The task of computing wavefunctions that are accurate, yet simple enough mathematical objects to use for reasoning has long been a challenge in quantum chemistry. The difficulty in drawing physical conclusions from a wavefunction is often related to the generally large number of configurations with similar weights. In Tensor Product Selected CI, we use a locally correlated tensor product state basis, which has the effect of concentrating the weight of a state onto a smaller number of physically interpretable degrees of freedom. In this paper, we apply TPSCI to a series of three molecular systems ranging in separability, one of which is the first application of TPSCI to an open-shell bimetallic system. For each of these systems, we obtain accurate solutions to large active spaces, and analyze the resulting wavefunctions through a series of different approaches including (i) direct inspection of the TPS basis coefficients, (ii) construction of Bloch effective Hamiltonians, and (iii) computation of cluster correlation functions.
The task of computing wavefunctions that are accurate, yet simple enough mathematical objects to use for reasoning has long been a challenge in quantum chemistry. The difficulty in drawing physical conclusions from a wavefunction is often related to the generally large number of configurations with similar weights. In Tensor Product Selected CI, we use a locally correlated tensor product state basis, which has the effect of concentrating the weight of a state onto a smaller number of physically interpretable degrees of freedom. In this paper, we apply TPSCI to a series of three molecular systems ranging in separability, one of which is the first application of TPSCI to an open-shell bimetallic system. For each of these systems, we obtain accurate solutions to large active spaces, and analyze the resulting wavefunctions through a series of different approaches including (i) direct inspection of the TPS basis coefficients, (ii) construction of Bloch effective Hamiltonians, and (iii) computation of cluster correlation functions.
The task of computing wavefunctions that are accurate, yet simple enough mathematical objects to use for reasoning, has long been a challenge in quantum chemistry. The difficulty in drawing physical conclusions from a wavefunction is often related to the generally large number of configurations with similar weights. In Tensor Product Selected Configuration Interaction (TPSCI), we use a locally correlated tensor product state basis, which has the effect of concentrating the weight of a state onto a smaller number of physically interpretable degrees of freedom. In this paper, we apply TPSCI to a series of three molecular systems ranging in separability, one of which is the first application of TPSCI to an open-shell bimetallic system. For each of these systems, we obtain accurate solutions to large active spaces, and analyze the resulting wavefunctions through a series of different approaches including (i) direct inspection of the TPS basis coefficients, (ii) construction of Bloch effective Hamiltonians, and (iii) computation of cluster correlation functions.
6098], we introduced a new approach for accurately approximating full CI ground states in large electronic active-spaces called Tensor Product Selected CI (TPSCI). In TPSCI, a large orbital active space is first partitioned into disjoint sets (clusters) for which the exact, local many-body eigenstates are obtained. Tensor products of these locally correlated many-body states are taken as the basis for the full, global Hilbert space. By folding correlation into the basis states themselves, the low-energy eigenstates become increasingly sparse, creating a more compact selected CI expansion. While we demonstrated that this approach can improve accuracy for a variety of systems, there is even greater potential for applications to excited states, particularly those which have some excited-state character. In this paper, we report on the accuracy of TPSCI for excited states, including a far more efficient implementation in the Julia programming language. In traditional SCI methods that use a Slater determinant basis, accurate excitation energies are obtained only after a linear extrapolation and at a large computational cost. We find that TPSCI with perturbative corrections provides accurate excitation energies for several excited states of various polycyclic aromatic hydrocarbons with respect to the extrapolated result (i.e., near exact result). Further, we use TPSCI to report highly accurate estimates of the lowest 31 eigenstates for a tetracene tetramer system with an active space of 40 electrons in 40 orbitals, giving direct access to the initial bright states and the resulting 18 doubly excited (biexcitonic) states.
Transition metal complexes present significant challenges for electronic structure theory due to strong electron correlation arising from partially filled d-orbitals. We compare our recently developed Tensor Product Selected Configuration Interaction (TPSCI) with Density Matrix Renormalization Group (DMRG) for computing exchange coupling constants in six transition metal systems, including dinuclear Cr, Fe, and Mn complexes and a tetranuclear Ni-cubane. TPSCI uses a locally correlated tensor product state basis to capture electronic structure efficiently while maintaining interpretability. From calculations on active spaces ranging from (22e,29o) to (42e,49o), we find that TPSCI consistently yields higher variational energies than DMRG due to truncation of local cluster states, but provides magnetic exchange coupling constants (J) generally within 10–30 cm –1 of DMRG results. Key advantages include natural multistate capability enabling direct J extrapolation with smaller statistical errors, and computational efficiency for challenging systems. However, cluster state truncation represents a fundamental limitation requiring careful convergence testing, particularly for large local cluster dimensions. We identify specific failure cases where current truncation schemes break down, highlighting the need for improved cluster state selection methods and distributed memory implementations to realize TPSCI’s full potential for strongly correlated systems.
