One-particle Orbitals Research Articles

A Fully Linear Response G0W0 Method That Scales Linearly up to Tens of Thousands of Cores.

We present a G0W0 approach that is based on the evaluation of the linear response of the actions of the G0 and W0 operators. In this way we avoid sums over empty one-particle orbitals and do not have to explicitly develop the screened Coulomb interaction W0 on a dedicated basis. For a given orbital, the self-energy is found by summing terms relative to a set of points in the real-space simulation cell. This permits us to easily control the ratio of the accuracy to the computational cost. A trivial parallelization strategy allows strong linear scaling up to tens of thousands of computing cores.

Exploring one-particle orbitals in large many-body localized systems

Strong disorder in interacting quantum systems can give rise to the phenomenon of Many-Body Localization (MBL), which defies thermalization due to the formation of an extensive number of quasi local integrals of motion. The one particle operator content of these integrals of motion is related to the one particle orbitals of the one particle density matrix and shows a strong signature across the MBL transition as recently pointed out by Bera et al. [Phys. Rev. Lett. 115, 046603 (2015); Ann. Phys. 529, 1600356 (2017)]. We study the properties of the one particle orbitals of many-body eigenstates of an MBL system in one dimension. Using shift-and-invert MPS (SIMPS), a matrix product state method to target highly excited many-body eigenstates introduced in [Phys. Rev. Lett. 118, 017201 (2017)], we are able to obtain accurate results for large systems of sizes up to L = 64. We find that the one particle orbitals drawn from eigenstates at different energy densities have high overlap and their occupations are correlated with the energy of the eigenstates. Moreover, the standard deviation of the inverse participation ratio of these orbitals is maximal at the nose of the mobility edge. Also, the one particle orbitals decay exponentially in real space, with a correlation length that increases at low disorder. In addition, we find a 1/f distribution of the coupling constants of a certain range of the number operators of the OPOs, which is related to their exponential decay.

Parameterized effective potential for excited electronic states

The method of the parameterized effective potential (PEP) is developed for constructing the potential curves of excited electronic states of the same symmetry. Unlike the traditional methods of the density functional theory, the PEP is constructed as the direct mapping of an external potential. The PEP generates the one-particle orbitals such that the corresponding single-determinant Kohn-Sham function constructed on their basis is orthogonal to the determinant functions of lower-lying states. A comparison of the potential curves constructed for the HeH and H2 molecules by the PEP method with the precision data obtained based on the time-consuming configuration interaction methods shows that the PEP method can provide good accuracy for the geometry of an equilibrium configuration and the vertical excitation energies.

Continuum mechanics for quantum many-body systems: Linear response regime

We derive a closed equation of motion for the current density of an inhomogeneous quantum many-body system under the assumption that the time-dependent wave function can be described as a geometric deformation of the ground-state wave function. By describing the many-body system in terms of a single collective field we provide an alternative to traditional approaches, which emphasize one-particle orbitals. We refer to our approach as continuum mechanics for quantum many-body systems. In the linear response regime, the equation of motion for the displacement field becomes a linear fourth-order integro-differential equation, whose only inputs are the one-particle density matrix and the pair correlation function of the ground-state. The complexity of this equation remains essentially unchanged as the number of particles increases. We show that our equation of motion is a hermitian eigenvalue problem, which admits a complete set of orthonormal eigenfunctions under a scalar product that involves the ground-state density. Further, we show that the excitation energies derived from this approach satisfy a sum rule which guarantees the exactness of the integrated spectral strength. Our formulation becomes exact for systems consisting of a single particle, and for any many-body system in the high-frequency limit. The theory is illustrated by explicit calculations for simple one- and two-particle systems.

