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
?