Primitive Orbits Research Articles

The holonomy inverse problem

Let (M,g) be a smooth Anosov Riemannian manifold and \mathcal{C}^\sharp the set of its primitive closed geodesics. Given a Hermitian vector bundle \mathcal{E} equipped with a unitary connection \nabla^{\mathcal{E}} , we define \mathcal{T}^\sharp(\mathcal{E}, \nabla^{\mathcal{E}}) as the sequence of traces of holonomies of \nabla^{\mathcal{E}} along elements of \mathcal{C}^\sharp . This descends to a homomorphism on the additive moduli space \mathbb{A} of connections up to gauge \mathcal{T}^\sharp: (\mathbb{A}, \oplus) \to \ell^\infty(\mathcal{C}^\sharp) , which we call the primitive trace map . It is the restriction of the well-known Wilson loop operator to primitive closed geodesics. The main theorem of this paper shows that the primitive trace map \mathcal{T}^\sharp is locally injective near generic points of \mathbb{A} when \dim(M) \geq 3 . We obtain global results in some particular cases: flat bundles, direct sums of line bundles, and general bundles in negative curvature under a spectral assumption which is satisfied in particular for connections with small curvature. As a consequence of the main theorem, we also derive a spectral rigidity result for the connection Laplacian. The proofs are based on two new ingredients: a Livšic-type theorem in hyperbolic dynamical systems showing that the cohomology class of a unitary cocycle is determined by its trace along closed primitive orbits, and a theorem relating the local geometry of \mathbb{A} to the Pollicott–Ruelle resonance near zero of a certain natural transport operator.

Complete dynamical evaluation of the characteristic polynomial of binary quantum graphs

We evaluate the variance of coefficients of the characteristic polynomial for binary quantum graphs using a dynamical approach. This is the first example where a spectral statistic can be evaluated in terms of periodic orbits for a system with chaotic classical dynamics without taking the semiclassical limit, which here is the limit of large graphs. The variance depends on the sizes of particular sets of primitive pseudo orbits (sets of distinct primitive periodic orbits): the set of primitive pseudo orbits without self-intersections and the sets of primitive pseudo orbits with a fixed number of self-intersections, all of which consist of two arcs of the pseudo orbit crossing at a single vertex. To show other pseudo orbits do not contribute we give two arguments. The first is based on a reduction of the variance formula from a sum over pairs of primitive pseudo orbits to a sum over pseudo orbits where no bonds are repeated. The second employs a parity argument for the Lyndon decomposition of words. For families of binary graphs, in the semiclassical limit, we show the pseudo orbit formula approaches a universal constant independent of the coefficient of the polynomial. This is obtained by counting the total number of primitive pseudo orbits of a given length.

Periodic orbit scar in wavepacket propagation

This study analyzed the scar-like localization in the time-average of a time-evolving wavepacket on a desymmetrized stadium billiard. When a wavepacket is launched along the orbits, it emerges on classical unstable periodic orbits as a scar in stationary states. This localization along the periodic orbit is clarified through the semiclassical approximation. It essentially originates from the same mechanism of a scar in stationary states: piling up of the contribution from the classical actions of multiply repeated passes on a primitive periodic orbit. To achieve this, several states are required in the energy range determined by the initial wavepacket.

Lyndon word decompositions and pseudo orbits on q-nary graphs

A foundational result in the theory of Lyndon words (words that are strictly earlier in lexicographic order than their cyclic permutations) is the Chen–Fox–Lyndon theorem which states that every word has a unique non-increasing decomposition into Lyndon words. This article extends this factorization theorem, obtaining the proportion of these decompositions that are strictly decreasing. This result is then used to count primitive pseudo orbits (sets of primitive periodic orbits) on q-nary graphs. As an application we obtain a diagonal approximation to the variance of the characteristic polynomial coefficients of q-nary quantum graphs.

Evaluating balance on social networks from their simple cycles

Signed networks have long been used to represent social relations of amity (+) and enmity (-) between individuals. Group of individuals who are cyclically connected are said to be balanced if the number of negative edges in the cycle is even and unbalanced otherwise. In its earliest and most natural formulation, the balance of a social network was thus defined from its simple cycles, cycles which do not visit any vertex more than once. Because of the inherent difficulty associated with finding such cycles on very large networks, social balance has since then been studied via other means. In this article we present the balance as measured from the simple cycles and primitive orbits of social networks. We specifically provide two measures of balance: the proportion $R_\ell$ of negative simple cycles of length $\ell$ for each $\ell\leq 20$ which generalises the triangle index, and a ratio $K_\ell$ which extends the relative signed clustering coefficient introduced by Kunegis. To do so, we use a Monte Carlo implementation of a novel exact formula for counting the simple cycles on any weighted directed graph. Our method is free from the double-counting problem affecting previous cycle-based approaches, does not require edge-reciprocity of the underlying network, provides a gray-scale measure of balance for each cycle length separately and is sufficiently tractable that it can be implemented on a standard desktop computer. We observe that social networks exhibit strong inter-edge correlations favouring balanced situations and we determine the corresponding correlation length $\xi$. For longer simple cycles, $R_\ell$ undergoes a sharp transition to values expected from an uncorrelated model. This transition is absent from synthetic random networks, strongly suggesting that it carries a sociological meaning warranting further research.

