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Unformatted text preview: Physical Chemistry Course Number: C362 15 Group Theory Basics 1. A good reference: Group Theory and Quantum Mechanics by Michael Tinkham. 2. We said earlier that we will go looking for the set of operators that commute with the molecular Hamiltonian. We had earlier encountered the angular momentum operators that commute with the full Hamiltonian. We will en- counter a new family of operators: molecular symmetry operators that also commute with the Hamiltonian. 3. Why is that so? Because if you exchange any two atoms that are identi- cal the molecule does not change. The probability given by the electronic wavefunction must be unchanged with respect to such exchanges or symme- try operators as they are called and all observables should be the same with respect to interchange of identical atoms. For example, if I replace the posi- tion on hydrogen atom in water, with that of the other, without telling you I did so, you (or for that matter any experimental detector) would be unable to tell the difference. Hence the Hamiltonian has to be invariant with respect to such operations. 4. In such cases that the Hamiltonian commutes with the operations that leave the molecule invariant. 5. Kind of symmetry operations that we look for in a molecule are (a) n-fold rotation axis. (b) mirror planes or reflection planes. (c) inversion center. (d) rotation-reflection axis. (e) identity (means do nothing). 6. Consider for example the case of ammonia. Chemistry, Indiana University 98 c circlecopyrt 2011, Srinivasan S. Iyengar (instructor) Physical Chemistry Course Number: C362 7. It has the following symmetry operations that leave the molecule invariant: E, C 3 , C 2 3 v , v , v 8. Like all the other operators that commute with the Hamiltonian, the symme- try operators also make things easier by providing additional information on the nature of the eigenfunctions. This is similar to the additional informa- tion we had due to commuting operators like the angular momentum opera- tors (where we obtained additional quantum numbers such as the azimuthal quantum number, l , and the magnetic quantum number, m ). The symmetry operators that commute with the Hamiltonian also provide addition quantum numbers but these are given a different name; irreducible representations as they are called have this additional information. We will see more on this. 9. In all cases these additional quantum numbers provide information about the wavefunction. 10. But before we get that far some examples where symmetry can make our life easier without doing any work: Consider a molecule that has an inversion center. Clearly, the probability density of the electrons has to be symmetric about this inversion center....
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- Winter '11