Variational Monte Carlo Optimization and Excited States

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1 Variational Monte Carlo Optimization and Excited States Eric Neuscamman August 9, 2018

2 motivation charge transfer core spectroscopy double excitations

3 the menu aperitif: number counting Jastrows main course: excited states with QMC dessert: optimization methods

4 number counting Jastrow factors C " r = 1 if r in region ' C " r = 0 otherwise Van Der Goetz & Neuscamman, JCTC 2017 Van Der Goetz & Neuscamman, unpublished

5 number counting Jastrow factors C " r = 1 if r in region ' C " r = 0 otherwise exp -. / 0 C " r 0 1 modifies the balance between these configs: C C C 2 C 3 C 3 C 2 Van Der Goetz & Neuscamman, JCTC 2017 Van Der Goetz & Neuscamman, unpublished

6 number counting Jastrow factors C " r = 1 if r in region ' C " r = 0 otherwise exp -. / general form: 0 C " r 0 1 modifies the balance between these configs: C C C 2 C 3 C 3 C 2 exp / 4506 F "8 C " r 0 C 8 r 6 Van Der Goetz & Neuscamman, JCTC 2017 Van Der Goetz & Neuscamman, unpublished

7 number counting Jastrow factors C " r = 1 if r in region ' C " r = 0 otherwise exp -. / 0 C " r 0 1 modifies the balance between these configs: C C C 2 C 3 C 3 C 2 general form: linear combinations? exp / 4506 F "8 C " r 0 C 8 r 6 Van Der Goetz & Neuscamman, JCTC 2017 Van Der Goetz & Neuscamman, unpublished

8 clean additivity -1.0 energy +12 (Hartrees) C-C distance (Angstroms) Van Der Goetz & Neuscamman, JCTC 2017 Van Der Goetz & Neuscamman, unpublished

9 clean additivity -1.0 energy +12 (Hartrees) C-C distance (Angstroms) Van Der Goetz & Neuscamman, JCTC 2017 Van Der Goetz & Neuscamman, unpublished

10 clean additivity -1.0 curvature at edges of counting regions! energy +12 (Hartrees) C-C distance (Angstroms) Van Der Goetz & Neuscamman, JCTC 2017 Van Der Goetz & Neuscamman, unpublished

11 clean additivity -1.0 curvature at edges of counting regions! energy +12 (Hartrees) clean additivity? C-C distance (Angstroms) Van Der Goetz & Neuscamman, JCTC 2017 Van Der Goetz & Neuscamman, unpublished

12 clean additivity -1.0 curvature at edges of counting regions! energy +12 (Hartrees) yes, trivially!! " $! " $ '! ' $ clean additivity? C-C distance (Angstroms) Van Der Goetz & Neuscamman, JCTC 2017 Van Der Goetz & Neuscamman, unpublished

13 clean additivity works energy +12 (Hartrees) RHF MRCI+Q C-C distance (Angstroms) Van Der Goetz & Neuscamman, in preparation

14 clean additivity works energy +12 (Hartrees) RHF +Jastrows MRCI+Q C-C distance (Angstroms) Van Der Goetz & Neuscamman, in preparation

15 clean additivity works -0.6 energy +12 (Hartrees) DMC RHF +Jastrows MRCI+Q C-C distance (Angstroms) Van Der Goetz & Neuscamman, in preparation

16 the menu aperitif: number counting Jastrows main course: excited states with QMC dessert: optimization methods

17 variational principles ground state Ψ " Ψ any eigenstate Ψ " $ % Ψ Umrigar, Filippi, others state near & Ψ & " % Ψ Choi, Lebeda, & Messmer, CPL 1970 Van Voorhis et al, JCP 2017 state above & Ψ (& ") Ψ Ψ & " % Ψ Zhao & Neuscamman, JCTC 2016, 2017 Shea & Neuscamman, JCTC 2017

18 variational principles ground state Ψ! Ψ evaluate! " any eigenstate Ψ! % " Ψ Umrigar, Filippi, others by Monte Carlo integration state near & Ψ &! " Ψ Choi, Lebeda, & Messmer, CPL 1970 Van Voorhis et al, JCP 2017 state above & Ψ (&!) Ψ Ψ &! " Ψ Zhao & Neuscamman, JCTC 2016, 2017 Shea & Neuscamman, JCTC 2017

