Energies, Fine Structures, and Hyperfine Structures of the 1𝑠²2𝑠𝑛𝑝³𝑃(𝑛=2−4) States for the Beryllium Atom

Abstract

Energies and wave functions of the 1𝑠²2𝑠𝑛𝑝³𝑃(𝑛=2−4) states for the beryllium atom are calculated with the full-core plus correlation wave functions. Fine structures and hyperfine structures are calculated with the first-order perturbation theory. For the 1𝑠²2𝑠2𝑝³𝑃 state, the calculated energies, fine structure, and hyperfine structure parameters are in good agreement with the latest theoretical and experimental data in the literature; it is shown that atomic parameters of the low-lying excited states for the beryllium atom can be calculated accurately using this theoretical method. For the 1𝑠²2𝑠𝑛𝑝³𝑃 (𝑛=3,4) states, the present calculations may provide valuable reference data for future theoretical calculations and experimental measurements.

1. Introduction

In recent years, studies of energies, fine structures, and hyperfine structures of the low-lying excited states for the beryllium atom [1–10] have been of great interest to spectroscopists because there are many strong optical transitions suitable for spectral and hyperfine structure measurements. On the other hand, studies of the low-lying excited states for the beryllium atom play an important role in developing the excited state theory of multielectron atoms and better understanding the complicated correlation effects between electrons. The fine structure comes from the spin-orbit, spin-other-orbit, and spin-spin interactions. The hyperfine structure of atomic energy levels is caused by the interaction between the electrons and the electromagnetic multipole moments of the nucleus. The leading terms of this interaction are the magnetic dipole and electric-quadrupole moments. The fine and hyperfine structure is sensitive to the correlation effects among electrons. Experimentally, some properties of the atomic nucleus can be obtained by investigating the hyperfine structure of the atomic energy levels. The nuclear electric-quadrupole moment, which is difficult to measure directly with nuclear physics techniques, can be determined using the measured hyperfine structure and the accurate theoretical results.

The 1𝑠²2𝑠2𝑝³𝑃 state of the beryllium atom is of interest since it is the lowest excited state in which hyperfine effects can occur, and the ground state has no hyperfine splitting because it is 𝐽=0. It is generally a very demanding task to calculate hyperfine structure accurately. Polarization of the closed shells in the 1𝑠²  core, due to the Coulomb interaction with open shells, can have a large effect on the hyperfine structure. Up till now, the most sophisticated theoretical calculations of the hyperfine structure parameters for the 1𝑠²2𝑠2𝑝³𝑃 state of the Be atom have been carried out using linked-cluster many-body perturbation (LC MBPT) theory [5, 6], Hartree-Fock and CI allowing all SD excitations to correlation orbitals of Slater type by Beck and Nicolaides [7], as well as multiconfiguration Hartree-Fock (MCHF) method [8, 9]. Experimentally, the magnetic dipole and electric-quadrupole hyperfine constants have been determined very accurately with the atomic-beam magnetic-resonance technique [10] for the 1𝑠²2𝑠2𝑝³𝑃 state in beryllium. To the best of our knowledge, few results on energies, fine structures, and hyperfine structures have been investigated for the 1𝑠²2𝑠𝑛𝑝³𝑃(𝑛≥3) states of the beryllium atom due to the restriction of resolution from experiments and the numerical unsteadiness in theoretical calculations.