Inner projection techniques for the low-cost handling of two-electron integrals in quantum chemistry

ABSTRACTThe density-fitting technique for approximating electron-repulsion integrals relies on the quality of auxiliary basis sets. These are commonly obtained through data fitting, an approach that presents some shortcomings. On the other hand, it is possible to derive auxiliary basis sets by removing elements from the product space of both contracted and primitive orbitals by means of a particular form of inner projection technique that has come to be known as Cholesky decomposition (CD). This procedure allows for on-the-fly construction of auxiliary basis sets that may be used in conjunction with any quantum chemical method, i.e. unbiased auxiliary basis sets. One key feature of these sets is that they represent the electron-repulsion integral matrix in atomic orbital basis with an accuracy that can be systematically improved. Another key feature is represented by the fact that locality of fitting coefficients is obtained even with the long-ranged Coulomb metric, as result of integral accuracy. Here we report on recent advances in the development of the CD-based density fitting technology. In particular, the implementation of analytical gradients algorithms is reviewed and the present status of local formulations – potentially linear scaling – is analysed in detail.

Dynamical zeta functions for Anosov flows via microlocal analysis

The purpose of this paper is to give a short microlocal proof of the meromorphic continuation of the Ruelle zeta function for C∞ Anosov flows. More general results have been recently proved by Giulietti–Liverani–Pollicott [GiLiPo] but our approach is different and is based on the study of the generator of the flow as a semiclassical differential operator. The purpose of this article is to provide a short microlocal proof of the meromorphic continuation of the Ruelle zeta function for C∞ Anosov flows on compact manifolds: Theorem. Suppose X is a compact manifold and φt : X → X is a C∞ Anosov flow with orientable stable and unstable bundles. Let {γ} denote the set of primitive orbits of φt, with T ] γ their periods. Then the Ruelle zeta function, ζR(λ) = ∏

Orbital structure of vibrations of nanoparticles, clusters, and coordination polyhedra

The necessity of the development of the orbital structure of vibrations of nanoparticles, clusters, and coordination polyhedra is dictated by synthesis of clusters, supermolecules, and other structures of nanoscale dispersion for which translational symmetry is absent and the crystal system is inapplicable. The composition of complicated molecules, polynuclear complexes, and clusters is described, in addition to the chemical formula, by the composition equations derived from analysis of the symmetry properties of molecular structures. This analysis enables the derivation of analytical formulas for the types of molecular orbitals of structures with arbitrary groups of symmetry. Here, we use the representation of nanoscale structures described by the orbital system as a set of concentric nested spherical orbits of atoms, orbits of faces of different order, and orbits of edges. The orbits are grouped into shells shaped as polyhedra with vertices, edges, or faces accommodating atoms with different types of packing. In such a way, the sets of molecular orbitals of all high-, intermediate-, and low-symmetry groups have been determined depending on the number of atoms in the axial, planar, and primitive orbits.

Strong approximation in the Apollonian group

The Apollonian group is a finitely generated, infinite index subgroup of the orthogonal group O Q ( Z ) fixing the Descartes quadratic form Q. For nonzero v ∈ Z 4 satisfying Q ( v ) = 0 , the orbits P v = A v correspond to Apollonian circle packings in which every circle has integer curvature. In this paper, we specify the reduction of primitive orbits P v mod any integer d > 1 . We show that this reduction has a multiplicative structure, and that mod primes p ⩾ 5 it is the full cone of integer solutions to Q ( v ) ≡ 0 for v ≢ 0 . This analysis is an essential ingredient in applications of the affine linear sieve as developed by Bourgain, Gamburd and Sarnak.

