Properties and Representations of Groups

FIGURE 4.14 Symmetry Operations for Ammonia. (Top view) NEi3 is of point group 1 H 1 1 H 

Each group must contain an identity operation that commutes (in other words, EA = AE) with all other members of the group and leaves them unchanged EA =AE = A). 

Each operation must have an inverse that, when combined with the operation, yields the identity operation (sometimes a symmetry operation may be its own inverse). Note By convention, we perform sequential symmetry operations/rom right to left as written. 

The product of any two group operations must also be a member of the group. This includes the product of any operation with itself. 

The associative property of combination must hold. In other words, A(BC) = AB)C. 

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We first note that all types of A orbitals (in D3h) have the same symmetry properties with respect to the rotations constituting the subgroup C3 also, both and " orbitals have the same properties with respect to these rotations. Thus we can use the group C3 to set up some linear combinations that will be correct to this extent. Since these rotations about the C3 axis do not interchange any of the orbitals 0, 02, 03 with those of the set 4, 05, 6, we can, temporarily, treat the two sets separately. We thus first write down linear combinations corresponding to the A and representations of C3. As shown in Section 7.3 for such cyclic systems, the characters are the correct coefficients, and we can thus write, by inspection of the character table for the group C3 ... 

To make these notions precise, the transformation properties of the wavefunction x under spatial and time translations as well as under spatial rotations and pure Lorentz transformations must be specified and it must be shown that the generators of these transformations form a unitary representation of the group of translations and proper Lorentz transformations. This can in fact be shown5 but will not be here. 

Invariance principle, 664 Invariance properties of quantum electrodynamics, 664 Inventory problem, 252,281,286 Inverse collisions, 11 direct and, 12 Inverse operator, 688 Investment problem, 286 Irreducible representations of crystallographic point groups, 726 Isoperimetric problems, 305 Iteration for the inverse, 60... 

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