Construction of CASCI-type wave functions for very large active spaces

Error Control and Automatic Detection of Reference Active Spaces in Many-Body Expanded Full Configuration Interaction

We present a wide-reaching revamp of the generalized many-body expanded full configuration interaction (MBE-FCI) method. First, we outline how to automatize the selection of reference active spaces, whereby the inherent bias introduced through a manual identification is reduced, also within the context of traditional complete active space methods. Second, we allow for the use of compact orbital clusters as expansion objects, which works to circumvent the unfavorable scaling with the number of orbitals included in the space complementary to the reference orbitals. Finally, we present a new algorithm for ensuring that many-body expansions can be efficiently terminated while conservatively accounting for resulting errors. These developments are all tested on a variety of molecular systems and different orbital representations to illustrate the abilities of our algorithm to produce correlation energies within predetermined error bounds, significantly broadening the overall applicability of the MBE-FCI method.

Vibrational Entanglement through the Lens of Quantum Information Measures

We introduce a quantum information analysis of vibrational wave functions to understand complex vibrational spectra of molecules with strong anharmonic couplings and vibrational resonances. For this purpose, we define one- and two-modal entropies to guide the identification of strongly coupled vibrational modes and to characterize correlations within modal basis sets. We evaluate these descriptors for multiconfigurational vibrational wave functions which we calculate with the n-mode vibrational density matrix renormalization group algorithm. Based on the quantum information measures, we present a vibrational entanglement analysis of the vibrational ground and excited states of CO2, which display strong anharmonic effects due to the symmetry-induced and accidental (near-) degeneracies. We investigate the entanglement signature of the Fermi resonance and discuss the maximally entangled state arising from the two degenerate bending modes.

Flexible DMRG-Based Framework for Anharmonic Vibrational Calculations

We present a novel formulation of the vibrational density matrix renormalization group (vDMRG) algorithm tailored to strongly anharmonic molecules described by general, high-dimensional model representations of potential energy surfaces. For this purpose, we extend the vDMRG framework to support vibrational Hamiltonians expressed in the so-called n-mode second-quantization formalism. The resulting n-mode vDMRG method offers full flexibility with respect to both the functional form of the PES and the choice of the single-particle basis set. We leverage this framework to apply, for the first time, vDMRG based on an anharmonic modal basis set optimized with the vibrational self-consistent field algorithm on an on-the-fly constructed PES. We also extend the n-mode vDMRG framework to include excited-state-targeting algorithms in order to efficiently calculate anharmonic transition frequencies. We demonstrate the capabilities of our novel n-mode vDMRG framework for methyloxirane, a challenging molecule with 24 coupled vibrational modes.

New Density Matrix Renormalization Group Approaches for Strongly Correlated Systems Coupled with Large Environments

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.

Assessing the Accuracy of Tailored Coupled Cluster Methods Corrected by Electronic Wave Functions of Polynomial Cost

Tailored coupled cluster theory represents a computationally inexpensive way to describe static and dynamical electron correlation effects. In this work, we scrutinize the performance of various coupled cluster methods tailored by electronic wave functions of polynomial cost. Specifically, we focus on frozen-pair coupled cluster (fpCC) methods, which are tailored by pair-coupled cluster doubles (pCCD), and coupled cluster theory tailored by matrix product state wave functions optimized by the density matrix renormalization group (DMRG) algorithm. As test system, we selected a set of various small- and medium-sized molecules containing diatomics (N₂, F₂, C₂, CN⁺, CO, BN, BO⁺, and Cr₂) and molecules (ammonia, ethylene, cyclobutadiene, benzene, hydrogen chains, rings, and cuboids) for which the conventional single-reference coupled cluster singles and doubles (CCSD) method is not able to produce accurate results for spectroscopic constants, potential energy surfaces, and barrier heights. Most importantly, DMRG-tailored and pCCD-tailored approaches yield similar errors in spectroscopic constants and potential energy surfaces compared to accurate theoretical and/or experimental reference data. Although fpCC methods provide a reliable description for the dissociation pathway of molecules featuring single and quadruple bonds, they fail in the description of triple or hextuple bond-breaking processes or avoided crossing regions.

Excited-State DMRG Made Simple with FEAST

We introduce DMRG[FEAST], a new method for optimizing excited-state many-body wave functions with the density matrix renormalization group (DMRG) algorithm. Our approach applies the FEAST algorithm, originally designed for large-scale diagonalization problems, to matrix product state wave functions. We show that DMRG[FEAST] enables the stable optimization of both low- and high-energy eigenstates, therefore overcoming the limitations of state-of-the-art excited-state DMRG algorithms. We demonstrate the reliability of DMRG[FEAST] by calculating anharmonic vibrational excitation energies of molecules with up to 30 fully coupled degrees of freedom.

