This review discusses progress in efficient solvers which have as their foundation a representation in real space, either through finite-difference or finite-element formulations. The relationship of real-space approaches to linear-scaling electrostatics and electronic structure methods is first discussed. Then the basic aspects of real-space representations are presented. Multigrid techniques for solving the discretized problems are covered; these numerical schemes allow for highly efficient solution of the grid-based equations. Applications to problems in electrostatics are discussed, in particular numerical solutions of Poisson and Poisson-Boltzmann equations. Next, methods for solving self-consistent eigenvalue problems in real space are presented; these techniques have been extensively applied to solutions of the Hartree-Fock and Kohn-Sham equations of electronic structure, and to eigenvalue problems arising in semiconductor and polymer physics. Finally, real-space methods have found recent application in computations of optical response and excited states in time-dependent density functional theory, and these computational developments are summarized. Multiscale solvers are competitive with the most efficient available plane-wave techniques in terms of the number of self-consistency steps required to reach the ground state, and they require less work in each self-consistency update on a uniform grid. Besides excellent efficiencies, the decided advantages of the real-space multiscale approach are 1) the near-locality of each function update, 2) the ability to handle global eigenfunction constraints and potential updates on coarse levels, and 3) the ability to incorporate adaptive local mesh refinements without loss of optimal multigrid efficiencies.

An understanding of molecular statistical thermodynamic theory is fundamental to the appreciation of molecular solutions. This complex subject has been simplified by the authors with down-to-earth presentations of molecular theory. Using the potential distribution theorem (PDT) as the basis, the text provides an up-to-date discussion of practical theories in conjunction with simulation results. The authors discuss the field in a concise and simple manner, illustrating the text with useful models of solution thermodynamics and numerous exercises. Modern quasi-chemical theories that permit statistical thermodynamic properties to be studied on the basis of electronic structure calculations are given extended development, as is the testing of those theoretical results with ab initio molecular dynamics simulations. The book is intended for students undertaking research problems of molecular science in chemistry, chemical engineering, biochemistry, pharmaceutical chemistry, nanotechnology, and biotechnology.

Simulations by molecular dynamics of 13-particle clusters of argon display distinct nonrigid, liquid-like and near-rigid, solid-like "phases." The simulations, conducted at constant total energy, display a low-energy region in which only the solid-like form appears, a high-energy region in which only the liquid-like form appears, and an intermediate band of energy-a "coexistence region"-in which clusters exhibit both forms. The intervals of time spent in each phase in the two-form coexistence region are long compared with the intervals required to establish equilibrium-like properties distinctive of each form, such as mean square displacement and power spectrum, so that well-defined phases can be said to exist. The fraction of time spent in each phase is a function of the energy. When a long simulation is separated into regions ofsolidlike and liquid-like behavior, the curve of the derived caloric equation of state is double valued in the two-phase range of energy, forming two well-defined, smooth branches. When, instead, the caloric curve is constructed from averages over all of a long run, its form is smooth and monotonic showing no trace of the "loop" that had been reported for earlier treatments with much shorter molecular dynamics runs, and which we could also reproduce with short runs.

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Simulations by constant energy molecular dynamics have been performed for numerous clusters in the size range N=7–33. Detailed investigations have been conducted on the portions of the caloric curves in which the transition between rigid and nonrigid behavior occurs, to study the N dependence of the solid–liquid phase change. Clusters of several sizes display a coexistence of forms, each with a characteristic mean temperature, over a well-defined energy range in the transition region, as had been observed for the Ar13 cluster. Within the coexistence region, the high temperature form is solid-like and the low temperature form behaves in a liquid-like fashion. The caloric curves of state for these clusters take on two-valued forms when averages are calculated for each of the two ‘‘phases’’ separately; the two branches are smooth extensions of the curves from the single phase regions. Clusters of other sizes do not display this clear coexistence of phases, but appear to pass through a ‘‘slush-like’’ state during the melting transition. The coexistence behavior is not a smooth function of N. The clusters Ar13 and Ar19, and to a certain extent Ar7, display high stability properties indicative of magic number behavior.

Free energy partitioning analysis is employed to explore the driving forces for ions interacting with the water liquid-vapor interface using recently optimized point charge models for the ions and SPC/E water. The Na(+) and I(-) ions are examined as an example kosmotrope/chaotrope pair. The absolute hydration free energy is partitioned into cavity formation, attractive van der Waals, local electrostatic, and far-field electrostatic contributions. We first compute the bulk hydration free energy of the ions, followed by the free energy to insert the ions at the center of a water slab. Shifts of the ion free energies occur in the slab geometry consistent with the SPC/E surface potential of the water liquid-vapor interface. Then the free energy profiles are examined for ion passage from the slab center to the dividing surface. The profiles show that, for the large chaotropic I(-) ion, the relatively flat total free energy profile results from the near cancellation of several large contributions. The far-field electrostatic part of the free energy, largely due to the water liquid-vapor interface potential, has an important effect on ion distributions near the surface in the classical model. We conclude, however, that the individual forms of the local and far-field electrostatic contributions are expected to be model dependent when comparing classical and quantum results. The substantial attractive cavity free energy contribution for the larger I(-) ion suggests that there is a hydrophobic component important for chaotropic ion interactions with the interface.

Solute retention in reversed phase liquid chromatography is driven by the free energy gradient on passing from the polar mobile phase into the nonpolar stationary phase. Only a partial understanding of retention exists currently. We present large scale molecular dynamics simulations of the transfer of a simple nonpolar solute from a water/methanol solvent mixture into a C 18 stationary phase at room temperature. In addition to a detailed examination of the local environment of the solute, we compute the excess chemical potential profile. The overall free energy change is consistent in magnitude with that expected for hydrophobic transfer from water rich to oil phases, but specific interfacial effects draw into question bulk partitioning models.

A theoretical study of the structural and electronic properties of the chloride ion and water molecules in the first hydration shell is presented. The calculations are performed on an ensemble of configurations obtained from molecular dynamics simulations of a single chloride ion in bulk water. The simulations utilize the polarizable AMOEBA force field for trajectory generation, and MP2-level calculations are performed to examine the electronic structure properties of the ions and surrounding waters in the external field of more distant waters. The ChelpG method is employed to explore the effective charges and dipoles on the chloride ions and first-shell waters. The Quantum Theory of Atoms in Molecules (QTAIM) is further utilized to examine charge transfer from the anion to surrounding water molecules. The clusters extracted from the AMOEBA simulations exhibit high probabilities of anisotropic solvation for chloride ions in bulk water. From the QTAIM analysis, 0.2 elementary charges are transferred from the ion to the first-shell water molecules. The default AMOEBA model overestimates the average dipole moment magnitude of the ion compared with the estimated quantum mechanical value. The average magnitude of the dipole moment of the water molecules in the first shell treated at the MP2 level, with the more distant waters handled with an AMOEBA effective charge model, is 2.67 D. This value is close to the AMOEBA result for first-shell waters (2.72 D) and is slightly reduced from the bulk AMOEBA value (2.78 D). The magnitude of the dipole moment of the water molecules in the first solvation shell is most strongly affected by the local water-water interactions and hydrogen bonds with the second solvation shell, rather than by interactions with the ion.