Tests for Convergence

It is difficult to conclude how the numerical effort grows with the size of the system from our numerical tests since there are several parameters that control the accuracy. It is also not too informative to present convergence tests for any particular energy because the convergence is different for different quantities and different energy ranges (see the figures). 

All Fourier series have to be made finite when performed numerically the choice of the number of waves used in any calculation is a compromise between the computational effort and the errors caused by the truncation they are difficult to estimate and one usually resorts to numerical testing. An example of a convergence test for a, B - calculated from p(a) as described above (Fig. 3.1) - is s iown in Tab. 3.1 the behavior of truncation errors is typical for many similar situations. Whereas the absolute values of both pressure and energy vary considerably with increasing number of waves, the a, B calculated from them evolve only slowly. Apparently a large part of the truncation error is systematic. Detailed convergence tests for the different potentials used can be found e.g. in Ref. 24 so far the most detailed study... 

Table 2. Convergence tests for bulk bcc iron using single ( (SZSP) and double (DZSP) basis. Lattice parameter a (bohrs), bulk modulus B (Mbar), and magnetic moment M (pb) are presented. Other ab-initio calculations and experimental results are given for...

Distorted Born expansion eigenvalue convergence test for H+H2 Distorted Born expansion eigenvalue convergence test for H+O2... 

In order to conserve the total energy in molecular dynamics calculations using semi-empirical methods, the gradient needs to be very accurate. Although the gradient is calculated analytically, it is a function of wavefunction, so its accuracy depends on that of the wavefunction. Tests for CH4 show that the convergence limit needs to be at most le-6 for CNDO and INDO and le-7 for MINDO/3, MNDO, AMI, and PM3 for accurate energy conservation. ZINDO/S is not suitable for molecular dynamics calculations. 

The convergence or divergence of an infinite series is unaffected by the removal of a finite number of finite terms. This is a trivial theorem but useful to remember, especially when using the comparison test to be described in the subsection Tests for Convergence and Divergence. ... 

Partial Sums of InBriite Series, and How They Grow Calculus textbooks devote much space to tests for convergence and divergence of series that are of little practical value, since a convergent... 

Test for convergence. If it fails, begin the next iteration. If it succeeds, go on to perform other parts of the calculation (such as population analysis). 

The usual method of testing for convergence and of extrapolating to zero step size assumes that the step size is halved in successive calculations. Example 2.4 quarters the step size. Develop an extrapolation technique for this procedure. Test it using the data in Example 2.15 in Appendix 2. 

The interval of convergence for each of the series solutions u and ui may be determined by applying the ratio test. For convergence, the condition... 

A typical test for convergence is Ak°H> < TOL where TOL is a user-specified tolerance. This test is suitable only when the unknown parameters are of the same order of magnitude. A more general convergence criterion is... 

A procedure to test for convergence to check if a satisfactory solution has been achieved. 

Rule (iii) is particularly important in the tests for series convergence that will be described in Section 2.11. 

The most useful test for the convergence of a series is called Cauchy s ratio test It can be summarized as follows for a series defined by Bq. (32). 

So far, the only approximation in our description of the FMS method has been the use of a finite basis set. When we test for numerical convergence (small model systems and empirical PESs), we often do not make any other approximations but for large systems and/or ab //i/Y/o-determined PESs (AIMS), additional approximations have to be made. These approximations are discussed in this subsection in chronological order (i.e., we begin with the initial basis set and proceed with propagation and analysis of the results). 

Step 4. Test for convergence to the minimum of /(x). If convergence is not attained, return to step 3. 

Because of the physical equilibrium, the association in the liquid phase is determinded by that in the vapour phase. Therefore no additional association constants are required for the liquid phase. In the case of liquid-liquid equilibrium calculations, an analogous procedure was adopted using convergence test (5), with y. referring to the second liquid phase. 

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