Polarizability quadrupole

Just as the dipole changes in an external field, so do all the other moments, and we can develop a set of equations for the quadrupole polarizability (and hyperpolarizabilities), the octupole polarizability, and so on. These esoteric quantities are rarely met in chemistry. 

Calculations of Dipole and Quadrupole Polarizability Radial Functions for LiH and HF A Comparison of Different Linear Response Methods... 

The calculation of polarizabilities is one of the research topics Jens Oddershede is working on since the beginning of his career [1-21], Already in one of his first papers he discussed the dipole polarizability of HF [1] and returned to it several times later [3,6,13,14,18]. Therefore, we decided to contribute to this special issue with a study of static dipole and quadrupole polarizabilities which are still one of the most studied electromagnetic properties. 

In the response function terminology [47] the i, j component of the frequency-dependent dipole polarizability tensor — w) (or the ij, kl component of the traceless quadrupole polarizability tensor is defined through... 

The isotropic dipole polarizability a, the dipole polarizability anisotropy Aa and the isotropic traceless quadrupole polarizability C are defined as... 

Calculations of Dipole and Quadrupole Polarizability Radiai Functions... 

The dipole and quadrupole polarizability tensor components of LiH were calculated by MCSCF linear response theory with the basis set of Roos and Sadlej [57] which consists of 13s-, 8p-, 6d-, and 2f-type sets of uncontracted Gaussian functions on Li and 12s-, 8p-, and 5d-type sets of uncontracted Gaussians on H. Due to the small size of the molecule we could perform MCSCF calculations over the whole range of internuclear distances with a very large CAS 0000 520,10,10,4 p g present the tensor components, isotropic, and anisotropic values of the dipole polarizability tensor a as function... 

Table 2. LiH dipole and quadrupole polarizability (in atomic units) for the vibrational ground state u = 0 calculated with different response theory methods. P(Pe) is the value at the minimum of the potential energy curve, Pq o is the value in the vibrational ground state and ZPVC = Pq o P(Pe) is the corresponding zero-point-vibrational correction...

Our best results for the dipole and quadrupole polarizability tensor radial functions of HF, obtained with the daug-cc-pVQZ and the wavefunction, are presented in Fig. 3. 

Fig. 5. HF dependence of the dipole and quadrupole polarizability tensor (in atomic units) on the vibrational quantum number v. Calculated from the ioo > 9552 polarizability radial function and the potential energy curve.

Very accurate values of the dipole and quadrupole polarizability for the equilibrium internuclear distance of HF can be found in a review article by Maroulis [71], calculated with finite-field Mpller-Plesset perturbation theory at various orders and coupled cluster theory using a carefully selected basis set. 

In Table 7 we compare the ZPVCs for the dipole and quadrupole polarizabilities of HF. In the same way as for LiH, we have calculated the vibrational averages for each method with two different wavefunctions - one obtained from the PEC of the same or related method as used in the calculation of the property curve and the other obtained from the loo CAS PEC. Compared with the equivalent results for LiH we observe significant differences between the calculations on the two molecules. Eirst of all the vibrational corrections are smaller than in LiH but roughly in the same ratio as the polarizabilities. The influence of the PEC is larger than in LiH. 

Values of (r-4) and (r-6) for hydrogenic wavefunctions have been calculated analytically, and the expressions are given in Table 2.3. Examining the forms of Table 2.3, it is apparent that both (r-4) and (r-6) exhibit n-3 scalings, due to the normalization of the radial wavefunction. However, they exhibit different i dependences (r-4) scales as 5, and (r-6) scales as 8. As a result of the very different i scalings the contributions of the dipole and quadrupole polarizabilities to the quantum defect are easily separated by measurements of the intervals between several i series. Furthermore, for high t, (r -4> , and as a result, for high ,... 

Fig. 16.6 Plot of AWIAP vs APQ/AP used to extract Cs+ effective dipole and quadrupole polarizabilities, ad and aq. a d is the y intercept of the line through the data points while...

Table 17.3. Ba+ Dipole and quadrupole polarizabilities, ad and aq, determined from an analysis of measured 6s n intervals and calculated using coulomb wavefunctions.a...

While the agreement of the measured and calculated Ba+ quadrupole polarizabilities is not very good, compared to an analysis based on adiabatic core polarization the agreement in Table 17.3 is superb. The adiabatic core polarization model leads to ad = 146flo and aq = —5800. The ground state of Ba+ cannot have a negative quadrupole polarizability. Taken together, the Ca and Ba experiments show clearly that the nonadiabatic effects in core polarization in the alkaline earth atoms are important and may be calculated with some accuracy. 

Kar and Ho [196] have estimated the oscillator strengths for different transitions, dipole and quadrupole polarizabilities of He for a wide range of the Debye screening parameters using explicitly correlated wavefunctions. Results presented by Kar and Ho [196] are very accurate and may be of substantial use for comparison with those from laser plasma experiments. The behavior of several singly and doubly excited states of He under screened potential was also accurately estimated by Kar and Ho [197] using correlated basis functions. Variation of the transition wavelength as a... 

For oriented samples, the rotation of the plane-polarized light becomes a tensor - that is, the optical rotation becomes directionally dependent - and includes a contribution from the electric dipole-electric quadrupole polarizability tensor, which is traceless and thus vanishes for freely rotating molecules [30], The term arising from these quadrupolar interactions can be expressed as [30]... 

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