Radial rescaling approach for the eigenvalue problem of a particle in an arbitrarily shaped box

In the present work we introduce a methodology for solving a quantum billiard with Dirichlet boundary conditions. The procedure starts from the exactly known solutions for the particle in a circular disk, which are subsequently radially rescaled in such a way that they obey the new boundary conditions. In this way one constructs a complete basis set which can be used to obtain the eigenstates and eigenenergies of the corresponding quantum billiard to a high level of precision. Test calculations for several regular polygons show the efficiency of the method which often requires one or two basis functions to describe the lowest eigenstates with high accuracy.

Mechanisms of spin-mixing instabilities in antiferromagnetic molecular wheels

The microscopic theory of field-induced spin-mixing instabilities in antiferromagnetic molecular wheels CsFe8 is proposed. The basic features of magnetic torque measurements [O. Waldmann, Phys. Rev. Lett. 96, 027206 (2006)] are well explained by the interplay of three basic ingredients: the spin-mixing vibronic interaction with field-dependent vibronic constants, cooperative elastic interactions, and spin-mixing interactions independent from vibrations. The main contribution to spin mixing comes from second-order zero-field splitting mechanisms. At variance with previous interpretations, we find that the observed anomalies are not associated with a phase transition.

Effect of the metal environment on the ferromagnetic interaction in the Co-NC-W pairs of octacyanotungstate(V)-Cobalt(II) three-dimensional networks

State of the art CASSCF and CASPT2 calculations have been performed to elucidate the nature of ferromagnetism of CoII-NC-WV pairs in the three-dimensional compound [[WV(CN)2]2[(micro-CN)4CoII(H2O)2]3.4H2O]n, which has been recently synthesized and investigated by a number of experimental techniques (Herrera, J. M.; Bleuzen, A.; Dromzée, Y.; Julve, M.; Lloret, F.; Verdaguer, M. Inorg. Chem. 2003, 42, 7052-7059). In this network, the Co ions are in the high-spin (S = 3/2) state, while the single unpaired electron on the W centers occupies the lowest orbital of the dz2 type of the 5d shell. In agreement with the suggestion made by Herrera et al., we find that the ferromagnetism is due to a certain occupation scheme of the orbitals from the parent octahedral t2g shell on CoII sites, in which the orbital accommodating the unpaired electron is orthogonal to the dz2 orbitals of the surrounding W ions. We investigate the stabilization of such an orbital configuration on the Co sites and find that it cannot be achieved in the ground state of isolated mononuclear fragments [CoII(NC)4(OH2)2]2- for any conformations of the coordinated water molecules and Co-N-C bond angles. On the other hand, it is stabilized by the interaction of the complex with neighboring W ions, which are simulated here by effective potentials. The calculated exchange coupling constants for the CoII-NC-WV binuclear fragments are in reasonable agreement with the measured Curie-Weiss constant for this compound. As additional evidence for the inferred electronic configuration on the Co sites, the ligand-field transitions, the temperature-dependent magnetic susceptibility, and the field-dependent low-temperature magnetization, simulated ab initio for the mononuclear Co fragments, are in agreement with the available data for another compound [WIV[(micro-CN)4-CoII(H2O)2]2.4H2O]n containing diamagnetic W and high-spin Co ions in an isostructural environment.

Effect of the Metal Environment on the Ferromagnetic Interaction in the Co−NC−W Pairs of Octacyanotungstate(V)−Cobalt(II) Three-Dimensional Networks

State of the art CASSCF and CASPT2 calculations have been performed to elucidate the nature of ferromagnetism of CoII-NC-WV pairs in the three-dimensional compound [[WV(CN)2]2[(micro-CN)4CoII(H2O)2]3.4H2O]n, which has been recently synthesized and investigated by a number of experimental techniques (Herrera, J. M.; Bleuzen, A.; Dromzée, Y.; Julve, M.; Lloret, F.; Verdaguer, M. Inorg. Chem. 2003, 42, 7052-7059). In this network, the Co ions are in the high-spin (S = 3/2) state, while the single unpaired electron on the W centers occupies the lowest orbital of the dz2 type of the 5d shell. In agreement with the suggestion made by Herrera et al., we find that the ferromagnetism is due to a certain occupation scheme of the orbitals from the parent octahedral t2g shell on CoII sites, in which the orbital accommodating the unpaired electron is orthogonal to the dz2 orbitals of the surrounding W ions. We investigate the stabilization of such an orbital configuration on the Co sites and find that it cannot be achieved in the ground state of isolated mononuclear fragments [CoII(NC)4(OH2)2]2- for any conformations of the coordinated water molecules and Co-N-C bond angles. On the other hand, it is stabilized by the interaction of the complex with neighboring W ions, which are simulated here by effective potentials. The calculated exchange coupling constants for the CoII-NC-WV binuclear fragments are in reasonable agreement with the measured Curie-Weiss constant for this compound. As additional evidence for the inferred electronic configuration on the Co sites, the ligand-field transitions, the temperature-dependent magnetic susceptibility, and the field-dependent low-temperature magnetization, simulated ab initio for the mononuclear Co fragments, are in agreement with the available data for another compound [WIV[(micro-CN)4-CoII(H2O)2]2.4H2O]n containing diamagnetic W and high-spin Co ions in an isostructural environment.

