# Dynamic average

Ion-atom and ion-molecule collisions at energies in the kiloelectron volts range are common in studies of energy deposition and stopping of swift particles in various materials. Theoretical treatments of such processes often employ stationary electronic states and their potential energy surfaces. At such elevated energies the relevant state vector of the system is an evolving state, which may be expressed as a superposition of a number of such energy states. In fact, the system moves on an effective PES, which is the dynamical average of a number of adiabatic smfaces and should in principle also include effects of the nonadiabatic coupling terms. [c.221]

Hi) The use of quantum methods to obtain correct statistical static (but not dynamic) averages for heavy quantum particles. In this category path-integral methods were developed on the basis of Feynman s path [c.4]

Averages or plotted values at regular time intervals. You specify an Average/Graph period in the Molecular Dynamics Averages dialog box. [c.80]

You choose the values to average in the Molecular Dynamics Averages dialog box. As you run a molecular dynamics simulation, HyperChem stores data in a CSV file. This file has the same name as the HIN file containing the molecular system, plus the extension. CSV. If the molecular system is not yet stored in a HIN file, HyperChem uses the filename chem.csv. [c.86]

Subsequent to equilibration, averages over the trajectory can be accumulated to describe statistical mechanical properties. For example, to calculate an average bond length, the bond should first be selected, prior to collecting molecular dynamics data or playing back snapshots, and made a named selection with the Select/Name Selection menu item. Then, the named selection should be placed in the Average only or Avg. graph column of the Molecular Dynamics Averages dialog box invoked by the Averages button of the Molecular Dynamics Options dialog box. A molecular dynamics simulation will then average the bond length. The average may be viewed after the sampling by re-opening the [c.316]

Molecular Dynamics Averages dialog box and selecting the desired property, so that it is outlined, in the Average only or Avg. graph column. If a graph was requested, the average appearing here is equivalent to the average over the simulation or playback period. [c.317]

To average or plot a structural quantity, the structural quantity must first be selected and named by the normal process for creating named selections (select the atoms and then use the menu item Select/Name Selection to give the selected atoms a name). Erom then on the Molecular Dynamics Averages dialog box will show these named selections as possible candidates to be averaged or plotted in addition to energetic quantities described above. [c.321]

To average a torsion, select the four atoms of the torsion, name the torsion tor, for example, and then select tor as the quantity to be averaged from the Molecular Dynamics Averages dialog box. [c.321]

The single average value that is reported for this quantity in the Molecular Dynamics averages dialog box is the limit reached by the plotted values at i=N, i.e. the RMS value of x [c.322]

You can request a plot by selecting a quantity from the Averages Only column of the Molecular Dynamics Averages dialog box and passing it to the right (to the Avg. graph column) by clicking the Add button. Only four quantities can be plotted as indicated by their presence in this last column of the dialog box. A molecular [c.323]

With any iterative approach it is necessary to have convergence criteria in order to judge when to exit an iterative loop. In the case of the optimization of empirical force field parameters it is difficult to define rigorous criteria due to the often poorly defined and system-dependent nature of the target data however, guidelines for such criteria are appropriate. In the case of the geometries, it is expected that bond lengths, angles, and torsion angles of the fully optimized model compound should all be within 0.02 A, 1.0°, and 1.0° respectively, of the target data values. In cases where both condensed phase and gas-phase data are available, the condensed phase data should be weighted more than the gas-phase data, although in an ideal simation both should be accurately fit. It should be noted that, because of the harmonic nature of bonds and angles, values determined from energy minimization are generally equivalent to average values from MD simulations, simplifying the optimization procedure. This, however, is less true for torsional angles and for non-bond interactions, for which significant differences in minimized and dynamic average values can exist. With respect to vibrational data, generally a root-mean-square (RMS) difference of 10 cm or an average difference of 5% between the empirical and target data should be considered satisfactory. Determination of these values, however, is generally difficult, owing to problems associated with unambiguous assignment of the normal modes to the individual frequencies. Typically the low frequency modes (below 500 cm ) associated with torsional deformations and out-of-plane wags and the high frequency modes (above 1500 cm ) associated with bond stretching are easy to assign, but significant mixing of different angle bending and ring deformation modes makes assignments in the intermediate range difficult. What should be considered when optimizing force constants to reproduce vibrational spectra is which modes will have the greatest impact on the final application for which the parameters are being developed. If that final application involves MD simulations, then the low frequency modes, which involve the largest spatial displacements, are the most important. Accordingly, efforts should be made to properly predict both the frequencies and assignments of these modes. For the 500-1500 cm region, efforts should be made to ensure that frequencies dominated by specific normal modes are accurately predicted and that the general assignment patterns are similar between the empirical and target data. Finally, considering the simplicity of assigning stretching frequencies, the high frequency modes should be accurately assigned, although the common use of the SHAKE algorithm [95] to constrain covalent bonds during MD simulations, especially those involving hydrogens, leads to these modes often having no influence on the results of MD simulations. [c.32]

