Many-electron Atoms Research Articles

Theoretical investigation of the inverse Faraday effect via a stimulated Raman scattering process

We study theoretically the origin and mechanism of the ultrafast inverse Faraday effect, which is a magneto-optical effect, attracting much interest nowadays. Laser-induced subpicosecond spin dynamics in hydrogenlike systems and isolated many-electron atoms are investigated in order to get insight into this process. We show that the stimulated Raman scattering process leads to a change of the magnetic state of a system. We obtain the time evolution of the induced magnetization, its dependencies on laser properties, and the connection with the spin-orbit coupling of a system.

New procedures and techniques for the construction and evaluation of atomic wave functions and spectra

A rotating coordinate system through the center of mass of two electrons and the related (D) functions are defined. The product of two-single particle functions is expressed in terms of the new coordinates, and a general expression for the two-particle matrix elements are obtained and compared with the traditional method of expression in spherical harmonics. For many-electron atoms, a group theoretic method is developed for step-wise construction of wave functions of N-electrons from those of (N-1) – electrons. Functions are characterized with specific quantum numbers, namely, permutational symmetry, angular momenta, & parity. Examples are given to illustrate the effectiveness of the method and compare results with experimental data.

Basis set effects on the Hartree–Fock description of confined many-electron atoms

In this work, the basis sets designed by Clementi, Bunge and Thakkar, for atomic systems, have been used to obtain the electronic structure of confined many-electron atoms by using Roothaan's approach in the Hartree–Fock context with a new code written in C, which uses the message-passing interface library. The confinement was imposed as Ludeña suggested to simulate walls with infinity potential. For closed-shell atoms, the Thakkar basis set functions give the best total energies (TE) as a function of the confinement radius, obtaining the following ordering: TE(Thakkar) < TE(Bunge) < TE(Clementi). However, for few open-shell atoms this ordering is not preserved and a trend, for the basis sets, is not observed. Although there are differences between the TE predicted by these basis set functions, the corresponding pressures are similar to each other; it means that changes in the total energy are described almost in the same way by using any of these basis sets. By analysing the total energy as a function of the inverse of the volume we propose an equation of state; for regions of small volumes, this equation predicts that the pressure is inversely proportional to the square of the volume.

Free–free matrix elements for a many-electron atom

Analytical expressions for both the dipole and the overlap matrix elements between two states of the continuous spectrum of an electron of a many-electron atom are derived. We have used the velocity form for the dipole matrix element. The derivation of these formulae is based directly on the use of the nonrelativistic Hartree–Fock equations. The formulae do not have phase shift dependence and are expressed only through the direct and exchange potentials of the many-electron atom. Furthermore, the expressions derived here can be used to calculate the phase shift.

Rank restriction for the variational calculation of two-electron reduced density matrices of many-electron atoms and molecules

Variational minimization of the ground-state energy as a function of the two-electron reduced density matrix (2-RDM), constrained by necessary $N$-representability conditions, provides a polynomial-scaling approach to studying strongly correlated molecules without computing the many-electron wavefunction. Here we introduce a new route to enhancing necessary conditions for $N$-representability through rank restriction of the 2-RDM. Rather than adding computationally more expensive $N$-representability conditions, we directly enhance the accuracy of two-particle (2-positivity) conditions through rank restriction, which removes degrees of freedom in the 2-RDM that are not sufficiently constrained. We select the rank of the particle-hole 2-RDM by deriving the ranks associated with model wavefunctions including both mean-field and antisymmetrized geminal power (AGP) wave functions. Because the 2-positivity conditions are exact for quantum systems with AGP ground states, the rank of the particle-hole 2-RDM from the AGP ansatz provides a minimum for its value in variational 2-RDM calculations of general quantum systems. To implement the rank-restricted conditions, we extend a first-order algorithm for large-scale semidefinite programming. The rank-restricted conditions significantly improve the accuracy of the energies; for example, the percentages of correlation energies recovered for HF, CO, and N$_{2}$ improve from 115.2%, 121.7%, and 121.5% without rank restriction to 97.8%, 101.1%, and 100.0% with rank restriction. Similar results are found at both equilibrium and non-equilibrium geometries. While more accurate, the rank-restricted $N$-representability conditions are less expensive computationally than the full-rank conditions.

On the completeness of one-particle states of a many-electron atom

A method for constructing the complete set of one-particle states of a many-electron atom is proposed in the framework of the one-configuration Hartree-Fock approximation.

Atoms as many-body systems

Contrary to common wisdom, not everything is clear and simple in the structure of many-electron atoms. Complexity in atoms is mainly a result of interelectron interaction that leads to rather unusual behaviour. Most transparently this is manifested in photo-ionization processes of many-electron atoms and some multi-atomic objects e.g. endohedrals. Particular attention will be given to the approach describing the interaction of photons with many-electron atoms in the frame of the many-body theory based on the Feynman diagrams technique. As a suitable one-electron approximation the Hartree - Fock (HF) approach will be presented. On its ground we will include the so-called electron correlation effects and discuss the frequently used Random Phase Approximation with Exchange - RPAE. Some results of recent calculations will be presented.

