Table Of ContentTheoretical energies of low-lying states of light helium-like ions
Vladimir A. Yerokhin
Center for Advanced Studies, St. Petersburg State Polytechnical University,
Polytekhnicheskaya 29, St. Petersburg 195251, Russia
Krzysztof Pachucki
Institute of Theoretical Physics, University of Warsaw, Hoz˙a 69, 00–681 Warsaw, Poland
Rigorousquantumelectrodynamicalcalculationispresentedforenergylevelsofthe11S,21S,23S,
0 21P1,and 23P0,1,2 statesof helium-likeionswith thenuclearcharge Z =3...12. Thecalculational
1 approachaccountsforallrelativistic,quantumelectrodynamical,andrecoileffectsuptoordersmα6
0 and m2/Mα5, thusadvancing the previously reported theory of light helium-like ions by one order
2
in α.
n
PACSnumbers: 12.20.Ds,31.30.J-,06.20.Jr,31.15.-p
a
J
1
I. INTRODUCTION of nuclear charges around Z = 12 is the most difficult
3
one for theory, as contributions not (yet) accounted by
] either of these methods have their maximal value there.
h Atomic helium and light helium-like ions have long In order to provide accurate predictions for the whole
p been attractive subjects of theoretical and experimen-
isoelectronic sequence, it is necessary to combine these
- tal investigations. From the theoretical point of view,
m two approaches.
helium-likeatomsarethesimplestfew-bodysystems. As
Forthe firsttime acombinationofthecomplementary
o such, they are traditionally used as a testing ground for
approaches was made by Drake [5]. His results for en-
t different methods of the description of atomic structure.
a ergies of helium-like ions comprise all effects up to order
. On the experimental side, small natural linewidths of
s mα5 in the low-Z region, whereas in the high-Z region,
c transitions between the metastable 3P and 3S states of
they are complete up to the next-to-the-leading order
i helium-like ions permit spectroscopic measurements of
s in 1/Z for nonradiative effects and to the leading or-
y high precision. For atomic helium, experimental investi-
der, for radiative effects. Since then, significant progress
h gations are nowadays carried out with the relative accu-
was achieved in theoretical understanding of energy lev-
p racy up to 7 10−12 [1]. An advantage of the helium-
[ × els of atomic helium, whose description is now complete
like ions as comparedto,e.g., hydrogen-likeones, is that
2 thetransitionfrequencyincreasesslowlywiththenuclear through order mα6 [6, 7]. Also in the high-Z region,
theoretical energies have recently been significantly im-
v chargenumberZ ( Z). Thisfeatureensuresthatwave-
∼ provedby a rigoroustreatment of the two-electronQED
6 lenghts of a significant part of the helium isoelectronic
corrections [8], which completed the O(1/Z) part of the
0 sequence fall in the region suitable for accurate experi-
4 radiative effects.
mental determination.
0 In the present investigation we aim to improve theo-
. There are presently two main theoretical approaches reticalpredictionsofthen=1andn=2energylevelsof
1
0 thatallowonetosystematicallyaccountfortheelectron- lighthelium-like ions. To this end,we performacalcula-
0 correlation, relativistic, and quantum electrodynamical tionthatincludesallQEDandrecoileffects uptoorders
1 (QED) effects in few-electron systems. The first one, mα6 and m2/Mα5 (M is the nuclear mass). In order to
: traditionally used for light systems, relies on an expan- establish a basis for merging the current approach with
v
i sion of the relativistic and QED effects in terms of α the all-order calculations, we perform an extensive anal-
X and Zα (α is the fine-structure constant) and treats the ysis of the 1/Z expansion of individual corrections. This
r nonrelativisticelectron-electroninteractionnonperturba- analysis also provides an effective test of consistency of
a tively. This approach started with the pioneering works our calculational results and of the 1/Z-expansion data
of Araki [2] and Sucher [3], who derived the expression available in the literature.
for the Lamb shift in many-electron systems complete
through the order mα5. The other approach aims pri-
marily at high-Z ions. It does not use any expansion II. THEORY OF THE ENERGY LEVELS
in the binding-strength parameter Zα (and thus is of-
ten referred to as the all-order approach) but treats the
Inthis section,wepresentasummaryofcontributions
electron-electron interaction within the perturbative ex-
totheenergylevelsoftwo-electronatomscompleteupto
pansion with the parameter 1/Z. A systematic formula-
orders mα6 and m2/Mα5.
tion of this method is presented in Ref. [4].
