Statistical weight factors conformations

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 ... 

Construct the partition function by summing over aU conformations, with appropriate products of statistical-weighting factors assign to each conformation. 

The conformational energies required in the Boltzmann expression for the statistical weight factors have been estimated by the semiempirical energy calculations, and the results are summarized in Table 7. Reliability of these parameters has been confirmed by testing against existing NMR data on 2,4-dimethoxy pentane. ... 

For a particular conformation c of a molecule, the positions of all (united) atoms in space as well as the chain conformers are known. The potential energy of this conformation is therefore just the sum of the contributions, as given by equation (9) for all the united atoms and a particular energy quantity per gauche bond in the chain. The statistical weight for this conformation is proportional to the Boltzmann factor containing this segment potential ... 

Because of the dependence of the NOE with the r-6 the transcription of the observed NOE to the conformation of the glycosidic bond is not unequivocally possible. Conformations with a low statistical weight and eventually a large NOE are highly overrepresented in the time averaged interpretation of the NOEs. An increase in the distance of two protons by just 12% causes the NOE between these two protons to decrease by a factor of two. Thus the deduction of a single conformer from NOE experiments usually leaves some uncertainty because the determination of the distance between the two protons involved overestimates the conformations with short contact between them. 

Edwards couched his theory in terms of G v(r,r ), which is the statistical weight of all conformations whereby a chain of N bonds starts at r and ends at r. To derive the differential equation for GA,(r,r ), the statistical weight for a conformation where all beads have prescribed positions r i(=r), ri, ts,..., r i, rw(=i ) is written down. This is simply the product of the appropriate normalized Boltzmann factors... 

In the self-consistent mean field approximation, /(r) describes the average repulsion experienced by one bead situated at r, due to all other beads. The Edwards approximation for excluded volume is obtained by recognizing that in the statistical weight of a particular conformation (ri, T2,. .., r, . ..), we must insert the factors... 

Z depends on T because the elements of the statistical weight matrices are Boltzmann factors. The conformational entropy is obtained from this result and In Z . 

Palau and co workers proposed a sdieme with elements of purely statistical methods (conformational preference parameters) and structure-stabilizing factors ( weighting factors ) The wei ting factors modify the conformational preference parameters by taking into account e.g. hydrophobic interactions with )-structural regions or the occurrence of hydrophobic triplets in the helical positions 1-2-5 and 1-4-5. Additional parameters can be introduced into the prediction scheme. 

The rotational isomeric state model can be used to rationalize the temperature-dependence of conformation-dependent properties, through the dependence of the statistical weights on temperature. How strong is the temperature dependence of the dimensions of the chains in a polyethylene melt, assuming that we remain above the temperature at which crystallization occurs Calculations of r )o from equations (2), (4), and (6), assuming that the only temperature dependence is from the Boltzmann factors a = expi-E /kT) and co = exp(-EJkT), yields 9 In (r )o/9 T = -0.001 deg. ... 

This factor gives a statistical weight for each conformation given as a continuous line, r(n). 

Each statistical weight is formulated as a Boltzmann factor with the appropriate energy and temperature. Sometimes there is inclusion of a preexponential factor that accounts for the differences in conformational entropy of the various rotational isomeric states. 

A conformation of the chain is specified in the rotational isometric state approximation by stipulation of the states for all internal bonds 2 to n-1 inclusive e.g., by g ttg-g , etc. Owing to the three-fold symmetry of the terminal methyl groups of the alkane chain, rotations about the terminal bonds are inconsequential and hence are ignored. The statistical weight for the specified conformation of the chain is obtained by selecting the appropriate factor for each bond from the array (15) according to the state of this bond and of its predecessor, and taking the product of such factors for all bonds 2 to n-1. In the example above this product is u ug t u t u8 8 , etc. It will be... 

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