On the completeness of one-particle states of a many-electron atom in an extended Hilbert space

The completeness of the set is a fundamental requirement of quantum mechanics. The violation of this requirement can lead to significant uncertainty of the physical interpretation of research results in the field of spectral characteristics of many-electron atoms, atomic ions and molecules. In the context of a nonrelativistic one-configurational Hartree–Fock approximation we suggested the solution of the problem of receiving the total complete set of one-particle states of many-electron atoms, including the states of a continuous spectrum. As an accompanying result we introduced the conception of an extended Hilbert space with reflection into mathematical formalism of quantum-mechanical atom description.

Quantum electrodynamic corrections for the valence shell in heavy many-electron atoms

We present quantum electrodynamic (QED) calculations within the picture of bound-state QED for the frequency-dependent Breit interaction between electrons, the vacuum polarization, and the electron self-energy correction starting from the Dirac-Coulomb Hamiltonian for the ionization potentials of the group 1, 2, 11, 12, 13, and 18 elements of the periodic table, and down to the superheavy elements up to nuclear charge Z=120. The results for the s-block elements are in very good agreement with earlier studies by Labzowsky et al. [Phys. Rev. A 59, 2707 (1999)]. We discuss the influence of the variational versus perturbative treatment of the Breit interaction for valence-space ionization potentials. We argue that the lowest-order QED contributions become as important as the Breit interaction for ionization potentials out of the valence s shell.

Explicitly-correlated Gaussian geminals in electronic structure calculations

Explicitly correlated functions have been used since 1929, but initially only for two-electron systems. In 1960, Boys and Singer showed that if the correlating factor is of Gaussian form, many-electron integrals can be computed for general molecules. The capability of explicitly correlated Gaussian (ECG) functions to accurately describe many-electron atoms and molecules was demonstrated only in the early 1980s when Monkhorst, Zabolitzky and the present authors cast the many-body perturbation theory (MBPT) and coupled cluster (CC) equations as a system of integro-differential equations and developed techniques of solving these equations with two-electron ECG functions (Gaussian-type geminals, GTG). This work brought a new accuracy standard to MBPT/CC calculations. In 1985, Kutzelnigg suggested that the linear r 12 correlating factor can also be employed if n-electron integrals, n > 2, are factorised with the resolution of identity. Later, this factor was replaced by more general functions f (r 12), most often by , usually represented as linear combinations of Gaussian functions which makes the resulting approach (called F12) a special case of the original GTG expansion. The current state-of-art is that, for few-electron molecules, ECGs provide more accurate results than any other basis available, but for larger systems the F12 approach is the method of choice, giving significant improvements over orbital calculations.

Theorem about electron energy in many-electron atoms in biological molecules

Theorem about electron energy in many-electron atoms in biological molecules

Colloquium: Ettore Majorana and the birth of autoionization

In some of the first applications of modern quantum mechanics to the spectroscopy of many-electron atoms, Ettore Majorana in 1931 solved several outstanding problems by developing the theory of autoionization. Later literature makes only sporadic references to this accomplishment. After reviewing his work in its contemporary context, we describe subsequent developments in understanding the spectra treated by Majorana and extensions of his theory to other areas of physics. Several puzzles are found concerning the treatment of Majorana's work in the subsequent literature and the way in which the modern theory of autoionization was developed.

On the alleged non-existence of orbitals

I argue that the claim made by Scerri that in many-electron atoms, orbitals do not exist according to quantum mechanics, is incorrect, for it relies on the view that orbitals are entities. Orbitals are states, not entities, and their use in describing many-electron atoms should be seen as an approximation. The writings by Scerri and others on the issue of realism that are based on the claim therefore lead astray. I furthermore disentangle two issues that Scerri discusses in arguing for his claim: that of electron correlation and that of the Pauli principle. Finally, I point out that more generally there is a misconception in chemistry of what quantum states are.

Construction of minimizing sequence of multiparticle functions for calculating the spectrum of an atom

A method for calculating the states of a many-electron atom is proposed based on the representation of the wave function of the atom as a linear combination of multideterminant functions. Determinants are constructed from single-particle hydrogen-like functions in which the positive nuclear charge is replaced by a certain effective charge. For each single-particle function, the effective charge is calculated such that to take into account the interaction of a given electron with the spherically symmetric part of the screening charge distribution of remaining electrons. Examples of constructing approximating sequence of functions are given and results of calculations of energies of stationary states of the He and Li atoms are presented.

