Operator tensor operators

The reduced matrix element of a vector operator (tensor operator of the first rank) is expressed in terms of the 67-symbol as follows... 

If we introduce the electromagnetic field tensor operator F ix), which can be decomposed as follows ... 

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... 

For low enough symmetries, both Bk and B q coefficients will be present in Equation 1.15, so that Equation 1.17 will be, in those cases, complex quantities. We finally note that the coefficients are transformed into CF parameters by multiplying them by the radial parts of the wave functions, represented by Rn/(r), on which the tensor operators do not act. 

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 ... 

Quantum mechanically the combination of two angular momenta is more complicated since the angular momenta are operators. (They are tensor operators of rank one.) The law of combination of angular momenta can be expressed, in general, by the so-called tensor product of two operators, indicated by the symbol x,... 

Another important property of the angular momentum is the Wigner-Eckart theorem.2 This theorem states that the matrix elements of any tensor operator can be separated into two parts, one containing the m dependence and one independent of m,... 

The algebra of U(4) can be written in terms of spherical tensors as in Table 2.1. This is called the Racah form. The square brackets in the table denote tensor products, defined in Eq. (1.25). Note that each tensor operator of multipolarity X has 2X+ 1 components, and thus the total number of elements of the algebra is 16, as in the uncoupled form. 

On account of the hermiticity of the operators, only even values of L are allowed. Since the D operators are tensor operators of rank 1, the only allowed values are L = 0,2. The L = 0 contribution has been treated in type (1). Hence, only the L = 2 contribution must be considered here. The matrix elements of the operators (4.122) with L = 2 are difficult to evaluate. Nonetheless, by making use of the angular momentum algebra, they can be evaluated in explicit... 

A tensor operator under the algebra G 3 G, T, is defined as that operator satisfying the commutation relations... 

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... 

As in Eq. (64), the electron spin spectral densities could be evaluated by expanding the electron spin tensor operators in a Liouville space basis set of the static Hamiltonian. The outer-sphere electron spin spectral densities are more complicated to evaluate than their inner-sphere counterparts, since they involve integration over the variable u, in analogy with Eqs. (68) and (69). The main simplifying assumption employed for the electron spin system is that the electron spin relaxation processes can be described by the Redfield theory in the same manner as for the inner-sphere counterpart (95). A comparison between the predictions of the analytical approach presented above, and other models of the outer-sphere relaxation, the Hwang and Freed model (HF) (138), its modification including electron spin... 

Tx are components of a tensor operator, the explicit expression for them being... 

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. 

We propose [15] a set of basis tensor algebra subroutines or btas. Tensors and tensor operators arise in many fields in the computational sciences, including computational quantum chemistry. The nomenclature BTAs(m,n), with m > n, where m and n are the respective ranks of the tensors, is proposed to establish a high level classification of tensor operations. The BTAS can be classified as follows -BTAS(1,0) BTAS(1,1)... 

Following are some of the important vector and tensor operations which are used in the derivations in Sec. II of this review these relations are given for Cartesian coordinates. The quantities v and F are vectors, v is a second-order symmetric tensor, and T is a scalar. 

In cartesian coordinates, the vector-tensor operator can be readily seen by inspection. In other coordinate systems, however, terms like the ones in the third row of the equation above result physically from the fact that control-surface areas vary, and mathematically from the fact that the derivatives of the unit vectors do not all vanish. We have recovered the expression in the previous section, which was developed entirely from vector-tensor manipulations ... 

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