Orbital Representation Research Articles

Holistic Approach to Partial Covalent Interactions in Protein Structure Prediction and Design with Rosetta

Partial covalent interactions (PCIs) in proteins, which include hydrogen bonds, salt bridges, cation-π, and π-π interactions, contribute to thermodynamic stability and facilitate interactions with other biomolecules. Several score functions have been developed within the Rosetta protein modeling framework that identify and evaluate these PCIs through analyzing the geometry between participating atoms. However, we hypothesize that PCIs can be unified through a simplified electron orbital representation. To test this hypothesis, we have introduced orbital based chemical descriptors for PCIs into Rosetta, called the PCI score function. Optimal geometries for the PCIs are derived from a statistical analysis of high-quality protein structures obtained from the Protein Data Bank (PDB), and the relative orientation of electron deficient hydrogen atoms and electron-rich lone pair or π orbitals are evaluated. We demonstrate that nativelike geometries of hydrogen bonds, salt bridges, cation-π, and π-π interactions are recapitulated during minimization of protein conformation. The packing density of tested protein structures increased from the standard score function from 0.62 to 0.64, closer to the native value of 0.70. Overall, rotamer recovery improved when using the PCI score function (75%) as compared to the standard Rosetta score function (74%). The PCI score function represents an improvement over the standard Rosetta score function for protein model scoring; in addition, it provides a platform for future directions in the analysis of small molecule to protein interactions, which depend on partial covalent interactions.

Analytic second-order energy derivatives in natural orbital functional theory

The analytic energy gradients in the atomic orbital representation have recently been published (J. Chem. Phys. 146, 014102, 2017) within the framework of the natural orbital functional theory (NOFT). We provide here an alternative expression for them in terms of natural orbitals, and use it to derive the analytic second-order energy derivatives with respect to nuclear displacements in the NOFT. The computational burden is shifted to the calculation of perturbed natural orbitals and occupancies, since a set of linear coupled-perturbed equations obtained from the variational Euler equations must be solved to attain the analytic Hessian at the perturbed geometry. The linear response of both natural orbitals and occupation numbers to nuclear geometry displacements need only specify the reconstruction of the second-order reduced density matrix in terms of occupation numbers.

Kinetic energy classification and smoothing for compact B-spline basis sets in quantum Monte Carlo.

Quantum Monte Carlo calculations of defect properties of transition metal oxides have become feasible in recent years due to increases in computing power. As the system size has grown, availability of on-node memory has become a limiting factor. Saving memory while minimizing computational cost is now a priority. The main growth in memory demand stems from the B-spline representation of the single particle orbitals, especially for heavier elements such as transition metals where semi-core states are present. Despite the associated memory costs, splines are computationally efficient. In this work, we explore alternatives to reduce the memory usage of splined orbitals without significantly affecting numerical fidelity or computational efficiency. We make use of the kinetic energy operator to both classify and smooth the occupied set of orbitals prior to splining. By using a partitioning scheme based on the per-orbital kinetic energy distributions, we show that memory savings of about 50% is possible for select transition metal oxide systems. For production supercells of practical interest, our scheme incurs a performance penalty of less than 5%.

A single-electron picture based on the multiconfiguration time-dependent Hartree–Fock method: application to the anisotropic ionization and subsequent high-harmonic generation of the CO molecule

The mechanisms of anisotropic near-IR tunnel ionization and high-order harmonic generation (HHG) in a CO molecule are theoretically investigated by using the multiconfiguration time-dependent Hartree–Fock (MCTDHF) method developed for the simulation of multielectron dynamics of molecules. The multielectron dynamics obtained by numerically solving the equations of motion (EOMs) in the MCTDHF method is converted to a single orbital picture in the natural orbital representation where the first-order reduced density matrix is diagonalized. The ionization through each natural orbital is examined and the process of HHG is classified into different optical paths designated by a combinations of initial, intermediate and final natural orbitals. The EOMs for natural spin-orbitals are also derived within the framework of the MCTDHF, which maintains the first-order reduced density matrix to be a diagonal one throughout the time propagation of a many-electron wave function. The orbital dependent, time-dependent effective potentials that govern the dynamics of respective time-dependent natural orbitals are deduced from the derived EOMs, of which the temporal variation can be used to interpret the motion of the electron density associated with each natural spin-orbital. The roles of the orbital shape, multiorbital ionization, linear Stark effect and multielectron interaction in the ionization and HHG of a CO molecule are revealed by the effective potentials obtained. When the laser electric field points to the nucleus O from C, tunnel ionization from the C atom side is enhanced; a hump structure originating from multielectron interaction is then formed on the top of the field-induced distorted barrier of the HOMO effective potential. This hump formation, responsible for the directional anisotropy of tunnel ionization, restrains the influence of the linear Stark effect on the energy shifts of bound states.

