PhD defense

This is my PhD dissertation in octobre 2014.
This PhD was done at the Laboratoire de Chimie et Physique
Quantique in Toulouse, under the supervision of Michel Caffarel
and Anthony Scemama.
The members of the jury were Julien Toulouse, Ali Alavi,
Emmanuel Fromager, Jean Paul Malrieu and my PhD supervisors.

Transcript

1. logos Introduction WF methods FN-DMC with CIPSI trial wave functions Coupling Configuration Interaction and quantum Monte Carlo methods: The best of both worlds E. Giner1 1Laboratoire de Chimie et Physique Quantiques / IRSAMC, Toulouse, France Monday, 20th Octobre 2014 E. Giner Coupling CI and FN-DMC
2. logos Introduction WF methods FN-DMC with CIPSI trial wave functions FN-DMC in a few words FN-DMC = Solve the Schrödinger equation stochastically Solve it in real space ⇔ infinite basis set Needs a Trial wave function |ψT The Schrödinger equation is solved by imposing the nodes (zeros) of |ψT : ψT (r1 , r2 , ..., rN ) = 0 ⇔ φFN−DMC 0 (r1 , r2 , ..., rN ) = 0 Only approximation in FN-DMC : the nodes of |ψT E. Giner Coupling CI and FN-DMC
3. logos Introduction WF methods FN-DMC with CIPSI trial wave functions Typical Trial wave function Jastrow Slater type |ψT ψT (r1 , r2 , ..., rN ) = e i,j J(ri,rj,rij) Ndet k ck |Dk Optimization of all the parameters of |ψT : Jastrow parameters : J(ri , rj , rij ) Slater determinants coefficient : ck Molecular orbitals : φl (r) Optimization in practice allows to : Decrease the statistical error (not critical) Improve the nodes : challenging E. Giner Coupling CI and FN-DMC
4. logos Introduction WF methods FN-DMC with CIPSI trial wave functions Alternative Trial wave function ? CI expansions allow simple and systematic improvement of wf |CI = k ck |Dk Optimization of ck : diagonalize the CI matrix. Question : can CI wf be used in practice for FN-DMC ? Do they lead to improved nodes ? E. Giner Coupling CI and FN-DMC
5. logos Introduction WF methods FN-DMC with CIPSI trial wave functions Practical bottleneck CI expansions include a large number of determinants (> 106) In practice too large for FN-DMC ! Only a small number of determinants are relevant in the FCI How to select them in the best way : CIPSI CIPSI : quasi FCI with tiny fraction of FCI space E. Giner Coupling CI and FN-DMC
6. logos Introduction WF methods FN-DMC with CIPSI trial wave functions 1 WF methods : CIPSI The FCI method The Perturbatively selected CI algorithm (CIPSI) Applications of CIPSI 2 Use of CIPSI in FN-DMC E. Giner Coupling CI and FN-DMC
7. logos Introduction WF methods FN-DMC with CIPSI trial wave functions The FCI method Given a finite molecular orbital basis set : Set of all possible Slater determinants : {|Dk , k = 1, NFCI } FCI = Diagonalization of H in {|Dk } : H|ψFCI = EFCI|ψFCI |ψFCI = NFCI Dk=1 cFCI Dk |Dk Main problem : size of the Hilbert space grows factorially ! E. Giner Coupling CI and FN-DMC
8. logos Introduction WF methods FN-DMC with CIPSI trial wave functions The FCI method Given a finite molecular orbital basis set : Set of all possible Slater determinants : {|Dk , k = 1, NFCI } FCI = Diagonalization of H in {|Dk } : H|ψFCI = EFCI|ψFCI |ψFCI = NFCI Dk=1 cFCI Dk |Dk Main problem : size of the Hilbert space grows factorially ! E. Giner Coupling CI and FN-DMC
9. logos Introduction WF methods FN-DMC with CIPSI trial wave functions The FCI method Two main diagonalization approaches : Deterministic : Davidson type method typical FCI space ≈ 109 Stochastic (recently) : diffusion process in FCI space. FCIQMC introduced by Alavi and coll. : treat much bigger FCI space All WF methods are approximations of the FCI E. Giner Coupling CI and FN-DMC
10. logos Introduction WF methods FN-DMC with CIPSI trial wave functions The FCI method Two main diagonalization approaches : Deterministic : Davidson type method typical FCI space ≈ 109 Stochastic (recently) : diffusion process in FCI space. FCIQMC introduced by Alavi and coll. : treat much bigger FCI space All WF methods are approximations of the FCI E. Giner Coupling CI and FN-DMC
11. logos Introduction WF methods FN-DMC with CIPSI trial wave functions Selection schemes Approximation of FCI wf ⇔ selection of a subspace of FCI space. Two big classes of selection : Selection "a priori" (excitation class and/or active space) CISD, CCSD, CCSD(T), CASSCF, MRCI, CAS-PT2 ... Selection without "a priori" CIPSI, FCI-QMC, iFCI-QMC ... ⇒ converge to the FCI E. Giner Coupling CI and FN-DMC
12. logos Introduction WF methods FN-DMC with CIPSI trial wave functions Selection schemes Approximation of FCI wf ⇔ selection of a subspace of FCI space. Two big classes of selection : Selection "a priori" (excitation class and/or active space) CISD, CCSD, CCSD(T), CASSCF, MRCI, CAS-PT2 ... Selection without "a priori" CIPSI, FCI-QMC, iFCI-QMC ... ⇒ converge to the FCI E. Giner Coupling CI and FN-DMC
13. logos Introduction WF methods FN-DMC with CIPSI trial wave functions Selection schemes Approximation of FCI wf ⇔ selection of a subspace of FCI space. Two big classes of selection : Selection "a priori" (excitation class and/or active space) CISD, CCSD, CCSD(T), CASSCF, MRCI, CAS-PT2 ... Selection without "a priori" CIPSI, FCI-QMC, iFCI-QMC ... ⇒ converge to the FCI E. Giner Coupling CI and FN-DMC
