Electronic structure quantum Monte Carlo methods and variable spins: beyond fixedphase/node

Transcription

1 Electronic structure quantum Monte Carlo methods and variable spins: beyond fixedphase/node approximations Cody Melton, M. Chandler Bennett, L. Mitas, with A. Ambrosetti, F. Pederiva North Carolina State University Universities of Trento and Padova INT Workshop, August 1, 2018

2 electronic structure qmc - ground and excited states, T=0 - energy differences ~ evs, accuracy target 0.05 ev (Hartree-Fock as reference, E_corr = E_exact - E_HF) interest in: 1) so far spins were just static labels (up, down) but we need spin-orbit, etc, varying spins 2) maybe, unify static and variable spins formulations 3) beyond the fixed-node/phase

3 projector QMC and variational fixed-node standard model FNDMC (90-95% of E_corr) Hamiltonian: interacting electrons in ionic potentials (or ECP) QMC/DMC: ψ 0 (r 1, r 2,..., r N )=lim τ exp( τ H )ψ T (rial) H ψ 0 =E 0 ψ 0 ψ T ψ 0, FN = node Γ(ψ)={R ; ψ( R)=0} dim(γ(ψ))=3n 1 codimension 1 fixed-node (FN) approx.! Γ(ψ 0, FN )=Γ(ψ T ) ψ T ψ 0, FN 0 trial function Slater-Jastrow: ψ T = k c k det k [ϕ α ]det k [ϕ β ]exp[u corr ]

4 if eigenstate is inherently complex (eg, stationary current): fixed-phase approximation write ψ=ρe i Φ ; ρ 0 ρ 0 =lim τ exp[ τ(h +( Φ 0 ) 2 /2)]ρ T (rial) (H +( Φ 0 ) 2 /2)ρ 0 =E 0 ρ 0 ρ T ρ 0, FP = dim(γ(ρ))=3n 2 codimension 2 fixed-phase (FP) approx.! Φ 0 =Φ T V eff,t =( Φ T ) 2 / 2 ψ T = k c k det k [ϕ α ]det k [ϕ β ]exp[u corr ]=ρ T e i Φ T

5 fixed-phase special case of fixed-node (sketch) let ψ T (R) be real, fermionic, with nodes at R node,t Γ(ψ T ) construct ψ=ψ T +ia ψ symm,>0 ϕ=arctan [(R ψ)/ ψ 2 ] then the limit of potential from the phase node lim a 0 ( ϕ) 2 C (1/a)δ[ R R node,t ] ie, can write also the fixed-node as effective singular potential H H +V (R node,t )

6 from spatial orbitals to spinorbitals spinless electrons-ions Hamiltonian spatial-only problem, spin channels factorized: ψ T = k c k det k [ϕ α (r i )]det k [ϕ β (r j )]exp[u corr ] now, include spin-orbit ϕ n (r i, s i )=α ϕ (r i )χ (s i )+βϕ (r i )χ (s i ) determinant of spinors spin functions and coordinates : ψ Trial =ψ Trial (R, S)=det [ϕ n (r i, s i )]exp(u corr ) χ (1/2)=χ ( 1/2)=1 χ ( 1/2)=χ (1/2)=0 - wf complex, good quantum number J

7 projection is more involved and less straightforward some ideas: - work in 80s on nuclei (Kalos, Carlson, Schmidt, others) - sample the spinors (Pederiva, Gandolfi, Ambrosetti 2000s) with spinor updates ( stochastic rotations of spinors ) - smooth out spin configurations + fixed-phase approximation (Melton, Ambrosetti, Pederiva, LM et al, 2016) -...

8 we smooth out spin configurations/paths - continuous (overcomplete) representation, ie, coordinates, possible choice: χ (s)=exp(+is), χ (s)=exp( is); s (0,2 π) different from rotating spinors, here: spinors are fixed why this choice in particular? ( later)

9 how can you do that? atomic spin-orbit acting on a valence electron i can be recast as L i S i l, j, m j l, j, m j > v lj (r i ) <l, j, m j correct action of SO and expectations need matrix elements l, j, m j χ = a χ +b χ c χ +d χ I s=1/2, 1/2 0 2 π ds

10 sample the spin configurations as free d.o.f. fixed-phase spinorbit DMC (FPSODMC) effective free-particle Hamiltonian (kinetic term) for spins H H +H spin, H spin (s i )= 1 2μ s [ 2 s i 2 +1 ] H spin annihilates arbitrary spinor H spin (s i )[α ϕ (r i )χ (s i )+β ϕ (r i )χ (s i )]=0 therefore, to the leading order no contribution to the energy (subleading contribution overshadowed by the fixed-phase bias since SO is small) FPSODMC method: tests on atomic and molecular systems

11 total energies: Pb atom valence only, vary effective mass, proportional to 1/(spin time step) 0.1 ev (small spin effective mass large)

12 total energies: Pb atom with valence 6s 2 6p 2 FPSODMC(.) vs CI with ccpvxz basis( ) j 1, j 2 = 3 2 j 1, j 2 = 1 2 Arxiv:...

