Size-extensive wave functions for QMC A linear-scaling GVB approach
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1 Size-extensive wave functions for QMC A linear-scaling GVB approach Claudia Filippi, University of Twente, The Netherlands Francesco Fracchia, University of Pisa, Italy Claudio Amovilli, University of Pisa, Italy ES12 Workshop, Wake Forest University, June 5-8, 2012
2 Wave functions and QMC: A healthy obsession Advantages of QMC Favorable scaling: N 4 versus for instance N 7 in CCSD(T) Freedom on the functional form of the wave functions... Inclusion of r ij terms, non-orthogonal orbitals etc. However No optimal recipe for the construction of the trial function Multi-determinant wave function is often solution Computational cost remains high even for small systems
3 In search of different functional forms Active area of research Geminals, pfaffians, backflow... The trial wavefunction should meet the following requirements Accurate Compact functional form Good scaling with respect to the size of the system Size-consistency and size-extensivity
4 Size-extensivity and size-consistency Size-extensivity (Bartlett, Ann. Rev. Phys. Chem. 1981) Computed energy scales linearly with number of particles lim (E n+1 E n ) = C n Size-consistency: Additivity of separate chemical units E A+B = E A + E B
5 Common choice in QMC Jastrow-Slater wave functions Ψ(r 1,..., r N ) = J (r 1,..., r N ) i c i D i (r 1,..., r N ) J Jastrow correlation factor Positive function of inter-particle distances Explicit dependence on electron-electron distances r ij ci D i Determinants of single-particle orbitals Few and not millions of determinants Determines the nodal surface
6 Determinants: A complete active space expansion CAS wave functions Good for isomerization, excitations,...!"#$%&' ()*"&' +++' In principle, size-consistent and size-extensive but scales as N! Small active space Not all electrons correlated Small portion of PES described Large active space Truncation Not size-extensive/consistent... dangerous
7 CAS wave functions and atomization energies G2 set with full-valence CAS (Petruzielo et al. JCP, 2012) Nearly chemical accuracy on atomization energies MAD of 1.2 kcal/mol in DMC Very careful truncation (small thereshold + inclusion of doubles) However, C 2 H determinants!
8 CAS wave functions and excitations Cyanines dyes (Send, Valsson, Filippi, JCTC 2011) Starting determinantal wave function: Already compromises CAS (4,6) (6,10) (8,14) (10,9) (12,11) Det QMC setup: Truncation Thr Det
9 A localized description of a many-body wave function Desire From molecular orbital to local description of molecule Bonding (over 2 or more centers), antibonding, lone-pairs...! "#$%&'#(%#)!!!! *$+,! &'#(%#)!!!!! In the context of a multi-determinant wave function!
10 GVB wave functions as inspiration Inspiration from multi-determinantal expression of GVB-PP Ψ GVB PP = Φ N i=1 Limited variational flexibility Our starting point q i Φ a i a i b i b i + Ψ LGVB1 = c 0 Φ N i<j 1 2 N i=1 q i q j Φ a i a i,a j a j b i b i,b j b j +... c i Φ a i a i b i b i
11 Our starting wave function: Emerging of localization in MCSCF LGVB1 wave function Ψ LGVB1 = c 0 Φ N i=1 c i Φ a i a i b i b i
12 Our starting wave function: Emerging of localization LGVB1 wave function Ψ LGVB1 = c 0 Φ 0 + Correlated all valence electrons 1 2 N i=1 c i Φ b i b i a i a i Good description of single-bond dissociation in a molecule Compact and scales linearly with system size Localization emerges naturally in MCSCF optimization
13 Improving on our starting LGVB1 wave function Tests: Use these localized orbitals in CISDTQ 1) Correlation mainly among close pairs (one atom in common)... Blocks of CAS(4,4) wave functions
14 Improving on our starting LGVB1 wave function Tests: Use these localized orbitals in CISDTQ 2) Hierarchy of excitations within close pair, completing CAS(4,4) E1 E2
15 Hierarchy of excitations: Correlating adjacent electron pairs E3 E4 E5 E6
16 Hierarchy of excitations: Correlating adjacent electron pairs E7 E8 E9 E10
17 Modular truncation Only excitations that couple adjacent electron pairs Approach is size-extensive Number of determinants scales linearly with size system Additional hierarchy within excitations of each pair
18 Building the LGVB wave function Correlating all 18 adjacent electron pairs
19 Building the LGVB wave function CAS(4,4) Correlating all 18 adjacent electron pairs
20 Building the LGVB wave function CAS(4,4) Correlating all 18 adjacent electron pairs
