Symmetry, tetrahedral

The thiosulfate ion has tetrahedral symmetry and the six fundamental modes are both infrared and Raman active. The calculated frequencies (3) are in good agreement with experimental values (4). 

Fig. 2. Structures for the solid (a) fee Cco, (b) fee MCco, (c) fee M2C60 (d) fee MsCeo, (e) hypothetical bee Ceo, (0 bet M4C60, and two structures for MeCeo (g) bee MeCeo for (M= K, Rb, Cs), and (h) fee MeCeo which is appropriate for M = Na, using the notation of Ref [42]. The notation fee, bee, and bet refer, respectively, to face centered cubic, body centered cubic, and body centered tetragonal structures. The large spheres denote Ceo molecules and the small spheres denote alkali metal ions. For fee M3C60, which has four Ceo molecules per cubic unit cell, the M atoms can either be on octahedral or tetrahedral symmetry sites. Undoped solid Ceo also exhibits the fee crystal structure, but in this case all tetrahedral and octahedral sites are unoccupied. For (g) bcc MeCeo all the M atoms are on distorted tetrahedral sites. For (f) bet M4Ceo, the dopant is also found on distorted tetrahedral sites. For (c) pertaining to small alkali metal ions such as Na, only the tetrahedral sites are occupied. For (h) we see that four Na ions can occupy an octahedral site of this fee lattice.

Cationic cyclopentadienyl complexes are not common in this group, but recent examples whose structures have been determined include [Nb ( j -C5H5)2Cl2]BF4< > and [Nb( j -C5H5)2L2](BF4)2 (L = CNMe and NCMe),< > which have pseudo-tetrahedral symmetry. 

Perhaps only slightly less common than octahedral symmetry is tetrahedral symmetry. We now examine the d orbital splitting in this environment. The story is much the same as above, except that it is now convenient to place the four point charges of the tetrahedron as shown in Fig. 3-6. Here ligands are put at alternate... 

Figure 3-7. Barycentre splitting of d orbitals in tetrahedral symmetry.

An S term, like an s orbital, is non-degenerate. Therefore, while the effect of a crystal field (of any symmetry) will be to shift its energy, there can be no question of its splitting. The ground term for the configuration is S. In an octahedral crystal field, this is relabelled Aig, in tetrahedral symmetry, lacking a centre of inversion, it is labelled M]. 

Strictly, L is defined only as a quantum number for a spherical environment - the free ion. The use of L ff = 0 for A terms or Leff = 1 for L terms on the grounds that (2Leff + 1) equals the degeneracy of these terms is, however, legitimate as used here. There is a close parallel between the quantum mechanics of T terms in octahedral or tetrahedral symmetry on the one hand, and of P terms in spherical symmetry on the other. 

D = dihedral (rotation plus dihedral rotation axes) I = inversion symmetry T = tetrahedral symmetry O = octahedral S5mimetry... 

Two types of species have been detected in the /rSR spectrum of Ceo- One shows an unreacted or meta-stable muonium state which may well correspond to an internal state, muonium is trapped inside the cage Mu Ceo in the current notation [2]. This may be compared with normal muonium (Mu ) in diamond and many other elemental and compound semi-conductors, where the trapping site is in one of the cavities of tetrahedral symmetry. This state of CeoMu is not discussed here, but it does exhibit all the characteristics expected of the internal chemistry of Ceo-The anomalous muonium state. Mu, observed in semi-conductors and generally accepted to arise from muonium being trapped within one of the chemical bonds of the crystal, is unknown in molecules [5,6]. The constraints of the crystal lattice are necessary for the bond-centred state to be stable. 

High-valent iron can occur in a wide variety of electronic configurations. Figure 8.25 (a-c, e-i) presents a summary of the corresponding one-electron crystal-field states for the 3(/, 3J, and 3J electron configurations, allocated to HS and LS states in distorted octahedral and tetrahedral symmetry. Part d, in addition, depicts the case of low-low-spin iron(IV) found in some trigonal... 

Silicon exists in nature only in the most thermodynamically stable form in a oxygen-containing atmosphere. Each silicon atom is surrounded by four oxygen atoms in tetrahedral symmetry. Mankind has used the special stability of such compounds to prepare glass, chinaware, ceramics, and building materials like concrete, etc. Silicates today are still one of the most important materials. 

Although the effect on the d orbitals produced by a field of octahedral symmetry has been described, we must remember that not all complexes are octahedral or even have six ligands bonded to the metal ion. For example, many complexes have tetrahedral symmetry, so we need to determine the effect of a tetrahedral field on the d orbitals. Figure 17.5 shows a tetrahedral complex that is circumscribed in a cube. Also shown are lobes of the dz- orbital and two lobes (those lying along the x-axis) of the dx> y> orbital. 

The band at 2223 cm-1 was deduced to have tetrahedral symmetry from the splitting that occurs upon a partial isotopic substitution of D for H (Bai et al., 1985) as was discussed in Section III.3. This band also shows a stress splitting pattern that can be fit with a tetrahedral model (Bech Nielsen et al., 1989). The suggestion (Bai et al., 1985) that this center may be a SiH4 or VH4 complex has been retained as a possible explanation of the symmetry determined by uniaxial stress techniques. 

The splitting of the C- and D-lines of the elemental acceptor aluminum is consistent with the tetrahedral symmetry Td of this impurity in a substitutional site. The stress behavior of A(D,C), on the other hand, can be described with a trigonal acceptor whose ground state is presplit along one of the four [111] directions. The ls-like ground states of the acceptors A(H,Si) and A(D,Si) are presplit by 1.07 meV. Their uniaxial stress behavior is the same as of A(H,C). It is convenient to assume that the internal splitting of the ls-state is caused by an internal short-range stress S that affects the ls-state but leaves the /7-like bound excited states undisturbed. 

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