Precise theory of levels of hydrogen and deuterium:

The one-photon self energy correction

Ulrich D. Jentschura,1, 2 Svetlana Kotochigova,1 Eric-Olivier Le Bigot,1, 3 and Peter J. Mohr1

1National Institute of Standards and Technology, Mail Stop 8401, Gaithersburg, MD 20899-8401, USA

2Max–Planck–Institut f ̈ur Kernphysik, Saupfercheckweg 1, 69117 Heidelberg, Germany

3Laboratoire Kastler Brossel, Ecole Normale Sup ́erieure et Universit ́e Pierre et Marie Curie, Case 74  ́

4, pl. Jussieu, 75005 Paris, France

(Dated: April 19, 2005)

The one-photon self energy is the dominant quantum electrodynamic correction to the energy

levels of hydrogen and deuterium atoms. Calculations of this correction are surveyed, and numerical values, based on extrapolations from higher values of the charge of the nucleus and plausible

approximations, are provided for states with n ≤ 200. The results of an evaluation of the energy

levels and transition frequencies in hydrogen and deuterium, based on values of the self energy given

here, are available on the NIST Physics Laboratory Web site at physics.nist.gov/hdel.

PACS numbers: 12.20.Ds, 31.30.Jv, 06.20.Jr, 31.15.-p

I. INTRODUCTION

Energy levels in hydrogen and deuterium are determined primarily by the eigenvalues of the Dirac equation, but to obtain accurate values for the levels, it is

necessary to include many additional corrections including those due to quantum electrodynamic (QED) effects.

In hydrogen and deuterium, the largest QED effect is

the self-energy, due to the emission and reabsorption of

a virtual photon by the electron.

II. SELF ENERGY

The level shift due to the one-photon self energy is

given by

E

(2)

SE =

α

π

(Zα)

4

n3

F(Zα) mec

2

, (1)

where

F(Zα) = A41 ln(Zα)

−2 + A40 + A50 (Zα)

+A62 (Zα)

2

ln2(Zα)

−2 + A61 (Zα)2

ln(Zα)

−2

+GSE(Zα) (Zα)

2

, (2)

with [1]

A41 =

4

3

δl0

A40 = −

4

3

ln k0(n, l) + 10

9

δl0 −

1

2κ(2l + 1)(1 − δl0)

A50 =



139

32

− 2 ln 2π δl0 (3)

A62 = −δl0

A61 =



4



1 +

1

2

+ · · · +

1

n



+

28

3

ln 2 − 4 ln n

−

601

180

−

77

45n2



δl0 +



1 −

1

n2

 

2

15

+

1

3

δj

1

2



δl1

+

96n

2 − 32l(l + 1)

3n2(2l − 1)(2l)(2l + 1)(2l + 2)(2l + 3)(1 − δl0) .

In Eqs. (1)–(3), n is the principal quantum number

of the state, κ is the Dirac angular momentum-parity

quantum number, l = |κ +

1

2

| −

1

2

is the nonrelativistic

angular momentum, and j = |κ| +

1

2

is the total angular

momentum. Bethe logarithms ln k0(n, l) that appear in

Eq. (3) needed for this work are given in an accompanying

paper.

III. THE COEFFICIENT A60

The next coefficient in the series in Eq. (2) is A60,

which is given by

lim

Zα→0

GSE(Zα) = A60 (4)

There is a long history of calculations of A60 [2, 3], leading up to the most accurate values of A60 for the 1S and

2S states [4]. These values are given in Tables I and

II, along with the results of subsequent calculations for

states with higher angular momentum. The references

for the data in Tables I and II are listed in Table III.

