Dipole terms

Hfi includes a nuclear Zeeman term, a nuclear dipole-dipole term, an electron-nuclear dipole term and a term describing the interaction between the nuclear dipole and the electron orbital motion. 

The interaetion with a magnetic field may similarly be written in term of magnetie dipole, quadrupole ete. moments (there is no magnetie monopole, eorresponding to eleetrie eharge). Sinee the magnetie interaetion is substantially smaller in magnitude than the eleetrie, only the dipole term is normally considered. 

In connection with electronic strucmre metlrods (i.e. a quantal description of M), the term SCRF is quite generic, and it does not by itself indicate a specific model. Typically, however, the term is used for models where the cavity is either spherical or ellipsoidal, the charge distribution is represented as a multipole expansion, often terminated at quite low orders (for example only including the charge and dipole terms), and the cavity/ dispersion contributions are neglected. Such a treatment can only be used for a qualitative estimate of the solvent effect, although relative values may be reasonably accurate if the molecules are fairly polar (dominance of the dipole electrostatic term) and sufficiently similar in size and shape (cancellation of the cavity/dispersion terms). 

The key element in London s approach is the expansion of the electrical potential energy in multipole series. Since neutral molecules or portions of molecules are involved, the leading term is that for dipole-dipole interaction. While attention has been given to higher-order terms, these are usually small, and the greater need seems to be for improved treatment of the dipole-dipole terms. London used second order perturbation theory in his treatment, but Slater and Kirkwood38,21 soon followed with a variation method treatment which yielded similar results. Other individual papers will be mentioned later, but the excellent review of Mar-genau26 should not be overlooked. 

Another problem arises from the presence of higher terms in the multipole expansion of the electrostatic interaction. While theoretical formulas exist for these also, they are even more approximate than those for the dipole-dipole term. Also, there is the uncertainty about the exact form of the repulsive interaction. Quite arbitrarily we shall group the higher multipole terms with the true repulsive interaction and assume that the empirical repulsive term accounts for both. The principal merit of this assumption is simplicity the theoretical and experimental coefficients of the R Q term are compared without adjustment. Since the higher multipole terms are known to be attractive and have been estimated to amount to about 20 per cent of the total attractive potential at the minimum, a rough correction for their possible effect can be made if it is believed that this is a preferable assumption. 

Potential functions induced-dipole terms, 84-85 minimization, 113-116 nonbonded interactions, 84-85 Potential of mean force, 43, 144 Potential surfaces, 1,6-11, 85, 87-88, 85 for amide hydrolysis, 176-181,178,179, 217-220, 218... 

Under some circumstances the rotationally anisotropy may be even further simplified for T-R energy transfer of polar molecules like HF (41). To explore this quantitatively we performed additional rigid-rotator calculations in which we retained only the spherically symmetric and dipole-dipole terms of the AD potential, which yields M = 3 (see Figures 1, 3, and 4). These calculations converge more rapidly with increasing N and usually yield even less rotationally inelastic scattering. For example Table 2 compares the converged inelastic transition probabilities... 

The second modification acts on the monopole-dipole term. 

The last (dipole) term is supposed to be approximated by the truncated Taylor series... 

Developed into a power series in R 1, where R is the intermolecular separation, H exhibits the dipole-dipole, dipole-quadrupole terms in increasing order. When nonvanishing, the dipole-dipole term is the most important, leading to the Forster process. When the dipole transition is forbidden, higher-order transitions come into play (Dexter, 1953). For the Forster process, H is well known, but 0. and 0, are still not known accurately enough to make an a priori calculation with Eq. (4.2). Instead, Forster (1947) makes a simplification based on the relative slowness of the transfer process. Under this condition, energy is transferred between molecules that are thermally equilibriated. The transfer rate then contains the same combination of Franck-Condon factors and vibrational distribution as are involved in the vibrionic transitions for the emission of the donor and the adsorptions of the acceptor. Forster (1947) thus obtains... 

Orbiting collisions, in the sense that the polar angle changes >71 and the azimuth changes >2it, are not found. These types of collisions are not found with molecules that are only polarizable and also they are unimportant when the dipole term dominates. 

Suppose now that the system is subjected to an oscillating electromagnetic field with a representative Fourier component of the electric field F0( >] cos cot. The predominant term in the interaction energy V is usually the electric dipole term Ei , e.g. for an electron in an atom... 

The first three terms, stretch, bend and torsion, are common to most force fields although their explicit form may vary. The nonbonded terms may be further divided into contributions from Van der Waals (VdW), electrostatic and hydrogen-bond interactions. Most force fields include potential functions for the first two interaction types (Lennard-Jones type or Buckingham type functions for VdW interactions and charge-charge or dipole-dipole terms for the electrostatic interactions). Explicit hydrogen-bond functions are less common and such interactions are often modeled by the VdW expression with special parameters for the atoms which participate in the hydrogen bond (see below). 

The intermolecular distance was calculated here relative to the center of mass (C. M.) of the water molecule. For this particular choice of the origin the quadruple term is rather small and the total electrostatic energy of interaction is reproduced fairly well by the first term of the expansion, the ion-dipole term alone. The difference between zIEcou and zIEcou of course is not exclusively... 

As an alternate way to measure the gradients of structural and chemical composition, we intend to use the method of SH microscopy. In an isotropic environment, which has a center of symmetry, the second harmonic generation is not allowed in a dipole approximation, and is predominantly governed by weak quadrupole and surface dipole terms, which are orders of magnitude weaker. However, as we have already pointed... 

There are some notable differences apparent in Fig. 11.14 between the extinction curves for aluminum spheres and those for water droplets. For example, av is still constant for sufficiently small aluminum particles but the range of sizes is more restricted. The large peak is not an interference maximum aluminum is too absorbing for that. Rather it is the dominance of the magnetic dipole term bx in the series (4.62). Physically, this absorption arises from eddy current losses, which are strong when the particle size is near, but less than, the skin depth. At X = 0.1 jam the skin depth is less than the radius, so the interior of the particle is shielded from the field eddy current losses are confined to the vicinity of the surface and therefore the volume of absorbing material is reduced. 

As noted in Chapter 2, computation of charge-charge (or dipole-dipole) terms is a particularly efficient means to evaluate electrostatic interactions because it is pairwise additive. However, a more realistic picture of an actual physical system is one that takes into account the polarization of the system. Thus, different regions in a simulation (e.g., different functional groups, or different atoms) will be characterized by different local polarizabilities, and the local charge moments, by adjusting in an iterative fashion to their mutual interactions, introduce many-body effects into a simulation. 

Figure 4. Recovery of the cube-root law activity coefficients calculated from polarized sphere model using coulombic and induced dipole terms.

In this expansion the dipole-dipole term is the most prominent if donor-acceptor distance R is not too small. The dipole-dipole term represents the interaction between the transition dipole moments Md and MA of donor and acceptor molecules, respectively. The square of these transition dipoles is proportional to the oscillator strengths fy> and fA for radiative transitions in the individual donor and acceptor molecules (equation 3.73). Higher order terms such as electric dipole-electric quadrupole, electric-dipole-magnetic dipole, become important at close approach and may be effective in crystals and highly ordered array of chromophores. 

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