Self-consistent Wave Functions Research Articles

Matrix methods for opacity calculations

Atomic structure calculations are the first step in an opacity calculation. They provide the self-consistent electronic potential and electron wavefunctions that are needed to evaluate the Slater integrals and dipole matrix elements. In this paper, we show that simple matrix methods can be used to perform Hartree–Fock–Slater or LDA calculations on standard atomic structure grids, including relativistic corrections, yielding bound and continuum wavefunctions that can be used, in conjunction with statistical broadening techniques, to obtain realistic average-atom opacities.

Satellite and hypersatellite structure of Mα1,2 x‐ray transitions in thorium excited by fast oxygen ions

AbstractThe high‐resolution spectra of thorium Mα1,2 (M5 → N6,7) and Mβ1 (M4 → N6) x‐ray lines induced by an impact of O ions with energy 376.0 MeV were measured using a von Hamos crystal spectrometer in order to study the x‐ray emission from multiple ionized atoms. The observed x‐rays have been interpreted in terms of the relativistic multiconfiguration Dirac‐Fock (MCDF) calculations performed for dominating vacancy configurations expected to contribute to the measured spectra. Due to the complexity of such calculations for selected multi‐vacancy configurations the simplified average <MCDF> model, combining the MCDF calculations and binomial distribution of vacancies has been used. The measured spectra clearly exhibit a rich structure of Mα1,2 x‐ray satellites of thorium, including the transitions from the multivacancy initial configurations M−1N−n as well as the hypersatellites corresponding to the transitions in complex multivacancy initial configurations M−mN−n (m,n ≤ 4). The present interpretation of the measured satellite/hypersatellite structure of Mα1,2 (M5 → N6,7) x‐ray transitions allows us to determine the ionization probabilities for the M‐ and N‐shell in central collisions. The measured ionization probabilities are compared with the theoretical predictions of the geometrical model (GM), and the semiclassical approximation (SCA) using both the Dirac hydrogenic and Dirac‐Hartree‐Fock selfconsistent wave functions.

Advanced modeling of materials with PAOFLOW 2.0: New features and software design

Recent research in materials science opens exciting perspectives to design novel quantum materials and devices, but it calls for quantitative predictions of properties which are not accessible in standard first principles packages. PAOFLOW, is a software tool that constructs tight-binding Hamiltonians from self-consistent electronic wavefunctions by projecting onto a set of atomic orbitals. The electronic structure provides numerous materials properties that otherwise would have to be calculated via phenomenological models. In this paper, we describe recent re-design of the code as well as the new features and improvements in performance. In particular, we have implemented symmetry operations for unfolding equivalent k-points, which drastically reduces the runtime requirements of first principles calculations, and we have provided internal routines of projections onto atomic orbitals enabling generation of real space atomic orbitals. Moreover, we have included models for non-constant relaxation time in electronic transport calculations, doubling the real space dimensions of the Hamiltonian as well as the construction of Hamiltonians directly from analytical models. Importantly, PAOFLOW has been now converted into a Python package, and is streamlined for use directly within other Python codes. The new object oriented design treats PAOFLOW’s computational routines as class methods, providing an API for explicit control of each calculation.

EMPIRE: a highly parallel semiempirical molecular orbital program: 3: Born-Oppenheimer molecular dynamics

Direct NDDO-based Born-Oppenheimer molecular dynamics (MD) have been implemented in the semiempirical molecular orbital program EMPIRE. Fully quantum mechanical MD simulations on unprecedented time and length scales are possible, since the calculation of self-consistent wavefunctions and gradients is performed in a massively parallel manner. MD simulations can be performed in the NVE and NVT ensembles, using either deterministic (Berendsen) or stochastic (Langevin) thermostats. Furthermore, dynamics for condensed-phase systems can be performed under periodic boundary conditions. We show three exemplary applications: the dynamics of molecular reorganization upon ionization, long timescale dynamics of an endohedral fullerene, and calculation of the vibrational spectrum of a nanoparticle consisting of more than eight hundred atoms.Graphical A snapshot from an MNDO-H simulation of NH4+@C60 at 4000 K shortly before a proton crosses the fullerene wall to give NH3@C60H+.

