Average dipole moment

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. 

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 )... 

At a given moment CO2 can possess a dipole moment due to some of its vibrational motions. However, at the next instant the dipole moment changes sign because the vibrational motion reverses its direction. Over time (for example, the time it takes to make a dipole moment measurement), the net dipole moment averages to zero and the molecule is nonpolar. 

In addition to the pear-shaped molecules, bent-shaped molecules were used to illustrate the dipolar origin of the flexoelectric effects in nematic liquid crystals. It was assumed that the constituent molecules of the nematic liquid crystals are free to rotate around their axes, and in the absence of electric fields, their dipole moments average out so the net polarization of the material is zero. However, when liquid crystals made from polar pear- or banana-shaped molecules are subjected to splay or bend deformations, respectively, they can become macroscopically polar, because the polar structures correspond to a more efficient packing of the molecules. It follows from symmetry considerations that the deformation-induced fiexo-electric polarization Pa can be written as ... 

The London interaction is a quantum mechanical phenomenon however, its physical origin can be understood as foUows Consider two atoms separated by only a few Angstroms. The electron clouds on each atom are not static objects but are constantly fluctuating. At any time, it is possible that there is slightly more electron density on one side of an atom than the other, which generates a small dipole moment, called an instantaneous dipole moment. Averaged over time (that is, the time it takes to make a dipole moment measurement), however, the dipole moment of the atom is zero because the instantaneous dipoles all cancel one another. When an instantaneous dipolJ f[Pg.270]

This is almost a trivial case, but it shows how we would verify mathematically that the dipole moment averages to zero when we evenly distribute the electron around the nucleus. 

Thus the number of independent components of the permittivity tensor will depend on the symmetry of the liquid crystal phase. The frequency dependence of the permittivity is described in terms of real and imaginary parts, and these also will be tensor quantities. Apart from complications of anisotropic internal fields, the static or low frequency part of the permittivity tensor can be related to the molecular polarizability and dipole moment averaged over the appropriate orientational distribution functions. 

The average cloud is spherically synnnetric with respect to the nucleus, but at any instant of time there may be a polarization of charge givmg rise to an instantaneous dipole moment. This instantaneous dipole induces a corresponding instantaneous dipole in the other atom and there is an interaction between the instantaneous dipoles. The dipole of either atom averages to zero over time, but the interaction energy does not because the instantaneous and induced dipoles are correlated and... 

Figure Bl.2.1. Schematic representation of the dependence of the dipole moment on the vibrational coordinate for a heteronuclear diatomic molecule. It can couple with electromagnetic radiation of the same frequency as the vibration, but at other frequencies the interaction will average to zero.

Measurements of Stark splittings in microwave and radiofrequency spectra allow tliese components to be detennined. The main contribution to tire dipole moment of tire complex arises from tire pennanent dipole moment vectors of tire monomers, which project along tire axes of tire complex according to simple trigonometry (cosines). Thus, measurements of tire dipole moment convey infonnation about tire orientation of tire monomers in tire complex. It is of course necessary to take account of effects due to induced dipole moments and to consider whetlier tire effects of vibrational averaging are important. 

In Ih e quail tiiin mechanical description of dipole moment, the charge is a continuous distribution that is a I linction of r. and the dipole moment man average over the wave function of the dipole moment operator, p ... 

I quantities x and y are different, then the correlation function js sometimes referred to ross-correlation function. When x and y are the same then the function is usually called an orrelation function. An autocorrelation function indicates the extent to which the system IS a memory of its previous values (or, conversely, how long it takes the system to its memory). A simple example is the velocity autocorrelation coefficient whose indicates how closely the velocity at a time t is correlated with the velocity at time me correlation functions can be averaged over all the particles in the system (as can elocity autocorrelation function) whereas other functions are a property of the entire m (e.g. the dipole moment of the sample). The value of the velocity autocorrelation icient can be calculated by averaging over the N atoms in the simulation ... 

For linear moleeules, the vibrationally averaged dipole moment pave lies along the moleeular axis henee its orientation in the lab-fixed eoordinate system ean be speeified in terms of the same angles (0 and ([)) that are used to deseribe the rotational funetions Yl,m (0,(1)). Therefore, the three eomponents of the <(l)ir Pave I (1)6 integral ean be written as ... 

The result of all of the vibrational modes contributions to la (3 J-/3Ra) is a vector p-trans that is termed the vibrational "transition dipole" moment. This is a vector with components along, in principle, all three of the internal axes of the molecule. For each particular vibrational transition (i.e., each particular X and Xf) its orientation in space depends only on the orientation of the molecule it is thus said to be locked to the molecule s coordinate frame. As such, its orientation relative to the lab-fixed coordinates (which is needed to effect a derivation of rotational selection rules as was done earlier in this Chapter) can be described much as was done above for the vibrationally averaged dipole moment that arises in purely rotational transitions. There are, however, important differences in detail. In particular. 

In effect, i is replaced by the vibrationally averaged electronic dipole moment iave,iv for each initial vibrational state that can be involved, and the time correlation function thus becomes ... 

State averaging gives a wave function that describes the first few electronic states equally well. This is done by computing several states at once with the same orbitals. It also keeps the wave functions strictly orthogonal. This is necessary to accurately compute the transition dipole moments. 

The electric field of a molecule however is not static but fluctuates rapidly Although on average the centers of positive and negative charge of an alkane nearly coincide at any instant they may not and molecule A can be considered to have a temporary dipole moment... 

When you perform a single point semi-empirical or ab initio calculation, you obtain the energy and the first derivatives of the energy with respect to Cartesian displacement of the atoms. Since the wave function for the molecule is computed in the process, there are a number of other molecular properties that could be available to you. Molecular properties are basically an average over the wave function of certain operators describing the property. For example, the electronic dipole operator is basically just the operator for the position of an electron and the electronic contribution to the dipole moment is... 

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