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Solvent Extraction through the Lens of Advanced Modeling and Simulation
Published in Bruce A. Moyer, Ion Exchange and Solvent Extraction: Volume 23, 2019
Aurora E. Clark, Michael J. Servis, Zhu Liu, Ernesto Martinez-Baez, Jing Su, Enrique R. Batista, Ping Yang, Andrew Wildman, Torin Stetina, Xiaosong Li, Ken Newcomb, Edward J. Maginn, Jochen Autschbach, David A. Dixon
The type of complexes that are relevant in transition metal (TM), lanthanide (Ln), and actinide (An) separations may have open-shell metal centers, namely unpaired electrons that are formally localized at one or several metal centers. The electronic structure may, in this case, require two or more electron configurations, for even a qualitatively correct description. This is referred to as a multi-configuration (MC) or multi-reference case (not to be confused with multiple resonance structures), or static correlation as opposed to the dynamic correlation, which describes the explicit avoidance of two electrons in the wave function. The approximate correlation functionals in DFT and low-order truncated CC WFT, for instance, are good at describing the dynamic correlation of single-configuration systems, but they can have severe difficulties with MC cases. The full CC (or full configuration interaction, CI) wave function treats both static and dynamic correlation, but in practice, as usual, a compromise must be made. For an MC system, the first priority is to get its description qualitatively correct, that is, to treat the static correlation. Among the more frequently applied methods for MC systems is complete active space26 (CAS) WFT and its variants. In a CAS calculation, an active space of orbitals and electrons, comprising the open shells and often additional orbitals, is selected, and a full CI calculation in this active space is performed, usually with simultaneous orbital optimization. Factorial scaling with the active space size severely limits these calculations, but in recent years CAS variants based on electronic states from density-matrix renormalization group (DMRG) calculations with polynomial scaling are showing much promise for large MC problems.27–29 CAS-type calculations can be performed with relativistic spinors,30 also with DMRG,31 but often only SR effects are treated variationally while the SO interaction is introduced via a CI-like state interaction.32,33The dynamic correlation in CAS-type calculations is usually treated approximately by perturbation theory (PT).34,35 There is also truncated multi-reference configuration interaction (MR-CI),36 which remains in use in particular for calculations of electronic spectra and to introduce dynamic correlation in a MC ground state. Promising singlet-paired CC methods37,38 and multi-configuration pair-density-functional methods39 have also been developed in recent years that may allow routine calculations of MC systems with the inclusion of dynamic correlation, and progress in MC–CC theory40 has been reported.
Effects of low-lying excitations in pentalene and its derivatives in singlet fission: a model exact and density matrix renormalisation group study
Published in Molecular Physics, 2021
As Configuration Interaction (CI) space grows exponentially with the increase in the number of carbon atoms, an efficient approximate technique is required to adopt to calculate the excited energy states of these molecules with comparatively larger carbon atoms (≥ 16). In this context, Density Matrix Renormalization Group (DMRG) technique is found to be a very efficient and an effective for studying such correlated one-dimensional as well as quasi-one-dimensional conjugated systems [31]. Symmetrized Density Matrix Renormalization Group (SDMRG) technique within model PPP Hamiltonian has been proven to be very efficient in calculating vertical excited states of polycyclic aromatic hydrocarbons (PAH) in literature in recent times. Theoretical study on low-lying excited states of graphene nanoribbons and polycyclic aromatic hydrocarbons are reported through SDMRG technique within the model PPP Hamiltonian approach and calculated excitation energy gaps are in very good agreement with experimental observations [32–34].
The time-dependent density matrix renormalisation group method
Published in Molecular Physics, 2018
The density matrix renormalisation group (DMRG) method, invented by White in 1992 [1], has nowadays been considered as a well-established numerical method of studying one-dimensional (1D) strongly correlated quantum systems. DMRG uses the eigenvalues of the subsystem's reduced density matrix as the decimation criterion of the Hilbert space and keeps a fixed number (M) of renormalised states during the enlargement of the system, and thus it can be efficiently applied to very large 1D systems. Low-energy equilibrium properties of both bosonic and fermionic 1D and quasi-1D many-body systems can be calculated at almost machine precision and comparatively low computational cost by virtue of using DMRG. Recent efforts have advanced DMRG successfully working in some two-dimensional (2D) systems. For example, the ground state of 2D spin-1/2 Kagome lattice was evidenced to be fully gapped [2], and the composite fermions on a 2D electron gas under a strong magnetic field are determined to be massless Dirac particles [3] by DMRG calculations. In the recent two decades, the applicability of DMRG has gone successfully beyond theoretical condensed matter physics, and was extended to a wide range of fields ranging from statistical mechanics [4] to quantum chemistry [5–11] as well as quantum information theory [12,13]. The readers can find extensive reviews on DMRG in Refs. [14–17].