Asymptotic formulas for A60 for the P and D states

for values of n higher than those in Table I are given in

Ref [9] as

A60(P1/2) = −1.249(9) + 0.0(2)

n

+

0.87(45)

n2

+ · · ·

A60(P3/2) = −0.69(2) + 0.15(5)

n

+

0.25(25)

n2

+ · · ·

A60(D3/2) = 0.011(1) −

0.032(7)

n

−

0.05(9)

n2

+ · · ·

A60(D5/2) = 0.034(2) + 0.025(5)

n

−

0.18(4)

n2

+ · · ·

(5)

NIST Technical Note 1469
2

TABLE I: Calculated values of the constant A60 for S, P, and D states.

n S1/2 P1/2 P3/2 D3/2 D5/2

1 −30.924 15(1)

2 −31.840 47(1) −0.998 904 402(1) −0.503 373 464(1)

3 −1.148 189 956(1) −0.597 569 388(1) 0.005 551 575(1) 0.027 609 989(1)

4 −1.195 688 142(1) −0.630 945 796(1) 0.005 585 985(1) 0.031 411 862(1)

5 −1.216 224 512(1) −0.647 013 509(1) 0.006 152 175(1) 0.033 077 571(1)

6 −1.226 702 391(1) −0.656 154 893(1) 0.006 749 745(1) 0.033 908 493(1)

7 −1.232 715 957(1) −0.662 027 568(1) 0.007 277 403(1) 0.034 355 926(1)

8 0.007 723 850(1) 0.034 607 492(1)

TABLE II: Calculated values of the constant A60 for F and G states.

n F5/2 F7/2 G7/2 G9/2

4 0.002 326 988(1) 0.007 074 961(1)

5 0.002 403 158(1) 0.008 087 020(1) 0.000 814 415(1) 0.002 412 929(1)

TABLE III: References for calculations of the coefficient A60.

States Reference

1S1/2, 2S1/2 Pachucki [4, 5]

2P1/2, 2P3/2 Jentschura and Pachucki [6]

3P1/2-4P1/2, 3P3/2-4P3/2 Jentschura et al. [7]

2P1/2-7P1/2, 2P3/2-7P3/2 Jentschura et al. [8]

3D3/2-8D3/2, 3D5/2-8D5/2 Jentschura et al. [8]

4F5/2-5F5/2, 4F7/2-5F7/2 Le Bigot et al. [9]

5G7/2, 5G9/2 Le Bigot et al. [9]

TABLE IV: Values of the constant A60 for P and D states

given by Eq. (5).

n P1/2 P3/2 D3/2 D5/2

5 −1.21(5) −0.65(3) 0.003(9) 0.032(8)

6 −1.22(4) −0.66(2) 0.004(5) 0.033(5)

7 −1.23(3) −0.66(2) 0.005(4) 0.034(4)

8 −1.24(3) −0.67(2) 0.006(3) 0.034(3)

9 −1.24(2) −0.67(2) 0.007(2) 0.035(3)

10 −1.24(2) −0.67(2) 0.007(2) 0.035(2)

11 −1.24(2) −0.67(2) 0.008(2) 0.035(2)

12 −1.24(2) −0.68(2) 0.008(1) 0.035(2)

13 −1.24(2) −0.68(2) 0.008(1) 0.035(2)

14 −1.24(2) −0.68(2) 0.008(1) 0.035(2)

15 −1.25(2) −0.68(2) 0.009(1) 0.035(2)

These expansions are obtained by making numerical extrapolations of the data in Table I. Numerical results

given by the asymptotic formulas are listed in Table IV.

The numbers in that table include a component of uncertainty of 1/n3

to account for the omitted higher order

terms in Eq. (5). The values of A60 provide a first approximation to the remainder function GSE(Zα) in the

cases where it has not been calculated numerically.

IV. THE REMAINDER FUNCTION GSE(Zα)

The function GSE(Zα) in Eq. (2) gives the complete

remainder of the self energy. Values for GSE(α) used

in the 2002 least-squares adjustment of the fundamental

constants are listed in Table V [10]. In that table, the

values for states with n = 1 and n = 2 are based on

direct numerical evaluations [11, 12]. More recently an

evaluation for the 3S and 4S states has been done with

the results [13]

GSE(α) = −31.047 7(9) 3S state (6)

GSE(α) = −30.912(4) 4S state (7)

These results are consistent with the values in Table V,

however they are not used in the current work in order

to maintain consistency of the results with the values of

the fundamental constants determined in the 2002 adjustment.

