Imaginary response functions

In the limit of small pressure perturbations, any kinetic equation modeling the response of a catalyst surface can be reduced to first order. Following Yasuda s derivation C, the system can be described by a set of functions which describe the dependence of pressure, coverage amplitude, and phase on T, P, and frequency. After a mass balance, the equations can be separated Into real and Imaginary terms to yield a real response function, RRF, and an Imaginary response function, IRF ... 

Figure 3. Keal and imaginary response functions for H2 on non-SMSI Rh/Ti02 at 0.4 Torr and 158 C.

Figure 4, Imaginary response functions for H2 on SMSI Rh/T102 at 0.34 Torr and 158 C, and at 0.4 Tort and 49 C.

For each EA spectrum, the transmission T was measured with the mechanical chopper in place and the electric field off. The differential transmission AT was subsequently measured without the chopper, with the electric field on, and with the lock-in amplifier set to detect signals at twice the electric-field modulation frequency. The 2/ dependency of the EA signal is due to the quadratic nature of EA in materials with definite parity. AT was then normalized to AT/T, which was free of the spectral response function. To a good approximation [18], the EA signal is related to the imaginary part of the optical third-order susceptibility ... 

From the above discussion it becomes clear that in order to eliminate the spin-orbit interaction in four-component relativistic calculations of magnetic properties one must delete the quaternion imaginary parts from the regular Fock matrix and not from other quantities appearing in the response function (35). It is also possible to delete all spin interactions from magnetic properties, but this requires the use of the Sternheim approximation [57,73], that is calculating the diamagnetic contribution as an expectation value. 

Static charge-density susceptibilities have been computed ab initio by Li et al (38). The frequency-dependent susceptibility x(r, r cd) can be calculated within density functional theory, using methods developed by Ando (39 Zang-will and Soven (40 Gross and Kohn (4I and van Gisbergen, Snijders, and Baerends (42). In ab initio work, x(r, r co) can be determined by use of time-dependent perturbation techniques, pseudo-state methods (43-49), quantum Monte Carlo calculations (50-52), or by explicit construction of the linear response function in coupled cluster theory (53). Then the imaginary-frequency susceptibility can be obtained by analytic continuation from the susceptibility at real frequencies, or by a direct replacement co ico, where possible (for example, in pseudo-state expressions). 

Adiabatic-connection fluctuation-dissipation theorem allows one to express the exchange-correlation energy-functional by means of imaginary-frequency density response function ( A) of the system with the scaled Coulomb potential (A/ r — r )11,13 ... 

The spectral response function x(ffl), a function of the angular frequency co, is the sum of an in-phase real part Xi(even function of co, and an out-of-phase (dissipative) imaginary part 2(ft)), which is an odd function... 

The time-gated heterodyne signal obtained in a phase-locked measurement gives the real and the imaginary parts of the response function ... 

When written with the help of the Tl matrix as in (19), from (20) the OR parameter and other linear response properties are seen to afford singularities where co = coj, just like in the SOS equation (2). Therefore, at and near resonances the solutions of the TDDFT response equations (and response equations derived for other quantum chemical methods) yield diverging results that cannot be compared directly to experimental data. In reality, the excited states are broadened, which may be incorporated in the formalism by introducing dephasing constants 1 such that o, —> ooj — iT j for the excitation frequencies. This would lead to a nonsingular behavior of (20) near the coj where the real and the imaginary part of the response function varies smoothly, as in the broadened scenario at the top of Fig. 1. 

Another way of probing dynamic xc effects experimentally is by inelastic X-ray scattering from bulk metals [201-203]. In this way, the so-called dynamical structure factor S q, to) can be measured which is proportional to the imaginary part of the full response function in reciprocal space. With this information at hand and with a first-principles calculation of the non-interacting response function, the connection (159) between and the response functions can be used to deduce information about / [204]. 

Figure 2 The imaginary part of frequency response function, Im (Au ). obtained by the OHD-OKE (dots) and the light scattering measurements by the double monochromator (dashed line). In the insert is shown Im.R(Au ) obtained by the tandem interferomator (solid line). A small peaic appearing at 3.3 cm (shown as a symbol of ) is the ghost due to the tandem interferometer.

Consider for example the paramagnons of an anisotropic 2D Fermi liquid as described by the one-electron spectrum (2) with = 0 the imaginary part of the retarded spin response function at low frequency is found to be... 

Equation (11) allows to interpret the electronic excitation in the Auger process as the medium response to a dynamic fluctuation of charge between the states i(r) and

linear response theory to an external pertirrbation which is the electrostatic potential v i(r) created by such charge fluctuation. In general terms, the response function r, m) completely determines the behavior of the system in response to an external perturbation, provided that the latter is sufficiently small and thus linear theory is applicable. The imaginary part of contains... 

Fig. 6.2 (a) Instantaneous normal modes in room temperature water as obtained from molecular dynamics simulations. The negative frequency axis is used to show the density of imaginary frequencies. (b) The solvation response function (see Chapter 15) for electron solvation in water, calculated from direct classical MD simulations (full line), from the instantaneous normal mode representation of water (dash-dotted line), and from a similar instantaneous normal mode representation in which the imaginary frequency modes were excluded (dashed line). The inset in Fig. 6.2 shows the short time behavior of the same data. (From C.-Y. Yang, K. F. Wong, M. S. Skaf, and P. J. Rossky, J. Chem. Phys. 114, 3598 (2001).)... 

The transformation defined by (11.56) is called Hilbert transform, and we have found that the real and imaginary parts of a function that is analytic in half of the complex plane and vanishes at infinity on that plane are Hilbert transforms of each other. Thus, causality, by which response functions have such analytical properties, also implies this relation. On the practical level this tells us that if we know the real (or imaginary) part of a response function we can find its imaginary (or real) part by using this transform. 

In Figure 2.13, an example is given of the basic simplex procedure. Consider the imaginary response surface of a method, representing the response as a function of two factors (xi and X2) and shown as contour plot (dotted lines). Suppose the highest response value is considered to be the optimum. 

In section 5.2 the concepts of hermiticity and time-reversal where introduced in the discussion of the first order response of the wave function. In this section we shall see that these concepts allows us to determine whether the linear response function is real or imaginary. The linear response function is given by (51), but using the first-order response equation (49) it may be simplified to (54). This may reduce the precision in the numerical evaluation, but is of no consequence for the following arguments. In the notation of section 5.2 the linear response function at the closed-shell HF level of theory is accordingly written... 

The quadratic response function is imaginary. In the static case to = 0 the quadratic response function is zero, as there are then no anti-Hermitian contribution to the first order response vectors. 

---