Density Functional Theory (DFT)
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1 Density Functional Theory (DFT) An Introduction by A.I. Al-Sharif Irbid, Aug, 2 nd, 2009
2 Density Functional Theory Revolutionized our approach to the electronic structure of atoms, molecules and solid materials. Plethora of publications reporting very successful applications in a wide range of fields in physics, chemistry, biology and others. Its appeal is in the simplicity and accuracy over the traditional methods: HF and CI
3 Walter Kohn: Founder of DFT 1998 Nobel Prize in Chemistry for the development of DFT. P. Hohenberg and W. Kohn, Phys. Rev. 136, B864(1964). W. Kohn and L.J. Sham, Phys. Rev. 140, A1133(1965). Thousands of citations.
4 Milestones Reviews
5 References: Texts
6 References: Texts Downloadable ebooks Info/Physics.php?id=30 Presentations, tutorials and more see the web
7 The DFT acronym Take care when searching the web for DFT, for you might hit: Discrete Fourier Transform Decision Field Theory. The UK's Department for Transport Design For Test DFT Digital Film Technology Deep Flow Technique and many others. Advise: Use the full name
8 Function vs Functional f ( x) 2 = x ; f (2) = 4 F[ f ] 1 = f ( x) dx; F[ x 0 2 ] = 1/ 3
9 The Basic Many Body Problem Ground state properties of a system of N charged fermions (electrons, positrons, protons, ) moving in an external field.
10 The Hamiltonian T: Diff. one body operator. V: Multiplicative one body operator U: Multiplicative two body operator.
11 Atomic Units Confusion! We use atomic Hartree units: Not the atomic Rydberg units: E o (Hdrgn) = ev = 1 Ryd = 0.5 Hart
12 T.I.S.E: HΨ = EΨ To compute any property, we need the many body wave function Ψ o (x 1,, x N ), which is a function of 4N independent variables. O = O[ Ψ] = Ψ O Ψ
13 No rigorous analytic solution No! there isn t, even for N=2 (Helium atom) What about N = The bottle neck is the interaction term U. Approximate treatment is needed.
14 One Body Operator Non-differential (multiplicative) one body operator (e.g. the external potential) V = V[n]
15 Can we generalize? Is the total energy a functional of the density? E = E[ Ψ] ok E = E[ n]?
16 Thomas Fermi Model (1927) Is an approximation! Basic postulate: electrons are distributed uniformly in phase space with 2e/h 3. T TF r [ n ] n 5/ 3 ( r ) r dr
17 The Density Functional Approach Ψ o (r 1,, r N ) has much more information than actually needed. All what we need is the ground state density n(r) which is a function of 3 variables only!
18 Fundamental Statement (Hohenberg and Kohn) n o r r ( r ) v( r ) E o = min ( E[ n])
19 Formal Proof Consider two different Hamiltonians Corresponding to two different wave functions
20 HK Proof reductio ad absurdum. If we assume that n = n we get a contradiction!
21 SE vs. DFT O = O[ n] =? O = O[ Ψ] = Ψ O Ψ DFT is a theory of existence but NOT a calculation recipe. The explicit functional form is, so far, unknown!!
22 The Kohn-Sham Scheme A further simplification towards this problem is introduced in the second cornerstone paper by Kohn and Sham. Since n o (r) is the thumbnail of any system: Two systems with the same ground state density should have the same physical properties.
23 The KS Scheme Now consider the two systems A and B: E A = E B => T + V + U = T KS + V KS
24 The KS scheme T + V + U = T KS + V KS V KS = V + U + (T T KS ) = V + J + Ũ + (T T KS ) = V + J + E xc
25 The KS potential The dilemma is now how to evaluate v xc!
26 The Local Density Approximation (LDA) is the exchange-correlation energy per particle in a uniform electron gas of density n
27 LDA Example
28 LDA: Success and failure! The simplest and lightest! Designed for a slowly varying densities but it works well beyond this limit! Fails to reproduce atomic Rydberg series (wrong 1/r potential tail) E gap, E c, B, a: Estimation errors! Workaround: SIC
29 The Generalized Gradient Approximation (GGA)
30 LDA Complete failure (bcc Iron) LDA: Fe is nonmagnetic fcc. GGA: Fe is a ferromagnetic bcc. Experiment: GGA prediction is correct!
31 Quantum Monte Carlo (QMC) and Exact Exchange (EXX) Elaborate and Heavy computations. Nearly exact atomic KS potentials (QMC). Excellent band gap (EXX). Recent developments: - Generating excellent atomic KS potentials using (LDA + SIC) - Reformulating EXX in terms of ground state orbitals. (approximate correlation!)
32 The KS equations The KS Hamiltonian is separable into N independent Hamiltonians {h i } each satisfying the one particle SE: Physical significance of {ϕ i } and {ε i }?
33 Basis Set Expansion Numerically it is more efficient to expand {ϕ i } in a suitable basis set (depending on the symmetry): - STO s (Slater Type Orbitals) or atomic orbitals. - Gaussians. -Plane waves. ϕ i = {c 1, c 2,,, c m }; }; Complex Numbers. Increase m to achieve convergence!
34 The Self-Consistent cycle ϕ i = {c 1, c 2,,, c m };
35 The Solids Complicated Problem v(r) depends on the moving nuclei: Coupled electronic and Nuclear degrees of freedom. (Born-Oppenheimer approximation) Very deep in the vicinity of the nuclei => wiggling orbitals => m is large. (Pseudopotentials) Infinite number of electrons => infinite number of extended wave functions. (Bloch s Theorem)
36 The Born-Oppenheimer Approximation
37 Solids: The Periodic Matter An ideal periodic solid contains infinite numbers of atoms and electrons. We need infinite number of wave functions that extend over the entire solid (infinite space). Using the periodicity character (symmetry) Block resolved those singularities!
38 Bloch s Theorem Infinite number of wave functions => Finite number of wave functions at an infinite number of k points.
39 The Monkhorst-Pack mesh For adjacent k points ψ n,k is almost the same => Divide the k-space into small volumes and chose one point in each volume to represent ψ n,k over this volume. How many k-points? k Check convergence!
40 Supercells: : Enforcing periodicity
41 Acknowledgment The flash animations are from: list.htm Some illustrations are from: e introduction-dft-goal-describeproperties-matter-theoretical-methods-firmlyrooted-fundamental-equations-as- Entertainment-ppt-powerpoint/
42 Thank you
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