Fermionic Orbital Optimization in Tensor Network States

What can quantum information theory offer to quantum chemistry?

It is the ultimate goal of this work to foster synergy between quantum chemistry and the flourishing field of quantum information theory. For this, we first translate quantum information concepts, such as entanglement and correlation, into the context of quantum chemical systems. In particular, we establish two conceptually distinct perspectives on 'electron correlation', leading to a notion of orbital and particle correlation. We then demonstrate that particle correlation equals total orbital correlation minimized over all orbital bases. Accordingly, particle correlation resembles the minimal, thus intrinsic, complexity of many-electron wave functions, while orbital correlation quantifies their complexity relative to a basis. We illustrate these concepts of intrinsic and extrinsic correlation complexity in molecular systems, which also manifests the crucial link between the two correlation pictures. Our results provide theoretical justification for the long-favored natural orbitals for simplifying electronic structures, and open new pathways for developing more efficient approaches towards the electron correlation problem.

Quantum Information Orbitals (QIO): Unveiling Intrinsic Many-Body Complexity by Compressing Single-Body Triviality

The simultaneous treatment of static and dynamic correlations in strongly correlated electron systems is a critical challenge. In particular, finding a universal scheme for identifying a single-particle orbital basis that minimizes the representational complexity of the many-body wave function is a formidable and longstanding problem. As a contribution toward its solution, we show that the total orbital correlation actually reveals and quantifies the intrinsic complexity of the wave function, once it is minimized via orbital rotations. To demonstrate the power of this concept in practice, an iterative scheme is proposed to optimize the orbitals by minimizing the total orbital correlation calculated by the tailored coupled cluster singles and doubles (TCCSD) ansatz. The optimized orbitals enable the limited TCCSD ansatz to capture more nontrivial information on the many-body wave function, indicated by the improved wave function and energy. An initial application of this scheme shows great improvement of TCCSD in predicting the singlet ground state potential energy curves of the strongly correlated C2 and Cr2 molecule.

Predicting the FCI Energy of Large Systems to Chemical Accuracy from Restricted Active Space Density Matrix Renormalization Group Calculations

We theoretically derive and validate with large scale simulations a remarkably accurate power law scaling of errors for the restricted active space density matrix renormalization group (DMRG-RAS) method [J. Phys. Chem. A 126, 9709] in electronic structure calculations. This yields a new extrapolation method, DMRG-RAS-X, which reaches chemical accuracy for strongly correlated systems such as the chromium dimer, dicarbon up to a large cc-pVQZ basis and even a large chemical complex such as the FeMoco with significantly lower computational demands than those of previous methods. The method is free of empirical parameters, performed robustly and reliably in all examples we tested, and has the potential to become a vital alternative method for electronic structure calculations in quantum chemistry and more generally for the computation of strong correlations in nuclear and condensed matter physics.

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.

Galván I, Alavi A, Aleotti F, Aquilante F, Autschbach J, Avagliano D, Baiardi A, Bao JJ, Battaglia S, Birnoschi L, Blanco-González A, Bokarev SI, Broer R, Cacciari R, Calio PB, Carlson RK, Carvalho Couto R, Cerdán L, Chibotaru LF, Chilton NF, Church JR, Conti I, Coriani S, Cuéllar-Zuquin J, Daoud RE, Dattani N, Decleva P, de Graaf C, Delcey M, De Vico L, Dobrautz W, Dong SS, Feng R, Ferré N, Filatov(Gulak) M, Gagliardi L, Garavelli M, González L, Guan Y, Guo M, Hennefarth MR, Hermes MR, Hoyer CE, Huix-Rotllant M, Jaiswal VK, Kaiser A, Kaliakin DS, Khamesian M, King DS, Kochetov V, Krośnicki M, Kumaar AA, Larsson ED, Lehtola S, Lepetit MB, Lischka H, López Ríos P, Lundberg M, Ma D, Mai S, Marquetand P, Merritt ICD, Montorsi F, Mörchen M, Nenov A, Nguyen VHA, Nishimoto Y, Oakley MS, Olivucci M, Oppel M, Padula D, Pandharkar R, Phung QM, Plasser F, Raggi G, Rebolini E, Reiher M, Rivalta I, Roca-Sanjuán D, Romig T, Safari AA, Sánchez-Mansilla A, Sand AM, Schapiro I, Scott TR, Segarra-Martí J, Segatta F, Sergentu DC, Sharma P, Shepard R, Shu Y, Staab JK, Straatsma TP, Sørensen LK, Tenorio BNC, Truhlar DG, Ungur L, Vacher M, Veryazov V, Voß TA, Weser O, Wu D, Yang X, Yarkony D, Zhou C, Zobel JP, Lindh R. The OpenMolcas Web: A Community-Driven Approach to Advancing Computational Chemistry

