Abstract: We introduce bicircular high-harmonic spectroscopy as a new method to probe dynamical symmetries of atoms and molecules and their evolution in time. Our approach is based on combining a circularly polarized femtosecond fundamental field of frequency ω with its counterrotating second harmonic 2ω. We demonstrate the ability of bicircular high-harmonic spectroscopy to characterize the orbital angular momentum symmetry of atomic orbitals. We further show that breaking the threefold rotational symmetry of the generating medium-at the level of either the ensemble or that of a single molecule-results in the emission of the otherwise parity-forbidden frequencies 3qω (q∈N), which provide a background-free probe of dynamical molecular symmetries.

Abstract: We report the quantitative assessment of nuclear quantum effects on the strength of a single hydrogen bond formed at a water-salt interface, using tip-enhanced inelastic electron tunneling spectroscopy based on a scanning tunneling microscope. The inelastic scattering cross section was resonantly enhanced by "gating" the frontier orbitals of water via a chlorine-terminated tip, so the hydrogen-bonding strength can be determined with high accuracy from the red shift in the oxygen-hydrogen stretching frequency of water. Isotopic substitution experiments combined with quantum simulations reveal that the anharmonic quantum fluctuations of hydrogen nuclei weaken the weak hydrogen bonds and strengthen the relatively strong ones. However, this trend can be completely reversed when a hydrogen bond is strongly coupled to the polar atomic sites of the surface.

Abstract: The computational bottleneck of the extremely fast simplified Tamm-Dancoff approximated (sTDA) time-dependent density functional theory procedure [S. Grimme, J. Chem. Phys. 138, 244104 (2013)] for the computation of electronic spectra for large systems is the determination of the ground state Kohn-Sham orbitals and eigenvalues. This limits such treatments to single structures with a few hundred atoms and hence, e.g., sampling along molecular dynamics trajectories for flexible systems or the calculation of chromophore aggregates is often not possible. The aim of this work is to solve this problem by a specifically designed semi-empirical tight binding (TB) procedure similar to the well established self-consistent-charge density functional TB scheme. The new special purpose method provides orbitals and orbital energies of hybrid density functional character for a subsequent and basically unmodified sTDA procedure. Compared to many previous semi-empirical excited state methods, an advantage of the ansatz is that a general eigenvalue problem in a non-orthogonal, extended atomic orbital basis is solved and therefore correct occupied/virtual orbital energy splittings as well as Rydberg levels are obtained. A key idea for the success of the new model is that the determination of atomic charges (describing an effective electron-electron interaction) and the one-particle spectrum is decoupled and treated by two differently parametrized Hamiltonians/basis sets. The three-diagonalization-step composite procedure can routinely compute broad range electronic spectra (0-8 eV) within minutes of computation time for systems composed of 500-1000 atoms with an accuracy typical of standard time-dependent density functional theory (0.3-0.5 eV average error). An easily extendable parametrization based on coupled-cluster and density functional computed reference data for the elements H-Zn including transition metals is described. The accuracy of the method termed sTDA-xTB is first benchmarked for vertical excitation energies of open- and closed-shell systems in comparison to other semi-empirical methods and applied to exemplary problems in electronic spectroscopy. As side products of the development, a robust and efficient valence electron TB method for the accurate determination of atomic charges as well as a more accurate calculation scheme of dipole rotatory strengths within the Tamm-Dancoff approximation is proposed.

Abstract: Fundamental understanding of support effects and metal–support interaction is critical in heterogeneous catalysis. In this work, theoretical investigations are carried out to study the nature of support effects of different tetravalent-metal dioxides of MO2 (M = Ti, Zr, Ce, Hf, Th) in single-atom gold catalysts using density functional theory with on-site Coulomb interactions (DFT+U). The properties of gold adatom on the stoichiometric (MO2) and reduced (MO2–x) surfaces as well as CO adsorption on Au1/MO2 and Au1/MO2–x have been investigated systematically. Our calculations indicate that the fundamental quantum primogenic effect that causes the radial contraction and low orbital energies of 3d and 4f orbitals in these MO2 oxides plays a vital role in determining the valence states and charge distribution of single-atom gold as well as the adsorption modes of CO on various MO2 supports. We find that gold atoms supported on different surfaces exhibit oxidation states from Au(−I) to Au(0) to Au(I), depending...

