A very short introduction to density functional theory (DFT)

FKA091 - Condensed matter physics Lecturer: Paul Erhart, Department of Physics Materials and Surface Theory Division, Origo, R7111 Overview

• What is density functional theory (DFT) good for?

• Basic considerations HK vext(r) n0(r) • Hohenberg-Kohn theorems

( r ) ( r ) n { i} 0 { i}

• Kohn-Sham equations and the exchange-correlation functional 1 2 + v (r)+v (r)+v (r) (r)=✏ (r) 2r ext H xc i i i ✓ ◆ • Shortcomings of DFT

Erhart, FKA091 11/2016 Some possible applications of DFT

Structure prediction and phase stability: Platinum series nitrides

Forces à Vibrations

N dimer

Total energies Pt Erhart, FKA091 11/2016 Some possible applications of DFT

Structure prediction and phase stability: Platinum series nitrides

Electronic structure High pressure Low pressure (single bonded nitrogen) (triple bonded nitrogen)

Erhart, FKA091 11/2016 Some possible applications of DFT Defects: Thermodynamics, kinetics, electronic properties Energetics – Which defects and how many

Vacancy level

Electronic structure Total energies and forces à optical properties Erhart, FKA091 11/2016 Some possible applications of DFT Property prediction and screening: Substituted norbornadienes

donor

acceptor

Substitutions affect position of absorption band à can be tuned to solar spectrum HOMO LUMO

Erhart, FKA091 11/2016 Some possible applications of DFT Property prediction and screening: Substituted norbornadienes

Balance between computational effort and accuracy/predictivness

Erhart, FKA091 11/2016 Some possible applications of DFT

• Enables calculation of total energies and forces under very general conditions – Structure prediction One of the standard – Thermodynamics and kinetics computational tools in condensed matter, chemistry, and biochemistry in both • Provides description of electronic structure academia and industry – starting point for e.g., optical properties

• Good balance between accuracy/predictiveness and computational cost – Scales N3 or better (N=number of electrons) – Enables calculations with thousands of electrons

• Very well established technique – Capabilities and limitations reasonably understood – Efficient implementations available Erhart, FKA091 11/2016 FKA091: Density functional theory Basic considerations The Schrödinger equation @ i = Hˆ nuclear R @t electronic r ~2 ~2 Hˆ = 2 2 mixed 2M rn 2m ri n i e X X ˆ TR Tˆr Kinetic terms | 1{z1 } |Z Z {ze2 } 1 1 e2 + n m + 4⇡✏0 2 Rn Rm 4⇡✏0 2 ri rj n=m i=j X6 | | X6 | |

Vˆr Vˆr | 1 {z Z e2 } | {z } + n 4⇡✏ r R 0 i,n i n X | |

Vˆr,R Potential terms | {z } Erhart, FKA091 11/2016 The Schrödinger equation @ i = Hˆ (r , r ,...,R , R ,...; t) @t 1 2 1 2 Wavefunction is a function of coordinates of all electrons and ions (and depends parametrically on time)

Hˆ = Tˆr + TˆR + Vˆr + Vˆr,R + VˆR

= TˆR + VˆR + Tˆr + Vˆr + Vˆr,R nuclear electronic mixed

Notational convenience:

(r1, r2,...,R1, R2,...; t) Density functional theory in a nut shell: Reduce 3N-dimensional problem =( r , R ; t) { i} { i} to a 3-dimensional one =(r, R; t) Erhart, FKA091 11/2016 Born-Oppenheimer approximation

Expand many body wave function in a complete electronic basis set that depends on positions of both nuclei and electrons

(r, R; t)= n(R; t) n(r, R) n Insert wave function ansatz into Schrödinger equationX @ i =Hˆ @t … and project onto the electronic basis: m(r, R) @ i = Tˆ + Vˆ + (r, R) Tˆ + Vˆ + Vˆ (r, R) + Vˆ @t m R R m m r r R,r n mn n ⇣ ⌘ n hD E i 2 X 2 ˆ ~ ~ 2 Vmn = m(r, R) Rk n(r, R) Rk + m(r, R) Rk n(r, R) Mk h |r | i · r 2Mk |r | ⌦ ↵ Vˆmn depends on derivatives of electronic states with respect to nuclear positions and velocities Erhart, FKA091 11/2016 Born-Oppenheimer approximation