21 Building the LGVB wave function CAS(4,4) Correlating all 18 adjacent electron pairs
22 Building the LGVB wave function CAS(4,4) Correlating all 18 adjacent electron pairs
23 LGVBn wave functions: Correlating adjecent electron pairs with n carbon atoms Ψ CSFs N(CSF) LGVB1 E0 + E1 3n + 2 LGVB2 LGVB1 + E2 15n + 2 LGVB3 LGVB2 + E3 18n + 3 LGVB4 LGVB3 + E4 30n + 3 LGVB5 LGVB4 + E5 36n + 3 LGVB6 LGVB5 + E6 48n + 3 LGVB7 LGVB6 + E7 60n + 3 LGVB8 LGVB7 + E8 72n + 3 LGVB9 LGVB8 + E9 84n + 3 LGVB10 LGVB9 + E10 96n + 3
24 Size of LGVBn wave functions Ψ N(CSF) LGVB1 11 LGVB2 47 LGVB4 93 LGVB CAS(20,20) >10 8
25 Back to quantum Monte Carlo: Jastrow-LGVBn Ψ(r 1,..., r N ) = J (r 1,..., r N ) LGVBn Generation of localized orbitals in MCSCF with LGVB1 Optimization of orbitals, CI, Jastrow in J-LGVB1 Further increase the wave function as J-LGVBn (n> 1)
26 Fragmentation tests Homolytic bond breaking Breaking of one electron pair Difficult to balance description of correlation in various fragments We must respect size-consistency and size-extensivity Set of fragmentation reactions: N 2 H 4 2NH 2 HNO 2 OH + NO CH 3 NH 2 CH 3 + NH 2 CH 3 OH CH 3 + OH
27 N 2 H 4 : VMC total energy VMC energy (Hartree) J-LGVB1 J-LGVB J-LGVBn order
28 N 2 H 4 : VMC total energy det VMC energy (Hartree) J-LGVB1 J-LGVB J-LGVBn order
29 N 2 H 4 : VMC total energy VMC energy (Hartree) J-LGVB1 1 det CAS(2,2) J-LGVB J-LGVBn order
30 N 2 H 4 : VMC total energy VMC energy (Hartree) J-LGVB1 1 det CAS(2,2) CAS(6,6) J-LGVB J-LGVBn order
31 N 2 H 4 : VMC total energy VMC energy (Hartree) J-LGVB1 1 det CAS(2,2) CAS(6,6) CAS(14,14) J-LGVB J-LGVBn order
32 N 2 H 4 : VMC total energy VMC energy (Hartree) J-LGVB det CAS(2,2) CAS(6,6) CAS(14,14) J-LGVBn order J-LGVB10
33 DMC fragmentation energy Exp. DMC binding energy (kcal/mol) J-LGVB1 J-LGVB J-LGVBn order
34 DMC fragmentation energy + DMC binding energy (kcal/mol) Exp. J-LGVB1 1 det J-LGVB J-LGVBn order
35 DMC fragmentation energy + DMC binding energy (kcal/mol) Exp. J-LGVB1 CAS(2,2) 1 det J-LGVB J-LGVBn order
36 DMC fragmentation energy Exp. DMC binding energy (kcal/mol) J-LGVB1 CAS(2,2) CAS(6,6) 1 det J-LGVB J-LGVBn order
37 DMC fragmentation energy + DMC binding energy (kcal/mol) Exp. J-LGVB1 CAS(14,14) CAS(2,2) CAS(6,6) 1 det J-LGVB J-LGVBn order
38 Fragmentation energy: DMC and CC at comparison Binding energy (kcal/mol) J-LGVB1 Exp. J-LGVB J-LGVBn order
39 Fragmentation energy: DMC and CC at comparison Binding energy (kcal/mol) J-LGVB1 Exp. J-LGVB CCSD(T)/cc-pVTZ J-LGVBn order
40 Fragmentation energy: DMC and CC at comparison Binding energy (kcal/mol) J-LGVB1 Exp. CCSD(T)/cc-pVQZ J-LGVB CCSD(T)/cc-pVTZ J-LGVBn order
41 Fragmentation energy: DMC and CC at comparison Binding energy (kcal/mol) J-LGVB1 Exp. CCSD(T)/cc-pV5Z CCSD(T)/cc-pVQZ J-LGVB CCSD(T)/cc-pVTZ J-LGVBn order
42 Fragmentations: Summary Method MAD MAX (kcal/mol) DMC 1 det DMC J-LGVB DMC J-LGVB CCSD(T)/cc-pVTZ CCSD(T)/cc-pVQZ CCSD(T)/cc-pV5Z B3LYP PBE M06-2X M08-HX B2PLYP DMC/pVTZ; DFT/aug-cc-pVQZ
43 Barrier heigths in reactions: In progress (1) Example: CH 4 + NH 2 CH 3 + NH 3 New orbital type: Bonding/antibonding of 2 elec on 3 centers Transition state Method BH for BH rev E DMC a 1 det J-LGVB J-LGVB G CCSD(T) a Reference value ? a DMC/aug-pVTZ; CC/aug-cc-pVTZ
44 Barrier heigths in reactions: In progress (2) Example: NH 2 + H 2 O NH 3 + OH Again: Bonding/antibonding of 2 elec on 3 centers Transition state Method BH for BH rev E DMC a 1 det J-LGVB J-LGVB G CCSD(T) a Reference value ??? a DMC/aug-pVTZ; CC/aug-cc-pVTZ
45 Challenges: The classical case of benzene 1) Multiple resonances J-(M)LGBV wave functions? antibonding bonding Problematic for MCSCF but not for QMC
46 Challenges: The classical case of benzene 2) Hybrid localized-delocalized description Localized description would be preferable
47 Conclusions Development of Jastrow-Slater Linear-GVB wave functions based on localized orbital, allow modular truncation correlate all valence electrons, compact, and size-extensive number of determinants scales linearly with system size Excellent performance on fragmentation of homolytic bonds J-LGVB2 satisfies chemical accuracy Performs as well as CCSD(T) with large cc-pv5z Barrier heights in progress Open challenge: Multiple resonances
48 Collaborators Francesco Fracchia, University of Pisa, Italy Claudio Amovilli, University of Pisa, Italy Fracchia, Filippi, Amovilli, JCTC (2012)
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