Moment traceless

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... 

Though the traceless moments can be derived from the unabridged moments, the converse is not the case because the information on the spherically averaged moments is no longer contained in the traceless moments. The general relations between the traceless moments and the unabridged moments follow from Eq. (7.3). For the quadrupole moments, we obtain with Eq. (7.2) ... 

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 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... 

As the traceless quadrupole moments are linear combinations of the spherical harmonic quadrupole moments, the corresponding expressions follow directly... 

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 relation between the second moments pxP of the deformation density and the traceless moments QxP can be illustrated as follows. From Eq. (7.2a), we may write... 

In the derivation of the traceless quadrupole moments from the electrostatic moments, the spherical components are subtracted. Thus, the quadrupole moments can be derived from the second moments, but the opposite is not the case. Spackman (1992) notes that the subtraction introduces an ambiguity in the comparison of quadrupole moments from theory and experiment. The spherical component subtracted is not that of the promolecule, but is based on the distribution itself. It is therefore generally not the same in the two densities being compared. On the other hand, the moments as defined by Eq. (7.1) are based on the total density without the intrusion of a reference state. 

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. 

When a nucleus with spin I is placed in a static magnetic field, the energy splits into 21 + 1 equally spaced levels, and a 2J-fold degenerate resonance line can be observed in an NMR experiment (16). Nuclei with spin I > 1/2 possess an electric quadrupole moment Q which may interact with the gradient of the electric crystal field at the site of the nucleus. This field gradient is a traceless tensor (17). 

Nuclei with spin I > j possess a quadrupole moment eQ and may interact with electric field gradients (EFG) present in the solid. The EFG is described by the traceless symmetrical tensor (72)... 

Hm describes the hyperfine interaction with the 57Fe nucleus. A is the magnetic hyperfine tensor and Hq describes the interaction of the quadrupole moment Q of the 7=3/2 nuclear excited state with the (traceless) electric field gradient (EFG) tensor V (the nuclear ground state has 7= 1/2 and lacks a spectroscopic quadrupole moment). In the absence of magnetic effects (for instance, for S 0, or S = integer for B = 0), the Mossbauer spectrum consists of a doublet with quadrupole splitting ... 

Among the molecular properties introduced above are the permanent electric dipole moment /xa and traceless electric quadrupole moment a(8, the electric dipole polarizability aajg(—w to) [aiso(to) = aaa(—or, o>)], the magnetizability a(8, the dc Kerr first electric dipole hyperpolarizability jBapy(—(o a>, 0) and the dc Kerr second electric-dipole hyperpolarizability yapys(— ( >, 0,0). The more exotic mixed hypersusceptibilities are defined, with the formalism of modern response theory [9]... 

An electrostatic quadrupole moment is a second-rank tensor characterized by three components in its principal-axis system. Since the trace of the quadrupole moment tensor is equal to zero, and atomic nuclei have an axis of symmetry, there is only one independent principal value, the nuclear quadrupole moment, Q. This quadrupole moment interacts with the electrostatic field-gradient tensor arising from the charge distribution around the nucleus. This tensor is also traceless but it is not necessarily cylindrically symmetrical. It therefore needs in general to be characterized by two independent components. The three principal values of the field-gradient tensor are represented by the symbols qxx, qyy and qzz with the convention ... 

Thus finally, the free energy contributions to G show explicitly that the electrostatic charge q interacts with the potential, the electric dipole moment vector m (Fig. 2.10) interacts with the external electric field E, the traceless electric quadrupole moment Q(/ interacts with the external field gradient, and so on ... 

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... 

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 ... 

The interaction between the orbital magnetic moment of the electrons and the nuclear moment can also be separated into an isotropic part (the chemical shift) and a traceless symmetric part (the shift anisotropy). The equation for the shift tensor contains two terms, usually called the dia- and paramagnetic contribution, but only the sum of the two corresponds to a physical quantity. Actually, the theory (76) is concerned with the shielding important difference between the two... 

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. 

Tables III-XVII give calculated permanent moments. Selected comparisons with experimental values or calculations of others are also listed. All values are in atomic units, and traceless rather than Cartesian forms are distinguished with Greek letters, 6 (quadrupole) and G (octupole). Coordinates for the atomic centers are listed. These specify the geometry used, which were equilibrium geometries, and implicitly the multipole expansion center (x = 0, y = 0, z = 0). The moments are given at both the SCF level and at the well-correlated level of coupled-cluster theory [95-102]. ACCD [103-106] was the particular coupled-cluster approach, and the moments were evaluated by expectation [102] with the cluster expansion truncated at single and double substitutions.

In the equations above, /i and indicate the permanent dipole and ( traceless) quadrupole moments, respectively. In the case of non-dipolar molecules the birefringence becomes... 

Explicit expressions for the traceless electric quadrupole, octupole and hexade-capole moments are... 

Most often, we first compute the moments and then use them to calculate the traceless multipole moments (cf. Table on p. 562). 

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