Symmetry Breaking Solution Research Articles

Revisiting the Mössbauer Isomer Shifts of the FeMoco Cluster of Nitrogenase and the Cofactor Charge.

Despite decades of research, the structure-activity relationship of nitrogenase is still not understood. Only recently was the full molecular structure of the FeMo cofactor (FeMoco) revealed, but the charge and metal oxidation states of FeMoco have been controversial. With the recent identification of the interstitial atom as a carbide and the more recent revised oxidation-state assignment of the molybdenum atom as Mo(III), here we revisit the Mössbauer properties of FeMoco. By a detailed error analysis of density functional theory-computed isomer shifts and computing isomer shifts relative to the P-cluster, we find that only the charge of [MoFe7S9C]1- fits the experimental data. In view of the recent Mo(III) identification, the charge of [MoFe7S9C]1- corresponds to a formal oxidation-state assignment of Mo(III)3Fe(II)4Fe(III), although due to spin delocalization, the physical oxidation state distribution might also be interpreted as Mo(III)1Fe(II)4Fe(2.5)2Fe(III), according to a localized orbital analysis of the MS = 3/2 broken symmetry solution. These results can be reconciled with the recent spatially resolved anomalous dispersion study by Einsle et al. that suggests the Mo(III)3Fe(II)4Fe(III) distribution, if some spin localization (either through interactions with the protein environment or through vibronic coupling) were to take place.

Symmetry-Breaking Global Bifurcation in a Surface Continuum Phase-Field Model for Lipid Bilayer Vesicles

We study a model for lipid bilayer membrane vesicles exhibiting phase separation, incorporating a phase field together with membrane fluidity and bending elasticity. We prove the existence of a plethora of equilibria in the large, corresponding to symmetry-breaking solutions of the Euler--Lagrange equations, via global bifurcation from the spherical state. To the best of our knowledge, this constitutes the first rigorous existence results for this class of problems. We overcome several difficulties in carrying this out. Due to inherent surface fluidity combined with finite curvature elasticity, neither the Eulerian (spatial) nor the Lagrangian (material) description of the model lends itself well to analysis. This is resolved via a singularity-free radial-map description, which effectively eliminates the grossly underdetermined midplane deformation. The resulting governing equations comprise a quasi-linear elliptic system with nonlinear constraints. We show the equivalence of the problem to that of findin...

Bifurcation for a free-boundary tumor model with angiogenesis

We consider a free boundary tumor model with vasculature which supply nutrients to the tumor, so that ∂σ∂n+β(σ−σ¯)=0 holds on the boundary, where a positive constant β is the rate of nutrient supply to the tumor and σ¯ is the nutrient concentration outside the tumor. The tumor cells proliferate at a rate μ. We show that for each μ2,μ4,μ6,μ8,⋯, symmetry-breaking stationary solutions bifurcate from the radially symmetric stationary solutions.

Polarizabilities of π-Conjugated Chains Revisited: Improved Results from Broken-Symmetry Range-Separated DFT and New CCSD(T) Benchmarks.

We present a detailed analysis of nonempirically tuned range-separated functionals, with both short- and long-range exchange, for calculating the static linear polarizability and second hyperpolarizabilities of various polydiacetylene (PDA) and polybutatriene (PBT) oligomers. Contrary to previous work on these systems, we find that the inclusion of some amount of short-range exchange does improve the accuracy of the computed polarizabilities and second hyperpolarizabilities. Most importantly, in contrast to prior studies on these oligomers, we find that the lowest-energy electronic states for PBT are not closed-shell singlets, and enhanced accuracy with range-separated DFT can be obtained by allowing the system to relax to a lower-energy broken-symmetry solution. Both the computed polarizabilities and second hyperpolarizabilities for PBT are significantly improved with these broken-symmetry solutions when compared to previously published and current benchmarks. In addition to these new analyses, we provide new large-scale CCSD(T) and explicitly correlated CCSD(T)-F12 benchmarks for the PDA and PBT systems, which comprise the most complete and accurate calculations of linear polarizabilities and second hyperpolarizabilities on these systems to date. These new CCSD(T) and CCSD(T)-F12 benchmarks confirm our DFT results and emphasize the importance of broken-symmetry effects when calculating polarizabilities and hyperpolarizabilties of π-conjugated chains.

