Quantum Monte Carlo tutorial. Lucas K. Wagner Dept. of Physics; University of Illinois at Urbana-Champaign

Transcription

1 Quantum Monte Carlo tutorial Lucas K. Wagner Dept. of Physics; University of Illinois at Urbana-Champaign

2 QMC: what is it good for? Theoretical gap (ev) FN-DMC DFT(PBE) VO 2 (monoclinic) La 2 CuO 4 FeO ZnSe ZnO MnO NiO S(q) e/site sys. H(DMC) G (S-J) TB(RPA) Hubbard 4 e/site sys. G(Slater) G(DMC) G(IXS) G(S-J) G(RPA) 0 VO 2 (rutile) Experimental gap (ev) q [Å 1 ] Gaps Wagner, Ceperley Rep. Prog. Phys (2016) Correlation functions FIG. 2. S(q) ofdi erent systems obtained through di erent methods. G: graphene; G : electrons only graphene; H: hydrogen; Zheng, TB: -band tight-binding Gan, model with 1/r interactions; Hubbard: the Hubbard model with U/t = 1.6. Caption should be q Abbamonte, Wagner Let us first consider the S(q) results for ab-initio graphene, denoted by G in Fig 2. For comparison, we have plotted S(q) of a non-interacting Slater determinant of Kohn-Sham orbitals [G(Slater)] in that plot,

3 (a) DFT-PBE0 La 2 CuO 4 CaCuO 2 CuO 2 2 Tetragonal A 1g B 1g Cu Ni Co Fe Mn Cr (b) DMC Cu Ni Co Fe Mn Cr FM Col Flip Check Unp Bicol Energy (ev/f.u.) Magneto-phonon coupling Wagner, Abbamonte PRB (2014) Ground states of magnetic systems Narayan, Busemeyer, Wagner (in preparation)

4 7 Pressure (GPa) EDMC - Eref (ev/f.u.) (a) z DMC(PBE0) Se (b) z Exp Se Cell Volume (Å 3 ) DMC(PBE0) DMC(opt) bicollinear checkerboard collinear staggered dimer collinear, flip 1 ferromagnetic Pressure dependence of magnetic energies in FeSe Busemeyer, Dagrada, Sorella, Casula, Wagner Phys. Rev. B (2016) FIG : For each calculation (QMC, PBE, and PBE0): (Right) Total energies for 8 f.u. cell for various magnetic orderings, function of volume, choosing experimental [10] valuesofz Se. (Left) Same as right, but choosing optimized values of z Se. the top QMC plots, energies are referenced to the collinear energy at around 77 Å 3. The DFT calculations are referenced he z Se minimum energy for that type of calculation. The DMC paramagnetic energies are 0.85 ev/f.u. higher than the rence collinear energy. function z Se and pressure. In FN-DMC and PBE0, Water on boron nitride Wu, Aluru, Wagner (a) The contact angl J. Chem. Phys. 142, (2015), (2016) on bulk hbn predicted by mole Al-Hamdani, Ma, Alfè, Lilienfeld & Michaelides tions. Three nano droplets com 8000 water molecules are consid J. Chem. Phys. 142, (2015). to the contact angle of the ma θ Optical excitations and magnetism,isperformed. (b)thebindi

5 Fitting models t01 = 2.76(1) U = 10.92(4) V01 = 7.13(3) V02 = 5.41(2) V03 = 4.57 Fitted Energy (ev) Fitted Energy (ev) DMC Emax =0.28 Erms = Figure 6. Comparison of input and fitted energies using the N-AIDMD procedure. The top panels (a)-(c) show the VMC results and the bottom ones (d)-(f) show the DMC ones. (a)-(d) correspond to the on-site Hubbard model, (b)-(e) the long range Hubbard model and (c)-(f) with additional third-nearest neighbor hopping DMC Energy (ev) (e) Excitation spectrum very close to experiment. tted energies using the N-AIDMD procedure. The top panels (a)-(c) show the VMC results and the es. (a)-(d) correspond to the on-site Hubbard model, (b)-(e) the long range Hubbard model and (c)-(f) hopping. (a) (b) Figure 7. (a) Comparison of experimental energy gaps and reconstructed energy gaps of eigenstates, by solving the extended Hubbard model or Parisier-Pople-Parr (PPP) model obtained using VMC, DMC and extrapolated parameters. (b) Comparison of experimental energy gaps for different model Hamiltonians. All experimental values and associated errorbars are taken from Bursill et al. 60, who used these values to fit to a PPP model with density-density interactions of the Ohno form (28). Changlani, Zheng, Wagner. J. Chem. Phys. 143, (2015)

6 Telluride School on Stochastic Approaches to Electronic Structure Calculations Hands on every day Time for walks (quantum and classical) Code QMC algorithms: Variational Diffusion Path integral Auxiliary field Full Configuration interaction telluridescience.org

7 Starting up Find 2-3 friends. You are now a group git clone qmc_tutorial modify runqmc.py to point to your QWalk installation. if you want to run natively on Linux or Mac: git clone qwalk cd qwalk/src make install sudo pip3 install seaborn matplotlib numpy pandas

