Reformulation of All ONIOM-Type Molecular Fragmentation Approaches and Many-Body Theories Using Graph-Theory-Based Projection Operators: Applications to Dynamics, Molecular Potential Surfaces, Machine Learning, and Quantum Computing.

We present a graph-theory-based reformulation of all ONIOM-based molecular fragmentation methods. We discuss applications to (a) accurate post-Hartree-Fock AIMD that can be conducted at DFT cost for medium-sized systems, (b) hybrid DFT condensed-phase studies at the cost of pure density functionals, (c) reduced cost on-the-fly large basis gas-phase AIMD and condensed-phase studies, (d) post-Hartree-Fock-level potential surfaces at DFT cost to obtain quantum nuclear effects, and (e) novel transfer machine learning protocols derived from these measures. Additionally, in previous work, the unifying strategy discussed here has been used to construct new quantum computing algorithms. Thus, we conclude that this reformulation is robust and accurate.

Resource Optimization for Quantum Dynamics with Tensor Networks: Quantum and Classical Algorithms.

The exponential scaling of the quantum degrees of freedom with the size of the system is one of the biggest challenges in computational chemistry and particularly in quantum dynamics. We present a tensor network approach for the time-evolution of the nuclear degrees of freedom of multiconfigurational chemical systems at a reduced storage and computational complexity. We also present quantum algorithms for the resultant dynamics. To preserve the compression advantage achieved via tensor network decompositions, we present an adaptive algorithm for the regularization of nonphysical bond dimensions, preventing the potentially exponential growth of these with time. While applicable to any quantum dynamical problem, our method is particularly valuable for dynamical simulations of nuclear chemical systems. Our algorithm is demonstrated using ab initio potentials obtained for a symmetric hydrogen-bonded system, namely, the protonated 2,2'-bipyridine, and compared to exact diagonalization numerical results.

Rare Events Sampling Methods for Quantum and Classical Ab Initio Molecular Dynamics.

We provide an approach to sample rare events during classical ab initio molecular dynamics and quantum wavepacket dynamics. For classical AIMD, a set of fictitious degrees of freedom are introduced that may harmonically interact with the electronic and nuclear degrees of freedom to steer the dynamics in a conservative fashion toward energetically forbidden regions. A similar approach when introduced for quantum wavepacket dynamics has the effect of biasing the trajectory of the wavepacket centroid toward the regions of the potential surface that are difficult to sample. The approach is demonstrated for a phenol-amine system, which is a prototypical problem for condensed phase-proton transfer, and for model potentials undergoing wavepacket dynamics. In all cases, the approach yields trajectories that conserve energy while sampling rare events.

Quantum Computing with Dartboards.

We present a physically appealing and elegant picture for quantum computing, using rules constructed for a game of darts. A dartboard is used to represent the state space in quantum mechanics, and the act of throwing the dart is shown to have close similarities to the concept of measurement or collapse of the wave function in quantum mechanics. The analogy is constructed in arbitrary dimensional spaces, that is, using arbitrary dimensional dartboards, and for such arbitrary spaces this also provides us a "visual" description of uncertainty. Finally, connections between qubits and quantum computing algorithms are also made, opening the possibility to construct analogies between quantum algorithms and coupled dart throws.

Graph-|Q⟩⟨C|: A Quantum Algorithm with Reduced Quantum Circuit Depth for Electronic Structure.

The accurate determination of chemical properties is known to have a critical impact on multiple fundamental chemical problems but is deeply hindered by the steep algebraic scaling of electron correlation calculations and the exponential scaling of quantum nuclear dynamics. With the advent of new quantum computing hardware and associated developments in creating new paradigms for quantum software, this avenue has been recognized as perhaps one way to address exponentially complex challenges in quantum chemistry and molecular dynamics. In this paper, we discuss a new approach to drastically reduce the quantum circuit depth (by several orders of magnitude) and help improve the accuracy in the quantum computation of electron correlation energies for large molecular systems. The method is derived from a graph-theoretic approach to molecular fragmentation and enables us to create a family of projection operators that decompose quantum circuits into separate unitary processes. Some of these processes can be treated on quantum hardware and others on classical hardware in a completely asynchronous and parallel fashion. Numerical benchmarks are provided through the computation of unitary coupled-cluster singles and doubles (UCCSD) energies for medium-sized protonated and neutral water clusters using the new quantum algorithms presented here.

