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The degree distribution of network nodes exhibits scale-free features, which is a universal phenomenon in natural and social networks. It is commonly believed that growth and linear preferential attachment are the evolutionary mechanisms of scaling laws. However, the underlying cause behind these mechanisms remains unknown. In this paper, we propose the concept of the topology state of network nodes, and then define the network homomorphism ratio. Combining the total energy of the network defined in previous studies, we systematically investigate the connection between these two quantities in preferential attachment growth networks. Our results show that under the thermodynamic limit condition, the emergence of the scaling laws of the preferential attachment growth network is dominated by two principles. One principle is that the homomorphism ratio gradually tends to zero as the network grows, and the other principle is that the total energy of the system is the lowest under the condition of satisfying principle one. Interestingly, these two principles of the network are similar to the arrangement rules of electrons outside the nucleus which dominated by the Pauli exclusion principle and the principle of minimum energy.

Here, we provide evidence of the path-dependency of the energy components of the energy decomposition analysis scheme, EDA, by studying a set of thirty-one closed-shell model systems with D2h symmetry point group. For each system, we computed EDA components from nine different pathways and numerically showed that the relative magnitudes of the components differ substantially from one path to the other. Not surprisingly, yet unfortunately, the most significant variations in the relative magnitudes of the EDA components appear in the case of species with bonds within the grey zone of covalency and ionicity. We further discussed that the role of anions and their effect on arbitrary Pauli repulsion energy components affects the nature of bonding defined by EDA. The outcome variation by the selected partitioning scheme of EDA might bring arbitrariness when a careful comparison is overlooked J.P. and M.S. thank the Ministerio de Economía y Competitividad (MINECO) of Spain (projects CTQ2017-85341-P, PID2020-113711GB-I00, PID-2019-106830GB-I00 and MDM2017-0767) and the Generalitat de Catalunya (2017SGR39 and 2017SGR348). D.M.A. thanks ERC StG (EU805113) and University of Saarland for financial support. CFN thanks National Science Centre, Poland 2020/39/B/ST4/02022 for partially funding this research

We realize a suite of logical operations on a distance-two logical qubit stabilized using repeated error detection cycles. Logical operations include initialization into arbitrary states, measurement in the cardinal bases of the Bloch sphere, and a universal set of single-qubit gates. For each type of operation, we observe higher performance for fault-tolerant variants over non-fault-tolerant variants, and quantify the difference through detailed characterization. In particular, we demonstrate process tomography of logical gates, using the notion of a logical Pauli transfer matrix. This integration of high-fidelity logical operations with a scalable scheme for repeated stabilization is a milestone on the road to quantum error correction with higher-distance superconducting surface codes., Comment: 16 pages, 9 figures, 2 tables

We suggest a scale of dispersion energy donors (DEDs) that allows for direct comparisons with steric effects. This scale is based on the classic A-values and allows groups to reorient to minimize strain, thereby providing an advantage over raw group polarizabilities. The A-value can no longer be considered purely a steric factor. Even for groups that do not participate in charge transfer or electrostatic interactions, the A-value includes Pauli repulsion (steric hindrance) and attractive London dispersion (LD) interactions. Although the common assumption is that, at the distances found in monosubstituted cyclohexanes, steric demands are the key factors influencing conformer preferences, we show in this computational study that there is a non-negligible LD part. We use this system to build a DED scale and a complementary steric scale. These scales are quantitatively comparable, as they are based on the same system, and allow for comparison of the two competing interactions in experimentally relevant settings. In addition, we show that LD interactions can be used to explain puzzling data regarding relative group sizes.

Photons not welcomeTwo identical fermions cannot occupy the same quantum state, or so says the Pauli principle. For a cold gas of fermionic atoms, this means that all states up to the Fermi energy will be occupied, with only the atoms with the highest energy able to change their state. Such conditions have long been predicted to suppress light scattering off gases because the atoms receiving a kick from collisions with photons would have no state to move to. Debet al., Margalitet al., and Sanneret al. now describe this so-called Pauli blocking of light scattering. —JS

The Pauli exclusion principle forbids indistinguishable fermions to occupy the same quantum mechanical state. Its implications are profound and it for example accounts for the electronic shell structure of atoms. Here we perform measurements on the scattering of off-resonant light from ultracold gasses of fermionic atoms. For Fermi gases in the quantum degenerate regime, we observe a marked suppression in light scattering as compared to a similarly prepared thermal Bose gas. We attribute the observed increased transmission of light through the quantum degenerate Fermi gas to Pauli blocking, where Fermi-Dirac statistics causes atoms to occupy a large region of the momentum space limiting the number of accessible states for the scattered atom. Our work confirms a longstanding fundamental result in the theory of the optical response of quantum gases and is an important step towards novel cooling and thermometry mechanisms for degenerate Fermi gases.

Exchange anisotropy effects in equal-spin s(1) = s(2) dimeric single-molecule magnets exhibiting C2 point- group symmetry have been studied. The Hamiltonian, which is written in two-spin 4 × 4 Pauli matrixes representation in the appropriate noncollinear local coordinates systems, has been transformed to common coordinates and used to derive energy levels and spin eigenstates. An anomalous variation of the magnetic system average spin depending on the misalignment of the local anisotropy axes has been found. Particularly, a fully nonmagnetic state of an s = 1/2 spin dimer and “silent” EPR mode has been predicted. The angular boundaries of the EPR existence have been determined for any possible mutual deflections of the local ion axes for each of the 6 possible transitions between the levels of the spin system, both in zero and in a nonzero magnetic field.

