United Atom Limit Research Articles

Experimental low velocity proton and 28Siq+ induced X-ray production cross sections for select elements within 24 ≤ Z ≤ 83

Particle Induced X-ray Emission (PIXE) spectroscopy is an effectively well-established analytical technique in the low projectile mass (Z < 3) regime. This is in part due to the availability of fairly accurate theoretical models of fundamental interaction parameters such as the X-ray production cross section (XPCS). Continual development of widely used theoretical models such as the ECPSSR theory with adapted corrections in the United Atom limit (UA), Electron Capture (EC) and Multiple Ionisation (MI); requires a rich database of experimental X-ray production cross sections. Beyond the scope of proton PIXE, validated reliable theoretical models may be useful for the accurate approximation of heavy ion induced X-ray production cross sections for Heavy Ion PIXE. This requires persistent efforts aimed at contributing to the currently sparse heavy ion induced X-ray production cross section database. In this work, we report on a new experimental dataset of low energy XPCS in the 0.3 MeV/u – 1.0 MeV/u ion velocity range in Cr, Ni, Ge, Mo, Sn, W, Au and Bi due to proton and 28Siq + ion impact, with an aim at enriching and updating the current database.

Semi-empirical dipole moment of carbon monoxide and line lists for all its isotopologues revisited

The semi-empirical permanent electric dipole-moment function (DMF) for the ground state of the CO molecule has been reconstructed analytically in the entire range of the inter-nuclear distance r∈[0,+∞) by means of the simultaneous non-linear least-squares fitting (NLLSF) of the selected experimental intensities for the main 12C16O isotopologue (including those with sub-percent uncertainties) and the ab initio permanent dipole moment. The ab initio DMFs were evaluated using single-reference coupled cluster (SR-CCSD(T)) and multi-reference averaged coupled-pair functional (MR-ACPF) methods. The ab initio data were involved in the NLLSF procedure to propagating smoothly the semi-empirical DMF outside the local region covered by the experimental intensities for the lowest vibrational v′≤11 levels. The derived mass-invariant DMF possesses the physically correct asymptotic behavior in both the united-atom and dissociation limits as well as reproduces the vast majority of the measured intensities in v″=0→v′∈[0,6] and v″=1→v′=4,5 bands within their experimental uncertainties. The resulting DMF and the mass-corrected potential-energy function of Meshkov et al. (2018)[16] were used to upgrade line lists for all CO isotopologues in the wide range of vibrational and rotational quantum numbers v∈[0,41],Δv∈[0,6], J∈[0,150] (J∈[0,200] for the 0-0 and 1-0 bands). The predicted intensities are compared with their experimental counterparts, which were not involved in the present NLLSF, to highlight presumable random and systematic errors in the measured data; in particular, the intensities in the “abnormal” 0→5 band of the 12C16O isotopologue and some other bands of minor isotopologues should be revisited experimentally. The resultant line list should be considered superior to previous efforts in terms of accuracy.

An accurate potential model for the a3Σu+ state of the lithium dimer.

An accurate Tang-Toennies (TT) model potential is introduced to describe the interatomic potential of the lithium dimer in the a3Σu+ state. With only one well-known parameter, the ionization energy, the new model potential compares favorably with the experimentally fitted Morse/Long-range (MLR) potential of Dattani and Le Roy [J. Mol. Spectrosc., 2011, 268, 199] and is in excellent agreement with the state-of-the-art ab initio potential of Lesiuk et al. [Phys. Rev. A, 2020, 102, 062806]. With the known dispersion coefficients and the ionization energy, the new potential requires only two experimental parameters, namely the depth of the potential well De and its location Re. The new potential can be extended to the region of zero separation by the united atom limit.

