Transcript Document

Density Functional Theory (DFT)
DFT is an alternative approach to the theory of electronic structure; electron
density plays a central role in DFT.
Why a new theory?
HF method scales as
CI methods scale as
MPn methods scale as
CC methods scale as
K4
K6-K10
>K5
>K6
(K - # of basis functions)
Correlated methods are not feasible for medium and large sized molecules!
Alternative: DFT
The electron density
- it is the central quantity in DFT
- is defined as:
The electron density
Properties of the electron density
Function: y=f(x)
ρ= ρ(x,y,z)
Functional: y=F[f(x)]
E=F[ρ(x,y,z)]
Hohenberg–Kohn Theorems
ρ(r)
First HK Theorem:
The external potential Vext(r) is (to within a constant) a
unique functional of ρ(r).
Since, in turn Vext(r) fixes H, the full many particle ground state is
a unique functional of ρ(r).
Thus, the electron density uniquely determines the Hamiltonian
operator and thus all the properties of the system.
 ρ(r)dr  N
N
ν(r)

H
Hˆ Ψ  EΨ
E
Proof: by reductio ad absurdum
Ψ’ as a test function for H:
Ψ as a testfunction for H’:
Summing up the last two inequalities:
Contradiction!
Variational Principle in DFT
Second HK Theorem
The functional that delivers the ground state energy of the system,
delivers the lowest energy if and only if the input density is the true
ground state density.
- variational principle
For any trial density ρ(r), which satisfies the necessary boundary conditions such as:
ρ(r)0 and
and which is associated with some external potential Vext, the energy obtained from the
functional of FHK represents an upper bound to the true ground state energy E0.
Thomas-Fermi model (1927)
L.H. Thomas, Proc. Camb. Phil. Soc., 23, 542-548 (1927)

3
2 2/3
5/3  
E. Fermi, Rend. Acad., Lincei, 6, 602-607 (1927)
TTF [ρ(r )]  (3π )  ρ (r )dr
10



ρ
(
r
)ρ
(
r


3
ρ(r )  1
1
2)  
2 2/3
5/3 
ETF [ρ(r )] 
(3π )
 ρ (r )dr  Z  r dr  2  r 12 dr1dr2
10
Next step

 
E[ρ]  ENe [ρ]  T[ρ]  Eee [ρ]   ρ(r )VNe (r )dr  FHK [ρ]
with
FHK ???
FHK [ρ]  T[ρ]  Eee


ρ
(
r
)ρ
(
r
1
1
2)  
Eee [ρ]  
dr1dr2  Enon_cl [ρ]  J[ρ]  Enon_cl [ρ]
2
r 12
Only J[ρ] is known!
The explicit form of T[ρ] and Enon-cl[ρ] is the major challenge of DFT
T[ρ]
– kinetic energy of the system
Kohn and Sham proposed to calculate the exact kinetic energy of a non-interacting
system with the same density as for the real interacting system.
1 N
TKS    Ψi 2 Ψi
2 i1
TKS
Ψi
– kinetic energy of a fictitious non-interacting
system of the same density ρ(r)
- are the orbitals for the non-interacting system
(KS orbitals)
T=TKS+(T-TKS)
FHK [ρ]  TKS[ρ]  J[ρ]  Enon-cl[ρ]
E[ρ]  ENe [ρ]  TKS[ρ]  J[ρ]  Exc [ρ] 
N
2
ZA
i (r1 ) dr1
A1 r1A
M
-
i1
1 N
  i 2 i
2 i1
Exc[ρ] includes everything which is
unknown:
2
2 1
1 N N
   i (r1 )
 j (r2 ) dr1dr2
2 i1 j1
r12
- exchange energy
 Exc [ρ]
- correction of kinetic energy (T-TKS)
- correlation energy
Kohn-Sham Equations:
Minimize E[ρ] with the conditions:
 ρ(r)dr  N
i  j  δij
M
 1 2
ρ(r2 )
ZA 



dr

v
(r
)


2
xc 1

i  εii

r12
A1 r1A 
 2
with:
Kohn-Sham Formalism
Kohn-Sham
equations
 1 2

    v(r)  ρ(r') dr'  Ki (r)  j  εj j
 r  r'
 2

i


Hartree-Fock equations
Exc[ρ] = ??
Local Density Approximation (LDA)
Exc [ρ]   ρ(r)εxc (ρ(r))dr
εxc only depends on the density at r
For the correlation part:
Monte-Carlo simulations – Ceperly and Alder
Good for solids
Generalized Gradient Approximation (GGA)
Exc [ρ]   ρ(r)εxc (ρ(r), ρ(r),...)dr
(1)
εxc depends on the density and its
gradient at r
Adjust εxc such that it satises all (or most) known properties of the exchangecorrelation hole and energy.
PW91, PBE…
(2) Fit εxc to a large data-set own exactly known binding energies of atoms and molecules.
BLYP, OLYP, HCTH…
Meta-GGAs
No major improvements!
Hybrid Functionals
GGA
Exchyb [ρ]  αExKS  (1  α)Exc
EXKS-the exact exchange calculated with the exact KS wave function
α- fitting parameter
Exchange and Correlation Functionals
In practice: BLYP, B3LYP, BPW91, …
Different functionals for different properties
Atomization energies:
Ionization energy: - B3LYP – the best!
Electron afinities:
Vibrational frequencies: - (BLYP), B3LYP, …
MO5-2X - bond dissociation energies, stacking and hydrogen-bonding
interactions in nucleobase pairs
Kohn-Sham orbitals