Excitation energy of
9Be
Krzysztof Pachucki*Institute of Theoretical Physics, Warsaw University, Hoża 69, 00-681 Warsaw, Poland
Jacek Komasa†
Quantum Chemistry Group, Faculty of Chemistry, A. Mickiewicz University, Grunwaldzka 6, 60-780 Poznań, Poland
共Received 15 March 2006; published 9 May 2006兲
The high precision relativistic and radiative corrections to the energy of the excited 31S state of the beryllium atom are obtained. The nonrelativistic wave function, expanded in a basis of exponentially correlated Gaussian functions, yields the lowest upper bounds to the energy of 21S and 31S states. By means of the integral representation, a reference-quality Bethe logarithm has been obtained. The resulting theoretical 21S -31S transition energy amounts to 54 677.78共45兲 cm−1and differs from the known experimental value by about 0.5 cm−1.
DOI:10.1103/PhysRevA.73.052502 PACS number共s兲: 31.30.Jv, 31.15.Pf, 31.25.Jf
I. INTRODUCTION
Theoretical predictions of the energy levels in many-electron atoms with an accuracy competitive to that achiev-able from measurements require two conditions to be ful-filled. First of all, the interparticle correlation has to be fully incorporated into the calculations. This goal can be achieved by using large expansions of the nonrelativistic wave func-tion in the basis of explicitly correlated funcfunc-tions with opti-mized nonlinear parameters. For the helium atom, the best examples of the application of such functions are computa-tions by Drake et al.关1兴, who employed the Hylleraas basis function and computations by Korobov关2兴 performed using a correlated exponential basis. The Hylleraas basis has been successfully employed also in the calculations on the lithium atom by Yan and Drake关3兴 and, most recently, by Puchalski and Pachucki关4兴 who predicted the nonrelativistic energy of the ground state with an accuracy exceeding 12 significant figures. For the beryllium atom, though, since the classical work by Sims and Hagstrom关5兴, some progress towards con-structing an accurate wave function has been observed 关6–10兴; difficulties in computing matrix elements in the Hyl-leraas basis prevents one from a full utilization of this basis functions. Hitherto, the most accurate energy for the ground state of the berylliumlike atoms has been obtained using the exponentially correlated Gaussian共ECG兲 basis sets 关11,12兴.
The second condition indispensable for obtaining accurate energy predictions is an adequate theoretical description of the beyond-nonrelativistic effects. For the many-electron at-oms, the routinely used methods rely on the Dirac-Coulomb Hamiltonian which is a sum of the one-electron Dirac Hamil-tonians with Coulomb interactions between electrons. This approach, though acceptable as the first approximation, can-not be used in high-accuracy calculations as it is inconsistent with quantum electrodynamics 共QED兲. A very convenient and well-founded approach, valid for light systems with few
electrons, relies on expansion of the total binding energy in powers of␣—the fine structure constant. Each coefficient of this expansion can be expressed as the expectation value of some operator, to be called an effective Hamiltonian 关13兴. These operators are known for the leading relativistic and QED corrections关14–17兴. An evaluation of these corrections for 31S excited state of a beryllium atom is the main subject
of the present work.
