One-electron Functions Research Articles

Scalar and vectorized implementations of the density functional theory via Geant4 and GeantV project

In this paper, we present the implementation of the Density Functional Theory (DFT) method using the Geant4-DNA framework in the Single Instruction Single Data (SISD) mode. Furthermore, this implementation is improved in terms of execution time within the GeantV project with vectorization techniques such as Single Instruction Multiple Data (SIMD). Within this framework, a set of SIMD strategies in molecular calculation algorithms such as one-electron operators, two-electron operator, quadrature grids, and functionals, was implemented using the VecCore library. The applications developed in this work implement two DFT functionals, the Local Density Approximation (LDA) and the General Gradient Approximation (GGA), to approximate the molecular ground-state energies of small molecules and amino acids. To assess the performance of the implementations, a standard test simulation was performed in multiple CPU platforms. The SIMD vectorization strategy significantly accelerates DFT calculations, leading to time ratios ranging from 1.6 to 5.4 in either individual steps or entire implementations when compared with the scalar process within Geant4.

Roadmap of electrons from donor side to the reaction center of photosynthetic purple bacteria with mutated cytochromes

In photosynthetic bacteria, the absorbed light drives the canonical cyclic electron transfer between the reaction center and the cytochrome bc1 complexes via the pools of mobile electron carriers. If kinetic or structural barriers hinder the participation of the bc1 complex in the cyclic flow of electrons, then the pools of mobile redox agents must supply the electrons for the multiple turnovers of the reaction center. These conditions were achieved by continuous high light excitation of intact cells of bacterial strains Rba. sphaeroides and Rvx. gelatinosus with depleted donor side cytochromes c2 (cycA) and tetraheme cytochrome subunit (pufC), respectively. The gradual oxidation by ferricyanide further reduced the availability of electron donors to pufC. Electron transfer through the reaction center was tracked by absorption change and by induction and relaxation of the fluorescence of the bacteriochlorophyll dimer. The rate constants of the electron transfer (~ 3 × 103 s‒1) from the mobile donors of Rvx. gelatinosus bound either to the RC (pufC) or to the tetraheme subunit (wild type) were similar. The electrons transferred through the reaction center dimer were supplied entirely by the donor pool; their number amounted to about 5 in wild type Rvx. gelatinosus and decreased to 1 in pufC oxidized by ferricyanide. Fluorescence yield was measured as a function of the oxidized fraction of the dimer and its complex shape reveals the contribution of two competing processes: the migration of the excitation energy among the photosynthetic units and the availability of electron donors to the oxidized dimer. The experimental results were simulated and rationalized by a simple kinetic model of the two-electron cycling of the acceptor side combined with aperiodic one-electron redox function of the donor side.

Phenomenological description of the acidity of the citric acid and its deprotonated species: informational-theoretical study.

In spite of the fact that molecular acidity is a fundamental physicochemical property of molecular systems, the vast majority of theoretical studies have focused attention on monoprotic acids and on the prediction of pKa's. Polyprotic acids, represent a challenge for electronic structure calculations since the multiple acidic sites result in a vast group of species with different conformations and reactivities. In this work, Information-theoretic (IT) concepts of localizability, order and uniformity are applied to the Citric Acid and its deprotonated species through the one-electron density functionals: Shannon entropy (S), Fisher information (I) and Disequilibrium (D), respectively. We pursue the goal of characterizing the acidity of the aforementioned species with the aim to associate the IT concepts to chemical features such as the polarizability of the protonated/deprotonated species, the liability of the acidic sites, atomic electrostatic potentials, covalent bonding. IT analyses looks very promising for future studies on the acidity of specific deprotonation-sites of polyprotic acids. Density functional theory (DFT) calculations were performed with Gaussian 09 program. A sensitivity analysis of the IT-measures was performed for the citric acid and the citrate using B3LYP, B3PW91, BPW91, M05-2X, M06-2X and PBEPBE functionals with the 6-311++g(3df,2p), 6-311++g(d,p), 6.311+g(d,p) and aug-cc-pVDZ basis sets. The rest of the analysis was performed with the M05-2X/6-311+G(d,p) level of theory. Additionally, aqueous media was considered by use of the SMD solvent model. The IT-measures were calculated using a suite of programs developed in our laboratory jointly with the DGRID software package.