An alternative strategy for calculating systemwide averages is to follow the motion of a single point tlirough phase space instead of averaging over the whole phase space all at once. That is, in this approach the motion of a single point (a single molecular state) through phase space is followed as a function of time, and the averages are calculated only over those points that were visited during the excursion. Averages calculated in this way are called dynamic averages. The motion of a single point through phase space is obtained by integrating the system s equation of motion. Starting from a point r(0), p(0), the integration procedure yields a trajectory that is the set of points r(t), p(it) [c.41]

Dynamic average. An average over a single point in phase space at all times. [c.42]

It is hoped that the point that is being dynamically followed will eventually cover all of phase space and that the dynamic average will converge to the desired thermodynamic average. A key concept that ties the two averaging strategies together is the ergodic hypothesis. This hypothesis states that for an infinitely long trajectory the thermodynamic ensemble average and the dynamic average become equivalent to each other, [c.42]

For the silica gel (Figure 3A), the solution was removed slightly less effectively, and more Cs was left (ca. 0.0020 atoms/A2). The spectral behavior is quite similar to that of boehmite, except that there is a peak due to surface Cs coordinated by only water molecules and not in contact with the surface oxygens (so-called outer sphere complexes)at 30% RH. Complete dynamical averaging among sites at frequencies greater than ca. 10 kHz occurs at 70% RH and greater. [c.162]

The spectral behavior with varying temperature, sample washing and varying solution concentration provides strong confirming evidence for the assignments and also additional dynamical information. Variable temperature data for illite at 100% RH (Figure 4) show that dynamical averaging at freguencies > ca. 1 kHz ceases near -20°C. At lower temperatures more Cs is transferred to inner sphere complexes (the more negative peak) as few water molecules remain on the surface due to freezing. The peaks also become broad, indicating statically asymmetric sites. [c.164]

The dependencies of viscoelastic coefficients Ki, K2 and t) against total doses of the irradiation and times of heating are shown in Fig. 5 and 6. Figures 7 and 8 show the dependencies of viscoelastic coefficients Ki, K2 and ti versus the elongation at break. A plot indicates the average value for ten cases of data resulting from impact tests that were performed with dynamic indentation method (Fig 1). A vertical width in the figure illustrates the amount of scattering. For impact tests the maximum value of contact force was ranging within 7-10 N and maximum depth of penetration increases from 50 to 90 mcm. The dependencies of viscoelastic coefficients Ki, K2 and r] versus the elongation at break present a rather good correlation between dynamic and static properties of PVC materials. [c.244]

A pulse-repetition frequency of 15 shots/sec was sufficient for measurement and determination of eccentricity at the highest drawing speed. This was a reasonable value although it was not obtainable when digital output was based on averaging several shots. Another important question concerning measuring speed was the dynamic range in the measurements. Was every single measurement correct, when the wall thickness was changing from one point to the next A measuring fixture was made to control this (Figure 4). A piece of tube with correct diameter and wall thickness, and with a representative eccentricity was [c.897]