II. THE METHOD
The total energy of a bound system can be represented by an expansion in powers of␣:
E共␣兲 = ENR+␣2EREL+␣3EQED+␣4␦EQED 共1兲 with the expansion coefficients having a transparent physical interpretation. A particular form of these coefficients, valid for the beryllium atom in the singlet S state, is described below. We use units where m =ប=c=1 and a common pref-actor m␣2 is pulled out from the binding energy. E
NR is a sum of E0, the nonrelativistic energy of the atom correspond-ing to the clamped nuclei Hamiltonian
H0= −
兺
i冉
i2 2 + Z ri冊
+兺
i⬎j 1 rij , 共2兲and the finite nuclear mass correction EFM. This correction, given by EFM= 1 2M具⌿兩
冉
兺
i i冊
2 兩⌿典, 共3兲where M is the nuclear mass, has been computed by sum-ming up the normal, ENMS= −E0/ M, and the specific, ESMS =具⌿兩兺i⬎ji·j兩⌿典/M mass shifts 关18兴. EREL=具⌿兩HREL兩⌿典 is the leading relativistic correction expressed as the nonrel-ativistic expectation value of the Breit-Pauli Hamiltonian
HREL, which for the closed shell atom in the nonrecoil limit reads
*Electronic address: krp@fuw.edu.pl †
Electronic address: komasa@man.poznan.pl
PHYSICAL REVIEW A 73, 052502共2006兲
HREL=
兺
i冋
−i 4 8 + Z 2 ␦共ri兲册
−兺
i⬎j共
1 +83si· sj兲
␦共rij兲 +兺
i⬎j 1 2冉
1 rij i·j+ 1 rij 3rij·共rij·i兲j冊
. 共4兲The third coefficient of the expansion共1兲, EQED, represents the leading radiative correction
EQED=
兺
i⬎j再
关
164 15 + 14 3 ln␣兴
具⌿兩␦共rij兲兩⌿典 − 14 3 具⌿兩 1 4P冉
1 rij 3冊
兩⌿典冎
+兺
i关
19 30+ ln共␣ −2兲 − ln k 0兴
4Z 3 具⌿兩␦共ri兲兩⌿典. 共5兲 This expression, apart from the expectation value of the Dirac delta distributions ␦, contains the Araki-Sucher term defined as 具兩P冉
1 r3冊
兩典 ⬅ lima→0冕
dr*共r兲共r兲冋
1 r3⌰共r − a兲 + 4␦共r兲共␥+ ln a兲册
, 共6兲 with⌰ being the step function and ␥—the Euler constant. Eq.共5兲 contains also the many-electron Bethe logarithm ln k0 defined byln k0= − 1
D具⌿兩共H0− E0兲ln关2共H0− E0兲兴 兩⌿典, 共7兲
where the following symbols are incorporated =
兺
i i, 共8兲 D = 2Z具⌿兩兺
i ␦共ri兲兩⌿典. 共9兲Since the last coefficient in Eq.共1兲, ␦EQED, is quite compli-cated关13兴, we approximate it by its leading term
␦EQED⬇ 4Z2
冉
139 128+ 5 192− ln共2兲 2冊
具⌿兩兺
i ␦共ri兲兩⌿典, 共10兲 which is the known correction to the Lamb shift in hydro-genlike systems. The remaining ␣4 contributions involve second-order terms which are relatively difficult to compute. The␣4␦EQEDcomponent gives a rough estimate of the error made by cutting off the expansion in Eq.共1兲.
III. EVALUATION OF MATRIX ELEMENTS The nonrelativistic wave function employed in this project is represented as a K-term linear expansion in a four-electron spatial basis兵l共r兲其 multiplied by an appropriate
sin-glet spin eigenfunction,⌶S,MS,
⌿共r,兲 = Aˆ
冉
⌶S,MS共兲兺
l=1 Kcll共r兲
冊
. 共11兲The expansion is antisymmetrized by applying the four-electron projector Aˆ. As the basis functions l共r兲 we used
exponentially correlated Gaussian functions introduced by Singer关19兴 l共r兲 = exp关− rTAlr兴 共12兲 and ˜ l共r兲 = riexp关− rTA˜lr兴 共13兲
of S and P symmetry, respectively. The positive definite ma-trices Al are built of nonlinear parameters determined in a
variational optimization process关20兴. The quality of the final results depends primarily on the effectiveness of the optimi-zation of these nonlinear parameters and this optimioptimi-zation is the most time consuming part of the project.
Some of the operators involved in the nonrelativistic ex-pansion共1兲 are of a singular nature. Expectation values of such operators computed in the Gaussian type basis converge very slowly even if the wave functions are energetically of high quality. One way to circumvent this problem is a refor-mulation of the expectation values in terms of other, less singular operators. This idea, for the first time applied by Drachman 关21兴, was already explored successfully in the atomic calculations reported in Refs. 关11,22兴, where a de-tailed description of the method and numerical convergence results can be found. Such regularized calculations were per-formed for the expectation values of␦共r兲, p4, and P共1/r3兲 operators. Convergence of the remaining operators, including the Breit operator, is less problematic and their expectation values were computed conventionally.
