logos Selected Configuration Interaction for QMC E. Giner1, A.Scemama1, M. Caffarel1 1Laboratoire de Chimie et Physique Quantiques / IRSAMC, Toulouse, France 28 Décembre 2013 E. Giner, A.Scemama, M. Caffarel CIPSI, perturbation theory and DMC 

logos Fixed Node Diffusion Monte Carlo (FN-DMC) in a few words Input of QMC : Trial wave function : |ψT > (ex : |HF >, |DFT > ,|CASSCF >) Stochastic solution of the Schrödinger equation : H|φFN 0 >= EFN 0 |φFN 0 > Fixed Node approximation : imposition of the nodes of |ψT > φFN 0 (r1, r2, ..., rN) = 0 when ψT(r1, r2, ..., rN) = 0 EFN 0 is variational : EFN 0 ≥ Eexact 0 E. Giner, A.Scemama, M. Caffarel CIPSI, perturbation theory and DMC 

logos How to improve the nodes of |ψT > in practice First : choosing a trial wave function ψT Standard choice : ψT = exp(J(r1, r2, ..., rN)) n i=1 ci|Di > (Jastrow Slater) Other choices : VB Jastrow (with a Valence Bond determinantal part) JAGP (use of geminals) Pfaffian use of backflow etc ... E. Giner, A.Scemama, M. Caffarel CIPSI, perturbation theory and DMC 

logos How to improve the nodes of |ψT > in practice Improving nodes by large-scale optimization of the parameters of |ψT > Two main problems : Keeping a coherent description for each points of the potential energy surface Optimizing many (thousands) of linear and non linear parameters within a MC frame work E. Giner, A.Scemama, M. Caffarel CIPSI, perturbation theory and DMC 

logos The path we chose Our main objectives : Coherence along a reaction path Automatic construction of the trial wave functions (usable by non experts) Control of the nodal quality It has been shown that near FCI wf have good nodes (Monari and all) The path we chose : Avoid stochastic optimization : analytically integrable functions → Configuration Interaction Compact trial wave functions → Selected CI E. Giner, A.Scemama, M. Caffarel CIPSI, perturbation theory and DMC 

logos The CIPSI-like ideas (1) Some remarks about a FCI space : Exponentially large size but very few important reference determinants Selecting by class of excitation (CISD,CISDTQ etc ...) introduces too many non contributing determinants Why not try to select them by their energy contribution ? → Selection by perturbation Ref : B. Huron, J. P. Malrieu, and P. Rancurel J. Chem. Phys. 58, 5745 (1973) E. Giner, A.Scemama, M. Caffarel CIPSI, perturbation theory and DMC 

logos The CIPSI-like algorithm (1) Starting with : |ψ(0) >= N(0) i=1 ci|Di > do k = 0, Niterations 1 look at all the determinants |Dj > for which < Dj|H|ψ(k) >= 0 (connected) 2 for each |Dj > compute the perturbative energy contribution j 3 Sort all the |Dj > by energy contribution 4 Select the n most important ones 5 Diagonalize H in the new set of determinants : N(k+1) = N(k) + n 6 you have a new reference wf : |ψ(k+1) >= N(k+1) i=1 ci|Di > end do E. Giner, A.Scemama, M. Caffarel CIPSI, perturbation theory and DMC 

logos The perturbative and variational energy CIPSI calculation : wave function |ψ >= N i=1 ci|Di > Two energetic quantity associated to |ψ > Variational energy : EVar = <ψ|H|ψ> <ψ|ψ> Second order Perturbative energy : EPT2 = j/ ∈|ψ> j j = 2 EVar− CIPSI ENERGY ≡ EVar + EPT2 approximation of the FCI energy E. Giner, A.Scemama, M. Caffarel CIPSI, perturbation theory and DMC 

logos Numerical results : the Oxygen ground state in cc-pVDZ basis set -74.94 -74.92 -74.9 -74.88 -74.86 -74.84 -74.82 -74.8 -74.78 0 100 200 300 400 500 600 700 800 900 1000 Energy/Hartree Number of determinants Convergence of the Energy for the oxygen atom (cc-pVDZ) E(Variational) FCI DZ E(CIPSI) E. Giner, A.Scemama, M. Caffarel CIPSI, perturbation theory and DMC 

