Linear response theories

The linear response theory of quantum mechanics allows to describe how an interacting system of electron respond to an external perturbation [5, 11]. The inverse of the longitudinal dielectric response function (e ) is defined from  

The quantity (r, r, ) is called longitudinal because it is related to the electrostatic potential and not the electric field. An electric field can always be written as  

The non-local operator fl is a quite complex object, which describes the response of an interacting system of electrons to an external potential. A related definition connects the 8n to the total potential 8 

The exact linear density response can be related to the linear density response of a non-interacting reference system of electrons fts using the Dyson equation  

For a homogeneous bulk solid one can use the Fourier transform in the space, i.e. assuming a spatial dependent of the t)q)e e and thus we have  

This free will that we have, wrested from fate. 

Lucretius (c.99-c.55 bce The way things are translated by Rolfe Humphries, Indiana University Press, 1968 

Equilibrium statistical mechanics is a first principle theory whose fundamental statements are general and independent of the details associated with individual systems. No such general theory exists for nonequilibrium systems and for this reason we often have to resort to ad hoc descriptions, often of phenomenological nature, as demonstrated by several examples in Chapters 1 and 8. Equilibrium statistical mechanics can however be extended to describe small deviations from equilibrium in a way that preserves its general nature. The result is Linear Response Theory, a statistical mechanical perturbative expansion about equilibrium. In a standard application we start with a system in thermal equilibrium and attempt to quantify its response to an applied (static- or time-dependent) perturbation. The latter is assumed small, allowing us to keep only linear terms in a perturbative expansion. This leads to a linear relationship between this perturbation and the resulting response. 

Let us make these statements more quantitative. Consider a system characterized by the Hamiltonian Hq. An external force acting on this system changes the Hamiltonian according to 

It should be noted that in addition to mechanical forces such as electric or magnetic fields that couple to charges and polarization in our system, other kinds of forces exist whose effect cannot be expressed by Eq. (11.2). For example, temperature or chemical potential gradients can be imposed on the system and thermal or material fluxes can form in response. In what follows we limit ourselves first to linear response to mechanical forces whose effect on the Hamiltonian is described by Eqs (11.2) or (11.3). 

If cause forever follows after cause In infinite, imdeviating sequence, 

And a new motion always has to come Out of an old one by fixed law if atoms Do not, by swerving, cause new moves which break The Laws of fate if cause forever follows, 

In infinite sequence, cause—where would we get This free will that we have, wrested fi-om fate, 

Let us consider a classical mechanical system of N particles at equilibrium. Let the Hamiltonian of the system be HoiJQ. The system s microscopic states evolve in time according to Hamilton s equations of motion. As we discussed in Chapter 4, an equivalent description is attained by the Liouville equation for the phase space probability distribution, D(p, q), written as 

Let us assume that the system has been at equilibrium since t = —oo and let F(t) be a field applied to a system at time t = 0. Consider a mechanical property A that is a function of particle momenta and coordinates, A(p, q). Assume that A p, q) couples and responds to the 

The average value of the mechanical property A changes from its equilibrium value in response to the external force F(t). We can again consider an ensemble of systems under the influence of the external force and denote the ensemble averaged perturbed value of. 4 as 

We limit ourselves to examining the linear response of the system to the external perturbation. This means that F(t) is sufficiently small so that 

Property A will change in time until it reaches a new steady state. There are numerous, important questions related to this process what is this new state And how can the dynamic response of the system be described in the linear response regime And if the external force is turned off after the system has been at the steady state for a long time, how does the system return to equilibrium A central theorem of non-equilibrium statistical mechanics that addresses these questions is 

---

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 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... 

We first examine the reiationship between particie dynamics and the scattering of radiation in the case where both the energy and momentum transferred between the sampie and the incident radiation are measured. Linear response theory aiiows dynamic structure factors to be written in terms of equiiibrium flucmations of the sampie. For neutron scattering from a system of identicai particies, this is [i,5,6]... 

The first-order term in this expansion renormalizes the potential V Q) while the bilinear term is analogous to the last term in (5.38). This is the linear-response theory for the bath. In fact, it shows... 

When the MFA is used in absence of the external field (J,- = 0) the Lagrange multipliers //, are assumed to give the actual density, p, known by construction. In presence of the field the MFA gives a correction Spi to the density p,. By using the linear response theory we can establish a hnear functional relation between J, and 8pi. The fields Pi r) can be expressed in term of a new field 8pi r) defined according to Pi r) = pi + 8pi + 8pi r). Now, we may perform a functional expansion of in terms of 8pi f). If this expansion is limited to a quadratic form in 8pj r) we get the following result [32]... 

When the linear response theory is applied to electric conductivity in an ionic melt, the total charge current J t) can be defined as... 

Saue and Jensen used linear response theory within the random phase approximation (RPA) at the Dirac level to obtain static and dynamic dipole polarizabilities for Cu2, Ag2 and Au2 [167]. The isotropic static dipole polarizability shows a similar anomaly compared with atomic gold, that is, Saue and Jensen obtained (nonrelativ-istic values in parentheses) 14.2 for Cu2 (15.1 A ), 17.3 A for Ag2 (20.5 A ), and 12.1 A for Au2 (20.2 A ). They also pointed out that relativistic and nonrelativistic dispersion curves do not resemble one another for Auz [167]. We briefly mention that Au2 is metastable at 5 eV with respect to 2 Au with a barrier to dissociation of 0.3 eV [168, 169]. 

This correlation can be quantitatively accounted for on the basis of the linear response theory and the RPA... 

It can be shown that the assumption of a weak perturbation central to linear response theory can be relaxed in this case [9]. The equations presented in this section relating the kinetic coefficients with the microscopic dynamics of the system remain valid for arbitrarily strong perturbations. 

Linear Response Theory and Free Energy Calculations... 

The dielectric response of a solvated protein to a perturbing charge, such as a redox electron or a titrating proton, is related to the equilibrium fluctuations of the unperturbed system through linear response theory [49, 50]. In the spirit of free energy... 

Linear Response Theory Application to Proton Binding and )Ka Shifts... 

As mentioned, this equivalence is a consequence of the fluctuation-dissipation theorem (the general basis of linear response theory [51]). In (12.68), we have dropped nonlinear terms and we have not indicated for which state Variance (rj) is computed (because the reactant and product state results only differ by nonlinear terms). We see that A A, AAstat, and AAr x are all linked and are all sensitive to the model parameters, with different computational routes giving a different sensitivity for AArtx. 

The above theory is usually called the generalized linear response theory because the linear optical absorption initiates from the nonstationary states prepared by the pumping process [85-87]. This method is valid when pumping pulse and probing pulse do not overlap. When they overlap, third-order or X 3 (co) should be used. In other words, Eq. (6.4) should be solved perturbatively to the third-order approximation. From Eqs. (6.19)-(6.22) we can see that in the time-resolved spectra described by x"( ), the dynamics information of the system is contained in p(Af), which can be obtained by solving the reduced Liouville equations. Application of Eq. (6.19) to stimulated emission monitoring vibrational relaxation is given in Appendix III. 

The Spectral Density Within the Linear Response Theory... 

The linear response theory [50,51] provides us with an adequate framework in order to study the dynamics of the hydrogen bond because it allows us to account for relaxational mechanisms. If one assumes that the time-dependent electrical field is weak, such that its interaction with the stretching vibration X-H Y may be treated perturbatively to first order, linearly with respect to the electrical field, then the IR spectral density may be obtained by the Fourier transform of the autocorrelation function G(t) of the dipole moment operator of the X-H bond ... 

---