According to QED theory, energy levels of atoms are
These two approaches can be considered as comple- represented by an expansion in powers of α of the form
mentary,the firstbeingclearlypreferableforlightatoms
and the second, for heavy ions. The intermediate region E(α)=E(2)+E(4)+E(5)+E(6)+E(7)+..., (1)
2
where E(n) mαn (n) is a contributionoforderαn and [9],
may include≡powerEs of lnα. Each of (n) is in turn ex-
E p~4 πZ Z ~r
panded in powers of the electron-to-nucleus mass ratio H(4) = a + δ3(r )+ ~σ a p~
m/M − 8 2 a 4 a· r3 × a
a (cid:20) a (cid:21)
X
1 δij ri rj
E(n) =E∞(n)+EM(n)+EM(n2)+... , (2) +a<b(cid:26)−πδ3(rab)− 2pia(cid:18)rab + arba3bab(cid:19)pjb
X
awnhderEeM(nE2)M(ni)s dtheenosteecsonthde-ocrodrerrecctoirornecotfiofinr.stToorndoetrei,nfomr/tMhe − 23π~σa·~σbδ3(rab)+ σ4aira3σbbj (cid:18)δij −3rairba2rbajb(cid:19)
nonrelativistic energy, it is more natural to expand in 1
+ 2 ~σ ~r p~ ~σ ~r p~
mr/M (where mr is the reduced mass) rather than in 4ra3b a· ab× b− b· ab× a
m/M, since such expansion has smaller coefficients. For (cid:2) (cid:0) (cid:1)
the relativistic corrections, however, the natural recoil + ~σ ~r p~ ~σ ~r p~ . (7)
b ab b a ab a
· × − · ×
expansion parameter is m/M, so for consistency we use (cid:27)
(cid:0) (cid:1)(cid:3)
it for the nonrelativistic energy as well. The finite nuclear mass correction to the Breit contri-
The terms of the double perturbation expansion (1) bution (4) is conveniently separated into the mass scal-
EM
and(2)areexpressedasexpectationvaluesofsomeeffec- ing, the mass polarization, and the operator parts. The
tive Hamiltonians (in some cases, of nonlocal operators) mass scaling prefactor is (m /m)4 for the first term in
r
andassecond-andhigher-orderperturbationcorrections Eq. (7) and (m /m)3, for all the others. The mass po-
r
induced by these Hamiltonians (operators). It is note- larization part represents the first-order perturbation of
worthy that the expansion (1) is employed also for the (4) by the mass-polarization operator (6). The opera-
∞
states that are mixed by the relativistic effects, namely E
tor part is given by the expectation value of the recoil
21P and 23P . The mixing effects are treated pertur-
1 1 addition to the Breit-Pauli Hamiltonian,
batively. (So, the leading effect due to the 21P 23P
1 1
mixing appears naturally as the second-order mα−6 cor- Zm ~r δij rirj
H(4) = a ~p ~σ pi + a a pj .
rection, together with contributions from other interme- rec 2M r3 × b· a− a r r3 b
diate states.) This differs from the approach used, e.g., Xab (cid:20) a (cid:18) a a (cid:19) (cid:21)
(8)
in Ref. [5], where a two-by-two matrix was constructed
for this pair of states and the energies were obtained by (5) is the leading QED correction[2, 3]. We divide it
a diagonalization. E∞
intothelogarithmicandthenonlogarithmicparts, (5) =
TheleadingcontributiontotheenergyE∞(2) ≡E0 isthe (5)(log)+ (5)(nlog), which are given by E∞
eigenvalue of the nonrelativistic Hamiltonian, E∞ E∞
14
H(2) H = p~a2 Z + 1 . (3) E∞(5)(log) = 3 ln(Zα) hδ3(rab)i
≡ 0 2 − r r Xa<b
a (cid:18) a(cid:19) a<b ab 4Z
X X + ln (Zα)−2 δ3(r ) , (9)
a
3 h i
The first-andsecond-orderrecoilcorrectionsto the non- a
(cid:2) (cid:3) X
relativistic energy are given by
and
m 164 14
EM(2) =−M E∞(2)+ Hr(e2c) , (4) E∞(5)(nlog) = 15 hδ3(rab)i− 3 Qab
D E Xa<b Xa<b
19 k 4Z e
+ ln 0 δ3(r ) + H(5) ,
m 2 m 30 − Z2 3 h a i h fs i
EM(2)2 = M E∞(2)−2M Hr(e2c) (cid:20) (cid:18) (cid:19)(cid:21) Xa (10)
(cid:16) (cid:17) 1 D E
+ H(2) H(2) , (5) where
rec( H )′ rec
(cid:28) E0− 0 (cid:29)
1
Q = +δ3(r ) lnZ , (11)
where ab 4πr3 ab
(cid:28) ab (cid:29)
H(2) = m p~ ~p (6) and the singuelar operator r−3 is defined by
rec M a· b
Xa<b 1
lim d3rφ∗(~r)φ(~r)
is the mass polarization operator. (cid:28)r3(cid:29) ≡a→0Z
(4) 1
The leading relativistic correction ∞ is given by the Θ(r a)+4πδ3(r)(γ+lna) , (12)
expectation value of the Breit-PauliEHamiltonian H(4) × r3 −
(cid:20) (cid:21)
3
where γ is the Euler constant. The Bethe logarithm is nonvanishing contribution to the fine-structure splitting
defined as only. It is given by
p~ (H ) ln 2(H ) ~p
ln(k0)= DPa a 02−πEZ0 (cid:2) δ3(r0−) E0 (cid:3) Pb bE. Hf(s5,)rec = Mm 4Zπ ~rra3 ×~pb·~σa. (18)
c c ab a
X
D E (13)
P
We note that the last term in Eq. (16) was omitted in
The operator H(5) is the anomalous magnetic moment the original derivation of Ref. [10].
fs
correction to the spin-dependent part of the Breit-Pauli Thecompleteresultforthemα6 correction (6) tothe
∞
Hamiltonian. H(5) does not contribute to the energies energylevels wasderivedby one of the authorsE(K.P.)in
fs
of the singletstates and to the spin-orbitaveragedlevels a series of papers [6, 7, 11, 12]
but it yields the mα5 contribution to the fine structure
splitting. It is given by
(6) = ln(Zα)π δ3(r ) +E
E∞ − h ab i sec
Z ~r
H(5) = ~σ a p~ Xa<b
fs 4π a a· ra3 × a + H(6) +H(6)+H(6)+H(6)+H(6) .
