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

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

Quantum Information-Assisted Complete Active Space Optimization (QICAS).

We propose an effective quantum information-assisted complete active space optimization scheme (QICAS). What sets QICAS apart from other correlation-based selection schemes is (i) the use of unique measures from quantum information that assess the correlation in electronic structures in an unambiguous and predictive manner and (ii) an orbital optimization step that minimizes the correlation discarded by the active space approximation. Equipped with these features, QICAS yields, for smaller correlated molecule, sets of optimized orbitals with respect to which the complete active space configuration interaction energy reaches the corresponding complete active space self-consistent field (CASSCF) energy within chemical accuracy. For more challenging systems such as the chromium dimer, QICAS offers an excellent starting point for CASSCF by greatly reducing the number of iterations required for numerical convergence. Accordingly, our study validates a profound empirical conjecture: the energetically optimal nonactive spaces are predominantly those that contain the least entanglement.

Refining and relating fundamentals of functional theory.

To advance the foundation of one-particle reduced density matrix functional theory (1RDMFT), we refine and relate some of its fundamental features and underlying concepts. We define by concise means the scope of a 1RDMFT, identify its possible natural variables, and explain how symmetries could be exploited. In particular, for systems with time-reversal symmetry, we explain why there exist six equivalent universal functionals, prove concise relations among them, and conclude that the important notion of v-representability is relative to the scope and choice of variable. All these fundamental concepts are then comprehensively discussed and illustrated for the Hubbard dimer and its generalization to arbitrary pair interactions W. For this, we derive by analytical means the pure and ensemble functionals with respect to both the real- and complex-valued Hilbert space. The comparison of various functionals allows us to solve the underlying v-representability problems analytically, and the dependence of its solution on the pair interaction is demonstrated. Intriguingly, the gradient of each universal functional is found to always diverge repulsively on the boundary of the domain. In that sense, this key finding emphasizes the universal character of the fermionic exchange force, recently discovered and proven in the context of translationally invariant one-band lattice models.

1-Matrix functional for long-range interaction energy of two hydrogen atoms.

The leading terms in the large-R asymptotics of the functional of the one-electron reduced density matrix for the ground-state energy of the H2 molecule with the internuclear separation R are derived thanks to the solution of the phase dilemma at the R → ∞ limit. At this limit, the respective natural orbitals (NOs) are given by symmetric and antisymmetric combinations of "half-space" orbitals with the corresponding natural amplitudes having the same amplitudes but opposite signs. Minimization of the resulting explicit functional yields the large-R asymptotics for the occupation numbers of the weakly occupied NOs and the C6 dispersion coefficient. The highly accurate approximates for the radial components of the p-type "half-space" orbitals and the corresponding occupation numbers (that decay like R-6), which are available for the first time thanks to the development of the present formalism, have some unexpected properties.

Excitations of Quantum Many-Body Systems via Purified Ensembles: A Unitary-Coupled-Cluster-Based Approach.

State-average calculations based on a mixture of states are increasingly being exploited across chemistry and physics as versatile procedures for addressing excitations of quantum many-body systems. If not too many states should need to be addressed, calculations performed on individual states are also a common option. Here we show how the two approaches can be merged into one method, dealing with a generalized yet single pure state. Implications in electronic structure calculations are discussed and for quantum computations are pointed out.

Foundation of One-Particle Reduced Density Matrix Functional Theory for Excited States.

In Phys. Rev. Lett. 2021, 127, 023001 a reduced density matrix functional theory (RDMFT) was proposed for calculating energies of selected eigenstates of interacting many-Fermion systems. Here, we develop a solid foundation for this so-called w-RDMFT and present the details of various derivations. First, we explain how a generalization of the Ritz variational principle to ensemble states with fixed weights w in combination with the constrained search would lead to a universal functional of the one-particle reduced density matrix. To turn this into a viable functional theory, however, we also need to implement an exact convex relaxation. This general procedure includes Valone's pioneering work on ground state RDMFT as the special case w = (1,0, ···). Then, we work out in a comprehensive manner a methodology for deriving a compact description of the functional's domain. This leads to a hierarchy of generalized exclusion principle constraints which we illustrate in great detail. By anticipating their future pivotal role in functional theories and to keep our work self-contained, several required concepts from convex analysis are introduced and discussed.