The Shape of Full Configuration Interaction to Come

We present a Perspective on what the future holds for full configuration interaction (FCI) theory, with an emphasis on conceptual rather than technical details. Upon revisiting the early history of FCI, a number of its key contemporary approximations are compared on as equal a footing as possible, using a recent blind challenge on the benzene molecule as a testbed [Eriksen et al., J. Phys. Chem. Lett., 2020 11, 8922]. In the process, we review the scope of applications for which FCI continues to prove indispensable, and the required traits in terms of robustness, efficacy, and reliability its modern approximations must satisfy are discussed. We close by conveying a number of general observations on the merits offered by the state-of-the-art alongside some of the challenges still faced to this day. While the field has altogether seen immense progress over the years-the past decade, in particular-it remains clear that our community as a whole has a substantial way to go in enhancing the overall applicability of near-exact electronic structure theory for systems of general composition and increasing size.

Portably parallel construction of a configuration-interaction wave function from a matrix-product state using the Charm++ framework

The construction of configuration-interaction (CI) expansions from a matrix product state (MPS) involves numerous matrix operations and the skillful sampling of important configurations in a large Hilbert space. In this work, we present an efficient procedure for constructing CI expansions from MPS employing the parallel object-oriented Charm++ programming framework, upon which automatic load-balancing and object migrating facilities can be employed. This procedure was employed in the MPS-to-CI utility (Moritz et al., J. Chem. Phys. 2007, 126, 224109), the sampling-reconstructed complete active-space algorithm (SR-CAS, Boguslawski et al., J. Chem. Phys. 2011, 134, 224101), and the entanglement-driven genetic algorithm (EDGA, Luo et al., J. Chem. Theory Comput. 2017, 13, 4699). It enhances productivity and allows the sampling programs to evolve to their population-expansion versions, for example, EDGA with population expansion (PE-EDGA). Further, examples of 1,2-dioxetanone and firefly dioxetanone anion (FDO^- ) molecules demonstrated the following: (a) parallel efficiencies can be persistently improved by simply by increasing the proportions of the asynchronous executions and (b) a sampled CAS-type CI wave function of a bi-radical-state FDO^- molecule utilizing the full valence (30e,26o) active space can be constructed within a few hours with using thousands of cores.

Near-Linear Scaling in DMRG-Based Tailored Coupled Clusters: An Implementation of DLPNO-TCCSD and DLPNO-TCCSD(T)

We present a new implementation of density matrix renormalization group based tailored coupled clusters method (TCCSD), which employs the domain-based local pair natural orbital approach (DLPNO). Compared to the previous local pair natural orbital (LPNO) version of the method, the new implementation is more accurate, offers more favorable scaling, and provides more consistent behavior across the variety of systems. On top of the singles and doubles, we include the perturbative triples correction (T), which is able to retrieve even more dynamic correlation. The methods were tested on three systems: tetramethyleneethane, oxo-Mn(Salen), and iron(II)-porphyrin model. The first two were revisited to assess the performance with respect to LPNO-TCCSD. For oxo-Mn(Salen), we retrieved between 99.8 and 99.9% of the total canonical correlation energy which is an improvement of 0.2% over the LPNO version in less than 63% of the total LPNO runtime. Similar results were obtained for iron(II)-porphyrin. When the perturbative triples correction was employed, irrespective of the active space size or system, the obtained energy differences between two spin states were within the chemical accuracy of 1 kcal/mol using the default DLPNO settings.

Semiclassical Dispersion Corrections Efficiently Improve Multiconfigurational Theory with Short-Range Density-Functional Dynamic Correlation

Multiconfigurational wave functions are known to describe the electronic structure across a Born-Oppenheimer surface qualitatively correct. However, for quantitative reaction energies, dynamic correlation originating from the many configurations involving excitations out of the restricted orbital space, the active space, must be considered. Standard procedures involve approximations that eventually limit the ultimate accuracy achievable (most prominently, multireference perturbation theory). At the same time, the computational cost increases dramatically due to the necessity to obtain higher-order reduced density matrices. It is this disproportion that leads us here to propose an MC-srDFT-D hybrid approach of semiclassical dispersion (D) corrections to cover long-range dynamic correlation in a multiconfigurational (MC) wave function theory, which includes short-range (sr) dynamic correlation by density functional theory (DFT) without double counting. We demonstrate that the reliability of this approach is very good (at negligible cost), especially when considering that standard second-order multireference perturbation theory usually overestimates dispersion interactions.

