Orbital Sets Research Articles

Reduced Scaling Correlated Natural Transition Orbitals for Multilevel Coupled Cluster Calculations.

Multilevel coupled cluster theory offers reduced scaling computation of intensive properties in systems that are too large for standard coupled cluster calculations. A significant benefit of the multilevel coupled cluster framework is the possibility of calculating intensive properties that are not tightly localized if an appropriate set of active orbitals is used. Correlated natural transition orbitals (CNTOs) are tailored to describe excitation processes. For multilevel coupled cluster singles and doubles (MLCCSD) and singles and perturbative doubles (MLCC2) calculations, the construction of CNTOs generally becomes the computational bottleneck. Here, we demonstrate how CNTOs can be obtained with operations, eliminating the -scaling steps involved in the original approach. This reduction in scaling moves the bottleneck of MLCC2 and MLCCSD calculations from the active orbital space preparation to the MLCC2 and MLCCSD equations with -scaling.

Stochastic Resolution of Identity to CC2 for Large Systems: Ground State and Triplet Excitation Energy Calculations.

An implementation of stochastic resolution of identity to the CC2 (sRI-CC2) ground state energy followed by triplet excitation energy calculations is presented. A set of stochastic orbitals is introduced to further decouple the expensive 4-index electron repulsion integrals on the basis of RI approximation. A Laplace transformation of the orbital energy difference denominators into numerical summations is adopted to obtain a third-order overall scaling. We select a series of hydrogen dimer chains with nearly thousands of electrons, as well as some other molecules, for sRI-CC2 energies and test the accuracy and time consumption in comparison with those of RI-CC2. Our sRI-CC2 results reproduce a modest agreement with the RI-CC2 in Q-Chem program package and it allows a steep scaling reduction from O(N5) to O(N3). Besides, the unrestricted sRI-CC2 calculations also fit well with the restricted results. Thus, our sRI-CC2 implementation of ground state energy and triplet excitation energy provides a cost-efficient alternative approach, especially for some large-sized systems.

Evaluating the Use of Unoccupied Aircraft Systems (UASs) for Planetary Exploration in Mars Analog Terrain

Planetary analog mission simulations are essential for testing science operations strategies and technologies. They also teach us how to use terrestrial analogs to inform studies of extraterrestrial environments. Unoccupied aircraft systems (UASs) have great potential for planetary surface exploration as demonstrated by the Mars 2020 Ingenuity helicopter and the in-development Dragonfly mission to Saturn’s moon Titan. Although applications of UAS technology for planetary exploration remain largely unexplored, simulated missions in planetary analog terrains can inform operational best practices. As part of the Rover–Aerial Vehicle Exploration Network project, we simulated a 12 sol UAS mission on Mars in the Holuhraun region of Iceland. The UAS had airborne imaging capability, as well as imaging, sampling, and geochemical analysis capabilities while landed. The mission evaluated the use of these instruments and developed operational strategies for using UASs to explore a planetary surface. Oblique airborne images were essential for mission planning and were used to scout large areas to identify both potential landing sites and targets for focused investigations. The airborne and landed data collected by the UAS allowed for detailed observations and interpretations not possible with analog orbital data sets, resulting in an improved scientific return for the simulated UAS mission compared to a premission analysis of only the analog orbital data. As a planetary exploration vehicle, a UAS is most advantageous for exploring large areas (many square kilometers) and is particularly useful when the terrain may be impassable to ground-based traverses (e.g., by rovers or humans).

Fragment quantum embedding using the Householder transformation: A multi-state extension based on ensembles

