Tensor operator definitions

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. 

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... 

Here Bk s stand for the crystal field parameters (CFP), and Ck(m) are one-electron spherical tensor operators acting on the angular coordinates of the mth electron. Here and in what follows the Wyboume notation (Newman and Ng, 2000) is used. Other possible definitions of CFP and operators (e.g. Stevens conventions) and relations between them are dealt with in a series of papers by Rudowicz (1985, 2000,2004 and references therein). Usually, the Bq s are treated as empirical parameters to be determined from fitting of the calculated energy levels to the experimental ones. The number of non-zero CFP depends on the symmetry of the RE3+ environment and increases with lowering the symmetry (up to 27 for the monoclinic symmetry), the determination of which is non-trivial (Cowan, 1981). As a result, in the literature there quite different sets of CFP for the same ion in the same host can be found (Rudowicz and Qin, 2004). 

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) ... 

Definitions of operators for angular momenta of range higher than two are presented in Table D.2. A similar table given in [95] contains misprints which are corrected in [250]. Various papers use various notations for tensor operators and polarization moments. A summary of the various ways of denoting tensor operators may be found in [73, 136, 304], and of denoting polarization moments may be found in [304]. 

The disadvantages of the cogredient form of definition are connected with the presence of the factor (—1) in the normalizing coefficient 2Nk (D.32), which is a result of defining the reduced matrix element of the tensor operator (D.ll) into which this factor is introduced, in our opinion, without particular necessity. 

From this, it is a short step to the alternative definition of a spherical tensor operator... 

We shall first recall some definitions about q-deformed tensor operators within the framework of the so, (3) algebra. 

On the other hand, the modified boson operators and B are not components of a tensor operator in the sense of the above definitions. However, as shown in [31], one can use them to define a vector operator TjJ, as ... 

The result of a calculation must not depend on the coordinate system. Therefore, the result of a tensor operation must in itself be a tensor (of the appropriate order). All rules in this section fulfil this condition. A product definition of the form c = aj6j, directly multiplying the components, is not physically meaningful because the value of (ci) would depend on the coordinate system. 

The deformed charge distribution is generally axially symmetrical and this fact has an important consequence. It permits to characterize the charge distribution asymmetry by means of only one quantity, Q (called the quadrupolar moment), even if the quadrupolar operator, is a 3x3 matrix (in classical physics, a quadrupole is a second rank tensor). The definition of Q and the explicit form of are given in references 2 and 3. 

The purpose of this contribution is to give an overview of the results which center around the atomic density function and the recovery of the periodicity. Since all the calculations are based on atomic density functions, it is appropriate to revisit the construction of these densities in some depth. First a workable definition of the density function is established in the framework of the multi-configuration Hartree-Fock method (MCHF) [2] and the spherical harmonic content of the density function is discussed. A spherical density function is established in a natural way, by using spherical tensor operators. The proposed expression can be evaluated for any multi-configuration state function corresponding to an atom in a particular well-defined state and a recently developed extension of the MCHF code [3] is used for that purpose. Three illustrative examples are given. In the next section relativistic density functions for the relativistic Dirac-Hartree-Fock method [4] are defined. The latter will be used for a thorough analysis of the influence of relativistic effects on electron density functions later on in this paper. 

The incorporation of spin in second quantization leads to operators with different spin synunetry properties as demonstrated in Section 2.2. Thus, spin-free interactions are represented by operatOTs that are totally symmetric in spin space and thus expressed in terms of orbital excitation operators that affect alpha and beta electrons equally, whereas pure spin interactions are represented by excitation operators that affect alpha and beta electrons differently. For the efficient and transparent manipulation of these operators, we shall apply the standard machinery of group theory. More specifically, we shall adopt the theory of tensor operators for angular momentum in quantum mechanics and develop a useful set of tools for the construction and classification of states and operators with definite spin symmetry properties. 

Spin tensor operators play an important role in the second-quantization treatment of electronic systems since they may be used to generate states with definite spin properties. In the remainder... 

With these definitions the creation operators a, rcj) transform as spherical tensors under rotation. The annihilation operators do not. However, it is easy to construct operators that do transform as spherical tensors [Eq. (1.23)]. These will be denoted by a tilde and written as... 

An element of an electrostatic moment tensor can only be nonzero if the distribution has a component of the same symmetry as the corresponding operator. In other words, the integrand in Eq. (7.1) must have a component that is invariant under the symmetry operations of the distribution, namely, it is totally symmetric with respect to the operations of the point group of the distribution. As an example, for the x component of the dipole moment to be nonzero, p(r)x must have a totally symmetric component, which will be the case if p(r) has a component with the symmetry of x. The symmetry restrictions of the spherical electrostatic moments are those of the spherical harmonics given in appendix section D.4. Restrictions for the other definitions follow directly from those listed in this appendix. 

We will not try to give a definite description or classification of mathematical objects here. This section should be regarded merely as a collection of useful facts and nomenclature. We will cover the most common terms regarding continuous spaces in general and vector spaces, operators and matrices. We will not touch upon spinors, nor on tensors. 

A vectorial product will be defined below by (5.14), and V as a tensor of first rank is defined by (2.12). Operator L may be defined also in a more general way by the commutation relations of its components. Such a definition is applicable to electron spin s, as well. Therefore, we can write the following commutation relations between components of arbitrary angular momentum j ... 

Let us present the main definitions of tensorial products and their matrix or reduced matrix (submatrix) elements, necessary to find the expressions for matrix elements of the operators, corresponding to physical quantities. The tensorial product of two irreducible tensors and is defined as follows ... 

According to the foregoing discussion two-shell operators are expressed in terms of irreducible products of tensors W KkK lt, lj) and W Kklc li). Using a standard technique, we can represent the submatrix elements of these products in terms of relevant quantities defined for these tensors. We shall, therefore, confine ourselves to consideration of the submatrix element of tensor W KkK h,l2)- By its definition (17.58),... 

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