Spatial Symmetry Simplifications

Spatial symmetry plays a role in a large number of the examples in Part II of this book. This can arise in a number ways, but the two most important are simplification of the calculations and labeling of the energy states. We have devoted considerable time and space in Chapter 5 to the effects of identical particle symmetry and spin. In this chapter we look at some of the ways spatial symmetry interacts with anti-symmetrization. 

For the simple one-electron reduction that we study here, it is possible to completely omit the boundary columns and rows without adversely affecting the simulation results as long as a reasonably dense spatial grid is employed. For the Z-sweep, we may simply ignore the first and last columns (f = 0 and i = n — I respectively), and for the R-sweep, the first and last rows (j = 0 and j = m — I respectively). This removes the need for extra code to deal with the altered explicit (A) terms that are necessary when considering these boundary rows and columns. However, for simulations of more complicated situations, such as homogeneous kinetics, this simplification may not be appropriate when considering the axial symmetry boundary, therefore we provide implementation details here. 

Eqs. (8.2) and (8.3) is that the effective interaction between two sites on the polymer depends on their instantaneous position, and only on the entire macromolecule conformation in an average (implicit) sense via the direct and collective pair correlations. This simplification does not preclude describing situations of broken conformational symmetry, such as polymer collapse, solvated electron localization, or spatially inhomogeneous conformational characteristics such as occur in star polymers (see Section IX)... 

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