The developments of the open-source OpenMolcas chemistry software environment since spring 2020 are described, with a focus on novel functionalities accessible in the stable branch of the package or via interfaces with other packages. These developments span a wide range of topics in computational chemistry and are presented in thematic sections: electronic structure theory, electronic spectroscopy simulations, analytic gradients and molecular structure optimizations, ab initio molecular dynamics, and other new features. This report offers an overview of the chemical phenomena and processes OpenMolcas can address, while showing that OpenMolcas is an attractive platform for state-of-the-art atomistic computer simulations.

Quantum machine learning for chemistry and physics

Machine learning (ML) has emerged as a formidable force for identifying hidden but pertinent patterns within a given data set with the objective of subsequent generation of automated predictive behavior. In recent years, it is safe to conclude that ML and its close cousin, deep learning (DL), have ushered in unprecedented developments in all areas of physical sciences, especially chemistry. Not only classical variants of ML, even those trainable on near-term quantum hardwares have been developed with promising outcomes. Such algorithms have revolutionized materials design and performance of photovoltaics, electronic structure calculations of ground and excited states of correlated matter, computation of force-fields and potential energy surfaces informing chemical reaction dynamics, reactivity inspired rational strategies of drug designing and even classification of phases of matter with accurate identification of emergent criticality. In this review we shall explicate a subset of such topics and delineate the contributions made by both classical and quantum computing enhanced machine learning algorithms over the past few years. We shall not only present a brief overview of the well-known techniques but also highlight their learning strategies using statistical physical insight. The objective of the review is not only to foster exposition of the aforesaid techniques but also to empower and promote cross-pollination among future research in all areas of chemistry which can benefit from ML and in turn can potentially accelerate the growth of such algorithms.

On the fly swapping algorithm for ordering of degrees of freedom in density matrix renormalization group

Density matrix renormalization group (DMRG) and its time-dependent variants have found widespread applications in quantum chemistry, includingab initioelectronic structure of complex bio-molecules, spectroscopy for molecular aggregates, and charge transport in bulk organic semiconductors. The underlying wavefunction ansatz for DMRG, matrix product state (MPS), requires mapping degrees of freedom (DOF) into a one-dimensional topology. DOF ordering becomes a crucial factor for DMRG accuracy. In this work, we propose swapping neighboring DOFs during the DMRG sweeps for DOF ordering, which we term 'on the fly swapping' (OFS) algorithm. We show that OFS is universal for both static and time-dependent DMRG with minimum computational overhead. Examples are given for one dimensional antiferromagnetic Heisenberg model,ab initioelectronic structure of N₂molecule, and the S₁/S₂internal conversion dynamics of pyrazine molecule. It is found that OFS can indeed improve accuracy by finding better DOF ordering in all cases.

The Effect of Geometry, Spin, and Orbital Optimization in Achieving Accurate, Correlated Results for Iron-Sulfur Cubanes

Iron-sulfur clusters comprise an important functional motif in the catalytic centers of biological systems, capable of enabling important chemical transformations at ambient conditions. This remarkable capability derives from a notoriously complex electronic structure that is characterized by a high density of states that is sensitive to geometric changes. The spectral sensitivity to subtle geometric changes has received little attention from correlated, large active space calculations, owing partly to the exceptional computational complexity for treating these large and correlated systems accurately. To provide insight into this aspect, we report the first Complete Active Space Self Consistent Field (CASSCF) calculations for different geometries of the [Fe(II/III)₄S₄(SMe)₄]^-2 clusters using two complementary, correlated solvers: spin-pure Adaptive Sampling Configuration Interaction (ASCI) and Density Matrix Renormalization Group (DMRG). We find that the previously established picture of a double-exchange driven magnetic structure, with minute energy gaps (<1 mHa) between consecutive spin states, has a weak dependence on the underlying geometry. However, the spin gap between the singlet and the spin state 2S + 1 = 19, corresponding to a maximal number of Fe-d electrons being unpaired and of parallel spin, is strongly geometry dependent, changing by a factor of 3 upon slight deformations that are still within biologically relevant parameters. The CASSCF orbital optimization procedure, using active spaces as large as 86 electrons in 52 orbitals, was found to reduce this gap compared to typical mean-field orbital approaches. Our results show the need for performing large active space calculations to unveil the challenging electronic structure of these complex catalytic centers and should serve as accurate starting points for fully correlated treatments upon inclusion of dynamical correlation outside the active space.