14. logos Introduction WF methods FN-DMC with CIPSI trial wave functions The CIPSI-like algorithm Starting with : |ψ(0) = N(0) i=1 ci |Di do k = 0, Niterations 1 for all |Dj with Dj |H|ψ(k) = 0 compute the perturbative energy contribution j 2 Select the n determinants |Dj with largest j 3 Diagonalize H in the new set of determinants : N(k+1) = N(k) + n 4 you have a new reference wf : |ψ(k+1) = N(k+1) i=1 ci |Di end do E. Giner Coupling CI and FN-DMC
15. logos Introduction WF methods FN-DMC with CIPSI trial wave functions The variational and perturbative energy 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 : ECIPSI = EVar + EPT2 ⇒ approximation of the FCI energy E. Giner Coupling CI and FN-DMC
16. logos Introduction WF methods FN-DMC with CIPSI trial wave functions Applications of the CIPSI algorithm 1 Total energies of atoms and cations : from B to Ar in large basis set (up to aug-cc-pV5Z ) 2 Dissociation energy curve of F2 : change of geometry, static/dynamic correlation effects 3 The CuCl2 molecule : 63 electrons, near degeneracy between 2Πu and 2Σu E. Giner Coupling CI and FN-DMC
17. logos Introduction WF methods FN-DMC with CIPSI trial wave functions 1) Total energies of atoms and cations Oxygen ground state in cc-pVXZ 100 101 102 103 104 105 Number of determinants −75.04 −74.98 −74.92 −74.86 −74.80 Energy (Hartree) EVar cc-pVDZ ECIPSI cc-pVDZ FCI cc-pVDZ E. Giner Coupling CI and FN-DMC
18. logos Introduction WF methods FN-DMC with CIPSI trial wave functions 1) Total energies of atoms and cations Oxygen ground state in cc-pVXZ 100 101 102 103 104 105 Number of determinants −75.04 −74.98 −74.92 −74.86 −74.80 Energy (Hartree) EVar cc-pVDZ ECIPSI cc-pVDZ FCI cc-pVDZ EVar cc-pVTZ ECIPSI cc-pVTZ FCI cc-pVTZ E. Giner Coupling CI and FN-DMC
19. logos Introduction WF methods FN-DMC with CIPSI trial wave functions 1) Total energies of atoms and cations Oxygen ground state in cc-pVXZ 100 101 102 103 104 105 Number of determinants −75.04 −74.98 −74.92 −74.86 −74.80 Energy (Hartree) EVar cc-pVDZ ECIPSI cc-pVDZ FCI cc-pVDZ EVar cc-pVTZ ECIPSI cc-pVTZ FCI cc-pVTZ EVar cc-pVQZ ECIPSI cc-pVQZ E. Giner Coupling CI and FN-DMC
20. logos Introduction WF methods FN-DMC with CIPSI trial wave functions 1) Total energies of atoms and cations Oxygen ground state in cc-pVXZ 100 101 102 103 104 105 Number of determinants −75.04 −74.98 −74.92 −74.86 −74.80 Energy (Hartree) EVar cc-pVDZ ECIPSI cc-pVDZ FCI cc-pVDZ EVar cc-pVTZ ECIPSI cc-pVTZ FCI cc-pVTZ EVar cc-pVQZ ECIPSI cc-pVQZ EVar cc-pV5Z ECIPSI cc-pV5Z E. Giner Coupling CI and FN-DMC
21. logos Introduction WF methods FN-DMC with CIPSI trial wave functions 1) Total energies of atoms and cations In black = converged CIPSI results In blue = converged FCI-QMC results of Booth and Alavi aug-cc-pVDZ aug-cc-pVTZ aug-cc-pVQZ aug-cc-pV5Z B -24.59241 -24.60665 -24.62410 -24.63023 -24.59242(1) -24.60665(2) -24.62407(11) -24.63023(2) B+ -24.29449 -24.30366 -24.32004 -24.32553 -24.29450(1) -24.30366(2) -24.32005(2) -24.32553(9) C -37.76657 -37.79161 -37.81300 -37.82000 -37.76656(1) -37.79163(2) -37.81301(2) -37.82001(4) C+ -37.35959 -37.37969 -37.39992 -37.40633 -37.35960(1) -37.37967(2) -37.39991(1) -37.40605(1) N -54.48881 -54.52794 -54.55422 -54.56300 -54.48881(2) -54.52797(1) -54.55423(3) -54.56303(2) N+ -53.96106 -53.99535 -54.02040 -54.02862 -53.96106(10) -53.99535(1) -54.01838(1) -54.02865(2) O -74.92769 -74.99077 -75.02528 -75.03744 -74.92772(2) -74.99077(4) -75.02534(4) -75.03749(6) O+ -74.44419 -74.49702 -74.52796 -74.53862 -74.444194(6) -74.49701(1) -74.52799(4) -74.53869(6) F -99.55221 -99.64032 -99.68442 -99.70012 -99.55223(1) -99.64036(2) -99.68460(10) -99.70029(5) F+ -98.92300 -99.00542 -99.04646 -99.06079 -98.923015(6) -99.00542(1) -99.04599(3) -99.06082(4) Ne -128.71149 -128.82579 -128.88057 -128.90026 -128.71145(3) -128.82577(5) -128.88065(6) Ne+ -127.92407 -128.03688 -128.08887 -128.10740 -127.92411(2) -128.03691(2) -128.08816(11) E. Giner Coupling CI and FN-DMC
22. logos Introduction WF methods FN-DMC with CIPSI trial wave functions 2) Dissociation energy curve of F2 1.0 1.5 2.0 2.5 3.0 3.5 4.0 F–F distance (˚ A) −199.04 −198.96 −198.88 −198.80 −198.72 Energy (a.u.) 500 103 5 103 104 5 104 7.5 104 105 1.301.351.401.451.501.551.60 −199.095 −199.090 −199.085 −199.080 −199.075 −199.070 FIGURE: Convergence of the variational energy of the F2 molecule Potential Energy Surface (PES) as a function of the number of determinants in the CIPSI wf. Basis set : cc-pVDZ E. Giner Coupling CI and FN-DMC
23. logos Introduction WF methods FN-DMC with CIPSI trial wave functions 2) Dissociation energy curve of F2 1.0 1.5 2.0 2.5 3.0 3.5 4.0 F–F distance (˚ A) −199.12 −199.04 −198.96 −198.88 −198.80 Energy (a.u.) 500 103 5 103 104 5104 7.5 104 105 1.301.351.401.451.501.551.60 −199.102 −199.100 −199.098 −199.096 −199.094 FIGURE: Convergence of the CIPSI energy of the F2 molecule Potential Energy Surface (PES) as a function of the number of determinants in the CIPSI wf. Basis set : cc-pVDZ E. Giner Coupling CI and FN-DMC
