There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

The Weighted Histogram Analysis Method (WHAM), an extension of Ferrenberg and Swendsen's Multiple Histogram Technique, has been applied for the first time on complex biomolecular Hamiltonians. The method is presented here as an extension of the Umbrella Sampling method for free-energy and Potential of Mean Force calculations. This algorithm possesses the following advantages over methods that are currently employed: (1) It provides a built-in estimate of sampling errors thereby yielding objective estimates of the optimal location and length of additional simulations needed to achieve a desired level of precision; (2) it yields the “best” value of free energies by taking into account all the simulations so as to minimize the statistical errors; (3) in addition to optimizing the links between simulations, it also allows multiple overlaps of probability distributions for obtaining better estimates of the free-energy differences. By recasting the Ferrenberg–Swendsen Multiple Histogram equations in a form suitable for molecular mechanics type Hamiltonians, we have demonstrated the feasibility and robustness of this method by applying it to a test problem of the generation of the Potential of Mean Force profile of the pseudorotation phase angle of the sugar ring in deoxyadenosine. © 1992 by John Wiley & Sons, Inc.

The weighted histogram analysis method for free-energy calculations on biomolecules. I: The method

Nonphysical sampling distributions in Monte Carlo free-energy estimation: Umbrella sampling

http://experimentationlab.berkeley.edu/sites/default/files/AFMImages/butt_AFM.pdf

Force measurements with the atomic force microscope: Technique, interpretation and applications

Implicit Solvation Models: Equilibria, Structure, Spectra, and Dynamics.

Isotropic and anisotropic hypernetted-chain (HNC) integral equation theories are used to obtain the interaction of solutes both near and far from the solvent liquid-vapor coexistence. Spherically symmetrical and chemically patterned (patched) solutes are considered, and the influences of particle and patch sizes are investigated. Solvophilic and solvophobic solutes (or patches) are examined. Near coexistence, in the solvophobic case dryinglike behavior occurs for solutes (patches) of sufficient size. This gives rise to relatively long ranged attractive forces that are strongly orientation dependent for the patched solute particles. We also report grand canonical Monte Carlo results for a pair of spherically symmetric solutes. This demonstrates that the anisotropic HNC theory gives qualitatively correct solvent structure in the vicinity of the solutes. Comparison with previous simulations also shows that the solute-solute potentials of mean force given by the anisotropic theory are more accurate (particula...

Solvent phase behavior and the interaction of uniform and patterned solutes.

In this work, a general theory to account for any kind of polarization arising from polar as well as ions induced interactions in fluid mixtures is proposed. The general treatment is based on the self-consistent mean field theory (SCMF) that was originally proposed and applied for pure components using integral equation theories and molecular simulation studies. The extended SCMF is consistent with theory-based equations of state applied to hard chain mixtures. The theory is extended to mixtures and compared to molecular simulation data. The comparison to molecular simulation data shows good to excellent results for phase co-existence properties, energy and effective dipole moment. The validity of the theory is demonstrated by studying VLE of highly non-ideal mixtures of aldehyde and ketones using statistical association fluid theory.

A general treatment of polar-polarizable systems for an equation of state

Mean solute-solute forces and solute-induced solvent structure are investigated for pairs of chemically patterned (patched) solutes in binary mixtures near demixing coexistence. The isotropic and anisotropic hypernetted-chain integral equation theories as well as a superposition approximation are solved and compared. The patched solutes consist of one end that favors the majority species in the mixture while the other end favors the minority species. A wide range of patch sizes is considered. The isotropic and anisotropic theories are found to be in good agreement for most orientations, including the most attractive and most repulsive configurations. However, some differences arise for asymmetrical orientations where unlike ends of the solute particles face each other. In contrast, superposition often gives a rather poor approximation to the mean force, even though the results obtained for the solvent densities agree qualitatively with the anisotropic theory. The mean force is sensitive to small differenc...

The interaction of patterned solutes in binary solvent mixtures.

On the theoretical calculation of the dielectric constant of dilute electrolyte solutions

Density‐functional theory is applied to the problem of salt crystallization from solution and explicit results are given for model aqueous alkali‐halide systems. Both direct‐ and Fourier‐space methods of calculation are considered and it is found that only the direct (i.e., r space) method converges sufficiently rapidly to provide reliable results for ionic crystals at 25 °C. It is shown that the density‐functional method is capable of predicting crystallization, but that the solid‐state parameters and, for some salts, the crystal structures obtained are in poor agreement with experiment or computer simulations. The calculated crystal/solution coexistence concentrations are found to be extremely sensitive to the short‐range part of the interionic pair potentials. This is consistent with earlier observations that the activity coefficients of model aqueous alkali‐halide solutions are very strongly dependent upon the short‐range ion–ion interactions. Therefore, we do not believe that this sensitivity to details of the short‐range interionic potentials is an artifact of theoretical approximations, but rather a real effect significantly influencing crystallization.

G. N. Patey

Papers

Solvent phase behavior and the interaction of uniform and patterned solutes.

A general treatment of polar-polarizable systems for an equation of state

The interaction of patterned solutes in binary solvent mixtures.

On the theoretical calculation of the dielectric constant of dilute electrolyte solutions

The crystallization of alkali halides from aqueous solution: An application of density‐functional theory