Orbital Sets Research Articles

Description of the KLL Auger-Meitner decay spectra of argon following primary and satellite core-ionized states.

The K-edge photoelectron and KLL Auger-Meitner decay spectra of Argon have been investigated computationally at the restricted active space perturbation theory to the second order level using biorthonormally transformed orbital sets. Binding energies were computed for the Ar 1s primary ionization, as well as for satellite states originated from shake-up and shake-off processes. Based on our calculations, the contributions of shake-up and shake-off states to the KLL Auger-Meitner spectra of Argon have been completely elucidated. Our results are compared with recent state-of-the-art experimental measurements on Argon.

Wada boundaries on a hyperbolic pair of pants

In this paper, the geodesics of an open multiply connected hyperbolic manifold are presented from the dynamical system point of view. The approach is completely numerical. Similar to the closed hyperbolic case there is a zero-measure set of periodic orbits. The difference is that now, most of the geodesics escape to infinity through one of the topologically distinct channels, and initial conditions are identified to which channel the orbit escapes. The initial condition mesh reveals basins of attraction to each channel. We verified that the basin boundary points are of the Wada type by two independent methods. We have also calculated the basin entropy of the system and verified that it depends on the opening angle of the escape channels. The manifold chosen is the pair of pants together with the leaves, so there are 3 distinct exit channels.

Dependency of the positive and negative long-time behaviors of flows on surfaces

Long-time behavior is one of the most fundamental properties in dynamical systems. The limit behaviors of flows on surfaces are captured by the Poincaré-Bendixson theorem using the ω-limit sets. This paper demonstrates that the positive and negative long-time behaviors are not independent. In fact, we show the dependence between the ω-limit sets and the α-limit sets of points of flows on surfaces, which partially generalizes the Poincaré-Bendixson theorem. Applying the dependency result to solve what kinds of the ω-limit sets appear in the area-preserving (or, more generally, non-wandering) flows on compact surfaces, we show that the ω-limit set of any non-closed orbit of such a flow on a compact surface is either a subset of singular points or a locally dense Q-set. Moreover, we show the wildness of surgeries to add totally disconnected singular points and the tameness of those to add finitely many singular points for flows on surfaces.

Arnol'd cat map lattices.

We construct Arnol'd cat map lattice field theories in phase space and configuration space. In phase space we impose that the evolution operator of the linearly coupled maps be an element of the symplectic group, in direct generalization of the case of one map. To this end we exploit the correspondence between the cat map and the Fibonacci sequence. The chaotic properties of these systems also can be understood from the equationsof motion in configuration space. These describe inverted harmonic oscillators, where the runaway behavior of the potential competes with the toroidal compactification of the phase space. We highlight the spatiotemporal chaotic properties of these systems using standard benchmarks for probing deterministic chaos of dynamical systems, namely, the complete dense set of unstable periodic orbits, which, for long periods, lead to ergodicity and mixing. The spectrum of the periods exhibits a strong dependence on the strength and the range of the interaction.

Matrix of orthogonalized atomic orbital coefficients representation for radicals and ions.

Chemical (molecular, quantum) machine learning relies on representing molecules in unique and informative ways. Here, we present the matrix of orthogonalized atomic orbital coefficients (MAOC) as a quantum-inspired molecular and atomic representation containing both structural (composition and geometry) and electronic (charge and spin multiplicity) information. MAOC is based on a cost-effective localization scheme that represents localized orbitals via a predefined set of atomic orbitals. The latter can be constructed from such small atom-centered basis sets as pcseg-0 and STO-3G in conjunction with guess (non-optimized) electronic configuration of the molecule. Importantly, MAOC is suitable for representing monatomic, molecular, and periodic systems and can distinguish compounds with identical compositions and geometries but distinct charges and spin multiplicities. Using principal component analysis, we constructed a more compact but equally powerful version of MAOC-PCX-MAOC. To test the performance of full and reduced MAOC and several other representations (CM, SOAP, SLATM, and SPAHM), we used a kernel ridge regression machine learning model to predict frontier molecular orbital energy levels and ground state single-point energies for chemically diverse neutral and charged, closed- and open-shell molecules from an extended QM7b dataset, as well as two new datasets, N-HPC-1 (N-heteropolycycles) and REDOX (nitroxyl and phenoxyl radicals, carbonyl, and cyano compounds). MAOC affords accuracy that is either similar or superior to other representations for a range of chemical properties and systems.

