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Journal of Physics Conference Series

Computational issues of configuration interaction frameworks describing open quantum systems

N. Michel

Published Aug 12, 2013• 0 citations
Publisher

Abstract

Open quantum systems (OQS), extended in space (halo nuclei) or even unbound, differ from closed quantum systems (CQS), for which the methods of standard shell model (SM) [1] can be utilized in order to expand their wave function in a configuration interaction framework. Configuration interaction methods based on the use of Berggren bases [2], comprising bound, resonant and scattering states, which have the ability to generate the very long range asymptotic behavior of OQSs, are used instead for that matter. This demands the introduction of new computational techniques, including the optimization and discretization of the Berggren basis [3], the development of an algorithm to efficiently calculate their two-body matrix elements [4], and an overall optimization of memory storage absent from SM, where, for instance, all data related to proton and neutron spaces only can be precalculated and stored [1]. In order to diagonalize the very large induced matrices, the Density Matrix Renormalization Group (DMRG) method [5] extended to OQSs has been developed [6, 7]. A renormalization procedure which generates more and more correlated many-body basis states iteratively is used therein, so that the Hamiltonian matrix to diagonalize is very small compared to that occurring with a many-body basis of independent particles [6, 7]. Parallelization of presented methods will also be discussed.

DOI
10.1088/1742-6596/454/1/012065

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Open quantum systems (OQS), extended in space (halo nuclei) or even unbound, differ from closed quantum systems (CQS), for which the methods of standard shell model (SM) [1] can be utilized in order to expand their wave function in a configuration interaction framework. Configuration interaction methods based on the use of Berggren bases [2], comprising bound, resonant and scattering states, which have the ability to generate the very long range asymptotic behavior of OQSs, are used instead for that matter. This demands the introduction of new computational techniques, including the optimization and discretization of the Berggren basis [3], the development of an algorithm to efficiently calculate their two-body matrix elements [4], and an overall optimization of memory storage absent from SM, where, for instance, all data related to proton and neutron spaces only can be precalculated and stored [1]. In order to diagonalize the very large induced matrices, the Density Matrix Renormalization Group (DMRG) method [5] extended to OQSs has been developed [6, 7]. A renormalization procedure which generates more and more correlated many-body basis states iteratively is used therein, so that the Hamiltonian matrix to diagonalize is very small compared to that occurring with a many-body basis of independent particles [6, 7]. Parallelization of presented methods will also be discussed.
Despite the success of modern quantum chemistry in predicting properties of organic molecules, the treatment of inorganic systems, which have many close lying states, remains out of quantitative reach for current methods. To treat non-dynamic correlation, we take advantage of the density matrix renormalization group (DMRG) method that has become very successful in the field of solid state physics. We present a detailed study of the DMRG method, and we pay special attention to the evolution of the understanding behind the mathematical structure of the DMRG wave function. Our primary goal is to develop a density matrix renormalization group self--consistent--field (DMRG-SCF) approach, analogous to the complete active space self--consistent field (CASSCF) method, but dealing with large active spaces that are too demanding for the full configuration interaction (FCI) method. \nAs a first step towards such a DMRG-SCF procedure, we present a spin-adapted DMRG algorithm designed to target spin- and spatial-symmetry states that are hard to obtain while using an unrestricted algorithm. \nOur next step is a modification of the DMRG algorithm to obtain decreasing energy at every step during the sweep. This monotonically convergent DMRG scheme lets us obtain the two-body density matrix as a by--product of the existing procedure without any additional cost in storage. Additionally, the two-body density matrix produced at convergence using this scheme is free from the N-representability problem that is present when the two--body density matrix is produced with the two-site DMRG scheme without additional storage cost. Finally, taking advantage of the modifications developed herein, we present results obtained from our DMRG-SCF method. Lastly, we discuss possible ways of incorporating dynamical correlation into the DMRG scheme, in order to obtain a modern multireference approach.
