Valence Electron Wave Functions Research Articles

Description of covalent bond in terms of generalized charges: potential and dissociation energy of homonuclear compound

The theory of generalized charges is an asymptotic approximation of quantum mechanics for interatomic forces. It is based on the model of multicomponent electron gas, which extends the Thomas–Fermi model of inhomogeneous electron gas to pair electronic states. The present research takes in consideration the participation of the field of generalized charges in interatomic bonds. Herein, the definition of generalized charges (GC) by means of the overlap integral of the wave functions of valence electrons is given, and properties of GC are found out. Equations are derived for the potential and length of a homopolar bond, and dependence of these parameters on the generalized charges is shown. The dissociation energy of a diatomic molecule is expressed in terms of the nuclear charges, the function and multiplicity of the covalent bond, the angular momentum, and the vibrational energy of the binding electrons. The parametrization rules are substantiated and an a priori calculation of the dissociation energy of diatomic homonuclear molecules is carried out against the electronic configuration of the bond electrons and the nuclear charge. The calculation results for homonuclear compounds of the elements of the first four rows of the periodic table are shown to be of satisfactory accuracy.

ARC 3.0: An expanded Python toolbox for atomic physics calculations

ARC 3.0 is a modular, object-oriented Python library combining data and algorithms to enable the calculation of a range of properties of alkali and divalent atoms. Building on the initial version of the ARC library (Šibalić et al., 2017), which focused on Rydberg states of alkali atoms, this major upgrade introduces support for divalent atoms. It also adds new methods for working with atom–surface interactions, for modelling ultracold atoms in optical lattices and for calculating valence electron wave functions and dynamic polarisabilities. Such calculations have applications in a variety of fields, e.g., in the quantum simulation of many-body physics, in atom-based sensing of DC and AC fields (including in microwave and THz metrology) and in the development of quantum gate protocols. ARC 3.0 comes with an extensive documentation including numerous examples. Its modular structure facilitates its application to a wide range of problems in atom-based quantum technologies. Program summaryProgram Title: ARC 3.0CPC Library link to program files:https://doi.org/10.17632/c4z4n2cdf7.1Licencing provisions: BSD-3-ClauseProgramming language: PythonExternal Routines: NumPy [1], SciPy [1], Matplotlib [2], SymPy [3], LmFit [4]Nature of problem: The calculation of atomic properties of alkali and divalent atoms including energies, Stark shifts and dipole–dipole interaction strengths using matrix elements evaluated through a variety of means.Solution method: Dipole matrix elements are calculated using an analytical semi-classical approximation or wave functions obtained by numerical integration of the radial Schrödinger equation for a one-electron model potential. Interaction energies and shifts due to external fields are calculated using second order degenerate perturbation theory or exact diagonalisation of the interaction Hamiltonian, yielding results valid even at large external fields or small interatomic separation.Additional comments including restrictions and unusual features: External electric and magnetic field must be parallel to the quantisation axis. The accuracy of short range (≲1μm) atom - atom interaction potentials is limited by the truncation of the basis. Only weak magnetic fields are supported as only linear Zeeman shifts are taken into account. Calculations for divalent atoms use a single-electron approximation and calculation of their wave functions is not supported.

Model operator approach to the relativistic Lamb shift calculations in many-electron atoms and highly charged molecular ions

The quantum electrodynamics (QED) corrections to the alkali metal ionization potentials and to the ground-state energy of the U92+-U91+ one-electron quasi-molecule are evaluated with the use of the model operator approach. The QED contributions to the ionization energies in the alkali metals are obtained both by averaging the model operator with the valence electron wave functions and by incorporating the QED operator into the Dirac–Coulomb–Breit Hamiltonian. The two-center calculations of the vacuum-polarization and self-energy contributions to the Lamb shift in the one-electron quasi-molecule are performed for the different values of the internuclear distances (from the chemical distances up to the critical ones), the more simplified evaluation in the monopole approximation is also considered. The results of the calculations are compared with the theoretical predictions obtained previously in the framework of the ab initio approaches.

