Linear response general formalism

Time-dependent response theory concerns the response of a system initially in a stationary state, generally taken to be the ground state, to a perturbation turned on slowly, beginning some time in the distant past. The assumption that the perturbation is turned on slowly, i.e. the adiabatic approximation, enables us to consider the perturbation to be of first order. In TD-DFT the density response dp, i.e. the density change which results from the perturbation dveff, enables direct determination of the excitation energies as the poles of the response function dP (the linear response of the KS density matrix in the basis of the unperturbed molecular orbitals) without formally having to calculate a(co). 

In what follows, we present in this short review, the basic formalism of TDDFT of many-electron systems (1) for periodic TD scalar potentials, and also (2) for arbitrary TD electric and magnetic fields in a generalized manner. Practical schemes within the framework of quantum hydrodynamical approach as well as the orbital-based TD single-particle Schrodinger-like equations are presented. Also discussed is the linear response formalism within the framework of TDDFT along with a few miscellaneous aspects. 

The important step of identifying the explicit dynamical motivation for employing centroid variables has thus been accomplished. It has proven possible to formally define their time evolution ( trajectories ) and to establish that the time correlations ofthese trajectories are exactly related to the Kubo-transformed time correlation function in the case that the operator 6 is a linear function of position and momentum. (Note that A may be a general operator.) The generalization of this concept to the case of nonlinear operators B has also recently been accomplished, but this topic is more complicated so the reader is left to study that work if so desired. Furthermore, by a generalization of linear response theory it is also possible to extract certain observables such as rate constants even if the operator 6 is linear. 

Because there is no general microscopic theory of liquids, the analysis of inelastic neutron scattering experiments must proceed on the basis of model calculations. Recently1 we have derived a simple interpolation model for single particle motions in simple liquids. This derivation, which was based on the correlation function formalism, depends on dispersion relation and sum rule arguments and the assumption of simple exponential decay for the damping function. According to the model, the linear response in the displacement, yft), satisfies the equation... 

In practical applications of the linear-response formalism, it is often more convenient to express the response in terms of a displacement V rather than a flux J, and we will therefore focus on this case here. The response of a gel to an applied stress is, for instance, conveniently expressed as a strain, and likewise, the response of a dielectric material to an applied electric field is often expressed as a polarization [which is closely related to the so-called electric displacement field (89)]. As a mechanical analogue would indicate, a flux is generally proportional to a velocity, and the displacement is therefore computed as the time integral of the flux. 

The continuum dielectric theory used above is a linear response theory, as expressed by the linear relation between the perturbation T> and the response , Eq. (15.1b). Thus, our treatment of solvation dynamics was done within a linear response framework. Linear response theory of solvation dynamics may be cast in a general form that does not depend on the model used for the dielectric environment and can therefore be applied also in molecular (as opposed to continuum) level theories. Here we derive this general formalism. For simplicity we disregard the fast electronic response of the solvent and focus on the observed nuclear dielectric relaxation. 

The derivation by Dirac [13] of TDHF can be expressed in terms of orbital functional derivatives [12]. This derivation can be extended directly to a formally exact time-dependent orbital-functional theory (TDOFT). The Hartree-Fock operator H is replaced by the OFT operator Q = TL + vc throughout Dirac s derivation. In the linear-response limit, this generalizes the RPA equations [14] to include correlation response [17]. 

The linear response of a system is determined by the lowest order effect of a perturbation on a dynamical system. Formally, this effect can be computed either classically or quantum mechanically in essentially the same way. The connection is made by converting quantum mechanical commutators into classical Poisson brackets, or vice versa. Suppose that the system is described by Hamiltonian H + where denotes an external perturbation that may depend on time and generally does not commute with H. The density matrix equation for this situation is given by the Bloch equation [32]... 

If one accepts the continuum, linear response dielectric approximation for the solvent, then the Poisson equation of classical electrostatics provides an exact formalism for computing the electrostatic potential (r) produced by a molecular charge distribution p(r). The screening effects of salt can be added at this level via an approximate mean-field treatment, resulting in the so-called Poisson-Boltzmann (PB) equation [13]. In general, this is a second order non-linear partial differential equation, but its simpler linearized form is often used in biomolecular applications ... 

We will first show how one can obtain the time-correlation function expression for the susceptibility in a classical statistical ensemble of particles which exhibits a linear response to an externally applied perturbation. This will be followed by an outline of the argument that leads to the generalized Langevin equation for the time-dependence of an arbitrary function of the molecular canonical coordinates. In both cases, derivations with minor modifications have been presented previously in numerous reviews, monographs, etc. However, the results are employed in a large fraction of current descriptions of dynamical processes in dense phases and thus, it seems worthwhile to again show the basic ideas underlying the formalism. 

---