Nuclear mass corrections for atoms and ions

Toshikatsu Koga *, Hisashi Matsuyama

Department of Applied Chemistry, Muroran Institute of Technology, Muroran, Hokkaido 050-8585, Japan

Received 31 August 2002; in final form 27 September 2002

Abstract

To take the nuclear motion contribution into account, we traditionally evaluate the normal mass enm and specific

mass (mass polarization) esm corrections separately and add them to the total electronic energy of atoms and ions

calculated in the fixed-nucleus approximation. We point out that the sum of these mass corrections emass ¼ enm þ esm is

immediately and rigorously obtained from the electron-pair moments in momentum space. The Hartree–Fock limit

values of the mass correction sum emass are reported for the 102 neutral atoms He through Lr, singly charged 53 cations

Liþ through Csþ, and 43 stable anions H� through I� in their experimental ground states.

� 2002 Elsevier Science B.V. All rights reserved.

1. Introduction

The non-relativistic total electronic energies E

of atoms and ions are customary calculated in the

fixed-nucleus (or infinite nuclear mass) approxi-

mation. When we discuss problems such as theisotope shift [1,2], however, we have to take the

contribution of the nuclear motion into account.

To do this, we usually evaluate two nuclear mass

corrections [3–5] separately by first-order pertur-

bation theory and add them to E: The first is the

normal mass correction enm given by

enm ¼ 1

2

1

l

�� 1

m

� XNi¼1

jpij2

* +; ð1Þ

and the second is the specific mass (or mass po-

larization) correction esm given by

esm ¼ 1

M

XNi<j¼1

pi � pj

* +; ð2Þ

where M and m are the nuclear and electron

masses, respectively, l ¼ mM=ðmþMÞ the elec-

tron reduced mass, N the number of electrons, and

pi is the momentum operator of the electron i. Thebrackets in Eqs. (1) and (2) stand for the expec-

tation values over the zero-order wavefunction for

the infinitely heavy nuclear mass. The normal mass

correction enm corrects the difference between mand l in the calculation of E, while the specific

mass correction esm arises from the separation of

the center-of-mass motion and internal motion

and reflects the fact that the individual electrons

affect one another�s motion.

As noted in the literature [3–5], however, the

sum emass of these mass corrections is rewritten as

Chemical Physics Letters 366 (2002) 601–605

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* Corresponding author. Fax: +81-143-46-5701.

E-mail address: koga@mmm.muroran-it.ac.jp (T. Koga).

0009-2614/02/$ - see front matter � 2002 Elsevier Science B.V. All rights reserved.

PII: S0009-2614 (02 )01646-9

emass ¼ enm þ esm ¼ 1

2M

XNi¼1

pi

����������2* +

: ð3Þ

In the present Letter, we first point out that the

mass correction sum emass can be immediately ob-

tained from the second electron-pair moments in

momentum space, without calculating the normal

enm and specific esm mass components separately.

Numerical results are then reported systematicallyfor 102 neutral atoms with 26N 6 103 and 96

singly charged ions with 26N 6 54 in their ex-

perimental ground states. Hartree atomic units are

used throughout.

2. Theoretical ground and computational outline

For N-electron (N P 2) atoms and molecules a

rigorous equality,

2ðN � 1ÞXNi¼1

pi

����������2* +

¼ ð2� NÞhv2i þ 4NhP 2i;

ð4Þhas been proved [6] very recently, where hv2i and

hP 2i are the second moments of the electron-pair

intracule (relative motion) and extracule (center-

of-mass motion) densities in momentum space

(see, e.g., [7]), respectively. Combining Eqs. (3) and

(4), we have a simple but exact relation,

emass ¼1

Mð2� NÞhv2i þ 4NhP 2i

4ðN � 1Þ ; ð5Þ

between the mass correction sum and the electron-

pair moments. For a particular case of N ¼ 2, Eq.

(5) is simplified to emass ¼ 2hP 2i=M , which clarifiesthat emass is proportional to the average center-of-

mass momentum squared of the electron pair.

