Statistical analysis bond angles

The use of constraints and restraints arise from unrealistic bond lengths and angles. It is a reasonable use of statistical analysis to include prior information in a refinement procedure, and geometry is one such restraint/constraint. Poor geometries are often a consequence of missing data down one axis, invariably the result of the missing cone or lack of an epitaxially grown crystal. This lack of data has a profound effect on the refinement process... 

Mean-square unperturbed dimensions a and their temperature coefficient, d tin 0) I d T, are calculated for ethylene-propylene copolymers by means of the RIS theory. Conformational energies required in the analysis are shown to be readily obtained from previous analyses of PE and PP, without additional approximations. Results thus calculated are reported as a function of chemical composition, chemical sequence distribution, and stereochemical composition of the PP sequences. Calculations of 0 / nP- are earned out using ( ) r r2 = 0.01, 1.0, 10.0, and 100.0, (ii) p, = 0.95, 0.50, and 0.05, liii) bond length of 153 pm and bond angles of 112°for all skeletal bonds, iv) = 0 and 10°, and (v) statistical weight factors appropriate for temperatures of 248, 298, and 348 K. Matrices used are ... 

In these models, the potential energy function is based on the molecular mechanics all-atom force field and includes the bond, angle, dihedral and non-bonded energy terms. The parameterization is based on the statistical analysis of sets of experimental structures. If a variable q describes a degree of freedom in the system (e.g., bond distances, angles, dihedrals) then, P(q), the probability distribution associated with this degree of freedom, is related to the potential of mean force, W(q), by the following equation... 

Force field parameters can be derived from stmctural parameters, such as bond lengths and angles, obtained from quantum chemical calculations. A detailed statistical analysis of the calculated model compounds was performed. Table 2 shows an example of the comparison between calculated and experimental structures. Please notice that the calculated model compound contains dimethylamino groups whereas the molecule from the X-ray structure has IV-methylanilino groups. Therefore deviations in bond angles and the conformation of the molecules were expected. 

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