Darwin Corrections Research Articles

Various methods for describing energies and transition properties in H-like ions under Debye and quantum plasmas

In this paper, we propose two kinds of theoretical methods: (i) a self-consistent fully relativistic model, Flexible Atomic Code (FAC) and (ii) an analytical form based on the tensor expression of the Breit-Pauli Hamiltonian, for describing electronic structure of highly charged ions under external confinement. The effect of confinement due to Debye plasma is incorporated through a standard Debye-Hückel potential, yet that of quantum plasma is treated under an exponential cosine screened Coulomb potential. For the latter model, the relativistic corrections arise from the mass and Darwin corrections are derived. As a test case, the effects of the confinement produced by the above two kinds of plasmas on the energies and transition rates of H-like Mg11+ and Fe25+ ions are investigated. Some characteristics of the respective properties in regard to the plasma shielding strength are analyzed. The present results are compared with each other and other available values for the same system, show a good quantitative agreement.

Study of energies and radiative properties of He-like ions within a dense plasma

On the basis of the tensor expression for the Breit-Pauli Hamiltonian and the irreducible tensor theory for many-electron atoms, we have proposed a general analytical method for calculating atomic structures of highly charged ions under a dense plasma environment. In this method, the standard ion-sphere (IS) potential is used to represent the interaction between the charge particles. The Racah wave functions are obtained as the linear combinations of the multi-Slater wave functions. The angular interactions and spin sums are worked out using an irreducible theory of complex systems. Analytical formulas for the calculation of relativistic corrections, such as the mass correction, the one-body Darwin correction, the two-body Darwin correction, and the spin-spin contact interaction, are derived. Energy levels and radiative properties of He-like Al11+ and Ni26+ ions are considered for illustrative purposes. Systematic trends with respect to both the nuclear charge Z and the electron density Ne are observed for all calculated quantities. Self-consistent calculations of dense plasma effects on the atomic structure, based on the same IS potential, are also presented using fully relativistic multiconfiguration Dirac-Fock (MCDF) wave functions in the active space approximation with the inclusion of a finite nuclear size, the Breit interaction, self-energy, and vacuum polarization for comparison purposes. Comparison and analysis are made between our two sets of results and the available results in the literature. The effects of relativistic correlations are investigated in detail for the first time and are found to play an important role in the stability of these systems. The results of this study should be of great help in the modeling and diagnostics of astrophysical and laboratory plasmas.

Practical theoretical expressions for ions embedded in Debye and quantum plasmas

An accurate theoretical expression is proposed for the atomic structure under external confinement. The analysis is based on the tensor expression for the Breit-Pauli Hamiltonian in which the Racah wave functions are expanded in terms of the linear combinations of the multi-Slater wave functions. The variables are separated into radial and angular parts, where the variational parameters in the trial wave functions are obtained by solving the radial Schrödinger equation and the expressions of the angular part are worked out using an irreducible theory of complex system. Relativistic corrections are derived directly, which are treated as a sum of five terms: mass correction, one-body Darwin correction, two-body Darwin correction, spin-spin contact interaction correction, and orbit-orbit interaction correction. Energies and radiative decay rates of Be-like Fe22+ and Kr32+ ions in the presence of two kinds of plasma environments are presented for demonstration purposes, one is the Debye plasma which is described by a standard Debye-Hückel potential and the other is the quantum plasma which is treated under an exponential cosine screened Coulomb potential. Independent self-consistent calculations within the fully relativistic frame by incorporating the above two potentials are also performed using the Flexible Atomic Code to verify the validity of the proposed expressions. Results are given over a wide range of screening lengths. Relativistic effects in energy spectra are studied for the first time and are found to be rather important, especially in the high-Z system. Comparisons between our two sets of results and other theoretical predictions when available are made. Some behavior of the respective properties with respect to the plasma shielding strength is discussed. The present study should be beneficial for the analysis of spectra in astrophysical and fusion plasmas.

