Traceless operators

R5. If S S, then trace (S) = 0 This follows directly from R4 since S is a matrix representation of the zero operator, which is traceless. 

In a different form, the traceless moment operators can be written as the Cartesian spherical harmonics c,mp multiplied by r, which defines the spherical harmonic electrostatic moments ... 

The EFG tensor elements can be obtained by differentiation of the operator in expression (8.5) for Ea to each of the three directions / . This procedure removes the spherical component, which does not affect the electric field, and yields the traceless result... 

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. 

To this end, we resort to a novel general approach to the control of arbitrary multidimensional quantum operations in open systems described by the reduced density matrix p(t) if the desired operation is disturbed by linear couplings to a bath, via operators S B (where S is the traceless system operator and B is the bath operator), one can choose controls to maximize the operation fidelity according to the following recipe, which holds to second order in the system-bath coupling (i) The control (modulation) transforms the system-bath coupling operators to the time-dependent form S t) (S) B(t) in the interaction picture, via the rotation matrix e,(t) a set of time-dependent coefficients in the operator basis, (Pauli matrices in the case of a qubit), such that ... 

The first term is a tensor of rank zero involving only spin variables. It does not contribute to the multiplet splitting of an electronic state but yields only a (small) overall shift of the energy and is, henceforth, neglected. The operator 7 is a traceless (irreducible) second-rank tensor operator, the form of which in Cartesian components is... 

For scalar /> and vector a, however, serious contradictions arise (Problem 3.6.1), which were "fixed" by defining ax, ay, az, and j8 as anticommuting operators, representable by the following traceless 4x4 matrices ... 

As usual, its isotropic portion, corresponding to the average pressure, is first removed as being without consequence for the incompressible flows of interest to us here. Its antisymmetric part, which is related to any external-body couples exerted on the suspended particles, is also separated out. These operations result in the symmetric and traceless average deviatoric stress,... 

The electric multipoles are sometimes redefined to be traceless in any two suffixes [6, 21]. For instance, the electronic contribution to the traceless electric quadrupole operator is... 

If the traceless form (34) of quadrupole operator is retained, then the corresponding Hamiltonian becomes... 

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 )... 

Thus, the computation of the SSC Hamiltonian matrix element requires calculations of six (five if one employs the fact that the SSC operator is traceless) Cartesian components of two-electron integrals. Those two-electron SSC integrals can be evaluated using an RI approximation [113]. 

In the static case, electric multipoles are often redefined to give forms that are traceless on any two indices. In this manner, the above restrictions (154) can be incorporated [49]. The general form of such traceless multipole operators is... 

As defined in Equation 3.4, the spin part of i is a traceless symmetric tensor operator that can have nonvanishing matrix elements between states of spin multiplicity different by up to four, such as singlet and quintet or triplet and septet, etc., but in first order we now consider only its elements in the basis of the three sublevels of the T state. JZ is diagonalized in this subspace by rotation of the coordinate system into the principal system of axes x, y, and z in the molecular frame. In principle, the molecular magnetic axes defined in this fashion are different for every triplet state of the molecule. In the presence of symmetry, some or all of them are constrained to the molecular symmetry axes. 

Traceless forms of the higher multipole operators have been given by Buckingham.12 Much of the work reviewed will refer to the case of a spatially uniform applied field, when the perturbed hamiltonian reduces to... 

A non-traceless form is used for the operator (6) [56]. The electronic interaction Hamiltonians are written... 

It is readily seen that the 2" rank Cartesian tensor that we have derived in equation (7) is symmetric and traceless and so Fj j is an irreducible Cartesian tensor of symmetry type 11 2. We have already discussed how to generate a tensor with a specified symmetry type from a given tensor using Young operators, but the question remains how to project out the traceless component of a tensor The example... 

Because of chain inextensibility, the shear rate of any slip system is not dependent on the normal-stress component in the chain direction (Parks and Ahzi 1990). This renders the crystalline lamellae rigid in the chain direction. To cope with this problem operationally, and to prevent global locking-up of deformation, a special modification is introduced to truncate the stress tensor in the chain direction c. Thus, we denote by S° this modification of the deviatoric Cauchy stress tensor S in the crystalline lamella to have a zero normal component in the chain direction, i.e., by requiring that 5 c,c = 0, where c,- and c,- are components of the c vector (Lee et al. 1993a). The resolved shear stress in the slip system a can then be expressed as r = where R is the symmetrical traceless Schmid tensor of stress resolution associated with the slip system a. The components of the symmetrical part of the Schmid tensor / , can be defined as = Ksfw" + fs ), where if and nj are the unit-vector components of the slip direction and the slip-plane normal of the given slip system a, respectively. 

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