Hybrid gausslet/Gaussian basis sets

We introduce hybrid gausslet/Gaussian basis sets, where a standard Gaussian basis is added to a gausslet basis in order to increase accuracy near the nuclei while keeping the spacing of the grid of gausslets relatively large. The Gaussians are orthogonalized to the gausslets, which are already orthonormal, and approximations are introduced to maintain the diagonal property of the two electron part of the Hamiltonian so that it continues to scale as the second power of the number of basis functions rather than the fourth. We introduce several corrections to the Hamiltonian designed to enforce certain exact properties, such as the values of certain two-electron integrals. We also introduce a simple universal energy correction that compensates for the incompleteness of the basis stemming from the electron-electron cusps based on the measured double occupancy of each basis function. We perform a number of Hartree Fock and full configuration interaction (full-CI) test calculations on two electron systems and Hartree Fock on a ten-atom hydrogen chain to benchmark these techniques. The inclusion of the cusp correction allows us to obtain complete basis set full-CI results for the two electron cases at the level of several micro-Hartrees, and we see similar apparent accuracy for Hartree Fock on the ten-atom hydrogen chain.

Density Matrix Renormalization Group with Dynamical Correlation via Adiabatic Connection

The quantum chemical version of the density matrix renormalization group (DMRG) method has established itself as one of the methods of choice for calculations of strongly correlated molecular systems. Despite its great ability to capture strong electronic correlation in large active spaces, it is not suitable for computations of dynamical electron correlation. In this work, we present a new approach to the electronic structure problem of strongly correlated molecules, in which DMRG is responsible for a proper description of the strong correlation, whereas dynamical correlation is computed via the recently developed adiabatic connection (AC) technique which requires only up to two-body active space reduced density matrices. We report the encouraging results of this approach on typical candidates for DMRG computations, namely, n-acenes (n = 2 → 7), Fe(II)-porphyrin, and the Fe₃S₄ cluster.

Measurement-Based Variational Quantum Eigensolver

Variational quantum eigensolvers (VQEs) combine classical optimization with efficient cost function evaluations on quantum computers. We propose a new approach to VQEs using the principles of measurement-based quantum computation. This strategy uses entangled resource states and local measurements. We present two measurement-based VQE schemes. The first introduces a new approach for constructing variational families. The second provides a translation of circuit- to measurement-based schemes. Both schemes offer problem-specific advantages in terms of the required resources and coherence times.

Concept of Orbital Entanglement and Correlation in Quantum Chemistry

A recent development in quantum chemistry has established the quantum mutual information between orbitals as a major descriptor of electronic structure. This has already facilitated remarkable improvements in numerical methods and may lead to a more comprehensive foundation for chemical bonding theory. Building on this promising development, our work provides a refined discussion of quantum information theoretical concepts by introducing the physical correlation and its separation into classical and quantum parts as distinctive quantifiers of electronic structure. In particular, we succeed in quantifying the entanglement. Intriguingly, our results for different molecules reveal that the total correlation between orbitals is mainly classical, raising questions about the general significance of entanglement in chemical bonding. Our work also shows that implementing the fundamental particle number superselection rule, so far not accounted for in quantum chemistry, removes a major part of correlation and entanglement seen previously. In that respect, realizing quantum information processing tasks with molecular systems might be more challenging than anticipated.

Topologically Protected Correlated End Spin Formation in Carbon Nanotubes

For most chiralities, semiconducting nanotubes display topologically protected end states of multiple degeneracies. We demonstrate using density matrix renormalization group based quantum chemistry tools that the presence of Coulomb interactions induces the formation of robust end spins. These are the close analogs of ferromagnetic edge states emerging in graphene nanoribbons. The interaction between the two ends is sensitive to the length of the nanotube, its dielectric constant, and the size of the end spins: for S=1/2 end spins, their interaction is antiferromagnetic, while for S>1/2, it changes from antiferromagnetic to ferromagnetic as the nanotube length increases. The interaction between end spins can be controlled by changing the dielectric constant of the environment, thereby providing a possible platform for two-spin quantum manipulations.