On the Notion of Metaplectic Barbasch–Vogan Duality

Abstract In analogy with the Barbasch–Vogan duality for real reductive linear groups, we introduce a duality notion useful for the representation theory of the real metaplectic groups. This is a map on the set of nilpotent orbits in a complex symplectic Lie algebra, whose range consists of the so-called metaplectic special nilpotent orbits. We relate this duality notion with the theory of primitive ideals and extend the notion of special unipotent representations to the real metaplectic groups. We also interpret the duality map in terms of double cells of Weyl group representations.

Painlevé equations, integrable systems and the stabilizer set of Virasoro orbit

We study a geometrical formulation of the nonlinear second-order Riccati equation (SORE) in terms of the projective vector field equation on [Formula: see text], which in turn is related to the stability algebra of Virasoro orbit. Using Darboux integrability method, we obtain the first integral of the SORE and the results are applied to the study of its Lagrangian and Hamiltonian descriptions. Using these results, we show the existence of a Lagrangian description for SORE, and the Painlevé II equation is analyzed.

Applicability of noninteger bessel type orbital basis sets: numerical and analytical approaches

Recently, usefulness of the noninteger principal quantum numbers for Bessel type orbitals was discussed by Weniger in (Adv Quant Chem 83:209–237, 2021). In this study, we analyzed the applicability and numerical accuracy of basis sets of noninteger Bessel type orbitals to electronic structure calculations. Both numerical and analytical approaches are applied to two-electron atomic systems. The results of the numerical test demonstrated the potential of the noninteger values of principal quantum number for the improving of Bessel type functions approach in use of LCAO methods. Nevertheless, the analytical approach is still not suitable and in development and needs to be investigated further.The performance of the presented basis functions is also compared to the numerical Hartree–Fock results.

Analytical approximation to the local softness and hypersoftness and to their applications as reactivity indicators.

Local density functional theory derivatives of the electron density have been calculated analytically for the set of canonical hydrogenic orbitals; original solutions have been obtained using the novel density gradient theorem. Results for the first and second derivatives of electron density over N (number of electrons) and over μ (chemical potential) have been demonstrated. Calculations of the state functions ΔN, ΔE, and Δμ disturbed by an external potential Δv(r) have been obtained via the concept of alchemical derivatives. The local softness s(r) and local hypersoftness [ds(r)/dN]v have been proved to provide crucial chemical information on the sensitivity of orbital density to the disturbance of the external potential Δv(r), leading to electron exchange ΔN and the corresponding changes of the state functions ΔE, Δμ. The results are fully compatible with the well-understood character of atomic orbitals in chemistry and open a perspective to applications to atoms, free or bonded.

Symmetrization of Localized Molecular Orbitals.

We present a novel algorithm for (i) detecting approximate symmetries inherently present among spatially localized molecular orbitals and (ii) enforcing these in numerically exact manners by means of unitary optimization techniques. The vast potential of our algorithm to compress a full set of molecular orbitals into only a minimal set of symmetry-unique orbitals is demonstrated, starting from localized bases of either Pipek-Mezey or Foster-Boys orbitals. Comparisons of results based on either of these two localization procedures indicate that Foster-Boys molecular orbitals can be spanned by a smaller number of symmetry-unique orbitals on average, making these outstanding candidates for the exploitation of general, (non-)Abelian point-group symmetries in a range of local correlation methods. As an illustration of said compressibility, our algorithm is capable of identifying a mere 14 symmetry-unique orbitals for the buckminsterfullerene in the highly symmetric Ih molecular point group, corresponding to only 1.7% of its total 840 molecular orbitals in a standard double-ζ basis set. The present work thus marks an important advancement in the exploitation of point-group symmetry within local correlation methods, for which the appropriate adaption of symmetry uniqueness among orbitals has the potential to yield unprecedented speed-ups.

