© Imperial College LondonPage 1 The Surface Energy of the Electron Gas Nicholas Hine Sunday 23 rd July 2006 Quantum Monte Carlo in the Apuan Alps II The

© Imperial College LondonPage 1

The Surface Energy of the Electron Gas

Nicholas Hine

Sunday 23rd July 2006

Quantum Monte Carlo in the Apuan Alps II

The Towler Institute, Vallico Sotto, Tuscany

Resolving a Long-Standing Contradiction

N.D.M. Hine, B. Wood, W.M.C. Foulkes, P. Garcia-Gonzales


The System

What is it?

• Jellium: uniform electron gas with positive background charge

• Surface produced by abruptly terminating background charge

• Smoothly decaying potential (with discontinuity in gradient at surface)

• Electron density spills slightly outside the slab


The System

Why is it interesting?

• Simplest possible model of metal surfaces (not particularly good as it predicts a negative surface energy for eg aluminium!) used as a first order approximation for things like catalysis, interaction with light, and more.

• Benchmark of many body methods, from Fermi Hypernetted Chain (1950s) and DFT - first ‘real-world’ LDA calculation was on Jellium surface (1960s) - to RPA, QMC, better DFT functionals, WVI, ...

• Test of behaviour of density functionals (LDA vs GGAs vs meta-GGA vs … ) in the situations they’re meant to be good at.

• If anything can give the ‘right’ answer it should be DMC.


Controversy

• Different methods disagree significantly


Controversy


• Even different calculations with the same method differ significantly…

For one density (rs=2.07), some total surface energies calculated in different methods:

AuthorsMethod (ergs/cm2)

Kurth and Perdew (1999)RPA & GGA combiniation-587

Kurth and Perdew (1999)RPA & LDA combination-553

Perdew (1999)meta-GGA-567

Perdew (1999)GGA-690

Yan, Perdew (2000)GGA-based WVI-533

Yan, Perdew (2000)LDA-610

Li, Needs, Martin, Ceperley (1992)Fixed Node DMC-465 ± 50

Pitarke (2004)Correction to above-554 ± 80

Acioli and Ceperley (1996)Fixed Node DMC-420 ± 80


Controversy


• Even different calculations with the same method differ significantly…

• Very long list of publications on exact same subject!

etc etc (more in 2004,2005,2006)


Controversy

• What makes it a hard calculation?

• Different reasons in different methods:

• Comparison of bulk and slab often produces non-cancelling systematic errors

• Poor performance of density functionals in regions of rapidly varying density

• Finite size effects of various kinds

• Geometry & Periodicity effects

• Large individual contributions el, S, XC nearly-cancel to give relatively small T : important to account for everything properly and consistently

• Not comparing like with like (!)


Many Body Methods

• In DFT one can study a surface terminating a semi-infinite bulk

• In many body methods we are constrained to working with slab geometries

• In QMC, additionally constrained to finite in-plane size (DFT etc can do in-plane integration analytically)

∞

∞


Supercell geometry

With finite supercells come periodic boundary conditions:

…

…

…

…

slab width s

In plane length L

L

Periodic boundary conditions in-plane but not in out of plane direction – previous studies used 3D periodic Ewald giving interactions between periodic array of slabs

Quasi-2D MPC is both more realistic and faster


Surface Energies

Deep inside an infinitely wide slab are bulk-like states with energy bulk

In a slab of finite width, we can still define surface energy by:

Energy per electron in slab systemEnergy per electron in bulk

andSlab volume

Volume per electron

Two surfaces of slab

Number of electrons in slab

Surface area

so


High accuracy required• If we were to use

an error of 1% in either of the components would lead to around a 10% error in the surface energy. Typically, slab ~ -10 to -80 mHa/elec, so accuracy required is ~ 0.1 mHa/elec = 0.0027 eV per electron for this level of precision.

• Quality of nodal surface may be vital! Bulk and slab calculations may well have different quality nodal surfaces (Previous study used release-node bulk calculations and fixed-node slab calculation!)

• Avoid comparison of bulk and slab results by using slab results only:

• Fit gradient of slab vs 1/s and ignore bulk

• Still requires high accuracy as range of accessible s values is small


Finite size errors: slab width s

• ‘Cross talk’ between surfaces produces oscillations in surface energy components with s

• Cusps in individual components as new subbands fall below Fermi level, and new subband can have wildly different contribution to ES, EXC, Eel

• No cusps in total surface energy because new eigenvalue is always at Fermi level.


Finite size errors: slab width s

• Work at ‘special’ slab widths where

s1 s2 s3

Subbands in z-direction

~ 30 erg/cm2 at accessible s


• Vagaries of subband filling with discretized in-plane k values from finite in-plane cells produce fluctuating energies as a function of L.