An elegant and complete variation approach, namely, the full core plus correlation (FCPC) method, has been developed by Chung [11, 12]. This method has been successfully applied to three- and four-electron systems, with the 1𝑠² core. Many elaborate calculations, especially for the dipole polarizabilities [13], quadrupole and octupole polarizabilities [14], and total atomic scattering factors [15], show that FCPC wave functions have a reasonable behavior over the whole configuration space for three-electron systems. This method has also been used to calculate the hyperfine structure of the 1𝑠²𝑛𝑠²𝑆 and 1𝑠²𝑛𝑝²𝑃 states (𝑛=2−5) for the lithium isoelectronic sequence; the results are in good agreement with the Hylleraas calculations and with the experiment data [16]. As is well known, theoretical calculations of the hyperfine structure parameters depend sensitively on the behavior of the wave function in the proximity of the nucleus. In addition, core polarization effects for the low 𝑙 states need to be included in the nonrelativistic wave function. It would be interesting to find out whether the FCPC wave function can also be successful for calculating hyperfine structure parameters of low-lying excited states for the beryllium atom. In this work, the FCPC wave functions are carried out on the 1𝑠²2𝑠𝑛𝑝³𝑃 (𝑛=2−4) states of the beryllium atom. The energies, fine structures, and hyperfine structures are calculated and compared with the data available in the literature. The purpose of this work is to explore the capacity of the FCPC wave function to calculate the atomic parameters of the low-lying excited states for the beryllium atom and provide more reliable theoretical data to stimulate further experimental measurements.

2. Theory

According to the FCPC method [11, 12], the wave function for the four-electron 1𝑠²2𝑠𝑛𝑝³𝑃 state can be written as Ψ()Φ1,2,3,4=𝐴_1𝑠1𝑠(1,2)Φ_{2𝑠𝑛𝑝}(3,4)+_𝑖𝐶_𝑖Φ_{𝑛(𝑖),𝑙(𝑖)}(1,2,3,4), (1) where 𝐴 is an antisymmetrization operator. Φ_1𝑠1𝑠 is a predetermined 1𝑠²-core wave function which is represented by a CI basis set, Φ_1𝑠1𝑠(1,2)=𝐴_{𝑘,𝑛,𝑙}𝐶_𝑘𝑛𝑙𝑟^𝑘₁𝑟^𝑛₂exp−𝛽_𝑙𝑟₁−𝜌_𝑙𝑟₂𝑌_𝑙(1,2)𝜒(1,2), (2) the angular part is 𝑌_𝑙(1,2)=_𝑚⟨𝑙,𝑚,𝑙,−𝑚∣0,0⟩𝑌_𝑙𝑚𝜃₁,𝜑₁𝑌_𝑙−𝑚𝜃₂,𝜑₂. (3)𝜒 (1,2) is a two-electron singlet spin function. The linear and nonlinear parameters in (2) are determined by optimizing the energy of the two-electron core. The factor Φ_{2𝑠𝑛𝑝}(3,4) represents the wave function of the two outer electrons which is given by Φ_{2𝑠𝑛𝑝}(3,4)=𝐴_{𝑘,𝑛,𝑙}𝑑_𝑘𝑛𝑙𝑟^𝑘₃𝑟^𝑛₄exp−𝜆_𝑙𝑟₃−𝜂_𝑙𝑟₄𝑌_𝑙(3,4)𝜒(3,4), (4) the angular part is 𝑌_𝑙(3,4)=_𝑚⟨𝑙,𝑚,𝑙+1,−𝑚∣0,0⟩𝑌_𝑙𝑚𝜃₃,𝜑₃𝑌_{𝑙+1−𝑚}𝜃₄,𝜑₄. (5) The latter wave function of (1) describes the core relaxation and the intrashell electron correlation in the four-electron system. It is given by Φ_{𝑛(𝑖),𝑙(𝑖)}(1,2,3,4)=𝜑_{𝑛(𝑖),𝑙(𝑖)}(𝑅)𝑌^LM_𝑙(𝑖)(Ω)𝜒_SS𝑍, (6) where 𝜑_{𝑛(𝑖),𝑙(𝑖)}(𝑅)=⁴_𝑗=1𝑟^𝑛𝑗_𝑗exp−𝛼_𝑗𝑟_𝑗. (7) A different set of 𝛼_𝑗  is used for each 𝑙(𝑖). The angular part is 𝑌^LM_𝑙(𝑖)𝑅=_𝑚𝑗𝑙₁𝑙₂𝑚₁𝑚₂∣𝑙₁₂𝑚₁₂×𝑙₁₂𝑙₃𝑚₁₂𝑚₃∣𝑙₁₂₃𝑚₁₂₃×𝑙₁₂₃𝑙₄𝑚₁₂₃𝑚₄∣LM⁴_𝑗=1𝑌_{𝑙𝑗𝑚𝑗}Ω_𝑗. (8)

