Unit Force Method computations

The efficient computation of fl and A was discussed in detail in Chapt 4. The most efficient method known for the computation of both fl and A for iV < 21 is the Unit Force Method (Method II), which is O(AT ) for an A/ degree-of-freedom manipulator with revolute and/or prismatic joints. For N > 21, the 0(N) Force Propagation Method (Method III) is the most efficient. The use of these two methods will be discussed further in Section 5.1. 

The two tables differ only in the algorithm used to compute the inverse operational space inertia matrix, A and the coefficient fl. In Chapter 4, the efficient computation of these two quantities was discussed in some detail. It was detomined that the Unit Force Method (Method II) is the most efficient algorithm for these two matrices together for N < 21. The Force Propagation Method (Method ni) is the best solution for and fl for AT > 21. The scalar opmtions required for Method II are used in Table 5.1, while those required for Method III are used in Table 5.2. 

The simulated spectrum is computed by the use of eq. (25), wherein the integrals are converted into discrete sums. It is clear from (25) that, in particular, one needs to know the resonant field values for the various transitions, as well as their transition probabilities for numerous orientations of the external magnetic field over the unit sphere over the unit sphere. A considerable saving of computer time can be accomplished if one uses numerical techniques to minimize the number of required diagonalizations of the SH matrix in the brute-force method. That is, when one uses the known resonant-field value at angle (0,(p) to calculate the one at an infinitesimally close orientation, (0 -i- 80, (p + 8(p), known as the method of homo-... 

Force fields split naturally into two main classes all-atom force fields and united atom force fields. In the former, each atom in the system is represented explicitly by potential functions. In the latter, hydrogens attached to heavy atoms (such as carbon) are removed. In their place single united (or extended) atom potentials are used. In this type of force field a CH2 group would appear as a single spherical atom. United atom sites have the advantage of greatly reducing the number of interaction sites in the molecule, but in certain cases can seriously limit the accuracy of the force field. United atom force fields are most usually required for the most computationally expensive tasks, such as the simulation of bulk liquid crystal phases via molecular dynamics or Monte Carlo methods (see Sect. 5.1). 

The molecular dynamics unit provides a good example with which to outline the basic approach. One of the most powerful applications of modem computational methods arises from their usefulness in visualizing dynamic molecular processes. Small molecules, solutions, and, more importantly, macromolecules are not static entities. A protein crystal structure or a model of a DNA helix actually provides relatively little information and insight into function as function is an intrinsically dynamic property. In this unit students are led through the basics of a molecular dynamics calculation, the implementation of methods integrating Newton s equations, the visualization of atomic motion controlled by potential energy functions or molecular force fields and onto the modeling and visualization of more complex systems. 

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