Kubo formalism

It is straightforward to show that this behavior also holds for fc-mers which perform elementary jumps of lenght equal to one lattice constant in one dimension. From the Green-Kubo formalism, the chemical diffusion coefficient is D(Q) = Dj(Q)Th(Q), where... 

Figure 7 (a) Vibrational autocorrelation function in the Abragam-Kubo formalism... 

A more sophisticated version of basically this same idea is associated with the Green-Kubo formalism in which the relevant transport coefficient is seen as an average of the appropriate velocity autocorrelation function. In particular, the diffusion coefficient may be evaluated as... 

Another theme often explored with model alkanes is the relative merits of equilibrium MD (EMD) and non-equilibrium MD (NEMD) as methods for obtaining transport coefficients. Dysthe et al. explored some aspects of the methodology used to obtain transport coefficients by MD. They applied the Green-Kubo formalism to flexible multicentre models fi om linear and branched... 

Transport properties, such as diffusion coefficients, shear viscosity, fhermal or electrical conductivity, can be determined from the time evolution of the autocorrelation function of a particular microscopic flux in a system in equilibrium based on the Green-Kubo formalism [217, 218] or the Einstein equations [219], Autocorrelation functions give an insight into the dynamics of a fluid and their Fourier transforms can be related to experimental spectra. The general Green-Kubo expression for an arbitrary transport coefficient y is given by ... 

The gas diffusion coefficient D may be extracted from a simulation in various ways. In a Molecular Dynamics simulation, if the hydrodynamk limit is reached (Le., the simulation time is long enough), D may be calculated either from the evolution of the penetrants instantaneous velocities by means of the Green-Kubo formalism or, alternatively, with the aid of the Einstein relation from their spatial positions. For TSA, there is only the Hnstein-route, via the mean-square displacement =< r(t) — r(0) > which is plotted against time, best in a log-log plot. The portion of the curve that represents oc t is used to fit a least-squares line, the slope of which (in a vs t plot) or the ptmtion of which (in a log [Pg.211]

Let us first consider nonequilibrium properties of dense fluids. Linear response theory relates transport coefficients to the decay of position and velocity correlations among the particles in an equilibrium fluid. For example, the shear viscosity ti can be expressed in Green-Kubo formalism as a time integral of a particular correlation function ... 

The dielectric spectroscopy of anisotropic fluids started in the 1970s by the extension of the Debye model from isotropic media (described in Appendix D) to uniaxial systems based on statistical mechanical Kubo formalism/ but no quantitative estimates about the critical frequencies or the susceptibilities were obtained. Quantitative estimates were given first on molecules with dipole moments along the long axis/ then for general dipole directions using the rotational Brownian picture in Maier-Saupe mean-field potential. This theory was subsequently refined in the 1990s.i ... 

According to the Kubo formalism of nonequilibrium statistical ihcnnodynamics [2J22], the respoose R(t) of a statistical system to any time-dependent external action A(/) can be generally expressed as... 

The frequency dependence of As can be expressed with the help of the Kubo formalism [20]. It can be shown that quite generally the following relation holds... 

Fig. 11. Shear viscosity for the [C2-mim][Q] ionic Hquid at different temperatures. Sohd symbols are results obtained by the rigid model, while open symbols are the values from the flexible model, using the Green-Kubo formalism in the NVE simulations. Crosses represent experimental (Earle Seddon, 2002). The hnes are just a guide to the eyes.

Andreu, J.S. Vega, L.F. (2010). On the Transport properties of [emimjCl through the Green Kubo formalism. Submitted. 

T. Ihle and D. M. Kroll, Stochastic rotation dynamics. I. Formalism, Galilean invariance, and Green—Kubo relations, Phys. Rev. E 67, 066705 (2003). 

An alternative approach involves integrating out the elastic degrees of freedom located above the top layer in the simulation.76 The elimination of the degrees of freedom can be done within the context of Kubo theory, or more precisely the Zwanzig formalism, which leads to effective (potentially time-dependent) interactions between the atoms in the top layer.77-80 These effective interactions include those mediated by the degrees of freedom that have been integrated out. For periodic solids, a description in reciprocal space decouples different wave vectors q at least as far as the static properties are concerned. This description in turn implies that the computational effort also remains in the order of L2 InL, provided that use is made of the fast Fourier transform for the transformation between real and reciprocal space. The description is exact for purely harmonic solids, so that one can mimic the static contact mechanics between a purely elastic lattice and a substrate with one single layer only.81... 

Fig. 9. Variable temperature 0 NMR results (Bq = 14.1 T) for solutions containing Tb(C104)3 (0,DX Mg(C104)2 and Tb(C104)3 ( , ), and Mg(C104)2 and Mn(C104)2 (relaxation agent) (A). Full lines result from non-linear fitting using Kubo-Sack formalism and short-dashed lines were calculated by an approximate 3-site Swift and Connick method [Ref. (40)].

A novel 1,8-photoaddition of dimethyl 1,4-naphthalenedicarboxylate and 1,4-dicyanonaphthalene to alkenes, a formal [3-1-2] cycloaddition, was reported by Kubo and co-workers. Although yields of the desired products were low, Cu(OAc)2 showed a profound impact on the reaction by increasing the yields considerably [85]. 

