Altered systems, perturbation theory

The Mpller-Plesset (MP) treatment of electron correlation [84] is based on perturbation theory, a very general approach used in physics to treat complex systems [85] this particular approach was described by M0ller and Plesset in 1934 [86] and developed into a practical molecular computational method by Binkley and Pople [87] in 1975. The basic idea behind perturbation theory is that if we know how to treat a simple (often idealized) system then a more complex (and often more realistic) version of this system, if it is not too different, can be treated mathematically as an altered (perturbed) version of the simple one. Mpller-Plesset calculations are denoted as MP, MPPT (M0ller-Plesset perturbation theory) or MBPT (many-body perturbation theory) calculations. The derivation of the Mpller-Plesset method [88] is somewhat involved, and only the flavor of the approach will be given here. There is a hierarchy of MP energy levels MPO, MP1 (these first two designations are not actually used), MP2, etc., which successively account more thoroughly for interelectronic repulsion. 

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. 

Perturbation theory formulations are conventionally used for calculating the effect of perturbations introduced to a reference system on its properties. GPT techniques can also be used to derive perturbation theory formulations for calculating the effects of perturbations in altered systems as well as for calculating the effect of different alterations on the effect of a perturbation. Examples for such formulations are presented in Section V,F. 

Perturbation theory for altered systems provides formulations for calculat-... 

An exact perturbation theory expression for the reactivity in an altered system is... 

The generalized perturbation theory expressions presented in this section for systems described by the homogeneous Boltzmann equation (excluding Section V,B,2) are in the form proposed by Stacey (40, 41). Had we assumed that the overall alteration in the reactor retains criticality, we would have achieved the Usachev-Gandini version of GPT. Stacey s version is often associated (41, 46, 48, 62) with the variational perturbation theory as distinguished from the GPT of Usachev-Gandini. Does the variational approach provide a different perturbation theory than the GPT derived (35,39) from physical considerations Is one of these versions of perturbation theory more general or more accurate than the other What does the term GPT stand for ... 

The advent of perturbation theory for altered systems (see Section V,F) opens a new field for the application of perturbation theory—the field of perturbation sensitivity studies. This is the study of changes in effects of perturbations, or system alterations, caused by uncertainties or variations in input parameters. Examples are (1) the uncertainty of the change in an integral parameter (such as the breeding ratio) resulting from design variations due to uncertainties in cross sections, (2) nonlinear effects of cross-section uncertainties, and (3) the effects of data uncertainties, approximations in computational models, or design variations on the detector response in a deep-penetration problem that is solved with a flux-difference or an adjoint-difference method (see Section V,E). 

Much work is still required before many of the perturbation theory formulations reviewed and presented here can be implemented in practice. If full benefit is to be drawn from perturbation theory techniques for a wide variety of problems concerned with the design, analysis, and optimization of nuclear systems, computer code systems of the future must include basic and specific modules for calculations based on perturbation theory formulations. The basic modules are intended for the calculation of different sensitivity functions. Specific modules should enable performanee of dilTerent studies sueh as cross-section sensitivity studies, analysis of alterations in the design or operating conditions of nuclear systems, and optimization studies. 

In principle we can perform some sort of molecular orbital calculation on molecules of almost any sort of complexity. It is, however, often extremely profitable in terms of understanding the orbital structure to relate the level arrangement in a complex system to that of a simpler one. 3.1-3.3 show three examples of different types of relationships which we will frequently use. We will be interested in seeing how the levels of the species at the left-hand side of these figures are altered electronically during these processes by using the powerful techniques of perturbation theory. We shall not derive the elements of the theory itself but will make use of its mathematical results/" which will very quickly show a striking resemblance to the orbital interaction results of Chapter 2. 

Valence bend theery 1 an appllcatlen ef a general qnantnm-mechanical appreximatlen methed called perturbation theory. In perturbatlen theery, a cemplex system (such as a melecule) Is viewed as a simpler system (such as twe stems) that Is slightly altered or perturbed by some additional force or Interaction (such as the Interaction between the two atoms). 

A particularly convenient improved approximation to this end can be obtained by use of self-consistent, first order, time-dependent perturbation theory. The essential physics to be included is that the external field distorts the atomic charge cloud (by admixture of excited orbitals) which in turn creates an electrostatic potential acting on the system, The self-consistent response of the electrons produces a mean field which reflects the atomic dielectric properties and alters the photoionization amplitudes. If this linear response approach is applied to the HFA one obtains precisely the RPAE, In what follows we consider the same approximation applied to the LDA. Given this parallelism, emphasis will be placed on direct comparisons with the RPAE,... 

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