Further investigations into a Laplace MP2 method using range separated Coulomb potential and orbital selective virtuals: Multipole correction, OSV extrapolation, and critical assessment.

We report further investigations to aid the development of a Laplace MP2 (second-order Møller Plesset) method with a range separated Coulomb potential partitioned into short- and long-range parts. The implementation of the method extensively uses sparse matrix algebra, density fitting techniques for the short-range part, and a Fourier transformation in spherical coordinates for the long-range part of the potential. Localized molecular orbitals are employed for the occupied space, whereas virtual space is described by orbital specific virtual orbitals (OSVs) associated with localized molecular orbitals. The Fourier transform is deficient for very large distances between localized occupied orbitals, and a multipole expansion for widely separated pairs is introduced for the direct MP2 contribution, which is applicable also to non-Coulombic potentials that do not satisfy the Laplace equation. For the exchange contribution, an efficient screening of contributing localized occupied pairs is employed, which is discussed more completely here. To mitigate errors due to the truncation of OSVs, a simple and efficient extrapolation procedure is used to obtain results close to MP2 for the full basis set of atomic orbitals Using a suitable set of default parameters, the accuracy of the approach is demonstrated. The current implementation of the approach is not very efficient, and the aim of this paper is to introduce and critically discuss ideas that can have more general applicability beyond MP2 calculations for large molecules.

Reachable set analysis of practical trans-lunar orbit via a retrograde semi-analytic model

Different from the Apollo flight mode, a safer trans-lunar flight mode for the crews is preferred. The previous-arrived lunar module rendezvouses with the crew exploration vehicle at a low lunar destination orbit, and then the crews ride the lunar module to descend the lunar surface sampling destination. The lunar module, which includes the descent and ascent stages, flies from a low Earth orbit to the low lunar destination orbit with two tangential impulses. The low lunar destination orbital reachable set of the practical trans-lunar orbit limits the feasible lunar surface sampling region. Therefore, this paper addresses the low lunar destination orbital reachable set of the practical trans-lunar orbit. A retrograde semi-analytic model is proposed for rapidly computing the practical two-impulse trans-lunar orbit firstly, which refers the ephemeris table twice for more precision perilune orbital elements. The reachable set is generated using the multiple-level traversal searching approach with this retrograde semi-analytic mode. Its envelope is re-checked by the continuation theory with the high-precision orbital model. Besides, some factors that affect the reachable set are also measured. The results show that neither the Earth–Moon distance nor the trans-lunar duration affects the reachable set. However, if the trans-lunar injection inclination is smaller than the inclination of the moon’s path, the reachable set becomes smaller or even reduces into an empty set. In brief, the proposed retrograde semi-analytic model for computing the reachable set provides a helpful and fast tool for selecting an applicable lunar surface sampling site for the manned lunar mission overall design.Graphical

Local Potentials Reconstructed within Linearly Independent Product Basis Sets of Increasing Size.

Given a matrix representation of a local potential v(r) within a one-electron basis set of functions that form linearly independent products (LIP), it is possible to construct a well-defined local potential that is equivalent to v(r) within that basis set and has the form of an expansion in basis function products. Recently, we showed that for exchange-correlation potentials vXC(r) defined on the infinite-dimensional Hilbert space, the potentials reconstructed from matrices of vXC(r) within minimal LIP basis sets of occupied Kohn-Sham orbitals bear only qualitative resemblance to the originals. Here, we show that if the LIP basis set is enlarged by including low-lying virtual Kohn-Sham orbitals, the agreement between and vXC(r) improves to the extent that the basis function products are appropriate as a basis for vXC(r). These findings validate the LIP technology as a rigorous potential reconstruction method.

Downward quantum learning from element 118: Automated generation of Fermi-Löwdin orbitals for all atoms.

