Using Higher-Order Singular Value Decomposition To Define Weakly

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Using higher-order singular value decomposition to define weakly coupled and strongly correlated cluster states: the n-body Tucker approximation Nicholas J. Mayhall J. Chem. Theory Comput., Just Accepted Manuscript • DOI: 10.1021/acs.jctc.7b00696 • Publication Date (Web): 22 Aug 2017 Downloaded from http://pubs.acs.org on August 22, 2017

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Journal of Chemical Theory and Computation

Using higher-order singular value decomposition to define weakly coupled and strongly correlated clusters: the n-body Tucker approximation Nicholas J. Mayhall∗ Department of Chemistry, Virginia Tech, Blacksburg, VA 24060, USA An approximate wavefunction ansatz is presented which describes low-energy states of a highly clustered molecular system as a linear combination of multiple reduced-rank tensors. Using the Tucker decomposition as a way to obtain local clusters states, the exact solution is solved for in the space spanned by a small number of states on each cluster, with complete correlation occuring between limited numbers of clusters at a time. In this initial study, we report the implementation for a Heisenberg spin Hamiltonian with numerical examples of regular grid spin lattices, and ab initio-derived spin-Hamiltonians used to analyze the approximation. From these results, we find that the proposed method works well when the Hamiltonian interactions within a cluster are larger than between a cluster, and when this is not true, the method is not effective.

I.

mode product:

INTRODUCTION

Tensors, or multilinear arrays, are useful mathematical objects for storing and manipulating higher-dimensional datasets where data entries carry several attributes. However, due to the fact that each additional dimension significantly increases the number of possible data entries, mathematical techniques for efficient storage and analysis of tensors is becoming increasingly important. One particularly important group of techniques are referred to as tensor decompositions,1–4 which has been applied to many data intensive problems, i.e., data mining,5,6 facial recognition,7 signal processing.8 These are attempts to generalize the techniques of matrix factorizations, such as the eigen-decomposition or singular value decomposition (SVD), so as to obtain a rank-exposing orthogonal subspace representation of the tensor. Although, there is no complete SVD generalization to n-dimensional tensors which retains all of the desirable properties of the SVD, there are a multiple useful decompositions which have been studied. Most of these fall into one of two categories, 1) PARAFAC/CANDECOMP3 or 2) Tucker decompositions (Higher-Order SVD, HOSVD).4 In PARAFAC, one attempts to find a compact representation of a d-mode input tensor T as a sum of rank-1 tensors (i.e., a sum of kronecker products of d vectors), T ≈

R X

(2) (d) λr u(1) r ⊗ ur ⊗ · · · ⊗ ur ,

(1)

r

where R is the tensor rank, and λr is a scalar coefficient. (d) Whereas with the Tucker decomposition, the vectors ur are replaced with matrices and the coefficient becomes another tensor, C: T ≈ C ×1 U (1) ×2 U (2) · · · ×d U (d) ,

(2)

where C is a smaller “core tensor”, U (d) is a unitary matrix in the vector space associated with mode d of T called a “Tucker factor”, and C ×d U (d) denotes the d-

C ×n U (n) ⇔Ci1 ,i2 ,...,in−1 ,in ,in+1 ,... Uin ,j = Ci1 ,i2 ,...,in−1 ,j,in+1 ,...

(3)

The Tucker decomposition can also be expressed with index notation: Ti,j,...,d ≈ Cα,β,...,γ Ui,α Uj,β · · · Ud,γ

(4)

As is easily seen in Eq. 4, the Tucker decomposition is essentially a change of basis, from i to α, from j to β, etc. Without detailing the various computational algorithms for performing PARAFAC and Tucker decompositions (which is outside the scope of this paper), we simply emphasize that it is often desirable to have accurate approximations to the original tensor, but which contain much less data, resulting in more efficient storage, algebraic manipulation, and analysis. Although much of the literature pertaining to tensor decompositions exist outside of the physical sciences, there are many areas of science which benefit from efficient tensor decomposition strategies. Due to the high dimensionality of the associated quantities, quantum chemistry is an ideal example of such an area of science. During the past several years, tensor decomposition techniques have become increasingly valuable for increasing the efficiency of several quantum chemistry algorithms. Most notably perhaps, has been the significant acceleration of the two-electron repulsion integrals based on Resolution of the Identity (RI) approximations or Cholesky Decompositions (CD).9–16,16–22 More recently, the Tensor HyperContraction (THC) methodology23–26 (and related methods),27 have been developed to decompose the 4-mode two-electron integral tensor, which can reduce both storage requirements and overall scaling of methods like MP2. The MP2 or coupled cluster amplitude tensors have also been factorized via SVD,28,29 , PARAFAC,30,31 , THC,24 and Tucker decompositions,32 to understand the effect correlation has on orbitals, and to improve computational efficiency. Compression techniques have also been used to obtain optimized orbital active spaces for aggressively truncating configuration expansions in corre-

