Four mini-talks on ground-state DFT Kieron Burke UC
Four mini-talks on ground-state DFT Kieron Burke UC Irvine Chemistry and Physics General ground-state DFT Semiclassical approach Potential functional approximations PERSISTENCE OF CHEMISTRY IN THE LIMIT OF LARGE ATOMIC NUMBER http://dft.uci.edu Jan 24, 2011 BIRS 1 General ground-state DFT Kieron Burke & John Perdew
Jan 24, 2011 BIRS 2 John Intro I KOHN-SHAM THEORY FOR THE GROUND STATE ENERGY E AND SPIN DENSITIES n (r ),n (r ) OF A MANY-ELECTRON SYSTEM. THE MOST WIDELY-USED METHOD OF ELECTRONIC STRUCTURE CALCULATION IN QUANTUM CHEMISTRY, CONDENSED MATTER PHYSICS, & MATERIALS ENGINEERING. NOT AS POTENTIALLY ACCURATE AS MANY-ELECTRON WAVEFUNCTION METHODS, BUT COMPUTATIONALLY MORE
EFFICIENT, ESPECIALLY FOR SYSTEMS WITH VERY MANY ELECTRONS. 3 John Intro 2 MANY-ELECTRON HAMILTONIAN N H i 1 1 i2 2 N
i 1 1 v i ( ri ) 2 i j i 1 ri r j GROUND-STATE WAVEFUNCTION
(r1 , 1 , r2 , 2 ,..., rN , N ) GROUND-STATE SPIN DENSITIES (= OR )= OR ) 2 3 3 n (r ) N d r2 ...d rN r , , r2 , 2 ,..., rN , N n n n 2 ... N
SPIN DENSITY FUNCTIONAL FOR G. S. ENERGY 1 n ( r )n(r `) E[n , n ] Ts [n , n ] d 3 rv (r )n (r ) d 3 r d 3 r ` E xc [n , n ] 2 r ` r Ts KINETIC ENERGY FOR NON-INTERACTING ELECTRONS WITH G. S. SPIN DENSITIES n , n E xc EXCHANGE-CORRELATION ENERGY 4
(r KOHN-SHAM METHOD: INTRODUCE ORBITALS ) John Intro 3 FOR THE NON-INTERACTING SYSTEM occup n ( r ) ( r ) 2 occup
Ts [n , n ] 1 2 2 THE EULER EQUATION TO MINIMIZE E[n , n ] AT FIXED N IS THE KOHNSHAM SELF-CONSISTENT ONE-ELECTRON EQUATION 1 2 v
n , n ; r ( r ) ( r ) s
2 OCCUPIED ORBITALS HAVE (= OR )AUFBAU PRINCIPLE) E xc n ( r `) 3 v s n , n ; r v (r ) d r ` r ` r n (r ) 5
LOCAL AND SEMI-LOCAL APPROX.` FOR E xc [n , n ] John Intro 4 LOCAL SPIN DENSITY APPROXIMATION (= OR )LSDA) E xcLSDA d 3 rn xcunif (n , n ) xcunif (n , n ) XC ENERGY PER PARTICLE OF AN ELECTRON GAS OF UNIFORM n , n . GENERALIZED GRADIENT APPROX. (= OR )GGA) E xcGGA d 3 rnf (n , n , n , n ) GIVES A BETTER DESCRIPTION OF STRONGLY INHOMOGENOUS SYSTEMS (= OR )E.G., ATOMS & MOLECULES) PERDEW-BURKE-ERNZERHOF 1996 (= OR )PBE) GGA: CONSTRUCTED NON-EMPIRICALLY TO SATISFY EXACT CONSTRAINTS. 6
First ever KS calculation with exact EXC[n] Used DMRG (= OR )density-matrix renormalization group) 1d H atom chain Miles Stoudenmire, Lucas Wagner, Steve White Jan 24, 2011 BIRS 7 Some important challenges in ground-state DFT
Systematic, derivable approximations to EXC[n] Deal with strong correlation (= OR )Scuseria, Prodan, Romaniello) Systematic, derivable, reliable, accurate, approximations to TS[n] Jan 24, 2011 BIRS 8 Functional approximations Original approximation to EXC[n] : Local density approximation (= OR )LDA) Nowadays, a zillion different approaches to constructing improved approximations Culture wars between purists (= OR )non-empirical) and pragmatists. This is NOT OK. Jan 24, 2011
BIRS 9 Too many functionals Jan 24, 2011 BIRS Peter Elliott 10 Things users despise about DFT No simple rule for reliability No systematic route to improvement If your property turns out to be inaccurate, must wait several decades for solution
Complete disconnect from other methods Full of arcane insider jargon Too many functionals to choose from Can only be learned from another DFT guru Oct 14, 2010 Sandia National Labs 11 Things developers love about DFT No simple rule for reliability No systematic route to improvement If a property turns out to be inaccurate, can take several decades for solution Wonderful disconnect from other methods Lots of lovely arcane insider jargon So many functionals to choose from Must be learned from another DFT guru Oct 14, 2010
