Regular tetrahedron

The point groups T, and /j. consist of all rotation, reflection and rotation-reflection synnnetry operations of a regular tetrahedron, cube and icosahedron, respectively. 

Figure Bl.8.4. Two of the crystal structures first solved by W L Bragg. On the left is the stnicture of zincblende, ZnS. Each sulphur atom (large grey spheres) is surrounded by four zinc atoms (small black spheres) at the vertices of a regular tetrahedron, and each zinc atom is surrounded by four sulphur atoms. On the right is tire stnicture of sodium chloride. Each chlorine atom (grey spheres) is sunounded by six sodium atoms (black spheres) at the vertices of a regular octahedron, and each sodium atom is sunounded by six chlorine atoms.

Methane, CH4, for example, has a central carbon atom bonded to four hydrogen atoms and the shape is a regular tetrahedron with a H—C—H bond angle of 109°28, exactly that calculated. Electrons in a lone pair , a pair of electrons not used in bonding, occupy a larger fraction of space adjacent to their parent atom since they are under the influence of one nucleus, unlike bonding pairs of electrons which are under the influence of two nuclei. Thus, whenever a lone pair is present some distortion of the essential shape occurs. 

When the ammonium ion NH is formed the lone pair becomes a bonding pair and the shape becomes a regular tetrahedron. 

The compounds of carbon and silicon with hydrogen would be expected to be completely covalent according to these models, but the dhectionality of the bonds, which is towards the apices of a regular tetrahedron, is not explained by these considerations. Another of Pauling s suggestions which accounts for this type of directed covalent bonding involves so-called hybrid bonds. 

Figure 2.3 The shapes of orbitals for the s electron pair, the three pairs of p electrons with obitals mutally at right angles, and the sp orbitals which have the major lobes pointing towards the apices of a regular tetrahedron.

In all the groups along the chain, the bond angle is fixed. It is determined by considering a carbon atom at the centre of a regular tetrahedron and the four covalent bonds are in the directions of the four comers of the tetrahedron. This sets the bond angle at 109° 28 as shown in Fig. A.4 and this is called the tetrahedral angle. 

The 12 even permutations of the 4 vertices correspond to the rotations which are deck transformations of the regular tetrahedron. 

This puts the four electron pairs as far AX4 results. The four bonds are directed toward the comers of a regular tetrahedron. All the... 

Four-coordinate metal complexes may have either of two different geometries (Figure 15.3). The four bonds from the central metal may be directed toward the comers of a regular tetrahedron. This is what we would expect from VSEPR model (recall Chapter 7). Two common tetrahedral complexes are Zn(NH3)42+ and C0CI42. ... 

C atom has one unpaired electron in each of its four sp hybrid orbitals and can therefore form four cr-bonds that point toward the corners of a regular tetrahedron. The C—C bond is formed by spin-pairing of the electrons in one sp hybrid orbital of each C atom. We label this bond hybrid orbital composed of 2s- and 2/t-orbitals on a carbon atom, and the parentheses show which orbitals on each atom overlap (Fig. 3.15). Each C—H bond is formed by spin-pairing of an electron in one of the remaining sp hybrid orbitals with an electron in a 1 s-orbital of an H atom (denoted His). These bonds are denoted cr(C2s/ Hls). 

The holes in the close-packed structure of a metal can be filled with smaller atoms to form alloys (alloys are described in more detail in Section 5.15). If a dip between three atoms is directly covered by another atom, we obtain a tetrahedral hole, because it is formed by four atoms at the corners of a regular tetrahedron (Fig. 5.30a). There are two tetrahedral holes per atom in a close-packed lattice. When a dip in a layer coincides with a dip in the next layer, we obtain an octahedral hole, because it is formed by six atoms at the corners of a regular octahedron (Fig. 5.30b). There is one octahedral hole for each atom in the lattice. Note that, because holes are formed by two adjacent layers and because neighboring close-packed layers have identical arrangements in hep and ccp, the numbers of holes are the same for both close-packed structures. 

This eigenfunction is equivalent to and orthogonal to pi, and has its maximum value of 2 at 6 = 19°28

angle between the lines drawn from the center to two corners of a regular tetrahedron. The third and fourth best bond eigenfunctions... 

The Tetrahedral Carbon Atom.—We have thus derived the result that an atom in which only s and p eigenfunctions contribute to bond formation and in which the quantization in polar coordinates is broken can form one, two, three, or four equivalent bonds, which are directed toward the corners of a regular tetrahedron (Fig. 4). This calculation provides the quantum mechanical justification of the chemist s tetrahedral carbon atom, present in diamond and all aliphatic carbon compounds, and for the tetrahedral quadrivalent nitrogen atom, the tetrahedral phosphorus atom, as in phosphonium compounds, the tetrahedral boron atom in B2H6 (involving single-electron bonds), and many other such atoms. 

Oil examining the KHF2 structure we observe that each potassium ion is surrounded by eight fluorine ions at the corners of a twisted cube, no four of them forming an approximately regular tetrahedron. If, however, we make each (001) layer of HF2 ions like the one above (so that the unit of structure has c = 3.53 A rather than 7.06 A), the... 

Each oxygen atom is surrounded by four metal atoms, two of which are nearer than the other two. These atoms are not at the corners of a regular tetrahedron the angle between the line connecting atoms B and C and that connecting D and E is about 60°, instead of 90° as in a regular tetrahedron. 

The framework of the structure consists of silicon tetrahedra (four oxygen ions coordinated about a silicon ion at the comers of an approximately regular tetrahedron) and aluminium octahedra and (or) tetrahedra, with Si-0 = 1.59 A, Al-0 = 1.89 A in octahedra, 1.75 A in tetrahedra, as observed in other aluminosilicate crystals. 

The atomic arrangement found for sulvanite is a new type, shown in Figure 3. Each copper atom is surrounded by four sulfur atoms at the corners of a nearly regular tetrahedron. Each vanadium atom is surrounded by four sulfur atoms at the comers of a regular tetrahedron. Each sulfur atom is surrounded by three copper atoms at three of the comers of a nearly regular tetrahedron, and a vanadium atom not at the fourth comer of the tetrahedron, but in the negative position to this that is, in the pocket formed by the three copper atoms. 

There are four equivalent orbitals, each called sp, which point to the comers of a regular tetrahedron (Fig. 1.4). The bond angles of methane (CH4) would thus be expected to be 109° 28, which is the angle for a regular tetrahedron. 

The carbon atoms in a diamond are connected in a three-dimensional network, each atom connected to four others. Each atom is at the center of a regular tetrahedron, as shown above. We describe this geometry, which occurs in many compounds of carbon, in Chapter 9. The three-dimensional connections result in a solid that is transparent, hard, and durable. The diamond structure forms naturally only at extremely high temperature and pressure, deep within the Earth. That s why diamonds are rare and precious. 

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