Semiclassical analysis of the Efimov energy spectrum in the unitary limit

We demonstrate that the (s-wave) geometric spectrum of the Efimov energy levels in the unitary limit is generated by the radial motion of a primitive periodic orbit (and its harmonics) of the corresponding classical system. The action of the primitive orbit depends logarithmically on the energy. It is shown to be consistent with an inverse-squared radial potential with a lower cut-off radius. The lowest-order WKB quantization, including the Langer correction, is shown to reproduce the geometric scaling of the energy spectrum. The (WKB) mean-squared radii of the Efimov states scale geometrically like the inverse of their energies. The WKB wavefunctions, regularized near the classical turning point by Langer's generalized connection formula, are practically indistinguishable from the exact wave functions even for the lowest ($n=0$) state, apart from a tiny shift of its zeros that remains constant for large $n$.

Compact contracted gaussian-type basis sets from Li to Ne

For the second-row atoms, Li through Ne, four minimal Gaussian basis sets are generated. The first one consists of three-term contraction of primitive Gaussian-type orbitals for 1s, 2s, and 2p atomic orbitals. The convenient shorthand notation would be (3,3) for Li-Be and (3,3/3) for B-Ne. The expansion terms for the other sets are (3,3/4), (4,3/3), and (4,3/4), respectively. Although the four basis sets are minimal type, they give the valence shell orbital energies which are close to those of double-zeta sets. These four and other sets derived from them are tested for some organic molecules and the F2 molecule. They are found to give the orbital energies which agree well with those given by extended calculations. Atomization energies and other spectroscopic constants are also calculated and compared with those of extended calculations. The results clearly indicate that the present basis sets can be used very effectively in the molecular calculations.

Multielectron structure and dynamics in scaled-energy Stark photoabsorption and recurrence spectra

We use the $4s{[3/2]}_{2}^{o}$ metastable state of atomic argon created by charge-transfer collisions between a 5 keV ${\text{Ar}}^{+}$ beam and K vapor to perform laser scaled-energy Stark photoabsorption and recurrence spectroscopy of even-parity Rydberg states, detected by field ionization and forced autoionization, in a region of the spectrum that contains two distinct perturbations. We apply a uniform electric field that changes with the frequency of the laser to maintain a constant scaled energy relative to the first ionization limit of the atom, associated with the ion core in a ${^{2}P}_{3/2}$ configuration. Local perturbations to the Rydberg series $(15\ensuremath{\le}n\ensuremath{\le}28)$ occur due to ${n}^{\ensuremath{'}}=8$ and 10 Stark states that belong to the series converging to the second ionization threshold with the ion core in a ${^{2}P}_{1/2}$ configuration. The resulting absorption spectra show multielectron effects due to configuration interaction and angular momentum coupling. These effects have dynamical implications in the recurrence spectrum. In particular, the variations in the Stark structure of the absorption spectrum result in a recurrence spectrum that shows two prominent multielectron features. One is the presence of recurrence peaks with scaled actions less than that of the hydrogenic primitive orbit. We attribute this to excitation in which both the ion core and the Rydberg electron absorb energy during photoexcitation. The second multielectron effect observed is recurrence peaks that occur at the sum of scaled actions of the primitive orbit (and its repetitions) associated with the pair of perturbations and a hydrogenic closed classical orbit. We attribute these peaks to core-changing inelastic scattering of the Rydberg electron with the ionic core due to angular momentum coupling. For comparison, we measured the regular autoionizing Rydberg series of argon between the first and second ionization thresholds, where no perturbing resonances exist.

Study on the maximum accuracy of the pseudopotential density functional method with localized atomic orbitals versus plane-wave basis sets

A detailed study on the accuracy attainable with numerical atomic orbitals in the context of pseudopotential first-principles density functional theory is presented. Dimers of first- and second-row elements are analyzed: bond lengths, atomization energies, and Kohn-Sham eigenvalue spectra obtained with localized orbitals and with plane-wave basis sets are compared. For each dimer, the cutoff radius, the shape, and the number of the atomic basis orbitals are varied in order to maximize the accuracy of the calculations. Optimized atomic orbitals are obtained following two routes: (i) maximization of the projection of plane wave results into atomic orbital basis sets and (ii) minimization of the total energy with respect to a set of primitive atomic orbitals as implemented in the OPENMX software package. It is found that by optimizing the numerical basis, chemical accuracy can be obtained even with a small set of orbitals.

