Electronic structure quantum Monte Carlo methods with variable spins and fixed-phase/node approximations

Transcription

1 Electronic structure quantum Monte Carlo methods with variable spins and fixed-phase/node approximations C. Melton, M.C. Bennett, L. Mitas with A. Ambrosetti, F. Pederiva North Carolina State University Universities of Trento and Padova SIGN 2017, INT, U. Washington, Seattle

2 projector QMC, Slater-Jastrow trial functions (single, multi-ref) and the fixed-node approximation standard model FNDMC Hamiltonian with (valence) electrons and ions QMC/DMC: ϕ0 =lim τ exp( τ H ) ψt (rial ) sampled in coord. space ϕ0 (r 1, r 2,...) = - trial function H ϕ0 =E 0 ϕ0 with fixed-node approx. ψt ϕ0, FN 0 ψt = k c k det k [ϕα ]det k [ϕβ ]exp[u corr ]

3 EoS FeO solid at high pressures: FNDMC transition at ~ 65 GPa (exper ); also agreement for cohesion, gaps, bulk moduli, etc ~ 95(2)% of E_corr J. Kolorenc & LM, PRL '08

4 but so far static spins only, while we need spins to vary spinless electrons-ions Hamiltonian spatial-only problem, ψ = c det [ϕ (r )]det spin channels factorized: α T k k k i k [ϕβ (r j )]exp[u corr ] now, include spin-orbit ϕ n (r i, si )=α ϕ (r i ) χ (s i )+β ϕ (r i ) χ (si ) determinant of spinors ψtrial =ψtrial (R, S )=det [ϕ n (r i, si )]exp(u corr ) spin coordinates : χ (1/ 2)=χ ( 1/ 2)=1 χ ( 1/ 2)=χ (1/ 2)=0

5 what is the problem then? variationally ok just sampling of a larger space d r 1... d r N σ... σ d r 1... d r N 1 N e.g., A. Ambrosetti, F. Pederiva, LM,, Phys. Rev. B 85, ('12) (also already in 1985 by J. Carlson and M. Kalos for nuclei...) VMC [ev] Exper. [ev] Tl J=3/2/g.s. J=1/2 0.85(5) 0.95 Pb J=1 /g.s J=0 0.88(7) 0.95 Pb J=2 /g.s. J=0 1.23(6) 1.31 Bi J=3/2 /g.s. J=3/ (8) 1.42

6 however, projection QMC (DMC etc) is less straightforward discrete moves between 2 N points: - moves of fixed length (no time step) - no smooth importance guiding - increased local energy fluctuations } inefficient with N more issues : - inherently complex wave functions - nonlocal SO pseudopots (PP) for heavy elements suggested ideas: - sample the spinors (Pederiva, Gandolfi, Ambrosetti 2000s) with gradual updates ( stochastic rotations of spinors ) - smooth out spin configurations + fixed-phase approximation (Melton, Ambrosetti, Pederiva, LM et al, 2016)

7 step one: smooth out spin configurations/paths we make spin configurations non-discrete by using continuous (overcomplete) and compact representation, possible choice: χ (s)=exp(+is), χ (s)=exp( is); s (0,2 π) different from rotating spinors, here: spinors are fixed since the spinors are fixed one can use projection onto trial function to eliminate nonlocality of PP and SO terms W SORPP = i [W (i)arpp + i W (iso) ] (i ) W (iso) = j, l v jl (r i ) P (i) l s P jl i i jl nonlocal SORPP term locality approximation (LM et al '91) W ℜ W Tℜ=ℜ[ ψ 1 T W ψt ]

8 step two: complex wave function fixed-phase (FP) ψ=ρ( R, S )exp[i ϕ( R, S )] the Schrodinger equation breaks into Re and Im t ρ=[t +V +W ℜ +(1/ 2)( ϕ)2 ]ρ t ϕ=[t ϕ ( ln ρ) ϕ+w ℑ ] the first equation gives the energy eigenvalue and we invoke the fixed-phase (FP) approximation (Ortiz et al '92) ϕ ϕt 2 V eff =(1/ 2)( ϕt ) FP seems like a step into an unknown territory, but it is not: fixed-node is a limit/special case of the fixed-phase for real wfs 2 ( ϕt ) C δ[ R R node,t ]