Abstract: The orbital degree of freedom plays a fundamental role in understanding the unconventional properties in solid state materials. Experimental progress in quantum atomic gases has demonstrated that high orbitals in optical lattices can be used to construct quantum emulators of exotic models beyond natural crystals, where novel many-body states such as complex Bose-Einstein condensates and topological semimetals emerge. A brief introduction of orbital degrees of freedom in optical lattices is given and a summary of exotic orbital models and resulting many-body phases is provided. Experimental consequences of the novel phases are also discussed.

Abstract: Despite recent advances, systematic quantitative treatment of the electron correlation problem in extended systems remains a formidable task. Systematically improvable Green’s function methods capable of quantitatively describing weak and at least qualitatively strong correlations appear as promising candidates for computational treatment of periodic systems. We present a periodic implementation of temperature-dependent self-consistent 2nd-order Green’s function (GF2) method, where the self-energy is evaluated in the basis of atomic orbitals. Evaluating the real-space self-energy in atomic orbitals and solving the Dyson equation in k-space are the key components of a computationally feasible algorithm. We apply this technique to the one-dimensional hydrogen lattice — a prototypical crystalline system with a realistic Hamiltonian. By analyzing the behavior of the spectral functions, natural occupations, and self-energies, we claim that GF2 is able to recover metallic, band insulating, and at least qualitatively Mott regimes. We observe that the iterative nature of GF2 is essential to the emergence of the metallic and Mott phases.

Abstract: A theory of above-threshold ionization of atoms by a strong laser field is formulated. Two versions of the strong-field approximation (SFA) are considered, the direct SFA and the improved SFA, which do not and do, respectively, take into account rescattering of the freed electron off the parent ion. The atomic bound state is included in two different ways: as an expansion in terms of Slater-type orbitals or as an asymptotic wave function. Even though we are using the single-active-electron approximation, multielectron effects are taken into account in two ways: by a proper choice of the ground state and by an adequate definition of the ionization rate. For the case of the asymptotic bound-state wave functions, using the saddle-point method, a simple expression for the $T$-matrix element is derived for both the direct and the improved SFA. The theory is applied to ionization by a bicircular field, which consists of two coplanar counterrotating circularly polarized components with frequencies that are integer multiples of a fundamental frequency $\ensuremath{\omega}$. Special emphasis is on the $\ensuremath{\omega}\text{\ensuremath{-}}2\ensuremath{\omega}$ case. In this case, the threefold rotational symmetry of the field carries over to the velocity map of the liberated electrons, for both the direct and the improved SFA. The results obtained are analyzed in detail using the quantum-orbit formalism, which gives good physical insight into the above-threshold ionization process. For this purpose, a specific classification of the saddle-point solutions is introduced for both the backward-scattered and the forward-scattered electrons. The high-energy backward-scattering quantum orbits are similar to those discovered for high-order harmonic generation. The short forward-scattering quantum orbits for a bicircular field are similar to those of a linearly polarized field. The conclusion is that these orbits are universal, i.e., they do not depend much on the shape of the laser field.

