Javier Junquera Fundamentals: the quantum-mechanical many-electron problem and the Density Functional Theory approach

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Most important reference followed in the tutorial:

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Most important reference followed in the tutorial: comprehensive review of DFT, including most relevant references and exercises

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Other reference books Rigurous and unified account of the fundamental principles of DFT
More intended for researchers and advanced students

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Other references: original milestones reviews and papers

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Other interesting references: Nobel lectures by W. Kohn and J. A. Pople Nobel prize in Chemistry 1998

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PROPERTIES structural electronic magnetic vibrational optical Goal: Describe properties of matter from theoretical methods firmly rooted in fundamental equations

Goal: Describe properties of matter from theoretical methods firmly rooted in fundamental equations:

Goal: Describe properties of matter from theoretical methods firmly rooted in fundamental equations

The Schrödinger equation (differential) must be solved subject to appropriate boundary conditions:

The Schrödinger equation (differential) must be solved subject to appropriate boundary conditions Atoms and molecules Regular infinite solid Appropriate periodic boundary conditions 0 at infinity

The electrons are fermions, the solution must satisfy the Pauli exclusion principle:

The electrons are fermions, the solution must satisfy the Pauli exclusion principle A many electron wave function must be antisymmetric with respect to the interchange of the coordinate (both space and spin) of any two electrons

Once the many-body wave function is known, we compute the expectation values of observables:

Once the many-body wave function is known, we compute the expectation values of observables A particular measurement give particular eigenvalue of Â
Many measurements average to < Â >

Minimization of the energy functional, totally equivalent to diagonalize the eigenvalue problem:

Minimization of the energy functional, totally equivalent to diagonalize the eigenvalue problem Since the eigenstates of the many-body hamiltonian are stationary points (saddle points or the minimum) The normalization condition can be imposed using Lagrange multipliers This must holds for any variation in the bra, so this can be satisfied if the ket satisfies

A closer look to the hamiltonian: A difficult interacting many-body system.:

A closer look to the hamiltonian: A difficult interacting many-body system. Kinetic energy operator for the electrons Potential acting on the electrons due to the nuclei Electron-electron interaction Kinetic energy operator for the nuclei Nucleus-nucleus interaction

This hamiltonian can not be solved exactly: practical and fundamental problems:

This hamiltonian can not be solved exactly: practical and fundamental problems Fundamental problem:
Schrödinger’s equation is exactly solvable for
- Harmonic oscillator (analytically)
- Two particles (analytically)
- Very few particles (numerically) Practical problem:
The number of electrons and nuclei in a pebble is of the order of 1023

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A macroscopic solid contains a huge number of atoms Au atomic weight: 196.966569 200
Number of moles in 1 kg of Au Atoms of Au in interaction

If the problem can not be solved exactly, how can we work it out from first-principles?:

If the problem can not be solved exactly, how can we work it out from first-principles? Use a set of “accepted” approximations
to solve the corresponding equations on a computer NO EMPIRICAL INPUT Chemical composition
Number of atoms
Type
Position Properties
Equilibrium structure
Band structure
Vibrational spectrum
Magnetic properties
Transport properties
… IDEAL AB-INITIO CALCULATION

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What are the main approximations? Born-Oppenhaimer
Decouple the movement of the electrons and the nuclei.
Density Functional Theory
Treatment of the electron ─ electron interactions.
Pseudopotentials
Treatment of the (nuclei + core) ─ valence.
Basis set
To expand the eigenstates of the hamiltonian.
Numerical evaluation of matrix elements
Efficient and self-consistent computations of H and S.
Supercells
To deal with periodic systems

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What are the main approximations? Born-Oppenhaimer
Decouple the movement of the electrons and the nuclei.
Density Functional Theory
Treatment of the electron ─ electron interactions.
Pseudopotentials
Treatment of the (nuclei + core) ─ valence.
Basis set
To expand the eigenstates of the hamiltonian.
Numerical evaluation of matrix elements
Efficient and self-consistent computations of H and S.
Supercells
To deal with periodic systems

Adiabatic or Born-Oppenheimer approximation decouple the electronic and nuclear degrees of freedom:

Adiabatic or Born-Oppenheimer approximation decouple the electronic and nuclear degrees of freedom At any moment the electrons will be in their ground state for that particular instantaneous ionic configuration.

