Frequency-dependent properties

The generality of a simple power series ansatz and an open-ended formulation of the dispersion formulas facilitate an alternative approach to the calculation of dispersion curves for hyperpolarizabilities complementary to the point-wise calculation of the frequency-dependent property. In particular, if dispersion curves are needed over a wide range of frequencies and for several optical proccesses, the calculation of the dispersion coefficients can provide a cost-efficient alternative to repeated calculations for different optical proccesses and different frequencies. The open-ended formulation allows to investigate the convergence of the dispersion expansion and to reduce the truncation error to what is considered tolerable. 

The CCS, CC2, CCSD, CC3 hierarchy has been designed specially for the calculation of frequency-dependent properties. In this hierarchy, a systematic improvement in the description of the dynamic electron correlation is obtained at each level. For example, comparing CCS, CC2, CCSD, CC3 with FCI singlet and triplet excitation energies showed that the errors decreased by about a factor 3 at each level in the coupled cluster hierarchy [18]. The CC3 error was as small as 0.016 eV and the accuracy of the CC3 excitation energies was comparable to the one of the CCSDT model [18]. 

A simple theory of the concentration dependence of viscosity has recently been developed by using the mode coupling theory expression of viscosity [197]. The slow variables chosen are the center of mass density and the charge density. The final expressions have essentially the same form as discussed in Section X the structure factors now involve the intermolecular correlations among the polyelectrolyte rods. Numerical calculation shows that the theory can explain the plateau in the concentration dependence of the viscosity, if one takes into account the anisotropy in the motion of the rod-like polymers. The problem, however, is far from complete. We are also not aware of any study of the frequency-dependent properties. Work on this problem is under progress [198]. 

In practice ultrasound is usually propagated through materials in the form of pulses rather than continuous sinusoidal waves. Pulses contain a spectrum of frequencies, and so if they are used to test materials that have frequency dependent properties the measured velocity and attenuation coefficient will be average values. This problem can be overcome by using Fourier Transform analysis of pulses to determine the frequency dependence of the ultrasonic properties. 

The dielectric properties of tissues and cell suspensions will be summarized for the total frequency range from a few Hz to 20 GHz. Three pronounced relaxation regions at ELF, RF and MW frequencies are due to counterion relaxation and membrane invaginations, to Maxwell-Wagner effects, and to the frequency dependent properties of normal water at microwave frequencies. Superimposed on these major dispersions are fine structure effects caused by cellular organelles, protein bound water, polar tissue proteins, and side chain rotation. 

The ratio of I to V is defined as the admittance Y(cu) of the sample, and is written as frequency-dependent to emphasize its implicit dependence on the frequency-dependent properties of the medium. 

Table 3-12. Static and frequency-dependent properties of pNA in gas phase and 1,4-dioxane solution...

Frequency-dependent properties are in fact usually defined by a time-dependent generalization of this equation. For example, when a periodic homogeneous electric field is applied, then the time development of defines the frequency-dependent properties. In the limit of a time-independent field the same equation then serves as a definition of frequency-independent properties. 

A large number of reports have been published of calculations on medium sized organic molecules, often using ab initio methods (with basis sets roughly of a size inversely proportional to the size of the molecule) and sometimes supplemented by semi-empirical calculations, especially for frequency dependent properties -... 

The present contribution reviews recent advances in the highly accurate calculation of frequency-dependent properties of atoms and small molecules, electronic struc-mre methods, basis set convergence and extrapolation techniques. Reported applications include first and second hyperpolarizabilities, Faraday, Buckingham and Cotton-Mouton effects as well as Jones and magneto-electric birefringence... 

The approach outlined above combines the calculation of response functions (i.e. of frequency-dependent properties) with the theory of analytic derivatives developed for static higher-order properties. In the limit of a static perturbation all equations above reduce to the usual equations for (unrelaxed) coupled cluster energy derivatives. This is an invaluable advantage for the implementation of frequency-dependent properties in quantum chemistry programs. 