Abstract: In the context of high-accuracy computational thermochemistry, the valence coupled cluster with all singles and doubles (CCSD) correlation component of molecular atomization energies presents the most severe basis set convergence problem, followed by the (T) component. In the present paper, we make a detailed comparison, for an expanded version of the W4-11 thermochemistry benchmark, between, on the one hand, orbital-based CCSD/AV{5,6}Z + d and CCSD/ACV{5,6}Z extrapolation, and on the other hand CCSD-F12b calculations with cc-pVQZ-F12 and cc-pV5Z-F12 basis sets. This latter basis set, now available for H–He, B–Ne, and Al–Ar, is shown to be very close to the basis set limit. Apparent differences (which can reach 0.35 kcal/mol for systems like CCl4) between orbital-based and CCSD-F12b basis set limits disappear if basis sets with additional radial flexibility, such as ACV{5,6}Z, are used for the orbital calculation. Counterpoise calculations reveal that, while total atomization energies with V5Z-F12 basis sets are nearly free of BSSE, orbital calculations have significant BSSE even with AV(6 + d)Z basis sets, leading to non-negligible differences between raw and counterpoise-corrected extrapolated limits. This latter problem is greatly reduced by switching to ACV{5,6}Z core-valence basis sets, or simply adding an additional zeta to just the valence orbitals. Previous reports that all-electron approaches like HEAT (high-accuracy extrapolated ab-initio thermochemistry) lead to different CCSD(T) limits than “valence limit + CV correction” approaches like Feller-Peterson-Dixon and Weizmann-4 (W4) theory can be rationalized in terms of the greater radial flexibility of core-valence basis sets. For (T) corrections, conventional CCSD(T)/AV{Q,5}Z + d calculations are found to be superior to scaled or extrapolated CCSD(T)-F12b calculations of similar cost. For a W4-F12 protocol, we recommend obtaining the Hartree-Fock and valence CCSD components from CCSD-F12b/cc-pV{Q,5}Z-F12 calculations, but the (T) component from conventional CCSD(T)/aug’-cc-pV{Q,5}Z + d calculations using Schwenke’s extrapolation; post-CCSD(T), core-valence, and relativistic corrections are to be obtained as in the original W4 theory. W4-F12 is found to agree slightly better than W4 with ATcT (active thermochemical tables) data, at a substantial saving in computation time and especially I/O overhead. A W4-F12 calculation on benzene is presented as a proof of concept.

Abstract: Nitrogen-doped graphene quantum dots (N-GQDs) hold promising application in electronics and optoelectronics because of their excellent photo-stability, tunable photoluminescence and high quantum yield. However, the absorption and emission mechanisms have been debated for years. Here, by employing time-dependent density functional theory, we demonstrate that the different N-doping types and positions give rise to different absorption and emission behaviors, which successfully addresses the inconsistency observed in different experiments. Specifically, center doping creates mid-states, rendering non-fluorescence, while edge N-doping modulates the energy levels of excited states and increases the radiation transition probability, thus enhancing fluorescence strength. More importantly, the even hybridization of frontier orbitals between edge N atoms and GQDs leads to a blue-shift of both absorption and emission spectra, while the uneven hybridization of frontier orbitals induces a red-shift. Solvent effects on N-GQDs are further explored by the conductor-like screening model and it is found that strong polarity of the solvent can cause a red-shift and enhance the intensity of both absorption and emission spectra.

Abstract: Here, we characterize the three-orbital Hubbard model using state-of-the-art determinant quantum Monte Carlo (DQMC) simulations with parameters relevant to the cuprate high-temperature superconductors. The simulations find that doped holes preferentially reside on oxygen orbitals and that the (π,π) antiferromagnetic ordering vector dominates in the vicinity of the undoped system, as known from experiments. The orbitally-resolved spectral functions agree well with photoemission spectroscopy studies and enable identification of orbital content in the bands. A comparison of DQMC results with exact diagonalization and cluster perturbation theory studies elucidates how these different numerical techniques complement one another to produce a more complete understanding of the model and the cuprates. Interestingly, our DQMC simulations predict a charge-transfer gap that is significantly smaller than the direct (optical) gap measured in experiment. Most likely, it corresponds to the indirect gap that has recently been suggested to be on the order of 0.8 eV, and demonstrates the subtlety in identifying charge gaps.

