Lecture 31Reducing Reducible Representations - PDF document

1 Physical ChemistryLecture 31Reducing Reducible Representations Reducible representations In degenerate groupsProducts of irreducible representations may give reducible representationsProducts of wave functions may be represented by reducible representations Direct sums representation is A direct sum is given by the representation with characters that are sum of characters 2 C3 3 C2 2 S3 1 2 1 2 2 C3 3 C2 2 S3 12 1 Direct product Direct products are Examples in C representation of a multi-electron wave Gives a means to decompose a multi-electron wave function into terms 2 C3 1 2 AAEE21 2 C3 E 12E Definitions in group theory Dimension of the group the characters of the operations in the Number of elements in a class (Number of similar operations in a class 2 C3 1 2 263vNNhC Projection operators Projection operator produces the number of representations in a reducible Mulliken symbols convenientForm projection operator for an irreducible representation by multiplying the number of operations in the class times the character“Normalized” by the dimension of the group ,2,26103),1(2,21613,2,161)1(3,12,11613,2,16113,12,116121EAAPP 2 C3 1 2 2 Reducing a representation with projection operators Inner product of the projection operator with the reducible representation gives the number of present in the reducible Example in C    66001242610,1,40,2,261)(166031241610,1,43,2,161)(166031241610,1,43,2,161)(21EEPEEPEEPEAAEAAEE21 Applications to H Configuration results in a direct product of one-electron wave functions Want multi-electron wave functions that conform to known symmetry Perform a direct product to find the term Reduce the direct product to a direct sumGives all terms arising from that configutation Example: ground term of 10 electrons 212122212113121babaa11111221111ABBAABBAAAA Excited state of H Find first excited state by promoting a single Consider only partially Filled shells give totally symmetric representation as a product electrons may be paired or unpairedSingletTriplet Multi-electron wave functions found as direct products of one-electron wave functionsCan be classified as reducible or irreducible representations Reducible representations can be expressed as a direct sum of irreduUse projection operators to determine the direct sum The direct sum gives the terms that arise from a configuration Use Pauli’s principle to determine possible spin of wave functions Terms symbols use irreducible representations of the