✏ (R)= (r; R) Tˆ + Vˆ + Vˆ (r; R) m h m | r r R,r| n i n X Rearrange terms @ i (R; t)=[Tˆ + Vˆ + ✏ (R)] (R; t)+ V (R; t) @t m R R m m mn n n X … and choose a representation (the so-called adiabatic representation), in which electronic wave function depends only parametrically on the nuclear positions, i.e. (r, R) (r; R) m ! m

Born-Oppenheimer approximation = neglect of cross terms

Erhart, FKA091 11/2016 Born-Oppenheimer approximation

The Born-Oppenheimer approximation enables separation into two separate problems

1. The electron-structure problem: Regard the nuclei as fixed in space and solve for the electronic degrees of freedom.

Hˆe m(r; R)=✏m(R) m(r; R) 2 ˆ 1 2 1 e Zn He = i + 2r 2 ri rj ri Rn i i=j i,n X X6 | | X | | ˆ ˆ Tr Vˆr Vr;R | {z } | {z } | {z } 2. The nuclear problem: Solve for the nuclear degrees of freedom using the derived potential energy surface (PES). @ i (R; t)=[Tˆ + Vˆ + ✏ (R)] (R; t) @t m R R m m Erhart, FKA091 11/2016 FKA091: Density functional theory The Hohenberg-Kohn theorems Schematic of Hohenberg-Kohn theorems

Hohenberg and Kohn, 1964 (HK): a density functional theory as an exact theory of many-body systems

Hˆ = Tˆr + Vˆr + Vˆr;R Reduction of dimensionality!

given an HK the ground state external potential vext(r) n0(r) density f=3 f=3

a solution to the the ground state Schrödinger solution n( ri ) 0( ri ) solution (lowest energy) {f=3N} {f=3N} Erhart, FKA091 11/2016 Hohenberg-Kohn theorem #1

Theorem #1 • For any electronic system in an external potential vext(r) said potential is determined uniquely, safe for a constant, by the ground state density n0(r)

Corollary #1 • Since the Hamiltonian is thus fully determined, it follows that the many-body wave functions for all states (ground and excited) are determined. • Thus all properties of the system are completely determined given only the ground state density.

Erhart, FKA091 11/2016 Hohenberg-Kohn theorem #1 – Proof

Let vext(r) be the external potential of a system, with associated ground state density n(r), the total number of particles N, Hamiltonian H, and the ground state wave function ψ and ground state energy E. vext : H, n(r), ,E,N

Similarly consider a second system of N particles v0 : H0,n0(r), 0,E0,N where and hence vext = v0 + C = 0 ext 6 ext 6

Then by the variational principle

E = H < 0 H 0 = 0 H0 0 + [v (r) v0 (r)] n0(r)dr h | | i h | | i h | | i ext ext Z = E0 + [v (r) v0 (r)] n0(r)dr ext ext Z

H = Tˆr + Vr + vext(R)n0(r) Z ˆ H0 = Tr + Vr + vext0 (R)n0(r) Z Erhart, FKA091 11/2016 Hohenberg-Kohn theorem #1 – Proof

Thus E

E0

E + E0

Thus any potential exceptvext(r) vext(r)+C leads to n0(r) = n(r) Q.E.D. 6

Erhart, FKA091 11/2016 Hohenberg-Kohn theorem #2 Theorem #2 • A universal functional of the energy can be defined E[n] in terms of the density , which is valid n(r) for any external potential vext(r) For any particular the vext(r) exact ground state of the system is determined by the global minimum value of this functional.

Corollary #2 • Said functional alone is sufficient to determine the ground state energy and density. • Excited states have to be determined by other means. • DFT is a ground state theory

Thus by means of minimization, the ground-state electron density can be found as

E0[n]=min Hˆ =min, min E[n] n˜ | | n˜ n˜ N ! D E ! 2 Erhart, FKA091 11/2016 Hohenberg-Kohn theorem #2

For convenience separate the energy functional into • and a universal part (independent on N, mass and charge of nuclei, etc) • a system dependent part, i.e. depends on vext(r)

Energy functional:

E[n(r)] = F [n(r)] + vext(r)n(r)dr universal Z system dependent

The universal functional contains the individual contributions of kinetic energy, classical Coulomb interaction, and the non-classical self-interaction correction

1 n(r)n(r0) true F [n(r)] = T [n(r)] + drdr0 + E 2 r r xc Z | 0|

Erhart, FKA091 11/2016 Some remarks

• The HK theorems can be generalized to include spin (not magnetic fields)

• … reformulate the many-body problem in terms of the density but do not actually solve it

• The HK functional is a priori unknown! There is no direct link from the density to e.g., the kinetic energy.

