Introduction to Density Functional Theory

Motivation Formalism 3G DFT + example

Introduction to Density Functional Theory S. Sharma Institut f¨ ur Physik Karl-Franzens-Universit¨ at Graz, Austria

19th October 2005

S. Sharma

Introduction to Density Functional Theory


Synopsis

1 2

Motivation : where can one use DFT Formalism : 1

Elementary quantum mechanics 1 2 3

2

Schr¨ odinger equation Born-Oppenheimer approximation Variational principle

Solving Schrödinger equation 1 2 3

Wave function methods: Hartree-Fock method Modern DFT : Kohn-Sham Exchange correlation functionals (LDA, GGA ....)

3

Third generation DFT

4

Example

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Motivation Walter Kohn was awarded with the Nobel Prize in Chemistry in 1998 for his development of the DFT.

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Motivation The following figure shows the number of publications where the phrase DFT appears in the title or abstract (taken from the ISI Web of Science).

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Motivation Where can DFT be applied: 1

DFT is presently the most successful (and also the most promising) approach to compute the electronic structure of matter.

2

Its applicability ranges from atoms, molecules and solids to nuclei and quantum and classical fluids.

3

Chemistry: DFT predicts a great variety of molecular properties: molecular structures, vibrational frequencies, atomization energies, ionization energies, electric and magnetic properties, reaction paths, etc

4

The original DFT has been generalized to deal with many different situations: spin polarized systems, multicomponent systems, systems at finite temperatures, superconductors, time-dependent phenomena, Bosons, molecular dynamics ...........

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Elementary QM Solving SE: HF Solving SE: DFT

Elementary quantum mechanics

Schrödinger equation ˆ i (x1 , x2 ...xN , R1 , R2 ...RM ) = Ei Φi (x1 , x2 , ...xN , R1 , R2 ...RM ) HΦ x ≡ {r, σ}

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Schrödinger equation

Kinetic energy ˆ = −1 H 2

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N X i

∇2i




Schrödinger equation Coloumb interaction ˆ = −1 H 2

N X i

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∇2i

N N X X 1 + rij i

j6=i





Nuclear electron interaction ˆ = −1 H 2

N X i

∇2i

N N X M N X X X 1 ZA + − rij riA

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i

j6=i

i

A





Nuclear terms ˆ = −1 H 2 −

N X i

∇2i +

N X M N X N X X ZA 1 − rij riA i

j6=i

i

A

M M M 1 X 1 2 X X ZA ZB ∇A + 2 MA RAB i

A B6=A

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Born-Oppenheimer approximation Due to their masses the nuclei move much slower than the electrons. So we can consider the electrons as moving in the field of fixed nuclei. ˆ = −1 H 2

N X i

∇2i +

N X N N X M X X 1 ZA − = Tˆ + VˆN e + Vêe rij riA i

j6=i

i

A

ˆ i (x1 , x2 ...xN ) = Ei Ψi (x1 , x2 , ...xN ) HΨ

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Variational principle The variational principle states that the energy computed from a guessed wave function Ψ is an upper bound to the true ground-state energy E0 . Full minimization of E with respect to all allowed N-electrons wave functions will give the true ground state. E0 [Ψ0 ] = min E[Ψ] = minhΨ|Tˆ + VˆN e + Vêe |Ψi Ψ

Ψ

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Elementary quantum mechanics Energy ˆ E0 [Ψ] = hΨ|H|Ψi =

Z

ˆ Ψ∗ (x)HΨ(x)dx

Wave function : example of two fermions Pauli exclusion principle results in an antisymmetric wave function Ψ(x1 , x2 ) = φ1 (x1 )φ2 (x2 ) − φ1 (x2 )φ2 (x1 ) This looks like a determinant ¯ ¯ φ1 (x1 ) φ2 (x1 ) ¯ ¯ φ1 (x2 ) φ2 (x2 ) S. Sharma

¯ ¯ ¯ ¯




Wave function method: Hartree-Fock The ground state wave function is approximated by a Slater determinant:

¯ ¯ φ1 (x1 ) φ2 (x1 ) · · · ¯ 1 ¯¯ φ1 (x2 ) φ2 (x2 ) · · · ΨHF (x1 , x2 ....xN ) = √ ¯ .. .. .. . N ! ¯¯ . . ¯ φ1 (xN ) φ2 (xN ) · · ·

¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ φN (xN ) ¯ φN (x1 ) φN (x2 ) .. .

