We have previously proposed the MPCC static embedding framework for quantum chemistry that self-consistently couples a high-level coupled cluster (CC) treatment of the fragment (active region) with a lower level, Møller-Plesset perturbation treatment of the environment. Our initial implementation was limited to single and double (SD) substitutions, with CCSD for the fragment and first-order perturbative SDamplitudes for the environment. Here, we extend the MPCC embedding treatment to triple substitutions, which is essential for achieving chemical accuracy in energy differences. To this end, we employ a CCSDT solver for the fragment subsystem. For the environment subsystem, we construct a perturbative estimate of the triples amplitudes, explicitly accounting for feedback from all fragment amplitudes. The resulting approach is denoted MPCCSDT(pt). We further introduce a more complete formulation in which feedback from the environment amplitudes to the fragment amplitudes is also included. This scheme involves an iterative treatment of the environment triples amplitudes and is denoted MPCCSDT(it). In addition, we assess the accuracy of the previously proposed low-level method by introducing a modified low-level approach that incorporates a lowest-order treatment of selected long-range effects, including spin fluctuations and charge polarization. All resulting approaches may be viewed as post-CCSD(T) methods. We therefore consider test cases for which CCSD(T) exhibits substantial deviations from CCSDT. These include (i) single- and triple-bond stretching in F2 and N2, (ii) bond dissociation energies of selected molecules from the W4-11 dataset, and (iii) total atomization energies of transition metal hydrides. Our results demonstrate that inclusion of triples amplitudes at the fragment level alone is insufficient; a perturbative treatment of the environment triples amplitudes is required. For many energy-difference applications, feedback from the environment triples amplitudes to the fragment amplitudes is not essential, but it does play a role in the very challenging CoH and FeH molecules. A very interesting finding from our study is that in some challenging cases, we need an improved (second-order) perturbative method for the SDamplitudes, going beyond the first-order one used in our earlier work. Considering both cost and accuracy, the MP2CCSDT(pt) model is the most promising for future applications among the candidates considered here.

We present a finite-temperature extension of density matrix embedding theory (FT-DMET) for realistic crystalline systems. We describe a practical framework for constructing extended bath orbitals, solving the embedding problem, and performing DMET self-consistency at finite temperature. To reduce computational cost, we introduce strategies based on mutual-information-guided bath truncation, controlled treatment of the thermal electron number without explicit optimization, and the use of low-temperature impurity solvers and one-shot FT-DMET in the low-temperature regime. We apply this approach to periodic hydrogen chains and square lattices to characterize their finite-temperature phases. We observe the Pomeranchuk-like effect in one dimension and enhanced stability of long-range order in two dimensions.

The Just Right State program (JRSP) is an occupational therapy-based, sensory intervention designed to support self-regulation. Qualitative methodology was used to explore the experiences of five members of staff working in a secure children’s home in training in and implementing the JRSP under the guidance of occupational therapy staff. Using thematic analysis four themes emerged: Sensory regulation training enhanced staff wellbeing, Sensory regulation as a new lens in a unique environment, Embedding theory in practice, and Considerations for the experiential JRSP training. The findings provide insights into effective training and implementation of sensory modulation interventions.

Consciousness science has generated diverse theoretical frameworks, each offering insights into different aspects of conscious experience. However, this diversity has created a fractured landscape: theories operate at different explanatory levels, and a principled account of how conscious phenomena arise from specific neural computations remains largely absent. This work argues that State Space Theory (SST) can serve as a unifying mechanistic framework for consciousness science. SST proposes that consciousness arises from hierarchical delay coordinate embedding (DCE) - the reconstruction of dynamical system structure from time-delayed signals - implemented through recurrent cortical circuits ('DCE engines'), with gain modulation determining which reconstructions achieve system-wide influence. SST identifies these dynamics with consciousness itself, not merely as correlates. We draw on recent empirical and theoretical work to demonstrate the feasibility of this proposal, including empirical demonstrations that recurrent networks learn via embedding and mathematical results linking recurrent dynamics to embedding theory. We identify how major cognitive theories map onto this architecture mechanistically: parallel DCE engines correspond to Dennett's competing "drafts," global broadcasting reflects gain-amplified propagation, recurrent processing enables the temporal integration DCE requires, and the attention schema emerges as a higher-order reconstruction of gain modulation dynamics. SST's fundamentally process-based character provides immunity to the unfolding argument and resolves the temporal paradox facing causal structure theories. The framework generates a number of falsifiable predictions related to topological structure of perceptual dynamics, temporal vulnerability windows, and selective disruption of recurrent timing. SST thus offers a computational foundation for consciousness research that grounds existing theories mechanistically while generating empirical commitments.