On the multi-layer multi-configurational time-dependent Hartree approach for bosons and fermions.

A multi-layer multi-configurational time-dependent Hartree (MCTDH) approach using a second quantization representation (SQR) based on optimized time-dependent orbitals is introduced. The approach combines elements of the multi-layer MCTDH-SQR approach of Wang and Thoss, which employs a preselected time-independent orbital basis, and the MCTDH for bosons and multi-configuration time-dependent Hartree-Fock approaches, which do not use multi-layering but employ time-dependent orbital bases. In contrast to existing MCTDH-type approaches, the results of the present approach for a given number of configurations are not invariant with respect to unitary transformations of the time-dependent orbital basis. Thus a natural orbital representation is chosen to achieve fast convergence with respect to the number of configurations employed. Equations of motion for the present ansatz, called (multi-layer) MCTDH in optimized second quantization representation, are derived. Furthermore, a scheme for the calculation of optimized unoccupied single-particle functions is given which can be used to avoid singularities in the equations of motion.

Dynamics of local symmetry correlators for interacting many-particle systems.

Recently [P. A. Kalozoumis et al. Phys. Rev. Lett. 113, 050403 (2014)] the concept of local symmetries in one-dimensional stationary wave propagation has been shown to lead to a class of invariant two-point currents that allow to generalize the parity and Bloch theorem. In the present work, we establish the theoretical framework of local symmetries for higher-dimensional interacting many-body systems. Based on the Bogoliubov-Born-Green-Kirkwood-Yvon hierarchy, we derive the equations of motion of local symmetry correlators which are off-diagonal elements of the reduced one-body density matrix at symmetry related positions. The natural orbital representation yields equations of motion for the convex sum of the local symmetry correlators of the natural orbitals as well as for the local symmetry correlators of the individual orbitals themselves. An alternative integral representation with a unique interpretation is provided. We discuss special cases, such as the bosonic and fermionic mean field theory, and show in particular that the invariance of two-point currents is recovered in the case of the non-interacting one-dimensional stationary wave propagation. Finally we derive the equations of motion for anomalous local symmetry correlators which indicate the breaking of a global into a local symmetry in the stationary non-interacting case.

A Nonorthogonal State-Interaction Approach for Matrix Product State Wave Functions.

We present a state-interaction approach for matrix product state (MPS) wave functions in a nonorthogonal molecular orbital basis. Our approach allows us to calculate, for example, transition and spin-orbit coupling matrix elements between arbitrary electronic states, provided that they share the same one-electron basis functions and size of the active orbital space, respectively. The key element is the transformation of the MPS wave functions of different states from a nonorthogonal to a biorthonormal molecular orbital basis representation, by exploiting a sequence of nonunitary transformations, following a proposal by Malmqvist [Int. J. Quantum Chem. 1986, 30, 479]. This is well-known for traditional wave function parametrizations but has not yet been exploited for MPS wave functions.

Electronic Properties of Cyclacenes from TAO-DFT.

Owing to the presence of strong static correlation effects, accurate prediction of the electronic properties (e.g., the singlet-triplet energy gaps, vertical ionization potentials, vertical electron affinities, fundamental gaps, symmetrized von Neumann entropy, active orbital occupation numbers, and real-space representation of active orbitals) of cyclacenes with n fused benzene rings (n = 4–100) has posed a great challenge to traditional electronic structure methods. To meet the challenge, we study these properties using our newly developed thermally-assisted-occupation density functional theory (TAO-DFT), a very efficient method for the study of large systems with strong static correlation effects. Besides, to examine the role of cyclic topology, the electronic properties of cyclacenes are also compared with those of acenes. Similar to acenes, the ground states of cyclacenes are singlets for all the cases studied. In contrast to acenes, the electronic properties of cyclacenes, however, exhibit oscillatory behavior (for n ≤ 30) in the approach to the corresponding properties of acenes with increasing number of benzene rings. On the basis of the calculated orbitals and their occupation numbers, the larger cyclacenes are shown to exhibit increasing polyradical character in their ground states, with the active orbitals being mainly localized at the peripheral carbon atoms.

Comment about the use of Monte-Carlo methodology for the representation of atomic electronic densities

Representations of atomic orbitals based on Monte-Carlo (MC) approaches are not always correct when using various sets of orthogonal coordinates other than Cartesian coordinates. The analysis proposed here gives elements for a proper use of MC methodology. It can be very useful for students and for teachers.

Natural ionization orbitals for interpreting electron detachment processes.