Ground and excited states of even-numbered Hubbard ring at half-filling: comparison of the extended Gutzwiller approach with exact diagonalization

It remains a great challenge in condensed matter physics to develop a method to treat strongly correlated many-body systems with balanced accuracy and efficiency. We introduce an extended Gutzwiller (EG) method incorporating a manifold technique, which builds an effective manifold of the many-body Hilbert space, to describe the ground-state (GS) and excited-state (ES) properties of strongly correlated electrons. We systematically apply an EG projector onto the GS and ES of a non-interacting system. Diagonalization of the true Hamiltonian within the manifold formed by the resulting EG wavefunctions gives the approximate GS and ES of the correlated system. To validate this technique, we implement it on even-numbered fermionic Hubbard rings at half-filling with periodic boundary conditions, and compare the results with the exact diagonalization (ED) method. The EG method is capable of generating high-quality GS and low-lying ES wavefunctions, as evidenced by the high overlaps of wavefunctions between the EG and ED methods. Favorable comparisons are also achieved for other quantities including the total energy, the double occupancy, the total spin and the staggered magnetization. With the capability of accessing the ESs, the EG method can capture the essential features of the one-electron removal spectral function that contains contributions from states deep in the excited spectrum. Finally, we provide an outlook on the application of this method on large extended systems.

Time-dependent multiconfiguration self-consistent-field and time-dependent optimized coupled-cluster methods for intense laser-driven multielectron dynamics

We reviewed the time-dependent multiconfiguration self-consistent-field (TD-MCSCF) method and the time-dependent optimized coupled-cluster (TD-OCC) method for first-principles simulations of high-field phenomena such as tunneling ionization and high-order harmonic generation in atoms and molecules irradiated by a strong laser field. These methods provide a flexible and systematically improvable description of the multielectron dynamics by expressing the all-electron wavefunction by configuration interaction expansion or coupled-cluster expansion, using time-dependent one-electron orbital functions. The time-dependent variational principle plays a key role in deriving these methods while satisfying gauge invariance and Ehrenfest theorem. The real-time/real-space implementation with an absorbing boundary condition enables the simulation of high-field processes involving multiple excitation and ionization. We present a detailed, comprehensive discussion of such features of the TD-MCSCF and TD-OCC methods.

Multiplicative potentials for kinetic energy and exact exchange.

Harriman showed that within finite basis sets of one-electron functions that form linearly independent products (LIP), differential and integral operators can be represented exactly and unambiguously by multiplicative (local) potentials. Although almost no standard basis sets of quantum chemistry form LIPs in a numerical sense, occupied self-consistent field (SCF) orbitals routinely do so. Using minimal LIP basis sets of occupied SCF orbitals, we construct multiplicative potentials for electronic kinetic energy and exact exchange that reproduce the Hartree-Fock and Kohn-Sham Hamiltonian matrices and electron densities for atoms and molecules. The results highlight fundamental differences between local and nonlocal operators and suggest a practical possibility of developing exact kinetic energy functionals within finite basis sets by using effective local potentials.

Reconstruction of Exchange-Correlation Potentials from Their Matrix Representations.

Within a basis set of one-electron functions that form linearly independent products (LIPs), it is always possible to construct a unique local (multiplicative) real-space potential that is precisely equivalent to an arbitrary given operator. Although standard basis sets of quantum chemistry rarely form LIPs in a numerical sense, occupied and low-lying virtual canonical Kohn-Sham orbitals often do so, at least for small atoms and molecules. Using these principles, we construct atomic and molecular exchange-correlation potentials from their matrix representations in LIP basis sets of occupied canonical Kohn-Sham orbitals. The reconstructions are found to imitate the original potentials in a consistent but exaggerated way. Since the original and reconstructed potentials produce the same ground-state electron density and energy within the associated LIP basis set, the procedure may be regarded as a rigorous solution to the Kohn-Sham inversion problem within the subspace spanned by the occupied Kohn-Sham orbitals.

