Density functional theory – An introduction

Presentation for the Praktikum in Theoretical Chemistry

Institut für Theoretische ChemieUniversity of Stuttgart

Matthias Ernst

Tutors: E. Goll, H. Stoll

Structure

1. Introduction to Density Functional Theorya) Theoryb) Functionalc) Densityd) Density – What for?

2. Basics of DFTa) The Hohenberg-Kohn Theoremsb) Kohn-Sham equationsc) Results

3. Practical approachesa) LSDb) Jacob's ladderc) GGAd) PBE as examplee) Hybrid functionalsf) Performance of DFT

4. Outlook

3

Theory:

In science, a theory is a proposed description, explanation, or model of the manner of interaction of a set of natural phenomena, capable of predicting future occurrences or observations of the same kind, and capable of being tested through experiment or otherwise falsified through empirical observation.

www.en.wikipedia.org

1. Introductiona)

The

ory

4

Functional:The initial meaning is a function that takes functions as its argument

Function: f:=x->f(x)

Functional: F:=f(x)->F[f]

examples: integral as functional

expectation value of an operator Ô

F [ f ]=∫−∞

∞

f xdx

⟨O ⟩=⟨∣O∣ ⟩=∫ ...∫*Odx1dx2 ...dxN

1. Introductionb)

Fun

ctio

nal

5

Density:here electron density or exactly electron probability density

Defined as probability to find an electron with spin σ in volume element dr at r:

n r , also written as r

n r=N ∑2...N

∫dr2 ...∫drN∣r ,r22,... ,rNN∣2

∑∫n rdr=N

n r∞=0

The electron density is an observable and thus can be measured experimentally

Additional properties: electron density has cusps at the positions of the nuclei,the charge of the nuclei can be determined by Kato's cusp condition

⟨Vext⟩=⟨∣∑i=1

N

v r i∣⟩=∫nrvextrdr

1. Introductionc)

Den

sity

6

Why using electron density?

„Convential approach“: Construction Hamiltonian, solving Schrödinger's equation

=> wavefunction

=> energy, molecular properties, frequencies, ...

Problems: - in most cases, the solution must be approximated- complicated, as there are 3N spatial + N spin variables- even more complicated if ee-interaction is taken care of- large computanional demand, impossible for large systems

But: Hamiltonian acts only on one or two particles and thus its form is independent of the size of the system

=> electron density, depends only on 3 spatial variables

Needed: proof, that n(r) determines the Hamiltonian and thus the entire system

plausible argument (not proof): n(r) contains all information to construct the Hamiltonian (number of electrons, position (and so number) and charge of nuclei)

r11, r21,... ,rNN

1. Introductionc)

Den

sity

– W

hat f

or?

7

The Hohenberg-Kohn Theorems

Given an external potential vext(r).

1) The external potential vext(r) is (to within a constant) a unique functional of the ground-state density n0(r). Since vext(r) determines the Hamiltonian, the ground-state wave function is also a unique functional of n0(r):

Thus, the expectation value of any observable Ô is also a functional of n0(r).2) Especially, the ground-state energy is a functional of n0(r):

3) For a given vext(r), the correct density n0(r) minimizes the true ground-state energy E0 (variational principle, proof also via wave-function variatonal principle):

0=[n0]

Ev ,0=Ev[n0 ]=⟨[n0]∣H∣[n0]⟩

Ev [n0 ]≤Ev[n' ]

Ev [n']=T[n' ]Vext [n']Vee[n']

2. Basics of DFTa)

Hoh

enbe

rg-K

ohn

Theo

rem

s

8

Proof:Original proof via reductio ad absurdumAlternate, more constructive proof: Constrained Search (Levy)

Recall: Hamiltonion for N-electron system in Born-Oppenheimer-Approximation:

Helec=−12∑i=1

N

∇ i2

Te

−∑i=1

N

∑A=1

M ZA

r i−RAVNe

∑i=1

N

∑ji

N 1r ij

Vee

=T Vext Vee

E0=minN

⟨∣H∣ ⟩=min⟨∣TVextVee∣ ⟩Variational principle:

Split into 2 steps:

minn

⟨∣H∣ ⟩=minn

⟨∣TVee∣ ⟩F [n ]

∫v r nr drVext

=F[n]∫ vextr nr dr

F [n]=minn

⟨∣TVee∣ ⟩=⟨nmin∣TVee∣n

min ⟩=T [n]Eee[n]

E0=minn

Ev [n]=minn{F[n]∫vextr n rdr }

„universal functional“: if known exactly, it would solve SE

2. Basics of DFTa)

Hoh

enbe

rg-K

ohn

Theo

rem

s

E0=minnN min

n⟨∣TVextV ee∣ ⟩

9

Definition: F[n]= TS[n]non−interacting

EH[n]Hartree /Coulomb energy

Exc [n]Exchange−correlation functional

Eee [n ]