9 fixed-phase special case of fixed-node, sketch of a demonstration let ψt ( R) be real, fermionic, with nodes at subset R node,t construct ψ=ψ T +ia ψsymm,>0 ψ 2 ] ϕ =arctan [(ℜ ψ)/ then the limit of potential from the phase node 2 lima 0 ( ϕ ) C δ[ R Rnode,T ]

10 step three: sampling of the spin configurations fixed-phase spinorbit DMC (FPSODMC) effective free-particle Hamiltonian (kinetic term) for spins H H +H spin, 1 H spin (si )= 2μ s [ 2 +1 si2 ] H spin annihilates arbitrary spinor H spin (si )[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 overshadowed by the fixed-phase bias since SO is small) - effective spin mass time step on the spin subspace (overall, basically two time steps, spatial and spin) FPSODMC method: tests on atomic and molecular systems

11 total energies: Pb atom with valence 6s26p2 FPSODMC(.) vs CI with ccpvxz basis( ) Arxiv:... J =0 ; J =0 ; 11 22

12 Cr and Mo atoms ground states 7S3 (d5s1) W atom is isovalent, what is its ground state? averaged SO, any method (DFT, CI, QMC) 7 S3 (5d56s1) explicit SO two-component, open-shell only CI 7 explicit SO two-component, full CI or FPSODMC/rCI 5 S3 (5d56s1) D0 (5d46s2) both SO and correlation needed to flip the state!

13 W atom SO splitted sd-manifold of excitations Config. State 5d46s2 5 5d56s1 7 5d46s2 D1 COSCI DMC/COSCI CISD DMC/rCISD Exp (1) (1) 0.21 S (1) (1) D (1) (1) d46s (1) (1) d46s (1) (1) 0.77 D3 D4 FPSODMC agrees with experiment ok, but better RPPs needed!

14 FPSODMC applied to the PbH molecule (the averaged SO treatment off by 1 ev!) Method E_bind [ev] spin-average CCSD(T) 2.66 spin-average FNDMC 2.58(1) MRCIS/SO+pert. spin-average CCSD(T) FPSODMC 1.63(1) Exper. ~ 1.69(5)

15 Sn2 dimer should be simple, it is the fourth row but SO correction is ~ 0.5 ev! (small core SORPP, 44 val. e-) Exp. A. Ambrosetti et al, to appear in PRB

16 interestingly, one can go back to fixed-node, ie, recover the spin-labeled fixed-node trial form... consider spinors is χ α (r, s)=ϕα (r)e, χβ (r, s)=ϕβ (r)e is set variables to two values: {up }={s i } s, {down }={s j } s ', s s'

17 fixed-node trial form but with a complex twist spins factorize out of the determinant and we get up/down product: ψt =det [χ j ] ψt =const [sin (s s ' ) N/2 ] det [ϕi ] det [ϕ j ] - the most interesting regime: {up }={s i } s {down }={s j } s ', - basically the fixed-node limit but complexified, ie, has properties of the fixed-phase - this can be achieved by the choice of spin variables and by adjusting the time step for spin variables

18 two limits: slow spins fixed-node fast spins full fixed-phase - group the spins to two distinct values up, down and run FP C atom, all e- ~ 5 % E_corr fast spins limit: bias from complex phase in full space indep. calculated FN value (~ 95% of E_corr) 0 spin time step large

19 fixed-node vs fixed-phases biases from: independent FN real wf vs FP at the FN limit

20 fixed-phase: some considerations - has a form of effective (many-body) potential/field V ph =(1/ 2)( ϕ) 2 τ ρ=[ T +V +V ph ]ρ - ρ(r) 0, its zeros are codimension 2 (unlike FN codimension 1) - ergodicity generically ok (no artificial nodal domains important for calculations of other properties than energy) - smaller fluctuations and easier sampling (no recrossing)

21 fixed-phase amplitude zeros: codimension harmonic electrons, P(sp) state ψexact = g (r 1, r 2, r 12 )det [1,Y 11 ] fixed-node: V FN =V δ( x 1 x 2 ) fixed-phase: V ph = 4 2 Γ={( x 1= x 2 ) R } d =5 line ( x 12 + y 12 ) 4 Π0={( x 12+ y 12 =0) R } d =4 point ix iy ψt =det [1, e, e ] - three periodic electrons ix iy fixed-node: ψt =ℜ {det [1, e, e ]} 2 nodal domains fixed-phase: ρt = ψt one domain

