Symmetry Breaking Solution Research Articles

Weak Antiferromagnetic Coupling via a Superexchange Interaction between Mn(II)–Mn(II) Ions: A QM/MM Study of the Active Site of Human Cytosolic X-Propyl Aminopeptidase P

We investigate the dinuclear manganese, Mn(II)-Mn(II), active site of human cytosolic X-propyl aminopeptidase (XPNPEP1) employing the QM/MM method. The optimized structure supports two manganese atoms at the active site and excludes the possibility of a single Mn(II) atom or other combination of divalent metal ions: Ca(II), Fe(II), Mg(II). A broken symmetry solution verifies an antiferromagnetically coupled state between the Mn(II)-Mn(II) pair, which is the ground state. From the energy difference between the high spin state (HS) and the broken symmetry state (BS), we estimate the exchange coupling constant, J, to be 5.15 cm(-1). Also, we observe multiple bridges (p orbitals) from solvent and two carboxylate linking to the Mn(II)-Mn(II), which leads to the weakly antiferromagnetic interaction of d(5)-d(5) electrons through superexchange coupling.

A Recipe for Geometry Optimization of Diradicalar Singlet States from Broken-Symmetry Calculations

The equilibrium geometries of the singlet and triplet states of diradicals may be somewhat different, which may have an influence on their magnetic properties. The single-determinantal methods, such as Hartree-Fock or Kohn-Sham density functional theory, in general rely on broken-symmetry solutions to approach the singlet-state energy and geometry. An approximate spin decontamination is rather easy for the energy of this state but is rarely performed for its geometry optimization. We suggest simple procedures to estimate the optimized geometry and energy of a spin-decontaminated singlet, the accuracies of which are tested on a few organic diradicals. This technique can be generalized to interactions between higher-spin units or to multispin systems.

Similarities of artificial photosystems by ruthenium oxo complexes and native water splitting systems

The nature of chemical bonds of ruthenium(Ru)-quinine(Q) complexes, mononuclear [Ru(trpy)(3,5-t-Bu(2)Q)(OH(2))](ClO(4))(2) (trpy = 2,2':6',2''-terpyridine, 3,5-di-tert-butyl-1,2-benzoquinone) (1), and binuclear [Ru(2)(btpyan)(3,6-di-Bu(2)Q)(2)(OH(2))](2+) (btpyan = 1,8-bis(2,2':6',2''-terpyrid-4'-yl)anthracene, 3,6-t-Bu(2)Q = 3,6-di-tert-butyl-1,2-benzoquinone) (2), has been investigated by broken-symmetry (BS) hybrid density functional (DFT) methods. BS DFT computations for the Ru complexes have elucidated that the closed-shell structure (2b) Ru(II)-Q complex is less stable than the open-shell structure (2bb) consisting of Ru(III) and semiquinone (SQ) radical fragments. These computations have also elucidated eight different electronic and spin structures of tetraradical intermediates that may be generated in the course of water splitting reaction. The Heisenberg spin Hamiltonian model for these species has been derived to elucidate six different effective exchange interactions (J) for four spin systems. Six J values have been determined using total energies of the eight (or seven) BS solutions for different spin configurations. The natural orbital analyses of these BS DFT solutions have also been performed in order to obtain natural orbitals and their occupation numbers, which are useful for the lucid understanding of the nature of chemical bonds of the Ru complexes. Implications of the computational results are discussed in relation to the proposed reaction mechanisms of water splitting reaction in artificial photosynthesis systems and the similarity between artificial and native water splitting systems.

Spin-spin and spin-orbit interactions in nanographene fragments: A quantum chemistry approach

The relativistic behavior of graphene structures, starting from the fundamental building blocks--the poly-aromatic hydrocarbons (PAHs) along with other PAH nanographenes--is studied to quantify any associated intrinsic magnetism in the triplet (T) state and subsequently in the ground singlet (S) state with account of possible S-T mixture induced by spin-orbit coupling (SOC). We employ a first principle quantum chemical-based approach and density functional theory (DFT) for a systematic treatment of the spin-Hamiltonian by considering both the spin-orbit and spin-spin interactions as dependent on different numbers of benzene rings. We assess these relativistic spin-coupling phenomena in terms of splitting parameters which cause magnetic anisotropy in absence of external perturbations. Possible routes for changes in the couplings in terms of doping and defects are also simulated and discussed. Accounting for the artificial character of the broken-symmetry solutions for strong spin polarization of the so-called "singlet open-shell" ground state in zigzag graphene nanoribbons predicted by spin-unrestricted DFT approaches, we interpolate results from more sophisticated methods for the S-T gaps and spin-orbit coupling (SOC) integrals and find that these spin interactions become weak as function of size and increasing decoupling of electrons at the edges. This leads to reduced electron spin-spin interaction and hence almost negligible intrinsic magnetism in the carbon-based PAHs and carbon nanographene fragments. Our results are in agreement with the fact that direct experimental evidence of edge magnetism in pristine graphene has been reported so far. We support the notion that magnetism in graphene only can be ascribed to structural defects or impurities.

