Lorentz-Berthelot rule

Lorentz-Berthelot rule. It is a foundational combining rule used in computational chemistry and statistical mechanics to estimate interaction parameters between dissimilar atoms or molecules, primarily for the Lennard-Jones potential. The rule provides simple arithmetic and geometric means for the potential's energy and size parameters, respectively, enabling the simulation of complex mixtures. Its simplicity has made it a standard in early molecular dynamics and Monte Carlo method simulations, despite known systematic deviations for many real substances.

Definition and mathematical formulation

The rule is applied to the parameters of a pairwise potential, most famously the Lennard-Jones potential. For two interacting particles *i* and *j*, the characteristic energy depth ε_ij and the collision diameter σ_ij are defined using the parameters for like interactions. The Lorentz rule states that the size parameter is the arithmetic mean: σ_ij = (σ_ii + σ_jj)/2. The Berthelot rule states that the energy parameter is the geometric mean: ε_ij = √(ε_ii * ε_jj). These formulations are central to the One-fluid theory approximation in perturbation theory. This approach is implemented in force fields like OPLS and early versions of AMBER and CHARMM for simulating noble gas mixtures or simple hydrocarbon systems.

Historical development and context

The origins of the combining rules lie in early kinetic theory and equations of state. The size-averaging rule is attributed to Hendrik Lorentz, who derived it from the collision of hard spheres in his studies on the theory of electrons. The energy-averaging rule is credited to Daniel Berthelot, who proposed it in the context of his modifications to the van der Waals equation. Their integration became standard with the rise of computer simulation in the mid-20th century, championed by researchers like Aneesur Rahman and Loup Verlet in seminal molecular dynamics studies. The rule's adoption was propelled by the work at institutions like Los Alamos National Laboratory and IBM Research.

Applications in molecular simulation

The rule's primary application is in constructing force field parameters for heterogeneous systems, enabling the study of solvation, diffusion, and phase equilibria. It has been extensively used in simulations of biological macromolecules in water using models like SPC and TIP3P, and in modeling ionic liquids and polymer blends. Major simulation packages, including GROMACS, NAMD, and LAMMPS, historically used it as a default. Its computational efficiency facilitated large-scale studies of protein folding and lipid bilayer properties, contributing to projects like the Folding@home initiative.

Limitations and deviations

Significant systematic deviations from the rule are well-documented, particularly for mixtures involving components with large differences in polarizability or dipole moment. For instance, it fails for aqueous solutions of alkanes, carbon dioxide mixtures, and systems with hydrogen bonding like chloroform and acetone. These failures are addressed in advanced equation of state models like SAFT and CPA. Empirical corrections, such as the Waldman-Hagler rules or the use of a non-unity scaling factor for ε_ij, are often implemented in modern force fields like TraPPE and COMPASS.

Related combining rules

Numerous alternative rules have been developed to overcome its limitations. The Kong rules combine specific combining rules for both energy and size parameters. The Fender-Halsey rule modifies the energy mean with an exponent. For charged systems, rules like those in the DRESP method are used. Other important approaches include the Hudson-McCoubrey rule, which incorporates ionization potential, and the Good-Hope rule, used in surface tension calculations. These are often integrated into more sophisticated molecular models for industrial chemistry applications by companies like Aspen Technology.

Category:Computational chemistry Category:Statistical mechanics Category:Physical chemistry