Experimental Term Values Research Articles

New values for someHe4I1snlenergy levels, ionization energies, and Lamb shifts

Recent experimental determinations of energy separations within the $1snl$ term system ($n=2\ensuremath{-}6$) have been used to reevaluate 35 levels. Most of the levels have estimated errors less than 0.001 ${\mathrm{cm}}^{\ensuremath{-}1}$ relative to the $2 ^{3}P$ levels. Addition of accurate theoretical term values (ionization energies) available for several $1snl$ levels to the corresponding experimental level values gives generally consistent values for the principal ionization energy (${E}_{I}$). The theoretical energies are further confirmed by the agreement of the weighted average of seven of these ${E}_{I}$ values with a value obtained by fitting Ritz formulas to three accurately determined $1snl$ series; the suggested new ${E}_{I}$ is 198 310.7745(40) ${\mathrm{cm}}^{\ensuremath{-}1}$ on an energy scale fixed by the value 171 135.0000 ${\mathrm{cm}}^{\ensuremath{-}1}$ for $2 ^{1}P$. Lamb shifts are derived for the $2$, $3$, $4 ^{3}S_{1}$, $2 ^{1}S_{0}$, $2 ^{3}P_{1}$, and $2 ^{1}P_{1}$ levels as differences between experimental term values obtained with the new ${E}_{I}$ and corresponding calculated term values not including Lamb shifts. The experimental and calculated values for the ${1s}^{2} ^{1}S_{0}$ ground level relative to the present $1snl$ excited-level system are 0.00\ifmmode\pm\else\textpm\fi{}0.15 and 0.073\ifmmode\pm\else\textpm\fi{}0.009 ${\mathrm{cm}}^{\ensuremath{-}1}$, respectively, so that a \ensuremath{\sim}20-fold increase in the experimental accuracy would be required to test the calculated ground-level Lamb shift.

Potential models and local mode vibrational eigenvalue calculations for acetylene

Two potential models for acetylene are developed and tested by comparison between variational calculations for the stretching vibrational term values and available spectroscopic data. The first model based on local bond potentials with harmonic interbond coupling gives root mean square deviations of 6 cm−1 for C2H2 and 3 cm−1 for C2D2. The second model is more ambitious, being designed to reproduce the dissociation characteristics of the molecules, and the calculated root mean square deviations from the experimental vibrational term values are larger, 32 cm−1 for C2H2 and 24 cm−1 for C2D2. The eigenvalue spectrum of C2H2 is shown to differ from that of C2D2 in showingmarked local mode features and this difference in behaviour is underlined by means of a correlation diagram. Finally it is shown how the known normal mode frequencies and anharmonic constants may be introduced into a simple model in order to predict the excited term values of C2H2, again with a root mean square deviation of 6 cm−1.

Model Potential Method in Hartree-Fock-Slater Formalism

A new formulation of the model potential method is developed within the Hatree-Fock-Slater (HFS) scheme. For the purpose of handy applications, the numerically-generated effective potentials are fitted to Bonifacic-Huzinaga type model potentials. Taking advantage of the intrinsically local property of the potentials in HFS formalism, the parameters involved in the model potentials are optimized using the least-square criterion for analytical functional fits without requiring any reference such as orbital energies or experimental term values etc. The frozen-core approximation which is the basic underlying notion of the theory is examined. It is shown that the model potential calculation of the electron population and orbital energies for carbon monoxide almost closely reproduces the all-electron calculation.

A study of potential curve crossing in X-Ar complexes (X=Mg, Ca, Sr, Ba)

Semiempirical pseudopotentials which are determined by fitting to experimental term values of Mg+, Ca+, Sr+, Ba+and Ar7+, are tested for ground and various excited states of the corresponding neutral atoms, using a spin-density functional approximation for valence correlation energies. Experimental triplet excitation energies are reproduced with an average error of 0.1 eV. The method is then applied to study adiabatic level crossing between ground and excited states in X-Ar complexes (X=Mg, Ca, Sr, Ba). Estimates of deactivation energies are given and are compared with results from deactivation cross section measurements.

K-shell excitation of HF and F2studied by electron energy-loss spectroscopy

The fluorine K-shell (1s) excitation spectra of HF and F2 have been studied by small-angle inelastic scattering of 2.5 keV electrons. The F2 spectrum exhibits a dramatic 1s to sigma * shape resonance 14.5 eV below the F 1s ionisation threshold which is qualitatively different from the 1s to sigma * transitions observed in all other first-row diatomics. An empirical correlation between experimental term values for both sigma * and pi * K-shell shape resonances and the sum of the atomic numbers of the heavy atoms in a series of first-row molecules is demonstrated and tested with respect to the energy of the 1s to sigma * transition in F2. The sigma * and pi * energies of a variety of molecules have been predicted using this relationship. In contrast to F2, the HF spectrum is more atom-like, exhibiting a moderately intense 1s to sigma * transition followed by the np Rydberg series. The equivalent-core analogy is applied to the HF and F2 results to predict excitation and ionisation energies of NeH and NeF.