Oscillatory bubbles induced by geometrical constraint

We show that a simple change in pore geometry can radically alter the behavior of a fluid-displacing air finger, indicating that models based on idealized pore geometries fail to capture key features of complex practical flows. In particular, partial occlusion of a rectangular cross section can force a transition from a steadily propagating centered finger to a state that exhibits spatial oscillations formed by periodic sideways motion of the interface at a fixed distance behind the moving finger tip. We characterize the dynamics of the oscillations, which suggest that they arise from a global homoclinic connection between the stable and unstable manifolds of a steady, symmetry-broken solution.

Continuation Along Bifurcation Branches for a Tumor Model with a Necrotic Core

We consider a free boundary problem for a system of partial differential equations, which arises in a model of tumor growth with a necrotic core. For any positive number R and 0<?<R, there exists a radially-symmetric stationary solution with tumor free boundary r=R and necrotic free boundary r=?. The system depends on a positive parameter μ, which describes tumor aggressiveness, and for a sequence of values μ 2<μ 3<?, there exist branches of symmetry-breaking stationary solutions, which bifurcate from these values. Upon discretizing this model, we obtain a family of polynomial systems parameterized by tumor aggressiveness factor μ. By continuously changing μ using a homotopy, we are able to compute nonradial symmetric solutions. We additionally discuss linear and nonlinear stability of such solutions.

Towards a low-spin configuration in extended metal atom chains. Theoretical study of trimetallic systems with 22 metal electrons

Different electronic configurations of a series of trinuclear heterometallic chains with 22 metallic electrons, MM'M(dpa)(4)X(2) (M = Co, Rh; M' = Ni, Pd; X = Cl, NCS), have been modelled in search of new systems with novel electrical properties. For this purpose, we explore the possibility of obtaining low-spin (extensively closed-shell) states by introducing chemical changes to the reference compound CoPdCo(dpa)(4)Cl(2) (1), isoelectronic to the herein studied systems, but possessing magnetically coupled localized electrons. The discussion is based on the orbital energies obtained by the DFT methodology. Among the systems herein analysed, CoNiCo(dpa)(4)(NCS)(2) has only two unpaired electrons vs. six in the case of 1, its closed-shell configuration appearing at high energies. For Rh(2)M-based chains, changes go a step further and the RhPdRh(dpa)(4)Cl(2) and RhPdRh(dpa)(4)(NCS)(2) molecules present a closed-shell ground state in close competition with the broken symmetry solution with S = ½ on each Rh(II). One-electron reduction of the latter compounds has been computed with marked structural changes. Our calculations show that the two lowest 23-electron states are separated by 7-8 kcal mol(-1) in favour of the state with an unpaired localized electron on the δ(Pd-N)* orbital instead of the delocalized one (σ(nb))(2)(σ*)(1).

A theoretical study on photomagnetic fluorescent protein chromophore coupled diradicals and their possible applications