• HK-DFT is in principle exact (no approximations) but impractical à In practice one uses the Kohn-Sham approach

Erhart, FKA091 11/2016 FKA091: Density functional theory The Kohn-Sham equations Kohn and Sham ansatz The Kohn-Sham ansatz • Kohn-Sham: replace the original many-body problem with an auxiliary systemà Replace of non-interactingthe original many particles.-body problem with an auxiliarysystem of non-interacting particles.

Non-interacting Interacting electrons auxiliary + real potential particles in an effective potential

Assumption: • K-S assumes that the ground stateGround densitystate density of of the originalinteracting interacting Walter Kohnsystem is equalLu Jeu toSham that of some chosensystem non-interactingis equal to that of some system thatnon is- exactly soluble, with all the difficult partsinteracting system (exchange andthat is exactly correlation)soluble, included in some approximate functionalwith all the difficult of the density.parts (exchange and correlation) included in some approximate functional of the density. 13. October 2010 6. The density functional theory 18

1998 (chemistry) Erhart, FKA091 11/2016

Lu Jeu Sham Independent particles approximations

• Hartree ansatz (r1, r2,...,rN )= 1(r1) 2(r2) ... N (rN )

• Fock: does not satisfy anti-symmetry requirement (fermions) à need to take Slater determinant

V. A. Fock

D. R. Hartree Erhart, FKA091 11/2016 Kohn-Sham ansatz

• In principle, the solution of the independent KS problem determines all properties of the full many-body system.

HK Kohn-Sham vext(r) n0(r) n0(r) vKS(r)

( r ) ( r ) n { i} 0 { i} i=1...N (r)

The auxiliary Hamiltonian contains the usual kinetic energy term and a local effective potential acting on the electrons

Erhart, FKA091 11/2016 The Kohn-Sham energy functional

• In KS decomposition the total energy of a many electron system in an external potential is

E[n]=T0[n(r)] + drn(r)vext(r)+EH [n(r)] + Exc(r) Z

N.B.: Here, we use atomic units. Erhart, FKA091 11/2016 Hohenberg-Kohn theorem #2

For convenience separate the energy functional into • and a universal part (independent on N, mass and charge of nuclei, etc) • a system dependent part, i.e. depends on vext(r)

Energy functional:

E[n(r)] = F [n(r)] + vext(r)n(r)dr universal Z system dependent

The universal functional contains the individual contributions of kinetic energy, classical Coulomb interaction, and the non-classical self-interaction correction

1 n(r)n(r0) true F [n(r)] = T [n(r)] + drdr0 + E 2 r r xc Z | 0|

Erhart, FKA091 11/2016 The Kohn-Sham energy functional

• In KS decomposition the total energy of a many electron system in an external potential is

E[n]=T0[n(r)] + drn(r)vext(r)+EH [n(r)] + Exc(r) kinetic external potential class. Coul. self-int. “0” à non-inter- Z acting particles occ 2 n(r)= (r) electron density | i | i X Zn v = potential from the nuclei ext r R n I X 1 n(r)n(r0) E [n]= drdr0 classical Coulomb energy H 2 r r ZZ | 0|

Exc[n]= drn(r)"xc[n(r)] exchange and correlation Z N.B.: Here, we use atomic units. Erhart, FKA091 11/2016 Kohn-Sham functional • The KS decomposition separates contributions that are (in principle) possible to compute from contributions that are much more complicated.

Jones, Gunnasson (1989)

kinetic Coulombic exchange-correlation (XC)

XC is on an absolute scale a minor contribution to the total energy. It is crucial that the kinetic energy is evaluated to a high accuracy (this is not done in Thomas-Fermi theory).

