Generalized perturbation theory formulations

Perturbation theory and techniques are coming of age. They provide increasing support for the design and analysis of nuclear systems, and for the evaluation of nuclear data. This is evidenced by the large number of perturbation theory based computer codes developed within the last few years. These trends characterize the new codes (1) the extension of conventional perturbation techniques to multidimensional systems and to high-order approximations of the Boltzmann equation (2) the development of methods for implementing new perturbation theory formulations, such as the generalized perturbation theory formulations and (3) the application of perturbation theory formulations to new fields, such as sensitivity studies and the solution of deep-penetration problems. 

In fact, our interest in the present formulation, the use ofNSS s andLKD s, has been aroused when studying the integrals over Cartesian Exponential Type Orhitals [la,b] and Generalized Perturbation Theory [ld,ej. The use of both symbols in this case has been extensively studied in the above references, so we will not repeat here the already published arguments. Instead we will show the interest of using nested sums in a wide set of Quantum Chemical areas, which in some way or another had been included in our research interests [Ic]. 

The traditional role of perturbation theory in reactor physics has been to estimate, with a first-order accuracy, the effect of an alteration in the reactor on its reactivity. Lately, application of perturbation theory techniques has increased significantly in both scope and volume. Two general trends characterize these developments (1) improvement of the accuracy of reactivity calculation, and (2) extension of the use of second-order perturbation theory formulations for estimating the effect of a perturbation on integral parameters other than reactivity, and to nuclear systems other than reactors. These trends reflect two special features of perturbation theory. First, it provides exact expressions for the effect of an alteration in the reactor on its reactivity. For small, and especially local alterations, these perturbation expressions are easier and cheaper to apply than other approaches. Second, second-order perturbation theory formulations can be applied with distribution functions pertaining only to the unperturbed system. 

The recent expansion of the application of perturbation theory formulations is mainly due to the development of the generalized perturbation theory (GPT). Several versions of GPT formulations have been described. They are characterized by their form and their method of derivation. They are also distinguished by the form of the integral parameters to which they apply and by the method they use to allow for the flux and adjoint perturbation. A unified presentation of GPT is given in Section V, together with an elucidation of problems of accuracy and range of applicability of different formulations. Also outlined in Section V is a perturbation theory for altered systems. 

The development and application of generalized perturbation theory (GPT) has made considerable progress since its introduction by Usachev (i(S). Usachev developed GPT for a ratio of linear flux functionals in critical systems. Gandini 39) extended GPT to the ratio of linear adjoint functionals and of bilinear functionals in critical systems. Recently, Stacey (40) further extended GPT to ratios of linear flux functionals, linear adjoint functionals, and bilinear functional in source-driven systems. A comprehensive review of GPT for the three types of ratios in systems described by the homogeneous and the inhomogeneous Boltzmann equations is given in the book by Stacey (41). In the present review we formulate GPT for composite functionals. These functionals include the three types of ratios mentioned above as special cases. The result is a unified GPT formulation for each type of system. 

Generalized perturbation theory for two special cases of composite functionals are presented and discussed in some detail GPT for reactivity (Section V,B), and GPT for a detector response in inhomogeneous systems (Section V,E). The GPT formulation for reactivity is equivalent to a high-order perturbation theory, in the sense that it allows for the flux perturbation, GPT for a detector response in inhomogeneous systems 42, 43) is, in fact, the second-order perturbation theory known from other derivations I, 44, 45). These perturbation theory formulations provide the basis for new methods for solution of deep-penetration problems. These methods are reviewed in Section V,E,2. 

It might be useful if a unified terminology were established for what is becoming an important field of perturbation theory. We propose that the term generalized perturbation theory be used for all perturbation theory formulations in which the flux and adjoint perturbations are allowed for... 

Relations (2.46) and (2.47) are equivalent formulations of the fact that, in a dense medium, increase in frequency of collisions retards molecular reorientation. As this fact was established by Hubbard within Langevin phenomenology [30] it is compatible with any sort of molecule-neighbourhood interaction (binary or collective) that results in diffusion of angular momentum. In the gas phase it is related to weak collisions only. On the other hand, the perturbation theory derivation of the Hubbard relation shows that it is valid for dense media but only for collisions of arbitrary strength. Hence the Hubbard relation has a more general and universal character than that originally accredited to it. 

This paper presents an account of the dynamics of electric charges coupled to electromagnetic fields. The main approximation is to use non-relativistic forms for the charge and current density. A quantum theory requires either a Lagrangian or a Hamiltonian formulation of the dynamics in atomic and molecular physics the latter is almost universal so the main thrust of the paper is the development of a general Hamiltonian. It is this Hamiltonian that provides the basis for a recent demonstration that the S-matrix on the energy shell is gauge-invariant to all orders of perturbation theory. 

As before, a general thermodynamic theory of stability formulation is quadratic in the perturbations of 8T. 8 V, and 8Nk, because the forces and flows vanish at equilibrium... 

We begin this paper by shortly recapitulating the concept of extremal pair functions (Sect. 2). Then we consider extremal pair functions in the context of Moller-Plesset perturbation theory (Sect. 3) and coupled-cluster theory (Sect. 4). We then come to the main topic of this paper, the use of extremal pairs in R12-methods. To this end we formulate a new access to R12-theory starting with two-electron systems (Sect. 5) and generalizing it to n-electron systems (Sect. 6). We show then how extremal pairs arise in a natural way in R12-methods (Sect. 7). We finish (Sect. 8) by giving numerical examples which demonstrate the gain in numerical stability by using extremal pairs in Recalculations. 

For nonautonomous systems, an additional term involving the time derivatives of W(p, q s) must be included in Eq. (A.66) [45, 46, 53]. In this Appendix, we have described how Lie transforms provide us with an important breakthrough in the CPT free from any cumbersome mixed-variable generating function as one encounters in the traditional Poincare-Von Zeipel approach. After the breakthrough in CPT by the introduction of the Lie transforms, a few modifications have been established in the late 1970s by Dewar [56] and Drag and Finn [47]. Dewar established the general formulation of Lie canonical perturbation theories for systems in which the transformation is not... 

---