24. logos Introduction WF methods FN-DMC with CIPSI trial wave functions 2) Dissociation energy curve of F2 Spectroscopic constant of F2 in the cc-pVDZ, comparaison with other methods Req D0 k Estimated FCI (Bytautas et al.) 1.460 45.14 0.80 FCI-QMC - 45.00(11) - CIPSI (105 determinants) 1.463 45.17 0.76 Exact NR 1.412 62.0 1.121 E. Giner Coupling CI and FN-DMC
25. logos Introduction WF methods FN-DMC with CIPSI trial wave functions 3) The CuCl2 molecule The lowest electronic states of the CuCl2 molecule : Very small gap 2Πu →2 Σu : 900 cm−1 DFT methods and post HF methods give many different results ([−2500 : 6500] cm−1) Cl Cu Cl FIGURE: Singly occupied orbital of CuCl2 in its fundamental state 2Πu . E. Giner Coupling CI and FN-DMC
26. logos Introduction WF methods FN-DMC with CIPSI trial wave functions 3) The CuCl2 molecule 100 101 102 103 104 105 106 Number of determinants −2558.2 −2558.1 −2558.0 −2557.9 −2557.8 −2557.7 −2557.6 Energy (a.u.) HF orbitals Natural orbitals B3LYP orbitals HF orbitals + PT2 FIGURE: CIPSI, CuCl2 in 6-31G : Convergence study of the variational and CIPSI energies as a function of the number of determinants, for the fundamental state 2Π, for different molecular orbitals. E. Giner Coupling CI and FN-DMC
27. logos Introduction WF methods FN-DMC with CIPSI trial wave functions 3) The CuCl2 molecule 100 101 102 103 104 105 106 Number of determinants −4500 −3000 −1500 0 1500 3000 4500 ∆E (cm−1) Variational Variational+PT2 900 cm−1 FIGURE: CIPSI, CuCl2 in 6-31G : Convergence study of the excitation energy between the fundamental state 2Π and the first excited state 2Σ as a function of the number of determinants E. Giner Coupling CI and FN-DMC
28. logos Introduction WF methods FN-DMC with CIPSI trial wave functions Summary of the results with CIPSI For all the systems studied : The Variational energy converges fast : CIPSI wf ⇒ compact approximation of the FCI wf The CIPSI energy converges even faster Near FCI quality (≤ 1 mEh) ECIPSI ≈ EFCI ψCIPSI ≈ ψFCI E. Giner Coupling CI and FN-DMC
29. logos Introduction WF methods FN-DMC with CIPSI trial wave functions CuCl2 excitation energy at near FCI level Good results for CuCl2 : CIPSI : 700 cm−1 vs Est Exact : 900 cm−1 Why such good results with a tiny basis set (6-31G) ? Near FCI : Exact solution in the basis set (static/dynamic correlation) 2 valence states : Same quality of description for both states Same Spin multiplicity Same Geometry Same number of interacting particles E. Giner Coupling CI and FN-DMC
30. logos Introduction WF methods FN-DMC with CIPSI trial wave functions CuCl2 excitation energy at near FCI level Good results for CuCl2 : CIPSI : 700 cm−1 vs Est Exact : 900 cm−1 Why such good results with a tiny basis set (6-31G) ? Near FCI : Exact solution in the basis set (static/dynamic correlation) 2 valence states : Same quality of description for both states Same Spin multiplicity Same Geometry Same number of interacting particles E. Giner Coupling CI and FN-DMC
31. logos Introduction WF methods FN-DMC with CIPSI trial wave functions CuCl2 excitation energy at near FCI level Good results for CuCl2 : CIPSI : 700 cm−1 vs Est Exact : 900 cm−1 Why such good results with a tiny basis set (6-31G) ? Near FCI : Exact solution in the basis set (static/dynamic correlation) 2 valence states : Same quality of description for both states Same Spin multiplicity Same Geometry Same number of interacting particles E. Giner Coupling CI and FN-DMC
32. logos Introduction WF methods FN-DMC with CIPSI trial wave functions CuCl2 excitation energy at near FCI level Good results for CuCl2 : CIPSI : 700 cm−1 vs Est Exact : 900 cm−1 Why such good results with a tiny basis set (6-31G) ? Near FCI : Exact solution in the basis set (static/dynamic correlation) 2 valence states : Same quality of description for both states Same Spin multiplicity Same Geometry Same number of interacting particles E. Giner Coupling CI and FN-DMC
33. logos Introduction WF methods FN-DMC with CIPSI trial wave functions CuCl2 excitation energy at near FCI level Good results for CuCl2 : CIPSI : 700 cm−1 vs Est Exact : 900 cm−1 Why such good results with a tiny basis set (6-31G) ? Near FCI : Exact solution in the basis set (static/dynamic correlation) 2 valence states : Same quality of description for both states Same Spin multiplicity Same Geometry Same number of interacting particles E. Giner Coupling CI and FN-DMC
34. logos Introduction WF methods FN-DMC with CIPSI trial wave functions CuCl2 excitation energy at near FCI level Good results for CuCl2 : CIPSI : 700 cm−1 vs Est Exact : 900 cm−1 Why such good results with a tiny basis set (6-31G) ? Near FCI : Exact solution in the basis set (static/dynamic correlation) 2 valence states : Same quality of description for both states Same Spin multiplicity Same Geometry Same number of interacting particles E. Giner Coupling CI and FN-DMC