Ensemble Reduced Density Matrix Functional Theory for Excited States and Hierarchical Generalization of Pauli's Exclusion Principle.

We propose and work out a reduced density matrix functional theory (RDMFT) for calculating energies of eigenstates of interacting many-electron systems beyond the ground state. Various obstacles which historically have doomed such an approach to be unfeasible are overcome. First, we resort to a generalization of the Ritz variational principle to ensemble states with fixed weights. This in combination with the constrained search formalism allows us to establish a universal functional of the one-particle reduced density matrix. Second, we employ tools from convex analysis to circumvent the too involved N-representability constraints. Remarkably, this identifies Valone's pioneering work on RDMFT as a special case of convex relaxation and reveals that crucial information about the excitation structure is contained in the functional's domain. Third, to determine the crucial latter object, a methodology is developed which eventually leads to a generalized exclusion principle. The corresponding linear constraints are calculated for systems of arbitrary size.

Toward an Atomic-Level Understanding of Ceria-Based Catalysts: When Experiment and Theory Go Hand in Hand.

ConspectusBecause ceria (CeO2) is a key ingredient in the formulation of many catalysts, its catalytic roles have received a great amount of attention from experiment and theory. Its primary function is to enhance the oxidation activity of catalysts, which is largely governed by the low activation barrier for creating lattice O vacancies. Such an important characteristic of ceria has been exploited in CO oxidation, methane partial oxidation, volatile organic compound oxidation, and the water-gas shift (WGS) reaction and in the context of automotive applications. A great challenge of such heterogeneously catalyzed processes remains the unambiguous identification of active sites.In oxidation reactions, closing the catalytic cycle requires ceria reoxidation by gas-phase oxygen, which includes oxygen adsorption and activation. While the general mechanistic framework of such processes is accepted, only very recently has an atomic-level understanding of oxygen activation on ceria powders been achieved by combined experimental and theoretical studies using in situ multiwavelength Raman spectroscopy and DFT.Recent studies have revealed that the adsorption and activation of gas-phase oxygen on ceria is strongly facet-dependent and involves different superoxide/peroxide species, which can now be unambiguously assigned to ceria surface sites using the combined Raman and DFT approach. Our results demonstrate that, as a result of oxygen dissociation, vacant ceria lattice sites are healed, highlighting the close relationship of surface processes with lattice oxygen dynamics, which is also of technical relevance in the context of oxygen storage-release applications.A recent DFT interpretation of Raman spectra of polycrystalline ceria enables us to take account of all (sub)surface and bulk vibrational features observed in the experimental spectra and has revealed new findings of great relevance for a mechanistic understanding of ceria-based catalysts. These include the identification of surface oxygen (Ce-O) modes and the quantification of subsurface oxygen defects. Combining these theoretical insights with operando Raman experiments now allows the (sub)surface oxygen dynamics of ceria and noble metal/ceria catalysts to be monitored under the reaction conditions.Applying these findings to Au/ceria catalysts provides univocal evidence for ceria support participation in heterogeneous catalysis. For room-temperature CO oxidation, operando Raman monitoring the (sub)surface defect dynamics clearly demonstrates the dependence of catalytic activity on the ceria reduction state. Extending the combined experimental/DFT approach to operando IR spectroscopy allows the elucidation of the nature of the active gold as (pseudo)single Au+ sites and enables us to develop a detailed mechanistic picture of the catalytic cycle. Temperature-dependent studies highlight the importance of facet-dependent defect formation energies and adsorbate stabilities (e.g., carbonates). While the latter aspects are also evidenced to play a role in the WGS reaction, the facet-dependent catalytic performance shows a correlation with the extent of gold agglomeration. Our findings are fully consistent with a redox mechanism, thus adding a new perspective to the ongoing discussion of the WGS reaction.As outlined above for ceria-based catalysts, closely combining state-of-the-art in situ/operando spectroscopy and theory constitutes a powerful approach to rational catalyst design by providing essential mechanistic information based on an atomic-level understanding of reactions.

Concept of Orbital Entanglement and Correlation in Quantum Chemistry.

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

Correlation Paradox of the Dissociation Limit: A Quantum Information Perspective.