The momentum-space electron-pair moments

hv2i and hP 2i are available in the literature [8–11]

at the Hartree–Fock limit level in a systematic

manner: For neutral atoms, the literature [8–10]

covers the 102 species from He ðN ¼ 2Þ to Lr

ðN ¼ 103Þ all in their experimental ground states[12,13]. For singly charged ions, the electron-pair

moments were compiled [11] for the 53 cations Liþ

ðN ¼ 2Þ through Csþ ðN ¼ 54Þ and the 43 anions

H� ðN ¼ 2Þ through I� ðN ¼ 54Þ, except the

anions of the group-2, -12, and -18 atoms, also in

their experimental ground states [12,14]. Two ex-

ceptions are Sc� and Pd� anions, for which the

second lowest states were examined [11] since

meaningful Hartree–Fock solutions were not ob-

tained [15] to the ground states.We have used these moment data in Eq. (5) to

obtain the Hartree–Fock limit values of emass for

the 102 neutrals and the 96 ions. As clear from Eq.

(5), however, the mass correction sum emass origi-

nates from the cancellation of the negative intra-

cule and positive extracule contributions.

Therefore, we used more significant figures for the

electron-pair moments than those tabulated in[8–11]. The accuracy and consistency of these

moment data were checked for each species by a

sum rule [16]

hv2i þ 4hP 2i � 2ðN � 1Þhp2i ¼ 0; ð6Þwhere hp2i is the single-electron second moment in

momentum space.

Since hp2i is twice the electronic kinetic energy

T , we note that alternative forms of Eq. (5) are

also derived by substitution of Eq. (6) into Eq. (5):

emass ¼1

Mð2�

� NÞT þ 2hP 2i�; ð7aÞ

emass ¼1

MNT

� 1

2hv2i

ð7bÞ

in which T can further be replaced with )E when

the virial theorem is satisfied.

3. Numerical results

The present numerical results are summarized

in Table 1 for the atoms and ions with 26N 6 54

and in Table 2 for the atoms with 556N 6 103.

Table 1 also includes the mass correction contri-butions to the ionization potentials and electron

affinities. We have reportedMemass values in Tables

1 and 2, because the values of M may improve as

time passes. We find in the tables that the Memass

values increase monotonically with increasing N or

atomic number Z. In fact, we find Memass is ap-

proximated by a quadratic function of Z. For the

102 neutral atoms, our regression analysis gives

602 T. Koga, H. Matsuyama / Chemical Physics Letters 366 (2002) 601–605

Table 1

Mass correction sums emass multiplied by nuclear masses M (in atomic units)a

Z Cation Neutral Anion Cation–neutral Neutral–anion

1 0.487930

2 2.86168

3 7.23642 7.43273 7.42823 )0.19631 0.00450

4 14.2774 14.5730 )0.29565 24.2376 24.1242 23.9835 0.1134 0.1407

6 36.4114 36.2944 36.1035 0.1170 0.1909

7 51.3383 51.2215 50.9530 0.1168 0.2685

8 69.1557 68.9597 68.6625 0.1960 0.2972

9 89.8882 89.6969 89.3708 0.1913 0.3261

10 113.753 113.569 0.184

11 140.888 141.056 141.051 )0.168 0.005

12 171.785 172.004 )0.21913 206.286 206.284 206.211 0.002 0.073

14 244.010 244.023 243.933 )0.013 0.090

15 285.263 285.294 285.145 )0.031 0.149

16 330.115 330.089 329.940 0.026 0.149

17 378.539 378.536 378.390 0.003 0.146

18 430.685 430.709 )0.02419 486.623 486.719 486.714 )0.096 0.005

20 546.393 546.516 )0.12321 608.931 609.084 608.384 )0.153 0.700

22 674.959 675.124 674.308 )0.165 0.816

23 743.515 744.723 743.813 )1.208 0.910

24 816.603 816.971 816.907 )0.368 0.064

25 894.552 894.741 893.679 )0.189 1.062

26 974.994 975.226 974.088 )0.232 1.138

27 1057.78 1059.39 1058.18 )1.61 1.21

28 1145.56 1147.28 1145.99 )1.72 1.29

29 1237.09 1237.50 1237.54 )0.41 )0.0430 1333.95 1334.34 )0.3931 1435.19 1435.19 1435.10 0.00 0.09