Single-photon double K-shell ionization of low-Z atoms

The photon energy dependence of the double K-shell ionization of light atoms is reported. Experimental double-to-single photoionization cross section ratios for Mg, Al, Si and Ca were obtained from measurements of high-resolution x-ray emission spectra. The double photoionization (DPI) cross-sections for K-shell hollow atom production are compared to convergent close-coupling calculations (CCC) for neutral atoms and He-like ions. The relative importance of the initial-state and final-state electron-electron interactions to the K-shell DPI in many-electron atoms and two-electron ions is addressed. Physical mechanisms and scaling laws of the K-shell double photoionization are examined. A semiempirical universal scaling of the DPI cross sections with the effective nuclear charge for neutral atoms 2≤Z≤47 is established.

Atoms in boxes: From confined atoms to electron-atom scattering

We show that both confined atoms and electron-atom scattering can be described by a unified basis set method. The central idea behind this method is to place the atom inside a hard potential sphere, enforced by a standard Slater type basis set multiplied by a cutoff factor. For confined atoms, where the wall is placed close to the atomic nucleus, we show how the energy of the highest occupied atomic orbital and the static polarizability of helium and neon atoms evolve with the confinement radius. To our knowledge, these are the first confined atom polarizability calculations that include correlation for many-electron atoms, through the use of time-dependent density-functional theory. By placing the atom in a large spherical box, with a wall outside the electron density, we obtain scattering phase shifts using a recently developed method [M. van Faassen, A. Wasserman, E. Engel, F. Zhang, and K. Burke, Phys. Rev. Lett. 99, 043005 (2007)]. We show that the basis set method gives identical results to previously obtained phase shifts for e-H and e-He+ scattering.

Two-Particle Coulomb Green Function Method with Projected Potential: Application to He Double Photoionization

A new method to compute fully differential double photoionization cross sections of atoms has been devised and fully developed for two-electron systems. The method exploits the Green function for two noninteracting electrons in the field of a nuclear charge to infer the effects of the residual potential projected on a set of L(2)-basis functions. Test calculations on helium at 100 eV excess energy indicate that, as long as the relevant part of the interaction potential is accounted for, the fully differential cross sections calculated in acceleration and velocity gauges converge in absolute value and reproduce measured angular distributions with a tunable accuracy. Generalization of the method to treat double photoionization of many-electron atoms is sketched.

Z-expansion calculations of energy levels and transition probabilities in many-electron atoms

This paper reviews recent calculations of energy levels and transition probabilities in many-electron atoms, based on expansions in inverse powers of the nuclear charge. Emphasis is given to simple approximation methods that apply to a wide class of atomic states. It is shown that the variational screening approximation often yields an accurate estimate of energy contributions of order Z−1 and that the relativistic contributions to the energy levels of light and moderately light atoms are comparable to or greater than the nonrelativistic contributions of order Z−1 when the degree of ionization exceeds 3 or 4. A new method is described for normalizing Coulomb wave functions that figure in the approximate evaluation of electric-dipole transition integrals. It employs variational screening parameters and makes use of double perturbation theory.

Relativistic effects on information measures for hydrogen-like atoms

Position and momentum information measures are evaluated for the ground state of the relativistic hydrogen-like atoms. Consequences of the fact that the radial momentum operator is not self-adjoint are explicitly studied, exhibiting fundamental shortcomings of the conventional uncertainty measures in terms of the radial position and momentum variances. The Shannon and Rényi entropies, the Fisher information measure, as well as several related information measures, are considered as viable alternatives. Detailed results on the onset of relativistic effects for low nuclear charges, and on the extreme relativistic limit, are presented. The relativistic position density decays exponentially at large r , but is singular at the origin. Correspondingly, the momentum density decays as an inverse power of p . Both features yield divergent Rényi entropies away from a finite vicinity of the Shannon entropy. While the position space information measures can be evaluated analytically for both the nonrelativistic and the relativistic hydrogen atom, this is not the case for the relativistic momentum space. Some of the results allow interesting insight into the significance of recently evaluated Dirac–Fock vs. Hartree–Fock complexity measures for many-electron neutral atoms.

An arbitrary order Douglas–Kroll method with polynomial cost

A new Douglas-Kroll transformation scheme up to arbitrary order is presented to study the convergence behavior of the Douglas-Kroll series and the influence of different choices of parametrization for the unitary transformation. The standard approach for evaluating the Douglas-Kroll Hamiltonian suffers from computational difficulties due to the huge number of matrix multiplications, which increase exponentially with respect to the order of truncation. This makes it prohibitively expensive to obtain results for very high order Douglas-Kroll Hamiltonians. The highest order previously presented is 14th order, but it is not enough to obtain accurate results for systems containing heavy elements, where the Douglas-Kroll series converges very slowly. In contrast, our approach dramatically reduces the number of matrix multiplications, which only increase with a polynomial scaling. With the new method, orders greater than 100 and machine accuracy are possible. This fast method is achieved by employing a special transformation to all Douglas-Kroll operators and our algorithm is very simple. We demonstrate the performance of our implementation with calculations on one-electron systems and many-electron atoms. All results show very good convergence behavior of the Douglas-Kroll series. Very small differences are found between the different parametrizations, and therefore the exponential form, which is the simplest and fastest, is recommended.