Quantum mechanics (QM) proposes that a quantum system measurement does not register a pre-existing reality but rather establishes reality from the superposition of potential states. Measurement reduces the quantum state according to a probability function, the Born rule, realizing one of the potential states. Consequently, a classical reality is observed. The strict randomness of the measurement outcome is well-documented (and theoretically predicted) and implies a strict indeterminacy in the physical world’s fundamental constituents. Wolfgang Pauli, with Carl Gustav Jung, extended the QM framework to measurement outcomes that are meaningfully related to human observers, providing a psychophysical theory of quantum state reductions. The Pauli-Jung model (PJM) proposes the existence of observer influences on quantum measurement outcomes rooted in the observer’s unconscious mind. The correlations between quantum state reductions and (un)conscious states of observers derived from the PJM and its mathematical reformulation within the model of pragmatic information (MPI) were empirically tested. In all studies, a subliminal priming paradigm was used to induce a biased likelihood for specific quantum measurement outcomes (i.e., a higher probability of positive picture presentations; Studies 1 and 2) or more pronounced oscillations of the evidence than expected by chance for such an effect (Studies 3 and 4). The replicability of these effects was also tested. Although Study 1 found strong initial evidence for such effects, later replications (Studies 2 to 4) showed no deviations from the Born rule. The results thus align with standard QM, arguing against the incompleteness of standard QM in psychophysical settings like those established in the studies. However, although no positive evidence exists for the PJM and the MPI, the data do not entirely falsify the model’s validity.

Among the isoelectronic ligands CN-, CO, and NO+, an oblique bonding to the metal is well-established for the nitrosyl ligand, with M-N-O angles down to ≈120°. In the last decades, the nitrosyl community got into the habit of addressing a bent-bonded nitrosyl ligand as 1NO-. Thus, because various redox forms of a nitrosyl ligand seem to exist, the ligand is considered to be "noninnocent" because of the obvious ambiguity of an oxidation state (OS) assignment of the ligand and metal. Among the bent-bonded species, the low-spin {CoNO}8 class is prototypic. From this class, some 20 new nitrosyl compounds, the X-ray structure determinations of which comply with strict quality criteria, were analyzed with respect to the OS issue. As a result, the effective OS method shows a low-spin d8 CoI-NO+ couple instead of a negative OS of the ligand at the BP86/def2-TZVP (+D3, +CPCM with infinite permittivity) level of theory. The same holds for some new members of the linear subclass of {CoNO}8 compounds. For all compounds, a largely invariable "real" charge of ≈ -0.3 e was obtained from population analyses. All of these electron-rich d8 species strive to manage Pauli repulsion between the metal electrons and the lone pair at the nitrosyl's nitrogen atom, with the bending of the CoNO unit as the most frequent escape.

Polymer self-consistent field theory techniques are used to derive quantum density functional theory without the use of the theorems of density functional theory. Instead, a free energy is obtained from a partition function that is constructed directly from a Hamiltonian, so that the results are, in principle, valid at finite temperatures. The main governing equations are found to be a set of modified diffusion equations, and the set of self-consistent equations are essentially identical to those of a ring polymer system. The equations are shown to be equivalent to Kohn-Sham density functional theory, and to reduce to classical density functional theory, each under appropriate conditions. The obtained non-interacting kinetic energy functional is, in principle, exact, but suffers from the usual orbital-free approximation of the Pauli exclusion principle in additional to the exchange-correlation approximation. The equations are solved using the spectral method of polymer self-consistent field theory, which allows the set of modified diffusion equations to be evaluated for the same computational cost as solving a single diffusion equation. A simple exchange-correlation functional is chosen, together with a shell-structure-based Pauli potential, in order to compare the ensemble average electron densities of several isolated atom systems to known literature results. The agreement is excellent, justifying the alternative formalism and numerical method. Some speculation is provided on considering the time-like parameter in the diffusion equations, which is related to temperature, as having dimensional significance, and thus picturing point-like quantum particles instead as non-local, polymer-like, threads in a higher dimensional thermal-space. A consideration of the double-slit experiment from this point of view is speculated to provide results equivalent to the Copenhagen interpretation., 32 pages, 5 figures

Asynchronicity in Diels–Alder reactions plays a crucial role in determining the height of the reaction barrier. Currently, the origin of asynchronicity is ascribed to the stronger orbital interaction between the diene and the terminal carbon of an asymmetric dienophile, which shortens the corresponding newly formed C–C bond and hence induces asynchronicity in the reaction. Here, we show, using the activation strain model and Kohn–Sham molecular orbital theory at ZORA-BP86/TZ2P, that this rationale behind asynchronicity is incorrect. We, in fact, found that following a more asynchronous reaction mode costs favorable HOMO–LUMO orbital overlap and, therefore, weakens (not strengthens) these orbital interactions. Instead, it is the Pauli repulsion that induces asynchronicity in Diels–Alder reactions. An asynchronous reaction pathway also lowers repulsive occupied–occupied orbital overlap which, therefore, reduces the unfavorable Pauli repulsion. As soon as this mechanism of reducing Pauli repulsion dominates, the reaction begins to deviate from synchronicity and adopts an asynchronous mode. The eventual degree of asynchronicity, as observed in the transition state of a Diels–Alder reaction, is ultimately achieved when the gain in stability, as a response to the reduced Pauli repulsion, balances with the loss of favorable orbital interactions., Quantum chemical activation strain analyses reveal that asynchronicity in Diels–Alder reactions reduces both destabilizing Pauli repulsion as well as stabilizing orbital interactions, and occurs if the former dominates.

TBD

Real stabilizer operators, which are also known as real Clifford operators, are generated, through composition and tensor product, by the Hadamard gate, the Pauli Z gate, and the controlled-Z gate. We introduce a normal form for real stabilizer circuits and show that every real stabilizer operator admits a unique normal form. Moreover, we give a finite set of relations that suffice to rewrite any real stabilizer circuit to its normal form., In Proceedings QPL 2021, arXiv:2109.04886

Nonlinear propagation of stationary large-amplitude electromagnetic (EM) solitary waves in a magnetized electron–positron (EP) plasma is studied using a fully relativistic two-fluid hydrodynamic model. In the first step, we wrote the one-body particle and N-body particle fluid equations for the electron–positron fluid, taking into account the spin effects and regardless of the thermal effects. These equations are the first to be set for electron and positron fluids. The particle-antiparticle nature of the relativistic particles is explicit in this fluid theory. The nonrelativistic limit is shown to be in agreement with previous attempts to develop a spin plasma theory derived from the Pauli Hamiltonian. In the second step, we investigated the thermal effects without considering the spin effects and according to the enthalpy of the system. In the last step, we studied the electromagnetic mode propagation for this type of plasma and determined the range of wave transition.