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lated calculations.33–41 A nice overview of the role of tensor decompositions in quantum chemistry can be found in Ref. 42. Tensor decompositions also have intimate connections to several emerging techniques for low-entanglement wavefunctions for strongly correlated systems.43 These are in contrast to the conventional methods for strong correlation, such as complete active space self consistent field (CASSCF),44 which this is effectively a bruteforce procedure and only possible when the number of strongly correlated orbitals is small. Although originally developed as a way to exploit scale separation in spin lattices,45–47 density matrix renormalization group (DMRG) soon became recognized as an algorithm which variationally optimizes a wavefunction having a specific decomposed tensor format known as matrix product state (MPS).43,48–52 Efficient codes have since demonstrated that DMRG can solve quantum chemistry problems which would otherwise be intractable.53–70 Because the MPS is a one-dimensional factorization of the FCI coefficient tensor, ideal performance only occurs for molecular systems with quasi one-dimensional geometry, although this has been found to not be overly restricting in many cases. Several new techniques such as complete graph tensor network states (CGTN)71 , tree tensor networks (TTN)72 , and multiscale renormalization ansatz (MERA)73 have been developed to exploit low-entanglement in higher dimensional systems, and many other strong correlation techniques are also being explored.50,74–84 While the approaches mentioned above attempt to find low-rank approximations by disentangling the indices of the FCI tensor, other approaches seek to find local subspaces which result in compact representations. Some of the approaches most relevant to this current work are the Active-Space Decomposition (ASD) of Shiozaki and coworkers,85–87 Block-Correlated Coupled Cluster (BCCC) of Li and coworkers,88–90 the cluster Mean-Field (cMF) method of Scuceria and coworkers,91 the Renormalized Exciton Method (REM) of Ma and coworkers,92–95 and the ab initio Frenkel Davydov Exciton Model (AIFDEM) of Herbert and coworkers.96–98 In ASD, the FCI space of a clustered system is decomposed as a direct product of low-energy states on each cluster. This is essentially an approximate Tucker decomposition of the FCI tensor. It is approximate in the sense that the “Tucker factors” are chosen to be eigenfunctions of the local cluster Hamiltonian, as opposed to the eigenvectors of a reduced density matrix (i.e., HOSVD). Parker et al. have shown86 that a benzene dimer’s lowenergy spectrum can be nearly converged by treating only tens of states per cluster. However, for larger numbers of clusters, even tens of states per cluster will quickly become intractable. In order to avoid this exponential increase in dimension, the DMRG algorithm was then used to approximate the ASD method for arbitrary numbers of clusters,85 with the increased correlation within a cluster leading to a decreased number of important renormalized

states compared to conventional DMRG. The BCCC method is a generalization of the coupled cluster ansatz to the case of a direct product reference wavefunction. The direct product wavefunction can capture important strong correlation effects within a block. Then, in order to capture the missing dynamical correlation, Li and coworkers88–90 use genaralized creation/annihilation operators which act to destroy a cluster’s reference state and replace it with a higher energy state (states which were discarded in the original definition of the direct product reference state). Similar to conventional coupled-cluster, these generalized excitation operators are then exponentiated and applied to the reference, with the resulting matrix elements derived in a similar (but vastly more complicated) manner as that used in single-reference coupled cluster. This has been very successful in quantum chemistry applications using both a CAS-CI reference,89,90,99,100 and as a perturbative correction to a GVB reference.101 In the cMF method, a direct product wavefunction is again assumed to describe the basic qualitative features of a given strongly correlated state. However, rather than choosing an arbitrary direct product state, Scuceria and Jiménez-Hoyos variationally optimized the molecular orbitals by minimizing the energy of the direct product wavefunction.91 This orbital optimization was found to be critical for obtaining accurate results. After optimizing the orbitals, a second-order correction (referred to as cPT2) was included which recovered a significant amount of the missing correlation for a wide range of correlation strengths of a Hubbard model. The REM approach of Malrieu and coworkers,92 can also be considered as a tensor decomposed wavefunction. Here, the large system’s wavefunction is approximated as a linear combination of block excitations, in which only a single block is excited at a time, while all others remain in their ground states. Optimizing the wavefunction in this space requires the Hamiltonian matrix elements between the various block excited states. However, instead of using the bare Hamiltonian, Malrieu and coworkers use Bloch’s effective Hamiltonian theory102,103 to obtain an effective interaction between the block excited states. This implicitly folds in the effects of both higher energy block states and cooperative multi-block excitations into the model space Hamiltonian matrix. This has been shown to be effective on gapped spin-lattices,92,104 in addition to the excited states of chromophore aggregates using the ab initio Hamiltonian.93–95 Targeting similar problems as REM, the ab initio Frenkel Davydov Exciton Model (AIFDEM) of Herbert and coworkers also seeks to approximate the low-lying singly-excited states of aggregates via a direct product wavefunction ansatz. However, unlike REM, AIFDEM uses the bare Hamiltonian in the evaluation of the exciton couplings. Although this prevents the implicit inclusion of multiblock states, (i.e., partial hybridization between Frenkel and charge-transfer excitations), AIFDEM does include some explicit 2-block charge-transfer states by