We have designed and theoretically studied three different pairs of green fluorescent protein chromophores and their different homologue-based diradicals coupled with imino nitroxides. To begin with, the geometries of all these diradicals have been optimized at high spin (HS) state in the gas phase, in a water medium and in a blood plasma medium. The process of calculations is straightforward and well-established in the case of the gas phase. However, for calculations in water, we have adopted our own N-layer integrated molecular orbital and molecular mechanics (ONIOM) method. Similarly for the blood phase calculations, the polarized continuum model (PCM) method has been adopted. With these optimized geometries the magnetic exchange coupling constant (J) values are estimated for these diradicals in different media using the broken symmetry (BS) approach in an unrestricted DFT framework. In order to obtain the BS solutions in the ONIOM method, we have carried out ONIOM-BS, where the BS calculations are done for the inner high-level layer (diradical system) keeping the outer water layer at low level. In a similar fashion, a PCM-BS technique has also been adopted for the BS calculations in the PCM method. We have found that these diradicals have an ability to change their magnetic nature from antiferromagnetic in the trans form to ferromagnetic in the cis form upon irradiation of light with the appropriate wavelength. Using a time-dependent DFT (TDDFT) technique, the required wavelengths of light by which non-fluorescent dark trans diradicals turn into their corresponding bright fluorescent cis isomers are determined for each pair of diradicals for the gas and water media. This color change is indeed a signature of the change in magnetic state of the diradicals concerned. Here, we have also calculated the zero field splitting (ZFS) parameter (D), rhombic ZFS parameter (E) and ZFS magnitude (a2). From our calculations we ambitiously expect that if these diradicals are synthesized then they might be used as a successful, non-hazardous magnetic resonance imaging contrast agent (MRICA) in place of other metal-based contrast agents.

Full correspondence between asymmetric filling of slits and first-order phase transition lines

Adsorption on single planar walls and filling of slits with identical planar walls are investigated in the frame of the density functional theory. In this sort of slits the external potential is symmetric with respect to its central plane. Calculations were carried out by applying both the canonical and grand canonical ensembles (CE and GCE, respectively). The behavior is analyzed by varying the strength of the adsorbate-substrate attraction, the temperature T, and the coverage Γℓ. Results obtained for physisorption of Xe on alkaline surfaces are reported in the present work. Prewetting (PW) lines and wetting temperatures, Tw, are determined from the analysis of adsorption on single walls. The filling of slits is analyzed for temperatures T &amp;gt; Tw. It is found that whenever for a given Xe-substrate combination the adsorption on a single wall exhibits a first-order wetting transition then asymmetric profiles that break the left-right symmetry of the external potential appear in the filling of an equivalent slit. These spontaneously symmetry breaking (SSB) solutions occur in a restricted range of Γℓ with a T-dependent width. In the case of closed slits analyzed in the CE scheme, the obtained asymmetric profiles exhibit lower Helmholtz free energies than the symmetric species and, therefore, could be stabilized in this geometry. For open slits, the GCE scheme yields all the symmetric and SSB states in the corresponding convex regimes of the free energy. It is shown that both the CE and the GCE frames yield three coexistent states, two symmetric and one asymmetric twofold degenerate. Both a PW line and the related SSB effect terminate at the same temperature. For rather strongly attractive surfaces reentrant SSB states are found at a fixed value of T.

Replica analysis of the generalized p-spin interaction glass model

We investigate the stability of replica symmetry breaking solutions in generalized p-spin models. It is shown that the kind of the transition to the one-step replica symmetry breaking state depends not only on the presence or absence of the reflection symmetry of the generalized ‘spin’-operators but on the number of interacting operators and their individual characteristics.

Symmetry breaking in a T-shaped photonic waveguide coupled with two identical nonlinear cavities

We consider light transmission in a T-shaped photonic waveguide coupled with two identical symmetrically positioned nonlinear microcavities. We present two types of symmetry breaking. The first one is a result of mixing of the symmetric input wave with antisymmetric bound states in the Fabry-P\'erot interferometer architecture. Similarly, the second mechanism of the symmetry breaking is the result of mixing the symmetrical input wave with the antibonding bound state in a straight waveguide coupled with two cavities positioned perpendicular to the waveguide. In both cases the mixing is due to nonlinearity. In turn, the symmetry-breaking solutions give rise to nonsymmetrical outputs in the T-shape waveguide. These effects are directly demonstrated by the electromagnetic field solutions which are complimented by coupled mode theory for the light transmission.

Bifurcation for a free boundary problem modeling the growth of a tumor with a necrotic core

We consider a free boundary problem for a system of partial differential equations, which arises in a model of tumor growth with a necrotic core. For any positive numbers ρ < R , there exists a radially symmetric stationary solution with tumor boundary r = R and necrotic core boundary r = ρ . The system depends on a positive parameter μ , which describes the tumor aggressiveness. There also exists a sequence of values μ 2 < μ 3 < ⋯ for which branches of symmetry-breaking stationary solutions bifurcate from the radially symmetric solution branch.