Fluctuations of observables from their average values, unless the observables are constants of motion, are especially important, since they are related to the response fiinctions of the system. For example, the constant volume specific heat of a fluid is a response function related to the fluctuations in the energy of a system at constant N, V and T, where A is the number of particles in a volume V at temperature T. Similarly, fluctuations in the number density (p = N/V) of an open system at constant p, V and T, where p is the chemical potential, are related to the isothemial compressibility iCp which is another response fiinction. Temperature-dependent fluctuations characterize the dynamic equilibrium of themiodynamic systems, in contrast to the equilibrium of purely mechanical bodies in which fluctuations are absent. [c.437]

Predicting the solvent or density dependence of rate constants by equation (A3.6.29) or equation (A3.6.31) requires the same ingredients as the calculation of TST rate constants plus an estimate of and a suitable model for the friction coefficient y and its density dependence. While in the framework of molecular dynamics simulations it may be worthwhile to numerically calculate friction coefficients from the average of the relevant time correlation fiinctions, for practical purposes in the analysis of kinetic data it is much more convenient and instructive to use experimentally detemiined macroscopic solvent parameters. [c.849]

I o averageu torsion, select the fouratoms of the torsion, name the torsion tor, for exam pie, and then select tor as Lh e quan tity Lo be averaged from the Molecular Dynamics. Averages dialog box. [c.321]

You can request the computation of average values by clicking the Averages button in the Molecular Dynamics Options dialog box to display the Molecular Dynamics Averages dialog box. The energetic quantities that can be averaged appear in the left Selection column. When you select one or more of these energetic quantities (EKIN, EPOT, etc.) and click Add, the quantity moves to the Average Only column on the right. You can move quantities back to the left column by selecting them and clicking Del. Quantities residing in the Average Only column are not plotted but are written to a CSV file and averaged over the molecular dynamics trajectory. If you return to this dialog box after generating the trajectory and select one of the quantities so that the outline appears around it in the Average Only column, the average value over the last trajectory is displayed at the bottom of the column next to the word Value . [c.320]

When a molecular dynamics average is requested over a trajectory, certain quantities can be plotted as previously described. It is desirable, however, to allow the user to collect a full set of appropriate data values for possible custom use. Since HyperChem is not a full plotting package, it cannot allow you every desired plot. Therefore, HyperChem collects the basic data used for averaging and plotting into a comma-separated-values file, with the extension xsv, for general use in other applications. For example, the values can be readily brought into Microsoft Excel and plotted and manipulated in innumerable ways using the convenient plotting and statistical analysis features of Excel. [c.322]

For kaolinite the sample permeability was very low and the solution was poorly removed. The spectra (Figure 3C) are consequently complex, containing peaks for inner and outer sphere complexes, CsCl precipitate from resMual solution (near 200 ppm) and a complex spinning sideband pattern. Spectral resolution is poorer, but at 70% RH for instance, inner sphere complexes resonate near 16 ppm and outer sphere complexes near 31 ppm. Dynamical averaging of the inner and outer sphere complexes occurs at 70% RH, and at 100% RH even the CsCl precipitate is dissolved in the water film and averaged. [c.163]

The structural environments and dynamical behavior of surface phosphate species contrasts strongly with those of Cs and Na described above. For Al-containing substrates, phosphate forms strongly bonded inner sphere complexes through P-O-Al linkages. Our work on the P NMR behavior of kaolinite, boehmite, and 7-alumina reacted at 25°C for 24 hrs with KH2P0 solutions with concentrations from 10" to 10" M and pHs from 3 to 11 are consistent with this idea but also indicate that amorphous K-H-aluminophosphate precipitates form under all conditions examined). The absence of dynamical averaging is clearly indicated by the spectrum of a moist 7-alumina sample (Figure 7). The narrow peak near +2 ppm is due to phosphate in the solution, that near -5 ppm to inner sphere complexes, and that near -12 ppm to amorphous precipitates. The chemical shifts of the inner sphere complexes become less negative with increasing pH, consistent with progressive deprotonation with increasing pH and rapid H-exchange among surface phosphate sites. [c.166]