EH[n]=12∫∫

nr1nr2

∣r1−r2∣dr1dr2

So

and

E [n]=F [n]∫v[r ]n[r ]Vext

dr=TS[n]V extEH[n]Exc [n]

EXC[n]= E [n]T [n]Vext [n ]Eee [n]

−TS[n]−V ext−EH[n]=T [n]−TS[n]TC

Eee[n]−EH[n]Encl

TS can be treated exactly, but EXC must be approximated

with Hartree-electrostatic self-repulsion

Furthermore: EXC = EX + EC

T[n] = TS[n] + TC[n] (TC is part of EC)

Prerequisites:

2. Basics of DFTb)

Koh

n-Sh

am e

quat

ions

10

N fermions which do not interact (no Coulomb repulsion) moving in a potential vS and residing in single-particle-orbitals

EX[i [n]]=−12∑jk ∫dr1∫dr2

j *r1k * r2 jr1k r2∣r−r '∣

TS[n]=−12∑i

N

∫dri * r∇2ir =−

12 ⟨∣∇

2∣⟩Using :

Now: E [n]=TS[i[n]]VextEH[n]Exc [n]

E[n] can no longer be minimized directly, as it's now a functional of φi

For a non-interacting system, a Slater-determinant delivers the exact wave function(as in Hartree-Fock), so are constructed as Slater-determinants.

2. Basics of DFTb)

Koh

n-Sh

am e

quat

ions

Kohn-Sham (KS) non-interacting system:

i ⇒ Vee=0

i

i

11

Indirect approach to solve E[n]:

Orbitals can be determined by self-consistent solution of Kohn-Sham-equation:

−12∇2vSr jr= j jr

n r≡nS r =∑j=1

N

∣ jr∣2

For KS-system in vS(r): 0= EnS

=TS

nSvSr

nsr =nr if vSr =vext r vHrvxcr

0= En

=TS

nVext

nEH

nExc

n=TS

nvextr vHr vxcr

and

For a converged solution n0:

E0=∑i

N

i−12∫dr1∫dr2

n0r1nor2∣r1−r2∣

−∫dr vext r n0 rExc [n0]

„direct“ solution impossible as vs depends on n

2. Basics of DFTb)

Koh

n-Sh

am e

quat

ions

i

12

IF Exc and vxc were known, the Kohn-Sham system would deliver the exactground state energy, so the correct eigenvalue of the Hamiltonian andthus, it would exactly solve Schrödinger's equation!

Unfortunately, vxc is unknown, and so is Exc, as vxc=Exc

n

Finding good and always better approximations for Exc and vxc is the goal of modern DFT.

−12∇2vSr jr= j jr

n r≡nS r =∑j=1

N

∣ jr∣2

vSr =v r vHr vxcr

Iteration until self-consistency is reacheddelivers

2. Basics of DFTc)

Res

ults

Results:

j , n r and

13

Cx=34 3

1 /3

ExLDA [n]=−CX∫n4 /3 r dr

xLDA[n]=−CXn1 /3

ExcLDA[n]=∫nr xcn rdr

xc :exchange−correlation energy per particle in a uniformelectron gas

exchange part:

with

often called Slater exchange functional S

correlation part fitted to QMC calculation of Ceperley&Alder, quite complexe.g. Vosko-Wilk-Nusair with 12 fitting constants

xc nr =Xnr C nr

3. Practical approachesa)

LD

A

Practical approaches: LDA

Assumption: uniform or slowly varying electron gas

14

Earth („Hartree world“)

Heaven (chemical accuracy)(5) + explicit dependence on

unoccupied orbitals (exact partial correlation)

(4) hyper-GGA + explicit dependence on occupied orbitals (exact exchange energy density)example: hybrid functionals

(3) meta-GGA + explicit dependence on kinetic energy density

(2) GGA + explicit dependence on gradients of the density

(1) LDA local density only

3. Practical approachesb)

Jac

ob's

ladd

er

Practical approaches: “Jacob's Ladder“

15

EXCGGA [n]=∫ fGGA nr ,∇nr dr

Ansatz: adding gradient expansion

EXCGGA [n]=EX

GGAECGGA

EXGGA [n]=EX

LDA−∑∫Fsn

4/3 r dr

with reduced density gradient s r =∣∇n r ∣n

4/3r

Depending on the method of construction of f, very different GGAs are obtained.

