# Proyecto Sabático Ikerbasque Prof. Vignale

#### Basque Government and UPV/EHU

*Status: finished project*

**Starting date**

**Ending date**

Atomic shell structure, electron pair bonds, lone electron pairs, π-electron subsystems, etc., are some of the most basic concepts in descriptive chemistry. These concepts enable us to visualize the bonding between the atoms and the overall molecular geometry in terms of small groups of localized electrons (e.g. two electrons of opposite spin in a simple covalent bond) and play an important role in guessing new molecular structures and in describing the time evolution of bonds (chemical reactions). Naturally, there have been many attempts to put these concepts on firm quantitative grounds, but none is completely satisfactory so far. Molecular orbitals and electronic densities miss the relevant information, while the laplacian of the density and the more recent "electron localization function" (ELF) have a mixed record of success and failure, and, more importantly, lack a sound physical justification.

The aim of this proposal is to develop the application of a new descriptive tool, the stress tensor – a symmetric 3´3 matrix field, which has a solid foundation in the classical mechanics of continuous media and can be generalized to quantum electronic systems. As its name suggests, the stress tensor gives a compact description of the forces per unit area (stresses) acting across variously oriented surfaces in the electron liquid. Its use in the present context is strongly suggested by the fact that the average of its diagonal elements -- the quantum pressure -- is similar to (but not identical with) the electron localization function. Our recent calculations ["Quantum Stress Focusing in Descriptive Chemistry", J.Tao, G. Vignale, and I. V. Tokatly, Phys. Rev. Lett. 100, 206405 (2008)] demonstrate that indeed an appropriately normalized quantum pressure reveals the well-known shell structure of atoms more clearly than the ELF, whereas the density and to a certain extent the Laplacian of the density fail (the latter in heavier atoms). Because of its physical and geometrical significance, the quantum pressure appears to be a more fundamental and more powerful indicator of atomic structure, than any of the previously proposed ones. Preliminary calculations for molecules, however, have shown that the quantum pressure is not sufficiently subtle to reveal and distinguish different types of bonds (e.g. double vs triple carbon-carbon bonds) because it gives excessive weight to the structure of the individual atoms. This led us to the idea of looking for different invariants, e.g. the eigenvalues of the stress tensor and the ratios between them, or even visualizing the whole stress tensor (eigenvalues+principal axes) in search for distinctive patterns that would allow us to recognize different types of chemical bond. Once this is accomplished, we shall move on to the problem of visualizing the time evolution of chemical bonds in chemical reactions, optically induced ionization, and dissociation.

Our interest in these ideas comes after years of work in density functional theory (DFT), a powerful theoretical method that allows, among other things, an efficient calculation of the stress tensor and a natural application of continuum mechanics to quantum many-body systems. We believe that the stress tensor, if properly represented, has the potential to supersede previous indicators and to become the standard tool for the visualization of molecular structures. But we definitely need to do more ground work to identify the correct representation of the stress tensor and to establish the credibility of the proposed approach. In particular, we must clarify the role played by shear stresses, which are completely ignored in the naive Thomas-Fermi theory. We (Ilya Tokatly and Giovanni Vignale) have a strong record of productive collaboration during the past few years, and each of us has made important individual contributions to density functional theory. We believe that during the few months of Vignale’s visit to San Sebastian we will be able to make much progress toward the achievement of the objectives outlined above.

### Participants

### Visitors

- Dr. Giovanni Vignale