Abstract: We present a global ground-state triplet potential energy surface for the N2O2 system that is suitable for treating high-energy vibrational-rotational energy transfer and collision-induced dissociation. The surface is based on multi-state complete-active-space second-order perturbation theory/minimally augmented correlation-consistent polarized valence triple-zeta electronic structure calculations plus dynamically scaled external correlation. In the multireference calculations, the active space has 14 electrons in 12 orbitals. The calculations cover nine arrangements corresponding to dissociative diatom-diatom collisions of N2, O2, and nitric oxide (NO), the interaction of a triatomic molecule (N2O and NO2) with the fourth atom, and the interaction of a diatomic molecule with a single atom (i.e., the triatomic subsystems). The global ground-state potential energy surface was obtained by fitting the many-body interaction to 54 889 electronic structure data points with a fitting function that is a permutationally invariant polynomial in terms of bond-order functions of the six interatomic distances.

Abstract: We have designed an energy decomposition analysis (EDA) to gain a deeper understanding of single chemical bonds, that is, those in which the interacting fragments are doublet open-shell systems but the supersystem is closed-shell. The method is a spin-pure extension of the absolutely localized molecular orbital (ALMO) EDA to the one-pair perfect pairing energy (equivalently to an active space of two electrons in two orbitals). The total interaction energy is broken up into four terms: frozen interactions, spin-coupling, polarization, and charge-transfer. A variety of single bonds are analyzed and, in addition, we use this method to show how solvation changes the nature of bonds, producing different results in the gas-phase and with explicit solvent molecules.

Abstract: In this work, we present a linear-response formalism of the complex two-component Hartree–Fock Hamiltonian that includes relativistic effects within the Douglas–Kroll–Hess and the Exact-Two-Component frameworks. The method includes both scalar and spin relativistic effects in the variational description of electronic ground and excited states, although it neglects the picture-change and explicit spin–orbit contributions arising from the two-electron interaction. An efficient direct formalism of solving the complex two-component response function is also presented in this work. The presence of spin–orbit couplings in the Hamiltonian and the two-component nature of the wave function and Fock operator allows the computation of excited-state zero-field splittings of systems for which relativistic effects are dominated by the one-electron term. Calculated results are compared to experimental reference values to assess the quality of the underlying approximations. The results show that the relativistic two-comp...

Abstract: A new perspective into correlation effects in electronically excited states is provided through quantum information theory. The entanglement between the electron and hole quasiparticles is examined, and it is shown that the related entanglement entropy can be computed from the eigenvalue spectrum of the well-known natural transition orbital (NTO) decomposition. Non-vanishing entanglement is obtained whenever more than one NTO pair is involved, i.e., in the case of a multiconfigurational or collective excitation. An important implication is that in the case of entanglement it is not possible to gain a complete description of the state character from the orbitals alone, but more specific analysis methods are required to decode the mutual information between the electron and hole. Moreover, the newly introduced number of entangled states is an important property by itself giving information about excitonic structure. The utility of the formalism is illustrated in the cases of the excited states of two interacting ethylene molecules, the conjugated polymer para-phenylene vinylene, and the naphthalene molecule.

Abstract: The electronic and magnetic properties of n- and p-type impurities by means of group V and VII atoms substituting sulfur in a MoS2 monolayer were investigated using first-principles methods based on density functional theory. Numerical results show that group V and VII atoms can induce magnetic properties, and the magnetic moment mainly originates from the dopant’s p orbital and neighbor Mo atoms’ d orbital. N-, P-, F-, and I-doped (or As-doped) MoS2 exhibits magnetic nanomaterial (or metallic) features, and Cl- and Br-doped systems exhibit half-metallic ferromagnetism (HMF). Moreover, the formation energy calculations also indicate that it is energetically favorable and relatively easier to incorporate group V and VII atoms into a MoS2 monolayer under Mo-rich experimental conditions. The formation energy of the F-doped system is the lowest, the next lowest formation energy is obtained in the Cl-doped system. By comparison, Cl-doped MoS2 is more suitable for spin injection. This finding is important for the achievement of spin devices on MoS2 nanostructures.

