Group of symmetry operations

Under the action of the electric field of the 0 environment, the state of the ( ) 3d configuration will split up into two states. The orbital degeneracy of a D state is 2 X 2 -f- 1 = 5. From the above discussion each of the resulting states must belong to one of the irreducible representations of Oh given in Table I. The state W corresponds to an irreducible representation of the group of symmetry operations of a sphere, i.e., the full rotation group R(3). 

In many extremely important cases, the analogy between a group of symmetry operations and a group of real numbers is more than superficial. For example, consider the molecule a—chloronaphthalene ... 

What we have just done is to substitute the algebraic process of multiplying matrices for the geometric process of successively applying symmetry operations. The matrices multiply together in the same pattern as do the symmetry operations it is clear that they must, since they were constructed to do just that. It will be seen in the next section that this sort of relationship between a set of matrices and a group of symmetry operations has great importance and utility. 

The set ( bj) therefore closes. The other necessary group properties are readily proved and so G is a group. Direct product (DP) without further qualification implies the outer direct product. Notice that binary composition is defined for each group (e.g. A and B) individually, but that, in general, a multiplication rule between elements of different groups does not necessarily exist unless it is specifically stated to do so. However, if the elements of A and B obey the same multiplication rule (as would be true, for example, if they were both groups of symmetry operators) then the product at bj is defined. Suppose we try to take (a,-, bj) as a, bj. This imposes some additional restrictions on the DP, namely that... 

A group-theoretical treatment of this symmetry contraint leads to the requirement that an MO must belong to an irreducible representation of the point group. A representation is a set of matrices - one for each symmetry operation - which constitutes a group isomorphous with the group of symmetry operations and can be used to represent the symmetry group. When we say that a function belongs to (or transforms as , or forms a basis for ) a particular representation, we mean that the matrices which constitute the representation act as operators which transform the function in the same way as the symmetry operations of the molecule. (The reader who knows little about matrices and their application as transformation operators can skip over such remarks.) An irreducible representation is one whose matrices cannot be simplified to sets of lower order. 

If ra is an irreducible representation of dimension k, and if F, F2, is a set of degenerate eigenfunctions that form the basis for theyth irreducible representation of the group of symmetry operations, these eigenfunctions transform according to the relation... 

For a symmetrical atomic or molecular system, these considerations place a severe restriction on the possible eigenfunctions of the system. All possible eigenfunctions must form bases for some irreducible representation of the group of symmetry operations. The form of the possible eigenfunctions is also determined to a large extent since they must transform in a quite definite way under the operations of the group. 

The number of inequivalent irreducible representations is equal to the number of classes in the group of symmetry operators. 

The group of symmetry operations form classes , is a class by itself A, B, C form a reflexion class D, F form a rotation class. Two elements P and Q satisfying the relation X PX=P or Q, where X is any element of the group and X its reciprocal, belong to the same class. This is seen to lead to the above results e.g. ... 

Suppose that G is the group of symmetry operations of a polyhedron or polygon, with vertices corresponding to the atomic positions in a particular molecular structure. The division of the structure into orbits, as sets of vertices equivalent under the actions of the group symmetry operations and the calculation of associated permutation representations/characters were described in Chapter 2. In this chapter, the identity between the permutation representa-tion/character on the labels of the vertices of an orbit and the a representation/character on sets of local s-orbitals or a-oriented local functions is exploited to constmct the characters of the representations that follow from the transformation properties of higher order local functions. 

Every molecule can be characterized as belonging to some group of symmetry operations from the above list, under which it can be transformed into indistinguishable copies of itself. We cannot, however, have arbitrary combinations of symmetry operations. For example, a molecule with a C axis of rotation can have only mirror planes which cither contain the axis or are perpendicular to... 

Point group A group of symmetry operations that leave unmoved at least one point within the object to which they apply. Symmetry elements include simple rotation and rotatory-inversion axes the latter include the center of symmetry and the mirror plane. Since one point remains invariant, all rotation axes must pass through this point and all mirror planes must contain it. A point group is used to describe isolated objects, such as single molecules or real crystals. 

The symmetry operations which commute with the non-relativistic (electrostatic) Hamiltonian Hes of a given system do not necessarily commute with the Breit-Pauli operator Hso. It is therefore appropriate to analyse the group of symmetry operators of Hso for each particular point-group symmetry of the electrostatic Hamiltonian. 

Using the methods discussed in the tutorial (Sect. 2), the group of symmetry operations of Hso can be constructed explicitly. As is shown in Appendix B, the symmetry group of Hso of (53) is T, the spin double group of the tetrahedral rotation group, is of order 48. T also is the symmetry group of the total Hamiltonian H = Hes + Hso-... 

Space group The group of symmetry operators which describe the symmetry of the crystal. Since biological molecules are optically active, their crystals belong to one of the 65 non-cen-trosymmetric space groups. 

Symmetry elements are operators. That is, each one describes an operation, such as reflection. When these operations are applied to the crystal, the external form is reproduced. It was found that all crystals fell into one or another of 32 different groups of symmetry operations. These were called crystal classes. Each crystal class could be allocated to one of the six crystal families. These symmetry elements and the resulting crystal classes are described in detail in Chapters 3 and 4. 

Let R be an element of the group of symmetry operations which leave the one-electron Hamiltonian operator h invariant. Then... 

The matrices that represent a group of symmetry operators have the same effect as the symmetry operators, so they must multiply together in the same way. We can show this for our present representation by carrying out the matrix multiplications. The matrices for the C3 group are... 

Show that the 1 by 1 matrices (scalars) in Eq. (9.78) obey the same multiplication table as does the group of symmetry operators. 

The four properties just enumerated are of fundamental importance. They are the properties—and the only properties—that any collection of symmetry operations must have in order to constitute a mathematical group. Groups consisting of symmetry operations are called symmetry groups or sometimes point groups. The latter term arises because all of the operations leave the molecule fixed at a certain point in space. This is in contrast to other groups of symmetry operations, such as those that may be applied to crystal structures in which individual molecules move from one location to another. 

For the Hamiltonian //we identify a symmetry group, and this is a group of symmetry operations of //a symmetry operation being defined as an operation that leaves //invariant (i.e., that commutes with //). In our example, the symmetry group is K (spatial). 

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