Sandia National Labs 12 Modern DFT development It must have sharp steps for stretched bonds Oct 14, 2010 It keeps H2 in singlet state as R
Sandia National Labs Its tail must decay like -1/r 13 Semiclassical underpinnings of density functional approximations Peter Elliott, Donghyung Lee, Attila Cangi UC Irvine, Chemistry and Physics Jan 24, 2011 BIRS
14 Difference between Ts and Exc Pure DFT in principle gives E directly from n Original TF theory of this type Need to approximate TS very accurately Thomas-Fermi theory of this type Modern orbital-free DFT quest. Misses quantum oscillations such as atomic shell structure KS theory uses orbitals, not pure DFT Made things much more accurate Much better density with shell structure in there. Only need approximate EXC[n]. Jan 24, 2011 BIRS 15
The big picture We show local approximations are leading terms in a semiclassical approximation This is an asymptotic expansion, not a power series Leading corrections are usually NOT those of the gradient expansion for slowly-varying gases Ultimate aim: Eliminate empiricism and derive density functionals as expansion in . Jan 24, 2011 BIRS 16 More detailed picture Turning points produce quantum oscillations Shell structure of atoms Friedel oscillations There are also evanescent regions
Each feature produces a contribution to the energy, larger than that of gradient corrections For a slowly-varying density with Fermi level above potential everywhere, there are no such corrections, so gradient expansion is the right asymptotic expansion. For everything else, need GGAs, hybrids, meta-GGAs, hyper GGAs, non-local vdW, Jan 24, 2011 BIRS 17 What we might get We study both TS and EXC For TS: Would give orbital-free theory (= OR )but not using n) Can study atoms to start with Can slowly start (= OR )1d, box boundaries) and work
outwards For EXC: Improved, derived functionals Integration with other methods Jan 24, 2011 BIRS 18 A major ultimate aim: EXC[n] Explains why gradient expansion needed to be generalized (= OR )Relevance of the slowly-varying electron gas to atoms, molecules, and solids J. P. Perdew, L. A. Constantin, E. Sagvolden, and K. Burke, Phys. Rev. Lett. 97, 223002 (= OR )2006).) Derivation of b parameter in B88 (= OR )Non-empirical 'derivation' of B88 exchange functional P. Elliott and K. Burke, Can. J. Chem. 87, 1485 (= OR )2009).). PBEsol
Restoring the density-gradient expansion for exchange in solids and surfaces J.P. Perdew, A. Ruzsinszky, G.I. Csonka, O.A. Vydrov, G.E. Scuseria, L.A. Constantin, X. Zhou, and K. Burke, Phys. Rev. Lett. 100, 136406 (= OR )2008)) explains failure of PBE for lattice constants and fixes it at cost of good thermochemistry Gets Au- clusters right Jan 24, 2011 BIRS 19 Structural and Elastic Properties Errors Errors in in LDA/GGA(PBE)-DFT LDA/GGA(PBE)-DFT computed computed lattice
lattice constants constants and and bulk bulk modulus modulus with with respect respect to to experiment experiment Fully Fully converged converged results results (basis (basis set, set, k-sampling, k-sampling, supercell
supercell size) size) Error Error solely solely due due to to xc-functional xc-functional GGA GGA does does not not outperform outperform LDA LDA characteristic
characteristic errors errors of of <3% <3% in in lat. lat. const. const. << 30% 30% in in elastic elastic const. const. LDA LDA and and GGA GGA provide provide bounds
bounds to to exp. exp. data data provide provide ab ab initio initio error error bars bars Blazej Grabowski, Dusseldorf Inspection of several xc-functionals is critical to estimate Janpredictive 24, 2011 BIRS power and error bars!
