Electrons in Materials Density Functional Theory Richard M. Martin

d orbitals Electron density in La 2 CuO 4 - difference from sum of atom densities - J. M. Zuo (UIUC)

Comp. Mat. Science School 2001 1

• Many Body Problem!

Outline

• Density Functional Theory Kohn-Sham Equations allow in principle exact solution for ground state of many-body system using independent particle methods Approximate LDA, GGA functionals • Examples of Results from practical calculations • Pseudopotentials - needed for plane wave calculations • Next Time - Bloch Theorem, Bands in crystals, Plane wave calculations, Iterative methods Comp. Mat. Science School 2001 2

Ab Initio Simulations of Matter

• Why is this a hard problem? • Many-Body Problem - Electrons/ Nuclei • Must be Accurate --- Computation • Emphasize here: Density Functional Theory – Numerical Algorithms – Some recent results Comp. Mat. Science School 2001 3

Eigenstates of electrons

• For optical absortion, etc., one needs the spectrum of excited states • For thermodynamics and chemistry the lowest states are most important • In many problems the temperature is low compared to characteristic electronic energies and we need only the ground state – Phase transitions – Phonons, etc.

Comp. Mat. Science School 2001 4

The Ground State

• General idea: Can use minmization methods to get the lowest energy state • Why is this difficult ?

• It is a Many-Body Problem • Y i ( r 1 , r 2 , r 3 , r 4 , r 5 , . . . ) • How to minimize in such a large space Comp. Mat. Science School 2001 5

The Ground State

• How to minimize in such a large space – Methods of Quantum Chemistry- expand in extremely large bases - Billions - grows exponentially with size of system • Limited to small molecules – Quantum Monte Carlo - statistical sampling of high-dimensional spaces • Exact for Bosons (Helium 4) • Fermion sign problem for Electrons Comp. Mat. Science School 2001 6

Quantum Monte Carlo

• Variational - Guess form for Y ( r 1 , r 2 , …) • Minimize total energy with respect to all parameters in Y E 0 =  dr 1 dr 2 dr 3 … Y H Y • Carry out the integrals by Monte Carlo • Diffusion Monte Carlo - Start with VMC and apply operator e -H t Y improved ground state Y 0 • Exact for Bosons (Helium 4) to project out an • Fermion sign problem for Electrons Comp. Mat. Science School 2001 7

Density Functional Theory

• 1998 Nobel Prize in Chemistry to Walter Kohn and John Pople • Key point - the ground state energy for the hard many-body problem can in principle be found by solving non-interacting electron equations in an effective potential determined only by the density H Y i (x,y,z) = E i Y i (x,y,z) , H = h 2 D 2 + V(x,y,z) 2 m • Recently accurate approximations for the functionals of the density have been found Comp. Mat. Science School 2001 8

Density Functional Theory

• Must solve N equations, I = 1, N with a self-consistent potential V(x,y,z) that depends upon the density of the electrons h 2 D H Y i (x,y,z) = E i Y i (x,y,z) , H = 2 m • Text-Book - Find the eigenstates 2 + V(x,y,z) • More efficient Modern Algorithms – Minimize total energy for N states subject to the condition that they must be orthonormal • Conjugate Gradient with constraints – Recent “Order N” Linear scaling methods Comp. Mat. Science School 2001 9

Examples of Results

• Hydrogen molecules - using the LSDA (from O. Gunnarsson) Comp. Mat. Science School 2001 10

Examples of Results

• Phase transformations of Si, Ge • from Yin and Cohen (1982) Needs and Mujica (1995) Comp. Mat. Science School 2001 11

Enthalpy vs pressure

• H = E + PV - equilibrium structure at a fixed pressure P is the one with minimum H • Transition pressures slightly below experiment 80 kbar vs ~100kbar Needs and Mujica (1995) Cubic Diamond Comp. Mat. Science School 2001 Simple Hexagonal 12

Graphite vs Diamond

• A very severe test • Fahy, Louie, Cohen calculated energy along a path connecting the phases • Most important - energy of grahite and diamond essentially the same!

~ 0. 3 eV/atom barrier Comp. Mat. Science School 2001 13

A new phase of Nitrogen

• Published in Nature this week. Reported in the NY Times Dense, metastable semiconductor • Predicted by theory ~10 years ago!

Molecular form Mailhiot, et al 1992 “Cubic Gauche” Polymeric form with 3 coordination Comp. Mat. Science School 2001 14

The Great Failures

• Excitations are NOT well-predicted by the “standard” LDA, GGA forms of DFT The “Band Gap Problem” Orbital dependent DFT is more complicated but gives improvements treat exchange better, e.g, “Exact Exchange” Ge is a metal in LDA!

M. Staedele et al, PRL 79, 2089 (1997) Comp. Mat. Science School 2001 15

Conclusions

• The ground state properties are predicted with remarkable success by the simple LDA and GGAs. Structures, phonons (~5%), …. • Excitations are NOT well-predicted by the usual LDA, GGA forms of DFT The “Band Gap Problem” Orbital dependant functionals increase the gaps - agree well with experiment now a research topic Comp. Mat. Science School 2001 16