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The exact coefficient tensor can therefore be written as: ˆ A PˆB + λQ ˆ B PˆC + λQ ˆC C =C PÂ + λQ (9)

increase the number of degrees of freedom, it is necessary to avoid the construction of the full Hamiltonian matrix by only evaluating the action of the Hamiltonian onto each of the Tucker block core tensors (σ vector). This is =C PÂ PˆB PˆC 0th Order essentially the same approach taken in conventional CI ˆ A PˆB PˆC + C PÂ Q ˆ B Pˆf C + C PÂ PˆB Q ˆ C 1st Order calculations using Davidson’s algorithm.106 The contri+ CQ ÂQ ˆ B PˆC + C Q ˆ A PˆB Q ˆ C + C PÂ Q ˆB Q ˆ C 2nd Order bution to the σ vector will include a sum over all one + CQ and two cluster Hamiltonian terms. For example, the ÂQ ˆB Q ˆC + CQ 3rd Order operator coupling clusters A and B will generate the following contribution to the σ vector, thus leading to a definition of the exact coefficient tensor k ˆ AB |Aα′ Bβ ′ Cγ ′ i Cαk′ β ′ γ ′ (13) σαβγ ⇐ hAα Bβ Cγ | H in terms of a sum of decreasingly important compressed th X X tensors. Our aim is to truncate this sum at n order and Jab hAα Bβ | Sâ+ Sˆb− + Sâ− Sˆb+ = solve for the associated core tensors. a∈A b∈B As a simple example, suppose we were to truncate

