Convergence of many-body wavefunction expansions using a plane wave basis: From the homogeneous electron gas to the solid state eserved@d = *@let@token TCM Electronic Structure Discussion GroupConvergence of many-body wavefunction
expansions using a plane wave basis: From the homogeneous electron gas
to the solid state
James Shepherd
Introduction MP2 Ek -cutoff Eg -cutoffs FCIQMC Solids Concluding Remarks
Determinantal Expansions
Introduction MP2 Ek -cutoff Eg -cutoffs FCIQMC Solids Concluding Remarks
What is the Simulation Cell HEG?
H = ∑
α
− 1
• N electrons in a periodic (cubic) box of length L
• Uniform density: ρ (r) = N L3
• Characteristic lengthscale: rs =
Introduction MP2 Ek -cutoff Eg -cutoffs FCIQMC Solids Concluding Remarks
Problems with Determinantal Expansions
• Perturbative approaches - MP series and CCSD(T) - diverge in T.L.
• Size extensivity important - truncated CI will retrieve zero energy in T.L.
Introduction MP2 Ek -cutoff Eg -cutoffs FCIQMC Solids Concluding Remarks
Electronic Cusps
2 electron MP2 wavefunction for UEG, with rs = 5 a.u.
Hartree-Fock
• (DFT) The density does not have e-e cusps
• (VMC) Jastrow factor
Introduction MP2 Ek -cutoff Eg -cutoffs FCIQMC Solids Concluding Remarks
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)
Introduction MP2 Ek -cutoff Eg -cutoffs FCIQMC Solids Concluding Remarks
Second Order Møller-Plesset Theory (MP2)
• H = ∑
• Perturbing Hamiltonian couples in double excitations from the full N-fold excited states
EMP2 =
occ ∑
i + j − a − b (spinorbitals)
Introduction MP2 Ek -cutoff Eg -cutoffs FCIQMC Solids Concluding Remarks
Double Excitations in k-space
EMP2 = occ ∑
i>j
L3(ki−ka) 2
• These two terms are called 1) direct and 2) cross or exchange-like terms
• Momentum is conserved: ki + kj = ka + kb
• Can define a momentum transfer vector: g = ka − ki
Introduction MP2 Ek -cutoff Eg -cutoffs FCIQMC Solids Concluding Remarks
Double Excitations in k-space (2)
An excitation in the discrete regime
j
i
b
a
An excitation in the continuous regime (direct and cross shown)
Introduction MP2 Ek -cutoff Eg -cutoffs FCIQMC Solids Concluding Remarks
Introducing a Finite Basis Set
Use a kinetic energy 1 2k2
c cutoff: ∑
ab∈virt → ∑
kf ≤ka≤kc
kf ≤kb≤kc
Allowed excitations now fall within a certain radius in k-space. Call this Ek -cutoff.
k2
k1
k4
k3
Introduction MP2 Ek -cutoff Eg -cutoffs FCIQMC Solids Concluding Remarks
Analysis of the Limiting Behaviour
• Ek -cutoff: radius of kc in k-space
• EMP2(kc ) = EMP2(∞) − EMP2(kc ) [basis set incompleteness error]
• EMP2(kc ) = ∑
∑
: assume same power-law for each electron pair
• ∝ k2 a , numerator ∝ 1/k4
a
• EMP2(kc ) ∝ ∑
Introduction MP2 Ek -cutoff Eg -cutoffs FCIQMC Solids Concluding Remarks
1/k 3 → 1/M
Ek -cutoff
CBS ± 1% CBS result
Introduction MP2 Ek -cutoff Eg -cutoffs FCIQMC Solids Concluding Remarks
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
• Originates from basis set convergence work in solid state systems.
