Frequency-dependent polarizability, linear

The last method used in this study is CCSD linear response theory [37]. The frequency-dependent polarizabilities are again identified from the time evolution of the corresponding moments. However, in CCSD response theory the moments are calculated as transition expectation values between the coupled cluster state l cc(O) and a dual state... 

The second approach to calculating MCD starts from its definition in terms of the real part of first-order correction to the frequency-dependent polarizability in the presence of a magnetic field (Section II.A.6). This definition can be used to consider all types of MCD linear in the magnetic field (9). Our current implementation is restricted to systems with a closed-shell ground state. We shall therefore only consider the calculation of A and terms by this method. 

Self-Consistent Field Linear Response Theory and Application to Ceo. Excitation Energies, Oscillator Strengths and Frequency-Dependent Polarizabilities. 

Eshuis et al have implemented fully propagated time-dependent Hartree-Fock theory to calculate the real time electronic dynamics of closed- and open-shell molecules in strong oscillating electric fields. This method has been illustrated on the determination of the frequency-dependent polarizability of ethylene and is shown to converge, in the weak field limit, to the same results as the linearized TDHF method. 

Comparing this with the classical expansion of a time-dependent dipole moment in Eq. (7.18) we can identify the frequency-dependent polarizability tensor as a linear response function or polarization propagator... 

In the previous sections it was shown that frequency-dependent linear response prop>-erties, such as frequency-dependent polarizabilities, can be obtained as the value of the polarization propagator for the appropriate operators. Furthermore, all static second-order properties discussed in Chapters 4 and 5 can be calculated as the value of a polarization propagator for zero frequency. 

The second approach is used by Baerends and co-workers. They use linear response theory, but instead of calculating the full linear response function they use the response function of the noninteracting Kohn-Sham system together with an effective potential. This response function can be calculated from the Kohn-Sham orbitals and energies and the occupation numbers. They use the adiabatic local density approximation (ALDA), and so their exchange correlation kernel, /xc (which is the functional derivative of the exchange correlation potential, Vxc, with respect to the time-dependent density) is local in space and in time. They report frequency dependent polarizabilities for rare gas atoms, and static polarizabilities for molecules. 

The third approach is that used by Salahub and co-workers. They initially used DFT RPA but recendy have reported an implementation of time-dependent density functional response theory (TD DFRT). Their Kohn-Sham linear response function involves a coupling matrix, K, which in the RPA case contains only the response to coulomb terms, but in their present implementation contains exchange and correlation response terms. Their K is time independent as they work within the adiabatic approximation. They calculate the frequency dependent polarizability from a sum-over-states (SOS) formula, and hence have to calculate the excitation spectrum. 

In the last chapter, starting from the time-dependent Sdirodinger equation (which is the equation of motion for the wavefunction), a number of new ideas were introduced. In particular, the linear response of a system to a time-dependent perturbation was characterized by (i) a time-correlation function K (BA t) and (ii) a frequency-dependent polarizability (FDP), i7(BA to), each being a Fourier transform of the other. 

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. 

Saue, T. and Jensen, H.J.Aa. (2003) Linear response at the 4-component relativistic level Application to the frequency-dependent dipole polarizabilities of the coinage metal dimers. Journal of Chemical Physics, 118, 522-536. 

Thus, just as linear polarizabilities are frequency dependent, so are the nonlinear polarizabilities. Perhaps it is not surprising that most organic materials, with almost exclusively electronic nonlinear optical polarization, have similar efficiencies for SHG and the LEO effect. In contrast, inorganic materials, such as lithium niobate, in which there is a substantial vibrational component to the nonlinear polarization, are substantially more efficient for the LEO effect than for SHG. 

R. Cammi and J. Tomasi, Nonequilibrium solvation theory for the polarizable continuum model - a new formulation at the SCF level with application to the case of the frequency-dependent linear electric-response function, Int. J. Quantum Chem., (1995) 465-74 B. Mennucci, R. Cammi and J. Tomasi, Excited states and solvatochromic shifts within a nonequilibrium solvation approach A new formulation of the integral equation formalism method at the self-consistent field, configuration interaction, and multiconfiguration self-consistent field level, J. Chem. Phys., 109 (1998) 2798-807 R. Cammi, L. Frediani, B. Mennucci, J. Tomasi, K. Ruud and K. V. Mikkelsen, A second-order, quadratically... 

A surface is illuminated with a high-intensity laser, and photons are generated at the second-harmonic frequency through non-linear optical process. For many materials only the surface region has the appropriate symmetry to produce a SHG signal. The non-linear polarizability tensor depends on the nature and geometry of adsorbed atoms and molecules. 

The frequency dependence of the polarizability governs the change of refractive index with the frequency of the light source. In Eq. (11), F becomes Fa and we can write a either as a(o)) or a(-03o Oi) with 0) = 0) and 0)1 = ox The theory for a(0)) was developed early on and there have been many calculations this is the linear effect. Computation of y(co), the nonlinear property, is a fairly recent departure and it is spurred on by the potential that non-linear processes have for commercial exploitation. 

The dipole polarizability is related to the frequency-dependent linear density response i(r, co) via... 

This polarizability involves a set of characteristic times jlkT, Id(, illoil, and (IcIIodI), between any two of which the correlation function may take a fairly simple form. However, this example indicates that for a linear system which falls to show normal mode behaviour the frequency-dependent admittance may be much more straightforward than the corresponding correlation function. 

Working to similar levels of accuracy, Pawlowski et al have calculated the static and frequency-dependent linear polarizability and second hyperpolarizability of the Ne atom using coupled-cluster methods with first order relativistic corrections. Good agreement with recent experimental results is achieved. Klopper et al.s have applied an implementation of the Dalton code that enables... 

The application of the Lorentz-Lorenz equation gives a convincing demonstration of the general similarity of the linear response in gas and liquid but its application in the liquid introduces an approximation which has not yet been quantified. A more precise objective for the theory would be to calculate the frequency dependent susceptibility or refractive index directly. For a continuum model this may lead to a polarizability rigorously defined through the Lorentz-Lorenz equation as shown in treatments of the Ewald-Oseen theorem (see, for example Born and Wolf, plOO),59 but the polarizability defined in this way need not refer to one molecule and would not be precisely related to the gas parameters. 

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). 

For a long time the finite oligomer approach was the only method available for determining linear and nonlinear polarizabilities of infinite stereoregular polymers. Recently, however, the problem of carrying out electronic band structure (or crystal orbital) calculations in the presence of static or frequency-dependent electric fields has been solved [115, 116]. A related discretized Berry phase treatment of static electric field polarization has also been developed for 3D solid state systems... 

We saw in Section III that the polarization propagator is the linear response function. The linear response of a system to an external time-independent perturbation can also be obtained from the coupled Hartree-Fock (CHF) approximation provided the unperturbed state is the Hartree-Fock state of the system. Thus, RPA and CHF are the same approximation for time-independent perturbing fields, that is for properties such as spin-spin coupling constants and static polarizabilities. That we indeed obtain exactly the same set of equations in the two methods is demonstrated by Jorgensen and Simons (1981, Chapter 5.B). Frequency-dependent response properties in the... 

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