In this work, we present a series of benchmark variational calculations for the ground and 19 lowest bound excited singlet S and P states of the beryllium atom. The nonrelativistic wave functions of the states that represent the motion of the nucleus and the four electrons around the center of mass of the atom are expanded in terms of up to 17 000 all-particle explicitly correlated Gaussians. The Gaussians are optimized independently for each state. The leading relativistic corrections to the energy levels are computed in the framework of the perturbation theory and they explicitly include the nuclear recoil effects. We also calculate the leading quantum electrodynamics (QED) corrections for each considered state. Using the obtained energy levels and the corresponding wave functions, we compute the transition frequencies, transition dipole moments, and oscillator strengths. A comparison with the available experimental data shows very good agreement. The results of this most comprehensive set of calculations of spectroscopic accuracy for Be to date may open up new applications pertinent to the precision tests of QED, determination of the nuclear charge radius, and modeling matter-radiation equilibria of the beryllium gas that has relevance to the physics of interstellar media.

Atomic spectra recorded from astrophysical observations provide rich information on the composition of interstellar objects. Properties such as abundances of elements, their densities, temperature, and other physical parameters can be inferred from these spectra.1,2 Based on these properties, models of the chemical and physical transformations can be constructed to explain the phenomena observed in interstellar media. Various space missions that have carried out instruments for very precise measurements of the incoming interstellar radiation have provided a wealth of spectral data for the analysis. However, for the analysis to be carried out with sufficient accuracy, it has to be aided by high-resolution atomic spectra obtained through precision measurements performed in the laboratory and by spectra obtained with high-accuracy theoretical calculations. As more of such new reference data become available, the probing of the physical and chemical conditions of astrophysical objects can be extended to a wider range of interstellar domains.

The present work is concerned with the first-principle quantum-mechanical calculations of the oscillator strengths of atomic inter-state transitions. Accurate values of inter-state-transition oscillator strengths to be used in astrophysical analysis are needed to carry out various astrophysical applications and for the modeling of interstellar media. Data provided so far have often been insufficiently complete and not accurate enough for the analysis of, for example, stellar atmospheres and plasmas. Continuous progress in the development of new computational methods provides the capability to perform more accurate oscillator strength calculations. However, it takes the combined efforts of theorists and experimentalists to generate data sufficiently accurate for astrophysical applications, as the values obtained from the calculations have to be checked against the laboratory measurements for accuracy and precision. It also happens that the oscillator strengths obtained in current high-precision quantum-mechanical calculations are often considerably more accurate than the measured values, which provides ground for the interplay between the theory and experiment and may lead to re-measurements. Moreover, the present calculations of the oscillator strengths of Be take into account the finite nuclear mass and, thus, can clearly discriminate between various Be isotopes. Previous calculations for lithium, beryllium, and boron atoms (see Refs. 3–6) show that the total uncertainty of the transition energies calculated with explicitly correlated wave functions is smaller than the isotope shifts. Thus, oscillator strengths can be reliably calculated for different isotopes. While, to the best of our knowledge, it is not yet possible to experimentally measure the isotopic shifts of the oscillator strengths for any of the few-electron atoms or ions, such measurements may become possible in the future.

The accuracy of the calculations of atomic oscillator strengths is primarily dependent on the quality of the wave functions used in the calculations. One needs to go beyond the simplest Hartree–Fock model and account for a large fraction of the electron correlation effects to obtain quality oscillator strengths. Wave functions obtained with various methods have been used to calculate the oscillator strengths. In principle, by including the majority of the electron correlation effects, one should be able to obtain almost exact wave functions in the calculations. However, as obtaining the exact solution of the Schrödinger equation is only possible for one-electron atomic systems, some inaccuracy is always present in the results of the calculations.

Among the methods used to calculate the oscillator strengths for the atomic electronic transitions, the MCHF method has been most frequently used. For example, Fleming et al.7–9 implemented the multi-configurational Dirac–Fock/Dirac–Hartree–Fock (MCDF/MCDHF) method in the CIV3 code10 and used it in their atomic oscillator-strength calculations. For the beryllium atom, oscillator-strength calculations were performed by Tachiev and Froese Fischer11 using the aforementioned method. Their results were confirmed by Ynnerman and Froese Fischer12 in their independent MCDF/MCDHF calculations.

The calculations presented in this work focus on the electronic transitions involving 1S and 1P states of the beryllium atom. The transition energies and the corresponding oscillator strengths are calculated with high accuracy using variationally optimized Gaussian functions that explicitly depend on the distances between the electrons, i.e., the so-called explicitly correlated Gaussian functions (ECGs). The calculations include the leading relativistic and quantum-electrodynamics effects and are performed with an approach where the finite mass of the nucleus is directly incorporated in the Hamiltonian representing the system. This Hamiltonian is used to generate the nonrelativistic wave function and the corresponding energy for each considered state of the system. The nonrelativistic wave function is expanded in terms of all-particle ECGs, where the term all-particle refers to both electrons and the nucleus.

Different types of explicitly correlated basis functions have been used in the calculations of atomic ground and excited states for several decades now. It has been shown that these types of functions have a clear advantage over single-particle orbitals in calculations aiming to generate almost exact solutions to the Schrödinger equation. For two- and three-electron atomic systems, the best results have been obtained with Hylleraas-type explicitly correlated functions.13 For example, for the ground state of helium, the calculations with the Hylleraas basis, in particular, its variant that includes the logarithmic terms that help describe the proper analytic behavior of the wave function at the three-particle coalescence point, yielded the total nonrelativistic ground-state energy with the accuracy exceeding 40 decimal figures.14–19 Results obtained with Hylleraas-type basis sets for lithium are accurate up to 15 digits.4,20–22

The most accurate results for atoms and ions with four electrons, including beryllium, have been obtained with the all-electron explicitly correlated Gaussian functions.23–27 In our recent two works, we used ECGs to calculated some lowest 1S and 1P states of the beryllium atom. The calculations included the leading relativistic and quantum electrodynamics (QED) corrections. The corrections were calculated using the perturbation theory at the first-order level with the zero-order wave function being the nonrelativistic wave function obtained variationally with the finite-nuclear-mass (FNM) approach mentioned before. With that, the relativistic corrections included contributions from the so-called recoil effects, i.e., the effects due to the dependency of the relativistic corrections on the finite mass of the nucleus. The frequencies of the inter-state transitions calculated for the four lowest 1S states considered in Ref. 26 agreed with the experimental values to 0.02–0.09 cm−1. Similar accuracy was achieved in the calculations for the 1P states in Ref. 27.

The main task carried out in the present work is the development of the algorithm for calculating the oscillator strengths for atomic SP transitions. It is applied to calculate the oscillator strengths for the 1S1P and 1P1S transitions in beryllium. In the first step of the calculations, ECG basis sets for 1S and 1P states are generated in an iterative process, in which the size of the basis for each state is increased gradually while performing a thorough optimization of both the added basis functions and the functions already included in each basis. The procedure for enlargement of the basis set and variational optimization of the ECG exponential parameters was described previously in Refs. 26 and 27. Due to the use of the analytical energy gradient determined with respect to the parameters of the procedure, the variational energy minimization can be carried out very efficiently. The availability of the analytical gradient in our ECG atomic variational calculations is an important feature that allows us to achieve very high accuracy of the results.

The 2s2 1S0 → 2s2p 1P1 transition has been calculated with rather high accuracy before. Puchalski et al.23 used ECG to perform high-precision calculations of the 2s2 1S0 → 2s2p 1P1 transition energy and obtained the value of 42 565.441(11) cm−1. The calculations included the leading relativistic, QED, and finite-nuclear-mass corrections. Their calculated transition energy agrees with the recent saturated absorption measurement of this transition at 42 565.4501(13) cm−1 performed by Cook et al.28 The oscillator strength of the transition, however, was not reported by Puchalski et al. On the experimental side, one should mention the precise measurements of the energy levels of the 9Be isotope of beryllium by Bozman et al.29 and the measurements of several beryllium transitions performed by Johansson.30 The energy for the 2s2 1S0 → 2s2p 1P1 transition obtained in Johansson’s experiment was 42 565.35(18) cm−1. An important conclusion that emerged from comparing the calculated 2s2 1S0 → 2s2p 1P1 transitions energy by Puchalski et al. and the measured values was that the high-precision atomic transition-energy calculations have to account for the QED effects in addition to the relativistic and finite-nuclear-mass (FNM) effects in order to achieve an accuracy approaching that of high-precision experiments. For example, the QED contribution to the 2s2 1S0 → 2s2p 1P1 excitation energy determined by Puchalski et al. was equal to 1.048(9) cm−1. The FNM, relativistic, and QED effects are included in the calculations performed in this work.

This work is structured as follows. In Sec. 2, we provide a brief description of the approach used in the calculations. This includes the formulas for the FNM nonrelativistic Hamiltonian used in the calculations, the ECG basis function used for expanding the wave functions of the S and P states considered in this work, and the oscillator strength calculated for the considered interstate SP and PS transitions. Some details concerning the computational implementation of the formulas, as well as the strategy used in the optimization of the ECG non-linear parameters, are also described in Sec. 2. In Sec. 3, the results of the calculations performed in this work are presented. The results concern the ten lowest 1S states and the ten lowest 1P states of 9Be and Be (i.e., the beryllium atom with an infinitely heavy nucleus). The presented results include the nonrelativistic total energies and relativistic and QED corrections. These quantities are used to calculate the interstate transition energies and the corresponding oscillator strengths. These latter results are compared with the available experimental values. It should be noted that the present results are of benchmark quality, which required several months of continuous computing with hundreds of cores on a parallel computer system. The bulk of the computational time has gone into the optimization of the non-linear parameters of the ECGs, which, as mentioned, are performed variationally with the aid of the analytically calculated energy gradient. The total variational energies for all considered states are notably lower than those obtained in our previous calculations and the calculations performed by others.

In order to put the present calculations in perspective of over 80 years of theoretical works devoted to the calculations of S and P bound states of the beryllium atom, we include Table 1 that shows the progress achieved over the years in calculating the energies of the ground and excited S and P states of the beryllium atom with an infinite-nuclear-mass (INM). Results obtained with different methods are surveyed in the table; most of the methods are based on the variational principle. As one can see, the best results to date have been obtained using ECG expansions of the wave functions. It should be noted that some of the energy values listed in Table 1 have reported uncertainties that are considerably smaller than the actual difference between those values and the corresponding nonrelativistic limit. The reasons for this discrepancy range from overly optimistic assessment of the uncertainty due to the basis truncation to reporting only the statistical errors in the calculations.

TABLE 1.

Comparison of nonrelativistic energies of Be obtained with various theoretical methods: Hartree–Fock (HF), configuration interaction method (CI), poly-detor variational method with exponential functions (PDVM), many-body perturbation theory (MBPT), Hylleraas-type functions (Hy), Hylleraas-CI method (Hy-CI), multiconfiguration Hartree–Fock method (MCHF), estimated exact method (EE), explicitly correlated Gaussian functions (ECG), density matrix renormalization group (DMRG), explicitly correlated coupled cluster (CCSDT1-R12), diffusion Monte Carlo (DMC), local Schrödinger equation over free iterative-complement-interaction wave function (LSE-ICI), and explicitly correlated factorizable coupled-cluster method (ECFCC). Some of the quoted values represent an extrapolation to the infinite basis set limit. All energies are in atomic units

WorkYearStateMethodEnergy
Hartree and Hartree31  1935 1S HF −14.57 
Boys32  1950 1S CI −14.58 
Boys33  1953 1S PDVM −14.637 
 1P  −14.434 
Brigman et al.34  1958 1S CI −14.581 5 
Watson35  1960 1S CI −14.657 40 
Weiss36  1961 1S CI −14.660 90 
Kelly37  1963 1S MBPT −14.663 11 
Szasz and Byrne38  1967 1S Hy −14.656 5 
Gentner and Burke39  1968 1S Hy −14.657 9 
Bunge40  1968 1S CI −14.664 19 
Sims and Hagstrom41  1971 1S Hy-CI −14.666 547 
Perkins42  1973 1S Hy −14.661 1 
Froese Fischer and Saxena43  1974 1S MCHF −14.665 870 
Bunge44  1976 1S CI −14.666 902 
Clementi et al.45  1991 1S CI −14.666 96 
Mårtensson-Pendrill et al.46  1991 1S MCHF −14.667 37 
Davidson et al.47  1991 1S EE −14.667 36 
Froese Fischer48  1993 1S MCHF −14.667 113 
Chakravorty et al.49  1993 1S EE −14.667 36 
Komasa et al.50  1995 1S ECG −14.667 360(2) 
Noga et al.51  1995 1S CCSDT1-R12 −14.667 261 
Jitrik and Bunge52  1997 1S CI −14.667 275 57 
Busse and Lüchow53  1997 1S Hy −14.667 354 7 
Komasa54  2001 1S ECG −14.667 355 536 
 1P  −14.473 444 33 
 1P  −14.393 113 93 
 1P  −14.361 789 21 
Pachucki and Komasa55  2004 1S ECG −14.667 355 627 
Pachucki and Komasa56  2006 1S ECG −14.667 355 748 
Nakatsuji et al.57  2007 1S LSE-ICI −14.667 300 
Toulouse and Umrigar58  2008 1S DMC −14.667 27(1) 
Stanke et al.24  2009 1S ECG −14.667 356 486 
 1S  −14.418 240 328 
 1S  −14.370 087 876 
 1S  −14.351 511 654 
 1S  −14.342 403 552 
Verdebout et al.59  2010 1S MCHF −14.667 114 52 
Bunge60  2010 1S CI −14.667 355(1) 
King et al.61  2011 1S Hy −14.667 02 
Sims and Hagstrom62  2011 1S Hy-CI −14.667 356 411 
Komasa et al.63  2013 1S ECG −14.667 356 4 
Puchalski et al.23  2013 1S ECG −14.667 356 498 
 1P  −14.473 451 370 
Sharma et al.64  2014 1S DMRG −14.667 207 
Sims and Hagstrom65  2014 1S Hy-CI −14.667 356 407 951 
Bubin and Adamowicz66  2014 1P ECG −14.473 451 378 
 1P  −14.393 143 528 
 1P  −14.361 938 388 
 1P  −14.347 876 275 
 1P  −14.340 470 145 
 1P  −14.336 115 562 
 1P  −14.333 344 814 
Przybytek and Lesiuk67  2018 1S ECFCC −14.667 351 
Hornyák et al.26  2019 1S ECG −14.667 356 508(1) 
 1S  −14.418 240 368(2) 
 1S  −14.370 087 938(4) 
 1S  −14.351 511 736(7) 
Stanke et al.27  2019 1P ECG −14.473 451 3882 
 1P  −14.393 143 5385 
 1P  −14.361 938 3998 
 1P  −14.347 876 2953 
 1P  −14.340 470 194 4 
 1P  −14.336 115 706 0 
 1P  −14.333 345 316 8 
 1P  −14.331 475 953 7 
 10 1P  −14.330 154 912 7 
 11 1P  −14.329 185 241 1 
Nasiri and Zahedi68  2020 1S DMC −14.667 34(5) 
Sims69  2020 1S Hy-CI −14.418 240 346 
 1S  −14.370 087 890 
 1S  −14.351 511 676 
 1S  −14.342 403 578 
 1S  −14.337 266 500 
WorkYearStateMethodEnergy
Hartree and Hartree31  1935 1S HF −14.57 
Boys32  1950 1S CI −14.58 
Boys33  1953 1S PDVM −14.637 
 1P  −14.434 
Brigman et al.34  1958 1S CI −14.581 5 
Watson35  1960 1S CI −14.657 40 
Weiss36  1961 1S CI −14.660 90 
Kelly37  1963 1S MBPT −14.663 11 
Szasz and Byrne38  1967 1S Hy −14.656 5 
Gentner and Burke39  1968 1S Hy −14.657 9 
Bunge40  1968 1S CI −14.664 19 
Sims and Hagstrom41  1971 1S Hy-CI −14.666 547 
Perkins42  1973 1S Hy −14.661 1 
Froese Fischer and Saxena43  1974 1S MCHF −14.665 870 
Bunge44  1976 1S CI −14.666 902 
Clementi et al.45  1991 1S CI −14.666 96 
Mårtensson-Pendrill et al.46  1991 1S MCHF −14.667 37 
Davidson et al.47  1991 1S EE −14.667 36 
Froese Fischer48  1993 1S MCHF −14.667 113 
Chakravorty et al.49  1993 1S EE −14.667 36 
Komasa et al.50  1995 1S ECG −14.667 360(2) 
Noga et al.51  1995 1S CCSDT1-R12 −14.667 261 
Jitrik and Bunge52  1997 1S CI −14.667 275 57 
Busse and Lüchow53  1997 1S Hy −14.667 354 7 
Komasa54  2001 1S ECG −14.667 355 536 
 1P  −14.473 444 33 
 1P  −14.393 113 93 
 1P  −14.361 789 21 
Pachucki and Komasa55  2004 1S ECG −14.667 355 627 
Pachucki and Komasa56  2006 1S ECG −14.667 355 748 
Nakatsuji et al.57  2007 1S LSE-ICI −14.667 300 
Toulouse and Umrigar58  2008 1S DMC −14.667 27(1) 
Stanke et al.24  2009 1S ECG −14.667 356 486 
 1S  −14.418 240 328 
 1S  −14.370 087 876 
 1S  −14.351 511 654 
 1S  −14.342 403 552 
Verdebout et al.59  2010 1S MCHF −14.667 114 52 
Bunge60  2010 1S CI −14.667 355(1) 
King et al.61  2011 1S Hy −14.667 02 
Sims and Hagstrom62  2011 1S Hy-CI −14.667 356 411 
Komasa et al.63  2013 1S ECG −14.667 356 4 
Puchalski et al.23  2013 1S ECG −14.667 356 498 
 1P  −14.473 451 370 
Sharma et al.64  2014 1S DMRG −14.667 207 
Sims and Hagstrom65  2014 1S Hy-CI −14.667 356 407 951 
Bubin and Adamowicz66  2014 1P ECG −14.473 451 378 
 1P  −14.393 143 528 
 1P  −14.361 938 388 
 1P  −14.347 876 275 
 1P  −14.340 470 145 
 1P  −14.336 115 562 
 1P  −14.333 344 814 
Przybytek and Lesiuk67  2018 1S ECFCC −14.667 351 
Hornyák et al.26  2019 1S ECG −14.667 356 508(1) 
 1S  −14.418 240 368(2) 
 1S  −14.370 087 938(4) 
 1S  −14.351 511 736(7) 
Stanke et al.27  2019 1P ECG −14.473 451 3882 
 1P  −14.393 143 5385 
 1P  −14.361 938 3998 
 1P  −14.347 876 2953 
 1P  −14.340 470 194 4 
 1P  −14.336 115 706 0 
 1P  −14.333 345 316 8 
 1P  −14.331 475 953 7 
 10 1P  −14.330 154 912 7 
 11 1P  −14.329 185 241 1 
Nasiri and Zahedi68  2020 1S DMC −14.667 34(5) 
Sims69  2020 1S Hy-CI −14.418 240 346 
 1S  −14.370 087 890 
 1S  −14.351 511 676 
 1S  −14.342 403 578 
 1S  −14.337 266 500 