35. logos Introduction WF methods FN-DMC with CIPSI trial wave functions Analysis of convergence of the Ionization Potential at near FCI level B C N O F Ne aug-cc-pVDZ 0.297 92 0.406 98 0.527 75 0.483 50 0.629 21 0.787 42 aug-cc-pVTZ 0.302 99 0.411 92 0.532 59 0.493 75 0.635 01 0.789 12 aug-cc-pVQZ 0.304 06 0.413 08 0.533 82 0.497 32 0.637 98 0.791 65 aug-cc-pV5Z 0.304 70 0.413 67 0.534 38 0.498 82 0.638 43 0.792 86 ‘exact NR’ 0.304 98 0.414 08 0.534 89 0.500 41 0.641 13 0.794 64 TABLE: Convergence of the ionization potentials at the converged CIPSI level as the function of the basis set for the atoms of the second row. Energies in Hartree. B C N O F Ne aug-cc-pVDZ 0.00706 0.00710 0.00714 0.01691 0.01192 0.00722 aug-cc-pVTZ 0.00199 0.00216 0.00229 0.00666 0.00612 0.00552 aug-cc-pVQZ 0.00092 0.00100 0.00107 0.00309 0.00315 0.00299 aug-cc-pV5Z 0.00028 0.00041 0.00051 0.00159 0.00270 0.00178 FCI space AV5Z 108.0 1010 1011 1013 1014 1016 TABLE: Convergence of the absolute errors of the IP respect to the estimated exact NR of Davidson as a function of the basis set. Energies in Hartree. All the IP are underestimated, slow convergence with basis set. E. Giner Coupling CI and FN-DMC
36. logos Introduction WF methods FN-DMC with CIPSI trial wave functions Analysis of convergence of the Ionization Potential at near FCI level B C N O F Ne aug-cc-pVDZ 0.297 92 0.406 98 0.527 75 0.483 50 0.629 21 0.787 42 aug-cc-pVTZ 0.302 99 0.411 92 0.532 59 0.493 75 0.635 01 0.789 12 aug-cc-pVQZ 0.304 06 0.413 08 0.533 82 0.497 32 0.637 98 0.791 65 aug-cc-pV5Z 0.304 70 0.413 67 0.534 38 0.498 82 0.638 43 0.792 86 ‘exact NR’ 0.304 98 0.414 08 0.534 89 0.500 41 0.641 13 0.794 64 TABLE: Convergence of the ionization potentials at the converged CIPSI level as the function of the basis set for the atoms of the second row. Energies in Hartree. B C N O F Ne aug-cc-pVDZ 0.00706 0.00710 0.00714 0.01691 0.01192 0.00722 aug-cc-pVTZ 0.00199 0.00216 0.00229 0.00666 0.00612 0.00552 aug-cc-pVQZ 0.00092 0.00100 0.00107 0.00309 0.00315 0.00299 aug-cc-pV5Z 0.00028 0.00041 0.00051 0.00159 0.00270 0.00178 FCI space AV5Z 108.0 1010 1011 1013 1014 1016 TABLE: Convergence of the absolute errors of the IP respect to the estimated exact NR of Davidson as a function of the basis set. Energies in Hartree. All the IP are underestimated, slow convergence with basis set. E. Giner Coupling CI and FN-DMC
37. logos Introduction WF methods FN-DMC with CIPSI trial wave functions Analysis of convergence of the Ionization Potential at near FCI level B C N O F Ne aug-cc-pVDZ 0.297 92 0.406 98 0.527 75 0.483 50 0.629 21 0.787 42 aug-cc-pVTZ 0.302 99 0.411 92 0.532 59 0.493 75 0.635 01 0.789 12 aug-cc-pVQZ 0.304 06 0.413 08 0.533 82 0.497 32 0.637 98 0.791 65 aug-cc-pV5Z 0.304 70 0.413 67 0.534 38 0.498 82 0.638 43 0.792 86 ‘exact NR’ 0.304 98 0.414 08 0.534 89 0.500 41 0.641 13 0.794 64 TABLE: Convergence of the ionization potentials at the converged CIPSI level as the function of the basis set for the atoms of the second row. Energies in Hartree. B C N O F Ne aug-cc-pVDZ 0.00706 0.00710 0.00714 0.01691 0.01192 0.00722 aug-cc-pVTZ 0.00199 0.00216 0.00229 0.00666 0.00612 0.00552 aug-cc-pVQZ 0.00092 0.00100 0.00107 0.00309 0.00315 0.00299 aug-cc-pV5Z 0.00028 0.00041 0.00051 0.00159 0.00270 0.00178 FCI space AV5Z 108.0 1010 1011 1013 1014 1016 TABLE: Convergence of the absolute errors of the IP respect to the estimated exact NR of Davidson as a function of the basis set. Energies in Hartree. All the IP are underestimated, slow convergence with basis set. E. Giner Coupling CI and FN-DMC
38. logos Introduction WF methods FN-DMC with CIPSI trial wave functions Analysis of convergence of the Ionization Potential at near FCI level B C N O F Ne aug-cc-pVDZ 0.297 92 0.406 98 0.527 75 0.483 50 0.629 21 0.787 42 aug-cc-pVTZ 0.302 99 0.411 92 0.532 59 0.493 75 0.635 01 0.789 12 aug-cc-pVQZ 0.304 06 0.413 08 0.533 82 0.497 32 0.637 98 0.791 65 aug-cc-pV5Z 0.304 70 0.413 67 0.534 38 0.498 82 0.638 43 0.792 86 ‘exact NR’ 0.304 98 0.414 08 0.534 89 0.500 41 0.641 13 0.794 64 TABLE: Convergence of the ionization potentials at the converged CIPSI level as the function of the basis set for the atoms of the second row. Energies in Hartree. B C N O F Ne aug-cc-pVDZ 0.00706 0.00710 0.00714 0.01691 0.01192 0.00722 aug-cc-pVTZ 0.00199 0.00216 0.00229 0.00666 0.00612 0.00552 aug-cc-pVQZ 0.00092 0.00100 0.00107 0.00309 0.00315 0.00299 aug-cc-pV5Z 0.00028 0.00041 0.00051 0.00159 0.00270 0.00178 FCI space AV5Z 108.0 1010 1011 1013 1014 1016 TABLE: Convergence of the absolute errors of the IP respect to the estimated exact NR of Davidson as a function of the basis set. Energies in Hartree. All the IP are underestimated, slow convergence with basis set. E. Giner Coupling CI and FN-DMC
39. logos Introduction WF methods FN-DMC with CIPSI trial wave functions Ionization Potential at near FCI level Ionization Potential with FCI : IPFCI = EFCI(A+) − EFCI(A) A+ has one electron less than A A+ not the same multiplicity A+ has less interacting pairs of electrons A+ always better described than A in the same finite basis set ⇒ Underestimated Ionization Potential ! E. Giner Coupling CI and FN-DMC
