chem516.02.origins of character tablesxuv.scs.illinois.edu › 516 › lectures › chem516.02.pdf · Origin of Character Tables • Symmetry operations obey the rules of group theory.
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• Matrix representations of symmetry operations can often be reduced into block matrices. Similarity transformations may help to reduce representations further. The goal is to find the irreducible representation, the only representation that can not be reduced further.
• The same ”type” of operations (rotations, reflections, etc.) belong to the same class. Formally R and R’ belong to the same class if there is a symmetry operation S such that R’=S-1RS. Symmetry operations of the same class will always have the same character.
Even if the matrices have different basis (e.g., axes),ICBST, the trace of the matrices are the same.
Character (trace) of the matrices
The symmetry operations gives various classes that can bedistinguished by their characters.
In addition, there are only a certain number of distinct waysin which a function can behave when subjectedto the symmetry operations of a particular point group.
These possibilities are called the irreducible representations:the characters for each possible irr rep under each sym opmakes up the character table for each point group.
• If a matrix representing a symmetry operation is transformed into block diagonal form then each little block is also a representation of the operation since they obey the same multiplication laws.
• When a matrix can not be reduced further we have reached the irreducible representation. The number of reducible representations of symmetry operations is infinite but there is a small finite number of irreducible representations.
• The number of irreducible representations is always equal to the number of classes of the symmetry point group.
• As stated before, all representations of a certain symmetry operation have the same character (trace of their matrix representation). So we will work with them rather than the matrices themselves.
• The characters of different irreducible representations of point groups are found in character tables. Character tables can easily be found in textbooks.
• Reducing big matrices to block diagonal form is always possible but not easy. Fortunately we do not have to do this ourselves.
Reducible to Irreducible RepresentationFrom the orthonormalization of the IRRs (also known as the ”Grand Orthogonalization Theorem”), we can see that
where (i.e., the red. rep. is a lin comb.)
where ai is the number of times the irreducible representation irr appears in red , h the order of the group, l an operation of the group, g(c) #of sym ops in class l,red the character of the operation l in the red. rep., and irr the character of l in the irreducible representation.
The projection operator takes a non-symmetry-adapted basis set of a representation and projects it along new directions,so that it belongs to a specific irr. rep. of the group.
R
ll RRh
P ˆ)(1ˆ )(
where P l is the projection operator of the irr rep l, (l) is the character of the operation R for the rep l, and R means application of R to our original basis component.
Within Dirac Equations (relativistic wave functions), the functionsthat represent half-integer spins are treated as double valuedwithin a single 2 rotation (sorta like a Möbius strip), “spinors”.
So the usual identity element (C1) does not return the function, but rather the negative of the function!
Solution:In addition to the usual rotations, we add the rotation operation
R (= C2 = original E) and all the operations formed by taking the product of R with the original sym ops.
For angular momentum basis functions with half-integer values ONLY, these new sym ops, R (i.e., C2) can have characters different than the “old” ops.
Twice the number of sym ops, but not twice as many classes. This also give us a new identity element E = C4