Polarizabilities static

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. 

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 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. 

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. 

Idrobo, J.C., Walkosz, W, Yip, S.F., Ogiit, S Wang, J. and Jellinek, J. (2007) Static polarizabilities and optical absorption spectra of gold clusters (Aun, n = 2-14 and 20) from first principles. Physical Review B - Condensed Matter, 76, 205422-1-205422-12. 

Where a, P and y are the linear polarizability, the first- and second-hyper polarizabilities, respectively, and are represented by second, third and fourth rank tensors, respectively, and is a static polarizability. 

Adamo, C., Cossi, M., Scalmani, G., Barone, V., 1999, Accurate Static Polarizabilities by Density Functional Theory Assessment of the PBE0 Model , Chem. Phys. Lett., 307, 265. 

Tozer, D. J., Handy, N. C., 1998, Improving Virtual Kohn-Sham Orbitals and Eigenvalues Apphcation to Excitation Energies and Static Polarizabilities , 7. Chem. Phys., 109, 10180. 

Let us now consider the errors in the CC3 S( — 2) Cauchy moment (the static polarizability). From the monotonic convergence of the CCSD doubly augmented basis-set calculations of S( — 2) in Table 1 it appears that the difference between the 5Z and 6Z results should give a good estimate of the CCSD basis-set error at the 6Z level. CC3/d-aug-pV5Z 5 ( —2) Cauchy moment is 2.670 a.u. Using the... 

In the present variation-perturbation calculations the first order corrections were expanded in 600-term ECG basis defined in equations (15) and (16). The components of the polarizability were computed from equation (11) using the optimized The optimization was performed separately for each component and intemuclear distance. The values of aj, (co) are arithmetic sums of the plus and minus components (equation (12)) computed from two separate first-order corrections. For a given component v (either or ), and are expanded in the same basis but, because they are solutions to two different equations (equation (9)) they differ in the linear expansion coefficients. The computed components of the static polarizability an(/ ) and a R) are drawn in Fig. 2 and their numerical values at selected intemuclear distances are listed in Table 1. 

Table 1.22 lists the values of optical polarizability a and static polarizability calculated by Lasaga and Cygan (1982) for various silicates on the basis of... 

Fig. 2. Basis set dependence of the static polarizability anisotropy (in a.u.) for the eight reference molecules evaluated at the SOS/CIS level of approximation.

Here a0 is the static polarizability where the perturbation is time-independent. For time-dependent fields (e.g. optical fields) the perturbing hamiltonian is H0> = — a(F,f)Fa(0. and the problem is treated in a similar way using time-dependent perturbation theory. 

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