Evidence accumulation models (EAMs) explain and predict human choices and response times in a way that maps more directly to cognitive processes than traditional analyses. For example, EAMs can separate the speed-accuracy trade-off from processing capacity. However, little guidance is available regarding how to use EAMs to instantiate cognitive process theories, which often involve complex mappings of parameters to experimental designs. This tutorial illustrates how to embed such theories using the R package EMC2. We show how the effects of cognitive processes can be estimated by mapping EAM parameters to experimental designs using an augmented linear model language. We demonstrate with two examples. The first instantiates a theory of prospective memory. The second instantiates a theory of how humans integrate advice from automated decision aids into their choices. We then show how to combine these two different theories in a unified framework. We conclude by discussing further directions for theory embedding, including non-linear mappings from stimulus values to EAM parameters and the incorporation of trial-by-trial dynamics.

Even after decades of electronic structure theory development, accurate modeling of reactions at metallic surfaces remains challenging. The most widely used method, i.e., density functional theory, can yield quantitatively and qualitatively inaccurate descriptions of electronic structures and reaction kinetics, motivating the use of higher-level correlated wavefunction (CW) theories, which better capture electron correlation effects. Quantum embedding theories allow advanced CW methods to be applied to a local region that interacts with its environment through, for example, an embedding potential optimized within density functional embedding theory (DFET). To ensure the accuracy of embedded electronic structure calculations, the local region of interest in the presence of the optimized embedding potential must reproduce the behavior of the original full system. Here, we introduce a polarizable embedding scheme that couples an external local potential to provide attractive and repulsive interactions, i.e., similar to the foundation of DFET, with a dielectric bath to reproduce polarization in response to the electric field of the cluster. Particularly, the time-consuming step within DFET, i.e., the optimized effective potential process, is replaced by a physics-informed approach to generate the embedding potential, significantly reducing the computational cost. We evaluate our polarizable embedding scheme using the Cu(111) surface and confirm that our approach outperforms the standard DFET in predicting the Fermi level, charge states, and binding strength of multiple adsorbates. We anticipate that this more efficient and robust embedding scheme could accelerate the mainstream use of embedded correlated wavefunction theory in the heterogeneous catalysis community.

By extending Takens' embedding theorem [Dynamical Systems and Turbulence, Warwick 1980, edited by D. Rand and L.-S. Young (Springer, Berlin, 1981), pp. 366-381], Deyle and Sugihara [PLoS One 6, 1-8 (2011)] provided a theoretical justification for using parallel measurement time series to reconstruct a system's attractor. Building on Takens' framework, Brunton et al. [Nat. Commun. 8, 19 (2017)] introduced the Hankel alternative view of Koopman (HAVOK) algorithm, a data-driven approach capable of linearizing chaotic systems through delay embeddings. In this work, a modified version of the original algorithm (mHAVOK) is presented, a practical realization of Deyle and Sugihara's generalized embedding theory. mHAVOK extends the original algorithm from one to multiple input time series and introduces a systematic approach to separating linear and nonlinear terms. An R2-informed quality score is introduced and shown to be a reliable guide for the selection of the reduced rank. The algorithm is tested on the familiar Lorenz system, as well as the more sophisticated Sprott system, which features different behaviors depending on the initial conditions. The quality of the reconstructions is assessed with the Chamfer distance, validating how mHAVOK allows for a more accurate reconstruction of the system dynamics. The new methodology generalizes HAVOK by allowing the analysis of multivariate time series, fundamental in real-life data-driven applications.

Regardless of how the electron correlation is treated, all methods based on frozen-density embedding theory rely on approximations to the non-additive kinetic potential bi-functional ṽtnad[ρA,ρB](r)≈vtnad[ρA,ρB](r). Open shell systems, in which the spin is localized on a specific molecular fragment, are particularly prone to incorrect redistribution of charge depending on the used ṽtnad[ρA,ρB]. In this work, we present a systematic analysis of spin densities obtained with several semi-local approximations to vtnad[ρA,ρB], with the aim of delimiting their respective domains of applicability. We show that spin distributions obtained using decomposable semi-local ṽtnad[ρA,ρB] fall into two distinct categories: they are either qualitatively incorrect or reasonably accurate and consistent with trends previously observed for other properties computed using the same approximants. In neither case do gradient-dependent corrections, although crucial for improving the corresponding energy bi-functional (Tsnad[ρA,ρB]), resolve the deficiencies observed for spin densities. We propose a simple criterion based on orbital energies that allows one to identify a priori the situations in which a given approximant is likely to fail. Finally, we show that a recently developed non-decomposable approximant ṽtnad(NDCS)[ρA,ρB] extends the range of applicability of FDET-based methods to embedded radicals that are inaccessible to semi-local approximants. Moreover, ṽtnad(NDCS)[ρA,ρB] yields improved spin densities even in cases where decomposable semi-local approximants already perform reasonably well.

