Choice of origin

The multipole moment of rank n is sometimes called the 2"-pole moment. The first non-zero multipole moment of a molecule is origin independent but the higher-order ones depend on the choice of origin. Quadnipole moments are difficult to measure and experimental data are scarce [17, 18 and 19]. The octopole and hexadecapole moments have been measured only for a few highly syimnetric molecules whose lower multipole moments vanish. Ab initio calculations are probably the most reliable way to obtain quadnipole and higher multipole moments [20, 21 and 22]. 

There are higher multipole polarizabilities tiiat describe higher-order multipole moments induced by non-imifonn fields. For example, the quadnipole polarizability is a fourth-rank tensor C that characterizes the lowest-order quadnipole moment induced by an applied field gradient. There are also mixed polarizabilities such as the third-rank dipole-quadnipole polarizability tensor A that describes the lowest-order response of the dipole moment to a field gradient and of the quadnipole moment to a dipolar field. All polarizabilities of order higher tlian dipole depend on the choice of origin. Experimental values are basically restricted to the dipole polarizability and hyperpolarizability [21, 24 and 21]. Ab initio calculations are an imponant source of both dipole and higher polarizabilities [20] some recent examples include [26, 22] ... 

Only if the total charge on the system (q) equals zero will the dipole moment be unchanged Similar arguments can be used to show that if both the charge and the dipole moment ar zero then the quadrupole moment is independent of the choice of origin. For convenience the origin is often taken to be the centre of mass of the charge distribution. 

Magnetic properties should be independent of the choice of coordinate origin. The term choice of origin is often translated into choice of gauge, and so we say that physical properties should be gauge-invariant (for a discussion, see Hameka, 1965). 

An alternative choice of origin would be midway between A and B planes. Let us say that the first plane at x>0 is then a B plane. It must be weighted with 3/4. The following A plane has the weight 1/4. That completes the termination, which this time is truly tapered. The excess energy of a stoichimetric unit in the perfect crystal is zero. Thus the excess energy of 3B/4+A/4 is equal to the excess energy of B/2, which was our first choice of termination. 

The structrrre factors of reflections in different cubic stractures are shown in Table 4.1. Note that the stmctme factors may be complex, but this depends upon the choice of origin of the rmit cell. The intensity formtrlae always contain the modulus... 

Expressions (7.8) and (7.9) demonstrate that the first nonvanishing moment is origin independent. Thus, the dipole moment of a neutral molecule, but not that of an ion, is independent of origin the quadrupole moment of a neutral molecule without dipole moment is not dependent on the choice of origin, and so on. 

Note, however, that the dipole moment for a charged molecule depends on choice of origin, and therefore is not unique. Dipole moments for ions cannot be measured and calculated dipole moments for charged molecules are meaningless. 

The choice of originating versus adopting a project may of course not be open to you. Junior meinagers in some organizations, or young managers at the outset of their career may not be allowed to take much initiative. 

Draw the symmetry elements for a P2,/c unit cell in its other two projections. The correct results as shown draw attention to several points. First, for the be projection, the choice of origin (upper right, not left) is dictated by the... 

The superscript EQC in the entries of Table 2.7 related to BE indicates that the origin-dependent quantities to which they are associated refer to the so-called effective quadrupolar centre [14], R(EQC co), a particular frequency-dependent vector in the coordinate space defined with respect to a given choice of origin of the coordinates, or . 

Figure 9.3.3 illustrates the derivation of coordinates for equipoints related by a 4-, 3-, and 6-axis along c through the origin. Coordinates related by a 4-, 3-, and 6-axis can be readily written down using this information. If a different choice of origin is made, a different set of coordinates will be obtained. 

The same approach may be applied to ionic systems using what is known as the center of charge framework [181]. In this case, the first (Coulomb) term in Eq. (7) is nonzero and dominates electrostatic interactions. It is the dipole term that is coordinate dependent and is eliminated by the choice of origin. This is accomplished by placing the origin Rcq... 

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