Challenges in constructing accurate methods for hydrogen transfer reactions in large biological assemblies: rare events sampling for mechanistic discovery and tensor networks for quantum nuclear effects.

We present two methods that address the computational complexities arising in hydrogen transfer reactions in enzyme active sites. To address the challenge of reactive rare events, we begin with an ab initio molecular dynamics adaptation of the Caldeira-Leggett system-bath Hamiltonian and apply this approach to the study of the hydrogen transfer rate-determining step in soybean lipoxygenase-1. Through direct application of this method to compute an ensemble of classical trajectories, we discuss the critical role of isoleucine-839 in modulating the primary hydrogen transfer event in SLO-1. Notably, the formation of the hydrogen bond between isoleucine-839 and the acceptor-OH group regulates the electronegativity of the donor and acceptor groups to affect the hydrogen transfer process. Curtailing the formation of this hydrogen bond adversely affects the probability of hydrogen transfer. The second part of this paper deals with complementing the rare event sampled reaction pathways obtained from the aforementioned development through quantum nuclear wavepacket dynamics. Essentially the idea is to construct quantum nuclear dynamics on the potential surfaces obtained along the biased trajectories created as noted above. Here, while we are able to obtain critical insights on the quantum nuclear effects from wavepacket dynamics, we primarily engage in providing an improved computational approach for efficient representation of quantum dynamics data such as potential surfaces and transmission probabilities using tensor networks. We find that utilizing tensor networks yields an accurate and efficient description of time-dependent wavepackets, reduced dimensional nuclear eigenstates and associated potential energy surfaces at much reduced cost.

Exotic electronic structures of SmxCe3-xOy (x = 0-3; y = 2-4) clusters and the effect of high neutral density of low-lying states on photodetachment transition intensities.

Lanthanide (Ln) oxide clusters have complex electronic structures arising from the partially occupied Ln 4f subshell. New anion photoelectron (PE) spectra of SmxCe3-xOy- (x = 0-3; y = 2-4) along with supporting results of density functional theory (DFT) calculations suggest interesting x and y-dependent Sm 4f subshell occupancy with implications for Sm-doped ionic conductivity of ceria, as well as the overall electronic structure of the heterometallic oxides. Specifically, the Sm centers in the heterometallic species have higher 4f subshell occupancy than the homonuclear Sm3Oy-/Sm3Oy clusters. The higher 4f subshell occupancy both weakens Sm-O bonds and destabilizes the 4f subshell relative to the predominantly O 2p bonding orbitals in the clusters. Parallels between the electronic structures of these small cluster systems with bulk oxides are explored. In addition, unusual changes in the excited state transition intensities, similar to those observed previously in the PE spectra of Sm2O- and Sm2O2- [J. O. Kafader et al., J. Chem. Phys. 146, 194310 (2017)], are also observed in the relative intensities of electronic transitions to excited neutral state bands in the PE spectra of SmxCe3-xOy- (x = 1-3; y = 2, 4). The new spectra suggest that the effect is enhanced with lower oxidation states and with an increasing number of Sm atoms, implying that the prevalence of electrons in the diffuse Sm 6s-based molecular orbitals and a more populated 4f subshell both contribute to this phenomenon. Finally, this work identifies challenges associated with affordable DFT calculations in treating the complex electronic structures exhibited by these systems, including the need for a more explicit treatment of strong coupling between the neutral and PE.

Proton relays in anomalous carbocations dictate spectroscopy, stability, and mechanisms: case studies on C2H5+ and C3H3.