40. logos Introduction WF methods FN-DMC with CIPSI trial wave functions Ionization Potential at near FCI level Ionization Potential with FCI : IPFCI = EFCI(A+) − EFCI(A) A+ has one electron less than A A+ not the same multiplicity A+ has less interacting pairs of electrons A+ always better described than A in the same finite basis set ⇒ Underestimated Ionization Potential ! E. Giner Coupling CI and FN-DMC
41. logos Introduction WF methods FN-DMC with CIPSI trial wave functions Ionization Potential at near FCI level Ionization Potential with FCI : IPFCI = EFCI(A+) − EFCI(A) A+ has one electron less than A A+ not the same multiplicity A+ has less interacting pairs of electrons A+ always better described than A in the same finite basis set ⇒ Underestimated Ionization Potential ! E. Giner Coupling CI and FN-DMC
42. logos Introduction WF methods FN-DMC with CIPSI trial wave functions F2 spectroscopic constants at near FCI level Req D0 k Estimated FCI (Bytautas et al.) 1.460 45.14 0.80 FCI-QMC - 45.00(11) - CIPSI (105 determinants) 1.463 45.17 0.76 Exact NR 1.412 62.0 1.121 E. Giner Coupling CI and FN-DMC
43. logos Introduction WF methods FN-DMC with CIPSI trial wave functions Atomization energies at near FCI level For the Atomization Energies : atomic centered basis set A. . . ∞ . . .B has much less less interacting pairs of electrons than A–B A. . . ∞ . . .B always better described than A–B in the same finite basis set ⇒ Underestimated Atomization Energies ! E. Giner Coupling CI and FN-DMC
44. logos Introduction WF methods FN-DMC with CIPSI trial wave functions Atomization energies at near FCI level For the Atomization Energies : atomic centered basis set A. . . ∞ . . .B has much less less interacting pairs of electrons than A–B A. . . ∞ . . .B always better described than A–B in the same finite basis set ⇒ Underestimated Atomization Energies ! E. Giner Coupling CI and FN-DMC
45. logos Introduction WF methods FN-DMC with CIPSI trial wave functions How can we go further ? The problem of FCI : Finite basis set ... FN-DMC works in an infinite basis set ... Combine FCI with FN-DMC ! Use of CIPSI wf as Trial wf for FN-DMC ! E. Giner Coupling CI and FN-DMC
46. logos Introduction WF methods FN-DMC with CIPSI trial wave functions How can we go further ? The problem of FCI : Finite basis set ... FN-DMC works in an infinite basis set ... Combine FCI with FN-DMC ! Use of CIPSI wf as Trial wf for FN-DMC ! E. Giner Coupling CI and FN-DMC
47. logos Introduction WF methods FN-DMC with CIPSI trial wave functions How can we go further ? The problem of FCI : Finite basis set ... FN-DMC works in an infinite basis set ... Combine FCI with FN-DMC ! Use of CIPSI wf as Trial wf for FN-DMC ! E. Giner Coupling CI and FN-DMC
48. logos Introduction WF methods FN-DMC with CIPSI trial wave functions How can we go further ? The problem of FCI : Finite basis set ... FN-DMC works in an infinite basis set ... Combine FCI with FN-DMC ! Use of CIPSI wf as Trial wf for FN-DMC ! E. Giner Coupling CI and FN-DMC
49. logos Introduction WF methods FN-DMC with CIPSI trial wave functions FN-DMC in few words FN-DMC needs a Trial wave function |ψT : ψT (r1 , . . . , rN ) Now consider a positive function F : F(r1 , . . . , rN ) > 0 ∀ r1 , . . . , rN ˜ ψ = F × ψT Variational energy of ˜ ψ : E[ ˜ ψ] = ψT F|H|FψT ψT F|FψT = E[F, ψT ] E. Giner Coupling CI and FN-DMC
50. logos Introduction WF methods FN-DMC with CIPSI trial wave functions FN-DMC in few words FN-DMC needs a Trial wave function |ψT : ψT (r1 , . . . , rN ) Now consider a positive function F : F(r1 , . . . , rN ) > 0 ∀ r1 , . . . , rN ˜ ψ = F × ψT Variational energy of ˜ ψ : E[ ˜ ψ] = ψT F|H|FψT ψT F|FψT = E[F, ψT ] E. Giner Coupling CI and FN-DMC
51. logos Introduction WF methods FN-DMC with CIPSI trial wave functions FN-DMC in few words FN-DMC needs a Trial wave function |ψT : ψT (r1 , . . . , rN ) Now consider a positive function F : F(r1 , . . . , rN ) > 0 ∀ r1 , . . . , rN ˜ ψ = F × ψT Variational energy of ˜ ψ : E[ ˜ ψ] = ψT F|H|FψT ψT F|FψT = E[F, ψT ] E. Giner Coupling CI and FN-DMC
52. logos Introduction WF methods FN-DMC with CIPSI trial wave functions FN-DMC in few words The FN-DMC algorithm extracts the energy of the best function F∗ for a given |ψT : EFN−DMC = E[F∗, ψT ] = min F E[F, ψT ] without any analytical knowledge on F∗ ! F∗ ⇔ infinite basis set ! E. Giner Coupling CI and FN-DMC
53. logos Introduction WF methods FN-DMC with CIPSI trial wave functions FN-DMC in few words The FN-DMC algorithm extracts the energy of the best function F∗ for a given |ψT : EFN−DMC = E[F∗, ψT ] = min F E[F, ψT ] without any analytical knowledge on F∗ ! F∗ ⇔ infinite basis set ! E. Giner Coupling CI and FN-DMC
54. logos Introduction WF methods FN-DMC with CIPSI trial wave functions FN-DMC in few words The FN-DMC algorithm extracts the energy of the best function F∗ for a given |ψT : EFN−DMC = E[F∗, ψT ] = min F E[F, ψT ] without any analytical knowledge on F∗ ! F∗ ⇔ infinite basis set ! E. Giner Coupling CI and FN-DMC