The aim of this work is to consider the Pauli–Fibonacci quaternions and to present some properties involving this sequence, including the Binet’s formula and generating functions. Furthermore, the Honsberger identity, the generating function, d’Ocagne’s identity, Cassini’s identity, Catalan’s identity for these quaternions are given. The matrix representations for Pauli–Fibonacci quaternions are introduced.

Variational energy decomposition analyses have been presented to quantify the σ-dative, ligand-to-metal forward charge transfer (CT) and the π-conjugative, metal-to-ligand backward charge delocalization on a series of isolelectronic transition-metal carbonyl complexes M(CO)6, including M = Ti2-, V-, Cr, Mn+, and Fe2+. Although the qualitative features of these energy terms are understood, well-defined quantitative studies have been scarce. Consistent with early findings, electrostatic and Pauli exchange effects play a key role in σ-donation, resulting in blue shifts in ligand vibrational frequency in the complex geometries. Excluding chemical bonding interactions between the CO ligand and the metal fragments in the energy decomposition analysis, we found that loosely bound electrostatic complexes can be formed at a longer metal-to-ligand distance due to the exponential decay of Pauli exchange. In all complexes, the overall binding stabilization can be attributed to CT effects, with opposing trends between σ-donation and π-back bonding that follows an order of Ti2- (4.4) > V1- (2.6) > Cr (1.5) > Mn1+ (1.1) > Fe2+ (0.5) in π-to-σ CT ratio. These electronic and energetic features are mirrored in the vibrational frequency shifts induced by different factors. The present investigation may help stimulate the use of energy decomposition techniques to understand the structure and activity of metallocatalysts using density functional theory.

It has long been known that there is a fundamental difference in the electronic structures of CH5- and SiH5-, two isoelectronic molecules. The former is a saddle point for the SN2 exchange reaction H- + CH4 → [CH5-]‡ → CH4 + H-, while the latter is a stable molecule that is bound relative to SiH4 + H-. SCGVB calculations indicate that this difference is the result of a dramatic difference in the nature of the axial electron pairs in these anions. In SiH5-, the axial pairs represent two stable bonds-a weak recoupled pair bond dyad. In CH5-, the axial electron pairs represent an intermediate transition between the electron pairs in the reactants and those in the products. Furthermore, the axial orbitals at the saddle point in CH5- are highly overlapping, giving rise to strong Pauli repulsion and a high barrier for the SN2 exchange reaction.

Atoms may repel or attract each other, but bound systems get virtually all of the attention in the classroom. To address the imbalance, instructors can explore the basis of repulsion in a few simple systems and highlight the important role played by the Pauli principle. A good example is the first triplet excited state of H2 wherein both electrons have the same spin. In light of the Hellmann–Feynman theorem, the net interaction is repulsive due to the low electron density that results in the region between the two nuclei. Analogous destabilization occurs in σ-bonding systems involving 3 electrons because two-thirds of the electrons have the same spin. Symmetrical systems like He2+ are exceptions and actually do form bonds because the minority spin is able to delocalize over both centers. In contrast, the ground state of the neutral He2 system entails 2 α and 2 β electrons and is strictly repulsive. Finally, the lowest energy triplet state of ethylene twists toward a D2d structure to avoid a destabilizing interaction akin to that identified in the triplet state of H2.

The α‐effect is a term used to explain the dramatically enhanced reactivity of α‐nucleophiles (R−Y−X:−) compared to their parent normal nucleophile (R−X:−) by deviating from the classical Brønsted‐type reactivity‐basicity relationship. The exact origin of this effect is, however, still heavily under debate. In this work, we have quantum chemically analyzed the α‐effect of a set of anionic nucleophiles, including O‐, N‐ and S‐based normal and α‐nucleophiles, participating in an SN2 reaction with ethyl chloride using relativistic density functional theory at ZORA‐OLYP/QZ4P. Our activation strain and Kohn–Sham molecular orbital analyses identified two criteria an α‐nucleophile needs to fulfill in order to show α‐effect: (i) a small HOMO lobe on the nucleophilic center, pointing towards the substrate, to reduce the repulsive occupied–occupied orbital overlap and hence (steric) Pauli repulsion with the substrate; and (ii) a sufficiently high energy HOMO to overcome the loss of favorable HOMO–LUMO orbital overlap with the substrate, as a consequence of the first criterion, by reducing the HOMO–LUMO orbital energy gap. If one of these two criteria is not fulfilled, one can expect no α‐effect or inverse α‐effect., The α‐effect explained! Quantum chemical activation strain analyses reveal the intrinsic factors that α‐nucleophiles need to exhibit α‐effect. The origin is traced back to a small HOMO lobe to reduce the destabilizing Pauli repulsion, and a sufficiently high energy HOMO to overcome the loss of favorable orbital overlap.

Moire quantum matter has emerged as a materials platform in which correlated and topological phases can be explored with unprecedented control. Among them, magic-angle systems constructed from two or three layers of graphene have shown robust superconducting phases with unconventional characteristics1-5. However, direct evidence of unconventional pairing remains to be experimentally demonstrated. Here we show that magic-angle twisted trilayer graphene exhibits superconductivity up to in-plane magnetic fields in excess of 10 T, which represents a large (2-3 times) violation of the Pauli limit for conventional spin-singlet superconductors6,7. This is an unexpected observation for a system that is not predicted to have strong spin-orbit coupling. The Pauli-limit violation is observed over the entire superconducting phase, which indicates that it is not related to a possible pseudogap phase with large superconducting amplitude pairing. Notably, we observe re-entrant superconductivity at large magnetic fields, which is present over a narrower range of carrier densities and displacement fields. These findings suggest that the superconductivity in magic-angle twisted trilayer graphene is likely to be driven by a mechanism that results in non-spin-singlet Cooper pairs, and that the external magnetic field can cause transitions between phases with potentially different order parameters. Our results demonstrate the richness of moire superconductivity and could lead to the design of next-generation exotic quantum matter.

