Non-parallelity error

For sake of comparison, in all studied cases, we run calculations for those geometries and basis sets with a FCI (or near FCI) available. The methods we deal with are CCSD, CAS-SDCI, (SC)2CAS-SDCI and ec-CCSD corrected from both CAS-SDCI and (SC) CAS-SDCI. The performance of the methods is examined from two aspects the total energy and the quality of the potential energy surface (PES), being this quality measured by the so-called non-parallelity error (NPE). For a given set of calculations in a dissociative curve, the NPE is defined as the difference between the maximal and minimal deviation from the exact FCI PES. 

Table 1 Calculations on HF with DZ+P basis. The total energies are reported as -(E + 99) hartree. Dimension should be understood as number of determinants (number of spin-adapted configurations in italics). NPE (non-parallelity error) is the difference between the maximal and minimal deviation from FCI...

Consequently, the RMR CCSD potential is almost parallel to the FCI potential. The so-called non-parallelism error (NPE) (34), defined as the difference between the maximal and minimal deviations from the exact FCI potential, is only about 0.23 mhartree (see also Section 5). Although the absolute differences between the RMR CCSD and FCI energies are further decreased when enlarging the (2,2) model space to (4,4), (particularly for a > 3 a.u.), this is not the case for the NPE. In all cases, however, we are within a 2 mhartree error, even for this very demanding model. 

As an overall measure of the quality of the shape of the computed potentials, two of the co-authors introduced the above mentioned non-parallelism error (NPE) (34), which is defined as the difference between the maximal and minimal deviations from the exact FCI potential. Clearly, NPE=0 when the computed potential differs from the FCI one by a constant shift. The NPE s of the computed potentials, given in Tables 1-5, are summarized in Table 6. We see that the CCSD and AL-CCSD NPE s, as well as the RMR CCSD and AL-RMR CCSD ones, are very similar, implying a similar quality of the resulting potentials. The RMR CCSD or AL-RMR CCSD NPE s are usually one order of magnitude smaller than the CCSD or AL-CCSD ones. For the H4 and H8 models, which are essentially two-reference cases, we indeed obtain similar results with either RMR CCSD or AL-RMR CCSD when using the (2,2) reference space. In the case of the S4 model and of the symmetric stretching of H2O, the AL-RMR CCSD potentials are slightly inferior to those obtained with RMR CCSD. However, the quality of the AL-RMR CCSD potentials improves when we employ a (4,4) reference space. 

Non-parallelism error (NPE, in mhartree) of resulting potentials for systems given in Tables 1-5. 

As it can be clearly seen in Fig. 1, for the dissociation of the HF molecule the CASPT2 calculations provide the most accurate results. The non-parallelity error (NEP) of the potential energy surface (PES) is an order of magnitude smaller than that for the other presented methods. The accuracy of the NEVPT2 and the QMBPT2 is comparable. 

Once a method is established, precision may be determined by suitable replicated experiments. However it is in inter-laboratory trails that the problems with environmental methods often show up. It is accepted that for trace analysis RSD values of tens of percent are likely. In studies conducted in Western Australia on pesticide residues in bovine fat RSD values for dieldrin were 12% and for dia-zonium were 28%. It is typical to see a quarter of the laboratories in such a trial producing values that could be termed outliers. In the previously mentioned study, 5 laboratories out of 26 had z> 3 for aldrin. In a parallel study RSD values for petroleum in dichloromethane and water were 40% and 25%, respectively. The conclusions of these studies was that there was poor comparability because of the different methods used, that accreditation apparently made no difference to the quality of results, and that a lack of understanding of definitions of the quantities to be analysed (for example gasoline range organics ) caused non-method errors. In relation to methods, this is contrary to the conclusion of van Nevel et al. who asserted that the results of the IMEP round of analyses of trace elements in natural and synthetic waters showed no dependence on method [11]. If properly validated methods do yield comparable results, then one conclusion from the range of studies around the world is that many environmental methods are not validated. It may be that validated methods are indeed used, but not for exactly the systems for which they were validated. 

Safety precantions have traditionally been oriented towards the removal or engineering control of potentially damaging energy, but the use of techniques for reducing non-citlpable error is also potentially fruitful. These parallel approaches are combined into a simple model from which is derived a conceptiral outline to gitide in the selection of coimter measitres. From this model a coitrse of formal study can be constructed in a manner siritable for incorporation into the citrricitlitm of those who will be professionally concerned with the prevention of accidental injmy. 

This should come as no surprise, since the physical behavior of materials is non-linear and unpredictable, especially when materials are formulated or in combination. Two examples will suffice high temperature ceramic superconductors and insulators above their critical temperatures or at non-ideal stoichiometries composite structures may show several times the strength or impact resistance than would be expected from their component materials. Materials discovery will always require a good deal of trial and error, factors that may be mitigated by techniques that permit the simultaneous synthesis of large numbers of materials, followed by rapid or parallel screening for desired properties. 

In pulse-echo-based techniques, the time of flight in a sample cannot be determined simply from the observation of the time span between adjacent echoes in the echo pattern if plane parallel transducers operated at resonant frequencies are employed. Transducers introduce substantial errors if the velocity is derived from such measurements, especially if relatively short samples are used. Various correction approaches have so far been developed in order to consider the influence of resonant transducers and the effects of diffraction [31-33]. The need for corrections can be avoided and a broad operational bandwidth obtained by using short pulses of duration equal to or shorter than the transduction [34] this requires a time resolution better than the transit time in the transducer. This short-pulse excitation (e.g. the maximum for a 10-MHz transducer is 50 ns) requires a high-power wide-band ultra-linear amplifier to ensure the detection of US signals with sufficient resolution under non-resonant conditions. 

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