logos Summary of the numerical results : CIPSI calculations on the Oxygen ground state in cc-pVXZ -75.05 -75 -74.95 -74.9 -74.85 -74.8 -74.75 1 10 100 1000 10000 100000 Energy/Hartree Number of determinants in the CIPSI wf E Var DZ E(CIPSI) DZ E FCI DZ E. Giner, A.Scemama, M. Caffarel CIPSI, perturbation theory and DMC 

logos Summary of the numerical results : CIPSI calculations on the Oxygen ground state in cc-pVXZ -75.05 -75 -74.95 -74.9 -74.85 -74.8 -74.75 1 10 100 1000 10000 100000 Energy/Hartree Number of determinants in the CIPSI wf E Var DZ E(CIPSI) DZ E FCI DZ E var TZ E(CIPSI) TZ E FCI TZ E. Giner, A.Scemama, M. Caffarel CIPSI, perturbation theory and DMC 

logos Summary of the numerical results : CIPSI calculations on the Oxygen ground state in cc-pVXZ -75.05 -75 -74.95 -74.9 -74.85 -74.8 -74.75 1 10 100 1000 10000 100000 Energy/Hartree Number of determinants in the CIPSI wf E Var DZ E(CIPSI) DZ E FCI DZ E var TZ E(CIPSI) TZ E FCI TZ E var QZ E(CIPSI) QZ E. Giner, A.Scemama, M. Caffarel CIPSI, perturbation theory and DMC 

logos Summary of the numerical results : CIPSI calculations on the Oxygen ground state in cc-pVXZ -75.05 -75 -74.95 -74.9 -74.85 -74.8 -74.75 1 10 100 1000 10000 100000 Energy/Hartree Number of determinants in the CIPSI wf E Var DZ E(CIPSI) DZ E FCI DZ E var TZ E(CIPSI) TZ E FCI TZ E var QZ E(CIPSI) QZ E var 5Z E(CIPSI) 5Z E. Giner, A.Scemama, M. Caffarel CIPSI, perturbation theory and DMC 

logos Introducing our FN-DMC with CIPSI wf To make a FN-DMC calculation we do 1 Run a CIPSI calculation stopping at a given number of determinants |ψT >= |CIPSI > 2 Run a FN-DMC calculation without any stochastic optimization Emmanuel Giner, Anthony Scemama, Michel Caffarel. Canadian Journal of Chemistry , 1-26, 2013 E. Giner, A.Scemama, M. Caffarel CIPSI, perturbation theory and DMC 

logos CIPSI wf for DMC : numerical results, the Oxygen ground state -75.065 -75.06 -75.055 -75.05 -75.045 -75.04 1 10 100 1000 10000 100000 DMC energy (a.u.) Number of determinants in the reference wave function cc-pVDZ cc-pVTZ cc-pVQZ E. Giner, A.Scemama, M. Caffarel CIPSI, perturbation theory and DMC 

logos Numerical results : total energies Hartree-Fock cc-pV5Z CIPSI cc-pV5Z Full CI aug-cc-pV5Z CIPSI cc-pVDZ CIPSI cc-pVTZ CIPSI cc-pVQZ CIPSI cc-pCVQZ Exact non-relativistic FN-DMC -74.80 -74.85 -74.90 -74.95 -75.00 -75.05 -75.10 } } CI Methods -75.050 -75.070 -75.060 i-FCIQMC : Booth, G. H. ; Alavi, A. J. Chem. Phys. 2010, 132, 174104 E. Giner, A.Scemama, M. Caffarel CIPSI, perturbation theory and DMC 

logos The Copper atom doublet ground state in cc-pVDZ : CIPSI study -1639.6 -1639.55 -1639.5 -1639.45 -1639.4 -1639.35 -1639.3 0 50000 100000 150000 200000 Total energy (a.u.) Number of determinants Variational energy Variational energy+PT2 E. Giner, A.Scemama, M. Caffarel CIPSI, perturbation theory and DMC 

logos The Copper atom doublet ground state in cc-pVDZ : CIPSI + DMC -1640.45 -1640.44 -1640.43 -1640.42 -1640.41 -1640.4 -1640.39 -1640.38 0 2000 4000 6000 8000 10000 Total energy (a.u.) Number of determinants FN-DMC energy E. Giner, A.Scemama, M. Caffarel CIPSI, perturbation theory and DMC 