The values of GSE(α) for higher-n states are based on

the low-Z limit of this function, GSE(0) = A60, in the

cases where it is known, together with extrapolations of

the results of complete numerical calculations of F(Zα)

[see Eq. (2)] at higher Z [9, 14]. Extensive numerical

evaluations of F(Zα) at higher Z provide information

for the determination of GSE(Zα) [15–18].

For this work, Table V is extended to all states with

n ≤ 200. Table VI gives a partial listing of the larger

set of values for S, P, and D states, based on updated

extrapolations of numerical calculations and calculated

and estimated values of the coefficient A60. The values

in Table VI are used in the present calculation, unless

results are available from Table V, in which case, the

latter values are employed in order maintain consistency

with the 2002 least-squares adjustment.

For S states with n ≥ 13, we employ an approximate

extrapolation given by

GSE(α) = −30.7 −

0.7

n

± 0.2, (8)
3

TABLE V: Values of the function GSE(α) used in the 2002 least-squares adjustment of the fundamental constants.

n S1/2 P1/2 P3/2 D3/2 D5/2

1 −30.290 24(2)

2 −31.185 15(9) −0.973 5(2) −0.486 5(2)

3 −31.01(6)

4 −30.87(5) −1.165(2) −0.611(2) 0.031(1)

6 −30.82(8) 0.034(2)

8 −30.80(9) 0.008(5) 0.034(2)

12 0.009(5) 0.035(2)

TABLE VI: Enlarged table of values of the function GSE(α).

n S1/2 P1/2 P3/2 D3/2 D5/2

1 −30.290 24(2)

2 −31.185 15(9) −0.973 5(2) −0.486 5(2)

3 −31.047 7(9) −1.118(2) −0.578(2) 0.005 73(15) 0.027 79(18)

4 −30.912(4) −1.165(2) −0.608(2) 0.005 80(9) 0.031 63(22)

5 −30.80(10) −1.183(3) −0.625(2) 0.006 37(9) 0.033 32(25)

6 −30.80(10) −1.193(3) −0.634(2) 0.006 97(9) 0.034 17(26)

7 −30.79(10) −1.199(3) −0.640(2) 0.007 50(9) 0.034 57(22)

8 −30.78(10) −1.22(5) −0.65(3) 0.007 94(9) 0.034 84(22)

9 −30.78(15) −1.22(5) −0.65(3) 0.008(2) 0.035(3)

10 −30.77(15) −1.22(5) −0.65(3) 0.008(2) 0.035(3)

11 −30.77(15) −1.22(5) −0.65(3) 0.008(2) 0.035(3)

12 −30.76(15) −1.22(5) −0.65(3) 0.008(2) 0.035(3)

and for P and D states with n ≥ 13, we employ the

approximation

GSE(α) = A60 ± ∆, (9)

where A60 is the value given by Eq. (5), including the

uncertainty indicated there, with an additional contribution to the uncertainty ∆ given by 0.05, 0.05, 0.005, 0.005

for the P1/2, P3/2, D3/2, and D5/2 states, respectively.

For F5/2, F7/2, G7/2, and G9/2 states with n ≤ 5,

we use Eq. (9) with values for A60 from Table II with

uncertainties ∆ given by 0.0002, 0.0008, 0.00008, and

0.0002, respectively. For the corresponding states with

n ≥ 6, we use the values for n = 5 with uncertainties ∆

given by 0.001, 0.005, 0.0005, and 0.001, respectively.

For all states with l ≥ 5, i.e., H or higher, we use

GSE(α) = 0, with an uncertainty of 0.01.

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