Influence of plasma shielding effect on ground state and excited state energies of Ar<sup>16+</sup>

A systematical knowledge of the atomic properties in plasma is of great interest for various research areas, such as the explanation of the X-ray radiation from universe, plasma diagnostics, extreme ultraviolet (EUV) and X-ray sources and so on. Among these researches, the detailed information about how the plasma influences the atomic energy level and transition spectrum are crucial for understanding the X-ray emission mechanism and the state of plasma. An analytic calculation method of treating the non-relativistic energy and its relativistic corrections for the multi-electron atoms embedded in weakly coupled plasma is developed based on the Rayleigh-Ritz variation method. The systematical investigations are performed for the ground state 1s<sup>2</sup> <sup>1</sup>S, single excited states 1sns <sup>1,2</sup>S (<i>n</i> = 2−5), 1s<i>n</i>p <sup>1,3</sup>P (<i>n</i> = 2−5) and double excited state 2s2p <sup>1</sup>P of Ar<sup>16+</sup> ion in weak coupled plasma. The analytic formulas for calculating the non-relativistic energy and its relativistic correction energy are derived, which include mass correction, one and two-body Darwin correction, spin-spin contact interaction and orbit-orbit interaction. All the angular integration spin sums involved in the problem are worked out explicitly by using the irreducible theory. The influence of plasma on non-relativistic energy and relativistic correction energy are discussed. The results show that the mass correction and the one-body Darwin correction are the main ones among the terms of relativistic correction, and are three orders of magnitude greater than the other relativistic terms. The plasma shielding effect mainly affects the non-relativistic energy, and has little effect on the relativistic correction. At the same time, it has a more significant selectivity for the electronic configuration. Further research shows that the influence of plasma on the energy of the outer shell electron is greater than that of the inner shell electron. With the increase of the plasma shielding parameters, the outer shell electron extends outward, and the higher the excited state, the greater the degree of extension is. This work should be useful for astrophysical applications where such a plasma environment exists.

The Effect of Relativistic Interactions on the Spectral Characteristics of the Ground State of Carbon Monoxide

Using high-precision nonempirical methods of modern quantum chemistry, the effect of the weak relativistic interactions on the potential energy and the permanent dipole moment of the ground electronic state of the CO molecule is studied. The relativistic energy is calculated by the following three optional methods: within the first-order perturbation theory using the Cowan–Griffin operator containing the sum of the mass-velocity and Darwin corrections, within the framework of the approximate Douglas–Kroll–Hess scalar Hamiltonian, and the most rigid “four-component” relativistic Dirac–Coulomb–Gaunt Hamiltonian. The relativistic correction obtained by different methods agrees within a few percents and equals about 55–60 cm–1 in the region of an equilibrium internuclear distance of $${{R}_{e}} = 1.128$$ A. The addition of the relativistic correction decreases the equilibrium bond length by about 0.0002 A. The magnitude of the Lamb shift estimated by the semiempirical scaling of the one-electron Darwin’s term does not exceed several inverse centimeters near $${{R}_{e}}$$ . The relativistic correction to the dipole moment function is in the range from –0.001 to +0.003 D, which does not exceed 1% of the nonrelativistic component of the dipole moment.

Relativistic effects on the energy levels and radiative properties of He-like ions immersed in Debye plasmas

Systematic investigations are performed for the energy levels and radiative properties for selected He-like C4+, Ne8+, Ar16+, and Kr34+ ions embedded in weakly coupled plasmas. For the conditions in which the Coulomb coupling parameter is small, the standard Debye model is adopted to describe the plasma screening effects. Within the relativistic framework, the modified version of the Flexible Atomic Code computations is carried out by considering a Debye-Hückel potential, in which the plasma screening is taken into account for both the electron-nucleus and electron-electron (e-e) interactions. An independent calculation for various Debye lengths is also presented using the multiconfiguration Dirac-Fock method for comparison purposes. For the nonrelativistic treatment, the analytical solution of the Schrödinger equation with the Debye screened potential is proposed. The variation method is developed with Slater wave function as a trial wave function that contains the variational parameters. An exact analytical expression of relativistic corrections such as the mass-velocity correction, the one/two-body Darwin correction, the spin-spin contact interaction correction, and the orbit-orbit interaction correction is derived. Differences among our three kinds of calculated energy levels and transition properties are analyzed in terms of the nuclear charge and/or the Debye length. Systematic trend is observed for all the properties under study with respect to increased screening. The influence of relativistic effects is also investigated in detail and found to play an important role in these systems. Our results are compared with available results from other theoretical calculations and the experimental values in the literature, and a good agreement is achieved. This work should be useful for astrophysical applications where such plasma environments exist.