Following the evolution of glassy states under external perturbations: the full replica symmetry breaking solution

.The state-following technique allows the study of metastable glassy states under external perturbations. Here we show how this construction can be used to study the behavior of glassy states of Hard Spheres in infinite dimensions under compression or shear strain. In Rainone et al (2015 Phys. Rev. Lett. 114 015701) it has been shown that in both cases, when the external perturbation is sufficiently strong, glassy states undergo a second-order transition, called the Gardner transition, whereupon a hierarchical structure of marginal micro-states manifests within the original glass state. The purpose of this work is to study the solution of the state-following construction in this marginal phase. We show that upon compression, close to the jamming transition, the metastable states are described by a scaling solution characterized by a set of non-trivial critical exponents that agree with the results of Charbonneau et al (2014 Nat. Commun. 5 3725), and we compute the value of the jamming density for various glassy states. Moreover we show that under the action of the shear strain, beyond the Gardner point, the metastable states can be followed in the marginal phase and we detect an overshoot in the stress-strain curve in agreement with numerical and experimental observations. Finally we further characterize the Gardner transition point by computing both the susceptibility and the exponent parameter λ that characterize the critical slowing down of the dynamics within a glassy state close to the transition.

Improved approaches to the exact solution of the machine covering problem

For the basic problem of scheduling a set of n independent jobs on a set of m identical parallel machines with the objective of maximizing the minimum machine completion time--also referred to as machine covering--we propose a new exact branch-and-bound algorithm. Its most distinctive components are a different symmetry-breaking solution representation, enhanced lower and upper bounds, and effective novel dominance criteria derived from structural patterns of optimal schedules. Results of a comprehensive computational study conducted on benchmark instances attest to the effectiveness of our approach, particularly for small ratios of n to m.

The Ground State of Monolayer Graphene in a Strong Magnetic Field.

Experiments indicate that the ground state of graphene in a strong magnetic field exhibits spontaneous breaking of SU(4) symmetry. However, the nature of the corresponding emergent state is unclear because existing theoretical methods approximate the broken-symmetry solutions, yielding nearly-degenerate candidate ground states having different emergent orders. Resolving this ambiguity in the nature of the strong-field ground state is highly desirable, given the importance of graphene for both fundamental physics and technical applications. We have discovered a new SO(8) symmetry that recovers standard graphene SU(4) quantum Hall physics, but predicts two new broken-SU(4) phases and new properties for potential ground states. Our solutions are analytical; thus we capture the essential physics of spontaneously-broken SU(4) states in a powerful yet solvable model useful both in correlating existing data and in suggesting new experiments.

Quantum Chemical Investigation of Light-Activated Spin State Change in Pyrene Coupled to Oxoverdazyl Radical Center.

Low-spin ground states and low-lying excited states of higher spin were investigated for four pyrene oxoverdazyl monoradicals 1-4 and eight pyrene dioxoverdazyl diradicals 5-12. The ground states for quartet and quintet spin symmetries that are in reality excited states were found in the region of 565-775 nm above the respective electronic ground states. We calculated the "adiabatic" magnetic exchange coupling constant in the electronic ground state of each isolated biradical (5-12) by unrestricted density functional theory. A number of hybrid functionals such as B3LYP, PBE0, M06, and M06-2X were used. We also used range-separated functionals such as LC-ωPBE and ωB97XD to compare their effects on the coupling constant and the relative energy of the high-spin state. Molecular geometries were optimized for the doublet and quartet spin states of every monoradical (1-4), and the broken symmetry and triplet solutions were optimized for every biradical (5-12), by systematically using 6-311G, 6-311G(d,p), and 6-311++G(d,p) basis sets with each functional. The geometry of each quintet diradical (5-12) was optimized using 6-311G basis set. B3LYP produced the best spin values. The excited state (quartet or quintet)-ground state energy difference (ΔE) increases in the presence of para-phenylene connectors. These energy differences were predicted here. The nature of spin coupling and consequently the ground state spin agree with spin alternation rule and the calculated atomic spin population. The adiabatic coupling constants were predicted for the biradicals (5-12) in their electronic ground states. Electron paramagnetic resonance parameters were determined at 6-311++G** level for the ground state and the quartet state of 1 and compared with the available experimental data. Low-lying excited states were found for the radical center (oxoverdazyl), pyrene, molecule 1, and diradical 5 by time-dependent density functional theory (TDDFT) method using B3LYP hybrid, 6-311++G(d,p) basis set, and the molecular geometry in the electronic ground state. Data from these calculations were used to discuss possible mechanisms for the achievement of the high-spin (excited) states in 1 and 5 and to predict a similar outcome for radicals 2-4 and 6-12 upon excitation. A comprehensive mechanism for the first excitation is proposed here. In particular, we show that the initial excitation of 1 involves large contributions from mixed transitions between pyrene and oxoverdazyl moieties, whereas the initial excitation of 5 is basically that of only the pyrene fragment. Subsequent internal conversion and intersystem crossing are likely to lead to the high-spin states of lower energy. Sample spin-flip TDDFT calculations were also done to confirm the energetic location and composition of the quartet state of 1 and the quintet state of 5.

