IPAM 1

Topics relevant to program

Kieron Burke UC Irvine Chemistry and Physics

with Lots of other folks

March 15, 2011

Many papers available at http://dft.uci.edu

IPAM 2

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, 2011

IPAM 3

Rise of empiricism

March 15, 2011

CECAM workshop: How to Speed Up Progress and Reduce Empiricism in Density Functional Theory

Location : ACAM, Dublin, IrelandJune 20, 2011 - June 24, 2011

IPAM 4

Things users despise about DFTNo simple rule for reliabilityNo systematic route to improvementIf your property turns out to be inaccurate, must wait several decades for solutionComplete disconnect from other methodsFull of arcane insider jargonToo many functionals to choose fromCan only be learned from a DFT guru

March 15, 2011

IPAM 5

Things developers love about DFTNo need to be reliableNo route to systematic improvementIf a property turns out to be inaccurate, can spend several decades looking for solutionNo need to connect other methodsLots of lovely arcane insider jargonOh so many functionals to choose fromEveryone needs their own DFT guru

March 15, 2011

IPAM 6

Alphabet soup

March 15, 2011

IPAM 7

Semiclassical derivations

March 15, 2011

CalTech 8

Kieron’s conjecture

All success of DFT approximations stems from exactness of local

approximations as N=Z→∞

March 1, 2011

IPAM 9

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 10

Structural and Elastic Properties of solidsErrors in LDA/GGA(PBE)-DFT computed lattice constants and bulk modulus with respect to experiment

Þ 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

→ 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, 2011

IPAM 11

I along first row

March 15, 2011alkalis

Noble gases

s-group p-group

Li

Ne

IPAM 12

I along first and second rows

March 15, 2011alkalis

Noble gases

s-group p-group

Na

Ar

IPAM 13

Extrapolation of Z→∞ by column

March 15, 2011

Lucian Constantin

Using code of Eberhard Engel

IPAM 14

I along last row

March 15, 2011alkalis

Noble gases

s-group p-groupFirst row

Second row

Infinith row

IPAM 15

I along last row

March 15, 2011alkalis

Noble gases

s-group p-group

HF

IPAM 16

I along last row

March 15, 2011alkalis

Noble gases

s-group p-group

HF

IPAM 17

I along last row

March 15, 2011alkalis

Noble gases

s-group p-group

XC

HF

IPAM 18

I along last row

March 15, 2011alkalis

Noble gases

s-group p-group

XC

HF

ETF

IPAM 19

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 20

Orbital-free theory

March 15, 2011

IPAM 21

Potential functional theory

March 15, 2011

IPAM 22

Orbital-free potential-functional for C density (Dongyung Lee)

March 15, 2011

4pr2n(r)

r•I(LSD)=11.67eV• I(PFT)=11.43 eV•I(expt)=11.26eV

IPAM 23

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 seriesSemiclassical connection with other methodsSemiclassical explanation of DFT effectsLimited non-empirical set of functionalsDoes not require explanation from DFT guru

March 15, 2011

IPAM 24

Road to partition

• Atoms in molecules• O(N)• QM/MM• Effective charges

March 15, 2011

IPAM 25

Basic partition theory• Consider fragments as isolated and minimize

their energies, but requiring sum of densities equal molecular density:

March 15, 2011

IPAM 26

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 27

Example: Partition potential• Each fragment

density is the ground-state density in effective fragment potential, va(r)+vp(r)

March 15, 2011

IPAM 28

12-atom chain

• Construct chain of Eckhardt potentials• Peter Elliott solved 12 single-atom fragment

problems.

March 15, 2011

IPAM 29

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.DOI: 10.1021/ct100247qCopyright © 2010 American Chemical Society

March 15, 2011

IPAM 30

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, 2011

IPAM 31

Embedding exact calc

March 15, 2011

IPAM 32

Aside: First ever KS calculation with exact EXC[n]

• Used DMRG (density-matrix renormalization group)

• 1d H atom chain• Miles

Stoudenmire, Lucas Wagner, Steve White

March 15, 2011 x

density

IPAM 33

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