Theoretical study of Au4 thymine, Au20 and Ag20 uracil and thymine complexes for surface enhanced Raman scattering

A Density Functional Theory (DFT) study using the relativistic effective core potentials (RECPs) is performed with an aim to understand the relative energies and gain deeper insight of the preferential binding of the sites of nucleobases towards particular sites of tetrahedral twenty atom Au and Ag clusters and Au4 cluster using reactivity descriptors. Such calculations carry high values towards understanding the relativistic effects of the Au atom towards the Surface Enhanced Raman spectrum (SERS) of small molecules like uracil and thymine when they are bound to the heavy metal clusters, viz., Au and Ag. The relative stabilities based on the prediction of the reactivity descriptors are analyzed. For Au4, scattering activities exhibit further enhancement in absence of relativistic mass-velocity contractions and Darwin corrections as implemented in LANL2DZ ECPS for Au in complexes, indicating the importance of relativistic effects in Au for proper evaluation of SERS and explains Ag acting as a better surface than Au for SERS.

Band structures of atomic chains of group IV, III–V, and II–VI elements

The relativistic band structures of AN and ANB8–N chains have been calculated by the linear augmented-cylindrical-wave method, which is an extension of the augmented-plane-wave method for cylindrical polyatomic systems. The band structures of covalent monatomic chains of Group IV elements are characterized by σ(s), π+, and π–, and σ(pz)* bands. The C, Si, Ge, and Sn chains are metallic. There is a considerable difference between the relativistic and nonrelativistic band structures. Because of the cylindrical symmetry of chains in the nonrelativistic model, the π bands crossing the Fermi level are orbitally doubly degenerate (i.e., the π+ and π– band energies are exactly the same). The spin and orbital motion of electrons are coupled in the chains to split π bands, but each π+ and π- band is doubly spin-degenerate. The spin–orbit splitting energy for C and Sn chains varies from 1.7 meV to 0.67 eV. The mass–velocity correction reduces all valence band levels: the level shifts are 2–5 meV for C and up to 2.2 eV for Sn. The Darwin corrections are several-fold lower than the mass–velocity contributions. A sharp change in the band structure is observed in going from covalent to partially ionic chains. The carbon chain has a metallic band structure with a zero gap in the center of the Brillouin zone, and a boron nitride chain is an insulator with an optical gap of 8 eV and optical transitions between the occupied π and vacant π* states at the edge of the Brillouin zone (this is explained by the existence of the antisymmetric component of the electron potential in the BN wire, which mixes even bonding and odd antibonding π states). Going from the BN chain to the AlP, GaAs, and InSb chains is accompanied by a decrease in the chemical bond ionicity, which leads to a gradual decrease in the π–π* gaps.

Relativistic effects in the one-bond spin-spin coupling constants involving selenium.

One-bond spin-spin coupling constants involving selenium of seven different types, (1) J(Se,X), X = (1) H, (13) C, (15) N, (19) F, (29) Si, (31) P, and (77) Se, were calculated in the series of 14 representative compounds at the SOPPA(CCSD) level taking into account relativistic corrections evaluated both at the RPA and DFT levels of theory in comparison with experiment. Relativistic corrections were found to play a major role in the calculation of (1) J(Se,X) reaching as much as almost 170% of the total value of (1) J(Se,Se) and up to 60-70% for the rest of (1) J(Se,X). Scalar relativistic effects (Darwin and mass-velocity corrections) by far dominate over spin-orbit coupling in the total relativistic effects for all (1) J(Se,X). Taking into account relativistic corrections at both random phase approximation and density functional theory levels essentially improves the agreement of theoretical results with experiment. The most 'relativistic' (1) J(Se,Se) demonstrates a marked Karplus-type dihedral angle dependence with respect to the mutual orientation of the selenium lone pairs providing a powerful tool for stereochemical analysis of selenoorganic compounds.

Prediction of 1P Rydberg energy levels of beryllium based on calculations with explicitly correlated Gaussians

Benchmark variational calculations are performed for the seven lowest 1s(2)2s np ((1)P), n = 2...8, states of the beryllium atom. The calculations explicitly include the effect of finite mass of (9)Be nucleus and account perturbatively for the mass-velocity, Darwin, and spin-spin relativistic corrections. The wave functions of the states are expanded in terms of all-electron explicitly correlated Gaussian functions. Basis sets of up to 12,500 optimized Gaussians are used. The maximum discrepancy between the calculated nonrelativistic and experimental energies of 1s(2)2s np ((1)P) →1s(2)2s(2) ((1)S) transition is about 12 cm(-1). The inclusion of the relativistic corrections reduces the discrepancy to bellow 0.8 cm(-1).

