The rate of convergence of expansions in the basis (1.2) has received little attention except for purely numerical studies [3,7,8,9,16] which indicated that the convergence is at least (unlike for bais set of type) not frustratingly slow. Rather detailed studies were performed for the even-tempered basis set, i.e. for exponents constructed from two parameters and /di (for each /) [Pg.80]

The relative rate of convergence, pre/> for a given synthesis is given by the ratio of slopes of the lines R2P and R2Pmcr [Pg.112]

Thus, provided the rate of convergence is not less than that for L2 jDt — n, all terms other than the first in the series may be neglected. Equation 10.139 will converge more rapidly than this for L2/Dt > n, and equation 10.141 will converge more rapidly for L2/Dt < n. [Pg.616]

There are hints [9,10,18] that the rate of convergence for basis sets of type (1.2) is even better than (1.4), if one uses better optimized basis sets than those of even tempered type (1.3), [Pg.81]

So far we have established an estimate for the rate of convergence in a very simple problem. It is possible to obtain a similar result for this problem by means of several other methods that might be even much more simpler. However, the indisputable merit of the well-developed method of energy inequalities is its universal applicability it can be translated without essential changes to the multidimensional case, the case of variable coefficients, difference schemes for parabolic and hyperbolic equations and other situations. [Pg.114]

On convergence and accuracy. The results obtained in the preceding two sections may be of help in establishing the rate of convergence for scheme [Pg.166]

Figure 1 The course of energy minimization of a DNA duplex with different choices of coordinates. The rate of convergence is monitored by the decrease of the RMSD from the final local minimum structure, which was very similar in all three cases, with the number of gradient calls. The RMSD was normalized by its initial value. CC, IC, and SG stand for Cartesian coordinates, 3N internal coordinates, and standard geometry, respectively. |

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. [Pg.97]

For quantum chemistry the expansion of e in a Gaussian basis is, of course, much more important than that of 1/r. The formalism is a little more lengthy than for 1/r, but the essential steps of the derivation are the same. For an even-tempered basis one has a cut-off error exp(—n/i) and a discretization error exp(-7//i), such that results of the type (2.15) and (2.16) result. Of course, e is not well represented for r very small and r very large. This is even more so for 1/r, but this wrong behaviour has practically no effect on the rate of convergence of a matrix representation of the Hamiltonian. This is very different for basis set of type (1.1). Details will be published elsewhere. [Pg.88]

See also in sourсe #XX -- [ Pg.43 , Pg.44 ]

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