A new algorithm based on a rigorous theorem and quantum data computationally mined from element 118 guarantees automated construction of initial Fermi-Löwdin-Orbital (FLO) starting points for all elements in the Periodic Table. It defines a means for constructing a small library of scalable FLOs for universal use in molecular and solid-state calculations. The method can be systematically improved for greater efficiency and for applications to excited states such as x-ray excitations and optically silent excitations. FLOs were introduced to recast the Perdew-Zunger self-interaction correction (PZSIC) into an explicit unitarily invariant form. The FLOs are generated from a set of N quasi-classical electron positions, referred to as Fermi-Orbital descriptors (FODs), and a set of N-orthonormal single-electron orbitals. FOD positions, when optimized, minimize the PZSIC total energy. However, creating sets of starting FODs that lead to a positive definite Fermi orbital overlap matrix has proven to be challenging for systems composed of open-shell atoms and ions. The proof herein guarantees the existence of a FLOSIC solution and further guarantees that if a solution for N electrons is found, it can be used to generate a minimum of N - 1 and a maximum of 2N - 2 initial starting points for systems composed of a smaller number of electrons. Applications to heavy and super-heavy atoms are presented. All starting solutions reported here were obtained from a solution for element 118, Oganesson.

Sparsity of the electron repulsion integral tensor using different localized virtual orbital representations in local second-order Møller-Plesset theory.

Utilizing localized orbitals, local correlation theory can reduce the unphysically high system-size scaling of post-Hartree-Fock (post-HF) methods to linear scaling in insulating molecules. The sparsity of the four-index electron repulsion integral (ERI) tensor is central to achieving this reduction. For second-order Møller-Plesset theory (MP2), one of the simplest post-HF methods, only the (ia|jb) ERIs are needed, coupling occupied orbitals i, j and virtuals a, b. In this paper, we compare the numerical sparsity (called the "ragged list") and two other approaches revealing the low-rank sparsity of the ERI. The ragged list requires only one set of (localized) virtual orbitals, and we find that the orthogonal valence virtual-hard virtual set of virtuals originally proposed by Subotnik et al. gives the sparsest ERI tensor. To further compress the ERI tensor, the pair natural orbital (PNO) type representation uses different sets of virtual orbitals for different occupied orbital pairs, while the occupied-specific virtual (OSV) approach uses different virtuals for each occupied orbital. Our results indicate that while the low-rank PNO representation achieves significant rank reduction, it also requires more memory than the ragged list. The OSV approach requires similar memory to that of the ragged list, but it involves greater algorithmic complexity. An approximation (called the "fixed sparsity pattern") for solving the local MP2 equations using the numerically sparse ERI tensor is proposed and tested to be sufficiently accurate and to have highly controllable error. A low-scaling local MP2 algorithm based on the ragged list and the fixed sparsity pattern is therefore promising.

Characterization of excited states in time-dependent density functional theory using localized molecular orbitals.

Localized molecular orbitals are often used for the analysis of chemical bonds, but they can also serve to efficiently and comprehensibly compute linear response properties. While conventional canonical molecular orbitals provide an adequate basis for the treatment of excited states, a chemically meaningful identification of the different excited-state processes is difficult within such a delocalized orbital basis. In this work, starting from an initial set of supermolecular canonical molecular orbitals, we provide a simple one-step top-down embedding procedure for generating a set of orbitals, which are localized in terms of the supermolecule but delocalized over each subsystem composing the supermolecule. Using an orbital partitioning scheme based on such sets of localized orbitals, we further present a procedure for the construction of local excitations and charge-transfer states within the linear response framework of time-dependent density functional theory (TDDFT). This procedure provides direct access to approximate diabatic excitation energies and, under the Tamm-Dancoff approximation, also their corresponding electronic couplings-quantities that are of primary importance in modeling energy transfer processes in complex biological systems. Our approach is compared with a recently developed diabatization procedure based on subsystem TDDFT using projection operators, which leads to a similar set of working equations. Although both of these methods differ in the general localization strategies adopted and the type of basis functions (Slaters vs Gaussians) employed, an overall decent agreement is obtained.