This work extends the orbital-free density functional theory to the field of quantum crystallography. The total electronic energy is decomposed into electrostatic, exchange, Weizsacker and Pauli components on the basis of physically grounded arguments. Then, the one-electron Euler equation is re-written through corresponding potentials, which have clear physical and chemical meaning. Partial electron densities related with these potentials by the Poisson equation are also defined. All these functions were analyzed from viewpoint of their physical content and limits of applicability. Then, they were expressed in terms of experimental electron density and its derivatives using the orbital-free density functional theory approximations, and applied to the study of chemical bonding in a heteromolecular crystal of ammonium hydrooxalate oxalic acid dihydrate. It is demonstrated that this approach allows the electron density to be decomposed into physically meaningful components associated with electrostatics, exchange, and spin-independent wave properties of electrons or with their combinations in a crystal. Therefore, the bonding information about a crystal that was previously unavailable for X-ray diffraction analysis can be now obtained.

Connection of Pauli’s principle with the nontrivial structure of the fermion supertime is shown. When supersymmetry is localized as supergravitation, fields of gravitational and exchange interaction carriers arise. The exchange interaction quantum of free fermions, being a superpartner of graviton (gravitino), is interpreted as a paulino – the particle responsible for the effect of mutual avoidance of identical fermions.

Ultracold ions provide one of the most promising platforms for quantum computing and make it possible to reach record coherence times, the fidelity of preparation and readout operations, and single- and two-qubit operations. Encoding quantum information in an optical qubit based on the $$^{2}{{S}_{{1/2}}}(F = 0,{{m}_{F}} = 0)$$ → $$^{2}{{D}_{{3/2}}}(F = 2,{{m}_{F}} = 0)$$ quadrupole transition in the 171Yb+ ion at a wavelength of 435.5 nm, which has potential advantages over similar systems in the scaling of the number of qubits and their sensitivity to fluctuations of the magnetic field, is proposed and experimentally studied. The proposed optical qubit in ytterbium is compared to other most widespread types of ion qubits. Experimental results on the implementation of the Pauli-X single-qubit operation are given. The fidelity of the operation is 96% after correction to the error of preparation and readout and is limited by the temperature of the ion.

Bonding to a quantum corral Chemical bonds generally form between electronic states of atoms; in principle, other electronic states could also form bonds. Stilp et al. found that the electronic states created within quantum corrals, large rings of iron atoms on a copper surface, can form chemical bonds with metal atoms on an atomic force microscope tip. The corral states form from many electrons but have a large spatial extent compared with an atomic orbital. The covalent bond to a 48-atom corral state had an energy of just 5 millielectron volts. Science , abe2600, this issue p. 1196

We present measurements of the magnetic torque, specific heat and thermal expansion of the bulk transition metal dichalcogenide (TMD) superconductor NbS2 in high magnetic fields, with its layer structure aligned strictly parallel to the field using a piezo rotary positioner. The upper critical field of superconducting TMDs in the 2D form is known to be dramatically enhanced by a special form of Ising spin orbit coupling. This Ising superconductivity is very robust to the Pauli paramagnetic effect and can therefore exist beyond the Pauli limit for superconductivity. We find that superconductivity beyond the Pauli limit still exists in bulk single crystals of NbS2 for a precisely parallel field alignment. However, the comparison of our upper critical field transition line with numerical simulations rather points to the development of a Fulde-Ferrell-Larkin-Ovchinnikov state above the Pauli limit as a cause. This is also consistent with the observation of a magnetic field driven phase transition in the thermodynamic quantities within the superconducting state near the Pauli limit. © 2021, The Author(s). We thank U. Lampe for technical support. This work was supported by grants from the Research Grants Council of the Hong Kong Special Administrative Region, China (GRF-16302018, GRF-16303820, C6025-19G-A, SBI17SC14). M.A.-H. acknowledges the financial support from the Swedish Research Council (VR) under the project no. 2018-05393 and by the Arab-German Young Academy of Sciences and Humanities (AGYA). M.A.-H and D.C. acknowledge the financial support by the Megagrant 2020-220-08-6358 of the Government Council on Grants of the Russian Federation.

We give a broad generalisation of the mapping, originally due to Dennis, Kitaev, Landahl and Preskill, from quantum error correcting codes to statistical mechanical models. We show how the mapping can be extended to arbitrary stabiliser or subsystem codes subject to correlated Pauli noise models, including models of fault tolerance. This mapping connects the error correction threshold of the quantum code to a phase transition in the statistical mechanical model. Thus, any existing method for finding phase transitions, such as Monte Carlo simulations, can be applied to approximate the threshold of any such code, without having to perform optimal decoding. By way of example, we numerically study the threshold of the surface code under mildly correlated bit-flip noise, showing that noise with bunching correlations causes the threshold to drop to $p_{\textrm{corr}}=10.04(6)\%$, from its known iid value of $p_{\text{iid}}=10.917(3)\%$. Complementing this, we show that the mapping also allows us to utilise any algorithm which can calculate/approximate partition functions of classical statistical mechanical models to perform optimal/approximately optimal decoding. Specifically, for 2D codes subject to locally correlated noise, we give a linear-time tensor network-based algorithm for approximate optimal decoding which extends the MPS decoder of Bravyi, Suchara and Vargo., Comment: 23 pages, comments welcome

We present the One-orbital Ensemble Self-Consistent Field (OE-SCF) method, an {alternative} orbital-free DFT solver that extends the applicability of DFT to system sizes beyond the nanoscale while retaining the accuracy required to be predictive. OE-SCF is an iterative solver where the (typically computationally expensive) Pauli potential is treated as an external potential and updated after each iteration. Because only up to a dozen iterations are needed to reach convergence, OE-SCF dramatically outperforms current orbital-free DFT solvers. Employing merely a single CPU, we carried out the largest ab initio simulation for silicon-based materials to date. OE-SCF is able to converge the energy of bulk-cut Si nanoparticles as a function of their diameter up to 16 nm, for the first time reproducing known empirical results. We model polarization and interface charge transfer when a Si slab is sandwiched between two metal slabs where lattice matching mandates a very large slab size. Additionally, OE-SCF opens the door to adopt even more accurate functionals in orbital-free DFT simulations while still tackling systems sizes beyond the nanoscale.