Hybrid Gaussian–discrete-variable representation for describing molecular double-ionization events

A hybrid method that combines Gaussian basis functions typically used in bound-state molecular electronic structure calculations with a grid-based discrete-variable representation with finite elements suitable for a general electronic continuum representation is used to fully describe the double ionization of molecular ${\mathrm{H}}_{2}$ by a single photoabsorption. This work expands the hybrid method, previously applied to single-ionization events, to double photoionization. Constructing the full two-electron operator encoding the electron correlation necessary to doubly ionize the target via the action of a single photon requires all classes of mixed integrals between combinations of the different basis-function types. Comparison of the present results with benchmark theoretical calculations and experimental results shows excellent agreement for both molecular ${\mathrm{H}}_{2}$ and its united-atom limit, atomic helium; the triply differential cross sections that relate the angular distribution and energy sharing of all of the particles in the frame of the molecule are compared. The two-electron results computed using this hybrid basis hint at application of this general descriptive scheme beyond this simplest molecular target towards describing double ionization in more complicated and experimentally relevant molecules.

Empirical normal intensity distribution for overtone vibrational spectra of triatomic molecules

Theoretical calculations are contributing a significantly higher proportion of data to contemporary spectroscopic databases, which have traditionally relied on experimental observations and semi-empirical models. It is now a common procedure to extend calculated line lists to include ro-vibrational transitions between all bound states of the ground electronic state up to the dissociation limit. Advanced ab initio methods are utilized to calculate the potential energy and dipole moment surfaces (PESs and DMSs), and semi-empirical PESs are then obtained by combining ab initio and experimental data. The objective is to reach high accuracy in the calculated transition intensities for all parts of spectrum, i.e. to increase the predictive power of the model. We show that in order to perform this task, one needs, in addition to the standard improvements of the PES and DMS in the spectroscopically accessible regions, to extend the ab initio calculations of the PES towards the united-atom limit along the stretching coordinates. The argument is based on the correlation between the intensities of high-overtone transitions and the repulsive potential wall that has previously been theoretically established for diatomic molecules and is empirically extended here to linear and nonlinear triatomic molecules. We generate partial line lists for water and ozone, and together with an already available line list for carbon dioxide, we derive the normal intensity distribution, which is a direct consequence of this correlation. The normal distribution is not an instrument to compute highly accurate intensities, rather it is a means to analyse the intensities computed by the traditional methods.

Physics beyond PIXE

The paper reveals geometrical effects that govern the accuracy of PIXE measurements: distinction between thin and thick targets, misorientation of the sample, surface roughness and contribution of the invisible matrix. It shows an importance of the mean X-ray production distance and its application in differential measurements. It further reviews the data base needed for thick target calculations and the importance of developing more accurate theoretical models based on the semiclassical picture and coupled channel approach. An example of transitions in a perturbed harmonic oscillator demonstrates that the united atom limit does not exist at adiabatic collisions and that the cross section likely exhibits an oscillatory behaviour.

Electron correlation in chemical bonds II. LiH and LiH+

The correlation energy in LiH and LiH+ molecules is studied as a function of bond distance, from the united atom limit to the complete bond dissociation, using customized basis sets and Full Configuration Interaction. The analysis reveals an unusual complexity of the correlation energy dependence on the bond distance. The obtained accurate correlation energy profiles are used to study the performance of the established correlation models. The density functional models are found to be reasonably accurate in the description of LiH molecule, and are deficient in LiH+.

Semi-empirical ground-state potential of carbon monoxide with physical behavior in the limits of small and large inter-atomic separations

