Symmetry Subspaces Research Articles

Reduced Gutzwiller formula with symmetry: Case of a Lie group

We consider a classical Hamiltonian H on R 2 d , invariant by a Lie group of symmetry G, whose Weyl quantization H ˆ is a selfadjoint operator on L 2 ( R d ) . If χ is an irreducible character of G, we investigate the spectrum of its restriction H ˆ χ to the symmetry subspace L χ 2 ( R d ) of L 2 ( R d ) coming from the decomposition of Peter–Weyl. We give semi-classical Weyl asymptotics for the eigenvalues counting function of H ˆ χ in an interval of R , and interpret it geometrically in terms of dynamics in the reduced space R 2 d / G . Besides, oscillations of the spectral density of H ˆ χ are described by a Gutzwiller trace formula involving periodic orbits of the reduced space, corresponding to quasi-periodic orbits of R 2 d .

A Gutzwiller type formula for a reduced Hamiltonian within the framework of symmetry

For a classical Hamiltonian with a finite group of symmetries, we give semi-classical asymptotics in a neighbourhood of an energy E of a regularized spectral density of the quantum Hamiltonian restricted to symmetry subspaces of Peter–Weyl defined by irreducible characters of the group. If we suppose that the energy level Σ E is compact, non-critical, and that its periodic orbits are non-degenerate, we get a Gutzwiller type formula for the reduced Hamiltonian, whose oscillating part involves the symmetry properties of closed trajectories of Σ E . To cite this article: R. Cassanas, C. R. Acad. Sci. Paris, Ser. I 340 (2005).

On the discrete symmetric localization problem

Workpiece localization has many applications in manufacturing automation. Symmetric localization plays an important role in it, since many commonly used features in engineering design are symmetric. In this paper, we discuss some new problems in symmetric localization. We find that multiple solutions exist even for localizing a cube, while it was known that a unique solution should be found. Then, we re-explore the symmetry subspaces for symmetric features. After that, we find that some symmetric features have different configuration spaces from those known before. This is actually the cause of the problem. A symmetric localization algorithm and simulation results are given to fix the existing problems.

Spectral theory for the dihedral group

Let \( H = -\Delta + V \) be a two-dimensional Schrodinger operator defined on a bounded domain \( \Omega \subset {\Bbb R}^2 \) with Dirichlet boundary conditions on \( \partial \Omega \). Suppose that H commutes with the actions of the dihedral group \( \Bbb D_{2n} \), the group of the regular n-gon. We analyze completely the multiplicity of the groundstate eigenvalues associated to the different symmetry subspaces related to the irreducible representations of \( \Bbb D_{2n} \). In particular we find that the multiplicities of these groundstate eigenvalues equal the degree of the corresponding irreducible representation. We also obtain an ordering of these eigenvalues. In addition we analyze the qualitative properties of the nodal sets of the corresponding ground state eigenfunctions.

Electromagnetic transitions of the helium atom in a strong magnetic field

We investigate the electromagnetic-transition probabilities for the helium atom embedded in a strong magnetic field. In total, approximately 12 000 transitions have been calculated covering a grid of 20 different field strengths in the complete regime $B=0\ensuremath{-}100 \mathrm{a}.\mathrm{u}.$ The changes of the oscillator strengths with increasing field strength are discussed in detail, addressing both individual transitions and sets of transitions among certain symmetry subspaces. A complete reorganization of the oscillator strengths in the intermediate-field regime is observed.

Nonsequential triple ionization in strong fields

We consider the final stage of triple ionization of atoms in a strong linearly polarized laser field. We propose that for intensities below the saturation value for triple ionization the process is dominated by the simultaneous escape of three electrons from a highly excited intermediate complex. We identify within a classical model two pathways to triple ionization, one with a triangular configuration of electrons and one with a more linear one. Both are saddles in phase space. A stability analysis indicates that the triangular configuration has the larger cross sections and should be the dominant one. Trajectory simulations within the dominant symmetry subspace reproduce the experimentally observed distribution of ion momenta parallel to the polarization axis.

Classical analysis of correlated multiple ionization in strong fields

We discuss the final stages of the simultaneous ionization of two or more electrons due to a strong laser pulse. An analysis of the classical dynamics suggests that the dominant pathway for non-sequential escape has the electrons escaping in a symmetric arrangement. Classical trajectory models within and near to this symmetry subspace support the theoretical considerations and give final momentum distributions in close agreement with experiments.