In this work, we are concerned with the quantum bound states of the atom. These states represent the motion of the particles forming the atom, i.e., the nucleus and the electrons, around the center of mass of the atom. To calculate the wave functions and the corresponding energies of such states, one needs to first derive a Hamiltonian operator that describes the intrinsic motion of all particles in the atom. In our approach, such a Hamiltonian is derived by starting with the standard nonrelativistic laboratory frame Hamiltonian representing the kinetic and potential energies of the nucleus and the electrons and partitioning this Hamiltonian into an operator representing the kinetic energy of the center-of-mass (COM) motion and the remaining part of the Hamiltonian that represents the “internal” state of the system. We call this latter part of the Hamiltonian “the internal Hamiltonian.” It is possible to rigorously make this partitioning by a transformation of the coordinates from the laboratory frame coordinates, which can be, for example, the Cartesian coordinates of the particles forming the atom defined with respect to a particular chosen fixed point in space, to a new set of coordinates. This new set can be chosen to consist, for example, of three coordinates representing the position of the center of mass in the Cartesian laboratory coordinate systems and 3N − 3 = 3n “internal” coordinates. There are different ways these internal coordinates can be chosen. One possibility is to use the textbook approach employed in solving the Schrödinger equation for the hydrogen atom, where the internal coordinates are the coordinates of the vector with the origin at the proton and the end at the electron. Generalizing this approach to an atom with n electrons, the internal coordinates can be chosen as a superposition of n sets of the coordinates, ri, i = 1, , n, where the ri is a vector consisting of the xi, yi, and zi coordinates with the origin at the nucleus and with the end at electron i. Thus, the new coordinate system consists of the lab-frame coordinates of the center of mass, XCM, YCM, and ZCM and the internal coordinates, ri, i = 1, , n.

To separate the full nonrelativistic laboratory frame Hamiltonian into the center-of-mass kinetic-energy operator and the internal operator, the Hamiltonian is expressed in terms of the new coordinates. This facilitates the separation. The internal Hamiltonian is only a function of the ri, i = 1, , n coordinates and has the following form (atomic units are assumed throughout):

Hnrint=12i=1n1μiri2+i=1njin1m0rirj+i=1nq0qiri+i=1nj<inqiqjrij,
(1)

where q0 is charge of the nucleus, qi = −1 (i = 1, …, n) are the electron charges, m0 is the nuclear mass (m0 = 16 424.2055me for 9Be), mi = 1 are the electron masses, μi = m0mi/(m0 + mi) is the reduced mass of electron i, ri is the distance between the nucleus and electron i, and rij = |rjri| is the distance between electrons i and j. In this work, we use the following 9Be nuclear mass: m0(9Be) = 16 424.205 5 me,70 where me is the mass of the electron. By setting m0 to infinity in Hnrint, one gets the INM Hamiltonian that is used in the standard calculations based on the Born–Oppenheimer approximation. Both FNM and INM Hamiltonians are used in the present calculations. When the FNM Hamiltonian is used, both the energy and the wave function depend on the mass of the nucleus. In this work, we report both the finite-mass and infinite-mass results.

Hamiltonian (1) can also be written in a compact matrix form71 as

Hnrint=rMr+i=1nq0qiri+i=1nj<inqiqjrij,
(2)

where

r=r1rn

is a n-component gradient vector and M = MI3 is the Kronecker product of a n × n matrix M and 3 × 3 identity matrix I3. Matrix M has diagonal elements 1/(2μ1), …, and 1/(2μn), while all off-diagonal elements are equal to 1/(2m0). The prime symbol denotes the matrix/vector transpose. Hamiltonian (2) is used in the present variational calculations of the nonrelativistic internal energy and the corresponding internal wave function. The internal wave function describing a state of the atom is a function of internal coordinates ri, i = 1, , n.

The basis functions of all-electron explicitly correlated Gaussians are used in the present calculations to construct the spatial parts of the wave functions for the P and S states considered in this work. The S-type Gaussians have the following form:

ϕk=exprAkI3r,
(3)

where Ak is an n × n real symmetric matrix, ⊗ is the Kronecker product, and I3 is a 3 × 3 identity matrix. The P-type Gaussians have the following form:

ϕk=zikexprAkI3r,
(4)

where zik is the z-coordinate of the ith electron and ik is the electron label, which varies in the (1, , n) range and is an adjustable parameter in our calculations. The parameter is specific for each basis function, ϕk, and its value is determined variationally when the ECG basis set is being extended. The above ECG contains an angular pre-exponential factor zik that makes it suitable for expanding the wave functions of L = 1 states. The factor is a Cartesian spherical harmonic corresponding to L = 1 and ML = 0. A proper L = 1 ECG is obtained regardless of the value of the electron-label index ik of the pre-exponential z factor. In order to make the initial random choice and subsequent optimization of ECGs more efficient, we treat the index as an additional variational parameter. This is optional, as choosing a fixed value of the electron index of z and not optimizing it, but only optimizing the non-linear exponential ECG parameters, should, in principle, lead to identical results.

The Gaussian basis functions [(3) and (4)] have to be square integrable to represent wave functions of bound S and P states of the atom. This happens when the matrix Ak is positive definite. To fulfill this requirement, Ak is represented in the Cholesky-factored form ad Ak=LkLk, where Lk is a lower triangular matrix. The Ak matrix given in the Cholesky-factored form is always positive definite regardless of the values of the Lk matrix elements. Thus, if these matrix elements are used as the variational parameters of the Gaussians and optimized by the minimization of the total energy of the state under the consideration, they can be varied without any constrains from − to . This is convenient because any constraint imposed on the variational parameters would make the optimization more cumbersome.

Before either S or P Gaussians given by expressions (3) or (4) are used in expanding the wave function of a state, they have to be appropriately symmetry adapted. In the present approach, we use the spin-free formalism to ensure the correct permutational symmetry properties of each matrix element. For this purpose, an appropriate permutational symmetry projector is constructed and applied to basis functions (3) and (4). To construct the permutation-symmetry projector, the standard procedure involving the Young operators is used.72,73 In the case of the singlet S and P states of beryllium, the permutation operator, Y, can be chosen in the form Y = (1 − P13)(1 − P24)(1 + P12)(1 + P34), where Pij denotes the permutation of the spatial coordinates of the ith and jth electrons (assuming particle 0 is the nucleus). More details about the generation of the wave function and its variational optimization can be found in Refs. 74 and 75. It should be noted that the variational optimization of the exponential parameters of the S and P basis functions and the electron indices, ik, of the zik factors in the P basis functions is only carried out for the wave functions of the 9Be isotope and then reused in the Be calculations without reoptimization. Our extensive experience with atomic calculations has shown that just rediagonalizing the Hamiltonian matrix by adjusting the linear variational parameters provides a sufficiently accurate way to account for a relatively small change of the wave function caused by the change in the nuclear mass.

In order to determine atomic transition energies with the accuracy that matches the accuracy of the most precise experiments, the state energies have to be calculated with account for the leading relativistic and quantum-electrodynamics (QED) effects. The most practical way for calculating these effects for few-electron light atoms is to use the standard perturbation theory and to expand the total energy in powers of the fine-structure constant, α.76,77 The first term in this expansion is the nonrelativistic energy of the considered state, Enr,

Etot=Enr+α2Erel(2)+α3EQED(3)+α4EHQED(4)+,
(5)

the second term, α2Erel(2), represents the leading relativistic corrections, the third term, α3EQED(3), represents the leading QED corrections, and the fourth term, α4EHQED(4), represents higher-order QED corrections.

The corrections are evaluated as expectation values of the effective operators representing some physical effects using the nonrelativistic wave function. This wave function can be obtained with either the FNM or INM approach. The α2Erel(2) term is calculated as the expectation value of the Dirac–Breit Hamiltonian in the Pauli approximation, Hrel.78,79 In the present calculations of the 1S and 1P states of beryllium, the Hamiltonian for the relativistic correction, Hrel, contains the following terms:

Hrel=HMV+HD+HOO+HSS,
(6)

where operators HMV, HD, HOO, and HSS represent the mass-velocity, Darwin, orbit–orbit, and spin–spin corrections, respectively. As all states considered in this work are singlet states, the spin–orbit interaction vanishes. The explicit form of the operators in the right-hand side of (6) in the internal coordinates, ri, i = 1, , n, is given in our previous publications.26,27 It should be mentioned that, when the finite-nuclear-mass approach is used in the calculations of the nonrelativistic energy and the corresponding wave function and of the relativistic corrections, these corrections explicitly depend on the nuclear mass. Thus, the so-called recoil effects are directly accounted for in the calculations.

The leading QED effects in (5) are represented by the α3EQED(3) term. For an atom, this term accounts for the two-photon exchange, the vacuum polarization, and the electron self-energy effects. The operator has the following form in the internal coordinates:

HQED(3)=i,j=1j>in16415+143lnαδrij76πP1rij3+i=1n19302lnαlnk04q03δri,
(7)

where the first term represents the so-called Araki–Sucher correction.80–84 The correction involves the principal value P1/rij3 defined as

P1rij3=lima01rij3Θrija+4πγ+lnaδrij.
(8)

Here, Θ(⋯) is the Heaviside step function and γ = 0.577 215… is the Euler–Mascheroni constant.

The last term in HQED(3) represents the electron self-energy. Its dominant contribution is the term involving the so-called Bethe logarithm, ln k0. The main obstacle in computing the QED correction accurately for a multi-electron atomic system comes from ln k0, which is difficult to calculate. However, ln k0 is known to mostly depend on the contributions from the core electrons. In Sec. 2.4, the procedure used to determine the ln k0 values for the considered states is described.

The last term in expansion (5) is the EHQED(4) term. It can be calculated as the expectation value of the following approximate operator derived by Pachucki et al.:55,85

HHQED(4)=πq02427962ln2i=1nδri.
(9)

EHQED(4) includes the dominating electron-nucleus one-loop radiative correction but neglects the two-loop radiative, electron–electron radiative, and the higher-order relativistic corrections. The expectation value of (9) only provides a rough approximation to EHQED(4) for light atoms with the overall error being of the order of 50%.

The expectation values of the HQED(3) and HHQED(4) Hamiltonians are calculated in this work with the INM wave functions. This is because the formulas used in the calculations were derived for the clamped nucleus.55,85 Thus, the EQED(3) and EHQED(4) corrections computed in this work do not include the recoil effects.

Some of the operators used in the calculations of the relativistic and QED effects include singular terms. Examples of such terms are the ri4 operator in HMV and the one- and two-electron Dirac delta functions, δ(ri) and δ(rij), in HD, HSS, HQED, and HHQED. The convergence of the expectation values of operators involving singular terms with the number of basis functions used to expand the wave function of the atom is usually much slower than for non-singular operators—the number of converged significant figures is typically about twice smaller. However, there have been studies that proposed neat workarounds to this problem.86–91 One way to improve the convergence is to employ expectation value identities, which involve certain global operators whose expectation values coincide with the expectation values of the singular operators in the case of the exact wave function. When an approximate wave function is used instead, the expectation values of those global operators typically converge to the infinite basis set limit at a much faster rate. The original idea was laid out by Drachman91 based on the work of Trivedi.90 Drachman’s approach has been adopted in several works published in recent decades with a good level of success. It has been demonstrated in comparative studies that the convergence of the expectation values of the operators found in the leading relativistic and QED corrections is significantly accelerated by using Drachman’s approach.26,92 In this work, we also adopt an approach in the same spirit to compute the expectation values of δ(ri), δ(rij), and ri4. More details on this can be found in Refs. 5 and 26.

Expression (7) contains a term that includes Bethe logarithm, ln k0. This term represents the dominant part of the electron self-energy. Accurate calculation of this quantity for multi-electron systems represents a major difficulty. In the case of the beryllium atom, Puchalski et al.4,23 studied the three lowest singlet states (2 1S, 2 1P, and 3 1S) and reported the Bethe logarithms for them. Those are currently the most accurate values available for Be in the literature. For other states, the ln k0 values have been either computed less accurately or not computed at all. Drake and Goldman93 showed that the value of the Bethe logarithm for atomic Rydberg states has the following asymptotic behavior: A + B/n3, where n is the principal quantum number and A and B are constants. In this work, we employ a fitting procedure with the above expression to estimate the values of Bethe logarithm for the S and P states using the available ln k0 values for S (2 1S, 3 1S) and P (2 1P) states of the Be atom. In our fitting, the Bethe logarithm value for the ground 22S state of Be+ ion94 is used as the asymptotic value when n. The values of ln k0 we adopted for the S and P states of Be considered in this work are shown in Table 2. To show how those differ from the ground-state value of the hydrogen-like atom, in Table 2, we also list the values of lnk0/q02 (where q0 is the nuclear charge).

TABLE 2.

Approximate values of the Bethe logarithm used in the calculations of the QED corrections for the lowest ten 1S and 1P states of beryllium. All values are in atomic units

StateReferenceln k0ln(k0/q02)
1S 4  5.750 46 2.977 871 
1S 4  5.751 49 2.978 901 
1S  5.751 698 2.979 109 
1S  5.751 783 2.979 195 
1S  5.751 821 2.979 232 
1S  5.751 840 2.979 252 
1S  5.751 851 2.979 262 
1S  5.751 858 2.979 269 
10 1S  5.751 862 2.979 273 
11 1S  5.751 865 2.979 276 
1P 4  5.752 320 2.979 731 
1P  5.751 989 2.979 401 
1P  5.751 909 2.979 320 
1P  5.751 880 2.979 291 
1P  5.751 867 2.979 279 
1P  5.751 861 2.979 272 
1P  5.751 857 2.979 269 
1P  5.751 855 2.979 266 
10 1P  5.751 854 2.979 265 
11 1P  5.751 853 2.979 264 
22SBe+ 94  5.751 849 2.979 260 
12S93  2.984 128 2.984 128 
StateReferenceln k0ln(k0/q02)
1S 4  5.750 46 2.977 871 
1S 4  5.751 49 2.978 901 
1S  5.751 698 2.979 109 
1S  5.751 783 2.979 195 
1S  5.751 821 2.979 232 
1S  5.751 840 2.979 252 
1S  5.751 851 2.979 262 
1S  5.751 858 2.979 269 
10 1S  5.751 862 2.979 273 
11 1S  5.751 865 2.979 276 
1P 4  5.752 320 2.979 731 
1P  5.751 989 2.979 401 
1P  5.751 909 2.979 320 
1P  5.751 880 2.979 291 
1P  5.751 867 2.979 279 
1P  5.751 861 2.979 272 
1P  5.751 857 2.979 269 
1P  5.751 855 2.979 266 
10 1P  5.751 854 2.979 265 
11 1P  5.751 853 2.979 264 
22SBe+ 94  5.751 849 2.979 260 
12S93  2.984 128 2.984 128 

In the length formalism, the absorption oscillator strength fif for a transition between an initial state i and a final state f is defined as95,96

fif=23giZrZpΔEifψiμψf2,
(10)

where gi = 2Ji + 1 is the statistical weight of the lower level, ΔEif = |EiEf| is the transition energy, Zr=q0me+m0nme+m0 and Zp=q0me+m0m0 are effective radiative charges (q0 is the charge of the nucleus, m0 is the nuclear mass, me is the mass of the electron, and n is the number of electrons), and μ is the electric dipole-moment operator. For an atom containing n electrons, μ=i=1nqiri, where qi and ri is the charge of electron i and its position in the internal coordinate system, respectively. Wave functions ψi and ψf are nonrelativistic wave functions obtained in variational calculations using Hamiltonian (2). As the Hamiltonian explicitly depends on the mass of the nucleus, ψi and ψf also depend on the nuclear mass, i.e., they are slightly different for 9Be and Be. It is worth mentioning that for a charge-neutral system, μ has the same form in the laboratory Cartesian coordinate system as in the coordinate system where the nucleus is placed at the origin of the internal coordinate system. The transition dipole moment associated with the if transition can be written in the following form:

|μif|2=ψiμψf2=ψiμxψf2+ψiμyψf2+ψiμzψf2.
(11)

For oscillator strengths, only the transition dipole moments between the S (L = 0, ML = 0) and P (L = 1, ML = 0) states need to be evaluated [in this case, only the last of the three terms in Eq. (11) is non-zero]. Restricting the calculations to the transitions between the S (L = 0, ML = 0) and P (L = 1, ML = 0) states only is possible because of symmetry. Effectively, one can obtain the transition dipole moments for all P states with ML = ±1 by knowing the corresponding values for ML = 0 (for more information, see Ref. 97).