Recently, several empirical models of the potential energy function (PEF) of the ground electronic state of CO have been developed by Coxon and Hajigeorgiou using the direct-potential-fit (DPF) analysis. These models reproduce more than twenty thousand of observed ro-vibrational line positions within their experimental accuracies and follow the theoretically predicted long-range asymptotic behavior of the potential beyond Le Roy's radius. However, the applicability of these models to calculations of the intensities of weak transitions is not obvious due to limitations discussed in this work. Here, we make the next step by imposing an additional requirement that the model has to display physical behavior outside the region to which the available spectroscopic data pertain. We perform ab initio calculations of the potential in the short-range region r = 0.3–0.8 Å where the potential increases well above the dissociation limit and include these data along with the estimated uncertainties in the DPF analysis. At smaller r, our PEF tends to the well-known united-atom limit. In the long-range region, it follows the ab initio data with their estimated accuracies available in the literature. On top of the above, the model is analytical, i.e. it has continuous derivatives of all orders. The model constructed is a linear function of twenty three adjustable parameters, and a non-linear function of additional five parameters. The properties of the new PEF are investigated in comparison with the previous empirical model due to Coxon and Hajigeorgiou (2004). Our PEF provides reproduction of all observed lines at the same level of accuracy. The new potential changes the intensities of high overtone transitions dramatically. In addition, we find that the anomalies in the purely rotational and ro-vibrational spectra change their intensities between the two PEFs even for the low-v bands.

Analytical Correlation Rules between States in the United Atom and Separated Atom Limits for Diatomic Screened Orbitals

The analytic correlation rule of Gershtein and Krivchenkov between states in the united atom and separated atom limits for one‐electron diatomic molecule is extended to diatomic molecules with more than one electron for diabatic screened molecular orbitals, which are solutions of the Hartree‐like Schrödinger equation that is separable in prolate spheroidal coordinates. The rule is interpreted with the WKB method.

Adiabatic representation in the Coulomb three-body problem in the united-atom limit: Nuclear widths of the energy levels of the muonic molecule ttµ

We study the asymptotic behavior of the wave function of the system of three Coulomb particles in the united-atom limit in the adiabatic representation of the three-body problem. This result is used to calculate the nuclear widths of muonic-molecule energy levels. We discuss features of the approach with regard to excited states of the muonic molecule ttµ with a nonzero orbital angular momentum.

Adiabatic representation in the Coulomb three-body problem in the united-atom limit: Nuclear widths of the energy levels of the muonic molecule $tt\mu$

Исследуется асимптотика волновой функции системы трех кулоновских частиц в пределе объединенного атома в адиабатическом представлении задачи трех тел. Этот результат используется при расчете ядерных ширин уровней энергии мезомолекул. Обсуждаются особенности подхода применительно к возбужденным состояниям мезомолекулы $tt\mu$ с ненулевым орбитальным моментом.

Communication: A simple full range analytical potential for H2b(3)∑u (+), H-He (2)∑(+), and He2 (1)∑g (.).

The Tang-Toennies potential for the weakly interacting systems H2b(3)Σu (+), H-He (2)Σ(+), and He2 (1)Σg (+) is extended down to the united atom limit of vanishing internuclear distance. A simple analytic expression connects the united atom limiting potential with the Tang-Toennies potential in the well region. The new potential model is compared with the most recent ab initio calculations for all three systems. The agreement is better than 20% (H2 and He2) or comparable with the differences in the available ab initio calculations (H-He) over six orders of magnitude corresponding to the entire range of internuclear distances.

Rydberg series of 1Σu+ and 1Δu states of the Li2 molecule studied by the promotion model