logos Summary of the numerical results for atoms CIPSI : Good convergence of the variational energy Even better convergence of the perturbative energy Good approximation for FCI energies (even for large FCI space) CIPSI + DMC : Systematic improvement of the nodes Compact enough wave functions (≈ 1000 dets) Extremely good total energies No stochastic optimization Question : energy differences ? Coherence along the reaction path ? E. Giner, A.Scemama, M. Caffarel CIPSI, perturbation theory and DMC 

logos A CIPSI study of F2 in the cc-pVDZ : the variational energy -199.1 -199 -198.9 -198.8 -198.7 -198.6 -198.5 -198.4 1 1.5 2 2.5 3 3.5 4 Energy/Hartree inter atomic distance (Angstrom) Convergence of the CIPSI variational energy for F2 in cc-pVDZ CAS 100 dets 500 dets 1 K dets 5 K dets 10 K dets 50 K dets E. Giner, A.Scemama, M. Caffarel CIPSI, perturbation theory and DMC 

logos A CIPSI study of F2 in the cc-pVDZ : the perturbative energy -199.15 -199.1 -199.05 -199 -198.95 -198.9 -198.85 -198.8 1 1.5 2 2.5 3 3.5 4 Energy/Hartree inter atomic distance (Angstrom) Convergence of the CIPSI perturbative energy for F2 in cc-pVDZ 100 dets + PT2 500 dets + PT2 1 K dets + PT2 5 K dets + PT2 10 K dets + PT2 50 K dets + PT2 E. Giner, A.Scemama, M. Caffarel CIPSI, perturbation theory and DMC 

logos DMC with CIPSI WF -199.5 -199.48 -199.46 -199.44 -199.42 -199.4 -199.38 1.5 2 2.5 3 3.5 4 4.5 Energy/Hartree Interatomic distance/Angs. Fixed-node Energy CAS 100 Dets 500 Dets E. Giner, A.Scemama, M. Caffarel CIPSI, perturbation theory and DMC 

logos DMC with CIPSI WF -199.5 -199.48 -199.46 -199.44 -199.42 -199.4 -199.38 1.5 2 2.5 3 3.5 4 4.5 Energy/Hartree Interatomic distance/Angs. Fixed-node Energy CAS Dets 500 Dets 1000 Dets 5000 Dets 10000 Dets E. Giner, A.Scemama, M. Caffarel CIPSI, perturbation theory and DMC 

logos Numerical results for F2 : atomization energies and equilibrium distances Fixed-node DMC R ∆E(millihartree) k CAS nodes 1.431 50.5(8) 1.052 CIPSI-100-dets nodes 1.428 59.5(2) 1.054 CIPSI-500-dets nodes 1.430 55.0(3) 0.911 CIPSI-1000-dets nodes 1.423 57.4(2) 0.995 CIPSI-5000-dets nodes 1.424 56.3(2) 0.944 CIPSI-10000-dets nodes 1.428 55.3(3) 1.117 DZ FCI 1.461 45.0 0.777 Exact 1.412 62.0 1.121 TABLE: DZ basis set : FN-DMC with CIPSI as a function of the number of dets E. Giner, A.Scemama, M. Caffarel CIPSI, perturbation theory and DMC 

logos Stopping criterions for CIPSI calculations Need a criterion to have more coherent wave functions in CIPSI ... Remarks about perturbation and CIPSI calculation : Each size N of |ψ > corresponds to a precise EPT2 The bigger is |ψ > the smaller is EPT2 ECIPSI = EVar + EPT2 is a good approximation of the FCI energy −→ EPT2 is a good mesure of what is missing in |ψ > Impose a constant EPT2 E. Giner, A.Scemama, M. Caffarel CIPSI, perturbation theory and DMC 

logos PT2 constant mode -199.15 -199.1 -199.05 -199 -198.95 -198.9 -198.85 -198.8 -198.75 1 1.5 2 2.5 3 3.5 4 Energy/Hartree Inter atomic distance CISPI with constant perturbation : -0.05 E VAR E VAR+E PT2 E. Giner, A.Scemama, M. Caffarel CIPSI, perturbation theory and DMC 

