Report

Tensor Network States: Algorithms and Applications Dec. 5, 2014 @ Beijing Tensor Network in Chemistry: Recent DMRG/TTNS Studies and Perspectives for Catalysis Research Naoki Nakatani Catalysis Research Center, Hokkaido University, Japan Catalysts Without Catalyst Ea E'a Catalyst promotes chemical reaction by lowering the energy barrier However, reaction mechanism becomes much complicated… With Catalyst What’s happened? Reaction Coordinate Reaction rate is given by k e Ea RT Theoretical understanding plays an important role for designing new catalysts What Catalyst is desired? H He Li Be B C N O F Ne Na Mg Al Si P Se Cl Ar K Ca Sc Ti V Cr Mn Fe Co Ni Cu Zn Ga Ge As Se Br Kr Rb Sr Y Zr Nb Mo Tc Ru Rh Pd Ag Cd In Sn Sb Te I Xe Cs Ba La Hf Ta W Re Os Ir Pt Au Hg Tl Pb Bi Po At Rn Fr Ra Ac Difficult to use but very cheap Widely used but very expensive Ce Pr Nd Pm Sm Eu Gd Tb Dy Ho Er Tm Yb Lu Th Pa U Np Pu Am Cm Bk Cf Es Fm Md No Lr Designing a new iron catalyst: You can get much money!! DMRG-related Activity in Chemistry • Why we need DMRG? • Providing a convenient tool to doing DMRG for Chemistry • Get starting practical applications Single-reference Ansatz (post-HF approach) virtual Common molecules, Mean-field approximation (i.e. Hartree-Fock theory) works very well 1 electron configuration covers more than 99% of electronic energy for the ground state The rest can be treated by a small perturbation occupied Many-body wavefunction can be spanned by taking particle excitations from a reference configuration Ψ Ψ H F c i E i Ψ H F c ij E i E j Ψ H F a Ψ HF dominant E i Ψ HF a ab a b a dominant small perturbation Multi-reference Ansatz (CASSCF approach) Secondary Transition metal complex, Mean-field approximation doesn’t work anymore Active Space Defining “Active Space” (AS), orbital subspace which can describe important contributions Inactive Computational cost: O(N!) for AS AS is limited up to 14~16 orbs. Ψ CA SSCF n1 Inside AS: dominant Ψ CA SSCF Outside AS: small perturbation C n1 nk nk 1, , k A S n1 nk Practical DMRG Calculations in Quantum Chemistry DMRG can be applied to solve CASSCF wavefunction C 12 34 β α β α T r A1 A 2 A 3 A 4 σ where A i : M M Computational cost: O(M3k3+M2k4)/sweep α Memory cost: O(M2k2) β 1234 Disk storage: O(M2k3) QC-DMRG code can approach ca 100 sites (orbitals) calculation AS limitation is considerably eased DMRG-CASSCF Implementations Several implementations of DMRG-CASSCF are available Used by only few groups Self-involved packages: • ORZ package: DMRG-CASSCF/CASPT2 in T. Yanai group (obtain upon request) • ORCA package: DMRG-CASSCF/NEVPT2 in F. Neese group (free for academic use) • Most users (including me) don’t like to use a new code because reading the manual is tough!! • It’s so hard for small groups to implement everything from scratch (e.g. integrals, SCF, geometry opt., relativistic effect, etc…) Integration to the conventional (i.e. well-established) package is important DMRG on Molcas package It’s not free software, but they already have a lot of users &rasscf symmetry= 1 spin= 1 nactel= 16 0 0 inactive= 49 0 ras2= 0 16 lumorb ciroot= 1 1 1 dmrg= 100 Normal CASSCF input in Molcas Only 1 line is required to carry out DMRG-CASSCF calc. All other settings are automatically determined for novice users Hopefully, we can get many DMRG/TNS people in near future!! Benchmark: DMRG-CASSCF/cu4-CASPT2 NC dependency of polyene ground state (AS : full π-valence) 1.E+05 1.00 only CASSCF 0.80 0.60 EDMRG − ECAS / mEh Wall time / sec. 1.E+04 CASSCF/CASPT2 1.E+03 DMRG-CASSCF/cu4-CASPT2 (M = 100) 1.E+02 0.40 DMRG-CASSCF (M = 100) 0.20 0.00 -0.20 -0.40 cu4-CASPT2 -0.60 -0.80 1.E+01 -1.00 6 8 10 12 14 16 18 NC 20 6 8 10 12 14 16 18 NC DMRG-CASSCF/CASPT2 shows polynomial scaling Energy agrees very well with CAS DMRG-CASSCF scales better for (14e, 14o) and larger Negative error in CASSCF might be a numerical error 20 Benchmark: DMRG-CASSCF Geometry Optimization [Fe2S2(CH3)4]2− Density Functional Theory (BP86) S Fe Fe S HS Fe-Fe = 3.003 Å / S-S = 3.514 Å LS Fe-Fe = 2.716 Å / S-S = 3.526 Å HS-LS Gap = 0.86 eV CASSCF(10 elec., 10 orbs.) HS Fe-Fe = 3.076 Å / S-S = 3.567 Å LS Fe-Fe = 3.045 Å / S-S = 3.575 Å HS-LS Gap = 0.06 eV CASSCF(22 elec., 16 orbs.) – performed by DMRG-CASSCF HS Fe-Fe = 3.065 Å / S-S = 3.551 Å LS Fe-Fe = 3.011 Å / S-S = 3.557 Å HS-LS Gap = 0.12 eV Tree Tensor Network States for QC • What’s the entanglement structure of molecule? • Efficient QC-DMRG algorithm on Tree graph lattice • Illustrative calculations TTNS: Note for Structure Two different types of TTNS were proposed: Here, focused on TPS type TTNS “Layered” TTNS “TPS-type” TTNS TTNS where each site has physical index Ψ n1 nk tr A n1 A ni A nk n1 nk Entanglement Structure of Molecule H2O / cc-pVDZ 24 1 23 2 Plotted exchange interaction b/w two orbitals 3 22 (main source of entanglement) 4 21 5 20 6 7 19 Hamiltonian is highly non-local Hˆ ij σ 18 † h ij aˆ i σ aˆ j σ 1 2 † † ij lk aˆ i σ aˆ j σ aˆ k σ aˆ l σ ijkl σ σ 8 9 17 10 16 15 14 13 12 11 Hard to map entanglement onto 1D-lattice! Somehow to map onto 1D-lattice for DMRG Heuristic algorithm to minimize a cost function F K 2 ij R ij ij 7 17 6 13 12 8 24 9 18 22 3 19 10 4 2 23 16 21 14 5 15 20 11 Exchange interactions to be closer as possible Even though, there still be a lot of non-local interaction!! Can any Tensor Network approach give the better computational scaling? 1 TTNS: Renormalization along with Tree Graph Biparticity of tree graph (i.e. no cycles) is useful to keep “Canonical Form” of wavefunction MPS O(M3k3+M2k4) per sweep Having the same super-block structure TTNS O(M4k3+M2k5) per sweep LEFT RIGHT TTNS is one of natural ND-generalizations of MPS • Non-local interaction can be considered without much increase of computational scaling • Many algorithms for MPS can be reused for TTNS DMRG Algorithm on TTNS DMRG on MPS aˆ i aˆ j DMRG on TTNS aˆ i aˆ j aˆ j aˆ i O(M3k2) per site O(M4k2) per site aˆ i aˆ j 2-Site Algorithm Half-Renormalization (HR): To reduce the variational space to be searched for each step M 4 4 n1 n2 M M M M M2 → 4M M2 → 4M 4 4 n1 n2 4M 4M M Reduced computational cost without loss of accuracy! What “shape of tree” should be used? Minimum Spanning Tree (MST) Minimum Entangled(?) Tree (MET) Mmax = 410 21 Mmax = 48 18 14 12 19 15 24 1 7 23 3 24 11 20 1 12 7 6 23 2 5 9 22 4 11 5 2 16 4 3 10 8 13 9 19 17 20 18 22 E = −76.243652 (M = 200) CPU: 1210 sec. 16 8 21 15 6 10 14 13 17 E = −76.243491 (M = 200) CPU: 827 sec. N2 molecule: Case where TTNS works better Large entanglement due to triple bond breaking R = 1.1208 Å R = 1.4288 Å R = 1.9050 Å The same accuracy can be obtained from TTNS with less than half # states in MPS Computational scaling is also better in TTNS Cr2 molecule: Case where MPS works better Strong correlation from sextuple bond in Cr2 Number of states can be reduced in TTNS but the total performance scales better in MPS Densritic molecule: Case where TTNS works extremely fine Stilbenoid dendrimer: Model for Light-Harvesting system in Photosystem M 100 MPS ΔE / Eh CPU / swp. sec. -1.9437 43146 M 50 TTNS ΔE / Eh CPU / swp. sec. -2.0214 19162 2-4 times faster in TTNS g = 2 (110e, 110o) Summary DMRG on Molcas package • • • • DMRG code (G. Chan) has integrated into conventional QC program so that novice users can carry out DMRG calculation much easier Large active space (~ 100 orbs.) can be taken into account Geometry optimization, perturbation treatment, etc. can be used without further coding Hoping we can get a lot of DMRG users in Chemistry • For me, now I’ve got the useful tool to do catalysis research Tree Tensor Network States (TTNS) • • • Introduced and developed efficient DMRG algorithm on TTNS for Quantum Chemistry Performance is highly depending on system So far, it can introduce another freedom to chose a lattice where entanglement structure of molecule is to be mapped