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A relativistic Thomas-Fermi model of compressed atoms
Volume 138, number 4,5
PHYSICS LETTERS A
26 June 1989
A RELATIVISTIC T H O M A S - F E R M I M O D E L OF COMPRESSED ATOMS Katsumi YONEI Department of Physics, Okayama University, Okayama 700,Japan Received 22 February 1989; revised manuscript received 25 April 1989; accepted for publication 2 May 1989 Communicated by B. Fricke
A relativistic Thomas-Ferrni-Dirac-Weizs~cker theory is applied to Fe and Au atoms under various compressions. The calculated values of electron density, energy, and pressure are compared with nonrelativistic values, and it is shown that even for lighter atoms the relativistic effects become significant when the compression is very high.
1. Introduction
During the last decade considerable interest has revived in Thomas-Fermi and related theories for compressed atoms. In particular, More [ 1 ] and Perrot [ 2 ], independently of each other, developed the Thomas-Fermi-Dirac theory with the modified Weizs~icker correction (TFDW) of compressed atoms and applied it to the calculation of the equation of state of metals. The same model can also be applied to the calculation of various other properties of high density matter [3] and furnishes knowledge useful for the study of fusion or astrophysical processes. Although the TFDW theory provides useful information on the electronic states of compressed atoms, it seems that there is still room for improvements. Among various improvements on the TFDW theory of compressed atoms, the inclusion of the relativistic correction will be one of the most important because its significance quite rapidly increases with the electron density. Even for a lighter atom, to which the relativistic effects are small when it is free, they can be quite large when the atom is highly compressed. In this article, therefore, we propose a simple version of the relativistic TFDW model of compressed atoms, apply it to the calculation of electronic states of atoms under various compressions, and see how the relativistic correction affects the behavior of several important quantities. The basic idea of the present model is the same as
that of the relativistic TFDW theory proposed by Tomishima [4] for free atoms. In this model the relativistic effects are taken into account only for the kinetic energy part of the local density approximation. All the other relativistic corrections, including those to the Weizs~cker and the exchange terms, are neglected. In spite of the absence of the relativistic corrections to these terms, this model can give fairly good total energies for free atoms when the Weizs~icker gradient term is weighted by a factor 2 = 1/5 [4 ]. In fact, the total energies of free atoms obtained by Tomishima [4 ] are in agreement with the DiracFock values obtained by Desclaux [5 ], with deviations of 1.5% or less for atomic numbers ranging from 10 to 100. Also, it has been shown that the effective potential determined from this model gives quite good one-electron energies of atoms [ 6 ]. Apart from all these merits, the present study will at least serve as a first step toward the investigation of the relativistic effects in compressed atoms.
2. Formulation
Suppose a spherical neutral atom of atomic number Z and radius ro embedded in a neutral medium of chemical potential # at the temperature T = 0. In accordance with Tomishima [ 4 ], we assume the total energy of the atom to be expressed in terms of the electron density n (r) as (we use atomic units except when otherwise stated)
0375-9601/89/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division )
193
Volume 138, number 4,5
PHYSICS LETTERS A
E=fek(n(r))dr+XKif(Vn)2dr-fZn(r)dr +½ f I n(r)n(r'l
dr'
26 June 1989
V= rZ- f ~n(r' )
dr'.
(2.4c)
Eq. (2.4c) may be replaced by the Poisson equation
dr-~a f n4/3(r) dr
V2V=4n[n(r) -Z~(r) (2.1)
] .
(2.4d)
where xi=~ and Ka=~(3/~) 1/3. In eq. (2.1) gk(n) of the first term means the relativistic kinetic energy density in the local density approximation and has the following form [7,8 ]:
The boundary conditions to be imposed on the solutions of eqs. (2.4a) and (2.4d) are as follows: First of all, the variational principle (2.3), besides the Euler equation (2.4a), requires that the electron density satisfies the condition [ 1,2,9 ]
~k(n)=3 (31tz)2/3nS/3A(n) ,
[Vn l,=,~, = 0 .
