Molecular error statistics

Heats of formation, molecular geometries, ionization potentials and dipole moments are calculated by the MNDO method for a large number of molecules. The MNDO results are compared with the corresponding MINDO/3 results on a statistical basis. For the properties investigated, the mean absolute errors in MNDO are uniformly smaller than those in MINDO/3 by a factor of about 2. Major improvements of MNDO over MINDO/3 are found for the heats of formation of unsaturated systems and molecules with NN bonds, for bond angles, for higher ionization potentials, and for dipole moments of compounds with heteroatoms. 

This equation means that when there is a free energy difference of a few fcs T the probability P( ) is reduced considerably, that is, those conformations with large A( ) are sampled very rarely. This is a very important observation in terms of numerical efficiency. At the transition region for example, the free energy is maximum and typically very few sample points are obtained during the course of molecular dynamics simulation. In turn this results in very large statistical errors. Those errors can only be reduced by increasing the simulation time, sometimes beyond what is practically feasible. 

Fig. 6.10. Comparison of overlap sampling and FEP calculation results for the free energy change along the mutation of an adenosine in aqueous solution (between A = 0.05 and 0.45) in a molecular dynamics simulation. The results represent the average behavior of 14 independent runs. (MD time step.) The sampling interval is 0.75 ps. The upper half of the plot presents the standard deviation of the mean (with gives statistical error) for AA as a function of sample size N the lower half of the plot gives the estimate of A A - for comparison of the accuracy, the correct value of AA is indicated by the bold horizontal line...

S. K. Schiferl and D. C. Wallace, Statistical Errors in Molecular Dynamics Averages, J. 

The estimate of (U X > will also contain statistical error. However, at high Reynolds numbers, the molecular transport term in (6.178) will be small, and thus noise in this term is less problematic. 

A third source of error is associated with the fragmentation pattern caused by dissociation of the molecular ions formed in the source region of the spectrometer. Under severe conditions these processes may proceed with substantial isotopic fractionation, and this obscures the measurements of isotopic composition at the collector. To some extent careful standardization of the instrumental conditions may ensure that errors from fragmentation are systematic, and thus cancel (at least to some extent). Alternatively, softer ionization methods can be used to prevent most or all of the fragmentation. The bottom spectrum in Fig. 7.7 illustrates this approach it shows the mass spectrum of chlorobenzene obtained by photoionization. Only the parent molecular ions are observed. It should be kept in mind, however, that softer ionization usually yields smaller ion currents and consequently statistical counting errors increase. 

In conclusion, the sequential U test is useful for the judgement of identity between MWDs of a pair of polymer samples whose molecular weight averages are Identical within the experimental errors. Identity of MWDs of the two polymer samples was established with more than four pairs of parallel measurements, and the disagreement of MWDs with two to four pairs of parallel measurements. Though this statistical treatment is useful for the identification or differentiation of the MWDs of the pair of polymers, it can not detect small differences in shapes of the both chromatograms. 

BB. Most molecule-based descriptors, such as logP, PSA, and molecular electronic properties, were used to construct models with a variety of statistical tools. The best performing models can approach the limit of experimental error, which was estimated to be 0.3 log units. 

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