logos PT2 constant mode 850 900 950 1000 1050 1100 1150 1 1.5 2 2.5 3 3.5 4 Num of Dets in the wf inter-atomic distance Number of determinants in PT2 cst mode for E PT2 = -0.05 E. Giner, A.Scemama, M. Caffarel CIPSI, perturbation theory and DMC 

logos CIPSI calculations with PT2 constant : variational energy of F2 -199.1 -199.05 -199 -198.95 -198.9 -198.85 -198.8 -198.75 -198.7 -198.65 -198.6 1 1.5 2 2.5 3 3.5 4 Energy/Hartree Inter atomic distance Variational energies of a CIPSI Wf in the PT2 constant mode PT2 = -0.2 PT2 = -0.1 PT2 = -0.05 PT2 = -0.02 PT2 =-0.01 PT2 = -0.008 E. Giner, A.Scemama, M. Caffarel CIPSI, perturbation theory and DMC 

logos Numerical results for F2 : atomization energies and equilibrium distances Fixed-node DMC R ∆E(millihartree) k CAS nodes 1.431 50.5± 1.052 PT2= 0.2 (∼ 160 dets) 1.442 50.1(3) 0.839 PT2= 0.1 (∼ 500 dets) 1.433 56.7(3) 1.190 PT2= 0.05 (∼ 1100 dets) 1.431 59.7(7) 1.131 PT2= 0.02 (∼ 5000 dets) 1.430 56.1(6) 1.129 PT2= 0.012 (∼ 10000 dets) 1.430 55.9(7) 1.129 Exact 1.412 62.0 1.121 TABLE: DZ basis set : FN-DMC with CIPSI as a function of constant PT2 E. Giner, A.Scemama, M. Caffarel CIPSI, perturbation theory and DMC 

logos Fit of the PT2 constant gap with FN-DMC 50 52 54 56 58 60 62 64 0 0.05 0.1 0.15 0.2 Bonding Energy/mHartree - EPT2 Hartree FN-DMC Bonding Energy with PT2 stopping criterion Exact FN-DMC E. Giner, A.Scemama, M. Caffarel CIPSI, perturbation theory and DMC 

logos Work in progress and perspectives Current work : Multi Reference Size-Consistant Spin-free PT2 (with Jean Paul Malrieu) Use of local orbitals Study of F2 in a bigger basis set Perspectives : Dressed Hamiltonians matrix Excited states (avoid crossing of Retinal molecule) Diffusion of our CI code E. Giner, A.Scemama, M. Caffarel CIPSI, perturbation theory and DMC 

logos Thank you for your attention ! -3612.3 -3612.2 -3612.1 -3612 -3611.9 -3611.8 -3611.7 -3611.6 -3611.5 -3611.4 0 500 1000 1500 2000 Total energy (a.u.) Number of determinants 29 e− in 49 MOs 227 e− in 276 MOs E. Giner, A.Scemama, M. Caffarel CIPSI, perturbation theory and DMC 

logos FN-DMC for the peptide + Cu -3612.3 -3612.2 -3612.1 -3612 -3611.9 -3611.8 -3611.7 -3611.6 -3611.5 -3611.4 0 500 1000 1500 2000 Total energy (a.u.) Number of determinants E. Giner, A.Scemama, M. Caffarel CIPSI, perturbation theory and DMC 

logos FN-DMC Cu -> Cu2+ E. Giner, A.Scemama, M. Caffarel CIPSI, perturbation theory and DMC 

logos PT2 constant mode -199.15 -199.1 -199.05 -199 -198.95 -198.9 -198.85 -198.8 1 1.5 2 2.5 3 3.5 4 Energy/Hartree Inter atomic distance Perturbative energies of a CIPSI Wf in the PT2 constant mode PT2 = -0.2 PT2 = -0.1 PT2 = -0.05 PT2 = -0.02 PT2 =-0.01 PT2 = -0.008 E. Giner, A.Scemama, M. Caffarel CIPSI, perturbation theory and DMC 

logos PT2 constant mode -199.5 -199.48 -199.46 -199.44 -199.42 -199.4 -199.38 1.5 2 2.5 3 3.5 4 4.5 Energy/Hartree Interatomic distance/Angs. Fixed-node Energy PT2 = -0.02 PT2 = -0.05 PT2 = -0.1 PT2 = -0.2 E. Giner, A.Scemama, M. Caffarel CIPSI, perturbation theory and DMC 