One of the most difficult quantities to be determined is the four-electron Bethe logarithm, Eq.共7兲. An efficient technique of evaluation of ln k0 for multielectron systems was de-scribed in Refs.关11,23兴. In this approach the Bethe logarithm is represented as a one-dimensional integral over the variable
t = 1 /
冑
1 + 2 ln k0= 1 D冕
0 1 f共t兲 − f0− f2t2 t3 dt, 共14兲 where f共t兲 = − 具⌿兩 H0− E0+ 兩⌿典
, 共15兲 and whereis a frequency corresponding to the energy of a virtual photon. It is crucial that f共t兲 is computed to a high accuracy. An important factor deciding on the final accuracy of ln k0 is the small t asymptotics of the integrand in Eq. 共14兲. As shown by Schwartz 关24兴, at very low values of t the integrand has the following expansion:f共t兲 = f0+ f2t2+ f
3t3+ f4t4ln共t兲 + o共t4兲, 共16兲 in which the lowest-order coefficients are known to be f0 = −具⌿兩2兩⌿典, f
2= −2D, f3= 8ZD, and f4= 16Z2D, and the higher-order coefficients were fit to f共t兲 of Eq. 共15兲. A feature
K. PACHUCKI AND J. KOMASA PHYSICAL REVIEW A 73, 052502共2006兲
of this approach which is not to be overestimated is that it can be applied to many-electron atoms as well as to mol-ecules. To evaluate the Bethe logarithm in the integral form of Eq. 共14兲 we need the first-order perturbation correction function⌿˜ , which is a solution of the following equation:
共H0− E0+兲⌿˜ = − ⌿. 共17兲 The operator defined in Eq. 共8兲, couples the unperturbed S states with the intermediate P states, hence⌿˜ is to be ex-panded in the basis of the form共13兲. Assuming ⌿ is known, the perturbation equation can be solved variationally by minimizing the Hylleraas functional at different:
J关⌿˜ 兴 = 具⌿˜ 兩H0− E0+兩⌿˜ 典 + 2具⌿˜ 兩兩⌿典. 共18兲 It is important that the same ⌿˜ minimizes the above func-tional in a broad range of, not only at the static case 共 = 0兲. To reach this effect, we separately optimize Jin Eq. 共18兲 using four 1200-term basis sets at four different fre-quencies= 0, 10, 100, and 1000 and glue them together to form a final 4800-term basis set for⌿˜ .
IV. RESULTS AND DISCUSSION
In this work, the wave functions for both 21S and 31S
states were expanded in a common set of basis functions of type共12兲. The advantage of using a single set of basis func-tions for both states is that the eigenvectors obtained in a single diagonalization of the Hamiltonian are perfectly or-thogonal. The final basis set was constructed from four smaller sets optimized separately with respect to eigenvalues of the four lowest states of the symmetry S. The subsets corresponding to the states 21S and 31S were composed of
1600 basis functions, whereas to the 41S and 51S states—of
600 and 800 functions, respectively, yielding a total of 4600 basis functions. The nonrelativistic wave functions obtained for the 21S and 31S states were subsequently employed to
compute the expectation values of several operators appear-ing in the expressions 共2兲–共10兲. Numerical values of these expectation values are listed in Table I. We note that the 21S
column of the table contains values corrected and slightly improved over those obtained from a smaller expansion共K = 3600兲 and reported previously 关11兴. The nonrelativistic clamped nucleus energies E0=具⌿兩H0兩⌿典 are the lowest upper bounds available to date. In particular, the new ground state energy is slightly improved with respect to the best previous energy estimate关11兴. The new upper bound to 31S energy is
lower by 14⫻10−6a.u. than previous estimations obtained from the 3600-term ECG wave function in Ref.关25兴 and by 3.7⫻10−3a.u. than the upper bound computed by Chung and Zhu关26兴. It is worth noting that the extrapolated energy ob-tained by Chung and Zhu using their full-core plus correla-tion 关27兴 method differs from our result by merely 5.6 ⫻10−7a.u.
The computed energy difference ⌬E between the levels 31S and 21S is presented in Table II. The total value of⌬E
is obtained as a sum of the components originating from Eq. 共1兲. The finite mass correction together with the relativistic
and radiative corrections are listed separately in Table II, which permits an assessment of the contributions coming from different physical effects. The total correction to the nonrelativistic excitation energy amounts to 2.45共23兲 cm−1 which is merely 45 ppm of the total⌬E. The uncertainties, given in the last column of the table, are transferred from the errors estimated for the 31S state. The final theoretical value
⌬E=54 677.78共45兲 cm−1 can be confronted with the experi-mentally derived value of 54 677. 26 cm−1, which comes from the measurements performed in 1962 by Johansson 关28兴. His results were included in the compilation of energy levels of Be by Kramida and Martin 关29兴. The anticipated experimental uncertainty displayed in Table II is based on Johansson’s opinion that “the estimated errors in the level values are, in general, less than ±0.05 cm−1.” Since the spac-ing between 31S and 21S levels has been determined
indi-rectly from two nS→mP transitions, we get an experimental uncertainty of 0.10 cm−1.
