Traced Cartesian moments

Laplace s equation, V V = 0, means that the number of unique elements needed to evaluate an interaction energy can be reduced. For the second moment this amounts to a transformation into a traceless tensor form, a form usually referred to as the quadrupole moment [5]. Transformations for higher moments can be accomplished with the conditions that develop from further differentiation of Laplace s equation. With modern computation machinery, such reduction tends to be of less benefit, and on vector machines, it may be less efficient in certain steps. We shall not make that transformation and instead will use traced Cartesian moments. It is still appropriate, however, to refer to quadrupoles or octupoles rather than to second or third moments since for interaction energies there is no difference. Logan has pointed out the convenience and utility of a Cartesian form of the multipole polarizabilities [6], and in most cases, that is how the properties are expressed here. 

Since a second-rank cartesian tensor Tap transforms in the same way as the set of products uaVfj, it can also be expressed in terms of a scalar (which is the trace T,y(y), a vector (the three components of the antisymmetric tensor (1 /2 ) Tap — Tpaj), and a second-rank spherical tensor (the five components of the traceless, symmetric tensor, (I /2)(Ta/= + Tpa) - (1/3)J2Taa). The explicit irreducible spherical tensor components can be obtained from equations (5.114) to (5.118) simply by replacing u vp by T,/ . These results are collected in table 5.2. It often happens that these three spherical tensors with k = 0, 1 and 2 occur in real, physical situations. In any given situation, one or more of them may vanish for example, all the components of T1 are zero if the tensor is symmetric, Yap = Tpa. A well-known example of a second-rank spherical tensor is the electric quadrupole moment. Its components are defined by... 

Each field index corresponds to a set of cartesian directions, and so each derivative has the symmetry of a power of the set x, y, z, symmetrized and with the trace(s) subtracted. The symmetries spanned by the moments and polarisabilities can therefore be deduced from then-indices as, for the moments... 

In this expression, C is a constant specific for each interaction, depending on gyromag-netic ratios and nuclear electric quadrupole moments. la is a Cartesian component of the nuclear spin, with the sum extended to all three Cartesian coordinates. Rap represents the components of a 3 x 3 Cartesian tensor of second rank that specifies the detailed nature of each interaction, some examples of which appeared in the expressions given previously (a, V, and J). Finally, Ap is a Cartesian component of a vector that can be the same nuclear spin vector (quadmpolar interaction), another nuclear spin vector (dipolar interaction or /-coupling), or the external magnetic field (chemical shiff interaction). Some of the tensors represented by Rap are traceless (cases of dipolar and quadmpolar interactions), whereas others (cases of chemical shift and /-coupling) posses non-vanishing trace. 

Cartesian electric quadrupole moment operator. The electric dipole - magnetic dipole and electric dipole - electric quadrupole polarizabilities govern the optical rotation of chiral molecules. For samples of randomly oriented molecules, the contribution from the electric dipole - electric quadrupole polarizability averages to zero, and only the trace (the sum of the diagonal elements) of the electric dipole - magnetic dipole polarizability contributes to the optical rotation. The... 

For infrared absorption is the operator for the electric dipole moment which is a polar vector transforming like the Cartesian coordinates x, y, z. The corresponding character is given by the trace of the matrix (Eqn [70]). 

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