Pressure‐Induced Bandgap Optimization in Lead‐Based Perovskites with Prolonged Carrier Lifetime and Ambient Retainability

Bond length and bond angle exhibited by valence electrons is essential to the core of chemistry. Using lead-based organic–inorganic perovskite compounds as an exploratory platform, it is demonstrated that the modulation of valence electrons by compression can lead to discovery of new properties of known compounds. Yet, despite its unprecedented progress, further efficiency boost of lead-based organic–inorganic perovskite solar cells is hampered by their wider bandgap than the optimum value according to the Shockley–Queisser limit. By modulating the valence electron wavefunction with modest hydraulic pressure up to 2.1 GPa, the optimized bandgap for single-junction solar cells in lead-based perovskites, for the first time, is achieved by narrowing the bandgap of formamidinium lead triiodide (HC(NH2)2PbI3) from 1.489 to 1.337 eV. Strikingly, such bandgap narrowing is partially retained after the release of pressure to ambient, and the bandgap narrowing is also accompanied with double-prolonged carrier lifetime. With First-principles simulation, this work opens a new dimension in basic chemical understanding of structural photonics and electronics and paves an alternative pathway toward better photovoltaic materials-by-design.

Anti-Stokes-enhanced tunnelling ionization of polar molecules

We generalize the correct account for molecule vibrational motion in tunnelling ionization, proposed earlier (Kornev and Zon 2012 Phys. Rev. A 86 043401), to the case of polar molecules. We consider the tunnel effect in both dc and ac fields, taking into account perturbation of vibrational motion by an external field. We develop a method for accounting for dipole moment on the base of the electronic wave function expansion over dipole-spherical functions instead of ordinary spherical functions. The change of polar molecule ionization probability compared to non-polar molecule ionization is mainly due to constant change in the asymptotic form of the valence electron wave function, both in the dc and ac fields. We present numerical results for HF and HCl molecules of various isotopic proportions for specific orientations of the molecule axis relative to the electric field strength. The maximum rate of the tunnel effect for these molecules is achieved if the internuclear axis is oriented perpendicularly to the electric field.

Grain Boundary Controlled Electron Mobility in Polycrystalline Titanium Dioxide

Grain Boundary Controlled Electron Mobility in Polycrystalline Titanium Dioxide

Influence of Orbital Hybridization on Kerr Nonlinearity of a Heavy Metal Borate Glass: Scaling of Polarizability and the Imaginary Contribution of Optical Susceptibility

Photonics properties of glasses can be designed by controlling their complex Kerr nonlinearity. Chemical structure and bonding properties are considered as the origin of glass third-order susceptibilities. Investigation of the role of orbital hybridization on the glass electronic polarizability and third-order susceptibility is carried out. Thus, series of heavy metal lead borate glass of the composition 0.25B2O3–0.75PbO is prepared by melt quenching technique. Orbital hybridization, as a linear combination for valence electron wave functions of p- and d-block elements, is obtained through structural co-substitution of very small contents of Cr2O3 and/or SeO2, by B2O3. It get succeed to tune the glass nonlinear optical characteristics such as; the complex components of third-order susceptibility. Scaling roles describing the relations between oxide ion polarizability and index of refraction and between imaginary part of third-order susceptibility and band gap energy are proposed. The glasses exhibit zero-dispersion wavelength at 1.55 μm band which is needed for telecommunication devices. The polarizability approach is applied to analyze and explain the obtained glass properties.

