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The Group Theory
Published in Mikhail G. Brik, Chong-Geng Ma, Theoretical Spectroscopy of Transition Metal and Rare Earth Ions, 2019
Mikhail G. Brik, Chong-Geng Ma
Since the point groups of all physical objects include the symmetry operations defined in the three-dimensional space, there can be only one-, two-, and three-dimensional irreducible representations (strictly speaking it is true for the point groups only). A standard practice is to denote the one-dimensional irreducible representations by letters A and B, two-dimensional—by letter E, and three-dimensional—by letter T. The letter “A” is used for the representations, the basis functions of which are symmetric with respect to the rotation about the main axis of the highest order, whereas the letter “B” is used for the representations with the functions, which are antisymmetric with respect to the same rotation. It is also clear that the character of an irreducible representation for an identity element is just the dimension of this irreducible representation. If there are several irreducible representations of the same dimension, they can be distinguished by a subscript, like A1, T1, T2, etc. This is the so-called Mulliken notation. Another notation, where the irreducible representations are denoted by the capital Greek letter Γ with subscripts (like Γ1, Γ2, etc.) was introduced by Bethe.
Symmetries and Group Theory
Published in Mattias Blennow, Mathematical Methods for Physics and Engineering, 2018
As we also saw, it is sometimes possible to further reduce these representations into direct sums of representations of lower dimension. Naturally, this cannot go on forever. At the very least, the process must stop when we reach one-dimensional representations, since one dimensional vector spaces cannot be split into direct sums of lower dimensional ones. We call representations that cannot be reduced irreducible representations or, the more common short-hand, irreps. For every finite group, we will find that there are only a finite number of (inequivalent) irreps and it will be possible to write any representation as a direct sum of these.
Linear Algebra and Mathematical Physics
Published in Leslie Hogben, Richard Brualdi, Anne Greenbaum, Roy Mathias, Handbook of Linear Algebra, 2006
Symmetry plays a big role in quantum mechanics. Both vectors and operators decompose into representations of the rotation groups SO (3) and SU (2). The irreducible representations are finite-dimensional, so the study of rotations (and angular momentum) often reduces to a study of finite matrices.
On extending and optimising the direct product decomposition
Published in Molecular Physics, 2019
The point group of a molecule is composed of the set of symmetry operations such as reflections, rotations, inversion, etc. that leave the molecular framework unchanged. These operations form a group in the mathematical sense, with multiplication defined as the composition of symmetry operations. When an object (such as a molecular orbital) is subjected to one of these operations, its behaviour can be described by its character, that is the overlap of the object before and after the operation. The collection of the characters for each operation gives the representation. For combinations of objects, such as a charge distribution defined by a pair of orbitals, the individual characters of the overall object are the product of the individual characters. Taking the representations as a whole, this element-wise multiplication to give a new combined representation is called the direct product (⊗), not to be confused with the outer or direct product used in other areas of mathematics. In order to describe and classify arbitrary representations, the irreducible representations (irreps) of the group form a unique basis: the way in which any particular object transforms (its representation) can be uniquely decomposed into a linear combination of the irreps. Each group has an irrep called the totally-symmetric irrep which has characters of unity. In order for an integral to be non-zero, the combined representation of the integrands must include (have non-zero projection onto) this totally-symmetric irrep. Individual components of the integrand (integrals, amplitudes, etc.) are generally chosen to be symmetry-adapted to a particular irrep, meaning that e.g. the direct product of the orbital representations for a given integral or amplitude tensor element must include the chosen overall irrep in order for that element to be non-zero. This sparsity is the origin of reduced computational cost when using point group symmetry.