Error of the difference approximation

It turns out that there is an uncountable set of difference expressions approximating Lv = v and this is something one might expect. The following question is of significant importance what is the error of one or another difference approximation and how does the difference fp(x) = Lh v x) — Lv[x) behave at a point x as h 0 The quantity tpi ) — Lh (2 ) — Lv x) refers to the error of the difference approximation to Lv at a point x. We next develop (a ) in the series by Taylor s formula... 

So far we have studied the error of the difference approximation on the functions v from certain classes V. In particular, in the preceding examples the class of sufficiently smooth functions stands for V. 

Thns, the accnrate account of the error of the difference approximation on a solution of the differential equation helps raise the order of approximation. 

The error of approximation on a grid. So far we have considered the local difference approximation meaning the approximation at a point. Just in this sense we spoke about the order of approximation in the preceding section. Usually some estimates of the difference approximation order on the whole grid are needed in various constructions. 

Finally in the third state the two approximations give very similar energy values, both with higher energy than the FCI one. In each approximation, the error of the different terms compensate each other to a certain extent. 

These behaviors and the different performance of the different approximations in this respect are well known. Nevertheless, research over the last 20 years has shown that, despite these large errors in the determination of gaps and bandwidths, these methods perform well in predicting a large variety of observables within an error bar that is in most cases acceptable and helping to draw conclusions about interesting physical and chemical properties of matter in the solid state. 

The statement of the difference problem and calculations of the approximation error. In this section we study the equation of vibrations of a string... 

Composition of discrete (difference) approximations to equations of mathematical physics and verifying a priori quality characteristics of these approximations, mainly the error of approximation, stability, convergence, and accuracy of the difference schemes obtained ... 

Core electrons are highly relativistic and DFT methods may show systematic errors in calculating the charge density at the nucleus because of the inherent approximations. Fortunately, this does not hamper practical calculations of isomer shifts of unknown compounds, because only differences of li//(o)P are involved. In practice, the reliability of the results depends more on the number of compounds used for calibration and how wide the spread of their isomer shift values was. The isomer shift scale for several Mossbauer isotopes has been calibrated by this approach, among which are Au [1], Sn [4], and Fe [5-9]. For details on practical calculation of Mossbauer isomer shifts, see Chap. 5. 

A description of the different terms contributing to the correlation effects in the third order reduced density matrix faking as reference the Hartree Fock results is given here. An analysis of the approximations of these terms as functions of the lower order reduced density matrices is carried out for the linear BeFl2 molecule. This study shows the importance of the role played by the homo s and lumo s of the symmetry-shells in the correlation effect. As a result, a new way for improving the third order reduced density matrix, correcting the error ofthe basic approximation, is also proposed here. 

In this paper we present the first application of the ZORA (Zeroth Order Regular Approximation of the Dirac Fock equation) formalism in Ab Initio electronic structure calculations. The ZORA method, which has been tested previously in the context of Density Functional Theory, has been implemented in the GAMESS-UK package. As was shown earlier we can split off a scalar part from the two component ZORA Hamiltonian. In the present work only the one component part is considered. We introduce a separate internal basis to represent the extra matrix elements, needed for the ZORA corrections. This leads to different options for the computation of the Coulomb matrix in this internal basis. The performance of this Hamiltonian and the effect of the different Coulomb matrix alternatives is tested in calculations on the radon en xenon atoms and the AuH molecule. In the atomic cases we compare with numerical Dirac Fock and numerical ZORA methods and with non relativistic and full Dirac basis set calculations. It is shown that ZORA recovers the bulk of the relativistic effect and that ZORA and Dirac Fock perform equally well in medium size basis set calculations. For AuH we have calculated the equilibrium bond length with the non relativistic Hartree Fock and ZORA methods and compare with the Dirac Fock result and the experimental value. Again the ZORA and Dirac Fock errors are of the same order of magnitude. 

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