^{1}and Garnet Kin-Lic Chan

^{1}

### Abstract

We investigate tree tensor network states for quantum chemistry. Tree tensor network states represent one of the simplest generalizations of matrix product states and the density matrix renormalization group. While matrix product states encode a one-dimensional entanglement structure, tree tensor network states encode a tree entanglement structure, allowing for a more flexible description of general molecules. We describe an optimal tree tensor network state algorithm for quantum chemistry. We introduce the concept of half-renormalization which greatly improves the efficiency of the calculations. Using our efficient formulation we demonstrate the strengths and weaknesses of tree tensor network states versus matrix product states. We carry out benchmark calculations both on tree systems (hydrogen trees and π-conjugated dendrimers) as well as non-tree molecules (hydrogen chains, nitrogen dimer, and chromium dimer). In general, tree tensor network states require much fewer renormalized states to achieve the same accuracy as matrix product states. In non-tree molecules, whether this translates into a computational savings is system dependent, due to the higher prefactor and computational scaling associated with tree algorithms. In tree like molecules, tree network states are easily superior to matrix product states. As an illustration, our largest dendrimer calculation with tree tensor network states correlates 110 electrons in 110 active orbitals.

This work was supported by the National Science Foundation (NSF) through Grant No. NSF-OCI-1148287 and NSF-CHE-1213933.

I. INTRODUCTION

II. OVERVIEW OF THE DMRG ALGORITHM BASED ON MPS

III. TREE TENSOR NETWORK STATES (TTNS)

A. Half-renormalization

B. Two-site TTNS algorithm

C. Tree shape and site ordering

IV. ILLUSTRATIVE CALCULATIONS

V. CONCLUSIONS

### Key Topics

- Tensor methods
- 48.0
- Wave functions
- 13.0
- Quantum entanglement
- 11.0
- Renormalization
- 9.0
- Chromium
- 7.0

## Figures

Examples of trees. The left panel shows a tree with degree *Z* = 3 and depth Δ = 4, and the right panel shows a tree with degree *Z* = 4 and depth Δ = 3.

Examples of trees. The left panel shows a tree with degree *Z* = 3 and depth Δ = 4, and the right panel shows a tree with degree *Z* = 4 and depth Δ = 3.

Graphical representations of MPS. (a) one-site coefficient vector is a rank-3 tensor, (b) the MPS wavefunction is given by contracting all horizontal bonds, (c) the overlap of two MPS wavefunctions can be efficiently computed recursively.

Graphical representations of MPS. (a) one-site coefficient vector is a rank-3 tensor, (b) the MPS wavefunction is given by contracting all horizontal bonds, (c) the overlap of two MPS wavefunctions can be efficiently computed recursively.

Graphical representations of DMRG wavefunction. Canonical form of MPS wavefunction (top) can be re-written as a block diagram in DMRG language (bottom).

Graphical representations of DMRG wavefunction. Canonical form of MPS wavefunction (top) can be re-written as a block diagram in DMRG language (bottom).

Graphical representations of MPO. The Hamiltonian operator is a rank-2*k* tensor (top panel). This can be divided into a contracted product of site-independent tensors similar to a MPS, leading to a Matrix Product Operator representation (MPO, bottom panel), in which each site tensor is a rank-4 tensor.

Graphical representations of MPO. The Hamiltonian operator is a rank-2*k* tensor (top panel). This can be divided into a contracted product of site-independent tensors similar to a MPS, leading to a Matrix Product Operator representation (MPO, bottom panel), in which each site tensor is a rank-4 tensor.

Graphical representation of TTNS. Left panel shows overall structure of TTNS as described in Eq. (15) , and right panel shows the one-site wavefunction spanned by the renormalized basis, as described in Eq. (16) . Note that physical indices (vertical bonds) are omitted in the right panel.

Half-renormalization (HR) algorithm on TTNS. (a) One-site algorithm: *Z* − 1 system blocks are mapped into one system block. (b) Two-site algorithm: additionally, *Z* − 1 environment blocks are mapped into one environment block. The half renormalized block contains only 4*M* states for any *Z* > 2.

Half-renormalization (HR) algorithm on TTNS. (a) One-site algorithm: *Z* − 1 system blocks are mapped into one system block. (b) Two-site algorithm: additionally, *Z* − 1 environment blocks are mapped into one environment block. The half renormalized block contains only 4*M* states for any *Z* > 2.

Sweep algorithm on a tree (*Z* = 3), by depth-first search with backtracking, where the labels indicate the order of the search. (a) One-site algorithm: starting from the center site, 1 sweep contains 18 one-site optimization steps, and (b) Two-site algorithm: starting from the center site and one adjacent site, 1 sweep contains 12 two-site optimization steps.

Sweep algorithm on a tree (*Z* = 3), by depth-first search with backtracking, where the labels indicate the order of the search. (a) One-site algorithm: starting from the center site, 1 sweep contains 18 one-site optimization steps, and (b) Two-site algorithm: starting from the center site and one adjacent site, 1 sweep contains 12 two-site optimization steps.

