Formulation with Addition Theorem

In the present analysis we will derive the expression of the transition matrix by using the translation properties of the vector spherical wave functions. The completeness property of the vector spherical wave functions on two enclosing surfaces, which is essential in our analysis, is established in Appendix D. 

Given the external excitation E., iJg as an entire solution to the Maxwell equations, find the scattered field Eg, Hs and the internal fields Ei i, Hi and Ei 2, Hifi satisfying the Maxwell equations 

Considering the general null-field equation (2.68), we use the vector spherical wave expansions of the incident field and of the dyad gl on a sphere enclosed in Di, to obtain 

For the general null-field equation (2.69) in Dg, we proceed analogously but restrict ri to lie on a sphere enclosing ),. We then have 

The surface fields ei i and hi i are the tangential components of the electric and magnetic fields in the domain bounded by the closed surfaces Sx and 52. Taking into account the completeness property of the system of regular and radiating vector spherical wave functions on two enclosing surfaces 

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The derivation [2] starts with an expansion of 1 /r.., just as the Cartesian formulation does, but this time we use the expansion in terms of spherical harmonics (the spherical harmonic addition theorem) [3]. As before, we write R = B — A, and obtain... 

The principal subject of this chapter is the calculation of the intramolecular distribution (ID) [109]. The computational procedure utilizes the generating functions derived in the previous chapter. We begin in Section 4.1 with the derivation of expressions (in closed form) of the one-dimensional ID in the special (zero temperature) case w = 0. Then, on this basis we discuss some important properties that we will formulate in terms of addition theorems. 

It is helpful to contrast the view we adopt in this book with the perspective of Hill (1986). In that case, the normative example is some separable system such as the polyatomic ideal gas. Evaluation of a partition function for a small system is then the essential task of application of the model theory. Series expansions, such as a virial expansion, are exploited to evaluate corrections when necessary. Examples of that type fill out the concepts. In the present book, we establish and then exploit the potential distribution theorem. Evaluation of the same partition functions will still be required. But we won t stop with an assumption of separability. On the basis of the potential distribution theorem, we then formulate additional simplified low-dimensional partition function models to describe many-body effects. Quasi-chemical treatments are prototypes for those subsequent approximate models. Though the design of the subsequent calculation is often heuristic, the more basic development here focuses on theories for discovery of those model partition functions. These deeper theoretical tools are known in more esoteric settings, but haven t been used to fill out the picture we present here. 

For the case of a purely electrostatic external potential, P = (F , 0), the complete proof of the relativistic HK-theorem can be repeated using just the zeroth component f (x) of the four current (in the following often denoted by the more familiar n x)), i.e. the structure of the external potential determines the minimum set of basic variables for a DFT approach. As a consequence the ground state and all observables, in this case, can be understood as unique functionals of the density n only. This does, however, not imply that the spatial components of the current vanish, but rather that j(jc) = < o[w]liWI oM) has to be interpreted as a functional of n(x). Thus for standard electronic structure problems one can choose between a four current DFT description and a formulation solely in terms of n x), although one might expect the former approach to be more useful in applications to systems with j x) 0 as soon as approximations are involved. This situation is similar to the nonrelativistic case where for a spin-polarised system not subject to an external magnetic field B both the 0 limit of spin-density functional theory as well as the original pure density functional theory can be used. While the former leads in practice to more accurate results for actual spin-polarised systems (as one additional symmetry of the system is take into account explicitly), both approaches coincide for unpolarized systems. 

A theorem exists in Fourier mathematics that when transforms can be sampled at least one additional time between normally allowed reciprocal lattice points, phases for the allowed reflections can be determined. Use of this non-integral sampling approach is steeped in complex mathematical formulations and processes, but it has nonetheless been put to good use in a number of cases, particularly with viruses and other highly redundant complexes whose symmetry arrangement exceeds the space group symmetry. 

In the paper [209], the theorem formulated above is proposed for A = m = 1. The concluding part of the proof makes use of the separatrix splitting method, in line with [167], but as is seen from the further analysis (see [210]), this step in the proof requires additional motivations. 

To prove the above formulated uniqueness theorem, we consider an arbitrary but loop-free network N and assume that it has a unique steady state for arbitrary but fixed values of its external variables A, . .. A. As mentioned earlier, an external variable can be visualized as an infinitely large capacitance. Now we convince ourselves that the uniqueness remains valid if one of the external variables, say Ap is replaced by a 0-junction to which an internal variable X and an additional reaction 2-port element R are connected. The new element R may possibly involve further internal or external variables Y and Z, but its connection to N is assumed to generate no closed loop together with the elements of N ... 

Fortunately, the additional complications associated with nonvariational energies are not as severe as we might first suspect since, in most cases, it is rather easy to modify the energy function of a nonvariational wave function in such a way that the optimized energy becomes stationary with respect to the variables of this new function, hi this variational formulation, then, the conditions of the Hellmann-Feynman theorem are indeed fulfilled and molecular properties may be calculated by a procedure that is essentially the same as for variational wave functions. 

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