1
Progress of Theoretical Physics, Vol. 22, No.1, July 1959
Thomas~Fermi-Dirac
Theory with Correlation Correction*
Yasuo TOMISHIMA
Department of Physics, Faculty of Science,
Okayama University, Okaymna
(Received February 24, 1959)
Using the formula of the correlation energy due to Pines, we modify the Thomas-FermiDirac model with the Fermi-Amaldi correction for free atom or ion to include correlations
between electrons. By the use of the present model, electron distributions for free Rb+, Kr
and Be are calculated, the results are shown graphically, and further the energy components
of the total energy are given numerically.
§ 1.
.~
r
I
Introd uction
In the theory of ionic crystals it becomes sometimes necessary for us to know
the electronic density distribution in the constituent ions. We have the Hartree or
Hartree-Fock solutions not for all the ions at present, therefore we must be contented
with some approximate solutions for the electronic density distributions. In many
approximate methods, the statistical one founded by Thomas1 ) and Fermi2) and
later modified by Dirac3) is very useful, especially for the atoms or ions with large
atomic number. The full account of this statistical method with its many applications
is shown in the text book by Gombas 4 ,5) or the review article by March G).
As is well known, the so-called Thomas-Fermi-Dirac model (abbreviated as
TFD hereafter) has two main failures. First, the interactions between electrons
are not correctly taken into account: A mean screening effect of the electron
cloud for the potential field by the nucleus is corrected by the fact that the electrons
with parallel spin have a tendency to keep away from each other due to their
statistical nature, in other words, by including the exchange energy. However, it
is not taken into account that whether the spins of the electrons being parallel or
antiparallel, there is a tendency to keep away from each other by the Coulomb
repulsion between them. The energy depression due to this tendency is well
known as the correlation energy. Second, the electrons unavoidably interact with
theIr own field in the formulation of the TFD theory.
Although the statistical model of the atom or ion is a very crude approximation
in its character, it is, naturally desirable to correct such fundamental failures in the
theory. An approach to attack at the latter has been made by Fermi and Amaldi7),
* The results were presented at the Hiroshima Meeting of the Chugoku-Shikoku Branch of
the Physical Society of Japan, in February 1959.
2
Y. Tomishima
and another to correct the former* by Gombas 8) using the expression for the correlation energy obtained by \Vigner 9) and taking Fermi-Amaldi correction into account
too.
Recently, the correlation energy for a free electron gas has been calculated by
many investigators. Especially, Nozieres and Pines lO ) have discussed the limit of
validity of the various correlation energy calculation and showed that the high
density calculation of Gell-Mann and Bruecknerll),
cc= -0.096+0.0622 In rs
IS
(Ry),
(1·1)**
valid for rs:$I, while the low density calculation of Wigner 9),
(1· 2)
is valid for rs~20.
Further for the region of actual metallic densities (I.8:S;r s:S;5.6),
the result obtained by Pines12 ),
cc= -0.115+0.0313 In rs
CRy),
C1· 3)
on the basis of the collective description of electron gas due to Bohm and Pines13 )
is useful.
On the other hand, the statistical calculation of the correlation energy by
Cowan and Kir kwood14 ) is not practicable owing to the fact that the results are
given only numerically and further the correlation energy cannot be broken up
from the exchange energy.
In determining the electronic density distribution in atoms, the correlation
energy is effective only near the atomic surface and the density at the surface of
the atoms in the TFD model is finite (r,,---4.2, d. eq. (2 ·15». The latter fact
has been considered to be more favourable for various problems than the TF model
for which the electron density decreases as inverse six power of the distance from
the nucleus as the distance increases. Therefore, we will use here Pines' expression
of the correlation energy.