Single-atom alloys (SAAs) are an increasingly popular platform for heterogeneous catalysis because of their distinct electronic structures and ability to break catalytic linear scaling relationships. This popularity has led to a proliferation of computational studies probing SAA reactivity at the density functional theory (DFT) level. However, some phenomena such as photo- and electrocatalysis require use of electronic structure methods beyond DFT; such studies are both rare and fundamentally challenging. Density functional embedding theory (DFET)/embedded correlated wavefunction (ECW) studies of reactions on metal surfaces have been shown to provide a reliable way to correct for DFT-related errors. DFET/ECW studies of chemistry involving SAAs, however, could require active spaces beyond the capabilities of traditional multireference methods when transition-metal dopants give rise to many degenerate states. To overcome this limitation, we combined our DFET/ECW methodology with the density matrix renormalization group (DMRG) complete active space self-consistent field (DMRGSCF) and DMRG N-electron valence state second-order perturbation theory (DMRG-NEVPT2) methods in the PySCF code. Using embedded DMRGSCF and embedded DMRG-NEVPT2, we analyze CO adsorption on Ni-, Rh-, Pd-, and Pt-doped Ag(100) with different active spaces. We show that the active spaces approachable with conventional multireference methods lead to overbinding of CO due to an inability to treat all of the dopant d-orbitals on equal footing. Larger active spaces, which are easily treated by both DMRGSCF and DMRG-NEVPT2, yield much more reasonable adsorption free energies. Our findings suggest that future multireference calculations of these systems should similarly employ active spaces containing all of the dopant d-orbitals along with sp-band orbitals of the host metal near the Fermi level. Emb-DMRG-NEVPT2 is a method that can be broadly applied to study catalytic reactions on metal surfaces when large active spaces are required.

The projection-based wave function in density functional theory (WF-in-DFT) embedding enables an efficient description of both the energetics and properties of large and complex chemical systems, with accuracy exceeding that of pure DFT. Recently, we have proposed using the density matrix renormalization group (DMRG) as the WF method for molecules containing strongly correlated fragments [Beran, P. J. Phys. Chem. Lett. 2023, 14, 716-722]. In this work, we demonstrate that the accuracy of the DMRG-in-DFT approach is primarily limited by the approximate treatment of the coupling between the active component and its environment through nonadditive exchange-correlation functionals. To address this issue, we combine exact exchange to reduce the nonadditive exchange error with a multireference adiabatic connection (AC) scheme to recover nonadditive correlation. The performance of the improved DMRG-in-DFT embedding is illustrated on two prototypical strongly correlated systems: the dissociation of the H20 chain and the cleavage of a triple CN bond in propionitrile.

Diamond defects are among the most promising qubits. Modeling their properties through accurate quantum mechanical simulations can further their development into robust units of information. We use the recently developed capped density functional embedding theory (capped-DFET) with the multiconfigurational n-electron valence second-order perturbation theory to characterize the electronic excitation energies for different spin manifolds of the well-characterized negatively charged substitutional N defect adjacent to a vacancy (VC) in diamond (NCVC−). We successfully reproduce vertical excitation energies for both triplet and singlet states of NCVC− with errors < 0.1 eV. Unlike other embedding methods, capped-DFET exhibits robust predictions that are approximately independent of the embedded cluster size: it only requires a cluster to contain the defect atoms and their nearest neighbors (as small as a 40-atom capped cluster). Furthermore, our method is free from slowly converging Coulomb interactions between charged defects, and thus also only weakly dependent on supercell size.

Accurately and efficiently predicting the equilibrium geometries of large molecules remains a central challenge in quantum computational chemistry, even with hybrid quantum-classical algorithms. Two major obstacles hinder progress: the large number of qubits required and the prohibitive cost of conventional nested optimization. In this work, we introduce a co-optimization framework that combines Density Matrix Embedding Theory (DMET) with Variational Quantum Eigensolver (VQE) to address these limitations. This approach substantially reduces the required quantum resources, enabling the treatment of molecular systems significantly larger than previously feasible. We first validate our framework on benchmark systems, such as H4 and H2O2, before demonstrating its efficacy in determining the equilibrium geometry of glycolic acid (C2H4O3)─a molecule of a size previously considered intractable for quantum geometry optimization. Our results show the method achieves high accuracy while drastically lowering computational cost. This work thus represents a significant step toward practical, scalable quantum simulations, moving beyond the small, proof-of-concept molecules that have historically dominated the field. More broadly, our framework establishes a tangible path toward leveraging quantum advantage for the in silico design of complex catalysts and pharmaceuticals.