20 Test system for 1d Ts v(= OR )x)=-D sinp(= OR )mx)x) Jan 24, 2011 BIRS 21 Semiclassical density for 1d box TF Classical momentum: Classical phase: Fermi energy: Classical transit time:
Elliott, Cangi, Lee, KB, PRL 2008 Jan 24, 2011 BIRS 22 Density in bumpy box Exact density: TTF[n]=153.0 Thomas-Fermi density: TTF[nTF]=115 Semiclassical density: TTF[nsemi]=151.4 DN < 0.2% Jan 24, 2011 BIRS
23 A new continuum Consider some simple problem, e.g., harmonic oscillator. Find ground-state for one particle in well. Add a second particle in first excited state, but divide by 2, and resulting density by 2. Add another in next state, and divide by 3, and density by 3 Jan 24, 2011 BIRS 24 Continuum limit
Leading corrections to local approximations Attila Cangi, Donghyung Lee, Peter Elliott, and Kieron Burke, Phys. Rev. B 81, 235128 (= OR )2010). Attila Cangi Jan 24, 2011 BIRS 25 Getting to real systems Include real turning points and evanescent regions, using Langer uniformization Consider spherical systems with Coulombic potentials (= OR )Langer modification) Develop methodology to numerically calculate
corrections for arbitrary 3d arrangements Jan 24, 2011 BIRS 26 Classical limit for neutral atoms For interacting systems in 3d, increasing Z in an atom, keeping it neutral, approaches the classical continuum, ie same as 0 (= OR )Lieb 81)
Jan 24, 2011 BIRS 27 Non-empirical derivation of density- and potential- functional approximations Attila Cangi UC Irvine Physics and Chemistry & Peter Elliott, Hunter College, NY Donghyung Lee, Rice, Texas E.K.U. Gross, MPI Halle http://dft.uci.edu Jan 24, 2011 BIRS
28 New results in PFT Universal functional of v(= OR )r): Direct evaluation of energy: Jan 24, 2011 BIRS 29 Coupling constant: New expression for F: Jan 24, 2011 BIRS
30 Variational principle Necessary and sufficient condition for same result: Jan 24, 2011 BIRS 31 All you need is n[v](r) Any approximation for the density as a functional of v(= OR )r) produces immediate selfconsistent KS potential and density Jan 24, 2011
BIRS 32 Evaluating the energy With a pair TsA[v] and nsA[v](= OR )r), can get E two ways: Both yield same answer if Jan 24, 2011 BIRS 33 Coupling constant formula for energy Choose any reference (= OR )e.g., v0(= OR )r)=0) and write Do usual Pauli trick
Yields Ts[v] directly from n[v]: Jan 24, 2011 BIRS 34 Accuracy and minimization For box problems, v(= OR )x)=-D sin2px, D=5 Use wavefunctions at different D to calculate E[v] CC results much more accurate CC has minimum at given potential Jan 24, 2011