Two simulation methods—Monte Carlo and molecular dynamics—allow calculation of the density profile and pressure difference of Eq. III-44 across the vapor-liquid interface [64, 65]. In the former method, the initial system consists of N molecules in assumed positions. An intermolecule potential function is chosen, such as the Lennard-Jones potential, and the positions are randomly varied until the energy of the system is at a minimum. The resulting configuration is taken to be the equilibrium one. In the molecular dynamics approach, the N molecules are given initial positions and velocities and the equations of motion are solved to follow the ensuing collisions until the set shows constant time-average thermodynamic properties. Both methods are computer intensive yet widely used. [c.63]

Scaiming probe microscopies have afforded incredible insight into surface processes. They have provided visual images of surfaces on the atomic scale, from which the atomic stnicture can be observed in real time. All of the other surface tecimiques discussed above involve averaging over a macroscopic region of the surface. From STM images, it is seen that many surfaces are actually not composed of an ideal single domain, but rather contain a mixture of domains. STM has been able to provide direct infomiation on the structure of atoms in each domain, and at steps and defects on surfaces. Furthemiore, STM has been used to monitor the movement of single atoms on a surface. Refinements to the instruments now allow images to be collected over temperatures ranging from 4 to 1200 K, so that dynamical processes can be directly investigated. An [c.310]

There is, in essence, no limitation, other than the computing time, to the accuracy and predictive capacity of molecular dynamic and Monte Carlo methods, and, although the derivation of realistic potentials for water is a fomiidable task in its own right, we can anticipate that accurate simulations of water will have been made relatively soon. However, there remain major theoretical problems in deriving any analytical theory for water, and indeed any other highly-polar solvent of the sort encountered in nomial electrochemistry. It might be felt, therefore, that the extension of the theory to analytical descriptions of ionic solutions was a well-nigh hopeless task. However, a major simplification of our problem is allowed by the possibility, at least in more dilute solutions, of smoothing out the influence of the solvent molecules and reducing their influence to such average quantities as the dielectric pemiittivity, of the medium. Such a viewpoint is developed within the [c.564]

In two of the simplest Langevin models, the order parameter (ji is the only relevant macrovariable in model A it is non-conserved and in model B it is conserved. (The labels A, B, etc have historical origin from the Langevin models of critical dynamics the scheme is often referred to as the Hohenberg-Halperin classification scheme.) For model A, the Langevin description assumes that, on average, the time rate of change of the order parameter is proportional to (the negative of) the themiodynamic force that drives the phase transition. For this single variable case, the themiodynamic force is canonically conjugate to the order parameter i.e. in a themiodynamic description, if < ) is a state variable, then its canonically conjugate force is dfldo (see figure A3.3.5), where/is the free energy. [c.735]

There are two different aspects to these approximations. One consists in the approximate treatment of the underlying many-body quantum dynamics the other, in the statistical approach to observable average quantities. An exlmistive discussion of different approaches would go beyond the scope of this introduction. Some of the most important aspects are discussed in separate chapters (see chapter A3.7. chapter A3.11. chapter A3.12. chapter A3.131. [c.774]

It is worth discussing the fact that a free energy can be directly relevant to the rate of a dynamical process such as a chemical reaction. After all, a free energy function generally arises from an ensemble average over configurations. On the other hand, most condensed phase chemical rate constants are indeed thennally averaged quantities, so this fact may not be so surprising after all, although it should be quantified in a rigorous fashion. Interestingly, the free energy curve for a condensed phase chemical reaction (cf figure A3.8.1) can be viewed, in effect, as a natural consequence of Onsager s Imear regression hypothesis as it is applied to condensed phase chemical reactions, along with some additional analysis and simplifications [7]. [c.884]

See pages that mention the term

**Dynamic average**:

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Computational biochemistry and biophysics (2001) -- [ c.41 , c.42 , c.270 ]