To simplify notation, this angular function is simply denoted as 𝑙(𝑖)=𝑙₁,𝑙₂𝑙₁₂,𝑙₃𝑙₁₂₃,𝑙₄, (9) with the understanding that 𝑙₁₂₃ and 𝑙₄ couple into 𝐿, the total orbital angular momentum. There are three possible spin functions for the 1𝑠²2𝑠𝑛𝑝³𝑃 state, namely, 𝜒₁=𝑠₁,𝑠₂0,𝑠₃12,𝑠₄,𝜒₂=𝑠₁,𝑠₂1,𝑠₃12,𝑠₄,𝜒₃=𝑠₁,𝑠₂1,𝑠₃32,𝑠₄. (10)

For the radial basis functions of each angular-spin component, a set of linear and nonlinear parameters is chosen. These parameters are determined in the energy optimization process. For each set of  𝑙₁, 𝑙₂, 𝑙₃, and 𝑙₄, we try all possible 𝑙(𝑖) and 𝜒 and keep the ones which make significant contribution to the energy in (1).

The fine structure perturbation operators [1, 2] are given by 𝐻_FS=𝐻_so+𝐻_soo+𝐻_ss, (11) where the spin-orbit, spin-other-orbit, and spin-spin operators are 𝐻_SO=𝑍2𝑐²⁴_𝑖=1𝑙_𝑖⋅𝑠_𝑖𝑟³_𝑖,𝐻_SOO1=−2𝑐²⁴_{𝑖,𝑗=1𝑖≠𝑗}1𝑟³_𝑖𝑗𝑟_𝑖−𝑟_𝑗×𝑝_𝑖⋅𝑠_𝑖+2𝑠_𝑗,𝐻_SS=1𝑐²⁴_{𝑖,𝑗=1𝑖<𝑗}1𝑟³_𝑖𝑗𝑠_𝑖⋅𝑠_𝑗−3𝑠_𝑖⋅𝑟_𝑖𝑗𝑠_𝑗⋅𝑟_𝑖𝑗𝑟²_𝑖𝑗. (12)

To calculate the fine structure splitting, the 𝐿𝑆𝐽 coupling scheme is used: Ψ_{𝐿𝑆𝐽𝐽𝑍}=_{𝑀,𝑆𝑍}𝐿𝑆𝑀𝑆_𝑍∣𝐽𝐽_𝑍Φ_{𝐿𝑆𝑀𝑆𝑍}. (13)

The fine structure energy levels are calculated by first-order perturbation theory Δ𝐸_FS𝐽=Φ_{𝐿𝑆𝐽𝐽𝑍}𝐻_so+𝐻_soo+𝐻_ssΦ_{𝐿𝑆𝐽𝐽𝑍}. (14)

For an 𝑁-electron system, the hyperfine interaction Hamiltonian can be represented as follows [17, 18]: 𝐻_ℎ𝑓𝑠=_𝑘=1𝑇^(𝑘)⋅𝑀^(𝑘), (15) where 𝑇^(𝑘) and 𝑀^(𝑘) are spherical tensor operators of rank 𝑘 in the electronic and nuclear spaces, respectively. The 𝑘=1 term represents the magnetic-dipole interaction between the magnetic field generated by the electrons and nuclear magnetic dipole moments, and the 𝑘=2 term the electric quadrupole interaction between the electric-field gradient from the electrons and the nonspherical charge distribution of the nucleus. The contributions from higher-order terms are much smaller and can often be neglected.