Determination of diatomic molecular constants using an inverted perturbation approach: Application to the [formula omitted] system of Mg 2

The measurements of the A 1 Σ u +-X 1 Σ g + system of the Mg 2 molecule of Balfour and Douglas have been reevaluated and molecular constants have been determined using a variational procedure which is based on an inverted perturbation approach. The method provides a leastsquares fit of the experimental term values to the calculated quantum mechanical energy eigenvalues of a rotating vibrator. All measured line positions of the system are fitted with two rotationless potentials U o which yield G v and B v values and which are independent of any centrifugal distortion parameters. The latter are determined in a separate least-squares fit in which the previously determined G v and B v values are held fixed. A comparison of the quantum mechanical U o and the semiclassical first-order RKR potential is carried out. The method provides dissociation energies which are more accurate than previous extrapolation procedures, and allows a self-consistency test of the experimental term values. The 24Mg 2 and 26Mg 2 molecules have been fitted using mass reduced variables, and an electronic isotope shift of 0.17 cm −1 has been obtained.

The term systems of Be I converging to the Be II 2s limit

In previous analyses of the Be I spectrum it has been assumed that the quantum defects mu ( epsilon ) can be fitted to low-order polynomials in the energy variable epsilon . Norcross and Seaton (1976) find the theoretical quantum defects, mu (theor mod epsilon ), for Be I 2snl levels are perturbed by states Be I 2pn'l' and, in consequence, cannot be fitted accurately to low-order polynomials. A new analysis of the Be I spectrum is made using calculated quantum defects mu (calc mod epsilon )= mu (theor mod epsilon )+ Delta mu (calc mod epsilon ) where the small correction Delta mu is assumed to be a linear or quadratic function, and where the parameters in Delta mu and the series limit Tinfinity are adjusted to give a least-squares fit to the experimental term values of Johansson (1974). A revised value (Tinfinity =75192.5+or-0.1 cm-1) of the limit is obtained and formulae are given for the accurate calculation of term values in the series 1S, 3S, 1P0, 3P0, 1D, 3D, 1F0 and 3F0. With the revised value of Tinfinity the experimental quantum defects for the 2sns 1S series show clearly the perturbation produced by the resonance state 2p2 1S above the Be II 2s limit.

Ultraviolet Absorption Spectrum of the CaH Molecule I. Rotational analysis of the 3 060 Å band system

Four new violet degraded bands of CaH have been observed in absorption in the region 2 900 to 3 200 Å by heating calcium and hydrogen to 1 200-1 400°C. Rotational analysis indicates that the bands belong to two close-lying systems K2Σ+ - X2Σ+ and L2Π - X2Σ+ which are separated by similar 50 cm-1 for the 0-0 bands. The rotational levels for the excited states approximate to Hund's case (d) coupling with L = 1. A Hamiltonian matrix has been constructed for the two states and constants for the "unperturbed" states evaluated. These constants reproduce the experimental term values to within 0.2 cm-1, except for levels involved in local perturbations.

On the calculation of molecular parameters for triplet states in diatomic molecules. TheG3ΔgandH3Φustates of N2

An exact numerical method for reducing the experimental term values (eigenvalues) of diatomic molecules to molecular parameters is discussed for triplet states with particular emphasis upon those for which Λ ≧ 2. Besides considering the well-known rotational, spin-orbit and spin-spin contributions, the theoretical treatment also includes the spin-rotation interaction, the spin-other-orbit interaction, various forms of the centrifugal distortion and second-order terms arising from triplet-triplet and triplet-singlet interactions. The present method of calculation including the theoretical refinements is applied to the G 3Δ g and H 3Φ u states of N2.

Dipole polarizabilities from the nf term values

Experimental term values of the nf series have been used for an empirical determination of the dipole polarizabilities α d for a number of ions in the Ne I, Na I, and A I isoelectronic series. The core charge densities obtained by the relativistic Hartree-Fock-Slater method were used for the calculation of the penetration correction. Other corrections related to the incomplete separation of the electron from the core are also discussed, α d, unlike the coefficient of 〈 r −6〉 (usually interpreted as quadrupole polarizability α q) is relatively insensitive to the uncertainties of the procedure.

Calculations on Rydberg Terms of the Beryllium Atom

Terms of P1,3 and S1,3 Rydberg series of Be are calculated up to n = 10. An effective Hamiltonian which contains two-electron operators for the core electrons is used. Trial Rydberg functions are used in a linear variation method, and the functions are optimized with respect to orbital exponents. The agreement with experimental term values is very encouraging. Penetration and exchange energies are calculated independently from the final Rydberg functions and are discussed for the different series.

