Perturbation theory formulation

A perturbation theory treatment known as the method of contact transformations [162] provides a convenient approach in defining the anharmonic potential energy and transitional dipole moment [158-163]. Both mechanical and electrical anhaimonicities influence the intensities of overtone and combination bands. 

In general terms, since an exact solution of the vibrational equation in terms of anharmonic wave function is not possible, use is made of the fact at for finite displacements at each step of the potential energy or transitional dipole moment expansion, the higher terms are much smaller than the respective lower terms. A perturbation theoiy treatment becomes, therefore, feasible. The potential energy may be expressed in the form [3] 

X is an ordering parameter defining die degree of smallness of the terms appearing in die expressions. 

The method of contact transformations simplifies the perturbation theory treatment. The perturbation Hamiltonian can be expressed as 

In applying contact transformations to the Hamiltonian, terms referring to a particular order can be removed from the expression with Hq remaining unchanged. Two successive contact transformations provide a new form of the Hamiltonian 

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Correlation can be added as a perturbation from the Hartree-Fock wave function. This is called Moller-Plesset perturbation theory. In mapping the HF wave function onto a perturbation theory formulation, HF becomes a hrst-order perturbation. Thus, a minimal amount of correlation is added by using the second-order MP2 method. Third-order (MP3) and fourth-order (MP4) calculations are also common. The accuracy of an MP4 calculation is roughly equivalent to the accuracy of a CISD calculation. MP5 and higher calculations are seldom done due to the high computational cost (A time complexity or worse). 

Note that the choice of non-orthogonal versus orthogonal basis functions has no consequence for the numerical variational solutions (cf. Coulson s treatment of He2, note 76), but it undermines the possibility of physical interpretation in perturbative terms. While a proper Rayleigh-Schrodinger perturbative treatment of the He- He interaction can be envisioned, it would not simply truncate at second order as assumed in the PMO analysis of Fig. 3.58. Note also that alternative perturbation-theory formulations that make no reference to an... 

By contrast the alternative approach of perturbation theory formulates an approximation for A in terms of the original single well wavefunctions as... 

A. Perturbation Theory Formulated as an Iteration of Unitary Transformations— Nonresonant Case KAM Techniques... 

In (7.58) the exciton-photon interaction as well as scattering by phonons, is taken into account. The expression (7.58) is finite for an arbitrary ui. It corresponds to a result which would be obtained by summation of probabilities of photon absorption in all orders of perturbation theory, formulated with respect to the constant of the exciton-photon interaction. If in (7.58) we let the light velocity go to infinity, and thus if we neglect the retardation, the expression of (7.58) attains the form (7.55). 

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. 

Several of the perturbation theory formulations reviewed require the solution of inhomogeneous Boltzmann equations with singular operators. The properties of these equations and techniques for their solution are reviewed in Section VIII. 

This review is essentially organized into four parts (1) the definition of reactivity (Section II), (2) perturbation theory formulations for reactivity as well as for other integral and differential parameters (Sections III-V), (3) applications for some of the new formulations (Section VI and Section V, E), and (4) problems encountered in practical implementation of some of the perturbation theory formulations (Sections VII and VIII). 

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. 

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. 

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... 

Perturbation theory formulations can be applied for predicting changes... 

Sensitivity functions provide the basis for a large variety of sensitivity studies. Sensitivity studies are becoming an important field in the application of perturbation theory. This is evidenced by the increasing number of papers published on this subject, which reached a high point in 1974 47, 48, 62, 66,68-80). This section sets out to describe (1) the principles of sensitivity and optimization methods that utilize sensitivity functions, and (2) potential uses for the application of perturbation-based sensitivity and optimization methods to fission reactors, fusion reactors and radiation transport problems. This is not intended to be a comprehensive review of either sensitivity or optimization methods, but rather an illustration of fields of application of perturbation theory formulations presented in Section V. Sensitivity and optimization studies not based on perturbation theory formulations are not discussed. 

A sensitivity function describes the functional relationship between the change in an integral parameter caused by a fractional change in some input parameter, when the latter is expressed as a function of independent variables. For most applications a linear functional relationship is desirable. Perturbation theory formulations provide such a linear relationship. A sensitivity function can be defined for any integral parameter it can correspond to variations in any of the input parameters and it can be expressed in terms of any of the independent variables. Thus, the total number of sensitivity functions for a given system can be very large, and can be expressed in terms of different combinations of the independent variables. When the input parameter has discrete variations only, we shall refer to the sensitivity functions as sensitivity coefficients. 

This section outlines the principles of optimization methods that are based on material density perturbations with the purpose of (1) illustrating another area for the application of perturbation theory formulations, and (2) promoting the use of these potentially powerful perturbation-based optimization methods. The perturbation theory foundations of optimization methods, and their relation with the variational formulation of these methods, have already been described in previous reviews (/, 56). Our presentation is restricted to a specific type of control variable—the material densities— and is given in terms of sensitivity functions. Moreover, we present only the conditions for the optimum and do not consider optimization algorithms. 

Several of the perturbation theory formulations reviewed in this article... 

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. 

Despite the significant progress made in perturbation theory since 1968, additional development is necessary before perturbation-based methods become reliable standard tools for the nuclear engineer. These developments can be grouped into four categories (1) resolution of fundamental questions, (2) development of formulations for new applications, (3) acquisition of practical experience on the range of applicability and relative merits of alternative perturbation theory formulations, and (4) development of computer code systems. 

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. 

Another approach is to do a nonrelativistic calculation, using, for example, the Hartree-Fock method, and then use perturbation theory to correct for relativistic effects. For perturbation-theory formulations of relativistic Hartree-Fock calculations and relativistic KS DFT calculations, see W. Kutzelnigg, E. Ottschofski, and R. Franke, J. Chem. Phys., 102,1740 (1995) and C. van Wiillen, J. Chem. Phys., 103,3589 (1995) 105,5485 (1996). 

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