Going beyond the frozen core approximation: Development of coordinate-dependent pseudopotentials and application to ${\rm Na}_2^+$ Na 2+

Mixed quantum/classical (MQC) simulations treat the majority of a system classically and reserve quantum mechanics only for a few degrees of freedom that actively participate in the chemical process(es) of interest. In MQC calculations, the quantum and classical degrees of freedom are coupled together using pseudopotentials. Although most pseudopotentials are developed empirically, there are methods for deriving pseudopotentials using the results of quantum chemistry calculations, which guarantee that the explicitly-treated valence electron wave functions remain orthogonal to the implicitly-treated core electron orbitals. Whether empirical or analytically derived in nature, to date all such pseudopotentials have been subject to the frozen core approximation (FCA) that ignores how changes in the nuclear coordinates alter the core orbitals, which in turn affects the wave function of the valence electrons. In this paper, we present a way to go beyond the FCA by developing pseudopotentials that respond to these changes. In other words, we show how to derive an analytic expression for a pseudopotential that is an explicit function of nuclear coordinates, thus accounting for the polarization effects experienced by atomic cores in different chemical environments. We then use this formalism to develop a coordinate-dependent pseudopotential for the bonding electron of the sodium dimer cation molecule and we show how the analytic representation of this potential can be used in one-electron MQC simulations that provide the accuracy of a fully quantum mechanical Hartree-Fock (HF) calculation at all internuclear separations. We also show that one-electron MQC simulations of Na(2)(+) using our coordinate-dependent pseudopotential provide a significant advantage in accuracy compared to frozen core potentials with no additional computational expense. This is because use of a frozen core potential produces a charge density for the bonding electron of Na(2)(+) that is too localized on the molecule, leading to significant overbinding of the valence electron. This means that FCA calculations are subject to inaccuracies of order ~10% in the calculated bond length and vibrational frequency of the molecule relative to a full HF calculation; these errors are fully corrected by using our coordinate-dependent pseudopotential. Overall, our findings indicate that even for molecules like Na(2)(+), which have a simple electronic structure that might be expected to be well-treated within the FCA, the importance of including the effects of the changing core molecular orbitals on the bonding electrons cannot be overlooked.

Band dispersion in the deep 1s core level of graphene

Chemical bonding in molecules and solids arises from the overlap of valence electron wave functions, forming extended molecular orbitals and dispersing Bloch states, respectively. Core electrons with high binding energies, on the other hand, are localized to their respective atoms and their wave functions do not overlap significantly. Here we report the observation of band formation and considerable dispersion (up to 60 meV) in the $1s$ core level of the carbon atoms forming graphene, despite the high C $1s$ binding energy of $\approx$ 284 eV. Due to a Young's double slit-like interference effect, a situation arises in which only the bonding or only the anti-bonding states is observed for a given photoemission geometry.

Probing the Angular Momentum Character of the Valence Orbitals of Free Sodium Nanoclusters

Although many properties of polyatomic metal clusters have been rationalized by an electron shell model resembling that used for free atoms, it remained unclear how reliable this analogy is with respect to the angular momentum eigenstate character of the electronic wave functions. We studied free size-selected negatively charged clusters of sodium atoms (Nan-) of approximately spherical shape (n = 19, 40, 55, 58, 147) by angle-resolved photoelectron spectroscopy over a broad range of photon energies (1.5 to 5 electron volts). Highly anisotropic, state- and energy-dependent angular distributions emerged for all sizes. Well-defined classes of energy dependence related to the approximate angular momenta of the bound-state orbitals indicate that the overall character of the valence electron wave functions is not appreciably influenced by the interaction with the ion background. The measured distributions nevertheless deviate strongly from the predictions of single-electron models, hinting at a distinct role of correlated multielectron effects in the photoemission process.

A computationally efficient exact pseudopotential method. I. Analytic reformulation of the Phillips-Kleinman theory