Role of dynamic pairing correlations in fission dynamics

We study the role of dynamic pairing correlations in fission dynamics by considering intrinsic Hartree-Fock-Bogoliubov wave functions that are obtained by minimizing the particle number projected energy. For the restricted variational space, the set of self-consistent wave functions with different values of proton and neutron number particle fluctuations are considered. The particle number projected energy is used to define potential energy surface for fission whereas collective inertias are computed within the traditional formulas for the intrinsic states. The results show that the effect of the restricted variation after particle number projection in the potential energy surface is small while collective inertias substantially decrease. On the other hand, we show that this quenching is strongly mitigated when Coulomb antipairing is considered. In the light of these beyond mean-field calculations, the validity of traditional fission calculations is discussed.

Variational Relativistic Two-Component Complete-Active-Space Self-Consistent Field Method.

The accurate description of the electronic structure of transition metals and their compounds can be complicated by both the large number of close-lying states that often have multiconfigurational nature and significant relativistic effects. In order to address these challenges we present a two-component complete-active-space self-consistent field method that includes scalar relativistic effects and one-electron spin-orbit coupling during the self-consistent wave function optimization procedure. These relativistic effects are included via an "exact two-component" transformation of the solution of the one-electron modified Dirac equation. The method is applied to the study of spin-orbit splitting of ground and low-lying excited states of main group elements and selected transition metals.

Tetrel bonding interaction: an analysis with the block-localized wavefunction (BLW) approach

In this study, fifty-one iconic tetrel bonding complexes were studied using the block localized wave function (BLW) method which can derive the self-consistent wavefunction for an electron-localized (diabatic) state where charge transfer is strictly deactivated. Energy decompositions based on the BLW method (BLW-ED in short) show that the frozen energy, which is mostly composed of the Pauli repulsion and electrostatic components, plays the dominating role in thirty-one out of these fifty-one systems. A further quantitative study of electrostatic potential (ESP) was carried out by analyzing the variation of ESP along the binding direction for each monomer. In general, a Lewis base with the stronger ESP leads to higher stabilizing frozen and polarization energies. In addition, the relative strengths for complexes constructed with different carbon group elements are also examined qualitatively using the ESP model. The positive contribution of the charge transfer interaction is confirmed by both the BLW-ED results and the weakening of bond strengths after the charge transfer interaction is disabled. Finally, by replacing all atoms of a monomer with σ-holes (Lewis acid) with empirical point charges, attractive electrostatic interaction was observed in all cases, providing us a simple electrostatic explanation for the tetrel bonding, though the anisotropy of electron density associated with the σ-holes is unconsidered.

Extended Møller-Plesset perturbation theory for dynamical and static correlations

We present a novel method that appropriately handles both dynamical and static electron correlations in a balanced manner, using a perturbation theory on a spin-extended Hartree-Fock (EHF) wave function reference. While EHF is a suitable candidate for degenerate systems where static correlation is ubiquitous, it is known that most of dynamical correlation is neglected in EHF. In this work, we derive a perturbative correction to a fully spin-projected self-consistent wave function based on second-order Møller-Plesset perturbation theory (MP2). The proposed method efficiently captures the ability of EHF to describe static correlation in degeneracy, combined with MP2's ability to treat dynamical correlation effects. We demonstrate drastic improvements on molecular ground state and excited state potential energy curves and singlet-triplet splitting energies over both EHF and MP2 with similar computational effort to the latter.

