Average square dipole moment

An average square dipole moment is defined in the same manner as the average square displacement of the molecule 18... 

Chapter E is devoted to the mean-square dipole moment and mean rotational relaxation time derived from dielectric dispersion measurements. Typical data, both in helieogenic solvents and in the helix-coil transition region, are presented and interpreted in terms of existing theories. At thermodynamic equilibrium, helical and randomly coiled sequences in a polypeptide chain are fluctuating from moment to moment about certain averages. These fluctuations involve local interconversions of helix and random-coil residues. Recently, it has been shown that certain mean relaxation times of such local processes can be estimated by dielectric dispersion experiment. Chapter E also discusses the underlying theory of this possibility. 

Polypeptides are electrically polar, carrying permanent dipoles at the planar CO-NH groups of the backbone chain and generally at some atomic groups of the side-chains. Because of the vector nature of dipoles, we must speak of the mean-square dipole moment, averaged over all possible conformations of the backbone chain and all accessible orientations of the side-chains when the dipolar nature of a polypeptide in solution is considered. The of a polypeptide thus may depend on what conformation the molecule assumes in a given solvent. 

The mean-square dipole moments of PPCS chains are calculated as a function of stereochemical composition using the RIS analysis recently published for PS. The calculations are in good agreement with the average of experimental results for atactic PPCS chains estimated to contain ca. 35% meso diads. The temperature coefficient is calculated to be negative in agreement with available experiments. 

Methods are presented for calculating mean-square dipole moments, , of polypeptide chains, averaged over all configurations of the chain skeleton. They are applicable to chains of any number (x+ U of residues, the residues being In any specified sequence. Dipole moments of glycine peptides are calculated and compared with experimental determinations. The effects of stereosequence on the dipole moment are well reproduced by the calculations. Ip is taken to be 380 pm. 

Most usually, polymer molecules will not be in a single, fixed conformation, and the experimentally observable quantity, the mean-square dipole moment, is an average over many different conformations. At any given instant the... 

The most frequently calculated property is the mean square unperturbed end-to-end distance, (r )o. Other properties susceptible to rapid computation include the average of the end-to-end vector, (r)o, and the mean square unperturbed radius of gyration, 5 )0. The viscosity of a dilute solution in a solvent can be estimated from 5 )0 or (r )o via the equivalent sphere model for hydrodynamic properties. Several higher even moments, (r )o and (s )o, p = 2,3,..., provide information about the shape (width, skewness) of the distribution functions for and When combined with information about the electronic charge distribution within individual rigid units, the RIS model can calculate the mean square dipole moment, (m )o-Also accessible are optical properties that depend on the anisotropy of the... 

The incompatibility of polymer pairs has been correlated with the dipole moment fi of two polymers. As it is known, the value of p of each polymer [69] is given per monomer repeating unit and is given by the relation p = p /DP, where DP is the number average degree of polymerization and jJ is the mean square dipole moment of the long-chain molecules. 

The first term in the brackets is the expectation value of the square of the dipole moment operator (i.e. the second moment) and the second term is the square of the expectation value of the dipole moment operator. This expression defines the sum over states model. A subjective choice of the average excitation energy As has to be made. 

Consider a monolayer of water molecules arranged in a square lattice with a lattice constant of 3 A. The dipole moment of a single molecule is 6.24 x 10 30 C m. (a) Calculate the potential drop across the monolayer if all dipole moments are parallel and perpendicular to the lattice plane, (b) If the potential drop across the layer is 0.1 V, what is the average angle of the dipole moment with the lattice plane ... 

Equation 5.9 is in essence the ensemble average of the total dipole moment squared. It is given in a form suitable for numerical computation [315], The computation of the spectral moment yi, on the other hand, begins with the integration of Eq. 5.1 over all frequencies,... 

In the case of the IR process, the dipole moment, p = p, is given by Equation 4.16 in the case of the Raman process, the dipole moment is given by p = aE. For molecules in the gas phase with a haphazard orientation and where the average of the square of the angular part is one, the probability for the absorption or emission of electromagnetic radiation per unit time for the dipole transition can be written as follows ... 

Succinonitrile. An excellent discussion of this system has been presented by Braun, Stockmayer, and Orwoll, who were the first to propose studying the dipole moment of succinonitrile. They show that the average square of the dipole moment is given by... 

As already mentioned, one of the main weaknesses of the simple reflection method is the fact that the electronic transition dipole moment, (or the transition dipole moment surface, TDMS for polyatomic molecules in Section 4) is assumed to be constant. This weakness will remain in the Formulae (12), (27) and (29) derived below. The average value of the square of the TDM (or TDMS) is then included in amplitude A and A = A /V. In Formulae (3), (3 ) and (3") the mass (or isotopologue) dependent parameters are p and the ZPE. In contrast, W and V., which define the upper potential, are mass independent. This Formula (3) is already known even if different notations have been used by various authors. As an example, Schinke has derived the same formula in his book [6], pages 81, 102 and 111. Now, the model will be improved by including the contribution of the second derivative of the upper potential at Re- The polynomial expansion of the upper potential up to second order in R - Re) can be expressed as ... 

Equation (85) shows that the dipole moment of the molecule, jx, is of great importance for EB. For a worm-like chain the square of this dipole moment averaged over all chain conformations by analogy with the expression (3) (p. 98) is determined from the equation )... 

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