2P1,3,3P1,3, and4P1,3States of He and the2P1State ofLi+

A method is outlined for calculating nonrelativistic eigenvalues and wave functions for a two-electron $P$ state of odd parity, and for evaluating the mass polarization and all of the relativistic corrections, apart from the radiative terms. Calculations have been made for the low-lying $P$ states of He and the $2^{1}P$ state of ${\mathrm{Li}}^{+}$ using up to 560 terms in expansion of the wave function. The nonrelativistic eigenvalues converge to within an accuracy of from ${10}^{\ensuremath{-}4}$ ${\mathrm{cm}}^{\ensuremath{-}1}$ to ${10}^{\ensuremath{-}2}$ ${\mathrm{cm}}^{\ensuremath{-}1}$. The values of the mass polarization and of the relativistic corrections converge more rapidly than this, so that the total theoretical ionization energy is estimated to be correct to within an error of not more than ${10}^{\ensuremath{-}2}$ ${\mathrm{cm}}^{\ensuremath{-}1}$, i.e., considerably less than the experimental error, in the case of all of the states considered. The difference between the theoretical and experimental term values is in no case greater than 0.1 ${\mathrm{cm}}^{\ensuremath{-}1}$ in absolute magnitude, and is presumed to be due to the contributions from the radiative terms, which have not been included in the calculation. The term value obtained for the $2^{1}P$ state of ${\mathrm{Li}}^{+}$ confirms the identification of the line at 9581.42 \AA{} in the ${\mathrm{Li}}^{+}$ spectrum as belonging to the $2^{1}S\ensuremath{-}2^{1}P$ transition.

Statistical Theory of Many-Electron Systems. Discrete Bases of Representation

The density matrix for a many-electron system has been examined. A formalism has been arrived at which facilitates its evaluation in terms of a basis corresponding to a discrete spectrum of eigenvalues. This allows a representation to be employed associated with a suitably chosen approximation to the Hamiltonian. Thereby, reasonably accurate estimates of the properties of many-electron systems may be anticipated for low orders of approximation.In developing the formalism, attention was focused upon a means for approximating the quantity $\mathrm{exp}(z\mathrm{H})$, where H is the many-particle Hamiltonian and $z$ is a complex number. It was found possible to represent this quantity exactly as an infinite product of exponential factors, each of which depends upon a portion of the Hamiltonian alone. The relation of this result to statistical mechanical applications is indicated. An approximation procedure is described in terms of which various finite products of exponential factors approach the desired quantity in the limit of indefinitely increasing numbers of factors; this limit is approached in a manner that permits any given approximation to contain all the terms of the previous approximation.The first order of approximation was employed to approximate the many-particle density matrix. The resulting theory was applied to the helium-like systems: He, ${\mathrm{Li}}^{+}$, ${\mathrm{Be}}^{++}$. The energies of the ground $^{1}S$ state of helium and the three lowest states of ${\mathrm{Li}}^{+}$ and ${\mathrm{Be}}^{++}$ were calculated. The agreement with experimental term values is reasonably satisfactory, the discrepancy between calculations and experiments not exceeding one percent. The triplet-singlet splitting for ${\mathrm{Li}}^{+}$ was in the order observed; the magnitude calculated was about one-half that observed.The relation between the present theory and the Thomas-Fermi theory is discussed. It is pointed out that the former is the analog of the latter when the density matrix is expressed in other than eigenfunctions of momentum. The quasi-classical character of the approximation is discussed.

New Approach to the Quantum-Mechanical Analysis of the Electronic Structures of Molecules. The Method of Deformed Atoms in Molecules

A modification of Moffit's method of atoms in molecules is here proposed. In this treatment, the modified atomic functions X are introduced, which are made up from the exact eigenfunctions of atoms modified by correction factors so as to take account of the deformation of atoms in molecules. The advantage of these basic functions X is as follows. Firstly, the difficulty caused by the correlation energy in atoms can be removed by the use of the experimental atomic term values just as in Moffitt's method. Secondly, the common effective charges for neutral and ionic structures can be applied for the convenience of calculating the interaction operators, which sometimes produces serious errors in Moffitt's method. Moreover the improvement on the original Heitler-London treatment—corresponding to Wang's or Weinbaum's work for the hydrogen molecule—can be expected. Then, it is discussed how to determine these modified atomic functions in order that the results appertain to the good range in spite of the rather simple calculation. The energy loss of atoms in molecules, induced by the deformation, is corrected, and the interatomic interaction energy is calculated by the use of the approximate functions composed of the orbital functions, where the molecular orbital method can be utilized for saving the difficulty of the higher order permutations. Then the unsatisfactory results, obtained in the calculation of Li2 by utilizing the common effective charges in Moffitt's method, are improved.