Transcript pptx - Christian B. Mendl

Electronic Structure of 3d
Transition Metal Atoms
Christian B. Mendl
TU München
joint work with Gero Friesecke
Oberwolfach Workshop
“Mathematical Methods in Quantum Chemistry”
June 26th – July 2nd, 2011
Outline
• Schrödinger equation for an N-electron atom,
asymptotics-based (𝑍 → ∞) FCI model
• this talk: algorithmic framework, up to 𝑁 = 30
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 Framework
time-independent, (non-relativistic, Born-Oppenheimer)
Schrödinger equation
with
single particle Hamiltonian: kinetic
energy and external nuclear potential
N number of electrons
Z nuclear charge
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:
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→ symmetry quantum numbers (corresponding to eigenvalues)
Asymptotics-Based CI Models
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Main idea: resolve gaps and wavefunctions correctly in the large-Z limit, at
fixed finite model dimension
Gero Friesecke and
finite-dimensional projection of the Schrödinger equation
Benjamin D. Goddard,
SIAM J. Math. Anal. (2009)
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Ansatz space V: obtained via perturbation theory in 1/𝑍, 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)
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
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goal: decompose FCI space into simultaneous eigenspaces of
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before touching the Hamiltonian → huge cost reduction
tensor product structure (no antisymmetrization needed between subshells)
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→ can iteratively employ Clebsch-Gordan formulae
→ key point: computing time linear in number of subshells at fixed angular
momentum cutoff, e.g., 1𝑠, 2𝑠, … , 𝑛 𝑠 ∼ 𝑂(𝑛)
tensor product ↔ lexicographical enumeration of Slaters
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still need simultaneous diagonalization on each
Christian B. Mendl and Gero
Friesecke, Journal of Chemical
Physics 133, 184101 (2010)
(next slide)
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
𝑚ℓ ≡ 0, 𝑚𝑠 ≡ 𝑠
(can traverse the 𝐿𝑧 and 𝑆𝑧 eigenstates by ladder operators 𝐿± and 𝑆± )
• example: Chromium with configurations
𝐴𝑟 3𝑑 𝑗 4𝑠 4𝑝𝑘 4𝑑 ℓ such that 𝑗 + 𝑘 + ℓ = 5
• full CI dimension:
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7S
26
5
= 65780
symmetry level (i.e., ℓ = 0, 𝑠 = 3, parity 𝑝 = 1)
14 states only
Asymptotic LS Dimensions
• 𝐿𝑧 -𝑆𝑧 eigenvalue multiplicities of ⋀4 𝑉8
• dimension of „central“𝐿𝑧 -𝑆𝑧 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 operations
Christian B. Mendl, Computer
Physics Communications 182
1327–1337 (2011)
http://sourceforge.net/projects/fermifab
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
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