In recent studies by Yalouz et al. [J. Chem. Phys. 157, 214112 (2022)] and Sekaran et al. [Phys. Rev. B 104, 035121 (2021) and Computation 10, 45 (2022)], density matrix embedding theory (DMET) has been reformulated through the use of the Householder transformation as a novel tool to embed a fragment within extended systems. The transformation was applied to a reference non-interacting one-electron reduced density matrix to construct fragments’ bath orbitals, which are crucial for subsequent ground state calculations. In the present work, we expand upon these previous developments and extend the utilization of the Householder transformation to the description of multiple electronic states, including ground and excited states. Based on an ensemble noninteracting density matrix, we demonstrate the feasibility of achieving exact fragment embedding through successive Householder transformations, resulting in a larger set of bath orbitals. We analytically prove that the number of additional bath orbitals scales directly with the number of fractionally occupied natural orbitals in the reference ensemble density matrix. A connection with the regular DMET bath construction is also made. Then, we illustrate the use of this ensemble embedding tool in single-shot DMET calculations to describe both ground and first excited states in a Hubbard lattice model and an ab initio hydrogen system. Finally, we discuss avenues for enhancing ensemble embedding through self-consistency and explore potential future directions.

Exploiting Non-Abelian Point-Group Symmetry to Estimate the Exact Ground-State Correlation Energy of Benzene in a Polarized Split-Valence Triple-Zeta Basis Set.

Local electronic-structure methods in quantum chemistry operate on the ability to compress electron correlations more efficiently in a basis of spatially localized molecular orbitals than in a parent set of canonical orbitals. However, many typical choices of localized orbitals tend to be related by selected, near-exact symmetry operations whenever a molecule belongs to a point group, a feature which remains largely unexploited in most local correlation methods. The present Letter demonstrates how to leverage a recent unitary protocol for enforcing symmetry properties among localized orbitals to yield a high-accuracy estimate of the exact ground-state correlation energy of benzene (D6h) in correlation-consistent polarized basis sets of both double- and triple-ζ quality. Through an initial application to many-body expanded full configuration interaction (MBE-FCI) theory, we show how molecular point-group symmetry can lead to computational savings that are inversely proportional to the order of the point group in a manner generally applicable to the acceleration of modern local correlation methods.

Bundled matrix product states represent low-energy excitations faithfully

Abstract We consider a set of density matrices. All of which are written in the same orbital basis, but the orbital basis size is less than the total Hilbert space size. We ask how each density matrix is related to each of the others by establishing a norm between density matrices based on the truncation error in a partial trace for a small set of orbitals. We find that states with large energy differences must have large differences in their density matrices. Small energy differences are divided into two groups, one where two density matrices have small differences and another where they are very different, as is the case of symmetry. We extend these ideas to a bundle of matrix product states and show that bond dimension of the wavefunction ansatz for two states with large energy differences are larger. Meanwhile, low energy differences can have nearly the same bond dimensions for similar states.

Closest Wannier functions to a given set of localized orbitals

Closest Wannier functions to a given set of localized orbitals

Single-site DFT+DMFT for vanadium dioxide using bond-centered orbitals

We present a combined density-functional theory and single-site dynamical mean-field theory (DMFT) study of vanadium dioxide (VO2) using an unconventional set of bond-centered orbitals as the basis of the correlated subspace. VO2 is a prototypical material undergoing a metal-insulator transition (MIT), hosting both intriguing physical phenomena and the potential for industrial applications. With our choice of correlated subspace basis, we investigate the interplay of structural dimerization and electronic correlations in VO2 in a computationally cheaper way compared to other state-of-the-art methods, such as cluster DMFT. Our approach allows us to treat the rutile and M1 monoclinic VO2 phases on an equal footing and to vary the dimerizing distortion continuously, exploring the energetics of the transition between the two phases. The choice of basis presented in this work hence offers a complementary view on the long-standing discussion of the MIT in VO2 and suggests possible future extensions to other similar materials hosting molecular-orbital-like states. Published by the American Physical Society 2024

Class Symbolic Regression: Gotta Fit ’Em All

We introduce “Class Symbolic Regression” (Class SR), the first framework for automatically finding a single analytical functional form that accurately fits multiple data sets—each realization being governed by its own (possibly) unique set of fitting parameters. This hierarchical framework leverages the common constraint that all the members of a single class of physical phenomena follow a common governing law. Our approach extends the capabilities of our earlier Physical Symbolic Optimization (Φ-SO) framework for symbolic regression, which integrates dimensional analysis constraints and deep reinforcement learning for unsupervised symbolic analytical function discovery from data. Additionally, we introduce the first Class SR benchmark, comprising a series of synthetic physical challenges specifically designed to evaluate such algorithms. We demonstrate the efficacy of our novel approach by applying it to these benchmark challenges and showcase its practical utility for astrophysics by successfully extracting an analytic galaxy potential from a set of simulated orbits approximating stellar streams.