We present a detailed analysis of the anomalous carbocations: C2H5+ and C3H3+. This work involves (a) probing electronic structural properties, (b) ab initio dynamics simulations over a range of internal energies, (c) analysis of reduced dimensional potential surfaces directed along selected conformational transition pathways, (d) dynamically averaged vibrational spectra computed from ab initio dynamics trajectories, and (e) two-dimensional time-frequency analysis to probe conformational dynamics. Key findings are as follows: (i) as noted in our previous study on C2H3+, it appears that these non-classical carbocations are stabilized by delocalized nuclear frameworks and "proton shuttles". We analyze this nuclear delocalization and find critical parallels between conformational changes in C2H3+, C2H5+, and C3H3+. (ii) The vibrational signatures of C2H5+ are dominated by the "bridge-proton" conformation, but also show critical contributions from the "classical" configuration, which is a transition state at almost all levels of theory. This result is further substantiated through two-dimensional time-frequency analysis and is at odds with earlier explanations of the experimental spectra, where frequencies close to the classical region were thought to arise from an impurity. While this is still possible, our results here indicate an additional (perhaps more likely) explanation that involves the "classical" isomer. (iii) Finally, in the case of C3H3+ our explanation of the experimental result includes the presence of multiple, namely, "cyclic", "straight", and propargyl, configurations. Proton shuttles and nuclear delocalization, reminiscent of those seen in the case of C2H3+, were seen all through and have a critical role in all our observations.

The electron shuffle: Cerium influences samarium 4f orbital occupancy in heteronuclear Ce-Sm oxide clusters.

The anion photoelectron (PE) spectra along with supporting results of density functional theory (DFT) calculations on SmO-, SmCeOy-, and Sm2Oy- (y = 1, 2) are reported and compared to previous results on CeO- [M. Ray et al., J. Chem. Phys. 142, 064305 (2015)] and Ce2Oy- (y = 1, 2) [J. O. Kafader et al., J. Chem. Phys. 145, 154306 (2016)]. Similar to the results on CexOy- clusters, the PE spectra of SmO-, SmCeOy-, and Sm2Oy- (y = 1, 2) all exhibit electronic transitions to the neutral ground state at approximately 1 eV e-BE. The Sm centers in SmO and Sm2O2 neutrals can be described with the 4f56s superconfiguration, which is analogous to CeO and Ce2O2 neutrals in which the Ce centers can be described with the 4f 6s superconfiguration (ZCe = ZSm - 4). The Sm center in CeSmO2, in contrast, has a 4f6 occupancy, while the Ce center maintains the 4f 6s superconfiguration. The less oxidized Sm centers in both Sm2O and SmCeO have 4f6 6s occupancies. The 4f6 subshell occupancy results in relatively weak Sm-O bond strengths. If this extra 4f occupancy also occurs in bulk Sm-doped ceria, it may play a role in the enhanced O2- ionic conductivity in Sm-doped ceria. Based on the results of DFT calculations, the heteronuclear Ce-Sm oxides have molecular orbitals that are distinctly localized Sm 4f, Sm 6s, Ce 4f, and Ce 6s orbitals. The relative intensity of two electronic bands in the PE spectrum of Sm2O- exhibits an unusual photon energy-dependence, and the PE spectrum of Sm2O2- exhibits a photon energy-dependent continuum signal between two electronic transitions. Several explanations, including the high magnetic moment of these suboxide species and the presence of low-lying quasi-bound anion states, are considered.

A Grotthuss-like proton shuttle in the anomalous C2H3+ carbocation: energetic and vibrational properties for isotopologues.

We probe the structure, stability and vibrational properties of the fundamental C2H3+ carbocation that exists with preference in a bridged hydrogen conformation. Our computational study includes electronic structure treatment, incorporation of nuclear motion through classical and quantum paradigms, the effect of temperature, and the associated sampling of the potential surface, and the effect of single H/D isotopic substitution (i.e., C2H2D+). We find that while the non-classical, "Bridged" isomer is most stable, the "Classical" form does have a small presence under ambient conditions since the zero point level straddles the barrier between the Classical and Bridged isomers in a reduced dimensional analysis of the Bridged ↔ Classical transfer coordinate. But the probability of the classical structure is too low and hence may remain undetected from the vibrational properties of the system. For the deuterated counterpart, the deuterium preferentially occupies the terminal instead of bridge position, in the more stable bridged isomeric structure. This preference is noted from nuclear dynamics. In all cases, at higher temperatures, an orbiting phenomenon is observed where the hydrogen atom density is distributed as an oblate ellipsoid surrounding the carbon-carbon bond. This is not observed at lower temperatures and the orbiting phenomenon is probed here by computing two-dimensional, time-frequency vibrational spectra, which show the spectral evolution in time and temperature, and the development of the system from one kind of isomer to another. New experiments that may probe this isomeric multiplicity are suggested, and these involve a combination of infra-red multiple photon dissociation (IRMPD) and argon-tagged action spectroscopy.

