The configuration interaction (CI) is a versatile wave function theory for interacting fermions, but it involves an extremely long CI series. Using a symmetric tensor decomposition 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 by using small molecules. Provided that the length of the symmetric tensor decomposition 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.
The configuration interaction (CI) is a versatile wave function theory for interacting fermions, but it involves an extremely long CI series. Using a symmetric tensor decomposition 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 by using small molecules. Provided that the length of the symmetric tensor decomposition 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.
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.
To avoid the combinatorial computational cost of configuration interaction (CI), we previously introduced the symmetric tensor decomposition CI (STD-CI) method, which takes advantage of the antisymmetric nature of the electronic wave function and expresses the CI coefficients compactly as a series of Kronecker product states (STD series) [W. Uemura and O. Sugino, Phys. Rev. Lett. 109, 253001 (2012)]. Here we extend the variational degrees of freedom by using different molecular orbitals for different terms in the STD series. This scheme is equivalent to the linear combination of the Hartree-Fock-Bogoliubov state or the antisymmetrized geminal powers (AGPs). The total energy converges very rapidly within 0.72 $\ensuremath{\mu}\mathrm{hartree}$ taking only 10 terms for the water molecule, and the convergence is likewise fast for Hubbard tetramers. The computational cost scales as the fifth power of the number of electrons and the square of the number of terms in the STD series, indicating the promise of this AGP-based scheme for highly accurate and efficient computation of quantum 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.
The configuration interaction (CI) method can be regarded as a generalization of the Hartree-Fock theory. The CI method includes multiple determinantal states to approximate the full many-body ground state, as opposed to just one in Hartree-Fock theory. In this work, we introduce an efficient and stable algorithm to obtain the optimal single-particle basis by formulating orbital searching in terms of a multivariable minimization problem. The algorithm is iterative and is based on imaginary-time dynamics. It is numerically efficient as the Hessian (second derivative of the energy with respect to the orbital variables) is not required; it is also stable as a reasonable initial orbital choice is not necessary. The algorithm can deal with Hamiltonians with complex coefficients. We demonstrate the power of the proposed algorithm by applying it to impurity problems, with the most complicated one including multiple correlated orbitals and spin-orbit coupling. Energies and spectral functions are computed to demonstrate the convergence of the CI method.
We analyze wave functions constructed as a sum of product of two-electron functions, or as a polynomial of geminals, to investigate their ability to represent the ground state of a strongly correlated few-body system. The known difficulty associated with variational determination of the total energy is overcome by applying a tensor decomposition method called Waring decomposition. Convergence speed of the total energy is compared for various polynomial types. The result provides information bridging between geminal product wave functions and the full-CI in the strongly correlated regime, thereby enriching knowledge on the hierarchy of molecular orbital theories of electron pairs.
The full configuration interaction (full-CI) method is capable of providing the numerically best wave functions and energies of atoms and molecules within basis sets being used, although it is intractable for classical computers. Quantum computers can perform full-CI calculations in polynomial time against the system size by adopting a quantum phase estimation algorithm (QPEA). In the QPEA, the preparation of initial guess wave functions having sufficiently large overlap with the exact wave function is recommended. The Hartree-Fock (HF) wave function is a good initial guess only for closed shell singlet molecules and high-spin molecules carrying no spin-β unpaired electrons, around their equilibrium geometry, and thus, the construction of multiconfigurational wave functions without performing post-HF calculations on classical computers is highly desired for applying the method to a wide variety of chemistries and physics. In this work, we propose a method to construct multiconfigurational initial guess wave functions suitable for QPEA-based full-CI calculations on quantum computers, by utilizing diradical characters computed from spin-projected UHF wave functions. The proposed approach drastically improves the wave function overlap, particularly in molecules with intermediate diradical characters.