Magnetic multipoles, molecular

Magnetic multipoles of rank higher than one become active in spin systems with I > 5 and their contribution to relaxation depends on dynamics. The appearance of multipole terms complicates the relaxation description and supports the multiexponential behavior of relaxation. Nosel et al. presented the effects of high rank multipoles on lineshape and longitudinal relaxation of 7=3 systems. Results obtained from both numerical simulation and experimental data show that longitudinal and transverse relaxation are strongly influenced by these multipole terms, especially at lower temperatures where, due to molecular mobility, the extreme narrowing condition is not fulfilled. 

Terms of higher order in the field amplitudes or in the multipole expansion are indicated by. . . The other two tensors in (1) are the electric polarizability ax and the magnetizability The linear response tensors in (1) are molecular properties, amenable to ab initio computations, and the tensor elements are functions of the frequency m of the applied fields. Because of the time derivatives of the fields involved with the mixed electric-magnetic polarizabilities, chiroptical effects vanish as a> goes to zero (however, f has a nonzero static limit). Away from resonances, the OR parameter is given by [32]... 

Electric and magnetic properties of microsystems. Definition of multipoles electrostatics of permanent multipoles interaction energies for two multipoles induced molecular multipoles interaction energies of induced multipoles. Tables of point groups tensor elements of multipoles vector elements of multipoles tensor elements of polarizabilities. 

The permanent electric multipoles (m = 0) are defined by equation (40), the electric multipoles of first order w = 1 by equation (72), and those of the second order w = 2 by equation (79). Similarly, magnetic potential energy of order m — I can be defined, on replacing in equation (83) E by B and p by m. From the general expression [equation (83)] one immediately and quite easily derives all the energies dealt with in the theory of non-linear molecular processes. ... 

Numerical Values of Molecular Electric Multipoles 219 Magnetic (Natural) Multipoles 231... 

To some purposes, one can define new molecular tensors that are independent of the origin [40]. At any rate, it can be easily proven that the induced moments (69)-(72) and fields (73)-(74) are, order by order, independent of the origin chosen for the multipole expansion, provided that all the terms of the same order of magnitude are retained. Thus, within the quadrupole approximation, both the magnetic field and the electric field gradient must be taken... 

In 1996, Munn extended the microscopic theory of bulk second-harmonic generation from molecular crystals to encompass magnetic dipole and electric quadrupole effects [96] and included all contributions up to second order in the electric field or bilinear in the electric field and the electric field gradient or the magnetic field. This was accomplished by replacing the usual polarization of Refs. 72 and 84 by an effective polarization as well as by defining an effective quadrupole moment. Consequently, the self-consistently evaluated local electric field and electric field gradient were expressed in terms of various molecular response coefficients and lattice multipole tensor sums (up to octupole). In this... 

Some arguments that imply limitations on the concept of molecular multipole moments due to the requirements of gauge invariance are presented. The discussion is based on a pair of polarization field vectors which are natural generalizations of multipole series. A class of transformations that mix some of the components of the polarization field vectors, so spoiling a separation into electric and magnetic types, is identified. The results are related to the gauge-invariant transition amplitudes. 

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. 

The condition for this quantity to be non-zero is that the chromophore of interest must have a non-zero magnetic and electric transition dipole moment along the same molecular diiectioa In the absence of pertuibing external fields, this is only trae for molecules that are chiral. Expressions that include higher order multipole contributions to eqs. (8), (9), and (11) can be found in previous theoretical descriptions of CPU theory (Riehl and Richardson, 1976a, 1986). 

Other postprocessing techniques allow the computation of the density of states/molecular orbital structure, local charges on atoms or fragments of the system, dipole and multipoles, magnetic properties, and the electrostatic potential. Energy minimization can also be performed in the presence of perturbations, such as external fields or imposed electrode... 

In contrast to magnetic properties, the theory of electric-field-like properties is much easier to cast into a set of working equations. One of them has attracted particular interest, and that is the electric field gradient (EFG). This property is of decisive importance to Mossbauer spectroscopy, i.e., to the spectroscopy of excited nuclear states whose energies are modulated by the molecular structure (the chemical environment ). In order to see how this property arises, we study the electrostatic electron-nucleus interaction of extended, not spherically symmetric charge distributions. For this we apply a multipole expansion in order to generate the properties term by term. 

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. 

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