Teoía del Funcional de la Densidad (Density Functional Theoy) Motivation: limitations of the standad appoach based on the wave function. The electonic density n() as the key vaiable: Functionals & Thomas-Femi theoy. Density functional theoy (DFT): Hohenbeg-Kohn (HK) Theoem Kohn-Sham equations. Exchange-Coelation functional: Local density appoximation (LDA): Limitations. Genealized Gadients Appoximations (GGA) and beyond. Física del Estado Sólido II, Cuso 2010/2011
Motivation
Time-independent Schodinge equation (SE) Hamiltonian fo an -electon system: Whethe it is an atom, a molecule o a solid depends only on v( i ) Two possible stategies: diect solution o minimization
Quantum Chemisty appoach to solve SE The geneal theoy of quantum mechanics is now almost complete. The undelying physical laws necessay fo the mathematical theoy of a lage pat of physics and the whole of chemisty ae thus completely known, and the difficulty is only that the exact application of these laws leads to equations much too complicated to be soluble. (Diac,1929)
Condensed Matte Physics Appoach: the density n() as the key vaiable QC DFT DFT povides a viable altenative, less accuate pehaps, but moe vesatile DFT ecognizes that systems diffe only by thei potential v() and povides a pesciption to deal with T and V ee maps the many-body poblem with V ee onto a single-body poblem without V ee. Knowledge of n() implies knowledge of and v(), and hence of all othe obsevables. Pactical implementation?
Thomas-Femi theoy fo atoms (1927-28) t Kinetic enegy density in a unifom electon gas with n = n() 3 10 2 2 3 ( ) (3 ) n( ) 5 3 Local appoximation GS minimization of enegy functional with the constaint: d n( ) = chemical potential (- =electonegativity) basic desciption of chage density & electostatic potential. It does not epoduce the atom shell stuctue!!!
Hohenbeg-Kohn Theoem (1)
Hohenbeg-Kohn Theoem (2)
Hohenbeg-Kohn Th.: Consequences (1) (1) (2) (and, thus, the excited states!!) ((2) not tue in spin-dft,.. )
Hohenbeg-Kohn Th.: Consequences (2) Fundamental equation in DFT : Minimization of E v0 [n] with the nomalization constaint
Thomas-Femi vs Hatee: A hint... Thomas-Femi (1927-28) Hatee (1928) Hatee descibes GS of atoms much bette than TF (epoduces the shell stuctue)
Kinetic enegy fo independent electons ) ( ) ( ) ( ) (! 1 ) ( det,, ( 1 1 1 1 1 i ),, ( 2 1 ),, ( ),, ( ˆ ),, ( 1 1 2 1 1 1 i i T Independent electons ) ( 2 1 ) ( ),, ( ˆ ),, ( 1 2 1 1 T i i i Geneal expession fo the Kinetic enegy
DFT as an effective single-body theoy: Kohn-Sham equations We know how to elate KE and the density fo a non-inteacting system!! ( 1,, ) det ( ) i
What is the local effective potential?
Kohn-Sham equations
Total Enegy in the Kohn-Sham scheme + E ion-ion + E ion-ion
HF vs LSD
The fist convincing DFT-LDA calculation M.T. Yin & M. Cohen, PRL 45, 1004 (1980)
Making DFT pactical: Appoximations Building the E xc functional. Local density appoximation (LDA) Genealized Gadient appoximation (GGA)
Local density appoximation (LDA) fo EXC
Local density appoximation (LDA) fo EXC
LDA fo E XC including spin (LSD)
LDA exchange enegy Simple agument: spheical hole of constant depth n/2 aound the electon n/2 R x V x atactive due to the e - chage deficit
LDA coelation enegy
LSD: pefomance E X : 5% smalle ; E C : 100% lage (E XC << T, V H, V ne ; but E XC 100% bonding enegy) Cohesive (atomization) enegies: 15% lage ( 1.3 ev ovebinding) bond lengths: 1% smalle ; bulk moduli (elastic constants) 5 % Favos close-packed stuctues Enegy baies: 100% too low (no chemical accuacy ) wong desciption of magnetic systems: Fe LDA is fcc paamagnetic (exp: BCC feomagnetic) Poo desciption of weak bonding (van de Waals, hydogen bonds). Atoms & Clustes V XC LSD : exponetial decay with n()**(1/3) instead of -1/ negative ions: geneally unstable (electon affinities: 20% eo)
n X n C n XC = n X + n C Femi hole Coulomb hole
Genealized Gadient Appoximation (PBE) Foced to etain the coect unifom electon gas limit (good apox. to a & Al metals, n XC of a eal system). Built fom the n GEA XC, emoving the spuious long-ange pats with a eal-space cutoff, to ecove the hole nomalization popeties. spin scaling: Satisfy constaints fom scaling laws and othe independent bounds Pedew, Buke & Enzehof, PRL 77, 3865 (1996) (Olde vesion: PW91; Pedew & Wang, PRB 46, 6671 (1992))
GGA (GGS) pefomance E X : 0.5% ; E C : 5% lage (LDA: E X : 5% ; E C : 100%) Cohesive enegies: 4% lage ( 0.3 ev ) (LDA: 15% l ( 1.3 ev)) bond lengths: 1% lage ; (LDA: 1% shote) impoved desciption of stuctual popeties Enegy baies: 30% too low (LDA: 100% too low) magnetic systems: Fe GGA is BCC feomagnetic!! impoved desciption of weak bonding (hydogen bonds). Atoms & Clustes V XC LSD : still wong exponential decay negative ions: impoved electon affinities (10% eo) GGA: majo impovement ove LDA, chemical accuacy not too fa away
DFT as a tool in anotechnology: Manipulation of a Si vacancy with an Atomic Foce Micoscope Tip-assisted themal diffusion mechanism Si-vacancy manipulation induced by the tip poximity Tip-sample inteaction educes baies to the limit that enables themally activated diffusion Diectionality imposed by the tip global displacement Baie eduction Manipulation Y. Sugimoto et al Phys. Rev. Lett. 98, 106104 (2007).