Convergence of many-body wavefunction expansions using a plane wave basis: From the homogeneous electron gas to the solid state

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1 Convergence of many-body wavefunction expansions using a plane wave basis: From the homogeneous electron gas to the solid state TCM Electronic Structure Discussion Group James Shepherd (CUC3, Alavi Group) February 14, 2012

2 Determinantal Expansions basis set limit heirarchy one-electron expansions exact solution exact solution finite basis heirarchy N-electron models

3 What is the Simulation Cell HEG? Ĥ = α α + α =β 1 2ˆv αβ Nv M N electrons in a periodic (cubic) box of length L Uniform density: ρ(r) = N L 3 ( Characteristic lengthscale: r s = Plane waves: ψ js (r,σ) = L πn ) L 3 e ik j r δ sσ. Thermodynamic limit: N, N L 3 = const.

4 Problems with Determinantal Expansions Ψ = i c i D i Perturbative approaches - MP series and CCSD(T) - diverge in T.L. Size extensivity important - truncated CI will retrieve zero energy in T.L.

5 Electronic Cusps 2 electron MP2 wavefunction for UEG, with rs = 5 a.u. Lines show (2n + 1) 3 k-points (from 27 to 16,000) (DFT) The density does not have e-e cusps (VMC) Jastrow factor Hartree-Fock x [credit: George Booth]

6 Overview 1) Simple theory to model basis set incompleteness: MP2 Drawbacks: low-level, diverges, unphysical Benefits: analytic matrix elements, cheap, qualitative short-range cusp behaviour 2) Introduce momentum transfer vector based basis set truncation 3) Apply this to FCIQMC by re-summation of the projected energy 4) Apply to plane-wave solid-state systems (if time)

7 Second Order Møller-Plesset Theory (MP2) H = α F α + H Perturbing Hamiltonian couples in double excitations from the full N-fold excited states occ virt E MP2 = i>j a>b ij ˆv ab ij ˆv ba 2 ǫ i + ǫ j ǫ a ǫ b (spinorbitals)

8 Double Excitations in k-space Transform MP2 expression into k-space representation: FT: v ki k a = occ virt E MP2 = i>j a>b 4π L 3 (k i k a) 2 δ ki k a,k j k b 2 v 2 k i k a v ki k a v kj k a ǫ These two terms are called 1) direct and 2) cross or exchange-like terms Momentum is conserved: k i + k j = k a + k b Can define a momentum transfer vector: g = k a k i

9 Double Excitations in k-space (2) b i j An excitation in the discrete regime a An excitation in the continuous regime (direct and cross shown)

10 Introducing a Finite Basis Set Use a kinetic energy 1 2 k2 c cutoff: ab virt k f k a k c k f k b k c Allowed excitations now fall within a certain radius in k-space. Call this E k -cutoff.

11 Analysis of the Limiting Behaviour E k -cutoff: radius of k c in k-space E MP2 (k c ) = E MP2 ( ) E MP2 (k c ) [basis set incompleteness error] E MP2 (k c ) = 0 k i k f k a>k c δ ki k a,k j k b 0 k j k f k b >k c 2 v 2 k i ka v k i ka v k j ka ǫ 0 k i k f 0 k j k f : assume same power-law for each electron pair ǫ a, numerator 1/k4 a k a k c k b k c δ ki k a,k j k b : single sum over k a E MP2 (k c ) 1 k a k c ka 6 E MP2 (k c ) 1 k c dk a ka 6 ka 2 1. kc 3

12 1/ 1/M Correlation energy / Eh E k -cutoff CBS result CBS ± 1% M electrons, r s =5.0 a.u.

13 Momentum Transfer Vectors It is possible to define two momentum transfer vectors for an excitation. Seek to define basis sets using this, to see if there are better extrapolation properties. Show there are three sensible choices for basis sets Originates from basis set convergence work in solid state systems. gʹ g gʹ g

14 Momentum Transfer Vectors (2) E MP2 = FS V δ (2 v g,g g 2 vgv ) g +k i k j ǫ gʹ g g gʹ excite anywhere here (FS) to here (V) [disclaimer: these diagrams do not show volume excluded due to occupation effects]

15 Local E g -cutoff E g -cutoff: radius of g c in k-space, but bounds momentum transfer vector g { gʹ} g c { + g}

16 Local E g -cutoff (2) Correlation energy (a.u.) E k -cutoff Local E g -cutoff CBS result CBS ± 1% E k Local M electrons, r s =5.0 a.u.

