Asymptotic region of a Thomas–Fermi atom
S. H. Patil
Citation: J. Chem. Phys. 80, 5073 (1984); doi: 10.1063/1.446576
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Asymptotic region of a Thomas-Fermi atom
s. H. Patil
Department of Physics, Indian Institute of Technology, Bombay 400 076, Maharashtra, India
(Received 27 October 1983; accepted 14 December 1983)
Thomas-Fermi model for an atom, is modified to include the correct asymptotic behavior of the
charge density and electrostatic potential, in terms of its ionization energy. The predictions of the
model, for the diamagnetic susceptibility, are in good agreement with the experimental values.
The Thomas-Fermi model provides a fairly useful description of average atomic properties of atoms and ions. Its
importance lies in its transparent simplicity. However, the
model in its simplest form is not accurate in the regions near
and far away from the nucleus. It is relatively easy to improve the description near the nucleus. I On the other hand,
the outer regions of the atom are much more troublesome
and difficult to handle. In a recent series of papers,203-4
Schwinger and his co-workers have proposed an approach to
understand the outer regions of a Thomas-Fermi atom.
They have included exchange and first quantum kinetic energy corrections, 2 and calculated molar diamagnetic susceptibilities which depend critically on the properties of the outer regions of the atom. Their first attempe was not
particularly successful though it produced significant improvements over the model without the corrections. In their
second attempt,4 they introduce a sharp boundary. The
boundary is chosen in an experimental way, with one of their
boundary conditions yielding quite satisfactory results for
atoms (and ions) with 18, 36, and 54 electrons, but not for
those with ten electrons.
In the present paper, we propose an alternate approach
to incorporate the correct asymptotic behavior of charge
density of the electrons in the Thomas-Fermi model. This is
done in terms of the ionization energy of the atom or ion and
is based on the observation that the asymptotic behavior of
an electronic wave function is governed by its binding energy. The model then predicts quite satisfactory values for the
susceptibilities of atoms and ions with 10, 18, 36, and 54
electrons. Their accuracy is comparable with that of Hartree-Fock results. We give a brief outline of the derivation of
charge density in an atom and then use it to obtain the electrostatic potential and diamagnetic susceptibility.
I. WKB CHARGE DENSITY
The charge density in the Thomas-Fermi model is most
conveniently obtained in terms of the WKB wave functions
as outlined by Kirzhnitz et al. I The WKB radial wave function in the classically allowed region is given by
"'n
= CPn- 1I2 sin [i>~ dr' + :]
Pn
=
[2(En
+ c,6) -
(1)
Pn dr = 1T(n
+ !)
J. Chern. Phys. 80 (10).15 May 1984
Pr
= 2S(21 + 1)d1 S dnl"'n 12.
(2)
(3)
(4)
Using Eqs. (1)-(3), one then obtains for the density (see
Kirzhnitz et al. in Ref. 1)
(5)
where E j is the highest energy of the electrons. The break in
the density at c,6 = - E j has come in because of the inadequacy of the WKB wave function near the turning points.
Therefore, whenever it is admissible we will take
(6)
For large r, when one is outside the classically allowed
region, the WKB wave function is
(7)
Proceeding as before, the leading r dependence of the density
is determined by the states with energies near E j , and comes
out to be
~A1" e - br for r> 1,
a=(Z-N+
1)I~J/2 -2,
(8)
b = 212Ej 1 1/2,
where Z is the charge of the nucleus, N is the number of
electrons, and A is a constant which is to be determined so as
to give the correct value for the potential c,6 near the nucleus.
The asymptotic result in Eq. (8) is a general, well-known
result. s We have given its derivation within the WKB approximation which is the framework of our analysis.
The Poisson equation for the electrostatic potential is
~=~-
~
which leads to
d,z = 41Trptot, 1] = rc,6.
r,
ro
C- 2 _ r'~
- Jo 2Pn
The radial density is given by
d 21]
(I + ~f/,zr12,
where c,6 is the electrostatic potential and we are using atomic
units. Here, the energy is determined by the condition
1
with ro and r1 being the turning points. For sufficiently large
quantum numbers, the normalization condition gives
(10)
II. NEUTRAL ATOMS
The equation for neutral atoms is taken to be
d 21]
(21])312 + Ac/' + Ie - br
d,z
31T r 1l2
'
=...±...
0021·9606/84/105073-03$02.10
@ 1984 American Institute of PhYSics
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(11)
5073
5074
S. H. Patil: Asymptotic region of a Thomas-Fermi atom
1.0
included. While the effect of the asymptotic term is small for
large Z atoms, it is quite substantial for small Z atoms, e.g.,
Z = 10. Our model implies a higher electron density at
shorter distances than is predicted by the Thomas-Fermi
atom.
The large r behavior of the density may be expected to
be important for diamagnetic susceptibility which (in units
of - 4.75 X 10- 6 ) is given by
0.6
""t-
0.5
21T
~IN 0.4
0.3
0.2
0.1
a
a
2
x
4
3
5
FIG. 1. Our values of(lIZ)rt,6 (r)forNe(Z = 10) and Xe(Z = 54) plotted as a
function of x = rZ 1/3, and compared with the universal Thomas-Fermi
plot.
where a and b are given in Eq. (8) and Ao = 41rA. It is to be
noted that while the second term is dominant for large r, it is
relatively quite small for small r. This of course is necessary
for the analysis to be meaningful. The asymptotic behavior
of TJ is given by
1
TJ-+ - 2A of' + Ie -
(12)
TJ(O) = Z.
