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The European Physical Journal A

The finite amplitude method for nuclear collective motion

Takashi Nakatsukasa, Nobuo Hinohara

Published Jul 3, 2026• 0 citations
Publisher

Abstract

Abstract The linear response theory is a powerful tool to study properties of quantum many-particle systems. It uses a one-body perturbation field (as a pump) and measures a change in another (or the same) one-body observable (as a probe). In the context of time-dependent density functional theory (TDDFT), the primary target of the linear response is the density response. The finite amplitude method (FAM) significantly reduces both computational cost and coding tasks, and succeeds in facilitating the linear response calculations in a variety of fields in nuclear physics. We start with a review of the linear response theory and present the essence of the FAM theory and various applications, especially in studies of nuclear collective dynamics.

DOI
10.1140/epja/s10050-026-01862-z

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Abstract The linear response theory is a powerful tool to study properties of quantum many-particle systems. It uses a one-body perturbation field (as a pump) and measures a change in another (or the same) one-body observable (as a probe). In the context of time-dependent density functional theory (TDDFT), the primary target of the linear response is the density response. The finite amplitude method (FAM) significantly reduces both computational cost and coding tasks, and succeeds in facilitating the linear response calculations in a variety of fields in nuclear physics. We start with a review of the linear response theory and present the essence of the FAM theory and various applications, especially in studies of nuclear collective dynamics.
We present the basic concepts and our recent developments in density functional approaches with Skyrme functionals for describing nuclear dynamics at low energy.Time-dependent density functional theory (TDDFT) is utilized for the exact linear response with an external perturbation.For a description of collective dynamics beyond the perturbative regime, we present a theory of a decoupled collective submanifold to describe a slow motion based on TDDFT.Selected applications are shown to demonstrate the quality of their performance and feasibility.The advantages and disadvantages of the numerical aspects are also discussed.. . . . . .
Introduction There are countless situations in physics when one is interested in calculating the response of a system to a small time-dependent perturbation acting on it. With some luck the response can be expanded in a power series of the strength of the perturbation, so that, to first order, it is a linear function of the latter. To compute this function is the objective of the linear response theory (LRT). Linear response theory has many important applications to the study of electronic matter. Virtually all interactions of electrons with experimental probes (electromagnetic fields, beams of particles) can be regarded as small perturbations to the system: if they were not, one would not be probing the system, but the system modified by the probe. Consequently, the results of these experiments can be expressed in terms of linear response functions, which are properties of the unperturbed system. In particular it will turn out that the analytic structure of these functions is entirely determined by the eigenvalues and eigenfunctions of the unperturbed system. Conversely, a measure of the linear response as a function of the frequency of the perturbation enables us to determine the excitation energies of the system. Beside being a cornerstone for the theory of single-particle properties to be developed in Chapter 8, the linear response functions also provide invaluable information in their own right. For example, as we will discuss in Chapter 5, the extent to which an external electrostatic potential is reduced by screening is controlled by the dynamical dielectric function which, in turn, is determined by the density–density response function.
This chapter discusses the linear-response framework of time-dependent density-functional theory (TDDFT). It begins with a thorough and self-contained exposition of general linear-response theory for many-body systems. The time-dependent as well as the frequency-dependent response are discussed. Several exact representations and properties of the density-density response function are reviewed, such as the Lehmann representation and fluctuation-dissipation theorem, and the Kramers-Kronig relations. It is shown how several important spectroscopic observables are obtained from linear-response theory. Then, the TDDFT formulation of linear-response theory is derived. The key quantity in this formalism is the so-called exchange-correlation kernel. The remainder of the chapter demonstrates how this linear-response formalism can be used to calculate excitation energies in principle exactly with TDDFT. The Casida equation is derived, as well as several approximate schemes. A comparison is made with time-dependent Hartree-Fock theory.
For the calculation of neutral excitations, time-dependent density functional theory (TDDFT) is an exact reformulation of the many-body time-dependent Schrödinger equation, based on knowledge of the density instead of the many-body wavefunction. The density can be determined in an efficient scheme by solving one-particle non-interacting Schrödinger equations—the Kohn–Sham equations. The complication of the problem is hidden in the—unknown—time-dependent exchange and correlation potential that appears in the Kohn–Sham equations and for which it is essential to find good approximations. Many approximations have been suggested and tested for finite systems, where even the very simple adiabatic local-density approximation (ALDA) has often proved to be successful. In the case of solids, ALDA fails to reproduce optical absorption spectra, which are instead well described by solving the Bethe–Salpeter equation of many-body perturbation theory (MBPT). On the other hand, ALDA can lead to excellent results for loss functions (at vanishing and finite momentum transfer). In view of this and thanks to recent successful developments of improved linear-response kernels derived from MBPT, TDDFT is today considered a promising alternative to MBPT for the calculation of electronic spectra, even for solids. After reviewing the fundamentals of TDDFT within linear response, we discuss different approaches and a variety of applications to extended systems.
Aiming at exploring microscopic dynamics responsible for the dissipative large-amplitude collective motion, the dynamic response and correlation functions are introduced within the general theory of nuclear coupled-master equations. The theory is based on the microscopic theory of nuclear collective dynamics which has been developed within the time-dependent Hartree-Fock (TDHF) theory for disclosing the complex structure of the TDHF manifold. A systematic numerical method for calculating the dynamic response and correlation functions is proposed. By performing numerical calculation for a simple model Hamiltonian, it is pointed out that the dynamic response function gives important information in understanding the large-amplitude dissipative collective motion which is described by an ensemble of trajectories within the TDHF manifold.