Algebra of symmetry operations

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. 

Algebraic description of symmetry operations is based on the following simple notion. Consider a point in a three-dimensional coordinate system with any (not necessarily orthogonal) basis, which has coordinates x, y, z. This point can be conveniently represented by the coordinates of the end of the vector, which begins in the origin of the coordinates 0, 0, 0 and ends at x,y, z. Thus, one only needs to specify the coordinates of the end of this vector in order to fully characterize the location of the point. Any symmetrical transformation of the point, therefore, can be described by the change in either or both the orientation and the length of this vector. 

A molecule is much more than an algebraic expression, but we can define symmetry operators in the same way that rearrange or compare similar regions of the molecule. The study of these sets of symmetry operators is a branch of group theory, and tools from this area of mathematics can greatly strengthen our understanding of molecular properties. 

A representation of a set of symmetry operations allows the operations to be manipulated algebraically. The representation must give the same multiplication table as the operations themselves. 

So, as we found for the case of the simpler basis in H2O, the matrix representation of symmetry operations allows products of operations to be considered algebraically. This matrix product approach can be extended to matrices of any size and so for any size of basis. 

Operator algebra and matrix algebra are similar to each other. A set of matrices can be a representation of a group of symmetry operators if there is a matrix corresponding to each of the symmetry operators and if the matrices obey the same multiplication table as the symmetry operators. We now show how one such representation can be constructed. Equation (13.21) represents the action of a general symmetry operator. A, on the location of a point. We rewrite this equation in the form ... 

Dynamic symmetry corresponds to an expansion of the Hamiltonian in terms of Casimir operators. The Casimir operator of U(2) plays no role, since it is a given number within a given representation of U(2) and thus can be reabsorbed in a constant term E(). The algebra U(l) has a linear invariant... 

Sylow, 6. symbol list, xv. symmetric matrix, 59. symmetry element, 7. symmetry operations, 7, 10 algebra of, 15 for symmetric tripod, 27. symmetry operator, 10. symmetry orbitals, 203, 206, 207, 210, 212, 246. 

Any set of algebraic functions or vectors may serve as the basis for a representation of a group. In order to use them for a basis, we consider them to be the components of a vector and then determine the matrices which show how that vector is transformed by each symmetry operation. The resulting matrices, naturally, constitute a representation of the group. We have previously used the coordinates jc, y, and z as a basis for representations of groups C2r (page 78) and T (page 74). In the present case it will be easily seen that the matrices for one operation in each of the three classes are as follows ... 

A systematic investigation of the application of Lie algebra to NMR was presented.29 The symmetry properties of the nuclear spin systems were naturally included in selection of the sets of the basis operators. With this theoretical framework, the existing sets of basis operators used for various specific purposes can be treated in a unified manner and their respective advantages and disadvantages can be evaluated. A number of 2H MAS spectra calculated on the basis of that theoretical framework are shown in Fig. 2. The... 

Symmetry operations may be represented by algebraic equations. The position of a point (an atom of a molecule) in a Cartesian coordinate system is described by the vector r with the components x, y, z. A symmetry operation produces a new vector r with the components t, /, z. The algebraic expression representing a symmetry operation is a matrix. A symmetry operation is represented by matrix multiplication. 

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