For nonlinear (magneto-) optical properties, calculations of an accuracy close to that of modern gas phase experiments require - similar to what has also been found for other properties like structures [79, 109], reaction enthalpies [79, 110, 111], vibrational frequencies [112, 113], NMR chemical shifts [114], etc. - at least an approximate inclusion of connected triple excitations in the wavefunction. This has been known for years now from calculations of static hyperpolarizabilities with the CCSD(T) approximation [9-13]. CCSD(T) accounts rather efficiently for connected triples through a perturbative correction on top of CCSD. For the reasons pointed out in Section 2.1 CCSD(T) is, as a two-step approach, not suitable for the calculation of frequency-dependent properties. Therefore, the CC3 model has been proposed [56, 58] as an alternative to CCSD(T) especially designed for use in connection with response theory. CC3 is an approximation to CCSDT - alike CCSDT-la and related methods - where the triples equations are truncated such that the scaling of the computational efforts with system size is reduced to as for CCSD(T),... 

The situation is somewhat different for the convergence with the wavefunction model, i.e. the treatment of electron correlation. As an anisotropic and nonlinear property the first dipole hyperpolarizability is considerably more sensitive to the correlation treatment than linear dipole polarizabilities. Uncorrelated methods like HF-SCF or CCS yield for /3 results which are for small molecules at most qualitatively correct. Also CC2 is for higher-order properties not accurate enough to allow for detailed quantitative studies. Thus the CCSD model is the lowest level which provides a consistent and accurate treatment of dynamic electron correlation effects for frequency-dependent properties. With the CC3 model which also includes the effects of connected triples the electronic structure problem for j8 seems to be solved with an accuracy that surpasses that of the latest experiments (vide infra). 

Coupled cluster response calculaAons are usually based on the HF-SCF wave-function of the unperturbed system as reference state, i.e. they correspond to so-called orbital-unrelaxed derivatives. In the static limit this becomes equivalent to finite field calculations where Aie perturbation is added to the Hamiltonian after the HF-SCF step, while in the orbital-relaxed approach the perturbation is included already in the HF-SCF calculation. For frequency-dependent properties the orbital-relaxed approach leads to artificial poles in the correlated results whenever one of the involved frequencies becomes equal to an HF-SCF excitation energy. However, in Aie static limit both unrelaxed and relaxed coupled cluster calculations can be used and for boAi approaches the hierarchy CCS (HF-SCF), CC2, CCSD, CC3,... converges in the limit of a complete cluster expansion to the Full CI result. Thus, the question arises, whether for second hyperpolarizabilities one... 

It is apparent that non-linear-optical processes rely on a dynamic or frequency-dependent property. I will therefore, in general, restrict this article to calculations made at this level and, with one or two exceptions, I will consider only ab initio theory. Much work has been done on the static hyper-polarizabilities (as well as the static and dynamic polarizability a) but in order to make a direct connection with experiment, I have chosen to exclude this work. Also, again with one or two exceptions, I will only deal with molecules and exclude atoms. 

Some types of frequency-dependent quantity can be calculated from finite field procedures, provided they are based on frequency-dependent properties obtained by other methods, such as those discussed later in the chapter. For example, a computer program that calculates the frequency-dependent polarizability can be modified to do so in the presence of a static electric field to yield pEOPE. yOKE values via the expansion- ... 

A third procedure for calculating correlated frequency-dependent properties is the coupled-cluster, equations-of-motion (CC-EOM) method. This approach is formally equivalent to solving the sum-over-states expression using a similarity transformed Hamiltonian, H = e He, where the transformation is obtained from the coupled cluster choice for a reference state,... 

Here, P" denotes the nuclear relaxation part of P, and the subscript oo -> oo invokes the infinite optical frequency approximation, which is generally considered to be good approximation to the frequency-dependent properties at the usual laser frequencies applied in experiments [75]. Due to the high computational cost of these calculations, in general the rather small 6-3IG basis was used. For some control calculations, the 6-31-fG basis set was used. 

Soscun et al. report ab initio and DFT studies of the static and frequency-dependent a and y tensors for ethyne. The coupled perturbed Hartree-Fock (CPHF) method has been used for the static calculations and TDHF for the frequency-dependent properties, which have been calculated at 633 nm. New basis sets have been proposed. 

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