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.

In this paper we describe a general method for the numerical solution of the full hypernetted‐chain (HNC) theory for fluids characterized by angle‐dependent pair potentials. This method is also applicable to the closely related reference hypernetted‐chain (RHNC) approximation. The only formal restriction is that the pair potential and correlation functions must be expandable in a basis set of rotational invariants. We present explicit numerical solutions of the RHNC theory for dense dipolar hard sphere fluids and detailed comparisons are made with previous theories and computer simulation results. It is found that the full RHNC theory generally improves upon the previous reference linearized and quadratic HNC approximations. The values given by the RHNC theory for the static dielectric constants are smaller than those given by these earlier approximations and are in much better agreement with computer simulations.

The solution of the hypernetted‐chain approximation for fluids of nonspherical particles. A general method with application to dipolar hard spheres

A Monte Carlo method for obtaining the solvent‐averaged interionic potential of mean force is described. The potential of mean force is obtained for two charged hard spheres immersed in a dipolar hard sphere solvent. The ions and solvent particles have the same hard sphere diameters and the ions bear single charges of opposite sign. The solvent particles are characterized by a reduced dipole moment μ*=1.0 which corresponds to a dielectric constant e?7.8. The Monte Carlo result is compared with the ’’primitive model’’ and other approximations which have been suggested for the potential of mean force. These approximations are all found to be inadequate for small ionic separations. Some implications with respect to ’’ion pairing’’ are mentioned.

A Monte Carlo method for obtaining the interionic potential of mean force in ionic solution

This paper describes a theoretical study of the thermodynamic, dielectric, and structural properties of model aqueous electrolyte solutions. The model considered consists of hard sphere ions immersed in a hard polarizable dipole tetrahedral–quadrupole solvent with water‐like parameters. The calculations involve the solution of the reference hypernetted‐chain (RHNC) approximation for ion–solvent mixtures at finite concentration and some details of the general method are discussed. The influence of the molecular polarizability of the solvent particles is treated at the self‐consistent mean field (SCMF) level and, surprisingly, the mean dipole moment of the solvent is found to be nearly independent of the salt concentration. Numerical results are reported for model alkali halide solutions and other 1:1 electrolytes, and comparisons are made with experimental results at 25 °C. The agreement obtained between theory and experiment is variable depending upon the particular property and solution considered. In ad...

On the molecular theory of aqueous electrolyte solutions. I. The solution of the RHNC approximation for models at finite concentration

Molecular-dynamics simulations are used to show that strongly interacting dipolar spheres can form a ferroelectric nematic phase. This is the first demonstration that dipolar forces alone can create an orientationally ordered liquid state. It is also the first time that the existence of a ferroelectric nematic phase has been established for a model fluid

Orientational order in simple dipolar liquids: Computer simulation of a ferroelectric nematic phase.

The mean spherical, linearized hypernetted chain and quadratic hypernetted chain approximations are solved for a fluid of hard spheres with embedded point dipoles and tetrahedral quadrupoles and this system is shown to be quite similar to the dipole-linear quadrupole case previously studied. However, tetrahedral quadrupoles have a larger influence upon the structural and thermodynamic properties and are slightly more effective in decreasing the dielectric constant from the purely dipolar value. Also we describe a simple self-consistent mean field theory which allows molecular polarizability to be taken into account. This approximation together with the integral equation methods is applied to a polarizable dipole-tetrahedral quadrupole fluid with water-like parameters. The dielectric constant of this system is found to be in good agreement with the experimental results for liquid water for temperatures ranging from 25°C to 300°C. The influence of molecular polarizability is shown to be very large. At 25°C ...

G. N. Patey

Papers

The solution of the hypernetted‐chain approximation for fluids of nonspherical particles. A general method with application to dipolar hard spheres

A Monte Carlo method for obtaining the interionic potential of mean force in ionic solution

On the molecular theory of aqueous electrolyte solutions. I. The solution of the RHNC approximation for models at finite concentration

Orientational order in simple dipolar liquids: Computer simulation of a ferroelectric nematic phase.

Fluids of polarizable hard spheres with dipoles and tetrahedral quadrupoles Integral equation results with application to liquid water