First-principles prediction and partial characterization of the vibrational states of water up to dissociation

A new, accurate, global, mass-independent, first-principles potential energy surface (PES) is presented for the ground electronic state of the water molecule. The PES is based on 2200 energy points computed at the all-electron aug-cc-pCV6Z IC-MRCI(8,2) level of electronic structure theory and includes the relativistic one-electron mass-velocity and Darwin corrections. For H 2 16O, the PES has a dissociation energy of D 0 = 41 109 cm −1 and supports 1150 vibrational energy levels up to 41 083 cm −1. The deviation between the computed and the experimentally measured energy levels is below 15 cm −1 for all the states with energies less than 39 000 cm −1. Characterization of approximate vibrational quantum numbers is performed using several techniques: energy decomposition, wave function plots, normal mode distribution, expectation values of the squares of internal coordinates, and perturbing the bending part of the PES. Vibrational normal mode labels, though often not physically meaningful, have been assigned to all the states below 26 500 cm −1 and to many more above it, including some highly excited stretching states all the way to dissociation. Issues to do with calculating vibrational band intensities for the higher-lying states are discussed.

Wavefunction and energy of the 1s22sns configuration in a beryllium atom

A new set of trial functions for 1s22sns configurations in a beryllium atom is suggested. A Mathematica program based on the variational method is developed to calculate the wavefunctions and energies of 1s22sns (n = 3–6) configurations in a beryllium atom. Non-relativistic energy, polarization correction and relativistic correction which include mass correction, one-and two-body Darwin corrections, spin-spin contact interaction and orbit-orbit interaction, are calculated respectively. The results are in good agreement with experimental data.

Fine-structure energy levels, oscillator strengths and lifetimes in Co XV

Excitation energies from ground state for 97 fine-structure levels as well as oscillator strengths and radiative decay rates for all electric-dipole-allowed and intercombination transitions among the fine-structure levels of the terms belonging to the (1s 22s 22p 6)3s 23p, 3s 3 p 2, 3s 23d, 3p 3, 3s3p3d, 3p 23d, 3s3d 2, 3s 24s, 3s 24p, 3s 24d, 3s 24f, and 3s3p4s configurations of Co XV are calculated, using extensive configuration-interaction (CI) wave functions, obtained with the CIV3 computer code of Hibbert. The important relativistic effects in intermediate coupling are included through the Breit-Pauli approximation via spin-orbit, spin-other-orbit, spin-spin, Darwin and mass correction terms. Small adjustments to the diagonal elements of the Hamiltonian matrices have been made. Our calculated excitation energies, including their ordering, are in excellent agreement with the experimental results and the experimentally compiled energy values of the National Institute for Standards and Technology (NIST) wherever available. The mixing among several fine-structure levels is found to be very strong, with most of the strongly mixed levels belonging to the (1s 22s 22p 6)3p 23d and 3s3d 2 configurations. The strong mixing among several fine-structure levels makes it very difficult to identify them uniquely. Perhaps, that may be the reason for the lack of both experimental and theoretical results for these levels. We believe that our extensive calculated values can guide experimentalists in identifying the fine-structure levels in their future work. From our radiative decay rates we have also calculated radiative lifetimes of some fine-structure levels. In this calculation we also predict new data for several fine-structure levels where no other theoretical and/or experimental results are available.

Energy and relativistic correction of the 1sns configuration in helium

A Mathematica program is developed to calculate the energy and relativistic correction of helium-like atoms in 1sns configuration. A new set of trial functions for 1sns configuration in helium has been suggested. By virtue of the Rayleigh-Ritz variational method, the non-relativistic energies of 1sns configuration in helium (n=2—5) have been calculated, the relativistic corrections, which include mass correction, one- and two-body Darwin correction and spin-spin contact interaction, have been further calculated with the theory of perturbation. The calculated results are in good agreement with experimental data.

Lowest vibrational states ofHe4He+3: Non-Born-Oppenheimer calculations

Very accurate quantum mechanical calculations of the first five vibrational states of the {sup 4}He{sup 3}He{sup +} molecular ion are reported. The calculations have been performed explicitly including the coupling of the electronic and nuclear motions [i.e., without assuming the Born-Oppenheimer (BO) approximation]. The nonrelativistic non-BO wave functions were used to calculate the {alpha}{sup 2} relativistic mass velocity, Darwin, and spin-spin interaction corrections. For the lowest vibrational transition, whose experimental energy is established with high precision, the calculated and the experimental results differ by only 0.16 cm{sup -1}.