Satellite and hypersatellite structures ofLα1,2andLβ1x-ray transitions in mid-Zatoms multiply ionized by fast oxygen ions

A detailed investigation of the $L{\ensuremath{\alpha}}_{1,2}$ (${L}_{3}\ensuremath{\rightarrow}{M}_{4,5}$) and $L{\ensuremath{\beta}}_{1}$ (${L}_{2}\ensuremath{\rightarrow}{M}_{4}$) x-ray satellite and hypersatellite structures in zirconium, molybdenum, and palladium atoms multiply ionized by impact with 278.6-MeV oxygen ions is reported. The x-ray spectra were measured with a high-resolution von Hamos bent crystal spectrometer. For the interpretation of the complex spectral features, relativistic multiconfiguration Dirac-Fock calculations were performed for all multivacancy configurations expected to contribute to the observed spectra. The data analysis clearly demonstrates that the spectra are dominated by structures originating from (${L}^{\ensuremath{-}1}{M}^{\ensuremath{-}m}{N}^{\ensuremath{-}n}$) satellite and (${L}^{\ensuremath{-}2}{M}^{\ensuremath{-}m}{N}^{\ensuremath{-}n}$) hypersatellite transitions corresponding to the radiative decay of the excited multivacancy configurations. The ionization probabilities of the $L$ and $M$ shell were determined from the data and compared with theoretical predictions from the geometrical model and the semiclassical approximation, using in the latter case both relativistic hydrogenlike and self-consistent Dirac-Hartree-Fock wave functions. The results support the independent electron picture of the multiple ionization. They also show the importance of using relativistic and self-consistent electronic wave functions for the $L$ and $M$ shells.

Parameterization of tight-binding models from density functional theory calculations

We present a rigorous bottom-up approach for the derivation of the electronic structure part of tight-binding (TB) models from density functional theory (DFT) calculations. The approach is based on a simultaneous optimization and projection of atomic-like orbitals on self-consistent DFT wave functions and is universally applicable to elements and compounds in arbitrary structural arrangements. The quality and transferability of the derived TB bond and overlap integrals are demonstrated for the examples of a covalent semiconductor (carbon), a transition metal (titanium), and a binary compound with mixed metallic-covalent bonding (TiC). The method can serve as a transparent and physically justified coarse-graining scheme for the construction of nonorthogonal and orthogonal TB total-energy models as well as the closely related bond-order potentials.

A Simple Algorithm for Determining Orthogonal, Self-Consistent Excited-State Wave Functions for a State-Specific Hamiltonian: Application to the Optical Spectrum of the Aqueous Electron.

We recently introduced a mixed quantum/classical model for the hydrated electron that includes electron/water polarization in a self-consistent fashion, using a polarizable force field for the water molecules [ J. Chem. Phys. 2010 , 133 , 154506 ]. Calculation of the electronic absorption spectrum for this model is not straightforward, owing to the state-specific nature of the Hamiltonian, the high density of electronic states, and the large solvent polarization response upon electronic excitation. Together, these properties make it difficult or impossible to converge the polarizable solvent dipoles self-consistently for each excited-state wave function. Here, we overcome this problem by means of an extended Lagrangian procedure for performing constrained annealing in the space of electronic variables. By construction, this algorithm affords self-consistent, mutually orthogonal solutions for any state-specific Hamiltonian, and we illustrate this approach by computing the optical spectrum of our polarizable model for the aqueous electron. The spectrum thus obtained affords better agreement with experiment than previous, perturbative calculations of solvent dipole relaxation. Strengths, weaknesses, and possible generalizations of this procedure are discussed.

