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**Unformatted text preview: **© M. S. Shell 2009 1/14 last modified 6/2/2009 Other free energy techniques ChE210D Today's lecture: various other methods for computing free energies, including absolute free energies, beyond histogram techniques Perturbation approaches to free energies Perturbation techniques have a long history in statistical mechanics and were among the earliest methods used to compute free energy changes in molecular simulations. They were pioneered by Born and Kirkwood in theory in the 1920s and 1930s. In the 1950s, Zwanzig introduced the free energy perturbation (FEP) method in the context of Monte Carlo and molecular dynamics simulations. The basic idea of the method is to compute the free energy between a reference state and some perturbed state. The perturbed state may include an additional particle, a slightly differ- ent potential energy function, or a small change in temperature, for example. The FEP approach is highly general and can be applied to compute many kinds of free energies. Early implementations of the FEP method do not have as good statistical accuracy as multiple histogram reweighting techniques, although subsequent re-formulations have yielded ap- proaches that are equally as accurate and, in some cases, identical to the Ferrenberg-Swendsen reweighting equations. Basic formalism In the following example, we will consider the free energy change as one perturbs the potential energy function in the canonical ensemble. Initially the energy function is g G ¡¢ £ ¤ and we perturb it to g ¥ ¡¢ £ ¤ . The earlier notes on histograms and free energies provided some exam- ples of kinds of perturbations that might be considered. One could also derive an expression in which we perturbed the temperature rather than the potential. The free energy difference between states 1 and 2 stems from a ratio of partition functions: ¦§ ¥ ¨ ¦§ G © ¨ ln ª G ª ¥ © ¨ ln « ¬ ¥ Λ¡¤ ®£ ¯! ° « ¬ G Λ¡¤ ®£ ¯! ° © ¨ ln ± ² ³´µ ¶ ·¢ ¸ ¹ º¢ £ ± ² ³´µ » ¡¢ ¸ ¤ º¢ £ © M. S. Shell 2009 2/14 last modified 6/2/2009 We can now re-express the top integral with the following identity: gG ¡ ¢ gG £ ¤ ¢ln ¥ ¦ §¨© ª «¬ ®¯¨© ° «¬ ®§¨© ° «¬ ® ±¬ ² ¥ ¦ §¨© ° ³¬ ´ ±¬ ² ¤ ¢ln ¥ ¦ §¨µ©«¬ ®§¨© ° «¬ ® ±¬ ² ¥ ¦ §¨© ° ³¬ ´ ±¬ ² Here, Δ¶³¬ ² ´ ¤ ¶ ¡ ³¬ ² ´ ¢ ¶ £ ³¬ ² ´ . Notice that this expression is reminiscent of the configura- tional distribution in state 0: · £ ³¬ ² ´ ¤ ¦ §¨© ° «¬ ® ¥ ¦ §¨© ° ³¬ ´ ±¬ ² Making this substitution, gG ¡ ¢ gG £ ¤ ¢ ln¸ · £ ³¬ ² ´¦ §¨µ©«¬ ® ±¬ ² We can re-express this as an ensemble average in state 0: gG ¡ ¢ gG £ ¤ ¢ ln¹¦ §¨µ© º £ This important result shows that we can compute the free energy difference between the two states by performing an average over configurations in state 0. Practically, we can perform a...

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