Perturbation theory sensitivity functions

Temperature dependence proton relaxivity 188 relaxation rate 144-5 Temperature-sensitive contrast agents 218-19 Th -" 368 Ti "" 347 Tilt angle 242 Time constant 14 Time correlation functions 76 Time-dependent mechanism 14 Time-dependent perturbation methods 23 Time-dependent perturbation theory 45-8 Titanium(III) 115,134-5,161 TPEN 224 TPPS 219... 

CCSD is similarly sensitive to multireference character, aldiough it is less obvious that this should be so based on the formalism presented above. However, inclusion of triples in the CCSD wave function is usually very effective in correcting for a single-reference treatment of a weakly to moderately multireference problem. Of course, die most common way to include die triples is by perturbation theory, i.e., CCSD(T), and as noted above, this level too can be unstable if singles amplitudes are large. In such an instance, BD(T) calculations, which eliminate die singles amplitudes, can be efficacious. 

Coupling also influences the relative intensity of the absorptions by mixing vibrational excitations of the two molecules (first-order perturbation theory gives a mixing coefficient of C/A). If M denotes the hypothetical intrinsic intensity ratio of the individual molecules (a function of IR polarization), and r denotes the observed intensity ratio, the following relationship allows more sensitive determination of small coupling constants. 

Fig. 15 Sensitivity of the real propagation constant (—) and effective index (---) of a surface plasmon on a metal-dielectric interface to a bulk refractive index change as a function of wavelength calculated rigorously from eigenvalue equation and using the perturbation theory. Waveguiding structure gold-dielectric (nj = 1.32)...

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. 

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. 

Generalized-function formulations of GPT for homogeneous systems are the source of sensitivity functions for different integral parameters Equation (189) for reactivity worths, and Eq. (162) for ratios of linear and bilinear functionals. The first-order perturbation theory expression for reactivity [Eq. (132)] can also be used for sensitivity studies. 

Cross-section sensitivity functions are very useful also for the adjustment of nuclear data so as to provide a best fit for measured integral parameters. Systematic adjustment of cross sections began in the middle sixties 103, 104) and has developed continuously. In fact, cross-section adjustment was one of the first applications of cross-section sensitivity functions (59). Descriptions of the activities in cross-section adjustment, including the application of perturbation theory, have been recently reported 76, 105-107). 

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. 

In this section we have tried to indicate the usefulness of sensitivity functions for the large number of applications. Considerable development is still required, however, before full benefit can be derived from perturbation theory methods for all these applications. Cross-section sensitivity studies, for example, will be more useful and reliable when cross-section error files supplement the cross-section files in present use. New computer code systems that can process these error files and perform sensitivity studies, allowing... 

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. 

The difficulty with perturbation theory is primarily with the initial assumption if the perturbation is not small compared to the zero-order Hamiltonian, convergence to an accurate set of energies and wavefunctions can be an excruciating process. Another option, not quite as sensitive to the complexities of the Hamiltonian, employs the variational principle the correct ground state wave-function of any system is the wavefunction that yields the lowest possible value of the energy. To rephrase it from a practical perspective, we start off with a guess wavefunction, and adjust that wavefunction to get the lowest-energy we can. 

Nonetheless, care needs to be exercised in calculations of the one-electron scalar relativistic correction by perturbation theory using these operators. It has been found that the magnitude of the correction is quite sensitive to the contraction of the basis set. This follows from the fact that the operators weight the region near the nucleus. If a basis function is taken out of the contraction the nonrelativistic wave funetion may not change very much, but a small change in the coefficient of the eore function can have a big effect on the relativistic correction. 

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