Handling Magnetic Coupling in Trinuclear Cu(II) Complexes.

The problem of deriving three different two-body magnetic couplings in three electrons/three centers in a general geometric arrangement is investigated using the trinuclear Cu(II) HAKKEJ complex as a real case example. In these systems, one quartet and two doublet low lying electronic states exist, which define the magnetic spectra. However, the two possible linearly independent energy differences do not provide enough information to extract the three magnetic coupling constants. Here, we show how to obtain these parameters without making any assumption on the symmetry of the system from a combination of density functional- and wave function-based calculations. The density functional calculations explore various broken symmetry solutions and relate the corresponding energy to the expectation value of the Heisenberg Hamiltonian. This allows one to obtain all magnetic couplings, although their magnitude strongly depends on the exchange-correlation functional. Interestingly, a constant ratio between the magnetic coupling constants along a series of investigated functionals is found. This provides an additional equation to be used when relying on energy differences between spin states, which in turn allow solving the Heisenberg spectrum. The magnetic couplings thus obtained are compared to the experiment. Implications for the appropriate interpretation of the experiment and for the study of more complex systems are discussed.

The Cu2O2 torture track for a real-life system: [Cu2(btmgp)2O2](2+) oxo and peroxo species in density functional calculations.

Density functional theory (DFT) calculations of the equilibrium geometry, vibrational modes, ionization energies, electron affinities, and optical response of [Cu2(btmgp)2(μ-O)2](2+) (oxo) and [Cu2(btmgp)2(μ-η(2):η(2)-O2)](2+) (peroxo) are presented. Comprehensive benchmarking shows that the description of the oxo-peroxo energetics is still a torture track for DFT, but finds the molecular geometry to be comparatively robust with respect to changes in the exchange-correlation functionals and basis sets. Pure functionals favor the oxo core found experimentally, whereas hybrid functionals shift the bias toward the peroxo core. Further stabilization of peroxo core results from relaxing the spin degrees of freedom using the broken-symmetry (BS) approach. Dispersion effects, conversely, tend to favor the oxo configuration. Triple-zeta basis sets are found to represent a sensible compromise between numerical accuracy and computational effort. Particular attention is paid to the modification of the electronic structure, optical transitions, and excited-state energies along the transition path between the oxo and peroxo species. The excited-state potential energy surface calculations indicate that two triplet states are involved in the transition that stabilize the BS solution. Charge decomposition and natural transition orbital analyses are used for obtaining microscopic insight into the molecular orbital interactions. Here, the crucial role of guanidine π-interactions is highlighted for the stabilization of the Cu2O2 core.

Selection of active spaces for multiconfigurational wavefunctions.