(2.2a)
where
A(n) =
(2.5a)
The other boundary conditions are [ 1,2,9 ] 10x-5{ ~ [x( 1 "+'2X 2 ) ( 1 +X 2) 1/2
V(ro) = 0 ,
(2.5b) (2.5c)
-- sinh - ' x] - }x 3} ,
(2.2b)
n ( 0 ) = a finite value,
x = (3/¢ 2 ) l/3~n 1/3,
(2.2c)
and
a being the fine structure constant. The second term of eq. (2.1) is the gradient correction to the kinetic energy, and here we assume the same form as that in the nonrelativistic case. In particular, the weighting factor 2 is taken to be 1/5 because this value of 2 yields quite good energies for free atoms as mentioned in section 1. As to the other terms, there will be no need of detailed explanations; the third and fourth terms of eq. (2.1) are the electron-nucleus and the electron-electron Coulombic interaction energies, and the last term is the exchange energy. The finite nuclear size effect is neglected and is left as a future problem although it might exhibit some significance for highly compressed atoms. In exactly the same way as in the nonrelativistic case [ 1,2 ], the variational principle, i.e.
a(E-# f n(r)dr)=O,
f n(r) dr=Z.
(2.5d)
3. Results and discussion Now let us proceed to the results of the present calculation. In the following the Fe and the Au atoms are chosen as an example. First, to see how the relativistic correction affects the electron density distribution, we give in fig. 1 a plot of the quantity
I y rn 1/2 (In a.u.) =
!
'~
^u
(2.3)
readily leads to the following set of fundamental equations:
2"0f t0
-1- ~O0~k - n -- 4K'an 1/3-- ( VJI- ]~) --~0 ,
(2.4a)
oo
(r/ro)l/3
og~
O---~=a-2{[l+(3n2)~/3a2n2/3]l/2-1},
194
(2.4b)
Fig. 1. Plot ofy=rn ~/2for Au at the normal density (Po= 10.5 g/cm 3, ro= 3.01 au).
Volume 138, number 4,5
PHYSICS LETTERSA
26 June 1989
Table l The total energiesof Fe and Au for various values of compressionp/po (in au). In the table ER and ENRdenote relativistic and nonrelativistic energies,respectively,while PD means the percentage deviation, i.e. PD = 100(ER--ENR)/IEsRi.
P/Po
1 5 l0 50 100 200 300 500 1000
Au (Z= 79)
Fe (Z=26) ER
ENR
PD
gR
ENR
PD
-1276.75 - 1273.64 - 1266.66 - 1187.90 - 1081.30 -872.845 -673.889 -299.650 543.165
-
-1.107 - 1.Ill - 1.120 - 1.233 - 1.421 - 1.952 -2.813 -8.011 -5.735
-
- 17816.8 - 17811.6 -17797.5 - 17593.3 - 17271.9 - 16580.9 -15878.0 - 14487.5 - 11166.4
-8.254 -8.258 -8.266 -8.386 -8.584 -9.045 -9.565 - 10.77 - 13.03
1262.77 1259.64 1252.63 1173.43 1066.15 -856.132 -655.450 -277.425 576.215
y= rn ~/2 versus (r/r0)t/3 at its n o r m a l density a n d compare it with the nonrelativistic result. Note that the relativistic correction causes a significant increase of the electron density near the nucleus a n d a slight reduction of it near the surface. The cause of this behavior in the electron distribution may eventually be ascribed to the fact that the magnitude of the relativistic correction, I A ( n ) - l l (cf. eq. ( 2 . 2 b ) ) , is an increasing function of the electron density n. The significant increase of electron density in the i n n e r part of the atom a n d the less significant decrease of it near the surface is one of the most f u n d a m e n t a l features of the relativistic correction a n d will more or less be reflected in various atomic properties, as seen below. To see the relativistic effects on the energy, we tabulate the relativistic a n d nonrelativistic total energies of the Fe a n d the Au atoms for various compressions P/Po in table 1 and make a comparison between them. Here p means the density of matter in a n arbitrary condition, while Po is the n o r m a l density (po= 7.85 g / c m 3 for Fe a n d 10.5 g / c m s for Au). The quantity P/Po is used in place of ro because it is widely used in the literature. The relation between the density p ( g / c m 3) a n d the corresponding atomic radius r0 ( c m ) may be described as
]xrgp = AW/NA,
(3.1)
where Aw a n d Ark are the atomic weight a n d Avogadro's n u m b e r , respectively. Reflecting the behavior of the electron distribution, a considerable lowering of the total energy is observed in the relativistic
19287.4 19282.5 19268.7 19068.7 18754.5 18080.6 17396.7 16047.2 12839.2
case. Note that even for the lighter atom the deviations of the relativistic energies from the nonrelativistic become quite remarkable as the atom is highly compressed. This clearly indicates the importance of the relativistic effects in the study of highly compressed atoms. Next, as another interesting quantity, we give a tabulation of pressure values at atomic surface in table 2. In the same m a n n e r as in the nonrelativistic case [ 9,10 ], the pressure can be calculated from the following formula: P=
1
dE
4~ro2 dro
=-[~(n)-tcan4/3-(V+#)n]r=,o.
