including Nodal Pulay Terms in Diffusion Monte Carlo - Accurate Forces 25. July 2007 The Towler Institut
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1 Accurate Forces in iffusion Monte Carlo - incluing Noal Pulay erms Alexaner Bainski, Peter Haynes,, Richar Nees Cavenish Laboratory, University of Cambrige, UK Physics epartment, Imperial College Lonon, UK QMC in the Apuan Alps III 5. July 007 he owler Institut
2 actual MC Wave Function Fixe-noe approximation to solve the fermion sign problem infinite potential barrier where 0 iling heorem noal pockets of the exact groun state are equivalent nee to sample only one pocket is trial wave function Ω is a single noal pocket Γ is bounary of Ω where 0 MC wave function for one noal pocket vali throughout space is ifferentiable where an Θ( for > 0 an Γ Ω Γ Θ( 0 for < 0 an 0 Note: an are parallel (.M. Ceperley, J. Stat. Phys. ol (99
3 3 ransfer results to Hamiltonian Ĥ Applying the Laplacian onto δ - Function erm ( Ĥ ( Ĥ δ Θ + ( ' ( ( ( δ δ δ Θ Note: one coul efine an effective Hamiltonian that inclues the δ fct. term interprete δ fct. term as infinite noal potential as require to force to be zero on Γ but not relevant here the stanar ientity x ( x x ( x δ δ (3 an L Hôpital in multi-im. on Γ ( gives ( ( δ Θ + + Using from q. (, (4
4 MC energy: MC nergy Graient Ĥ Ω Ω for mixe MC ( an pure MC ( the δ function term oes not contribute to no ifference whether region of integration in expressions for is over Ω over all space choose all space! ake the erivative of wrt parameter (nucleus coorinate, electric fiel, etc. Ĥ ( Ĥ ( Ĥ + + HF (Hellmann- (olume erm Feynman erm N (Noa erm Note that the integration limit was inepenent of HF : is mixe estimator ( or pure estimator ( 4
5 Rate of Change of Γ wrt We can write the total ifferential of as (, r + r We set 0 an ivie by r 0 + Choose r parallel an r _ orthogonal to normal of Γ: Hence, r r r lhs can be interprete as the rate of change of the NS of raial to Γas moves his rate of change must be the same for an for i.e., on Γ (5 5
6 olume erm For our specific form of, one can show that Ĥ is Hermitian in all our expressions, i.e. ( Ĥ ( Ĥ N his is not obvious as Ĥ is not Hermitian when integrals are over Ω For mixe MC ( use Reynols approximation N (mix ( Ĥ For pure MC ( N (pure 0 P. Reynols, et al. Internat. J. Quant. Chem (986 6
7 Noal erm N Inserting q. (4 an using Ĥ gives ( ( δ + Ĥ N δ ( r-r' Using stanar ientity δ ( S Γ S Γ N (mix Γ N (pure Interpretation of N (pure is kinetic energy ensity an S an L Hôpital is rate of change of Γ wrt N is rate of change of kinetic energy arising from kink in at Γ (lin. response of kinetic energy wrt variations of NS N is a single particle expression! 7
8 Calculate Noal erm (I It is not straightforwar to evaluate mixe & pure MC Noal erms! o fin an expression for the mixe MC noal term, use that is Hermitian (proof omitte, i.e. N N ( Ĥ Γ o calculate the pure MC noal term, we note that the pure an mixe MC Noal erms N can be rewritten using q.(3 an (5, for mixe MC ( an pure MC ( Average quantity Q is same for mixe an pure MC only contains S ( Ĥ Ĥ 8
9 Calculate Noal erm (II Use extrapolation formula which is exact to n orer in error of Q Q Q pure mixe MC with Q (which is also vali when integrals are over surface he average of Q in MC is zero Γ N S Γ 0 Since is continuous on Γ, the surface integrals of two borering pockets cancel. integral over Γ of all pockets simultaneously is zero surface integral over each pocket is zero (tiling theorem (We varifie this numerically, too! Hence we can calculate the pure MC noal term as N (pure MC N (mixe MC S Γ S 9