A Universal Power Law Governing the Accuracy of Wave Function-Based Electronic Structure Calculations

A universal power law governing the accuracy of wave function-based electronic structure calculations is derived from first principles. The resulting expression ΔE(N,N)N≳19π2gNN, where g is a system-specific factor assuming values between zero and one and ≳ stands for asymptotic inequality at the limit of N→∞, allows facile estimation of the error ΔE(N,N) in the electronic energy of a singlet state of an N-electron system computed with a basis set of N one-electron functions. Several approaches to the estimation of the factor g, which depends on the on-top two-electron density, are presented.

Errors in approximate ionization energies due to the one-electron space truncation of the EKT eigenproblem.

Unless the approximate wavefunction of the parent system is expressed in terms of explicitly correlated basis functions, the finite size of the generalized Fock matrix is unlikely to be the leading source of the truncation error in the ionization energy E produced by the EKT (extended Koopmans' theorem) formalism. This conclusion is drawn from a rigorous analysis that involves error partitioning into the parent- and ionized-system contributions, the former being governed by asymptotic power laws when the underlying wavefunction is assembled from a large number of spinorbitals and the latter arising from the truncation of the infinite-dimensional matrix V whose elements involve the 1-, 2-, and 3-matrices of the parent system. Quite surprisingly, the decay of the second contribution with the number n of the natural spinorbitals (NOs) employed in the construction of the truncated V turns out to be strongly system-dependent even in the simplest case of the 1S states of two-electron systems, following the n-5 power law for the helium atom while exhibiting an erratic behavior for the H- anion. This phenomenon, which stems from the presence of the so-called solitonic natural spinorbitals among the NOs, renders the extrapolation of the EKT approximates of E to the complete-basis-set limit generally unfeasible. However, attaining that limit is not contingent upon attempted reproduction of the ill-defined one-electron function known as "the removal orbital," which does not have to be invoked in the derivation of EKT and whose expansion in terms of the NOs diverges.

Fast and Accurate Quantum Mechanical Modeling of Large Molecular Systems Using Small Basis Set Hartree-Fock Methods Corrected with Atom-Centered Potentials.

There has been significant interest in developing fast and accurate quantum mechanical methods for modeling large molecular systems. In this work, by utilizing a machine learning regression technique, we have developed new low-cost quantum mechanical approaches to model large molecular systems. The developed approaches rely on using one-electron Gaussian-type functions called atom-centered potentials (ACPs) to correct for the basis set incompleteness and the lack of correlation effects in the underlying minimal or small basis set Hartree-Fock (HF) methods. In particular, ACPs are proposed for ten elements common in organic and bioorganic chemistry (H, B, C, N, O, F, Si, P, S, and Cl) and four different base methods: two minimal basis sets (MINIs and MINIX) plus a double-ζ basis set (6-31G*) in combination with dispersion-corrected HF (HF-D3/MINIs, HF-D3/MINIX, HF-D3/6-31G*) and the HF-3c method. The new ACPs are trained on a very large set (73 832 data points) of noncovalent properties (interaction and conformational energies) and validated additionally on a set of 32 048 data points. All reference data are of complete basis set coupled-cluster quality, mostly CCSD(T)/CBS. The proposed ACP-corrected methods are shown to give errors in the tenths of a kcal/mol range for noncovalent interaction energies and up to 2 kcal/mol for molecular conformational energies. More importantly, the average errors are similar in the training and validation sets, confirming the robustness and applicability of these methods outside the boundaries of the training set. In addition, the performance of the new ACP-corrected methods is similar to complete basis set density functional theory (DFT) but at a cost that is orders of magnitude lower, and the proposed ACPs can be used in any computational chemistry program that supports effective-core potentials without modification. It is also shown that ACPs improve the description of covalent and noncovalent bond geometries of the underlying methods and that the improvement brought about by the application of the ACPs is directly related to the number of atoms to which they are applied, allowing the treatment of systems containing some atoms for which ACPs are not available. Overall, the ACP-corrected methods proposed in this work constitute an alternative accurate, economical, and reliable quantum mechanical approach to describe the geometries, interaction energies, and conformational energies of systems with hundreds to thousands of atoms.

