Representations of the rotation group

We next seek the irreducible representations of the full rotation group, formed by the infinite number of finite rotations R(ait). Because all such rotations can be expressed in terms of the infinitesimal rotation operators Jx, Jy and Jz (or equivalently J+, J and Jz), we start from these. 

It is a well-known piece of bookwork which we do not reproduce here (see, for example, Mandl [8]) to show that, starting from the commutation relations, the simultaneous eigenfunctions of J2 and Jz are  

There is a phase convention implicit in these two equations, the so-called Condon and Shortley convention [9], which is universally adopted. 

The (2 / + 1) eigenfunctions /. m) for a given value of j and for m values ranging from j in integer steps down to — j are transformed among themselves and with no other functions by the operators Jz and. J, and hence by rotations in general. They thus form the basis for an irreducible representation of dimension (2j + 1) which must be an integer. From this we see that j can take the possible values 0,1 /2,1,3/2,2,  

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Hyperspherical harmonics are now explicitly considered as expansion basis sets for atomic and molecular orbitals. In this treatment the key role is played by a generalization of the famous Fock projection [5] for hydrogen atom in momentum space, leading to the connection between hydrogenic orbitals and four-dimensional harmonics, as we have seen in the previous section. It is well known that the hyperspherical harmonics are a basis for the irreducible representations of the rotational group on the four-dimensional hypersphere from this viewpoint hydrogenoid orbitals can be looked at as representations of the four-dimensional hyperspherical symmetry [14]. 

Combining this last result with our knowledge of the classification of the irreducible representations of the group 50(3), we can show that the representation of the rotation group on homogeneous harmonic polynomials of any fixed degree is irreducible. 

Fano and Racah [5] defined the concept of the irreducible tensorial set as a set of 2k + 1 quantities (k is an integer or half-integer) transforming through each other according to the irreducible tensorial representations of the rotation group... 

The matrices DAB(afiy) provide a representation of the rotation group. 

La + l)-dimcnsional matrix D Wa) which represents the rotation Qa molecuk A. The set of these matrices forms a (2La + l)-dimensional irredudWe representation of the rotation group SO(3) In the active rotation convention, which wc are using, the rotation matrices are givMi by ... 

The quantities relevant to the rotationally averaged situation of randomly oriented species in solution or the gas phase must necessarily be invariants of the rotational symmetry. Accordingly, they must transform under the irreducible representations of the rotation group in three dimensions (without inversion), R3, just like the angular momentum functions of an atom. The polarisability, po, is a second-rank cartesian tensor and gives rise to three irreducible tensors (5J), (o), a(i),o(2), corresponding in rotational behaviour to the spherical harmonics, with / = 0,1,2 respectively. The components W, - / < m < /, of the irreducible tensors are given below. 

For the mathematically inclined Volkmer s Chapter 29 on Lame functions in the NIST Handbook of Mathematical Functions [50], and the article "A new basis for the representations of the rotational group. Lame and Heun polynomials" [51] are parts of the toolbox. 

The Wigner matrices multiply just like the rotations themselves. There is a one-to-one correspondence between the Wigner matrices of index l and the rotations R. These matrices form a representation of the rotation group. In fact, since the 2/ + 1 spherical harmonics of order / form an invariant subspace of Hilbert space with respect to all rotations, it follows that the matrices D ,m(R) form a (21 + 1) dimensional irreducible representation of the rotation R. Explicit formulas for these matrices can be found in books on angular momentum (notably Edmunds, 1957). 

The matrices Rab Py) provide a representation of the rotation group. However, this representation is reducible and thus its decomposition into... 

Each subset transforms only within itself under rotation and thus these subsets are irreducible tensors, they provide bases for irreducible representations of the rotation group of dimension 1, 3 and 5, respectively (see Table 1.13). 

The dimension of the vector representation is 2j I this result coincides with our earlier result of 2Z -f- 1. It is of great interest to note that since the dimension of the vector space g must be a whole number, the condition in Eq. 6.56 implies that,/ must be either integral or half-integral. Thus by this method we obtain both the single and double-valued representations of the rotation group. Each value of j (0, 1, 2, )... 

According to Racah, a unique characterization of the terms of these configurations is achieved by introducing only one quantum number in addition to the set (nd) SL. For this purpose the irreducible representations of the rotation group in five dimensions R can be used, but it is more common to use the seniority number v. In the latter case the states are classified according to the eigenvalues of the seniority operator... 

For the purpose of characterizing the terms of Racah used the irreducible representations of the rotation group in seven dimensions, Ry, and of the special... 

The Hamiltonian (O Eq. 2.23) maintains full symmetry and is invariant under electronic permutations and under rotation-reflections of the electronic coordinates. Trial functions are usually constructed from atomic orbitals and from their spin-orbitals. Permutational antisymmetry is achieved by forming Slater determinants from the spin-orbitals. Rotational symmetry is usually realized by vector coupling of orbitals that form bases for representations of the rotation group SO(3). Spin-eigenfunctions too are achieved by vector coupling. ... 

The Electronic Adiabatic-to-Diabatic Transformation Matrix and the Irreducible Representation of the Rotation Group... 

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