An essentially exact ground-state calculation algorithm for few-electron systems based on superposition of nonorthogonal Slater determinants (SDs) is described, and its convergence properties to ground states are examined. A linear combination of SDs is adopted as many-electron wave functions, and all one-electron wave functions are updated by employing linearly independent multiple correction vectors on the basis of the variational principle. The improvement of the convergence performance to the ground state given by the multi-direction search is shown through comparisons with the conventional steepest descent method. The accuracy and applicability of the proposed scheme are also demonstrated by calculations of the potential energy curves of few-electron molecular systems, compared with the conventional quantum chemistry calculation techniques.

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A practical algorithm for many-electron systems based on the path-integral renormalization group (PIRG) method is proposed in the real-space finite-difference (RSFD) approach. The PIRG method, developed for investigating strongly correlated electron systems, has been successfully applied to some models such as Hubbard models. However, to apply this method to more realistic systems of electrons with long-range Coulomb interactions within the RSFD formalism, the one-body Green's function, which requires large computational resources, is to be replaced with an alternative. For the same reason, an efficient algorithm for computing the Fock matrix is needed. The newly proposed algorithm is free of the one-body Green's function and enables us to compute the Fock matrix efficiently. Our result shows a significant reduction in CPU time and the possibility of using the present algorithm as a practical numerical tool. KEYWORDS: ab initio calculation; strongly correlated electrons; long-range Coulomb interactions I TRODUCTIOUp to now, many numerical algorithms for strongly correlated electron systems have been proposed and applied to various systems, e.g., quantum Monte Carlo method [1] , density matrix renormalization group (DMRG) method [2] , configuration interaction method [3] , and coupled cluster method [4] . However, even now, the nature of the ground state still remains a challenge because of the immaturity of numerical tools.The path-integral renormalization group (PIRG) method [5,6] was proposed to obtain the many-body ground state. Unlike the quantum Monte Carlo method, the PIRG method does not suffer from the sign problem [7] . Furthermore, unlike the DMRG method, the PIRG method does not limit the dimensionality of systems because numerical renormalization is carried out in an imaginary time space. In the PIRG method, the ground-state wave function is expressed by a linear combination of basis states, e.g., Slater determinants, in a truncated Hilbert space. While retaining the size of the truncated Hilbert space, the optimized basis states and the ground state are projected out numerically.To make the PIRG method applicable to more realistic systems, we extend the PIRG method with the real-space finite-difference (RSFD) approach in which every physical quantity is defined only on grid points in the discretized space [8][9][10][11] . In this endeavor, The process of "choosing more preferable basis states" becomes the main drawback with respect to the computational cost. In particular, because more basis states tend to be required in the RSFD scheme, one-body Green's functions and the Fock matrix dominate the computational cost and prevent us from applying this method to realistic systems.The aim of this work is to introduce a new algorithm within the framework of the RSFD approach to overcome the above-mentioned problems and to show its applicability to a realistic quantum system. METHODOLOGY A D UMERICAL APPLICATIO Wave Function RepresentationLet us expand the many-body wave function

The path-integral renormalization group and direct energy minimization method of practical first-principles electronic structure calculations for multi-body systems within the framework of the real-space finite-difference scheme are introduced. These two methods can handle higher dimensional systems with consideration of the correlation effect. Furthermore, they can be easily extended to the multicomponent quantum systems which contain more than two kinds of quantum particles. The key to the present methods is employing linear combinations of nonorthogonal Slater determinants (SDs) as multi-body wavefunctions. As one of the noticeable results, the same accuracy as the variational Monte Carlo method is achieved with a few SDs. This enables us to study the entire ground state consisting of electrons and nuclei without the need to use the Born-Oppenheimer approximation. Recent activities on methodological developments aiming towards practical calculations such as the implementation of auxiliary field for Coulombic interaction, the treatment of the kinetic operator in imaginary-time evolutions, the time-saving double-grid technique for bare-Coulomb atomic potentials and the optimization scheme for minimizing the total-energy functional are also introduced. As test examples, the total energy of the hydrogen molecule, the atomic configuration of the methylene and the electronic structures of two-dimensional quantum dots are calculated, and the accuracy, availability and possibility of the present methods are demonstrated.

We develop a biorthogonal linear-scaling algorithm for a transcorrelated method based on the localized nature of transformed orbitals. The transcorrelated method, which employs a similaritytransformed Hamiltonian referred to as a transcorrelated Hamiltonian, enables highly accurate first-principles condensed-matter calculations in principle. Meanwhile, the transcorrelated Hamiltonian numerically prevents us from applying it to large systems because the transcorrelated Hamiltonian is a non-Hermitian operator and contains a 3-body electron-electron interaction term. Non-Hermiticity means that left and right wave functions of the total energy expectation value are different from each other. Namely, a biorthogonal form is required. Our new method allows us to handle the non-Hermitian operator and exhibits a linear-scaling behavior.