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.
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.
We investigated some variational methods to compute a wavefunction based on antisymmetric product of geminals (APG). The Waring decomposition on the APG wavefunction leads a finite sum of antisymmetrized geminal power (AGP) wavefunctions, each for which the variational principle can be applied. We call this as AGP-CI method which provides a variational solution of the APG wavefunction efficiently. However, number of AGP wavefunctions in the exact AGP-CI formalism become exponentially large in case of many-electron systems. Therefore, we also investigate the low-rank APG wavefunction, in which the coefficient matrices of geminals are factorized by the Schur decomposition. Interestingly, only a few non-zero eigenvalues (up to half number of electrons) were found from the Schur decomposition on the APG wavefunction. We developed some methods to approximate the APG wavefunction by lowering the ranks of coefficient matrices of geminals, and demonstrate their performance. Our new geminal method based on the low-rank decomposition can drastically reduce the number of variational parameters, although no efficient formalism has been elaborated so far, due to the difficulty of optimization caused by the inability to compute analytical derivatives.
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.
The singular value decomposition of the n-particle excitation operator as determined by coupled cluster or perturbation theory is used to extract the dominant and interesting electron-electron correlations from complex molecular wave functions. As an example of the very general formalism, the decomposition of the T(2) operator obtained from coupled cluster doubles calculations is used to analyze the strength and character of pair correlations in a variety of molecules with interesting electronic structure. The magnitude of the largest singular value(s) determines the strength of the correlation(s), and the corresponding right- and left-hand singular vectors characterize the physical and spatial nature of the correlations. The primary advantage of this tool over natural orbital analysis is that it provides direct associations between the occupied and virtual geminals involved in the correlations.
The Lewis electron-pair picture pervades all of chemistry and a substantialdomain of materials science. “Geminals” are electron-pair quantum states whichare used to express approximate solutions of the Schrödinger equation of polyelec-tronic systems. In quantum chemistry, geminal-based methods are potentially moreeffective than the traditional ones based on the orbital picture: electrons occupyorbitals and experience only the average effect of each other. This is especially truefor strongly correlated-systems. However, the computational cost of using arbitrarygeminal wave functions is non-polynomial in the system size.It is therefore an important challenge to find additional constraints on geminalmodels which preserve the accuracy of the wavefunction description while reducingthe calculations to polynomial cost. The Ayers’ group at Mc Master has developedone such successful geminal model called AP1roG because the wavefunction is anantisymmetric product of geminals with one distinct reference orbital occupied ineach geminal [1]. This ansatz relaxes the so-called “strong-orthogonality” constraintbetween geminals. The latter makes wavefunction matrix element computationsvery easy but appears too drastic from the physical point of view, since the geminalelectron pairs are then distinguishable.We have followed a different path to relax the strong-orthogonality constraint. Wehave defined the geometrical concept of n-orthogonality [2] which can be viewed asa graded indistinguishability measure for electronic states: 1-orthogonality coincideswith strong orthogonality, the larger n the less distinguishable the n-orthogonal elec-tronic states will be. We have studied geminal models constrained by n-orthogonalityrelations for different values of n, imposed to different combinations of geminal prod-uct states [3]. However, in this work the scaling of matrix element computations wasstill unsatisfactory. We will present new ideas developed since then to remedy thisproblem.References[1] P. A Limacher, P. W. Ayers, P. A. Johnson, S. De Baerdemacker, D. Van Neck and P. Bultinck, J. of Chem. Theory and Computation 9, 1394-1401 (2013).[2] P. Cassam-Chenaı̈, Phys. Rev. A77, 032103 (2008).[3] P. Cassam-Chenaı̈, V. Rassolov, Chem. Phys. Lett. 487, 147-152 (2010).
Variational quantum Monte Carlo (VMC) method applied to interacting electron systems has played a pivotal role for the studies of electronic states of atoms, molecules and condensed matters. First, the many-body wave functions optimized by the VMC method provide good guiding functions in more accurate, diffusion quantum Monte Carlo (DMC) calculations for identical systems. Second, the VMC calculations themselves can give a good upper bound for the exact ground-state energy of many-electron systems when an appropriate form for the variational wave function is employed. In addition to a conventional form for the variational wave function that is expressed by the product of the Jastrow factor and the single Slater determinant, recent investigations have found that the expression in terms of many Slater determinants can provide a very accurate description of electronic states, especially for intrinsically multi-reference systems, which corresponds to the configuration interaction (CI) expansion method in conventional quantum chemistry. Although the CI method needs an extremely large number of expansion coefficients as variational parameters, the number of parameters can be reduced efficiently by using an idea of (molecular) orbital pair correlations. The purpose of the present paper is to examine the feasibility and usefulness of this idea within the framework of VMC method with the Jastrow–Slater wave function, where the ground state of a water molecule is investigated as a test case. In this study we rely on a variational wave function composed of a product of the Jastrow and Slater parts as 1⁄4 J S; ð1Þ
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.