Abstract: An early triumph of quantum mechanics was the explanation of metallic and insulating behavior based on the filling of electronic bands. A complementary, classical picture of insulators depicts electrons as occupying localized and symmetric Wannier orbitals that resemble atomic orbitals. We report the theoretical discovery of band insulators for which electron filling forbids such an atomic description. We refer to them as filling-enforced quantum band insulators (feQBIs) because their wave functions are associated with an essential degree of quantum entanglement. Like topological insulators, which also do not admit an atomic description, feQBIs need spin-orbit coupling for their realization. However, they do not necessarily support gapless surface states. Instead, the band topology is reflected in the insulating behavior at an unconventional filling. We present tight binding models of feQBIs and show that they only occur in certain nonsymmorphic, body-centered cubic crystals.

Abstract: We present a simple and general formalism to compute efficiently the derivatives of a multi-determinant Jastrow-Slater wave function, the local energy, the interatomic forces, and similar quantities needed in quantum Monte Carlo. Through a straightforward manipulation of matrices evaluated on the occupied and virtual orbitals, we obtain an efficiency equivalent to algorithmic differentiation in the computation of the interatomic forces and the optimization of the orbital paramaters. Furthermore, for a large multi-determinant expansion, the significant computational gain recently reported for the calculation of the wave function is here improved and extended to all local properties in both all-electron and pseudopotential calculations.

Abstract: Recent theories of charge-density-wave (CDW) order in high-temperature superconductors have predicted a primarily d CDW orbital symmetry Here, we report on the orbital symmetry of CDW order in the canonical cuprate superconductors La1875Ba0125CuO4 (LBCO) and YBa2Cu3O667 (YBCO), using resonant soft X-ray scattering and a model mapped to the CDW orbital symmetry From measurements sensitive to the O sublattice, we conclude that LBCO has predominantly s' CDW orbital symmetry, in contrast to the d orbital symmetry recently reported in other cuprates Furthermore, we show for YBCO that the CDW orbital symmetry differs along the a and b crystal axes and that these both differ from LBCO This work highlights CDW orbital symmetry as an additional key property that distinguishes the different cuprate families We discuss how the CDW symmetry may be related to the '1/8-anomaly' and to static spin ordering

Abstract: We study orbital and spin-orbit proximity effects in graphene adsorbed to the Cu(111) surface by means of density functional theory (DFT). The proximity effects are caused mainly by the hybridization of graphene $\ensuremath{\pi}$ and copper $d$ orbitals. Our electronic structure calculations agree well with the experimentally observed features. We carry out a graphene-Cu(111) distance dependent study to obtain proximity orbital and spin-orbit coupling parameters, by fitting the DFT results to a robust low energy model Hamiltonian. We find a strong distance dependence of the Rashba and intrinsic proximity induced spin-orbit coupling parameters, which are in the meV and hundreds of $\ensuremath{\mu}\mathrm{eV}$ range, respectively, for experimentally relevant distances. The Dirac spectrum of graphene also exhibits a proximity orbital gap, of about 20 meV. Furthermore, we find a band inversion within the graphene states accompanied by a reordering of spin and pseudospin states, when graphene is pressed towards copper.

Abstract: Angular displacement from linear overlap of but a few degrees in the transition state of the enzyme- substrate complex has been postulated to be of great kinetic significance ("orbital steering"). The concept of orbital steering is shown to have evolved from the orienta- tion parameters of an equation previously proposed to evaluate the kinetic importance of propinquity. This equation is shown to be naive. Arguments provided against the concept of orbital steering include: (a) force constants predicted from orbital steering are about 100 times those experimentally determined from displacement of nuclei in a direction normal to the axis of a covalent bond (for example, at room temperature vibrational bending ampli- tudes of +5? or more are common); (b) because of the lessened directionality of orbitals containing nonbonded electron pairs, the force constants in transition states should be even smaller than in the case of a covalent bond; and (c) molecular orbital calculations predict shallow total energy minima for orbital alignment. The experi- mental rate data offered as a basis for the concept of orbital steering are shown to find rationalization in the previously observed dependence of AST on kinetic order and the energy requirements for the freezing-out of single bonds in the transition state leading to the formation of medium-size ring compounds from extended ground states. It is concluded that if orbital steering does exist, experi- mental and theoretical evidence to support this concept have yet to be presented. A task of the physical organic chemist has been one of provid- ing useful models for enzymatic processes. Any proposed model must be consistent with what we know to be true in terms of the fundamental theories and principles of basic physics and chemistry. In this paper we shall present a discus- sion of the implications of a recently proposed model, termed orbital steering (1), whose purpose is to explain the large en- hancement of the rates of a particular set of intramolecular reactions over their intermolecular counterparts. We will in addition present an alternative interpretation of the experi- mental observations in question, based on thermodynamics and reaction rate theory.