Topography‐Enhanced Ultra‐Cold Trapping at the LCROSS Impact Site

AbstractSmall topographic features below the resolution of existing orbital data sets may create “micro ultra‐cold traps” within the larger permanently shadowed regions that are present at the lunar poles. These ultra‐cold traps are protected from the major primary and secondary illumination sources, and thus would create surfaces that are much colder than lower‐resolution temperature maps would indicate. We examine this effect by creating a high resolution (1 m pix−1) terrain map based on upscaled data from the Lunar Orbiter Laser Altimeter. This map is illuminated by scattered sunlight and infrared emissions from sunlit terrain, which are then run through a thermal model to determine temperatures. We find that while most of the terrain experiences maximum temperatures around 50 K, there are a number of 1–30 m‐scale ultra‐cold traps with maximum temperatures as low as 20–30 K. By comparing our modeled ultra‐cold trapping area to volatile abundances measured by Lunar Crater Observation and Sensing Satellite (LCROSS), we reveal a diverse environment where the surficial abundances necessary to explain the LCROSS results are strongly dependent on precisely where the impact occurred.

Water‐Limited Hydrothermalism and Volcanic Resurfacing of Eridania Basin, Mars

AbstractThe Eridania Basin system, located in the martian southern highlands, may have hosted long‐lived lacustrine hydrothermalism. Despite its multi‐basin scale and suggested longevity that should pervasively alter the landscape, evidence for consistent compositional signatures of lacustrine hydrothermalism across varying spatial scales is poorly known. Here we synthesize orbital data sets (derived from the Gamma Ray Spectrometer, GRS; Thermal Emission Spectrometer, TES; and Compact Reconnaissance Imaging Spectrometer for Mars, CRISM) to characterize the composition of Eridania's landscape from regional to outcrop scales. The bulk geochemistry representing decimeter depths is broadly on par with mafic igneous compositions seen elsewhere on Mars. The landscape‐scale mineralogy of Eridania suggests a compositional distinction between Eridania's northernmost basin and the remaining basins. Meanwhile, chloride mapping and analyses of amorphous silica and acid‐altered clay phases at the outcrop scale both show that mineralogical evidence for hydrothermal alteration and putative lacustrine activity is broadly limited to the northernmost basin, and that most other amorphous silica in Eridania is consistent with low‐to‐minimal water/rock weathering. This suggests either limited regional hydrothermalism or extensive mafic resurfacing occurred in Eridania, the latter of which is consistent with previous studies. Our findings collectively suggest that Eridania underwent local crystal hydrothermal alteration, perhaps confined to the northernmost basin, while much of the remainder of the Eridania's basins either experienced low‐temperature weathering or extensive resurfacing to at least meter scale depths by later volcanism. This supports prior inferences of spatially and temporally limited surface hydrology on Mars and extensive volcanic resurfacing of basins, as in the Gusev Crater.

Analysis and Assessment of Knowles' Partitioning in Many-Body Perturbation Theory.

A detailed analysis of a new partitioning in many-body perturbation theory recently proposed by Knowles (J. Chem. Phys. 156, 011101, 2022), termed "perturbation adapted partitioning" (PAPT), is presented. Level shift and orbital rotation effects are identified as gears of the zero-order Hamiltonian. These two components are examined separately, revealing that, in themselves, neither of the two is competitive with the combined effect. The success of PAPT can be attributed to determining a set of molecular orbitals and corresponding orbital energies that can systematically outperform the canonical orbitals and Koopmans' energy-based Møller-Plesset partitioning. The self-consistent version of the method is also tested in terms of energy and convergence. Previous numerical studies are further complemented with an application to an inherent multireference example and an investigation of van der Waals interaction energies. In addition, a rigorous mathematical analysis of the consequence of the linear dependence of projection functions on the solution of the Knowles' equations is provided.

Stochastic Resolution of Identity to CC2 for Large Systems: Excited State Properties.

