The search for the nodes of the fermionic ground state

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The search for the nodes of the fermionic ground state ψ ( R) = 0 + - + - - + - + Some believe that unicorns are just shy and difficult to find.

1 The search for the nodes of the fermionic ground state ψ ( R) = Some believe that unicorns are just shy and difficult to find. However, most people believe blue ones do not exist. We all believe that the nodes of a many body fermionic wavefunctions exist. However, many think they are as hard to find as blue unicorns.

2 (1975) Notation: mama = DMC algorithm life = time fantasy = imaginary man, boy = walker landslide = trial wave function monstrosity, nothing = node DMC for poets

3 DMC for poets So I see you a think little silhouetto you can stone of a man, me and spit in my eye So Scaramouch, you think you scaramouch can love me will and you leave do the me Fandango to die Oh Thunderbolt Baby - Can't and do Lightning this to me - Babyvery frightening me- Just Gallileo, gottagallileo, get out- just gotta get right outta here - Gallileo, gallileo, Gallileo Figaro - Magnifico - Random walk over all iε nt Lyrics space continues Random walk over all ihe t Mama, Is just this killed the real a walker, time e ψ e({ R}) = c ψ n n({ R j}) e Put a gun against his head, n The Random Is this just Walk imaginary is described now Pulled my trigger, now he's dead, For imaginary time : t = iτ lyrics space lim He τ ε 0 Caught in a wave function Re ψ e({ Ri}) = c ψ 0 0({ Ri}) e τ 2 2 d Mama, life had just begun, i ψ e({ Ri}) = h + V{ Ri} ε 0 No escape from reality dτ i 2m Too late, my time has come, - I'm just a poor boy nobody loves me Debate on whether a walker He's just a poor boy froma poor family must be killed or rejectede Spare him his life from Tthis monstrosity when crossing a node - + Sends shivers + down my spine Easy + Open Body's come, + your aching easy eyes all go the - will time, you let me go Persistent or stack - walker Bismillah! Look Goodbye No, up to everybody - we will the skies - -not and I've let see got you to go - let him go - solved by Bismillah! I'm Gotta just leave We a walker, you will not I need behind let you no and go - sympathy face Let him the truth go Nothing really matters C. Umrigar et al. JPC (1993) Bismillah! - + Because Mama, - ooo We I'm -will not let you go - Let him go easy come, + easy go, M. Casula + et al. PRL (2005) Will Little I don't Anyone not let high, want you little to go can die, - Let see me go Will low + Anyway I sometimes not + let you the wind - wish go - Let blows, I'd never me go Nothing really matters, doesn't been born really nothing at matter all - No, no, no, no, no, no, noto me really - to mematters - to me Mama mia, mama mia, mama mia let me go - Freddie Mercury The node Population control problems are discussed now τ ( ) ψ ({ R }) - Equilibrium population ε 0 ε 0 i 3 i N{ r}

Hood (LLNL) Research performed at the Materials Science and Technology Division and the Center of

4 A self-healing DMC algorithm (a systematic correction of nodal surfaces in small systems) Fernando A. Reboredo (ORNL) Paul R. C. Kent (ORNL) R. Q. Hood (LLNL) Research performed at the Materials Science and Technology Division and the Center of Nanophase Material Sciences at Oak Ridge National Laboratory sponsored the Division of Materials Sciences and the Division of Scientific User Facilities U.S.

5 DMC run E best 420 E Local Total Energy Exact (Full CI ) energy Number of DMC Steps

6 DMC run E best E Local 420 Total Energy Exact (Full CI ) energy Number of DMC Steps

7 DMC run E best E Local E= <ψ T H ψ T > l 420 Total Energy Exact (Full CI ) energy Number of DMC Steps

8 DMC run E best E Local E= <ψ T H ψ T > l 420 Total Energy Exact (Full CI ) energy Number of DMC Steps

9 A Theoretical Blue Unicorn : finding the blue nodes DMC run Total Energy E best E Local E= <ψ T H ψ T > l NO DFT NO HF NO VMC NO Jastrow Exact (Full CI ) energy Number of DMC Steps

