Histograms

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Unformatted text preview: © M. S. Shell 2009 1/29 last modified 6/2/2009 Histograms and free energies ChE210D Today's lecture: basic, general methods for computing entropies and free ener- gies from histograms taken in molecular simulation, with applications to phase equilibria. Overview of free energies Free energies drive many important processes and are one of the most challenging kinds of quantities to compute in simulation. Free energies involve sampling at constant temperature, and ultimately are tied to summations involving partition functions. There are many kinds of free energies that we might compute. Macroscopic free energies We may be concerned with the Helmholtz free energy or Gibbs free energy. We might com- pute changes in these as a function of their natural variables. For single-component systems: gG¡,¢,£¤ ¥G¡,¦,£¤ For multicomponent systems, gG¡, ¢,£ § ,…,£ ¨ ¤ ¥G¡,¦,£ § ,…,£ ¨ ¤ Typically we are only interested in the dependence of these free energies along a single para- meter, e.g., gG¢¤,¥G¦¤,¥G¡¤, etc. for constant values of the other independent variables. Free energies for changes in the interaction potential It is also possible to define a free energy change associated with a change in the interaction potential. Initially the energy function is © ª G« ¬ ¤ and we perturb it to © § G« ¬ ¤ . If this change happens in the canonical ensemble, we are interested in the free energy associated with this perturbation: Δg ­ g § G¡,¢, £¤ ® g ª G¡,¢,£¤ © M. S. Shell 2009 2/29 last modified 6/2/2009 g G¡ ¢ £ ln ¤ ¥¦ §¨© ª «¬ ­ ® ¯¬ ° ¥ ¦ §¨© ± ²¬ ­ ³ ¯¬ ° ´ What kinds of states 1 and 0 might we use to evaluate this expression? Here is a small number of sample applications: • electrostatic free energy – charging of an atom or atoms in a molecule, in which state 0 has zero partial charges and state 1 has finite values • dipolar free energy – adding a point dipole to an atom between states 0 and 1 • solvation free energy – one can “turn on” interactions between a solvent and a solute as a way to determine the free energy of solvation • free energy associated with a field – states 0 and 1 correspond to the absence and presence, respectively, of a field, such as an electrostatic field • restraint free energy – turning on some kind of restraint, such as confining a molecule to have a particular conformation or location in space. Such restraints would corres- pond to energetic penalties for deviations from the restrained space in state 1. • free energies of alchemical transforms – we convert one kind of molecule (e.g., CH 4 ) to another kind (e.g., CF 4 ). This gives the relative free energies of these two kinds of mole- cules in the system of interest (e.g., solvation free energies in solution)....
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This note was uploaded on 12/29/2011 for the course CHE 210d taught by Professor Shell during the Fall '09 term at UCSB.

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Histograms - © M S Shell 2009 1/29 last modified...

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