X nrad R1 R2 fs fs,amm
+ 1 σai σbj δij 3raibrajb D E(19)
4π r3 − r2
a<b(cid:26) ab (cid:18) ab (cid:19)
X
1 The firstterm in the aboveexpressioncontains the com-
+ 2 ~σ ~r ~p ~σ ~r p~
4πr3 a· ab× b− b· ab× a plete logarithmic dependence of the mα6 correction.
ab
(cid:2) (cid:0) (cid:1) The part of it proportional to lnα was first obtained in
+ ~σ ~r ~p ~σ ~r p~ . (14) Ref. [13]. The remaining logarithmic part proportional
b ab b a ab a
· × − · ×
(cid:27) to lnZ was implicitly present in formulas reported in
(cid:0) (cid:1)(cid:3)
Ref.[6,7](itoriginatesfromtheexpectationvalueofthe
We note that despite the presence of terms with lnZ in
operator 1/r3 ). In Eq. (19), we group all logarithmic
Eq. (10), the correction (5)(nlog) does not have loga- ab
E∞ terms together so that the remaining part does not have
rithmic terms in its 1/Z expansion.
any logarithms in its 1/Z expansion.
The recoil correction (5) consists of four parts [10],
EM ThetermEsecinEq.(19)isthesecond-orderperturba-
m tion correction induced by the Breit-Pauli Hamiltonian.
(5) = + + + H(5) , (15)
EM M E1 E2 E3 h fs,reci (More specifically, it is the finite residual after separat-
ing divergent contributions that cancel out in the sum
(cid:0) (cid:1)
where withthe expectationvalue ofthe effective mα6 Hamilto-
4Z 14 nian.) The firstpartof the effective Hamiltonian, H(6) ,
E1 =−3E∞(5)+ 3 hδ3(ra)i− 3 hδ3(rab)i, (16) originates from the non-radiative part of the electnrroand-
a a<b nucleus and the electron-electron interaction. The next
X X
(6) (6)
two terms, H and H , are due to the one-loop and
R1 R2
two-loop radiative effects, respectively. The last two
= Z2 2ln(Zα)+ 62 8ln k0 δ3(r ) parts H(6) and H(6) are the spin-dependent opera-
E2 "−3 9 − 3 (cid:18)Z2(cid:19)# a h a i torsfirstfsderivedbyfs,DamomuglasandKroll[14]. Theydonot
X
14Z2 contributetothe energiesofthe singletstatesandto the
Q , (17)
a spin-orbit averaged levels. Expressions for these opera-
− 3
Xa tors are well known and are given, e.g., by Eqs. (3) and
e (7)ofRef.[15]. Thenon-radiativepartofthemα6 effec-
with Q defined analogously to Eq. (11), and (m/M)
a E3 tive Hamiltonian is rather complicated. For simplicity,
is the first-order perturbation of ∞(5) due to the mass- we present it specifically for a two-electron atom. The
E
polariezation operator (6). The operator H(5) yields a corresponding expression reads [6, 7]
fs,rec
4
3 3 1 2Z Zπ
H(6) = E0 + + p~2+ − δ3(r )+(1 2)
nrad − 2 −E0 2 2 r 4 1 ↔
(cid:20)(cid:18) 2 (cid:19) (cid:21)
P~2 (3+~σ ~σ ) Z Z π
+ πδ3(r) 1· 2 π~pδ3(r)p~ + δ3(r)
6 − 24 − r r 2
(cid:18) 1 2(cid:19)
13 8 3 39ζ(3) 2+2 (4)
+ + ln(2) πδ3(r)+ E0 E
12 π2 − 2 − 4π2 4r
(cid:18) (cid:19)
2
(31+5~σ ~σ ) Z Z Z Z
0 1 2 0 0
E · E + + E +
−r2 32 − 2r r r 4 r r
(cid:18) 1 2(cid:19) (cid:18) 1 2(cid:19)
2
1 Z Z 1 (23+5~σ ~σ ) 1 Z Z
1 2
+ · +
−r2 r r − r 32 − 4r r r
(cid:18) 1 2 (cid:19) (cid:18) 1 2(cid:19)
Z2 Z Z 1 ~r ~r ~r (13+5~σ ~σ )
1 2 1 2
+ + + Z ·
2r r E0 r r − r − r3 − r3 · r3 64
1 2 (cid:18) 1 2 (cid:19) (cid:18) 1 2(cid:19)
Z ~r ~r ~r Z2 ri (rirj 3δijr2) rj
+ 1 2 1 − 2
4 r3 − r3 · r2 − 8 r3 r r3
(cid:18) 1 2(cid:19) 1 2
Z2 1 Z2 1 1 (47+5~σ ~σ )
+ p~2+ p~ ~p +p~ p~ 1· 2 +(1 2)
8 r2 2 8 1 r2 1 1 r2 1 64 ↔
(cid:20) 1 1 (cid:21)
1 Z Z (rirj +δijr2) (3rirj δijr2) ( 3+~σ ~σ )
+ pi + pj +Pi − Pj − 1· 2
4 1 r r r3 2 r5 192
(cid:18) 1 2(cid:19)
Z ri ri rj rk rirjrk
pk 1 δjk δik δij pj +(1 2)
− 8 2 r3 r − r − r − r3 2 ↔
(cid:20) 1 (cid:18) (cid:19) (cid:21)
1 Z Z 1 1
E0 p2p2 p2 + p2+ ~p p~ p~ p~
− 8 1 2− 4 1 r r 2 4 1× 2 r 1× 2
(cid:18) 1 2(cid:19)
1 rirk rjrl rirjrkrl
+ pkpl δjl δik +3 pi pj +ln(Z)πδ3(r), (20)
8 1 2 − r3 − r3 r5 1 2
(cid:18) (cid:19)
where P~ = p~ +p~ , ~p = (p~ p~ )/2, ~r = ~r ~r . We The second-order correction can be represented as
1 2 1 2 1 2
− −
intosteextpheactttahteioonpvearalutoesrHdon(e6ra)sdnisotdecfionnetadininasnuychloagwaraiythtmhaict Esec = Hn(4fs)′(E 1H )′Hn(4fs)′
terms in the 1/Z expansion, as the last term of Eq. (20) (cid:28) 0− 0 (cid:29)
1
is compensated by the corresponding contribution from +2 H(4) H(4)
the 1/r3 operator. (cid:28) nfs(E0−H0)′ fs (cid:29)
The effective Hamiltonians induced by the radiative 1
+ H(4) H(4) , (23)
effects are [6, 16, 17] fs (E H )′ fs
(cid:28) 0− 0 (cid:29)
427 where H(4) and H(4) are the spin-independent and spin-
HR(61) =Z2 96 −2 ln(2) π δ3(r1)+δ3(r2) dependenntfsparts offsthe Breit-Pauli Hamiltonian (7), re-
6ζ(cid:20)(3) 697 (cid:21) (cid:2) 1099 (cid:3) spectively. TheoperatorHn(4fs)′ isobtainedfromHn(4fs) bya
+ 8 ln(2)+ πδ3(r), (21) transformationthateliminatesdivergencesinthesecond-
π2 − 27π2 − 72
(cid:20) (cid:21) order matrix elements [6]. The transformed operator is
given by
and
H(4)′ = 1( V)2 pi 1 δij + rirj pj