• Apply same idea as with slab width: pick N’s and hence L’s such that finite-cell DFT result approximately agrees with infinite-cell DFT result.

• Standard independent particle finite size correction:

takes care of the rest

Finite size errors: in-plane size L


The system

Need to account for finite size effects while working with slabs only

s

L

s1

s2

s3

Need even integer N per cell and small FS so we cannot choose same L values for each s


Reliable Surface Energies in QMC – recap

• Step by Step Method

• Infinite in-plane cell DFT simulations for a range of slab widths.

• Pick three ‘special’ slab widths where

• For each s, perform DFT simulations and pick a range of values of N for which the LDA result closely matches the infinite-cell limit.

• Optimize Jastrow separately for each (s,N) pairing and perform VMC and DMC simulations with LDA orbitals.

• Plot slab vs s/N (proportional to 1/L2), find separation of lines at fixed L, which is hopefully independent of the value of L.

• Plot eslab vs 1/s at fixed L. Gradient of this gives surface energy with associated statistical error.


Wavefunctions• Lots of work (BW) on plasmon-like wavefunctions: ultimately not used but

suggested the form of the short range term of the Jastrow:

• Started before new Jastrows (same problems with varmin that we spent a lot of time discussing last year). Instead, manual optimization of was used.

• Best Jastrow term comes from cancelling out the density change produced by the u-term (cf. Malatesta, Fahy, Bachelet, Phys. Rev. B 56, 12201 (1997))

• Trial wavefunctions relatively poor as evidenced by VMC energies.


Wavefunctions

• Two potential sources of orbitals for determinant: LDA and GGA. Methods give very different surface energies because of different Hamiltonian but not very different orbitals – but provide a test of the significance of fixed node error.

• Include image potential in VXC to match real decay of VXC in vacuum region


Results – DFT & RPA

• Surface energies by extrapolation of slab results agree very accurately with infinite slab bulk vs slab comparison

• Same in GGA, although surface energies are very much lower

• RPA has moderate finite-s error in extrapolation method.

Can be confident about the extrapolation method even at small s

-689.0

-689.6

-689.4

-688.2

-7.532

-7.532

-7.532

bulk LDA

-609.3

(interpolation)

-608.8-10.324-9.99911.7783

-608.5-9.707-9.45315.1317

-609.0-9.312-9.10318.4851

(difference)slab GGAslab LDAs (bohr)


Results - VMC

• VMC total energies not that great due to varying performance of Jastrow factor as L increases.

• Surface energies, extracted from spacing of lines, are not bad though: in-plane finite-size error is nearly independent of s.

• Minor differences in surface energy with L possibly due to variation in performance of Jastrow with slab width – should be gone in DMC.


Results - VMC

• VMC total energies not that great due to varying performance of Jastrow factor as L increases.

• Surface energies, extracted from spacing of lines, are not bad though: in-plane finite-size error is nearly independent of s.

• Minor differences in surface energy with L possibly due to variation in performance of Jastrow with slab width – should be gone in DMC.


Results - DMC

• 400-1000 CPU hours per (s,N) point• 12 points per value of rs

• 5 values of rs = a lot of cpu time!

• Lines for slab(1/L2) not as straight as in VMC, so harder to fit to.


Results - DMC

• Difficulties of non-straight lines less severe at larger rs

• Agreement with LDA and RPA becomes even closer

• Thankfully, DMC with GGA orbitals agrees pretty much with DMC-LDA and thus LDA, rather than GGA (nodal surface from orbitals not dominating!)


Behaviour of (rs)

Suggestion from recent papers in the field is that ‘correct’ result should lie between RPA and LDA

DMC results seem instead to lie pretty much in line with LDA and slightly below RPA and RPA+.

GGA is significantly lower still, but is already known to be poor in this situation.


Conclusions

• Close agreement with LDA, poor agreement with GGA (at least, PBE GGA – the many improvements since then such as Meta-GGA are designed to get closer to the RPA answer and do. RPA answer not necessarily correct though…).

• Method for surface energies works: Limited to relatively thin slabs as errors in gradient grow as 1/s shrinks (correction for finite width slabs is small in RPA, however, which is encouraging).

• Controversy resolved: Accioli & Ceperley results were unfairly comparing release-node bulk with fixed node slab results and did not account for in-plane finite size effects properly.

• Provides a template for future work on real surfaces.

Documents

© Imperial College LondonPage 1 The Surface Energy of the Electron Gas Nicholas Hine Sunday 23 rd July 2006 Quantum Monte Carlo in the Apuan Alps II The