Multipole moments classical

The two-center two-electron repulsion integrals ( AV Arr) represents the energy of interaction between the charge distributions at atom Aand at atom B. Classically, they are equal to the sum over all interactions between the multipole moments of the two charge contributions, where the subscripts I and m specify the order and orientation of the multipole. MNDO uses the classical model in calculating these two-center two-electron interactions. 

Conceptually, the self-consistent reaction field (SCRF) model is the simplest method for inclusion of environment implicitly in the semi-empirical Hamiltonian24, and has been the subject of several detailed reviews24,25,66. In SCRF calculations, the QM system of interest (solute) is placed into a cavity within a polarizable medium of dielectric constant e (Fig. 2.2). For ease of computation, the cavity is assumed to be spherical and have a radius ro, although expressions similar to those outlined below have been developed for ellipsoidal cavities67. Using ideas from classical electrostatics, we can show that the interaction potential can be expressed as a function of the charge and multipole moments of the solute. For ease... 

Consider tlie mutual approach of two noble gas atoms. At infinite separation, there is no interaction between them, and this defines die zero of potential energy. The isolated atoms are spherically symmetric, lacking any electric multipole moments. In a classical world (ignoring the chemically irrelevant gravitational interaction) there is no attractive force between them as they approach one another. When tliere are no dissipative forces, the relationship between force F in a given coordinate direction q and potential energy U is... 

A rotation of the H2 molecule through 180° creates an identical electric field. In other words, for every full rotation of a H2 molecule, the dipole induced in the collisional partner X oscillates twice through the full cycle. Quadrupole induced lines occur, therefore, at twice the (classical) rotation frequencies, or with selection rules J — J + 2, like rotational Raman lines of linear molecules. Orientational transitions (J — J AM 0) occur at zero frequency and make up the translational line. Besides multipole induction of the lowest-order multipole moments consistent with... 

Asymptotic formulae. For a discussion of induced dipoles in highly polarizable species, it is often sufficient to consider the so-called classical multipole induction approximation in its simplest form (i.e., neglecting field gradients and hyperpolarizabilities). In such a case, one needs to know only the vibrational matrix elements of the multipole moments,... 

Thus, in this section we have described the manner in which absorption of light by a molecule leads to polarization of the angular momenta of the absorbing level. We have also shown how to calculate the multipole moments created on the lower level. It is important to stress that the adopted model of description enables us to obtain precise analytical expressions for the multipole moments, including both cases, namely those for arbitrary values of angular momenta and those for the classic limit J — oo. Our subsequent discussion will concern problems connected with the manifestation of ground state angular momenta anisotropy in experimentally observable quantities. 

In order to describe a signal by this method we will first use the classical approach. At the beginning we will ascertain how either probability density Pb(9, multipole moments ipq of the excited state 6, entering into the fluorescence intensity expressions (2.23) or (2.24), are connected to the corresponding magnitudes pa(9, ground state a. The respective kinetic balance equation for probability density and its stationary solution, assuming that the conditions supposed to hold in Eq. (3.4) are in force, is very simple indeed ... 

Thus, we have attempted to give, in the present appendix, an idea of the various methods of determining classical and quantum mechanical polarization moments and some related coefficients. We have considered only those methods which are most frequently used in atomic, molecular and chemical physics. An analysis of a great variety of different approaches creates the impression that sometimes the authors of one or other investigation find it easier to introduce new definitions of their own multipole moments, rather than find a way in the rather muddled system of previously used ones. This situation complicates comparison between the results obtained by various authors considerably. We hope that the material contained in the present appendix might, to some extent, simplify such a comparison. 

Equation (1-239) relates the interaction-induced part of the dipole moment of the complex AB to the distortion of the electron density associated with the electrostatic, exchange, induction, and dispersion interactions between the monomers. The polarization contributions to the dipole moment through the second-order of perturbation theory (A/a, A/a, and A/a ) have an appealing, partly classical, partly quantum, physical interpretation. The first-order multipole-expanded polarization contribution (F) is due to the interactions of permanent multipole moments on A with moments induced on B by the external field F, and vice versa. The terms... 

Solvent continuum models are now routinely used in quantum mechanical (QM) studies to calculate solvation effects on molecular properties and reactivity. In these models, the solvent is represented by a dielectric continuum that in the presence of electronic and nuclear charges of the solute polarizes, creating an electrostatic potential, the so-called reaction field . The concept goes back to classical electrostatic schemes by Martin [1], Bell [2] and Onsager [3] who made fundamental contributions to the theory of solutions. Scholte [4] and Kirkwood [5] introduced the use of multipole moment distributions. The first implementation in QM calculations was reported in a pioneer work by Rivail and Rinaldi [6,7], Other fundamental investigations were carried out by Tapia and Goscinski [8], Hilton-McCreery et al. [9] and Miertus et al. [10], Many improvements have been made since then (for a review,... 

At first glance the induction energy is a very classical term, and conceptually a simple one. A multipole moment on A induces another multipole moment on B, and eventually they interact electrostatically. Different mechanisms behind induction interaction in a variety of molecular settings were elucidated and composed into an elegant theory [7]. 

The electrostatic interaction, which is defined as the classical Coulombic interaction between the undistorted charge distributions of the isolated molecules, is the easiest to derive from wavefunctions. When there is no overlap of the charge distributions of the molecules, all that is required is a representation of the molecular charge density. The traditional, and simplest, representation of the molecular charge distribution is in terms of the total multipole moments. The first nonvanishing multipole moment could often be derived from experi-... 

Electric polarization, dipole moments and other related physical quantities, such as multipole moments and polarizabilities, constitute another group of both local and molecular descriptors, which can be defined either in terms of classical physics or quantum mechanics. They encode information about the charge distribution in molecules [Bbttcher et al, 1973]. They are particularly important in modelling solvation properties of compounds which depend on solute/solvent interactions and in fact are frequently used to represent the -> dipolarity/polarizability term in - linear solvation energy relationships. Moreover, they can be used to model the polar interactions which contribute to the determination of the -> lipophilicity of compounds. 

Higher and mixed multipole interactions are also always present and may play a dominant role in the absence of lower multipole moments, especially in the absence of mono- and dipoles. Note also that these electrostatic multipole interactions are purely classic and typical quantum mechanical effects (like Pauli repulsion etc.) are not captured. 

The dispersion energy is a pure quantum effect, more difficult to interpret than those terms for which one can apply classical concepts. In general terms, the dispersion energy is due to the correlation of the electronic motions in different monomers. Using the time-dependent point of view, one can loosely say that instantaneous multipole moments in monomer X—created by fluctuations of electronic density—induce... 

The multipole moments of the classical models and QM results [55] are compared in Table 9.1. In examining the QM results, the moments increase from 10 to 30% from the gas phase to the liquid phase. The classical models have moments that are generally... 

Since this interaction is very weak, it causes the low solubility of polar molecules in nonpolar solvents. This interaction is also incorporated in Kohn-Sham SCF calculations. The dispersion interaction is a universal interaction, which acts even between bodies with neither charge nor multipole moment. As a classical expression, the potential function between two heterogeneous bodies, which London developed using perturbation theory, is used (London 1930),... 

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