Analogy between Boltzmann Machines and Feynman Path Integrals.

Machine learning has had a significant impact on multiple areas of science, technology, health, and computer and information sciences. Through the advent of quantum computing, quantum machine learning has developed as a new and important avenue for the study of complex learning problems. Yet there is substantial debate and uncertainty in regard to the foundations of machine learning. Here, we provide a detailed exposition of the mathematical connections between a general machine learning approach called Boltzmann machines and Feynman's description of quantum and statistical mechanics. In Feynman's description, quantum phenomena arise from an elegant, weighted sum over (or superposition of) paths. Our analysis shows that Boltzmann machines and neural networks have a similar mathematical structure. This allows the interpretation that the hidden layers in Boltzmann machines and neural networks are discrete versions of path elements and allows a path integral interpretation of machine learning similar to that in quantum and statistical mechanics. Since Feynman paths are a natural and elegant depiction of interference phenomena and the superposition principle germane to quantum mechanics, this analysis allows us to interpret the goal in machine learning as finding an appropriate combination of paths, and accumulated path-weights, through a network, that cumulatively captures the correct properties of an x-to-y map for a given mathematical problem. We are forced to conclude that neural networks are naturally related to Feynman path-integrals and hence may present one avenue to be considered as quantum problems. Consequently, we provide general quantum circuit models applicable to both Boltzmann machines and Feynman path integrals.

Synthesis of Hidden Subgroup Quantum Algorithms and Quantum Chemical Dynamics.

We describe a general formalism for quantum dynamics and show how this formalism subsumes several quantum algorithms, including the Deutsch, Deutsch-Jozsa, Bernstein-Vazirani, Simon, and Shor algorithms as well as the conventional approach to quantum dynamics based on tensor networks. The common framework exposes similarities among quantum algorithms and natural quantum phenomena: we illustrate this connection by showing how the correlated behavior of protons in water wire systems that are common in many biological and materials systems parallels the structure of Shor's algorithm.

Capturing Weak Interactions in Surface Adsorbate Systems at Coupled Cluster Accuracy: A Graph-Theoretic Molecular Fragmentation Approach Improved through Machine Learning.

The accurate and efficient study of the interactions of organic matter with the surface of water is critical to a wide range of applications. For example, environmental studies have found that acidic polyfluorinated alkyl substances, especially perfluorooctanoic acid (PFOA), have spread throughout the environment and bioaccumulate into human populations residing near contaminated watersheds, leading to many systemic maladies. Thus, the study of the interactions of PFOA with water surfaces became important for the mitigation of their activity as pollutants and threats to public health. However, theoretical study of the interactions of such organic adsorbates on the surface of water, and their bulk concerted properties, often necessitates the use of ab initio methods to properly incorporate the long-range electronic properties that govern these extended systems. Notable theoretical treatments of "on-water" reactions thus far have employed hybrid DFT and semilocal DFT, but the interactions involved are weak interactions that may be best described using post-Hartree-Fock theory. Here, we aim to demonstrate the utility of a graph-theoretic approach to molecular fragmentation that accurately captures the critical "weak" interactions while maintaining an efficient ab initio treatment of the long-range periodic interactions that underpin the physics of extended systems. We apply this graph-theoretical treatment to study PFOA on the surface of water as a model system for the study of weak interactions seen in the wide range of surface interactions and reactions. The approach divides a system into a set of vertices, that are then connected through edges, faces, and higher order graph theoretic objects known as simplexes, to represent a collection of locally interacting subsystems. These subsystems are then used to construct ab initio molecular dynamics simulations and for computing multidimensional potential energy surfaces. To further improve the computational efficiency of our graph theoretic fragmentation method, we use a recently developed transfer learning protocol to construct the full system potential energy from a family of neural networks each designed to accurately model the behavior of individual simplexes. We use a unique multidimensional clustering algorithm, based on the k-means clustering methodology, to define our training space for each separate simplex. These models are used to extrapolate the energies for molecular dynamics trajectories at PFOA water interfaces, at less than one-tenth the cost as compared to a regular molecular fragmentation-based dynamics calculation with excellent agreement with couple cluster level of full system potential energies.

Quantum Computation of Hydrogen Bond Dynamics and Vibrational Spectra.