logos CIPSI wf for DMC : numerical results (2) Basis set Ndets E0 (FN-DMC) Correlation Energy (%) cc-pVDZ 1(HF) -75.0418(5) 90.1(2) cc-pVDZ 5 000 -75.0519(4) 94.0(2) cc-pVTZ 1(HF) -75.0457(4) 91.6(2) cc-pVTZ 1 000 -75.0591(5) 96.8(2) cc-pVQZ 20 000 -75.0642(2) 98.8(1) cc-pCVQZ 100 000 -75.0658(1) 99.4(1) Other works FCIQMC -75.03749(6) 88.40(2) FN-DMC -75.0654(1) 99.2(1) r12-MR-ACPF -75.066960 99.83 Exact NR -75.0674 100.0 TABLE: FN-DMC energies and other methods for the 3P ground-state of the oxygen atom. E. Giner, A.Scemama, M. Caffarel CIPSI, perturbation theory and DMC 

logos Method R ∆E(millihartree) k CAS 1.531 22.1 0.430 CIPSI 100 dets 1.445 57.4 0.875 100 dets + PT2 1.451 61.4 0.812 500 dets 1.403 55.1 1.352 500 dets + PT2 1.450 52.1 0.845 1k dets 1.442 48.2 0.927 1k dets + PT2 1.458 49.9 0.800 5k dets 1.465 40.1 0.733 5k dets + PT2 1.460 45.7 0.784 10k dets 1.464 41.9 0.748 10k dets + PT2 1.460 45.6 0.782 50k dets 1.463 43.8 0.765 50k dets + PT2 1.462 45.3 0.774 DZ FCI ? ? 43.8 ? ? Exact 1.412 62.0 1.121 TABLE: DZ CASSCF orbitals E. Giner, A.Scemama, M. Caffarel CIPSI, perturbation theory and DMC 

logos DMC with CIPSI WF -199.485 -199.48 -199.475 -199.47 -199.465 -199.46 -199.455 -199.45 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 Energy/Hartree Number of determinants Convergence of the DMC energies with a CIPSI WF : near equilibrium R = 1.45 DMC Energy E. Giner, A.Scemama, M. Caffarel CIPSI, perturbation theory and DMC 

logos DMC with CIPSI WF -199.43 -199.425 -199.42 -199.415 -199.41 -199.405 -199.4 -199.395 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 Energy/Hartree Number of determinants Convergence of the DMC energies with a CIPSI WF : near dissociation R = 4 DMC Energy E. Giner, A.Scemama, M. Caffarel CIPSI, perturbation theory and DMC 

logos Comparaison with CCSD energy -199.15 -199.1 -199.05 -199 -198.95 -198.9 -198.85 -198.8 1 1.5 2 2.5 3 3.5 4 Energy/Hartree inter atomic distance (Angstrom) Convergence of the CIPSI perturbative energy for F2 in cc-pVDZ comparaison with CISD and CCSD 100 dets + PT2 500 dets + PT2 1 K dets + PT2 5 K dets + PT2 10 K dets + PT2 50 K dets + PT2 CCSD E. Giner, A.Scemama, M. Caffarel CIPSI, perturbation theory and DMC 

logos Zoom on the crucial points -199.11 -199.1 -199.09 -199.08 -199.07 -199.06 -199.05 -199.04 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 Energy/Hartree inter atomic distance (Angstrom) Convergence of the CIPSI perturbative energy for F2 in cc-pVDZ 100 dets + PT2 500 dets + PT2 1 K dets + PT2 5 K dets + PT2 10 K dets + PT2 50 K dets + PT2 E. Giner, A.Scemama, M. Caffarel CIPSI, perturbation theory and DMC 

logos Zoom on the crucial points -199.07 -199.065 -199.06 -199.055 -199.05 -199.045 -199.04 2 2.5 3 3.5 4 Energy/Hartree inter atomic distance (Angstrom) Convergence of the CIPSI perturbative energy for F2 in cc-pVDZ 100 dets + PT2 500 dets + PT2 1 K dets + PT2 5 K dets + PT2 10 K dets + PT2 50 K dets + PT2 FCI F * 2 E. Giner, A.Scemama, M. Caffarel CIPSI, perturbation theory and DMC