High-resolution atomic force microscopy (HR-AFM) has emerged as a transformative technique for imaging and manipulating matter with atomic precision. By functionalizing the scanning probe with a CO molecule, HR-AFM enables direct visualization of chemical bonds, intermolecular interactions, charged states, and electron orbital signatures. We provide an overview of HR-AFM from both experimental and theoretical perspectives. The operational principles of frequency-modulation AFM and the role of tip functionalization are described, together with methods that combine AFM and STM for enhanced imaging and spectroscopy. Theoretical approaches, such as the virtual tip method, full density functional theory, frozen density embedding theory, and tip-tilting correction methods, enable the quantitative interpretation of tip-sample interactions and image contrast. These developments support applications of HR-AFM in resolving bond orders, functional groups, heteroatoms, and orbital fingerprints in single molecules, as well as in characterizing complex industrial hydrocarbons. Beyond imaging, HR-AFM also serves as a platform for controlled bond rupture and manipulation at the atomic scale. The benchmark Si(111)-(7 × 7) surface is revisited with recent insights into tip-induced contrast dynamics arising from B doping. Extensions of HR-AFM to state-resolved imaging of quantum defects in two-dimensional materials are also discussed. By combining high-resolution imaging with first-principles modeling, HR-AFM demonstrates a unique capability to reveal previously inaccessible surface phenomena, thereby further decoding the atomic landscapes of matter at the single-atom and molecular scale.

Abstract Space–time projection (STP) is introduced as a data-driven forecasting approach for high-dimensional, time-resolved data. The method computes extended space–time proper orthogonal modes from training data spanning a prediction horizon comprising both hindcast and forecast intervals. Forecasts are generated by projecting the hindcast portion of these modes onto new data, leveraging their orthogonality and optimal correlation with the forecast extension. Rooted in proper orthogonal decomposition (POD) theory, dimensionality reduction and time-delay embedding are intrinsic to the approach. The only tunable parameters are the truncation rank and the hindcast length; no additional hyperparameters are required. Hindcast accuracy serves as a reliable indicator for short-term forecast accuracy. The method’s efficacy is demonstrated using two datasets: transient, highly anisotropic simulations of supernova explosions in a turbulent interstellar medium and experimental velocity fields of a turbulent high-subsonic engineering flow. In a comparative study with standard dynamic mode decomposition (DMD) and long short-term memory (LSTM) networks (acknowledging that alternative architectures or training strategies may yield different outcomes) STP achieved the lowest errors at short and long lead times and was comparable at intermediate horizons. Considering its simplicity and robust performance, STP offers an interpretable and competitive baseline for forecasting high-dimensional transient chaotic processes, relying purely on spatio-temporal correlation.

Domain adaptation methods have been extensively studied for rolling bearing fault diagnosis under various conditions. However, some existing methods only consider the one-way embedding of original space into a low-dimensional subspace without backward validation, which leads to inaccurate embeddings of data and poor diagnostic performance. In this paper, a rolling bearing fault diagnosis method based on multi-source domain joint structure preservation transfer with autoencoder (MJSPTA) is proposed. Firstly, similar source domains are screened by inter-domain metrics; then, the high-dimensional data of both the source and target domains are projected into a shared subspace with different projection matrices, respectively, during the encoding stage. Finally, the decoding stage reconstructs the low-dimensional data back to the original high-dimensional space to minimize the reconstruction accuracy. In the shared subspace, the difference between source and target domains is reduced through distribution matching and sample weighting. Meanwhile, graph embedding theory is introduced to maximally preserve the local manifold structure of the samples during domain adaptation. Next, label propagation is used to obtain the predicted labels, and a voting mechanism ultimately determines the fault type. The effectiveness and robustness of the method are verified through a series of diagnostic tests.

Decoding the connectivity patterns of complex networks from time series measurements is crucial for understanding and controlling their dynamics. Although network inference algorithms have advanced significantly in identifying both pairwise and higher-order interactions, they often rely on the availability of full-state measurements, an assumption that is difficult to satisfy in practice. In this article, we address this limitation by introducing Network Inference from Partial States (NIPS), a framework for network reconstruction from partial-state observations of network units. Focusing initially on networks coupled through observable states, we model coupling inputs as external forcing and utilize forced-delay embedding theory to establish a map that describes the evolution of the node observables as a function of observable state components. Specifically, the dynamics of the observable state of a node depends only on delayed observations of that node itself, not on delayed observations of other nodes. This enables accurate network reconstruction with limited data, which is demonstrated using both simulated and experimental data obtained from a wide range of networks. We evaluate the robustness of NIPS to noisy data and hidden network nodes and subsequently extend the framework to networks coupled through unobservable states.

