Ground State Energy Of Electron Research Articles

Augmented Lagrangian method for order-N electronic structure

Molecular electronic ground-state theories, whether ab initio, or semiempirical are most often formulated as a variational principle, where the electronic ground-state energy, considered a linear or nonlinear functional of a reduced density matrix, obtains a constrained minimum. In this communication, we present a Lagrangian analysis of the self-consistent-field electronic structure problem, which does not resort to the concept of orthogonal molecular orbitals. We also develop a method of constrained minimization efficiently applicable to nonlinear energy functional minimization, as well as to linear models such as tight-binding. The method is able to treat large molecules with an effort that scales linearly with the system size. It has built-in robustness and leads directly to the desired minimal solution. Performance is demonstrated on linear alkane and polyene chains.

Self-consistent field algorithms for Kohn–Sham models with fractional occupation numbers

The calculations of electronic ground state energies, following either the Hartree–Fock or the Kohn–Sham schemes, are major issues in quantum chemistry. In a recent publication, we have proposed a new numerical method, namely the relaxed constrained algorithms (RCA), to solve the Hartree–Fock problem. The purpose of the present paper is to discuss the extension of this method to the case of the Kohn–Sham problem. It is shown that RCA seems to be more robust than other self-consistent field algorithms currently used and that they provide in addition a natural way to solve the extended Kohn–Sham problem, obtained by allowing fractional occupancy of the single-particle orbitals.

Quantum-confined Stark effects of InAs/GaAs self-assembled quantum dot

Quantum-confined Stark effects in InAs/GaAs self-assembled quantum dots are investigated theoretically in the framework of effective-mass envelope function theory. The electron and hole energy levels and optical transition energies are calculated in the presence of perpendicular and parallel electric field. In our calculation, the effect of finite offset, valence band mixing, and strain are all taken into account. The results show that the perpendicular electric field weakly affects the electron ground state and hole energy levels. The energy levels are affected strongly by the parallel electric field. For the electron, the energy difference between the ground state and the first excited state decreases as electric field increases. The optical transition energies have clear redshifts in electric field. The theoretical results agree well with the available experimental data. Our calculated results are useful for the application of quantum dots to photoelectric devices.

Influence of Heat on Impurity States in an Artificial Semiconductor Atom

semiconductor as the potential barrier material of the artificial atom. Different heat dependent effective masses and different heat dependent dielectric constants are used for the two semiconductors constitute the ASA. The lowest energy and the binding energy of the ground state are calculated. The calculations have shown that the lowest electron energy decreases by increasing the ASA radius. For very large radii the lowest energies corresponding to different aluminum contents approach a certain value which is the bulk limit. At a constant ASA radius the lowest energy increases as a function of temperature. A pronounced deviation is obtained when the calculations of the present work are compared with those of Montenegro and Merchancano [18] in which the effects of heat and mismatches for both effective masses and dielectric constants were neglected. It is found that decreasing the temperature and shrinking the radius of the ASA lead to more binding of the donor electron. Increasing the aluminum concentration also enhances the donor electron binding energy. Therefore the electron ground state energy and its associated binding energy are both mainly functions of temperature, ASA radius, and aluminum content.

Dependence of the energy spectrum of a strained ZnSe/ZnS superlattice on the charge-carrier concentration

The model of self-consistent electron-deformation coupling was used to show that additional periodic local electron-deformation wells and barriers arise in the vicinity of heterocontact in a strained ZnSe/ZnS superlattice within the main quantum well (ZnSe) and above the main barrier (ZnS). It is found that, for the thickness of the ZnSe overgrown layer in the range from 10 to 20 A, the electron ground-state energy E0c decreases steadily with increasing conduction-electron concentration nc; however, for the layer thickness exceeding 20 A, the concentration dependence E0c=f(nc) becomes nonmonotonic and features a maximum, which shifts to lower concentrations nc as the layer thickness increases.

