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arXiv:1708.07748 · Physical Review C

Power series expansion method in tensor-optimized antisymmetrized molecular dynamics beyond the Jastrow correlation method

Takayuki Myo, Hiroshi Toki, Kiyomi Ikeda, Hisashi Horiuchi, Tadahiro Suhara

Published Sep 8, 2017• 21 citations
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Abstract

We developed a new variational method for tensor-optimized antisymmetrized molecular dynamics (TOAMD) for nuclei. In TOAMD, the correlation functions for the tensor force and the short-range repulsion are introduced and used in the power series form of the wave function, which is different from the Jastrow method. Here, nucleon pairs are correlated in multisteps with different forms, while they are correlated only once including all pairs in the Jastrow correlation method. Each correlation function in every term is independently optimized in the variation of total energy in TOAMD. For $s$-shell nuclei using the nucleon-nucleon interaction, the energies in TOAMD are better than those in the variational Monte Carlo method with the Jastrow correlation function. This means that the power series expansion using the correlation functions in TOAMD describes the nuclei better than the Jastrow correlation method.

DOI
10.1103/physrevc.96.034309

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Abstract We study 5^55He variationally as the first ppp-shell nucleus in the tensor-optimized antisymmetrized molecular dynamics (TOAMD) using the bare nucleon–nucleon interaction without any renormalization. In TOAMD, the central and tensor correlation operators promote the AMD’s Gaussian wave function to a sophisticated many-body state including the short-range and tensor correlations with high-momentum nucleon pairs. We develop a successive approach by applying these operators successively with up to double correlation operators to get converging results. We obtain satisfactory results for 5^55He, not only for the ground state but also for the excited state, and discuss explicitly the correlated Hamiltonian components in each state. We also show the importance of the independent optimization of the correlation functions in the variation of the total energy beyond the condition assuming common correlation forms used in the Jastrow approach.
We propose a new many-body method based on the correlation functions, in which the multiple products of the correlation functions are expanded into the many-body diagrams using the cluster expansion method and every diagram is independently optimized in the total-energy variation. We apply this idea to the tensor-optimized antisymmetrized molecular dynamics (TOAMD) using the bare nucleon-nucleon interaction and show the results of the sss-shell nuclei within the triple products of the correlation functions of tensor and central types. We evaluate the effect of the independent optimization of the many-body diagrams on the solutions. It is found that the triple products provide the sizable effect in the present scheme, which results in the good reproduction of the total energy and the Hamiltonian components of nuclei with respect to the few-body calculations.
We develop a new formalism to treat nuclear many-body systems using the bare nucleon–nucleon interaction. It has become evident that the tensor interaction plays an important role in nuclear many-body systems due to the role of the pion in strongly interacting systems. We take the antisymmetrized molecular dynamics (AMD) as a basic framework and add a tensor correlation operator acting on the AMD wave function using the concept of the tensor-optimized shell model. We demonstrate a systematical and straightforward formulation utilizing the Gaussian integration and differentiation method and the antisymmetrization technique to calculate all the matrix elements of the many-body Hamiltonian. We can include the three-body interaction naturally and calculate the matrix elements systematically in the progressive order of the tensor correlation operator. We call the new formalism “tensor-optimized antisymmetrized molecular dynamics”.
Fermionic Molecular Dynamics (FMD) models a system of fermions by means of many-body states which are composed of antisymmetrized products of single-particle states. These consist of one or several Gaussians localized in coordinate and momentum space. The parameters specifying them are the dynamical variables of the model. As the repulsive core of the nucleon-nucleon interaction induces short range correlations which cannot be accommodated by a Slater determinant, a novel approach, the unitary correlation operator method (UCOM), is applied. The unitary correlator moves two particles away from each other whenever their relative distance is within the repulsive core. The time-dependent variational principle yields the equations of motion for the variables. Energies of the stationary ground states are calculated and compared to exact many-body results for nuclei up to Ca 48. Time-dependent solutions are shown for collisions between nuclei.
Abstract The tensor-optimized Fermi sphere (TOFS) theory is applied first for the study of the property of nuclear matter using the Argonne V4′^\prime′NNNNNN potential. In the TOFS theory, the correlated nuclear matter wave function is taken to be a power-series type of the correlation function FFF, where FFF can induce central, spin–isospin, tensor, etc. correlations. This expression has been ensured by a linked cluster expansion theorem established in the TOFS theory. We take into account the contributions from all the many-body terms arising from the product of the nuclear matter Hamiltonian H\mathcal{H}H and FFF. The correlation function is optimally determined in the variation of the total energy of nuclear matter. It is found that the density dependence of the energy per particle in nuclear matter is reasonably reproduced up to the nuclear matter density ρ≃0.20\rho \simeq 0.20ρ≃0.20 fm−3^{-3}−3 in the present numerical calculation, in comparison with other methods such as the Brueckner–Hartree–Fock approach.
quantum Monte Carlo (QMC) is a state-of-the-art numerical approach for evaluating accurate expectation values of many-body wave functions. However, one of the major drawbacks that still hinders widespread QMC applications is the lack of an affordable scheme to compute unbiased atomic forces. In this study, we propose an efficient method to obtain unbiased atomic forces and pressures in the variational Monte Carlo (VMC) framework with the Jastrow-correlated Slater determinant ansatz or the Jastrow antisymmetrized geminal power ansatz, exploiting the gauge-invariant and locality properties of their geminal representation. We demonstrate the effectiveness of our method for H 2 and Cl 2 molecules and for the cubic boron nitride crystal. Our framework has a better algorithmic scaling with the system size than the traditional finite-difference method and, in practical applications, is as efficient as single-point VMC calculations. Thus, it paves the way to study dynamical properties of materials, such as phonons, and is beneficial for pursuing more reliable machine-learning interatomic potentials based on unbiased VMC forces. Published by the American Physical Society 2024