ConspectusOrganic chemistry has undoubtedly had a profound impact on humanity. Day in and day out, we find ourselves constantly surrounded by organic compounds. Pharmaceuticals, plastics, fuels, cosmetics, detergents, and agrochemicals, to name a few, are all synthesized by organic reactions. Very often, these reactions require a catalyst in order to proceed in a timely and selective manner. Lewis acids and organocatalysts are commonly employed to catalyze organic reactions and are considered to enhance the frontier molecular orbital (FMO) interactions. A vast number of textbooks and primary literature sources suggest that the binding of a Lewis acid or an iminium catalyst to a reactant (R1) stabilizes its LUMO and leads to a smaller HOMO(R2)-LUMO(R1) energy gap with the other reactant (R2), thus resulting in a faster reaction. This forms the basis for the so-called LUMO-lowering catalysis concept. Despite the simplicity and popularity of FMO theory, a number of deficiencies have emerged over the years, as a consequence of these FMOs not being the operative factor in the catalysis. LUMO-lowering catalysis is ultimately incomplete and is not always operative in catalyzed organic reactions. Our groups have recently undertaken a concerted effort to generate a unified framework to rationalize and predict chemical reactivity using a causal model that is rooted in quantum mechanics. In this Account, we propose the concept of Pauli repulsion-lowering catalysis to understand the catalysis in fundamental processes in organic chemistry. Our findings emerge from state-of-the-art computational methods, namely, the activation strain model (ASM) of reactivity in conjunction with quantitative Kohn-Sham molecular orbital theory (KS-MO) and a matching energy decomposition analysis (EDA). The binding of the catalyst to the substrate not only leads to a stabilization of its LUMO but also induces a significant reduction of the two-orbital, four-electron Pauli repulsion involving the key molecular orbitals of both reactants. This repulsion-lowering originates, for the textbook Lewis acid-catalyzed Diels-Alder reaction, from the catalyst polarizing the occupied πorbital of the dienophile away from the carbon atoms that form new bonds with the diene. This polarization of the occupied dienophile πorbital reduces the occupied orbital overlap with the diene and constitutes the ultimate physical factor responsible for the acceleration of the catalyzed process as compared to the analogous uncatalyzed reaction. We show that this physical mechanism is generally applicable regardless of the type of reaction (Diels-Alder and Michael addition reactions) and the way the catalyst is bonded to the reactants (i.e., from pure covalent or dative bonds to weaker hydrogen or halogen bonds). We envisage that the insights emerging from our analysis will guide future experimental developments toward the design of more efficient catalytic transformations.

A detailed analysis of a complete set of the local potentials that appear in the Euler equation for electron density is carried out for noncovalent interactions in the crystal of a uracil derivative using experimental X-ray charge density. The interplay between the quantum theory of atoms in molecules and crystals and the local potentials and corresponding inner-crystal electronic forces of electrostatic and kinetic origin is explored. Partitioning of crystal space into atomic basins and atomic-like potential basins led us to the definite description of interatomic interaction and charge transfer. Novel physically grounded bonding descriptors derived within the orbital-free quantum crystallography provided the detailed examination of π-stacking and intricate C=O⋅⋅⋅π interactions and nonclassical hydrogen bonds present in the crystal. The donor-acceptor character of these interactions is revealed by analysis of Pauli and von Weizsäcker potentials together with well-known functions, e. g., deformation electron density and electron localization function. In this way, our analysis throws light on aspects of these closed-shell interactions hitherto hidden from the description.

The unique and often-weird properties of quantum mechanics allow an information carrier to propagate through multiple trajectories of quantum channels simultaneously. This ultimately leads us to quantum trajectories with an indefinite causal order of quantum channels. It has been shown that indefinite causal order enables the violation of bottleneck capacity, which bounds the amount of the transferable classical and quantum information through a classical trajectory with a well-defined causal order of quantum channels. In this treatise, we investigate this beneficial property in the realm of both entanglement-assisted classical and quantum communications. To this aim, we derive closed-form capacity expressions of entanglement-assisted classical and quantum communication for arbitrary quantum Pauli channels over classical and quantum trajectories. We show that by exploiting the indefinite causal order of quantum channels, we obtain capacity gains over classical trajectory as well as the violation of bottleneck capacity for various practical scenarios. Furthermore, we determine the operating region where entanglement-assisted communication over quantum trajectory obtains capacity gain against classical trajectory and where the entanglement-assisted communication over quantum trajectory violates the bottleneck capacity., Accepted for publication in IEEE Transactions on Wireless Communications

Quantum error correction is an essential ingredient for universal quantum computing. Despite tremendous experimental efforts in the study of quantum error correction, to date, there has been no demonstration in the realisation of universal quantum error-correcting code, with the subsequent verification of all key features including the identification of an arbitrary physical error, the capability for transversal manipulation of the logical state and state decoding. To address this challenge, we experimentally realise the [5, 1, 3] code, the so-called smallest perfect code that permits corrections of generic single-qubit errors. In the experiment, having optimised the encoding circuit, we employ an array of superconducting qubits to realise the [5, 1, 3] code for several typical logical states including the magic state, an indispensable resource for realising non-Clifford gates. The encoded states are prepared with an average fidelity of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} }{}$57.1(3)\%$\end{document} while with a high fidelity of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} }{}$98.6(1)\%$\end{document} in the code space. Then, the arbitrary single-qubit errors introduced manually are identified by measuring the stabilisers. We further implement logical Pauli operations with a fidelity of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} }{}$97.2(2)\%$\end{document} within the code space. Finally, we realise the decoding circuit and recover the input state with an overall fidelity of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} }{}$74.5(6)\%$\end{document}, in total with 92 gates. Our work demonstrates each key aspect of the [5, 1, 3] code and verifies the viability of experimental realisation of quantum error-correcting codes with superconducting qubits., Experimental demonstration of encoding, decoding, logical operation, and error detection of a five-qubit error correcting code on a superconducting quantum processor.