Convergent summation of Møller–Plesset perturbation theory

Rational and algebraic Padé approximants are applied to Møller–Plesset (MP) perturbation expansions of energies for a representative sample of atoms and small molecules. These approximants can converge to the full configuration–interaction result even when partial summation diverges. At order MP2 (the first order beyond the Hartree–Fock approximation), the best results are obtained from the rational [0/1] Padé approximant of the total energy. At MP3 rational and quadratic approximants are about equally good, and better than partial summation. At MP4, MP5, and MP6, quadratic approximants appear to be the most dependable method. The success of the quadratic approximants is attributed to their ability to model the singularity structure in the complex plane of the perturbation parameter. Two classes of systems are distinguished according to whether the dominant singularity is in the positive half plane (class A) or the negative half plane (class B). A new kind of quadratic approximant, with a constraint on one of its constituent polynomials, gives better results than conventional approximants for class B systems at MP4, MP5, and MP6. For CH3 with the C–H distance at twice the equilibrium value the quadratic approximants yield a complex value for the ground-state electronic energy. This is interpreted as a resonance eigenvalue embedded in the ionization continuum.

Electrical and optical properties of self-assembled InAs quantum dots in InP studied by space-charge spectroscopy and photoluminescence

In this paper, we report on a detailed investigation using junction space-charge techniques and photoluminescence of InAs quantum dots embedded in an InP matrix. From measurements of thermal emission rates we have determined the ground-state energy of electrons and holes bound to the InAs dots. In contrast to other dot systems the holes are found to be more strongly confined than the electrons. Corresponding optical emission rates have been measured for holes and the photoionization is found to be well described by a photothermal excitation process similar to what has previously been observed for deep impurities. Furthermore, we have performed photoluminescence measurements revealing excited hole states with energies in good agreement with theory.

Non-linear response and hydrogen bond dynamics for electron solvation in methanol

Non-equilibrium and equilibrium adiabatic mixed quantum-classical molecular dynamics computer simulations of the solvation dynamics of an excess electron in methanol are reported. We develop the connection between the multiple time scales reflected in solvent response and individual physical phenomena, such as the radial collapse of the electron and structural relaxation of the hydrogen-bonding network of the solvent. The significant role of the latter aspect appears responsible for the breakdown of the linear response approximation for the relaxation of the adiabatic ground state energy of the excess electron.

Electronic correlation energy in linear and cyclic carbon tetramers

Theoretical studies have predicted the ground electronic states of the linear and cyclic (rhombic) forms of the carbon tetramer to be nearly isoenergetic. Previous theoretical studies have not explicitly considered the effect of higher than triple excitations on the ground state electronic correlation energy. In the set of calculations presented in this Letter, configuration interaction (CI) calculations are performed and no restriction on excitation level is imposed. The effect of quadruple and higher excitations freely enter the calculations. No previous study has been conclusive in determining the relative stability of the two carbon tetramer isomers. The CI calculations presented predict the rhombic structure to be energetically more stable than the linear form for all levels of approximation considered. The effect of inclusion and exclusion of triple and quadruple excitations on the ground state energy of the carbon tetramer is discussed.

Direct ab initio calculation of ground-state electronic energies and densities for atoms and molecules through a time-dependent single hydrodynamical equation

By using an imaginary-time evolution technique, coupled with the minimization of an expectation value, ground-state electron densities and energies have been directly calculated for six atomic and molecular systems (He, Be++, Ne, H2, HeH+, He2++), from a single time-dependent (TD) quantum fluid dynamical equation of motion whose real-time solution yields the TD electron density. For all the systems, a local Wigner-type correlation functional has been employed. For Ne, a local exchange functional is used while, for all the other systems, the exchange energy is calculated exactly. The static (ground-state) results are of beyond-Hartree–Fock quality for all the species.