Tensor Network Representations of Parton Wave Functions

Tensor network states and parton wave functions are two pivotal methods for studying quantum many-body systems. This work connects these two subjects as we demonstrate that a variety of parton wave functions, such as projected Fermi sea and projected fermionic or bosonic paired states, can be represented exactly as tensor networks. The results can be compressed into matrix product states with moderate bond dimensions so various physical quantities can be computed efficiently. For the projected Fermi sea, we develop an excellent compression scheme with high fidelity using maximally localized Wannier orbitals. Numerical calculations on two parton wave functions demonstrate that our method exceeds commonly adopted Monte Carlo methods in some aspects. It produces energy and correlation function with very high accuracy that is difficult to achieve using Monte Carlo method. The entanglement measures that were almost impossible to compute before can also be obtained easily using our method.

Breaking the Entanglement Barrier: Tensor Network Simulation of Quantum Transport

The recognition that large classes of quantum many-body systems have limited entanglement in the ground and low-lying excited states led to dramatic advances in their numerical simulation via so-called tensor networks. However, global dynamics elevates many particles into excited states, and can lead to macroscopic entanglement and the failure of tensor networks. Here, we show that for quantum transport-one of the most important cases of this failure-the fundamental issue is the canonical basis in which the scenario is cast: When particles flow through an interface, they scatter, generating a "bit" of entanglement between spatial regions with each event. The frequency basis naturally captures that-in the long-time limit and in the absence of inelastic scattering-particles tend to flow from a state with one frequency to a state of identical frequency. Recognizing this natural structure yields a striking-potentially exponential in some cases-increase in simulation efficiency, greatly extending the attainable spatial and time scales, and broadening the scope of tensor network simulation to hitherto inaccessible classes of nonequilibrium many-body problems.

Quantum Chemistry in the Age of Quantum Computing

Practical challenges in simulating quantum systems on classical computers have been widely recognized in the quantum physics and quantum chemistry communities over the past century. Although many approximation methods have been introduced, the complexity of quantum mechanics remains hard to appease. The advent of quantum computation brings new pathways to navigate this challenging and complex landscape. By manipulating quantum states of matter and taking advantage of their unique features such as superposition and entanglement, quantum computers promise to efficiently deliver accurate results for many important problems in quantum chemistry, such as the electronic structure of molecules. In the past two decades, significant advances have been made in developing algorithms and physical hardware for quantum computing, heralding a revolution in simulation of quantum systems. This Review provides an overview of the algorithms and results that are relevant for quantum chemistry. The intended audience is both quantum chemists who seek to learn more about quantum computing and quantum computing researchers who would like to explore applications in quantum chemistry.

Quantum information-based analysis of electron-deficient bonds

Recently, the correlation theory of the chemical bond was developed, which applies concepts of quantum information theory for the characterization of chemical bonds, based on the multiorbital correlations within the molecule. Here, for the first time, we extend the use of this mathematical toolbox for the description of electron-deficient bonds. We start by verifying the theory on the textbook example of a molecule with three-center two-electron bonds, namely, diborane(6). We then show that the correlation theory of the chemical bond is able to properly describe the bonding situation in more exotic molecules which have been synthesized and characterized only recently, in particular, the diborane molecule with four hydrogen atoms [diborane(4)] and a neutral zerovalent s-block beryllium complex, whose surprising stability was attributed to a strong three-center two-electron π bond stretching across the C-Be-C core. Our approach is of high importance especially in the light of a constant chase after novel compounds with extraordinary properties where the bonding is expected to be unusual.

Numerical and Theoretical Aspects of the DMRG-TCC Method Exemplified by the Nitrogen Dimer

In this article, we investigate the numerical and theoretical aspects of the coupled-cluster method tailored by matrix-product states. We investigate formal properties of the used method, such as energy size consistency and the equivalence of linked and unlinked formulation. The existing mathematical analysis is here elaborated in a quantum chemical framework. In particular, we highlight the use of what we have defined as a complete active space-external space gap describing the basis splitting between the complete active space and the external part generalizing the concept of a HOMO–LUMO gap. Furthermore, the behavior of the energy error for an optimal basis splitting, i.e., an active space choice minimizing the density matrix renormalization group-tailored coupled-cluster singles doubles error, is discussed. We show numerical investigations on the robustness with respect to the bond dimensions of the single orbital entropy and the mutual information, which are quantities that are used to choose a complete active space. Moreover, the dependence of the ground-state energy error on the complete active space has been analyzed numerically in order to find an optimal split between the complete active space and external space by minimizing the density matrix renormalization group-tailored coupled-cluster error.