1. Electronic Structure of 3d Transition Metal Atoms Christian B. Mendl TU München Oberwolfach Workshop “Mathematical Methods in Quantum Chemistry” June 26 th – July 2 nd, 2011 Oberwolfach Workshop “Mathematical Methods in Quantum Chemistry” June 26 th – July 2 nd, 2011 joint work with Gero Friesecke

2. Outline

3. QM Framework time-independent, (non-relativistic, Born-Oppenheimer) Schrödinger equation with N number of electrons Z nuclear charge single particle Hamiltonian: kinetic energy and external nuclear potential inter-electron Coulomb repulsion

4. LS Symmetries invariance under simultaneous rotation of electron positions/spins, sign reversal of positions → angular momentum, spin and parity operators action on N-particle space pairwise commuting: → symmetry quantum numbers (corresponding to eigenvalues)

5. Asymptotics-Based CI Models Gero Friesecke and Benjamin D. Goddard, SIAM J. Math. Anal. (2009)

6. Configurations fix numbers of electrons in atomic subshells (occupation numbers) example: configurations (like 1s 2 2s 1 2p 3 ) invariant under the symmetry operators L, S, R (but not under the Hamiltonian) must allow for all Slater determinants with these occupation numbers, otherwise symmetry lost FCI space equals direct sum of relevant configurations

7. Fast Algorithm for LS Diagonalization Christian B. Mendl and Gero Friesecke, Journal of Chemical Physics 133, 184101 (2010)

8. Simultaneous Diagonalization of result: direct sum of irreducible LS representation spaces multiplicities of L z -S z eigenstates easily enumerable

9. Dimension Reduction via Symmetries 14 states only

10. Asymptotic LS Dimensions

11. Bit Representations of Slaters representation of (symbolic) fermionic wavefunctions via bit patterns 1 0 1 1 0 1 0 RDM formation creation/annihilation operators translated to efficient bit operations Christian B. Mendl, Computer Physics Communications 182 1327–1337 (2011) http://sourceforge.net/projects/fermifab

12. Results for Transition Metal Atoms green: experimental ground state symmetry blue: the lower of each pair of energies → symmetry in exact agreement with experimental data! goal: derive the anomalous filling order of Chromium from first principles quantum mechanics http://sourceforge.net/projects/fermifab additional ideas used: RDMs sparse matrix structure closed-form orthonormalization of STOs, Hankel matrices

13. Transition Metal Atoms, other Methods d

14. Conclusions Efficient algorithm for asymptotics-based CI Key point: fast symmetry decomposition via hidden tensor product structure and iteration of Clebsch-Gordan formula (linear scaling wrt. including higher radial subshells Correctly captures anomalous orbitals filling of transition metal atoms Christian B. Mendl and Gero Friesecke, Journal of Chemical Physics 133, 184101 (2010) Christian B. Mendl, Computer Physics Communications 182 1327–1337 (2011) http://sourceforge.net/projects/fermifab