Basis vectors symmetry-related

Fig. 10 Schematic drawing of the Ni-terminated octopolar reconstruction of NiO(l 11). Small circles stand for Ni atoms and the large ones for O atoms. The first and second planes are respectively 75% and 25% vacant. The arrows indicate the basis vectors of the p(2x2) surface lattice mesh. The symmetry related radial relaxations 5,5 and 5 are respectively represented around apex atoms labeled 1 and 2 by dashed and dotted arrows and applies to the second oxygen and the third nickel layers that are patterned. (Right) Side view of the octopolar reconstruction and of the two first relaxations and Q perpendicular to the surface plane.

The choice of unit cell shape and volume is arbitrary but there are preferred conventions. A unit cell containing one motif and its associated lattice is called primitive. Sometimes it is convenient, in order to realise orthogonal basis vectors, to choose a unit cell containing more than one motif, which is then the non-primitive or centred case. In both cases the motif itself can be built up of several identical component parts, known as asymmetric units, related by crystallographic symmetry internal to the unit cell. The asymmetric unit therefore represents the smallest volume in a crystal upon which the crystal s symmetry elements operate to generate the crystal. 

The projection operator method for obtaining a picture of the motion represented by each of the irreducible representations begins by considering the effect of each operation in the group on one, or a subset, of the basis vectors for the symmetry-related atoms. 

The projection operator results provide SALCs for symmetry-related basis vectors. Because it depends on the symmetry operations of the point group, the method does not provide information on the relative motion of symm tty-inequivalent atoms. For example, the basis of four C—H bond vectors shown in Figure 6.21 could be used to investigate the C—H stretch modes of the C2v molecule 1,2-difluorobenzene. The four basis vectors easily split into two subsets (f>i with b2 and bs with (>4) because none of the point-group operations interchange vectors between these pairs (e.g. bi and cannot be swapped by an operation). Projection of the b vector would give the two functions already seen with the simple H2O example ... 

These do not involve or because these basis vectors are not symmetry related to b. Another two functions can be obtained by projection of, say, b, to give... 

Figure 6.26 The facial (fac) isomer and meridian (mer) isomer of the general complex MLi(CO)-i. In each case the upper diagram shows a sketch in the normal orientation for octahedral complexes and the lower pictures use a view that should make the symmetry elements easier to see. The basis arrows along carbonyl bonds that are used in the vibrational analysis of carbonyl stretching modes are drawn slightly to the side of each ligand for clarity. Note that, for the C2V mer-isomer, the basis vectors bi and bi are symmetry related to each other, but not to b. ...

Problem 6.17 For the C2., isomer, the basis vectors bi and 2 are symmetry related. Use the projection operator method to show that the Ai and B2 SALCs have the form... 

When the basis contains subsets of symmetry-related vectors, separate SALCs will be obtained for each subset. These can be combined by taking further linear combinations within which each subset has the same irreducible representation. 

The weights are restricted by the same symmetries as the coefficients in the equilibrium distribution cf , but are not necessarily the same in the D3Q19 model there are then three independent values of (f . The normalization factors, Wk > 0, are related to the choice of basis vectors ... 

For a basis set of (re) atomic orbitals which are singly noded in the plane perpendicular to the radial vector (see Fig. 16b), the required Harmonics are the Vector Surface Harmonics146). The two p11 (or d") atomic orbitals at each cluster vertex (i) behave as a pair of orthogonal unit vectors which are tangential to the surface of the sphere, jrf and jif are defined as Jt-symmetry orbitals on the ith cluster atom, pointing in the direction of increasing 0 and cj> respectively, as shown in Fig. 18. At the poles of the cluster sphere these vectors may be related to Cartesian vectors as follows ... 

It is possible to be more general than this and state that every eigenfunction of Eq. 5.39 must form a basis for an irreducible representation of Civ if the operator X is only invariant under these four symmetry transformations. The phrase basis for an irreducible representation means that the functions in Eq. 5.40 generate the matrices of an irreducible representation under the point group linear operators. In the same sense a vector along the z axis of a Car system (with satisfies the relations... 

In Eq. (34), the phonon polarization vectors are complex numbers. However, in diamond-type materials, such as Si and Ge, with two similar atoms within the basis (S = 1,2) a simplification can be made. If we choose the origin to lie midway between the two atoms, we find that 2 1 - - (a/8)(111). By employing time reversal invariance and inversion symmetry, we find that the polarization vectors e . and ef o are related in the... 

The symmetry properties of basis functions with other vectors k can be determined with the use of the compatibility relations. 

The plane-wave expansion method described above is a general method that applies to any periodic structure. However, using the plane waves as basis functions leads to a large number of Fourier coefficients, and many of these coefficients are related by the symmetry of the system. For an ordered structure, it is possible to reduce the number of independent coefficients by exploiting the point group symmetry of the structure. The basic idea is that, due to the point group symmetry, the Fourier coefficients for the reciprocal lattice vectors within one star are related [26]. A set of new basis functions, which are linear combinations of the plane waves with wave vectors within one star, can be constructed using this observation. Each of these new basis functions is a linear combination of the form. 

---