Calculating observable properties of chemical systems is often classically intractable and widely viewed as a promising application of quantum information processing. Here, we introduce a new framework for solving generic quantum chemical dynamics problems using quantum logic. We experimentally demonstrate a proof-of-principle instance of our method using the QSCOUT ion-trap quantum computer, where we experimentally drive the ion-trap system to emulate the quantum wavepacket dynamics corresponding to the shared-proton within an anharmonic hydrogen bonded system. Following the experimental creation and propagation of the shared-proton wavepacket on the ion-trap, we extract measurement observables such as its time-dependent spatial projection and its characteristic vibrational frequencies to spectroscopic accuracy (3.3 cm-1 wavenumbers, corresponding to >99.9% fidelity). Our approach introduces a new paradigm for studying the chemical dynamics and vibrational spectra of molecules and opens the possibility to describe the behavior of complex molecular processes with unprecedented accuracy.

Influence of water on anharmonicity, stability, and vibrational energy distribution of hydrogen-bonded adducts in atmospheric reactions: case study of the OH + isoprene reaction intermediate using ab initio molecular dynamics.

The effect of water on the stability and vibrational states of a hydroxy-isoprene adduct is probed through the introduction of 1-15 water molecules. It is found that when a static nuclear harmonic approximation is invoked there is a substantial red-shift of the alcohol O-H stretch (of the order of 800 cm(-1)) as a result of introduction of water. When potential energy surface sampling and associated anharmonicities are introduced through finite temperature ab initio dynamics, this hydroxy-isoprene OH stretch strongly couples with all the water vibrational modes as well as the hydroxy-isoprene OH bend modes. A new computational technique is introduced to probe the coupling between these modes. The method involves a two-dimensional, time-frequency analysis of the finite temperature vibrational properties. Such an analysis not only provides information about the modes that are coupled as a result of finite-temperature analysis, but also the temporal evolution of such coupling.

"Pump-probe" atom-centered density matrix propagation studies to gauge anharmonicity and energy repartitioning in atmospheric reactive adducts: case study of the OH + isoprene and OH + butadiene reaction intermediates.

Time-resolved "pump-probe" ab initio molecular dynamics studies are constructed to probe the stability of reaction intermediates, the mechanism of energy transfer, and energy repartitioning, for moieties involved during the interaction of volatile organic compunds with hydroxyl radical. These systems are of prime importance in the atmosphere. Specifically, the stability of reaction intermediates of hydroxyl radical adducts to isoprene and butadiene molecules is used as a case study to develop novel computational techniques involving "pump-probe" ab initio molecular dynamics. Starting with the various possible hydroxyl radical adducts to isoprene and butadiene, select vibrational modes of each of the adducts are populated with excess energy to mimic the initial conditions of an experiment. The flow of energy into the remaining modes is then probed by subjecting the excited adducts to ab initio molecular dynamics simulations. It is found that the stability of the adducts arises directly due to the anhormonically driven coupling of the modes to facilitate repartitioning of the excess vibrational energy. This kind of vibrational repartitioning has a critical influence on the energy density.

Graph Theoretic Molecular Fragmentation for Multidimensional Potential Energy Surfaces Yield an Adaptive and General Transfer Machine Learning Protocol.

Over a series of publications we have introduced a graph-theoretic description for molecular fragmentation. Here, a system is divided into a set of nodes, or vertices, that are then connected through edges, faces, and higher-order simplexes to represent a collection of spatially overlapping and locally interacting subsystems. Each such subsystem is treated at two levels of electronic structure theory, and the result is used to construct many-body expansions that are then embedded within an ONIOM-scheme. These expansions converge rapidly with many-body order (or graphical rank) of subsystems and have been previously used for ab initio molecular dynamics (AIMD) calculations and for computing multidimensional potential energy surfaces. Specifically, in all these cases we have shown that CCSD and MP2 level AIMD trajectories and potential surfaces may be obtained at density functional theory cost. The approach has been demonstrated for gas-phase studies, for condensed phase electronic structure, and also for basis set extrapolation-based AIMD. Recently, this approach has also been used to derive new quantum-computing algorithms that enormously reduce the quantum circuit depth in a circuit-based computation of correlated electronic structure. In this publication, we introduce (a) a family of neural networks that act in parallel to represent, efficiently, the post-Hartree-Fock electronic structure energy contributions for all simplexes (fragments), and (b) a new k-means-based tessellation strategy to glean training data for high-dimensional molecular spaces and minimize the extent of training needed to construct this family of neural networks. The approach is particularly useful when coupled cluster accuracy is desired and when fragment sizes grow in order to capture nonlocal interactions accurately. The unique multidimensional k-means tessellation/clustering algorithm used to determine our training data for all fragments is shown to be extremely efficient and reduces the needed training to only 10% of data for all fragments to obtain accurate neural networks for each fragment. These fully connected dense neural networks are then used to extrapolate the potential energy surface for all molecular fragments, and these are then combined as per our graph-theoretic procedure to transfer the learning process to a full system energy for the entire AIMD trajectory at less than one-tenth the cost as compared to a regular fragmentation-based AIMD calculation.

