Multipole classical - model

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. 

Second moments have also been computed, both from first principles and on the basis of the classical multipole-induction model. These are found to be in close agreement with measurements where these exist. Second moments are of a special interest in connection with modeling of three-parameter line profiles from three spectral moments [52]. In analyses based on classical expressions, the second moment is expressible in terms of the first moment specified above, multiplied by 2kT/h. 

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... 

A classical description of M can for example be a standard force field with (partial) atomic charges, while a quantum description involves calculation of the electronic wave function. The latter may be either a semi-empirical model, such as AMI or PM3, or any of the ab initio methods, i.e. HF, MCSCF, CISD, MP2 etc. Although the electrostatic potential can be derived directly from the electronic wave function, it is usually fitted to a set of atomic charges or multipoles, as discussed in Section 9.2, which then are used in the actual solvent model. 

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... 

The results of energy partitioning in Li+... OH2 obtained with a number of different basis sets are listed in Table 3. Since intermolecular overlap is small in these kind of complexes (Table 1), we expect the electrostatic model to be a good approximation for classical contributions to the total energy of interaction. Indeed, ZlE cou is to a good approximation proportional to the dipole moment of the water molecule calculated with the same basis set. This can be seen even more clearly in Table 4 where zIEcou is compared with ion-dipole and ion-quadrupole energies obtained from the classical expression of the multipole expansion series 45,95-97) ... 

Table 3.7 also lists ternary spectral moments for a few systems other than H2-H2-H2. For the H2-He-He system, the pairwise-additive dipole moments are also known from first principles. The measured spectral moments are substantially greater than the ones calculated with the assumption of pairwise additivity - just as this was seen in pure hydrogen. For the other systems listed in the Table, no ab initio dipole surfaces are known and a comparison with theory must therefore be based on the approximate, classical multipole model. 

While exchange- and dispersion-induced dipole components are of a quantum nature, the multipole-induced dipole components can be modeled by classical relationships, if the quantum effects are small. For many systems of practical interest, multipolar induction generates the dominant dipole components. The classical multipole induction approximation has been very successful, except for the weakly polarizable partners (e.g., He atoms) [193]. It models the dipole induced in the collisional partner by polarization in the molecular multipole fields. 

K. L. C. Hunt. Classical multipole models Comparison with ab initio and experimental results. In G. Birnbaum, ed., Phenomena Induced by Intermolec. Interactions, p. 1, Plenum Press, New York, 1985. 

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. 

The two-electron integrals (Equation 6.32) are determined from atomic experimental data in the one-center case, and are evaluated from a semiempirical multipole model in the two-center case that ensures correct classical behavior at large distances and convergence to the correct one-center limit. Interestingly, this parameterization results in damped effective electron-electron interactions at small and intermediate distances, which reflects a (however less regular) implicit partial inclusion of electron correlation (Thiel, 1998). In this respect, semiempirical methods go beyond the HF level, and may accordingly be superior to HF ab initio treatments for certain properties that have a direct or indirect connection to the parameterization procedure. 

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,... 

In the water trimer induction nonadditivity provides a dominant contribution, which effectively overshadows all the other terms. Its mechanism is simple. For instance, in a cyclic water trimer the multipoles of A inductively alter the multipoles at B, which, in turn, inductively alter the multipoles at C, which then alter those on A, and so on, until the self-consistency is reached. Various formulations of this simple model were implemented in the simulations since the 1970s [84-87,63,64,50]. To include the many-body induction effects of point charges interacting with a set of polarizable atomic centers the following classical electrostatics equation is solved iteratively... 

Equations [22]-[24] illustrate why the derivation and programming of the forces and torques, and second derivatives for all the terms up to R in the atom-atom multipole expansion of the electrostatic energy is a nontrivial exercise in classical mechanics. It has been described in detail by Popelier and Stone,and, with the additional derivatives required for modeling molecular crystal structures, by Willock et al. ... 

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. 

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