this expansion after first order terms. The wavefunction (14) +2Sâz Sˆbz |Aα′ Bβ ′ i Cγ Cγ′ Cαk′ β ′ γ ′ would then be written as, XX + − − + Jab Sa,αα = ′ Sb,ββ ′ + Sa,αα′ Sb,ββ ′ |ψi =Cijk |Ai Bj Ck i (10) a∈A b∈B k z z ≈ (Cαβγ Uiα Ujβ Ukγ + Cαβγ V U U (11) ¯ iα ¯ jβ kγ (15) + 2Sa,αα ′ Sb,ββ ′ Cα′ β ′ γ k |A B C i + Cαβγ U V U + C U U V ¯ i j k iα j β¯ kγ αβ¯ γ iα jβ k¯ γ (16) =Hαβα′ β ′ Cα′ β ′ γ E E ˜ ˜ ˜ ˜ ˜ ˜ ≈Cαβγ Aα Bβ Cγ + Cαβγ Aα¯ Bβ Cγ ¯ where a and b correspond to lattice sites in clusters A E E and B, respectively. ˜ ˜ ˜ ˜ ˜ ˜ + Cαβγ (12) ¯ Aα Bβ¯ Cγ + Cαβ¯ γ Aα Bβ Cγ ¯ Because the Sˆz operator is diagonal we can easily use this quantity to project each Tucker block onto single Ms where Uiα and Viα¯ are the vectors in the P and Q spaces, block. In such a case, the dimer Hamiltonian in Eq. 16 respectively. Because these projection operators are decan be collapsed onto the given subspace before contractfined in terms of a HOSVD, we will refer to each of ing with the trial vectors. these sets Eof basis vectors as Tucker blocks. For instance, In the cases presented here, the non-interacting Hamil ˜ ˜ ˜ ˆ 1B = L H ˆ A , is used to precondition the Aα Bβ Cγ defines a Tucker block defined by the states tonian, H A Hamiltonian matrix in the Davidson algorithm. This spanned by α, β, γ. seems to be effective under the same conditions required In the simplest approach (and the one explored in this for the n-body Tucker method to be accurate (“highly initial paper) one could then use the variational principle clustered” systems). However, as inter-cluster couplings to optimize these coefficients. Although a variational opincrease relative to the gaps between the P and Q spaces timization does not provide a purely size-extensive theof the clusters, the preconditioner becomes less wellory, it does guarantee size-consistency in certain cases. suited, and other directions will need to be explored. For instance, a system comprised of an arbitrary number of clusters computed at infinity, is exactly equal to the sum of the individual cluster energies. This is a result B. Self-consistent cluster state optimization of the direct product reference. However, if one had four clusters arranged in two groups, with infinite distance beBy defining the cluster states in terms of the HOSVD tween the groups and non-zero inter-cluster interactions of the exact tensor, it is obvious that one does not diwithin a group, a 2-body Tucker approximation would rectly arrive at a computationally useful algorithm. The not yield size consistent results. In future work we will inexact solution is a prerequisite, so we need to define a vestigate the use of perturbation theory to determine the coefficient tensors, which would be exactly size-extensive. bottom-up strategy for obtaining accurate cluster states. In order to optimize the wavefunction, we will need If we were to use only the zeroth-order Tucker block, to evaluate matrix elements in this representation. In an HOSVD of the resulting tensor will always return the same states, since no mixing with the Q space is included. this first paper, we explore the implementation of a However, because the higher order Tucker blocks mix the model spin Hamiltonian to study the convergence properP and Q spaces, the sum of multiple tensors will allow the ties and behavior of this wavefunction ansatz, although occupation numbers to spill into the Q space, thus giving fully ab initio implementation will be the next step in us a new set of singular vectors, which changes the variathis work. Our implementation is inspired by Shiozaki’s tional space accessible within our truncated wavefunction Active-space Decomposition (ASD)85–87 in that we comapproximation. This naturally defines a self-consistency pute matrix elements using operators represented diloop between the cluster state partitioning and an eigenrectly in the associated local basis. However, because state of the full system Hamiltonian. In order to ensure the addition of Q space contributions will significantly


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puted individually and added together. This also has an interesting interpretation from an quantum embedding perspective. For example, in an incremental formulation, a 2-body interaction between clusters A and C would be obtained by diagonalizing the Hamiltonian in the basis spanned by all states on A and C, while the rest of the clusters are compressed into their respective P spaces. As such, each resulting n-body group of clusters would be computed with a quantum bath, enabling direct entanglement between the n-body collection and the rest of the system. Preliminary data is promising for this approach, and will be the focus of future work.

B.

H4 Clusters

In real systems, the interactions are not only between nearest neighbor sites as in the above 2-d spin lattices. More generally, all sites can interact with one another with a magnitude which decays with increasing distance. In order to explore the performance of the nbody Tucker expansion with systems which contain interactions between all sites, we have constructed an ab initio model consisting of 24 hydrogen atoms in space by starting with a methane cluster and deleting the carbon atoms. These hydrogen atoms are thus arranged into near-tetrahedral clusters, which themselves are arranged in a near-octahedron. In this study, each neartetrahedral group of four hydrogen atoms defines a cluster for the n-body Tucker decomposition. A potential energy surface scan is explored in which one of the clusters is pushed into the center of the octahedron. This system has no exact symmetry, so no interactions are necessarily zero. This ab initio system is then mapped onto a spin lattice, using the spin-flip CI plus Bloch effective Hamiltonian strategy, which has been presented previously,109,110 and then solved with the currently proposed n-body Tucker method. At each position of the PES scan, an ab initio spin-flip calculation is performed, the exchange coupling constants are extracted, and the resulting spin Hamiltonian is solved. Each ab initio spinflip calculation has been performed with a development version of QChem.111 These results are presented in Fig. 6. This ab initio parameterization of a spin Hamiltonian has been demonstrated to provide qualitative agreement with ab initio results for many cases. In this particular case, we do not expect quantitative accuracy, due to the strong coupling of the hydrogen atoms when they become close, for which the Bloch effective Hamiltonian projection has a norm of only 0.747 for the most compact geometry. Nonetheless, this does provide a physically motivated (and qualitatively meaningful) parameterization of a spin lattice to better understand this methodology. The extracted model Hamiltonian parameters are available in the supplementary information for each point on the PES. Moving from right to left on this PES in Fig. 6(b),