H(6) =Z 9ζ(3) 2179 + 3 ln(2) 10 π nfs −2 E0− − 1 2r (cid:18) r2 (cid:19) 2
R2 − 4π2 − 648π2 2 − 27 1 Z ~r Z ~r
(cid:20) (cid:21) + ~2 ~2 1 ~ 2 ~ , (24)
δ3(r )+δ3(r ) + 15ζ(3) + 631 4∇1∇2− 4 r13 ·∇1− 4 r23 ·∇2
× 1 2 2π2 54π2
(cid:20) where ~2 ~2 is understood as a differentiation of the
(cid:2) 29 (cid:3) ∇1∇2
5 ln(2)+ πδ3(r). (22) wavefunctionontherighthandsideasafunction(omit-
− 27 ting δ3(r)) and V = Z/r Z/r +1/r.
(cid:21) 1 2
− −
5
An intriguing feature ofthe formulas presentedin this thus [5]
section is that the logarithmic dependence of all of them
entersonlyinthe formofln(Zα). Thisisnotatallobvi- (7+) (21P )= (7+)(1s)+ 2 (7+)(2p )+ 1 (7+)(2p ),
ousapriorisinceln(Zα)appearsnaturallyonlyincontri- Erad,H 1 Erad 3Erad 3/2 3Erad 1/2
butionsinducedbytheelectron-nucleusinteraction. The (7+) (23P )= (7+)(1s)+ 1 (7+)(2p )+ 2 (7+)(2p ).
effects of the electron-electron interaction usually yield Erad,H 1 Erad 3Erad 3/2 3Erad 1/2
logarithms of α, whereas logarithms of Z are implicitly (29)
present in matrix elements of singular operators. The
In our calculation, the one-electron remainder func-
fact that logarithms of α and logarithms of Z have the
coefficients that match each other comes “accidentally” tion (7+)(nlj)includes allknowncontributions of order
from the derivation. mα7Earandd higher coming from (i) the one-loop radiative
The complete result for the corrections of order mα7 correction,(ii)thetwo-loopradiativecorrection,(iii)the
for the helium Lamb shift is not presently available (it three-loopradiativecorrection,seethereviewinRef.[19]
is known for the fine-structure splitting only [15, 18]). and Ref. [20] for an update on the two-loop remainder
One can, however, easily generalize some of the hydro- function.
genic results, namely those that are proportional to the Besides the finite-nuclear-size and radiative correc-
electron density at the nucleus. These are (i) the one- tions, there are also non-radiative effects, denoted as
loop radiative correction of order mα(Zα)6 ln2(Zα)−2, (7+) and estimated within the 1/Z expansion. More
Enrad
(ii)thetwo-loopradiativecorrectionofordermα2(Zα)5, specifically, we include the higher-order remainder due
and (iii) the nonrelativistic correction due to the finite to the one-electron Dirac energy and due to the one-
nuclear size. The first two effects yield the main contri- photon exchange correction. They enter at the order
butiontothehigher-orderremainderfunctionofS states mα8 only but are enhanced by factors of Z8 and Z7,
in light hydrogen-like ions. We expect that they domi- respectively. Despite this enhancement, numerical con-
nate for light helium-like ions as well. tributions of these effects are rather small for the ions
FollowingRef.[5],weapproximatethehigher-orderra- considered in the present work.
diative (“rad”) and the finite-nuclear-size (“fs”) correc-
tion to the energies of helium-like ions by
III. RESULTS AND DISCUSSION
δ3(r )
(7+) = (7+) i i , (25)
Erad Erad,H δ3(r ) A. Numerical results
(cid:10)Pi i (cid:11)H
E = E (cid:10)Piδ3(ri)(cid:11) , (26) Thenonrelativisticenergiesandwavefunctionsareob-
fs fs,H δ3(r )
(cid:10)Pi i (cid:11)H tained by minimizing the energy functional with the ba-
sis set constructed with the fully correlated exponential
(cid:10)P (cid:11)
where the subscript H corresponds to the “hydrogenic”
functions. The choice of the basis set and the general
limit, i.e., the limit of the non-interacting electrons, and
strategy of optimization of the nonlinear parameters fol-
low the main lines of the approach developed by Ko-
Z3 δ
δ3(r ) = 1+ l,0 . (27) robov [21, 22]. The calculational scheme is described in
i H π n3 previous publications [6, 7, 15] and will not be repeated
i (cid:18) (cid:19)
(cid:10)X (cid:11) here. Numerical values of the nonrelativistic energies of
The approximationof Eqs.(25)and (26) is exact for the helium-like ions with the nuclear charge Z =3...12 are
above-mentionedcorrectionsproportionaltotheelectron presented in Table I. The results were obtained with
density at the nucleus. It is expected also to provide N = 2000 basis functions and are accurate to about 18
a meaningful estimate for contributions that weakly de- decimals (more than shown in the table). The energy
pend on n (such as the nonlogarithmic radiative correc- levelsofthe heliumatomtraditionallyattractspecialat-
tion of order mα(Zα)6). Moreover, this approximation tention, so we present the corresponding results in full
is exact to the leading order in the 1/Z expansion, thus length below. Our numerical values of the upper varia-
providing a meaningful estimate for high-Z helium-like tional limit of the nonrelativistic energies of helium are
ions as well.