Abstract: An experiment towards the search for the interaction of the Schiff moment (S) of the (207)Pb nuclei with electrons in PbTiO3 crystal which violates the time reversal (T) and space parity (P) symmetries was proposed by Mukhamedjanov and Sushkov [Phys. Rev. A 72, 034501 (2005)]. The interpretation of the experiment in terms of the Schiff moment requires knowledge of an electronic density gradient parameter (usually designated as X) on the Pb nucleus in the crystal, which is determined by the electronic structure of the crystal. Here we propose a theoretical approach to calculate the properties in solids which are directly sensitive to the changes of valence electron densities in atomic cores but not in the valence spatial regions (Mossbauer parameters, hyperfine structure (HFS) constants, parameters of T,P-odd Hamiltonians, etc. [L. V. Skripnikov and A. V. Titov, Phys. Rev. A 91, 042504 (2015)]). It involves constructing the crystalline orbitals via the linear combination of atomic orbitals and employs a two-step concept of calculating such properties that was earlier proposed by us for the case of heavy-atom molecules. The application of the method to the PbTiO3 crystal results in the energy shift, Δe=0.82×10(6)S((207)Pb)eaB (3)eV, due to the T,P-odd interactions. The value is compared to the corresponding parameter in diatomic molecules (TlF, RaO, PbO), which have been proposed and used in the past decades in the search for the nuclear Schiff moment. We also present the calculation of the electric field gradient at the Pb nucleus in PbTiO3 for the comparison with other solid-state electronic structure approaches.

Abstract: For first-row transition metals, high-resolution 3d electronic structure information can be obtained using resonant inelastic X-ray scattering (RIXS). In the hard X-ray region, a K pre-edge (1s→3d) excitation can be followed by monitoring the dipole-allowed Kα (2p→1s) or Kβ (3p→1s) emission, processes labeled 1s2p or 1s3p RIXS. Here the restricted active space (RAS) approach, which is a molecular orbital method, is used for the first time to study hard X-ray RIXS processes. This is achieved by including the two sets of core orbitals in different partitions of the active space. Transition intensities are calculated using both first- and second-order expansions of the wave vector, including, but not limited to, electric dipoles and quadrupoles. The accuracy of the approach is tested for 1s2p RIXS of iron hexacyanides [Fe(CN)6]n− in ferrous and ferric oxidation states. RAS simulations accurately describe the multiplet structures and the role of 2p and 3d spin–orbit coupling on energies and selection rules. C...

Abstract: In quantum chemical computations the combination of Hartree–Fock or a density functional theory (DFT) approximation with relatively small atomic orbital basis sets of double-zeta quality is still widely used, for example, in the popular B3LYP/6-31G* approach. In this Review, we critically analyze the two main sources of error in such computations, that is, the basis set superposition error on the one hand and the missing London dispersion interactions on the other. We review various strategies to correct those errors and present exemplary calculations on mainly noncovalently bound systems of widely varying size. Energies and geometries of small dimers, large supramolecular complexes, and molecular crystals are covered. We conclude that it is not justified to rely on fortunate error compensation, as the main inconsistencies can be cured by modern correction schemes which clearly outperform the plain mean-field methods.

