Spherical tensor operator

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... 

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. 

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 same ideas can be readily extended to cover quantum mechanical operators. The formulation is slightly different because, although a transformation S turns a wave function irreducible spherical tensor operator (usually abbreviated to spherical tensor operator) T T) of rank k is defined as an entity with (2k + 1) components, Tkp(T), which transform under rotations as... 

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

The most obvious example of a spherical tensor operator is the angular momentum itself (a spherical tensor of rank one) ... 

Matrix elements of spherical tensor operators the Wigner-Eckart theorem... 

We now consider how to evaluate the matrix elements of a spherical tensor operator, written as (rj, j, m Tk(A) rj, /". m ) where r] and if denote any further quantum numbers required to characterise the states (for example, vibrational quantum numbers v). If we now rotate the bra, operator and ket through Euler angles o> using equations (5.44) and (5.102), the result must be unaffected. Thus ... 

Another immediate corollary of the Wigner-Eckart theorem is the replacement theorem, which allows one to write the matrix elements of one spherical tensor operator,... 

The rest of the derivation is performed by expressing the Clebsch-Gordan coefficients in terms of3-j symbols andmaking use of equation (5.92). The corresponding equation for a spherical tensor operator Tk(A2) which acts only on part 2 ofthe coupled scheme is... 

In molecular quantum mechanics, we often find ourselves manipulating expressions so that one of a pair of interacting operators is expressed in laboratory-fixed coordinates while the other is expressed in molecule-fixed. A typical example is the Stark effect, where the molecular electric dipole moment is naturally described in the molecular framework, but the direction of an applied electric field is specified in space-fixed coordinates. We have seen already that if the molecule-fixed axes are obtained by rotation of the space-fixed axes through the Euler angles (, 6, /) = >, the spherical tensor operator in the laboratory-fixed system Tkp(A) can be expressed in terms of the molecule-fixed components by the standard transformation... 

If the scalar product is formed from spherical tensor operators which both act on the same inner part of a coupled scheme, it is intuitively obvious that... 

This Hamiltonian has been generalised so that it applies to n states of any multiplicity. It may be readily rewritten in terms of spherical tensor operators as follows ... 

This form of the Hamiltonian, using the Frosch and Foley constants, is less useful than the alternative form, written in terms of spherical tensor operators. This is particularly true when the basis functions for the two electronic states are different. For the ground state we use the functions t], A S, I, G N, G, F) and for the excited state //. A N, S,./ ./. /. / ). As we have seen in chapters9 and 10, the appropriate effective Hamiltonian when the excited state basis functions are used is... 

The Wigner-Eckart theorem (Biedenharn and Louck, 1981a, p. 96) can be used to obtain Eq. (40) directly. According to this theorem the matrix elements in an angular momentum basis of a spherical tensor operator V (q = —k, — k + 1,k) have a particularly simple structure given by3... 

The time development of the spin system during the pulse sequences of fig. 2 and 3 is conveniently discussed in terms of a spherical tensor operator expansion of the density matrix. For any time,... 

T ,k are the irreducible spherical tensor operators for the quadrupole interaction given by... 

To gain some insight into the operation of Eq. (8) in practise, we shall write the time-dependent, perturbing operator in terms of spherical tensors and spherical tensor operators ... 

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