Tetrahedral molecule symmetry

As we proceed to molecules of higher symmetry the vibrational selection rules become more restrictive. A glance at the character table for the 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... 

All these compounds have (distorted) tetrahedral molecules, those of formula O2SX2 having C2v symmetry and the others Cj. Dimensions are in Table 15.15 the remarkably short O-S and S-F distances in O2SF2 should be noted (cf. above). Indeed, the implied strength of bonding in this molecule is reflected by the fact that it can be made by reacting the normally extremely inert compound SFg (p. 687) with the fluoro-acceptor SO3 ... 

Consider now spin-allowed transitions. The parity and angular momentum selection rules forbid pure d d transitions. Once again the rule is absolute. It is our description of the wavefunctions that is at fault. Suppose we enquire about a d-d transition in a tetrahedral complex. It might be supposed that the parity rule is inoperative here, since the tetrahedron has no centre of inversion to which the d orbitals and the light operator can be symmetry classified. But, this is not at all true for two reasons, one being empirical (which is more of an observation than a reason) and one theoretical. The empirical reason is that if the parity rule were irrelevant, the intensities of d-d bands in tetrahedral molecules could be fully allowed and as strong as those we observe in dyes, for example. In fact, the d-d bands in tetrahedral species are perhaps two or three orders of magnitude weaker than many fully allowed transitions. 

The theoretical reason is as follows. Although the placing of the ligands in a tetrahedral molecule does not define a centre of symmetry, the d orbitals are nevertheless centrosymmetric and the light operator is still of odd parity and so d-d transitions remain parity and orbitally Al = 1) forbidden. It is the nuclear coordinates that fail to define a centre of inversion, while we are considering a... 

Collectively, the symmetry elements present in a regular tetrahedral molecule consist of three S4 axes, four C3 axes, three C2 axes (coincident with the S4 axes), and six mirror planes. These symmetry elements define a point group known by the special symbol Td. 

Having seen the development of the molecular orbital diagram for AB2 and AB3 molecules, we will now consider tetrahedral molecules such as CH4, SiH4, or SiF4. In this symmetry, the valence shell s orbital on the central atom transforms as A, whereas the px, py, and pz orbitals transform as T2 (see Table 5.5). For methane, the combination of hydrogen orbitals that transforms as A1 is... 

The hydrogen group orbitals are referred to as symmetry adjusted linear combinations (SALC). Although their development will not be shown here, the molecular orbital diagrams for other tetrahedral molecules are similar. 

In these molecules, the boron atom has only six electrons surrounding it, so it interacts readily with species that can function as electron pair donors. For example, when l reacts with BF3, the product is BF4-, in which sp3 hybrids are formed, so such species are tetrahedral (7 ( symmetry). In most cases, molecules containing boron exhibit one of these types of bonding to boron. The boron hydrides represent a special situation that is described later. 

SnMe4 Snl4-like structure. Tetrahedral molecule with 3-fold axis of symmetry along a slightly shorter Sn—C bond. The different Sn—C bond lengths are in accord with NMR, INS and Mossbauer data (see Table 2). 131, 204... 

Figure 6. Measured molecular structure of methyl chloride (CH3CI), taken from Jensen (1981). CH3CI is a nearly tetrahedral molecule with symmetry. All C-H bond lengths, H-C-H angles and H-C-Cl angles are identical.

This is the operation of clockwise rotation by 2w/ about an axis followed by reflection in a plane perpendicular to that axis (or vice versa, the order is not important). If this brings the molecule into coincidence with itself, the molecule is said to have a n-fold alternating axis of symmetry (or improper axis, or rotation-reflection axis) as a symmetry element. It is the knight s move of symmetry. It is symbolized by Sn and illustrated for a tetrahedral molecule in Fig. 2-3.3.f... 

Symmetry plays an important role in localized MO theory since the orbitals used in the construction of the MOs y>A and xpB of eqn (11-1.1), must be symmetric about the bond axis (for the present we will limit our discussion to o-bonding). The most natural, though not mandatory, building blocks to use for tpk and tpB are the atomic orbitals (AOs) of the component atoms (A and B). Jn some cases there is available a single AO on A and a single AO on B, both of which are symmetric about the bond axis and therefore meet our requirements. But more often, and particularly when A has to form several bonds, there are not the required number of atomic orbitals with the appropriate symmetry and it is necessary to synthesize xpk (or tpB) from several AOs of A (or B). For example methane CH4 is a tetrahedral molecule with four equivalent C—H bonds pointing to the comers of a tetrahedron and each localized MO is made up of an orbital from the... 

A regular tetrahedral molecule possesses six planes of symmetry. Using the numbering system illustrated in Figure 3.1, we may specify these symmetry planes by stating the atoms they contain ... 

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). 

Fig. 2. The bending (Tbc) and stretching (tsc) vibrations of a tetrahedral molecule or complex, involved in the PJT-type Td — C2v vibronic process due to the presence of a lone pair - Tbc and tsc are the displacements according to combinations of the symmetry breaking components of the t2 modes with the angular s(6) component and the totally symmetric ax stretching vibration respectively.

This is a tetrahedral molecule with 7d symmetry. The derivation of the vibrational modes is summarized below. 

Nuclei are said to be chemically equivalent when they have the same chemical shift, usually as a result of molecular symmetry (e.g., the 2 and 6 protons or the 3 and 5 protons in phenol) but occasionally as a result of an accidental coincidence of shielding effects. Nuclei in a set are magnetically equivalent when they all possess the same chemical shift and all nuclei in the set are coupled equally with any other single nucleus in the molecule. Thus, in the tetrahedral molecule difluoromethane (I) H and H(, are magnetically equivalent because by symmetry they must have... 

Although the polyhedron in cubane, or in a similar molecule, may have the full Oh symmetry of a cube, the AtB type structures can have at best tetrahedral, Td, symmetry since they consist of two interpenetrating tetrahedra. 

Now let us take account of symmetry. The MOs of a tetrahedral molecule are transformed according to irreducible representations (IR) and of the group Tj. 

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