8 Approximate wave functions 1 2 For an up and down electron: a( 1 (r 1 ) 2 (r 2 )+ 2 (r 1 ) 1 (r 2 )) +b( 1 (r 1 ) 1 (r 2 )+ 2 (r 1 ) 2 (r 2 )) Position (Bohr) Limits: a=b: non-interacting b=0: no double occupancy

9 Non-interacting case a( 1 (r 1 ) 2 (r 2 )+ 2 (r 1 ) 1 (r 2 )) +b( 1 (r 1 ) 1 (r 2 )+ 2 (r 1 ) 2 (r 2 )) a=b. This is the Hartree-Fock (RHF) result. The probability of finding two electrons on a given site is exactly 0.25

10 Why Jastrow? 1 2 When the densities overlap a lot, then the electrons avoid each other on a much smaller scale Position (Bohr) The Jastrow factor includes that kind of correlation.

11 Measurements Total energy (Hartrees: ~27 ev) Double occupancy of atomic orbitals (2-RDM) Electron-electron radial distribution function

12 Wave function ansatz Restricted Hartree-Fock Multiple Slater determinants Multiple Slater-Jastrow diffusion Monte Carlo Comment No electron correlation at all Reduces double occupancies of orbitals Electrons also avoid one another at small distances Exact or nearly exact in this situation (everything else!) File runqmc.py scan.py plot.py, plot_gr.py, plot_double.py hubbard.py Purpose Interface to QWalk Run an ensemble of jobs - >.pickles Plot scripts Find best Hubbard model

13 Define the Hamiltonian Define the Hamiltonian (qwsinglet1.0.hf) SYSTEM { MOLECULE NSPIN { 1 1 ATOM { H 1 COOR ATOM { H 1 COOR One up, one down Position in Bohr Atomic charge

14 Multiple slater determinant TRIALFUNC { SLATER ORBITALS { CUTOFF_MO MAGNIFY 1 NMO 2 ORBFILE qwsinglet1.44.orb INCLUDE qwsinglet1.44.basis CENTERS { USEATOMS STATES { CSF { CSF{ OPTIMIZE_DET 1, 2 a( 1 (r 1 ) 2 (r 2 )+ 2 (r 1 ) 1 (r 2 )) +b( 1 (r 1 ) 1 (r 2 )+ 2 (r 1 ) 2 (r 2 ))

15 Procedure: evaluate wave function method { vmc nstep 1000 average { gr average { tbdm_basis npoints 1 ORBITALS { CUTOFF_MO MAGNIFY 1 NMO 2 ORBFILE qwsinglet1.44.orb INCLUDE qwsinglet1.44.basis CENTERS { USEATOMS Evaluate the wave function and average the radial distribution function and two body density matrix on the atomic basis.

16 TRIALFUNC { slater-jastrow wf1 { SLATER ORBITALS { CUTOFF_MO MAGNIFY 1 NMO 2 ORBFILE qwsinglet1.44.orb INCLUDE qwsinglet1.44.basis CENTERS { USEATOMS STATES { CSF { CSF{ OPTIMIZE_DET wf2 { JASTROW2 GROUP { TWOBODY_SPIN { FREEZE LIKE_COEFFICIENTS { UNLIKE_COEFFICIENTS { Define Jastrow factor exp 4 X i, EEBASIS { EE CUTOFF_CUSP GAMMA 24 CUSP 1 CUTOFF 7.5 EEBASIS { EE CUTOFF_CUSP GAMMA 24 CUSP 1 CUTOFF 7.5 GROUP { ONEBODY { COEFFICIENTS { H TWOBODY { COEFFICIENTS { EIBASIS { H POLYPADE RCUT 7.5 BETA0-0.4 NFUNC 3 EEBASIS { EE POLYPADE RCUT 7.5 BETA NFUNC 3 2 (onebody) X k c k a k (r i )+ X i,j (twobody) X Define a s and b s k 3 c k b k (r ij ) 5

17 Optimize: Optimize and then method { LINEAR total_nstep 250 method { vmc nstep 1000 average { gr average { tbdm_basis npoints 1 ORBITALS { CUTOFF_MO MAGNIFY 1 NMO 2 ORBFILE qwsinglet1.44.orb INCLUDE qwsinglet1.44.basis CENTERS { USEATOMS

18 0.4 Energy (Hartree) Slater,singlet Multiple Slater,singlet Multiple Slater-Jastrow,singlet DMC,singlet Slater,triplet Multiple Slater,triplet Multiple Slater-Jastrow,triplet DMC,triplet r (Bohr)

19 Discussion questions Where do the singlet and triplet state become degenerate for the different wave function ansatz? Why the differences? Where is the multiple slater without Jastrow good, and where is it poor? Why does DMC have the highest double occupancy in the singlet state around r=3 Bohr?

20 U (Hartree) dmc multislat sj slat Why does RHF always result in zero effective U? 0.6 t (Hartree) dmc multislat sj slat Why does multiple Slater result in larger U values than when short-range correlation is included? U/t Bond length (Bohr) dmc multislat sj slat