10 Mixe MC Pure MC + Summary so far Ĥ ( Ĥ Ĥ + + ( Ĥ ( Ĥ + O ( + O ( if NS is inepenent of 0 on Γ N is zero if exact N is zero (not obvious but can be shown using the tiling theorem 0
11 Computational etails Stuy Germanium hyrie imer (GeH no electron-electron interaction (noal terms arise from kinetic energy! use Slater et. with non-interacting MOs calculate using GAMSS-US 4 basis sets: very goo (9sp8, goo (8s4p, poor (8s4p, very poor (8sp cover range of errors in for practical calculations local pseuopotentials (constructe to emulate non-local pseuopotentials first erivative of from finite ifference use extrapolation an Future walking to calculate HF < A > A( r r r k k k q( q( k k with q ( z 0 lim f ( x,z 0 for reference, calculate energy graient from potential energy curve t,t x For convenience: convert changes in forces F to changes in bon length a: F 0.00 a.u. a 0.0 a.u. J. Casulleras an J. Boronat Phys. Rev. B ol (994
12 Potential nergy Curve sp8 8s4p 8s4p 8sp f Ge-H istance in a.u Orers of Fitte Polynomial nergy in a.u. nergy Graient in a.u. 9sp8 8s4p 8s4p 8sp evaluate at.99 a.u. (3.003 a.u. is expt. Bon length MC MC
13 Future Walking stimate (on H For goo/very goo basis sets For poor/very poor basis sets s4p 9sp sp 8s4p f Future Walking ime in a.u Future Walking ime in a.u. 3 Force on H in a.u. Force on H in a.u
14 Future Walking stimate (on Ge For goo/very goo basis sets For poor/very poor basis sets s4p 9sp f Future Walking ime in a.u sp 8s4p Future Walking ime in a.u. 4 Force on Ge in a.u. Force on Ge in a.u
15 Mixe MC Force Forces on H atom - energy graient Forces on Ge atom energy graient HF HF+(N HF+(N+( HF HF+(N Increasing basis set quality significant basis set epenence aing volume term ( significantly improves mixe HF force aing noal term (N slightly improves mixe HF force Remaining errors: Reynols approx. ( st orer in error of extrapolation approx. in noal term ( n orer in error of 5 F o rces in a.u. o n H Forces in a.u. on Ge HF+(N+( Increasing basis set quality
16 Pure MC Force Forces on H atom - energy graient Forces on Ge atom energy graient HF(extr. HF(futr. HF(extr.+*(N HF(extr. HF(futr. HF(extr.+*(N HF(futr.+*(N HF(futr.+*(N significant basis set epenence Future walking estimate of HF seem not better than extrapolation, but aing (N to HF(extr. overshoots correction for poor basis sets aing (N to HF(futr. gives exact total forces within (close to one stanar error pure HF forces (without noal term correction as goo as total mixe forces total pure MC force (with noal terms more accurate than total mixe MC force 6 F o rc e s in a.u. F o rces in a.u. Increasing basis set quality Increasing basis set quality
17 Conclusions We erive exact expressions for forces within mixe an pure MC incluing Pulay surface terms previously neglecte in pure MC calculations In mixe MC: Pulay surface terms can be calculate straightforwar In pure MC: Pulay surface terms are approximate as twice the mixe terms ests for the GeH inicate that mixe an pure MC noal terms are significant an incluing them significantly improves forces! Pure MC forces seem more accurate than mixe MC forces (in particular when the quality is less goo Pure MC forces are slighly more costly than mixe once, but seems worth! Outlook: apply to larger systems, use with non-local pseuopotentials 7
18 Acknowlegments Many thanks to John rail an Zoltan Ranai Financial Support: ngineering an Physical Sciences Research Council of the Unite Kingom 8
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