17 Local E g -cutoff (3) Treats electron coalescences on the same footing Different basis set per k-point/electron Spans further into k-space for the same number of functions per electron Loss of variationality w.r.t. CBS limit Breaks permutational symmetry Loss of size extensivity gʹ g gʹ g (E k ) (local E g )

18 Intersection E g -cutoff

19 Intersection E g -cutoff (2) Correlation energy (a.u.) E k -cutoff Local E g -cutoff Intersection E g -cutoff CBS result CBS ± 1% E k Local M Intersection 14 electrons, r s =5.0 a.u.

20 Union E g -cutoff

21 Union E g -cutoff (2) Correlation energy (a.u.) E k -cutoff Local E g -cutoff Intersection E g -cutoff Union E g -cutoff CBS result CBS ± 1% E k Local Intersection M Union 14 electrons, r s =5.0 a.u.

22 Determinantal Expansions basis set limit heirarchy one-electron expansions exact solution (FCIQMC+ extrapolation) exact solution finite basis (FCIQMC) heirarchy N-electron models

23 Application to FCIQMC Write wavefunction as a series of Slater Determinants Ψ = i c i D i Solve using a stochastic algorithm Energy can be found by projection onto a reference As before, FT: ij E corr = ab j {doubles} D j H D 0 c j c 0 occ virt ( ) c kak b k i k j E corr = δ ki k a,k j k vki b k a v kj k a c 0

24 Application to FCIQMC (2) occ virt ( ) c kak b k i k j E corr = δ ki k a,k j k vki b k a v kj k a c 0 ij ab Apply cutoffs as before Unlike before, this is now approximate Good enough for large M Extrapolation from a single calculation (SPE)

25 Application to FCIQMC (3) Correlation energy / Eh E g SPE E i-fciqmc M 1

26 Comparison with DMC Correlation energy (a.u. per elec.) DMC FCIQMC r s N = 14 (Somewhat different trend than for 54 electrons) DMC results: Pablo López Ríos Consistent with: K.M. Rasch, L. Mitas, Chem. Phys. Lett. (2012), doi: /j.cplett

27 Comparison with DMC (2) Fixed node error (a.u. per elec.) E DMC E FCIQMC r s 10 1 [DMC results: Pablo López Ríos]

28 Comparison with DMC (3) Correlation energy (a.u. per elec.) VMC FCIQMC N [VMC results: Pablo López Ríos]

29 LiH solid Correlation energy (Eh) Previous lit. scheme E k SPE E g SPE M

30 Conclusions 1) Analyzed basis set incompleteness error for UEG 2) Found 1/M power-law holds for MP2 3) Discussed alternative momentum transfer vector basis sets 4) Applied this to FCIQMC by re-summation of the projected energy

31 Looking Ahead Twist-averaging to reach the thermodynamic limit More accurate electron gas energies (benchmark DMC fixed-node error) Does the CC series diverge for metals?

32 Acknowledgements Prof. Ali Alavi Dr. Andreas Grueneis (CCSD and RPA calculations; solid state applications) & VASP Dr. George Booth (MP2 and FCIQMC code) Dr. Pablo López Ríos (VMC/DMC) & CASINO Alavi Group EPSRC References FCIQMC: GH Booth, AJW Thom, A Alavi, JCP 131 5, (2009) i-fciqmc: D Cleland, GH Booth, A Alavi, JCP 132 4, (2010) HEG/i-FCIQMC: JJS, GH Booth, A Gruneis, A Alavi, PRB 85, (R) (2012) HEG/i-FCIQMC/Extrapolation: JJS, GH Booth, A Alavi, arxiv: [physics.comp-ph] MP2/Extrapolation/Solid State: upcoming