10 What could we do if we new the Fixed Node wave function? Removing the kink in Y T moves the node in the right direction

11 One equation with three known quantities and two unknowns could be soluble Standard Importance Sampling Diffusion Monte Carlo Algorithm D. M. Ceperley and B. J. Alder, Phys. Rev. Lett. 45, 566 (1980) Number of configurations Trial Wave-Function Distribution of walkers Fixed-Node Ground State Wave-Function We know, and We want to know and

12 Expansion of the Fixed-Node Ground-State Wave-Function Every anti-symmetric function can be written as a product of a symmetric function (Jastrow) and a complete sum of antisymmetric functions Multi-Determinant Expansion

13 Expansion of the Fixed-Node Ground-State Wave-Function Every anti-symmetric function can be written as a product of a symmetric function (Jastrow) and a complete sum of antisymmetric functions.

14 Expansion of the Fixed-Node Walker Distribution Function Any expression of the trial wave-function Single determinant Multi determinant Back flow Pfaffian etc

15 Expansion of the Fixed-Node Walker Distribution Function

16 The Wave-Functions Projectors Reboredo & Proetto PRB 67, (2003).

17 The Wave-Functions Coefficients DMC Umrigar, Nightingale &Runge J.CP 99, (1993).

18 The Wave-Functions Coefficients Umrigar, Nightingale &Runge J.CP 99, (1993).

19 The Wave-Functions Coefficients Umrigar, Nightingale &Runge J.CP 99, (1993).

20 Errors in the Wave-Functions Coefficients

21 The Fixed Node Wave-Function Fixed-Node Ground State Wave-Function Projectors DMC sampling.can be obtained directly by sampling over the walker distribution.

22 Errors in the Wave-Functions Coefficients Fixed-Node Ground State The nodes move because of errors.

23 A Self-Healing DMC Algorithm Fixed-Node Ground State The nodes can move because of errors How do we take advantage of errors?

24 A Simple Self-Healing DMC Algorithm Ns = 200, starting trial wave function: anything standard DMC accumulation (sample of N s configurations) New trial wave-function º N s = a N s, a > 1

25 A Simple Self-Healing DMC Algorithm 1) Are non-interacting eigen-functions: ï increasing kinetic energy 2) = 0 if < x 4 else = 3) We set =

26 A Simple Self-Healing DMC Algorithm: Model The model test system: two interacting spinless fermions in a box Case (1) Model interaction V = 8γ π 2 cos[απ(x-x )] cos[απ(y-y )] α and γ control the shape and strength of V V is repulsive for α < 1/2 and g > 0 Full CI: H expanded in the first 300 noninteracting eigenfunctions with the symmetry of the ground state. All integrals done analytically. Converged results V The nodes are not trivial Case (2) Coulomb interaction V = 20 π 2 1/sqrt[(x-x ) 2 +(y-y ) 2 ] We will compare the algorithm against almost-analytical results

27 A Simple Self-Healing DMC Algorithm: Initial Y T Ns = 200, starting trial wave function: anything l 1 = - a 3 standard DMC accumulation l 3 = a 1 l n = 0 (for any other l n ) (sample of Ns configurations) New trial wave-function º a n (full CI coefficients) The starting trial wave function: is orthogonal to the exact CI ground state N s = a N s, a > 1 The standard DMC algorithm must fail with this trial wave-function choice = 1

28 A Simple Self-Healing DMC Algorithm: Results The DMC energy converges to the full CI even starting from the worst trial wavefunction

29 A Simple Self-Healing DMC Algorithm: Results Expansion of the Ground-State Wave-Function Determinant expansion Coeffient Full CI DMC (size = error bar) Determinant basis index An expansion of the Ground-State Wave-Function can be obtained from DMC with full CI quality

30 A Simple Self-Healing DMC Algorithm: Results Expansion of the Ground-State Wave-Function (time evolution) 1 (l) First trial Wave-function Steps in the Algorithm (n) Determinant basis index Systematic convergence of the wave-function towards the CI solution

31 A Self-Healing DMC Simple Algorithm: Results Projection of the SH-DMC WF on the CI Ground-State WF 1 (P) P = First trial Wave-function Number of DMC steps [(2 Log(2) Log (N DMC /200)]