This paper examines the integration of theoretical frameworks and studio-based practice in contemporary fine art education. Moving beyond the traditional separation of lectures and studio production, it proposes a holistic pedagogy that positions the studio as a site of inquiry, reflection, and critical engagement. Drawing on constructivism, experiential learning, and reflective practice, the paper presents a case-based approach through the “Art &amp; Identity” studio module. Findings suggest that embedding theory within studio processes enhances students' critical thinking, artistic authorship, and conceptual clarity, while also highlighting structural challenges in curriculum design.

Abstract We formulate a quantum embedding algorithm in real-space for the simultaneous theoretical treatment of nonlocal electronic correlations and disorder, the coherent cellular dynamical mean-field theory (C-CDMFT). This algorithm combines the molecular coherent potential approximation with the cellular dynamical mean-field theory. After a pedagogical review of quantum embedding theories for disordered and interacting electron systems, and a detailed discussion of its work flow, we present first results from C-CDMFT for the half-filled two-dimensional Anderson-Hubbard model on a square lattice: (i) the analysis of its Mott metal-insulator transition as a function of disorder strength, and (ii) the impact of different types of disorder on its magnetic phase diagram. For the latter, by means of a "disorder diagnostics", we are able to precisely identify the contributions of different disorder configurations to the system's magnetic response.

Abstract Recently discovered infinite-layer nickelates share a cuprate-like structure, thereby providing a promising platform for elucidating the mechanism of high-temperature superconductivity. Motivated by recent photoemission measurements on the La 0.8 Sr 0.2 NiO 2 , we carry out a systematic study of the infinite-layer nickelate using both dynamical mean-field theory and density matrix embedding theory. The renormalized electronic structure and Fermi surface of correlated La 0.8 Sr 0.2 NiO 2 are studied in an effective two-band model through the dynamical mean-field calculation. We find the correlation effects reflect mainly on the Ni d band, which is consistent with the experimental findings. We further study the ground state through the density matrix embedding theory. Within the experimental doping range and rigid-band approximation, we show that the d -wave superconductivity is the lowest energy state, while the static magnetism is absent except very close to zero doping.

Simulating electron transfer at reactive solid-liquid interfaces under constant electrochemical potentials of the constituents (electrons, ions, solvent etc.) is crucial to understanding the formation, functional and failure of electrochemical devices. Albeit being sufficiently accurate in decreasing breaking and formation of chemical bond at solid surfaces, existing methods based on Kohn-Sham density functional theory (DFT) are unsatisfactory in system consistency, namely, simulating the open solid-liquid interface under grand-canonical conditions, as well as in scaling up the simulation due to its high computational cost. Herein, to improve the system consistency and computational efficiency, we develop density-potential functional theoretic (DPFT) schemes out of Kohn-Sham DFT, drawing upon ideas of orbital-free DFT, frozen density embedding theory, and tight-binding DFT. The DPFT transforms an all-atom, Kohn-Sham DFT description of the nonreactive electrolyte solution into a coarse-grained, field-based description, while retaining a Kohn-Sham DFT description for the reactive subsystem. In the absence of surface reactions, the solid electrode can be described using orbital-free DFT, reducing the computational cost further. On the conceptual level, the physical meaning of potential in DPFT is examined. Recent applications of DPFT to simulate EDLs with mesoscopic roughness and EDLs at supported nanoparticles will be introduced. Related papers Zhang, Y., Binninger, T., Huang, J., & Eikerling, M. H. (2025). Theory of Electro-Ionic Perturbations at Supported Electrocatalyst Nanoparticles. Physical Review Letters, 134(6), 066201.Zhang, M., Chen, Y., Eikerling, M., & Huang, J. (2025). Structured solvent on a split electron tail: A semiclassical theory of electrified metal-solution interfaces. Physical Review Applied, 23(2), 024009.Huang, J., Domínguez-Flores, F., & Melander, M. (2024). Variants of surface charges and capacitances in electrocatalysis: Insights from density-potential functional theory embedded with an implicit chemisorption model. PRX Energy, 3(4), 043008. Acknowledgements Our research is supported financially by the Initiative and Networking Fund of the Helmholtz Association (no. VH-NG-1709), and European Research Council (ERC) Starting Grant (MESO-CAT, Grant agreement No. 101163405).