Density functional study of multiplicity-changing valence and Rydberg excitations of p-block elements: Delta self-consistent field, collinear spin-flip time-dependent density functional theory (DFT), and conventional time-dependent DFT

A database containing 17 multiplicity-changing valence and Rydberg excitation energies of p-block elements is used to test the performance of density functional theory (DFT) with approximate density functionals for calculating relative energies of spin states. We consider only systems where both the low-spin and high-spin state are well described by a single Slater determinant, thereby avoiding complications due to broken-symmetry solutions. Because the excitations studied involve a spin change, they require a balanced treatment of exchange and correlation, thus providing a hard test for approximate density functionals. We test three formalisms for predicting the multiplicity-changing transition energies. First is the ΔSCF method; we also test time-dependent density functional theory (TDDFT), both in its conventional form starting from the low-spin state and in its collinear spin-flip form starting from the high-spin state. Very diffuse basis functions are needed to give a qualitatively correct description of the Rydberg excitations. The scalar relativistic effect needs to be considered when quantitative results are desired, and we include it in the comparisons. With the ΔSCF method, most of the tested functionals give mean unsigned errors (MUEs) larger than 6 kcal/mol for valence excitations and MUEs larger than 3 kcal/mol for Rydberg excitations, but the performance for the Rydberg states is much better than can be obtained with time-dependent DFT. It is surprising to see that the long-range corrected functionals, which have 100% Hartree-Fock exchange at large inter-electronic distance, do not improve the performance for Rydberg excitations. Among all tested density functionals, ΔSCF calculations with the O3LYP, M08-HX, and OLYP functionals give the best overall performance for both valence and Rydberg excitations, with MUEs of 2.1, 2.6, and 2.7 kcal/mol, respectively. This is very encouraging since the MUE of the CCSD(T) coupled cluster method with quintuple zeta basis sets is 2.0 kcal/mol; however, caution is advised since many popular density functionals give poor results, and there can be very significant differences between the ΔSCF predictions and those from TDDFT.

Switching through symmetry breaking for transmission in a T-shaped photonic waveguidecoupled with two identical nonlinear micro-cavities

Using coupled mode theory we consider transmission in a T-shaped waveguide coupled withtwo identical symmetrically positioned nonlinear micro-cavities with mirror symmetry. Forinput power injected into the central waveguide we show the existence of a symmetrybreaking solution which is a result of mixing of the symmetrical input wave with anantisymmetric standing wave in the Fabry–Pérot interferometer. With growth of the inputpower, a feature in the form of loops arises in the solution which originates from bistabilityin the transmission in the output left/right waveguide coupled with the first/secondnonlinear cavity. The domains of stability of the solution are found. The breaking of mirrorsymmetry gives rise to nonsymmetrical left and right outputs. We demonstrate thatthis phenomenon can be explored for all-optical switching of light transmissionfrom the left output waveguide to the right one by application of input pulses.

Symmetry and broken symmetry in molecular orbital description of unstable molecules IV: comparison between single- and multi-reference computational results for antiaromtic molecules

First principle calculations of the effective exchange integrals (J) in the Heisenberg model for diradical species are presented for both symmetry-adapted multi-reference (MR) and single-reference broken-symmetry (BS) methods. The Mukherjee-type state-specific MR coupled cluster singles and doubles (MkCCSD) method with several different reference orbitals including BS natural orbitals is used to calculate the singlet–triplet energy gaps (S–T energy gap or 2J) and diradical characters for the antiaromatic molecules [a cyclopropenyl anion (CPA), b cyclobutadiene (CBD), and c cyclopentadienyl cation (CPC)], the cyclobutadiene derivatives with polar substitutents [d aminocyclobutadiene (ACBD), e formylcyclobutadiene (FCBD), and f 1-amino-2-formyl-cyclobutadiene (AFCBD)] and finally the cyclobutadine derivatives with radical substitutents [g 1,2-bis(methylene)cyclobutadiene (1,2-BMCBD) and h 1,3-bis(methylene)cyclobutadiene (1,3-BMCBD)]. For the BS methods, the spin-unrestricted Hartree–Fock based CCSD (UHF-CCSD), the CCD with the spin-unrestricted Brueckner determinant (UBD), and BS density functional theory (UDFT) computations are performed. Comparison between MkCCSD and the UHF-CCSD results indicates that spin-contamination of UHF-CCSD solutions still remains. In comparison with UHF-CCSD, the UBD results show that spin-contamination involved in BS solutions is greatly suppressed. To eliminate the spin contamination, an approximate spin-projection (AP) scheme is applied to the BS solutions. The AP procedure with the use of the expectation value of the total-spin operator corresponding to UHF-CCSD and UBD results yields good agreement with the MkCCSD results. As for the AP correction of the UDFT methods, three different computational schemes for predicting the expectation value of the total-spin operator are examined. Systematic comparisons between these methods are presented for the S–T energy gaps (2J). Implications of the present computational results have been discussed in relation to the design of magnetic oligomers and polymers.