Vortices and Their Relation to Ring Currents and Magnetic Moments in Nanographenes in High Magnetic Field

Much attention has been paid to the role of vortices in the magnetic response properties of superconductors, but less so for molecular systems. Here we present a theoretical analysis on nanographenes subject to a strong homogeneous magnetic field. The analysis is based on the simple Hückel-London model, for which we derive the topological definition of vorticity. The results are confirmed by a more elaborate model that includes nonnearest neighbor interaction, the explicit presence of nuclei and all terms due to the magnetic field. We find that due to frontier orbital intersections, large changes in magnetic dipole moments occur. Orbital energy minima and maxima can be related to change of vortex patterns with flux.

Multiquanta vortex entry and vortex-antivortex pattern expansion in a superconducting microsquare with a magnetic dot

We investigate the nucleation of superconductivity in a microsquare with a magnetic dot on top. The cusplike behavior of the calculated normal-superconducting phase boundaries, T(c)(H), shows a transition between short-period to long-period oscillations when going from positive to negative applied fields, H. Vorticity changes by more than 1, indicating multiquanta vortex entries, have been detected along this asymmetric T(c)(H) boundary. The dot also expands dramatically the symmetry-consistent vortex-antivortex patterns, thus facilitating their experimental observation.

[Mo2(CN)11]:5- A Detailed Description of Ligand-Field Spectra and Magnetic Properties by First-Principles Calculations

An ab initio multiconfigurational approach has been used to calculate the ligand-field spectrum and magnetic properties of the title cyano-bridged dinuclear molybdenum complex. The rather large magnetic coupling parameter J for a single cyano bridge, as derived experimentally for this complex by susceptibility measurements, is confirmed to a high degree of accuracy by our CASPT2 calculations. Its electronic structure is rationalized in terms of spin-spin coupling between the two constituent hexacyano-monomolybdate complexes. An in-depth analysis on the basis of Anderson's kinetic exchange theory provides a qualitative picture of the calculated CASSCF antiferromagnetic ground-state eigenvector in the Mo dimer. Dynamic electron correlations as incorporated into our first-principles calculations by means of the CASPT2 method are essential to obtain quantitative agreement between theory and experiment.

Magnetic Anisotropy of [Mo(CN)7]4- Anions and Fragments of Cyano-Bridged Magnetic Networks

Quantum chemistry calculations of CASSCF/CASPT2 level together with ligand field analysis are used for the investigation of magnetic anisotropy of [Mo(CN)7]4- complexes. We have considered three types of heptacyano environments: two ideal geometries, a pentagonal bipyramid and a capped trigonal prism, and the heptacyanomolybdate fragment of the cyano-bridged magnetic network K2[Mn(H2O)2]3[Mo(CN)7]2.6H2O. At all geometries the first excited Kramers doublet is found remarkably close to the ground one due to a small orbital energy gap in the ligand field spectrum, which ranges between a maximal value in the capped trigonal prism (800 cm(-1)) and zero in the pentagonal bipyramid. The small value of this gap explains (i) the axial form of the g tensor and (ii) the strong magnetic anisotropy even in strongly distorted complexes. Comparison with available experimental data for the g tensor of the mononuclear precursors reveals good agreement with the present calculations for the capped trigonal prismatic complex and a significant discrepancy for the pentagonal bipyramidal one. The calculations for the heptacyanomolybdate fragment of K2[Mn(H2O)2]3[Mo(CN)7]2.6H2O give g(perpendicular)/g(parallel) approximately 0.5 and the orientation of the local anisotropy axis close to the symmetry axis of an idealized pentagonal bipyramid. These findings are expected to be important for the understanding of the magnetism of anisotropic Mo(III)-Mn(II) cyano-bridged networks based on the [Mo(CN)7]4- building block.