19 poor properties ground state Ψ " Ψ any eigenstate Ψ " $ % Ψ Umrigar, Filippi, others state near & Ψ & " % Ψ Choi, Lebeda, & Messmer, CPL 1970 Van Voorhis et al, JCP 2017 state above & Ψ (& ") Ψ Ψ & " % Ψ Zhao & Neuscamman, JCTC 2016, 2017 Shea & Neuscamman, JCTC 2017

20 poor properties ground state Ψ " Ψ any eigenstate Ψ " $ % Ψ Umrigar, Filippi, others state near & Ψ & " % Ψ Choi, Lebeda, & Messmer, CPL 1970 Van Voorhis et al, JCP 2017 state above & Ψ (& ") Ψ Ψ & " % Ψ Zhao & Neuscamman, JCTC 2016, 2017 Shea & Neuscamman, JCTC 2017

21 poor properties ground state Ψ " Ψ any eigenstate Ψ " $ % Ψ Umrigar, Filippi, others state near & Ψ & " % Ψ Choi, Lebeda, & Messmer, CPL 1970 Van Voorhis et al, JCP 2017 state above & Ψ (& ") Ψ Ψ & " % Ψ Zhao & Neuscamman, JCTC 2016, 2017 Shea & Neuscamman, JCTC 2017

22 interconversion ground state Ψ " Ψ state near & Ψ & " % Ψ Choi, Lebeda, & Messmer, CPL 1970 Van Voorhis et al, JCP 2017 & $ any eigenstate Ψ " $ % Ψ Umrigar, Filippi, others & $ * state above & Ψ (& ") Ψ Ψ & " % Ψ Zhao & Neuscamman, JCTC 2016, 2017 Shea & Neuscamman, JCTC 2017

23 interconversion ground state Ψ % Ψ state near! Ψ! % ' Ψ Choi, Lebeda, & Messmer, CPL 1970 Van Voorhis et al, JCP 2017! # any eigenstate Ψ % # ' Ψ Umrigar, Filippi, others! # * state above! Ψ (! %) Ψ Ψ! % ' Ψ Zhao & Neuscamman, JCTC 2016, 2017 Shea & Neuscamman, JCTC 2017

24 interconversion ground state Ψ " Ψ state near & Ψ & " % Ψ Choi, Lebeda, & Messmer, CPL 1970 Van Voorhis et al, JCP 2017 & $ any eigenstate Ψ " $ % Ψ Umrigar, Filippi, others & $ * state above & Ψ (& ") Ψ Ψ & " % Ψ Zhao & Neuscamman, JCTC 2016, 2017 Shea & Neuscamman, JCTC 2017

25 interconversion ground state Ψ " Ψ state near # Ψ # " % Ψ Choi, Lebeda, & Messmer, CPL 1970 Van Voorhis et al, JCP 2017 # ) any eigenstate Ψ " ) % Ψ Umrigar, Filippi, others # ) * state above # Ψ (# ") Ψ Ψ # " % Ψ Zhao & Neuscamman, JCTC 2016, 2017 Shea & Neuscamman, JCTC 2017

26 having it both ways excited energy of CO ground non-specific optimization % excited state character in guess Shea & Neuscamman, JCTC 2017

27 having it both ways excited energy of CO interconverting optimization ground non-specific optimization % excited state character in guess Shea & Neuscamman, JCTC 2017

28 having it both ways CO + N 2 size consistency error fixed!: 2.60 ± 0.2 me h excited energy of CO adaptive!: 0.04 ± 0.2 me interconverting h optimization ground non-specific optimization % excited state character in guess Shea & Neuscamman, JCTC 2017

29 charge transfer orbital relaxation state-averaged CASSCF: 2.7 ev to 5.9 ev (yikes!) state-averaged CASPT2: intruder states state-specific CASSCF: 4.4 ev state-specific CASPT2: 3.5 ev [ ] - EOM-CCSD: 3.8 ev! " Ψ -matched VMC: $ = multi-slater from CISPI 3.8 ev Pineda Flores & Neuscamman, in preparation