Grazing-incidence small-angle X-ray scattering from a random rough surface: a self-consistent wavefunction approximation

An attempt is made to go beyond the distorted-wave Born approximation addressed to the grazing-incidence small-angle X-ray (GISAX) scattering from a random rough surface. The integral wave equation adjusted with the Green function formalism is applied. To find out an asymptotic solution of the non-averaged integral wave equation in terms of the Green function formalism, the theoretical approach based on a self-consistent approximation for the X-ray wavefunction is elaborated. Such an asymptotic solution allows one to describe the reflected X-ray wavefield everywhere in the scattering (θ, ϕ) angular range, in particular below the critical angle θ(cr) for total external reflection (θ is the grazing scattering angle with the surface, ϕ is the azimuth scattering angle; θ(0) is the grazing incidence angle). Analytical expressions for the reflected GISAX specular and diffuse scattering waves are obtained using the statistical model of a random Gaussian surface in terms of the r.m.s. roughness and two-point cumulant correlation function. For specular scattering the conventional Fresnel expression multiplied by the Debye-Waller factor is obtained. For the reflected GISAX diffuse scattering the intensity of the R(dif)(θ, ϕ) scan is written in terms of the statistical scattering factor eta(theta, theta0) and Fourier transform of the two-point cumulant correlation function. To be specific for isotropic solid surfaces, the statistical scattering factor eta(theta, theta0) and Fourier transform of the two-point cumulant correlation function parametrically depend on the root-mean-square roughness σ [eta(theta, theta0) = 0 for σ = 0] and cumulant correlation length ℓ, respectively. The reflected R(dif)(θ, ϕ) scans are numerically simulated for the typical-valued {θ(0), σ, ℓ} parameters array.

Ferromagnetic properties of selectively Mn-doped (Ga,Mn)As quantum wells

We investigate temperature dependence of spin-resolved subband structure and spontaneous magnetization for different doping profiles of magnetic impurities in diluted (Ga,Mn)As quantum well structures. The self-consistent hole subband energies and wave functions are determined by solving combined Schr\"odinger and Poisson equations numerically. We included the coupling effects among heavy-hole, light-hole, and split-off bands (six-band model) in our calculation and compared the result with the case of two-band model. We show that selective doping of magnetic ions in GaAs quantum well enhances the Curie temperature ${T}_{c}$ significantly, but holes occupying light-hole subbands result to reduce spontaneous magnetization and hence the Curie temperature of the system. This reduction effect reveals more clearly near the $T\ensuremath{\sim}{T}_{c}$ hindering an increase in the Curie temperature in a selectively Mn-doped system.

Ordinary field-theoretic methods for self-consistent wave functions which describe bond formation and dissociation. III. The commutative coupling approximation

The commutative coupling approximation of Hartree–Bogoliubov–Valatin theory is realized when the number-conserving renormalized one-body and number-nonconserving pairing operators are assumed to commute. In this case, a set of self-consistent one-particle equations which involves a single operator may be solved. The commutative coupling approximations and the exact solutions are compared here for the fragmentation of LiH and FH. The results suggest that the commutative coupling approximation provides a useful zeroth-order representation of fragmentation, and hence will induce suitable vacuua for applications' of Nambu's spinor representation of Feynman–Dyson–Goldstone diagrammatic perturbation theory to the problem of bond formation and dissociation.

Critical temperature forα-particle condensation within a momentum-projected mean-field approach

\ensuremath{\alpha}-particle (quartet) condensation in homogeneous spin-isospin symmetric nuclear matter is investigated. The usual Thouless criterion for the critical temperature is extended to the quartet case. The in-medium four-body problem is strongly simplified by the use of a momentum-projected mean-field ansatz for the quartet. The self-consistent single-particle wave functions are shown and discussed for various values of the density at the critical temperature. Excellent agreement of the critical temperature with a numerical solution of the Faddeev-Yakubovsky equation is obtained.

Iterative active-space selection for vibrational configuration interaction calculations using a reduced-coupling VSCF basis

We adapt a variation–perturbation method to perform vibrational configuration interaction (VCI) calculations on large molecular systems. Starting from a self-consistent vibrational wave function, we use the vibrational configuration interaction with perturbation selected interactions (VCIPSI) algorithm to select an active VCI space iteratively. We then extend this approach to use a reduced-coupling description of potential energy surfaces. The accuracy of the method is tested on methane and benzoic acid molecules. The significant reduction in the size of the VCI basis obtained using the VCIPSI scheme does not affect the quality of the computed vibrational frequencies and reduces computational time dramatically.

