QR method

Within the DFT framework, we apply two different approaches to deal with relativistic effects, the so-called quasi-relativistic (QR) method (73) and the more modem "Zeroth Order Regular Approximation for Relativistic Effects" (ZORA) (14-16). The QR method is also known as the Pauli approach. 

QR Method. The first relativistic method is the so-called quasi-relativistic (QR) method. It has been developed by Snijders, Ziegler and co-workers (13). In this approach, a Pauli Hamiltonian is included into the self-consistent solution of the Kohn-Sham equations of DFT. The Pauli operator is in a DFT framework given by... 

The older, Pauli-Hamiltonian based QR method has recently been compared to the more modem ZORA approach (10). The comparison has been done based on methane derivatives. We have summarized the results in Table II. Both the ZORA and QR methods agree well for molecules containing atoms no heavier than Cl. This shouldn t be surprising since relativistic effects are still small in such molecules. However, at least for this sample of molecules, the ZORA method is clearly superior for molecules containing heavy nuclei like Br or I. This is reflected in the mean absolute deviation between theory and experiment of 9.2 ppm (ZORA) and 15.6 ppm (QR), respectively (10). Note that some of the same systems have also been studied by other authors (35-37). 

The comparison between the ZORA and QR methods is still rather limited (70), Table II. We are currently in the process of carrying out a more thorough comparison of the two approaches. The ZORA approach is clearly superior to the QR method on theoretical grounds since it avoids - rather than circumvents - the fundamental stability problems of the Pauli operator. 

Skaggs and Kabala [59] employed the QR method for the same problem solved in their TR method. In the QR method, Skaggs and Kabala solved an equation that is close to the original equation and that is stable with a negative time step. The diffusion operator... 

A moving coordinate system was used to account for the velocity term of the ADE. A problem similar to the one presented in Skaggs and Kabala [58] is used for the QR study. The results are less accurate than that of the TR approach, but it is computationally less expensive. The authors claimed that it is much easier to incorporate heterogeneous parameters in the QR method. However, to date, heterogeneous parameters have not been incorporated either in the QR method or in the Tikhonov method by Skaggs and Kabala. 

When a dynamical system is nonhyperbolic, there exist time intervals where part of the finite-time Lyapunov exponents accumulate around zero. Hence the spectra of the exponents are (quasi-)degenerate. These degenerate spectra impede our ability to obtain accurate numerical values of finite-time Lyapunov exponents using the existing numerical methods, namely, the QR method and the SVD method [9,17] ... 

The QR methods, based on the matrix factorization of QR decomposition [18], are effective and thus are widely used algorithms for computing the Lyapunov exponents [1,19-22]. However, for the, finite-time Lyapunov exponents, these methods introduce errors that decrease only algebraically... 

To evade this numerical difficulty, the standard QR method evaluates Eq. (16) as follows [1,19] ... 

Figure 2. The finite-time errors in QR method, (r, 0) — X (l, 0), are plotted against t for...

The results shown in Table 6 seem to indicate that the different approximations for relativistic effects in all-electron calculations have a comparable accuracy. This is not the case. It has been found that the QR method using the PauU Hamiltonian can lead to significant errors in the bond energy (55). An example will be given in the following section, about substituted carbonyl complexes. 

The (iterative) use of the Pauli Hamiltonian, the so-called quasi-rela-tivistic (QR) method, must be regarded as obsolete, as the Pauli... 

Another method is based on introduction into polymerization of compounds (quenching agents) labeled by a radioactive isotope and by termination of polymerization in such a manner that this compound or its part joins a growing polymer chain (QR method). In the case of olefin polymerization on ZN catalysts, alcohol labeled with in the hydroxyl group was used as a quenching agent (QR RO H method) ... 

Data on the number of active centers and propagation rate constants for olefin polymerization on traditional ZN catalysts and highly active supported ZN catalysts, obtained by use of SF and QR methods, will be presented and discussed below. 

---