In this paper, the TFD equation for an atom or ion is derived by taking
account of the correlation correction with Pines' expression, as well as the FermiAmaldi correction. By approximately solving the above equation, the electron
density distribution for Br-, Kr and Rb+ in free state all having 36 electrons are
given, especially for Kr the electron density distribution due to the ordinary TFD
model and that due to the Hartree-Fock method are compared with the present
* When we had finished the numerical work of this paper, we found the work with respect
to this problem by H. W. Lewis in Physical Review 111 (1958), 1554, in. which numerical results
were not reported. He has used the expression for the correlation energy obtained by interpolation
from Gell-Mann and Brueckner's. Fermi-Amaldi correction has not been taken into account.
** Ce means the correlation energy per electron of a free electron gas. r {; is the mean interelectronic distance in a.u. (atomic unit), relating to the electronic density p in a.u. by the equation
r s = (3/4n) 1/3p -l/3.
Thomas-Fermi-Dirac Theory with Correlation Correction
3
result. Lastly, total kinetic, potential, exchange and correlation energies In these
Ions are calculated and compared with the results obtained by the ordinary TFD
method.
§ 2.
Basic equations
We shall treat a free atom or ion with atomic number Z and electron number
N. If we use atomic units throughout this paper, and express the electron density
at r measured from the nucleus by p (r), the total energy of the system, in the
TFD model, may be given by the following equation.
(2·1)
where
('
E1c= 1C1c
J
j
p5 3
dv,
(2·2)
J I04/3dv ,
(2· 3)
('
Ee=
-IC
e
\
('
El~= Epc=
Jp Vndv,
-~(l-l/N) ipVe.dv,
Ec=- ipg(p)dv,
(2·4)
(2· 5)
(2·6)
and
Vn(r) =Z/!r!,
(2·4a)
Veer) = - i p(r')/!r-r'!·dv',
(2·5a)
(2·6a)
further
ICk
= (3/10) (3172) 2/3= 2.871,
lCe
= (3/4) (3/17)1 /3=0.7386,
(2· 7)
a 1=0.065,
a 2 =0.0052.
E k , E e, Epn, Epe and Eo are the total kinetic, exchange, potential due to nucleus,
potential due to electrons and correlation energies respectively. In eq. (2·5),
(1-1/ N) is the Fermi-Amaldi correction factor7) to exclude the self-interaction of
the electrons. Here we use the correlation energy per electron given by Pines12 ),
cc= -g(p) (cf. eq. (1·3». In the interior of the atom where the electron density
Y. Tomishima
4
is very high, expression (2· 6a) is not correct, as has been pointed out by Nozieres
and Pines10 ). However, it would not bring a serious error into the determination
of electron distribution to use this approximate formula instead of a more exact
one even in the high density region, because the large kinetic energy almost
determines the distribution there. Further, if we estimate Ec by using this formula,
as we shall do in § 4, a correct order of magnitude of correlation energy will be
gIven. (d. eqs. (1·1) and (1·3»
At the absolute zero of temperature, the total energy of the system E must
be minimum, subject to the subsidiary condition
f'
N=
J pdv = constant.
(2·8)
Therefore, the variation equation with respect to p (r)
6(E+ VoN) =0,
that
(2 ·9)
IS,
(2 ·10)
where Vo IS the Lagrange multiplier, and
(2·11)
may determine p as a function of r, in connection with Poisson's equation
AV*= N-l 4 7rp.
N
iJ
(2·12)
Boundary conditions:
Now, we shall make the usual assumption of spherical symmetry of the electron
distribution in the atom. We should first determine the atomic radius ro and the
density P at r= ro which will be expressed as po hereafter, since the solution of the
simultaneous equations (2 ·10) and (2 ·12) ca'n be determined uniquely only for the
definite ro and po. Therefore, we shall determine ro and po, according to Jensen 15 ),
by the following minimum condition of the total energy,
dEldro=o,
(2 ·13)
which corresponds to the zero pressure at the surface of a free atom. In calculating
dEl dro, it should be noticed that all the integrals in E are to be extended only
in the sphere of radius ro, since outside this sphere p= 0. Eliminating 0 Veloro and
0ploro in the expression of dEl d ro by using (2· 5a) and (2·8), we have the
following equation,
(2 ·14)
Therefore,
(2 ·15)
5
Thomas-Fermi-Dirac Theory with Correlation Correction
For the determination of ro, we will use the following boundary conditions.
