Response, linear

Equation (5.1) described the vibrational response of a single particle to an applied forceF(t). In a (crystalline) system of many mobile particles (ensemble), the problem is analogous but the question now is how the whole system responds to an external force or perturbation Let us define the system s state (a) as a particular configuration of its particles and the probability of this state as pa. In a thermodynamic system, transitions from an a to a p configuration occur as thermally activated events. If the transition frequency a- /5 is copa and depends only on a and / (Markovian), the time evolution of the system is given by a master equation which links atomic and macroscopic parameters (dynamics and kinetics) 

The propagator Hpa t) is the (conditional) probability that the system will be found in state / at time t, given that it was in a at t = 0. 

If the (equilibrium) system (upper index °) is disturbed by an externally applied field E, we then assume that the (first order) changes of the system s thermodynamic (p) and kinetic (co) parameters are given by 

Let us assume (in accordance with transition state theory, see Section. 5.1.2) that in a linearized version 

Let us illustrate the simplest response approach by an example representing the many-particle system counterpart of Eqn. (5.1). Let F(t) stem from an (periodic) electric field E(t) acting upon an electric charge. The response of a dielectric with permittivity to the field E is the displacement 

We now discuss the linear response of the spin-unpolarized uniform electron gas to a weak, static, external potential dn(r). This is a well-studied problem [59], and a practical one for the local-pseudopotential description of a simple metal [60]. 

Because the unperturbed system is homogeneous, we find that, to first order in t (r), the electron density response is 

In other words, the density response function of the interacting uniform electron gas is 

These results are particularly simple in the long-wavelength q — 0) limit, in which 7xc( ) tends to a constant and 

s q) is a kind of dielectric function, but it is not the standard dielectric function e(g) which predicts the response of the electrostatic potential alone  

The response of a solid to a perturbation AH can be described straightforwardly by means of the Dyson equation. Restricting to a linear response, the Green function GB of the distorted system is given by 

Assuming that the perturbation AH stems from a coupling of an external magnetic field Bext = Bext z to the spin of the electrons we may write for the induced spin magnetization, 

Here S is the so-called Stoner enhancement factor, usually written as 

Calculating xw within the framework of plain spin density functional theory (SDFT), there is no modification of the electronic potential due to the induced orbital magnetization. Working instead within the more appropriate current density functional theory, however, there would be a correction to the exchange correlation potential just as in the case of the spin susceptibility giving rise to a Stoner-like enhancement. Alternatively, this effect can be accounted for by adopting Brooks s orbital polarization formalism (Brooks 1985). 

Within a nonrelativistic formalism the spin and orbital degrees of freedom are completely decoupled this means that the cross terms (azGlzG) and (lzGazG) vanish. However, Yasui and Shimizu (1985) pointed out that these cross terms lead to corresponding contributions xso and xos respectively, to the susceptibility if the spin-orbit coupling is accounted for. 

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Note the presence of the ra prefactor in the absorption spectrum, as in equation (Al.6.87) again its origm is essentially the faster rate of the change of the phase of higher frequency light, which in turn is related to a higher rate of energy absorption. The equivalence between the other factors in equation (Al.6.110) and equation (Al.6.87) under linear response will now be established. 

Linear response theory is an example of a microscopic approach to the foundations of non-equilibrium thennodynamics. It requires knowledge of tire Hamiltonian for the underlying microscopic description. In principle, it produces explicit fomuilae for the relaxation parameters that make up the Onsager coefficients. In reality, these expressions are extremely difficult to evaluate and approximation methods are necessary. Nevertheless, they provide a deeper insight into the physics. 

The linear response of a system is detemiined by the lowest order effect of a perturbation on a dynamical system. Fomially, this effect can be computed either classically or quantum mechanically in essentially the same way. The connection is made by converting quantum mechanical conmuitators into classical Poisson brackets, or vice versa. Suppose tliat the system is described by Hamiltonian where denotes an... 

The usual context for linear response theory is that the system is prepared in the infinite past, —> -x, to be in equilibrium witii Hamiltonian H and then is turned on. This means that pit ) is given by the canonical density matrix... 

Radiation probes such as neutrons, x-rays and visible light are used to see the structure of physical systems tlirough elastic scattering experunents. Inelastic scattering experiments measure both the structural and dynamical correlations that exist in a physical system. For a system which is in thennodynamic equilibrium, the molecular dynamics create spatio-temporal correlations which are the manifestation of themial fluctuations around the equilibrium state. For a condensed phase system, dynamical correlations are intimately linked to its structure. For systems in equilibrium, linear response tiieory is an appropriate framework to use to inquire on the spatio-temporal correlations resulting from thennodynamic fluctuations. Appropriate response and correlation functions emerge naturally in this framework, and the role of theory is to understand these correlation fiinctions from first principles. This is the subject of section A3.3.2. 