Influence of quantum defects on recurrence strengths of closed orbits

Experimentally obtained Stark-recurrence spectra taken at low principal quantum numbers show unusual degrees of orbit profile asymmetry. To clearly illustrate the semiclassical mechanisms behind this behavior a numerical experiment is performed where orbit profiles (recurrence strength as a function of scaled energy) are found from computed Stark spectra. These spectra are generated for a wide range of quantum defects assuming a highly simplified excitation and core structure which represents a semiclassical system restricted to s-wave scattering. It is noted that at low quantum numbers, the expected dominant nonhydrogenic feature of recurrence spectra is scattered orbits whose scaled actions are unresolved from existing hydrogenic orbits. The semiclassical orbit profiles obtained from absorption spectra are compared with semiclassical closed-orbit theory. Closed-orbit theory successfully predicts the systematic shifting of recurrence strength as a function of quantum defect. In the limited parameter space investigated it is found that the distribution of recurrence strength is influenced primarily by interference with scattered combinations containing a primitive orbit repetition. The systematic shifting of recurrence strength as a function of quantum defect is attributed to a relative phase shift between the contributing orbits.

Block-Localized Wavefunction (BLW) Method at the Density Functional Theory (DFT) Level

The block-localized wavefunction (BLW) approach is an ab initio valence bond (VB) method incorporating the efficiency of molecular orbital (MO) theory. It can generate the wavefunction for a resonance structure or diabatic state self-consistently by partitioning the overall electrons and primitive orbitals into several subgroups and expanding each block-localized molecular orbital in only one subspace. Although block-localized molecular orbitals in the same subspace are constrained to be orthogonal (a feature of MO theory), orbitals between different subspaces are generally nonorthogonal (a feature of VB theory). The BLW method is particularly useful in the quantification of the electron delocalization (resonance) effect within a molecule and the charge-transfer effect between molecules. In this paper, we extend the BLW method to the density functional theory (DFT) level and implement the BLW-DFT method to the quantum mechanical software GAMESS. Test applications to the pi conjugation in the planar allyl radical and ions with the basis sets of 6-31G(d), 6-31+G(d), 6-311+G(d,p), and cc-pVTZ show that the basis set dependency is insignificant. In addition, the BLW-DFT method can also be used to elucidate the nature of intermolecular interactions. Examples of pi-cation interactions and solute-solvent interactions will be presented and discussed. By expressing each diabatic state with one BLW, the BLW method can be further used to study chemical reactions and electron-transfer processes whose potential energy surfaces are typically described by two or more diabatic states.

Scaling group flow and Lefschetz trace formula for laminated spaces with p-adic transversal

In his approach to analytic number theory C. Deninger has suggested that to the Riemann zeta function ζ ˆ ( s ) (resp. the zeta function ζ Y ( s ) of a smooth projective curve Y over a finite field F q , q = p f )) one could possibly associate a foliated Riemannian laminated space ( S Q , F , g , ϕ t ) (resp. ( S Y , F , g , ϕ t ) ) endowed with an action of a flow ϕ t whose primitive compact orbits should correspond to the primes of Q (resp. Y). Precise conjectures were stated in our report [E. Leichtnam, An invitation to Deninger's work on arithmetic zeta functions, in: Geometry, Spectral Theory, Groups, and Dynamics, in: Contemp. Math. vol. 387, Amer. Math. Soc., Providence, RI, 2005, pp. 201–236] on Deninger's work. The existence of such a foliated space and flow ϕ t is still unknown except when Y is an elliptic curve (see Deninger [C. Deninger, On the nature of explicit formulas in analytic number theory, a simple example, in: Number Theoretic Methods, Iizuka, 2001, in: Dev. Math., vol. 8, Kluwer Acad. Publ., Dordrecht, 2002, pp. 97–118]). Being motivated by this latter case, we introduce a class of foliated laminated spaces ( S = L × R + ∗ q Z , F , g , ϕ t ) where L is locally D × Z p m , D being an open disk of C . Assuming that the leafwise harmonic forms on L are locally constant transversally, we prove a Lefschetz trace formula for the flow ϕ t acting on the leafwise Hodge cohomology H τ j ( 0 ⩽ j ⩽ 2 ) of ( S , F ) that is very similar to the explicit formula for the zeta function of a (general) smooth curve over F q . We also prove that the eigenvalues of the infinitesimal generator of the action of ϕ t on H τ 1 have real part equal to 1 2 . Moreover, we suggest in a precise way that the flow ϕ t should be induced by a renormalization group flow “à la K. Wilson”. We show that when Y is an elliptic curve over F q this is indeed the case. It would be very interesting to establish a precise connection between our results and those of Connes (p. 553 in [A. Connes, Noncommutative Geometry Year 2000, in: Special Volume GAFA 2000 Part II, pp. 481–559], p. 90 in [A. Connes, Symétries Galoisiennes et Renormalisation, in: Séminaire Bourbaphy, Octobre 2002, pp. 75–91]) and Connes–Marcolli [A. Connes, M. Marcolli, Q -lattices: quantum statistical mechanics and Galois theory, in: Frontiers in Number Theory, Physics and Geometry, vol. I, Springer-Verlag, 2006, pp. 269–350; A. Connes, M. Marcolli, From physics to number theory via noncommutative geometry. Part II: renormalization, the Riemann–Hilbert correspondence, and motivic Galois theory, in: Frontiers in Number Theory, Physics and Geometry, vol. II, Springer-Verlag, 2006, pp. 617–713] on the Galois interpretation of the renormalization group.

