Response functions perturbation theory formulation

There are several possible ways of deriving the equations for TDDFT. The most natural way departs from density-functional perturbation theory as outlined above. Initially it is assumed that an external perturbation is applied, which oscillates at a frequency co. The linear response of the system is then computed, which will be oscillating with the same imposed frequency co. In contrast with the standard static formulation of DFPT, there will be special frequencies cov for which the solutions of the perturbation theory equations will persist even when the external field vanishes. These particular solutions for orbitals and frequencies describe excited electronic states and energies with very good accuracy. 

Generalized perturbation theory for two special cases of composite functionals are presented and discussed in some detail GPT for reactivity (Section V,B), and GPT for a detector response in inhomogeneous systems (Section V,E). The GPT formulation for reactivity is equivalent to a high-order perturbation theory, in the sense that it allows for the flux perturbation, GPT for a detector response in inhomogeneous systems 42, 43) is, in fact, the second-order perturbation theory known from other derivations I, 44, 45). These perturbation theory formulations provide the basis for new methods for solution of deep-penetration problems. These methods are reviewed in Section V,E,2. 

A lower max response at resonance was noted for poly butadiene-acrylic acid-containing pro-pints compared with polyurethane-containing opaque proplnts. Comparison of the measured response functions with predictions of theoretical models, which were modified to consider radiant-heat flux effects for translucent proplnts rather than pressure perturbations, suggest general agreement between theory and expt. The technique is suggested for study of the effects of proplnt-formulation variations on solid-proplnt combustion dynamics... 

In the earlier sections of this chapter we reviewed the many-electron formulation of the symmetry-adapted perturbation theory of two-body interactions. As we saw, all physically important contributions to the potential could be identified and computed separately. We follow the same program for the three-body forces and discuss a triple perturbation theory for interactions in trimers. We show how the pure three-body effects can be separated out and give working equations for the components in terms of molecular integrals and linear and quadratic response functions. These formulas have a clear, partly classical, partly quantum mechanical interpretation. The exchange terms are also classified for the explicit orbital formulas we refer to Ref. (302). 

Taking these introductory comments as a motivation, we shall turn to the formalism of response theory. Response theory is first of all a way of formulating time-dependent perturbation theory. In fact, time-dependent and time-independent perturbation theory are treated on equal footing, the latter being a special case of the former. As the name implies, response functions describe how a property of a system responds to an external perturbation. If initially, we have a system in the state 0) (the reference state), as a weak perturbation V(t) is turned on, the average value of an operator A will develop in time according to... 

Analytic response theory, which represents a particular formulation of time-dependent perturbation theory, has constituted a core technology in much of the this development. Response functions provide a universal representation of the response of a system to perturbations, and are applicable to all computational models, density-functional as well as wave-function models, and to all kinds of perturbations, dynamic as well as static, internal as well as external perturbations. The analytical character of the theory with properties evaluated from analytically derived expressions at finite frequencies, makes it applicable for a large range of experimental conditions. The theory is also model transferable in that, once the computational model has been defined, all properties are obtained on an equal footing, without further approximations. 

We discussed some aspects of the responses of chemical systems, linear or nonlinear, to perturbations on several earlier occasions. The first was the responses of the chemical species in a reaction mechanism (a network) in a nonequilibrium stable stationary state to a pulse in concentration of one species. We referred to this approach as the pulse method (see chapter 5 for theory and chapter 6 for experiments). Second, we studied the time series of the responses of concentrations to repeated random perturbations, the formulation of correlation functions from such measurements, and the construction of the correlation metric (see chapter 7 for theory and chapter 8 for experiments). Third, in the investigation of oscillatory chemical reactions we showed that the responses of a chemical system in a stable stationary state close to a Hopf bifurcation are related to the category of the oscillatory reaction and to the role of the essential species in the system (see chapter 11 for theory and experiments). In each of these cases the responses yield important information about the reaction pathway and the reaction mechanism. 

Abstract This chapter presents the general aspects of the response theory for molecular solutes in the presence of time-dependent perturbing fields (i) the nonequilibrium solvation, (ii) the variational formulation of the time-dependent nonlinear QM problem, and (iii) the connection of the molecular response functions with their macroscopic counterparts. The linear and quadratic molecular response functions are described at the coupled-cluster level. 

The formulation of approximate response theory based on an exponential parame-trization of the time-dependent wave function, Eq. (11.36), and the Ehrenfest theorem, Eq. (11.40), can also be used to derive SOPPA and higher-order Mpller-Plesset perturbation theory polarization propagator approximations (Olsen et al., 2005). Contrary to the approach employed in Chapter 10, which is based on the superoperator formalism from Section 3.12 and that could not yet be extended to higher than linear response functions, the Ehrenfest-theorem-based approach can be used to derive expressions also for quadratic and higher-order response functions. In the following, it will briefly be shown how the SOPPA linear response equations, Eq. (10.29), can be derived with this approach. 

Response theory deviates in one crucial aspect from the formulation of time-dependent perturbation theory in most textbooks on quantum mechanics the response parameters are not explicitly expressed in terms of the excited states. As a consequence, knowledge of the excited state wave functions is not needed in response theory. Instead, we must solve the response equations (O Eq. 5.4) for each set of perturbation operators V (tu). This is a tremendous computational advantage as there are significantly fewer pertiubation operators, and hence response equations to solve, than excited states in virtually all cases of practical interest. 

Time-dependent density-functional response theory. An alternative approach to real-time TDDFT as described above is the application of linear-response theory. If the perturbation to the system in its ground-state—in our case, e.g., the exposure to a time-dependent electric field— is only small, the system will response linearly. The formulation of the resulting time-dependent density-functional response theory (TD-DFRT) has been given by Casida. " ... 

So far, the possibility of optimizing the orbitals in the presence of a perturbation (i.e. of making self-consistent property calculations) has been considered only at the Hartree-Fock level. In many cases, however, it is necessary to use a many-determinant wavefunction, either because the IPM ground state is degenerate or because electron-correlation effects are too important to be ignored and it is then desirable to optimize both Cl coefficients and orbitals as in MC SCF theory (Section 8.6). To formulate the perturbation equations, both coefficients and orbitals will be expanded in terms of a perturbation parameter and the orders will be separated the zeroth-order equations will be the MC SCF equations in the absence of the perturbation, while the first-order equations will determine the (optimized) response of the wavefunction, and will thus permit the calculation of second-order properties. Important progress had been made in this area (Jaszunski, 1978 Daborn and Handy, 1983), for particular types of perturbation and Cl function. In fact, however, the equations in their most general form have been known for many years (Moccia, 1974), and are implicit in the stationary-value... 

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