First, we insert the value po given by (2 ·15) into (2 ·10), then we have
(2 ·16)
where go is the value of 9 for p= po. Second, making use of the fact that at the
surface, r= ro, the electric field due to the atom or ion under consideration in the
spherical approximation must be just as the field by the point charge (Z - N) at
the origin, we may set up the boundary condition at the surface as follows,
(2 ·17)
Third, near the ongm, the fieJd due to the charge of the nucleus being absolutely
dominant, we have
lim rV*=Z.
(2 ·18)
The three conditions stated above, (2 ·16), (2 ·17) and (2 ·18), can determine uniquely
the solution p of the second order simultaneous differential equation (2,10) and
(2 ·12), together with the atomic radius roo
§ 3.
Simplification of the basic equation and its solution
It is very difficult to solve eqs. (2 ·10) and (2 ·12) exactly because of the
logarithmic term of p in (2 ·10). As stated before, however, the term originating
from the correlation correction is essential to the determination of p only near the
atomic surface, where the electron density is small, so we may replace 9 (p) by
another function which is easy to treat and varies with p similarly to 9 (p) near
the surface. After the manner of Gombiis 8) , we will expand 9 (p) as a power
series with respect to p1/3 around p=po, and take only up to the first order, then
we have an approximate function g* (p), such as
(7* (0)
=a1'+a'2 Inl /3,
I
(3·1)
•
where
a/=a1 +a2 1npo-3a2 =0.0195,
(3 ·la)
and
a/ = 3a 2/
p~/3= 0.1061.
(3 ·lb)
In Fig. 1, Y (p) and y* ((') are shown against
('1/3.
If we use g* ((') instead of
9 (p), the correlation correction can be included into the exchange correction term,
so the treatment of the differential equation becomes simpler. This simplification
would not bring a serious error into the determination of electron distribution,
while the correlation energy of the system should be calculated by using the
expression 9 (p) rather, as will be done in § 4.
Replacing 9 (p) by g* (p) in the expression of E e , (2· 6), and inserting it into
the variation equation (2·9), we have the following equation instead of (2 ·10) :
file
r?/3_:1
r1'3_
::I k f'
::I JC e' f'
where
(V*-
~0 +a1')
=0 ,
(3·2)
(3·2a)
6
Y. Tomishima
which, m connection with Poisson's equation
(2 -12), may determine p as a function of r.
Replacement of g (p) by g* (p) does not
completely change the value derived by the
condition (2 ·13), as may be verified by a
calculation similar to that in § 2. Therefore,
the expression of po, (2 -15), may be held
correctly in this case. Using the new notation
K~, we may write
r/ro
<--
1
0.08
for
0.8
0.06
/
/
Kr
0.4
0.6
/
/
/
/.
0.04
(3·3)
0.02
Accordingly, the boundary condition (2 ·16)
IS to be
(V*- Vo+a/)r=ro=-:!K:2/ICk'
o
(3·4)
The other two conditions (2 ·17) and (2 ·18)
should be held also for this case.
Equation (3·2) may be written as
o
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Fig. 1. Functions 9 (p) and 9* (p) versus
pl/3. The upper abscissa indicates the
ratio of the distance from the nucleus
to the atomic radius ro for Kr in free
state.
(3·2a)
Now, we shall introduce the new function wand the new variable y instead of
V* or p and r respectively by the following relations, which are a slight modification
of the ones due to Thomas16 ),
(3·5)
and
r= ro exp( - y),
(3 ·6)
where
v==
(N/N-l)2.