The transition dipole moment matrix elements with S and P ECG basis functions (3) and (4) can be evaluated in a similar way as the overlap matrix elements (Refs. 74 and 98 contain detailed derivations of the latter, as well as other relevant matrix elements). For the sake of consistency and convenience, below we adopt the same notation scheme as in Refs. 74 and 98. The z-component of the transition dipole-moment matrix element between S (L = 0, ML = 0) and P (L = 1, ML = 0) ECGs can be expressed as

P̂kϕk(0)ziP̂lϕl(1)=ϕ̃k(0)ziϕ̃l(1)=exp[r(ÃkI3)r]zizm̃l×exp[r(ÃlI3)r]dr=(vir)exp[rÃkr](ṽlr)exp[rÃlr]dr.
(12)

Here, P̂k and P̂l are the particle permutation operators for the bra and ket wave functions, respectively, AkAkI3, AkAkI3, and vlvlɛz, where ɛz ≡ (0, 0, 1). vl is a sparse n-component vector with all components equal to zero, except the mlth component. The scalar product of a 3n-component vector vl with another 3n-component vector r yields a single coordinate, zml=vlr. The tilde symbol denotes the action of the permutation matrices PkPkI3 and PlPlI3 corresponding to operators P̂k and P̂l on matrices Ak, Al, and vector vl,

Ãk=PkAkPk,Ãl=PlAlPl,ṽl=Plvl,zm̃k=ṽlr.

The integral in Eq. (12) is given by formula (28) in Ref. 74. In that formula, one needs to make a replacement vkvi. With that expression, (12) becomes

ϕ̃k(0)ziϕ̃l(1)=π3n22viÃkl1ṽlÃkl32,
(13)

where Ãkl=Ãk+Ãl.

The density of particle i in the center-of-mass (COM) coordinate frame is defined as ρiξ=δRiRcmξ, where i = 1, , N and Rcm is the position vector of the center of mass in the laboratory coordinate frame. In this work, the COM-frame density plots are generated for both the nucleus and the electrons. These density distributions provide a representation of the coupled motion of the nucleus and the electrons around the center of mass in the beryllium atom. If the atom is excited to an increasingly higher Rydberg state, the average radius of the electronic density increases, as manifested by the increasing value of the nucleus-electron average distance and by increasing diffuseness of the COM-frame electron density. At the same time, the electronic density becomes more oscillatory. The oscillations of the electronic density are mirrored by the oscillations of the density of the nucleus in the COM-frame. The matching number of the maxima in the electronic and nuclear densities for a given state occurs because only then the center-of-mass of the atom can remain immobile. However, due to the much larger mass of the nucleus in comparison with the electron mass, the average radius of the nuclear motion around the center of mass is orders of magnitude smaller than the average radius of the electronic motion. A pictorial comparison of the two motions using the electronic and nuclear COM-frame densities is presented Sec. 3. The feature to notice in the plots is the difference in the scales used to plot the electronic and nuclear densities.

For calculating atomic S and P states in the framework of the ECG method, we used our in-house parallel computer code written in FORTRAN and employing MPI (Message Passing Interface) for communication between parallel processes. The generation of large ECG basis sets together with high accuracy targeted in the calculations requires the use of extended precision (80-bits) arithmetic, which has a hardware implementation in floating-point modules on the ×86 architecture. The calculations performed with extended precision are typically slower by a factor of 2–3, yet they provide additional 12 bits (or about four decimal figures) of accuracy compared to the standard double precision. Ten singlet S states and ten singlet P states of beryllium have been calculated in this work. In the first step of the calculations, the nonrelativistic wave functions and the corresponding energies are obtained. The calculations are carried out using the standard variational method and involve the generation of sets of basis functions for each state with different sizes. The growing of the basis set for a particular state is performed independently from other states. It involves adding new functions to the set and variationally optimizing their non-linear parameters using a procedure that employs the analytical energy gradient determined with respect to the parameters. More details about the basis-set enlargement procedure can be found in our previous works.99,100 It should be noted that the generation of the basis set for each considered state is by far the most time-consuming step of the calculations. It required about a year of continuous computing using several 100 cores of parallel computer systems equipped with Intel Xeon E5-2695v3 and AMD EPYC 7642 central processing units (CPUs).

In generating the ECG basis sets, the internal Hamiltonian explicitly dependent on the mass of the nucleus of the 9Be isotope, i.e., the FNM Hamiltonian (2) is used. The basis sets are subsequently used in the calculations for Be. The nonrelativistic energies are shown in Table 3. The results for the 9Be isotope and for Be are shown as well. For each 1S and 1P state of 9Be, the nonrelativistic energy is reported for three to six basis set sizes to demonstrate the convergence with the number of basis functions. The nonrelativistic energy for Be is only shown for the largest basis set generated for each state, as the convergence of these values has essentially the same pattern as those for 9Be. These energies are compared with the best previously published results. Our largest basis sets range from 16 000 basis functions for the lower states to 17 000 for the top states. For all states, the values of the nonrelativistic 9Be and Be energies are notably lower than the values previously reported in the literature. For example, for the ground 1S state, the present energy value of Be, −14.667 356 508 8 a.u., is lower than the value of −14.667 356 498(3) a.u. reported in 2013 by Puchalski et al.23 by about 1 × 10−8 a.u. For the lowest 1P state, the analogous comparison is −14.473 451 389 5 a.u. (our present value) and −14.473451 37(4) a.u. (the value of Puchalski et al.23). Another example concerns our previous calculations of the five lowest 1S states.26 The 9Be and Be energies obtained in that work with 7000 ECGs for the ground state are −14.666 435 525 and −14.667 356 507 a.u., respectively, while the corresponding values obtained in the present calculations are −14.666 435 526 4 and −14.667 356 508 4 a.u., respectively. For 5 1S, the corresponding comparison is −14.350 610 425 9 and −14.351 511 733 9 a.u. (the present calculations) and −14.350 610 414 and −14.351 511 722 in the calculations of Hornyák et al.26 As one can notice, the improvement is larger for 5 1S than for the 2 1S state. In general, due to a larger number of radial nodes in the wave functions of higher excited states, the number of basis functions used in the calculations needs to be increased to maintain a similar level of accuracy for all states. In the present calculations, the number of the basis functions for lower states (up to n = 8) is kept constant and and then increased by 1000 for the higher states (n = 9–11). The main factor that limits the increase of the basis size in the present calculations is the CPU time needed to optimize a very large number of ECGs. The results presented in this work reflect a practical limit of the computational resources we have been able to allocate for our calculations at present.

TABLE 3.

Convergence of the nonrelativistic energies (Enr), expectation values of the mass-velocity correction (HMV), δ-functions dependent on interparticle distances, principal value P(1/rij3), and total energies (Etot) with the number of basis functions for the lowest ten 1S and 1P states of beryllium atom (9Be and Be). The numbers in parentheses are estimated uncertainties due to the basis truncation error. The tilde sign indicates that expectation value identities were employed to compute the corresponding quantities to improve their convergence. All values are in atomic units