If the nuclear positions are fixed (ignore nuclear velocities), the wave function can be decoupled :

If the nuclear positions are fixed (ignore nuclear velocities), the wave function can be decoupled Constant (scalar) Fixed potential “external” to e-

The next problem… how to solve the electronic equation:

The next problem… how to solve the electronic equation Exact solution only for one electron systems H, hydrogenoid atoms, H2+ Main difficulty: very complicate electron-electron interactions.

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What are the main approximations? Born-Oppenhaimer
Decouple the movement of the electrons and the nuclei.
Density Functional Theory
Treatment of the electron ─ electron interactions.
Pseudopotentials
Treatment of the (nuclei + core) ─ valence.
Basis set
To expand the eigenstates of the hamiltonian.
Numerical evaluation of matrix elements
Efficient and self-consistent computations of H and S.
Supercells
To deal with periodic systems

The many-electron problem in interaction: An 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 …) The many-electron problem in interaction: An old and extremely hard problem.

DFT: primary tool for calculation of electronic structure in condensed matter:

DFT: primary tool for calculation of electronic structure in condensed matter A special role can be assigned to the density of particles in the ground-state of a quantum many-body system Many electron wave function Undoubted merit: satisfies the many-electron Schrödinger equation Contains a huge amount of information 3N degrees of freedom for N electrons One electron density Integrates out this information One equation for the density is remarkably simpler than the full many-body Schrödinger equation All properties of the system can be considered as unique functionals of the ground state density

First theorem of Hohenberg-Kohn :

First theorem of Hohenberg-Kohn

Corollary of first theorem of Hohenberg-Kohn :

Corollary of first theorem of Hohenberg-Kohn

Second theorem of Hohenberg-Kohn :

Second theorem of Hohenberg-Kohn

Some definitions :

Some definitions Function: rule for going from a variable x to a number f(x) Functional: rule for going from a function to a number
A function of which the variable is a function -300 eV (a value for the energy) Universal means the same for all electron systems, independent of the external potential

The kinetic energy and the interaction energy of the particles are functionals only of the density:

The kinetic energy and the interaction energy of the particles are functionals only of the density PROBLEM: Functional is unkown Excited states for the electrons must be determined by other means.

The Kohn-Sham ansatz replaces the many-body problem with an independent-particle problem:

The Kohn-Sham ansatz replaces the many-body problem with an independent-particle problem But no prescription to solve the difficult interacting many-body hamiltonian

One electron or independent particle model:

One electron or independent particle model We assume that each electron moves independently in a potential created by the nuclei and the rest of the electrons. Actual calculations performed on the auxiliary independent-particle system

The independent-particle kinetic energy is given explicitly as a functional of the orbitals:

The independent-particle kinetic energy is given explicitly as a functional of the orbitals Equivalent to independent particles under the potential They rewrote the functional as Coulomb The rest:
Exchange-correlation

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The one-particle eigenstates are filled following the “Aufbau” principle: from lower to higher energies The ground state has one (or two if spin independent) in each of the orbitals with the lowest eigenvalues

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The Kohn-Sham equations must be solved self-consistently The potential (input) depends on the density (output)

The paper by Kohn-Sham contains an error… :

The paper by Kohn-Sham contains an error…

All the unknown terms below a carpet: the exchange-correlation functional:

All the unknown terms below a carpet: the exchange-correlation functional

DFT thanks to Claudia Ambrosch (Graz):

DFT thanks to Claudia Ambrosch (Graz) GGA follows LDA

All the unknown terms below a carpet: the exchange-correlation functional:

All the unknown terms below a carpet: the exchange-correlation functional Provide required accuracy for DFT to be adopted by the Chemistry Community
Problem: does not lead to consistent improvement over the LSDA

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Accuracy of the xc functionals in the structural and electronic properties LDA: crude aproximation but sometimes is accurate enough (structural properties, …).
GGA: usually tends to overcompensate LDA results, not always better than LDA.