In the nonrelativistic framework, the electronic tensor operators, in atomic units, can be written as 𝑇⁽¹⁾=𝛼²2⁴_𝑖=12𝑔_𝑙𝑟_𝑖⁻³𝑙_𝑖⁽¹⁾−√10𝑔_𝑠𝑠_𝑖⁽¹⁾𝐶_𝑖⁽²⁾⁽¹⁾𝑟_𝑖⁻³,+8𝜋3𝑔_𝑠𝑠_𝑖⁽¹⁾𝛿𝐫_𝑖𝑇⁽²⁾=−⁴_𝑖=1𝑟_𝑖⁻³𝐶_𝑖⁽²⁾, (16) where 𝑔_𝑙=(1−𝑚_𝑒/𝑀) is the orbital electron 𝑔 factor, and 𝑔_𝑠=2.0023193 is the electron spin 𝑔 factor. 𝑀 is the nuclear mass. The tensor 𝐶_𝑖⁽²⁾ is connected to the spherical harmonics 𝑌_𝑙𝑚(𝑖) by 𝐶_𝑚^(𝑙)=4𝜋2𝑙+1𝐘_𝑙𝑚. (17)

The hyperfine interaction couples the electronic angular momenta J and the nuclear angular momenta I to a total angular momentum F = I + J. The uncoupling and coupling hyperfine constants are defined in atomic units as [17, 18]: 𝑎_𝐶=𝛾𝐿𝑆𝑀_𝐿𝑀_𝑆^𝑁_𝑖=18𝜋𝛿³𝑟_𝑖s₀(𝑖)𝛾𝐿𝑆𝑀_𝐿𝑀_𝑆(Fermicontact),𝑎_SD=𝛾𝐿𝑆𝑀_𝐿𝑀_𝑆^𝑁_𝑖=12𝐶₀⁽²⁾(𝑖)𝑠₀(𝑖)𝑟_𝑖⁻³𝛾𝐿𝑆𝑀_𝐿𝑀_𝑆Spindipolar,𝑎_𝑙=𝛾𝐿𝑆𝑀_𝐿𝑀_𝑆^𝑁_𝑖=1𝑙₀(𝑖)𝑟_𝑖⁻³𝛾𝐿𝑆𝑀_𝐿𝑀_𝑆(orbital),𝑏_𝑞=𝛾𝐿𝑆𝑀_𝐿𝑀_𝑆^𝑁_𝑖=12𝐶₀⁽²⁾(𝑖)𝑟_𝑖⁻³𝛾𝐿𝑆𝑀_𝐿𝑀_𝑆electricquadrupole,𝐴_𝐽=𝜇_𝐼𝐼1[𝐽(𝐽+1)(2𝐽+1)]^1/2𝛾𝐽𝑇⁽¹⁾,𝐴𝛾𝐽_{𝐽−1,𝐽}=𝜇_𝐼𝐼1[𝐽(2𝐽−1)(2𝐽+1)]^1/2𝛾𝐽−1𝑇⁽¹⁾,𝐵𝛾𝐽_𝐽()=2𝑄2𝐽2𝐽−1(2𝐽+1)(2𝐽+2)(2𝐽+3)^1/2𝛾𝐽𝑇⁽²⁾𝛾𝐽, (18) where 𝑀_𝐿=𝐿 and 𝑀_𝑆=𝑆. In these expressions, 𝜇_𝐼 is the nuclear magnetic moment and 𝑄 is the nuclear electric quadrupole moment. 𝐼 is the nuclear spin, and 𝐽 is the atomic electronic angular moment.

3. Results and Discussions

In order to achieve accurate calculation results for various properties of the low-lying excited states for the beryllium atom, the choice of basis function with sufficiently high quality is critical and it is our major concern. The seven 𝑙 components (0,0), (1,1), (2,2), (3,3), (4,4), (5,5), (6,6) altogether 159 terms are used for the 1𝑠² core. The Φ_{2𝑠𝑛𝑝} in (1) has four angular components, 𝑙 is summed from 0 to 3 with the angular components (0,1), (1,2), (2,3), and (3,4), and the number of terms in Φ_{2𝑠𝑛𝑝} ranges from 36 to 15. Most of the other correlation effects are included in (6), which accounts for the intershell as well as the intrashell correlations. Many relevant angular and spin couplings are important for the energy, these basis functions are tried to include in Φ_{𝑛(𝑖),𝑙(𝑖)} (1, 2, 3, 4) with significant energy contribution. For each set of orbital angular momenta 𝑙₁, 𝑙₂, 𝑙₃, and 𝑙₄, there could be several ways to couple this set into the desired total orbital angular momentum. In this work, for 1𝑠²2𝑠𝑛𝑝³𝑃 states, the important angular series (𝑙₁, 𝑙₂, 𝑙₃, 𝑙₄) are (0,0,𝑙,(𝑙+1)), (0,1,𝑙,𝑙), (𝑙,𝑙,0,1), and so forth. In both cases, the value of 𝑙 is from 0 to 6, as the energy contribution from set with 𝑙>6 is small and negligible. In order to get the high-quality wave function, the number of angular-spin components in the Φ_{𝑛(𝑖),𝑙(𝑖)} wave functions ranges from 15 to 66, and the number of terms in the Φ_{𝑛(𝑖),𝑙(𝑖)} of (6) is about 790. The linear and nonlinear parameters are individually optimized in the energy minimization process. Using the Rayleigh-Ritz variational method, the basic wave function Ψ and the corresponding eigenvalue 𝐸 are determined.