BIRS 35 Different kinetic energy density CC formula gives DIFFERENT kinetic energy density (= OR )from any usual definitions) But approximation much more accurate globally and pointwise than with direct approximation Jan 24, 2011 BIRS 36
Not perfect Now make variations in p: V(= OR )x)=-D sinp px Still CC much more accurate Minimum not quite correct Generally, need to satisfy symmetry: Jan 24, 2011 nns(= OR )r)/nv(= OR )r )=nns(= OR )r )/nv(= OR )r) BIRS 37 PERSISTENCE OF CHEMISTRY IN THE LIMIT OF LARGE ATOMIC
NUMBER JOHN P. PERDEW PHYSICS TULANE UNIVERSITY NEW ORLEANS CO-AUTHORS FROM U. C. IRVINE: LUCIAN A. CONSTANTIN JOHN C. SNYDER KIERON BURKE Jan 24, 2011 BIRS 38 John B1 THE PERIODIC TABLE OF THE ELEMENTS SHOWS A QUASIPERIODIC VARIATION OF CHEMICAL PROPERTIES WITH ATOMIC NUMBER Z. THE IONIZATION ENERGY I=E+1 E0 OF
AN ATOM INCREASES ACROSS EACH ROW OR PERIOD, AS A SHELL IS FILLED, BUT DECREASES DOWN A COLUMN, AS THE ATOMIC NUMBER INCREASES AT FIXED ELECTRON CONFIGURATION. r THE VALENCE-ELECTRON RADIUS DECREASES ACROSS A PERIOD, BUT INCREASES DOWN A COLUMN. 39 John B2 DO THESE TRENDS PERSIST IN THE NON-RELATIVISTIC LIMIT OF LARGE ATOMIC NUMBER Z? EXPERIMENT CANNOT ANSWER THIS QUESTION, BUT KOHNSHAM THEORY CAN! WHAT IS KNOWN SO FAR ABOUT THE NON-RELATIVISTIC Z LIMIT?
TOTAL ENERGY E = -AZ7/3 +BZ2 +CZ5/3+ THE SIMPLE THOMAS-FERMI APPROX. (= OR )LSDA FOR TS, & NEGLECT OF EXC) GIVES THE CORRECT E = -AZ7/3 LEADING TERM. 40 John B3 THE Z LIMIT OF I IS THOMAS-FERMI APPROX. ITF = 1.3 eV EXTENDED TF APPROX. . IETF = 3.2 eV (= OR )TFSWD) PROVEN TO BE FINITE IN HF THEORY (= OR )Solovej) THE Z LIMIT OF THE VALENCE-ELECTRON RADIUS IS TF r 9bohr 5 THESE RESULTS SHOW NO PERSISTENCE OF CHEMICAL PERIODICITY.
BUT ARE THEY CORRECT? ONLY KOHN-SHAM THEORY CAN ACCOUNT FOR SHELL STRUCTURE. 41 John B4 WE HAVE PERFORMED KOHN-SHAM CALCULATIONS (= OR )LSDA, PBE-GGA, AND EXACT EXCHANGE OEP) FOR ATOMS WITH UP TO 3,000 ELECTRONS, FROM THE MAIN OR sp BLOCK OF THE PERIODIC TABLE. WE TOOK THE ELECTRON SHELL-FILLING FROM MADELUNG`S RULE: SUBSHELLS nl FILL IN ORDER OF INCREASING n+l, AND, FOR FIXED n+l, IN ORDER OF INCREASING n. 42
Ionization as Z Jan 24, 2011 BIRS 43 John B5 WE SOLVED THE KOHN-SHAM EQUATIONS ON A RADIAL GRID, USING A SPHERICALLY-AVERAGED KOHN-SHAM POTENTIAL. FOR EACH COLUMN, WE PLOTTED I vs. Z-1/3 FOR Z-1/3 > 0.07, AND FOUND A NEARLY-LINEAR BEHAVIOR FOR 0.07 < Z-1/3 < 0.2 Z=3000 Z=125
THEN WE EXTRAPOLATED QUADRATICALLY TO Z-1/3 =0 OR Z = . 44 LIMITING Z IONIZATION ENERGIES John Tab 1 (eV) GROUP OR COLUMN LSDA GGA (PBE) ns
I II 1.9 2.4 1.8 2.3 np III IV V VI VII VIII 3.3 3.8
