Order of approximation

To obtain the Hamiltonian at zeroth-order of approximation, it is necessary not only to exclude the kinetic energy of the nuclei, but also to assume that the nuclear internal coordinates are frozen at R = Ro, where Ro is a certain reference nucleai configuration, for example, the absolute minimum or the conical intersection. Thus, as an initial basis, the states t / (r,s) = t / (r,s Ro) are the eigenfunctions of the Hamiltonian s, R ). Accordingly, instead of Eq. (3), one has... 

The next higher order of approximation, the first-order approximation, is obtained by estimating molecular properties by the additivity of bond contributions. In the following, we will concentrate on thermochemical properties only. 

In principle, energy-analyzer systems can be designed such that their electron-optical properties do not limit the energy resolution attainable, i. e. their intrinsic energy resolution is much better than the energy width of the primary electron beam, which is of the order of approximately 1.5-2.5 eV for a tungsten hairpin cathode, approximately 1 eV for a LaBg cathode, approximately 0.7 eV for a Schottky field emitter, and 0.3-0.5 eV for a pure cold-field emitter. 

We consider first the polarizability of a molecule consisting of two or more polarizable parts which may be atoms, bonds, or other units. When the molecule is placed in an electric field the effective field which induces dipole moments in various parts is not just the external field but rather the local field which is influenced by the induced dipoles of the other parts. The classical theory of this interaction of polarizable units was presented by Silberstein36 and others and is summarized by Stuart in his monograph.40 The writer has examined the problem in quantum theory and finds that the same results are obtained to the order of approximation being considered. 

To the same order of approximation of the equations, that is, with only terms linear in / (v) kept, better approximations to the viscosity may be found by considering the equations of higher order than Eqs. (1-86) and (1-87). These new equations will, to this order of approximation, have zero on the left sides (since the higher order coefficients are taken equal to zero) on the right sides appears the factor (p/fii) multiplied by a series of terms like those in Eq. (1-110). Using these equations, and the first order terms of Eq. (1-86) for arbitrary v,... 

To the lowest order of approximation, which is used to evaluate the collision integrals for the perturbation terms (vzfx and we take m/M — 0. There is, thus, no interchange of energy between the electrons and neutral atoms, so that... 

Although this lowest order approximation is used in determining the first order corrections to the distribution function, it is necessary to go to a higher order of approximation in determining the collision integral of Eq. (1-140). If we keep terms to first order in the small quantity m/M, the collision integral may be evaluated to give 28... 

It is important to note that in all these methods, the first term in the series solution constitutes the so-called approximation of zero order. This is generally the solution of a simple linear problem e.g., the harmonic oscillator the second term appears as the first approximation, and so on. The amount of labor increases very rapidly with the order of approximation, but the additional information obtained from approximations of higher orders (beginning with the second) does not increase our knowledge from the qualitative point of view. It merely adds small quantitative corrections to the first approximation, and in most applied problems, these corrections are scarcely worth the considerable complication in calculations. For that reason the first approximation is generally sufficient in exploring a new problem, or in investigating the qualitative aspect of a phenomenon. 

The order of approximation which obtains may be seen from the following figures (A. Magnus, 1910) ... 

According to the classical theory, the effect of a magnetic field on a system composed of electrons in motion about a fixed nucleus is equivalent to the first order of approximation to the imposition on the system of a uniform rotation... 

The expansion of the approximation error = Lh v — Lv in powers of h is aimed at achieving the order of approximation as high as possible. Indeed, we might have... 

In principle one can continue further the process of raising the order of approximation further and achieve any order in the class of sufficiently smooth functions v V. During this process the pattern, that is, the total... 

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. 

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

We will show that its order of approximation on a solution u = u x) can be made higher without enlarging a pattern under a proper choice of d and (p. 

In this section we give several examples of raising the order of approximation for boundary and initial conditions without upgrading a pattern. 

It is worth emphasizing here that we have succeeded in raising the order of approximation without enlarging the total number of grid nodes which will be needed in this connection for approximating the boundary condition. 

In this case the condition u(a ,0) = Ug x) and the boundary conditions are approximated exactly. For instance, one of the schemes arising in Section 1.2 is good enough for the difference approximation of the initial equation. No doubt, we preassumed not only the existence and continuity of the derivatives involved in the equation on the boundary of the domain in view (at. r = 0 or f = 0), but also the existence and boundedness of the third derivatives of a solution for raising the order of approximation of boundary and initial conditions. 

With these relations established, we conclude that if the scheme is stable and approximates the original problem, then it is convergent. In other words, convergence follows from approximation and stability and the order of accuracy and the rate of convergence are connected with the order of approximation. 

We are going to show that A) 2 3.nd A12 have the first and second order of approximation, respectively ... 

The statement of the Dirichlet difference problem providing a higher-order approximation. On the basis of the cross scheme it is possible to construct a scheme with the error of approximation 0( h j ) or 0 h ) on a solution in the case of a square (cube) grid. In order to raise the order of approximation, we exploit the fact that u = u x) is a solution of Poisson s equation... 

We proceed to the estimation of the order of approximation for scheme (II) under the agreement that u = u(x,t) possesses a number of derivatives in X and t necessary in this connection for performing current and subsequent manipulations. Within the notations... 

So, not only the order of approximation, but also the stability of scheme (16) depend on the parameter a. 

If scheme (II) is stable with respect to the right-hand side and approximates problem (I), then it converges and the order of accuracy coincides with the order of approximation. 

In preparation for this, we agree to consider p = f = /(x,As before, we suppose once again that G is a parallelepiped and A is specified by formula (4). We investigate the order of approximation by appeal to the expression... 

Prove its absolute stability, find the order of approximation and point out the method for solving the problem. 

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