We applied a set of advanced bonding descriptors to establish the hidden electron density features and binding energy characteristics of intermolecular DH∙∙∙A hydrogen bonds (OH∙∙∙O, NH∙∙∙O and SH∙∙∙O) in 150 isolated and solvated molecular complexes. The exchange-correlation and Pauli potentials as well as corresponding local one-electron forces allowed us to explicitly ascertain how electron exchange defines the bonding picture in the proximity of the H-bond critical point. The electron density features of DH∙∙∙A interaction are governed by alterations in the electron localization in the H-bond region displaying itself in the exchange hole. At that, they do not depend on the variations in the exchange hole mobility. The electrostatic interaction mainly defines the energy of H-bonds of different types, whereas the strengthening/weakening of H-bonds in complexes with varying substituents depends on the barrier height of the exchange potential near the bond critical point. Energy variations between H-bonds in isolated and solvated systems are also caused the electron exchange peculiarities as follows from the corresponding potential and the interacting quantum atom analyses complemented by electron delocalization index calculations. Our approach is based on the bonding descriptors associated with the characteristics of the observable electron density and can be recommended for in-depth studies of non-covalent bonding.

Many-particle systems contained inside other particles (problem 1), as well as systems of particles delimited inside other particles, which, in turn, are delimited inside the third particles (problem 2), are considered. Under natural assumptions about conserved quantities such as the total number of all particles, the total energy, and a number of others, the most probable energy distributions of particles, true (internal) and external observables, are studied. The results of problems 1 and 2 can be interpreted as the separation of bottom level particles into kinds with different distributions observed in different experiments. In single-particle observations in the full probability problem 1, an effective Pauli principle arises in a consistent manner.

Water is ubiquitous on Earth and dominates chemical and biological processes in daily life. However, how water behaves under some critical conditions is not fully understood. In this paper, we employed quantum first-principles calculations and dynamics simulations to reveal the unexpectedly high mobility of water molecules in ultraconfined spaces. The water molecules rotated more freely in the (4, 4) carbon nanotube than in the (5, 5) carbon nanotube, which is induced by the Pauli repulsion from the wall of the narrower channel when reducing the size of the channel from general confinement to ultraconfinement. Moreover, this quantum effect facilitates the transport of water molecules into the space within their van der Waals diameter easily, which is in contrast to the general understanding. Thus, the conventional concept that the tighter the confined space, the more difficult the motion of the confined object is not always correct. This quantum-induced enhancement of water mobility by Pauli repulsion calls us to pay more attention to the existence and the function of water in neglected ultraconfined spaces (e.g., cells and the Earth's crust) in the future.

Relativistic effects are quite large for the actinide 5f and 6d open shells, and in the present study we present combinatorial enumeration of $$\upomega {{-}} \upomega $$ states of all 14 actinide (lanthanide) dimers using multinomial symmetric function (S-function) techniques. The techniques presented here enumerate all possible $$\upomega {{-}} \upomega $$ states that arise from relativistic 2-component molecular spinors composed of the 7s, 6d 5f, and possibly 7p shells of the actinide dimers Th2 through Lr2. The combinatorial enumerations techniques reveal that that for Americium dimer (Am2) there are 20,058,300 S-function terms of which only 616,227 terms satisfy the Pauli exclusion principle yielding 401,166,110 $$\upomega {{-}} \upomega $$ states (142.6 MB of output), all of which arise from primarily the 7s/6d/5f shells of Am. Explicit tables are constructed for the relativistic $$\upomega {{-}} \upomega $$ states for Th2 to Am2 with several restrictions imposed on the levels of excitations primarily to reduce the table sizes, and through the electron–hole equivalence $$\upomega {{-}} \upomega $$ states of the remaining dimers of the actinide row are enumerated. The computational and mathematical techniques also enumerate the $$\upomega {{-}} \upomega $$ states of Ce2 to Eu2 and by electron–hole equivalence Gd2 to Tm2, although spin–orbit splittings are smaller for lanthanides. A unique feature of the present combinatorial technique is that it facilitates any number of orbitals and any number of open shells including the possibility of 14 singly-occupied orbitals for actinide and lanthanide dimers, and thus providing for a technique to enumerate $$\upomega {{-}} \upomega $$ states arising from both very high spin open-shell states as well as low spin states exhaustively.

We investigate the dynamical behaviors of quantum-memory-assisted (QMA) entropic uncertainty for the central two-qubit system coupled to the N-site spin XY chain environment with z-component Dzyaloshinsky-Moriya (DM) interaction and external magnetic field, asymmetrically. The results show that the dynamics of QMA entropic uncertainty for a pair of Pauli observables are oscillatory and ascending gradually with evolution time, which is anti-correlated to the entanglement of the two-qubit system. We also find that QMA entropic uncertainty exhibits sudden change phenomena near the critical point of quantum phase transform both in the case of weak and strong coupling regimes, which can act as a new witness of the critical point of quantum phase transition. Meanwhile, the DM interactions of the spin chain environment have non-equivalent effects on the QMA entropic uncertainty when the external magnetic field strength is far from the critical point. Furthermore, the asymmetricallity of the qubit-environment couplings has slightly effects on QMA entropic uncertainty in the case of strong coupling regime of the two-qubit system. While, in the weak coupling regime of the two-qubit system, QMA entropic uncertainty is very sensitive to the asymmetricallity of qubit-environment couplings. Finally, the larger the number of spin sites of the environment, the more quickly the QMA entropic uncertainty increases to the maximal value both in the weak and strong coupling regimes of the two-qubit system.