The efficient and accurate description of the electronic structure of strongly correlated systems is still a largely unsolved problem. The usual procedures start with a multiconfigurational (usually a Complete Active Space, CAS) wavefunction which accounts for static correlation and add dynamical correlation by perturbation theory, configuration interaction, or coupled cluster expansion. This procedure requires the correct selection of the active space. Intuitive methods are unreliable for complex systems. The inexpensive black-box unrestricted natural orbital (UNO) criterion postulates that the Unrestricted Hartree-Fock (UHF) charge natural orbitals with fractional occupancy (e.g., between 0.02 and 1.98) constitute the active space. UNOs generally approximate the CAS orbitals so well that the orbital optimization in CAS Self-Consistent Field (CASSCF) may be omitted, resulting in the inexpensive UNO-CAS method. A rigorous testing of the UNO criterion requires comparison with approximate full configuration interaction wavefunctions. This became feasible with the advent of Density Matrix Renormalization Group (DMRG) methods which can approximate highly correlated wavefunctions at affordable cost. We have compared active orbital occupancies in UNO-CAS and CASSCF calculations with DMRG in a number of strongly correlated molecules: compounds of electronegative atoms (F2, ozone, and NO2), polyenes, aromatic molecules (naphthalene, azulene, anthracene, and nitrobenzene), radicals (phenoxy and benzyl), diradicals (o-, m-, and p-benzyne), and transition metal compounds (nickel-acetylene and Cr2). The UNO criterion works well in these cases. Other symmetry breaking solutions, with the possible exception of spatial symmetry, do not appear to be essential to generate the correct active space. In the case of multiple UHF solutions, the natural orbitals of the average UHF density should be used. The problems of the UNO criterion and their potential solutions are discussed: finding the UHF solutions, discontinuities on potential energy surfaces, and inclusion of dynamical electron correlation and generalization to excited states.

Explicit generation of the branching tree of states in spin glasses

We present a numerical method to generate explicit realizations of the tree of states in mean-field spin glasses. The resulting study illuminates the physical meaning of the full replica symmetry breaking solution and provides detailed information on the structure of the spin-glass phase. A cavity approach ensures that the method is self-consistent and permits the evaluation of sophisticated observables, such as correlation functions. We include an example application to the study of finite-size effects in single-sample overlap probability distributions, a topic that has attracted considerable interest recently.

On the NP-completeness of the Hartree-Fock method for translationally invariant systems.

The self-consistent field method utilized for solving the Hartree-Fock (HF) problem and the closely related Kohn-Sham problem is typically thought of as one of the cheapest methods available to quantum chemists. This intuition has been developed from the numerous applications of the self-consistent field method to a large variety of molecular systems. However, as characterized by its worst-case behavior, the HF problem is NP-complete. In this work, we map out boundaries of the NP-completeness by investigating restricted instances of HF. We have constructed two new NP-complete variants of the problem. The first is a set of Hamiltonians whose translationally invariant Hartree-Fock solutions are trivial, but whose broken symmetry solutions are NP-complete. Second, we demonstrate how to embed instances of spin glasses into translationally invariant Hartree-Fock instances and provide a numerical example. These findings are the first steps towards understanding in which cases the self-consistent field method is computationally feasible and when it is not.

CCSD(T) calculations of confined systems: in-crystal polarizabilities of F⁻, Cl⁻, O²⁻, and S²⁻.

We explore dipole polarizabilities of the singly and doubly charged anions F(-), Cl(-), O(2-), and S(2-) in an external, harmonic oscillator (HO) confining potential ∑(i)½ω(2)r(i)(2). We find that in contrast to F(-) and Cl(-) those for O(2-) and S(2-) are unrealistically high due to the instability of the corresponding restricted Hartree-Fock (RHF) solutions. Yet, already a relatively weak HO confining potential stabilizes their RHF solutions and eliminates any possible broken-symmetry solutions. The coupled-cluster theory with single, double and noniterative triple excitations (CCSD(T)) then yields considerably reduced polarizabilities for O(2-) and S(2-) relative to their unconfined values. We showed that polarizabilities of O(2-) and S(2-) are more sensitive to the strength of a confinement potential than are those for F(-) and Cl(-). This enables us to relate the confining parameter ω with the known experimental polarizabilities for selected crystals (our "training set") and to find a specific confining parameter ω for which the CCSD(T) polarizability equals the experimental in-crystal polarizability of an anion in the training set. The latter may then be used as an alternative approach for determining the in-crystal polarizabilities of anions by exploiting the fact that the characteristic ω values depend linearly on the ionic radius of a cation participating in specific crystals containing these anions. Using this method we then calculate the isotropic dipole polarizabilities for F(-), Cl(-), O(2-), and S(2-) embedded in the LiF, LiCl, NaF, NaCl, KF, KCl, ZnO, ZnS, MgO, MgS, CaO, CaS, SrO, SrS, BaO, BaS, and other crystals containing halogen, oxygen, or sulphur anions. We compare our results with those obtained via alternative models of the in-crystal anionic polarizabilities.

