] (25-30%) (30) and [Ru(DMAP)4(OH) (NO)] (30%) (38). As a comparison, recent calculations with [Fe(por) (Ml)(NO)] afforded a pair of LUMO and LUMO +1 orbitals with composition close to 68% Jt x,Jty and 27% dxz,dyz(69j. 

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