(3 ·7)
Then the simultaneous equations determining p, (3·2) and (2 ·12), can be reduced
to the following equation for w as a function of y,
: ; =i w +e-(5/2)Y (e Y/4w 1/2+r) 3,
(3 ·8)
(3 ·9)
where
And the boundary conditions (3·4), (2 ·17) and (2 ·18) change the form respectively as follows:
At the boundary y=O (r= ro)
( 'YVl/~) y=o =2
4
L
r,
(3 ·10)
Thomas-Fermi-Dirac Theory with Correlation Correction
7
w)' =1J-l/4_~~~ (~'='--(37r2)1/3_1, )3/\3/2(Z_N+1),
( _r!'!!!_+~
dy
y=o
97r ,,32
and
ICe
while as y
~
(3·11)
'
co (r ~ 0)
c;\ ·12)
Eq. (3·8), subject to the boundary conditions (3 ·10), (3 ·11) and (3 ·12),
may be solved directly by the method of Gauss, Jackson and Numerov17 ). However,
more simply, it would be allowed to determine w by the double interpolation with
respect to the initial value of 'W i.e. r and the order of ionization Z - N from the
solutions of the ordinary TFD equation due to Thomas16 ). F or a particular atom
50
\
---'"";3
20
'\~
()
(rJ
()
....c:
...,
·ct.\S
~
\\
(J)
'8
,.\
r\,
I • ~
,
1\
10
\\
"\.
bJJ
'\.
0
'0
", ~
'"U
;::J
CJ
-(rJ
0...
5
0..
Ci
~
-.::j<
2
""
1
\~ '-- ..
'\~'---. !!
\
Rb+\
Kr
\
I
i
\
o
I
I
I
o
Fig. 2.
1
2
r
2.7845 3
3.3009
Br-
4
4.2150
-~
Radial density distribution 4nr2 p versusr, for Be, Kr and Rb+ in free state.
(in a. u.)
Y. Tomishima
8
~',
',
~ \
50
\
I
Kr
-,\
V\,\:.'\.
. :,
\
20
r-----~~--T-----------_r----------_+----------~r_--------__4
'\
~VPresent
~~
calculation
10r---------\~~~--------r_--------~----------~--------~
\\.\.
'"
5
r-----------r_----~~--r_----------r_----------+_--------~
~"
'~,
~,
2~·--------~·--------~~--~~----~--------_+--------~
"'~
_.•i
lr-----------i-----~----_r---------~-~~'~,~--------~--------~
'\
--~
"
""- ___ .
',-
TFD
"'-, }-!!
o ~---------L----------L----------L--L-------L------~.~--~~~
o
Fig. 3.
1
2
3
3.3009
4
4.5275
5
Radial density distribution 4nr2 p versus r for Kr in free state. (in a. u.)
Present calculation.
- - - - TFD method due to Thomas 16l .
- - - - - Hartree-F ock method due to Worsley20l.
or lon, In other words for given Z and N, the value of r, for which the solution
w of eq. (3· 8), subject to the initial conditions (3 ·10) and (3 ·11), satisfies the
boundary condition (3 '12), gives the radius ro of this atom or ion in free state
according to (3·9), and further inserting the solution w (y) thus obtained into
(3·5), we may have the electron distribution p as a function of y. Such an
interpolation procedure would presumably be more correct than the perturbation like
a treatment due to Gombas. (d. T able I)
In Table I, ro-values in the present model for Rb+, Kr and Br- are compared
with the values due to Thomas16 ) (the ordinary TFD method), due to Gombas18)
(the TFD method including Wigner's correlation and the Fermi-Amaldi corrections)
and due to Pauling19 ) (the semi-empirical method). The fact that the ro-values by
the present calculation are very close to those by Pauling, would give a theoretical
basis to the hard ion model which is a fairly good approximation in the theory o~
ionic crystals.
9
Thomas-Fermi-Dirac Theory with Correlation Correction
Table 1.
I
ro values in a. u.
Rb+I
I
Kr
Br-
-~------------------------~------~-'----------- -----~---.-'- - - - - - - - -
Present calculation
2.7845
Thomas*
Gombas**
Pauling***
4.2150
3.4211
3.3009
4.5275
3.55
4.22
5.19
2.80
3.19
3.69
*
* Ordinary TFD method, in which negative ions cannot be treated.