StateIsotopeBasisReferenceEnrH̃MVHOOδ̃(ri)δ̃(rij)P(1/rij3)Etot
1S 9Be 14 000  −14.666 435 525 2 −270.636 666 −0.918 461 59 8.840 617 349 0.267 506 284 −1.222 66 −14.668 440 604 
15 000  −14.666 435 526 1 −270.636 645 −0.918 461 58 8.840 617 352 0.267 506 285 −1.222 43 −14.668 440 604 
16 000  −14.666 435 526 4 −270.636 556 −0.918 461 57 8.840 617 354 0.267 506 286 −1.221 35 −14.668 440 600 
  −14.666 435 526 8(2)      −14.66 844 059 7(18) 
7 000 26  −14.666 435 525 −270.636 610 −0.918 461 58 8.840 617 34 0.267 506 284   
 26  −14.666 435 526(1)       
Be 16 000  −14.667 356 508 4 −270.703 526 −0.891 823 69 8.842 251 649 0.267 550 917 −1.221 78 −14.669 361 498 
  −14.667 356 508 7(2)      −14.669 361 494(18) 
 4  −14.667 356 498(3) −270.703 76(11) −0.891 824 10(8) 8.842 250 65(15) 0.267 550 888(15) −1.221 127 7(5)  
7 000 26  −14.667 356 507 −270.703 579 −0.891 823 62 8.842 251 64 0.267 550 915 −1.222 52  
 26  −14.667 356 508(1)       
1P 9Be 14 600  −14.472 543 756 9 −266.598 045 −0.838 224 67 8.722 853 523 0.261 279 705 −1.184 89 −14.474 498 170 
15 500  −14.472 543 757 7 −266.598 039 −0.838 224 67 8.722 853 527 0.261 279 705 −1.184 83 −14.474 498 170 
16 400  −14.472 543 761 1 −266.598 021 −0.838 224 64 8.722 853 539 0.261 279 707 −1.184 62 −14.474 498 173 
  −14.472 543 764 7(19)      −14.474 498 176(13) 
16 400 27  −14.472 543 759 8  −0.838 224 6     
Be 16 400  −14.473 451 389 5 −266.664 445 −0.812 091 50 8.724 481 450 0.261 324 034 −1.185 06 −14.475 405 724 
  −14.473 451 393 3(19)      −14.475 405 726(13) 
 4  −14.473 451 37(4) −266.665 14(15) −0.812 092 9(3) 8.724 478 65(20) 0.261 323 93(3) −1.182 858(13)  
16 400 27  −14.473 451 388 2  −0.812 091 5     
1S 9Be 14 000  −14.417 335 140 0 −268.474 994 −0.926 539 44 8.780 376 771 0.263 820 372 −1.248 41 −14.419 312 274 
15 000  −14.417 335 142 9 −268.474 893 −0.926 539 39 8.780 376 781 0.263 820 375 −1.247 36 −14.419 312 272 
16 000  −14.417 335 143 5 −268.474 803 −0.926 539 37 8.780 376 790 0.263 820 377 −1.246 34 −14.419 312 269 
  −14.417 335 144 1(6)      −14.419 312 265(19) 
7 000 26  −14.417 335 139 −268.474 926 −0.926 539 41 8.780 376 76 0.263 820 371   
 26  −14.417 335 143(2)       
Be 16 000  −14.418 240 368 2 −268.541 128 −0.900 128 50 8.781 996 527 0.263 864 254 −1.246 77 −14.420 217 407 
  −14.418 240 369 4(6)      −14.420 217 403(19) 
7 000 26  −14.418 240 364 −268.541 251 −0.900 128 21 8.781 996 50 0.263 864 248 −1.248 22  
 26  −14.418 240 368(2)       
1P 9Be 14 600  −14.392 242 880 3 −267.390 569 −0.882 080 43 8.747 804 857 0.262 237 919 −1.227 10 −14.394 206 423 
15 500  −14.392 242 881 8 −267.390 566 −0.882 080 43 8.747 804 867 0.262 237 920 −1.227 07 −14.394 206 424 
16 400  −14.392 242 886 1 −267.390 546 −0.882 080 40 8.747 804 893 0.262 237 922 −1.226 80 −14.394 206 427 
  −14.392 242 890 4(22)      −14.394 206 431(17) 
16 400 27  −14.392 242 884 4  −0.882 080 40     
Be 16 400  −14.393 143 540 2 −267.456 802 −0.855 811 50 8.749 425 262 0.262 281 844 −1.227 23 −14.395 106 998 
  −14.393 143 544 6(22)      −14.395 107 001(17) 
16 400 27  −14.393 143 538 5  −0.855 811 50     
1S 9Be 14 000  −14.369 185 505 1 −268.315 505 −0.932 337 62 8.776 228 493 0.263 512 821 −1.252 93 −14.371  60 487 
15 000  −14.369 185 510 9 −268.315 475 −0.932 337 58 8.776 228 516 0.263 512 824 −1.252 49 −14.371 160 492 
16 000  −14.369 185 512 9 −268.315 411 −0.932 337 56 8.776 228 530 0.263 512 826 −1.251 82 −14.371 160 491 
  −14.369 185 515 1(15)      −14.371 160 493(10) 
7 000 26  −14.369 185 506 −268.315 685 −0.932 337 71 8.776 228 47 0.263 512 813   
 26  −14.369 185 514(4)       
Be 16 000  −14.370 087 936 7 −268.381 663 −0.905 938 87 8.777 846 410 0.263 556 605 −1.252 25 −14.372 062 828 
  −14.370 087 939 7(15)      −14.372 062 830(10) 
7 000 26  −14.370 087 930 −268.381 937 −0.905 938 65 8.777 846 35 0.263 556 591 −1.255 16  
 26  −14.370 087 938(4)       
1P 9Be 14 600  −14.361 037 794 6 −267.893 328 −0.913 549 58 8.763 470 236 0.262 900 115 −1.244 68 −14.363 007 512 
15 500  −14.361 037 796 1 −267.893 322 −0.913 549 56 8.763 470 248 0.262 900 116 −1.244 63 −14.363 007 513 
16 400  −14.361 037 802 2 −267.893 223 −0.913 549 52 8.763 470 279 0.262 900 121 −1.243 49 −14.363 007 515 
  −14.361 037 808 6(36)      −14.363 007 517(9) 
16 400 27  −14.361 037 799 9  −0.913 549 5     
Be 16 400  −14.361 938 402 2 −267.959 411 −0.887 204 01 8.765 087 272 0.262 943 867 −1.243 92 −14.363 908 028 
  −14.361 938 409 4(36)      −14.363 908 030(9) 
16 400 27  −14.361 938 399 8  −0.887 204 0     
1S 9Be 14 000  −14.350 610 411 8 −268.272 968 −0.934 164 35 8.775 161 626 0.263 430 584 −1.254 27 −14.352 584 783 
15 000  −14.350 610 424 7 −268.272 868 −0.934 164 26 8.775 161 679 0.263 430 591 −1.253 07 −14.352 584 791 
16 000  −14.350 610 425 9 −268.272 870 −0.934 164 26 8.775 161 683 0.263 430 592 −1.253 06 −14.352 584 793 
  −14.350 610 427 4(8)      −14.352 584 795(11) 
7 000 26  −14.350 610 414 −268.273 164 −0.934 164 45 8.775 161 59 0.263 430 571 −1.255 16  
 26  −14.350 610 428(7)       
Be 16 000  −14.351 511 733 9 −268.339 105 −0.907 768 57 8.776 779 118 0.263 474 347 −1.253 49 −14.353 486 014 
  −14.351 511 735 4(8)      −14.353 486 015(11) 
7 000 26  −14.351 511 722 −268.339 399 −0.907 768 18 8.776 779 03 0.263 474 325 −1.256 59  
 26  −14.351 511 736(7)       
1P 9Be 14 600  −14.346 975 854 1 −268.071 039 −0.924 932 25 8.769 034 122 0.263 137 388 −1.252 92 −14.348 947 726 
15 500  −14.346 975 856 8 −268.071 032 −0.924 932 23 8.769 034 146 0.263 137 390 −1.252 84 −14.348 947 729 
16 400  −14.346 975 867 2 −268.070 985 −0.924 932 17 8.769 034 198 0.263 137 397 −1.252 35 −14.348 947 737 
  −14.346 975 878 4(56)      −14.348 947 745(42) 
16 400 27  −14.346 975 863 8  −0.924 932 2     
Be 16 400  −14.347 876 298 8 −268.137 180 −0.898 561 42 8.770 650 913 0.263 181 123 −1.252 78 −14.349 848 081 
  −14.347 876 310 0(56)      −14.349 848 090(42) 
16 400 27  −14.347 876 295 3  −0.898 561 4     
1S 9Be 14 000  −14.341 502 883 9 −268.257 684 −0.93492953 8.774 781 459 0.263 400 264 −1.258 15 −14.343 477 035 
15 000  −14.341 502 912 6 −268.257 307 −0.934 929 21 8.774 781 652 0.263 400 285 −1.254 90 −14.343 477 046 
16 000  −14.341 502 922 5 −268.257 223 −0.934 929 13 8.774 781 679 0.263 400 291 −1.254 12 −14.343 477 052 
  −14.341 502 933 1(53)      −14.343 477 060(36) 
Be 16 000  −14.342 403 676 6 −268.323 452 −0.908 534 40 8.776 399 001 0.263 444 038 −1.254 55 −14.344 377 719 
  −14.342 403 691 8(53)      −14.344 377 725(36) 
1P 9Be 14 600  −14.339 569 897 0 −268.147 264 −0.929 839 34 8.771 437 062 0.263 240 458 −1.254 08 −14.341 542 673 
15 500  −14.339 569 904 8 −268.147 262 −0.929 839 27 8.771 371 04 0.263 240 462 −1.254 05 −14.341 542 680 
16 400  −14.339 569 932 7 −268.147 115 −0.929 839 12 8.771 437 280 0.263 240 478 −1.252 23 −14.341 542 702 
  −14.339 569 964(15)      −14.341 542 73(12) 
16 400 27  −14.339 569 924  −0.929 839 3     
Be 16 400  −14.340 470 2036 −268.213 318 −0.903 457 80 8.773 054 048 0.263 284 203 −1.252 66 −14.342 442 886 
  −14.340 470 233(15)      −14.342 442 91(12) 
16 400 27  −14.340 470 194  −0.903 457 8     
1S 9Be 14 000  −14.336 365 900 9 −268.251 719 −0.935 320 56 8.774 622 594 0.263 386 930 −1.265 45 −14.338 339 983 
15 000  −14.336 366 054 2 −268.251 428 −0.935 319 72 8.774 623 708 0.263 387 016 −1.262 35 −14.338 340 121 
16 000  −14.336 366 091 9 −268.251 267 −0.935 319 38 8.774 623 968 0.263 387 047 −1.260 89 −14.338 340 151 
  −14.336 366 132(20)      −14.338 340 18(15) 
Be 16 000  −14.337 266 532 0 −268.317 496 −0.908 924 95 8.776 241 280 0.263 430 793 −1.261 32 −14.339 240 504 
  −14.337 266 572(20)      −14.339 240 54(15) 
1P 9Be 14 600  −14.335 215 466 9 −268.185 790 −0.932 286 61 8.772 641 931 0.263 292 332 −1.258 24 −14.337 188 708 
15 500  −14.335 215 483 6 −268.185 794 −0.932 286 55 8.772 642 041 0.263 292 336 −1.258 21 −14.337 188 725 
16 400  −14.335 215 573 5 −268.185 654 −0.932 286 11 8.772 642 568 0.263 292 395 −1.255 95 −14.337 188 808 
  −14.335 215 667(50)      −14.337 188 89(42) 
16 400 27  −14.335 215 559 2  −0.932 286 5     
Be 16 400  −14.336 115 721 1 −268.251 863 −0.905 899 50 8.774 259 420 0.263 336 122 −1.256 38 −14.338 088 869 
  −14.336 115 820(50)      −14.338 088 95(42) 
16 400 27  −14.336 115 706  −0.905 899 7     
1S 9Be 14 000  −14.333 185 490 7 −268.248 468 −0.935 550 47 8.774 551 560 0.263 380 462 −1.266 85 −14.335 159 515 
15 000  −14.333 185 759 8 −268.247 953 −0.935 549 61 8.774 553 205 0.263 380 607 −1.262 41 −14.335 159 758 
16 000  −14.333 186 005 6 −268.247 983 −0.935 548 81 8.774 554 820 0.263 380 723 −1.260 05 −14.335 160 006 
  −14.333 186 26(16)      −14.335 160 3(15) 
Be 16 000  −14.334 086 251 0 −268.314 213 −0.909 154 49 8.776 172 170 0.263 424 469 −1.260 48 −14.336 060 164 
  −14.334 086 56(16)      −14.336 060 4(15) 
1P 9Be 14 600  −14.332 444 940 2 −268.207 584 −0.933 641 05 8.773 310 065 0.263 321 187 −1.265 04 −14.334 418 458 
15 500  −14.332 445 006 9 −268.207 593 −0.933 640 86 8.773 310 409 0.263 321 197 −1.264 88 −14.334 418 525 
16 400  −14.332 445 316 5 −268.207 487 −0.933 640 32 8.773 312 276 0.263 321 319 −1.261 84 −14.334 418 829 
  −14.332 445 63(20)      −14.334 419 1(17) 
16 400 27  −14.332 445 263 5  −0.933 641 1     
Be 16 400  −14.333 345 372 7 −268.273 702 −0.907 250 64 8.774 929 236 0.263 365 049 −1.262 28 −14.335 318 798 
  −14.333 345 77(20)      −14.335 319 1(17) 
16 400 27  −14.333 345 316 8  −0.907 251 0     
1S 9Be 15 000  −14.331 080 417 2 −268.245 947 −0.935 704 83 8.774 517 184 0.263 376 905 −1.270 61 −14.333 054 364 
16 000  −14.331 080 611 5 −268.246 125 −0.935 704 50 8.774 519 513 0.263 377 024 −1.269 88 −14.333 054 565 
17 000  −14.331 080 817 4 −268.246 315 −0.935 703 87 8.774 521 837 0.263 377 150 −1.269 07 −14.333 054 778 
  −14.331 081 18(18)      −14.333 055 3(28) 
Be 17 000  −14.331 980 933 9 −268.312 545 −0.909 308 87 8.776 139 224 0.263 420 896 −1.26951 −14.333 954 808 
  −14.331 981 33(18)      −14.333 955 1(28) 
1P 9Be 15 500  −14.330 575 490 3 −268.220 833 −0.934 452 69 8.773 710 101 0.263 338 312 −1.271 53 −14.332 549 183 
16 400  −14.330 575 988 0 −268.220 546 −0.934 451 25 8.773 713 099 0.263 338 531 −1.266 99 −14.3325 49 665 
17 000  −14.330 576 136 9 −268.220 503 −0.934 449 97 8.773 713 972 0.263 338 625 −1.265 58 −14.3325 49 811 
  −14.330 576 29(8)      −14.332 550 0(7) 
16 400 27  −14.330 575 973 2  −0.934 453 0     
Be 17 000  −14.331 476 124 8 −268.286 721 −0.908 058 38 8.775 331 042 0.263 382 359 −1.266 01 −14.333 449 712 
  −14.331 476 29(8)      −14.333 449 9(7) 
16 400 27  −14.331 475 953 7  −0.908 060 7     
10 1S 9Be 15 000  −14.329 610 717 9 −268.246 720 −0.935 837 94 8.774 463 692 0.263 372 106 −1.294 41 −14.331 584 771 
16 000  −14.329 613 264 1 −268.246 831 −0.935 827 86 8.774 482 339 0.263 373 330 −1.286 64 −14.331 587 302 
17 000  −14.329 614 314 4 −268.246 463 −0.935 818 78 8.774 494 082 0.263 374 247 −1.278 20 −14.331 588 323 
  −14.329 616 5(11)      −14.331 591(14) 
Be 17 000  −14.330 514 341 4 −268.312 694 −0.909 422 21 8.776 111 519 0.263 417 990 −1.278 64 −14.332 488 262 
  −14.330 516 6(11)      −14.332 490(14) 
10 1P 9Be 15 500  −14.329 252 437 2 −268.231 374 −0.934 976 66 8.773 943 600 0.263 347 709 −1.295 66 −14.331 226 367 
16 400  −14.329 255 100 6 −268.230 280 −0.934 968 26 8.773 960 301 0.263 348 977 −1.281 86 −14.331 228 959 
17 000  −14.329 255 839 5 −268.230 087 −0.934 967 09 8.773 964 094 0.263 349 217 −1.279 28 −14.331 229 684 
  −14.329 256 4(3)      −14.331 230 5(40) 
16 400 27  −14.329 255 009 9  −0.934 974 5     
Be 17 000  −14.330 155 775 7 −268.296 311 −0.908 572 88 8.775 581 319 0.263 392 953 −1.279 72 −14.332 129 533 
  −14.330 156 3(3)      −14.332 130 3(40) 
16 400 27  −14.330 154 912 7  −0.908 577 6     
11 1S 9Be 15 000  −14.328 536 040 2 −268.250 503 −0.935 922 20 8.774 378 167 0.263 364 456 −1.334 33 −14.330 510 390 
16 000  −14.328 547 329 0 −268.251 558 −0.935 940 51 8.774 457 465 0.263 368 146 −1.321 39 −14.330 521 634 
17 000  −14.328 549 768 5 −268.250 488 −0.935 930 94 8.774 469 172 0.263 369 954 −1.311 73 −14.330 524 007 
  −14.328 556 5(34)      −14.330 530 4(32) 
Be 17 000  −14.329 449 730 7 −268.316 720 −0.909 530 06 8.776 086 711 0.263 413 679 −1.312 17 −14.331 423 881 
  −14.329 456 5(34)      −14.331 430 2(32) 
11 1P 9Be 15 000  −14.328 274 730 5 −268.236 832 −0.935 344 41 8.774 047 267 0.263 348 909 −1.320 58 −14.330 248 804 
16 000  −14.328 276 608 0 −268.236 811 −0.935 341 26 8.774 054 107 0.263 349 863 −1.318 40 −14.330 250 672 
17 000  −14.328 287 870 0 −268.236 605 −0.935 320 84 8.774 118 432 0.263 354 974 −1.295 65 −14.330 261 848 
  −14.328 2909(15)      −14.330 265 9(20) 
16 400 27  −14.328 285 422 3  −0.935 344 2     
Be 17 000  −14.329 187 765 9 −268.302 834 −0.908 922 79 8.775 735 832 0.263 398 707 −1.296 08 −14.331 161 656 
  −14.329 190 8(15)      −14.331 164 7(20) 
16 400 27  −14.329 185 241 1  −0.908 9403     
StateIsotopeBasisReferenceEnrH̃MVHOOδ̃(ri)δ̃(rij)P(1/rij3)Etot
1S 9Be 14 000  −14.666 435 525 2 −270.636 666 −0.918 461 59 8.840 617 349 0.267 506 284 −1.222 66 −14.668 440 604 
15 000  −14.666 435 526 1 −270.636 645 −0.918 461 58 8.840 617 352 0.267 506 285 −1.222 43 −14.668 440 604 
16 000  −14.666 435 526 4 −270.636 556 −0.918 461 57 8.840 617 354 0.267 506 286 −1.221 35 −14.668 440 600 
  −14.666 435 526 8(2)      −14.66 844 059 7(18) 
7 000 26  −14.666 435 525 −270.636 610 −0.918 461 58 8.840 617 34 0.267 506 284   
 26  −14.666 435 526(1)       
Be 16 000  −14.667 356 508 4 −270.703 526 −0.891 823 69 8.842 251 649 0.267 550 917 −1.221 78 −14.669 361 498 
  −14.667 356 508 7(2)      −14.669 361 494(18) 
 4  −14.667 356 498(3) −270.703 76(11) −0.891 824 10(8) 8.842 250 65(15) 0.267 550 888(15) −1.221 127 7(5)  
7 000 26  −14.667 356 507 −270.703 579 −0.891 823 62 8.842 251 64 0.267 550 915 −1.222 52  
 26  −14.667 356 508(1)       
1P 9Be 14 600  −14.472 543 756 9 −266.598 045 −0.838 224 67 8.722 853 523 0.261 279 705 −1.184 89 −14.474 498 170 
15 500  −14.472 543 757 7 −266.598 039 −0.838 224 67 8.722 853 527 0.261 279 705 −1.184 83 −14.474 498 170 
16 400  −14.472 543 761 1 −266.598 021 −0.838 224 64 8.722 853 539 0.261 279 707 −1.184 62 −14.474 498 173 
  −14.472 543 764 7(19)      −14.474 498 176(13) 
16 400 27  −14.472 543 759 8  −0.838 224 6     
Be 16 400  −14.473 451 389 5 −266.664 445 −0.812 091 50 8.724 481 450 0.261 324 034 −1.185 06 −14.475 405 724 
  −14.473 451 393 3(19)      −14.475 405 726(13) 
 4  −14.473 451 37(4) −266.665 14(15) −0.812 092 9(3) 8.724 478 65(20) 0.261 323 93(3) −1.182 858(13)  
16 400 27  −14.473 451 388 2  −0.812 091 5     
1S 9Be 14 000  −14.417 335 140 0 −268.474 994 −0.926 539 44 8.780 376 771 0.263 820 372 −1.248 41 −14.419 312 274 
15 000  −14.417 335 142 9 −268.474 893 −0.926 539 39 8.780 376 781 0.263 820 375 −1.247 36 −14.419 312 272 
16 000  −14.417 335 143 5 −268.474 803 −0.926 539 37 8.780 376 790 0.263 820 377 −1.246 34 −14.419 312 269 
  −14.417 335 144 1(6)      −14.419 312 265(19) 
7 000 26  −14.417 335 139 −268.474 926 −0.926 539 41 8.780 376 76 0.263 820 371   
 26  −14.417 335 143(2)       
Be 16 000  −14.418 240 368 2 −268.541 128 −0.900 128 50 8.781 996 527 0.263 864 254 −1.246 77 −14.420 217 407 
  −14.418 240 369 4(6)      −14.420 217 403(19) 
7 000 26  −14.418 240 364 −268.541 251 −0.900 128 21 8.781 996 50 0.263 864 248 −1.248 22  
 26  −14.418 240 368(2)       
1P 9Be 14 600  −14.392 242 880 3 −267.390 569 −0.882 080 43 8.747 804 857 0.262 237 919 −1.227 10 −14.394 206 423 
15 500  −14.392 242 881 8 −267.390 566 −0.882 080 43 8.747 804 867 0.262 237 920 −1.227 07 −14.394 206 424 
16 400  −14.392 242 886 1 −267.390 546 −0.882 080 40 8.747 804 893 0.262 237 922 −1.226 80 −14.394 206 427 
  −14.392 242 890 4(22)      −14.394 206 431(17) 
16 400 27  −14.392 242 884 4  −0.882 080 40     
Be 16 400  −14.393 143 540 2 −267.456 802 −0.855 811 50 8.749 425 262 0.262 281 844 −1.227 23 −14.395 106 998 
  −14.393 143 544 6(22)      −14.395 107 001(17) 
16 400 27  −14.393 143 538 5  −0.855 811 50     
1S 9Be 14 000  −14.369 185 505 1 −268.315 505 −0.932 337 62 8.776 228 493 0.263 512 821 −1.252 93 −14.371  60 487 
15 000  −14.369 185 510 9 −268.315 475 −0.932 337 58 8.776 228 516 0.263 512 824 −1.252 49 −14.371 160 492 
16 000  −14.369 185 512 9 −268.315 411 −0.932 337 56 8.776 228 530 0.263 512 826 −1.251 82 −14.371 160 491 
  −14.369 185 515 1(15)      −14.371 160 493(10) 
7 000 26  −14.369 185 506 −268.315 685 −0.932 337 71 8.776 228 47 0.263 512 813   
 26  −14.369 185 514(4)       
Be 16 000  −14.370 087 936 7 −268.381 663 −0.905 938 87 8.777 846 410 0.263 556 605 −1.252 25 −14.372 062 828 
  −14.370 087 939 7(15)      −14.372 062 830(10) 
7 000 26  −14.370 087 930 −268.381 937 −0.905 938 65 8.777 846 35 0.263 556 591 −1.255 16  
 26  −14.370 087 938(4)       
1P 9Be 14 600  −14.361 037 794 6 −267.893 328 −0.913 549 58 8.763 470 236 0.262 900 115 −1.244 68 −14.363 007 512 
15 500  −14.361 037 796 1 −267.893 322 −0.913 549 56 8.763 470 248 0.262 900 116 −1.244 63 −14.363 007 513 
16 400  −14.361 037 802 2 −267.893 223 −0.913 549 52 8.763 470 279 0.262 900 121 −1.243 49 −14.363 007 515 
  −14.361 037 808 6(36)      −14.363 007 517(9) 
16 400 27  −14.361 037 799 9  −0.913 549 5     
Be 16 400  −14.361 938 402 2 −267.959 411 −0.887 204 01 8.765 087 272 0.262 943 867 −1.243 92 −14.363 908 028 
  −14.361 938 409 4(36)      −14.363 908 030(9) 
16 400 27  −14.361 938 399 8  −0.887 204 0     
1S 9Be 14 000  −14.350 610 411 8 −268.272 968 −0.934 164 35 8.775 161 626 0.263 430 584 −1.254 27 −14.352 584 783 
15 000  −14.350 610 424 7 −268.272 868 −0.934 164 26 8.775 161 679 0.263 430 591 −1.253 07 −14.352 584 791 
16 000  −14.350 610 425 9 −268.272 870 −0.934 164 26 8.775 161 683 0.263 430 592 −1.253 06 −14.352 584 793 
  −14.350 610 427 4(8)      −14.352 584 795(11) 
7 000 26  −14.350 610 414 −268.273 164 −0.934 164 45 8.775 161 59 0.263 430 571 −1.255 16  
 26  −14.350 610 428(7)       
Be 16 000  −14.351 511 733 9 −268.339 105 −0.907 768 57 8.776 779 118 0.263 474 347 −1.253 49 −14.353 486 014 
  −14.351 511 735 4(8)      −14.353 486 015(11) 
7 000 26  −14.351 511 722 −268.339 399 −0.907 768 18 8.776 779 03 0.263 474 325 −1.256 59  
 26  −14.351 511 736(7)       
1P 9Be 14 600  −14.346 975 854 1 −268.071 039 −0.924 932 25 8.769 034 122 0.263 137 388 −1.252 92 −14.348 947 726 
15 500  −14.346 975 856 8 −268.071 032 −0.924 932 23 8.769 034 146 0.263 137 390 −1.252 84 −14.348 947 729 
16 400  −14.346 975 867 2 −268.070 985 −0.924 932 17 8.769 034 198 0.263 137 397 −1.252 35 −14.348 947 737 
  −14.346 975 878 4(56)      −14.348 947 745(42) 
16 400 27  −14.346 975 863 8  −0.924 932 2     
Be 16 400  −14.347 876 298 8 −268.137 180 −0.898 561 42 8.770 650 913 0.263 181 123 −1.252 78 −14.349 848 081 
  −14.347 876 310 0(56)      −14.349 848 090(42) 
16 400 27  −14.347 876 295 3  −0.898 561 4     
1S 9Be 14 000  −14.341 502 883 9 −268.257 684 −0.93492953 8.774 781 459 0.263 400 264 −1.258 15 −14.343 477 035 
15 000  −14.341 502 912 6 −268.257 307 −0.934 929 21 8.774 781 652 0.263 400 285 −1.254 90 −14.343 477 046 
16 000  −14.341 502 922 5 −268.257 223 −0.934 929 13 8.774 781 679 0.263 400 291 −1.254 12 −14.343 477 052 
  −14.341 502 933 1(53)      −14.343 477 060(36) 
Be 16 000  −14.342 403 676 6 −268.323 452 −0.908 534 40 8.776 399 001 0.263 444 038 −1.254 55 −14.344 377 719 
  −14.342 403 691 8(53)      −14.344 377 725(36) 
1P 9Be 14 600  −14.339 569 897 0 −268.147 264 −0.929 839 34 8.771 437 062 0.263 240 458 −1.254 08 −14.341 542 673 
15 500  −14.339 569 904 8 −268.147 262 −0.929 839 27 8.771 371 04 0.263 240 462 −1.254 05 −14.341 542 680 
16 400  −14.339 569 932 7 −268.147 115 −0.929 839 12 8.771 437 280 0.263 240 478 −1.252 23 −14.341 542 702 
  −14.339 569 964(15)      −14.341 542 73(12) 
16 400 27  −14.339 569 924  −0.929 839 3     
Be 16 400  −14.340 470 2036 −268.213 318 −0.903 457 80 8.773 054 048 0.263 284 203 −1.252 66 −14.342 442 886 
  −14.340 470 233(15)      −14.342 442 91(12) 
16 400 27  −14.340 470 194  −0.903 457 8     
1S 9Be 14 000  −14.336 365 900 9 −268.251 719 −0.935 320 56 8.774 622 594 0.263 386 930 −1.265 45 −14.338 339 983 
15 000  −14.336 366 054 2 −268.251 428 −0.935 319 72 8.774 623 708 0.263 387 016 −1.262 35 −14.338 340 121 
16 000  −14.336 366 091 9 −268.251 267 −0.935 319 38 8.774 623 968 0.263 387 047 −1.260 89 −14.338 340 151 
  −14.336 366 132(20)      −14.338 340 18(15) 
Be 16 000  −14.337 266 532 0 −268.317 496 −0.908 924 95 8.776 241 280 0.263 430 793 −1.261 32 −14.339 240 504 
  −14.337 266 572(20)      −14.339 240 54(15) 
1P 9Be 14 600  −14.335 215 466 9 −268.185 790 −0.932 286 61 8.772 641 931 0.263 292 332 −1.258 24 −14.337 188 708 
15 500  −14.335 215 483 6 −268.185 794 −0.932 286 55 8.772 642 041 0.263 292 336 −1.258 21 −14.337 188 725 
16 400  −14.335 215 573 5 −268.185 654 −0.932 286 11 8.772 642 568 0.263 292 395 −1.255 95 −14.337 188 808 
  −14.335 215 667(50)      −14.337 188 89(42) 
16 400 27  −14.335 215 559 2  −0.932 286 5     
Be 16 400  −14.336 115 721 1 −268.251 863 −0.905 899 50 8.774 259 420 0.263 336 122 −1.256 38 −14.338 088 869 
  −14.336 115 820(50)      −14.338 088 95(42) 
16 400 27  −14.336 115 706  −0.905 899 7     
1S 9Be 14 000  −14.333 185 490 7 −268.248 468 −0.935 550 47 8.774 551 560 0.263 380 462 −1.266 85 −14.335 159 515 
15 000  −14.333 185 759 8 −268.247 953 −0.935 549 61 8.774 553 205 0.263 380 607 −1.262 41 −14.335 159 758 
16 000  −14.333 186 005 6 −268.247 983 −0.935 548 81 8.774 554 820 0.263 380 723 −1.260 05 −14.335 160 006 
  −14.333 186 26(16)      −14.335 160 3(15) 
Be 16 000  −14.334 086 251 0 −268.314 213 −0.909 154 49 8.776 172 170 0.263 424 469 −1.260 48 −14.336 060 164 
  −14.334 086 56(16)      −14.336 060 4(15) 
1P 9Be 14 600  −14.332 444 940 2 −268.207 584 −0.933 641 05 8.773 310 065 0.263 321 187 −1.265 04 −14.334 418 458 
15 500  −14.332 445 006 9 −268.207 593 −0.933 640 86 8.773 310 409 0.263 321 197 −1.264 88 −14.334 418 525 
16 400  −14.332 445 316 5 −268.207 487 −0.933 640 32 8.773 312 276 0.263 321 319 −1.261 84 −14.334 418 829 
  −14.332 445 63(20)      −14.334 419 1(17) 
16 400 27  −14.332 445 263 5  −0.933 641 1     
Be 16 400  −14.333 345 372 7 −268.273 702 −0.907 250 64 8.774 929 236 0.263 365 049 −1.262 28 −14.335 318 798 
  −14.333 345 77(20)      −14.335 319 1(17) 
16 400 27  −14.333 345 316 8  −0.907 251 0     
1S 9Be 15 000  −14.331 080 417 2 −268.245 947 −0.935 704 83 8.774 517 184 0.263 376 905 −1.270 61 −14.333 054 364 
16 000  −14.331 080 611 5 −268.246 125 −0.935 704 50 8.774 519 513 0.263 377 024 −1.269 88 −14.333 054 565 
17 000  −14.331 080 817 4 −268.246 315 −0.935 703 87 8.774 521 837 0.263 377 150 −1.269 07 −14.333 054 778 
  −14.331 081 18(18)      −14.333 055 3(28) 
Be 17 000  −14.331 980 933 9 −268.312 545 −0.909 308 87 8.776 139 224 0.263 420 896 −1.26951 −14.333 954 808 
  −14.331 981 33(18)      −14.333 955 1(28) 
1P 9Be 15 500  −14.330 575 490 3 −268.220 833 −0.934 452 69 8.773 710 101 0.263 338 312 −1.271 53 −14.332 549 183 
16 400  −14.330 575 988 0 −268.220 546 −0.934 451 25 8.773 713 099 0.263 338 531 −1.266 99 −14.3325 49 665 
17 000  −14.330 576 136 9 −268.220 503 −0.934 449 97 8.773 713 972 0.263 338 625 −1.265 58 −14.3325 49 811 
  −14.330 576 29(8)      −14.332 550 0(7) 
16 400 27  −14.330 575 973 2  −0.934 453 0     
Be 17 000  −14.331 476 124 8 −268.286 721 −0.908 058 38 8.775 331 042 0.263 382 359 −1.266 01 −14.333 449 712 
  −14.331 476 29(8)      −14.333 449 9(7) 
16 400 27  −14.331 475 953 7  −0.908 060 7     
10 1S 9Be 15 000  −14.329 610 717 9 −268.246 720 −0.935 837 94 8.774 463 692 0.263 372 106 −1.294 41 −14.331 584 771 
16 000  −14.329 613 264 1 −268.246 831 −0.935 827 86 8.774 482 339 0.263 373 330 −1.286 64 −14.331 587 302 
17 000  −14.329 614 314 4 −268.246 463 −0.935 818 78 8.774 494 082 0.263 374 247 −1.278 20 −14.331 588 323 
  −14.329 616 5(11)      −14.331 591(14) 
Be 17 000  −14.330 514 341 4 −268.312 694 −0.909 422 21 8.776 111 519 0.263 417 990 −1.278 64 −14.332 488 262 
  −14.330 516 6(11)      −14.332 490(14) 
10 1P 9Be 15 500  −14.329 252 437 2 −268.231 374 −0.934 976 66 8.773 943 600 0.263 347 709 −1.295 66 −14.331 226 367 
16 400  −14.329 255 100 6 −268.230 280 −0.934 968 26 8.773 960 301 0.263 348 977 −1.281 86 −14.331 228 959 
17 000  −14.329 255 839 5 −268.230 087 −0.934 967 09 8.773 964 094 0.263 349 217 −1.279 28 −14.331 229 684 
  −14.329 256 4(3)      −14.331 230 5(40) 
16 400 27  −14.329 255 009 9  −0.934 974 5     
Be 17 000  −14.330 155 775 7 −268.296 311 −0.908 572 88 8.775 581 319 0.263 392 953 −1.279 72 −14.332 129 533 
  −14.330 156 3(3)      −14.332 130 3(40) 
16 400 27  −14.330 154 912 7  −0.908 577 6     
11 1S 9Be 15 000  −14.328 536 040 2 −268.250 503 −0.935 922 20 8.774 378 167 0.263 364 456 −1.334 33 −14.330 510 390 
16 000  −14.328 547 329 0 −268.251 558 −0.935 940 51 8.774 457 465 0.263 368 146 −1.321 39 −14.330 521 634 
17 000  −14.328 549 768 5 −268.250 488 −0.935 930 94 8.774 469 172 0.263 369 954 −1.311 73 −14.330 524 007 
  −14.328 556 5(34)      −14.330 530 4(32) 
Be 17 000  −14.329 449 730 7 −268.316 720 −0.909 530 06 8.776 086 711 0.263 413 679 −1.312 17 −14.331 423 881 
  −14.329 456 5(34)      −14.331 430 2(32) 
11 1P 9Be 15 000  −14.328 274 730 5 −268.236 832 −0.935 344 41 8.774 047 267 0.263 348 909 −1.320 58 −14.330 248 804 
16 000  −14.328 276 608 0 −268.236 811 −0.935 341 26 8.774 054 107 0.263 349 863 −1.318 40 −14.330 250 672 
17 000  −14.328 287 870 0 −268.236 605 −0.935 320 84 8.774 118 432 0.263 354 974 −1.295 65 −14.330 261 848 
  −14.328 2909(15)      −14.330 265 9(20) 
16 400 27  −14.328 285 422 3  −0.935 344 2     
Be 17 000  −14.329 187 765 9 −268.302 834 −0.908 922 79 8.775 735 832 0.263 398 707 −1.296 08 −14.331 161 656 
  −14.329 190 8(15)      −14.331 164 7(20) 
16 400 27  −14.329 185 241 1  −0.908 9403     

