Centroid methods

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... 

Furthermore, the situation becomes even worse for an asymmetric potential like that in (3.18), because at low temperature nearly the entire period p is spent on dwelling in the potential well (see appendix A), so that lim -oo < >ins = 0- In other words, unless the potential is strictly symmetric, the transition state position x tends to the minimum of the initial state It is natural to expect that the centroid approximation will work well when x does not deviate too far from x. To summarize, the centroid method is an instructive way to describe in a unique TST-like manner both the high [T > T ) and fairly low [T < T ) temperature regions, but it does not give a reliable estimate for k. ... 

The method used here is called Forgy s method [25], This is one of the K-center or K-centroid methods, another well-known variant of which is MacQueen s K-means method [26]. Forgy s method involves the following steps. 

The most popular recipe for defuzzification, although not the fastest, is the centroid method. This finds the vertical line, which would divide the aggregated set determined in the previous step, into two equal portions. The center of gravity (cog) is defined by ... 

It is not possible to make a general statement as to whether the single- or centroid-link approach will perform better. The centroid method results in a better measure of the distance betw een clusters when the clusters are well separated. This is because it measures the distance between the centers of the clusters rather than the distances from the edges as in single link. On the other hand, the centroid method is sensitive to outliers because these unusual points... 

Further improvement of the centroid method came with the introduction of centroid dynamics.Here the fundamental idea is to construct a centroid Hamiltonian in the full phase space of the system and the bath. The Boltzman factor is then the one obtained from this centroid Hamiltonian while seal time dynamics is obtained by running classical trajectories. This method has been applied to realistic systems " and recently derived from first principles.244 The main advantage of the centroid methodology is that thermodynamic quantum effects can be computed numerically exactly as it is not too difficult to converge numerically the computation of the centroid potential. 

V.2.2 Quantum transition state theory. The centroid method is one way... 

In the last few years we have witnessed the successful development of several methods for the numerical solution of multi-dimensional quantum Hamiltonians Monte Carlo methods centroid methods,mixed quantum-classical methods, and recently a revival of semiclassical methods. We have developed another approach to this problem, the exponential resummation of the evolution operator. - The rest of this Section will explain briefly this method. 

G. A. Voth, Adv. Chem. Phys., 93,135 (1996). Path-Integral Centroid Methods in Quantum... 

VoTH, G. A., Path-integral centroid methods in quantum statistical mechanics and dynamics, Adv. Chem. Phys. 1996, 93, 135-218. 

The centroid method uses a simple algorithm which finds the geometric center of the portion of the SPR dip imder a certain threshold. Although the geometric center does not necessarily coincide with the minimum of the spectrum, as SPR sensing usually relies on relative measurements, the offset of the geometric center does not affect the final measurements. The centroid is calculated as follows ... 

Subsequently, a metric which can be associated with the angular or wavelength position of an SPR dip is calculated from angiflar or wavelength normalized spectrum. Numerous algorithms have been developed to calculate the sensor output. One of the most commonly utihzed algorithms is the centroid method, which calculates the centre-of-mass of a portion of SPR dip that... 

PATH-INTEGRAL CENTROID METHODS IN QUANTUM STATISTICAL MECHANICS AND... 

---