Canonical weight

The particular linear combinations of the X- euid F-variables achieving the maximum correlation are the so-called first canonical variables, say tj = Xw, and u.-Yq,. The vectors of coefficients Wj and q, in these linear combinations are the canonical weights for the X-variables and T-variables, respectively. For the data of Table 35.5 they are found to be Wj = [0.583, -0.561] and qj = [0.737,0.731]. The correlation between these first canonical variables is called the first canonical correlation, p,. This maximum correlation turns out to be quite high p, = 0.95 R = 0.90), indicating a strong relation between the first canonical dimensions of X and Y. 

We then need to consider a canonical weighted average of the contributions in Eq. (5.75) of all trajectories starting on the reactant side, and crossing the dividing surface in the direction from the reactant side to the product side. Thus, after introduction of P(p, q), Eq. (5.70) takes the form... 

For the canonical ensemble, the systems are constrained to constant N and constant V, and are in contact with a reservoir at constant temperature T. Hence it is an N,V,T ensemble. Since all the ensembles we will consider are at constant N and V, the notation of constant N and V will be suppressed. The canonical weight function is... 

If no better initial guess is available, one typically sets w (E) = 1 in the beginning. The zeroth iteration thus corresponds to a Metropolis run at infinite temperature, which generates the histogram muca( ) = can( ) = g E). Thus, the histogram is already an estimate for the density of states g E) such that s E) = In g E). Then, the first estimate for the multi canonical weight function W l aiE) can be obtained from the recursion (4.105)-(4.108), which is used to initiate the second recursion, etc. The recursion... 

Eor many systems the ensemble that is used in an MC simulation refers to the canonical ensemble, (N, F/ T). This ensemble permits a rise and fall in the pressure of the system, P, because the temperature and volume are held constant. Thus, the probabiUty that any system of N particles, in a volume H at temperature Tis found in a configuration x is proportional to the Boltzmann weighted energy at that state, E, and it is given by... 

An alternative method, proposed by Andersen [23], shows that the coupling to the heat bath is represented by stochastic impulsive forces that act occasionally on randomly selected particles. Between stochastic collisions, the system evolves at constant energy according to the normal Newtonian laws of motion. The stochastic collisions ensure that all accessible constant-energy shells are visited according to their Boltzmann weight and therefore yield a canonical ensemble. 

The N-Lim classification does not eliminate the possibility of borderline cases between these two categories, but it leads to the suggestion that no sharp distinction can be made between the possible intermediates in these mechanisms and that perhaps all solvolyses proceed via an intermediate. The mechanistic category of a particular solvolysis then depends upon the relative weights of the canonical structures 3, 4, and 5 to the transition state resonance hybrid. 

A possibly more accurate value for the double bond character of the bonds in benzene (0.46) id obtained by considering all five canonical structures with weights equal to the squares of their coefficients in the wave function. There is some uncertainty aS to the significance of thfa, however, because of- the noii -orthogOnality of the wave functions for the canonical structures, and foF chemical purposes it fa sufficiently accurate to follow the simple procedure adopted above. 

In the VB method, a wave equation is written for each of various possible electronic structures that a molecule may have (each of these is called a canonical form), and the total )/ is obtained by summation of as many of these as seem plausible, each with its weighting factor ... 

The two chief general methods of approximately solving the wave equation, discussed in Chapter 1, are also used for compounds containing delocalized bonds. In the VB method, several possible Lewis structures (called canonical forms) are drawn and the molecule is taken to be a weighted average of them. Each in Eq. (1.3),... 

A quantitative method for weighting canonical forms has been proposed by Gasteiger, J. Sailer, H. Angew. Chem., Int. Ed. Engl., 1985, 24, 687. 

This form of writing reveals what is meant by the canonical form of a weighted two-layer scheme. 

Upon substituting these expressions into (9) we write the weighted scheme in the canonical form (8), where... 

C.J.F. ter Braak, Interpreting canonical correlation analysis through biplots of structure correlations and weights. Psychometrika, 55 (1990) 519-531. 

The additional factor of Qi(V, T) in Eq. (21) makes the leading term in the sum unity, as suggested by the usual expression for the cluster expansion in terms of the grand canonical partition function. Note that i in the summand of Eq. (20) is not explicitly written in Eq. (21). It has been absorbed in the n , but its presense is reflected in the fact that the population is enhanced by one in the partition function numerator that appears in the summand. Equation (21) adopts precisely the form of a grand canonical average if we discover a factor of (9(n, V, T) in the summand for the population weight. Thus... 

Let us illustrate this procedure with the grand-canonical ensemble, and take the scenario in which we desire to achieve a uniform distribution in particle number N at a given temperature. In the weights formalism, we introduce the weighting factor r/(/V) into the microstate probabilities from (3.31) so that... 

Fig. 3.4. Evolution of the weights i],(N) and the histograms fi(N) in a grand-canonical implementation of the multicanonical method for the Lennard-lones fluid al V 125. The temperature is T = 1.2 and the initial chemical potential is /./ = —3.7. The weights are updated after each 10-million-step interval, and the numbers indicate the iteration number. The second peak in the weights at large particle numbers indicates that the initial chemical potential is close to its value at coexistence...

For Newtonian dynamics and a canonical distributions of initial conditions one can reject or accept the new path before even generating the trajectory. This can be done because Newtonian dynamics conserves the energy and the canonical phase-space distribution is a function of the energy only. Therefore, the ratio plz ]/p z at time 0 is equal to the ratio p[.tj,n ]/p z ° at the shooting time and the new trajectory needs to be calculated only if accepted. For a microcanonical distribution of initial conditions all phase-space points on the energy shell have the same weight and therefore all new pathways are accepted. The same is true for Langevin dynamics with a canonical distribution of initial conditions. 

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