We studied the Rydberg series for the 12 1Σu+, 7 1Δu, and 1 1Γu states. This set consists of the seventeen states derivable from the Li(2s) + Li(nl) (n = 2, 3, 4, 5) configuration, one 1Σu+ state and one 1Δu state derivable from Li(2s) + Li(6p) and Li(2s) + Li(6d), respectively, plus one ionic state. We used the multireference configuration interaction method, combined with the Stuttgart/Köln group’s effective core potential/core polarization potential method, to calculate their potential energy curves (PECs). Four Rydberg series, σuns (n = 3, 4, 5, 6), σunp (n = 2, 3, 4, 5), δund (n = 3, 4, 5, 6), and δunf (n = 4, 5), are identified near the potential energy minima (referred to as the ‘Rydberg region’). The promotion model is used to examine the behaviour of PECs and quantum defect curves (QDCs) by constructing diabatic PECs and QDCs. Besides QDCs, effective n and promotion curves are also used to directly examine the promotion. This reveals that the promotion model can be successfully applied to the singlet ungerade states of Li2, indicating that the Rydberg region has the same characteristics as the united atom (UA) limit. Thus, the Rydberg region in Li2 is much closer to the UA limit than the separated atoms (SA) limit. Correlation diagrams based on the promotion model show a big difference in the number of avoided crossings for the 1Σu+ and 1Δu states. The larger number of avoided crossings for 1Σu+ states produces features in the PECs of the 1Σu+ states such as shoulders, flattening, and grouping. The promotion model not only helps to identify Rydberg series, but also to explain all the major and minor aspects and subtle phenomena observed in the PECs and spectroscopic constants of the singlet ungerade Rydberg states of Li2.

Dynamical adiabatic theory of atomic collisions: The global structure of dynamical adiabatic potential energy curves

The concept of dynamical adiabatic states, originally proposed to describe one-electron atom(ion)-ion collision systems is developed and the properties of the corresponding dynamical adiabatic potential energy curves are studied for a complete range of internuclear separations $R$. The advantages of a dynamical adiabatic basis are threefold. First, it is compatible with the boundary conditions, whereas in standard adiabatic two-Coulomb center basis we have nonvanishing inelastic transitions when internuclear distance $R\ensuremath{\rightarrow}\ensuremath{\infty}$. Second, rotational transitions are transformed into radial transitions via a type of hidden crossings in contrast with the standard adiabatic basis, where these transitions could only be included by numerical close-coupling calculations. And third, the ionization process can be described using a basis of the complete discrete orthogonal wave packets, which is much more satisfactory for the process compared with the standard adiabatic approach where the continuum states which have no direct physical meaning are employed. Results of the calculation for the (HeH)${}^{2+}$ quasimolecular system are presented and discussed. Comparison is made with previous results derived by perturbation theory in the united-atom and separated atoms limits.

L-shell X-ray production of molybdenum and niobium induced by 1500–3500 keV Xe26+ ions

L-shell X-ray production cross sections are measured for molybdenum and niobium target induced by Xe26+ ions. The incident energy range varies from 1500 to 3500keV. The experimental results are well reproduced by the binding-energy-modified binary encounter approximation model in the united-atom limit. In addition to target L-shell X-ray spectra, we also observe a weak spectrum which corresponds to the forbidden transition 3d→2s from the projectiles.

The origins and later developments of molecular orbital theory

AbstractHund in 1926 speculated on the assignment of quantum numbers for the electronic states of diatomic molecules, basing his assignments on the correlation between the diatomic states and those of the united atom limit. Mulliken, who had earlier considered the problem of this assignment using old quantum theory, followed Hund in 1927, in the new quantum theory, correlating the states of a diatomic molecule from the united to the separated atoms. In 1932, he introduced the term orbital for the one‐electron states of an atom or molecule. He was the first to seriously explore the orbitals of polyatomic molecules (1932). Burrau in 1927 made the first successful attempt to solve the Schrodinger's equation for H, and Condon (1927) assigned two electrons to Burrau's orbital to obtain an estimate of the binding energy of H2. Lennard‐Jones (LJ) (1929) introduced the linear combination of atomic orbital model, and following the aufbau principle showed that this explained the paramagnetism of O2. Slater (1929) wrote many‐electron wave functions as determinants of spin‐orbitals, and LJ (1949) showed that with this formulation molecular orbitals (MOs) could be transformed into bond‐localized functions; this provided the link to the valence bond approach and the traditional view of the chemical bond. Huckel (1930) was the first to develop a semiempirical MO model for π‐electron hydrocarbons, and this was later extended by others for all‐electron wave functions. Boys (1950) saw the implication of Gaussian functions for calculating the electron repulsion integrals needed for ab initio calculations, and a later approach by Kohn and coworkers (1964 and 1965) produced a density functional MO theory in which electron repulsion is calculated from the whole electron density. © 2011 Wiley Periodicals, Inc. Int J Quantum Chem, 2011