32 A Self-Healing DMC Simple Algorithm: Results Projection of the SH-DMC WF on the CI Ground-State WF Systematic Improvement (bad nodes) -3x+c (R P ) Statistical Improvement (good nodes) -x+c Log(Nw T ) We observe first a fast improvement followed by a slow improvement regime

33 A Self-Healing DMC Simple Algorithm: Results Why does it work? = - (R P ) Log(Nw T ) Truncation of high energy components moves the nodes in right direction

34 1) A Self-Healing DMC Simple Algorithm: Results Why does it work? Are non-interacting eigen-functions: î increasing kinetic energy 2) = 0 if else = < x 4 3) We set = Truncation of poorly resolved high energy components moves the nodes in right direction

35 A Self-Healing DMC Simple Algorithm: Results Why does it work? + R - = 0 Statistical Improvement (good nodes) -x+c Noise in the coefficients plays the role of a temperature in a simulated annealing approach. Good fluctuations are reinforced bad ones are abandoned.

36 A Self-Healing DMC Simple Algorithm: Results Why does it work? R standard DMC accumulation (sample of Ns configurations) New trial wave-function = = 0 N s = a N s, a > 1 Noise in the coefficients plays the role of a temperature in a simulated annealing approach. Good fluctuations are reinforced bad ones are abandoned.

37 A Self-Healing DMC Simple Algorithm: Results Case Case (2) (2) Coulomb interaction interactionv = 20 πv 2 = 1/sqrt[(x-x ) 20 π 2 1/sqrt[(x-x ) 2 +(y-y ) 2 ] +(y-y ) 2 ] Expansion of the Ground-State Wave-Function Electronic energy Number of DMC steps [(2 Log(2) Log (N DMC /200)] We obtain a systematic reduction of the energy (beyond error bar)

38 A Self-Healing DMC Simple Algorithm: Results Case (2) Coulomb interaction V = 20 π 2 1/sqrt[(x-x ) 2 +(y-y ) 2 ] Expansion of the Ground-State Wave-Function (l) (n)

39 A Self-Healing DMC Simple Algorithm: Results Case (2) Coulomb interaction V = 20 π 2 1/sqrt[(x-x ) 2 +(y-y ) 2 ] Interacting Ground-State: Electronic Density = 1 Final Initial trial wave function density With the ground state wave function we can evaluate any observable

40 A Self-Healing DMC Simple Algorithm: Results Case Case (2) (2) Coulomb interaction interactionv = 20 πv 2 = 1/sqrt[(x-x ) 20 π 2 1/sqrt[(x-x ) 2 +(y-y ) 2 ] +(y-y ) 2 ] Expansion of the Ground-State Wave-Function Electronic energy Number of DMC steps [(2 Log(2) Log (N DMC /200)] We obtain a systematic reduction of the energy (beyond error bar)

41 A Self-Healing DMC Simple Algorithm: Results Case (2) Coulomb interaction V = 20 π 2 1/sqrt[(x-x ) 2 +(y-y ) 2 ] Non-Interacting Kohn-Sham Electronic Density 1 (a) (b) 8 Reboredo & Kent PRB (2008) DMC KS

42 A Self-Healing DMC Simple Algorithm: Results Case (2) Coulomb interaction V = 20 π 2 1/sqrt[(x-x ) 2 +(y-y ) 2 ] Kohn-Sham Potential Reboredo & Kent PRB (2008)

43 Summary The fixed node ground state wave function can be obtained directly from the walker distribution An iterative algorithm based on the update of the trial wave function leads to a systematic reduction of nodal errors The algorithm takes advantage of truncation and statistical errors to improve the trial wave function The Kohn-Sham potential can be obtained directly from a SH-DMC run Tests in larger systems are in progress

All-electron quantum Monte Carlo calculations for the noble gas atoms He to Xe

All-electron quantum Monte Carlo calculations for the noble gas atoms He to Xe A. Ma, N. D. Drummond, M. D. Towler, and R. J. Needs Theory of Condensed Matter Group, Cavendish Laboratory, University of