Selected configuration interaction (SCI) methods are currently enjoying a resurgence due to several recent developments which improve either the overall computational efficiency or the compactness of the resulting SCI vector. These recent advances have made it possible to get full CI (FCI) quality results for much larger orbital active spaces compared to conventional approaches. However, due to the starting assumption that the FCI vector has only a small number of significant Slater determinants, SCI becomes intractable for systems with strong correlation. This paper introduces a method for developing SCI algorithms in a way which exploits local molecular structure to significantly reduce the number of SCI variables. The proposed method is defined by first grouping the orbitals into clusters over which we can define many-particle cluster states. We then directly perform the SCI algorithm in a basis of tensor products of cluster states instead of Slater determinants. While the approach is general for arbitrarily defined cluster states, we find significantly improved performance by defining cluster states through a Tucker decomposition of the global (and sparse) SCI vector. To demonstrate the potential of this method, called tensor product selected configuration interaction (TPSCI), we present numerical results for a diverse set of examples: (1) modified Hubbard model with different inter- and intracluster hopping terms, (2) less obviously clusterable cases of bond breaking in N 2 and F 2, and (3) ground state energies of large planar π-conjugated systems with active spaces of up to 42 electrons in 42 orbitals. These numerical results show that TPSCI can be used to significantly reduce the number of SCI variables in the variational space, thus paving a path for extending these deterministic and variational SCI approaches to a wider range of physical systems.
There are a number of evidences and findings in quantum electronic systems in the last a few decades, such as the high Tc superconductivity in cuprates and the colossal magnetoresistance in perovskite oxides, etc.Much effort has been paid to understand the variety of challenging phenomena in condensed matter physics, but we still lack a unified and indisputable theoretical framework that can explain and predict most of these experimental facts.The situation of the theoretical understanding for quantum many body systems has been much refreshed after the discovery of Density Matrix Renormalization Group (DMRG) method in 1992.Since then, numerical and mathematical researches in the so called tensor network frameworks have attracted considerable attention.We focus on theoretical and numerical approaches toward the tensor network framework that make use of quantum information theory and are applicable to quantum chemistry.The purpose of this thesis is to give a tensor network framework that explicitly considers the antisymmetry of the electronic wavefunction contrary to most of the previous works.With a proper consideration of the antisymmetry of the wavefunction, we can dramatically reduce the number of states constituting the total wavefunction and improve the speed of numerical calculation and enhance the convergence of the resulting energy.In chapter 3, we introduce our recently published wavefunction theory where tensor network of the wavefunction is treated using the symmetric tensor decomposition framework with the antisymmetry of the wavefunction explicitly taken into account.This scheme is based on the configuration interaction (CI) which is a versatile wavefunction theory for interacting fermions but involving an extremely long CI series.Using a symmetric tensor decomposition (STD) method, we convert the CI series into a compact and numerically tractable form.The converted series encompasses the Hartree-Fock state in the first term and rapidly converges to the full-CI state, as numerically tested using small molecules.Provided that the length of the STD-CI series grows only moderately with the increasing complexity of the system, the new method will serve as one of the alternative variational methods to achieve full-CI with enhanced practicability.In Chapter 4, we introduce an improvement of STD-CI.In the Extended STD (ESTD) scheme, the variational degree of freedom in the wavefunction is dramatically extended.Therefore we can expect a considerable improvement in the convergence of the energy in ESTD.And also, in ESTD we can calculate the expectation value of the energy and the first derivatives with a calculation cost proportional to N 5 , while the original scaling in STD-CI is N 6 , where N is the number of electrons.This immediately means that we can dramatically extend the size of the systems that we can treat with ESTD, compared with STD-CI.In ESTD, the wavefunction is given as a linear combination of the Antisymmetrized Geminal Powers (AGP), and the norm and energy are given through a simple formalism based on Pfaffian.We provide a comparison of numerical results of the water molecule and the Hubbard model that indicates the numerical superiority of ESTD with respect to STD-CI.In future we expect to treat larger systems with ESTD, including three dimensional periodic systems.Such calculations will be propelled by the fact that in ESTD we can treat non orthogonal basis sets without taking orthonormalization, which is a procedure that significantly complicates large-scale calculations.
Quantum-mechanical calculations based on Kohn–Sham density functional theory (DFT) played a significant role in accurately predicting various aspects of materials behavior over the past decade. The Kohn–Sham approach to DFT reduces the many-body Schrodinger (eigen value) problem of interacting electrons into an equivalent problem of noninteracting electrons in an effective mean field that is governed by electron-density. Despite the reduced computational complexity of Kohn–Sham DFT, large-scale DFT calculations are still computationally very demanding with the resulting computational complexity scaling cubically with number of atoms in a given materials system. Numerical algorithms with reduced scaling behavior which are robust, computationally efficient and scalable on parallel computing architectures are always desirable to enable simulations at larger scales and on more complex systems. Following this line of thought, this study explores the use of tensor structured methods for ab-initio numerical solution of Kohn–Sham equations arising in DFT calculations. Earlier studies on tensor-structured methods have been quite successful in the accurate calculation of Hartree and the nonlocal exchange operators arising in the Hartree–Fock equations. A recent investigation of low-rank Tucker-type decomposition of the electron-density of large aluminum clusters (obtained from the finite-element discretization of orbital free DFT) shows the exponential decay of approximation error with respect to Tucker rank (number of tensor-basis functions in Tucker type representation). The results also indicate a smaller Tucker rank for the accurate representation of the electron density and is only weakly dependent on the system sizes studied. The promising success of tensor-structured techniques in resolving the electronic structure of material systems has enabled us to take a step further. In this study, we propose a systematic way of computing a globally adapted Tucker-type basis for solving the Kohn–Sham DFT problem by using a separable approximation of the Kohn–Sham Hamiltonian. Further, the resulting Kohn–Sham eigenvalue problem is projected into the aforementioned Tucker basis and is solved for ground-state energy using a self-consistent field iteration. The rank of the resulting Tucker representation and the computational complexity of the calculation are examined on representative benchmark examples involving metallic and insulating systems.