The wave functions calculated for the S and P states for 9Be and Be are used to calculate the expectation values of the operators representing the leading relativistic and QED corrections. The results of the calculations of these quantities are shown in Table 3. The results include the expectation values of the mass-velocity correction, ⟨HMV⟩, the orbit–orbit correction, ⟨HOO⟩, and the one- and two-electron δ-functions [⟨δ(ri)⟩ and ⟨δ(rij)⟩ respectively]. The P(1/rij3) expectation value is also shown. For the 9Be isotope, the results obtained using basis sets with different numbers of ECGs are shown to assess the convergence. For Be, we only show the results obtained with the largest basis sets. The sum of the nonrelativistic energy and the relativistic corrections for each state, Etot, is shown in the last column of Table 3 along with an estimated uncertainty of the result (for more details on uncertainty evaluations, see Ref. 6). Etot values are used to calculate the interstate transition energies.

The SP and PS transition energies for 9Be and Be calculated using Etot values taken from Table 3 are shown in Table 4. The values derived from experimental data are also included for comparison. As can be seen, the FNM effects provide a significant contribution to the transition energies particularly for lower states. The transition energies calculated for 9Be are in very good agreement with the experimental data available in the NIST Atomic Spectra Database (ver. 5.8).101 For the majority of states, the calculated transition-energy values are within the experimental error bars. However, there are states for which the discrepancies are slightly larger. These include, for example, the 9 1P state. The origin of the discrepancies may be related to the way the experimental values were obtained. It is stated in the NIST ASD that the experimental energy levels for n1P(n = 7–11) states listed there were determined by an extrapolation of the values obtained for the Rydberg states with lower n’s using a quantum-defect expansion formula. However, it seems that this formula can only provide sound estimates of the energy levels for states with larger n. Upon a closer look at the transition energies involving state 9 1P, one finds that all experimental and the calculated results differ by about 10 cm−1. Such a consistent difference suggests that there may be a typographical error in the NIST ASD for the 9 1P energy level.

TABLE 4.

Transition energies, ΔEif (in cm−1). The numbers in the parentheses are estimated uncertainties due to the basis set truncation and due to neglecting higher order corrections (both uncertainties are considered independent in our estimates). The experimental values, ΔEifExp, have been taken from NIST Atomic Spectra Database (ver. 5.8).101 It should be noted that the experimental energy levels for n1P(n = 7–11) states in NIST ASD were determined by extrapolation of values with lower n using a quantum-defect expansion formula along the Rydberg series