Block structure of the two-site algorithm. Top panel shows the two-site MPS which contains one system block, system site, environment site, and one environment block. Bottom panel shows the two-site TTNS which contains *Z* − 1 system blocks, system site, environment site, and *Z* − 1 environment blocks.

Block structure of the two-site algorithm. Top panel shows the two-site MPS which contains one system block, system site, environment site, and one environment block. Bottom panel shows the two-site TTNS which contains *Z* − 1 system blocks, system site, environment site, and *Z* − 1 environment blocks.

Approximate degree-fixed minimum spanning tree (MST, *Z* = 3) for 24 orbitals from a RHF/cc-pVDZ calculation of the water molecule. Left panel shows a representation of the exchange integral *K* _{ ij }, where solid lines denote *K* _{ ij } ⩾ 0.10, dashed lines denote *K* _{ ij } ⩾ 0.07, and dotted lines denote *K* _{ ij } < 0.07. Colored lines are connected lines in the right panel and gray lines are ignored interactions. For *K* _{ ij } < 0.07, only connected lines are shown. Labels in the right panel indicate the MOs (as indexed by energy).

Approximate degree-fixed minimum spanning tree (MST, *Z* = 3) for 24 orbitals from a RHF/cc-pVDZ calculation of the water molecule. Left panel shows a representation of the exchange integral *K* _{ ij }, where solid lines denote *K* _{ ij } ⩾ 0.10, dashed lines denote *K* _{ ij } ⩾ 0.07, and dotted lines denote *K* _{ ij } < 0.07. Colored lines are connected lines in the right panel and gray lines are ignored interactions. For *K* _{ ij } < 0.07, only connected lines are shown. Labels in the right panel indicate the MOs (as indexed by energy).

Minimum-entangled tree (MET, *Z* = 3) for 24 orbitals from a RHF/cc-pVDZ calculation of the water molecule. Left panel shows the stepwise construction of MET. First, 24 sites are divided into 1 center site and 3 branches containing 7, 8, and 8 sites. Second, 7 sites are divided into 1 site and 2 branches containing 3 sites for each, and 8 sites are divided into 2 sites and 2 branches containing 3 sites for each. Finally, we used a genetic algorithm to map the orbitals to the tree sites as shown in right panel. Labels in right panel indicate MO indices (as indexed by energy).

Minimum-entangled tree (MET, *Z* = 3) for 24 orbitals from a RHF/cc-pVDZ calculation of the water molecule. Left panel shows the stepwise construction of MET. First, 24 sites are divided into 1 center site and 3 branches containing 7, 8, and 8 sites. Second, 7 sites are divided into 1 site and 2 branches containing 3 sites for each, and 8 sites are divided into 2 sites and 2 branches containing 3 sites for each. Finally, we used a genetic algorithm to map the orbitals to the tree sites as shown in right panel. Labels in right panel indicate MO indices (as indexed by energy).

Hydrogen atoms on Cayley-trees (*Z* = 3, *g* = 2 (10 sites), 3 (22 sites), and 4 (46 sites)). Top panel shows the actual structures of the hydrogen trees, in which red atoms are the core region, orange atoms are for *g* = 2, yellow atoms are for *g* = 3, and green atoms are for *g* = 4. The distance between adjacent hydrogens is 2.0 bohrs and torsional angle between different generations is 30°. Bottom panel shows corresponding Cayley-tree diagrams. The red-dotted line denotes the site ordering used in the MPS.

Hydrogen atoms on Cayley-trees (*Z* = 3, *g* = 2 (10 sites), 3 (22 sites), and 4 (46 sites)). Top panel shows the actual structures of the hydrogen trees, in which red atoms are the core region, orange atoms are for *g* = 2, yellow atoms are for *g* = 3, and green atoms are for *g* = 4. The distance between adjacent hydrogens is 2.0 bohrs and torsional angle between different generations is 30°. Bottom panel shows corresponding Cayley-tree diagrams. The red-dotted line denotes the site ordering used in the MPS.

Energy convergence and total CPU time computed with normal two-site TTNS (denoted full) and half-renormalized two-site TTNS (denoted HR-TTNS).

Energy convergence and total CPU time computed with normal two-site TTNS (denoted full) and half-renormalized two-site TTNS (denoted HR-TTNS).

Energy convergence of hydrogen trees plotted with respect to the number of renormalized states *M* and CPU time (s) per one sweep; (a) and (b) show energy versus *M* and CPU time per sweep, respectively, for 10 sites, and (c) and (d) show energy versus *M* and CPU time per sweep, respectively, for 22 sites. All calculations are for the triplet ground state energy in an orthogonalized STO-3G basis.

Energy convergence of hydrogen trees plotted with respect to the number of renormalized states *M* and CPU time (s) per one sweep; (a) and (b) show energy versus *M* and CPU time per sweep, respectively, for 10 sites, and (c) and (d) show energy versus *M* and CPU time per sweep, respectively, for 22 sites. All calculations are for the triplet ground state energy in an orthogonalized STO-3G basis.