Learning from Stochastic Rules by Spherical Perceptrons under Finite Temperature –Optimal Temperature and Asymptotic Learning Curve–

In the problem of learning under external disturbance, there is a possibility that the existence of some tolerance or flexibility in the system weakens the effect of noise and helps the system to perform more efficiently. In a previous letter, we gave one example of such phenomena in learning from stochastic rules by spherical perceptrons adopting the Gibbs algorithm using statistical mechanical methods. By the replica method, we showed that, in the output noise model, there exists an optimal temperature at which the generalization error takes its minimum for the stable replica symmetric (RS) solution. On the other hand, for other types of noise including input noise, it was shown that no such temperature exists up to the one-step replica symmetry breaking (1RSB) solution. That is, it was shown that for the asymptotic region of a large number of training sets, the RS solution becomes unstable, and the asymptotic behavior is determined by the 1RSB solution, The asymptotic expressions for learning curves we...

Singlet–triplet energy gap for trimethylenemethane, oxyallyl diradical, and related species: single- and multireference computational results

We have investigated the through-bond exchange interactions in three non-Kekule hydrocarbon diradicals on the basis of single- and multireference coupled cluster and related broken-symmetry (BS) methods. The singlet–triplet energy gap (S-T gap) and diradical characters for these species are evaluated. It is found that the spin contamination involved in the BS solutions is non-negligible and the approximate spin-projection method greatly improves the usual BS solutions. As for Mukherjee’s state-specific multireference coupled cluster (MkMRCC) computations, the size-consistent correction with the UHF localized natural orbitals (ULO) is useful to obtain the qualitatively correct 2J values.

Sign-changing and symmetry-breaking solutions to singular problems

We consider the degenerate elliptic equation related to the Caffarelli–Kohn–Nirenberg inequality. We show that it possesses infinitely many solutions which are sign-changing and nonradial. The solutions are obtained by constrained minimization on subspaces consisting of functions which have certain prescribed symmetry properties. We also extend these results to higher order equations.

A spectral-Galerkin continuation method using Chebyshev polynomials for the numerical solutions of the Gross–Pitaevskii equation

We study an efficient spectral-Galerkin continuation method (SGCM) and two-grid centered difference approximations for the numerical solutions of the Gross–Pitaevskii equation (GPE), where the second kind Chebyshev polynomials are used as the basis functions for the trial function space. Some basic formulae for the SGCM are derived so that the eigenvalues of the associated linear eigenvalue problems can be easily computed. The SGCM is implemented to investigate the ground and first excited-state solutions of the GPE. Both the parabolic and quadruple-well trapping potentials are considered. We also study Bose–Einstein condensates (BEC) in optical lattices, where the periodic potential described by the sine or cosine functions is imposed on the GPE. Of particular interest here is the investigation of symmetry-breaking solutions. Sample numerical results are reported.

Multireference Character of 1,3-Dipolar Cycloaddition of Ozone with Ethylene and Acrylonitrile

In the present study, the concerted and stepwise reaction mechanisms for 1,3-dipole cycloaddition of ozone with ethylene (1) and acrylonitrile (2) are investigated. The stationary points are optimized by using four hybrid R(U)DFT methods. A geometry optimization method based on an approximate spin projection (AP-opt method) is applied to eliminate a spin contamination from the broken-symmetry (BS) solution. The AP-opt method reveals that a diradical intermediate for the stepwise pathway is spurious due to the spin contamination. The revised reaction profile with no diradical intermediate supports the stereospecificity. On the basis of the experimental data, the RCCSD(T) method outperforms AP-UCCSD(T), AP-UBD(T), and MkMRCCSD(4e,4o) for the systems, indicating that the RCCSD(T) method can describe the diradical character of ozone within a framework of single reference wave function. The subsequent single point energy calculations show that the highly synchronous transition state is much more favorable than the asynchronous one for 1. In the case of 2, there is not much difference between two transition states because of its asymmetric structure and charge separations in the transition states.