TransitionΔEifBeΔEif9BeΔEifExp
1S → 2 1P 42 568.372(9) 42 565.443(9) 42 565.4502(10) 
Ref. 4   42 565.441(11)  
1S → 3 1P 60 191.905(7) 60 187.444(7) 60 187.443(21) 
1S → 4 1P 67 039.288(6) 67 034.814(6) 67 034.80(3) 
1S → 5 1P 70 125.089(6) 70 120.578(6) 70 120.59(3) 
1S → 6 1P 71 750.342(6) 71 745.796(6) 71 746.17(6) 
1S → 7 1P 72 705.938(11) 72 701.365(11) 72 701.8(5) 
1S → 8 1P 73 313.90(3) 73 309.31(3) 73 309.7(5) 
1S → 9 1P 73 724.12(2) 73 719.51(2) 73 709.4(5) 
1S → 10 1P 74 013.86(8) 74 009.24(8) 74 009.2(5) 
1S → 11 1P 74 226.3(2) 74 221.7(2) 74 221.1(5) 
1P → 3 1S 12 112.435(4) 12 111.906(4) 12 111.898(21) 
1P → 4 1S 22 681.144(4) 22 680.000(4) 22 679.986(21) 
1P → 5 1S 26 758.283(4) 26 756.894(4) 26 756.84(7) 
1P → 6 1S 28 757.323(4) 28 755.812(4) 28 755.79(8) 
1P → 7 1S 29 884.811(5) 29 883.232(5) 29 882.92(18) 
1P → 8 1S 30 582.82(3) 30 581.19(3) 30 581.21(19) 
1P → 9 1S 31 044.89(2) 31 043.24(2) 31 043.1(3) 
1P → 10 1S 31 366.767 31 365.09(11) 31 365.0(3) 
1P → 11 1S 31 600.4(3) 31 598.7(3) 31 598.0(3) 
1S → 3 1P 5 511.098(3) 5 510.095(3) 5 510.094(30) 
1S → 4 1P 12 358.480(1) 12 357.465(1) 12 357.45(4) 
1S → 5 1P 15 444.282(2) 15 443.230(2) 15 443.24(4) 
1S → 6 1P 17 069.534(3) 17 068.447(3) 17 068.82(6) 
1S → 7 1P 18 025.131(10) 18 024.016(10) 18 024.4(5) 
1S → 8 1P 18 633.09(3) 18 631.96(3) 18 632.3(5) 
1S → 9 1P 19 043.31(2) 19 042.16(2) 19 032.0(5) 
1S → 10 1P 19 333.05(8) 19 331.89(8) 19 331.8(5) 
1S → 11 1P 19 545.5(2) 19 544.3(2) 19 543.7(5) 
1P → 4 1S 5 057.611(2) 5 057.998(2) 5 057.993(30) 
1P → 5 1S 9 134.750(2) 9 134.893(2) 9 134.856(73) 
1P → 6 1S 11 133.790(2) 11 133.811(2) 11 133.80(8) 
1P → 7 1S 12 261.278(4) 12 261.230(4) 12 260.93(18) 
1P → 8 1S 12 959.28(3) 12 959.19(3) 12 959.2(2) 
1P → 9 1S 13 421.35(2) 13 421.24(2) 13 421.1(3) 
1P → 10 1S 13 743.22(11) 13 743.09(11) 13 743.0(3) 
1P → 11 1S 13 976.8(3) 13 976.7(3) 13 976.0(3) 
1S → 4 1P 1 789.771 6(10) 1 789.371 4(10) 1 789.36(37) 
1S → 5 1P 4 875.573(1) 4 875.136(1) 4 875.15(37) 
1S → 6 1P 6 500.826(2) 6 500.353(2) 6 500.73(64) 
1S → 7 1P 7 456.422(9) 7 455.923(9) 7 456.3(5) 
1S → 8 1P 8 064.38(3) 8 063.86(3) 8 064.2(5) 
1S → 9 1P 8 474.60(2) 8 474.06(2) 8 463.9(5) 
1S → 10 1P 8 764.35(8) 8 763.80(8) 8 763.7(5) 
1S → 11 1P 8 976.8(2) 8 976.2(2) 8 975.6(5) 
1P → 5 1S 2 287.3679(9) 2 287.5231(9) 2 287.50(76) 
1P → 6 1S 4 286.4074(10) 4 286.4411(10) 4 286.45(85) 
1P → 7 1S 5 413.896(3) 5 413.861(3) 5 413.58(18) 
1P → 8 1S 6 111.90(3) 6 111.82(3) 6 111.87(19) 
1P → 9 1S 6 573.97(2) 6 573.87(2) 6 573.8(3) 
1P → 10 1S 6 895.84(11) 6 895.72(11) 6 895.7(3) 
1P → 11 1S 7 129.4(3) 7 129.3(3) 7 128.7(3) 
1S → 5 1P 798.4338(9) 798.2415(9) 798.29(76) 
1S → 6 1P 2 423.686(2) 2 423.459(2) 2 423.87(92) 
1S → 7 1P 3 379.283(9) 3 379.028(9) 3 379.5(5) 
1S → 8 1P 3 987.24(3) 3 986.97(3) 3 987.4(5) 
1S → 9 1P 4 397.46(2) 4 397.17(2) 4 387.1(5) 
1S → 10 1P 4 687.21(8) 4 686.90(8) 4 686.9(5) 
1S → 11 1P 4 899.630(2) 4 899.3(2) 4 898.8(5) 
1P → 6 1S 1 200.6057(5) 1 200.6764(5) 1 200.66(76) 
1P → 7 1S 2 328.094(2) 2 328.096(2) 2 327.79(18) 
1P → 8 1S 3 026.10(3) 3 026.06(3) 3 026.0(2) 
1P → 9 1S 3 488.17(2) 3 488.10(2) 3 488.01(3) 
1P → 10 1S 3 810.04(11) 3 809.95(11) 3 809.91(3) 
1P → 11 1S 4 043.6(3) 4 043.5(3) 4 042.91(3) 
1S → 6 1P 424.647(2) 424.541(2) 424.92(10) 
1S → 7 1P 1 380.243(9) 1 380.110(9) 1 380.5(5) 
1S → 8 1P 1 988.20(3) 1 988.05(3) 1988.4(5) 
1S → 9 1P 2 398.42(2) 2 398.25(2) 2 388.1(5) 
1S → 10 1P 2 688.17(8) 2 687.99(8) 2 687.9(5) 
1S → 11 1P 2 900.6(2) 2 900.4(2) 2 899.8(5) 
1P → 7 1S 702.8414(10) 702.8786(10) 702.21(19) 
1P → 8 1S 1 400.85(2) 1 400.84(2) 1 400.5(20) 
1P → 9 1S 1 862.92(2) 1 862.88(2) 1 862.4(3) 
1P → 10 1S 2 184.79(11) 2 184.73(11) 2 184.3(3) 
1P → 11 1S 2 418.4(3) 2 418.3(3) 2 417.3(3) 
1S → 7 1P 252.755(6) 252.691(6) 253.4(5) 
1S → 8 1P 860.72(3) 860.63(3) 861.3(5) 
1S → 9 1P 1 270.93(2) 1 270.83(2) 1 261.0(5) 
1S → 10 1P 1 560.68(8) 1 560.57(8) 1 560.8(5) 
1S → 11 1P 1 773.1(2) 1 773.0(2) 1 772.7(5) 
1P → 8 1S 445.25(2) 445.27(2) 444.87(5) 
1P → 9 1S 907.32(2) 907.31(2) 906.8(6) 
1P → 10 1S 1 229.19(11) 1 229.16(11) 1 228.7(6) 
1P → 11 1S 1 462.8(3) 1 462.8(3) 1 461.7(6) 
1S → 8 1P 162.711(6) 162.669(6) 163.0(5) 
1S → 9 1P 572.93(2) 572.87(2) 562.7(5) 
1S → 10 1P 862.67(8) 862.61(8) 862.5(5) 
1S → 11 1P 1 075.1(2) 1 075.0(2) 1 074.4(5) 
1P → 9 1S 299.36(2) 299.37(2) 298.9(6) 
1P → 10 1S 621.23(11) 621.22(11) 620.8(6) 
1P → 11 1S 854.8(3) 854.8(3) 853.8(6) 
1S → 9 1P 110.856(7) 110.827(7) 100.8(6) 
1S → 10 1P 400.60(6) 400.56(6) 400.6(6) 
1S → 11 1P 613.0(2) 613.0(2) 612.5(6) 
1P → 10 1S 211.01(10) 211.02(10) 221.1(6) 
1P → 11 1S 444.62(2) 445.13(2) 454.1(6) 
10 1S → 10 1P 78.73(3) 78.71(3) 78.7(6) 
10 1S → 11 1P 291.2(2) 291.1(2) 290.6(6) 
10 1P → 11 1S 154.9(2) 154.9(2) 154.3(6) 
11 1S → 11 1P 57.6(2) 57.5(2) 57.6(6) 
TransitionΔEifBeΔEif9BeΔEifExp
1S → 2 1P 42 568.372(9) 42 565.443(9) 42 565.4502(10) 
Ref. 4   42 565.441(11)  
1S → 3 1P 60 191.905(7) 60 187.444(7) 60 187.443(21) 
1S → 4 1P 67 039.288(6) 67 034.814(6) 67 034.80(3) 
1S → 5 1P 70 125.089(6) 70 120.578(6) 70 120.59(3) 
1S → 6 1P 71 750.342(6) 71 745.796(6) 71 746.17(6) 
1S → 7 1P 72 705.938(11) 72 701.365(11) 72 701.8(5) 
1S → 8 1P 73 313.90(3) 73 309.31(3) 73 309.7(5) 
1S → 9 1P 73 724.12(2) 73 719.51(2) 73 709.4(5) 
1S → 10 1P 74 013.86(8) 74 009.24(8) 74 009.2(5) 
1S → 11 1P 74 226.3(2) 74 221.7(2) 74 221.1(5) 
1P → 3 1S 12 112.435(4) 12 111.906(4) 12 111.898(21) 
1P → 4 1S 22 681.144(4) 22 680.000(4) 22 679.986(21) 
1P → 5 1S 26 758.283(4) 26 756.894(4) 26 756.84(7) 
1P → 6 1S 28 757.323(4) 28 755.812(4) 28 755.79(8) 
1P → 7 1S 29 884.811(5) 29 883.232(5) 29 882.92(18) 
1P → 8 1S 30 582.82(3) 30 581.19(3) 30 581.21(19) 
1P → 9 1S 31 044.89(2) 31 043.24(2) 31 043.1(3) 
1P → 10 1S 31 366.767 31 365.09(11) 31 365.0(3) 
1P → 11 1S 31 600.4(3) 31 598.7(3) 31 598.0(3) 
1S → 3 1P 5 511.098(3) 5 510.095(3) 5 510.094(30) 
1S → 4 1P 12 358.480(1) 12 357.465(1) 12 357.45(4) 
1S → 5 1P 15 444.282(2) 15 443.230(2) 15 443.24(4) 
1S → 6 1P 17 069.534(3) 17 068.447(3) 17 068.82(6) 
1S → 7 1P 18 025.131(10) 18 024.016(10) 18 024.4(5) 
1S → 8 1P 18 633.09(3) 18 631.96(3) 18 632.3(5) 
1S → 9 1P 19 043.31(2) 19 042.16(2) 19 032.0(5) 
1S → 10 1P 19 333.05(8) 19 331.89(8) 19 331.8(5) 
1S → 11 1P 19 545.5(2) 19 544.3(2) 19 543.7(5) 
1P → 4 1S 5 057.611(2) 5 057.998(2) 5 057.993(30) 
1P → 5 1S 9 134.750(2) 9 134.893(2) 9 134.856(73) 
1P → 6 1S 11 133.790(2) 11 133.811(2) 11 133.80(8) 
1P → 7 1S 12 261.278(4) 12 261.230(4) 12 260.93(18) 
1P → 8 1S 12 959.28(3) 12 959.19(3) 12 959.2(2) 
1P → 9 1S 13 421.35(2) 13 421.24(2) 13 421.1(3) 
1P → 10 1S 13 743.22(11) 13 743.09(11) 13 743.0(3) 
1P → 11 1S 13 976.8(3) 13 976.7(3) 13 976.0(3) 
1S → 4 1P 1 789.771 6(10) 1 789.371 4(10) 1 789.36(37) 
1S → 5 1P 4 875.573(1) 4 875.136(1) 4 875.15(37) 
1S → 6 1P 6 500.826(2) 6 500.353(2) 6 500.73(64) 
1S → 7 1P 7 456.422(9) 7 455.923(9) 7 456.3(5) 
1S → 8 1P 8 064.38(3) 8 063.86(3) 8 064.2(5) 
1S → 9 1P 8 474.60(2) 8 474.06(2) 8 463.9(5) 
1S → 10 1P 8 764.35(8) 8 763.80(8) 8 763.7(5) 
1S → 11 1P 8 976.8(2) 8 976.2(2) 8 975.6(5) 
1P → 5 1S 2 287.3679(9) 2 287.5231(9) 2 287.50(76) 
1P → 6 1S 4 286.4074(10) 4 286.4411(10) 4 286.45(85) 
1P → 7 1S 5 413.896(3) 5 413.861(3) 5 413.58(18) 
1P → 8 1S 6 111.90(3) 6 111.82(3) 6 111.87(19) 
1P → 9 1S 6 573.97(2) 6 573.87(2) 6 573.8(3) 
1P → 10 1S 6 895.84(11) 6 895.72(11) 6 895.7(3) 
1P → 11 1S 7 129.4(3) 7 129.3(3) 7 128.7(3) 
1S → 5 1P 798.4338(9) 798.2415(9) 798.29(76) 
1S → 6 1P 2 423.686(2) 2 423.459(2) 2 423.87(92) 
1S → 7 1P 3 379.283(9) 3 379.028(9) 3 379.5(5) 
1S → 8 1P 3 987.24(3) 3 986.97(3) 3 987.4(5) 
1S → 9 1P 4 397.46(2) 4 397.17(2) 4 387.1(5) 
1S → 10 1P 4 687.21(8) 4 686.90(8) 4 686.9(5) 
1S → 11 1P 4 899.630(2) 4 899.3(2) 4 898.8(5) 
1P → 6 1S 1 200.6057(5) 1 200.6764(5) 1 200.66(76) 
1P → 7 1S 2 328.094(2) 2 328.096(2) 2 327.79(18) 
1P → 8 1S 3 026.10(3) 3 026.06(3) 3 026.0(2) 
1P → 9 1S 3 488.17(2) 3 488.10(2) 3 488.01(3) 
1P → 10 1S 3 810.04(11) 3 809.95(11) 3 809.91(3) 
1P → 11 1S 4 043.6(3) 4 043.5(3) 4 042.91(3) 
1S → 6 1P 424.647(2) 424.541(2) 424.92(10) 
1S → 7 1P 1 380.243(9) 1 380.110(9) 1 380.5(5) 
1S → 8 1P 1 988.20(3) 1 988.05(3) 1988.4(5) 
1S → 9 1P 2 398.42(2) 2 398.25(2) 2 388.1(5) 
1S → 10 1P 2 688.17(8) 2 687.99(8) 2 687.9(5) 
1S → 11 1P 2 900.6(2) 2 900.4(2) 2 899.8(5) 
1P → 7 1S 702.8414(10) 702.8786(10) 702.21(19) 
1P → 8 1S 1 400.85(2) 1 400.84(2) 1 400.5(20) 
1P → 9 1S 1 862.92(2) 1 862.88(2) 1 862.4(3) 
1P → 10 1S 2 184.79(11) 2 184.73(11) 2 184.3(3) 
1P → 11 1S 2 418.4(3) 2 418.3(3) 2 417.3(3) 
1S → 7 1P 252.755(6) 252.691(6) 253.4(5) 
1S → 8 1P 860.72(3) 860.63(3) 861.3(5) 
1S → 9 1P 1 270.93(2) 1 270.83(2) 1 261.0(5) 
1S → 10 1P 1 560.68(8) 1 560.57(8) 1 560.8(5) 
1S → 11 1P 1 773.1(2) 1 773.0(2) 1 772.7(5) 
1P → 8 1S 445.25(2) 445.27(2) 444.87(5) 
1P → 9 1S 907.32(2) 907.31(2) 906.8(6) 
1P → 10 1S 1 229.19(11) 1 229.16(11) 1 228.7(6) 
1P → 11 1S 1 462.8(3) 1 462.8(3) 1 461.7(6) 
1S → 8 1P 162.711(6) 162.669(6) 163.0(5) 
1S → 9 1P 572.93(2) 572.87(2) 562.7(5) 
1S → 10 1P 862.67(8) 862.61(8) 862.5(5) 
1S → 11 1P 1 075.1(2) 1 075.0(2) 1 074.4(5) 
1P → 9 1S 299.36(2) 299.37(2) 298.9(6) 
1P → 10 1S 621.23(11) 621.22(11) 620.8(6) 
1P → 11 1S 854.8(3) 854.8(3) 853.8(6) 
1S → 9 1P 110.856(7) 110.827(7) 100.8(6) 
1S → 10 1P 400.60(6) 400.56(6) 400.6(6) 
1S → 11 1P 613.0(2) 613.0(2) 612.5(6) 
1P → 10 1S 211.01(10) 211.02(10) 221.1(6) 
1P → 11 1S 444.62(2) 445.13(2) 454.1(6) 
10 1S → 10 1P 78.73(3) 78.71(3) 78.7(6) 
10 1S → 11 1P 291.2(2) 291.1(2) 290.6(6) 
10 1P → 11 1S 154.9(2) 154.9(2) 154.3(6) 
11 1S → 11 1P 57.6(2) 57.5(2) 57.6(6) 

In Table 5, we show the calculated values of the transition dipole moments and the oscillator strengths for 9Be and Be for the SP and PS transitions involving all states considered in this work. The oscillator strengths are compared with available literature results. The oscillator strengths for all transitions considered in this work are shown in Fig. 1 in the form of a map that depicts their relative magnitude on a logarithmic scale.

TABLE 5.

The squares of the transition dipole moments, |μif|2, and oscillator strengths, fif, between 1S and 1P states. For |μif|2, the numbers in parentheses are estimated uncertainties due to the basis truncation. The oscillator strength uncertainties are taken as root mean squares of the uncertainties of |μif|2 and ΔEif. Numbers in square brackets, [s], denote a multiplication factor 10s

