Space focusing second-order

In this chapter we investigate the interaction between experimental design and information quality in two-factor systems. However, instead of looking again at the uncertainty of parameter estimates, we will focus attention on uncertainty in the response surface itself. Although the examples are somewhat specific (i.e., limited to two factors and to full second-order polynomial models), the concepts are general and can be extended to other dimensional factor spaces and to other models. 

The principle of these SHG correlation measurements is to split the intensity of the laser pulse (wj) into two replicas, each of which travel along an interferometer arm and are reflected back to be focused 2md superimposed in a nonlinear crystal. Only when the pulses overlap precisely in time and space is the SHG signal (at W2=2wi) observed. The intensity dependence of SHG is related to the second order, background free correlation function as follows ... 

TOF/RTOF MSI is second-order space focus. Ions are reaccelerated into a second-order reflectron Boesl et al." ... 

A computationally less expensive alternative to coupled-cluster theory is provided by multireference perturbation theory (MRPT). The currently most widely used form of MRPT in f element chemistry is second-order Complete Active Space Perturbation Theory (CASPT2) [79,80], and we shall focus on this method in this section. 

This process, according to the manufacturer,54 has been developed in such a way that space requirements are kept to a minimum. A BIOPAQ IC reactor is used as the initial step in the treatment process. The name of this anaerobic reactor is derived from the gas-lift driven internal circulation that is generated within a tall, cylindrical vessel. These reactors have been operational in the paper industry since 1996. The second step in the purification process is a mechanically mixed and aerated tank. The aerating injectors can be cleaned in a simple way without the need to empty the aeration tank. Potential scaling materials are combined into removable fine particles. At the same time, the materials that may cause an odor nuisance are oxidized into odorless components. The process can be completed by a third and a fourth step. The third step focuses on suspended solids recovery and removal. The fourth step is an additional water-softening step with lamella separation and continuous sand filters in order to produce fresh water substitute. The benefits claimed by the manufacturer are as follows54 ... 

The second step of the calculation involves the treatment of dynamic correlation effects, which can be approached by many-body perturbation theory (62) or configuration interaction (63). Multireference coupled-cluster techniques have been developed (64—66) but they are computationally far more demanding and still not established as standard methods. At this point, we will only focus on configuration interaction approaches. What is done in these approaches is to regard the entire zeroth-order wavefunc-tion Tj) or its constituent parts double excitations relative to these reference functions. This produces a set of excited CSFs ( Q) that are used as expansion space for the configuration interaction (Cl) procedure. The resulting wavefunction may be written as... 

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