Much of the recent literature on RDM reconstruction functionals is couched in terms of cumulant decompositions [13, 27-38]. Insofar as the p-RDM represents a quantum mechanical probability distribution for p-electron subsystems of an M-electron supersystem, the RDM cumulant formalism bears much similarity to the cumulant formalism of classical statistical mechanics, as formalized long ago by by Kubo [39]. (Quantum mechanics introduces important differences, however, as we shall discuss.) Within the cumulant formalism, the p-RDM is decomposed into connected and unconnected contributions, with the latter obtained in a known way from the lower-order -RDMs, q < p. The connected part defines the pth-order RDM cumulant (p-RDMC). In contrast to the p-RDM, the p-RDMC is an extensive quantity, meaning that it is additively separable in the case of a composite system composed of noninteracting subsystems. (The p-RDM is multiphcatively separable in such cases [28, 32]). The implication is that the RDMCs, and the connected equations that they satisfy, behave correctly in the limit of noninteracting subsystems by construction, whereas a 2-RDM obtained by approximate solution of the CSE may fail to preserve extensivity, or in other words may not be size-consistent [40, 42]. 

The combinatorial point of view is reminiscent of the classical cumulant formalism developed by Kubo [39], and indeed the structure of Eqs. (25) and (28) is essentially the same as the equations that define the classical cumulants, up to the use of an antisymmetrized product in the present context. In further analogy to the classical cumulants, the p-RDMC is identically zero if simultaneous p-electron correlations are negligible. In that case, the p-RDM is precisely an antisymmetrized product of lower-order RDMs. 

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. 

As regards the conductivity, in the Kubo—Greenwood formalism this will depend on a0 4, so... 

Spectral lineshapes were first expressed in terms of autocorrelation functions by Foley39 and Anderson.40 Van Kranendonk gave an extensive review of this and attempted to compute the dipolar correlation function for vibration-rotation spectra in the semi-classical approximation.2 The general formalism in its present form is due to Kubo.11 Van Hove related the cross section for thermal neutron scattering to a density autocorrelation function.18 Singwi et al.41 have applied this kind of formalism to the shape of Mossbauer lines, and recently Gordon15 has rederived the formula for the infrared bandshapes and has constructed a physical model for rotational diffusion. There also exists an extensive literature in magnetic resonance where time-correlation functions have been used for more than two decades.8... 

Basically the perturbative techniques can be grouped into two classes time-local (TL) and time-nonlocal (TNL) techniques, based on the Nakajima-Zwanzig or the Hashitsume-Shibata-Takahashi identity, respectively. Within the TL methods the QME of the relevant system depends only on the actual state of the system, whereas within the TNL methods the QME also depends on the past evolution of the system. This chapter concentrates on the TL formalism but also shows comparisons between TL and TNL QMEs. An important way how to go beyond second-order in perturbation theory is the so-called hierarchical approach by Tanimura, Kubo, Shao, Yan and others [18-26], The hierarchical method originally developed by Tanimura and Kubo [18] (see also the review in Ref. [26]) is based on the path integral technique for treating a reduced system coupled to a thermal bath of harmonic oscillators. Most interestingly, Ishizaki and Tanimura [27] recently showed that for a quadratic potential the second-order TL approximation coincides with the exact result. Numerically a hint in this direction was already visible in simulations for individual and coupled damped harmonic oscillators [28]. 

A complete and detailed analysis of the formal properties of the QCL approach [5] has revealed that while this scheme is internally consistent, inconsistencies arise in the formulation of a quantum-classical statistical mechanics within such a framework. In particular, the fact that time translation invariance and the Kubo identity are only valid to O(h) have implications for the calculation of quantum-classical correlation functions. Such an analysis has not yet been conducted for the ILDM approach. In this chapter we adopt an alternative prescription [6,7]. This alternative approach supposes that we start with the full quantum statistical mechanical structure of time correlation functions, average values, or, in general, the time dependent density, and develop independent approximations to both the quantum evolution, and to the equilibrium density. Such an approach has proven particularly useful in many applications [8,9]. As was pointed out in the earlier publications [6,7], the consistency between the quantum equilibrium structure and the approximate... 

But if we examine the localized near the donor or the acceptor crystal vibrations or intra-molecular vibrations, the electron transition may induce much larger changes in such modes. It may be the substantial shifts of the equilibrium positions, the frequencies, or at last, the change of the set of normal modes due to violation of the space structure of the centers. The local vibrations at electron transitions between the atomic centers in the polar medium are the oscillations of the rigid solvation spheres near the centers. Such vibrations are denoted by the inner-sphere vibrations in contrast to the outer-sphere vibrations of the medium. The expressions for the rate constant cited above are based on the smallness of the shift of the equilibrium position or the frequency in each mode (see Eqs. (11) and (13)). They may be useless for the case of local vibrations that are, as a rule, high-frequency ones. The general formal approach to the description of the electron transitions in such systems based on the method of density function was developed by Kubo and Toyozawa [7] within the bounds of the conception of the harmonic vibrations in the initial and final states. 

Timmermans et al. report that diastereoselectivity can be induced in the intramoleuclar meta photocycloaddition of ethenes to the benzene ring as a result of minimisation of steric interactions between substituents on the linking tether of the bichromophore and a methoxy group at the 2-position of the arene unity this type of photoprocess has also been used as a key step in a formal synthesis of crinipellin B (Wender and Dore). New polycyclic cage compounds 43 have been obtained by irradiation of the [3.3,3] (1,3,5) cyclophane 44 (Sakamoto et al.) and Kubo et al. have described the intramolecular [3 + 2] photocycloaddition of bichromophores such as 45 which gives rise to nine- to eleven-membered ring systems 46. 

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