Even with modern computers, it is still not possible to solve the Schrodinger equation exactly for systems with more than a handful of electrons. For many systems, the deeply bound core electrons serve merely as placeholders and only a few valence electrons participate in the chemical process of interest. Pseudopotential theory takes advantage of this fact to reduce the dimensionality of a multielectron chemical problem: the Schrodinger equation is solved only for the valence electrons, and the effects of the core electrons are included implicitly via an extra term in the Hamiltonian known as the pseudopotential. Phillips and Kleinman (PK) [Phys. Rev. 116, 287 (1959)]. demonstrated that it is possible to derive a pseudopotential that guarantees that the valence electron wave function is orthogonal to the (implicitly included) core electron wave functions. The PK theory, however, is expensive to implement since the pseudopotential is nonlocal and its computation involves iterative evaluation of the full Hamiltonian. In this paper, we present an analytically exact reformulation of the PK pseudopotential theory. Our reformulation has the advantage that it greatly simplifies the expressions that need to be evaluated during the iterative determination of the pseudopotential, greatly increasing the computational efficiency. We demonstrate our new formalism by calculating the pseudopotential for the 3s valence electron of the Na atom, and in the subsequent paper, we show that pseudopotentials for molecules as complex as tetrahydrofuran can be calculated with our formalism in only a few seconds. Our reformulation also provides a clear geometric interpretation of how the constraint equations in the PK theory, which are required to obtain a unique solution, are themselves sufficient to calculate the pseudopotential.

Influence of perturbations on the electron wavefunction inside the nucleus

The variation of the valence electron wavefunction inside a nucleus induced by aperturbative potential is expressed in simple explicit terms, as the zeroth and firstmomenta of the potential. As an application we consider the QED vacuumpolarization corrections to the parity non-conservation amplitude, caused by theUehling and Wichmann–Kroll potentials. The Uehling potential is found togive −0.47% for the amplitude of the 6s–7s transition in 133Cs, while thecontribution of the Wichmann–Kroll potential is proved to be negligible.

High-momentum components and temperature dependence of the Compton profile of beryllium

We present Compton-scattering studies on the high-momentum components of the valence electron momentum density in beryllium. The experiments were performed with incident photon energies of 29 and 56 keV with a momentum space resolution of 0.10 and 0.16 a.u. units of momentum, respectively. The temperature dependence of the Compton profile and the high-momentum components were studied within the temperature range of 40--850 K. The incident photon energy was 56 keV and the photon scattering vector was along the [110] reciprocal lattice vector. We compare the temperature dependence of the experimental Compton profiles to empirical local pseudopotential computations that take into account both thermal expansion and disorder. The position and intensity dependence of the high-momentum components as a function of temperature in solid beryllium were found to be quite small, which suggests that the valence electron wave functions in beryllium are not significantly affected by thermal disorder. This proves that the previously observed broadening of experimental Compton profiles in comparison to highly accurate theoretical profiles is not due to thermal disorder, and other reasons, e.g., correlation, need to be sought as the source of the broadening.

First-principles calculation of coincidence Doppler broadening of positron annihilation radiation

We report a first-principles method for calculating the momentum density of positron-electron pairs in materials, which can be accurately measured, in a wide momentum range, by means of coincidence Doppler broadening (CDB) of positron annihilation radiation. The calculation is based on the two-component density-functional theory within the local-density approximation. The electron and positron wave functions are calculated by means of the full-potential linearized-augmented-plane-wave method with use of semicore orbitals and of the pure plane-wave method, respectively. This hybrid basis set accurately determines the wave functions of core and valence electrons and is free from any shape or symmetry assumption for the positron wave function. The method is applied to two typical systems. i.e., Al and graphite having isotropic and anisotropic positron densities, respectively. The calculations agree well with experiments over the entire measurable momentum region. Especially, the calculations well reproduce the anisotropic high-momentum CDB tails of graphite, which originate from the quasi-two-dimensionally distributed positron. This reproduction suggests that the present method is applicable for a variety of materials.