We apply a stochastic resolution of identity approximation (sRI) to the CC2 method for the excitation energy calculations. A set of stochastic orbitals are employed to decouple the crucial 4-index electron repulsion integrals and optimize the contraction steps in CC2 response theory. The CC2 response for excitations builds upon sRI-CC2 ground-state calculations, which scales as O(N3), where N is a measure for the system size. Overall, the current algorithm for excited states also allows a sharp scaling reduction from original O(N5) to O(N3). We test the sRI-CC2 for different molecular systems and basis sets, and we show that our sRI-CC2 method can accurately reproduce the results of the deterministic CC2 approach. Our sRI-CC2 exhibits an experimental scaling of O(N2.59) for a series of olefin chains, allowing us to calculate systems with nearly thousands of electrons.

Leveraging Machine Learning Approaches to Predict Organic Carbon Abundance in Mars‐Analog Hypersaline Lake Sediments

AbstractModern advancements in laboratory and instrumental techniques in astrobiology have improved our life detection capabilities on both Earth and beyond. These advancements have also increased the complexity of data often resulting in data sets that are characterized by complex and non‐linear relationships. Machine learning methods are underutilized in astrobiology; however, these methods are extremely effective at revealing structure and patterns in complex data sets when paired with the right algorithms. Here, we employ a series of classification and regression algorithms to predict the abundance of organic carbon (OC) from X‐ray fluorescence (XRF) heavy element (>Mg) data in dynamic Mars‐analog hypersaline lake sediments. More specifically, we constructed models using the random forest, k‐nearest neighbors (KNN), support vector machine, and logistic regression algorithms. Overall, our trained models showed good performance with predicting the abundance of OC, with accuracies from 80% to 94%. Machine learning approaches such as classification and regression algorithms offer insight into complex data while providing agnostic insights, ultimately creating a more efficient search for OC. We applied our trained model on XRF data from Martian soil using rover‐based (PIXL) and orbital (Odyssey) data sets to produce probability predictions of OC abundance. Our predictions show a high probability that OC abundance is low which is comparable to OC data from recently landed missions. These results highlight the potential for predictive machine learning models to be trained on data from analog environments on Earth and then applied to extraterrestrial targets, ultimately, improving life detection efforts.

Recurrent flow patterns as a basis for two-dimensional turbulence: Predicting statistics from structures

A dynamical systems approach to turbulence envisions the flow as a trajectory through a high-dimensional state space [Hopf, Commun. Appl. Maths 1, 303 (1948)]. The chaotic dynamics are shaped by the unstable simple invariant solutions populating the inertial manifold. The hope has been to turn this picture into a predictive framework where the statistics of the flow follow from a weighted sum of the statistics of each simple invariant solution. Two outstanding obstacles have prevented this goal from being achieved: 1) paucity of known solutions and 2) the lack of a rational theory for predicting the required weights. Here, we describe a method to substantially solve these problems, and thereby provide compelling evidence that the probability density functions (PDFs) of a fully developed turbulent flow can be reconstructed with a set of unstable periodic orbits. Our method for finding solutions uses automatic differentiation, with high-quality guesses constructed by minimizing a trajectory-dependent loss function. We use this approach to find hundreds of solutions in turbulent, two-dimensional Kolmogorov flow. Robust statistical predictions are then computed by learning weights after converting a turbulent trajectory into a Markov chain for which the states are individual solutions, and the nearest solution to a given snapshot is determined using a deep convolutional autoencoder. In this study, the PDFs of a spatiotemporally chaotic system have been successfully reproduced with a set of simple invariant states, and we provide a fascinating connection between self-sustaining dynamical processes and the more well-known statistical properties of turbulence.