For allthe states under considerationexcept21P and (2)(11S) = 2.903724377034119598310+0, (30)
1 E∞ − −2
23P1, the “hydrogenic” remainder function is just the (2)(21S) = 2.145974046054417415799+0, (31)
sum of the corresponding remainders for the two elec- E∞ − −8
trons in the configuration, E∞(2)(23S) =−2.175229378236791305738977−+20,
(32)
Er(a7d+,H) =Er(a7d+)(1s)+Er(a7d+)(nlj). (28) E∞(2)(21P) =−2.123843086498101359246−+20, (33)
(2)(23P) = 2.13316419077928320514696+0 .
For the 21P1 and 23P1 states, the Dirac levels need to E∞ − −10
(34)
be first transformed from the jj to the LS coupling and
6
The value for the ground state is given only for com- thelinesdescribedinRefs.[6,7]. Hereweonlynotethat
pleteness,sincemuchmoreaccuratenumericalresultsare calculations for higher values of Z often exhibit a slower
availablein the literature [22, 23]. The numericalresults numericalconvergence(andnumericalstability)thanfor
fortheleadingrelativisticcorrection (4) aresummarized helium, especially so for the second-order corrections in-
E
in Table II. volving singular operators. The variationaloptimization
The leading QED correction (5) is given by Eqs. (9), of nonlinear parameters for the symmetric second-order
(10), and (15). ComputationalEly the most problematic corrections was performed in several steps with increas-
part of it is represented by the Bethe logarithm ln(k ) ing the number of basis functions on each step up to
0
and its mass-polarization correction ln(k ) . Accurate N = 1000 or 1200. The final values were obtained with
0 M
calculationsofln(k )wereperformedbyDrakeandGold- merging several basis sets and enlarging the number of
0
man [24] for helium-like like atoms with Z 6 and by functions up to N = 5000 7000. The nonsymmetric
Korobov[25]forZ =2. Calculationsofthere≤coilcorrec- second-order corrections we−re evaluated as described in
tion to the Bethe logarithm were reported by Pachucki Ref.[15]. Thecalculationswereperformedinthequadru-
and Sapirstein [10] for Z = 2 and by Drake and Gold- ple, sixtuple, and octuple arithmetics implemented in
man [24] for Z 6. In the present investigation,we per- Fortran 95 by V. Korobov [27].
form accurate e≤valuations of the Bethe logarithm ln(k ) TableVIIpresentstheresultsforthefinitenuclearsize
0
anditsrecoilcorrectionln(k ) forhelium-likeionswith correction and approximate values of the higher-order
0 M
Z 12. The calculational approach is described in Ap- (mα7 and higher) correction to the ionization energy.
pen≤dix A. The uncertainty of the total theoretical prediction origi-
nates from the higher-order radiative effects; it was esti-
Table III summarizes the numerical results obtained
matedbydividingtheabsolutevalueofthiscorrectionby
and gives a comparison with the previous calculations.
Z. The values for the root-mean-square radius of nuclei
Numerical values for the Bethe logarithm are presented
forthedifferenceln(k ) ln(Z2)sincethisdifferencehasa were taken from Ref. [28].
0
−
weakZ-dependenceanddoesnotcontainanylogarithms
in its 1/Z expansion. The table also lists the coefficients
of the 1/Z expansion of ln(k /Z2). The leading coef- B. Comparison with the all-order approach
0
ficient c is known from the hydrogen theory; accurate
0
numerical values can be found in Ref. [26]. The higher- In this subsection we discuss the calculational results
order coefficients were obtained by fitting our numerical obtained for the mα6 correction in more detail and
data. It is interesting to compare them with the analo- make a comparison with the results obtained previously
gous results reported previously by Drake and Goldman within the all-order, 1/Z-expansion approach. The log-
[24]. For the next-to-the-leading coefficient c , the re- arithmic part of the correction, (6)(log), behaves as
sults agree up to about 4-5 digits for S states a1nd about mα3(Zα)3 forlargeZ andthuscorErespondstodiagrams
3-4 digits for P states. For the higher-order coefficients, with three photon exchanges that have not yet been ad-
the agreement gradually deteriorates. However, the re- dressed within the all-order approach. The nonlogarith-
sults for the sum of the two expansions agree very well mic part (6)(nlog), however, contains some pieces that
E
witheachother. Morespecifically,the maximalabsolute are known and identified below.