4.2 4.3 4.7 5.2 3.1 3.7 4.2 4.1 4.6 5.1 AS Z DOWN A COLUMN, I DECREASES TO A COLUMNDEPENDENT LIMIT, WHICH INCREASES ACROSS A PERIOD. THE PERIODIC TABLE BECOMES PERFECTLY PERIODIC. 45 Z limit of ionization potential Shows even energy differences can be found Looks like LDA exact for EX
as Z. Looks like finite EC corrections Looks like extended TF (= OR )treated as a potential functional) gives some sort of average. Lucian Constantin, John Snyder, JP Perdew, and KB, JCP 2010 Jan 24, 2011 BIRS 46 Exactness for Z for Bohr atom Using hydrogenic orbitals to
improve DFT John C Snyder Jan 24, 2011 BIRS 47 John B6 THE AVERAGE OF I OVER COLUMNS, IN THE Z LIMIT, IS CLOSE TO THE EXTENDED TF LIMIT OF 3.2 eV. RADIAL IONIZATION DENSITY DnR ( Z , r ) 4pr 2 n0 ( Z , r ) n1 ( Z , r ) dr D n
R ( Z , r ) 1 0 WE EXTRAPOLATED THIS VERY CAREFULLY, THEN COMPUTED THE LIMITING VALENCE-ELECTRON RADIUS r drr DnR ( Z , r ) Z 0 48 bohr GROUP OR COLUMN
r r Z John Tab 2 GGA (PBE) Z ns I II 14.1 13.6 np
III IV V VI VII VIII 10.2 9.8 9.5 9.4 9.1 8.8 THE VALENCE-ELECTRON RADIUS INCREASES DOWN A COLUMN TO A COLUMN-DEPENDENT LIMIT THAT DECREASES ACROSS A PERIOD. THE AVERAGE OF r OVER COLUMNS IS CLOSE TO THE TF LIMITING VALUE OF 9 bohr. 49
Ionization density as Z Jan 24, 2011 BIRS 50 Ionization density as Z Jan 24, 2011 BIRS 51 John Conc CONCLUSIONS THE OBSERVED CHEMICAL TRENDS OF THE KNOWN
PERIODIC TABLE SATURATE IN THE NON-RELATIVISTIC Z LIMIT, IN WHICH THE PERIODIC TABLE BECOMES PERFECTLY PERIODIC. THE Z ATOMS HAVE LARGE VALENCE-ELECTRON RADII AND SMALL IONIZATION ENERGIES, SUGGESTING A LIMITING CHEMISTRY OF LONG WEAK BONDS. 52 John Conc 2 THE AVERAGES OF r Z AND I Z OVER COLUMNS ARE DESCRIBED RATHER WELL BY TF AND ETF. LSDA AND GGA AGREE CLOSELY IN THE Z LIMIT. AT THE EXCHANGE-ONLY (= OR )NO CORRELATION) LEVEL, LSDA AND GGA BECOME EXACT OR NEARLY EXACT FOR I AS Z .
(= OR )MORE NEARLY SO FOR THE np THAN FOR THE ns SUBSHELLS). 53 John future FUTURE WORK WE WILL CHECK IF THE MADELUNG`S-RULE CONFIGURATIONS SATISFY THE AUFBAU PRINCIPLE FOR LARGE Z. WE WILL CALCULATE THE LIMITING Z ELECTRON AFFINITIES. OUR CONCLUSIONS ARE BASED UPON NUMERICAL CALCULATION AND EXTRAPOLATION. CAN THEY BE PROVED RIGOROUSLY? 54 Orbital-free potential-functional for
C density (Dongyung Lee) 4pr2(= OR )r) r Jan 24, 2011 BIRS I(= OR )LSD)=11.67eV PFT:I=0.24eVI=0.24eV I(= OR )expt)=11.26eV 55 Simple math challenges Why do you study variational properties of approximate functionals? Give us mathematical rigor for PFT Prove results for large Z ionization potentials
Help us with asymptotic expansions Thanks to students and NSF Jan 24, 2011 BIRS 56
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