Quantum time correlation functions

The quantum mechanical analog of the equilibrium correlation function (6.6) is 

From the cyclic property of the trace and the fact that the equilibrium density operator p = Q exp(—commutes with the time evolution operator exp(— iHt/K) it follows that 

These identities are identical to their classical counterparts, for example, (6.28). 

A special case of the identity (6.64) is ( (Z) (O)) = (J[(—Z) (O)). This shows that (6.63) holds also for the autocorrelation function of a hermitian (not necessarily real) operator. 

Problem 6.5. F or the real and imaginary part of the quantum correlation function  

Time correlation functions The reader should notice the difference between Cq co) = and 

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The approach to the evaluation of vibrational spectra described above is based on classical simulations for which quantum corrections are possible. The incorporation of quantum effects directly in simulations of large molecular systems is one of the most challenging areas in theoretical chemistry today. The development of quantum simulation methods is particularly important in the area of molecular spectroscopy for which quantum effects can be important and where the goal is to use simulations to help understand the structural and dynamical origins of changes in spectral lineshapes with environmental variables such as the temperature. The direct evaluation of quantum time- correlation functions for anharmonic systems is extremely difficult. Our initial approach to the evaluation of finite temperature anharmonic effects on vibrational lineshapes is derived from the fact that the moments of the vibrational lineshape spectrum can be expressed as functions of expectation values of positional and momentum operators. These expectation values can be evaluated using extremely efficient quantum Monte-Carlo techniques. The main points are summarized below. 

Amides, alkaline hydrolysis, 215 Anharmonic systems, direct evaluation of quantum time-correlation functions, 93 Apollo DSP—160, CHARMM performance, 129/ simulations, solvent effects, 83... 

The classical time-correlation function, , does not obey the condition of detailed balance. Computer experiments provide detailed information about classical time-correlation functions. Is there any way to use the classical functions to predict quantum-mechanical time-correlation functions The answer to this question is affirmative. There exist approximations which enable the quantum-time-correlation functions to be predicted from the corresponding classical functions. Let us denote by v fn(t) the classical time-correlation function and by It(r) the one-sided quantum-mechanical correlation function ... 

A centroid trajectory for a given set of initial centroid conditions must contain some degree of dynamical information due to the nonstationarity of the ensemble created by the centroid constraints. It is therefore important to explore the correlations in time of these trajectories. In the centroid dynamics perspective, a general quantum time correlation function can be expressed as... 

R. Hernandez and G. Voth. Quantum time correlation functions and classical coherence. Chem. Phys., 223 243, 1998. 

D.F. Coker and S. Bonella, Linearized path integral methods for quantum time correlation functions, in Computer simulations in condensed matter From materials to chemical biology, eds. M. Ferrario and G. Ciccotti and K. Binder, Lecture Notes in Physics 703, (Springer-Verlag, Berlin), p. 553, 2006. 

Causo, M.S., Ciccotti, G., Montemayor, D., Bonella, S., Coker, D.F. An adiabatic linearized path integral approach for quantum time correlation functions electronic transport in metal-molten salt solutions. J. Phys. Chem. B 109 6855... 

The basic theoretical framework for understanding the rates of these processes is Fermi s golden rule. The solute-solvent Hamiltonian is partitioned into three terms one for selected vibrational modes of the solute, including the vibrational mode that is initially excited, one for all other degrees of freedom (the bath), and one for the interaction between these two sets of variables. One then calculates rate constants for transitions between eigenstates of the first term, taking the interaction term to lowest order in perturbation theory. The rate constants are related to Fourier transforms of quantum time-correlation functions of bath variables. The most common... 

There are also situations when one is not in the classical limit, and so Equation (13) would not seem applicable, and instead one would like to approximate one of the quantum mechanical expressions for Ti by relating the relevant quantum time-correlation function to its classical analog. For the sake of definiteness, let us consider the case where the oscillator is harmonic and the oscillator-bath coupling is linear in q, as discussed above. In this case k 0 can be written as... 

In a wider perspective analogies have been pointed out with respect to the calculation of quantum time correlation functions. ... 

D.F. Coker and S. Bonella Linearized Path Integral Methods for Quantum Time Correlation Functions, Lect. Notes Phys. 703, 553—590 (2006)... 

System-Bath Representation of Quantum Time Correlation Functions... 

The quantum time correlation function of two operators of the system is defined as... 

In (13.20) and (13.21) the subscripts c and q correspond respectively to the classical and the quantum time correlation functions and denotes a semiclassical approximation. We refer to the form (13.21) as semiclassical because it carries aspects of the quantum thermal distribution even though the quantum time correlation function was replaced by its classical counterpart. On the face of it this approximation seems to make sense because one could anticipate that (1) the time correlation functions involved decay to zero on a short timescale (of order 1 ps that characterizes solvent configuration variations), and (2) classical mechanics may provide a reasonable short-time approximation to quantum time correlation functions. Furthermore note that the rates in (f3.2f) satisfy detailed balance. 

Although CMD is a substantial breakthrough in the approximate computation of quantum time correlation functions, the determination of the centroid force in Eq. (3.59), as defined by Eq. (3.60), represents an algorithmic challenge for realistic many-body simulations. Equation... 

Our purpose is not to deal with the above accurate correlation functions. However, we want to obtain approximate quantum time correlation functions from the classical fluctuation functions, which should satisfy the detailed balance principle. Defining ... 

Here. .. )t is the quantum thermal average, (... )t = Tr[e ... ]/Tr[e" ], and Tr denotes a trace over the initial manifold z. We have thus identified the golden rule rate as an integral over time of a quantum time correlation function associated with the interaction representation of the coupling operator. 

We have thus found that the k. (-2 is given by a Fourier transform of a quantum time correlation function computed at the energy spacing that characterizes the... 

The computational approaches up to 2006 were reviewed by Perry et al. °" Briefly, these methods are based on representing the SFG spectrum by the Fourier transform of a polarizability-dipole quantum time correlation function (QTCF). A fully classical approach to computing the SFG spectrum is then obtained by replacing the QTCF by a classical expression including a harmonic correction factor ... 

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