55. logos Introduction WF methods FN-DMC with CIPSI trial wave functions CIPSI and FN-DMC : 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 FIGURE: Convergence of the FN-DMC energy as a function of the number of determinants selected in the CIPSI wave function in cc-pVXZ basis set (X=D,T,Q) E. Giner Coupling CI and FN-DMC
56. logos Introduction WF methods FN-DMC with CIPSI trial wave functions Summary of the total energies : Oxygen ground state 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 E. Giner Coupling CI and FN-DMC
57. logos Introduction WF methods FN-DMC with CIPSI trial wave functions Protocol In a given basis set 1 Run Restricted Hartree Fock FN-DMC with RHF nodes 2 Run a converged CIPSI FN-DMC with CIPSI nodes E. Giner Coupling CI and FN-DMC
58. logos Introduction WF methods FN-DMC with CIPSI trial wave functions CIPSI+FN-DMC : atoms and cations with CIPSI in cc-pVDZ 4 6 8 10 12 14 16 18 Nuclear Charge −0.025 −0.020 −0.015 −0.010 −0.005 0.000 0.005 Gain in FN energy (Hartree) Neutral First cation FIGURE: Gain in FN-DMC energy FN-DMC(CIPSI) - FN-DMC(HF) as a function of the nuclear charge for the neutral and first cation. CIPSI done in cc-pVDZ E. Giner Coupling CI and FN-DMC
59. logos Introduction WF methods FN-DMC with CIPSI trial wave functions CIPSI+FN-DMC : 3d transition metals Atom E0 (HF nodes) E0 (CIPSI nodes) Gain in FN energy Sc -760.5265(13) -760.5718(16) -0.0453(21) Ti -849.2405(14) -849.2841(19) -0.0436(24) V -943.7843(13) -943.8234(16) -0.0391(21) Cr -1044.3292(16) -1044.3603(17) -0.0311(23) Mn -1150.8880(17) -1150.9158(20) -0.0278(26) Fe -1263.5589(19) -1263.5868(21) -0.0279(28) Co -1382.6177(21) -1382.6377(24) -0.0200(32) Ni -1508.1645(23) -1508.1901(25) -0.0256(34) Cu -1640.4271(26) -1640.4328(29) -0.0057(39) Zn -1779.3371(26) -1779.3386(31) -0.0015(40) FIGURE: FN-DMC total energies for the 3d series of transition metal atoms. Energy in Hartree. Best variational energies of the literature. E. Giner Coupling CI and FN-DMC
60. logos Introduction WF methods FN-DMC with CIPSI trial wave functions Systematic improvement of the FN-DMC energy ... What about the differences of energy ? Ionization Potential F2 dissociation curve E. Giner Coupling CI and FN-DMC
61. logos Introduction WF methods FN-DMC with CIPSI trial wave functions Ionization Potential with cc-pVDZ nodes 4 6 8 10 12 14 16 18 20 Charge of the nucleus [4;100] −0.006 −0.003 0.000 0.003 0.006 Absolute error to the IP (Hartree) FN-DMC (HF cc-pVDZ) FN-DMC (CIPSI cc-pVDZ) Chemical accuracy Chemical accuracy FIGURE: Error to the estimated exact IP using FN-DMC with HF and CIPSI nodes in cc-pVDZ HF nodes : MAD = 2.29(23) mHartree CIPSI nodes : MAD = 1.68(30) mHartree E. Giner Coupling CI and FN-DMC
62. logos Introduction WF methods FN-DMC with CIPSI trial wave functions Ionization Potential with cc-pVTZ nodes 4 5 6 7 8 9 10 11 Charge of the nucleus −0.006 −0.004 −0.002 0.000 0.002 Absolute error to the IP (Hartree) FN-DMC (HF cc-pVTZ) FN-DMC (CIPSI cc-pVTZ) Chemical accuracy Chemical accuracy FIGURE: Error to the estimated exact IP using FN-DMC with HF and CIPSI nodes in cc-pVTZ HF nodes : MAD = 2.75(20) mHartree CIPSI nodes : MAD = 1.23(20) mHartree E. Giner Coupling CI and FN-DMC
63. logos Introduction WF methods FN-DMC with CIPSI trial wave functions Ionization potential at the FCI level 4 6 8 10 12 14 16 18 Charge of the nucleus 0.000 0.006 0.012 0.018 0.024 Absolute error to the IP (Hartree) FN-DMC (CIPSI cc-pVDZ) FCI cc-pVDZ FCI cc-pVTZ FCI cc-pVQZ Chemical accuracy Chemical accuracy FIGURE: Error to the estimated exact IP using FN-DMC with CIPSI nodes in cc-pVDZ, comparison with FCI the cc-pVXZ (X=D,T,Q) E. Giner Coupling CI and FN-DMC
64. logos Introduction WF methods FN-DMC with CIPSI trial wave functions CIPSI+FN-DMC The IP in CIPSI+FN-DMC : Good quality (≥ FCI in cc-pVQZ) Improved by increasing the basis set of the CIPSI calculation Quality depends weakly on the number of electrons CIPSI+FN-DMC are systematically better than those using CIPSI (≈FCI) Control of the quality of the trial wave function E. Giner Coupling CI and FN-DMC
65. logos Introduction WF methods FN-DMC with CIPSI trial wave functions CIPSI+FN-DMC : F2 potential energy curve 1.5 2.0 2.5 3.0 3.5 4.0 F–F distance (˚ A) −199.495 −199.480 −199.465 −199.450 −199.435 Energy (a.u.) 1 000 dets CAS(2,2) 5 000 dets 10 000 dets FIGURE: FN-DMC with CIPSI-cc-pVDZ nodes. Convergence of the FN-DMC energy curve as a function of the number of determinants selected in the trial wave function E. Giner Coupling CI and FN-DMC
66. logos Introduction WF methods FN-DMC with CIPSI trial wave functions FCI+FN-DMC : F2 spectroscopic constants Req D0 k CASSCF(cc-pVDZ) + FN-DMC 1.434(8) 48.3(7) 1.0(2) 104-det CIPSI(cc-pVDZ) + FN-DMC 1.428(20) 55.2(1.2) 1.0(4) FCI cc-pVDZ 1.460 45.14 0.8 FCI cc-pVTZ 1.416 56.7 1.075 Exact NR 1.412 62.0 1.121 Summary of results for F2 : CIPSI+FN-DMC results are better than CIPSI results CIPSI+FN-DMC results in cc-pVDZ are comparable with FCI in cc-pVTZ E. Giner Coupling CI and FN-DMC
