Frequency dependence response

We now proceed to the spectrum, or frequency-dependent response [44, 42]. The power, or rate of energy absorption, is given by... 

Table 2 Convergence of the Taylor series and the series of diagonal Fade approximants (CCSD response, t-aug-cc-pVDZ basis). The inifinite order results were calculated using the implementation for the frequency-dependent response function.

Here, the frequency-dependent response functions y(r, r oj) and y0(r, oj) correspond, respectively, to the actual interacting system and the equivalent Kohn-Sham noninteracting system. Using the expression of the effective potential, one can write... 

The imaginary and real parts of the frequency-dependent response of the electric dipole moment to a magnetic field ((3) are responsible for optical rotation and CD, respectively, while the imaginary and real parts of the first-order correction to the... 

The basis set representing the first order perturbed orbitals should also be chosen such that it satisfies the imposed finite boundary conditions and can be represented by a form like Equation (36) with the STOs having different sets of linear variation parameters and preassigned exponents. The coefficients of the perturbed functions are determined through the optimization of a standard variational functional with respect to, the total wavefunction . The frequency dependent response properties of the systems are analyzed by considering a time-averaged functional [155]... 

The next step is to consider the extra contributions related to the term involving the operator WQM/CM since they give rise to significant modifications of the terms that enter the procedure for solving the time-dependent response equations. Through the calculations of frequency-dependent response functions for the molecular subsystem we are able to investigate the effects of the structured environment on the molecular properties. 

Usually, the frequency-dependent response properties are of interest, so that the frequency-dependent density response Spa(ry,co) to the potential change <5vscr (r2, t>) is considered in TDDFRT. It is obtained with the Fourier transformation of 8pa(ri,t) of (12)... 

The preceding derivatives are evaluated at zero field, and there are several ways of evaluating them that are discussed in Ref. 38. If frequency dependence is to be included, that is, the field is E(o>) = E0 + E sin(wt), then the calculations become somewhat more complex. A summary of methods used to calculate frequency dependent responses is given in Ref. 38. 

The coefficient AHam itself varies with separation /. It takes the form of a sum over all frequencies at which fluctuations can occur wherein each term depends on the frequency-dependent responses of materials A, B, and m to electromagnetic fields. These responses are written in terms of "dielectric" functions sA, b, and em that are extracted from absorption spectra. It is the differences in these dielectric responses that create interactions. To first approximation,... 

When the environment is not stationary, response functions such as x,M, (t, t ) and Xvx(t. t1) depend separately on the two times t and t7 entering into play, and not only on the time difference or observation time z t f. However, the observation time continues to play an essential role in the description. Hence, it has been proposed to define time- and frequency-dependent response functions as Fourier transforms with respect to x of the corresponding two-time quantities [5,6,58]. The time f, which represents the waiting time or the age of the system, then plays the role of a parameter. 

Let us first briefly recall these definitions, and examine under which conditions the age—and frequency-dependent response functions share the analytic properties of the corresponding stationary quantities. 

The expression immediately gives an estimate of the enthalpy of adsorption in taking an atom from the gaseous (vacuum) state to a liquid, or to a composite medium like a zeolite, characterised by its measured dielectric frequency dependent response (o>). It is, exactly as for the electrostatic Bom self-energy in taking an ion from vacuum to water ... 

Norman and Jensen27 have implemented a method for obtaining second order response functions within the four component (relativistic) time-dependent Hartree-Fock scheme. Results are presented for the first order hyperpolarizabilities for second harmonic generation, />(—2o o),o ) for CsAg and CsAu. A comparison of the results with those of non-relativistic calculations implies that the nonrelativistic results are over-estimated by 18% and 66% respectively. In this method transitions that are weakly-allowed relativistically can lead to divergences in the frequency-dependent response, which would be removed if the finite lifetimes of the excited states could be taken into account. 

TD-DFT can also be used to get better approximations to the ground-state exchange-correlation energy based on frequency dependent response functions. For example, the long-range dispersion term between two well-separated systems can be obtained from the frequency dependent susceptibility of the two systems [228,229]. 

Turning back to the embedded systems, we recall that the embedded electron orbitals are derived from Eq. 31 and yield the electron density which minimizes the total energy under the condition that 5p = SpA (i.e. Spg = 0), where 5p is the variation of electron density. Assuming that the frequency-dependent response of the electron density of the whole system can be expressed by means of the embedded... 

The ab initio calculation of NLO properties has been a topic of research for about three decades. In particular, response theory has been used in combination with a number of electronic structure methods to derive so-called response functions [41 8], The latter describe the response of a molecular system for the specific perturbation operators and associated frequencies that characterize a particular experiment. For example, molecular hyperpolarizabilities can be calculated from the quadratic and cubic response functions using electric dipole operators. From the frequency-dependent response functions one can also determine expressions for various transition properties (e.g. for multi-photon absorption processes) and properties of excited states [42]. 

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... 

Because of its computational simplicity and other obvious qualities the random-phase approximation has been used in many calculations. Reviews of RPA calculations include one on chiroptical properties by Hansen and Bouman (1980), one on the equation-of-motion formulation of RPA (McCurdy et al, 1977) and my own review of the literature through 1977 (Oddershede, 1978, Appendix B). Ab initio molecular RPA calculations in the intervening period are reviewed in Table I. Coupled Hartree-Fock calculations have not been included in the table. Only calculations which require diagonalization of both A -I- B and A — B and thus may give frequency-dependent response properties and excitation spectra are included. In CHF we only need to evaluate either (A -I- B) or (A — B) Mn order to determine the (static) response properties. 

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