Abstract
The eigenfunction approach used for discrete symmetries is deduced from the concept of quantum numbers. We show that the irreducible representations (irreps) associated with the eigenfunctions are indeed a shorthand notation for the set of eigenvalues of the class operators (character table). The need of a canonical chain of groups to establish a complete set of commuting operators is emphasized. This analysis allows us to establish in natural form the connection between the quantum numbers and the eigenfunction method proposed by J.Q. Chen to obtain symmetry adapted functions. We then proceed to present a friendly version of the eigenfunction method to project functions.
Highlights
The importance of symmetry at the level of fundamental laws of nature is widely recognized [1,2,3,4,5]
In this work we have presented a deep insight into the importance and meaning of the irreducible representations used to label the states of molecular systems
We suggest an alternative way to show that the irreps are quantum numbers in the context of finite symmetry groups
Summary
The importance of symmetry at the level of fundamental laws of nature (e.g., in high energy physics, subnuclear physics, nuclear and atomic physics) is widely recognized [1,2,3,4,5]. For discrete systems like molecules and crystals, the general labeling scheme consists in assigning to the states an irrep of the symmetry group together with the Hamiltonian eigenvalue, since several eigenstates may carry the same irreducible representation It is not stressed enough, the fact that the irreps correspond in reality to a set of quantum numbers associated with eigenvalues of the class operators of the group. Known as the eigenfunction approach, which allowed in a efficient way to carry out the projection of a basis set of functions to the space of functions spanning irreps This approach is presented in detail in his book from a mathematical point of view, establishing the connection with the traditional theory of representations of groups [17].
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