The Hartree-Fock approximation is the method whereby the orthogonal orbitals φi are found that minimize the energy. The variational principle is used. EHF = min E[φHF ] φHF

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Problems with wave function methods

I. Visualization and probing The conventional wave function approaches use wave function as the central quantity, since it contains the full information of a system. However, is a very complicated quantity that cannot be probed experimentally and that depends on 3N variables, N being the number of electrons.

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Problems with wave function methods II. Time consuming The interactions are very difficult to calculate for a realistic system, in fact this is most time consuming part. Vêe =

N N X X 1 rij i

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j6=i




Problems with wave function methods III. Inefficient One Slater determinant does a rather bad job for expanding the many-body wave function. Ψ(x, y)

φ1 (x)φ2 (y)

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Two Slater determinants

Ψ(x, y)

φ1 (x)φ2 (y) + φ3 (x)φ4 (y)

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Eight Slater determinants

Ψ(x, y)

φ1 (x)φ2 (y) + φ3 (x)φ4 (y)......φ15 (x)φ16 (y)

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Density Functional Theory

Experiments probe density Z ρ(r1 ) = Ψ∗ (x1 , x2 , .....xN )Ψ(x1 , x2 , .....xN )dx2 ....dxN

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Hohenberg-Kohn theorems

I Hohenberg-Kohn theorem ˆ = Tˆ + VˆN e + Vêe H This first theorem states that the VˆN e is (to within a constant) a unique ˆ we see that the functional of density (ρ); since, in turn this potential fixes H full many particle ground state is a unique functional of density.

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Hohenberg-Kohn theorems

II Hohenberg-Kohn theorem The second H-K theorem states that 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. This is nothing but the variational principle,but this time with density and not wave function E0 [ρ0 ] = min E[ρ] = min T [ρ] + EN e [ρ] + Eee [ρ] ρ

ρ

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Modern DFT Energy as a functional of density E[ρ] = T [ρ] + EN e [ρ] + Eee [ρ] E[ρ] = T [ρ] + EN e [ρ] + EH [ρ] + Ex [ρ] + Ec [ρ]

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Modern DFT: Kohn-Sham The system is replaced by a fictitious non-interaction system with same density as the real system. Kohn-Sham non-interacting system E[ρ] = T [ρ] + EN e [ρ] + EH [ρ] + Ex [ρ] + Ec [ρ] 1 T [ρ] = − hφi |∇2 |φi i 2 Z Z ρ(r) 1 ρ(r1 )ρ(r2 ) EN e [ρ] + EH [ρ] = −Z dr1 dr2 dr − r 2 r12 Local Density Approximation Based on the properties of uniform electron gas XC potential is approximated as: Z 3 2 1/3 ρ4/3 (r)dr Exc [ρ] = − (3π ) 4π S. Sharma




Modern DFT: Kohn-Sham The system is replaced by a fictitious non-interaction system with same density as the real system. Kohn-Sham non-interacting system E[ρ] = T [ρ] + EN e [ρ] + EH [ρ] + Ex [ρ] + Ec [ρ] 1 T [ρ] = − hφi |∇2 |φi i 2 Z Z ρ(r) 1 ρ(r1 )ρ(r2 ) EN e [ρ] + EH [ρ] = −Z dr1 dr2 dr − r 2 r12 Local Density Approximation Based on the properties of uniform electron gas XC potential is approximated as: Z 3 2 1/3 ρ4/3 (r)dr Exc [ρ] = − (3π ) 4π S. Sharma



Third generation DFT: Exact exchange

Treating exchange term exactly E[ρ] = T [ρ] + EN e [ρ] + EH [ρ] + Ex [ρ] Using the Fock integral and its functional derivative the exchange term can be treated exactly . This method is called EXX or OEP. S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Phys. Rev. Lett. 95 136402 (2005)

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Phase transition in Ce using LDA

Energy (Ha)

-0.7

-0.705

-0.71

-0.715

140

160 3 volume (a.u.) S. Sharma

180

200



New physics captured with EXX

Energy (Ha)

-8845

-8846

α−Ce

γ − Ce

-8847

-8848

8.8

9

9.2

9.4 9.6 9.8 Lattice constant (a.u.) S. Sharma

10

10.2

10.4



Many thanks to

1

Funding agency : FWF and Exciting EU network

2

Dr. J. K. Dewhurst

3

D. Rankin and S. Hinchley

4

EXCITING code: http://exciting.sourceforge.net/

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