Tensor Network in Chemistry: Recent DMRG/TTNS Studies and Perspectives for Catalysis Research Naoki Nakatani Catalysis Research Center, Hokkaido University, Japan Tensor Network States: Algorithms and Applications Dec. 5, Beijing

Catalysts Reaction rate is given by Reaction Coordinate EaEa E' a Theoretical understanding plays an important role for designing new catalysts Catalyst promotes chemical reaction by lowering the energy barrier However, reaction mechanism becomes much complicated… With Catalyst Without Catalyst What’s happened?

What Catalyst is desired? Designing a new iron catalyst: You can get much money!! Widely used but very expensive Difficult to use but very cheap

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) 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 Many-body wavefunction can be spanned by taking particle excitations from a reference configuration virtual occupied dominant small perturbation

Secondary Inactive Active Space Inside AS: dominant Outside AS: small perturbation Defining “Active Space” (AS), orbital subspace which can describe important contributions Computational cost: O(N!) for AS AS is limited up to 14~16 orbs. Multi-reference Ansatz (CASSCF approach) Transition metal complex, Mean-field approximation doesn’t work anymore

where QC-DMRG code can approach ca 100 sites (orbitals) calculation AS limitation is considerably eased Computational cost: O(M 3 k 3 +M 2 k 4 )/sweep Memory cost: O(M 2 k 2 ) Disk storage: O(M 2 k 3 ) DMRG can be applied to solve CASSCF wavefunction Practical DMRG Calculations in Quantum Chemistry

Several implementations of DMRG-CASSCF are available 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…) Used by only few groups DMRG-CASSCF Implementations Integration to the conventional (i.e. well-established) package is important

It’s not free software, but they already have a lot of users &rasscf symmetry= 1 spin= 1 nactel= inactive= 49 0 ras2= 0 16 lumorb ciroot= dmrg= 100 Hopefully, we can get many DMRG/TNS people in near future!! DMRG on Molcas package 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

N C dependency of polyene ground state (AS : full π-valence) Energy agrees very well with CAS Negative error in CASSCF might be a numerical error DMRG-CASSCF/CASPT2 shows polynomial scaling DMRG-CASSCF scales better for (14e, 14o) and larger Benchmark: DMRG-CASSCF/cu4-CASPT2

Fe S S Benchmark: DMRG-CASSCF Geometry Optimization [Fe 2 S 2 (CH 3 ) 4 ] 2− Density Functional Theory (BP86) CASSCF(22 elec., 16 orbs.) – performed by DMRG-CASSCF CASSCF(10 elec., 10 orbs.) HS LS HS LS HS LS Fe-Fe = Å / S-S = Å Fe-Fe = Å / S-S = Å Fe-Fe = Å / S-S = Å Fe-Fe = Å / S-S = Å Fe-Fe = Å / S-S = Å Fe-Fe = Å / S-S = Å HS-LS Gap = 0.12 eV HS-LS Gap = 0.06 eV HS-LS Gap = 0.86 eV

Tree Tensor Network States for QC What’s the entanglement structure of molecule? Efficient QC-DMRG algorithm on Tree graph lattice Illustrative calculations

Two different types of TTNS were proposed: Here, focused on TPS type TTNS TTNS: Note for Structure “Layered” TTNS TTNS where each site has physical index “TPS-type” TTNS

H 2 O / cc-pVDZ Plotted exchange interaction b/w two orbitals Hamiltonian is highly non-local (main source of entanglement) Hard to map entanglement onto 1D-lattice! Entanglement Structure of Molecule

Even though, there still be a lot of non-local interaction!! Heuristic algorithm to minimize a cost function Exchange interactions to be closer as possible Somehow to map onto 1D-lattice for DMRG Can any Tensor Network approach give the better computational scaling?

Biparticity of tree graph (i.e. no cycles) is useful to keep “Canonical Form” of wavefunction LEFTRIGHT 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 MPS TTNS Having the same super-block structure TTNS: Renormalization along with Tree Graph O(M 3 k 3 +M 2 k 4 ) per sweep O(M 4 k 3 +M 2 k 5 ) per sweep

DMRG on MPS O(M 3 k 2 ) per site DMRG Algorithm on TTNS DMRG on TTNS O(M 4 k 2 ) per site

Half-Renormalization (HR): To reduce the variational space to be searched for each step Reduced computational cost without loss of accuracy! n1n1 n2n2 M 44 M 2 → 4M M MM M n1n1 n2n2 M 44 4M4M4M4M 2-Site Algorithm

Minimum Spanning Tree (MST)Minimum Entangled(?) Tree (MET) E = − (M = 200) CPU: 1210 sec. E = − (M = 200) CPU: 827 sec. M max = 4 10 M max = 4 8 What “shape of tree” should be used?

Large entanglement due to triple bond breaking The same accuracy can be obtained from TTNS with less than half # states in MPS Computational scaling is also better in TTNS R = ÅR = Å R = Å N 2 molecule: Case where TTNS works better

Strong correlation from sextuple bond in Cr 2 Number of states can be reduced in TTNS but the total performance scales better in MPS Cr 2 molecule: Case where MPS works better

Stilbenoid dendrimer: Model for Light-Harvesting system in Photosystem g = 2 (110e, 110o) 2-4 times faster in TTNS Densritic molecule: Case where TTNS works extremely fine

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