Electronic Structure of 3d Transition Metal Atoms

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

2. Outline • Schrödinger equation for an N-electron atom, asymptotics-based () FCI model • this talk: algorithmic framework, up to electrons • basic idea: efficient calculation of symmetry subspaces to escape “curse of dimensionality” • main ingredients: use tensor product structure, irreducible representations of angular momentum and spin eigenspaces

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 • Main idea: resolve gaps and wavefunctions correctly in the large-Z limit, at fixed finite model dimension • finite-dimensional projection of the Schrödinger equation • Ansatz spaceV: obtained via perturbation theory in , contains exact large-Z limits of low eigenstates • for example carbon: V = configurations • asymptotics-based → Slater-type orbitals (STOs) • corresponds to FCI in an active space for the valenceelectrons • retains LS symmetries of the atomic Schrödinger equation • orbital exponent relaxation after symmetry subspace decomposition and Hamiltonian matrix diagonalization (different fromusingHartree-Fock orbitals in CI methods) tayloredtoatoms (molecules: STOs inconvenient; noL2andLz) GeroFriesecke and Benjamin D. Goddard, SIAM J. Math. Anal. (2009)

6. Configurations • fix numbers of electrons in atomic subshells (occupation numbers) • example: • configurations (like 1s2 2s1 2p3) 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 GeroFriesecke, Journal of Chemical Physics 133, 184101 (2010) • goal: decompose FCI space into simultaneous eigenspaces of • before touching the Hamiltonian → huge cost reduction • tensor product structure (no antisymmetrization needed between subshells) → can iteratively employ Clebsch-Gordanformulae → key point: computing time linear in number of subshells at fixed angular momentum cutoff, e.g., • tensor product ↔ lexicographical enumeration of Slaters • still need simultaneous diagonalization on each (next slide)

8. Simultaneous Diagonalizationof result: direct sum of irreducible LS representation spaces multiplicitiesofLz-Szeigenstates easily enumerable

9. Dimension Reduction via Symmetries • diagonalize H within each LS eigenspace separately • representation theory → from each irreducible representation space, need only consider states with quantum numbers (can traverse the and eigenstates by ladder operators and ) • example: Chromium with configurations • full CI dimension: • 7S symmetry level (i.e., , , parity ) such that 14 states only

10. Asymptotic LS Dimensions • -eigenvalue multiplicities of • dimensionof „central“-eigenspace

11. Bit Representations of Slaters • representation of (symbolic) fermionicwavefunctions via bit patterns 1 0 1 1 0 1 0 • RDM formation • creation/annihilationoperatorstranslatedtoefficientbitoperations Christian B. Mendl, Computer Physics Communications 182 1327–1337 (2011) http://sourceforge.net/projects/fermifab

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

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 GeroFriesecke, Journal of Chemical Physics 133, 184101 (2010) Christian B. Mendl, Computer Physics Communications 182 1327–1337 (2011) http://sourceforge.net/projects/fermifab