(3.2)
Table 2 tells us that the effect of the relativistic correction on the pressure is less significant than on the energy. This may be understood from the smallness of the relativistic effect on the electron density near the atomic surface. Note, however, that the relativistic effect becomes quite significant as the compression increases. Therefore in the calculation of the pressure too, the inclusion of the relativistic correction becomes important for highly compressed atoms.
4. Conclusion I n the above we have observed some features of the relativistic T F D W model of compressed atoms. S u m m i n g up these observations, one may conclude 195
Volume 138, number 4,5
PHYSICS LETTERS A
26 June 1989
Table 2 The pressure values at the atomic surface for Fe and Au (in au). PR and PNR denote relativistic and nonrelativistic pressures, respectively, while PD means the percentage deviation, i.e. PD= 1 0 0 ( P R - - P N R ) / I P N R [ .
P/Po
1 5 10 50 100 200 300 500 1000
Fe (Z=26)
Au (Z=79)
PR
PNR
PD
PR
PNR
PD
2.3061E-3 3.6366E- 1 1.9728E+0 6.6231E+ 1 2.6669E+2 1.0198E + 3 2.1907E+3 5.6401E + 3 1.9803E + 4
2.3162E-3 3.6543E- 1 1.9828E+0 6.6669E+ 1 2.6881E+2 1.0300E + 3 2.2164E+3 5.7227E + 3 2.0210E + 4
-0.436 -0.484 -0.504 -0.657 -0.788 - 0.990 - 1.16 - 1.44 - 2.01
1.8780E- 3 4.6666E- 1 2.9005E+0 1.2851E+2 5.7190E+2 2.3847E + 3 5.3547E+ 3 1.4478E + 4 5.3691E + 4
1.9125E-3 4.7479E- 1 2.9541E+0 1.3142E+2 5.8644E+2 2.4544E + 3 5.5276E+ 3 1.5018E + 4 5.6564E + 4
- 1.70 - 1.71 - 1.81 -2.21 -2.47 - 2,84 -3.12 - 3.60 - 5.08
that e v e n for lighter a t o m s the relativistic c o r r e c t i o n c e r t a i n l y brings a b o u t a p p r e c i a b l e effects o n the electron density distribution and other important quantities o f a t o m s w h e n t h e y are highly c o m p r e s s e d . We c o n c l u d e this article by a d d i n g two r e m a r k s : ( 1 ) T h e e l e c t r o n d e n s i t y c h a n g e c a u s e d by the relativistic c o r r e c t i o n n a t u r a l l y gives rise to a change in the e f f e c t i v e p o t e n t i a l for electrons. T h e one-elect r o n energies for t h e e f f e c t i v e p o t e n t i a l are e x p e c t e d to e x h i b i t c o n s i d e r a b l e d e v i a t i o n s f r o m the n o n r e lativistic e v a l u a t i o n s . ( 2 ) A l t h o u g h we h a v e c o m p l e t e l y n e g l e c t e d the relativistic c o r r e c t i o n s to the g r a d i e n t a n d e x c h a n g e t e r m s here, t h e i r effects o f c o u r s e m u s t be e x a m i n e d in the n e x t step. R e c e n t l y a n e w d e v e l o p m e n t o f the relativistic T F D W m o d e l has b e e n a c h i e v e d by Engel a n d D r e i z l e r [ 11,12 ]. F r o m the theoretical p o i n t o f view, t h e i r m o d e l s e e m s to be q u i t e satisfactory, in that it takes t h e relativistic c o r r e c t i o n s to b o t h o f these two t e r m s i n t o a c c o u n t . It will be w o r t h w h i l e to e x a m i n e w h a t effects the relativistic c o r r e c t i o n s o f these two t e r m s give rise to. T h e s e two studies are n o w in progress a n d we h o p e that the result will be r e p o r t e d shortly.
196
Acknowledgement T h e a u t h o r w o u l d like to express his cordial thanks to P r o f e s s o r Y. T o m i s h i m a for his k i n d interest in this work. T h a n k s are also due to P r o f e s s o r F u r u tani, w h o k i n d l y g a v e the a u t h o r an o p p o r t u n i t y to give a talk a b o u t this subject.