The effect of protonation on the photodissociation processes in formamide – An ab initio surface hopping dynamics study

Photodeactivation processes of protonated formamide were investigated using the multireference configuration interaction calculations with singles and doubles (MR-CISD) method while dynamics were simulated by employing the mixed quantum-classical direct trajectory method with surface hopping based on multi-configurational self-consistent wave functions. The dynamics calculations from the first excited singlet state in O-protonated formamide resembled those found for the second valence excited state of the parent molecule. Two deactivation processes were found: the C–N (major) and C–O (minor) dissociations with very short lifetimes (33 fs). Similarly, the major process for photodecomposition in the first excited state of N-protonated formamide resembles that for the parent formamide, and involves C–N dissociation with a lifetime around 390 fs. However, 55% of trajectories remained undissociated and undeactivated until 1000 fs, indicating the existence of other deactivation processes on a longer time scale.

Numerical studies of free exciton and trion wave functions in high magnetic fields

Wave functions of excitons and negative and positive trions confined in GaAs quantum wells and subject to high magnetic fields are studied by exact numerical diagonalization in Haldane spherical geometry. Finite width of the quantum well and its asymmetry caused by one-sided doping are both fully taken into account by using self-consistent subband wave functions in the integration of Coulomb matrix elements and by inclusion of multiple electron and hole subbands and Landau levels in the configuration-interaction basis. The main results regard the interaction-induced mixing of electron and hole single-particle levels, normal charge density profiles, in-plane correlation functions, and optical oscillator strengths. These characteristics are calculated for the exciton and different trion states, in terms of their dependence on the quantum-well width, carrier concentration, and magnetic field.

Exact-diagonalization studies of trion energy spectra in high magnetic fields

Binding energies of negative and positive trions in doped GaAs quantum wells in high magnetic fields are studied by exact numerical diagonalization in spherical geometry. Compared to earlier calculations, finite width of the quantum well and its asymmetry caused by one-sided doping are both fully taken into account by using self-consistent subband wave functions in the integration of Coulomb matrix elements, and by inclusion of higher subbands along with several Landau levels in the Hilbert space. Detailed analysis of the accuracy and convergence of the exact diagonalization scheme is presented, including dependence on Landau level and subband mixing, sensitivity to the (not well known) single-particle spectrum in the valence band, and the estimate of finite-size errors. The main results are the exciton dispersion and trion binding energy spectrum calculated as a function of the magnetic field, quantum well width, electron concentration, and the presence of an ionized impurity. As a complementary approach, a combination of the exact diagonalization in the quantum well plane and the variational calculation in the normal direction is used as well.

Effect of the basis set selection on the electron momentum distribution in graphite

This work evaluates three spatial distribution functions of the Bernal graphite momentum: the linear electron momentum distribution LEMD, the Compton profile function CP, and the self-correlation function or reciprocal form function B( r). We used the CRYSTAL software with a restricted Hartree–Fock Hamiltonian to obtain an ab initio and self-consistent crystalline wavefunction, whose square module, integrated over the first Brillouin zone and added over the occupied orbitals, defines the electron momentum density. The effect of different basis sets on these functions is reviewed. We have used modified STO-3GM, 3-21GM, and 6-31GM basis sets and standard basis sets with the same designation. Our results reveal the preponderance of p z and s orbitals in the electron momentum in the 〈001〉 direction, and p x and s orbitals in the 〈100〉 direction. The Compton profile for the core and valence electrons presents significant differences in the limit of small vectors. The results obtained by using the standard basis sets differ substantially from those obtained in this work. They do so, basically because of the failure of the former to reproduce the adequate Fermi level, something that is underscored via anisotropy curves of the linear electron momentum density and the Compton profile in the directions studied. Our calculations overestimate the Compton profile anisotropy between directions 〈001〉 and 〈100〉.