Thanks to the high compression of the matrix product state (MPS) form of the wave function and the efficient site-by-site iterative sweeping optimization algorithm, the density matrix normalization group (DMRG) and its time-dependent variant (TD-DMRG) have been established as powerful computational tools in accurately simulating the electronic structure and quantum dynamics of strongly correlated molecules with a large number (10 1–2 ) of quantum degrees of freedom (active orbitals or vibrational modes). However, the quantitative characterization of the quantum many-body behaviors of realistic strongly correlated systems requires a further consideration of the interaction between the embedded active subsystem and the remaining correlated environment, e.g., a larger number (10 2–3 ) of external orbitals in electronic structure or infinite condensed-phase phononic modes in nucleus dynamics. To this end, we introduced three new post-DMRG and TD-DMRG approaches, namely (1) DMRG2sCI-MRCI and DMRG2sCI-ENPT by the reconstruction of selected configuration interaction (sCI) type of compact reference function from DMRG coefficients and the use of externally contracted MRCI (multireference configuration interaction) and Epstein–Nesbet perturbation theory (ENPT), without recourse to the expensive high order n -electron reduced density matrices ( n -RDMs). (2) DMRG combined with RR-MRCI (renormalized residue-based MRCI), which improves the computational accuracy and efficiency of internally contracted (ic) MRCI by renormalizing the contracted bases with small-sized buffer environment(s) of a few external orbitals as probes based on quantum information theory. (3) HM (hierarchical mapping)-TD-DMRG in which a large environment is reduced to a small number of renormalized environmental modes (which accounts for the most vital system–environment interactions) through stepwise mapping transformation. These advances extend the efficacy of highly accurate DMRG/TD-DMRG computations to the quantitative characterization of the electronic structure and quantum dynamics in realistic strongly correlated systems coupled with large environments and are reviewed in this paper.
Abstract We present a pedagogical introduction to the in-medium similarity renormalization group (IMSRG) framework for ab initio calculations of nuclei. The IMSRG performs continuous unitary transformations of the nuclear many-body Hamiltonian in second-quantized form, which can be implemented with polynomial computational effort. Through suitably chosen generators, it is possible to extract eigenvalues of the Hamiltonian in a given nucleus, or drive the Hamiltonian matrix in configuration space to specific structures, e.g., band- or block-diagonal form. Exploiting this flexibility, we describe two complementary approaches for the description of closed- and open-shell nuclei: the first is the multireference IMSRG (MR-IMSRG), which is designed for the efficient calculation of nuclear ground-state properties. The second is the derivation of non-empirical valence-space interactions that can be used as input for nuclear shell model (i.e., configuration interaction (CI)) calculations. This IMSRG+shell model approach provides immediate access to excitation spectra, transitions, etc, but is limited in applicability by the factorial cost of the CI calculations. We review applications of the MR-IMSRG and IMSRG+shell model approaches to the calculation of ground-state properties for the oxygen, calcium, and nickel isotopic chains or the spectroscopy of nuclei in the lower sd shell, respectively, and present selected new results, e.g., for the ground- and excited state properties of neon isotopes.
In these lecture notes, we present a pedagogical review of a number of related numerically exact approaches to quantum many‐body problems. In particular, we focus on methods based on the exact diagonalization of the Hamiltonian matrix and on methods extending exact diagonalization using renormalization group ideas, i.e., Wilson’s Numerical Renormalization Group (NRG) and White’s Density Matrix Renormalization Group (DMRG). These methods are standard tools for the investigation of a variety of interacting quantum systems, especially low‐dimensional quantum lattice models. We also survey extensions to the methods to calculate properties such as dynamical quantities and behavior at finite temperature, and discuss generalizations of the DMRG method to a wider variety of systems, such as classical models and quantum chemical problems. Finally, we briefly review some recent developments for obtaining a more general formulation of the DMRG in the context of matrix product states as well as recent progress in calculating the time evolution of quantum systems using the DMRG and the relationship of the foundations of the method with quantum information theory.