Decoupling procedures

This equation is exact, and its kernel is expanded into the series (2.82). The decoupling procedure resulting in (2.24) is equivalent to retention of the first term in this series. Using Eq. (2.86), one may develop a consistent perturbation theory which will take into consideration higher orders of... 

The problem of state-parameter estimation in dynamic systems is considered in terms of decoupling the estimation procedure. By using the extended Kalman filter (EKF) approach, the state-parameter estimation problem is defined and a decoupling procedure developed that has several advantages over the classical approach. 

The network is decoupled by successively replacing junction tetrahedra by two independent strands, each terminated by remaining junctions and one with a slow point in the middle. The order of decoupling and assignment of slow points must be performed in a rather special way the symmetry of remaining junctions must be preserved so that the same decoupling procedure can be applied to them in turn. 

The Chompff-Duiser procedure of symmetry-preserved decoupling does not appear to be applicable in an easy way to random networks in three dimensions. Junctions which anchor strands of unequal length can be decoupled, but with a considerable increase in difficulty (234). On the other hand, Chompff and Prins (235) have introduced a degree of randomness in the decoupling procedure (the distribution of slow points within each uncoupled chain of the equivalent system was taken to be random) without altering the results appreciably. Random decoupling is much easier to implement, as shown below. 

In the so-called electron propagators, one studies simultaneously the ionization potentials and the electron affinities of the system, whereas, in the so-called polarization propagators, one studies directly the excitation or de-excitation energies. By means of certain decoupling procedures, Linderberg and Ohm could construct approximate schemes for the practical computational evaluation of these propagators. 

In this way they could not only replace the previous decoupling procedures used in connection with the expansions (1.40) and (1.41) by more consistent approximations, but they could also show that the poles of the superoperator R iz) were given by the zero points of the determinant... 

Using Eq. (6) and decoupling procedures for irreducible spherical tensors [32], we easily derive the following general form of the autocorrelation function... 

Reference materials certified by international organizations are available for judging the accuracy of an analytical procedure. As far as NMR spectroscopy is concerned, systematic errors which are frequently related to the lack of accuracy arise mainly during the acquisition step and in the treatment of the FID s. In order to minimize bias in signal acquisition, oversampling and very stable decoupling procedures are recommended, and, to obtain very reproducible intensity values, an operator independent curve fitting procedure should be implemented in the analytical sequence. 

The Foldy-Wouthuysen (FW) transformation [67] offers a decoupling, which in principle is exact, but it is impractical and leads to a singular expansion in 1/c in the important case of a Coulomb potential [68]. Douglas and Kroll (DK) suggested an alternative decoupling procedure based on a series of appro-... 

Consider now what happens when we irradiate the proton transitions during the broad-band decoupling procedure. The irradiation of the protons causes the proton transitions to become saturated. In other words, the probability of an upward and a downward transition for these nuclei (the proton transitions shown in Fig. 4.6) now becomes equal. The population of level N4 becomes equal to the population of level N2, and the population of level N3 is now equal to the population of level Ni. The populations of the spin states can now be represented by the following expressions. 

Fig. 9.22 An illustration of the SNIFTIRS decoupling procedure. (A) Spectrum calculated for the slab of 5.2x10 ° mol cm of pyridine molecules evenly distributed within the thin cavity between the optical window and the electrode.

Propagator or Green s function methods are employed in this chapter to analyze the many-electron problem in planar unsaturated molecules as treated within the Pariser-Parr-Pople (PPP) model. A derivation of the model in many-electron theory serves to demonstrate the nature of the approximations involved. Applications are presented for the case of weakly interacting atoms. A decoupling procedure for Green s functions proposed by the authors is shown capable of yielding a correct description of this case. 

The expression (11.75) is also valid in the Hartree-Fock approximation, where all the Vkr vanish. In the more general case, this expression may not agree with the direct calculation of the hamiltonian expectation value. This is a difficulty not uncommon in the application of Green s functions. It should have been avoided by some auxiliary constraint on the decoupling procedure . 

The simultaneous validity of Eqs. (11.75) and (11.76) puts conditions on the relations of various expectation values and thereby on the decoupling procedure. There is no simple way to incorporate these conditions, and it is not given any further considerations in this chapter. 

For finite values of b, the E (e -b a) JT spectrum is generally no longer given by convolution of the individual-mode spectra. For sufficiently different values of the vibrational frequencies, a vibrationally adiabatic decoupling procedure has been formulated and investigated, where the coupling... 

The decoupling procedure was performed by the same standing press and the same fixtures, again with a (different) randomized order of the trials and with on-line monitoring and recording of the exerted release force. 

At this stage it is, however, not clear which of these parametrizations should be preferred over the others for application in decoupling procedures and whether they all yield identical block-diagonal Hamiltonians. Furthermore, the four possibilities given above to parametrize unitary transformations differ obviously in their radius of convergence Rc, which is equal to unity for the square root and the McWeeny form, whereas Rc = 2 for the Cayley parametrization, and Rc = co for the exponential form. 

Within any decoupling scheme there are only a few restrictions on the choice of the transformations U. First, they have to be unitary and analytic (holomorphic) functions on a suitable domain of the one-electron Hilbert space V, since any parametrization has necessarily to be expanded in a Taylor series around W = 0 for the sake of comparability but also for later application in nested decoupling procedures (see chapter 12). Second, they have to permit a decomposition of in even terms of well-defined order in a given expansion parameter of the Hamiltonian (such as 1/c or V). It is thus possible to parametrize U without loss of generality by a power-series ansatz in terms of an antihermitean operator W, where unitarity of the resulting power series is the only constraint. In the next section this most general parametrization of U is discussed. 

It is noteworthy that the same results were found by Chompff and Duiser using a different approach [70]. They proposed a decoupling procedure which, upon being applied to polymer chains cross-linked into a network,... 

The symmetry decoupling procedures of the previous section apply here as well without change. 

The portion that operates on the electron solutions is just the nonrelativistic Hamiltonian. This should be no surprise, because we are applying a decoupling procedure in powers of 1 /c. 

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