Exchange-correlation, dipole, and image charge potentials for electron sources: Temperature and field variation of the barrier height

Potential barrier profiles for large applied fields and/or high temperature are developed for the study of field and thermionic emission electron sources intended for radio frequency power tube applications. The numerical implementation provides a fast and flexible method to obtain the barriers which govern current density, and yet allows for complications such as nanoprotrusions, adsorbates, “internal” field emission, the sputtering of low work function emission sites, and so on. The model consists of (i) a modified form of the Wigner Lattice expansion of the electron ground state energy to evaluate the exchange and correlation potential, (ii) a simplified form of the ionic core potential to correct the “Jellium” model, (iii) a triangular representation of the barrier with a single adjustable parameter which enables both the solution of Schrödinger’s equation in terms of Airy functions and thus an exact evaluation of the electron density near the barrier, and (iv) a numerical integration of Poisson’s equation to evaluate the dipole potential and positive background boundary. An iterative calculation is performed such that the barrier used in the solution of Schrödinger’s equation becomes equivalent to the barrier predicted from the exchange-correlation and dipole potentials. As a test of the method, evaluations of the work function of various metals are made. A good correspondence is found between the potential profiles and an “analytic” image charge potential (which contains modifications to the standard image charge model). Modifications to the Richardson–Laue–Dushman and Fowler Nordheim equations, so as to obtain current density estimates, are described. The (only) adjustable parameter used to correlate theory and experimental work functions is the magnitude of the ionic core “radius,” which is often close to the actual radius of the metal ions in the test cases considered. The temperature and field dependence of the work function, which is dependent upon electron penetration of the barrier and its effect on the dipole potential, are investigated. The method is suggested to be suitable for the analysis of more complex potential barrier profiles that are encountered in actual (realistic) thermionic and field emission electron sources. The limitations of the model are discussed and methods to circumvent them are proposed.

Dimension‐dependent two‐electron Hamiltonian matrix elements

Since the birth of quantum mechanics the ground state electronic energy of the two‐electron atom has received special attention. This is because the two‐electron system is the simplest atom to include electron–electron interactions. These interactions are key to understanding many‐electron systems. This paper adds to the knowledge of two‐electron atoms by presenting closed form solutions for Hamiltonian matrix elements at arbitrary spatial dimension, \(D\). The basis functions are the \(D\)‐dependent hydrogenic wavefunctions: \(\left\{ {{\text{1s}}^{\text{2}} {\text{,2p}}^{\text{2}} {\text{,3d}}^{\text{2}} {\text{,4f}}^{\text{2}} } \right\}\). The electron–electron repulsion integrals are solved by the Fourier integral transform.

Energy States in Graded Cylindrical GaAs/AlxGa1?xAs Quantum Wires

The existence of graded interfaces is shown to blue shift the confined electron energy states in cylindrical GaAs/AlxGa1—xAs quantum wires. The shift strength depends on the wire radius and on the interface width. High excited electron states can be eliminated when the wire interface is thick enough. Numerical calculations performed for a cylindrical GaAs/Al0.3Ga0.7As quantum wire with radius 40 Å show that the electron ground state energy is blue shifted as much as 16 meV when its interface is only 10 Å wide, and that all excited states are precluded.

Minimal adiabatic approximation of molecular energies

The minimal necessary and sufficient condition for Hamiltonians to yield so-called adiabatic energies of molecules is determined and used to formulate the non-perturbative non-variational Minimal Adiabatic Approximation for obtaining them. No finite electronic energy eigenvalues which result from this theory can become accidentally degenerate at any localized configuration of their nuclei, thus establishing a general ‘non-crossing’ rule of overall molecular potentials, which obviates the Jahn-Teller effect entirely on an adiabatic basis. The resulting electronic energy eigenfunctions are single-valued continuous bounded appropriately differentiable functions of their parametric nuclear coordinates and cannot have any geometric phase factor ascribed to them. The ground-state electronic energy eigenfunctions cannot have any phase factors which depend on nuclear configuration affixed to them and still retain their ground-state status. The ground-state electronic energy eigenvalues are bounded from below by those of the Born-Oppenheimer approximation and from above by those of the BornHuang approximation, as are their corresponding ground-state total energies.

