Convergence of Averages

Given a collection of phase space points distributed according to the density po q,p), we know that its evolution (under Langevin dynamics) after a time t will be described by 

More precisely this relation should be interpreted in the variational sense, so that we have, for any suitable test function p, e.g. 0 C °, 

This is of course an inner product, and due to the properties of the inner product, we may interpret the right hand side as 

How rapidly an average converges will depend on the spectral properties of the operator as well as the choice of the initial distribution. For example, if we assume that Ai, pd) is some eigenvalue, eigenfunction pair with Re(Ai) 0 and take the initial distribution to be 

Since all the eigenvalues lie to the left of a vertical line in the left half plane, for any in C°° we have 

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The idea of defining a distribution in terms of an integral against another function is very appropriate in the setting of molecular dynamics. We are, after all, interested in the convergence of averages of smooth functions with respect to the evolving... 

The interaction with the solvent is of similar importance as the intramolecuiar energy contributions and a correct representation of the solvent is therefore es.sential. If an explicit solvent description is chosen, averaging over many different solvent configurations is necessary in order to obtain converged statistical averages. Advantageous in this respect is describing the solvent as... 

For purposes of exploring fluctuations and determining the convergence of these statistical averages the root mean square (RMSl deviation m x is also computed ... 

In deciding the convergence of these averages, the RMS deviation of a value from its average (i.e, Dxi may be a very useful indicator. 

In addition to bein g able to plot sim pie in stan tan eous values of a quantity x along a trajectory and reporting the average, , HyperChem can also report information about the deviation of x from its average value. Ihese RMS deviations may have particular sign ifican ce in statistical tn ech an ics or just represen t lh e process of convergence of the trajectory values. 

In the next section we derive the Taylor expansion of the coupled cluster cubic response function in its frequency arguments and the equations for the required expansions of the cluster amplitude and Lagrangian multiplier responses. For the experimentally important isotropic averages 7, 7i and yx we give explicit expressions for the A and higher-order coefficients in terms of the coefficients of the Taylor series. In Sec. 4 we present an application of the developed approach to the second hyperpolarizability of the methane molecule. We test the convergence of the hyperpolarizabilities with respect to the order of the expansion and investigate the sensitivity of the coefficients to basis sets and correlation treatment. The results are compared with dispersion coefficients derived by least square fits to experimental hyperpolarizability data or to pointwise calculated hyperpolarizabilities of other ab inito studies. 

The second approach, ABF, consists of estimating the average force at different and removing it from the system. This leads to a diffusive-like motion along and a much better convergence of the calculation. We will describe ABF in Sect. 4.6. 

Jarzynski, C., Rare events and the convergence of exponentially averaged work values, Phys. Rev. E 2006, 73, 046105... 

The Jarzynski identity (7.33) is an exact result and applies to transformations of arbitrary length From a computational point of view this property seems very attractive because it implies that free energy differences can be calculated from short and therefore computationally inexpensive trajectories. However, the convergence of the exponential average from (7.33) quickly deteriorates if the transformation (or the switching) is carried out too rapidly. 

In a recent development, Corcelli et al. [110] introduced a convenient bias function with general applicability that promises to accelerate the convergence of rate calculations in systems with large enthalpy barriers. They apply a puddle potential (used previously by the same group to enhance thermodynamic averaging [69]) that changes the potential energy surface from which the trajectories are initiated to become... 

Convergence of the FEP results was carefully studied. A typical mutation has 5-10 windows each consisting of 15 M configurations of equilibration and 10 M configurations of averaging. From the batch means procedure, the resulting statistical uncertainty in the computed AAGb values was ca. 0.15 kcal/mol. Two closed perturbation cycles (H —> F Cl - OH... 

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