ωB97X-3c: A composite range-separated hybrid DFT method with a molecule-optimized polarized valence double-ζ basis set.

A new composite density functional theory (DFT) method is presented. It is based on ωB97X-V as one of the best-performing density functionals for the GMTKN55 thermochemistry database and completes the family of "3c" methods toward range-separated hybrid DFT. This method is consistently available for all elements up to Rn (Z = 1-86). Its further key ingredients are a polarized valence double-ζ (vDZP) Gaussian basis set, which was fully optimized in molecular DFT calculations, in combination with large-core effective core potentials and a specially adapted D4 dispersion correction. Unlike most existing double-ζ atomic orbital sets, vDZP shows only small basis set superposition errors (BSSEs) and can compete with standard sets of triple-ζ quality. Small residual BSSE effects are efficiently absorbed by the D4 damping scheme, which overall eliminates the need for an explicit treatment or empirical corrections for BSSE. Thorough tests on a variety of thermochemistry benchmark sets show that the new composite method, dubbed ωB97X-3c, is on par with or even outperforms standard hybrid DFT methods in a quadruple-zeta basis set at a small fraction of the computational cost. Particular strengths of this method are the description of non-covalent interactions and barrier heights, for which it is among the best-performing density functionals overall.

On the derived Lusztig correspondence

Abstract Let G be a connected reductive group, T a maximal torus of G, N the normalizer of T and $W=N/T$ the Weyl group of G. Let ${\mathfrak {g}}$ and ${\mathfrak {t}}$ be the Lie algebras of G and T. The affine variety $\mathfrak {car}={\mathfrak {t}}/\!/W$ of semisimple G-orbits of ${\mathfrak {g}}$ has a natural stratification $$ \begin{align*} \mathfrak{car}=\coprod_{\lambda}\mathfrak{car}_{\lambda} \end{align*} $$ indexed by the set of G-conjugacy classes of Levi subgroups: the open stratum is the set of regular semisimple orbits and the closed stratum is the set of central orbits. In [17], Rider considered the triangulated subcategory $D_{\mathrm {c}}^{\mathrm {b}}([{\mathfrak {g}}_{\mathrm {nil}}/G])^{\mathrm {Spr}}$ of $D_{\mathrm {c}}^{\mathrm {b}}([{\mathfrak {g}}_{\mathrm {nil}}/G])$ generated by the direct summand of the Springer sheaf. She proved that it is equivalent to the derived category of finitely generated dg modules over the smash product algebra ${\overline {\mathbb {Q}}_{\ell }}[W]\# H^{\bullet }_G(G/B)$ where $H^{\bullet }_G(G/B)$ is the G-equivariant cohomology of the flag variety. Notice that the later derived category is $D_{\mathrm {c}}^{\mathrm {b}}(\mathrm {B}(N))$ where $\mathrm {B}(N)=[\mathrm {Spec}(k)/N]$ is the classifying stack of N-torsors. The aim of this paper is to understand geometrically and generalise Rider’s equivalence of categories: For each $\lambda $ we construct a cohomological correspondence inducing an equivalence of categories between $D_{\mathrm {c}}^{\mathrm {b}}([{\mathfrak {t}}_{\lambda }/N])$ and $D_{\mathrm {c}}^{\mathrm {b}}([{\mathfrak {g}}_{\lambda }/G])^{\mathrm {Spr}}$ .

Second-order superatoms: Au52-PAP featuring a three-dimensional cluster-of-clusters core.

The recent characterization of Au52-PAP cluster can be viewed as a three-dimensional arrangement featuring four Au13 motifs. As a result, a new set of superatomic orbitals are built up from the superatomic shell of each constituent unit, denoted by 1S'21P'62S'21D'102P'61F'6 and, thus, referred to as a second-order superatomic shell structure. This favors the rationalization of larger species toward the formation of cluster-assembled materials of different sizes.

Classification of Morse–Smale Diffeomorphisms with a Finite Set of Heteroclinic Orbits on Surfaces

Classification of Morse–Smale Diffeomorphisms with a Finite Set of Heteroclinic Orbits on Surfaces