In some cases, GGA is a must: DFT ground state of iron:

In some cases, GGA is a must: DFT ground state of iron LSDA
NM
fcc
in contrast to
experiment
GGA
FM
bcc
Correct lattice constant
Experiment
FM
bcc
GGA GGA LSDA LSDA Results obtained with Wien2k.
Courtesy of Karl H. Schwartz

Kohn-Sham fails in strongly correlated systems:

Kohn-Sham fails in strongly correlated systems CoO
in NaCl structure
antiferromagnetic: AF II
insulator
t2g splits into a1g and eg‘
Both LDA and GGA find them to be metals (although GGA almost splits the bands) LDA GGA gap Results obtained with Wien2k.
Courtesy of Karl H. Schwartz

The number of citations allow us to gauge the importance of the works on DFT:

11 papers published in APS journals since 1893 with >1000 citations in APS journals (~5 times as many references in all science journals) From Physics Today, June, 2005 The number of citations allow us to gauge the importance of the works on DFT

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What are the main approximations? Born-Oppenhaimer
Decouple the movement of the electrons and the nuclei.
Density Functional Theory
Treatment of the electron ─ electron interactions.
Pseudopotentials
Treatment of the (nuclei + core) ─ valence.
Basis set
To expand the eigenstates of the hamiltonian.
Numerical evaluation of matrix elements
Efficient and self-consistent computations of H and S.
Supercells
To deal with periodic systems

Treatment of the boundary conditions:

Treatment of the boundary conditions Isolated objects (atoms, molecules, clusters)
open boundary conditions
(defined at infinity) 3D periodic objects (crystals)
periodic boundary conditions
(might be considered as the repetition of a building block, the unit cell) Mixed boundary conditions
1D periodic (chains)
2D periodic (slabs and interfaces)

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NO exactly periodic systems in Nature
(periodicity broken at the boundary)
BUT
The great majority of the physical quantities are unaffected by the existence of a border Periodic systems are idealizations of real systems Conceptual problems

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In a periodic solid:
Number of atoms
Number and electrons
Number of wave functions ?? Bloch theorem will rescue us!! 2. Wave function will be extended over the entire solid () Periodic systems are idealizations of real systems Computational problems

A periodic potential commensurate with the lattice. The Bloch theorem:

A periodic potential commensurate with the lattice. The Bloch theorem

The wave vector k and the band index n allow us to label each electron (good quantum numbers) :

The wave vector k and the band index n allow us to label each electron (good quantum numbers) The Bloch theorem changes the problem Instead of computing an infinite number of electronic wave functions Finite number of wave functions at an infinite number of k-points.

Systems with open and mixed periodic boundary conditions are made artificially periodic: supercells :

Systems with open and mixed periodic boundary conditions are made artificially periodic: supercells M. C. Payne et al., Rev. Mod. Phys., 64, 1045 (1992) Defects Molecules Surfaces

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Recap Born-Oppenheimer approximation
Electron nuclear decoupling One electron problem in effective self-consistent potential (iterate) Extended crystals: periodic boundary conditions + k-sampling Many electron problem treated within DFT (LDA, GGA)

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Suplementary information

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K. Reuter, C. Stampfl, and M. Scheffler, cond-mat/0404510 Length and time scales:
More suitable methods for a particular problem

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In equilibrium Atomic positions Length of the springs A classical view of the Born-Oppenhaimer approximation

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A classical view of the Born-Oppenhaimer approximation

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The equation has non trivial solutions if and only if Assuming that so we can decompose A classical view of the Born-Oppenhaimer approximation

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Solution at first-order: A classical view of the Born-Oppenhaimer approximation

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By: tyagibarc (65 month(s) ago)

A nice introduction