Self-consistent theory of large-amplitude collective motion: applications to approximate quantization of nonseparable systems and to nuclear physics

Dang, G. Do; Klein, Abraham; Walet, Niels R.

doi:10.1016/s0370-1573(99)00119-2

Cited by 35 publications

(70 citation statements)

References 221 publications

(406 reference statements)

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“…Another set of applications concerns the adiabatic approximation to the time-dependent HFB (ATDHFB) [2,[5][6][7][8][9] wherein derivatives with respect to collective coordinates are often approximated by finitedifference expressions [10].…”

Section: Introductionmentioning

confidence: 99%

Augmented Lagrangian method for constrained nuclear density functional theory

et al. 2010

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Abstract. The augmented Lagrangiam method (ALM), widely used in quantum chemistry constrained optimization problems, is applied in the context of the nuclear Density Functional Theory (DFT) in the self-consistent constrained Skyrme Hartree-Fock-Bogoliubov (CHFB) variant. The ALM allows precise calculations of multi-dimensional energy surfaces in the space of collective coordinates that are needed to, e.g., determine fission pathways and saddle points; it improves the accuracy of computed derivatives with respect to collective variables that are used to determine collective inertia; and is well adapted to supercomputer applications.

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Section: Introductionmentioning

confidence: 99%

Augmented Lagrangian method for constrained nuclear density functional theory

et al. 2010

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“…a particular quadrupole moment). Once a set of collective variables has been chosen, either guided by the magical physical intuition of a given set of authors or by some other more formal set of rules [15], one has to determine for each particular value of each collective coordinate the optimal GSD, which will minimize the total energy of a nuclear many-body Hamiltonian. This goal is typically achieved by solving a constrained self-consistent meanfield problem.…”

mentioning

confidence: 99%

“…A serious drawback of this approach is the lack of a reliable measure of the theoretical error. This restricted representation of the many-body wave function could be exact if the many-body dynamics would be exactly separable by introducing a suitable set of intrinsic and the collective coordinates q k [15]. Since this is never the case, the representation (I.1) is of an unquantifiable quality.…”

mentioning

confidence: 99%

“…Apart from the fact that these collective variables are not optimized, as discussed for example in Ref. [15], in the sense that they are not optimally decoupled from the intrinsic degrees of freedom, the set of GSDs generated in this manner is also not complete and, moreover, there is a large degree of redundancy [20] (as one can easily establish by computing the spectrum of the norm matrix). If collective degrees of freedom would be indeed well decoupled from the intrinsic motion, there would be no need to even discuss whether one should perform calculations with variation before or after the projection for example.…”

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confidence: 99%

“…The only constructive procedure that I am aware of, which in principle could determine the relevant set of collective coordinates and also generate some measure of the accuracy of the representation (I.1) was developed by Klein and collaborators [15]. However, this approach is extremely difficult to implement in practice and the mathematical structure of the approach is not fully understood.…”

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

Bulgac

2010

J. Phys. G: Nucl. Part. Phys.

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At present there are two vastly different ab initio approaches to the description of the the manybody 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 observable. However, when applied to Large Amplitude Collective Motion (LACM) this approach needs to be extended in order to accommodate the phenomenon of surface-hoping, 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 straightforward endeavor. However, within a functional integral approach one seems to be able to evaluate in principle any kind of observables, such as the fragment mass and energy distributions in nuclear fission. These two radically approaches can likely be brought brought together by formulating a stochastic time-dependent DFT approach to many-body dynamics. We know a great deal by now about interactions between nucleons, although one might rightly argue that the link to quarks and gluons through the more fundamental theory Quantum Chromodynamics has not yet been fully established. It will still be a long time until this goal will be reached with the needed degree of accuracy and sophistication. In this respect the nuclear many-body theory is at a great disadvantage when compared to condensed matter and atomic theories or chemistry, where the dominant role of the Coulomb interaction and the link to Quantum Electrodynamics were clarified a long time ago. There is a general consensus by now however, that in order to describe nuclear many-body systems one probably can get away with the use of one set or another of reasonably well established two-and three-nucleon potentials (if only one would manage to use them in a convincing implementation) and that a non-relativistic approach is likely sufficient for now. The best example we have so far that such an ab initio approach can be brought to fruition is the Green Function Monte-Carlo (GFMC) approach implemented so successfully by V.R. Pandharipande and his many students and collaborators J. Carlson (truly the author of GFMC), S.C. Pieper, R. Schiavilla, K.E. Schmidt, R.B. Wiringa, and many others [1]. One of the most frustrating limitations of the GFMC approach is the relatively low particle number this method can deal with. Presently one cannot envision the study of nuclei heavier than oxygen within a GFMC framework. The Coupled-Cluster (CC) method [2] looks very promising in this respect, as it might be applied to medium size closed shell nuclei in the near future. The No-Core-Shell Model (NCSM) [3] will likely be successfully applied to light and medium size nuclei in the foreseeable future with a reasonable degree of accuracy.Heavy-nuclei seem out of reach from a direct frontal ab initio attack and th...

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Application of the SCC method to the multi-O(4) model: The collective Hamiltonian

Kobayasi

2009

Sci. China Ser. G-Phys. Mech. Astron.

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The collective Hamiltonian up to the fourth order for a multi-O(4) model is derived for the first time based on the self-consistent collective-coordinate (SCC) method, which is formulated in the framework of the time-dependent Hartree-Bogoliubov (TDHB) theory. This collective Hamiltonian is valid for the spherical case where the HB equilibrium point of the multi-O(4) model is spherical as well as for the deformed case where the HB equilibrium points are deformed. Its validity is tested numerically in both the spherical and deformed cases. Numerical simulations indicate that the low-lying states of the collective Hamiltonian and the transition amplitudes among them mimic fairly well those obtained by exactly diagonalizing the Hamiltonian of the multi-O(4) model. The numerical results for the deformed case imply that the "optimized RPA boundary condition" is also valid for the well-known η * , η expansion around the unstable HB point of the multi-O(4) model. All these illuminate the power of the SCC method.self-consistent collective-coordinate method, multi-O(4) model, time-dependent Hartree-Bogoliubov theory, collective Hamiltonian Atomic nuclei exhibit rich collective motions: rotational, vibrational, wobbling, fission, fusion and shape coexistence motions. They are the collective excitations of nucleons. A powerful tool for understanding both low-lying collective states and giant resonances is the random phase approximation(RPA), the approximation, which should be good for collective vibrations as long as their amplitudes are small. One has to go beyond the RPA in order to describe kinds of large amplitude collective motions (LACM)such as fission and shape coexistence. Microscopic understanding of the LACM is a long-standing fundamental subject of nuclear structure physics.A useful microscopic tool for describing the LACM is the generator coordinate method (GCM) [1−4] . The GCM wave function is usually taken as a combination of many intrinsic states, calculated self-consistently within constrained Hartree-Fock-Bogoliubov (HFB) theory. The constraining operators define collective degrees of freedom. The GCM wave function is rich enough to accommodate correlation absent in the mean field and is not limited to the adiabatic

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Self-consistent theory of large-amplitude collective motion: applications to approximate quantization of nonseparable systems and to nuclear physics

Cited by 35 publications

References 221 publications

Augmented Lagrangian method for constrained nuclear density functional theory

Augmented Lagrangian method for constrained nuclear density functional theory

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

Application of the SCC method to the multi-O(4) model: The collective Hamiltonian

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