Hypothetical scanning method

H. Meirovitch, /. Chem. Phys., 97,5816 (1992). Entropy, Pressure, and Chemical Potential of Multiple Chain Systems from Computer Simulation. 11. Application of the Metropolis and the Hypothetical Scanning Methods. 

Whereas the MC method selects configurations correctly with the Boltzmann PD, it does not provide the value of this PD, and therefore the absolute entropy cannot be obtained in a direct manner with Eq. [29]. The difficulty stems from the fact that if the simulation starts from configuration i and after t MC steps reaches /, one knows the probability density of the specific avenue i j that was chosen in the simulation. However, to obtain the PD of / one has to sum up the probability densities of all the large number of possible avenues / in t MC steps. Yet as mentioned in the Introduction, P (x) [like E(x)j is also written on the configurations x and can be deciphered approximately using the local states method or the hypothetical scanning method (discussed later). A direct estimation of S (hence F) by Eq. [29] becomes possible with these methods. 

Several methods have been suggested to overcome the attrition problem. One is the scanning method, which is an extension of the Rosenbluth and Rosenbluth procedure. The scanning method forms the basis of the local states method and the hypothetical scanning method, enabling one to estimate the entropy from MC and MD simulations. The fundamental concepts of these techniques are described next. 

Two methods are described here the hypothetical scanning method and the local states (LS) method. Both were developed originally for spin systems and are discussed in a later section, but for simplicity we illustrate how they have been applied to a model of SAWs on a square lattice. These methods enable one to extract the approximate entropy from a sample simulated by any technique, in particular by the MC or the MD procedures. They are based on the concept that two samples in equilibrium generated by different simulation methods are equivalent in the sense that they lead to the same estimates (within statistical error) of average properties, such as the entropy, energy, and their fluctuations. ... 

V], can be used only for short chains. Therefore, the scanning and hypothetical scanning methods are based on the approximate transition probabilities. .., Vj, / ) (Eq. [66]), where in practice N. In principle,... 

As for the hypothetical scanning method, one defines an entropy functional S b) by replacing Pf b) with (/") in Eq. [72] S b) and S ib) are defined correspondingly and have the same properties as those described for their counterparts defined for the hypothetical scanning method. For models with finite interactions, the average energy and each of these functionals lead to approximations for the free energy. For example, one can define b) by... 

It would be of interest to extend the LS method and the hypothetical scanning method to MC and MD samples of continuum models of fluids (e.g., argon and water) and peptides in aqueous solutions. These methods enable one to calculate differences in free energy between significantly different wide microstates directly—that is, from two samples—and in principle, can handle even small differences in free energy of such microstates because the free energy fluctuations decrease with enhancement of the quality of the approximation. 

A data set with 2°<20 140° CuKa was collected for a barrelshaped crystal on an Enraf Nonius CAD4 diffractometer with a 3.5° variable-speed 6-20 scan to accommodate the broad diffractions. From 196 unique diffractions above background (2o) out of 2170 measured diffractions, the structure was solved independently by the Multan direct method and a search of hypothetical structures. The resulting framework (Figure 2) is 81 predicted by J.V. Smith(4). Refinement to R 16% yielded the framework geometry, but the T-0 distances did not correspond satisfactorily to Al, P alternation, and the TPAOH was not located. Nevertheless the framework topology appeared correct, and the calculated powder pattern (Figure lb) was satisfactory. 

Assuming a known and fixed molecular conformation - either the molecule is rigid, or structure optimization has been carried out by some computational method - and that there is no more than one molecule in the asymmetric unit, the variables describing the crystal structure are three coordinates of the center of mass, three orientation angles, and the cell parameters, which range in number from one for a hypothetical cubic crystal to six for a triclinic crystal. These variables define the multidimensional space that must be scanned by the computational procedure. It is a relatively easy matter to instruct a computer to search this space and to provide computational polymorphs for a given molecular structure. 

---