Generalized perturbation theory accuracy

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 design and analysis of realistic power systems increasingly involves the representation of nonlinear models with progressively higher demands on the accuracy of computations. The difficulties of nonlinear problems are well known, and approximation methods, of which perturbation theory is one, are to be welcomed. Of course, perturbation theory has been applied to problems, such as fuel burnup, where the properties are a function of the neutron flux it has been customary, however, to linearize the problem around the unperturbed problem. When the accuracy obtained by this or similar devices is inadequate, there is a case for considering a more general perturbation theory for nonlinear systems. 

Doppler reactivity is the other important parameter in FBR cores because it assures the intrinsic safety characteristics of the core. The present cross-section adjustment method cannot treat the Doppler reactivity which is dominated by resonance-peak broadening of cross-sections. We have launched a new study to extend the applicability of the cross-section adjustment and design accuracy evaluation system to the Doppler effect. The basic method to evaluate sensitivity of self-shielding factors has been successfully derived from a generalized perturbation theory, and a prototype system to calculate the Doppler sensitivity is now under verification. 

To illustrate the accuracy of the perturbation theory these results are worth comparing with the well-known values of h and I4 for t = 1 rigorously found from first principles in [8]. It turns out that the second moment in Eq. (2.65a) is exact. The evaluation of I4, however, is inaccurate its first component is half as large as the true one. The cause of this discrepancy is easily revealed. Since M = / and (/) = J/xj, the second component in Ux) is linear in e. Hence, it is as exact in this order as perturbation theory itself. In contrast, the first component in IqXj is quadratic in A and its value in the lowest order of perturbation theory is not guaranteed. Generally speaking... 

During last decades the DFT based methods have received a wide circulation in calculations on TMCs electronic structure [34,85-88]. It is, first of all, due to widespread use of extended basis sets, allowing to improve the quality of the calculated electronic density, and, second, due to development of successful (so called - hybrid) parameterizations for the exchange-correlation functionals vide infra for discussion). It is generally believed, that the DFT-based methods give in case of TMCs more reliable results, than the HER non-empirical methods and that their accuracy is comparable to that which can be achieved after taking into account perturbation theory corrections to the HER at the MP2 or some limited Cl level [88-90]. 

A more widely used approach for organic molecules is based on second-order perturbation theory. Here the dipolar contribution to the field induced charge displacement is calculated by inclusion of the optical field as a perturbation to the Hamiltonian. Since the time dependence of the field is included here, dispersion effects can be accounted for. In this approach the effect of the external field is to mix excited state character into the ground state leading to charge displacement and polarization. The accuracy of this method depends on the parameterization of the Hamiltonian in the semi-empirical case, the extent to which contributions from various excited states are incorporated into the calculation, and the accuracy with which those excited states are described. This in turn depends on the nature of the basis set and the extent to which configuration interaction is employed. This method is generally referred to as the sum over states (SOS) method. 

Increasing chain length changes the general features of the fi, q plots rather drastically. For v = 10 the range of random replication appears to be substantially wider (Figure 11). The (fi, q curves are almost horizontal on both sides of the maximum irregularity condition at q = 0.5. In addition, the transitions from direct to random replication and from random to complementary replication are rather sharp. We are now in a position to compare the minimum accuracy of replication that we derived in Section III by perturbation theory with the exact population dependence on q. From Eqs. (III.l) and (III.4) we find (D = 0 k = 0,1,. . . , n)... 

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