Electric properties multipole moments

Typical properties of the charge distribution are summarized by its various electric multipole moments. The electric dipole moment p. induced in the system by the external field is obviously... 

Although 1 is one of the best investigated molecules, there is, apart from data concerning its electron density distribution, very little information available on its one-electron properties. In principle, accurate data could be obtained by correlation-corrected ab initio methods, but almost nothing has been done in this direction, which of course has to do with the fact that experimental data on one-electron properties of 1 are also rare, and therefore, it is difficult to assess the accuracy and usefulness of calculated one-electron properties such as higher multipole moments, electric field gradients, etc. 

The moments and polarizabilities of molecules can be determined by indirect means. In collision experiments, the nature of the interaction is governed by the potential energy surface, itself a function of the molecular properties of the colliding partners. Usually the potential energy is written in a multipole expansion whereby the electrical properties are displayed in the long-range terms [38]. The potential that is generated must satisfy simultaneously... 

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. 

As an example, explicit expressions of /3 can be given in the case of the dipole polarizability of the H atom and for a few simple VdW interactions which depend on the electrical properties of the molecules such as electric dipole moments and polarizabilities (Stone, 1996). As we have already said, these dipole moments, and the higher ones known generally as multipole moments, can be permanent (when they persist in absence of any external field) or induced (when due, temporarily, to the action of an external field and disappear when the field is removed). 

In the present review article 1985 s results obtained in applications of the concept of vibronic interactions to the investigation of electric properties of molecules (dipole and multipole moments and polarizabilities) are presented. Molecular aspects of these topics are almost untouched in the publications listed in the preceding paragraph. The idea of dipole instability was used first as a basis of the so-called vibronic theory of ferroelectricity (Bersuker, 1966 Bersuker and Vekhter, 1978). Meanwhile, the manifestation of the electronic or vibronic degeneracy in the electric responses of molecules, being no less essential than other vibronic effects, has some special features. 

In this last section we mention a few cases, where properties other than the energy of a system are considered, which are influenced in particular by the change from the point-like nucleus case (PNC) to the finite nucleus case (FNC) for the nuclear model. Firstly, we consider the electron-nuclear contact term (Darwin term), and turn then to higher quantum electrodynamic effects. In both cases the nuclear charge density distribution p r) is involved. The next item, parity non-conservation due to neutral weak interaction between electrons and nuclei, involves the nuclear proton and neutron density distributions, i.e., the particle density ditributions n r) and n (r). Finally, higher nuclear electric multipole moments, which involve the charge density distribution p r) again, are mentioned briefly. 

Some of the most important properties that a quantum mechanical calculation provides are the electric multipole moments of the molecule. The electric multipoles reflect the distribution of charge in a molecule. The simplest electric moment (apart from the total net charge on the molecule) is the dipole. The dipole moment of a distribution of charges located at positions r, is given by ij r/. If there are just two charges +q and -q separated by a distance... 

Calculation of properties, such as multipole moments, electrostatic potentials, electric fields and electric... 

A wide variety of molecular properties can be accurately obtained with ADF. The time-dependent DFT implementation " yields UV/Vis spectra (singlet and triplet excitation energies, as well as oscillator strengths), frequency-dependent (hyper)polarizabilities (nonlinear optics), Raman intensities, and van der Waals dispersion coefficients. Rotatory strengths and optical rotatory dispersion (optical properties of chiral molecules ), as well as frequency-dependent dielectric functions for periodic structures, have been implemented as well. NMR chemical shifts and spin-spin couplingsESR (EPR) f-tensors, magnetic and electric hyperfme tensors are available, as well as more standard properties like IR frequencies and intensities, and multipole moments. Relativistic effects (ZORA and spin-orbit coupling) can be included for most properties. 

