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Note that the growth has introduced a bias into the scheme. A weighting factor has to be calculated in order to take into account for the bias. This weighting factor is defined as
for a particlular selected bead, where i refers to a bead label (0,1,2,...,N) and B is the sum of the total Boltzmann factor for all trial beads for Bead i. The total weighting factor for the chain is therefore W0 X W1 X W2 X ... X WN.
In order to decide whether to accept the newly grown chain or not, the ratio of weighting factors for the new chain over the old chain, W(new)/W(old), is compared with a random number (a floating point number between 0 to 1). If the ratio is greater or equal to the random number the newly grown chain is accepted, otherwise, the old chain is retained and counted once more. If the new chain is accepted
then the current old chain is destroyed. This new chain now becomes old chain for the next MC cycle.
In this way chain structures with lower energies will be more likely to be accepted, although there is still a finite (although small) possibility for chains with higher energies to be accpeted. Such sampling scheme is called the Metropolis importance sampling. It ensures that the areas of phase space which contribute most to the averages are the regions which are sampled most frequently.
This is a superior scheme than the 'hit and miss' approach as given in the example on Page 2. In this simple sampling scheme a large number of structures may have to be generated in order to sample the few that contribute to the sum.
The procedure to obtain W(old) is similar to that of new chain. Basically, similar equation is used as shown above. However, only CHOICE - 1 number of links are generated with the last link being the Bead i itself. Once again, energies are calculated for all links to obtain Bi.
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