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.

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.

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.

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.

Weighted-Graph-Theoretic Methods for Many-Body Corrections within ONIOM: Smooth AIMD and the Role of High-Order Many-Body Terms.

We present a weighted-graph-theoretic approach to adaptively compute contributions from many-body approximations for smooth and accurate post-Hartree-Fock (pHF) ab initio molecular dynamics (AIMD) of highly fluxional chemical systems. This approach is ONIOM-like, where the full system is treated at a computationally feasible quality of treatment (density functional theory (DFT) for the size of systems considered in this publication), which is then improved through a perturbative correction that captures local many-body interactions up to a certain order within a higher level of theory (post-Hartree-Fock in this publication) described through graph-theoretic techniques. Due to the fluxional and dynamical nature of the systems studied here, these graphical representations evolve during dynamics. As a result, energetic "hops" appear as the graphical representation deforms with the evolution of the chemical and physical properties of the system. In this paper, we introduce dynamically weighted, linear combinations of graphs, where the transition between graphical representations is smoothly achieved by considering a range of neighboring graphical representations at a given instant during dynamics. We compare these trajectories with those obtained from a set of trajectories where the range of local many-body interactions considered is increased, sometimes to the maximum available limit, which yields conservative trajectories as the order of interactions is increased. The weighted-graph approach presents improved dynamics trajectories while only using lower-order many-body interaction terms. The methods are compared by computing dynamical properties through time-correlation functions and structural distribution functions. In all cases, the weighted-graph approach provides accurate results at a lower cost.

Efficient and Accurate Approach To Estimate Hybrid Functional and Large Basis-Set Contributions to Condensed-Phase Systems and Molecule-Surface Interactions.

We present a graph theoretic approach to adaptively compute contributions from many-body approximations in an efficient manner and perform accurate hybrid density functional theory (DFT) electronic structure calculations for condensed-phase systems. The salient features of the approach are ONIOM-like. (a) The full-system calculation is performed at a lower level of theory (pure DFT) by enforcing periodic boundary conditions. (b) This treatment is then improved through a correction term that captures many-body interactions up to any given order within a higher (in this case, hybrid DFT) level of theory. (c) In the spirit of ONIOM, contributions from the many-body approximations that arise from the higher level of theory [i.e., from (b) above] are included through extrapolation from the lower level calculation. The approach is implemented in a general, system-independent, fashion using the graph-theoretic functionalities available within Python. For example, the individual one-body components within the unit cell are designated as "nodes" within a graph. The interactions between these nodes are captured through edges, faces, tetrahedrons, and so forth, thus building a many-body interaction hierarchy. Electronic energy extrapolation contributions from all of these geometric entities are included within the above-mentioned ONIOM paradigm. The implementation of the method simultaneously uses multiple electronic structure packages. Here, for example, we present results which use both the Gaussian suite of electronic structure programs and the Quantum ESPRESSO program within a single calculation. Thus, the method integrates both plane-wave basis functions and atom-centered basis functions within a single structure calculation. The method is demonstrated for a range of condensed-phase problems for computing (i) hybrid DFT energies for condensed-phase systems at pure DFT cost and (ii) large, triple-zeta, multiply polarized, and diffuse atom-centered basis-set energies at computational costs commensurate with much smaller sets of basis functions. The methods are demonstrated through calculations performed on (a) homogeneous water surfaces as well as heterogeneous surfaces that contain organic solutes studied using two-dimensional periodic boundary conditions and (b) bulk simulations of water enforced through three-dimensional periodic boundary conditions. A range of structures are considered, and in all cases, the results are in good agreement with those obtained using large atom-centered basis and hybrid DFT calculations on the full periodic systems at significantly lower cost.

Efficiently Capturing Weak Interactions in ab Initio Molecular Dynamics with on-the-Fly Basis Set Extrapolation.

Weak interactions have a critical role in accurately portraying conformational change. However, the computational study of these often requires large basis electronic structure calculations that are generally cost-prohibitive within ab initio molecular dynamics. Here, we present a new approach to efficiently obtain AIMD trajectories in agreement with large, triple-ζ, polarized valence basis functions, at much reduced computational cost. For example, it follows from our studies that AIMD trajectories can indeed be constructed in agreement with basis sets such as 6-311++G(2df,2pd) with computational effort commensurate with those from much smaller basis sets such as 6-31+G(d), for polypeptide systems with 100+ atoms. The method is based on molecular fragmentation and allows a range-specified repartitioning of intramolecular (and potentially intermolecular) interactions where noncovalent interactions are selectively assembled using a piece-wise reconstruction based on a set-theoretic inclusion-exclusion principle generalization of ONIOM. Through a simplex decomposition of molecular systems the approach efficiently provides the necessary many-body interactions to faithfully represent noncovalent interactions at the large basis limit. Conformational stabilization energies are provided at close to the complete-basis limit at much reduced cost, and similarly AIMD trajectories (both Born-Oppenheimer and Car-Parrinello-type) are obtained in agreement with very large basis set sizes, in an extremely efficient and accurate manner. The method is demonstrated through simulations on polypeptide fragments of a variety of sizes.

Adaptive, Geometric Networks for Efficient Coarse-Grained Ab Initio Molecular Dynamics with Post-Hartree-Fock Accuracy.

We introduce a new coarse-graining technique for ab initio molecular dynamics that is based on the adaptive generation of connected geometric networks or graphs specific to a given molecular geometry. The coarse-grained nodes depict a local chemical environment and are networked to create edges, triangles, tetrahedrons, and higher order simplexes based on (a) a Delaunay triangulation procedure and (b) a method that is based on molecular, bonded and nonbonded, local interactions. The geometric subentities thus created, that is nodes, edges, triangles, and tetrahedrons, each represent an energetic measure for a specific portion of the molecular system, capturing a specific set of interactions. The energetic measure is constructed in a manner consistent with ONIOM and allows assembling an overall molecular energy that is purely based on the geometric network derived from the molecular conformation. We use this approach to obtain accurate MP2 energies for polypeptide chains containing up to 12 amino-acid monomers (123 atoms) and DFT energies up to 26 amino-acid monomers (263 atoms). The energetic measures are obtained at much reduced computational costs; the approach currently yields MP2 energies at DFT cost and DFT energies at PM6 cost. Thus, in essence the method performs an efficient "coarse-graining" of the molecular system to accurately reproduce the electronic structure properties. The method is comparable in principle to several fragmentation procedures recently introduced in the literature, including previous procedures introduced by two of the authors here, but critically differs by overcoming the computational bottleneck associated with adaptive fragment creation without spatial cutoffs. The method is used to derive a new, efficient, ab initio molecular dynamics formalism (both Born-Oppenheimer and Car-Parrinello-style extended Lagrangian schemes are presented) a critical hallmark of which is that, at each dynamics time-step, multiple electronic structure packages can be simultaneously invoked to assemble the energy and forces for the full system. Indeed, in this paper, as an illustration, we use both Psi4 and Gaussian09 simultaneously at every time-step to perform AIMD simulations and also the energetic benchmarks. The approach works in parallel (currently over 100 processors), and the computational implementation is object oriented in C++. MP2 and DFT based on-the-fly dynamics results are recovered to good accuracy from the coarse-grained AIMD methods introduced here at reduced costs as highlighted above.