Electrical multipoles defined

By a Taylor expansion of r —about r = 0 one obtains directly electric multipoles defined according to (151). The expansion point is, as above, taken to be the origin and within the charge distribution p(r). It is usually assumed that the sources of the field are at an appreciable distance from the expansion point a minimum requirement is that the charge density p generating the field should be zero at the expansion point, as well as all its derivatives [72]. In the general case this leads to the condition... 

Consider tlie mutual approach of two noble gas atoms. At infinite separation, there is no interaction between them, and this defines die zero of potential energy. The isolated atoms are spherically symmetric, lacking any electric multipole moments. In a classical world (ignoring the chemically irrelevant gravitational interaction) there is no attractive force between them as they approach one another. When tliere are no dissipative forces, the relationship between force F in a given coordinate direction q and potential energy U is... 

The highest probabilities are for transitions between configurations with i = n2 and h = h + 1. In the final state the coupling, close to LS, holds for neighbouring shells then the matrix element of electric multipole transition is defined by formulas (25.28), (25.30). Similar expressions for other coupling schemes may be easily found starting with the data of Part 6 and Chapter 12. 

Multipole Moments.—For any system of charges (1 < n) a set of electric multipoles can be defined. They are tensor quantities, the 2n-th pole being an n-rank tensor. For a set of charges a with position vectors n relative to some origin, the first few are ... 

Within such a stribution of charges, which in general can be rather complicated, we can discern electric sub-systems of spedfically defined configurations of charges. Electric sub-systems of this kind are jointly referred to as electric multipoles, and the quantities representing thdr properties are the multipole mommts. We shall now consider these multipoles one by one. 

The permanent electric multipoles (m = 0) are defined by equation (40), the electric multipoles of first order w = 1 by equation (72), and those of the second order w = 2 by equation (79). Similarly, magnetic potential energy of order m — I can be defined, on replacing in equation (83) E by B and p by m. From the general expression [equation (83)] one immediately and quite easily derives all the energies dealt with in the theory of non-linear molecular processes. ... 

Induced Interaction between Two Multipole Systems. Equation (58) defines in general form the classical electrostatic interaction of two electric systems having permanent multipoles and pj" , in conformity with the classical theory of Keesom. In the classical approach also, as shown by Debye and Falkenhagen, one has to take into consideration energies due to interactions between the permanent multipoles of the one system and electric multipoles induced in the other, and vice versa. Restricting the problem in a first approximation to the energy arising from the mutual interaction of dipoles, we can write ... 

In analogy to the definition of electric dipole moment, electric multipole moments are also defined. In particular, the quadrupole moment Q and the octupole moment U are defined as ... 

Full knowledge of the charge distribution of a molecule requires specification of the charge density at all points. For some purposes the charge density provides excess information thus, the potential outside a sodium ion is independent of the distribution of the electrons, and the interaction of a molecule with a uniform external field is determined by its dipole moment and dipole polarizabilities. The electric multipole moments characterize the charge distribution the first three are defined as follows ... 

The definition of the moments becomes unambiguous by specifying the orientation of the nucleus. By convention, in the definition of the electric multipole moments the nuclear angular momentum is assumed to be aligned along the z axis. The intrinsic electric quadrupole moment Qo is then defined as... 

Parameters Jafi(iJ) define not only the projections of the exchange interaction, but also the corresponding contributions of electric multipole interactions and the interaction via... 

An alternative approach is to start from the onset with a low resolution model, which is set up essentially by physical intuition, where a whole molecule or monomer is replaced by one or few beads. The typical aim of those simple empirical models is to examine the minimum set of molecular features needed to obtain a given molecular organization, for instance a certain anisotropy of shape or the presence of electric multipole moments, and the qualitative relatirm between variations in the microscopic model and macroscopic properties. This type of modeling does not necessitate a preliminary weU-defined and known chemical structure, but is more akin to a reverse molecular engineering process, where one guesses what key features are needed to achieve the desired macroscopic behavior before actually trying to write down and possibly synthesize a certain molecule. 

An important feature of the electric multipole moments, as defined in Eqs. (4.4)-(4.6) and (4.8), is that the first non-vanishing moment of a charge distribution is independent of the choice of the origin Rq- However, all the higher moments depend on this origin. Thus, the dipole moment of a neutral molecule or the quadrupole moment of a neutral and non-polar molecule are both independent of the origin Rq, whereas the dipole moment of an ion or the quadrupole moment of a neutral but polar molecule will depend on the origin Rq [see Exercise 4.2]. 

Contrary to the electric multipole expansion we want to denote the arbitrary origin with Rqo and call it the gauge origin as defined in Section 2.9. 

In the usual texts a multipole expansion involving spherical Bessel functions and spherical vector harmonics is also introduced [16,23,23,26]. The fields from electric and magnetic dipoles correspond to the lowest-order terms ( =1) in the expansion. If we define dipole by this expansion then our toroidal antenna is an electric dipole. In any event, the fields away from the source are the same. This is perhaps a matter of consistency in definitions. 

The general definition of the electron transition probability is given by (4.1). More concrete expressions for the probabilities of electric and magnetic multipole transitions with regard to non-relativistic operators and wave functions are presented by formulas (4.10), (4.11) and (4.15). Their relativistic counterparts are defined by (4.3), (4.4) and (4.8). They all are expressed in terms of the squared matrix elements of the respective electron transition operators. There are also presented in Chapter 4 the expressions for electric dipole transition probabilities, when the corresponding operator accounts for the relativistic corrections of order a2. If the wave functions are characterized by the quantum numbers LJ, L J, then the right sides of the formulas for transition probabilities must be divided by the multiplier 2J + 1. 

We firstly define the electric and magnetic multipoles, then show how the interaction energy with external fields can be expressed in terms of the induced moments and go on to see how these are related to the field-free moments via response functions. Most of our treatment will concern the electric case, since this is the simpler, and since magnetic effects can be treated by analogy. 

The quadmpoie operator transforms as WY V and makes the 1 s —> 3d transition electric quadmpoie allowed when the dY v2 orbital (in molecular coordinates) bisects the k and E vectors (which define the laboratory coordinates). Quadmpoie intensity is usually very low however, at —9000 eV the wavelength of light is — 1.4 A and in this case the long-wave approximation no longer holds and higher terms in the multipole expansion in Equation 1.2 become important. 

Electric hexadecapole. In microsystems of higher e.g. octahedral) symmetry, as we shall see further on, all the above defined moments vanish and it is necessary to define multipoles of higher orders, beginning by a hexadecapole system the permanent moment of which is defined by a symmetric tensor of rank 4 ... 

For a Gaussian beam, the fields of the radiating electric and magnetic multipoles satisfy the same boundary conditions (vanishing faster than 1/p as p oo) so that the fields in the plane(s) defined by the transverse E (H) field and the optical axis are symmetric. It is difficult to generate a balanced hybrid mode in conventional smooth-walled metallic waveguide instead, one may use a component called a scalar horn. 

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