Erhart, FKA091 11/2016 Kohn-Sham one-electron orbitals

• The one-electron orbitals (KS orbitals) are solutions to the KS equation (one-particle equation!!) 1 2 + v (r)+v (r)+v (r) (r)=✏ (r) 2r ext H xc i i i ✓ ◆

KS orbital • It is tempting to treat these states as KS eigenenergy proper electronic (quasi) particle states; but beware, this are fictitious states

• Nonetheless band structures often useful; although band gap underestimation is common

Erhart, FKA091 11/2016 DFT in practice: The exchange-correlation functional

✏ E [n]= drn(r)" [n(r)] v = xc xc xc xc n Z potential Local density approximation (LDA)

LDA HEG Exc [n]= drn(r)"xc (n(r)) Z HEG where is the exchange"xc (n) -correlation energy of a homogeneous electron gas of density n

Generalized gradient approximations (GGA; semi-local functionals)

GGA Exc [n]= drn(r)"xc(n(r), n(r)) Z

Erhart, FKA091 11/2016 DFT in practice: The self-consistency loop Initial guess (usually superposition of atomic densities for k=1) n(k)(r) k k +1 Evaluate effective potential ! (k) (k) ve↵(r)=vext(r)+vH [n (r)] + vxc[n (r)]

Solve KS equations (for given potential) 1 2 + v [n(k)(r)] (r)=✏ (r) 2r e↵ i i i ✓ ◆ occ 2 Evaluate actual density n(r)= (r) | i | i X

Compute energy, Yes Consistent? No n(k)(r) n(r) < forces, stresses … || ||2

Erhart, FKA091 11/2016 Implementation aspects

Beyond DFT

DFT

1 2 + v (r)+v (r) (r)=✏ (r) 2r ext Hxc i i i ✓ ◆ In practice • a lot of numerics related to basis sets, treatment of core/valence electrons, exchange-correlation, solving differential equations, finding eigen values etc. • Actual production DFT codes contain tens to hundreds of thousands of lines of code Kohn-Sham one-electron orbitals

• The one-electron orbitals (KS orbitals) are solutions to the KS equation (one-particle equation!!) 1 2 + v (r)+v (r)+v (r) (r)=✏ (r) 2r ext H xc i i i ✓ ◆

KS orbital • It is tempting to treat these states as KS eigenenergy proper electronic (quasi) particle states; but beware, this are fictitious states

• Nonetheless band structures often useful; although band gap underestimation is common

Erhart, FKA091 11/2016 Kohn-Sham one-electron orbitals

• Band gap underestimation

Solid State Physics (1970)

GW is a many body perturbation technique G: Green’s function W: Screened Coulomb interaction Erhart, FKA091 11/2016 Kohn-Sham one-electron orbitals

• Band structures

silicon

• Dispersion usually very good

Erhart, FKA091 11/2016 Beyond DFT

• DFT represents a very starting point for many-body perturbation theory

GW: LaBr3 Bethe-Salpeter equation (BSE)

Erhart, FKA091 11/2016 Take away messages

1) n(r) ( r ) Hohenberg-Kohn theorems { } • f=3 f=3N 1) All properties determined by density HK 2) A functional can be defined the minimum 2) vext(r) n0(r) of which yields the ground state

• Kohn-Sham equations n( ri ) 0( ri ) – Replace fully interacting system { } { } with set of fictitious effective single-particle problems vKS(r) n0(r) Kohn-Sham – The magic happens in the exchange-correlation functional Can be solve iteratively – i=1...N (r) DFT yields total energies, forces, and electronic structure à excellent starting point for many-body perturbation theory techniques (GW, BSE, also TD-DFT) Erhart, FKA091 11/2016 Take away messages Density functional theory (DFT) • … yields total energies, forces, and electronic structure and therefore allows computing e.g., – Structural relaxation and phase stability – Energy differences (input for thermodynamics, kinetics …) – Phonon dispersions (mechanical stability) – Property prediction and screening – Band structures (a bit more complicated, see above)

• … provides a reasonable balance between computational efficiency and accuracy (thousands of electrons, hundreds of atoms)

• … has limitations e.g., with respect to – Band gaps and optical properties (excitoniceffects) – Strongly correlated systems Erhart, FKA091 11/2016