Toward the efficient local tailored coupled cluster approximation and the peculiar case of oxo-Mn(Salen)

We introduce a new implementation of the coupled cluster method with single and double excitations tailored by the matrix product state wave functions (DMRG-TCCSD), which employs the local pair natural orbital (LPNO) approach. By exploiting locality in the coupled cluster stage of the calculation, we were able to remove some of the limitations that hindered the application of the canonical version of the method to larger systems and/or with larger basis sets. We assessed the accuracy of the approximation using two systems: tetramethyleneethane (TME) and oxo-Mn(Salen). Using the default cut-off parameters, we were able to recover over 99.7% and 99.8% of the canonical correlation energy for the triplet and singlet state of TME, respectively. In the case of oxo-Mn(Salen), we found that the amount of retrieved canonical correlation energy depends on the size of the complete active space (CAS)-we retrieved over 99.6% for the larger 27 orbital CAS and over 99.8% for the smaller 22 orbital CAS. The use of LPNO-TCCSD allowed us to perform these calculations up to quadruple-ζ basis set, amounting to 1178 basis functions. Moreover, we examined dependence of the ground state of oxo-Mn(Salen) on the CAS composition. We found that the inclusion of 4d_xy orbital plays an important role in stabilizing the singlet state at the DMRG-CASSCF level via double-shell effect. However, by including dynamic correlation, the ground state was found to be triplet regardless of the size of the basis set or the composition of CAS, which is in agreement with previous findings by canonical DMRG-TCCSD in smaller basis.

Externally-Contracted Multireference Configuration Interaction Method Using a DMRG Reference Wave Function

The recent development of the density matrix renormalization group (DMRG) method in multireference quantum chemistry makes it practical to evaluate static correlation in a large active space, while dynamic correlation provides a critical correction to the DMRG reference for strong-correlated systems and is usually obtained using multireference perturbation (MRPT) or configuration interaction (MRCI) methods with internal contraction (ic) approximation. These methods can use an active space scalable to relatively larger size references than has previously been possible. However, they are still hardly applicable to systems with an active space larger than 30 orbitals and/or a large basis set because of high computation and storage costs of high-order reduced density matrices (RDMs) and the crucial dependence of the MRCI Hamiltonian dimension on the number of virtual orbitals. In this work, we propose a new effective implementation of DMRG-MRCI, in which we use reconstructed CASCI-type configurations from DMRG wave function via the entropy-driving genetic algorithm (EDGA) [ Luo et al. J. Chem. Theory Comput. 2017 , 13 , 4699 - 4710 . ] and integrate it with MRCI by an external contraction (ec) scheme. This bypasses the bottleneck of computing high-order RDMs in traditional DMRG dynamic correlation methods with ic approximation, and the number of MRCI configurations is not dependent on the number of virtual orbitals. Therefore, the DMRG-ec-MRCI method is promising for dealing with a larger active space than 30 orbitals and large basis sets. We demonstrate the capability of our DMRG-ec-MRCI method in several benchmark applications, including the evaluation of the potential energy curve of Cr₂, single-triplet gaps of higher n-acene molecules, and the energy of the Eu-BTBP(NO₃)₃ complex.

Coupled Cluster Method with Single and Double Excitations Tailored by Matrix Product State Wave Functions

In the past decade, the quantum chemical version of the density matrix renormalization group (DMRG) method has established itself as the method of choice for calculations of strongly correlated molecular systems. Despite its favorable scaling, it is in practice not suitable for computations of dynamic correlation. We present a novel method for accurate "post-DMRG" treatment of dynamic correlation based on the tailored coupled cluster (CC) theory in which the DMRG method is responsible for the proper description of nondynamic correlation, whereas dynamic correlation is incorporated through the framework of the CC theory. We illustrate the potential of this method on prominent multireference systems, in particular, N₂ and Cr₂ molecules and also oxo-Mn(Salen), for which we have performed the first post-DMRG computations in order to shed light on the energy ordering of the lowest spin states.

Entanglement Measures for Single- and Multireference Correlation Effects

Electron correlation effects are essential for an accurate ab initio description of molecules. A quantitative a priori knowledge of the single- or multireference nature of electronic structures as well as of the dominant contributions to the correlation energy can facilitate the decision regarding the optimum quantum chemical method of choice. We propose concepts from quantum information theory as orbital entanglement measures that allow us to evaluate the single- and multireference character of any molecular structure in a given orbital basis set. By studying these measures we can detect possible artifacts of small active spaces.

Accurate ab Initio Spin Densities

We present an approach for the calculation of spin density distributions for molecules that require very large active spaces for a qualitatively correct description of their electronic structure. Our approach is based on the density-matrix renormalization group (DMRG) algorithm to calculate the spin density matrix elements as a basic quantity for the spatially resolved spin density distribution. The spin density matrix elements are directly determined from the second-quantized elementary operators optimized by the DMRG algorithm. As an analytic convergence criterion for the spin density distribution, we employ our recently developed sampling-reconstruction scheme [J. Chem. Phys.2011, 134, 224101] to build an accurate complete-active-space configuration-interaction (CASCI) wave function from the optimized matrix product states. The spin density matrix elements can then also be determined as an expectation value employing the reconstructed wave function expansion. Furthermore, the explicit reconstruction of a CASCI-type wave function provides insight into chemically interesting features of the molecule under study such as the distribution of α and β electrons in terms of Slater determinants, CI coefficients, and natural orbitals. The methodology is applied to an iron nitrosyl complex which we have identified as a challenging system for standard approaches [J. Chem. Theory Comput.2011, 7, 2740].