From our material exploration study in wide pressure and temperature conditions, we found a new 6H polymorph of BaRhO3 was stabilised under high pressure conditions from 14 to 22 GPa. The material crystallised in the monoclinic 6H hexagonal perovskite structure in space group C2/c. The 4H BaRhO3 polymorph was stabilised at lower pressures, but the 3C cubic BaRhO3 likely requires pressures greater than 22 GPa. Both 6H and 4H polymorphs contain Rh2O9 dimers and the large 4d Rh orbital spatial diffusivity in these dimers leads to Pauli paramagnetic and metallic ground states, which are also supported by first-principles electronic structure calculations. High Wilson ratios of approximately 2 for either compound indicate strong electron correlation.

Spin is an intrinsic form of angular momentum carried by elementary particles, composite particles, and atomic nuclei. It is wildly believed that spin is a purely quantum mechanical concept and has no classical analogue. In fact, elementary particles are conceived as point objects which have no axis to “spin” around. Therefore, there is no explaining how spin arises at the fundamental level, why particles have the values they do, and what underpins the Pauli Exclusion principle and Bose-Einstein behavior. However, spin is like a vector quantity; it has a definite magnitude, and it has a “direction”, in order to spin should be composite. In this paper we propose a physical explanation for spin of the electron at the sub-particle level, relying on the vortex model of the electron. The electron is described as a superfluid frictionless vortex which has a mass, angular momentum and spin to provide a complete explanation of all properties of the electron: it composite, spinning around its own axis, produces a tiny magnetic fields independent of those from its orbital motions. The classical hydrodynamic laws are used to describe the quantum properties of the electron, such as spin, angular momentum, magnetic momentum and a magnetic dipole. The circulation in the vortex is constant, and the angular momentum of the vortex is conserved and has the same value of Planck constant. The direction of the angular momentum of a spinning electron vortex is along the axis of rotation and determined by the direction of spin. The spin quantum number 1/2 has a fixed value which represents the gap between the circulation rate of the core of the vortex and the boundaries of the vortex. The changeable values +1/2 “spin-up” or -1/2 “spin-down” indicate the direction of the magnetic dipole of the vortex. The relation between spin and Planck constant is discussed and the origin h/4pi angular momentum units are revealed.

Recent experiments on sound waves in a unitary Fermi gas reveal many transport properties about strongly interacting fermions. Sound propagates through the coupling of momentum and heat transport, and attenuates strongly with the presence of a phase transition. In this work, focusing on the temperature regimes near and below the superfluid critical temperature $$T_c$$ T c in the BCS-BEC crossover, we present a Kubo-based microscopic calculation of thermal conductivity $$\kappa$$ κ , which has not attracted much attention compared to the shear viscosity. Our approach primarily addresses the contributions of the fermionic quasiparticles to thermal transport and our results return to the kinetic descriptions at high temperatures. $$\kappa$$ κ drops upon crossing the pseudogap temperature $$T^*$$ T ∗ , and its temperature dependence changes below $$T_c$$ T c . The drops become more pronounced on the weakly coupled BCS side, where the Pauli blocking causes the upturn of $$\kappa$$ κ above $$T^*$$ T ∗ . Our calculations fit well with the sound measurement on the damping rate.

An assumption that all the six flavour quarks are attributed to be the components of a same, a common isospin multiplets space named STS is proposed. Base on Pauli Exclusion Principle, every quark is assigned to different flavour marks in STS. Every flavour quark possesses its own colour spectral line array specially appointed. The collection of colour spectral line arrays of the six flavour quarks constructs together the CSDF, Colour Spectrum Diagram of Flavour, further baryons and mesons could be constructed from CSDF. STS, Spin Topological Space is a math frame with infinite dimensional matrix representation for spin angular momentum. Flavours is an isospin angular momentum coupling phenomena of the three-colour-quarks.

The mechanism and origin of stereoselectivity of silver-catalyzed hydroalkylation of alkynes were computationally investigated at the B3LYP-D3BJ/6-311+G(d,p)-SDD//B3LYP/6-31G(d)-LANL2DZ level. The complex of alkynyl trialkylboronate with cationic silver is a key intermediate, which triggers the rate- and stereoselectivity-determining 1,2-migration step. Energy decomposition analysis indicates that the difference of Pauli repulsion dominates the stereoselectivity.

In this paper we study the quantum graphs of mixed-unitary channels generated by tensor products of Pauli operators, which we call Pauli channels. We show that most quantum graphs arising from Pauli channels have non-trivial quantum cliques or quantum anticliques which are stabilizer codes. However, a reformulation of Nik Weaver's quantum Ramsey theorem in terms of stabilizer codes and Pauli channels fails. Specifically, for every positive integer $n$, there exists an $n$-qubit Pauli channel for which any non-trivial quantum clique or quantum anticlique fails to be a stabilizer code., 11 pages, final version. To appear in IEEE Transactions on Information Theory

Cage-like metal nanoclusters are rarely found due to the densely packed property of metals. Recently, single crystallography has unraveled for the first time that multi-shell golden cages are formed in large-size thiolate (SR) and alkynl (CCR) protected neutral Au144 nanoclusters, denoted as Au144(SR)60 and Au144(CCR)60. In this study, the origin of the structural stability of golden cage Au144 clusters is studied based on the density functional theory (DFT) energy calculation and energy decomposition analysis (EDA). The formation of hollow cages rather than centre-filled icosahedrons in the Au144 clusters is attributed to the significant Pauli repulsion between the central gold atom and the surrounding metal shell, which leads to the decrease of the averaged formation energy of the clusters. The present study also shows that the Au144 cluster is unique in size. The smaller size clusters Au133 and Au130 and the larger size cluster Au279 both preferred the centre-filled golden icosahedrons, decahedrons or octahedrons.

