The atomic polarization

Atoms exist in a molecule in the form of ions no matter whether they arc associated in ionic or co-valent bonds. Like the electrons moving under an electric field, ions can move along the direction of the electric field too. Ions of opposite charges are bound together through chemical bonds, forming a molecule. Under an electric field, those ions will move to the opposite direction, creating dipole moment. The polarizability of ions, a , can be expressed as [5]  

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The derivative of the dipole moment with respect to the coordinates determines the intensity of IR absorptions (Section 10.1.5). A central quantity in this respect is the Atomic Polar Tensor (APT), which for a given atom is defined as... 

Figure 1 3. Contour plot of the electron density of CO, showing the magnitudes and directions of atomic and charge transfer dipoles (arrow length is proportional to magnitude). Arrow heads point to the negative end. The molecular dipole moment is given by the vector sum of charge transfer terms (p.c.t.) and the atomic polarizations ( ra p). Values were obtained at the DFT level using the B3LYP functional and the 6-31 1+G(3df) basis set. The SCF molecular dipole = 0.096 D the computed molecular dipole ( Jtc.t.[0] + Aa.p.[0] + Hc.JC] + Aa.p.[C]) = 0.038 au = 0.096 D, close to the experimental value of 0.1 10 D (15).

An approximate form of the NMO model, the atomic polar tensor (APT) model, has also proved effective (42). Rather than considering local contributions, this model considers contributions from the derivative of the total molecular dipole moment with respect to Cartesian displacement of a given nucleus, (3p/9R )o. The latter ate the elements of the atomic polar tensor for atom n. 

This model has the advantage that the atomic polar tensor elements can be determined at the equilibrium geometry from a single molecular orbital calculation. Coupled with a set of trajectories (3R /3G)o obtained from a normal coordinate analysis, the IR and VCD intensities of all the normal modes of a molecule can be obtained in one calculation. In contrast, the other MO models require a separate MO calculation for each normal mode, since the (3p,/3G)o contributions for each unit are determined by finite displacement of the molecule along each normal coordinate. Both the APT and FPC models are useful in readily assessing how changes in geometry or refinements in the vibrational force field affect the frequencies and intensities of all the vibrational modes of a molecule. 

Atomic polarization contributes to the relative motion of atoms in the molecule affected by perturbation by the applied field of the vibrations of atoms and ions having a characteristic resonance frequency in the IR region. The atomic polarization is large in inorganic materials which contain low-energy conductive bonds and approaches zero for nonconductive polymers. The atomic polarization is rapid, and this, as well as the electronic polarization, constitutes the instantaneous polarization components. 

Ionic polarization occurs in ionic materials because an applied field acts to displace cations in the direction of the applied field while displacing anions in a direction opposite to the applied field. This gives rise to a net dipole moment per formula unit. For an ionic solid, the atomic polarization is given by... 

The atomic polarization is rapid, and this and the electronic polarizations constitute the instantaneous polarization components. The remaining types of polarization are absorptive types with characteristic relaxation times corresponding to relaxation frequencies. 

The dipole moment of a compound is a function of the distribution of charge within the molecule, and appears to be a sensitive test for the accuracy of the compound s molecular wave functions. The dipole moment of a molecule can be approximated for a given direction as the sum of two components, /iq, the contribution from net charge densities on the atoms, and for each atom A, ftsp (A), an atomic polarization moment produced by the distortion of the electronic cloud around the atom. The atomic polarization moment results essentially from the mixture of s and p orbitals and, for a heteroatom, includes mv, the lone pair moment. 

When Barriol begun with his laboratory, he inherited what had been left from the ancient laboratory of Donzelot, the laboratory of physical chemistry. [12] But because the laboratories were poorly equipped in this after-war time, the spectroscopic equipment had already been taken by other laboratories. There remained little at the laboratory of physical chemistry, only something such as a spiritual inheritance and a tradition that wanted to found theoretical speculations on experiment. Two topics were further pursued Raman spectrography, or, more generally, the studies of vibrations inside the molecule, and dielectric problems. The first assistants had to build their own installation for the measurement of dipolar moments, and a part of an installation for the measurement of the refractive index in order to determine the atomic polarization. 