13 Cr and Mo atoms electronic ground states 7 S 3 (d 5 s 1 ) W atom is isovalent, what is its ground state? averaged SO (CI, QMC) 7 S 3 (5d 5 6s 1 ) explicit SO two-component, open-shell only CI 7 S 3 (5d 5 6s 1 ) explicit SO two-component, full CI or FPSODMC/rCI 5 D 0 (5d 4 6s 2 ) both SO and correlation needed to flip the state!

14 W atom SO splitted sd-manifold of excitations: correct ground state in FPSODMC Config. State COSCI DMC/COSCI CISD DMC/rCISD Exp 5d 4 6s 2 5 D (1) (1) d 5 6s 1 7 S (1) (1) d 4 6s 2 5 D (1) (1) d 4 6s 2 5 D (1) (1) d 4 6s 2 5 D (1) (1) 0.77

15 W atom: also correct order of excitations! Config. State COSCI DMC/COSCI CISD DMC/rCISD Exp 5d 4 6s 2 5 D (1) (1) d 5 6s 1 7 S (1) (1) d 4 6s 2 5 D (1) (1) d 4 6s 2 5 D (1) (1) d 4 6s 2 5 D (1) (1) 0.77 FPSODMC agrees with experiment, higher accuracy needs better ECP

16 Sn 2 dimer should be simple, it is only the fourth row but SO correction is ~ 0.5 ev! (small cores, 44 val. e-) Exp. A. Ambrosetti et al, to appear in PRB

17 why this in particular? - similar to spatial coords but much smaller space - no divergencies, no jumps, importance sampling ok - simplifies dealing with pseudopotentials (effective cores) and generate similar bias, close to fixed-node regime but more χ (s)=exp(+is), χ (s)=exp( is); s (0,2 π) - enables to smoothly complexify also real eigenstates - and still more...

18 interestingly, from such spinor wf, one can recover the spin-labeled fixed-node trial form... in spinors χ α (r, s)=ϕ α (r)e is, χ β (r, s)=ϕ β (r)e is adjust to two values: {up }={s i } s, {down}={s j } s ', s s'

19 full sampling of all possible spin states and configurations: cartoon R 3N (2π) N N! [(N /2)!] 2 2N singlets N-electron continuous spin-position space

20 restricting spins into particular up and down subspace R 3N one (N/2)*(N/2) choice fixed-node

21 fixed-node trial wf form but with a complex twist spins factorize out of the determinant and we get up.down product: ψ T =det [χ j (r k, s k )] ψ T = fac(s, s ' )det [ϕ i (r k )]det [ϕ j (r k ' )] - the most interesting regime: {up }={s i } s {down }={s j } s ', - basically, the fixed-node limit but complexified, ie, it has properties of the fixed-phase, as can be achieved by: - the choice of spin variables (one assigns a set of particles as spin-up or -down, ie, particular subset of permutations) - explore how close/far to fixed-node by τ spin / τ space

22 fixed-node vs fixed-phase biases in atoms: FN real w.f. vs FP at the FN limit essentially the same C. Melton, LM, PRE, 96, or arxiv

23 similar for molecules now including nonlocal ECPs FN vs FP at the FN limit: binding curves of N 2

24 released-node

25 released-node: importance sampling with symmetric guiding function while projecting out the fermionic component antisymm. ψ T FNDMC ψ T ψ FN E FN = ψ FN ψ T [(H ψ T )/ ψ T ] ψ FN ψ T symm. ψ G RNDMC ψ G ψ RN = symm+antisymm E RN = ψ RN ψ G (ψ T /ψ G )[(H ψ T )/ ψ T ] ψ RN ψ G (ψ T / ψ G )

26 choice of guiding function ψ G =ρ T ψ T =ρ T (R, S )exp[i ϕ T ( R, S)] why? - amplitude is symmetric by definition - its node is codimension 2, ie, generically ergodic sampling ψ T - it is close to that implies close to optimal importance sampling local energy fluctuations almost the same

27 few electron system (all-el O atom): released-node and the well-known exponential noise FN exact

28 better tuned algorithm: released-node eliminates the bias fully FN exact

29 summary - unifying formalism FPSODMC, FN and FP, static/variable spins, sampling + nodes sampling + effective potential - wave functions with phase/spins are more general, more smooth, ergodic sampling (zeros codim 2) - new options for attacking fixed-node/phase bias - more variational freedom (?) PRA 2016, JCP 2016, PRE more coming