Electronic Structure of 3d Transition Metal Atoms

Christian B. MendlTU München

Oberwolfach Workshop“Mathematical Methods in Quantum Chemistry”

June 26th – July 2nd, 2011

joint work with Gero Friesecke

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

QM Frameworktime-independent, (non-relativistic, Born-Oppenheimer) Schrödinger equation

with

N number of electronsZ nuclear charge

single particle Hamiltonian: kinetic energy and external nuclear potential

inter-electron Coulomb repulsion

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)

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 space V: 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 valence electrons• retains LS symmetries of the atomic Schrödinger equation• orbital exponent relaxation after symmetry subspace decomposition and Hamiltonian

matrix diagonalization (different from using Hartree-Fock orbitals in CI methods)

taylored to atoms (molecules: STOs inconvenient; no L2 and Lz)

Gero Friesecke and Benjamin D. Goddard, SIAM J. Math. Anal. (2009)

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

Fast Algorithm for LS Diagonalization• 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-Gordan formulae→ 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)

Christian B. Mendl and Gero Friesecke, Journal of Chemical Physics 133, 184101 (2010)

Simultaneous Diagonalization of

result: direct sum of irreducible LS representation spaces

multiplicities of Lz-Sz eigenstates easily enumerable

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 )14 states only

𝑚ℓ≡0 ,𝑚𝑠≡𝑠

such that

Asymptotic LS Dimensions• - eigenvalue multiplicities of • dimension of „central“-eigenspace

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 operationsChristian B. Mendl, Computer Physics Communications 182 1327–1337 (2011) http://sourceforge.net/projects/fermifab

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

Transition Metal Atoms, other Methods

• d

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