Energy convergence of hydrogen chains with canonical molecular orbitals (CMOs) plotted by number of renormalized states *M* and CPU time (s) per one sweep; (a) and (b) showed those of *M* and CPU time per sweep, respectively, for 20 sites, and (c) and (d) showed those of *M* and CPU time per sweep, respectively, for 30 sites. Singlet ground state energy was computed with STO-3G basis sets.

Energy convergence of hydrogen chains with canonical molecular orbitals (CMOs) plotted by number of renormalized states *M* and CPU time (s) per one sweep; (a) and (b) showed those of *M* and CPU time per sweep, respectively, for 20 sites, and (c) and (d) showed those of *M* and CPU time per sweep, respectively, for 30 sites. Singlet ground state energy was computed with STO-3G basis sets.

Energy errors for bond dissociation of nitrogen dimer using (10e, 26o) active space and cc-pVDZ basis sets. Reference energies were from previous full-CI work in Ref. ^{ 41 } .

Energy errors for bond dissociation of nitrogen dimer using (10e, 26o) active space and cc-pVDZ basis sets. Reference energies were from previous full-CI work in Ref. ^{ 41 } .

Energy convergence of the chromium dimer at 1.5 Å using a (24e, 30o) active space and Ahlrichs’ split-valence plus polarization (SVP) basis set. (a) Plotted as a function of *M* and (b) as a function of CPU time per sweep. Reference energies were from previous DMRG work in Ref. ^{ 42 } .

Energy convergence of the chromium dimer at 1.5 Å using a (24e, 30o) active space and Ahlrichs’ split-valence plus polarization (SVP) basis set. (a) Plotted as a function of *M* and (b) as a function of CPU time per sweep. Reference energies were from previous DMRG work in Ref. ^{ 42 } .

Structures of stilbenoid dendrimers *g* = 0, 1, and 2. Geometries were optimized at the B3LYP/cc-pVDZ level of theory.

Structures of stilbenoid dendrimers *g* = 0, 1, and 2. Geometries were optimized at the B3LYP/cc-pVDZ level of theory.

Tree graphs for TTNS calculation of stilbenoid dendrimer (*g* = 1). (a) Selected molecular orbitals of the localized ethylene and benzene fragments, (b) Tree graph of the single-valence (STO-3G) calculation, and (c) tree graph of the double-valence (6-31G) calculation where the filled circles represent local π and π* orbitals and empty circles represent local 3*p* orbitals. Note that identical orbitals are omitted in the tree graphs.

Tree graphs for TTNS calculation of stilbenoid dendrimer (*g* = 1). (a) Selected molecular orbitals of the localized ethylene and benzene fragments, (b) Tree graph of the single-valence (STO-3G) calculation, and (c) tree graph of the double-valence (6-31G) calculation where the filled circles represent local π and π* orbitals and empty circles represent local 3*p* orbitals. Note that identical orbitals are omitted in the tree graphs.

Energy convergence of stilbenoid dendrimers *g* = 0 and *g* = 1 plotted by number of renormalized states *M* and CPU time (sec.) per sweep; (a) and (b) shows energy versus *M* and CPU time per sweep, respectively, for *g* = 0, single valence (14e, 14o), (c) and (d) shows energy versus *M* and CPU time per sweep, respectively, for *g* = 0, double valence (14e, 28o), (e) and (f) shows energy versus *M* and CPU time per sweep, respectively for *g* = 1, single valence (46e, 46o). All energies are for the singlet ground-state energy.

Energy convergence of stilbenoid dendrimers *g* = 0 and *g* = 1 plotted by number of renormalized states *M* and CPU time (sec.) per sweep; (a) and (b) shows energy versus *M* and CPU time per sweep, respectively, for *g* = 0, single valence (14e, 14o), (c) and (d) shows energy versus *M* and CPU time per sweep, respectively, for *g* = 0, double valence (14e, 28o), (e) and (f) shows energy versus *M* and CPU time per sweep, respectively for *g* = 1, single valence (46e, 46o). All energies are for the singlet ground-state energy.

## Tables

Complexity of optimal tensor contractions for per site. Multiplying by gives complexity per sweep.

Complexity of optimal tensor contractions for per site. Multiplying by gives complexity per sweep.

The ground state energies of water molecule calculated with two different trees.

The ground state energies of water molecule calculated with two different trees.

CPU times per sweep in s for MPS and TTNS calculations for the nitrogen dimer using (10e, 26o) active space and cc-pVDZ basis sets.

CPU times per sweep in s for MPS and TTNS calculations for the nitrogen dimer using (10e, 26o) active space and cc-pVDZ basis sets.

Complexity of optimal tensor contractions for TTNS renormalization per site. Multiplying by gives complexity per sweep.

Complexity of optimal tensor contractions for TTNS renormalization per site. Multiplying by gives complexity per sweep.

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