Density Functional Theory (DFT) 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 K 4 (K - # of basis functions) CI methods scale as K 6 -K 10 MPn methods scale as >K 5 CC methods scale as >K 6 Correlated methods are not feasible for medium and large sized Correlated methods are not feasible for medium and large sized molecules! molecules! The electron density - it is the central quantity in DFT - is defined as: Alternative: DFT
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 asK 4 (K - # of basis functions) CI methods scale asK 6 -K 10 MPn methods scale as>K 5 - PowerPoint PPT Presentation
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Density Functional Theory (DFT)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 K4 (K - # of basis functions)CI methods scale as K6-K10
MPn methods scale as >K5
CC methods scale as >K6
Correlated methods are not feasible for medium and large sized Correlated methods are not feasible for medium and large sized molecules!molecules!
The electron density- it is the central quantity in DFT- is defined as:
Alternative: DFT
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)]
ρ(r)
ν(r)
H
E
N
Nρ(r)dr
EΨΨH ˆ
First HK Theorem:
Hohenberg–Kohn Theorems
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.
Ψ’ as a test function for H:
Ψ as a testfunction for H’:
Summing up the last two inequalities:
Proof: by reductio ad absurdum
Contradiction!
Variational Principle in DFTVariational 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)Thomas-Fermi model (1927)
The explicit form of T[ρ] and Enon-cl[ρ] is the major challenge of DFT