Symmetry Reduction of Structures for Large Rotations

The reduction of the number of unknowns while capturing the essential physical features in a nonlinear analysis of large spatial structures has long been a challenging task for researchers. We approach the problem of large displacement and large rotation analysis of space frame structures by means of group theory and substructuring technique. An accurate nonlinear analysis requires an element that accurately reflects the nonlinear behavior of the structure being modeled. Therefore details of the element formulations and updating procedure for large rotation will be given. The present formulation is capable of modeling a structure (with small to moderate axial force in its members) using one element per member, so that the number of degrees of freedom could be kept to minimum. The present methodology is based on the approach in a paper by Healey and Treacy. Axial deformation only was considered and matrix-iteration was used. We extend their idea to deal with the analysis of symmetry structures under going large displacement and large rotation in bending. Instead of matrix-iteration, we find that the basis vectors required to project the solution space into its symmetry subspace can be easily determined from the projection matrix. Excellent reduction can be obtained using the present method. The reduction efficiency depends on the symmetry of the structures. The higher the degree of symmetry the structure possesses, the greater the reduction will be.

Substituent effects and the noncrossing rule: The importance of reduced symmetry subspaces. I. The quenching of OH(A 2Σ+) by H2

The effects of substituent substitution on the locus of a seam of conical intersection and the importance of conical intersections in the associated low symmetry subspaces are considered. For molecules with more than three atoms and with some symmetry the seam of conical intersection may well include an accidental symmetry-allowed portion involving two states of different symmetry. However, in regions of reduced point group symmetry, conical intersections involving two states of the same symmetry may exist. This later class of conical intersections is rarely considered although it could significantly alter the predicted outcome of a nonadiabatic process. The efficient quenching of OH(A 2Σ+)by H2, a consequence of OH–H2 conical intersections, is particularly compelling in this regard. Previous analyses have considered only the C2v2A1–2B2 accidental symmetry-allowed portion of the seam of conical intersection. It is demonstrated that when intersections of states of the same symmetry are considered conical intersections with Cs symmetry are found that are likely to play an important role in the quenching process.

Dynamical nonlinear optical coefficients from the symmetrized density-matrix renormalization-group method

We extend the symmetrized density-matrix renormalization-group method to compute the dynamic nonlinear optical coefficients for long chains. By computing correction vectors in the appropriate symmetry subspace, we obtain the dynamic polarizabilities, ${\ensuremath{\alpha}}_{\mathrm{ij}}(\ensuremath{\omega})$, and third-order polarizabilities ${\ensuremath{\gamma}}_{\mathrm{ijkl}}(\ensuremath{\omega},\ensuremath{\omega},\ensuremath{\omega})$ of the Hubbard and ``U-V'' chains in an all transpolyacetylene geometry, with and without dimerization. We rationalize the behavior of $\overline{\ensuremath{\alpha}}$ and $\overline{\ensuremath{\gamma}}$ on the basis of the low-lying excitation gaps in the system.

Spatio-temporal dynamics of forced periodic flows in a confined domain

Nonlinear dynamics of a localized linear array of vortices is investigated by numerical simulations. The setup consists of a thin fluid layer (electrolyte) enclosed in a rectangular box and driven by the injection of homogeneous electric currents in an alternating magnetic field. The spectral model simulates the Navier–Stokes equations in two dimensions with steady forcing and linear bottom friction. The model provides an accurate representation of the evolution of flow pattern. Fourier decomposition of the streamfunction shows that a subharmonic instability occurs in the same symmetry subspace as that of a basic flow in the primary instability regime and a shear flow mode appears in a different symmetry subspace in the secondary instability regime. The exploration of temporal behavior shows that the system produces Hopf-type bifurcations. Chaos and frequency locking due to the mutual interaction between the unstable modes are observed. A general scenario of the dynamics of forced periodic flows is discussed by the amplitude equations, which are modeled using formal group theoretical techniques.

Symmetrized density-matrix renormalization-group method for excited states of Hubbard models.

We extend the density-matrix renormalization-group (DMRG) method to exploit parity, ${\mathit{C}}_{2}$ (rotation by \ensuremath{\pi}), and electron-hole symmetries of model Hamiltonians. We demonstrate the power of this method by obtaining the lowest-energy states in all eight symmetry subspaces of Hubbard chains with up to 50 sites. The ground-state energy, optical gap, and spin gap of regular U=4t and U=6t Hubbard chains agree very well with exact results. This development extends the scope of the DMRG method and allows future applications to study of optical properties of low-dimensional conjugated polymeric systems. \textcopyright{} 1996 The American Physical Society.