Orbital Radii and Environment-Independent Transferable Atomic Length Scales

The importance of the orbital radii, r l , obtained from the classical turning point of the valence electron wave function of angular momentum, I, in obtaining interatomic distances for all bonding situations is demonstrated. Single bond interatomic distances may be expressed in terms of a universal multiplicative constant of a core s orbital radius and another universal additive term which is close to the interatomic distance in the hydrogen molecule. These relationships are obtained from the dependence of the radii of positive or negative singly charged ionic species, CR + and CR - , on r l . The shortening of the distances in multiple bond systems or in systems involving transition metal d electron elements is described by a simple universal function, F s , associated with the number of unpaired valence electrons. A principle of maximum mechanical hardness based on minimization of bond distances is proposed to obtain correct distances in heteropolar MX bonds. The application of these rules to a large number of compounds with ionic, covalent, metallic, and nonbonded interactions yield interatomic distances which are within 2% of the observed distances. A brief discussion is made on the physical significance of the transferable length scales CR + and CR - in the context of discrimination of structure based on radius ratio and the requirement of a universal equilibrium chemical potential for transferability.

Numerical Evaluation of Radiative Corrections in the Lithium Isoelectronic Sequence

A numerical evaluation of the radiative corrections for the energies of the 2 s, 2 p, and 1 s states of the Li isoelectronic sequence has been carried out. The results are based on the ab initio QED calculations made previously by two of the authors. Those for the 2 s-2 p Lamb shift are almost all in agreement with experiment. The main effect of screening involves a renormalization of the valence electron wave function at the origin. As expected, the smaller effects, originating in interactions between two electrons, play a significant role only for the lower range of Z values. Subtler QED effects appear in the p-state (rather than s-state) energies and will become more apparent in other atoms such as B.

Fundamentals of inelastically scattered X-ray synchrotron radiation as a probe for the electronic structure of condensed matter

Several interactions of X-ray photons with the electrons of condensed matter are included under the general term “inelastic scattering”. Each of these interactions can be established as the basis for the study of some electronic properties. X-ray Compton scattering is the only spectroscopical method in which the momentum and not the energy is of importance, while the X-ray resonant Raman spectrum contains information about the phase of the valence electron wavefunction, and finally the X-ray Raman spectra provide information similar to that of XANES and EXAFS but easier. Which inelastic interaction is dominant for a given sample depends on the combination of the energy of the incident photon and the scattering angle. The tunability of the synchrotron X-ray beam, the high flux and degree of polarization are the reasons that the synchrotron facilities are the only places where the study of condensed matter via inelastic scattering can be carried out. Presented here are the interactions with the common characteristic ω 1 > ω 2, and the advantages of employing the synchrotron radiation sources in the utilization of these interactions as tools in the solid-state investigation.

A non‐local pseudopotential in the FSGO model: Study of some organometallic systems

AbstractA non‐local core pseudopotential has been used in the framework of floating spherical Gaussian orbital (FSGO) model to study the equilibrium geometries and valence electronic structures of some organolithium and organoberyllium systems. The calculated equilibrium geometries, heats of hydrogenation, average electric polarizabilities, and magnetic susceptibilities are in good agreement with the results of the all‐electron FSGO model calculations. Valence electron wave functions obtained here have been used to predict the valence electron Compton profiles (CP) and electron momentum distributions (EMD) of the systems studied. A good correlation has been shown among the peak height of the CP (J(0)), valence electron energy (Ev), and number of valence electrons (Nv).

The asymptotic theory of resonance charge exchange between diatomics

The asymptotic theory of resonance charge exchange between a ground-state diatomic molecular ion and its neutral parent is presented. The parameters of the valence electron wavefunction and asymptotically precise exchange interaction potential are calculated. The role of rotational transitions is discussed. The vibrational excitation transfer is taken into account and the coupled equations, describing the charge exchange process between diatomics, are solved both in limiting cases and numerically. The total charge transfer cross sections are calculated for many diatomic systems and the results are compared with experimental data.

The properties of helium atoms and as impurities in metals

Abstract When the atoms of simple metals condense from the gas phase to give a solid, the valence electron wavefunctions are strongly modified by their overlap. At first sight this strong overlap might be expected to produce a complex situation, difficult to treat quantitatively. Yet the simple metals are perhaps the best understood of all solids and the reason is that the electron wavefunctions are plane-wave-like to a first approximation and deviations can be treated by perturbation theory.