Perturbation theory technique

In electronic structure theory, the single-configuration picture (e.g., the ls22s22p4 description of the Oxygen atom) forms the mean-field starting point the configuration interaction (Cl) or perturbation theory techniques are then used to systematically improve this level of description. 

The effectiveness of the method of Green function is largely determined by the existence of the appropriate representations. Since the analytical representation for the Green function are known only for the Coulomb held, the use of this approach is restricted to problems in which the difference of the potential and the Coulomb potential is insignihcant or can be taken into account by perturbation-theory techniques. 

Apart from these supramolecular approaches to the interaction energy, advanced perturbation theory techniques, which combine symmetry-adapted perturbation theory and DFT and are thereby able to capture dispersion interactions, are also under development [89-91]. 

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. 

Many predictive applications of perturbation theory techniques for the design, analysis, and optimization of various nuclear systems, as well as for the evaluation and adjustment of nuclear data, are based on sensitivity functions. Section VI presents the fundamentals of perturbation-based sensitivity studies, describes the origin of sensitivity functions, and discusses several areas of application for sensitivity studies. These include cross-section sensitivity studies and optimization studies which are emerging as important fields for the application of perturbation theory. 

A primary objective of this work is to provide the general theoretical foundation for different perturbation theory applications in all types of nuclear systems. Consequently, general notations have been used without reference to any specific mathematical description of the transport equation used for numerical calculations. The formulation has been restricted to time-independent and linear problems. Throughout the work we describe the scope of past, and discuss the possibility for future applications of perturbation theory techniques for the analysis, design and optimization of fission reactors, fusion reactors, radiation shields, and other deep-penetration problems. This review concentrates on developments subsequent to Lewins review (7) published in 1968. The literature search covers the period ending Fall 1974. 

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. 

Many experimental techniques now provide details of dynamical events on short timescales. Time-dependent theory, such as END, offer the capabilities to obtain information about the details of the transition from initial-to-final states in reactive processes. The assumptions of time-dependent perturbation theory coupled with Fermi s Golden Rule, namely, that there are well-defined (unperturbed) initial and final states and that these are occupied for times, which are long compared to the transition time, no longer necessarily apply. Therefore, truly dynamical methods become very appealing and the results from such theoretical methods can be shown as movies or time lapse photography. 

Most of the techniques described in this Chapter are of the ab initio type. This means that they attempt to compute electronic state energies and other physical properties, as functions of the positions of the nuclei, from first principles without the use or knowledge of experimental input. Although perturbation theory or the variational method may be used to generate the working equations of a particular method, and although finite atomic orbital basis sets are nearly always utilized, these approximations do not involve fitting to known experimental data. They represent approximations that can be systematically improved as the level of treatment is enhanced. 

Detailed reaction dynamics not only require that reagents be simple but also that these remain isolated from random external perturbations. Theory can accommodate that condition easily. Experiments have used one of three strategies. (/) Molecules ia a gas at low pressure can be taken to be isolated for the short time between coUisions. Unimolecular reactions such as photodissociation or isomerization iaduced by photon absorption can sometimes be studied between coUisions. (2) Molecular beams can be produced so that motion is not random. Molecules have a nonzero velocity ia one direction and almost zero velocity ia perpendicular directions. Not only does this reduce coUisions, it also aUows bimolecular iateractions to be studied ia intersecting beams and iacreases the detail with which unimolecular processes that can be studied, because beams facUitate dozens of refined measurement techniques. (J) Means have been found to trap molecules, isolate them, and keep them motionless at a predetermined position ia space (11). Thus far, effort has been directed toward just manipulating the molecules, but the future is bright for exploiting the isolated molecules for kinetic and dynamic studies. 

Another approach to electron correlation is Moller-Plesset perturbation theory. Qualitatively, Moller-Plesset perturbation theory adds higher excitations to Hartree-Fock theory as a non-iterative correction, drawing upon techniques from the area of mathematical physics known as many body perturbation theory. 

There are three main methods for calculating the effect of a perturbation derivative techniques, perturbation theory and propagator methods. The former two are closely related while propagator methods are somewhat different, and will be discussed separately. 

