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: [email protected] (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.
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