Teoría del Funcional de la Densidad (Density Functional Theory)
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1 Teoría del Funcional de la Densidad (Density Functional Theory) Motivation: limitations of the standard approach based on the wave function. The electronic density n(r) as the key variable: Functionals & Thomas-Fermi theory. Density functional theory (DFT): Hohenberg-Kohn (HK) Theorem Kohn-Sham equations. Exchange-Correlation functional: Local density approximation (LDA): Limitations. Generalized Gradients Approximations (GGA) and beyond. Física de Sistemas Complejos, Curso 2008/2009
2 Motivation
3 Born-Oppenheimer Approximation Hamiltonian for M nuclei and N electrons M α >> m ionic and (much faster) electronic motions can be decoupled electrons relaxed to GS for a given ionic configuration. nuclei move in a potential given by electronic GS energy
4 Time-independent Schrodinger equation (SE) Hamiltonian for an N-electron system: Whether it is an atom, a molecule or a solid depends only on v(r i ) Two possible strategies: direct solution or minimization
5 Quantum Chemistry approach to solve SE The general theory of quantum mechanics is now almost complete. The underlying physical laws necessary for the mathematical theory of a large part of physics and the whole of chemistry are thus completely known, and the difficulty is only that the exact application of these laws leads to equations much too complicated to be soluble. (Dirac,1929)
6 Condensed Matter Physics Approach: the density n(r) as the key variable QC DFT DFT provides a viable alternative, less accurate perhaps, but more versatile DFT recognizes that systems differ only by their potential v(r) and provides a prescription to deal with T and V ee maps the many-body problem with V ee onto a single-body problem without V ee. Knowledge of n(r) implies knowledge of Ψ and v(r), and hence of all other observables. Practical implementation?
7 Thomas-Fermi theory for atoms ( ) t Kinetic energy density in a uniform electron gas with n = n(r) ( r) = (3π ) n( r ) 5 3 Local approximation GS minimization of energy functional with the constraint: d rn( r) = µ = chemical potential (- µ =electronegativity) basic description of charge density & electrostatic potential. It does not reproduce the atom shell structure!!!
8 DFT basics: Hohenberg-Kohn Theorem
9 Practical DFT : Kohn-Sham Equations...We do not expect an accurate description of chemical bonding with the Local Density Approximation (LDA)... (Kohn & Sham, 1965) LDA: structural predictive power (e.g. transition pressure ZB β-sn in Si). GGA: not too far from chemical accuracy (1kcal/mole = ev/atom) (W. Kohn & J. Pople) Nobel Prize in Chemistry 1998
10 The first convincing DFT-LDA calculation M.T. Yin & M. Cohen, PRL 45, 1004 (1980)
11 R.P., M.C. Payne & A.D. Simpson, PRL 75, 4748 (1995)
12 Hohenberg-Kohn Theorem (1)
13 Hohenberg-Kohn Theorem (2)
14 Hohenberg-Kohn Th.: Consequences (1) (1) (2) (and, thus, the excited states!!) ((2) not true in spin-dft,.. )
15 Hohenberg-Kohn Th.: Consequences (2) Fundamental equation in DFT : Minimization of E v0 [n] with the normalization constraint
16 Thomas-Fermi vs Hartree: A hint... Thomas-Fermi ( ) Hartree (1928) Hartree describes GS of atoms much better than TF (reproduces the shell structure)
17 Kinetic energy for independent electrons ) ( ) ( r r φ φ L ),, ( 2 1 ),, ( ),, ( ˆ ),, ( i i r r r r r r T r r L L L L ψ ψ ψ ψ = = Independent General expression for the Kinetic energy { } ) ( ) (! 1 ) ( det,, ( 1 1 i r r r r r φ φ φ ψ L M O M L = = Independent electrons ) ( 2 1 ) ( ),, ( ˆ ),, ( r r r r T r r i i i φ φ ψ ψ = = L L
18 DFT as an effective single-body theory: Kohn-Sham equations We know how to relate KE and the density for a non-interacting system!! { φ( )} ψ ( r 1, L, r ) = det r i