Electronic structure of solid FeO at high pressures by quantum Monte Carlo methods

We determine equation of state of stoichiometric FeO by employing the diffusion Monte Carlo method. The fermionic nodes of the many-body wave function are fixed by a single Slater determinant of one-particle orbitals extracted from spin-unrestricted Kohn–Sham equations utilizing a hybrid exchange-correlation functional. The calculated ambient pressure properties agree very well with available experimental data. At approximately 65 GPa, the atomic lattice is found to change from the rocksalt B1 to the NiAs-type inverse B8 structure.

Two-electron quantum dot with soft-edge barrier

A variational formalism to calculate energies of 1s, 2 1p, and 2 3p levels of a spherical quantum dot containing two electrons is developed. We express correspondent wave functions as a product of combinations of exact one-particle orbitals and variationally determined envelope function that depends only on separation between particles. An Euler–Lagrange differential equation for this last function is solved by using the trigonometric sweep method. We present novel curves for 1s, 2 1p, and 2 3p energy states dependencies on the radius of the quantum dots with hard wall and soft-edge barrier potential shapes. The effect of the repulsive core on the energies of the low-lying states and the pair correlation function is also analyzed. It is shown that the probability distribution of the electron–electron separation in a quantum dot with repulsive core is modified essentially with the increase of the quantum dot dimension, making more probable the finding of two electrons on diametrically opposite sides of the repulsive core.

Approximate and exact nodes of fermionic wavefunctions: Coordinate transformations and topologies

A study of fermion nodes for spin-polarized states of a few-electron ions and molecules with $s,p,d$ one-particle orbitals is presented. We find exact nodes for some cases of two electron atomic and molecular states and also the first exact node for the three-electron atomic system in $^4S(p^3)$ state using appropriate coordinate maps and wavefunction symmetries. We analyze the cases of nodes for larger number of electrons in the Hartree-Fock approximation and for some cases we find transformations for projecting the high-dimensional node manifolds into 3D space. The node topologies and other properties are studied using these projections. We also propose a general coordinate transformation as an extension of Feynman-Cohen backflow coordinates to both simplify the nodal description and as a new variational freedom for quantum Monte Carlo trial wavefunctions.

Treatment of correlation effects in electron momentum density: density functional theory and beyond

Recent high resolution Compton scattering experiments clearly reveal that there are fundamental limitations to the conventional local density approximation (LDA) based description of the ground state electron momentum density (EMD) in solids. In order to go beyond the framework of the density function theory (DFT), we consider for the correlated system a BCS-like approach in which we start with a singlet pair wavefunction or a ‘geminal’ from which the many body wavefunction is then constructed by taking an antisymmetrized geminal product (AGP). A relatively simple practical implementation of the AGP method is developed where the one-particle orbitals are approximated by the Kohn–Sham solutions used in standard band computations, and the orbital-dependent BCS energy scale Δ i is determined through a readily computed exchange-type integral. The methodology is illustrated by considering EMD and Compton profiles in Li, Be and Al. It is found that in Li the present scheme predicts a substantial renormalization of the LDA result for the EMD; in Be, the computed correlation effect is anisotropic, while in Al, the deviations from the LDA are relatively small. These theoretical predictions are in qualitative accord with the corresponding experimental observations on Li, Be and Al, and indicate the potential of the AGP method for describing correlation effects on the EMD in wide classes of materials.

First-degree homogeneousN-particle noninteracting kinetic-energy density functionals

It is known in density-functional theory that the noninteracting kinetic-energy density functional T s [ρ] is not first-degree homogeneous in density scaling. However, it is shown here that, for every particle number N, there is an N-particle noninteracting kinetic-energy density functional T N [ρ], that is, a density functional that gives the noninteracting kinetic energy for N-particle densities, which is of first-degree homogeneity in the density ρ(r). This gives a powerful tool, a strong requirement, for constructing such functionals. A systematic procedure to obtain the real part of T N [ρ], the full T N [ρ] in one-dimension, for each N is also proposed. It is pointed out, further, that in the Euler-Lagrange equations that determine the one-particle orbitals that define T s [ρ], the Lagrange multiplier that forces the orbitals to yield ρ(r) is not other than the first derivative of T s [ρ], δT s [ρ]/δρ(r), which yields a natural derivation of the Kohn-Sham equations. Utilizing the same idea, it is shown for ground states how the Schrodinger equation can be derived from the basics of density-functional theory as well.