Two main classes of realisation of f:Always: Find a suitable functional formEither: Parameters are then fitted to sets of experimental data (e.g. G2) [Chemistry]Or: Exact physical constraints are taken into account, no fitting [Physics]

(local inhomogeneity parameter)

3. Practical approachesc)

GG

A

Practical approaches: GGA

16

GGA – example: Contraints for the construction of the PBE-functional(see „GGA made simple“, Perdew, 1996)

ECGGA [n ,n ]=∫n[C

LDArS ,H rS , , t]

With: local Seitz radius rS ( ), relative spin polarisation

Dimensionless density gradient with

and

n= 34 rs

3=kF

3

32 =n −n n n

t= ∣∇n∣22ksn

kS= 4kF

=1

2 [12 /31−2 /3]

- Slowly varying limit (t->0): with

- Rapidly varying limit (t->∞, no correlation):

- Uniform scaling to the high-density limit [ and ]:

EC must scale to a constant

HMB3 t2

H−CLDArS ,MB=0.066725

n r 3n r ∞

„Simple Ansatz“: H=c03 ln {1MB

c0t2 [ 1At2

1At2A2 t4 ]} A=MB

c0exp−cLDA rS,

c03 −1

Correlation term:

3. Practical approachesc)

Exa

mpl

e: G

GA

mad

e si

mpl

e

17

EXGGA[n]=∫nx

LDA Fxsdr

- Exact energy obeys spin scaling relationship:

- Linear response (small density variations around uniform density): LSD is a good approximation => try to keep the LSD linear response

- Lieb-Oxford bound:=

2

3≃0.21951s0: Fx s=1s2

xLDA=−3

43 3n

Fxs=1− 1s2/

Exchange term:

EX [n ,n ]=12EX [2n ]EX [2n ]

where

EX [n ,n ]≥Exc [n ,n ]≥−1.679∫n4 /3 dr

Choice of F to satisfy these constraints:

3. Practical approachesc)

Exa

mpl

e: G

GA

mad

e si

mpl

e GGA – example: Contraints for the construction of the PBE-functional(see „GGA made simple“, Perdew, 1996)

18

EXCMGGA=∫neXC

MGGAn r ,∇ nr ,∇2 nr ,r dr

With Kohn-Sham orbital kinetic energy density

r =12∑i

∣∇ir 2∣

Exchange-correlation potential becomes orbital dependentHybrid functionals are meta-GGAs

Also for mGGAs: either fitting to experimental data or derivating by physical constraints

3. Practical approachesc)

met

a-G

GA

Practical approaches: meta-GGA

19

EXC[n]=12∫∫

nr nxc r , r 'r−r '

dr dr '=∫0

1

Encl d

with coupling-constant average hole density

where is the coupling parameternxcr ,r '=∫0

1

nxc r , r 'd

At : No interaction (pure Kohn-Sham system)at : full interaction („real“ system)

Any non-interacting system can be described exactly using Hartree-Fock

If is linear in and HH:

=0

=1

Encl Exc

HH= 12

Exc=0

Exc

exact

12

Exc=1

Exchyb=aEx

exact1−aExGGAEc

GGAIf not:

with (here) 1 or more parameters to be determined (experimentally or theoretically)

3. Practical approachesc)

Hyb

rid fu

nctio

nals

Practical approaches: Hybrid functionals

20

Method G2 LSDA GGA BPW91 B3PW91 Type GGA Hybrid

Atomization energies 1,2 35,7 3,9 5,7 2,4 Ionization energies 1,4 6,3 11,2 4,1 3,8 Proton Affinities 1,0 5,6 2,4 1,5 1,2

Comparison of different DFT methods by mean absolute deviations (kcal/mol)

Type

G2 1,6 8,2G2(MP2) 2 10,1G2(MP2, SVP) 1,9 12,5SVWN LDA 90 228,7BLYP GGA 7,1 28,4BPW91 GGA 7,9 32,2B3LYP Hybrid 3,1 20,1B3PW91 Hybrid 3,5 21,8

Method Meanabs. dev.

Maximum abs. dev.

Different DFT methods (kcal/mol)LDA

113 106,08 108,38461,21 419,7 419,7336,68 302,09 297,48267,5 232,91 232,91

CO 299,78 269,81 258,28175,26 142,97 119,91

Molecule GGA(PBE)

Exact

H2

CH4

NH3

H2O

O2

Atomization energies (kcal/mol)

3. Practical approachesd)

Per

form

ance

Performance of DFT

21

- Find better functionals, improve the existing ones

- Taking next step(s) on Jacob's ladder: find new functionals

- apply time-dependent DFT (TDDFT)

- apply relativistic DFT

- combine DFT with wave-function based methods(e.g. MP2, CI or CC)

4. Outlook

What is possible to use DFT?