Function Vectors, Linear Operators, Representations

Function Vectors, Linear Operators, Representations be square iiitegrable that is, the integral... 

Proposition 6.11 implies that irreducible representations are the identifiable basic building blocks of all finite-dimensional representations of compact groups. These results can be generalized to infinite-dimensional representations of compact groups. The main difficulty is not with the representation theory, but rather with linear operators on infinite-dimensional vector spaces. Readers interested in the mathematical details ( dense subspaces and so on) should consult a book on functional analysis, such as Reed and Simon [RS],... 

Here mx is the column vector of the localized material property function m/,. d is the column vector of the field data, and G is the matrix representation of the corresponding linear operator Gxj or G/x-... 

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... 

By analogy with ordinary vectors, Uji is the projection of the transformed vector Uj onto the basis vector 4>i. The matrix u is the representation of the linear operator U in the space of functions 0. ... 

Projection operators are a technique for constructing linear combinations of basis functions that transform according to irreducible representations of a group. Projection operators can be used to form molecular orbitals from a basis set of atomic orbitals, or to form normal modes of vibration from a basis of displacement vectors. With projection operators we can revisit a number of topics considered previously but which can now be treated in a uniform way. 

The translational motion corresponds to a displacement of the molecule as a whole in an arbitrary direction it can be depicted by a single vector showing the displacement of the center of mass. Let this vector have components x,y,z. We showed in Section 9.3 that under any symmetry operation, each of the functions x,y,z is transformed into a linear combination of x,y, and z. Hence (Section 9.6) the set of functions x,y,z forms a basis for some three-dimensional representation of the molecular point group we shall call this representation rtran8. [The representation (9.25) is... 

This is more complex than in the H2O example, since vectors are now interchanged. This makes it difficult to write down exactly what the operation does in terms of single characters, because any of the new functions could be composed from any of the original set. The matrix representation allows for this each new vector is written as a linear combination of the original basis vectors. Figure 4.6 can be used to construct the relationships required by inspection ... 

Table 6.10 The result of each of the C2V operations on the generating vector h, and the resuiting functions using the characters from each irreducibie representation as coefficients in linear combinations.

The C=0 bond on the C2 axis, with basis vector bs, is separate from bi and 2 because it is not exchanged with them by any of the symmetry operations, b lies on the symmetry axis, so it can only have an Ai representation. This means it can only be involved in linear combinations with the Ai function found for the bi and 2 set, i.e. 

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