Polarization associated with the coupling to isoscalar dipole compression mode

The effect of core polarization on isoscalar dipole one-particle moments due to the coupling with isoscalar giant dipole resonance (compression mode) is estimated. Numerical calculations with the core nucleus ${}_{82}^{208}{\mathrm{Pb}}_{126}$ show that a small enhancement of one-particle moments is obtained, of which the magnitude depends sensitively on relevant one-particle orbitals. The results are important for a microscopic understanding of the electric dipole moments (Schiff moments) measured for nuclei in this mass region.

Orbital-free molecular dynamics simulations of melting in Na8 and Na20: Melting in steps

The melting-like transitions of Na8 and Na20 are investigated by ab initio constant energy molecular dynamics simulations using a variant of the Car–Parrinello method which employs an explicit electronic kinetic energy functional of the density, thus avoiding the use of one-particle orbitals. Several melting indicators are evaluated in order to determine the nature of the various transitions, and are compared with other simulations. Both Na8 and Na20 melt over a wide temperature range. For Na8, a transition is observed to begin at ∼110 K, between a rigid phase and a phase involving isomerizations among the different permutational isomers of the ground state structure. The “liquid” phase is completely established at ∼220 K. For Na20, two transitions are observed: the first, at ∼110 K, is associated with isomerization transitions among those permutational isomers of the ground state structure which are obtained by interchanging the positions of the surface-like atoms; the second, at ∼160 K, involves a structural transition from the ground state isomer to a new set of isomers with the surface molten. The cluster is completely liquid at ∼220 K.

Deformation and one-particle orbits in3671Kr35and3571Br36

Using recent experimental information on the ${\ensuremath{\beta}}^{+}$ decay of ${}^{71}\mathrm{Kr},$ as well as available data on ${}^{71}\mathrm{Br},$ we suggest the deformation and relevant one-particle orbitals of low-lying states in the $A=71$ nuclei. Prolate deformation and a weaker spin-orbit potential are suggested in order to be consistent with available data on ${}^{71}\mathrm{Br}.$

A general program for computing matrix elements in atomic structure with nonorthogonal orbitals

To solve some problems concerning the atomic structure and transition probabilities, one needs to evaluate the matrix elements of various operators with respect to the configuration wave functions formed from the nonorthogonal one-particle orbitals. The program presented performs the angular integration necessary for expressing such matrix elements as weighted sums of radial integrals. Any amount of nonorthogonality between the orbitals may be presented leading to overlap integrals for the matrix elements. The calculations follow the method based on the representation of configuration wave functions through the Slater determinants. The program, therefore, can also be used for obtaining the corresponding vector-coupling coefficients. A distinguishing feature of the present procedure is the preliminary generation of vector-coupling coefficients for the individual subshells, which reduces considerably the amount of calculations. The operators included in the present version of the program are the one- and two-particle electrostatic interaction along with the one-particle tensor operator and simple overlap. Matrix elements of these operators provide the basis for studying the formation or decay of the inner-shell vacancies.

Photoionization from 1sns 1,3Se states of helium.

We present theoretical photoionization cross sections from the 1sns 1,3 S e bound excited states of the helium atom based on a configuration-interaction procedure for the continuum, using a finite L 2 -basis set constructed from a nearly complete set of B-spline-based one-particle hydrogenic orbitals. In addition to the nonresonant spectra near the first ionization threshold, resonant structures associated with selected sp, 2n ± and 2pnd 1,3 P o autoionization series below the He + N=2 threshold are examined in detail. The widths and resonance energies of all 1,3 P autoionization series are also reported and compared with other available theoretical and experimental results

He photoionization to the doubly excited (2pnd) and (sp,2n-) 1Po series.