The Hitchhiker's Guide to the Wave Function∥.

The electronic wave function of molecules is 3N-dimensional and inseparable in the coordinates of the N electrons. Whereas molecular orbitals are often invoked to visualize the electronic structure, they are nonunique, with the same 3N-dimensional wave function being represented by an infinite number of 3-D, one-electron functions (orbitals). Furthermore, multireference wave functions cannot be described by an antisymmetrized product of a single set of occupied orbitals. What is required is a way to visualize the full dimensionality of the wave function, including the effects of correlation, as a 3N-dimensional being would be able to do. In the past 5 years, we have been developing a way to analyze and visualize highly dimensional wave functions by focusing on the structure of the repeating unit demanded by fermionic behavior. This 3N-dimensional repeating unit, the wave function "tile", can be projected onto the three dimensions of each electron, in turn, to reveal the complete electronic structure. It is found that the tile reproduces canonical chemical motifs such as core-electrons, single bonds and lone pairs. Multiple bonds emerge as the "banana" bonds favored by Pauling. As a function of the reaction coordinate, electron motions are visualized that correspond to the curly arrow notation of organic chemists. Excited states can also be inspected. Analyzing a wave function in terms of fermionic tiling allows for insight not facilitated by the inspection of orbitals or configuration interaction vectors: The wave function tiles of resonance structures reveal that electron correlation in benzene pushes opposing spin electrons to occupy alternate Kekulé structures, and in C2, the emerging structure supports the notion of a triply bonded structure with a weak, fourth bonding contribution.

Energy natural orbitals.

We propose and numerically demonstrate that highly correlated electronic wavefunctions such as those of configuration interaction, the cluster expansion, and so on, and electron wavepackets superposed thereof can be analyzed in terms of one-electron functions, which we call energy natural orbitals (ENOs). As the name suggests, ENOs are members of the broad family of natural orbitals defined by Löwdin, in that they are eigenfunctions of the energy density operator. One of the major characteristics is that the (orbital) energies of all the ENOs are summed up exactly equal to the total electronic energy of a wavefunction under study. Another outstanding feature is that the population of each ENO varies as the chemical reaction proceeds, keeping the total population constant though. The study of ENOs has been driven by the need for new methods to analyze extremely complicated nonadiabatic electron wavepackets such as those embedded in highly quasi-degenerate excited-state manifolds. Yet, ENOs can be applied to scrutinize many other chemical reactions, ranging from the ordinary concerted reactions, nonadiabatic reactions, and Woodward-Hoffman forbidden reactions, to excited-state reactions. We here present the properties of ENOs and a couple of case studies of numerical realization, one of which is about the mechanism of nonadiabatic electron transfer.

Time reversal symmetry and the structure of Cooper pair wavefunction in topological superconductor UTe2

Recently discovered topological superconductor UTe2 reveals unusual symmetry features, namely, chiral nonunitary triplet ESP (equal spin pairing) SOP (superconducting order parameter) with point nodes. However, the description of these symmetry features in a framework of crystal point group D2h is not sufficient. In the present work the space-group approach to the wavefunction of a Cooper pair, based on Mackey-Bradley theorem, is extended, making use of Anderson ansatz, to the pair functions in Shubnikov group symmetry and applied to UTe2. A nonunitary triplet SOP was obtained for Shubnikov group m′m′m on the basis of one-electron functions. Irreducible corepresentations of group m′m′m for triplet ESP Cooper pairs, whose nodal structure corresponds to all experimentally established symmetry features, namely point nodes, nonunitarity and chirality, of SOP of UTe2 are identified.