References [ 11 R.M. More, Phys. Rev. A 19 ( 1979 ) 1234. [2] F. Perrot, Physica A 98 (1979) 555. [3]R.M. More, Lawrence Livermore Laboratory preprint UCRL 84991 ( 1981 ). [4] Y. Tomishima, Prog. Theor. Phys. 42 (1969) 437. [ 5 ] J.P. Desclaux, At. Data Nucl. Data Tables 12 ( 1973 ) 311. [6] K. Yonei, J. Phys. Soc. Japan 54 (1985) 93. [7] E.K.U. Gross and R.M. Dreizler, Phys. Lett. A 81 (1981) 447. [8] E.K.U. Gross and R.M. Dreizler, in: Density functional methods in physics, eds. R.M. Dreizler and J. da Provincia (Plenum, New York, 1985) p. 81. [9] K. Yonei, J. Ozaki and Y. Tomishima, J. Phys. Soc. Japan 56 (1987) 2697. [ 10 ] P. Gombds, Die statistische Theorie des Atoms under ihre Anwendungen (Springer, Berlin, 1949) p. 37. [ 11 ] E. Engel and R.M. Dreizler, Phys. Rev. A 35 (1987) 3607. [ 12] E. Engel and R.M. Dreizler, Phys. Rev. A 38 (1988) 3909.
PHYSICS LETTERS A
26 June 1989
A RELATIVISTIC T H O M A S - F E R M I M O D E L OF COMPRESSED ATOMS Katsumi YONEI Department of Physics, Okayama University, Okayama 700,Japan Received 22 February 1989; revised manuscript received 25 April 1989; accepted for publication 2 May 1989 Communicated by B. Fricke
A relativistic Thomas-Ferrni-Dirac-Weizs~cker theory is applied to Fe and Au atoms under various compressions. The calculated values of electron density, energy, and pressure are compared with nonrelativistic values, and it is shown that even for lighter atoms the relativistic effects become significant when the compression is very high.
1. Introduction
During the last decade considerable interest has revived in Thomas-Fermi and related theories for compressed atoms. In particular, More [ 1 ] and Perrot [ 2 ], independently of each other, developed the Thomas-Fermi-Dirac theory with the modified Weizs~icker correction (TFDW) of compressed atoms and applied it to the calculation of the equation of state of metals. The same model can also be applied to the calculation of various other properties of high density matter [3] and furnishes knowledge useful for the study of fusion or astrophysical processes. Although the TFDW theory provides useful information on the electronic states of compressed atoms, it seems that there is still room for improvements. Among various improvements on the TFDW theory of compressed atoms, the inclusion of the relativistic correction will be one of the most important because its significance quite rapidly increases with the electron density. Even for a lighter atom, to which the relativistic effects are small when it is free, they can be quite large when the atom is highly compressed. In this article, therefore, we propose a simple version of the relativistic TFDW model of compressed atoms, apply it to the calculation of electronic states of atoms under various compressions, and see how the relativistic correction affects the behavior of several important quantities. The basic idea of the present model is the same as
that of the relativistic TFDW theory proposed by Tomishima [4] for free atoms. In this model the relativistic effects are taken into account only for the kinetic energy part of the local density approximation. All the other relativistic corrections, including those to the Weizs~cker and the exchange terms, are neglected. In spite of the absence of the relativistic corrections to these terms, this model can give fairly good total energies for free atoms when the Weizs~icker gradient term is weighted by a factor 2 = 1/5 [4 ]. In fact, the total energies of free atoms obtained by Tomishima [4 ] are in agreement with the DiracFock values obtained by Desclaux [5 ], with deviations of 1.5% or less for atomic numbers ranging from 10 to 100. Also, it has been shown that the effective potential determined from this model gives quite good one-electron energies of atoms [ 6 ]. Apart from all these merits, the present study will at least serve as a first step toward the investigation of the relativistic effects in compressed atoms.
2. Formulation
Suppose a spherical neutral atom of atomic number Z and radius ro embedded in a neutral medium of chemical potential # at the temperature T = 0. In accordance with Tomishima [ 4 ], we assume the total energy of the atom to be expressed in terms of the electron density n (r) as (we use atomic units except when otherwise stated)
0375-9601/89/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division )
193
Volume 138, number 4,5
PHYSICS LETTERS A
E=fek(n(r))dr+XKif(Vn)2dr-fZn(r)dr +½ f I n(r)n(r'l
dr'
26 June 1989
V= rZ- f ~n(r' )
dr'.