deviationbetweenthe values of the Bethe logarithmsfor Forallstatesexcept21P and23P ,theleadingtermof
1 1
Z >12obtainedwithour1/Z-expansioncoefficientsand the 1/Z expansion of (6)(nlog) is of order m(Zα)6 and
E
withthosebyDrakeandGoldmanis1 10−8forthe11S comes from the Zα expansion of the one-electron Dirac
state, 3 10−8 for the 21S state, 6 ×10−9 for the 23S energy. Forthe21P and23P states,theleadingtermis
1 1
state,1 ×10−7forthe21P state,and6× 10−8forthe23P ofthepreviousorderin1/Z,m(Zα)6Z,andisduetothe
× ×
state. So, the accuracy of these expansions is sufficient mixing of these levels. More specifically,the mixing con-
for most practical purposes. tribution is δE = 21P H(4) 23P 2/[E (21P )
mix 1 1 0 1
Another part of the calculation of (5) that needs a E (23P )] for the 21P shtate a|nd t|hat wiith the opposit−e
E 0 1 1(cid:12) (cid:12)
separate discussion is the evaluation of the expectation sign, for 23P . The co(cid:12)ntribution of ord(cid:12)er m(Zα)6 for
1
value of the singular operator 1/r3, which is defined by the mixing states comes from the expansion of the one-
Eq.(12). The calculationalapproachis describedin Ap- electron Dirac energy and from the expansion of δE .
mix
pendix B. Total results for the logarithmic and the non- The next term of the 1/Z expansion is of order
logarithmic partof the leading QED correctionare sum- mα(Zα)5 and comes from the one-electron one-loop ra-
marizedinTablesIVandV,respectively. Theresultsare diativecorrectionandfromtheone-photonexchangecor-
in good agreement with the previous calculations [5]. rection. The radiative part is well known [19]. The part
TableVIpresentsthenumericalvaluesofthemα6 cor- duetotheone-photonexchangewasobtainedforthe11S,
rection, the main result of this investigation. The corre- 23S, 23P , and 23P states analytically in Ref. [29] and
0 2
sponding calculations for atomic helium were reported for the other states numerically in this work. For the
in Refs. [6, 7]; our present numerical values agree with 21P and 23P states, there is a small additional mixing
1 1
the ones obtained previously. Calculations performed in contribution, which we were unable to determine unam-
this work for helium-like ions were accomplished along biguously.
7
The first two coefficients of the 1/Z expansion of C. Total energies
(6)(nlog) are listed in Table VI. A fit of our numeri-
E
cal data agrees well with these coefficients. The agree- Ourtotalresultsfortheionizationenergyofthen=1
mentobservedshowsconsistencyofournumericalresults and n = 2 states of helium-like atoms with the nuclear
with independent calculations at the level of the one- charge Z = 2...12 are listed in Table VIII. The fol-
photoneffects. We nowturntothe contributionoforder lowing values of fundamental constants were employed
mα2(Zα)4. This contribution is induced by nontrivial [19], R = 10973731.568527(73) m−1 and α−1 =
∞
two-photon effects, so that a comparison drawn for this 137.0359999679(94). The atomic masses were taken
part will yield a much more stringent test of consistency from Ref. [30].
of different approaches. The results for atomic helium presented in Table VIII
The part of (6)(nlog) that is of order mα2(Zα)4 is differ from those reported previously only because of
E
implicitly present in the two-electron QED contribution the different approximate treatment of the higher-order
calculated numerically in Ref. [8] to all orders in Zα. (mα7 and higher) contribution employed in this work.
This contribution can be represented as (see Eq. (72) of For the S states of helium, the present values are prac-
Ref. [8]) tically equivalent to those of Refs. [6, 7]. (The difference
is that now we include some contributions of order mα8
∆E2QeElD =mα2(Zα)3 a31 ln(Zα)−2+a30+(Zα)G(Z) , and higher, which are negligible for helium but become
h (35)i noticeable for higher-Z ions.) However, for the helium
P states,ourpresentestimateofthehigher-ordercontri-
where the remainder function G(Z) incorporates all bution is by about 1 MHz higher than that of Ref. [7].
higher orders in Zα. The two-electron QED correction The reason is that the one-electron radiative correction
comprisestheso-calledscreenedself-energyandvacuum- of the p electron state was previously not included into
polarizationcontributionsandthepartofthetwo-photon the approximation(25). Itis included now [see Eq.(29)]
exchangecorrectionthat is beyondthe Breitapproxima- in order to recover the correct asymptotic behaviour of
tion. the radiative correction in the high-Z limit.
The coefficients a and a in Eq. (35) correspond to A selection of our theoretical results for transition en-
31 30
thesecondtermofthe1/Z expansionoftheleadingQED ergies is compared with the theory by Drake [5, 31] and
correction (5). More specifically, a correspondsto the withexperimentaldatainTableIX. Agreementbetween
∞ 31
E theory and experiment is excellent in all cases studied.
coefficient c from Table IV and a , to that from Ta-
1 30
We observe a distinct improvement in theoretical accu-
ble V. The Z = 0 limit of the higher-order remainder
racy as comparedto the previous results by Drake. This
function G(Z) corresponds to the third coefficient of the
1/Z expansion of the correction (6)(nlog), G(0) = c , improvementisduetothecompletetreatmentofthecor-
for all states except 21P and 23EP . The values of c2 rectionsofordermα6 andm2/Mα5 accomplishedinthis
1 1 2
work.
obtained by fitting our numerical data are listed in Ta-
bleVI. Forthe 21P and23P states,the coefficientc is Theoretical results for the fine structure splitting in-
1 1 2 tervals 23P 23P and 23P 23P are not analyzed
notdirectlycomparablewiththeall-orderresultsbecause 0 1 1 2
− −
in the present work. This is because these intervals can
of the mixing effects.