30 solids in QMC, solids are molecules with funny electrons! = # + % && + % '& + % '' periodic sums, but still a finite number of particles

31 the wave function splined DFT orbitals

32 the wave function splined DFT orbitals DFT orbitals are not perfect

33 the wave function splined DFT orbitals DFT orbitals are not perfect Ψ = all singles + doubles on important singles (then cut off small singles to variance match)

34 optical gaps 14 LiF 12 VMC gap (ev) ZnO C LiH 2 0 Si Experimental gap (ev) Zhao & Neuscamman, arxiv

35 Zinc Oxide G 0 W 0 HF Gap (ev) G 0 W 0 PBE0 G 0 W 0 HSE Experiment BSE exciton binding ~ 0.06 ev G 0 W 0 PBE Bechstedt et al, PRB 2009 Pulci et al, PRB 2010 Kresse et al, PRB 2007 (2 papers) Zhao & Neuscamman, arxiv

36 Zinc Oxide G 0 W 0 HF Gap (ev) G 0 W 0 PBE0 G 0 W 0 HSE GW HSE GW PBE Experiment BSE exciton binding ~ 0.06 ev G 0 W 0 PBE Bechstedt et al, PRB 2009 Pulci et al, PRB 2010 Kresse et al, PRB 2007 (2 papers) Zhao & Neuscamman, arxiv

37 Zinc Oxide G 0 W 0 HF VMC-HF 90% HOMO LUMO Gap (ev) VMC LDA VMC PBE0 VMC HF G 0 W 0 PBE0 G 0 W 0 HSE GW HSE GW PBE Experiment VMC-PBE0 85% HOMO LUMO VMC-LDA 80% HOMO LUMO BSE exciton binding ~ 0.06 ev G 0 W 0 PBE Bechstedt et al, PRB 2009 Pulci et al, PRB 2010 Kresse et al, PRB 2007 (2 papers) Zhao & Neuscamman, arxiv

38 the menu aperitif: number counting Jastrows main course: excited states with QMC dessert: optimization methods

39 the linear method Ψ # a & Ψ # + (, )*+ - ).Ψ.# ) minimize / w.r.t. -, or, in other words, solve 0 - = / 2 - good: fast convergence well established bad: matrices get huge highly nonlinear Umrigar, Toulouse, Filippi, Sorella, Hennig, PRL 2007

40 accelerated descent remember the average movement and add momentum good: low memory nearly linear bad: first order method little VMC data so far Schwarz, Alavi, Booth, PRL 2017 Sabzevari & Sharma, arxiv

41 blocked linear method Ψ # with # = {# &,, # ), # )*&,, # +), # +)*&,, #,), } do linear method for first set Zhao & Neuscamman, JCTC 2017

42 blocked linear method Ψ # with # = {# &,, # ), # )*&,, # +), # +)*&,, #,), } do linear method for first set then second set Zhao & Neuscamman, JCTC 2017

43 blocked linear method Ψ # with # = {# &,, # ), # )*&,, # +), # +)*&,, #,), } do linear method for first set then second set then third set Zhao & Neuscamman, JCTC 2017

44 blocked linear method Ψ # with # = {# &,, # ), # )*&,, # +), # +)*&,, #,), } do linear method for first set then second set then third set now restrict variables to best directions from each set do LM on Ψ # Ψ / with / much shorter than # good: fast convergence low memory bad: extra samples little VMC data so far Zhao & Neuscamman, JCTC 2017

45 hybrid method? hard to couple stiff parameters w/o 2 nd order derivatives accelerated descent blocked linear method needs info about other blocks to get coupling right

46 comparing methods in CaO optimizing NCJF, orbitals, 2-body Jastrows strongly coupled linear method Energy (Hartree) blocked linear method hybrid method VMC Samples (millions)

47 future directions oo-msj in solids core excitations selective CI VMC DMC black-box hybrid optimizer

48 acknowledgements - Gas Phase Chemical Physics - Computational & Theoretical Chemistry

Time-dependent linear-response variational Monte Carlo.

Time-dependent linear-response variational Monte Carlo. Bastien Mussard bastien.mussard@colorado.edu https://mussard.github.io/ Julien Toulouse julien.toulouse@upmc.fr Sorbonne University, Paris (web)