Scaling limits of the electronic Schrödinger equation fordiatomic molecules: Asymptotic prediction of correlation structure,potential energy curves, and symmetry quantum numbers

• Applicant: Professor Dr. Gero Friesecke
• Subject Area: Mathematics
• Term: from 2013 to 2018
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 234732874

The electronic Schrödinger equation plays a central role in molecular physics and quantum chemistry as it provides a quantitative and chemically specific description of a molecule's electronic structure. However, rigorous mathematical work on this equation has hitherto focused on qualitative, universal aspects. Here we will develop methods for the rigorous extraction of quantitative and chemically specific predictions in natural scaling limits, (i) the limit of high nuclear charge at fixed electron number and (ii) the double-limit of high nuclear charge and large interatomic distance. Explicit asymptotic results on correlation structure, potential energy curves, and symmetry quantum numbers will be worked out for the homonuclear dimer series H_2, He_2, Li_2, Be_2, B_2, C_2, N_2, O_2, F_2, Ne_2. This will require generalizing the functional-analytic and symmetry-reduction methods of Friesecke and Goddard [1, 2] from atoms to dimers, careful use of the theory of the single-electron Schrödinger equation for diatomic molecules, and utilization of symbolic computer packages such as Mathematics to handle cases with six or more electrons.The asymptotic results will be compared to experimental data, to the semi-empirical picture of Hund-Mulliken molecular orbital theory (we expect, based on analogous findings for atoms [1, 2], that asymptotic ground states in regime (ii) are similar to this picture), and state-of-the-art computational methods (both of wave function and density functional type). The envisioned results will establish, for the first time, a rigorous link between the quantitative precision of the Schrödinger equation and semi-empirical concepts of chemical bonding, elucidate mathematically the mechanisms leading to the high chemical specificity of bond formation, and provide rare benchmark correlated many-electron data for the design and validation of computational methods.

• DFG Programme: Research Grants
• Participating Person: Privatdozent Dr. Heinz-Jürgen Flad