Linear response theory functions

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]... 

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 ... 

The strategy, usually adopted to achieve a theoretical description of this complex dynamics, is to describe the influence of the solvent environment on the electron-transfer reaction within linear response theory [5, 26, 196, 197] as linear coupling to a bath of harmonic oscillators. Within this model, all properties of the bath enter through a single function called the spectral density [5, 168]... 

The article is organized as follows. The main features of the linear response theory methods at different levels of correlation are presented in Section 2. Section 3 describes the calculation of the dipole and quadmpole polarizabilities of two small diatomic molecules LiH and HF. Different computational aspects are discussed for each of them. The LiH molecule permits very accurate MCSCF studies employing large basis sets and CASs. This gives us the opportunity to benchmark the results from the other linear response methods with respect to both the shape of the polarizability radial functions and their values in the vibrational ground states. The second molecule, HF, is undoubtedly one of the most studied molecules. We use it here in order to examine the dependence of the dipole and quadmpole polarizabilities on the size of the active space in the CAS and RASSCF approaches. The conclusions of this study will be important for our future studies of dipole and quadmpole polarizabilities of heavier diatomic molecules. 

The dipole and quadrupole polarizability tensor components of LiH were calculated by MCSCF linear response theory with the basis set of Roos and Sadlej [57] which consists of 13s-, 8p-, 6d-, and 2f-type sets of uncontracted Gaussian functions on Li and 12s-, 8p-, and 5d-type sets of uncontracted Gaussians on H. Due to the small size of the molecule we could perform MCSCF calculations over the whole range of internuclear distances with a very large CAS 0000 520,10,10,4 p g present the tensor components, isotropic, and anisotropic values of the dipole polarizability tensor a as function... 

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

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. 

The paper is organized as follows. In Section 2, derivation of the the SRPA formalism is done. Relations of SRPA with other alternative approaches are commented. In Sec. 3, the method to calculate SRPA strength function (counterpart of the linear response theory) is outlined. In Section 4, the particular SRPA versions for the electronic Kohn-Sham and nuclear Skyrme functionals are specified and the origin and role of time-odd currents in functionals are scrutinized. In Sec. 5, the practical SRPA realization is discussed. Some examples demonstrating accuracy of the method in atomic clusters and nuclei are presented. The summary is done in Sec. 6. In Appendix A, densities and currents for Skyrme functional are listed. In Appendix B, the optimal ways to calculate SRPA basic values are discussed. 

Using linear response theory and noting (according to the results at the end of Section 5.1.3) that the (complex) electrical conductivity a is the Fourier transform of the current density autocorrelation function, we obtain from Eqn. (5.75) (see the equivalent Eqn. (5.21))... 

Assuming further that linear response theory for the solute/solvent system applies, the dependence of the nonequilibrium free energies of the system (in the ground Fg and excited Fe states) are portrayed in Figure 1 as a function of the electrical polarization of the solvent (see below). In a transient fluores-... 

Linear response theory is reviewed in Section II in order to establish contact between experiment and time-correlation functions. In Section III the memory function equation is derived and applied in Section IV to the calculation of time-correlation functions. Section V shows how time-correlation functions can be used to guess time-dependent distribution functions and similar methods are then applied in Section VI to the determination of time-correlation functions. In Section VII a succinct review is given of other exact and experimental calculations of time-correlation functions. 

Thus the response of a spatially uniform system in thermodynamic equilibrium is always characterized by translationally invariant and temporaly stationary after-effect functions. This article is restricted to a discussion of systems which prior to an application of an external perturbation are uniform and in equilibrium. The condition expressed by Eq. (7) must be satisfied. Caution must be exercised in applying linear response theory to problems in double resonance spectroscopy where non-equilibrium initial states are prepared. Having dispensed with this caveat, we adopt Eq. (7) in the remainder of this review article. 

Each of these correlation functions appear in linear response theory. 

The paper [50] was written, when our general linear response-theory (ACF method) was still in progress, so the derivation of the spectral function described in Ref. 50 is more specialized than that given, for example, in VIG and GT. 

Validity of our formulas for the resonance lines, which express the complex susceptibility through the spectral function, could be confirmed as follows. We have obtained an exact coincidence of the equations (353), (370), (371), which were (i) directly calculated here in terms of the harmonic oscillator model and (ii) derived in GT and VIG (see also Section II, A.6) by using a general linear-response theory. 

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