Transition|μif|2Be|μif|29BefifBefif9BefifBeafifBebfifBec
1S → 2 1P 1.063 146 385(10) [+1] 1.063 255 365(10) [+1] 1.374 689 2(3) [0] 1.374 400 8(3) [0] 1.375 [0] 1.38 [0] 1.380 [0] 
1S → 3 1P 4.703 28(2) [−2] 4.695 34(2) [−2] 8.599 31(4) [−3] 8.582 06(4) [−3] 9.01 [−3] 8.98 [−3] 8.985 [−3] 
1S → 4 1P 1.319 61(4) [−3] 1.329 66(4) [−3] 2.687 20(9) [−4] 2.706 82(9) [−4] 2.30 [−4] 1.20 [−4]  
1S → 5 1P 4.026 52(7) [−3] 4.038 60(7) [−3] 8.576 84(14) [−4] 8.599 94(14) [−4] 8.10 [−4] 6.90 [−4]  
1S → 6 1P 3.622 30(13) [−3] 3.630 70(13) [−3] 7.894 7(3) [−4] 7.910 5(3) [−4] 7.50 [−4] 6.80 [−4]  
1S → 7 1P 2.760 62(14) [−3] 2.766 31(14) [−3] 6.096 8(3) [−4] 6.107 5(3) [−4] 5.90 [−4] 5.40 [−4]  
1S → 9 1P 1.524 52(2) [−3] 1.527 54(2) [−3] 3.414 03(4) [−4] 3.41974(4) [−4] 3.20 [−4] 3.07 [−4]  
1S → 10 1P 1.154(2) [−3] 1.156(2) [−3] 2.594(5) [−4] 2.599(5) [−4]    
1S → 11 1P 8.90(7) [−4] 8.92(7) [−4] 2.01(2) [−4] 2.01(2) [−4]    
1P → 3 1S 9.613 578 7(15) [0] 9.613 186 1(15) [0] 1.179 016 1(5) [−1] 1.178 629 3(5) [−1] 1.18 [−1] 1.15 [−1] 1.147 [−1] 
1P → 4 1S 4.280 48(6) [−1] 4.280 54(6) [−1] 9.830 15(14) [−3] 9.827 42(14) [−3] 9.82 [−3] 9.80 [−3]  
1P → 5 1S 1.340 7(2) [−1] 1.340 8(2) [−1] 3.632 4(6) [−3] 3.631 7(6) [−3] 3.62 [−3] 3.55 [−3]  
1P → 6 1S 6.776 3(2) [−2] 6.777 9(2) [−2] 1.973 1(5) [−3] 1.972 9(5) [−3] 1.96 [−3] 1.88 [−3]  
1P → 7 1S 4.223 04(9) [−2] 4.224 96(9) [−2] 1.277 85(3) [−3] 1.278 05(3) [−3] 1.23 [−3] 1.18 [−3]  
1P → 8 1S 2.943 73(5) [−2] 2.945 90(5) [−2] 9.115 45(15) [−4] 9.119 48(15) [−4] 9.08 [−4] 8.11 [−4]  
1P → 9 1S 2.193 74(2) [−2] 2.196 04(2) [−2] 6.895 71(8) [−4] 6.900 87(8) [−4] 6.94 [−4] 5.92 [−4]  
1P → 10 1S 1.706(2) [−2] 1.709(2) [−2] 5.420(7) [−4] 5.425(7) [−4] 5.62 [−4]   
1P → 11 1S 1.370(7) [−2] 1.372(7) [−2] 4.38(2) [−4] 4.39(2) [−4]    
1S → 3 1P 5.713 342 8(3) [+1] 5.713 911 1(3) [+1] 9.564 292(5) [−1] 9.561 174(5) [−1] 9.58 [−1] 9.57 [−1] 9.465 [−1] 
1S → 4 1P 2.762 51(5) [−1] 2.769 26(5) [−1] 1.037 04(2) [−2] 1.039 23(2) [−2] 9.82 [−2] 9.80 [−2]  
1S → 5 1P 3.153 21(2) [−1] 3.157 18(2) [−1] 1.479 26(8) [−2] 1.480 66(8) [−2] 1.45 [−2] 1.45 [−2]  
1S → 6 1P 2.103 53(5) [−1] 2.105 71(5) [−1] 1.090 67(2) [−2] 1.091 47(2) [−2] 1.07 [−2] 1.08 [−2]  
1S → 7 1P 1.389 96(2) [−1] 1.391 27(2) [−1] 7.610 34(14) [−3] 7.615 22(14) [−3] 7.51 [−3] 7.50 [−3]  
1S → 8 1P 9.503 6(5) [−2] 9.512 6(5) [−2] 5.379 0(3) [−3] 5.382 4(3) [−3] 5.29 [−3] 4.73 [−3]  
1S → 9 1P 6.738 1(14) [−2] 6.744 9(14) [−2] 3.897 7(8) [−3] 3.900 4(8) [−3] 3.79 [−3] 3.87 [−3]  
1S → 10 1P 4.934(10) [−2] 4.940(10) [−2] 2.897(6) [−3] 2.900(6) [−3]    
1S → 11 1P 3.72(3) [−2] 3.72(3) [−2] 2.21(2) [−3] 2.21(2) [−3]    
1P → 4 1S 4.060 230(3) [+1] 4.059 474(3) [+1] 2.079 214(2) [−1] 2.078 480(2) [−1] 2.09 [−1] 2.12 [−1]  
1P → 5 1S 2.630 541(4) [0] 2.630 732(4) [0] 2.433 016(3) [−2] 2.432 638(3) [−2] 2.43 [−2] 2.44 [−2]  
1P → 6 1S 7.612 7(7) [−1] 7.614 0(7) [−1] 8.582 0(7) [−3] 8.581 3(7) [−3] 8.57 [−3] 8.59 [−3]  
1P → 7 1S 3.407 8(5) [−1] 3.408 7(5) [−1] 4.230 8(6) [−3] 4.230 8(6) [−3] 4.23 [−3] 4.22 [−3]  
1P → 8 1S 1.874 3(3) [−1] 1.875 0(3) [−1] 2.459 3(4) [−3] 2.459 6(4) [−3] 2.47 [−3] 2.44 [−3]  
1P → 9 1S 1.161 85(10) [−1] 1.162 45(10) [−1] 1.578 89(13) [−3] 1.579 30(13) [−3] 1.60 [−3] 1.56 [−3]  
1P → 10 1S 7.791(9) [−2] 7.796(9) [−2] 1.084(13) [−3] 1.085(13) [−3] 1.14 [−3]   
1P → 11 1S 5.53(2) [−2] 5.54(2) [−2] 7.83(3) [−4] 7.83(3) [−4]    
1S → 4 1P 2.631 297 0(6) [+2] 2.631 708 6(6) [+2] 1.430 513 2(9) [0] 1.430 068 9(9) [0] 1.43 [0]   
1S → 5 1P 1.230 43(14) [−1] 1.237 64(14) [−1] 1.822 24(20) [−3] 1.832 31(20) [−3] 1.59 [−3] 1.67 [−3]  
1S → 6 1P 2.616 7(2) [−1] 2.622 2(2) [−1] 5.167 1(4) [−3] 5.176 4(4) [−3] 4.95 [−3] 5.10 [−3]  
1S → 7 1P 1.857 1(8) [−1] 1.860 0(8) [−1] 4.206(2) [−3] 4.212(2) [−3] 4.08 [−3] 4.17 [−3]  
1S → 8 1P 1.256 3(2) [−1] 1.258 0(2) [−1] 3.077 4(5) [−3] 3.080 8(5) [−3] 2.98 [−3] 2.71 [−3]  
1S → 9 1P 8.726(2) [−2] 8.738(2) [−2] 2.246(5) [−3] 2.249(5) [−3] 2.16 [−3] 2.23 [−3]  
1S → 10 1P 6.269(6) [−2] 6.278(6) [−2] 1.669(2) [−3] 1.671(2) [−3]    
1S → 11 1P 4.65(3) [−2] 4.66(3) [−2] 1.27(9) [−3] 1.27(9) [−3]    
1P → 5 1S 1.266 409(2) [+2] 1.266 156(2) [+2] 2.933 007(4) [−1] 2.931 905(4) [−1] 2.95 [−1]   
1P → 6 1S 8.287 77(7) [0] 8.288 28(7) [0] 3.596 95(3) [−2] 3.596 32(3) [−2] 3.60 [−3] 3.61 [−3]  
1P → 7 1S 2.312 0(5) [0] 2.312 4(5) [0] 1.267 4(3) [−2] 1.267 2(3) [−2] 1.27 [−2] 1.27 [−2]  
1P → 8 1S 1.009 2(2) [0] 1.009 5(2) [0] 6.245 3(11) [−3] 6.245 4(11) [−3] 6.27 [−3] 6.25 [−3]  
1P → 9 1S 5.459 4(6) [−1] 5.461 7(6) [−1] 3.634 0(4) [−3] 3.634 5(4) [−3] 3.69 [−3] 3.63 [−3]  
1P → 10 1S 3.348(3) [−1] 3.351(3) [−1] 2.338(2) [−3] 2.339(2) [−3] 2.45 [−3]   
1P → 11 1S 2.232(8) [−1] 2.234(8) [−1] 1.611(6) [−3] 1.612(6) [−3]    
1S → 5 1P 7.481 658(8) [+2] 7.482 926(8) [+2] 1.814 518(3) [0] 1.813 946(3) [0] 1.82 [0]   
1S → 6 1P 8.401(5) [−2] 8.496(5) [−2] 6.185(3) [−4] 6.253(3) [−4] 4.78 [−4]   
1S → 7 1P 3.272(4) [−1] 3.281(4) [−1] 3.359(4) [−3] 3.367(4) [−3] 3.16 [−3] 3.32 [−3]  
1S → 8 1P 2.420 7(9) [−1] 2.424 8(9) [−1] 2.931 8(11) [−3] 2.935 9(11) [−3] 2.79 [−3] 2.59 [−3]  
1S → 9 1P 1.673(3) [−1] 1.676(3) [−1] 2.235(4) [−3] 2.237(4) [−3] 2.10 [−3] 2.22 [−3]  
1S → 10 1P 1.179(2) [−1] 1.181(2) [−1] 1.679(3) [−3] 1.681(3) [−3]    
1S → 11 1P 8.57(5) [−2] 8.59(5) [−2] 1.28(7) [−3] 1.28(7) [−3]    
1P → 6 1S 3.157 64(2) [+2] 3.157 04(2) [+2] 3.838 54(3) [−1] 3.837 10(3) [−1] 3.86 [−1]   
1P → 7 1S 2.031 3(2) [+1] 2.031 5(2) [+1] 4.788 4(6) [−2] 4.787 6(6) [−2] 4.79 [−2]   
1P → 8 1S 5.498 6(8) [0] 5.499 8(8) [0] 1.684 8(3) [−2] 1.684 7(3) [−2] 1.69 [−2]   
1P → 9 1S 2.350(2) [0] 2.351(2) [0] 8.299(9) [−3] 8.300(9) [−3] 8.45 [−2] 8.32 [−3]  
1P → 10 1S 1.257(3) [0] 1.257(3) [0] 4.848(12) [−3] 4.849(12) [−3] 5.10 [−2]   
1P → 11 1S 7.66(3) [−1] 7.67(3) [−1] 3.14(13) [−3] 3.14(13) [−3]    
1S → 6 1P 1.691 681(10) [+3] 1.691 971(10) [+3] 2.182 081(16) [0] 2.181 379(16) [0] 2.19 [ 0]   
1S → 7 1P 4.85(2) [−2] 4.96(2) [−2] 2.03(9) [−4] 2.08(9) [−4] 1.12 [−4]   
1S → 8 1P 4.238(8) [−1] 4.251(8) [−1] 2.560(5) [−3] 2.567(5) [−3] 2.30 [−3]   
1S → 9 1P 3.257(13) [−1] 3.261(13) [−1] 2.373(9) [−3] 2.375(9) [−3] 2.12 [−3]   
1S → 10 1P 2.272(6) [−1] 2.276(6) [−1] 1.855(5) [−3] 1.858(5) [−3]    
1S → 11 1P 1.606(4) [−1] 1.608(4) [−1] 1.415(3) [−3] 1.417(3) [−3]    
1P → 7 1S 6.706 9(4) [+2] 6.705 8(4) [+2] 4.772 9(3) [−1] 4.771 2(3) [−1] 4.79 [−1]   
1P → 8 1S 4.231(2) [+1] 4.231(2) [+1] 6.001(2) [−2] 6.000(2) [−2] 6.03 [−2]   
1P → 9 1S 1.119 4(14) [+1] 1.119 7(14) [+1] 2.111 4(26) [−2] 2.111 4(26) [−2] 2.15 [−2]   
1P → 10 1S 4.705(3) [0] 4.708(3) [0] 1.041(7) [−2] 1.041(7) [−2] 1.10 [−2]   
1P → 11 1S 2.485(6) [0] 2.487(6) [0] 6.084(16) [−3] 6.088(16) [−3]    
1S → 7 1P 3.316 5(2) [+3] 3.317 0(2) [+3] 2.546 2(2) [0] 2.545 4(2) [0] 2.56 [0]   
1S → 8 1P 1.307(7) [−2] 1.387(7) [−2] 3.418(19) [−5] 3.624(19) [−5] 5.00 [−8] 5.00 [−8]  
1S → 9 1P 5.380(8) [−1] 5.384(8) [−1] 2.077(3) [−3] 2.078(3) [−3] 1.64 [−3]   
1S → 10 1P 4.31(3) [−1] 4.30(3) [−1] 2.04(1) [−3] 2.04(1) [−3]    
1S → 11 1P 3.00(6) [−1] 3.00(6) [−1] 1.62(3) [−3] 1.62(3) [−3]    
1P → 8 1S 1.269 9(3) [+3] 1.269 7(3) [+3] 5.725 2(13) [−1] 5.723 2(13) [−1] 5.75 [−1]   
1P → 9 1S 7.869(7) [+1] 7.869(7) [+1] 7.229(6) [−2] 7.228(6) [−2] 7.33 [−2]   
1P → 10 1S 2.045(5) [+1] 2.046(5) [+1] 2.546(6) [−2] 2.546(6) [−2] 2.68 [−2]   
1P → 11 1S 8.46(4) [0] 8.47(4) [0] 1.25(5) [−2] 1.25(5) [−2]    
1S → 8 1P 5.888 2(12) [+3] 5.888 9(12) [+3] 2.910 2(6) [0] 2.909 1(6) [0] 2.94 [0]   
1S → 9 1P 1.6(4) [−3] 1.3(4) [−3] 2.8(8) [−6] 2.3(8) [−6] 1.47 [−6]   
1S → 10 1P 6.59(4) [−1] 6.57(4) [−1] 1.73(1) [−3] 1.72(1) [−3]    
1S → 11 1P 5.42(14) [−1] 5.41(14) [−1] 1.77(5) [−3] 1.76(5) [−3]    
1P → 9 1S 2.207 0(8) [+3] 2.206 5(8) [+3] 6.689 5(24) [−1] 6.686 9(24) [−1] 6.73 [−2]   
1P → 10 1S 1.349(7) [+2] 1.349(7) [+2] 8.49(4) [−2] 8.48(4) [−2] 8.82 [−2]   
1P → 11 1S 3.47(5) [+1] 3.47(5) [+1] 3.00(5) [−2] 3.00(5) [−2]    
1S → 9 1P 9.719 9(11) [+3] 9.720 7(11) [+3] 3.273 0(4) [0] 3.271 6(4) [0] 3.34 [0]   
1S → 10 1P 5.(5) [−2] 5.(5) [−2] 6.(6) [−5] 6.(6) [−5]    
1S → 11 1P 8.(2) [−1] 8.(2) [−1] 1.(3) [−3] 1.(3) [−3]    
1P → 10 1S 3.583(13) [+3] 3.582(13) [+3] 7.656(27) [−1] 7.652(27) [−1] 7.80 [−1]   
1P → 11 1S 2.19(6) [+2] 2.19(6) [+2] 9.84(26) [−2] 9.85(26) [−2]    
10 1S → 10 1P 1.519(5) [+4] 1.519(5) [+4] 3.632(12) [0] 3.630(12) [0]    
10 1S → 11 1P 1.(2) [−1] 1.(2) [−1] 1.(2) [−4] 1.(2) [−4]    
10 1P → 11 1S 5.50(7) [+3] 5.49(7) [+3] 8.62(11) [−1] 8.61(11) [−1]    
11 1S → 11 1P 2.28(4) [+4] 2.28(4) [+4] 3.98(7) [0] 3.98(7) [0]    
Transition|μif|2Be|μif|29BefifBefif9BefifBeafifBebfifBec
1S → 2 1P 1.063 146 385(10) [+1] 1.063 255 365(10) [+1] 1.374 689 2(3) [0] 1.374 400 8(3) [0] 1.375 [0] 1.38 [0] 1.380 [0] 
1S → 3 1P 4.703 28(2) [−2] 4.695 34(2) [−2] 8.599 31(4) [−3] 8.582 06(4) [−3] 9.01 [−3] 8.98 [−3] 8.985 [−3] 
1S → 4 1P 1.319 61(4) [−3] 1.329 66(4) [−3] 2.687 20(9) [−4] 2.706 82(9) [−4] 2.30 [−4] 1.20 [−4]  
1S → 5 1P 4.026 52(7) [−3] 4.038 60(7) [−3] 8.576 84(14) [−4] 8.599 94(14) [−4] 8.10 [−4] 6.90 [−4]  
1S → 6 1P 3.622 30(13) [−3] 3.630 70(13) [−3] 7.894 7(3) [−4] 7.910 5(3) [−4] 7.50 [−4] 6.80 [−4]  
1S → 7 1P 2.760 62(14) [−3] 2.766 31(14) [−3] 6.096 8(3) [−4] 6.107 5(3) [−4] 5.90 [−4] 5.40 [−4]  
1S → 9 1P 1.524 52(2) [−3] 1.527 54(2) [−3] 3.414 03(4) [−4] 3.41974(4) [−4] 3.20 [−4] 3.07 [−4]  
1S → 10 1P 1.154(2) [−3] 1.156(2) [−3] 2.594(5) [−4] 2.599(5) [−4]    
1S → 11 1P 8.90(7) [−4] 8.92(7) [−4] 2.01(2) [−4] 2.01(2) [−4]    
1P → 3 1S 9.613 578 7(15) [0] 9.613 186 1(15) [0] 1.179 016 1(5) [−1] 1.178 629 3(5) [−1] 1.18 [−1] 1.15 [−1] 1.147 [−1] 
1P → 4 1S 4.280 48(6) [−1] 4.280 54(6) [−1] 9.830 15(14) [−3] 9.827 42(14) [−3] 9.82 [−3] 9.80 [−3]  
1P → 5 1S 1.340 7(2) [−1] 1.340 8(2) [−1] 3.632 4(6) [−3] 3.631 7(6) [−3] 3.62 [−3] 3.55 [−3]  
1P → 6 1S 6.776 3(2) [−2] 6.777 9(2) [−2] 1.973 1(5) [−3] 1.972 9(5) [−3] 1.96 [−3] 1.88 [−3]  
1P → 7 1S 4.223 04(9) [−2] 4.224 96(9) [−2] 1.277 85(3) [−3] 1.278 05(3) [−3] 1.23 [−3] 1.18 [−3]  
1P → 8 1S 2.943 73(5) [−2] 2.945 90(5) [−2] 9.115 45(15) [−4] 9.119 48(15) [−4] 9.08 [−4] 8.11 [−4]  
1P → 9 1S 2.193 74(2) [−2] 2.196 04(2) [−2] 6.895 71(8) [−4] 6.900 87(8) [−4] 6.94 [−4] 5.92 [−4]  
1P → 10 1S 1.706(2) [−2] 1.709(2) [−2] 5.420(7) [−4] 5.425(7) [−4] 5.62 [−4]   
1P → 11 1S 1.370(7) [−2] 1.372(7) [−2] 4.38(2) [−4] 4.39(2) [−4]    
1S → 3 1P 5.713 342 8(3) [+1] 5.713 911 1(3) [+1] 9.564 292(5) [−1] 9.561 174(5) [−1] 9.58 [−1] 9.57 [−1] 9.465 [−1] 
1S → 4 1P 2.762 51(5) [−1] 2.769 26(5) [−1] 1.037 04(2) [−2] 1.039 23(2) [−2] 9.82 [−2] 9.80 [−2]  
1S → 5 1P 3.153 21(2) [−1] 3.157 18(2) [−1] 1.479 26(8) [−2] 1.480 66(8) [−2] 1.45 [−2] 1.45 [−2]  
1S → 6 1P 2.103 53(5) [−1] 2.105 71(5) [−1] 1.090 67(2) [−2] 1.091 47(2) [−2] 1.07 [−2] 1.08 [−2]  
1S → 7 1P 1.389 96(2) [−1] 1.391 27(2) [−1] 7.610 34(14) [−3] 7.615 22(14) [−3] 7.51 [−3] 7.50 [−3]  
1S → 8 1P 9.503 6(5) [−2] 9.512 6(5) [−2] 5.379 0(3) [−3] 5.382 4(3) [−3] 5.29 [−3] 4.73 [−3]  
1S → 9 1P 6.738 1(14) [−2] 6.744 9(14) [−2] 3.897 7(8) [−3] 3.900 4(8) [−3] 3.79 [−3] 3.87 [−3]  
1S → 10 1P 4.934(10) [−2] 4.940(10) [−2] 2.897(6) [−3] 2.900(6) [−3]    
1S → 11 1P 3.72(3) [−2] 3.72(3) [−2] 2.21(2) [−3] 2.21(2) [−3]    
1P → 4 1S 4.060 230(3) [+1] 4.059 474(3) [+1] 2.079 214(2) [−1] 2.078 480(2) [−1] 2.09 [−1] 2.12 [−1]  
1P → 5 1S 2.630 541(4) [0] 2.630 732(4) [0] 2.433 016(3) [−2] 2.432 638(3) [−2] 2.43 [−2] 2.44 [−2]  
1P → 6 1S 7.612 7(7) [−1] 7.614 0(7) [−1] 8.582 0(7) [−3] 8.581 3(7) [−3] 8.57 [−3] 8.59 [−3]  
1P → 7 1S 3.407 8(5) [−1] 3.408 7(5) [−1] 4.230 8(6) [−3] 4.230 8(6) [−3] 4.23 [−3] 4.22 [−3]  
1P → 8 1S 1.874 3(3) [−1] 1.875 0(3) [−1] 2.459 3(4) [−3] 2.459 6(4) [−3] 2.47 [−3] 2.44 [−3]  
1P → 9 1S 1.161 85(10) [−1] 1.162 45(10) [−1] 1.578 89(13) [−3] 1.579 30(13) [−3] 1.60 [−3] 1.56 [−3]  
1P → 10 1S 7.791(9) [−2] 7.796(9) [−2] 1.084(13) [−3] 1.085(13) [−3] 1.14 [−3]   
1P → 11 1S 5.53(2) [−2] 5.54(2) [−2] 7.83(3) [−4] 7.83(3) [−4]    
1S → 4 1P 2.631 297 0(6) [+2] 2.631 708 6(6) [+2] 1.430 513 2(9) [0] 1.430 068 9(9) [0] 1.43 [0]   
1S → 5 1P 1.230 43(14) [−1] 1.237 64(14) [−1] 1.822 24(20) [−3] 1.832 31(20) [−3] 1.59 [−3] 1.67 [−3]  
1S → 6 1P 2.616 7(2) [−1] 2.622 2(2) [−1] 5.167 1(4) [−3] 5.176 4(4) [−3] 4.95 [−3] 5.10 [−3]  
1S → 7 1P 1.857 1(8) [−1] 1.860 0(8) [−1] 4.206(2) [−3] 4.212(2) [−3] 4.08 [−3] 4.17 [−3]  
1S → 8 1P 1.256 3(2) [−1] 1.258 0(2) [−1] 3.077 4(5) [−3] 3.080 8(5) [−3] 2.98 [−3] 2.71 [−3]  
1S → 9 1P 8.726(2) [−2] 8.738(2) [−2] 2.246(5) [−3] 2.249(5) [−3] 2.16 [−3] 2.23 [−3]  
1S → 10 1P 6.269(6) [−2] 6.278(6) [−2] 1.669(2) [−3] 1.671(2) [−3]    
1S → 11 1P 4.65(3) [−2] 4.66(3) [−2] 1.27(9) [−3] 1.27(9) [−3]    
1P → 5 1S 1.266 409(2) [+2] 1.266 156(2) [+2] 2.933 007(4) [−1] 2.931 905(4) [−1] 2.95 [−1]   
1P → 6 1S 8.287 77(7) [0] 8.288 28(7) [0] 3.596 95(3) [−2] 3.596 32(3) [−2] 3.60 [−3] 3.61 [−3]  
1P → 7 1S 2.312 0(5) [0] 2.312 4(5) [0] 1.267 4(3) [−2] 1.267 2(3) [−2] 1.27 [−2] 1.27 [−2]  
1P → 8 1S 1.009 2(2) [0] 1.009 5(2) [0] 6.245 3(11) [−3] 6.245 4(11) [−3] 6.27 [−3] 6.25 [−3]  
1P → 9 1S 5.459 4(6) [−1] 5.461 7(6) [−1] 3.634 0(4) [−3] 3.634 5(4) [−3] 3.69 [−3] 3.63 [−3]  
1P → 10 1S 3.348(3) [−1] 3.351(3) [−1] 2.338(2) [−3] 2.339(2) [−3] 2.45 [−3]   
1P → 11 1S 2.232(8) [−1] 2.234(8) [−1] 1.611(6) [−3] 1.612(6) [−3]    
1S → 5 1P 7.481 658(8) [+2] 7.482 926(8) [+2] 1.814 518(3) [0] 1.813 946(3) [0] 1.82 [0]   
1S → 6 1P 8.401(5) [−2] 8.496(5) [−2] 6.185(3) [−4] 6.253(3) [−4] 4.78 [−4]   
1S → 7 1P 3.272(4) [−1] 3.281(4) [−1] 3.359(4) [−3] 3.367(4) [−3] 3.16 [−3] 3.32 [−3]  
1S → 8 1P 2.420 7(9) [−1] 2.424 8(9) [−1] 2.931 8(11) [−3] 2.935 9(11) [−3] 2.79 [−3] 2.59 [−3]  
1S → 9 1P 1.673(3) [−1] 1.676(3) [−1] 2.235(4) [−3] 2.237(4) [−3] 2.10 [−3] 2.22 [−3]  
1S → 10 1P 1.179(2) [−1] 1.181(2) [−1] 1.679(3) [−3] 1.681(3) [−3]    
1S → 11 1P 8.57(5) [−2] 8.59(5) [−2] 1.28(7) [−3] 1.28(7) [−3]    
1P → 6 1S 3.157 64(2) [+2] 3.157 04(2) [+2] 3.838 54(3) [−1] 3.837 10(3) [−1] 3.86 [−1]   
1P → 7 1S 2.031 3(2) [+1] 2.031 5(2) [+1] 4.788 4(6) [−2] 4.787 6(6) [−2] 4.79 [−2]   
1P → 8 1S 5.498 6(8) [0] 5.499 8(8) [0] 1.684 8(3) [−2] 1.684 7(3) [−2] 1.69 [−2]   
1P → 9 1S 2.350(2) [0] 2.351(2) [0] 8.299(9) [−3] 8.300(9) [−3] 8.45 [−2] 8.32 [−3]  
1P → 10 1S 1.257(3) [0] 1.257(3) [0] 4.848(12) [−3] 4.849(12) [−3] 5.10 [−2]   
1P → 11 1S 7.66(3) [−1] 7.67(3) [−1] 3.14(13) [−3] 3.14(13) [−3]    
1S → 6 1P 1.691 681(10) [+3] 1.691 971(10) [+3] 2.182 081(16) [0] 2.181 379(16) [0] 2.19 [ 0]   
1S → 7 1P 4.85(2) [−2] 4.96(2) [−2] 2.03(9) [−4] 2.08(9) [−4] 1.12 [−4]   
1S → 8 1P 4.238(8) [−1] 4.251(8) [−1] 2.560(5) [−3] 2.567(5) [−3] 2.30 [−3]   
1S → 9 1P 3.257(13) [−1] 3.261(13) [−1] 2.373(9) [−3] 2.375(9) [−3] 2.12 [−3]   
1S → 10 1P 2.272(6) [−1] 2.276(6) [−1] 1.855(5) [−3] 1.858(5) [−3]    
1S → 11 1P 1.606(4) [−1] 1.608(4) [−1] 1.415(3) [−3] 1.417(3) [−3]    
1P → 7 1S 6.706 9(4) [+2] 6.705 8(4) [+2] 4.772 9(3) [−1] 4.771 2(3) [−1] 4.79 [−1]   
1P → 8 1S 4.231(2) [+1] 4.231(2) [+1] 6.001(2) [−2] 6.000(2) [−2] 6.03 [−2]   
1P → 9 1S 1.119 4(14) [+1] 1.119 7(14) [+1] 2.111 4(26) [−2] 2.111 4(26) [−2] 2.15 [−2]   
1P → 10 1S 4.705(3) [0] 4.708(3) [0] 1.041(7) [−2] 1.041(7) [−2] 1.10 [−2]   
1P → 11 1S 2.485(6) [0] 2.487(6) [0] 6.084(16) [−3] 6.088(16) [−3]    
1S → 7 1P 3.316 5(2) [+3] 3.317 0(2) [+3] 2.546 2(2) [0] 2.545 4(2) [0] 2.56 [0]   
1S → 8 1P 1.307(7) [−2] 1.387(7) [−2] 3.418(19) [−5] 3.624(19) [−5] 5.00 [−8] 5.00 [−8]  
1S → 9 1P 5.380(8) [−1] 5.384(8) [−1] 2.077(3) [−3] 2.078(3) [−3] 1.64 [−3]   
1S → 10 1P 4.31(3) [−1] 4.30(3) [−1] 2.04(1) [−3] 2.04(1) [−3]    
1S → 11 1P 3.00(6) [−1] 3.00(6) [−1] 1.62(3) [−3] 1.62(3) [−3]    
1P → 8 1S 1.269 9(3) [+3] 1.269 7(3) [+3] 5.725 2(13) [−1] 5.723 2(13) [−1] 5.75 [−1]   
1P → 9 1S 7.869(7) [+1] 7.869(7) [+1] 7.229(6) [−2] 7.228(6) [−2] 7.33 [−2]   
1P → 10 1S 2.045(5) [+1] 2.046(5) [+1] 2.546(6) [−2] 2.546(6) [−2] 2.68 [−2]   
1P → 11 1S 8.46(4) [0] 8.47(4) [0] 1.25(5) [−2] 1.25(5) [−2]    
1S → 8 1P 5.888 2(12) [+3] 5.888 9(12) [+3] 2.910 2(6) [0] 2.909 1(6) [0] 2.94 [0]   
1S → 9 1P 1.6(4) [−3] 1.3(4) [−3] 2.8(8) [−6] 2.3(8) [−6] 1.47 [−6]   
1S → 10 1P 6.59(4) [−1] 6.57(4) [−1] 1.73(1) [−3] 1.72(1) [−3]    
1S → 11 1P 5.42(14) [−1] 5.41(14) [−1] 1.77(5) [−3] 1.76(5) [−3]    
1P → 9 1S 2.207 0(8) [+3] 2.206 5(8) [+3] 6.689 5(24) [−1] 6.686 9(24) [−1] 6.73 [−2]   
1P → 10 1S 1.349(7) [+2] 1.349(7) [+2] 8.49(4) [−2] 8.48(4) [−2] 8.82 [−2]   
1P → 11 1S 3.47(5) [+1] 3.47(5) [+1] 3.00(5) [−2] 3.00(5) [−2]    
1S → 9 1P 9.719 9(11) [+3] 9.720 7(11) [+3] 3.273 0(4) [0] 3.271 6(4) [0] 3.34 [0]   
1S → 10 1P 5.(5) [−2] 5.(5) [−2] 6.(6) [−5] 6.(6) [−5]    
1S → 11 1P 8.(2) [−1] 8.(2) [−1] 1.(3) [−3] 1.(3) [−3]    
1P → 10 1S 3.583(13) [+3] 3.582(13) [+3] 7.656(27) [−1] 7.652(27) [−1] 7.80 [−1]   
1P → 11 1S 2.19(6) [+2] 2.19(6) [+2] 9.84(26) [−2] 9.85(26) [−2]    
10 1S → 10 1P 1.519(5) [+4] 1.519(5) [+4] 3.632(12) [0] 3.630(12) [0]    
10 1S → 11 1P 1.(2) [−1] 1.(2) [−1] 1.(2) [−4] 1.(2) [−4]    
10 1P → 11 1S 5.50(7) [+3] 5.49(7) [+3] 8.62(11) [−1] 8.61(11) [−1]    
11 1S → 11 1P 2.28(4) [+4] 2.28(4) [+4] 3.98(7) [0] 3.98(7) [0]    
a