Properly speaking the distortion polarization Pj is the sum of two parts, the atomic polarization P and the electronic polarization P. The atomic polarization results from distortion of the nuclear framework of the molecule in response to the electric field, while the electronic polarization results from distortion of the electron cloud on a time scale short compared to that in which the nuclei are able to move. 

If the low-freqnency dielectric constant has been determined at more than one temperature and the refractive index has also been measured with visible light, calculate the atomic polarization from... 

Let us now consider whether neglecting the atomic polarization P is justified. A simple classical treatment of atomic polarization of a gas molecule has been given by Van Vleck, Coop and Sutton, and Smyth. It is based on the approximation that the vibrations of the molecule are harmonic. Van Vleck argues that the result of this classical treatment is (to the same level of approximation) valid also under quantum mechanics. 

There is no sharp difference between covalent and ion types of bonds. In covalent compounds, electron density is spread almost symmetrically over the partners, its mass centre being in the middle of the distance between atoms, while in purely ionic compounds the maximum of binding electron orbital coincides with the centre of one of the atoms. Polar bonds exhibit asymmetry in the distribution of electron density, mass centre being shifted from the middle to one of the parmers. This shift is characterized by the degree of the bond s ionicity. 

Here the components of (dfildR ) are the elements of the atomic polar tensor for the n" atom in the molecule. 

The very-short-time behavior of micro-cavity quantum electrodynamics for characterizing micro-laser arrays for high speed applications, where highly nonlinear coupling of the vacuum fluctuations and the atomic polarizations exist, requires the full time-dependent quantum treatment discussed here. 

The atomic polarization cannot be measured directly it is equal to the difference of the polarizations P + Pjd — Pr-... 

A study of the contributions of the individual molecular orbital densities to the total force exerted on a nucleus in homonuclear diatomic molecules was made by Bader et al. (1967a). This study shows that the atomic polarizations binding the nuclei in these molecules do not arise primarily from the IcTg and l(Tu molecular orbitals whose densities correspond to slightly polarized Is-like atomic distributions. Indeed, in C2 these orbital densities exert small antibinding forces on the nuclei. It is the density of the 2atomic forces which bind the nuclei, while the densities of the 2cr and 3cTg orbitals are responsible for the oppositely directed atomic first moments. 

The second term in Eq. (12) has the form characteristic for atomic polarization, the summation in this term being performed over the vibronic states of the ground electronic term. The occurrence of this term is due to the Jahn-Teller effect, since in the absence of the effect the matrix elements < o o- ne) are identically equal to zero. Its contribution to the atomic polarization is determined by the magnitude of vibronic interaction and, generally speaking, it is not small. Visually, the vibronic contribution to the atomic polarization can be explained by the increase of the mobility and hence the polarizability of the vibrational system due to the Jahn-Teller effect. 

We consider the NMOR in coherent atomic media, where the basic mechanism of NMOR is the laser-induced coherence between the Zeeman sublevels of atomic ground state and, hence, the detected NFS is sensitive to the damping rate of atomic coherence. An atomic transition is chosen such that both A- and M-systems are created. Under usual conditions, the contributions of these systems to the Faraday signal cannot be separated, because their manifestations are similar. On the other hand, it is well known that for a given state the highest order atomic coherence is uniquely associated with the atomic polarization moment (PM) of the same order. This means that if we are able to detect the NMOR signal separately from different PM, the corresponding atomic coher-... 

For the chemical shift it turns out that it can be expressed in terms of the atomic polarization energy. Expanding it with respect to the chemical shift tensor around the experimental value and taking the gradient, the BPT pseudo-force can be deduced... 

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