This paper presents a new bidirectional quantum teleportation (BQT) protocol in which users can simultaneously transmit an arbitrary number of particles to each other via a four-qubit cluster state as quantum channel. With the use of auxiliary qubits and controlled-not gates, transformation circuits are introduced which transform n-qubit states into single superposition state and then teleport them over the channel by using Bell-state measurement and applying a proper Pauli operator. At last, with the aid of another transformation circuit, the initial states can be reconstructed. With this method, the BQT purpose is realized more advantageous compared to recent presented protocols.

In this work, the 2s+2s (face-to-face) prototypical example of a photochemical reaction has been re-examined to characterize the evolution of chemical bonding. The analysis of the electron localization function (as an indirect measure of the Pauli principle) along the minimum energy path provides strong evidence in support that CC bond formation occurs not in the excited state but at the ground electronic state after crossing the rhombohedral S1/S0 conical intersection.

Paramagnetism and diamagnetism of a material characterized by its magnetic susceptibility. When a material is exposed to an external magnetic field, magnetic susceptibility is defined as the ratio of the induced magnetization and the magnetic field. A paramagnetic material has magnetic susceptibility with positive sign. On the other hand, a diamagnetic material has magnetic susceptibility with negative sign. Atomically, paramagnetic materials consist of atoms that has orbital with unpaired electrons. Theoretical study of paramagnetic susceptibility and diamagnetic susceptibility are well described by Pauli paramagnetism and Landau diamagnetism, respectively. Although paramagnetism and diamagnetism are among the simplest magnetic properties of material that are studied in basic physics, theoretical derivations of Pauli paramagnetic and Landau diamagnetic susceptibility require second quantization formalism of quantum mechanics. We aim to discuss the paramagnetic and diamagnetic susceptibilities for simple three-dimensional quantum well using first quantization formalism.

Binding energies of the CH3O radical on hexagonal water ice (Ih) and amorphous solid water (ASW) were calculated using the ONIOM(QM:MM) method. A range of binding energies is found (0.10-0.50 eV), and the average binding energy is 0.32 eV. The CH3O radical binding on the ASW surfaces is stronger than on the Ih surfaces. The computed binding energies from the ONIOM(wB97X-D/def2-TZVP:AMBER) and wB97X-D/def2-TZVP methods agree quite well. Therefore, the ONIOM(QM:MM) method is expected to give accurate binding energies at a low computational cost. Binding energies from the ONIOM(wB97X-D/def2-TZVP:AMBER) and ONIOM(wB97X-D/def2-TZVP:AMOEBA09) methods differ noticeably, indicating that the choice of force field matters. According to the energy decomposition analysis, the electrostatic interactions and Pauli repulsions between the CH3O radical and ice play a crucial role in the binding energy. This study gives quantitative insights into the CH3O radical binding on interstellar ices.

Indefinite causal order has introduced disruptive procedures to improve the fidelity of quantum communication by introducing the superposition of { orders} on a set of quantum channels. It has been applied to several well characterized quantum channels as depolarizing, dephasing and teleportation. This work analyses the behavior of a parametric quantum channel for single qubits expressed in the form of Pauli channels. Combinatorics lets to obtain affordable formulas for the analysis of the output state of the channel when it goes through a certain imperfect quantum communication channel when it is deployed as a redundant application of it under indefinite causal order. In addition, the process exploits post-measurement on the associated control to select certain components of transmission. Then, the fidelity of such outputs is analysed to characterize the generic channel in terms of its parameters. As a result, we get notable enhancement in the transmission of information for well characterized channels due to the combined process: indefinite causal order plus post-measurement.

The modern state of the Pauli Exclusion Principle (PEP) is discussed. PEP can be considered from two viewpoints. On the one hand, it asserts that particles with half-integer spin (fermions) are described by antisymmetric wave functions, and particles with integer spin (bosons) are described by symmetric wave functions. This is the so-called spin-statistics connection (SSC). As we will discuss, the physical reasons why SSC exists are still unknown. On the other hand, according to PEP, the permutation symmetry of the total wave functions can be only of two types: symmetric or antisymmetric, both belong to one-dimensional representations of the permutation group, all other types of permutation symmetry are forbidden; whereas the solution of the Schrodinger equation may have any permutation symmetry. It is demonstrated that the proof in some textbooks on quantum mechanics that only symmetric and antisymmetric states can exist is wrong. However, the scenarios, in which arbitrary permutation symmetry (degenerate permutation states) is permitted, lead to contradictions with the concepts of particle identity and their independence. Thus, the existence in our Nature of particles only in nondegenerate permutation states (symmetric and antisymmetric) is not accidental and so-called symmetrization postulate should not be considered as a postulate, since all other symmetry options for the total wave function may not be realized. From this an important conclusion follows: we may not expect that in the future some unknown elementary particles can be discovered that are not fermions or bosons.

We investigate the propagation characteristics of nonlinear kinetic Alfven waves in quantum plasma with effects of Pauli paramagnetism and degeneracy temperature. In an arbitrary amplitude regime, we derive the energy integral equation by employing the quantum magnetohydrodynamic model. From the graphical analysis, it is found that the width of soliton structure decreases by enhancing the value of factor $$\chi \left[ =T_\mathrm{e}/T_\mathrm{F}\right] $$ for nearly degenerate plasma state and increases by increasing the value of factor $$z\left[ =\exp (\mu /k_\mathrm{B}T_\mathrm{e})\right] $$ for nearly non-degenerate plasma state, while there is no change in its amplitude. Moreover, it is also revealed that by changing the value of magnetic field strength H the width of the nonlinear structure is modified more in case of nearly degenerate plasma state as compared to a nearly non-degenerate plasma state with no change in its amplitude. The importance of the present work to astrophysical plasmas such as white dwarfs and neutron stars is also highlighted.