Graph-|Q⟩⟨C|, a Graph-Based Quantum/Classical Algorithm for Efficient Electronic Structure on Hybrid Quantum/Classical Hardware Systems: Improved Quantum Circuit Depth Performance.

We present a procedure to reduce the depth of quantum circuits and improve the accuracy of results in computing post-Hartree-Fock electronic structure energies in large molecular systems. The method is based on molecular fragmentation where a molecular system is divided into overlapping fragments through a graph-theoretic procedure. This allows us to create a set of projection operators that decompose the unitary evolution of the full system into separate sets of processes, some of which can be treated on quantum hardware and others on classical hardware. Thus, we develop a procedure for an electronic structure that can be asynchronously spawned onto a potentially large ensemble of classical and quantum hardware systems. We demonstrate this method by computing Unitary Coupled Cluster Singles and Doubles (UCCSD) energies for a set of [H2]n clusters, with n ranging from 4 to 128. We implement our methodology using quantum circuits, and when these quantum circuits are processed on a quantum simulator, we obtain energies in agreement with the UCCSD energies in the milli-hartree energy range. We also show that our circuit decomposition approach yields up to 9 orders of magnitude reduction in the number of CNOT gates and quantum circuit depth for the large-sized clusters when compared to a standard quantum circuit implementation available on IBM's Quantum Information Science kit, known as Qiskit.

Graph-Theoretic Molecular Fragmentation for Potential Surfaces Leads Naturally to a Tensor Network Form and Allows Accurate and Efficient Quantum Nuclear Dynamics.

Molecular fragmentation methods have revolutionized quantum chemistry. Here, we use a graph-theoretically generated molecular fragmentation method, to obtain accurate and efficient representations for multidimensional potential energy surfaces and the quantum time-evolution operator, which plays a critical role in quantum chemical dynamics. In doing so, we find that the graph-theoretic fragmentation approach naturally reduces the potential portion of the time-evolution operator into a tensor network that contains a stream of coupled lower-dimensional propagation steps to potentially achieve quantum dynamics with reduced complexity. Furthermore, the fragmentation approach used here has previously been shown to allow accurate and efficient computation of post-Hartree-Fock electronic potential energy surfaces, which in many cases has been shown to be at density functional theory cost. Thus, by combining the advantages of molecular fragmentation with the tensor network formalism, the approach yields an on-the-fly quantum dynamics scheme where both the electronic potential calculation and nuclear propagation portion are enormously simplified through a single stroke. The method is demonstrated by computing approximations to the propagator and to potential surfaces for a set of coupled nuclear dimensions within a protonated water wire problem exhibiting the Grotthuss mechanism of proton transport. In all cases, our approach has been shown to reduce the complexity of representing the quantum propagator, and by extension action of the propagator on an initial wavepacket, by several orders, with minimal loss in accuracy.

Quantum wavepacket ab initio molecular dynamics for extended systems.

In this paper, we introduce a symmetry-adapted quantum nuclear propagation technique that utilizes distributed approximating functionals for quantum wavepacket dynamics in extended condensed-phase systems. The approach is developed with a goal for implementation in quantum-classical methods such as the recently developed quantum wavepacket ab intio molecular dynamics (QWAIMD) to facilitate the study of extended systems. The method has been numerically benchmarked for extended electronic systems as well as protonic conducting systems that benefit from quantum nuclear treatment. Vibrational properties are computed for the case of the protonic systems through use of a novel velocity-flux correlation function. The treatment is found to be numerically accurate and efficient.