Numerical atomic basis orbitals from H to Kr

We present a systematic study for numerical atomic basis orbitals ranging from H to Kr, which could be used in large scale $\mathrm{O}(N)$ electronic structure calculations based on density-functional theories (DFT). The comprehensive investigation of convergence properties with respect to our primitive basis orbitals provides a practical guideline in an optimum choice of basis sets for each element, which well balances the computational efficiency and accuracy. Moreover, starting from the primitive basis orbitals, a simple and practical method for variationally optimizing basis orbitals is presented based on the force theorem, which enables us to maximize both the computational efficiency and accuracy. The optimized orbitals well reproduce convergent results calculated by a larger number of primitive orbitals. As illustrations of the orbital optimization, we demonstrate two examples: the geometry optimization coupled with the orbital optimization of a ${\mathrm{C}}_{60}$ molecule and the preorbital optimization for a specific group such as proteins. They clearly show that the optimized orbitals significantly reduce the computational efforts, while keeping a high degree of accuracy, thus indicating that the optimized orbitals are quite suitable for large scale DFT calculations.

Theoretical analysis of electronic delocalization

A block-localized wave function method is introduced to evaluate the electronic delocalization effect in molecules. The wave function for the hypothetical and strictly localized structure is constructed based on the assumption that all electrons and primitive basis functions can be divided into several subgroups; each localized molecular orbital is expanded in terms of primitive orbitals belonging to only one subgroup. The molecular orbitals belonging to the same subgroup are constrained to be mutually orthogonal, while those belonging to different subgroups are free to overlap. The final block-localized wave function at the Hartree–Fock level is expressed by a Slater determinant. In this manner, the energy difference between the Hartree–Fock wave function and the block-localized wave function can be generally defined as the electronic delocalization energy. The method is applied to two cases. The first concerns the resonance stabilization in the allyl ions. We find that the vertical resonance energies for the planar cation and anion are −45.7 (or −44.7) and −46.7 (or −48.2) kcal/mol at the HF/6-31G* (or 6-31+G*) level, respectively. Their rotational barriers are decomposed in terms of conjugation, hyperconjugation, steric effect, and pyramidalization. The n→σ* negative hyperconjugation in the staggered allyl anion is very strong and stabilizes the system by as much as −13 kcal/mol. The second concerns the hyperconjugation effect in propene. Our calculations suggest that the theoretical hyperconjugation energy in propene is about −5 kcal/mol, which is close to the experimental estimate (−2.7 kcal/mol) derived from the hydrogenation heats of propene and ethylene. Comparisons between the results based on the present block-localized wave function method and those based on the natural bond orbital method are presented and discussed. The examples demonstrate that the block-localized wave function method can be employed as a useful model to analyze chemical bondings and intuitive concepts.

Periodic Orbit Theory for Rydberg Atoms in External Fields

Although hydrogen in external fields is a paradigm for the application of periodic orbits and the Gutzwiller trace formula to a real system, the trace formula has never been applied successfully to other Rydberg atoms. We show that spectral fluctuations of general Rydberg atoms are given with remarkable precision by the addition of diffractive terms. Previously unknown features in atomic spectra are exposed: there are new modulations that are neither periodic orbits nor combinations of periodic orbits; ``core shadowing'' generally decreases primitive periodic orbit amplitudes but can also lead to increases.

Role of core-scattered closed orbits in nonhydrogenic atoms.

While both diamagnetic and Stark spectra of hydrogen can be analyzed accurately in terms of classical orbits, in nonhydrogenic atoms the multielectron core induces additional spectral modulations that cannot be analyzed reliably in terms of standard periodic orbit-type theories. However, by extending closed-orbit theory to include core-scattered waves consistently, both diamagnetic and Stark photoabsorption spectra of nonhydrogenic Rydberg atoms at constant scaled energy can be analyzed semiclassically using only the closed orbits of the corresponding hydrogenic systems. Frequencies and amplitudes of the core-scattered modulations, as well as corrected amplitudes for contributions from repetitions of primitive hydrogenic orbits, are found to be in excellent agreement with quantum results. We consider whether these nonhydrogenic systems correspond to quantum chaos. \textcopyright{} 1996 The American Physical Society.