Moments, induced

The quantity on the left side of the equation is called the molar polarization, and this expression is the Clausius—Mosotti equation. On the right side the quantity a is the polarizability, which measures the ease with which an induced moment is... 

In the AMOEBA force field the permanent atomic multipoles are determined from QM calculations [40], The prescription considers that the resulting multipoles on the atoms result from two components, the permanent and the induced moments ... 

The scaling factor Sj can take any value between 0 and 1 and is applied to site j. The superscripts p and m indicate permanent and mutual induction, respectively. Equation (9-19) can be solved iteratively using similar procedures to those used to solve Eq. (9-3). The formal permanent moments can be calculated by subtracting induced moments from moments from ab initio calculations. For any conformation of a given compound the atomic multipoles can be determined from Distributed Multipole Analysis (DMA) [51]. 

The actual evaluations of Edd, II, and EHH are complex. Note that for Edd, CKJ uses Buckingham s (1957) prescription for the number of dipoles in the first solvation shell and considers both the thermally averaged dipole moment and the induced moment. The polarization energy is obtained from Land and O Reilly (1967). 

Induced Moments, Polarizability Isotope Effects 12.3.1 The Polarizability... 

London-van der Waals forces generally are multipole (dipole-dipole or dipole-induced dipole) interactions produced by a correlation between fluctuating induced multipole (principal dipole) moments in two nearly uncharged polar molecules. Even though the time-averaged, induced multipole in each molecule is zero, the correlation between the two induced moments does not average to zero. As a result an attractive interaction between the two is produced at very small molecular distances. 

A simple calculation for urea by Spackman is instructive. Urea crystallizes in an acentric space group (it is a well-known nonlinear optical material), in which the symmetry axes of the molecules coincide with the two-fold axes of the space group. All molecules are lined up parallel to the tetragonal c axis. If the electric field is given by E, and the principal element of the diagonalized molecular polarizability tensor along the c axis by oc , the induced moment along the polar c axis is... 

This equation is not valid for liquids or crystals, but only for substances for which the dielectric constant is very close to unity, as for gases. For other substances an equation derived by consideration of the effect of the induced moments of neighboring molecules upon the molecule undergoing polarization must be considered. In a polarized medium each molecule is affected by the electric field in the region occupied by the molecule, called the local field. For many substances the local field is satisfactorily represented by the Clausius-Mossotti expression, derived in 1850. Each molecule is considered to occupy a spherical cavity. The part of the substance outside the spherical cavity undergoes polarization in the applied field. A simple calcula-... 

A sample of a diamagnetic substance placed in an inhomogeneous magnetic field is acted on by a force that tends to push it away from the strongfield region. This force is proportional to the diamagnetic susceptibility of the substance, which is defined as the ratio of the induced moment, /x, to the field strength, H ... 

We firstly define the electric and magnetic multipoles, then show how the interaction energy with external fields can be expressed in terms of the induced moments and go on to see how these are related to the field-free moments via response functions. Most of our treatment will concern the electric case, since this is the simpler, and since magnetic effects can be treated by analogy. 

In addition to R, electric dipole moment a-components are given in Table 7. Assuming that charge transfer can be neglected, this dipole moment is simply the projection of the hydrogen halide moment along the a-axis plus induced moment terms [Eqn. (8)],... 

The second term in Eq. (13-17) concerns particles and induced moments located in the MM system. The polarization energy for the QM system interacting with the MM polarization sites is given by the first term in Eq. (13-17) and it is related to the interaction operator, Hv° as... 

In a mean field and linear approximation, the induced dipole moments are related to the wave function through Eq. (13-14) and we introduce the polarizability and the electric fields in place of the induced moments by inserting this equation once into Eq. (13-18). Taking the expectation value of Eq. (13-8) we obtain the following expression for the QM/MM energy [24,54]... 

As might be expected, there is a very large variation in the magnitudes of these terms at some normal or standard concentration. Thus, at a monolayer coverage, these terms will have approximate upper limits of 104, 103, 102, and 101 cal. per mole, respectively, when the induced moments are taken into account along with the permanent ones. At low coverages, the relative differences will be even greater, and we may expect to be able to account for the observed interaction heats on the basis of pure dispersion-attraction and simple dipole interactions. 

In a particle with a polarizability a a dipole moment of size (Xj = aF is, by definition, induced in a field of strength F. [The induced moment is always directed parallel to the inducing field]. The interaction energy is here proportional to... 

The anisotropies of the electrostatic (e 0 ), SCF-deformation (AEj ) and perturbation induction ( r) terms are quite similar and qualitatively reciprocal to the anisotropy of the HL-exchange energy, see Fig.4.A. s0) is reduced for this complex to the penetration part only and has no long range component. . r, which includes the interaction of the quadrupole and higher CO2 moments with induced moments of Ar (restrained by overlap effects and... 

Finally, the induction interaction Vjn(j was cast into a similar form as the dispersion part with CJ,"l describing the interactions between permanent and induced moments of particular... 

The magnitude of the induced dipole moment that is produced when an indole molecule in its ground Sq and electronically excited Si states is polarized by the attachment of a hydrogen-bonded water molecule in the gas-phase complex indole-H20 have been determined as /i I(So) = 0.7 D atid /t I(Si) = 0.5 D <2005JCP1743011>. The permanent dipole moment values for the complex (/tlW(So) =4.4D and /tlW(Si) = 4.0 D) are substantially different from calculated values based on vector sums of the dipole moments of the component parts. The orientation of the induced moment is also significantly different in the two electronic states. 

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