Nonrelativistic energies of the 1𝑠²2𝑠𝑛𝑝³𝑃 (𝑛=2−4) states for the beryllium atom are given in Table 1. As Table 1 shows, for the 1𝑠²2𝑠𝑛𝑝 (𝑛=2,3) ³𝑃 states, the nonrelativistic energies in this work are lower and better than those of Hibbert and Weiss [1, 2], the improvement ranging from 0.0479 a.u. to 0.0473 a.u. Hibbert and Weiss reported a set of large-scale configuration interaction (CI) calculations for the 1𝑠²2𝑠𝑛𝑝 (𝑛=2,3) ³𝑃 states, which can give an accurate approximation for each state, but it may tend to obscure the global picture of the spectrum which is so transparent in the other approach. The work of Hibbert and of Weiss did not include any intrashell correlation in the 1𝑠 shell, as the calculations were of transitions in the outer subshells. The correlation energy of the 1𝑠 shell is almost independent of the nuclear charge and also of the number of additional electrons outside the 1𝑠 shell. For Be, it is about 0.0457 a.u. and this accounts for the main difference between earlier work and the present; more accurate results are presented in Table 1. Of course, for the calculation of hyperfine parameters, correlation within the 1𝑠 shell is crucial in obtaining accurate hyperfine parameters, and this has been achieved in the present work. For the 1𝑠²2𝑠4𝑝³𝑃 state, the present calculation from the FCPC method is also lower than the result of Weiss [2].

Table 1 

Nonrelativistic energies of the 1𝑠²2𝑠𝑛𝑝³𝑃(𝑛=2−4) states for the beryllium atom (in a.u.).

If including the effects of the spin-orbit, spin-other-orbit, and spin-spin interactions, the energies of the fine structure resolved 𝐽 levels are obtained. In this work, the fine structure splittings of the triplet states are calculated with the 𝐻_so, 𝐻_soo, and 𝐻_ss operators using the first-order perturbation theory. Table 2 gives the fine-structure splittings of the 1𝑠²2𝑠𝑛𝑝³𝑃_𝐽 (𝑛=2−4) states for the beryllium atom. The experimental Be 2𝑠2𝑝³𝑃_𝐽 splittings are 2.35 (𝐽=2→1) and 0.64 (𝐽=1→0) cm⁻¹ [3]. They agree with our prediction 2.36 and 0.64 cm⁻¹. Although many theoretical studies have been done on the BeI excited systems, the published theoretical fine structure results are scarce. One exception is Laughlin, Constantinides, and Victor [4]. They use a model potential calculation and predict the splittings to be 2.53 and 0.71 cm⁻¹ for the 1𝑠²2𝑠2𝑝³𝑃_𝐽 state, which should be considered as quite good in view of the simplicity in their computation and fall in experimental uncertainties. Present calculations for this state are more accurate due to correlation effect well described in this method. For the experiment, the splitting of 1𝑠²2𝑠3𝑝³𝑃_𝐽(𝐽=1,0) is not resolved. But the splitting from the 𝐽=2 state to the 𝐽=1, 0 is determined to be 0.37 cm⁻¹. In this work, the calculated splittings are 0.35 (𝐽=2→1) and 0.092 (𝐽=1→0) cm⁻¹. This implies that the predicted splitting from 𝐽=2 to the center of gravity of 𝐽=1 and 0 should be 0.373 cm⁻¹. It agrees with the experiment. The good agreement with experiment could be used as the indication of the accuracy of the wave function constructed here. For the 1𝑠²2𝑠4𝑝³𝑃_𝐽 state, our calculated splittings are hoped to offer reference for further experimental measurements.

