Time correlation formalism

As we strive here to understand motion of water molecules at a microscopic level, we need to use a certain formalism developed in the area of statistical mechanics. This formalism is broadly known as time correlation formalism (TCP). While many specialized texts exist in the literature on this important topic, we have included in Appendix 3.A a brief discussion on the time correlation functions necessary to understand the dynamics of rotation of a molecule in liquid. 

The slope of (7(f) in the time regime rmoi < f forward reaction rate constant. Thus, for the calculation of reaction rate constants it is sufficient to determine the time correlation function (7(f). In the following paragraphs we will show how to do that in the transition path sampling formalism. 

The theory of statistical mechanics provides the formalism to obtain observables as ensemble averages from the microscopic configurations generated by such a simulation. From both the MC and MD trajectories, ensemble averages can be formed as simple averages of the properties over the set of configurations. From the time-ordered properties of the MD trajectory, additional dynamic information can be calculated via the time correlation function formalism. An autocorrelation function Caa( = (a(r) a(t + r)) is the ensemble average of the product of some function a at time r and at a later time t + r. 

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. 

It was recently shown that a formal density expansion of space-time correlation functions of quantum mechanical many-body systems is possible in very general terms [297]. The formalism may be applied to collision-induced absorption to obtain the virial expansions of the dipole... 

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... 

In this article the memory function formalism has been used to compute time-correlation functions. It has been shown that a number of seemingly disparate attempts to account for the dynamical behavior of time correlation functions, such as those of Zwanzig,33,34 Mori,42,43 and Martin,16 are... 

The relaxation equations for the time correlation functions are derived formally by using the projection operator technique [12]. This relaxation equation has the same structure as a generalized Langevin equation. The mode coupling theory provides microscopic, albeit approximate, expressions for the wavevector- and frequency-dependent memory functions. One important aspect of the mode coupling theory is the intimate relation between the static microscopic structure of the liquid and the transport properties. In fact, even now, realistic calculations using MCT is often not possible because of the nonavailability of the static pair correlation functions for complex inter-molecular potential. 

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... 

In order to use this formal expression in a calculation of the rate constant we need to choose a representation. In the following we will determine the coordinate representation of the correlation function. We use the coordinate representation of the flux operator as derived above. It is introduced in the expression for the time-correlation function by introducing three unit operators like... 

We will present the topic by introducing the nuclear spins as probes of molecular information. Some basic formal NMR theory is given and connected to MD simulations via time correlation functions. A large number of examples are chosen to demonstrate different possible ways to combine MD simulations and experimental NMR relaxation studies. For a conceptual clarity, the examples of MD simulations presented and discussed in different sections, are arranged according to the specific relaxation mechanisms. At the end of each section, we will also specify some requirements of theoretical models for the different relaxation mechanisms in the light of the simulation results and in terms of which properties these models should be parameterized for conceptual simplicity and fruitful interpretation of experimental data. 

The second method is to compute the diffusion coefficient as the integral of the molecular CoM velocity autocorrelation function by using time correlation functions formalisms ... 

The linearized transport equations (7), the equations for the equilibrium time correlation functions (13), and the equation for collective mode spectrum (14) form a general basis for the study of the dynamic behavior of a multicomponent fluid in the memory function formalism. 

The time correlation function of two dynamical variables A and B can formally be defined by (see also Eq. (7.42a))... 

How do the definitions (6.3) and (6.6) relate to each other While a formal connection can be made, it is more important at this stage to understand their range of applicability. The definition (6.6) involves the detailed time evolution of all particles in the system. Equation (6.3) becomes useful in reduced descriptions of the system of interest. In the present case, if we are interested only in the mutual dynamics of the observables A and B we may seek a description in the subspace of these variables and include the effect of the huge number of all other microscopic variables only to the extent that it affects the dynamics of interest. This leads to a reduced space dynamics that is probabilistic in nature, where the functions P(B,t2, A,t].) and P(B, Z2 A, Zi) emerge. We will dwell more on these issues in Chapter 1. Common procedures for evaluating time correlation functions are discussed in Section 7.4.1. 

On the left side of (11.15) we have the time evolution of a prepared deviation from equilibrium of the dynamical variable B. On the right side we have a time correlation function of spontaneous equilibrium fluctuations involving the dynamical variables A, which defined the perturbation, and B. The fact that the two time evolutions are the same has been known as the Onsager regression hypothesis. (The hypothesis was made before the formal proof above was known.)... 

H. Kono, Y. Nomura, and Y. Fujimura, Adv. Chent. Phys. 80, 403 (1991). Here, the time-correlation function formalism is used to classify the RSR into the fluorescencelike, the Raman-like and the interference-like components. This classification corresponds to the nomenclature II in [6] and the interference-like component is not positive definite in general. 

In the previous sections, we derived general correlation function expressions for the nonlinear response function that allow us to calculate any 4WM process. The final results were recast as a product of Liouville space operators [Eqs. (49) and (53)], or in terms of the four-time correlation function of the dipole operator [Eq. (57)]. We then developed the factorization approximation [Eqs. (60) and (63)], which simplifies these expressions considerably. In this section, we shall consider the problem of spontaneous Raman and fluorescence spectroscopy. General formal expressions analogous to those obtained for 4WM will be derived. This will enable us to treat both experiments in a similar fashion and compare their information content. We shall start with the ordinary absorption lineshape. Consider our system interacting with a stationary monochromatic electromagnetic field with frequency w. The total initial density matrix is given by... 

The basic formulation of this problem was given by Van Hove [25] in the form of his space-time correlation functions, G ir, t) and G(r, t). He showed that the scattering functions, as defined above, for a diffusing system are given by the Fourier transformation of these correlation functions in time and space. Incoherent scattering is linked to the self-correlation function, Gs(r, t) which provides a full definition of tracer diffusion while coherent scattering is the double Fourier transform of the full correlation function which is similarly related to chemical or Fick s law diffusion. Formally the equations can be written ... 

Until now we have discussed only elementary methods for determining correlation functions, based on ad hoc models. In this chapter a powerful formalism for computing time-correlation functions is presented. As a by-product of this formalism several useful theorems emerge which result from symmetry considerations. Moreover some of the assumptions made in Chapter 10 are shown to be valid. Throughout this chapter we treat classical systems. The methods developed here can also be applied to quantum systems. This is shown in Appendix 11.A. The formalism of this chapter is applied in Chapter 12 to the calculation of the depolarized spectrum. 

Becuase of the formal similarity between C(f) and scalar products in quantum mechanics the mathematical techniques of quantum mechanics can be applied to a study of classical time-correlation functions. 

Figure 18 Time cross-correlation functions for three Raman transitions in iodobenzene (from the ground state to the B excited electronic state with v = 1, 2, 3 quanta in the vu vibrational mode. (Left) Computed for a harmonic B state potential and convoluted with a 125-fs-wide window function. The spectrum is computed from this cross-correlation function. (Right) The time correlation function determined from the Raman frequency spectrum (the excitation profile) via the maximum entropy formalism, as discussed in the text, using nine Lagrange multipliers kr. (From Ref. (102).)...

In Paper I, general imaginary-time correlation functions were expressed in terms of an averaging over the coordinate-space centroid density p (qj and the centroid-constrained imaginary-time-position correlation function Q(t, qj. This formalism was extended in Paper III to the phase-space centroid picture so that the momentum could be treated as an independent variable. The final result for a general imaginary-time correlation function is found to be given approximately by [5,59]... 

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