Traceless definition

The moments defined by Eq. (7.1) are referred to as the unabridged moments. For moments with / > 2, an alternative, traceless definition is often used. In the traceless definition, the quadrupole moment is given by... 

As noted above, in the traceless definition the /th-order multipoles are the sole contributors to the /th electrostatic moments. This implies that the traceless moments derived from the total density p(r) and from the deformation density Ap(r) are identical, that is, ,mp(p) = 0,mp(Ap) for / > 2. 

The operators for the potential, the electric field, and the electric field gradient have the same symmetry, respectively, as those for the atomic charge, the dipole moment, and the quadrupole moment discussed in chapter 7. In analogy with the moments, only the spherical components on the density give a central contribution to the electrostatic potential, while the dipolar components are the sole central contributors to the electric field, and only quadrupolar components contribute to the electric field gradient in its traceless definition. 

The linear relationships between the traceless moments 0 and the spherical harmonic moments lmp are obtained by use of the definitions of the functions clmp. For example, for the quadrupolar moment element xx, we obtain the equality (3x2 — l)/2 = a (x2 — y2)/2 + b(3z2 — 1). Solution for a and b for this and corresponding equations for the other moments leads to... 

One prominent class of monofunctional anchor that provides access to molecules having no attachment memory for solid-phase synthesis is called traceless or clean break linkers.7,16-18 Although this definition could be used for the classification of linkers, one would advantageously... 

Nuclear quadrupolar interaction arises from the coupling between the nuclear quadrupole moment Q and the EFG at the nuclear position. The EFG varies in space and is described by a traceless second-rank tensor. The EFG tensor is diagonal and its three principal components are VXXr Vyy and Rzz with the definition of VZZ > Vyy > VX < Such a principal-axis system for the EFG tensor is defined with the direction of the external magnetic field, as illustrated in Figure 3(A). It is convenient to express such quadrupolar interactions by using the following two parameters ... 

At first order, it can be shown that only the symmetrized part of the interaction tensor contributes to the frequency shift. The majority of second-order contributions arise from large EFG, the EFG tensor being symmetric by definition. Thus, only symmetric second-rank tensors T can be considered, which can be decomposed into two contributions T = isol3 + AT with D3 the identity matrix. The first term is the isotropic part so = l/3Tr(T) that is invariant by any local symmetry operation. The second term is the anisotropic contribution AT, a symmetric second-rank traceless tensor, which depends then on five parameters the anisotropy 8 and the asymmetry parameter rj that measures the deviation from axial symmetry, and three angles to orient the principal axes system (PAS) in the crystal frame. The most common convention orders the eigenvalues of AT such that IAzzL and defines 8 — Xzz and rj — fzvv i )... 

The normalization constant G is often conveniently defined by setting Qzz equal to unity in perfectly oriented system. By definition, Q is real, symmetric, and traceless. Q vanishes in the isotropic phase as per the requirement of its suitability for an order parameter to describe the I-N transition. The macroscopic tensor order parameter Q can always be diagonalized ... 

There are in fact only three distinct classes of non-zero traceless tensors with a definite symmetry type, and these are given below together with their more familiar quantum numbers [17] ... 

The interaction tensors are symmetric with respect to permutation of their suffixes, as is clear from the definition. Moreover /R satisfies Laplace s equation V (l// ) = 0. Consequently is traceless with respect to any pair of suffixes = 0. 

Now Qq/3 is traceless by definition. In the principal axis system of the susceptibility tensor, Qa/3 can be written in terms of two order parameters Q and P,... 

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