Table 2 

Fine structure splittings 𝜈_𝐽−𝐽 of the 1𝑠²2𝑠𝑛𝑝³𝑃_𝐽(𝑛=2−4) states for the beryllium atom (in cm⁻¹).

The hyperfine structure parameters of the 1𝑠²2𝑠𝑛𝑝³𝑃(𝑛=2−4) states for the beryllium atom are calculated in this work: Fermi contact 𝑎_𝑐, the spin dipolar 𝑎_SD, the orbital 𝑎_𝑙, and the electric quadrupole 𝑏_𝑞. In the present calculation, 𝑄=0.0530𝑏, 𝜇_𝐼 = −1.177492 nm, 𝐼=1.5 for Be are taken from [19]. The hyperfine interaction in the 1𝑠²2𝑠2𝑝³𝑃 state for the beryllium atom is of interest since it is the lowest excited state in which hyperfine effects can occur, which has been studied over the past four decades [5–10]. Table 3 gives the hyperfine structure parameters of the 1𝑠²2𝑠2𝑝³𝑃 state for the beryllium atom through the FCPC wave function to compare with data in the literature. As can be seen from Table 3, the present results for hyperfine structure parameters are better than the earlier theoretical results [5–7] in whole. The present calculations also agree with the results by FE MCHF (finite-element multiconfiguration Hartree-Fock) method [8] to two significant figures. The calculated Fermi contact term 𝑎_𝑐 in this work differs from the results from the latest calculation through MCHF method [9] by only 0.07%, and the differences for the other terms are on the order of a few parts in a thousand. This means that the wave function used in the present work is reasonable and accurate in the full configuration space. The hyperfine coupling constants 𝐴_𝐽 are also listed in Table 3 to compare with results from other calculations and experiments. Our calculated hyperfine coupling constants agree perfectly with the experimental value [10] to four significant figures. That is also true for the MCHF calculation of [9]. It is shown that hyperfine structure parameters of the low-lying excited states for the beryllium atom can be calculated accurately using the present FCPC wave function. For the 1𝑠²2𝑠𝑛𝑝³𝑃 (𝑛=3,4) states, to the best of our knowledge, there is no report on hyperfine structure parameters in the literature. The present predictions for the hyperfine structure parameters and coupling constants are listed in Table 4, which may provide valuable reference data for other theoretical calculations and experimental measurements in future.

Table 3 

The hyperfine structure parameters (in a.u.) and coupling constants (in MHz) of the 1𝑠²2𝑠2𝑝³𝑃 state for the beryllium atom.

Table 4 

The hyperfine structure parameters (in a.u.) and coupling constants (in MHz) of the 1𝑠²2𝑠𝑛𝑝³𝑃(𝑛=3,4) states for the beryllium atom.

4. Summary

In this work, energies, fine-structure splittings, and hyperfine structure parameters of the 1𝑠²2𝑠𝑛𝑝³𝑃 (𝑛=2−4) states for the beryllium atom are calculated with the FCPC wave functions. The obtained nonrelativistic energies are much lower than the previous published theoretical values. The calculated fine structure splittings are in good agreement with experiment. For the 1𝑠²2𝑠2𝑝³𝑃 state, the calculated hyperfine structure parameters are in good agreement with the latest theoretical and experimental data in the literature; it is shown that hyperfine constants of the low-lying excited states for the beryllium atom can be calculated accurately using this kind of wave function. For other states, the present predicted hyperfine structure parameters may provide valuable reference data for future theoretical calculations and experimental measurements.

Acknowledgments

The author is grateful to Dr. Kwong T. Chung for his computer code. The work is supported by National Natural Science Foundation of China and the Basic Research Foundation of Beijing Institute of Technology.