The interplay between electron interaction and geometry in a molecular system can lead to rather paradoxical situations. The prime example is the dissociation limit of the hydrogen molecule. While a significant increase of the distance r between the two nuclei marginalizes the electron-electron interaction, the exact ground state does, however, not take the form of a single Slater determinant. By first reviewing and then employing concepts from quantum information theory, we resolve this paradox and its generalizations to more complex systems in a quantitative way. To be more specific, we illustrate and prove that thermal noise due to finite, possibly even just infinitesimally low, temperature T will destroy the entanglement beyond a critical separation distance rcrit(T) entirely. Our analysis is comprehensive in the sense that we simultaneously discuss both total correlation and entanglement in the particle picture as well as in the orbital/mode picture. Our results reveal a conceptually new characterization of static and dynamical correlation in ground states by relating them to the (non)robustness of correlation with respect to thermal noise.

Reduced Density Matrix Functional Theory for Bosons.

Based on a generalization of Hohenberg-Kohn's theorem, we propose a ground state theory for bosonic quantum systems. Since it involves the one-particle reduced density matrix γ as a variable but still recovers quantum correlations in an exact way it is particularly well suited for the accurate description of Bose-Einstein condensates. As a proof of principle we study the building block of optical lattices. The solution of the underlying v-representability problem is found and its peculiar form identifies the constrained search formalism as the ideal starting point for constructing accurate functional approximations: The exact functionals F[γ] for this N-boson Hubbard dimer and general Bogoliubov-approximated systems are determined. For Bose-Einstein condensates with N_{BEC}≈N condensed bosons, the respective gradient forces are found to diverge, ∇_{γ}F∝1/sqrt[1-N_{BEC}/N], providing a comprehensive explanation for the absence of complete condensation in nature.

Diverging Exchange Force and Form of the Exact Density Matrix Functional.

For translationally invariant one-band lattice models, we exploit the ab initio knowledge of the natural orbitals to simplify reduced density matrix functional theory (RDMFT). Striking underlying features are discovered. First, within each symmetry sector, the interaction functional F depends only on the natural occupation numbers n. The respective sets P_{N}^{1} and E_{N}^{1} of pure and ensemble N-representable one-matrices coincide. Second, and most importantly, the exact functional is strongly shaped by the geometry of the polytope E_{N}^{1}≡P_{N}^{1}, described by linear constraints D^{(j)}(n)≥0. For smaller systems, it follows as F[n]=[under ∑]i,i^{'}V[over ¯]_{i,i^{'}}sqrt[D^{(i)}(n)D^{(i^{'})}(n)]. This generalizes to systems of arbitrary size by replacing each D^{(i)} by a linear combination of {D^{(j)}(n)} and adding a nonanalytical term involving the interaction V[over ^]. Third, the gradient dF/dn is shown to diverge on the boundary ∂E_{N}^{1}, suggesting that the fermionic exchange symmetry manifests itself within RDMFT in the form of an "exchange force." All findings hold for systems with a nonfixed particle number as well and V[over ^] can be any p-particle interaction. As an illustration, we derive the exact functional for the Hubbard square.

Communication: Relating the pure and ensemble density matrix functional.

A crucial theorem in Reduced Density Matrix Functional Theory (RDMFT) suggests that the universal pure and ensemble functionals coincide on their common domain of pure N-representable one-matrices. We refute this by a comprehensive analysis of the geometric picture underlying Levy's constrained search. Moreover, we then show that the ensemble functional follows instead as the lower convex envelop of the pure functional. It is particularly remarkable that the pure functional determines the ensemble functional even outside its own domain of pure N-representable one-matrices. From a general perspective, this demonstrates that relaxing pure RDMFT to ensemble RDMFT does not necessarily circumvent the complexity of the one-body pure N-representability conditions (generalized Pauli constraints). Instead, the complexity may simply be transferred from the underlying space of pure N-representable one-matrices to the structure of the universal one-matrix functional.

Experimental and Theoretical Study on the Nature of Adsorbed Oxygen Species on Shaped Ceria Nanoparticles.