Spin-vibronic superexchange in Mott-Hubbard fullerides

In the Mott-Hubbard cubic fulleride Li3(NH3)6C60 the superexchange energy is found to be much smaller than the rotational quantum for Jahn-Teller deformations at fullerene sites. This gives rise to a new type of superexchange interaction involving threefold degenerate vibronic ground states of C3-60 ions. In contrast with spin-orbital models, the spin-vibronic superexchange can be only antiferromagnetic and shows a significant vibronic reduction of the superexchange amplitude, in agreement with magnetic susceptibility data. As a function of the transfer parameters, two quadrupolar fully dynamical vibronic orders with quenched vibronic moments on sites develop in the ground state.

The E⊗e dynamic Jahn-Teller problem: A new insight from the strong coupling limit

Correct boundary conditions for the E x e dynamic Jahn-Teller problem are considered explicitly for the first time to obtain approximate analytical solutions in the strong coupling limit. Numerical solutions for the decoupled equations using the finite difference method are also presented. The numerical solutions for the decoupled equations exhibit avoided crossings in the weak coupling region, which explains the oscillating behavior of the solutions obtained by Longuet-Higgins et al. for the coupled equations. The obtained analytical energy expressions show improved agreement with the numerical calculations as compared with the previous treatment in which the potentials were assumed to be harmonic. We demonstrate that the pseudorotational energy j(2)/(2g(2)), where g is the dimensionless vibronic coupling constant, and j total angular momentum: j=+/-1/2,+/-3/2,..., in the conventional strong coupling expression for the vibronic levels of the lower sheet is exact. Non-Hermitian first-order perturbation theory gives the energy which is correct up to 1/g(4). The asymptotic behavior of the wave function at the origin does not influence the corrected energy up to order of 1/g(4). At the same time the treatment of the upper sheet with correct boundary conditions gives solutions which are entirely different from the corresponding Slonczewski's solutions. Besides, the correct boundary conditions enable us to evaluate the nonadiabatic coupling between the lower and upper potential sheets. The energy correction due to the nonadiabatic coupling is estimated to be of order 1/g(6).

Ab initio study of small graphitic cones with triangle, square, and pentagon apex

Accurate geometries of carbon nanocones of different sizes with a triangle, square or pentagon at the apex have been determined for the first time using a quantum chemical optimization method. The structure close to the apex is distorted from an ideal conical surface. The charging effect of the central defect is quite different from that predicted by tight-binding calculations. The symmetry behavior of the frontier orbitals and the size of the highest occupied molecular orbital-lowest unoccupied molecular orbital gap versus cone type and size is explained. The density of states quickly converges towards that of graphite when the size of the cone increases. In comparison to previous results in the literature it is found that the local densities of states of cones, that are locally different but belong to the same topo-combinatoric class, share common features.

Assignment of the Electronic Spectra of [Mo(CN)8]4- and [W(CN)8]4- by Ab Initio Calculations

CASPT2 calculations are performed on the dodecahedral and square antiprismatic isomers of the [Mo(CN)(8)](4)(-) and [W(CN)(8)](4)(-) complexes. The high-energy experimental bands above 40000 cm(-)(1) are assigned to MLCT transitions. The experimental observed trend of the extinction coefficients for the molybdenum and tungsten complex is reproduced by our CASSCF oscillator strengths. All bands below 40000 cm(-)(1) can be ascribed to ligand-field transitions, although small contributions from forbidden MLCT transitions cannot be excluded. In order to account for all experimental bands in the electronic spectrum of these octacyanocomplexes, a dynamic equilibrium in solution between the two isomeric forms must be hypothesized. Spin-orbit coupling effects are found to be more important for the square antiprismatic isomers; in particular, large singlet-triplet mixings are calculated for this isomer of [W(CN)(8)](4)(-). Ligand-field and Racah parameters as well as spin-orbit coupling constants are determined on the basis of the calculated transition energies. The obtained values for these parameters support the recently proposed model for exchange interactions in magnetic clusters and networks containing pentavalent octocyanometalates of molybdenum and tungsten.

