Electronic Structure of 3D Transition Metal Atoms Christian B. Mendl M7 (Gero Friesecke) TU München DMV Jahrestagung March 10, 2010 DMV Jahrestagung March 10, 2010

Outline basic setting: Schrödinger equation for an N- electron atom, configuration-interaction (CI) model this talk: algorithmic framework, up to N = 30 electrons naïve approch: main ingredient: efficient simultaneous angular momentum and spin diagonalization (prior to setting up the CI Hamiltonian) Gero Friesecke and Christian B. Mendl. An algorithmic framework for electronic structure calculations of 3D transition metal atoms. to be published (2010)

QM Framework time-independent, nonrelativistic Schrödinger equation with N number of electrons Z > 0 nuclear charge single particle Hamiltonian: kinetic energy and external nuclear potential inter-electron Coulomb repulsion

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

Configurations start from irreducible representation spaces for the joint spin and angular momentum algebra in the single-particle L 2 space (“atomic subshells”) abstract definition of “configuration” invariant under the angular momentum, spin and parity operators L, S, R example: Gero Friesecke and Benjamin D. Goddard. Asymptotics-based CI models for atoms. SIAM Journal on Multiscale Modeling and Simulation (2009)

Asymptotics-Based CI Models finite-dimensional projection of the Schrödinger equation retains the spin and angular momentum symmetries of the atomic Schrödinger equation asymptotics-based → Slater-type orbitals (STOs), correct asymptotic behavior for N fixed and Z→∞ under suitable choice of configurations

LS Diagonalization basic idea: calculate simultaneous angular momentum, spin and parity eigenspaces first before touching the Hamiltonian → huge cost reduction need tensor product structure to make use of Clebsch- Gordan coefficients, but have to take antisymmetrization constraint into account → solution via configurations isometry observation: isometry respects LS operators, i.e., Gero Friesecke and Christian B. Mendl. An algorithmic framework for electronic structure calculations of 3D transition metal atoms. to be published (2010)

→ LS Diagonalization (cont.) first step: simultaneous diagonalization of configurations → each V I is an invariant subspace of the LS-operators → diagonalization can be performed on each V I independently via Clebsch-Gordan coefficients, → obtain decomposition into irreducible LS eigenspaces Angelika Bunse-Gerstnert, Ralph Byers, and Volker Mehrmann. Numerical methods for simultaneous diagonalization. SIAM Journal on Matrix Analysis and Applications (1993) such that

Cost Analysis: Configurations algorithm of order, e.g. simultaneous matrix diagonalization Stirling’s formula → approximated gain factor (quotient) independent of particle number N! example: Chromium with three active subshells (3p,3d,4s) and N = 12 active orbitals, algorithmic order p = 3 → vs. approximated factor: “exact” factor:

Cost Analysis: Symmetries diagonalize H within each LS eigenspace separately representation theory → need only consider states with quantum numbers (can traverse the L z and S z eigenstates by ladder operators L ± and S ± ) example: Chromium with configurations full CI dimension: 7 S symmetry level 14 states only

Results green: experimental ground state symmetry blue: the lower of each pair of energies exact agreement with experimental data! goal: derive the anomalous filling order of Chromium from first principles quantum mechanics additional ideas used: RDMs sparse matrix structure closed-form orthnormalization, Hankel matrices