Two complementary molecular energy decomposition schemes: the Mayer and Ziegler-Rauk methods in comparison

In the present paper we discuss and compare two different energy decomposition schemes: Mayer's Hartree-Fock energy decomposition into diatomic and monoatomic contributions [Chem. Phys. Lett. 382, 265 (2003)], and the Ziegler-Rauk dissociation energy decomposition [Inorg. Chem. 18, 1558 (1979)]. The Ziegler-Rauk scheme is based on a separation of a molecule into fragments, while Mayer's scheme can be used in the cases where a fragmentation of the system in clearly separable parts is not possible. In the Mayer scheme, the density of a free atom is deformed to give the one-atom Mulliken density that subsequently interacts to give rise to the diatomic interaction energy. We give a detailed analysis of the diatomic energy contributions in the Mayer scheme and a close look onto the one-atom Mulliken densities. The Mulliken density ��A has a single large maximum around the nuclear position of the atom A, but exhibits slightly negative values in the vicinity of neighboring atoms. The main connecting point between both analysis schemes is the electrostatic energy. Both decomposition schemes utilize the same electrostatic energy expression, but differ in how fragment densities are defined. In the Mayer scheme, the electrostatic component originates from the interaction of the Mulliken densities, while in the Ziegler-Rauk scheme, the undisturbed fragment densities interact. The values of the electrostatic energy resulting from the two schemes differ significantly but typically have the same order of magnitude. Both methods are useful and complementary since Mayer's decomposition focuses on the energy of the finally formed molecule, whereas the Ziegler-Rauk scheme describes the bond formation starting from undeformed fragment densities

Friedman's Excess energy and the McMillan-Mayer theory of solutions:Thermodynamics

In his version of the theory of multicomponent systems, Friedman used the analogy which exists between the virial expansion for the osmotic pressure obtained from the McMillan-Mayer (MM) theory of solutions in the grand canonical ensemble and the virial expansion for the pressure of a real gas. For the calculation of the thermodynamic properties of the solution, Friedman proposed a definition for the"excess free energy" that is a reminder of the ancient idea for the"osmotic work". However, the precise meaning to be attached to his free energy is, within other reasons, not well defined because in osmotic equilibrium the solution is not a closed system and for a given process the total amount of solvent in the solution varies. In this paper, an analysis based on thermodynamics is presented in order to obtain the exact and precise definition for Friedman"s excess free energy and its use in the comparison with the experimental data.

El naturalismo cient��fico en la obra de John Tyndall: Fragments of science

El an��lisis de los ensayos de John Tyndall, Fragments of Science, permite identificar la teor��a at��mica, el principio de conservaci��n de la energ��a y el evolucionismo darwinista como los elementos constitutivos del andamiaje te��rico del naturalismo cient��fico. As��, en su ensayo ���On the Study of Physics��� se resumen sus brillantes facetas como educador y divulgador cient��fico, desarrolladas fundamentalmente en el seno de la Royal Institution. En la lectura ���On Force���, Tyndall da por finalizada la controversia Joule- Mayer sobre la primac��a del descubrimiento del principio de conservaci��n de la energ��a, a la vez que plantea algunas de las claves de la lucha por el liderazgo en el seno de la comunidad cient��fica. El discurso presidencial ante la British Association de 1874 en Belfast ejemplifica el coraje de Tyndall en su empe��o por demarcar los territorios de la ciencia y la religi��n, a la luz de los nuevos desaf��os cient��ficos. En el trasfondo subyacen los procesos de secularizaci��n de la sociedad y de profesionalizaci��n de una comunidad cient��fica heterog��nea. El compromiso c��vico que Tyndall demuestra en ���The Belfast Address��� es digno corolario de una vida y obra que permite situarlo como paradigma de lo que, en la terminolog��a de Turner, se ha dado en denominar cient��fico p��blico.

Effective atomic orbitals for fuzzy atoms

The method of extracting effective atomic orbitals and effective minimal basis sets from molecular wave function characterizing the state of an atom in a molecule is developed in the framework of the "fuzzy" atoms. In all cases studied, there were as many effective orbitals that have considerable occupation numbers as orbitals in the classical minimal basis. That is considered to be of high conceptual importance

Second-order M��ller-Plesset perturbation theory without basis set superposition error : II : open-shell systems

The basis set superposition error-free second-order M��ller-Plesset perturbation theory of intermolecular interactions was studied. The difficulties of the counterpoise (CP) correction in open-shell systems were also discussed. The calculations were performed by a program which was used for testing the new variants of the theory. It was shown that the CP correction for the diabatic surfaces should be preferred to the adiabatic ones

Energy partitioning for "fuzzy" atoms

The total energy of molecule in terms of 'fuzzy atoms' presented as sum of one- and two-atomic energy components is described. The divisions of three-dimensional physical space into atomic regions exhibit continuous transition from one to another. The energy components are on chemical energy scale according to proper definitions. The Becke's integration scheme and weight function determines realization of method which permits effective numerical integrations

Density functional energy decomposition into one- and two-atom contributions

The present work provides a generalization of Mayer's energy decomposition for the density-functional theory (DFT) case. It is shown that one- and two-atom Hartree-Fock energy components in Mayer's approach can be represented as an action of a one-atom potential VA on a one-atom density �� A or �� B. To treat the exchange-correlation term in the DFT energy expression in a similar way, the exchange-correlation energy density per electron is expanded into a linear combination of basis functions. Calculations carried out for a number of density functionals demonstrate that the DFT and Hartree-Fock two-atom energies agree to a reasonable extent with each other. The two-atom energies for strong covalent bonds are within the range of typical bond dissociation energies and are therefore a convenient computational tool for assessment of individual bond strength in polyatomic molecules. For nonspecific nonbonding interactions, the two-atom energies are low. They can be either repulsive or slightly attractive, but the DFT results more frequently yield small attractive values compared to the Hartree-Fock case. The hydrogen bond in the water dimer is calculated to be between the strong covalent and nonbonding interactions on the energy scale

One- and two-center physical space partitioning of the energy in the density functional theory

A conceptually new approach is introduced for the decomposition of the molecular energy calculated at the density functional theory level of theory into sum of one- and two-atomic energy components, and is realized in the "fuzzy atoms" framework. (Fuzzy atoms mean that the three-dimensional physical space is divided into atomic regions having no sharp boundaries but exhibiting a continuous transition from one to another.) The new scheme uses the new concept of "bond order density" to calculate the diatomic exchange energy components and gives them unexpectedly close to the values calculated by the exact (Hartree-Fock) exchange for the same Kohn-Sham orbitals

A Reply to comments by D. Bar-Yosef M. Stiner on "Prehistoric exploitation of marine resources in Southern Africa with particular reference to shellfish gathering: opportunities and continuities"

I am very grateful to Daniella E. Bar-Yosef Mayer and Mary Stiner for their comments on an article I published in the previous issue of Pyrenae. Having spent many years working with coastal sites in South Africa and now settling in the Mediterranean academic landscape, I value the feedback from these two well-known archaeologists who have dedicated years of hard work in this later part of the world. Their opinions are very much appreciated for they allow me to bring new contexts to some of the (old) questions I have pursued in South Africa, an exercise that would help me with the process of broadening my research interests to the Mediterranean region.