nowdays be calculated more accurately (complete up to
The higher-order remainder function G(Z) inferred
ordermα7),likeitwasrecentlydoneforhelium[15]. We
fromthe numericalresults ofRef.[8]is plottedin Fig.1,
intend to perform such a calculation in a subsequent in-
together with its limiting value at Z = 0 obtained by
vestigation.
a fit of our numerical data. It should be stressed that
Among the results listed in Table VIII for helium-like
the identification of the remainder implies a great deal
ions,the ground-stateenergyofthe carbonionis ofpar-
ofnumericalcancellations,especiallyforthe all-orderre-
ticular importance, because it is used in the procedure
sults. The comparison drawn in Fig. 1 provides a strin-
of the adjustment of fundamental constants [32] for the
gent cross-check of the two complementary approaches.
determination of the mass of 12C4+ and, consequently,
The visual agreement between the results is very good
of the proton mass from the Penning trap measurement
for the S states, whereas for the P states a slight dis-
by Van Dyck et al. [33]. Our result for the ground-state
agreement seems to be present.
ionization energy of helium-like carbon is
It is tempting to merge the all-order and the Zα-
expansion results by fitting the all-order data for G(Z) E(12C4+)= 3162423.60(32)cm−1, (36)
towards lower values of Z. However, we do not attempt −
to do this in the present work. The reasons are, first, which is in agreement with the previous result by Drake
that the numerical accuracy of the all-order results is [5] of 3162423.34(15)cm−1. We note that despite the
−
apparently not high enough and, second, that the ex- fact that our calculation is by an additional order of α
pansion of the remainder function G(Z) contains terms more complete than that by Drake, our estimate of un-
(Zα)ln2(Zα) and (Zα)ln(Zα), which cannot be reason- certainty is more conservative.
ably fitted with numerical data available in the high-Z In summary, significant progress has been achieved
region only. during last decades both in experimental technique and
8
theoretical calculations of helium-like atoms. In the andR(κ)isasimplefunctionobtainedbyintegratingout
present investigation, we performed a calculation of the the separated asymptotic expansion terms,
energy levels of the n = 1 and n = 2 states of light
helium-like atoms with the nuclear charge Z = 2...12, ~2 4√2Z 2Z2(lnκ+1)
R(κ)=κ ∇ +ln(2κ)+ .
within the approach complete up to orders mα6 and D κ1/2 − κ
(cid:10) (cid:11)
m2/Mα5. An extensive analysis of the 1/Z expansion (A5)
ofindividualcorrectionswascarriedoutandcomparison
The calculational scheme employed for the evaluation
with results of the complementary approach was made
ofEq.(A3) is similarto thatpreviously used[15] for the
whenever possible. Our general conclusion is that the
relativistic corrections to the Bethe logarithm. At the
results obtained within the approaches based on the Zα
first step, a careful optimization of nonlinear basis-set
and the 1/Z expansion are consistent with each other
parameterswascarriedoutforasequenceofscalesofthe
up to a high level of precision. However, further im- photon momentum: k = 10i and i = 1,...,i , with
i max
provement of numerical accuracy of the all-order, 1/Z-
i = 5 for the S states and i = 4 for the P states.
max max
expansion results and their extension into the lower-Z
The optimization was performed with incrementing the
regionisneededinordertosafelymergethetwocomple-
size of the basis until the prescribedlevel of convergence
mentary approaches.
isachievedforthefunctionw(k). Atthesecondstep,the
integrationsofthe photonmomentumk wereperformed.
For a given value of k, the function J(k) was calculated
Acknowledgments
witha basisobtainedby mergingtogether the optimized
bases for the two closest k points, thus essentially dou-
i
We wish to thank Tomasz Komorek for participation
bling the number of the basis functions. The function
in the beginning of the project. Computations reported w(k) was obtained from J(k) according to Eq. (A4).
in this work were performed on the computer clusters
Theintegraloverk [0,κ]wascalculatedanalytically,
of St. Petersburg State Polytechnical University and of by diagonalizing the H∈amiltonian matrix and using the
Institute of Theoretical Physics, University of Warsaw.
spectral representation of the propagator. The value of
The research was supported by NIST through Precision theauxiliarparameterκwassettoκ=100. Theintegral
Measurement Grant PMG 60NANB7D6153. V.A.Y. ac- over k [κ, ) was separated into two parts, k <10imax
knowledgesadditionalsupportfromthe“Dynasty”foun- and k ∈> 10∞imax. The first part was evaluated numeri-
dation and from RFBR (grant No. 10-02-00150-a).
callybyusingtheGauss-Legendrequadratures,afterthe
change of variables t = 1/k2. The second part was ob-
tained by integrating the asymptotic expansion of the
Appendix A: Bethe logarithm
function w(k). The coefficients of this expansion were
obtained by fitting the numerical data for w(k) to the
FollowingtheapproachofRefs.[34,35],theBethelog- form
arithm (13) is expressed in terms of an integral over the
1 lnk 1
momentum of the virtual photon, w(k)=pol + pol , (A6)
√k k k
(cid:18) (cid:19) (cid:18) (cid:19)
1 K
ln(k )= lim ~2 K+D ln(2K)+ dkkJ(k) , where pol(x) denotes a polynomial of x. The total num-
0
D K→∞" ∇ Z0 # ber of fitting parameterswas about9 11. The rangeof
(cid:10) (cid:11) (A1) k to be fitted and the exact form of t−he fitting function
wereoptimizedsoastoyieldthe bestpossibleresultsfor
where D =2πZ δ3(r )+δ3(r ) , ~ ~ + ~ , and
1 2 1 2 the known asymptotic expansion terms of J(k).
