Centroids

Voth G 1996 Path integral centroid methods Advances in Chemical Physics, New methods in Computational Quantum Mechanics vol XCIII, ed I Prigogine and S A Rice... 

As a result of several complementary theoretical efforts, primarily the path integral centroid perspective [33, 34 and 35], the periodic orbit [36] or instanton [37] approach and the above crossover quantum activated rate theory [38], one possible candidate for a unifying perspective on QTST has emerged [39] from the ideas from [39, 40, 4T and 42]. In this theory, the QTST expression for the forward rate constant is expressed as [39]... 

The quantity is the Feynman path integral centroid density [43] that is understood to be expressed asymptotically as... 

The key feature of A3.8.18 is that the centroids of the reaction coordinate Feymnan paths are constrained to be at the position q. The centroid g particular reaction coordinate path q(x) is given by the zero-frequency Fourier mode, i.e. 

Under most conditions, the sign of V" (q ) in (A3.8.17) is negative. In such cases, the centroid variable naturally appears in the theory 39, and the equation for the quantum thennal rate constant from (A3.8.14) -(A3.8.17) is tlien given by [39]... 

It should be noted that in the cases where y"j[,q ) > 0, the centroid variable becomes irrelevant to the quantum activated dynamics as defined by (A3.8.Id) and the instanton approach [37] to evaluate based on the steepest descent approximation to the path integral becomes the approach one may take. Alternatively, one may seek a more generalized saddle point coordinate about which to evaluate A3.8.14. This approach has also been used to provide a unified solution for the thennal rate constant in systems influenced by non-adiabatic effects, i.e. to bridge the adiabatic and non-adiabatic (Golden Rule) limits of such reactions. 

It seems that surface hopping (also called Molecular Dynamics with Quantum Transitions, MDQT) is a rather heavy tool to simulate proton dynamics. A recent and promising development is path integral centroid dynamics [123] that provides approximate dynamics of the centroid of the wavefunctions. Several improvements and applications have been published [123, 124, 125, 126, 127, 128). 

Cao, J., Voth, G.A. The formulation of quantum statistical mechanics based on the Feynman path centroid density. I. Equilibrium properties. J. Chem. Phys. 100 (1994) 5093-5105 II Dynamical properties. J. Chem. Phys. 100 (1994) 5106-5117 III. Phase space formalism and nalysis of centroid molecular dynamics. J. Chem. Phys. 101 (1994) 6157-6167 IV. Algorithms for centroid molecular dynamics. J. Chem. Phys. 101 (1994) 6168-6183 V. Quantum instantaneous normal mode theory of liquids. J. Chem. Phys. 101 (1994) 6184 6192. 

Levitt Warshel [17, 18] were the first to show that reduced representations may work they used Ca atoms and virtual atoms at side chain centroids. OOBATAKE Crippen [24] simplified further by only considering the Ca atoms. This is snfficient since there are reasonably reliable methods (Holm Sander [11, 12]) that compute a full atom geometry from the geometry of the Ca atoms. (All atom representations are used as well, but limited to the prediction of tiny systems such as enkephalin.)... 

The distance of a compound from a library can be given by the distance of this compound from a) its nearest neighbor in the library, b) its fe nearest neighbors in the library, and c) the centroid of the Ubraiy. 

This calibration model for the best-fit fit line requires that the line pass through the centroid of the points (X, Y). It can be shown that ... 

The curve (a) traces the outline of the peak obtained directly from the number of events recorded (Figure 31.5). The second curve (b) traces the outline of the peak obtained after correcting for coincidental events (dead time, shown by the shaded area). The centroids of peaks a and b are shown, and it can be seen that they occur at the same m/z value. Thus the deadtime correction alters only the abundances and not the m/z values of the ions. 

If digital voltage readings (V1-V9) are taken at time intervals (At = 0.0001 sec in the example of Figure 44.4), then the area of the true peak (dotted) can be (mathematically) closely approximated to give ion abundance and, similarly, the time (tj to the center of gravity (centroid) of the peak can be determined, thereby giving the m/z value. 

This reduction in information is achieved by a preprocessor, which uses the digital voltages corresponding to an ion peak to estimate the peak area (ion abundance) and centroid (mean arrival time of peak, equivalent to m/z value) these two pieces of information — plus a flag to identify the peak — are stored. 

Fig. 3. Location of traverse sampling points, (a) Cross section of stack showing location of traverse points ( ) on perpendicular diameters. The circular cross section is divided into three equal areas at 0.5774 r, 0.8165 r, and r, where ris the radius. Sampling points are at the centroids of these areas at 0.38 r, 0.70 r, and 0.911 r. (b) Cross section of rectangular stack divided into 12 equal areas having traverse points ( ) at the centroid of each area.

An x-ray study of the stmcture of Cp2Hf(CO)2 revealed the expected tetrahedral disposition of ligands with OC—Hf—CO and (centroid Cp)—Hf—(centroid Cp) angles of 89.3° and 141°, respectively, and mean bond lengths for both bond types of 0.216 nm (241). The Zr analogue is isomorphous with bond lengths of 0.2187 nm and a OC—Zr—CO bond angle of 89.2° (242). 

ISCC-NBS Centroid Color Charts, NBS Standard Reference Material No. 2106, National Institute of Standards and Technology, Washiagton, D.C.,... 

For each independent variable, form the average value at which it was run in the complex. Draw a line from the coordinates of the worst cake through the average point—called the centroid—and continue on that line a distance that is twice that between these two points. This point will be the next test point. First decide if it is feasible. If so, bake the cake and discover if it leads to a cake that is better than the worst cake from the last set of cakes. If it is not feasible or it is not better, then return half the distance toward the average values from the last... 

The force exerted on a submerged planar surface of area A is given by F = p A where p is the pressure at the geometrical centroid of the surface. The center of pressure, the point of application of the net force, is always lower than the centroid. For details see, for example. Shames, where may also be found discussion of forces on curved surfaces, buoyancy, and stability of floating bodies. 

FIG. 25-25 Example showing rectangular stack cross section divided into 12 equal areas, with a traverse point at centroid of each area. 

To show that this guess is actually consistent with the Im F approach and to see what happens to the velocity factor at low temperatures let us study the statistics of centroids. We introduce the centroid density... 

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