** TFD method including Wigner's correlation and Fermi-Amaldi corrections.
*** Semi-empirical method.
The radial density distributions 47rr2p's obtained above, are shown in Fig. 2
for Rb+, Kr and Br-, all having the identical electron number N=36. Especially
for Kr the present calculation is compared with that by the ordinary TFD method16)
and that by the Hartree-Fock method 20 ) in Fig. 3. From this figure, we see that
the electron distribution contracts toward the nucleus as a result of inclusion of
the correlation and Fermi-Amaldi corrections, and with increasing distance from
the nucleus the electron density decreases faster than in the case of the ordinary
TFD model, more similarly to that due to the Hartree-Fork metod. This means
that the introduction of these corrections weakens the screening effect of the electron
cloud as is expected.
§ 4.
Energy relations
We shall derive the relations realized between the energy components in
(2 ·1). After the Fock linear-scale-factor method 21 l, we shall assume that the distances of electron-electron as well as electron-nucleus are contracted by a factor 1/A
compared with the case of the minimum total energy, the total electron number
N being unchanged. We express the density at the distance r from the nucleus
in the contracted distribution by PA (r), then by the. normalization condition (2·8)
we have
(4 ·1)
where P is the density to minimize the total energy. vVe distinguish each energy
component for the contracted distribution by the upper suffix A, as
A+EA+EnA+Ee).
E "-E
k
e
p
"'p -LE).
, "'c ,
(4·2)
where
(4 ·3)
(4·4)
n
E PnA= "jE
.P'
(4 ·5)
Ee).=).E e
(4 ·6)
p
P'
Y. Tomishima
10
(4 ·7)
and
For the value A=l, the total energy EA should be mImmum, therefore
( ~EA)
dA
=0.
(4-8)
A=l
Inserting eqs. (4·2) to (4·7), into (4·8), we find the following energy relation,
(4 ·9)
The density distribution function p (r) obtained in the preceding section, and
eqs. (2·2), (2·3), (2·4) and (2·6) enable us to evaluate the values of Eft, E e,
E;: and E c, and then eq. (4·9) the value of El~' These values thus obtained are
columned in Table II.
Table II.
Values of energy components in a. u.
Kr
Be
3409.0
(3377.9)
3229.3
93.6
(89.6)
88.3
Rb+
Elc
3648.9
(3598.9)*
--Ee
96.8
(93.6)
-
----~.
---_.
--
-
- --
--
---_ ...
_-
----
----
- - ------ -- ---.-------- --
-El/'
8431.3
(8322.8)
7887.5
(7820.0)
7433.0
Epc
1230.8
(1218.6)
1163.7
(1153.8)
1063.3
----
------------------
*
-Eo
2.8 0
2.77
2.71
-E
3651.2
(3598.9)
3411.2
(3377.9)
3231.4
The values in brackets are the ones of the ordinary TFD model due to Thomas 16 ).
F or the sake of comparison the energy values of the ordinay TFD model due
to Thomas16 ) are also reproduced in the same table. From this table we see that
each energy value becomes slightly larger than the corresponding one due to
Thomas because of the contraction of the electron cloud. Further the correlation
energy Ec is very small compared with the other energies E 1c , Ec and E ln and
since we may neglect the last term 3N a 2 in (4·9), the Virial theorem for the
ordinary TFD theory 22) remains to hold in this case too.
§ 5. . Conclusions
In this paper, the basic equations, the Pines correlation and Fermi-Amaldi
corrections being taken into account, are simplified to the form which enables us to
determine the electron density distributions by the interpolation method from the
solutions of the ordinary TFD theory given by Thomas16 ), rather simply and
correctly for various atoms or ions.