Consequences for finite electronic systems of homogeneity properties of density-functional energy components

For finite electronic systems, the proposal is explored that the functional expansions of the universal Hohenberg–Kohn functionals T[ ρ] and V ee[ ρ] in terms of their own functional derivatives truncate at the first- and second-order levels, respectively. It is shown that then the universal functional F[ ρ]= T[ ρ]+ V ee[ ρ] can be expressed as the sum of functionals that are homogeneous of degrees one and two in the density, and the variational principle, which determines the ground-state density and total electronic energy E 0 for an N-electron system having external potential v 0, takes the form 〈 ρ v 0〉+〈 ρδ F/δ ρ〉−1/2〈 ρ ρδ 2 F/δ ρδ ρ〉≥ E 0. Knowledge of δ F[ ρ]/δ ρ( r ) and δ 2 F[ ρ]/δ ρ( r )δ ρ( r ′), without knowledge of F[ ρ] itself, accordingly may suffice for the variational determination of the electron density and total energy. A simple energy formula is derived if v 0 is Coulombic, namely, (3 N−1) E=( N−1)∑ ϵ i + NV ne[ ρ], where the ϵ i are Kohn–Sham orbital energies and V ne is the nuclear–electron attraction energy. Numerical tests are reported for atoms, and a small correction to the formula is suggested.

Relativistic one-electron Hamiltonians `for electrons only' and the variational treatment of the Dirac equation

A relation χ= Xϕ between the upper and lower components of the Dirac spinor ψ=( ϕ, χ), with X satisfying a nonlinear operator equation, guarantees that the expectation value 〈 D〉 ψ of the Dirac operator is bounded from below by the exact electronic ground state energy. Unfortunately X can, except for special cases, not be constructed in closed form. It is relatively easy to satisfy the condition χ= Xϕ if ϕ has the correct behaviour near a point nucleus, i.e. if it goes in the spherical average as r ν with ν slightly smaller than 0. To satisfy χ= Xϕ for trial functions regular at the position of the nuclei is possible in principle, but very difficult in practice. If one satisfies the condition χ= Xϕ only approximatively, one may still get a lower bound, i.e. a variationally stable approach, but the lower bound is below its exact counterpart, i.e. the exact electronic ground state. Examples of variationally stable approaches are analyzed, in particular the regularized stationary direct perturbation theory (SDPT), the so-called regular approximation (RA), and the Douglas-Kroll-Hess transformation (DKH), of which a few new features are revealed. Finally the possibility of a minimax principle for the Dirac equation is discussed.

The use of Gaussian spinors in relativistic electronic structure calculations: the effect of the boundary of the finite nucleus of uniform proton charge distribution

The ground-state electronic energies of hydrogen-like ions are computed by solving the matrix Dirac equation in large basis sets of Gaussian spinors to compare the accuracy of the expansion method with that of Desclaux's finite-difference and Matsuoka's matching equation methods. The nuclei are approximated as spheres of uniform charge distribution. We examine in detail the quality of the computed spinors in the vicinity of the nuclear boundary where relativistic dynamics is also important.

Studies of e − + CO 2 ⇌ CO 2− equilibria in hexamethyldisiloxane and bis(trimethylsilyl) methane

Electrons react reversibly with CO 2 in bis(trimethylsilyl)methane and hexamethyldisiloxane. This reaction is studied as a function of both temperature and pressure. The ground state energies of electrons are deduced and are quite low consistent with the observed high electron mobility. A linear dependence of the log of the attachment rate on the free energy of reaction is observed. The dependence of the equilibrium constant on pressure indicates the electrostriction volume of CO 2 − to be ≈−200 cm 3/mol in these liquids.

