Topics relevant to program Kieron Burke UC Irvine Chemistry and Physics with Lots of other folks Many papers available at http://dft.uci.edu March 15, 2011 IPAM 1 Outline Kierons 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, 2011 IPAM 2 Rise of empiricism CECAM workshop: How to Speed Up Progress and Reduce Empir icism in Density Functional Theory

Location : ACAM, Dublin, Ireland June 20, 2011 - June 24, 2011 March 15, 2011 IPAM 3 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, 2011 IPAM 4 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, 2011

IPAM 5 Alphabet soup March 15, 2011 IPAM 6 Semiclassical derivations March 15, 2011 IPAM

7 Kierons conjecture All success of DFT approximations stems from exactness of local approximations as N=Z March 1, 2011 CalTech 8 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, 2011 IPAM 9 Structural and Elastic Properties of solids 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 March 15, 2011 IPAM predictive power and error bars! 10

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

11 I along first and second rows p-group s-group Ar Na IPAM Noble gases alkalis

March 15, 2011 12 Extrapolation of Z by column Lucian Constantin Using code of Eberhard Engel March 15, 2011 IPAM 13

I along last row p-group s-group First row Second row Infinith row IPAM Noble gases alkalis

March 15, 2011 14 I along last row p-group s-group HF IPAM Noble gases alkalis

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

March 15, 2011 16 I along last row p-group s-group XC HF IPAM Noble gases

alkalis March 15, 2011 17 I along last row p-group s-group XC HF ETF IPAM Noble gases

alkalis March 15, 2011 18 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 (treated as a potential functional) gives average.

Constantin, Snyder, Perdew, and KB, J. Chem. Phys. 133, 241103 (2010) March 15, 2011 IPAM 19 Orbital-free theory March 15, 2011 IPAM

20 Potential functional theory March 15, 2011 IPAM 21 Orbital-free potential-functional for C density (Dongyung Lee) 4pr2n(r) r March 15, 2011

IPAM I(LSD)=11.67eV I(PFT)=11.43 eV I(expt)=11.26eV 22 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, 2011 IPAM 23 Road to partition Atoms in molecules O(N) QM/MM

Effective charges March 15, 2011 IPAM 24 Basic partition theory Consider fragments as isolated and minimize their energies, but requiring sum of densities equal molecular density: March 15, 2011 IPAM

25 Basics: Partition potential How to find minimum? Use Lagrange multipliers: Lagrange multiplier is called partition potential, vp(r), a global property of the molecule March 15, 2011 IPAM 26 Example: Partition potential

Each fragment density is the ground-state density in effective fragment potential, va(r) +vp(r) March 15, 2011 IPAM 27 12-atom chain Construct chain of Eckhardt potentials

Peter Elliott solved 12 single-atom fragment problems. March 15, 2011 IPAM 28 Fragment densities for the A-atom when ZA = 1.005 and: solid lines: ZB = 0.995, and R = 1.65; dotted lines: ZB = 0.895, and R = 1.80. These have been shifted and renormalized to test shape transferability (see text). Published in: Yu Zhang; Adam Wasserman; J. Chem. Theory Comput. 2010, 6, 3312-3318. March 15, 2011 DOI: 10.1021/ct100247q

Copyright 2010 American Chemical Society IPAM 29 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. Weve done it for model systems March 15, 2011

IPAM 30 Embedding exact calc March 15, 2011 IPAM 31 Aside: First ever KS calculation with exact EXC[n] Used DMRG (density-matrix

renormalization density group) 1d H atom chain Miles Stoudenmire, Lucas Wagner, Steve White March 15, 2011 IPAM x 32 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, 2011 IPAM 33