The difference between theory and experiment is close to the sum of both theoretical and experimental uncertainties. It
TABLE I. Expectation values共in a.u.兲 of various operators with nonrelativistic wave functions of a beryllium atom共K=4600兲 in the 21S and 31S states. The implicit summation over i and over pairs i⬎ j is assumed. 21S 31S 具⌿兩H0兩⌿典 −14.667 355 748 −14.418 236 555 具⌿兩2兩⌿典 −30.255 159共8兲 −29.737 482共30兲 具⌿兩␦共ri兲兩⌿典 35.368 92共4兲 35.127 90共12兲 具⌿兩␦共rij兲兩⌿典 1.605 302共4兲 1.583 070共12兲 具⌿兩pi· pj兩⌿典 0.460 224共4兲 0.450 512共10兲 具⌿兩pi 4兩⌿典/8 270.704 8共5兲 268.562共13兲 具⌿兩rij −3 rij共rij· pi兲pj +rij−1pi· pj兩⌿典/2 −0.891 825共1兲 −0.900 470共4兲 具⌿兩P共1/rij 3兲兩⌿典/4 −0.583 03共5兲 −0.594 08共13兲 ln k0 5.750 48共6兲 5.750 89共15兲
TABLE II. Components of the 21S→31S excitation energy for the 9Be atom. E0 is the nonrelativistic clamped nucleus energy; EFM,␣2EREL,␣3EQED, and␣4␦EQEDare the finite nuclear mass, the leading relativistic, the leading radiative, and the higher-order ra-diative corrections, respectively, defined by Eqs.共2兲–共5兲, 共10兲, and 共3兲. The mass of the9Be nuclei M = 16424.203m. Physical constants are from关30兴.
Be共21S兲 共a.u.兲 Be共31S兲 共a.u.兲 ⌬E 共cm−1兲
E0 −14.667 355 748 −14.418 236 555 54675.34共22兲 EFM 0.000 028 019 0.000 027 428 −3.459共0兲 ␣2E REL −0.002 360 312 −0.002 331 034 6.43共16兲 ␣3E QED 0.000 339 785 0.000 337 520 −0.497共1兲 ␣4␦E QED 0.000 015 435 0.000 015 330 −0.023共6兲 Total −14.669 332 811 −14.420 187 320 54677.78共45兲 Experimenta 54677.26共10兲 Diff. 0.52共55兲 a References关28,29兴
EXCITATION ENERGY OF9Be PHYSICAL REVIEW A 73, 052502共2006兲
is a justification of our computational approach, which is based on explicitly correlated Gaussian functions and the perturbative expansion of energy in Eq. 共1兲. However, the theoretical uncertainty is still 4–5 times larger than the ex-perimental one. The main contribution to the theoretical un-certainty in⌬E comes from the the nonrelativistic energy of the 31S state. This can be improved by using the Hylleraas
basis set, although computational difficulties with four-electron integrals have not yet been resolved. A relatively large uncertainty in the relativistic correction to⌬E is due to the relativistic kinetic energy of the 31S state. In spite of five
significant figures of具⌿兩pi
4兩⌿典 being obtained, the cancella-tion between one-electron contribucancella-tions in具⌿兩HREL兩⌿典 and contributions coming from 31S and 21S levels results in
about 2% uncertainty of the total relativistic correction. This work, as well as our previous work on the ionization potential of Be 关11兴, illustrate the present possibilities of ECG functions in predicting the energy levels of four-electron atoms. We can conclude that the main factor limit-ing the accuracy of the present day theoretical predictions of atomic levels is the nonrelativistic energy.
ACKNOWLEDGMENTS
This work was supported in part by the Polish State Com-mittee for Scientific Research Grant No. SPB/COST/T-09/ DWM 572. The support from the Poznań Networking and Supercomputing Center is also gratefully acknowledged.
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K. PACHUCKI AND J. KOMASA PHYSICAL REVIEW A 73, 052502共2006兲
