logos Selected Configuration Interaction for QMC E. Giner1, A. Scemama1, M. Caffarel1, T. Applencourt1 1Laboratoire de Chimie et Physique Quantiques / IRSAMC, Toulouse, France PAMO 2014 E. Giner, A. Scemama, M. Caffarel, T. Applencourt CIPSI, perturbation theory and DMC 

logos Outline 1 Basics of FN-DMC 2 Common practice of QMC 3 CIPSI algorithm 4 FN-DMC with CIPSI |ψT E. Giner, A. Scemama, M. Caffarel, T. Applencourt CIPSI, perturbation theory and DMC 

logos Fixed Node Diffusion Monte Carlo (FN-DMC) in a few words Stochastic solution of the Schrödinger equation : H|φFN 0 = EFN 0 |φFN 0 Input of FN-DMC : Trial wave function : |ψT (ex : |HF , |DFT ,|CASSCF ) 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 Excellent total energies (∼ 90% or correlation energy or more ) Quality of energy differences may depend on nodes E. Giner, A. Scemama, M. Caffarel, T. Applencourt 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, T. Applencourt CIPSI, perturbation theory and DMC 

logos How to improve the nodes of |ψT in practice Improving nodes by large-scale stochastic optimization of the parameters of |ψT Two main aspects : 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, T. Applencourt 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 The path we chose : Avoid stochastic optimization : analytically integrable functions → Configuration Interaction Compact trial wave functions → Selected CI E. Giner, A. Scemama, M. Caffarel, T. Applencourt 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, T. Applencourt 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, T. Applencourt 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 = ( Dj|H|ψ )2 EVar− Dj|H|Dj CIPSI ENERGY ≡ EVar + EPT2 ⇒ approximation of the FCI energy E. Giner, A. Scemama, M. Caffarel, T. Applencourt CIPSI, perturbation theory and DMC 

logos Applications done Oxygen atom : ψDMC ∼ 105 Dets Series of 3d metal atoms : ψDMC ∼ 104; 5 104 Dets F2 molecule : ψDMC ∼ 104 Dets CuCl2 molecule : ψCIPSI ∼ 106 Dets Cu − β -amyloid molecule : ψCIPSI ∼ 106 Dets , ψDMC ∼ 103 Dets E. Giner, A. Scemama, M. Caffarel, T. Applencourt 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, T. Applencourt 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, T. Applencourt 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, T. Applencourt 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, T. Applencourt 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, T. Applencourt 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 E. Giner, A. Scemama, M. Caffarel, T. Applencourt CIPSI, perturbation theory and DMC 

logos CIPSI wf for DMC : numerical results, the Oxygen ground state 100 101 102 103 104 105 Number of determinants −75.065 −75.060 −75.055 −75.050 −75.045 −75.040 Energy (a.u.) cc-pVDZ/DMC cc-pVTZ/DMC cc-pVQZ/DMC E. Giner, A. Scemama, M. Caffarel, T. Applencourt 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, T. Applencourt CIPSI, perturbation theory and DMC 

logos CIPSI+DMC wf for DMC : numerical results for transition metals Atom Literaturea HF/DMC CIPSI/DMC Ndet Sc -760.5288(6) -760.5265(13) -760.5718(16) 11389 Ti -849.2492(7) -849.2405(14) -849.2841(19) 14054 V -943.7882(6) -943.7843(13) -943.8234(16) 12441 Cr -1044.3289(6) -1044.3292(16) -1044.3603(17) 10630 Mn -1150.8897(7) -1150.8880(17) -1150.9158(20) 11688 Fe -1263.5607(6) -1263.5589(19) -1263.5868(21) 13171 Co -1382.6216(8) -1382.6177(21) -1382.6377(24) 15949 Ni -1508.1743(7) -1508.1645(23) -1508.1901(25) 15710 Cu -1640.4266(7) -1640.4271(26) -1640.4328(29) 48347 Zn -1779.3425(8) -1779.3371(26) -1779.3386(31) 44206 a Buendía, E. et al, A. Chem. Phys. Lett., 559(12 - 17), (2013) Atom EA Exp. (eV) EA DMC-HF (eV) Gain in DMC-CIPSI energy (eV) Sc 0.19 -0.07 1.35 Cr 0.68 0.33 0.81 E. Giner, A. Scemama, M. Caffarel, T. Applencourt CIPSI, perturbation theory and DMC 

logos Ionization potential E. Giner, A. Scemama, M. Caffarel, T. Applencourt CIPSI, perturbation theory and DMC 

logos Cu-β-amyloid CIPSI/DMC T rial wave function T otal energy ( a.u.) cc-pV T Z basis ( 1929 basis function) H F nodes -3612.1819( 38) B 3LY P nodes -3612.1401( 32) cc-pV D Z basis ( 794 basis functions) H F nodes -3611.4304( 64) C IP SI( 100 dets) nodes -3611.6839( 83) C IP SI( 500 dets) nodes -3612.0238( 68) C IP SI( 1000 dets) nodes -3612.163( 12) -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 227 e− in 276 MOs E. Giner, A. Scemama, M. Caffarel, T. Applencourt CIPSI, perturbation theory and DMC 

logos Summary of the numerical results 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 E. Giner, A. Scemama, M. Caffarel, T. Applencourt 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, T. Applencourt CIPSI, perturbation theory and DMC 

logos PT2 constant mode 1.0 1.5 2.0 2.5 3.0 3.5 4.0 Inter-nuclear distance(˚ A) −199.12 −199.04 −198.96 −198.88 −198.80 Energy (a.u.) Variational energy CIPSI energy 1000 1050 1100 1150 1200 Number of determinants Number of dets E. Giner, A. Scemama, M. Caffarel, T. Applencourt CIPSI, perturbation theory and DMC 

logos DMC with CIPSI WF at constant PT2 1.0 1.5 2.0 2.5 3.0 3.5 4.0 z (a.u.) −199.50 −199.48 −199.46 −199.44 −199.42 Total energy (a.u.) 0.2 0.1 0.05 0.02 E. Giner, A. Scemama, M. Caffarel, T. Applencourt CIPSI, perturbation theory and DMC 

logos Conclusion Automatic construction of |ψT Systematic improvment of the nodes Possible for large systems Ref : E. Giner, A. Scemama, and M. Caffarel, Can. J. Chem. 91, 879 (2013) M. Caffarel, E. Giner, A. Scemama, and A. Ramirez-Solis, Spin density distribution in open shell transistion metal system : A comparative post-HF, DFT and QMC study of the CuCl2 molecule ArXiv : 1405.4082 (2014) E. Giner, A. Scemama, and M. Caffarel FN-DMC potential energy curve of the fluorine molecule F2 using selected configuration interactions trial wave functions preprint July 2014 E. Giner, A. Scemama, M. Caffarel, T. Applencourt CIPSI, perturbation theory and DMC 

logos Work in progress and perspectives Current work : Implementation of even more compact wf (CISD+SC2) Use of local orbitals PhD redaction ... Perspectives : Dressed Hamiltonians matrix Excited states (avoid crossing of Retinal molecule) Diffusion of our CI+QMC code E. Giner, A. Scemama, M. Caffarel, T. Applencourt CIPSI, perturbation theory and DMC 

logos Thank you for your attention ! ! E. Giner, A. Scemama, M. Caffarel, T. Applencourt CIPSI, perturbation theory and DMC