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Physical Review A

Fully numerical complex-coordinate Hartree-Fock calculations for alkaline-earth resonance states

Murry Bentley

Published Aug 1, 1990• 8 citations
Publisher

Abstract

A fully numerical multiconfiguration Hartree-Fock program has been modified for performance of calculations on atomic quasibound states using the complex-coordinate (dilation) technique. We present single-configuration calculations for the lowest $^{2}$P\ifmmode^\circ\else\textdegree\fi{} states of ${\mathrm{Be}}^{\mathrm{\ensuremath{-}}}$, ${\mathrm{Mg}}^{\mathrm{\ensuremath{-}}}$, ${\mathrm{Ca}}^{\mathrm{\ensuremath{-}}}$, ${\mathrm{Sr}}^{\mathrm{\ensuremath{-}}}$, and ${\mathrm{Ba}}^{\mathrm{\ensuremath{-}}}$, along with results of some small-scale multiconfiguration investigations of the Mg resonance. An instability originating in difficulty satisfying orthogonality constraints prevents a full multiconfiguration treatment of this resonance. Single-configuration dilated Hartree-Fock calculations predict $^{2}$P\ifmmode^\circ\else\textdegree\fi{} shape resonances for all these systems, though recent extensive calculations show the Ca, Sr, and Ba ions to be bound. Nonetheless, we think it is significant that the dilated numerical method can easily carry out fully variational computations on these heavy atoms, and our results, in conjunction with those of other studies, shed light on the importance of electron correlation in its effect upon the resonance parameters and its variation with progressively heavier alkaline-earth atoms. We also establish the feasibility of accounting for important correlation effects through the use of multiconfiguration approximations, though we have not attempted an accurate calculation in this work. Our Be, Mg, and Ca results agree well with previously published single-configuration dilated basis-set calculations, but no such calculations have been carried out for Sr or Ba. Finally, we investigate the dilational stability of the numerical method since this has been an important consideration in basis-set calculations. A simple exercise based on Cauchy's theorem shows that analyticity of the Hartree-Fock radial wave functions implies that the energy and other physical quantities are independent of the dilation parameter, and a high degree of such independence is demonstrated in our numerical results.