67. logos Introduction WF methods FN-DMC with CIPSI trial wave functions FCI+FN-DMC : F2 spectroscopic constants Req D0 k CASSCF(cc-pVDZ) + FN-DMC 1.434(8) 48.3(7) 1.0(2) 104-det CIPSI(cc-pVDZ) + FN-DMC 1.428(20) 55.2(1.2) 1.0(4) FCI cc-pVDZ 1.460 45.14 0.8 FCI cc-pVTZ 1.416 56.7 1.075 Exact NR 1.412 62.0 1.121 Summary of results for F2 : CIPSI+FN-DMC results are better than CIPSI results CIPSI+FN-DMC results in cc-pVDZ are comparable with FCI in cc-pVTZ E. Giner Coupling CI and FN-DMC
68. logos Introduction WF methods FN-DMC with CIPSI trial wave functions Last Work : other CI-type wf For bigger systems : how can we get a better convergence ? CISD(SC)2 : CISD with Size consistent correction Selected version of the CISD(SC)2 MRPT2 based on determinant : alternative to the CIPSI algorithm All these approaches can be done in a uncontracted formalism ! ⇒ the perturbation can change the reference CI wf ⇒ the perturbation can change the nodes ! E. Giner Coupling CI and FN-DMC
69. logos Introduction WF methods FN-DMC with CIPSI trial wave functions Last Work : other CI-type wf For bigger systems : how can we get a better convergence ? CISD(SC)2 : CISD with Size consistent correction Selected version of the CISD(SC)2 MRPT2 based on determinant : alternative to the CIPSI algorithm All these approaches can be done in a uncontracted formalism ! ⇒ the perturbation can change the reference CI wf ⇒ the perturbation can change the nodes ! E. Giner Coupling CI and FN-DMC
70. logos Introduction WF methods FN-DMC with CIPSI trial wave functions Last Work : other CI-type wf For bigger systems : how can we get a better convergence ? CISD(SC)2 : CISD with Size consistent correction Selected version of the CISD(SC)2 MRPT2 based on determinant : alternative to the CIPSI algorithm All these approaches can be done in a uncontracted formalism ! ⇒ the perturbation can change the reference CI wf ⇒ the perturbation can change the nodes ! E. Giner Coupling CI and FN-DMC
71. logos Introduction WF methods FN-DMC with CIPSI trial wave functions Last Work : other CI-type wf For bigger systems : how can we get a better convergence ? CISD(SC)2 : CISD with Size consistent correction Selected version of the CISD(SC)2 MRPT2 based on determinant : alternative to the CIPSI algorithm All these approaches can be done in a uncontracted formalism ! ⇒ the perturbation can change the reference CI wf ⇒ the perturbation can change the nodes ! E. Giner Coupling CI and FN-DMC
72. logos Introduction WF methods FN-DMC with CIPSI trial wave functions Last Work : other CI-type wf For bigger systems : how can we get a better convergence ? CISD(SC)2 : CISD with Size consistent correction Selected version of the CISD(SC)2 MRPT2 based on determinant : alternative to the CIPSI algorithm All these approaches can be done in a uncontracted formalism ! ⇒ the perturbation can change the reference CI wf ⇒ the perturbation can change the nodes ! E. Giner Coupling CI and FN-DMC
73. logos Introduction WF methods FN-DMC with CIPSI trial wave functions Conclusion and perspective What we have learned about FN-DMC : It is possible to use near FCI wf in FN-DMC Systematic improvement of the nodes It is possible to avoid the stochastic optimization of |ψT Good results on various systems (atoms, molecule) For the future : Rationalize the approach Try simpler WF models (CISD(SC)2, dressed hamiltonian matrix, DDCI etc ...) Try bigger system (Poly peptide ) Try excited states, avoided crossing (Retinal chromophore) Build deterministic Jastrow factor for core electrons E. Giner Coupling CI and FN-DMC
74. logos Introduction WF methods FN-DMC with CIPSI trial wave functions Conclusion and perspective What we have learned about FN-DMC : It is possible to use near FCI wf in FN-DMC Systematic improvement of the nodes It is possible to avoid the stochastic optimization of |ψT Good results on various systems (atoms, molecule) For the future : Rationalize the approach Try simpler WF models (CISD(SC)2, dressed hamiltonian matrix, DDCI etc ...) Try bigger system (Poly peptide ) Try excited states, avoided crossing (Retinal chromophore) Build deterministic Jastrow factor for core electrons E. Giner Coupling CI and FN-DMC
75. logos Introduction WF methods FN-DMC with CIPSI trial wave functions E. Giner Coupling CI and FN-DMC
76. logos Introduction WF methods FN-DMC with CIPSI trial wave functions Example with Nelec = cst : atomization energies with iFCI-QMC cc-pVDZ cc-pVTZ cc-pVQZ exact N2 200.52(8) 216.86(9) 223.20(8) 227.6 O2 105.17(6) 114.35(8) 117.5(1) 120.4 F2 27.59(7) 35.4(1) 36.9(1) 39.0 TABLE: Convergence des energies d’atomisations (kcal.mol−1) en fonction de la base au niveau i-FCI-QMC E. Giner Coupling CI and FN-DMC