Electronic structure of f1 lanthanide and actinide complexes. Part 2. Non-relativistic and relativistic calculations of the ground state electronic structures and optical transition energies of [Ce(η-C5H5)3], [Th(η-C5H5)3] and [Pa(η-C8H8)2

Abstract Non-relativistic and relativistic discrete variational-X α calculations have been performed on [Ce( η -C 5 H 5 ) 3 ], [Th( η -C 5 H 5 ) 3 ] and [Pa( η -C 8 H 8 ) 2 ]. Metal—ligand covalent interactions in [Ce( η -C 5 H 5 ) 3 ] and [Th( η -C 5 H 5 ) 3 ] are dominated by metal d-orbital participation in the ( η -C 5 H 5 ) π 2 -based 2e and 3e molecular orbitals, and an f-orbital contribution to the la 2 level. The non-relativistic calculations predict formal ground configurations of 4f 1 and 5f 1 for [Ce( η -C 5 H 5 ) 3 ] and [Th( η -C 5 H 5 ) 3 ] respectively, while the destabilisation of the n f and slight stabilisation of the ( n + 1)d σ level on the incorporation of relativistic effects result in a 6d 1 electronic configuration for [Th( η -C 5 H 5 ) 3 ]. [Pa( η -C 8 H 8 ) 2 ] is found to have a 5f 1 ground configuration in both non-relativistic and relativistic calculations, and both the 6d and 5f metal orbitals participate significantly in metal-ligand covalent bonding. The 4f-based molecular orbitals of [Ce( η -C 5 H 5 ) 3 ] are found to be little altered from those of free Ce(III), and show a three-below-four ( j = 5/2 below j = 7/2) spin—orbit coupling pattern. The greater radial extension of the 5f atomic orbitals and their closer energy match with vacant carbocyclic ring levels destroys this splitting pattern in [Th( η -C 5 H 5 ) 3 ] and [Pa( η -C 8 H 8 ) 2 ]. The electronic absorption spectra of all three molecules have been calculated using the transition state method. In [Ce( η -C 5 H 5 ) 3 ] and [Pa( η -C 8 H 8 ) 2 ], the transitions are principally f → f in nature, although a low-lying f → d σ shift is predicted in both cases. In [Ce( η -C 5 H 5 ) 3 ], the calculated f → d σ transition is in good agreement with that found experimentally for [Ce( η -C 5 H 3 (SiMe 3 ) 2 ) 3 ]. The calculated spectrum of [Th( η -C 5 H 5 ) 3 ] consists of a series of d → f transitions, in agreement with the intense peaks observed in the experimental spectrum of [Th( η -C 5 H 3 (SiMe 3 ) 2 ) 3 ]. Comparison of the calculated transitions of [Pa( η -C 8 H 8 ) 2 ] with the experimental spectrum of [Pa( η -C 8 H 4 (CH 3 ) 4 ) 2 ] suggests that the latter is due to charge transfer transitions and that the f → d σ shift has not yet been experimentally observed.

Density-matrix-functional calculations for matter in strong magnetic fields: Ground states of heavy atoms.

We report on a numerical study of the density matrix functional introduced by Lieb, Solovej, and Yngvason for the investigation of heavy atoms in high magnetic fields. This functional describes exactly the quantum mechanical ground state of atoms and ions in the limit when the nuclear charge Z and the electron number N tend to infinity with N/Z fixed, and the magnetic field B tends to infinity in such a way that B/${\mathit{Z}}^{4/3}$\ensuremath{\rightarrow}\ensuremath{\infty}. We have calculated electronic density profiles and ground-state energies for values of the parameters that prevail on neutron star surfaces and compared them with results obtained by other methods. For iron at B=${10}^{12}$ G the ground-state energy differs by less than 2% from the Hartree-Fock value. We have also studied the maximal negative ionization of heavy atoms in this model at various field strengths. In contrast to Thomas-Fermi type theories atoms can bind excess negative charge in the density matrix model. For iron at B=${10}^{12}$ G the maximal excess charge in this model corresponds to about one electron. \textcopyright{} 1996 The American Physical Society.