Whereas the hfs can serve as a local probe for intramolecular fields and field gradients, information about the total charge-density distribution is given by the electric multipole moments, especially the dipole moment of the molecule. Both molecular properties complement each other in giving a more complete picture of the electronic structure and chemical bonding. In section IV we show a way to determine electric dipole moments from high precision Stark effect measurements. The techniques described in sections II-IV will be applicable to many other free radicals. 

In its broadest sense, spectroscopy is concerned with interactions between light and matter. Since light consists of electromagnetic waves, this chapter begins with classical and quantum mechanical treatments of molecules subjected to static (time-independent) electric fields. Our discussion identifies the molecular properties that control interactions with electric fields the electric multipole moments and the electric polarizability. Time-dependent electromagnetic waves are then described classically using vector and scalar potentials for the associated electric and magnetic fields E and B, and the classical Hamiltonian is obtained for a molecule in the presence of these potentials. Quantum mechanical time-dependent perturbation theory is finally used to extract probabilities of transitions between molecular states. This powerful formalism not only covers the full array of multipole interactions that can cause spectroscopic transitions, but also reveals the hierarchies of multiphoton transitions that can occur. This chapter thus establishes a framework for multiphoton spectroscopies (e.g., Raman spectroscopy and coherent anti-Stokes Raman spectroscopy, which are discussed in Chapters 10 and 11) as well as for the one-photon spectroscopies that are described in most of this book. 

An alternative approach is to start from the onset with a low resolution model, which is set up essentially by physical intuition, where a whole molecule or monomer is replaced by one or few beads. The typical aim of those simple empirical models is to examine the minimum set of molecular features needed to obtain a given molecular organization, for instance a certain anisotropy of shape or the presence of electric multipole moments, and the qualitative relatirm between variations in the microscopic model and macroscopic properties. This type of modeling does not necessitate a preliminary weU-defined and known chemical structure, but is more akin to a reverse molecular engineering process, where one guesses what key features are needed to achieve the desired macroscopic behavior before actually trying to write down and possibly synthesize a certain molecule. 

Only static and dynamic molecular properties involving electric dipole and quadrupole operators will be discussed below. However, electric properties related to higher-order electric multipole operators can also be determined in a similar manner to the properties described here, in terms of expectation values, linear and nonlinear response functions. Nevertheless, it should be kept in mind that although the same formalism is applied in the calculation of response functions involving octupole, hexadecapole, and higher moments, in practice it may... 

EFGs and other electric-field-related properties are dealt with in a somewhat different manner. The EFG and multipole moments are calculated as expectation values with the relevant operators and the electron charge density. To avoid PC errors, if the operators are the four-component versions this charge density has to be the four-component (Dirac) density. The latter differs from the two-component density [24,25] already in order which is the same leading order as the relativistic effects on the properties. In the so-called ZORA-4 (Z4) framework, the relevant operators are kept in their four-component form, and an approximate four-component electron charge density is reconstructed from the two-component ZORA density. As was shown by van Lenthe and Baerends [26], the Z4 method eliminates most of the PC errors in order c , with relatively small residual errors. In a Kohn-Sham (KS) DFT framework with two-component molecular orbitals y>, with occupations the ZORA two-component density is... 

When sufficiently weak electric fields are applied it is possible to extract the electric properties of the molecule from the above expansions. In previous work, we have applied various computational schemes based on the finite-field [22] approach to the calculation of electric properties from perturbed atomic/molecular energies and induced multipole moments [23-28]. 

Thus in an isotropic environment the number of different relaxation rates is reduced from (21 +1) to (2Jg+l), giving one relaxation rate for each multipole moment of the excited state. Experimentally the number of effective relaxation rates might still be embarrassingly large were it not for the fact that in resonance fluorescence we prepare and monitor the excited atoms through the absorption and emission of electric dipole radiation. The electric-dipole matrix elements have the properties associated with rank one tensors and consequently the observable multipole moments in these experiments are limited to those corresponding to tensors of rank 0, 1, and 2 respectively. 

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