B-spline CI with semi-empirical core potential (BCICP) method.102 

b

MCHF method.11 

c

B-spline CI (BCIBP) method.103 

FIG. 1.

The logarithmic map of calculated oscillator strengths for SP and PS transitions between states considered in this work.

FIG. 1.

The logarithmic map of calculated oscillator strengths for SP and PS transitions between states considered in this work.

Close modal

In general, agreement between the oscillator strengths calculated in the present work and the available literature values is fairly good. The present calculations include a considerably wider range of the oscillator strength values than it was calculated in prior works. Both the tabulated values of the oscillator strengths and their depiction in Fig. 1 show that the largest values of the strengths correspond, as expected, to transitions between states with the same principal quantum number, i.e., n1Sn1P transitions. However, the oscillator strength values are also quite sizable for the n1P → (n + 1)1S transitions. This indicates a possibility to use “cascade” excitations involving a sequence of the following transitions: 2 1S → 2 1P, 2 1P → 3 1S, 3 1S → 3 1P, 3 1P → 4 1S, etc., to prepare the atom in a particular Rydberg state.

In Table 6, we provide a more detailed comparison of our oscillator strength values for the lowest 2 1S → 2 1P transition with experimental and theoretical results from the literature. All theoretical values agree within the experimental uncertainties. It is interesting to note that there is a slight difference between the values of the oscillator strength calculated for 9Be and Be of 0.0003. However, this isotopic shift is too small to be experimentally verifiable at present.

TABLE 6.

Comparison of the oscillator strength values for the 2 1S → 2 1P transition in Be obtained with various experimental and theoretical approaches: Beam-foil (BF), Time-resolved laser-induced-fluorescence (TR-LIF), Time-dependent gauge invariant (TDGI), multiconfiguration Hartree–Fock (MCHF), B-spline CI with semi-empirical core potential (BCICP), B-spline CI (BCIBP), and CI+ core polarization (CICP)

Experimental 
BF104  1.34(4) 
BF105  1.40(4) 
TR-LIF106  1.34(3) 
Theoretical 
TDGI107,108 1.398 
BCICP102  1.375 
MCHF11  1.38 
BCIBP103  1.380 
CICP109  1.374 3 
this work (Be) 1.374 689 2(3) 
this work (9Be) 1.374 400 8(3) 
Experimental 
BF104  1.34(4) 
BF105  1.40(4) 
TR-LIF106  1.34(3) 
Theoretical 
TDGI107,108 1.398 
BCICP102  1.375 
MCHF11  1.38 
BCIBP103  1.380 
CICP109  1.374 3 
this work (Be) 1.374 689 2(3) 
this work (9Be) 1.374 400 8(3) 

The finite-nuclear-mass effects can also be observed in the 3D plots of the radial electronic and nuclear densities determined with respect to the center of mass of the atom. The plots of these densities for some selected 1S and 1P states are shown in Figs. 2 and 3.

FIG. 2.

The density of the nucleus for some of the 1S (left-column) and 1P states (right-column) in the center-of-mass coordinate frame for the beryllium atom.

FIG. 2.

The density of the nucleus for some of the 1S (left-column) and 1P states (right-column) in the center-of-mass coordinate frame for the beryllium atom.

Close modal
FIG. 3.

The density of the electrons for some of the 1S (left-column) and 1P states (right-column) in the center-of-mass coordinate frame for the beryllium atom.

FIG. 3.

The density of the electrons for some of the 1S (left-column) and 1P states (right-column) in the center-of-mass coordinate frame for the beryllium atom.

Close modal

As one can see, with the increasing excitation level, the number of the radial nodes in the densities, as expected, increases. In addition, in the densities of the 1P states, there is a nodal plane along the z plane. The most interesting is the comparison between the electronic and nuclear densities for a particular state. As one can see, the oscillatory patterns for the two densities for every state are almost identical. However, the scale of the coordinate axes in the plots of the electronic densities is about four orders of magnitude larger than in the nuclear densities. As mentioned, this is understandable because the average radius of the motion of the nucleus around the center of mass of the atom is much smaller than the average radius of the motion of the electrons (the radius scales as the mass inverse). This also explains why the nodal patterns of the two densities for a particular state are very similar. This happens because, for each maximum of the nuclear density, a maximum of the electronic density has to appear to balance out the centrifuge effects associated with the motions of the particles. Due to this balance, the center of mass of the atom does not move.

In this work, an algorithm for calculating the oscillator strength for an atomic inter-state spectral transition is implemented and used to calculate the SP and PS transitions in the beryllium atom. In the calculations, the nonrelativistic variational wave functions expanded in terms of all-electron explicitly correlated Gaussian functions are used. In the calculation of the corresponding transition energies, the nonrelativistic energies are augmented with the leading relativistic and QED corrections and high-accuracy results are obtained. The nonrelativistic energies are the best to date. The oscillator strengths for the transitions show an interesting pattern. As the strength values are the most sizable for the n1Sn1P and n1P → (n + 1)1S transitions, one can envision preparing a beryllium atoms in a particular excited Rydberg 1S or 1P state by a cascade of the following excitations: 2 1S → 2 1P, 2 1P → 3 1S, 3 1S → 3 1P, etc. In moving back to the ground 2S state from a particular excited Rydberg 1S or 1P state, the above excitation cascade can be followed in reverse.

The present results may also be employed in the modeling of light emission and absorption events involving beryllium atoms in the interstellar media. Such models usually require accurate values of the transition energies and the oscillator strengths, which the present work provides.

This work was supported by Nazarbayev University (Faculty Development Grant Nos. 090118FD5345 and 021220FD3651) and the National Science Foundation (Grant No. 1856702). The authors thank Professor Gordon Drake (Canterbury College, University of Windsor, Ontario, Canada) for useful discussions about the oscillator strengths.

The data that support the findings of this study are available within the article and from the corresponding author upon reasonable request.

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