In such cases the expression from fii st-order perturbation theory (10.18) yields a result identical to the first derivative of the energy with respect to A. For wave functions which are not completely optimized with respect to all parameters (Cl, MP or CC), the Hellmann-Feynman theorem does not hold, and a first-order property calculated as an expectation value will not be identical to that obtained as an energy derivative. Since the Hellmann-Feynman theorem holds for an exact wave function, the difference between the two values becomes smaller as the quality of an approximate wave function increases however, for practical applications the difference is not negligible. It has been argued that the derivative technique resembles the physical experiment more, and consequently formula (10.21) should be preferred over (10.18). 

In this section we consider how to express the response of a system to noise employing a method of cumulant expansions [38], The averaging of the dynamical equation (2.19) performed by this technique is a rigorous continuation of the iteration procedure (2.20)-(2.22). It enables one to get the higher order corrections to what was found with the simplest perturbation theory. Following Zatsepin [108], let us expound the above technique for a density of the conditional probability which is the average... 

The wavefunction corrections can be obtained similarly through a resolvent operator technique which will be discussed below. The n-th wavefunction correction for the i-th state of the perturbed system can be written in the same marmer as it is customary when developing some scalar perturbation theory scheme by means of a linear combination of the unperturbed state wavefunctions, excluding the i-th unperturbed state. That is ... 

These compounds have been the subject of several theoretical [7,11,13,20)] and experimental[21] studies. Ward and Elliott [20] measured the dynamic y hyperpolarizability of butadiene and hexatriene in the vapour phase by means of the dc-SHG technique. Waite and Papadopoulos[7,ll] computed static y values, using a Mac Weeny type Coupled Hartree-Fock Perturbation Theory (CHFPT) in the CNDO approximation, and an extended basis set. Kurtz [15] evaluated by means of a finite perturbation technique at the MNDO level [17] and using the AMI [22] and PM3[23] parametrizations, the mean y values of a series of polyenes containing from 2 to 11 unit cells. At the ab initio level, Hurst et al. [13] and Chopra et al. [20] studied basis sets effects on and y. It appeared that diffuse orbitals must be included in the basis set in order to describe correctly the external part of the molecules which is the most sensitive to the electrical perturbation and to ensure the obtention of accurate values of the calculated properties. 

Since the spin-orbit interaction energy is small, the solution of equations (7.43) to obtain E is most easily accomplished by means of perturbation theory, a technique which is presented in Chapter 9. The evaluation of E is left as a problem at the end of Chapter 9. 

However, due to the availability of numerous techniques, it is important to point out here the differences and equivalence between schemes. To summarize, two EDA families can be applied to force field parametrization. The first EDA type of approach is labelled SAPT (Symmetry Adapted Perturbation Theory). It uses non orthogonal orbitals and recomputes the total interaction upon perturbation theory. As computations can be performed up to the Coupled-Cluster Singles Doubles (CCSD) level, SAPT can be seen as a reference method. However, due to the cost of the use of non-orthogonal molecular orbitals, pure SAPT approaches remain limited... 

Perturbation theory is one of the oldest and most useful, general techniques in applied mathematics. Its initial applications to physics were in celestial mechanics, and its goal was to explain how the presence of bodies other than the sun perturbed the elliptical orbits of planets. Today, there is hardly a field of theoretical physics and chemistry in which perturbation theory is not used. Many beautiful, fundamental results have been obtained using this approach. Perturbation techniques are also used with great success in other fields of science, such as mathematics, engineering, and economics. 

In general, no simple, consistent set of analytical expressions for the resonance condition of all intradoublet transitions and all possible rhombicities can be derived with the perturbation theory for these systems. Therefore, the rather different approach is taken to numerically compute all effective g-values using quantum mechanics and matrix diagonalization techniques (Chapters 7-9) and to tabulate the results in the form of graphs of geff,s versus the rhombicity r = E/D. This is a useful approach because it turns out that if the zero-field interaction is sufficiently dominant over... 

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