32 1540.22 1540.23 1540.13 )0.01 0.10

33 1649.44 1649.47 1649.32 )0.03 0.15

34 1762.88 1762.86 1762.71 0.02 0.15

35 1880.49 1880.49 1880.35 0.00 0.14

36 2002.37 2002.40 )0.0337 2128.57 2128.65 2128.64 )0.08 0.01

38 2259.06 2259.16 )0.1039 2393.87 2393.35 2393.35 0.52 0.00

40 2531.49 2531.62 2531.09 )0.13 0.53

41 2673.19 2673.48 2673.45 )0.29 0.03

42 2819.75 2820.05 2820.00 )0.30 0.05

43 2971.40 2971.55 2970.78 )0.15 0.77

44 3125.50 3125.80 3125.79 )0.30 0.01

45 3284.74 3285.04 3285.05 )0.30 )0.0146 3448.26 3447.79 3448.58 0.47 )0.7947 3616.09 3616.39 3616.42 )0.30 )0.0348 3789.35 3789.64 )0.2949 3967.24 3967.23 3967.15 0.01 0.08

50 4149.29 4149.29 4149.20 0.00 0.09

51 4335.83 4335.84 4335.71 )0.01 0.13

52 4526.87 4526.84 4526.72 0.03 0.12

53 4722.38 4722.36 4722.25 0.02 0.11

T. Koga, H. Matsuyama / Chemical Physics Letters 366 (2002) 601–605 603

Memass ffi 2:15915Z2 � 29:3983Z þ 214:175; ð8Þwith correlation coefficient 0.999959. Analogous

results are obtained for the ionic cases.

When isoelectronic species are compared in

Table 1, we observe that the cations and anions

have larger and smaller Memass values, respectively,than the neutrals with no exceptions. On the other

hand, we find that the atoms and ions with the

same Z have similar Memass values, particularly for

a large Z. When the ions of the third and higher

periods are examined, the average relative devia-

tions of the 45 cationic and 36 anionic Memass from

the neutral values are only 0.03% in both cases.

The ionic Memass is well approximated by theneutral atom value with the same Z. As antici-

pated from Eq. (3), the mass correction emass is

predominantly governed by inner electrons with

larger momenta. Thus the above observation im-

plies that inner electron shells are hardly affected

by the ionization and electron attachment in va-

lence shells.

Table 3 compares the present results with theliterature values [17] for the group-18 atoms He,

Ne, Ar, Kr, Xe, and Rn, where the literature Mesmvalues in atomic units were derived from the ori-

ginal data [17] in cm�1 by the prescription

Table 3

Comparison of the mass corrections for the group-18 atomsa

Atom Fraga et al. [17] Present Experimental

Menm Mesm Sum Memass ð3=4ÞSðþ1Þ

He 2.8617 0 2.8617 2.86168 3.14b

Ne 128.55 )15 114 113.569 119.8c

Ar 526.82 )96.1 430.7 430.709 418.1d

Kr 2752.1 )749.3 2002.8 2002.40

Xe 7232.1 )2308 4924 4922.43

Rn 21 867 )8236 13 631 13622.6

a The first moment Sðþ1Þ, multiplied by 3/4, of the experimental oscillator strength density is also given when available. All values

are in atomic units.bRef. [21].c Ref. [22].dRef. [23].

Table 2

Mass correction sums emass multiplied by nuclear masses M (in

atomic units)a

Z Memass Z Memass

55 5127.09 80 11616.7

56 5336.28 81 11938.9

57 5549.67 82 12265.9

58 5764.33 83 12597.8

59 5980.09 84 12934.5

60 6202.97 85 13276.1

61 6430.08 86 13622.6

62 6661.38 87 13974.1

63 6896.91 88 14330.4

64 7139.95 89 14691.3

65 7380.97 90 15057.2

66 7629.39 91 15423.5

67 7882.10 92 15796.4

68 8139.14 93 16174.1

69 8400.48 94 16554.2

70 8666.15 95 16941.2

71 8939.66 96 17335.2

72 9218.02 97 17731.8

73 9501.19 98 18130.7

74 9789.13 99 18536.6

75 10081.9 100 18947.4

76 10379.3 101 19362.9

77 10681.5 102 19783.2

78 10987.6 103 20211.0

79 11299.2

a For neutral atoms with 556Z6 103.