k2
k1
k4
k3
–gg
Introduction MP2 Ek -cutoff Eg -cutoffs FCIQMC Solids Concluding Remarks
Momentum Transfer Vectors (2)
k2
k1
k4
k3
–gg
to here (V)
[disclaimer: these diagrams do not show volume excluded due to occupation effects]
Introduction MP2 Ek -cutoff Eg -cutoffs FCIQMC Solids Concluding Remarks
Local Eg -cutoff
Eg -cutoff: radius of gc in k-space, but bounds momentum transfer vector g
k2
k1
gc
Introduction MP2 Ek -cutoff Eg -cutoffs FCIQMC Solids Concluding Remarks
Local Eg -cutoff (2)
M−1
Ek
k2
k1
k4
k3
Local
k2
k1
k4
k3
Introduction MP2 Ek -cutoff Eg -cutoffs FCIQMC Solids Concluding Remarks
Local Eg -cutoff (3)
• 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
Introduction MP2 Ek -cutoff Eg -cutoffs FCIQMC Solids Concluding Remarks
Intersection Eg -cutoff
Introduction MP2 Ek -cutoff Eg -cutoffs FCIQMC Solids Concluding Remarks
Intersection Eg -cutoff (2)
M−1
Ek
k2
k1
k4
k3
Local
k2
k1
k4
k3
Intersection
k2
k1
k4
k3
Introduction MP2 Ek -cutoff Eg -cutoffs FCIQMC Solids Concluding Remarks
Union Eg -cutoff
Introduction MP2 Ek -cutoff Eg -cutoffs FCIQMC Solids Concluding Remarks
Union Eg -cutoff (2)
M−1
Ek
k2
k1
k4
k3
Local
k2
k1
k4
k3
Intersection
k2
k1
k4
k3
Union
k2
k1
k4
k3
Introduction MP2 Ek -cutoff Eg -cutoffs FCIQMC Solids Concluding Remarks
Determinantal Expansions
Introduction MP2 Ek -cutoff Eg -cutoffs FCIQMC Solids Concluding Remarks
Application to FCIQMC
|Ψ = ∑
i
• Energy can be found by projection onto a reference
Ecorr = ∑
Introduction MP2 Ek -cutoff Eg -cutoffs FCIQMC Solids Concluding Remarks
Application to FCIQMC (2)
• Good enough for large M
• Extrapolation from a single calculation (SPE)
Introduction MP2 Ek -cutoff Eg -cutoffs FCIQMC Solids Concluding Remarks
Application to FCIQMC (3)
Eg SPE
M−1
C or
re la
ti on
en er
Introduction MP2 Ek -cutoff Eg -cutoffs FCIQMC Solids Concluding Remarks
Comparison with DMC
−0.045
−0.040
−0.035
−0.030
−0.025
−0.020
c. )
DMC
FCIQMC
• N = 14 (Somewhat different trend than for 54 electrons) • DMC results: Pablo Lopez Ros
• Consistent with: K.M. Rasch, L. Mitas, Chem. Phys. Lett. (2012), doi:10.1016/j.cplett.2012.01.016
Introduction MP2 Ek -cutoff Eg -cutoffs FCIQMC Solids Concluding Remarks
Comparison with DMC (2)
( a .u . p e r e le c. )
EDMC−EFCIQMC
Introduction MP2 Ek -cutoff Eg -cutoffs FCIQMC Solids Concluding Remarks
Comparison with DMC (3)
−0.040
−0.035
−0.030
−0.025
−0.020
−0.015
Introduction MP2 Ek -cutoff Eg -cutoffs FCIQMC Solids Concluding Remarks
LiH solid
Ek SPE
∞−1
Introduction MP2 Ek -cutoff Eg -cutoffs FCIQMC Solids Concluding Remarks
Conclusions
• 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
Introduction MP2 Ek -cutoff Eg -cutoffs FCIQMC Solids Concluding Remarks
Looking Ahead
• More accurate electron gas energies (benchmark DMC fixed-node error)
• Does the CC series diverge for metals?
Introduction MP2 Ek -cutoff Eg -cutoffs FCIQMC Solids Concluding Remarks
Acknowledgements
• Prof. Ali Alavi
• Dr. Andreas Grueneis (CCSD and RPA calculations; solid state applications) & VASP
• Dr. George Booth (MP2 and FCIQMC code)
• Dr. Pablo Lopez Ros (VMC/DMC) & CASINO
• Alavi Group
• EPSRC
References
• FCIQMC: GH Booth, AJW Thom, A Alavi, JCP 131 5, 054106 (2009)
• i-FCIQMC: D Cleland, GH Booth, A Alavi, JCP 132 4, 041103 (2010)
• HEG/i-FCIQMC: JJS, GH Booth, A Gruneis, A Alavi, PRB 85, 081103(R) (2012)
• HEG/i-FCIQMC/Extrapolation: JJS, GH Booth, A Alavi, arXiv:1201.4691 [physics.comp-ph]
• MP2/Extrapolation/Solid State: upcoming