Continued fractions and the hydrogen molecular ion H2+

The Jaffé continued fraction solution for the hydrogen molecular ion has been the basis for most H2+ energy calculations at physically relevant nuclear separation R. I show that its slow convergence at small R can be overcome so it can also be used to obtain the power series expansion of the energy about the united atom limit R = 0. I restrict the discussion to the H2+ ground state and show, by a generalization of Wynn's accelerated convergence technique and an analytical calculation, that the continued fraction yields the O(R5) series term which is the first term in which log(R) makes an appearance. For higher order terms I apply the convergence method numerically to obtain very accurate energies at small R and, by fitting, numerical coefficients in the R series. The analytic representations of the coefficients are identified using the integer relation algorithm PSLQ. All terms to O(R14) are obtained exactly and to O(R20) in a mixed analytic/numeric form. Comparisons with previous work based on wavefunction perturbation theory are made.

X-ray study ofM-shell ionization of heavy atoms by 8.0–35.2-MeVOq+ions: The role of the multiple-ionization effects

The $M$-shell ionization in high-$Z$ atoms by ${\text{O}}^{q+}$ ions have been studied systematically in the energy range of 8.0--35.2 MeV in order to verify the available theoretical approaches describing the $M$-shell ionization by charged particles in asymmetric collisions. The measured $M$ x-ray spectra were analyzed taking into account the effects of x-ray line shifting and broadening caused by the multiple ionization in the $M$ and $N$ shells. The $M$-subshell ionization cross sections, derived by using the $M$-shell decay rates modified for the multiple ionization effects, have been compared with the theoretical predictions based on the plane-wave Born approximation (PWBA), the semiclassical approximation (SCA), and the binary-encounter approximation (BEA). In the PWBA approach two theoretical calculations were considered: the energy-loss Coulomb deflection perturbed stationary state relativistic (ECPSSR) theory and its recent modification called the energy-loss Coulomb deflection united and separated atoms relativistic (ECUSAR) theory, which corrects a description of the electron binding effect to account for the united and separated atoms (USA) electron binding energy limits. In the SCA calculations performed with relativistic hydrogenic wave functions the binding effect was included in the limiting cases of separated-atom (SA) and united-atom (UA) limits. The measured $M$-subshell ionization cross sections are the best reproduced by the SCA-UA calculations, with exception of the ${M}_{2,3}(3p)$-subshell cross sections which are strongly enhanced and cannot be reproduced by the discussed calculations.

Linear response theory of the dipole–dipole dispersion interaction between H(1s) atoms

A self-consistent theory of the linear response previously developed by one of us is applied to the dipole–dipole dispersion interaction of two ground state H atoms in the united atom (He) limit, using a Sellmeier representation of the polarizability in terms of pseudostates. Numerical calculations show that a truncated 12-term expansion out of the N = 22 H(1s) dipole pseudospectrum built from STOs with orbital exponent c = 0.985 gives a result exceeding by no more than 5% the variational result for the He dispersion energy.

Algebraic description of the inelastic collision between an atom and a Morse oscillator in one dimension

The 1D collision between an atom and a diatomic molecule is investigated using an algebraic approach. Closed expressions for the quantum mechanical excitation transition probabilities are obtained. A Morse potential for the diatomic molecule is used. The classical trajectories are obtained in the united atom limit by taking an average over a finite number of states using the matrix density formalism, which allows the problem to be reformulated in terms of a time-dependent Hamiltonian. The system is studied in the interaction picture, which permits us to carry out an algebraic description through the approximation of the interaction potential in terms of a linear combination of the generators of the su(2) algebra. Comparisons with exact quantum mechanical results for transition probabilities in different systems are presented.