Table 1 (continued)

Z Cation Neutral Anion Cation–neutral Neutral–anion

54 4922.42 4922.43 )0.0155 5127.04

a For atoms and ions with 26N 6 54, where N is the number of electrons. The symbol Z stands for atomic number.

604 T. Koga, H. Matsuyama / Chemical Physics Letters 366 (2002) 601–605

described in [18]. For the first three atoms, the

literature Menm þMesm values agree with the

present Memass in the reported significant figures.

For heavy atoms such as Xe and Rn, however, the

literature values are found to be insufficiently ac-

curate. The limited accuracy of the numericalHartree–Fock calculations in [17] was also pointed

out [19]. Dalgarno and Lynn [20] showed a rela-

tion Memass ¼ ð3=4ÞSðþ1Þ, where Sðþ1Þ is the first

moment of the oscillator strength density. For the

three atoms He, Ne, and Ar, we have compared in

Table 3 the present Memass with the experimental

[21–23] ð3=4ÞSðþ1Þ. A moderate agreement is ob-

served.

4. Summary

We have shown that the sum emass of the normal

and specific mass corrections is immediately ob-

tained from the electron-pair intracule and extra-

cule moments in momentum space. Numericalresults of emass have been reported for 102 neutral

atoms and 96 singly charged ions in their experi-

mental ground states.

Acknowledgements

This work was supported in part by a Grant-in-Aid for Scientific Research from the Ministry of

Education of Japan.

References

[1] J. Bauche, R.-J. Champeau, Adv. At. Mol. Phys. 12 (1976)

39.

[2] W.H. King, Isotope Shifts in Atomic Spectra, Plenum

Press, New York, 1984.

[3] D.S. Hughes, C. Eckart, Phys. Rev. 36 (1930) 694.

[4] J.P. Vinti, Phys. Rev. 56 (1939) 1120.

[5] J.P. Vinti, Phys. Rev. 58 (1940) 882.

[6] T. Koga, H. Matsuyama, J. Chem. Phys. 115 (2001) 3984.

[7] T. Koga, in: J. Cioslowski (Ed.), Many-Electron Densities

and Reduced Density Matrices, Kluwer Academic Pub-

lishers/Plenum Press, New York, 2000, p. 267.

[8] T. Koga, H. Matsuyama, J. Chem. Phys. 107 (1997) 8510.

[9] T. Koga, H. Matsuyama, J. Chem. Phys. 108 (1998) 3424.

[10] T. Koga, H. Matsuyama, J. Chem. Phys. 113 (2000) 10114.

[11] H. Matsuyama, T. Koga, Y. Kato, J. Phys. B 32 (1999)

3371.

[12] C.E. Moore, Ionization potentials and ionization limits

derived from the analysis of optical spectra, NSRDS-NBS

34, Nat. Bur. Stand. US, Washington, DC, 1970.

[13] H.L. Anderson, in: A Physicist�s Desk Reference, AIP

Press, New York, 1989, p. 94.

[14] T. Andersen, H.K. Haugen, H. Hotop, J. Phys. Chem. Ref.

Data 28 (1999) 1511.

[15] T. Koga, H. Tatewaki, A.J. Thakkar, J. Chem. Phys. 100

(1994) 8140.

[16] T. Koga, J. Chem. Phys. 114 (2001) 2511.

[17] S. Fraga, J. Karwowski, K.M.S. Saxena, Handbook of

Atomic Data, Elsevier, Amsterdam, 1976.

[18] H. Matsuyama, T. Koga, Theor. Chem. Acc. (in press).

[19] T. Koga, S. Watanabe, A.J. Thakkar, Int. J. Quantum

Chem. 54 (1995) 261.

[20] A. Dalgarno, N. Lynn, Proc. Phys. Soc. Lond. A 70 (1957)

802.

[21] U. Fano, J.W. Cooper, Rev. Mod. Phys. 40 (1968) 441.

[22] R.P. Saxon, Phys. Rev. A 8 (1973) 839.

[23] E. Eggarter, J. Chem. Phys. 62 (1975) 833.

T. Koga, H. Matsuyama / Chemical Physics Letters 366 (2002) 601–605 605