A compact orbital representation of ionization processes is described utilizing the difference of calculated one-particle density matrices. Natural orbital analysis involving this difference density matrix simplifies interpretation of electronic detachment processes and allows differentiation between one-electron transitions and shake-up/shake-off transitions, in which one-electron processes are accompanied by excitation of a second electron into the virtual orbital space.

Application of Semi-analytical Satellite Theory orbit propagator to orbit determination for space object catalog maintenance

Catalog maintenance for Space Situational Awareness (SSA) demands an accurate and computationally lean orbit propagation and orbit determination technique to cope with the ever increasing number of observed space objects. As an alternative to established numerical and analytical methods, we investigate the accuracy and computational load of the Draper Semi-analytical Satellite Theory (DSST). The standalone version of the DSST was enhanced with additional perturbation models to improve its recovery of short periodic motion. The accuracy of DSST is, for the first time, compared to a numerical propagator with fidelity force models for a comprehensive grid of low, medium, and high altitude orbits with varying eccentricity and different inclinations. Furthermore, the run-time of both propagators is compared as a function of propagation arc, output step size and gravity field order to assess its performance for a full range of relevant use cases. For use in orbit determination, a robust performance of DSST is demonstrated even in the case of sparse observations, which is most sensitive to mismodeled short periodic perturbations. Overall, DSST is shown to exhibit adequate accuracy at favorable computational speed for the full set of orbits that need to be considered in space surveillance. Along with the inherent benefits of a semi-analytical orbit representation, DSST provides an attractive alternative to the more common numerical orbit propagation techniques.

Comment on “Visualizing Three-Dimensional Hybrid Atomic Orbitals Using Winplot: An Application for Student Self Instruction”

The graphical representations of hybrid orbitals in “Visualizing Three-Dimensional Hybrid Atomic Orbitals Using Winplot: An Application for Student Self Instruction” do not accurately represent their isosurfaces.

A projection-free method for representing plane-wave DFT results in an atom-centered basis.

Plane wave density functional theory (DFT) is a powerful tool for gaining accurate, atomic level insight into bulk and surface structures. Yet, the delocalized nature of the plane wave basis set hinders the application of many powerful post-computation analysis approaches, many of which rely on localized atom-centered basis sets. Traditionally, this gap has been bridged via projection-based techniques from a plane wave to atom-centered basis. We instead propose an alternative projection-free approach utilizing direct calculation of matrix elements of the converged plane wave DFT Hamiltonian in an atom-centered basis. This projection-free approach yields a number of compelling advantages, including strict orthonormality of the resulting bands without artificial band mixing and access to the Hamiltonian matrix elements, while faithfully preserving the underlying DFT band structure. The resulting atomic orbital representation of the Kohn-Sham wavefunction and Hamiltonian provides a gateway to a wide variety of analysis approaches. We demonstrate the utility of the approach for a diverse set of chemical systems and example analysis approaches.

Double-Well Ultracold-Fermions Computational Microscopy: Wave-Function Anatomy of Attractive-Pairing and Wigner-Molecule Entanglement and Natural Orbitals.

"Bottom-up" approaches to the many-body physics of fermions have recently demonstrated precise number and site-resolved preparations with tunability of interparticle interactions in single-well, SW, and double-well, DW, nanoscale confinements created by manipulating ultracold fermionic atoms with optical tweezers. These experiments emulate an analogue-simulator mapping onto the requisite microscopic Hamiltonian, approaching realization of Feynmans' vision of quantum simulators that "will do exactly the same as nature". Here we report on exact benchmark configuration-interaction computational microscopy solutions of the Hamiltonian, uncovering the spectral evolution, wave function anatomy, and entanglement properties of the interacting fermions in the entire parameter range, including crossover from an SW to a DW confinement and a controllable energy imbalance between the wells. We demonstrate attractive pairing and formation of repulsive, highly correlated, ultracold Wigner molecules, well-described in the natural orbital representation. The agreement with the measurements affirms the henceforth gained deep insights into ultracold molecules and opens access to the size-dependent evolution of nanoclustered and condensed-matter phases and ultracold-atoms quantum information.

Orbital representations for the next generation of satellite-based augmentation systems

The landscape of the Global Navigation Satellite System (GNSS) is changing. New constellations are coming online, and a diversity of new signals are coming to the user space. Multi-frequency adds a means for ionospheric correction as well as robustness to jamming. Multi-constellation gives rise to better geometry and robustness to satellite failures. Systems which require a high degree of safety such as aviation require Satellite-Based Augmentation Systems (SBAS) to be used in conjunction with GNSS. As such, SBAS standards must be modernized to reflect the evolving GNSS environment. SBAS will deliver additional service on a new frequency at L5, giving the ideal opportunity to modernize the SBAS Minimum Operational Performance Standards (MOPS). Geostationary (GEO) satellites currently comprise the space segment of SBAS. However, GEOs remain at the equator limiting their visibility at the Poles. As activity in the Arctic is increasing, SBAS service in this region is of utmost importance to ensure safety. As such, it is desired that the next-generation L5 MOPS allow for orbit classes other than GEO. Orbital diversity for the delivery of SBAS corrections will allow for better visibility of this service on all places on earth. Here, we discuss the design and qualification of the L5 MOPS orbit messages, namely the ephemeris and almanac. These will support a multitude of orbit classes including all of those used today by both GNSS and SBAS.