Solving the Schrödinger equation of atoms and molecules using one- and two-electron integrals only

A variational theory called free-complement (FC) ${s}_{ij}$ theory for solving the Schr\odinger equation of atoms and molecules using only one- and two-electron integrals over Slater or Gaussian functions is proposed. It is derived from the scaled Schr\odinger equation [Phys Rev. Lett. 93, 030403 (2004)] by replacing the two-electron part ${r}_{ij}$ of the scaling $g$ operator with ${s}_{ij}={r}_{ij}^{2}$, both of which solve the divergence difficulty inherent to the original Schr\odinger equation by avoiding the interelectron collisions. The ${s}_{ij}$ function can be rewritten with only one-electron functions so that when the initial wave function of the FC theory is composed of only one-electron functions, the FC wave function including ${s}_{ij}$ can be rewritten with only one-electron functions. Therefore, the variational calculations of the FC ${s}_{ij}$ theory can be performed with only one- and two-electron integrals. However, in comparison with the ${r}_{ij}$ function, the ${s}_{ij}$ function behaves less efficiently when two electrons come close to each other: the electron-electron cusp condition is not satisfied with the FC ${s}_{ij}$ theory, though the wave function has explicit ${r}_{ij}$ dependence. On the other hand, the electron-nuclear cusp condition is satisfied, even with the Gaussian functions, for the presence of the electron-nuclear function ${r}_{iA}$ in the $g$ operator. Test applications of the FC ${s}_{ij}$ theory were done to He, Li, and the $^{5}S^{\mathrm{o}} {\mathit{sp}}^{3}$ state of carbon atom and to hydrogen molecule using the local-molecular orbital (MO)- and valence bond (VB)-type initial functions. We examined both Slater and Gaussian functions. As the order of the FC theory increased, the wave functions were improved and the energies approached from a few kcal/mol to less than 1 kcal/mol accuracies relative to the known exact values. Thus, with the FC ${s}_{ij}$ theory, the Schr\odinger equation was solved to less than 1 kcal/mol accuracy for all the systems examined. The costs for the ${s}_{ij}$ calculations were much smaller than those of the ${r}_{ij}$ theory. However, for the $^{5}S^{\mathrm{o}} {\mathit{sp}}^{3}$ state of the carbon atom, we observed that the convergence rate became slow when the calculation came close to $\ensuremath{\sim}1$ kcal/mol accuracy. This suggests that for the FC ${s}_{ij}$ theory, the calculations would become easier if the required accuracy is within a few kcal/mol for the absolute energy. Actually, what we need in chemical studies are mostly related to the difference energy, whose accuracy could be better than that of the absolute energy with the robust variational method. Two lines of future studies were suggested.

Role of matrix elements in the time-resolved photoemission signal

Time- and angle-resolved photoemission spectroscopy (TR-ARPES) provides access to the ultrafast evolution of electrons and many-body interactions in solid-state systems. However, the momentum- and energy-resolved transient photoemission intensity may not be unambiguously described by the intrinsic relaxation dynamics of photoexcited electrons alone. The interpretation of the time-dependent photoemission signal can be affected by the transient evolution of the electronic distribution, and both the one-electron removal spectral function as well as the photoemission matrix elements. Here we investigate the topological insulator Bi1.1Sb0.9Te2S to demonstrate, by means of a detailed probe-polarization dependent study, the transient contribution of matrix elements to TR-ARPES.