Electronic Structure of Linear Thiophenolate-Bridged Heteronuclear Complexes [LFeMFeL]n+ (M = Cr, Co, Fe; n = 1−3): A Combination of Kinetic Exchange Interaction and Electron Delocalization

The electronic properties of the isostructural series of heterotrinuclear thiophenolate-bridged complexes of the general formula [LFeMFeL](n)(+) with M = Cr, Co and Fe where L represents the trianionic form of the ligand 1,4,7-tris(4-tertbutyl-2-mercaptobenzyl)-1,4,7-triazacyclononane, synthesized and investigated by a number of experimental techniques in the previous work(1), are subjected now to a theoretical analysis. The low-lying electronic excitations in these compounds are described within a minimal model supported by experiment and quantum chemistry calculations. It was found indeed that various experimental data concerning the magnetism and electron delocalization in the lowest states of all seven compounds are completely reproduced within a model which includes the electron transfer between magnetic orbitals at different metal centers and the electron repulsion in these orbitals (the Hubbard model). Moreover, due to the trigonal symmetry of the complexes, only the electron transfer between nondegenerate orbital, a(1), originating from the t(2g) shell of each metal ion in a pseudo-octahedral coordination, is relevant for the lowest states. An essential feature resulting from quantum chemistry calculations, allowing to explain the unusual magnetic properties of these compounds, is the surprisingly large value and, especially, the negative sign of the electron transfer between terminal iron ions, beta'. According to their electronic properties the series of complexes can be divided as follows: (1). The complexes [LFeFeFeL](3+) and [LFeCrFeL](3+) show localized valences in the ground electronic configuration. The strong antiferromagnetic exchange interaction and the resulting spin 1/2 of the ground-state arise from large values of the transfer parameters. (2). In the complex [LFeCrFeL](+), due to a higher energy of the magnetic orbital on the central Cr ion than on the terminal Fe ones, the spin 3/2 and the single unpaired a(1) electron are almost localized at the chromium center in the ground state. (3). The complex [LFeCoFeL](3+) has one ground electronic configuration in which two unpaired electrons are localized at terminal iron ions. The ground-state spin S = 1 arises from a kinetic mechanism involving the electron transfer between terminal iron ions as one of the steps. Such a mechanism, leading to a strong ferromagnetic interaction between distant spins, apparently has not been discussed before. (4). The complex [LFeFeFeL](2+) is characterized by both spin and charge degrees of freedom in the ground manifold. The stabilization of the total spin zero or one of the itinerant electrons depends on beta', i.e., corresponds to the observed S = 1 for its negative sign. This behavior does not fit into the double exchange model. (5). In [LFeCrFeL](2+) the delocalization of two itinerant holes in a(1) orbitals takes place over the magnetic core of chromium ion. Although the origin of the ground-state spin S = 2 is the spin dependent delocalization, the spectrum of the low-lying electronic states is again not of a double exchange type. (6). Finally, the complex [LFeCoFeL](2+) has the ground configuration corresponding to the electron delocalization between terminal iron atoms. The estimated magnitude of the corresponding electron transfer is smaller than the relaxation energy of the nuclear distortions induced by the electron localization at one of the centers, leading to vibronic valence trapping observed in this compound.

Mechanism of a strongly anisotropic MoIII-CN-MnII spin-spin coupling in molecular magnets based on the [Mo(CN)(7)](4-) heptacyanometalate: a new strategy for single-molecule magnets with high blocking temperatures