77. logos Introduction WF methods FN-DMC with CIPSI trial wave functions FCI+FN-DMC : IP of atoms with CIPSI in cc-pVDZ EFN(RHF) EFN(1k) Gain in FN energy B -24.63878(70) -24.65133(54) -0.01255(63) B+ -24.32853(53) -24.34771(37) -0.01918(46) C -37.82902(41) -37.83991(71) -0.01089(58) C+ -37.40929(39) -37.42826(60) -0.01897(51) N -54.57358(36) -54.58244(60) -0.00886(49) N+ -54.03387(32) -54.04735(94) -0.01348(70) O -75.04876(21) -75.05492(77) -0.00616(56) O+ -74.54900(17) -74.55801(61) -0.00901(45) F -99.71129(64) -99.71864(87) -0.00735(76) F+ -99.07191(52) -99.07917(76) -0.00726(65) Ne -128.91538(49) -128.91875(66) -0.00337(58) Ne+ -128.12010(37) -128.12663(58) -0.00653(49) Na -162.23875(58) -162.23840(76) +0.00035(68) Na+ -162.05096(49) -162.04989(68) +0.00107(59) Mg -200.03001(90) -200.03636(75) -0.00635(83) Mg+ -199.75561(82) -199.75590(58) -0.00029(71) Al -242.31978(79) -242.32611(93) -0.00633(86) Al+ -242.10114(75) -242.10599(87) -0.00485(81) Si -289.32683(66) -289.33652(85) -0.00969(76) Si+ -289.02832(72) -289.03439(97) -0.00607(85) P -341.22212(96) -341.23354(81) -0.01142(89) P+ -340.83490(91) -340.84468(70) -0.00978(81) S -398.06549(59) -398.07936(72) -0.01387(66) S+ -397.68669(46) -397.70156(81) -0.01487(66) Cl -460.09575(55) -460.11382(85) -0.01807(72) Cl+ -459.61977(50) -459.63873(83) -0.01896(69) Ar -527.48190(62) -527.50362(97) -0.02172(81) Ar+ -526.89957(58) -526.92060(100) -0.02103(82) TABLE: Summary of the total energies obtained using CIPSI+FN-DMC in cc-pVDZ E. Giner Coupling CI and FN-DMC
78. logos Introduction WF methods FN-DMC with CIPSI trial wave functions Variance of CIPSI wf : Oxygen groud state 100 101 102 103 104 105 Number of determinants 6 9 12 15 18 Variance (Hartree) CIPSI cc-pVDZ CIPSI cc-pVTZ CIPSI cc-pVTZ CIPSI cc-pCVQZ E. Giner Coupling CI and FN-DMC
79. logos Introduction WF methods FN-DMC with CIPSI trial wave functions Variance of CIPSI wf : 3d transition metals Atom Ndet FN-DMC energy Variance Sc 1 -760.5265(13) 182.695(87) 11389 -760.5718(16) 134.15(16) Ti 1 -849.2405(14) 205.252(99) 14054 -849.2841(19) 150.58(34) V 1 -943.7843(13) 229.187(98) 12441 -943.8234(16) 169.04(21) Cr 1 -1044.3292(16) 254.579(79) 10630 -1044.3603(17) 189.49(59) Mn 1 -1150.8880(17) 282.08(13) 11688 -1150.9158(20) 210.66(32) Fe 1 -1263.5589(19) 311.58(12) 13171 -1263.5868(21) 233.23(22) Co 1 -1382.6177(21) 343.01(12) 15949 -1382.6377(24) 255.96(57) Ni 1 -1508.1645(23) 376.71(15) 15710 -1508.1901(25) 282.15(33) Cu 1 -1640.4271(26) 412.62(24) 48347 -1640.4328(29) 302.17(44) Zn 1 -1779.3371(26) 449.76(17) 44206 -1779.3386(31) 337.42(35) E. Giner Coupling CI and FN-DMC
80. logos Introduction WF methods FN-DMC with CIPSI trial wave functions New total energies for the Cu atom : gaussian basis set Basis set Ndet FN-DMC energy Slater(spdg) 1 -1640.4271(26) Slater(spdg) 48347 -1640.4328(29) gaussian (cc-pVDZ) 10000 -1640.4564(80) E. Giner Coupling CI and FN-DMC
81. logos Introduction WF methods FN-DMC with CIPSI trial wave functions Ionization Potential with HF in cc-pVDZ and cc-pVTZ 5 6 7 8 9 10 Charge of the nucleus −0.002 0.000 0.002 0.004 0.006 Absolute error to the IP (Hartree) FN-DMC (HF cc-pVDZ) FN-DMC (HF cc-pVTZ) Chemical accuracy Chemical accuracy FIGURE: Error to the estimated exact IP using FN-DMC with HF nodes in cc-pVDZ and cc-pVTZ E. Giner Coupling CI and FN-DMC
82. logos Introduction WF methods FN-DMC with CIPSI trial wave functions Ionization Potential with CIPSI in cc-pVDZ and cc-pVTZ 5 6 7 8 9 10 Charge of the nucleus −0.006 −0.003 0.000 0.003 0.006 0.009 Absolute error to the IP (Hartree) FN-DMC (CIPSI cc-pVDZ) FN-DMC (CIPSI cc-pVTZ) Chemical accuracy Chemical accuracy FIGURE: Error to the estimated exact IP using FN-DMC with CIPSI nodes in cc-pVDZ and cc-pVTZ E. Giner Coupling CI and FN-DMC
83. logos Introduction WF methods FN-DMC with CIPSI trial wave functions Benchmark study of Ionization Potentials at the FCI level : Na to Mg Na Mg Al Si P S Cl Ar cc-pVDZ 0.0060 0.0038 0.0057 0.0084 0.0113 0.0267 0.0234 0.0208 aug-cc-pVDZ 0.0059 0.0044 0.0045 0.0061 0.0077 0.0198 0.0154 0.0122 cc-pVTZ 0.0042 0.0025 0.0024 0.0026 0.0033 0.0101 0.0101 0.0112 aug-cc-pVTZ 0.0041 0.0023 0.0022 0.0022 0.0024 0.0084 0.0082 0.0083 cc-pVQZ 0.0027 0.0017 0.0014 0.0015 0.0016 0.0049 0.0046 0.0054 TABLE: Convergence of the aboslute errors in Hartree of the IP calculated at the converged CIPSI level E. Giner Coupling CI and FN-DMC
84. logos Introduction WF methods FN-DMC with CIPSI trial wave functions FCI+FN-DMC study DMC(CIPSI)DZ DMC(CIPSI)TZ B 0.00257(54) 0.00239(66) B+ 0.00118(37) 0.00162(27) C 0.00509(71) 0.00314(42) C+ 0.00269(60) 0.00252(43) N 0.00686(60) 0.00447(60) N+ 0.00725(94) 0.00385(55) O 0.01248(77) 0.00606(76) O+ 0.00889(61) 0.00441(81) F 0.01546(87) 0.00752(60) F+ 0.01383(76) 0.00819(57) Ne 0.01955(66) 0.01151(77) Ne+ 0.01707(58) 0.01005(70) Na 0.01700(76) Na+ 0.01681(68) Mg 0.01764(75) Mg+ 0.01730(58) Al 0.02089(93) Al+ 0.02101(87) Si 0.02348(85) Si+ 0.02561(97) P 0.02646(81) P+ 0.02732(70) S 0.03164(72) S+ 0.02944(81) Cl 0.03618(85) Cl+ 0.03427(83) Ar 0.04038(97) Ar+ 0.04040(100) TABLE: Erreurs absolues par rapport aux énergies totales estimées exactes non relativistes[?] des énergies totales FN-DMC obtenues avec des nœuds Hartree-Fock (RHF) et CIPSI en base VDZ et VTZ . E. Giner Coupling CI and FN-DMC
85. logos Introduction WF methods FN-DMC with CIPSI trial wave functions Error IP FCI 4 6 8 10 12 14 16 18 Charge of the nucleus 0.000 0.006 0.012 0.018 0.024 Absolute error to the IP (Hartree) FCI cc-pVDZ FCI cc-pVTZ FCI cc-pVQZ Chemical accuracy Chemical accuracy E. Giner Coupling CI and FN-DMC