Fine-structure energy levels, oscillator strengths and lifetimes in Mg-like chromium

Excitation energies from ground state for 86 fine-structure levels as well as oscillator strengths and radiative decay rates for all fine-structure transitions among the levels of the terms (1s22s22p6)3s2(1S), 3s3p(1,3Po), 3s3d(1,3D), 3s4s(1,3S), 3s4p(1,3Po), 3s4d(1,3D), 3s4f(1,3Fo), 3p2(1S, 3P, 1D), 3p3d(1,3Po, 1,3Do, 1,3Fo), 3p4s(1,3Po), 3p4p(1,3S, 1,3P, 1,3D), 3p4d(1,3Po, 1,3Do, 1,3Fo), 3p4f(1,3D, 1,3F, 1,3G) and 3d2(1S, 3P, 1D, 3F,1G) of Cr XIII are calculated using extensive configuration-interaction (CI) wave functions obtained with the CIV3 computer code of Hibbert. The important relativistic effects in intermediate coupling are included through the Breit-Pauli approximation via spin-orbit, spin-other-orbit, spin-spin, Darwin and mass correction terms. Small adjustments to the diagonal elements of the Hamiltonian matrices have been made. The mixing among several fine-structure levels is found to be very strong. Our excitation energies, including their ordering, are in excellent agreement (better than 0.5%) with the available experimental results. From our transition probabilities, we have also calculated radiative lifetimes of some fine-structure levels. Our calculated lifetime for the longer-lived level 3s3p(3P1) is found to be in excellent agreement with the experimental result of Curtis compared to other theoretical calculations.

Darwin and mass-velocity relativistic corrections in non-Born-Oppenheimer variational calculations

The Pauli approach to account for the mass-velocity and Darwin relativistic corrections has been applied to the formalism for quantum mechanical molecular calculations that does not assume the Born-Oppenheimer (BO) approximation regarding separability of the electronic and nuclear motions in molecular systems. The corrections are determined using the first order perturbation theory and are derived for the non-BO wave function of a diatomic system expressed in terms of explicitly correlated Gaussian functions with premultipliers in the form of even powers of the internuclear distance. As a numerical example we used calculations of the transition energies for pure vibrational states of the HD(+) ion.

Darwin and mass-velocity relativistic corrections in the non-Born-Oppenheimer calculations of pure vibrational states of H2

The Darwin and mass-velocity relativistic corrections have been calculated for all pure vibrational states of the H2 using the perturbation theory and very accurate variational wave functions obtained without assuming the Born-Oppenheimer (BO) approximation. Expansions in terms of explicitly correlated Gaussians with premultipliers in the form of even powers of the internuclear distance were used for the wave functions. With the inclusion of the two relativistic corrections to the non-BO energies the transition energies for the highest states agree more with the experimental results.

Convergence of Experiment and Theory on the Pure Vibrational Spectrum ofHeH+

Very accurate quantum mechanical calculations of the pure vibrational spectrum of the molecular ion are reported and compared with newly obtained pure vibrational transitions extracted from the available experimental data. The calculations are performed without assuming the Born-Oppenheimer approximation regarding separability of the nuclear and electronic motions and include the first order relativistic mass-velocity and Darwin corrections. For the two lowest transitions, whose experimental energies are established with the highest precision, the calculated and the experimental results show very good agreement.

Breit-Pauli energy levels and radiative lifetimes in neutral chlorine

Breit-Pauli Energy levels, oscillator strengths and transition probabilities for all the transitions in Cl I between the fine structure levels of 3s2 3p5, 3s2 3p43d, 3s2 3p44s and 3s23p44p states are calculated using extensive configuration interaction (CI) wave functions. We have also determined the lifetimes of 3s2 3p43d, 3s2 3p44s and 3s2 3p44p levels. The relativistic effects are included through Breit-Pauli approximation via spin-orbit, Darwin and mass correction terms. Prior to the calculations of the oscillator strengths and transition probabilities, we fine-tune the CI coefficient using experimental energies. Our results are compared with experimental and other available theoretical data. The calculated energy levels are in close agreement with most of the NIST compiled data. We predict new lifetime data for several levels where no other theoretical and/or experimental results are available, which will form the basis for the future experimental work.