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© 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 Towler Institute, Vallico Sotto, Tuscany Resolving a Long-Standing Contradiction N.D.M. Hine, B. Wood, W.M.C. Foulkes, P. Garcia-Gonzales

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© Imperial College LondonPage 2 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

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© Imperial College LondonPage 3 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.

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© Imperial College LondonPage 4 Controversy Different methods disagree significantly

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© Imperial College LondonPage 5 Controversy Different methods disagree significantly 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/cm 2 ) 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

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© Imperial College LondonPage 6 Controversy Different methods disagree significantly 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)

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© Imperial College LondonPage 7 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 (!)

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© Imperial College LondonPage 8 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) ∞ ∞

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© Imperial College LondonPage 9 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

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© Imperial College LondonPage 10 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 system Energy per electron in bulk and Slab volume Volume per electron Two surfaces of slab Number of electrons in slab Surface area so

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© Imperial College LondonPage 11 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

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© Imperial College LondonPage 12 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 E S, E XC, E el No cusps in total surface energy because new eigenvalue is always at Fermi level.

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© Imperial College LondonPage 13 Finite size errors: slab width s Work at ‘special’ slab widths where s1s1 s2s2 s3s3 Subbands in z-direction ~ 30 erg/cm 2 at accessible s

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© Imperial College LondonPage 14 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

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© Imperial College LondonPage 15 The system Need to account for finite size effects while working with slabs only s L s1s1 s2s2 s3s3 Need even integer N per cell and small FS so we cannot choose same L values for each s

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© Imperial College LondonPage 16 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/L 2 ), 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.

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© Imperial College LondonPage 17 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.

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© Imperial College LondonPage 18 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 V XC to match real decay of V XC in vacuum region

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© Imperial College LondonPage 19 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 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 GGA slab LDA s (bohr)

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© Imperial College LondonPage 20 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.

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© Imperial College LondonPage 21 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.

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© Imperial College LondonPage 22 Results - DMC 400-1000 CPU hours per (s,N) point 12 points per value of r s 5 values of r s = a lot of cpu time! Lines for slab (1/L 2 ) not as straight as in VMC, so harder to fit to.

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© Imperial College LondonPage 23 Results - DMC Difficulties of non-straight lines less severe at larger r s 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!)

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© Imperial College LondonPage 24 Behaviour of (r s ) 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.

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© Imperial College LondonPage 25 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.

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