Dipole moment defined

Recalling the expression for the induced dipole moment, defined in terms of the polarizability of the molecule (Equation 12-22), and then differentiating, we obtain our final expression for the electric field arising from the interaction of a small, single molecule with polarized light (Equation 12-25) ... 

Debye in 1912 extended Langevin s procedure to dielectrics, by attributing to the molecules a permanent (natural) electric dipole moment defined as in equation (34), The classical permanent magnetic dipole moment of a molecule is defined as follows (as an axial vector) ... 

One of the most powerful features of HPLC stems from the fact that the eluent is not merely a transport medium, it rather contributes significantly to the mechanism of separation. Retention as well as selectivity arise from the combined action of mobile and stationary phase on the solutes. Because of its distinct physicochemical characteristics, a given solvent interacts in a specific manner with the solutes. Its capacity to donate or accept protons or to induce a dipole moment defines the nature of its interaction with the solutes in solution. Slight differences in these interactions complemented by stationary-phase action are often enough to provide the desired selectivity in HPLC. Solvent classification based on their elution strength or polarity represents a classic area of investigation into the fundamentals of retention mechanisms in HPLC (2S-27), and the chapter is not yet closed on its development and refinement (28-30). Here we will present a practical and comprehensive way to exploit the effect that the nature of the solvent has on retention and selectivity in reversed phase HPLC. 

The p,nm electronic transition dipole moments, defined as matrix elements of the electronic dipole operator with diabatic states... 

As it stands, eq 3.29 is useful only for essentially non-polar gases but Tsono-poulos showed that the addition of a third term, B2, to the right-hand side of that equation permitted the results for polar gases to be correlated also. This third term is expressed in terms of a reduced dipole moment, defined by... 

Before substiUitmg everything back into equation B1.2.6, we define the transition dipole moment between states 1 and 2 to be the integral... 

In order to describe the second-order nonlinear response from the interface of two centrosynnnetric media, the material system may be divided into tlnee regions the interface and the two bulk media. The interface is defined to be the transitional zone where the material properties—such as the electronic structure or molecular orientation of adsorbates—or the electromagnetic fields differ appreciably from the two bulk media. For most systems, this region occurs over a length scale of only a few Angstroms. With respect to the optical radiation, we can thus treat the nonlinearity of the interface as localized to a sheet of polarization. Fonnally, we can describe this sheet by a nonlinear dipole moment per unit area, -P ", which is related to a second-order bulk polarization by hy P - lx, y,r) = y. Flere z is the surface nonnal direction, and the... 

C3.4.13)). The dimer has a common ground state and excitation may temrinate in eitlier tire or excited state (see tire solid arrows in figure C3.4.3). The transition dipole moments of tliese transitions are defined as ... 

In stead, the electrostatic con tribn tion conies from definin g a set of bond dipole moments associated woth polar bonds. These bond moments are defined in the m m psir.LxL(dbf) file along with the bond stretching parameters and are given in units of Debyes. The cen ter of th e dipole Is defined to be th e m Idpoint of the bond an d two dipoles p. and pj. separated by Rjj. as shown beltnv ... 

Energy, geometry, dipole moment, and the electrostatic potential all have a clear relation to experimental values. Calculated atomic charges are a different matter. There are various ways to define atomic charges. HyperChem uses Mulliken atomic charges, which are commonly used in Molecular Orbital theory. These quantities have only an approximate relation to experiment their values are sensitive to the basis set and to the method of calculation. 

The permanent dipole moment of an isolated molecule depends on the magnitude of the charge and on the distance separating the positive and negative charges. It is defined as... 

Measured at 1.907 lm unless otherwise iadicated. 1 is the dipole moment and P is the molecular first hyperpolarizabiUty defined in equation 1. Measured at 1.3 lm Ref. 9. 

A variety of methodologies have been implemented for the reaction field. The basic equation for the dielectric continuum model is the Poisson-Laplace equation, by which the electrostatic field in a cavity with an arbitrary shape and size is calculated, although some methods do not satisfy the equation. Because the solute s electronic strucmre and the reaction field depend on each other, a nonlinear equation (modified Schrddinger equation) has to be solved in an iterative manner. In practice this is achieved by modifying the electronic Hamiltonian or Fock operator, which is defined through the shape and size of the cavity and the description of the solute s electronic distribution. If one takes a dipole moment approximation for the solute s electronic distribution and a spherical cavity (Onsager s reaction field), the interaction can be derived rather easily and an analytical expression of theFock operator is obtained. However, such an expression is not feasible for an arbitrary electronic distribution in an arbitrary cavity fitted to the molecular shape. In this case the Fock operator is very complicated and has to be prepared by a numerical procedure. 

In the second type of interaction contributing to van der Waals forces, a molecule with a permanent dipole moment polarizes a neighboring non-polar molecule. The two molecules then align with each other. To calculate the van der Waals interaction between the two molecules, let us first assume that the first molecule has a permanent dipole with a moment u and is separated from a polarizable molecule (dielectric constant ) by a distance r and oriented at some angle 0 to the axis of separation. The dipole is also oriented at some angle from the axis defining the separation between the two molecules. Overall, the picture would be very similar to Fig. 6 used for dipole-dipole interaction except that the interaction is induced as opposed to permanent. 

The dipole moment p. is a molecular property defined as the product of charge (usually just a fraction of the electronic change, of course) and distance between the centers of positive and negative charge in the molecule. The dipole moment is usually expressed in debyes (D), where 1 D = 1(T esu in SI units 1 D = 3.3356 X 10 ° C-m. so, for example, the dipole moment of water is 1.84 D or 6.14 in units of 10 C-m. Again a rough correspondence is seen between this property of a molecule and its polarity, though e and p. are not precisely correlated. 

If we define the positive Z direction as up, the hydrogen atoms lie below this plane, and the fluorine atoms lie above it. The dipole moment points down, toward the hydrogen atoms, which is where we expect the positive charge to be. The same is true for the SS form. 

This is not a unique way of classifying molecular properties. For example, Dykstra et al. (1990) concentrate on the response of a system to an apphed external field the electric dipole moment can be defined as the first derivative of the energy with respect to the field, and so on. I will stick with the Boys and Cook nomenclature as a broad basis for discussion. 

In Figure 16.1, we see a pair of point charges Qp, at position vector Fa and <2b at position vector fb. <2a is not necessarily equal to <2b> and one is not necessarily the negative of the other. Their electric dipole moment pe is defined as... 

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. 

The derivative of the dipole moment with respect to the coordinates determines the intensity of IR absorptions (Section 10.1.5). A central quantity in this respect is the Atomic Polar Tensor (APT), which for a given atom is defined as... 

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