Density Functional Theory - Technionphycomp.technion.ac.il/~jeremie/Density Functional Theory.pdf · B- Density Functional Theory (DFT) ... If an exact functional of the correlation

Introduction to Density Functional Theory

Jeremie Zaffran2nd year-MSc. (Nanochemistry)

From Hartree-Fock model to DFT

From Hartree-Fock model to DFT

A- Hartree approximations

eN HHH^^^

Born- Oppenheimer approximation…

Let’s remind…

at

j

ext

ie

H

rR

ZV

T^

0

^

^

2

1

iji ji

ee

rrV

,

^ 1

2

1and

eeate VHH^^^

eeextee VVTH^^^^

Key-problem

The goal of computational chemistry… eH^

???

First Hartree approximation:

Instead of considering the operator , let’s consider each electron in the mean field experienced from all the other electrons .

eeV^

j

i

i

iee rVV^

rd

rr

rrV i

i3

2

where

Effective potential: eeexteff VVV

From Hartree-Fock model to DFT

Second Hartree approximation:

Due to the Pauli exclusion principle the system wave function is supposed to be a single Slater determinant:

NNNN

N

N

S

xxx

xxx

xxx

N

21

22221

11211

!

1

NiiNii xxxxxxxxxx ,,,,,,,,,,,, 121121

where srx iiiii

From Hartree-Fock model to DFT

Consequences…

From Hartree-Fock model to DFT

o Variationnal principle

o Orthonormalisation of the spin orbitals basis set

i jiatee jiijjjiiiHiHE

2

1^^

ijji Jxr

xjjii 2

2

12

2

1

1

ijijji Kr

jiij 211

2112

Coulomb integral

Exchange integral

i

iiHF KJV Hartree-Fock potential:

Limitations of the Hartree-Fock Model…

1. Hartree-Fock model deals with a non-interacting reference system, and thus the correlation energy is not taken in account.

3. The wave function has no physical sense, only its square has one!

2. The wave function is a functions relies on 3N variables, where N is the number of electrons in the system.

Very time-consuming, only small systems

N.B: The correlation energy could be reached using post Hartree-Fock methods expanding the wave function on a basis of several Slater determinants (Configuration Interaction-CI …), or perturbation method (MP2…)

From Hartree-Fock model to DFT

From Hartree-Fock model to DFT

B- Density Functional Theory (DFT)

Let’s set basis…

eeextee VVTH^^^^

Functional: Mathematical application going from the functions space to the scalars space.

Notation: F[f]=x which means xfF

Any chemical system is utterly defined provided one knows its electrons number N and its external potential , and thus ground state energy could be reached such that…

extV^

extVNEE ,0

From Hartree-Fock model to DFT

Hohenberg-Kohn Theorems (1964)

Consequence: Provided one knows the ground state density, one gets in turn the external potential and thus the hamiltonian, resulting in the ground state wave function and energy (and all the system properties).

00

^

0 ,, EHVN ext (and all other properties)

1) “The external potential is a unique functional of the ground state density ”

extV^

0

From Hartree-Fock model to DFT

2) “The ground state energy will be reached if and only if one use the ground state density in the energy functional.”…in other words, the well known variational principle!

00E

which means… 00, EEE trialtrial

or

EE min0

Hohenberg-Kohn Theorems (1964)

From Hartree-Fock model to DFT

DFT key points…

The electronic density becomes the fundamental variable!

Interest: • is only a function of 4 variables (x,y,z,s) and no more of 3N variables as with .

• is an observable.

Any DFT algorithm should aim to reach only the ground state and no excited state!

The energy minimisation algorithms have to take care about two main constraints lying on the density:

• must be N-representable, which means associated to an acceptable wave function :square integrable functions… The Slater determinant is only an example of such a set!

• must be Vext-representable, which means giving rise to a finite external potential.Note that to this date we don’t know what makes a density Vext-representable on the mathematical point of view.

Levy constrained search scheme

From Hartree-Fock model to DFT

Expression of the energy functional and limitations of the Hohenberg-Kohn theorems

Feature of the system

Universal functional

drVV eNext

HKF ???? ( T is not a functional of the density, and Eee is not completely known)

HKF

eeext ETVE

From Hartree-Fock model to DFT

Kohn-Sham approach

eeHK ETF

Idea…

The major part of the Hohenberg-Kohn functional is the kinetic energy, the remainder could be just approximated.

So let’s find a way to express T…

xcEJ

?Coloumbic repulsion (known)

From Hartree-Fock model to DFT

Owing to the Hartree-Fock theory, T is exactly known for a non-interacting reference system…

NNNN

N

N

S

xxx

xxx

xxx

N

21

22221

11211

!

1 : Kohn-Sham (KS) orbitals i

And thus i

i

2

(iterative resolution)

Kohn-Sham equations

iii

KS

f ^

Where the KS operator is j

jSj

KS

rVf2

1^

Effective or Sham potential eeextjS VVrV

From Hartree-Fock model to DFT

Highlights of KS approach…

KS-orbitals have no physical meaning! The target is only to reach the density.

The KS-orbitals could be expressed as atomic orbitals or as Bloch waves according to the calculation code.

An initial electronic density input is necessary…

Self-Consistent-Field (SCF)

From Hartree-Fock model to DFT

Initial input density

Sham potential calculation

Sham equations resolution

Density output extraction

Self-Consistent-Field?

OUTYES

NO

SCF scheme

From Hartree-Fock model to DFT

Focus on correlation energy

Correlation:

Mathematical definition: electron 1 at r1 and electron 2 at r2 are correlated if the following relation is NOT verified

Physical meaning: Classical and non classical effects due to the many-body interacting system.

2121 ~, rrrr

If an exact functional of the correlation energy was known, the Schrodinger equation could have been solved EXACTLY-without any approximation…

To this date, no useful expression of the correlation is known!

From Hartree-Fock model to DFT

… But unfortunately the only mathematical (and not so useful) formalism we have is…

The Kohn-Sham approach is exact only the exchange-correlation functional has to be approximated!

xcE

nclC

eeSxc

ET

JETTE

ncl

C

S

E

T

T

T : Kinetic energy of the real system: Kinetic energy of the reference system: Residual kinetic energy: Non-classical energy

with

From Hartree-Fock model to DFT

C- The exchange-correlation problem

• Local Density Approximation (LDA):

Based on the homogeneous electrons gas model. Exchange-correlation density functional is exactly known owing to the Thomas-Fermi model.

• Gradient Generalized Approximation (GGA): PBE…

Application of the gradient operator on the previous model.

• Meta-GGA: BB95…

Application also of the laplacian operator.

• Hybrid functional: HSE06, B3LYP…

Introduction of an exact Hartree-Fock part in the Exchange functional.

GGAC

GGAX

HFXxc EEEE %)1(%

How to approximate this functional?

From Hartree-Fock model to DFT

Jacob’s ladder (Pedrew metaphor)…

Earth: HF model

Heaven: Exact solution

LDA

GGA

MGGA

Hybrid

?