Unusual spin coupling between Mo(III) and Mn(II) cyano-bridged ions in bimetallic molecular magnets based on the [Mo(III)(CN)(7)](4-) heptacyanometalate is analyzed in terms of the superexchange theory. Due to the orbital degeneracy and strong spin-orbit coupling on Mo(III), the ground state of the pentagonal-bipyramidal [Mo(III)(CN)(7)](4-) complex corresponds to an anisotropic Kramers doublet. Using a specially adapted kinetic exchange model we have shown that the Mo(III)-CN-Mn(II) superexchange interaction is extremely anisotropic: it is described by an Ising-like spin Hamiltonian JS(z)(Mo) S(z)(Mn) for the apical pairs and by the J(z)S(z)(Mo) S(z)(Mn) + J(xy)(Sx(Mo) Sx(Mn) + Sy(Mo) Sy(Mn)) spin Hamiltonian for the equatorial pairs (in the latter case J(z) and J(xy) can have opposite signs). This anisotropy resulted from an interplay of several Ising-like (Sz(Mo) Sz(Mn)) and isotropic (S(Mo)S(Mn)) ferro- and antiferromagnetic contributions originating from metal-to-metal electron transfers through the pi and sigma orbitals of the cyano bridges. The Mo(III)-CN-Mn(II) exchange anisotropy is distinct from the anisotropy of the g-tensor of [Mo(III)(CN)(7)](4-); moreover, there is no correlation between the exchange anisotropy and g-tensor anisotropy. We indicate that highly anisotropic spin-spin couplings (such as the Ising-like JS(z)(Mo) S(z)(Mn)) combined with large exchange parameters represent a very important source of the global magnetic anisotropy of polyatomic molecular magnetic clusters. Since the total spin of such clusters is no longer a good quantum number, the spin spectrum pattern can differ considerably from the conventional scheme described by the zero-field splitting of the isotropic spin of the ground state. As a result, the spin reorientation barrier of the magnetic cluster may be considerably larger. This finding opens a new way in the strategy of designing single-molecule magnets (SMM) with unusually high blocking temperatures. The use of orbitally degenerate complexes with a strong spin-orbit coupling (such as [Mo(III)(CN)(7)](4-) or its 5d analogues) as building blocks is therefore very promising for these purposes.

An Ab initio study of the ligand field and charge-transfer transitions of Cr(CN)(6)(3-) and Mo(CN)(6)(3-)

High-level ab initio calculations on the excited states of Cr(CN)63- and Mo(CN)63- are reported. For the latter complex, a rather large 10 Dq value of 42 000 cm-1 is obtained, reflecting the increased covalency. The lowest lying charge-transfer transitions for both complexes are predicted to be of the type ligand-to-metal, an assignment in agreement with the photochemical behavior of Cr(CN)63-. A good correspondence between the well-known experimental spectrum of the chromium complex and the theoretical CASPT2 excitation energies is found.

The electronic structure and spectrum of Mo(CN)8(3-)

State of the art CASSCF and CASPT2 calculations have been performed to elucidate the nature of the electronic transitions observed in the experimental spectrum of the octacyanomolybdate(V) cation. Assuming a triangular dodecahedral structure for this complex gives a convincing agreement between theory and experiment. All absorption bands are assigned to low-lying charge-transfer transitions involving excitations from ligand orbitals to 4dx2-y2. The calculated molecular orbitals reveal weak pi interactions between the metal and ligand orbitals, compared to much stronger sigma interactions. This calculated electronic structure substantiates the previous hypothesis concerning the giant spin ground states of magnetic clusters and networks containing Mo(CN)8(3-) as a constituent part.

Vortex entry and nucleation of antivortices in a mesoscopic superconducting triangle

The nucleation of superconductivity in mesoscopic equilateral triangles is investigated by using the linearized Ginzburg-Landau equation (LGLE). The trigonal symmetry of the sample has a profound effect on the superconducting state in the presence of a magnetic field H leading, in particular, to the formation of antivortices in symmetry-consistent states. For the same given irreducible representation, vortices enter always by three via the middle of the edges, approach the center, and then are dispatched towards the corners of the triangle. The measured superconducting phase boundary T(c)(H) is in good agreement with the T(c)(H) line found from the LGLE.

Symmetry-induced formation of antivortices in mesoscopic superconductors

Recent progress in nanotechnology has stimulated interest in mesoscopic superconductors as components for quantum computing and cryoelectronics. The critical parameters for superconductivity (current and field) of a mesoscopic sample are determined by the pattern of vortices in it, which in turn is controlled by the symmetry imposed by the shape of the sample (see ref. 1 and references therein). Hitherto it has been unclear what happens when the number of vortices is not consistent with the natural symmetry. Here we show that additional vortex-antivortex pairs nucleate spontaneously so as to preserve the symmetry of the sample. For example, in a square with three vortices, the spontaneously generated pair, along with the original three vortices, distribute themselves so that the four vortices sit in the four corners, with the antivortex in the centre. The measured superconducting phase boundary (of superconducting transition temperature Tc versus magnetic field strength) is in very good agreement with the calculations, giving direct experimental evidence for these symmetry-induced vortex-antivortex pairs. Vortex entry into the sample is also changed: vortices enter a square in fours, with antivortices generated to preserve the imposed vorticity. The symmetry-induced nucleation of antivortices is not restricted to superconductors, but should also apply to symmetrically confined superfluids and Bose-Einstein condensates.