Theoretical calculations for low-lying adiabatic states of Be 2+

The lowest bound states in the 2Σ u,g and 2Π u,g symmetry subspaces of the molecular ion Be 2 + are studied within the adiabatic approximation. Using MRDCI procedure and large Gaussian basis sets, the corresponding potential energy curves are calculated; vibrational-rotational energy levels are then obtained by solving the rovibronic Schrödinger equation. The ground state of Be 2 + is found to have 2Σ u + symmetry with equilibrium distance R e=2.236 Å and dissociation energy D e=2.01 eV. Various possibilities for transitions between the computed adiabatic curves are discussed.

Location of essential spectrum of intermediate Hamiltonians restricted to symmetry subspaces

A theorem is presented on the location of the essential spectrum of certain intermediate Hamiltonians used to construct lower bounds to bound-state energies of multiparticle atomic and molecular systems. This result is an analog of the Hunziker–Van Winter–Zhislin theorem for exact Hamiltonians, which implies that the continuum of an N-electron system begins at the ground-state energy for the corresponding system with N−1 electrons. The work presented here strengthens earlier results of Beattie [SIAM J. Math. Anal. 16, 492 (1985)] in that one may now consider Hamiltonians restricted to the symmetry subspaces appropriate to the permutational symmetry required by the Pauli exclusion principle, or to other physically relevant symmetry subspaces. The associated convergence theory is also given, guaranteeing that all bound-state energies can be approximated from below with arbitrary accuracy.

Computation of symmetry modes and exact reduction in nonlinear structural analysis

The exact, primary response of a nonlinear structural system with symmetry can be obtained with a reduced number of degrees of freedom or symmetry modes, cf. Healey (A group theoretic approach to computational bifurcation problems with symmetry, Comput. Meth. appl. Mech. Engng, to appear). A group-theoretic approach shows that the set of symmetry transformations of the undeformed structure can be used to construct a projection matrix from configuration space onto the symmetry subspace spanned by the symmetry modes. A numerical approach to the computation of symmetry modes is presented in this paper. The technique exploits the fact that the symmetry modes are the eigenvectors of the projection matrix corresponding to a repeated eigenvalue of unity. The multiplicity of the unit eigenvalue is equal to the trace of the projection matrix. Thus, the symmetry modes can be extracted by inverse iteration combined wtih matrix deflation. Several examples are presented to illustrate the method.

Fluctuations in level spectra-role of range

Matrix calculations are carried out for different ranges of the Yukawa interaction for eight particles distributed in the N=2 harmonic oscillator shell. The eigenvalue spectra, corresponding to an exact symmetry subspace of dimensionality 287, are analysed for fluctuation properties such as near-neighbour spacing distributions and short- and long-range correlation measures, Q and Delta 3 respectively. Both short- and long-range ordering decrease as the range of the interaction is increased or decreased from the nuclear range.

The symmetry and Efimov's effect in systems of three-quantum particles

The finiteness of the discrete spectrum of three body Schrodinger operators restricted to certain symmetry subspaces is proved. The symmetry subspaces are those associated with nonzero angular momentum and those associated with two or three identical fermions.

A New Tamm-Dancoff Method Based on the SO (2N+1) Regular Representation of Fermion Many-Body Systems

A new Tamm-Dancoff method is developed on the basis of the SO(2N + 1) regular representation formalism of Fermion many-body systems. The Schrodinger equation in the SO(2N + 1) generator coordinate representation is decomposed into symmetry subspaces by means of a Peierls-Yoccoz, projection procedure. The decomposed Schrödinger equation is transformed to a quasi-particle frame by a transformation of the generator coordinate and expanded by the symmetry projected Tamm-Dancoff bases. The Tamm-Danco[f equation and wave function thus obtained satisfy all symmetry requirements at every order of the expansion. The first order Tamm-Dancoff wave function is neither an independent particle nor quasi-particle approximation but represents Bose condensed Fermion pairs (and an unpaired particle) performing a coherent motion. The higher order terms represent fluctuations of Fermion pairs (and the unpaired particle) from the Bose condensate.

Extended Hartree–Fock calculations for the helium ground state

AbstractThe extended Hartree–Fock (EHF) wave function of an n‐electron system is defined (Löwdin, Phys. Rev. 97, 1509 (1955)) as the best Slater determinant built on one‐electron spin orbitals having a complete flexibility and projected onto an appropriate symmetry subspace. The configuration interaction equivalent to such a wavefunction for the 1S state of a two‐electron atom is discussed. It is shown that there is in this case an infinite number of solutions to the variational problem with energies lower than that of the usual Hartree–Fock function, and with spin orbitals satisfying all the extremum conditions. Two procedures for obtaining EHF spin orbitals are presented. An application to the ground state of Helium within a basic set made up of 4(s), 3(p0), 2(d0) and 1 (f0) Slater orbitals has produced 90% of the correlation energy.