United calculation of the optical and EPR spectral data for Co2+-doped CdS crystal

This article reports a united calculation of the optical band positions and EPR parameters (g factors g//, g⊥ and zero-field splitting D) for Co2+ ion at the trigonal tetrahedral Cd2+ site of CdS crystal by using the complete diagonalization (of energy matrix) method built on the cluster approach. In the approach, the one-electron basis functions are the molecular orbitals composed of the d orbitals of central dn ion and p orbitals of ligands (rather than only the pure d orbitals in the conventional crystal-field theory), and so the contributions from not only the spin-orbit parameter of dn ion, but also that of ligand ion are contained. The calculated results are in rational agreement with the twenty-two (nineteen optical band positions and three EPR parameters) observed spectral values found in the literature. The defect structure (particularly, the angular distortion) of the trigonal Co2+ center in CdS: Co2+ is also acquired by the calculation. The outcomes are discussed.

Investigations of the optical and EPR spectral data for the trigonal Cr3+ center in ZnGa2O4: Cr3+ crystal with a unified method-complete diagonalization of energy matrix

The optical and EPR spectral data of the trigonal Cr3+ centers in ZnGa2O4: Cr3+ crystal are unifiedly calculated by applying the complete diagonalization of energy matrix method resting on the two-spin-orbit-parameter model, where the one-electron basis functions are the molecular orbitals and the contributions from both the spin-orbit parameters of dn ion and ligand ion are included. Thus, the Racah parameters B, C, two spin-orbit parameters and two orbit reduction factors can be evaluated from a adjustable parameter, the covalence factor, and so only three adjustable parameters are used in the calculation. The ten calculated spectral data (seven optical band positions and three EPR parameters, g factors g//, g⊥ and zero-field splitting D) are rationally consistent with the experimental values. The defect structure of the trigonal Cr3+ center is also given. These outcomes are discussed.

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 structure and correlations of vitamin B12 studied within the Haldane-Anderson impurity model

We study the electronic structure and correlations of vitamin B12 (cyanocobalamine) by using the framework of the multi-orbital single-impurity Haldane-Anderson model of a transition-metal impurity in a semiconductor host. The parameters of the effective Haldane-Anderson model are obtained within the Hartree-Fock (HF) approximation. The quantum Monte Carlo (QMC) technique is then used to calculate the one-electron and magnetic correlation functions of this effective model. We observe that new states form inside the semiconductor gap found by HF due to the intra-orbital Coulomb interaction at the impurity 3d orbitals. In particular, the lowest unoccupied states correspond to an impurity bound state, which consists of states from mainly the CN axial ligand and the corring ring as well as the Co e_g-like orbitals. We also observe that the Co(3d) orbitals can develop antiferromagnetic correlations with the surrounding atoms depending on the filling of the impurity bound states. In addition, we make comparisons of the HF+QMC data with the density functional theory calculations. We also discuss the photoabsorption spectrum of cyanocobalamine.

Uniform electron gases. III. Low-density gases on three-dimensional spheres.

By combining variational Monte Carlo (VMC) and complete-basis-set limit Hartree-Fock (HF) calculations, we have obtained near-exact correlation energies for low-density same-spin electrons on a three-dimensional sphere (3-sphere), i.e., the surface of a four-dimensional ball. In the VMC calculations, we compare the efficacies of two types of one-electron basis functions for these strongly correlated systems and analyze the energy convergence with respect to the quality of the Jastrow factor. The HF calculations employ spherical Gaussian functions (SGFs) which are the curved-space analogs of Cartesian Gaussian functions. At low densities, the electrons become relatively localized into Wigner crystals, and the natural SGF centers are found by solving the Thomson problem (i.e., the minimum-energy arrangement of n point charges) on the 3-sphere for various values of n. We have found 11 special values of n whose Thomson sites are equivalent. Three of these are the vertices of four-dimensional Platonic solids - the hyper-tetrahedron (n = 5), the hyper-octahedron (n = 8), and the 24-cell (n = 24) - and a fourth is a highly symmetric structure (n = 13) which has not previously been reported. By calculating the harmonic frequencies of the electrons around their equilibrium positions, we also find the first-order vibrational corrections to the Thomson energy.