∇≡∇ ∇
Thefirst-orderperturbationoftheBethelogarithmby
(cid:10) 1(cid:11)
J(k)= ~ ~ . (A2) themass-polarizationoperatorcanberepresentedas[10]
∇E H k ∇
(cid:28) 0− 0− (cid:29)
For the purpose of numerical evaluation, the integration m 1 κ
ln(k ) = R (κ)+ dkkJ (k)
0 M M M
overthe photonmomentumk isdividedinto tworegions M" D Z0
by introducing the auxiliar parameter κ,
∞ w (k)
1 κ ∞ w(k) + dk M , (A7)
ln(k0)=R(κ)+ D dkkJ(k)+ dk k2 , (A3) Zκ k2 #
Z0 Zκ
where
wherethefunctionw(k) representstheresidualobtained
from J(k) by removing all known terms of the large-k J (k)= 2 φ ~ 1 ~ δφ
asymptotics, M ∇E H k ∇
(cid:28) (cid:12) 0− 0− (cid:12) (cid:29)
w(k)= k3J(k)+ k2 ~2 +k 2√2Zk1/2+2Z2lnk, + φ ~ 1(cid:12)(cid:12)(cid:12) δE p~1 p~(cid:12)(cid:12)(cid:12)2 1 ~ φ ,
D D ∇ − (cid:28) (cid:12)∇E0−H0−k − · E0−H0−k ∇(cid:12) (cid:29)
(cid:10) (cid:11) (A4) (cid:12)(cid:12) (cid:2) (cid:3) (cid:12)(cid:12)(A8)
(cid:12) (cid:12)
9
and δE = ~p p~ . The perturbed wave function δφ is expand it in small ǫ. This way is possible, but we prefer
1 2
·
defined by to use a simpler procedure, which is also the closest to
(cid:10) (cid:11)
the method of evaluation of the regular integrals.
δ D
δφ = δ φ φ M , (A9) Werecallthatallregularintegralsareimmediatelyob-
M
| | −| D tained from the master integral
(cid:11) (cid:11) (cid:11)
whereδ standsforthe first-orderperturbationinduced
M
1 exp( αr βr γr)
JbyM(tkh)eiospmeruacthorsip~m1p·lep~r2.thaTnhethaastyomfpJt(okti)caenxdpawnMsi(okn)oisf 16π2 Z d3r1d3r2 − r11−r2r 2−
just 1
= (B2)
(α+β)(β+γ)(α+γ)
k3 k2
w (k)= J (k)+ 2 φ ~2 δφ . (A10)
M M
D D |∇ | by formal differentiation or integration with respect to
(cid:10) (cid:11) the corresponding parameters. The differentiation over
Correspondingly, the function R (κ) is
M α and β and an integration over γ delivers a result for
2κ the integral of the type 1/r2. This integral is conver-
RM(κ)= φ ~2 δφ . (A11) gent, so the result is exact. The second integration over
D |∇ |
γ (which would yield an integral of the type 1/r3) is di-
(cid:10) (cid:11)
The numerical evaluation of Eq. (A7) was performed in vergent. The simplest way to proceed is as follows. We
a way similar to that for the plain Bethe logarithm. In introduce a cutoff parameter for large values of γ, eval-
particular, the same sets of optimized nonlinear param- uate the integral over γ, and drop all cutoff-dependent
eters were used. Since a high accuracy is not needed terms. The expression obtained in this way differs from
for this correction, a somewhat simplified calculational the correct one by a γ-independent constant only, which
scheme was used in this case. The high-energy part of can be proved by differentiating with respect to γ.
the photon-momentum integral, k [100, ), was eval- The missing constant is most easily recovered by con-
∈ ∞
uated by integrating the fitted asymptotic expansion for sideringthe behaviorof the integral I when γ . For
→∞
w (k),whichwastakentobeoftheform(A6)with6 9 very large γ, only the region of very small r contributes
M
−
fitting parameters. and we have
∞ ∞ r1+r e−αr1−βr2−γr
Appendix B: Expectation value of 1/r3 Iǫ =2 drr dr1r1 dr2r2 r3
Zǫ Z0 Z|r1−r|
2 ∞ e−γr
The definition of the expectation value of the regular- dr . (B3)
ized operator 1/r3 is given by Eq. (12). With the basis- ≈ (α+β)3 Zǫ r
set representation of the wave function employed in this
Therefore,
work, a typical singular integral to be calculated is
1 exp( αr βr γr) γ→∞ 2
Iǫ = 16π2 d3r1d3r2 − 1r−3 2− Θ(r−ǫ). Ireg ≡ǫl→im0 Iǫ+γ+lnǫ = −(α+β)3 lnγ. (B4)
Z
(B1) (cid:2) (cid:3)
The above equation yields the necessary condition for
The straightforward way is to evaluate this integral an- determiningthe missingconstantterminthe generalex-
alytically for a finite value of the regulator ǫ and then pression for the regularized integral I .
reg
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-3.0 -0.94
-0.40
-3.2
-0.96
-3.4 -0.98 -0.45
Z) -3.6
G( -3.8 11S -1.00 21S -0.50 23S
-4.0 -1.02
-4.2 -1.04 -0.55
-4.4 -1.06 -0.60
0 10 20 30 40 50 60 70 80 90 100 0 10 20 30 40 50 60 70 80 90 100 0 10 20 30 40 50 60 70 80 90 100
Z Z Z
-0.6
-0.13
-0.7 23P
2
) -0.14
Z -0.8
G( 23P
-0.9 0 -0.15
-1.0 -0.16
0 10 20 30 40 50 60 70 80 90 100 0 10 20 30 40 50 60 70 80 90 100
Z Z
FIG.1: Thehigher-orderremainderfunctionG(Z)fromEq. (35)inferredfromtheall-ordernumericalresultsofRef.[8]forthe
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