Cubic-scaling iterative solution of the Bethe-Salpeter equation for finite systems

The Bethe-Salpeter equation (BSE) is currently the state of the art in the description of neutral electron excitations in both solids and large finite systems. It is capable of accurately treating charge-transfer excitations that present difficulties for simpler approaches. We present a local basis set formulation of the BSE for molecules where the optical spectrum is computed with the iterative Haydock recursion scheme, leading to a low computational complexity and memory footprint. Using a variant of the algorithm we can go beyond the Tamm-Dancoff approximation (TDA). We rederive the recursion relations for general matrix elements of a resolvent, show how they translate into continued fractions, and study the convergence of the method with the number of recursion coefficients and the role of different terminators. Due to the locality of the basis functions the computational cost of each iteration scales asymptotically as $O(N^3)$ with the number of atoms, while the number of iterations is typically much lower than the size of the underlying electron-hole basis. In practice we see that , even for systems with thousands of orbitals, the runtime will be dominated by the $O(N^2)$ operation of applying the Coulomb kernel in the atomic orbital representation

A semiclassical treatment of the ℓ-j correlation in atom-diatom collisions.

The explicit consideration of the vector correlations is an essential step when it comes to determining the mechanism of chemical reactions. Usual vector correlations involve initial and final relative velocity vectors and rotational angular momenta. However, the correlation between the orbital, ℓ, and rotational, j, angular momenta has seldom received any attention. In this article, we present a semiclassical methodology capable of describing the ℓ-j correlation that may serve as a connection between the quantum and quasiclassical treatments. Using the scattering matrix in the orbital angular momentum representation, the ℓ-j correlation is expressed as a probability density function of the angle formed by both vectors. This technique is exemplified through the H + D2 reaction and its accuracy is appraised by comparing with results derived from calculations based on quasiclassical trajectories.

Merging quantum chemistry packages with B-splines for the multichannel scattering problem

SynopsisTo study the photoionization of polyelectronic systems by attosecond pulses, we take advantage of existing quantum chemistry packages for the description of correlated electronic states, and of hybrid Gaussian- B-splines basis for the representation of continuum orbitals. In our approach, a short-range region, which can host all the interacting electrons and is described by commercial packages, is matched to a long-range region which describes single-ionization states in terms of a close-coupling expansion. We validate this approach showing multichannel ionization results for He.

Operator algebras and representations from commuting semigroup actions

Let $\sS$ be a countable, abelian semigroup of continuous surjections on a compact metric space $X$. Corresponding to this dynamical system we associate two operator algebras, the tensor algebra, and the semicrossed product. There is a unique smallest C$^*$-algebra into which an operator algebra is completely isometrically embedded, which is the C$^*$-envelope. We provide two distinct characterizations of the C$^*$-envelope of the tensor algebra; one developed in a general setting by Katsura, and the other using tools of projective and inductive limits, which gives the C$^*$-envelope as a crossed product C$^*$-algebra. We also study two natural classes of representations, the left regular representations and the orbit representations. The first is Shilov, and the second has a Shilov resolution.

Coupled-Perturbed SCF Approach for Calculating Static Polarizabilities and Hyperpolarizabilities with Nonorthogonal Localized Molecular Orbitals.

Coupled-perturbed self-consistent-field (CPSCF) approach has been broadly used for polarizabilities and hyperpolarizabilities computation. To extend this application to large systems, we have reformulated the CPSCF equations with nonorthogonal localized molecular orbitals (NOLMOs). NOLMOs are the most localized representation of electronic degrees of freedom. Methods based on NOLMOs are potentially ideal for investigating large systems. In atomic orbital representation, with a static external electric field added, the wave function and SCF operator of unperturbed NOLMO-SCF wave function/orbitals are expanded to different orders of perturbations. We have derived the corresponding equations up to the third order, which are significantly different from those of a conventional CPSCF method because of the release of the orthogonal restrictions on MOs. The solution to these equations has been implemented. Several chemical systems are used to verify our method. This work represents the first step toward efficient calculations of molecular response and excitation properties with NOLMOs.