Ceria is widely used in heterogeneous catalysis owing to its redox properties. Engineering the shape of ceria particles offers a powerful tool to develop materials with enhanced catalytic properties. In this study, we provide evidence for the shape-dependent dioxygen adsorption and activation of ceria nanoparticles with (111) and (100) facets by in situ Raman spectroscopy and relate these properties to unique adsorption sites employing density functional theory. Temperature- and gas-phase-dependent experiments demonstrate facilitated formation of peroxide, superoxide, and weakly bound dioxygen species on the (100) facets as rationalized by calculated vibrational frequencies of O22-, O2-, and O2 species on CeO2- x(100) surfaces. Our results show that localization of the excess charge, driving the Ce4+ → Ce3+ reduction, significantly affects the stretching vibrations. Our approach provides a powerful basis for future developments of ceria-based catalysts by bridging the materials gap between idealized and real catalytic systems.

Adaptive moment closure for parameter inference of biochemical reaction networks.

Continuous-time Markov chain (CTMC) models have become a central tool for understanding the dynamics of complex reaction networks and the importance of stochasticity in the underlying biochemical processes. When such models are employed to answer questions in applications, in order to ensure that the model provides a sufficiently accurate representation of the real system, it is of vital importance that the model parameters are inferred from real measured data. This, however, is often a formidable task and all of the existing methods fail in one case or the other, usually because the underlying CTMC model is high-dimensional and computationally difficult to analyze. The parameter inference methods that tend to scale best in the dimension of the CTMC are based on so-called moment closure approximations. However, there exists a large number of different moment closure approximations and it is typically hard to say a priori which of the approximations is the most suitable for the inference procedure. Here, we propose a moment-based parameter inference method that automatically chooses the most appropriate moment closure method. Accordingly, contrary to existing methods, the user is not required to be experienced in moment closure techniques. In addition to that, our method adaptively changes the approximation during the parameter inference to ensure that always the best approximation is used, even in cases where different approximations are best in different regions of the parameter space.

Prospective, comparative study of the On-Q® PainBuster® postoperative pain relief system and thoracic epidural analgesia after thoracic surgery.

OBJECTIVE: Pain after thoracotomy is associated with intense discomfort leading to impaired pulmonary function. DESIGN: Prospective, non-randomized trial from April 2009 to September 2011. SETTING: Department of Thoracic Surgery, single-center. PARTICIPANTS: Thoracic surgical patients. INTERVENTIONS: Comparison of thoracic epidural analgesia (TEA) with the On-Q® PainBuster® system after thoracotomy. MEASUREMENTS AND MAIN RESULTS: The TEA group (n=30) received TEA with continuous 0.2% ropivacaine at 4 mL-to-8 mL/h, whereas Painbuster® patients (n=32) received 0.75% ropivacaine at 5 mL/h until postoperative day 4 (POD4). Basic and on-demand analgesia were identical in both groups. Pain was measured daily on a numeric analog scale from 0 (no pain) to 10 (worst pain) at rest and at exercise. There were no significant differences regarding demographic and preoperative data between the groups, but PainBuster® patients had a slightly lower relative forced expiratory volume in 1 second (FEV1) (71±20% versus 86±21%; p=0.01). Most common surgical procedures were lobectomies (38.8%) and atypical resections (28.3%) via anterolateral thoracotomy. Most common primary diagnoses were lung cancer (48.3%) and tumor of unknown origin (30%). At POD1, median postoperative pain at rest was 2.1 (1; 2.8) in the TEA group and 2 (1.5; 3.8; p=0.62) in the PainBuster® group. At exercise, median pain was 4.3 (3.5; 3.8) in the TEA group compared to 5.0 (4.0; 6.5; p=0.07). Until POD 5 there were decreases in pain at rest and exercise but without significant differences between the groups. CONCLUSIONS: Sufficient analgesia after thoracotomy can be achieved with the intercostal PainBuster® system in patients, who cannot receive TEA.

Pinning of fermionic occupation numbers.

The Pauli exclusion principle is a constraint on the natural occupation numbers of fermionic states. It has been suspected since at least the 1970s, and only proved very recently, that there is a multitude of further constraints on these numbers, generalizing the Pauli principle. Here, we provide the first analytic analysis of the physical relevance of these constraints. We compute the natural occupation numbers for the ground states of a family of interacting fermions in a harmonic potential. Intriguingly, we find that the occupation numbers are almost, but not exactly, pinned to the boundary of the allowed region (quasipinned). The result suggests that the physics behind the phenomenon is richer than previously appreciated. In particular, it shows that for some models, the generalized Pauli constraints play a role for the ground state, even though they do not limit the ground-state energy. Our findings suggest a generalization of the Hartree-Fock approximation.