Static polarizability

KEY TERMS dispersion forces nonpolar polarizability static electricity... 

Polarizability (Static Dielectric Polarizability) is a measure of the linear response of the electronic cloud of a chemical species to a weak external electric field of particular strength. 

In linear, spherical and synnnetric tops the components of a along and perpendicular to the principal axis of synnnetry are often denoted by a and respectively. In such cases, the anisotropy is simply Aa = tty -If the applied field is oscillating at a frequency w, then the dipole polarizability is frequency dependent as well a(co). The zero frequency limit of the dynamic polarizability a(oi) is the static polarizability described above. 

These moments are related to many physical properties. The Thomas-Kulm-Reiche sum rule says that. S (0) equals the number of electrons in the molecule. Other sum rules [36] relate S(2),, S (1) and. S (-l) to ground state expectation values. The mean static dipole polarizability is md = e-S(-2)/m,.J Q Cauchy expansion... 

Flere, is the static polarizability, a is the change in polarizability as a fiinction of the vibrational coordinate, a" is the second derivative of the polarizability with respect to vibration and so on. As is usually the case, it is possible to truncate this series after the second tenn. As before, the electric field is = EQCOslnvQt, where Vq is the frequency of the light field. Thus we have... 

The dielectric constant is a property of a bulk material, not an individual molecule. It arises from the polarity of molecules (static dipole moment), and the polarizability and orientation of molecules in the bulk medium. Often, it is the relative permitivity 8, that is computed rather than the dielectric constant k, which is the constant of proportionality between the vacuum permitivity so and the relative permitivity. 

Formal Theory A small neutral particle at equihbrium in a static elecdric field experiences a net force due to DEP that can be written as F = (p V)E, where p is the dipole moment vecdor and E is the external electric field. If the particle is a simple dielectric and is isotropically, linearly, and homogeneously polarizable, then the dipole moment can be written as p = ai E, where a is the (scalar) polarizability, V is the volume of the particle, and E is the external field. The force can then be written as ... 

To give a simple classical model for frequency-dependent polarizabilities, let me return to Figure 17.1 and now consider the positive charge as a point nucleus and the negative sphere as an electron cloud. In the static case, the restoring force on the displaced nucleus is d)/ AtteQO ) which corresponds to a simple harmonic oscillator with force constant... 

The poles con espond to excitation energies, and the residues (numerator at the poles) to transition moments between the reference and excited states. In the limit where cj —> 0 (i.e. where the perturbation is time independent), the propagator is identical to the second-order perturbation formula for a constant electric field (eq. (10.57)), i.e. the ((r r))Q propagator determines the static polarizability. 

SA(A)1011]. Static dipole polarizabilities were eomputed up to the MP4(SDQ) level [94MP557]. A study of the eleetronie strueture of the and S" " states of 192 showed that inelusion of dynamie eleetron eorrelation effeets is very important [98JPC(A)8021]. The multiplieity of the 1,3,5-triazine dieation is predieted to be a high-spin triplet while the trieation is most likely a doublet. In hexahydro-... 

The static electric dipole polarizability a of the model system investigated is therefore... 

The quality of the ) states has been tested through their energy and also their transition moment. Moreover from the natural orbitals and Mulliken populations analysis, we have determined the predominant electronic configuration of each ) state and its Rydberg character. Such an analysis is particularly interesting since it explains the contribution of each ) to the calculation of the static or dynamic polarizability it allows a better understanding in the case of the CO molecule the difficulty of the calculation and the wide range of published values for the parallel component while the computation of the perpendicular component is easier. In effect in the case of CO ... 

Figure 1 Static parallel polarizability vs dipolar factor for CO. Solid(dashed) line with(without) dipolar factor, from Ref. 1...

Figure (1) gives an illustration of this extrapolation proeedure for the ealculation of the static parallel polarizability in CO. In this ease the extrapolated value (a = 14.96) was obtained with the following equation a —... 

Table 1 Partial sums of discrete series for static polarizabilities of hydrogen (from Ref. 13)...

On the other hand, the static polarizability can be calculated by a sum-over-istates on the spectral states W ) the discrete series converge on a value defined by... 

Table 2 Vibronic correction for the static polarizability of CO (from Ref. 5)...

Moreover, the values obtained for the dynamic polarizability by varying the wavelength (until A > 3511A) are in good agreement with experiment (1). Table 4 resumes the results obtained for the static polarizability of CO ... 

This calculation has shown the importance of the basis set and in particular the polarization functions necessary in such computations. We have studied this problem through the calculation of the static polarizability and even hyperpolarizability. The very good results of the hyperpolarizabilities obtained for various systems give proof of the ability of our approach based on suitable polarization functions derived from an hydrogenic model. Field—induced polarization functions have been constructed from the first- and second-order perturbed hydrogenic wavefunctions in which the exponent is determined by optimization with the maximum polarizability criterion. We have demonstrated the necessity of describing the wavefunction the best we can, so that the polarization functions participate solely in the calculation of polarizabilities or hyperpolarizabilities. 

Due to the relativistic 6s contraction in gold, the 6s shell becomes more compact (inert, hence the nobility of gold) and the (static dipole) polarizability an decreases substantially from 9.50 (NR) to 5.20 (R) [99], Table 4.3. The relativistic enhance-... 

Table 4.3 Nonrelativistic (NR) and relativistic (R) static dipole polarizabilities tto (in A ), relativistic effects Af.aD, and relativistic enhancementfactors Yaforthe Croup 11 elements ofthe periodic table.

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