Ensemble corrections

It is instructive to work out a few examples of the MD ensemble corrections. For the kinetic energy per particle, KE/N, equation (63) gives... 

Notice, Extreme caution must be observed when calculating the ensemble correction for a dynamical variable, such as KEqj, which depends on the ensemble parameters p and/or... 

For a further discussion of ensemble corrections, the student is referred to the paper of Wallace and Straub. have also derived the MD... 

Important point. The ensemble correction for a fluctuation average is of the same order as the fluctuation average itself. 

Exercise. Show that the MD ensemble corrections vanish for H and for Jj. ... 

Ensemble corrections. These result from the fact that... 

Boundary condition effects. This category is meant to include all finite-size effects not contained in the ensemble corrections. This is abbreviated BC effects, and PBC effects in the case of periodic boundary conditions. In the remainder of this subsection, we will discuss what is currently known (to the author) about BC effects. 

However, it is computationally much less expensive to follow the structural route explained before, which was based on the PI centroids and related OZ2 calculations (i.e., Eq. 114) plus grand ensemble corrections (BDH+GC method). By doing so, one corrects for the finite N effects and can obtain, at the same time, accurate information on k-space properties of the fluid under the action of an external constant-strength field. A complete discussion on this issue can be found in Ref. 96. With the use of these/j. estimates calculated along isotherms the pressure EOS is easily determined by integration, which can be summarized in the following standard equations ... 

In the calculation of ensemble averages, we correct for the weighting as follows... 

An ensemble of trajectory calculations is rigorously the most correct description of how a reaction proceeds. However, the MEP is a much more understandable and useful description of the reaction mechanism. These calculations are expected to continue to be an important description of reaction mechanism in spite of the technical difficulties involved. 

Due to the noncrystalline, nonequilibrium nature of polymers, a statistical mechanical description is rigorously most correct. Thus, simply hnding a minimum-energy conformation and computing properties is not generally suf-hcient. It is usually necessary to compute ensemble averages, even of molecular properties. The additional work needed on the part of both the researcher to set up the simulation and the computer to run the simulation must be considered. When possible, it is advisable to use group additivity or analytic estimation methods. 

Bl) The metrics effect is very significant in special theoretical examples, like a freely joined chain. In simulations of polymer solutions of alkanes, however, it only slightly affects the static ensemble properties even at high temperatures [21]. Its possible role in common biological applications of MD has not yet been studied. With the recently developed fast recursive algorithms for computing the metric tensor [22], such corrections became affordable, and comparative calculations will probably appear in the near future. 

Another principal difficulty is that the precise effect of local dynamics on the NOE intensity cannot be determined from the data. The dynamic correction factor [85] describes the ratio of the effects of distance and angular fluctuations. Theoretical studies based on NOE intensities extracted from molecular dynamics trajectories [86,87] are helpful to understand the detailed relationship between NMR parameters and local dynamics and may lead to structure-dependent corrections. In an implicit way, an estimate of the dynamic correction factor has been used in an ensemble relaxation matrix refinement by including order parameters for proton-proton vectors derived from molecular dynamics calculations [72]. One remaining challenge is to incorporate data describing the local dynamics of the molecule directly into the refinement, in such a way that an order parameter calculated from the calculated ensemble is similar to the measured order parameter. 

Eq. (1) would correspond to a constant energy, constant volume, or micro-canonical simulation scheme. There are various approaches to extend this to a canonical (constant temperature), or other thermodynamic ensembles. (A discussion of these approaches is beyond the scope of the present review.) However, in order to perform such a simulation there are several difficulties to overcome. First, the interactions have to be determined properly, which means that one needs a potential function which describes the system correctly. Second, one needs good initial conditions for the velocities and the positions of the individual particles since, as shown in Sec. II, simulations on this detailed level can only cover a fairly short period of time. Moreover, the overall conformation of the system should be in equilibrium. 

More rigorously, if we have an ensemble of n particles and their energy at a certain level of theory is related to the energy of a single particle (at the same level of theory) by e(n) = ne(l), then we say that the theory scales correctly. 

Another problem with microcononical-based CA simulations, and one which was not entirely circumvented by Hermann, is the lack of ergodicity. Since microcanoriical ensemble averages require summations over a constant energy surface in phase space, correct results are assured only if the trajectory of the evolution is ergodic i.e. only if it covers the whole energy surface. Unfortunately, for low temperatures (T << Tc), microcanonical-based rules such as Q2R tend to induce states in which only the only spins that can flip their values are those that are located within small... 

The 2D model was built from a wide array of descriptors, including also E-state indices, by Simulations Plus [89], The model is based on the associative neural network ensembles [86, 87] constructed from n=9658 compounds selected from the BioByte StarList [10] of ion-corrected experimental logP values. The model produced MAE = 0.24, r = 0.96 (R. Fraczkiewicz, personal communication). 

Are there any remedies in sight within approximate Kohn-Sham density functional theory to get correct energies connected with physically reasonable densities, i. e., without having to use wrong, that is symmetry broken, densities In many cases the answer is indeed yes. But before we consider the answer further, we should point out that the question only needs to be asked in the context of the approximate functionals for degenerate states and related problems outlined above, an exact density functional in principle also exists. The real-life solution is to employ the non-interacting ensemble-Vs representable densities p intro-... 

The strategy to first use broken symmetry solutions and later restore the correct (spin) density by employing ensembles can be applied successfully to solve many degeneracy related problems. However, in practice it is very rarely used because there are hardly any... 

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