Thomas-Fermi-Dirac Theory with Correlation Correction
11
In Fig. 3, the present calculation shows that the electron density distribution,
with the correlation and Fermi-Amaldi corrections taken into account, is more
contracted towards the nucleus than in the case of the ordinary TFD theory. This
means that the screening effect of the inner electrons for the electrostatic field of
the nucleus to the outer ones is overestimated in the statistical theory if these
corrections are not considered. This is a reasonable result. Since the correlation
energy is very small compared with the other energy components-even if we use"
a more exact formula of correlation energy density at high electron density region
of the atom, for example, Cell-Mann and Brueckner's instead of Pines' which is
used here, this conclusion would not be changed-the deviation of the density
distribution due to the present calculation from that due to the ordinary TFD'
method would predominantly be originated from the F ermi-Amaldi correetion.
Therefore, at the present stage of the statistieal theory, a correct formulation of
the Fermi-Amaldi correction would be most desirable. However, since the effect
of these correetions in eleetron distribution is conspicuous only near the atomic
surface, the TFD model is sufficient to treat the problem concerning bulky
properties of the atom. For the atomic nature in which the outer electrons play
an important role, a statistieal theory with sufficient corrections should be used.
As regards the energy values of the atom, the correlation energy may well beignored and the values of the energy components as well as the Virial relation are"
practically maintained as those of the ordinary TFD theory.
The author would like to express his sineere thanks to Professor K. U meda
for reading the manuseript and giving many valuable advices. He is also indebted
to Dr. S. Asano for his continuous interest and encouragement. This work was~
finaneed by the Scientifie Research Fund of the Ministry of Education.
References
1)
2)
3)
4)
5)
6)
7)
8)
9)
10)
11)
12)
13)
14)
15)
16)
17)
18)
19)
20)
21)
22)
L. H. Thomas, Proc. Cambridge Phil. Soc. 23 (1926), 542.
E. Fermi, Z. Physik 48 (1928), 73.
P. A. M. Dirac, Proc. Cambridge Phil. Soc. 26 (1930), 376.
P. Gombas, Statistische Behandlung des Atoms, H. B. der Physik Band XXXVI (Springer,"
Berlin, 1956) p. 109.
P. Gombas, Die statistische Theor'ie des Atoms und ihre Anwendungen, (Springer, Wien,
1949)
N. H. March, Advances in Phys. 6 (1957), 1.
E. Fermi and E. Amaldi, Mem. accad. sci. ist. Bologna 6 (1934), 117.
P. Gombas, Z. Physik 121 (1943), 523.
E. P. Wigner, Phys. Rev. 46 (1934), 1002.
P. Nozieres and D. Pines, Phys. Rev. 111 (1958), 442.
M. Gell-Mann and K. A. Brueckner, Phys. Rev. 106 (1957), 364.
K. Sawada, ibid. 106 (1957), 372.
D. Pines, Phys. Rev. 92 (1953), 626. Solid State Physics, edited by F. Seitz and D~"
Turnbull (Academic Press, Inc., New York, 1955) Vol. 1, p. 373.
D. Bohm and D. Pines, Phys. Rev. 85 (1952), 332, ibid. 92 (1953), 609.
R. D. Cowan and J. G. Kirkwood, Phys. Rev. 111 (1958), 1460.
H. Jensen, Z. Physik 93 (1935), 232.
L. H. Thomas, J. Chern. Phys. 22 (1954), 1758.
B. V. Numerov, Monthly Notices Roy. Astro. Soc. 84 (1924), 592, J. Jackson, ibid. 84"
(1924), 602.
P. Gombas, Reference 4) p. 146.
L. Pauling, Nature of the Chemical Bond (Cornell University Press, Ithaca, 1945) p. 346~
B. H. Worsley, Proc. Roy. Soc. A247 (1958), 390.
V. Fock, Phys. Z. Sowjet. 1 (1932), 747. N. H. March, Phil. Mag. VII 44 (1953), 1193.
H. Jensen, Z. Physik 89 (1934), 713. N. H. March, Phil. Mag. VII 43 (1952), 1042.