We present an application of a configuration-interaction method for the [ital continuum] [ital spectrum] to He photoionization from the ground state to the doubly excited (2[ital pnd]) and ([ital sp],2[ital n][sup [minus]]) [sup 1][ital P[ital o]] series, using a finite [ital L][sup 2]-basis set constructed from a nearly complete set of spline-based one-particle hydrogenic orbitals. Our calculated resonance structures are in good agreement with the recent high-resolution synchrotron-radiation spectra [M. Domke, G. Remmers, and G. Kaindl, Phys. Rev. Lett. 69, 1171 (1992)]. Our calculated widths, resonance energies, and peak photoionization cross sections are also in close agreement with the most accurate existing theoretical results.

Formulation of N- and v-representable density functional theory. VIII. Relationship between the density-driven and the local-scaling versions

It is shown that Cioslowski’s density-driven construction of one-particle orbitals corresponds to a finite basis representation of local-scaling transformations. Implications of this correspondence, with respect to the formulation and implementation of a variational principle for the density, are discussed.

The ΔN = 2 coupling between one-particle orbitals in rare-earth nuclei

One-particle transfer reactions have been extensively studied for nuclei in the rare-earth region. The experiments show a considerable mixing between certain single-particle states which originate from different major shells. It is shown that the strength of coupling depends very strongly on the hexadecapole deformation of the Woods-Saxon potential.

Projected Hartree Product Wavefunctions

A method for performing a restricted configuration interaction calculation on atoms or molecules based on the use of Young operators from the symmetric groups is discussed and its relation to some other types of calculations is pointed out. The method in its most general form can be said to be an extension of the independent particle approach which represents the wavefunction in terms of the n best possible one-particle orbitals for the n-particle system. Illustrations are given for the hydrogen molecule, the helium atom, and the allyl radical.

Decomposition property of the Slater determinant

The first step in solving the atomic many-body problem is the independent-particle, or Hartree-Fock method. This leads to the formulation of the system eigenfunction in terms of a single Slater determinant (neglecting correlations). By defining a matrix, the elements of which are the one-particle Orbitals occurring in the Slater determinant, a method is developed whereby the wave function is written as a polynomial in the eigenvalues of the matrix. This method, when used in conjunction with the variational principle, reduces the arithmetic tedium usually resulting in the Hartree-Fock method. The numerical results are in agreement with the Hartree-Fock method (to within 1.5 %) and within 1.56% of the exact values for a seven electron atom.

Summation Convention and the Density Matrix in Quantum Theory

Projection operators may be used instead of vectors to represent the states of a quantum-mechanical system and the appropriate equation of motion is obtained. When a basis is used the basic vectors may be oblique or orthogonal. A formulation appropriate to an oblique basis is developed with the help of tensor methods and the summation convention. This is generalized to be applicable to systems containing $N$ electrons subject to Pauli's exclusion principle. The projection operator corresponding to an arbitrary state is the density operator and is represented by an $N\mathrm{th}$-order tensor. The well-known tensor operation of contraction of indices applied to the density operator furnishes similar, reduced operators giving one-, two-, or more-particle properties of the system. The one-particle properties are of special interest in chemical systems, the corresponding reduced density matrix being called the charge and bond order matrix by L\"owdin. The full formulas for the reduction of the general density matrix are given, showing the origin of the Coulomb and exchange operators.When the symbolic equation of motion is expressed by means of components and the process of contraction (reduction) is applied, the equation of motion of the one-particle reduction is obtained as well as an expression for the energy of the system. In the special case of the Hartree-Fock approximation the time-dependent wave equations for the one-particle orbitals are shown to satisfy the general equation of motion, suggesting that the energies of the individual orbitals need not be stationary. This approximation is characterized by the idempotent property of the one-particle reduced density matrix. This condition alone leads to a close restriction on the possible values of the charge and bond orders of a molecule. Specific formulas for the molecule butadiene show that these can be expressed in terms of two parameters only, in this case.