19 What is the local effective potential?
20 Kohn-Sham equations
21 Kohn-Sham equations: Remarks
22 Total Energy in the Kohn-Sham scheme + E ion-ion + E ion-ion
23 HF vs LSD
24 Making DFT practical: Approximations Building the E xc functional. Local density approximation (LDA) Generalized Gradient approximation (GGA)
25 Local density approximation (LDA) for EXC
26 Local density approximation (LDA) for EXC
27 LDA for E XC including spin (LSD)
28 LDA exchange energy Simple argument: spherical hole of constant depth n/2 around the electron n/2 R x V x atractive due to the e - charge deficit
29 LDA correlation energy
30 LSD: performance E X : 5% smaller ; E C : 100% larger (E XC << T, V H, V ne ; but E XC 100% bonding energy) Cohesive (atomization) energies: 15% larger ( 1.3 ev overbinding) bond lengths: 1% smaller ; bulk moduli (elastic constants) 5 % Favors close-packed structures Energy barriers: 100% too low (no chemical accuracy ) wrong description of magnetic systems: Fe LDA is fcc paramagnetic (exp: BCC ferromagnetic) Poor description of weak bonding (van der Waals, hydrogen bonds). Atoms & Clusters V XC LSD : exponetial decay with n(r)**(1/3) instead of -1/r negative ions: generally unstable (electron affinities: 20% error)
31 Beyond LDA: Gradient expansions (GEA)
32 Relating G[n] to T S [n]. XC-hole (n XC ) Coupling constant integration technique relates the non-interacting (λ=0) system with the (λ=1) interacting one; g λ (r,r ) λ / r-r n XC describes the effect of e - e - repulsion: the presence of an e - in r reduces the probability of finding another e - in r electronic charge defect (effective positive charge) E XC : coulomb interaction (attractive) between an e - and its XC-hole
33 Properties of the XC-hole
34 n X n C n XC = n X + n C Fermi hole Coulomb hole
35 Why LSD works?
36 Generalized Gradient Approximations (GGA) Two different strategies to determine f(n, n)... Semiempirical (Becke): fitted to reproduce molecular results (but they fail for delocalized systems) Chemistry (BLYP) Non-empirical, based on general arguments and capable of describing different types of bonding (Perdew) Physics (PBE)
37 Generalized Gradient Approximation (PBE) Forced to retain the correct uniform electron gas limit (good aprox. to Na & Al metals, n XC of a real system). Built from the n XC GEA, removing the spurious long-range parts with a real-space cutoff, to recover the hole normalization properties. spin scaling: Satisfy constraints from scaling laws and other independent bounds Perdew, Burke & Ernzerhof, PRL 77, 3865 (1996) (Older version: PW91; Perdew & Wang, PRB 46, 6671 (1992))
38 Generalized Gradient Approximation (PBE)
39 Generalized Gradient Approximation (BLYP) E X from Becke (PRA 38, 3098 (1988)): functional form without the r divergence of the 2nd order expansion and β, γ fitted to reproduce HF atomic energies. E C from Lee, Yang & Parr (PRB 37, 785 (1988)): n C does not satisfied some basic constraints. The combination (BLYP) works extremely well for chemical applications (empirical)
40 GGA (GGS) performance E X : 0.5% ; E C : 5% larger (LDA: E X : 5% ; E C : 100%) Cohesive energies: 4% larger ( 0.3 ev ) (LDA: 15% l ( 1.3 ev)) bond lengths: 1% larger ; (LDA: 1% shorter) improved description of structural properties Energy barriers: 30% too low (LDA: 100% too low) magnetic systems: Fe GGA is BCC ferromagnetic!! improved description of weak bonding (hydrogen bonds). Atoms & Clusters V XC LSD : still wrong exponential decay negative ions: improved electron affinities (10% error) GGA: major improvement over LDA, chemical accuracy not too far away
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