22 coming back to spatial-only nodes: some properties - roots of 1D polynomial/function anchor its behavior - nodes are roots of multivariate polynomial/function (dn-1)-manifold a.e. given by ψ(r 1, r 2,..., r N )=ψ(r)=0 - in addition, nodes of eigenst. of Schrodinger equation are special: - nodal domain averages nda - nodal surface averages nsa

23 nda: nodal domain averages (not usual expectations, direct imprint from amplitudes) write the total energy as kinetic and potential components that are one-sided expectations, or, nodal domain averages (nda): nda nda E= E kin + E pot nda / E kin = Γ R ψ d R ψ d R E pot = V ( R) ψ d R / nda ψ d R nda E kin determined by R ψ solely on the node Γ={R ; ψ(r)=0 } - nda components enable to distinguish between degenerate states with different nodes (eg, different symmetries) - enables to show some unexpected equivalences, eg, fermionic and bosonic (excited) nodes equivalent, just rotated

24 different states, even different statistics, but equivalent nodes three atomic states, 2p2 occupation: 3P, 1S, 1D all have the same nda energy components 3 P, 1S, 1D (2p2) E tot E kin E pot E nda kin E nda pot -1/4 1/4-1/2 1/12-1/3 why? Consider the 5D node projected into 3D: 3 P : electron sees a plane defined by ang. mom. axis and the second el. 1 D : electron sees a plane which contains ang. mom. axis and is orthogonal to such plane defined by the second electron 1 S : electron sees a plane which is orthogonal to the position vector of the second electron in all three cases the node is a 5D hyperbolic surface in 6D arxiv:

25 nsa: nodal surface averages total energy as the following kinetic and potential components nsa nsa E= E kin +E pot E kin = Γ h( R) R ψ... expression is too long nsa 2 nsa / E pot = Γ h( R)V ( R) d R Γ h( R) d R h( R) is a weight function, all integrals only over the nodal surface - choose h( R) such that the average of potential over the node gives the eigenvalue (note: kinetic part vanishes) Γ h(r)[v ( R) E ] d R=0 - node is special: any other level set needs more information

26 summary - unifying formalism: FN and FP, static and variable spins - sampling advantages of codimension 2 (for excitations, especially) - wave functions with phase/spins are more general, possible additional variational freedom - nodal surfaces are unique properties can possibly reveal how to construct them more efficiently PRA 2016, JCP 2016, , to be submitted soon

27 total energies Bi atom with valence 6s26p3 FPSODMC/COS( _), FPSODMC/restr.CI(...) vs CI( Arxiv:... )

28 similar for molecules now including nonlocal ECPs and differences dimer atoms, example of N2 small difference but within the scale of FN biases

29 Bi atom excitations/splittings compared with experiment Bi atom SO splittings [ev] (w.r.t. to the g.s. J=3/2>) J; j1,j2,j3 > DF CI FPSODMC Exper. 3/2; > (1) /2; > (1) /2; > (1) /2; > (1) 1.42

30 key points about FPSODMC method - fixed-phase: no sign, basis or ergodicity problem ρ(r) is nonzero except for isolated points (ie, codimension 2) - zero variance property energy fluctuations decrease with ψt error squared (as for the fixed-node) ψt V eff one-to-one mapping for any state - treatment of SO terms natural to its nonlocality localization - reuse of much existing QMC methodology/codes from static spins

31 Green's function for spinor sampling y x x y example of Rashba SO in 2D*: V Rashba =λ j [ p j σ j p j σ j ] G spin j ( cos( γ δ r j ) = sin ( γ δ r j ) where γ λ and i δ y j +δ x j δrj δ r j,δ x j,δ y j sin ( γ δ r j ) i δ y j +δ x j δrj cos( γ δ r j ) ) are spatial displacements - similar to Hubbard-Stratonovitch approaches *A. Ambrosetti, F. Pederiva, E. Lipparini, S. Gandolfi, PRB 80, ('09)

32 2D fermion gas with Stoner and Rashba interactions: spin-polarization vs. interaction strength A. Ambrosetti, F. Pederiva, LM, et al, PRA 91, ('15)

33 FNDMC vs traditional correlated wf methods: nodes converge in basis very rapidly, augtzv or so (HF)2 dimer