one can see that all approaches work well for the system when they are separated. However, as the rightmost cluster moves closer to the others, the defined clustering pattern becomes non-ideal. The errors increase quickly for the 0-body Tucker curve and 2-body Tucker curve, while the 4-body Tucker curve remains nearly flat, with the largest error of 0.004 eV for the point at −0.65 ˚ A. The reason the 2-body Tucker method is unable to fully recover this interaction energy is easily seen by the exchange interaction graphs in Fig. 6(a). For this compressed geometry, the largest interactions in the system are between the translated cluster, and two other clusters. Thus, in order to contain the strongest interactions simultaneously, groups of at least three clusters would be needed, which are absent in the 2-body Tucker approximation. One way to improve the 2-body Tucker results would be to use a geometry dependent clustering. For example, if at r= −0.65 ˚ A a new clustering is chosen such that the strongest interactions are within a cluster (still with only four sites per cluster), the NB2 error is only 0.13 eV, compared to 1.72 eV with the original clustering. We plan on exploring better ways of clustering in the future, drawing inspiration from pair entanglement measures like the “mutual information” index from Reiher and Stein.112 Alternatively, one could increase the number of states in the P space for the clusters. If the three strongly coupled clusters are given four eigenstates each (one singlet and one triply degenerate triplet), a 1-body Tucker approximation can be used instead, providing an error of only 0.05 eV. In Fig. 6(c), the convergence of the cluster states is examined using the 2-body Tucker approximation for scan distance r=−0.65 ˚ A, which is the most compact geometry point. By comparing the energy convergence behavior of the DIIS extrapolated results, and the nonextrapolated results (simple iterations similar to taking Roothaan steps in Hartree-Fock), it is clearly seen that the uncoupled DIIS extrapolation is very effective in this case for improving the convergence of the cluster states (or Tucker factors), with the number of iterations required decreasing from over 1000 to 23, to obtain convergence in the energy change to within 10−9 eV. Without the DIIS extrapolations, the iterations only slowly change the energy for the first 100 iterations before rapidly decreasing the energy, before ultimately plateauing at the same point obtained with the DIIS extrapolation. The norms of the DIIS error vectors for each cluster is also shown in this figure, each represented with a grey line. Although DIIS is known to be very effective at accelerating self-consistency optimizations, the impressive performance here is perhaps surprising, due to the fact that the we are extrapolating each cluster’s reduced density matrix, which are local quantities, instead of a global quantity. As described above, this means that each cluster carries it’s own DIIS subspace, and that each extrapolation occurs independently of one another. While this seems to work well for the cases investigated so far, we do expect to find situations where this is less than ideal,


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IV.

CONCLUSIONS

gether using an incremental approach for expectation values, and finally an ab initio implementation for fermionic systems to allow direct quantum chemistry calculations to be performed with this methodology.

In this paper, we propose a new wavefunction approximation for strongly correlated but highly clustered molecular systems, which combines Tucker tensor decompositions with more commonly used n-body organizations. The exact coefficient tensor is approximated as a sum of low-rank tensors (in the Tucker sense), and the resulting compressed coefficient tensors are determined using the variational principle. To explore the suitability of this approximation under different conditions, we have implemented this method to solve the Heisenberg spin Hamiltonian, which is a good model of molecules in the strongly correlated limit. Numerical calculations are performed for regular grid spin lattices, in addition to ab initio-derived spin lattices of different dimensionalities. From the data in this paper, we see that the n-body Tucker method works well for well-clustered systems, and incurs increasingly large errors when the clusters become less and less defined. Future work will focus on using perturbation theory and coupled-electron pairbased methods instead of the variational principle to parameterize the wavefunction, (which will recover full size-extensivity), avoiding the full-system expansion alto-

The author would like to thank an anonymous reviewer for pointing out the similarity of the presented n-body Tucker decomposition with the REM92–95 and AIFDEM methods.96,97 Support for this work was provided by the Department of Chemistry at Virginia Tech. The authors acknowledge Advanced Research Computing at Virginia Tech for providing computational resources and technical support that have contributed to the results reported within this paper. URL: http://www.arc.vt.edu

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Supporting Information

All Cartesian coordinates for all structures used in this paper can be found in the Supporting Information.

Acknowledgement

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