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arXiv:1001.0396 · Journal of Physics G Nuclear and Particle Physics

The long journey fromab initiocalculations to density functional theory for nuclear large amplitude collective motion

Aurel Bulgac

Published Mar 29, 2010• 14 citations
arXivHTMLPDF

Abstract

At present there are two vastly different ab initio approaches to the description of the many-body dynamics: the density functional theory (DFT) and the functional integral (path integral) approaches. On one hand, if implemented exactly, the DFT approach can allow in principle the exact evaluation of arbitrary one-body densities. However, when applied to large amplitude collective motion (LACM), this approach needs to be extended in order to accommodate the phenomenon of surface-hopping, when adiabaticity is strongly violated and the description of a system using a single (generalized) Slater determinant is not valid anymore. The functional integral approach on the other hand does not appear to have such restrictions, but its implementation does not appear to be a straightforward endeavor. However, within a functional integral approach one seems to be able to evaluate in principle any kind of observable, such as the fragment mass and energy distributions in nuclear fission. These two radically different approaches can likely be brought together by formulating a stochastic time-dependent DFT approach to many-body dynamics.

DOI
10.1088/0954-3899/37/6/064006

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At present there are two vastly different ab initio approaches to the description of the many-body dynamics: the density functional theory (DFT) and the functional integral (path integral) approaches. On one hand, if implemented exactly, the DFT approach can allow in principle the exact evaluation of arbitrary one-body densities. However, when applied to large amplitude collective motion (LACM), this approach needs to be extended in order to accommodate the phenomenon of surface-hopping, when adiabaticity is strongly violated and the description of a system using a single (generalized) Slater determinant is not valid anymore. The functional integral approach on the other hand does not appear to have such restrictions, but its implementation does not appear to be a straightforward endeavor. However, within a functional integral approach one seems to be able to evaluate in principle any kind of observable, such as the fragment mass and energy distributions in nuclear fission. These two radically different approaches can likely be brought together by formulating a stochastic time-dependent DFT approach to many-body dynamics.
Since the early days of quantum mechanics, the wave function theory has proven to be very successful in describing many different quantum processes and phenomena. However, in many problems of quantum chemistry and solid-state physics, where the dimensionality of the systems studied is relatively high, ab initio calculations of the structure of atoms, molecules, clusters, and crystals, and their interactions are very often prohibitive. Hence, alternative formulations based on the direct use of the probability density, gathered under what is generally known as the density matrix theory [1], were also developed since the very beginning of the new mechanics. The independent electron approximation or Thomas-Fermi model, and the Hartree and Hartree-Fock approaches are former statistical models developed in that direction [2]. These models can be considered direct predecessors of the more recent density functional theory (DFT) [3], whose principles were established by Hohenberg, Kohn, and Sham [4,5] in the mid-1960s. According to this theory, the fundamental physical information about a many-body system is provided by single-particle densities in a three-dimensional space, which are obtained variationally within a time-independent framework. When compared with other previous formalisms, DFT presents two clear advantages: (1) it is able to treat many-body problems in a sufficiently accurate way and (2) it is computationally simple. This explains why it is one of the most widely used theories to deal with electronic structure-the electronic ground-state energy as a function of the position of the atomic nuclei determines molecular structures and solids, providing at the same time the forces acting on the atomic nuclei when they are not at their equilibrium positions. At present, DFT is used routinely to solve many problems in gas phase and condensed matter. Furthermore, it has made possible the development of accurate molecular dynamics schemes in which the forces are evaluated quantum mechanically ‘‘on the fly.’’ Nonetheless, DFT is a fundamental tool provided the systems studied are relatively large; for small systems, standard methods based on the use of the wave function render quite accurate results [6]. Moreover, it is also worth stressing that all practical applications of DFT rely on essentially uncontrolled approximations [7] (e.g., the local density approximation [4,5], the local spin-density approximation, or generalized gradient approximations [8]), and therefore the validity of DFT is conditioned to its capability to provide fairly good values of the experimental data.
We show that to any Lie algebra of observables in quantum mechanics there corresponds a system of densities, of which the one-body densities are just one example. These densities may correspond to a single pure state or to an ensemble of states. Like the one-body densities, they support a Hamiltonian dynamics of the time-dependent Hartree-Fock type, which can be applied, for example, to collective motions where the collective states of interest may be either more or less general than single Slater determinants. The dynamics can also be applied to the average dynamics of an ensemble of states.
This chapter addresses a range of problems for which functional integrals provide unique insights and approximations. It shows how to generate a variety of alternative functional integrals for the evolution operator which gives rise to different physical approximations in the stationary-phase approximation. The chapter aims to perform the general saddle point approximation around a static mean field solution and derive an expansion in terms of the fundamental vibrational excitations of the system. It also shows how time-dependent mean field solutions may be used to calculate transition amplitudes between specific states, approximate quantum eigenstates of large amplitude collective motion, study barrier penetration and spontaneous fission. The chapter evaluates the asymptotic behavior of large orders of perturbation theory and explores the freedom available in formulating functional integrals for the evolution operator and how to exploit this freedom in making physical approximations.
The time-dependent Hartree-Fock (TDHF) and density functional theory (DFT) are among the most useful approaches within mean-field theories for studying static and dynamic properties of complex many-body systems in different branches of physics. Despite the fact that they provide a good approximation for the average properties of one-body degrees of freedoms, they are known to fail to include quantal fluctuations of collective observables and they do not provide sufficient dissipation of collective motion. In order to incorporate these missing effects the stochastic mean-field (SMF) approach was proposed (Ayik 2008). In the SMF approach a set of stochastic initial one-body densities are evolved. Each stochastic one-body density matrix consists of a set of stochastic Gaussian random numbers that satisfy the first and second moments of collective one-body observables. Recent works indicate that the SMF approach provides a good description of the dynamics of the nuclear systems (Yilmaz et al. 2018; Ayik et al. 2019). In this work, the one-dimensional Fermi-Hubbard model is simulated with the SMF approach by using different distributions such as Gaussian, uniform, bimodal and two-point distributions. The dissipative dynamics are discussed and the predictive power of the SMF approach with different probability distributions are compared with each other and the exact dynamics. As a result it is shown that by considering different distributions, the predictive power of the SMF approach can be improved.
We present a general method for constructing path intergrals for the nuclear many-body problem. This method uses continuous and overcomplete sets of vectors in the Hilbert space. The state labels play the role of classical coordinates which are quantized as bosons. The equations of motion for the classical coordinates are obtained by calculating the functional integral in the saddle point approximation. In the particular case where the over-complete set considered is the set of all Slater determinants, the classical equations of motion are the time-dependent Hartree-Fock equations. The functional integral provides a way of requantizing these classical equations. This quantization involves boson degrees of freedom and is in some cases very similar to the method of boson expansion. It is shown that the functional integral formalism provides a unifying framework to describe various approaches to the nuclear many-body problem.NUCLEAR STRUCTURE Functional integrals on continuous over-complete sets. Time-dependent Hartree and Hartree-Fock theories. Boson representations for fermion systems.