DOI
10.1103/physreva.42.1192

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A fully numerical multiconfiguration Hartree-Fock program has been modified for performance of calculations on atomic quasibound states using the complex-coordinate (dilation) technique. We present single-configuration calculations for the lowest 2^{2}2P\ifmmode^\circ\else\textdegree\fi{} states of Be\ensuremath−{\mathrm{Be}}^{\mathrm{\ensuremath{-}}}Be\ensuremath−, Mg\ensuremath−{\mathrm{Mg}}^{\mathrm{\ensuremath{-}}}Mg\ensuremath−, Ca\ensuremath−{\mathrm{Ca}}^{\mathrm{\ensuremath{-}}}Ca\ensuremath−, Sr\ensuremath−{\mathrm{Sr}}^{\mathrm{\ensuremath{-}}}Sr\ensuremath−, and Ba\ensuremath−{\mathrm{Ba}}^{\mathrm{\ensuremath{-}}}Ba\ensuremath−, along with results of some small-scale multiconfiguration investigations of the Mg resonance. An instability originating in difficulty satisfying orthogonality constraints prevents a full multiconfiguration treatment of this resonance. Single-configuration dilated Hartree-Fock calculations predict 2^{2}2P\ifmmode^\circ\else\textdegree\fi{} shape resonances for all these systems, though recent extensive calculations show the Ca, Sr, and Ba ions to be bound. Nonetheless, we think it is significant that the dilated numerical method can easily carry out fully variational computations on these heavy atoms, and our results, in conjunction with those of other studies, shed light on the importance of electron correlation in its effect upon the resonance parameters and its variation with progressively heavier alkaline-earth atoms. We also establish the feasibility of accounting for important correlation effects through the use of multiconfiguration approximations, though we have not attempted an accurate calculation in this work. Our Be, Mg, and Ca results agree well with previously published single-configuration dilated basis-set calculations, but no such calculations have been carried out for Sr or Ba. Finally, we investigate the dilational stability of the numerical method since this has been an important consideration in basis-set calculations. A simple exercise based on Cauchy's theorem shows that analyticity of the Hartree-Fock radial wave functions implies that the energy and other physical quantities are independent of the dilation parameter, and a high degree of such independence is demonstrated in our numerical results.
Conditional probability distributions for the valence electrons of alkaline-earth atoms in their ground and valence excited states are constructed using large-basis Sturmlan configuration-interaction wave functions for two electrons in frozen, effective-core potentials. The distributions indicate that the ground states of Be, Mg, Ca, Sr, and Ba, and the valence excited states of Be, are much more like linear triatomic molecules, akin to doubly excited He, than like independent-particle systems such as the ground states of He and He-like ions.
We have calculated quadrupole moments, Qzz{\mathit{Q}}_{\mathit{z}\mathit{z}}Qzz​, of helium in several of its doubly excited states and in two of its singly excited Rydberg states, and of the alkaline-earth atoms Be, Mg, Ca, Sr, and Ba in their ground and low-lying excited states. The calculations use well-converged, frozen-core configuration-interaction (CI) wave functions and, for interpretive purposes, Hartree-Fock (HF) atomic wave functions and single-term, optimized, molecular rotor-vibrator (RV) wave functions. The quadrupole moments calculated using RV wave functions serve as a test of the validity of the correlated, moleculelike model, which has been used to describe the effects of electron correlation in these two-electron and pseudo-two-electron atoms. Likewise, the quadrupole moments calculated with HF wave functions test the validity of the independent-particle model. In addition to their predictive use and their application to testing simple models, the quadrupole moments calculated with CI wave functions reveal previously unavailable information about the electronic structure of these atoms. Experimental methods by which these quadrupole moments might be measured are also discussed. The quadrupole moments computed from CI wave functions are presented as predictions; measurements of Qzz{\mathit{Q}}_{\mathit{z}\mathit{z}}Qzz​ have been made for only two singly excited Rydberg states of He, and a value of Qzz{\mathit{Q}}_{\mathit{z}\mathit{z}}Qzz​ has been computed previously for only one of the states reported here. We present these results in the hope of stimulating others to measure some of these quadrupole moments.
Multiconfiguration-Hartree-Fock wave functions for 1S^{1}S1S and 1D^{1}D1D states in Ca I and Sr I are compared with wave functions based on a multichannel-quantum-defect analysis. Discrepancies are found for all of the four series studied. In all cases the discrepancies pertain to perturbers which probably are located in the continuum.
Studies of anions are inherently more challenging than investigations of neutrals and cations because of the diffuse and weakly bound character of an anionic electron. Here, we present a comprehensive computational examination of ground-state diatomic molecular anions composed of alkali-metal (Li, Na, K, Rb, Cs, Fr) and alkaline-earth-metal (Be, Mg, Ca, Sr, Ba, Ra) atoms. We study 21 alkali-metal diatomic anions in the X2Σ+^{2}Σ^{+}2Σ+ electronic state and 36 alkali-metal--alkaline-earth-metal diatomic anions in the X1Σ+^{1}Σ^{+}1Σ+ electronic state. The calculations employ a hierarchy of the coupled cluster methods, combined with large Gaussian basis sets and small-core relativistic energy-consistent pseudopotentials for heavier elements. We compute potential energy curves, permanent electric dipole moments, and static polarizabilities, and we assess convergence and uncertainties of our results. Additionally, using the multireference configuration interaction and equation-of-motion electron-attachment coupled cluster methods, we investigate excited electronic states of alkali-metal molecular anions, including valence-bound and dipole-bound states. We predict crossings between ground neutral and excited anionic states, which may enhance resonant electron attachment and subsequent anion dissociation. This finding may be relevant for experiments with mixtures of ultracold ground-state alkali-metal molecules and Rydberg atoms.
Apropos to the growing interest in the study of long-range interactions which for their applications in cold atom physics, we have performed theoretical calculation for the two-dipole C6C_6C6​ and three-dipole C9C_9C9​ dispersion coefficients involving alkaline-earth atoms with alkaline-earth atoms and alkaline-earth ions. The C6C_6C6​ and C9C_9C9​ coefficients are expressed in terms of the dynamic dipole polarizabilities, which are calculated using relativistic methods. Thereafter, the calculated C6C_6C6​ coefficients for the considered alkaline-earth atoms among themselves are compared with the previously reported values. Due to unavailability of any other earlier theoretical or experimental results, for the C6C_6C6​ coefficients for alkaline-earth atoms with alkaline-earth ions and the C9C_9C9​ coefficients, we have performed separate fitting calculations and compared. Our calculations match in an excellent manner with the fitting calculations. We have also reported the oscillator strengths for the leading transitions and static dipole polarizabilities for the ground states of the alkaline-earth ions, i.e., Mg+^++, Ca+^++, Sr+^++, and Ba+^++ as well as the alkaline-earth atoms, i.e., Mg, Ca, Sr, and Ba. These, when compared with the available experimental results, show good agreement.
Within the framework of the close-coupling method, ab initio calculations of parameters of multichannel quantum-defect theory have been carried out for several Rydberg series of Ca, Sr, and Ba. A local-density approximation, commonly used in solid-state calculations, including self-consistent, self-energy corrections, was made to the exchange-correlation potential. Good agreement with experimental values of multichannel quantum-defect theory parameters is obtained, permitting accurate prediction of highly excited-state binding energies.