Tensor operator method

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. 

It is well-known that the electron repulsion perturbation gives rise to LS terms or multiplets (also known as Russell-Saunders terms) which in turn are split into LSJ spin-orbital levels by spin-orbit interaction. These spin-orbital levels are further split into what are known as Stark levels by the crystalline field. The energies of the terms, the spin-orbital levels and the crystalline field levels can be calculated by one of two methods, (1) the Slater determinantal method [310-313], (2) the Racah tensor operator method [314-316]. 

Although in the case of f2, either method can be applied with ease, the determinantal method becomes cumbersome for more than two-electrons and it is necessary to take recourse to the tensor operator method for fN, N > 2. We will show the derivation of energy levels by both methods in what follows ... 

Note that the above energies are actually the diagonal elements. In order to calculate the off-diagonals, it is necessary to take recourse to either the Slater determinantal method or the tensor operator method. 

Crystal field energies by the tensor operator method We may expand the potential in terms of the tensor operators Cq to give... 

The alternative parameters P and P refer directly to an operator equivalent method 26), an irreducible tensor operator method (75) of handling the general first-order perturbation model. 

Each coupled spin level in the zero-field or correlation diagram had energy E(Si2, S34, S) for each wave function Si2, S34, S, M). The coupling scheme adopted was S12 = Sj + S2 S34 = S3 + S4 S = S12 + S34. Since the matrix of Eq. (14) cannot be diagonalized, matrix elements were worked out by tensor operator methods (50, 68). 

The most straightforward way of evaluating the angular matrix elements of Eqs. (17.12) and (17.13) is to use the method of Edmonds.5 The matrix elements are evaluated as the scalar products of tensor operators operating on the wave-functions of electrons 1 and 2. Using this approach we can write Wdas... 

Lanthanide complexes with axial symmetry (i.e., possessing at least a threefold axis, see sect. 2.4.2) are exclusively considered because the principal magnetic z axis coincides with the molecular symmetry axis (Forsberg et al., 1995) and the c 2 spherical tensor operators do not contribute to the crystal-field potentials (Gorller-Walrand and Binne-mans, 1996). The rhombic term of Bleaney s approach V6B Hi (eqs. (42), (46)) thus vanishes and the crystal-field independent methods (eqs. (51), (53)) can be used without complications. 

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) ... 

The desired coupled basis will be performed by the methods given by Racah and Wigner. Making use of the Wigner-Racah formalism and of the Wigner-Eckart theorem and observing some rules for the matrix elements of the products of tensor operators, we obtain for the matrix elements of the quadrupole interaction operator/Z ... 

Determinantal method In this method it is necessary to calculate the one electron matrix elements of an f electron. These are, in terms of tensor operator parameters... 

Although we have insured the correctness of our crystal field matrix elements by calculating them by two different methods (determinantal and tensor operators), there are three interesting checks that one can make to insure that there are no errors in the calculated crystal field matrix elements3. 

In calculations involving higher J multiplets, matrix elements of the crystal field Hamiltonian between states belonging to different J multiplets are needed. Although these can be calculated by the method of operator equivalents extended to elements non-diagonal in J, it is convenient to use a more general approach, utilizing Racah s tensor operator technique (26). In this method the crystal field interaction may be written as... 

We now consider this matter from the point of view of the irreducible tensor method. It is convenient to work in terms of the unit contra-standard tensor operators which are defined for each system i by the equation... 

Often basis functions are chosen which are bases for irreducible representations of the three-dimensional rotation-inversion group Rst, even though the physical system has a sub-group symmetry. In this case the tensorial methods exibit their particular potency because the tensor operators — also those representing constants of motion — can be expanded into components of irreducible representations of Rsi-... 

Giese, T. J., and York, D. M. (2008). Spherical tensor gradient operator method for integral rotation A simple, efficient, and extendable alternative to Slater-Koster tables,/. Chem. Phys. 129(1), 016102. 

Although the use of any one of these two methods completely solves the problem of calculating the matrix elements of Hi for configurations, several short-cuts can be achieved corresponding to various special cases. These also are due to Racah (1943, 1949). He defined unit tensor-operators such that... 

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. 

For strings containing two or more elementary operators, it is possible to constmct more than one tensor operator. We shall in Section 2.6.7 present a general method for the construction of tensor operators from strings of elementary operators. At present, we note that, by coupling the two doublet operators and it is possible to generate a singlet two-body creation... 

The various response tensors are identified as terms in these series and are calculated using numerical derivatives of the energy. This method is easily implemented at any level of theory. Analytic derivative methods have been implemented using self-consistent-field (SCF) methods for a, ft and y, using multiconfiguration SCF (MCSCF) methods for ft and using second-order perturbation theory (MP2) for y". The response properties can also be determined in terms of sum-over-states formulation, which is derived from a perturbation theory treatment of the field operator — [iE, which in the static limit is equivalent to the results obtained by SCF finite field or analytic derivative methods. 

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