Exact theory of dense amorphous hard spheres in high dimension. III. The full replica symmetry breaking solution

.In the first part of this paper, we derive the general replica equations that describe infinite-dimensional hard spheres at any level of replica symmetry breaking (RSB) and in particular in the fullRSB scheme. We show that these equations are formally very similar to the ones that have been derived for spin glass models, thus showing that the analogy between spin glasses and structural glasses conjectured by Kirkpatrick, Thirumalai and Wolynes is realized in a strong sense in the mean-field limit. We also suggest how the computation could be generalized in an approximate way to finite-dimensional hard spheres. In the second part of the paper, we discuss the solution of these equations and we derive from it a number of physical predictions. We show that, below the Gardner transition where the 1RSB solution becomes unstable, a fullRSB phase exists and we locate the boundary of the fullRSB phase. Most importantly, we show that the fullRSB solution predicts correctly that jammed packings are isostatic, and allows one to compute analytically the critical exponents associated with the jamming transition, which are missed by the 1RSB solution. We show that these predictions compare very well with numerical results.

Experimental investigation of chimera states with quiescent and synchronous domains in coupled electronic oscillators.

Chimera states, that is, dynamical regimes characterized by the existence of a symmetry-broken solution where a coherent domain and an incoherent one coexist, have been theoretically demonstrated and numerically found in networks of homogeneously coupled identical oscillators. In this work we experimentally investigate the behavior of a closed and an open chain of electronic circuits with neuron-like spiking dynamics and first neighbor connections. Experimental results show the onset of a regime that we call chimera states with quiescent and synchronous domains, where synchronization coexists with spatially patterned oscillation death. The whole experimental bifurcation scenario, showing how disordered states, synchronization, chimera states with quiescent and synchronous domains, and oscillatory death states emerge as coupling is varied, is presented.

Shear modulus of glasses: results from the full replica-symmetry-breaking solution.

We compute the shear modulus of amorphous hard and soft spheres, using the exact solution in infinite spatial dimensions that has been developed recently. We characterize the behavior of this observable in the whole phase diagram, and in particular around the glass and jamming transitions. Our results are consistent with other theoretical approaches, which are unified within this general picture, and they are also consistent with numerical and experimental results. Furthermore, we discuss some properties of the out-of-equilibrium dynamics after a deep quench close to the jamming transition, and we show that a combined measure of the shear modulus and of the mean square displacement allows one to probe experimentally the complex structure of phase space predicted by the full replica-symmetry-breaking solution.

Gerald Stanford Guralnik

Gerald Stanford Guralnik

Multi-Species Mean Field Spin Glasses. Rigorous Results

We study a multi-species spin glass system where the density of each species is kept fixed at increasing volumes. The model reduces to the Sherrington–Kirkpatrick one for the single species case. The existence of the thermodynamic limit is proved for all density values under a convexity condition on the interaction. The thermodynamic properties of the model are investigated and the annealed, the replica-symmetric and the replica symmetry breaking bounds are proved using Guerra’s scheme. The annealed approximation is proved to be exact under a high-temperature condition. We show that the replica-symmetric solution has negative entropy at low temperatures. We study the properties of a suitably defined replica symmetry breaking solution and we optimize it within a novel ziggurat ansatz. The generalized order parameter is described by a Parisi-like partial differential equation.

An efficient initial guess formation of broken-symmetry solutions by using localized natural orbitals

A novel procedure to generate initial broken-symmetry solutions is proposed. Conventional methods for the initial broken-symmetry solutions are the MO alter, HOMO–LUMO mixing and fragment methods. These procedures, however, are quite complex. Our new approach is efficient, automatic and highly practical especially for large QM systems. This approach, called the LNO method, is applied to the following four typical open-shell systems: H2, dicarbene and two iron–sulfur clusters of Rieske-type [2Fe–2S] and [4Fe–4S]. The performance and the efficiency as an electronic structural analysis are discussed. The LNO method will be applicable for general systems in the complicated broken symmetry states.