FUNDAMENTALS The quantum-mechanical many-electron problem and Density Functional Theory Emilio Artacho Department of Earth Sciences University of Cambridge.

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Presentation transcript:

FUNDAMENTALS The quantum-mechanical many-electron problem and Density Functional Theory Emilio Artacho Department of Earth Sciences University of Cambridge Summer school 2002 Linear-scaling ab initio molecular modelling of environmental processes

First-principles calculations Fundamental laws of physics Set of “accepted” approximations to solve the corresponding equations on a computer No empirical input PREDICTIVE POWER

Artillery F = m a Approximations Flat Earth Constant g (air friction: phenomenological)

Fundamental laws for the properties of matter at low energies Atomic scale (chemical bonds etc.) Yes BUT Electrons and nuclei (simple Coulomb interactions) => Quantum Mechanics

Many-particle problem Schroedinger’s equation is exactly solvable for - Two particles (analytically) - Very few particles (numerically) The number of electrons and nuclei in a pebble is ~10 => APPROXIMATIONS 23

Born-Oppenheimer (1) (2)   Nuclei are much slower than electrons electronic/nuclear decoupling

electronselectrons nucleinuclei Classical =>

Many-electron problem Old and extremely hard problem! Different approaches Quantum Chemistry (Hartree-Fock, CI…) Quantum Monte Carlo Perturbation theory (propagators) Density Functional Theory (DFT) Very efficient and general BUT implementations are approximate and hard to improve (no systematic improvement) (… actually running out of ideas …)

Density-Functional Theory 2. As if non-interacting electrons in an effective (self-consistent) potential 1. particle density

Hohenberg - Kohn For our many-electron problem (universal functional) (depends on nuclear positions) PROBLEM: Functional unknown! PROBLEM:

Kohn - Sham Independent particles in an effective potential They rewrote the functional as: Kinetic energy for system with no e-e interactions Hartree potential The rest: exchange correlation Equivalent to independent particles under the potential

E xc & V xc Local Density Approximation (LDA) (function parameterised for the homogeneous electron liquid as obtained from QMC) Generalised Gradient Approximation (GGA) (new terms parameterised for heterogeneous electron systems (atoms) as obtained from QC)

Independent particles

Self-consistency PROBLEM: The potential (input) depends on the density (output)

Solving: 1. Basis set Expand in terms of a finite set of known wave-functions Expand in terms of a finite set of known wave-functions unknown unknown ~ - ~- HC n =ɛ n SC n DefDef andand

Basis set: Atomic orbitals s p d f Strictly localised (zero beyond cut-off radius)

Solving: 2. Boundary conditions Isolated object (atom, molecule, cluster):Isolated object (atom, molecule, cluster): open boundary conditions open boundary conditions (defined at infinity) (defined at infinity) 3D Periodic object (crystal):3D Periodic object (crystal): Periodic Boundary Conditions Periodic Boundary Conditions Mixed: 1D periodic (chains) Mixed: 1D periodic (chains) 2D periodic (slabs) 2D periodic (slabs) Isolated object (atom, molecule, cluster):Isolated object (atom, molecule, cluster): open boundary conditions open boundary conditions (defined at infinity) (defined at infinity) 3D Periodic object (crystal):3D Periodic object (crystal): Periodic Boundary Conditions Periodic Boundary Conditions Mixed: 1D periodic (chains) Mixed: 1D periodic (chains) 2D periodic (slabs) 2D periodic (slabs)

k-point sampling Electronic quantum states in a periodic solid labelled by: Band index Band index k-vector: vector in reciprocal space within the first Brillouin k-vector: vector in reciprocal space within the first Brillouin zone (Wigner-Seitz cell in reciprocal space) zone (Wigner-Seitz cell in reciprocal space) Other symmetries (spin, point-group representation…) Other symmetries (spin, point-group representation…) Electronic quantum states in a periodic solid labelled by: Band index Band index k-vector: vector in reciprocal space within the first Brillouin k-vector: vector in reciprocal space within the first Brillouin zone (Wigner-Seitz cell in reciprocal space) zone (Wigner-Seitz cell in reciprocal space) Other symmetries (spin, point-group representation…) Other symmetries (spin, point-group representation…) Approximated by sums over selected k points

Some materials’ properties Exp.LAPWOther PWPWDZP a (Å) CB (GPa) E c (eV) a (Å) SiB (GPa) E c (eV) a (Å) NaB (GPa) E c (eV) a (Å) CuB (GPa) E c (eV) a (Å) AuB (GPa) E c (eV)

Absence of DC conductivity in -DNA P. J. de Pablo et al. Phys. Rev. Lett. 85, 4992 (2000) Effect of sequence disorder and vibrations on the electronic structure => Band-like conduction is extremely unlikely: DNA is not a wire

Pressing nanotubes for a switch M. Fuhrer et al. Science 288, 494 (2000) Y.-G. Yoon et al. Phys. Rev. Lett. 86, 688 (2001) Pushed them together, relaxed & calculated conduction at the contact: SWITCH

Pyrophyllite, illite & smectite Structural effects of octahedral cation substitutions C. I. Sainz-Diaz et al. (American Mineralogist, 2002) Organic molecules intercalated between layers M. Craig et al. (Phys. Chem. Miner. 2002) WET SURFACES

Recap Born-Oppenheimer: electron-nuclear decoupling Many-electron -> DFT (LDA, GGA) One-particle problem in effective self- consistent potential (iterate) Basis set => Solving in two steps: 1. Calculation of matrix elements of H and S 2. Diagonalisation Extended crystals: PBC + k sampling