Static response

In previous chapters it was sometimes useful to use different notations for an observable and the corresponding dynamical variable A r, p ). In this chapterwe will not make this distinction because it makes the presentation somewhat cumbersome. The difference between these entities should be clear from the text. 

Consider first the response to a static perturbation, that is, we take F = constant in Eq. (11.2). In this case we are not dealing with a nonequilibrium situation, only comparing two equilibrium cases. In this case we need to evaluate A B = B — B q where 

In (11.8) Xba is the response coefficient relating a static response in (5) to a static perturbation associated with a field F which couples to the system through an additive term H = —FA in the Hamiltonian. Consider next the dynamical experiment in which the system reached equilibrium with Hq - - H and then the field suddenly switched off. How does A(S), the induced deviation of B from its original equilibrium value (B)o, relax to zero The essential point in the following derivation is that the time evolution is carried out under the Hamiltonian Hq (after the field has 

Starting from Eq. (11.9) we again expand the exponential operators. Once exp(—/S(7/o + Hi)) is replaced by exp(—/S//o)(l — yNe get a form in which the time evolution and the averaging are done with the same Hamiltonian Hq. We encounter terms such as 

On the left side of (11.15) we have the time evolution of a prepared deviation from equilibrium of the dynamical variable B. On the right side we have a time correlation function of spontaneous equilibrium fluctuations involving the dynamical variables A, which defined the perturbation, and B. The fact that the two time evolutions are the same has been known as the Onsager regression hypothesis. (The hypothesis was made before the formal proof above was known.) 

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We only consider static response properties in this chapter, which arise from fixed external field. Their dynamic counterparts describe the response to an oscillating electric field of electromagnetic radiation and are of great importance in the context of non-linear optics. As an entry point to the treatment of frequency-dependent electric response properties in the domain of time-dependent DFT we recommend the studies by van Gisbergen, Snijders, and Baerends, 1998a and 1998b. 

For single arms neither the minimum in the dynamics nor the maximum in the static response are observed... 

For the star shell the situation is inversed compared with the core (maximum in the dynamic and minimum in the static response)... 

Shock Response Versus Quasi-Static Response for Internal Blast. We noted earlier that internal detonations of high explosives within structures caused both initial and reflected shock loadings, plus longer term gas pressure loads called quasi-static pressures. Figure 11 is a reproduction of a pressure trace showing both phases of the loading. 

A strength increase is also produced at ultimate strength (F ) for steels however, the ratio f dynamic to static strength is less than at yield. A typical stress-strain curve describing dynamic and static response of steel is shown in Figure 5.5. Elongation at failure is relatively unaffected by the dynamic response of the material. 

Ultimate strength for concrete is greater under dynamic loads. Though the modulus of elasticity is also greater, this difference is small and is usually ignored. Figure 5.6 describes the relationship between dynamic and static response for... 

In terms of the linear response theory the static scattering function (Q) relates to the static response function x(Q) by ... 

The static responses discussed above are determined for the complete equilibrium. For this reason, they are the easiest to calculate but not at all easy to observe. Indeed, at low temperatures the time needed to achieve the interwell equilibrium becomes exponentially large [55],... 

S. Yu. Savrasov, "A linearized direct approach for calculating the static response in solids," Solid State Commun. 74 (1990), 69-72. 

In the literature the response theory is largely described in the time-dependent form which requires a somewhat complicated technique of time ordering of the operators and Fourier transformations between time and frequency domains. The static responses which are largely needed in the present book appear as a result of subtle limit procedures for the frequencies flowing to zero. Here we have developed the necessary static results within their own realm. 

The softness kernels are relevant to the remaining cases of two or more interacting systems. However, they do not by themselves provide sufficient information to constitute a basis for a theory of chemical reactivity. Clearly, the chemical stimulus to one molecule in a bimolecular reaction is provided by the other. That being the case, an eighth issue arises. Both the perturbing system and the responding system have internal dynamics, yet the softness kernel is a static response function. Dynamic reactivities need to be defined. 

The static response functions utilised in Appendix C are then obtained by taking the zero-frequency limit,... 

Note, that (C.l) could be used as alternative definition for the static response functions. 

In Eq. (23), x(r) and Xo O ) are the static response function of the homogeneous liquid and the response function of the noninteracting electrons (namely the Lindhard function [47]). 

Nunes and Gonze [153] have recently extended DFPT to static responses of insulating ciystals for any order of perturbation theory by combining the variation perturbation approach with the modern theory of polarization [154]. There are evident similarities between this formalism and (a) the developments of Sipe and collaborators [117,121,123] within the independent particle approximation and (b) the recent work of Bishop, Gu and Kirtman [24, 155,156] at the time-dependent Hartree Fock level for one-dimensional periodic systems. 

Whereas the distinction between collective and cooperative effects can appear artificial, it is obvious that, since optical responses are gs properties, their nonadditivity cannot be ascribed to the delocalized nature of excited states. On the other hand, static responses can be calculated from sum-over-state (SOS) expressions involving excited state energies and transition dipole moments [35]. And in fact tlie exciton model has been recently used by several authors to calculate and/or discuss linear and non-linear optical responses of mm [36, 37, 38, 39, 40, 41, 42]. But tlie excitonic model hardly accounts for cooperativity and one may ask if there is any link between collective effects related to the delocalized nature of exciton states and cooperative effects in the gs, related to the self-consistent dependence of tlie local molecular gs on the surrounding molecules. 

Marcus theory is that the actual change in the electronic charge distribution of the ET system is fast relative to the nuclear motions underlying the static response, but is slow relative to the electronic motions which determine s. In other words the electron transfer occurs at constant nuclear polarization, or at fixed nuclear positions. This is an expression of the Franck-Condon principle in this continuum dielectric theoiy of electronic transitions. 

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. 

From our analysis we can further derive another property of the static response function. Since (r, r/) is a real Hermitian integral kernel it has an orthonormal set of eigenfunctions which can be chosen to be real. Let/(r) be such an eigenfunction, i.e.,... 

We therefore find that A < 0. However, we already know that A = 0 is only possible if/(r) is constant. We have therefore obtained the result that if a nonzero density variation is proportional to the potential that generates it, i.e., 8n = A 8v, then the constant of proportionality A is negative. This is exactly what one would expect on the basis of physical considerations. In actual calculations one indeed finds that the eigenvalues of x are negative. Moreover, it is found that there is a infinite number of negative eigenvalues arbitrarily close to zero, which causes considerable numerical difficulties when one tries to obtain the potential variation that is responsible for a given density variation. We finally note the invertability proof for the static response function can be extended to the time-dependent case. For a recent review we refer to Ref. [15]. 

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