Irreducible tensor operator

Here D(Q) = D(a,f, y), Euler angles a, (5 and y being chosen so that the first two coincide with the spherical angles determining orientation e = e(j], a). Using the theorem about transformation of irreducible tensor operators during rotation [23], we find... 

It is noteworthy that dq(e,t) does not satisfy this relation, as equality [J,x, dq] = 2 C q dq+ll (the definition of an irreducible tensor operator) does not hold for it [23]. Integration in (7.18), performed over the spherical angles of vector e, may be completed up to an integral over the full rotational group due to the axial symmetry of the Hamiltonian relative to the field. This, together with (7.19), yields... 

A complete decomposition of the ab initio computed CF matrix in irreducible tensor operators (ITOs) and in extended Stevens operators. The parameters of the multiplet-specific CF acting on the ground atomic multiplet of lanthanides, and the decomposition of the CASSCF/RASSI wave functions into functions with definite projections of the total angular momentum on the quantization axis are provided. 

Addressing first the heteronuclear case (i.e., the DCP experiment introduced by Schaefer and coworkers in 1984), the dipolar coupling may be expressed as in (10) that upon heteronuclear truncation of the transverse terms may be recast in irreducible tensor operator form (7Z = T[ 0, Sz = Tf 0) as... 

T%0 are irreducible tensor operators of rank X. The Hamiltonian of a static powder sample can be easily derived from the MAS Hamiltonian by setting a>r = 0 ... 

As discussed in Ref. [1], we describe the rotation of the molecule by means of a molecule-fixed axis system xyz defined in terms of Eckart and Sayvetz conditions (see Ref. [1] and references therein). The orientation of the xyz axis system relative to the XYZ system is defined by the three standard Euler angles (6, (j), %) [1]. To simplify equation (4), we must first express the space-fixed dipole moment components (p,x> Mz) in this equation in terms of the components (p. py, p along the molecule-fixed axes. This transformation is most easily done by rewriting the dipole moment components in terms of so-called irreducible spherical tensor operators. In the notation in Ref. [3], the space-fixed irreducible tensor operators are... 

The forced electric dipole mechanism was treated in detail for the first time by Judd (1962) through the powerful technique of irreducible tensor operators. Two years later it was proposed by Jorgensen and Judd (1964) that an additional mechanism of 4/-4/ transitions, originally referred to as the pseudo-quadrupolar mechanism due to inhomogeneities of the dielectric constant, could be as operative as, or, for some transitions, even more relevant than, the forced electric dipole one. 

An irreducible tensor operator of rank S in spin space is defined as a set of 2S+1 operators, T(S,M), with M running from -S to S fulfilling the relations... 

A general development of matrix elements of an irreducible tensor operator leads to the Wigner-Eckart theorem (see, for example, Tinkham [2] or Chisholm [7]), which relates matrix elements between specific symmetry species to a single reduced matrix element that depends only on the irrep labels, but this is beyond the scope of the present course. 

Since any operator can be written as the sum of Hermitian and anti-Hermitian operators, we can restrict our discussion to these two types only. Further, any operator can be written as a linear combination of irreducible symmetry operators, so we can restrict ourselves to irreducible tensor operators. An operator matrix 0(r, K) that transforms according to the symmetry (T, K) obeys the relationship... 

There is no paradox [112] in the use of e(3) as an operator as well as a unit vector. In the same sense [112], there is no paradox in the use of the scalar spherical harmonics as operators. The rotation operators in space are first-rank Toperators, which are irreducible tensor operators, and under rotations, transform into linear combinations of each other. The Toperators are directly proportional to the scalar spherical harmonic operators. The rotation operators, J, of the full rotation group are related to the T operators as follows... 

The irreducible tensor operators obey the following properties [35-38] ... 

As a consequence of the Wigner-Eckart theorem the replacement theorem holds true a matrix element of any irreducible tensor operator can be expressed with the help of the matrix elements formed of the angular momenta... 

Of frequent interest is the need of evaluation of matrix elements for a tensor product of two irreducible tensor operators... 

In this section we first present a set of general transformation formulae for tensor operators associated with SRMs. These then serve as a mathematical tool for the formulation of Wigner-Eckart theorems and selection rules for irreducible tensor operators associated with multipole transitions of SRMs. The concept of isometric groups will allow a formulation of selection rules in strict analogy to the group theoretical treatment of quasirigid molecules first presented by Wigner5. ... 

Ayl transforms as an irreducible tensor operator under operations of G, and as a rank-2 spinor in the angular momentum algebra generated by the quasispin operators. We form the quasispin generators as a coupled tensor in quasispin space Q(A) = i[AAAA]7V2, where [AB] = Y.qq lm q c/)AqBqi. In the Condon and Shortley spherical basis choice (with m = 1, 0, — 1) for the SO(3) Clebsch-Gordan coefficients [11-13,21-23] this takes the form [6,21] ... 

In second quantized notation, a single-particle irreducible-tensor operator Vn(aK, a L, r) is written... 

The unit tensor operators are irreducible-tensor operators with reduced matrix elements of unity. They are a valid choice to use as a basis in order to express any arbitrary tensor operator as a linear combination, since they are linearly independent. Attention is restricted to these for the sake of simplicity. Hence the definition of unit tensor single-particle operator U (aK, a L, r) is... 

To take advantage from the pseudo-angular momentum representation we shall employ the technique of the irreducible tensor operators as suggested in Ref. [10]. One can easily establish the following interrelations between the matrices Orr and the orbital angular momentum operators ... 

Then, using the matrix elements for the complex irreducible tensor operators (see Ref. [11]) we arrive at the following expression for the matrix elements of the vibronic interaction ... 

A number of techniques are known that can reduce the dimension of the involved matrices that either utilize symmetry arguments (in the case of the irreducible tensor operator method)22 or effectively limit the set of results to the lowest eigenvalues. 

The multipole (or polarization) moments introduced according to (2.14) present a classical analogue of quantum mechanical polarization moments [6, 73, 96,133, 304]. They are obtained by expanding the quantum density matrix [73, 139] over irreducible tensor operators [136, 140, 379] and will be discussed in Chapters 3 and 5. 

As has already been mentioned in Section 3.1, the simplest way of accounting for the symmetry properties of processes participating in optical pumping is the expansion of the density matrices /mm and over irreducible tensor operators Tq [136, 140, 304, 379] ... 

The choice of the phase and the normalization of the irreducible tensor operators is somewhat arbitrary [379]. Following [133], we will employ the following definition of matrix elements of tensor operators (other existing methods are discussed in Appendix D) ... 

Other forms of normalization, as well as forms denoting irreducible tensor operators may be found in [304] and in Appendix D. With the aid of the orthogonality relation one may easily express the quantum mechanical polarization moments fq and Pq through the elements of the density matrix /mm and... 

In order to determine whether orientation has taken place or not, it is convenient to expand the matrix elements /mm over irreducible tensor operators according to (5.17). As follows from Appendix D, reverse expansion (5.20) may be written conveniently in the following way ... 

In the paper by Fano [141] real irreducible tensor operators were first obtained ... 

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