Abstract: Generally, the long-range electronic communication between spatially orthogonal orbitals is inefficient and limited to field and inductive effects. In this work, we provide experimental evidence that such communication can be achieved via intramolecular electron transfer between two degenerate and mutually orthogonal frontier molecular orbitals (MOs) at the transition state. Interaction between orthogonal orbitals is amplified when the energy gap between these orbitals approaches zero, or at an "orbital crossing". The crossing between two empty or two fully occupied MOs, which do not lead to stabilization, can be "activated" when one of the empty MOs is populated (i.e., electron injection) or one of the filled MOs is depopulated (i.e., hole injection). In reductive cycloaromatization reactions, such crossings define transition states with energies defined by both the in-plane and out-of-plane π-systems. Herein, we provide experimental evidence for the utility of this concept using orbital crossings in reductive C1-C5 cycloaromatization reactions of enediynes. Communication with remote substituents via orbital crossings greatly enhances regioselectivity of the ring closure step in comparison to the analogous radical cyclizations. We also present photophysical data pertaining to the efficiency of electron injection into the benzannelated enediynes.

Abstract: The analytical energy gradient and Hessian of the two-component Normalized Elimination of the Small Component (2c-NESC) method with regard to the components of the electric field are derived and used to calculate spin-orbit coupling (SOC) corrected dipole moments and dipole polarizabilities of molecules, which contain elements with high atomic number. Calculated 2c-NESC dipole moments and isotropic polarizabilities agree well with the corresponding four-component-Dirac Hartree–Fock or density functional theory values. SOC corrections for the electrical properties are in general small, but become relevant for the accurate prediction of these properties when the molecules in question contain sixth and/or seventh period elements (e.g., the SO effect for At2 is about 10% of the 2c-NESC polarizability). The 2c-NESC changes in the electric molecular properties are rationalized in terms of spin-orbit splitting and SOC-induced mixing of frontier orbitals with the same j = l + s quantum numbers.

Abstract: High-temperature superconductivity emerges upon doping a state of matter that is insulating because of interactions. A widely studied model considers one orbital per ${\mathrm{CuO}}_{2}$ unit cell on a square lattice with a strong intraorbital repulsion that leads to a so-called Mott-Hubbard insulator. Here we solve a model that takes into account, within each unit cell, two oxygen orbitals where there is no electron-electron repulsion and a copper orbital with strong electron-electron repulsion. The insulating phase is a so-called charge-transfer insulator, not a Mott-Hubbard insulator. Using cluster dynamical mean-field theory with continuous-time quantum Monte Carlo as an impurity solver and 12 atoms per cluster, we report the normal and superconducting phase diagram of this model as a function of doping, interaction strength, and temperature. As expected, the three-orbital model is consistent with the experimental observation that doped holes are located predominantly on oxygens, a result that goes beyond the one-orbital model. Nevertheless, the phase boundary between pseudogap and correlated metal, the Widom line, and the origin of the pairing energy (kinetic vs potential) are similar to the one-orbital model, demonstrating that these are emergent phenomena characteristic of doped Mott insulators, independently of many microscopic details. Broader implications are discussed.

Abstract: We derive an exact formula of orbital susceptibility expressed in terms of Bloch wave functions, starting from the exact one-line formula by Fukuyama in terms of Green's functions. The obtained formula contains four contributions: (1) Landau-Peierls susceptibility, (2) interband contribution, (3) Fermi surface contribution, and (4) contribution from occupied states. Except for the Landau-Peierls susceptibility, the other three contributions involve the crystal-momentum derivatives of Bloch wave functions. Physical meaning of each term is clarified. The present formula is simplified compared with those obtained previously by Hebborn et al. Based on the formula, it is seen first of all that diamagnetism from core electrons and Van Vleck susceptibility are the only contributions in the atomic limit. The band effects are then studied in terms of linear combination of atomic orbital treating overlap integrals between atomic orbitals as a perturbation and the itinerant feature of Bloch electrons in solids are clarified systematically for the first time.