The GLE can be derived by invoking the linear response approximation for the response of the solvent modes coupled to the motion of the reaction coordinate. 

Up to this point, we have calculated the linear response of the medium, a polarization oscillating at the frequency m of the applied field. This polarization produces its own radiation field that interferes with the applied optical field. Two familiar effects result a change in tlie speed of the light wave and its attenuation as it propagates. These properties may be related directly to the linear susceptibility The index of... 

The simplifying approximation of a linear response is now made, by which it is assumed that rotations about different axes may be decoupled. This is only strictly valid for small rotations, but is surprisingly good for larger rotations too. This means that Mo(r)constant. Accordingly, at the end of the pulse the... 

Using the fluctuation-dissipation theorem [361, which relates microscopic fluctuations at equilibrium to macroscopic behaviour in the limit of linear responses, the time-dependent shear modulus can be evaluated [371 ... 

Equations (C3.4.5) and (C3.4.6) cover the common case when all molecules are initially in their ground electronic state and able to accept excitation. The system is also assumed to be impinged upon by sources F. The latter are usually expressible as tlie product crfjo, where cr is an absorjition cross section, is tlie photon flux and ftois tlie population in tlie ground state. The common assumption is tliat Jo= q, i.e. practically all molecules are in tlie ground state because n n. This is tlie assumption of linear excitation, where tlie system exhibits a linear response to tlie excitation intensity. This assumption does not hold when tlie extent of excitation is significant, i.e. 

Carlson, H. A., Jorgensen, W. L. An extended linear response method for determining free energies of hydration. J. Phys. Chem. 99 (1995) 10667-10673... 

Appendix 11.3 Expansion of Zwanzig Expression for the Free Energy Difference for the Linear Response Method... 

Carlson H A and W L Jorgensen 1995. An Extended Linear Response Method for Determining Free Energies of Hydration. Journal of Physical Chemistry 99 10667-10673. 

Jones-Hertzog D K and W L Jorgensen 1997. Binding Affinities for Sulphonamide Inhibitors witl Human Thrombin Using Monte Carlo Simulations with a Linear Response Method. Journal o Medicinal Chemistry 40 1539-1549. 

Directions for preparing a potentiometric biosensor for penicillin are provided in this experiment. The enzyme penicillinase is immobilized in a polyacrylamide polymer formed on the surface of a glass pH electrode. The electrode shows a linear response to penicillin G over a concentration range of 10 M to 10 M. 

This experiment describes the preparation and evaluation of two liquid-membrane Na+ ion-selective electrodes, using either the sodium salt of monensin or a hemisodium ionophore as ion exchangers incorporated into a PVG matrix. Electrodes prepared using monensin performed poorly, but those prepared using hemisodium showed a linear response over a range of 0.1 M to 3 X 10 M Na+ with slopes close to the theoretical value. 

Construct a calibration curve for the electrode, and report (a) the range of concentrations in which a linear response is observed, (b) the equation for the calibration curve in this range, and (c) the concentration of penicillin in a sample that yields a potential of 142 mV. 

When an analyte is too concentrated, it is easy to overload the column, thereby seriously degrading the separation. In addition, the analyte may be present at a concentration level that exceeds the detector s linear response. Dissolving the sample in a volatile solvent, such as methylene chloride, makes its analysis feasible. 

The final part of a gas chromatograph is the detector. The ideal detector has several desirable features, including low detection limits, a linear response over a wide range of solute concentrations (which makes quantitative work easier), responsiveness to all solutes or selectivity for a specific class of solutes, and an insensitivity to changes in flow rate or temperature. 

Figure 7.9 shows a schematic representation of this effect, in which the ratio of the two isotopes changes with time. To obtain an accurate estimate of the ratio of ion abundances, it is better if the relative ion yields decrease linearly (Figure 7.9) which can be achieved by adjusting the filament temperature continuously to obtain the desired linear response. An almost constant response for the isotope ratio can be obtained by slow evaporation of the sample, viz., by keeping the filament temperature as low as is consistent with sufficient sensitivity of detection (Figure 7.9). 

In Vivo Biosensing. In vivo biosensing involves the use of a sensitive probe to make chemical and physical measurements in living, functioning systems (60—62). Thus it is no longer necessary to decapitate an animal in order to study its brain. Rather, an electrochemical biosensor is employed to monitor interceUular or intraceUular events. These probes must be small, fast, sensitive, selective, stable, mgged, and have a linear response. 

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