(13)
br for r-+oo.
b
The constant Ao is determined so as to give
For a specific atom, the values of a and b are given in terms of
the ionization energy which we take to be the experimental
value. 6 We start with a suitable value of Ao and the asymptotic behavior in Eq. (12) for TJ{r) and trace the solution to
smaller values of r by using Eq. (11). We choose that value of
Ao for which Eq. (13) is satisfied. Two of the solutions for (11
Z)TJ are shown in Fig. 1 as a function of x:
x=rZ I / 3 ,
1= - S p(r)r4 dr.
(15)
3
The values of this integral for Z = 10, 18, 36, and 54 are
given in Table I. The predictions are seen to be in quite good
agreement with the experimental values. 7 The predicted values oscillate about the experimental values for low Z atoms
but tend essentially to the exact value for large Z atoms, i.e.,
Z = 54. The average absolute percent error is 8.5% which is
even smaller than the corresponding value of9.2% for Hartree-Fock calculations8 (Hartree-Fock predictions are in
excess by 9.9% for Ne, 5.3% for Ar, 8.7% for Kr, and
13.0% for Xe). The calculations of Schwinger and his coworkers give quite poor results for Z = 10, being offby nearly a factor of 2, but give good results4 for Z = 18, 36, and 54,
the average absolute percent error for these three atoms being 6.5% (our corresponding error is 6.8%). It should be
noted that the inclusion of asymptotic behavior in terms of
ionization energy brings in an improvement of nearly an order of magnitude over the values predicted by the ThomasFermi model without the asymptotic term.
III. IONS
In the case of ions, TJ tends to a nonzero value Z - N,
for r-+ 00. This means that the first term in Eq. (11) dominates at large r, and the density would be incorrectly represented by Eq. (11). For the ions, we use the density given in
Eq. (5) which leads to
~~ = 3~ ( ~ )1/2(TJ + Ejr)3/2(J (TJ + Ejr) + A of' + Ie - br.
(16)
(14)
The asymptotic behavior of TJ in this case is
along with the universal solution for the Thomas-Fermi
model without the asymptotic term. It is seen that the curve
for {lIZ)TJ is no longer universal if the asymptotic term is
TJ-+Z - N
+ ~Aor" + Ie - br for r-+ 00 •
(17)
b
As before, we start with a suitable value of Ao and the asymp-
TABLE I. Predictions I for diamagnetic susceptibility in terms of the integral in Eq. (15), along with the
experimental values Ie' the adjusted Hartree-Fock values I AHF and the predictions Ips of Englert and
Schwinger (Ref. 4).
N
10
18
36
54
Z
E, ineV
Ao
I
Ie
IAHF
Ips
10
21.56
47.29
80.12
15.76
31.81
51.21
14.0
27.50
12.13
25.1
44.0
179
481
35.5
121.5
295
40.0
128
32.4
121.5
1.61
1.09
0.76
3.51
2.73
2.10
6.34
5.26
9.14
7.53
1.42
1.28
0.91
4.12
3.07
2.25
6.06
4.63
9.24
7.39
1.42
0.97
0.72
4.12
3.09
2.44
6.06
4.88
9.24
7.75
2.71
1.75
1.22
4.24
3.04
2.30
6.91
5.42
9.01
7.37
11
12
18
19
20
36
37
54
55
J. Chern. Phys., Vol. 80, No. 10, 15 May 1984
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S. H. Patil: Asymptotic region of a Thomas-Fermi atom
totic behavior in Eq. (17) and trace the solution to smaller
values of r by using Eq. (16). The value Ao is selected so as to
satisfy the condition in Eq. (13). The susceptibilities calculated from Eq. (15) using these solutions are listed in Table I for
different values of Z and N.
For comparison, we note that the experimental values9
of ionic susceptibilities are rather uncertain. One may compare the predictions with the adjusted Hartree-Fock results4
which are obtained by assuming that the calculated HartreeFock numbers are in excess by the same fractional amounts
for a given N but different Z, and use the experimental results for the neutral atoms. Generally, our predictions are in
good agreement with the experimental and adjusted Hartree-Fock numbers.
The accurate predictions of susceptibilities are an indication of the importance of requiring the correct asymptotic
behavior for charge densities and potentials. The densities
and potentials we have calculated may be used to calculate
other atomic properties such as polarizability, etc. Apart
from the numerical agreement, we would like to make a
point that an accurate theory of atoms must incorporate the
5075
necessary asymptotic conditions on charge densities [Eq. (8)]
and potentials [Eqs. (12) and (17)].
ACKNOWLEDGMENT
It is a pleasure to thank Dr. G. Mukhopadhyay for interesting discussions.
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G. V. Shpatakovskaya, Usp. Fiz. Nauk 117, 3 (1975) [Sov. Phys.-Usp.18,
649 (1976)]; J. Schwinger, Phys. Rev. A 22, 1827 (1980).
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3L. DeRaad and J. Schwinger, Phys. Rev. A 25, 2399 (1982).
4B_G. Englert and J. Schwinger, Phys. Rev. A 26, 2322 (1982).
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A 23, 2106 (1981).
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3ll (1973).
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J. Chern. Phys., Vol. 80, No.1 0, 15 May 1984
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