Natural Orbitals and Targeted Non-Orthogonal Orbital Sets for Atomic Hyperfine Structure Multiconfiguration Calculations

Hyperfine structure constants have many applications, but are often hard to calculate accurately due to large and canceling contributions from different terms of the hyperfine interaction operator, and also from different closed and spherically symmetric core subshells that break up due to electron correlation effects. In multiconfiguration calculations, the wave functions are expanded in terms of configuration state functions (CSFs) built from sets of one-electron orbitals. The orbital sets are typically enlarged within the layer-by-layer approach. The calculations are energy-driven, and orbitals in each new layer of correlation orbitals are spatially localized in regions where the weighted total energy decreases the most, overlapping and breaking up different closed core subshells in an irregular pattern. As a result, hyperfine structure constants, computed as expectation values of the hyperfine operators, often show irregular or oscillating convergence patterns. Large orbital sets, and associated large CSF expansions, are needed to obtain converged values of the hyperfine structure constants. We analyze the situation for the states of the {2s22p3,2s22p23p,2s22p24p} odd and {2s22p23s,2s2p4,2s22p24s,2s22p23d} even configurations in N I, and show that the convergence with respect to the increasing sets of orbitals is radically improved by introducing separately optimized orbital sets targeted for describing the spin- and orbital-polarization effects of the 1s and 2s core subshells that are merged with, and orthogonalized against, the ordinary energy-optimized orbitals. In the layer-by-layer approach, the spectroscopic orbitals are kept frozen from the initial calculation and are not allowed to relax in response to the introduced layers of correlation orbitals. To compensate for this lack of variational freedom, the orbitals are transformed to natural orbitals prior to the final calculation based on single and double substitutions from an increased multireference set. The use of natural orbitals has an important impact on the states of the 2s22p23s configuration, bringing the corresponding hyperfine interaction constants in closer agreement with experiment. Relying on recent progress in methodology, the multiconfiguration calculations are based on configuration state function generators, cutting down the time for spin-angular integration by factors of up to 50, compared to ordinary calculations.

Orbital Tolerance and Intrinsic Orbital Capacity for Electric Propulsion Constellations

Large constellations are typically designed using sets of periodic orbits and satellite control boxes sized to ensure compatible phasing between coorbital constellation planes to prevent conjunction events. This work investigates how the control action for orbit maintenance influences feasible intrashell and intershell distances in a satellite constellation. Two-dimensional lattice flower constellations are simulated in an environment inclusive of orbital perturbations, with onboard sensors and an electric propulsion system constituting the satellite guidance, navigation, and control system. An analysis to define estimates for minimum separation distances is performed as a function of control, propulsion, orbit, environmental, and spacecraft characteristics. An estimate of the intrinsic (geometric) orbital capacity is proposed, based on current and advanced technologies, in order to quantify the number of admissible spacecraft and orbital shells in a selected altitude range. Simulation results are used to improve the fidelity of intrinsic orbital capacity estimates and to understand factors that influence the number of admissible satellite locations in low Earth orbit.

Distribution of periodic orbits in the homology group of a knot complement

AbstractConsider a transitive Anosov flow on a closed 3-manifold. After removing a finite set of null-homologous periodic orbits, we study the distribution of the remaining periodic orbits in the homology of the knot complement.

Projected Atomic Orbitals As Optimal Virtual Space for Excited State Projection-Based Embedding Calculations.

The projected atomic orbital (PAO) technique is presented for the construction of virtual orbital spaces in projection-based embedding (PbE) applications. The proposed straightforward procedure produces a set of virtual orbitals that are used in the final, high-level calculation of the embedded active subsystem. The PAO scheme is demonstrated on intermolecular potentials of bimolecular complexes in ground and excited states, including Rydberg excitations. The results show the outstanding performance of the PbE method when used with PAO virtual orbitals compared with those produced using common orbital localization techniques. The good agreement of the resulting PbE potential curves with those from high-level ab initio dimer calculations, also in diffuse basis sets, confirms that the PAO technique can be suggested for future applications using top-down embedding methods.

Elliptic Reeb orbit on some real projective three-spaces via ECH

We prove the existence of an elliptic Reeb orbit for some contact forms on the real projective three space \mathbb{R}P^{3} . The main ingredient of the proof is the existence of a distinguished pseudoholomorphic curve in the symplectization given by the U map on ECH. Also, we check that the first value on the ECH spectrum coincides with the smallest action of null-homologous orbit sets for 1/4 -pinched Riemannian metrics. This action coincides with twice the length of a shortest closed geodesic. In addition, we compute the ECH spectrum for the irrational Katok metric example.