Asymptotics of the ground state energy of heavy molecules in self-generated magnetic field

Victor Ivrii

Published 2013-03-19, updated 2013-12-31Version 4

We consider asymptotics of the ground state energy of heavy atoms and molecules in the self-generatedl magnetic field. Namely, we consider $$ H=((D-A)\cdot\boldsymbol{\sigma})^2-V $$ with $$V=\sum_{1\le m\le M} \frac{Z_m}{|x-y_m|}$$ and a corresponding Multiparticle Quantum Hamiltonian $$ \mathsf{H}=\sum_{1\le n\le N} H_{x_n} +\sum_{1\le n < n'\le N}|x_n-x_{n'}|^{-1} $$ on the Fock space $\wedge _{1\le n\le N} L^2(\mathbb{R}^3, \mathbb{C}^2)$. Here $A$ is a self-generated magnetic fiels. Then the ground state energy is given by $$ \mathsf{E}(A)=\inf \operatorname{Spec}(\mathsf{H})+\frac{1}{\alpha}\int |\nabla \times A|^2\,dx $$ where the last term is the energy of magnetic field. Under assumption $\alpha Z\le \kappa^*$ (with a small constant $\kappa^*$) we study the ground State Energy $$ \mathsf{E}^*=\inf _{A}\mathsf{E}(A). $$ We derive its asymptotics including Scott, and Schwinger and Dirac corrections. We also consider related topics: an excessive negative charge, ionization energy and excessive positive charge when atoms can still bind into molecules.