Topics relevant to program Kieron Burke UC Irvine Chemistry and Physics with Lots of other folks March 15, 2011IPAM1 Many papers available at

Outline Kieron’s conjecture – Rise of empiricism in DFT – Changing Z, keeping N=Z. – Orbital-free theory Embedding: – atoms in molecules – O(N) methods – QM/MM March 15, 2011IPAM2

Rise of empiricism March 15, 2011IPAM3 CECAM workshop: How to Speed Up Progress and Reduce Empiricism in Density Functional TheoryHow to Speed Up Progress and Reduce Empiricism in Density Functional Theory Location : ACAM, Dublin, Ireland June 20, June 24, 2011

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 a DFT guru March 15, 2011IPAM4

Things developers love about DFT No need to be reliable No route to systematic improvement If a property turns out to be inaccurate, can spend several decades looking for solution No need to connect other methods Lots of lovely arcane insider jargon Oh so many functionals to choose from Everyone needs their own DFT guru March 15, 2011IPAM5

Alphabet soup March 15, 2011IPAM6

Semiclassical derivations March 15, 2011IPAM7

Kieron’s conjecture All success of DFT approximations stems from exactness of local approximations as N=Z→∞ March 1, 2011CalTech8

Results so far Can show LDA is leading term in semiclassical expansion in terms of either potential or density Expansion is asymptotic Leading corrections are universal as functionals of the potential, not the density Derived parameter in B88. New conditions on kinetic energy functional. March 15, 2011IPAM9

Improvements of PBEsol Structural and Elastic Properties of solids Errors in LDA/GGA(PBE)-DFT computed lattice constants and bulk modulus with respect to experiment Errors in LDA/GGA(PBE)-DFT computed lattice constants and bulk modulus with respect to experiment  Inspection of several xc-functionals is critical to estimate predictive power and error bars! → Fully converged results (basis set, k-sampling, supercell size) → Error solely due to xc-functional → Fully converged results (basis set, k-sampling, supercell size) → Error solely due to xc-functional → GGA does not outperform LDA → characteristic errors of <3% in lat. const. < 30% in elastic const. → LDA and GGA provide bounds to exp. data → provide “ab initio error bars” → GGA does not outperform LDA → characteristic errors of <3% in lat. const. < 30% in elastic const. → LDA and GGA provide bounds to exp. data → provide “ab initio error bars” Blazej Grabowski, Dusseldorf March 15, IPAM

I along first row March 15, 2011IPAM11 alkalis Noble gases s-group p-group Li Ne

I along first and second rows March 15, 2011IPAM12 alkalis Noble gases s-group p-group Na Ar

Extrapolation of Z →∞ by column March 15, 2011IPAM13 Lucian Constantin Using code of Eberhard Engel

I along last row March 15, 2011IPAM14 alkalis Noble gases s-group p-group First row Second row Infinith row

I along last row March 15, 2011IPAM15 alkalis Noble gases s-group p-group HF

I along last row March 15, 2011IPAM16 alkalis Noble gases s-group p-group HF

I along last row March 15, 2011IPAM17 alkalis Noble gases s-group p-group XC HF

I along last row March 15, 2011IPAM18 alkalis Noble gases s-group p-group XC HF ETF

Z →∞ limit of ionization potential Shows even energy differences can be found Looks like LDA exact for E X as Z→∞. Looks like finite E C corrections Looks like extended TF (treated as a potential functional) gives average. Constantin, Snyder, Perdew, and KB, J. Chem. Phys. 133, (2010 ) March 15, 2011IPAM19

Orbital-free theory March 15, 2011IPAM20

Potential functional theory March 15, 2011IPAM21

Orbital-free potential-functional for C density (Dongyung Lee) March 15, 2011IPAM22 4  r 2 n(r) r I(LSD)=11.67eV I(PFT)=11.43 eV I(expt)=11.26eV

Things we hope to fix about DFT approximations Give rules for reliability Systematic route to improvement If your property turns out to be inaccurate, try a different resumming of asymptotic series Semiclassical connection with other methods Semiclassical explanation of DFT effects Limited non-empirical set of functionals Does not require explanation from DFT guru March 15, 2011IPAM23

Road to partition Atoms in molecules O(N) QM/MM Effective charges March 15, 2011IPAM24

Basic partition theory Consider fragments as isolated and minimize their energies, but requiring sum of densities equal molecular density: March 15, 2011IPAM25

Basics: Partition potential How to find minimum? Use Lagrange multipliers: Lagrange multiplier is called partition potential, v p (r), a global property of the molecule March 15, 2011IPAM26

Example: Partition potential Each fragment density is the ground-state density in effective fragment potential, v  (r)+v p (r) March 15, 2011IPAM27

12-atom chain Construct chain of Eckhardt potentials Peter Elliott solved 12 single-atom fragment problems. March 15, 2011IPAM28

Fragment densities for the A-atom when ZA = and: solid lines: ZB = 0.995, and R = 1.65; dotted lines: ZB = 0.895, and R = These have been shifted and renormalized to test shape transferability (see text). Published in: Yu Zhang; Adam Wasserman; J. Chem. Theory Comput. 2010, 6, DOI: /ct100247q Copyright © 2010 American Chemical Society March 15, IPAM Adams partitions

Basic statements Can perform your KS calculation as sum of atomic calculations, each atom in an effective field. This can be done exactly, but costs more than molecular/solid calculation. Should be possible to make simple ‘neighborhood’ approximation to get linear scaling Directly calculate the dissociation energy, without total energies. We’ve done it for model systems March 15, 2011IPAM30

Embedding exact calc March 15, 2011IPAM31

Aside: First ever KS calculation with exact E XC [n] Used DMRG (density-matrix renormalization group) 1d H atom chain Miles Stoudenmire, Lucas Wagner, Steve White March 15, 2011IPAM32 x density

Summary Conjecture – New way to think about DFT approximations – What about going down columns? – What about large Z: everything is continuous Partition – Formalism for answering many questions – Allows embedding of accurate calculation within KS-DFT calculation March 15, 2011IPAM33