Statistical Mechanical Averages

Statistical mechanical averages in a molecular dynamics run are obtained by simply averaging an energetic or structural value over time steps. Thus if the values x , i are being computed in a trajectory, the statistical mechanical average is just 

For purposes of exploring fluctuations and determining the convergence of these statistical averages the root mean square (RMS) deviation in x is also computed  

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A molecular dynamics (MD) calculation collects statistical information as it progresses. So, for example, if the calculated position vector of atom A at times ti,t2, -, tn is rA(f2), .., t-A(tn), then the statistical-mechanical average is... 

Given an approximation to V( 0s ) acceptable for the purposes at hand, one can proceed to compute equilibrium, i.e., statistically mechanically averaged, values

for properties P( 0s ) of interest using standard procedures which weight each conformation of the carbohydrate molecule by the Boltzmann factor of V( 0s ) normalized by the configuration integral given in eqn. (9). 

Exact methods are developed for calculating second and fourth moments of chain molecules. The RIS approximation is adopted to represent the effects of hinderance potentials affecting bond rotations. IMo other approximations are invoked in deriving the statistical-mechanical averages. 

The statistical mechanical average of the quantity enclosed, taken over all configuration of the chain. 

The expressions are particularly useftd in the evaluation of integrals over products of rotational matrices, as we shall see. They are widely used in many branches of physics and chemistry from multipole expansions through to statistical mechanical averaging. 

The calculation of the molar polarizabilities, often involves statistical mechanical averaging over orientational distributions of the molecules. An important example is the distribution function w caused by dipole orientation in an externally applied static electric field E° because it describes the process of electric poling of NLO-phores. To second order in the field, the dipolar contributions to this (normalized) function are given by (100),... 

We will also assume the common circumstance that Z(fl) is an unbiased estimator of Z in practical terms, this means that if the same estimation is performed repeatedly then the average of the estimates is the quantity desired, Z = [Z(n)], the expected value of the estimator. The alternative notation for expectation is used here to avoid potential confusion with the statistical mechanical averages used throughout this book. 

If is the statistical-mechanical average of the uniJerturbed end-to-end distance of a chain composed of n bonds of length 1, it follows that the ratio acquires the significance of a characteristic... 

PES the other is macroscopic and related to phenomenological interpretation of empirical kinetic data. The connection between a transition structure and a TS involves statistical-mechanical averaging. Of course, the same distinction could be made for a stable molecule between its minimum energy structure and its equilibrium state at a given temperature and pressure. 

By use of some advanced statistical mechanical averaging procedures it is possible to translate a simation which corresponds to discrete ions moving around in the solution into one... 

Step 3 involved smearing out of the discrete ions of the ionic atmosphere into a continuous cloud of charge so that the Poisson equation could be used. The problem in this step, however, is to find the actual distribution of the discrete ions of the ionic atmosphere around the central y -ion. Statistical mechanics in the form of the Maxwell-Boltzmann distribution is used. The derivation of the Maxwell-Boltzmann distribution automatically involves an implicit averaging process, but this statistical mechanical averaging is different from that used for the Poisson equation when smearing out has been done. 

This is an absolutely crucial part of the model and has been dealt with by statistical mechanical averaging procedures. But only spherical symmetry has been assumed. And so for large non-spherical ions a modification to the smearing out procedure is needed, but see Sections 10.17.3 and 10.19. 

Other quantities can be obtained from suitable derivatives of the free energy or direct statistical mechanics averages e.g. in the semi-grandcanonical ensemble of a symmetrical mixture (NA = NB = N) (where the chemical potential difference Ap = pA — pB per effective monomer is a given independent thermodynamical variable) the order parameter defined as the relative excess in the number of A-chains (nA) over the number of B-chains (nB) in the system... 

Evaluation of a statistical mechanics averaged steric energy on the stable orientations located by the SIMPLEX OPTIMIZATION PROCEDURE. 

Evaluation of a statistical mechanics averaged interaction energy (AH, AS, AG) on the resulting stable orientations. 

The subscript zero indicates an unperturbed state and the angle brackets denote a statistical mechanical average. The length of a fully extended chain is nb or n l. ... 

Osted, A., J. Kongsted, K. V. Mikkelson, P.-O. Astrand, and 0. Christiansen, Statistical mechanically averaged molecular properties of liquid water calculated using the combined coupled cluster/molecular dynamics method. Chem. Phys., 2006.124 124503-124516. 

The following two parameters are traditionally used to characterize the flexibility of an isolated alkyl chain 1) the rms end-to-end vector , a measure of chain deminsions 2) the RIS probabilities p. , a measure of the frequency of occurance of the trans state at a particular C-C bond. Both parameters involve an equilibrium statistical mechanical average (conformational average) over all of the conformations of the chain. They may be computed with a high degree of accuracy using the RIS scheme. 

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