(2.4c)
Eq. (2.4c) may be replaced by the Poisson equation
dr-~a f n4/3(r) dr
V2V=4n[n(r) -Z~(r) (2.1)
] .
(2.4d)
where xi=~ and Ka=~(3/~) 1/3. In eq. (2.1) gk(n) of the first term means the relativistic kinetic energy density in the local density approximation and has the following form [7,8 ]:
The boundary conditions to be imposed on the solutions of eqs. (2.4a) and (2.4d) are as follows: First of all, the variational principle (2.3), besides the Euler equation (2.4a), requires that the electron density satisfies the condition [ 1,2,9 ]
~k(n)=3 (31tz)2/3nS/3A(n) ,
[Vn l,=,~, = 0 .
(2.2a)
where
A(n) =
(2.5a)
The other boundary conditions are [ 1,2,9 ] 10x-5{ ~ [x( 1 "+'2X 2 ) ( 1 +X 2) 1/2
V(ro) = 0 ,
(2.5b) (2.5c)
-- sinh - ' x] - }x 3} ,
(2.2b)
n ( 0 ) = a finite value,
x = (3/¢ 2 ) l/3~n 1/3,
(2.2c)
and
a being the fine structure constant. The second term of eq. (2.1) is the gradient correction to the kinetic energy, and here we assume the same form as that in the nonrelativistic case. In particular, the weighting factor 2 is taken to be 1/5 because this value of 2 yields quite good energies for free atoms as mentioned in section 1. As to the other terms, there will be no need of detailed explanations; the third and fourth terms of eq. (2.1) are the electron-nucleus and the electron-electron Coulombic interaction energies, and the last term is the exchange energy. The finite nuclear size effect is neglected and is left as a future problem although it might exhibit some significance for highly compressed atoms. In exactly the same way as in the nonrelativistic case [ 1,2 ], the variational principle, i.e.
a(E-# f n(r)dr)=O,
f n(r) dr=Z.
(2.5d)
3. Results and discussion Now let us proceed to the results of the present calculation. In the following the Fe and the Au atoms are chosen as an example. First, to see how the relativistic correction affects the electron density distribution, we give in fig. 1 a plot of the quantity
I y rn 1/2 (In a.u.) =
!
'~
^u
(2.3)
readily leads to the following set of fundamental equations:
2"0f t0
-1- ~O0~k - n -- 4K'an 1/3-- ( VJI- ]~) --~0 ,
(2.4a)
oo
(r/ro)l/3
og~
O---~=a-2{[l+(3n2)~/3a2n2/3]l/2-1},
194
(2.4b)
Fig. 1. Plot ofy=rn ~/2for Au at the normal density (Po= 10.5 g/cm 3, ro= 3.01 au).
Volume 138, number 4,5
PHYSICS LETTERSA
26 June 1989
Table l The total energiesof Fe and Au for various values of compressionp/po (in au). In the table ER and ENRdenote relativistic and nonrelativistic energies,respectively,while PD means the percentage deviation, i.e. PD = 100(ER--ENR)/IEsRi.
P/Po
1 5 l0 50 100 200 300 500 1000
Au (Z= 79)
Fe (Z=26) ER
ENR
PD
gR
ENR
PD
-1276.75 - 1273.64 - 1266.66 - 1187.90 - 1081.30 -872.845 -673.889 -299.650 543.165
-
-1.107 - 1.Ill - 1.120 - 1.233 - 1.421 - 1.952 -2.813 -8.011 -5.735
-
- 17816.8 - 17811.6 -17797.5 - 17593.3 - 17271.9 - 16580.9 -15878.0 - 14487.5 - 11166.4
-8.254 -8.258 -8.266 -8.386 -8.584 -9.045 -9.565 - 10.77 - 13.03
1262.77 1259.64 1252.63 1173.43 1066.15 -856.132 -655.450 -277.425 576.215
y= rn ~/2 versus (r/r0)t/3 at its n o r m a l density a n d compare it with the nonrelativistic result. Note that the relativistic correction causes a significant increase of the electron density near the nucleus a n d a slight reduction of it near the surface. The cause of this behavior in the electron distribution may eventually be ascribed to the fact that the magnitude of the relativistic correction, I A ( n ) - l l (cf. eq. ( 2 . 2 b ) ) , is an increasing function of the electron density n. The significant increase of electron density in the i n n e r part of the atom a n d the less significant decrease of it near the surface is one of the most f u n d a m e n t a l features of the relativistic correction a n d will more or less be reflected in various atomic properties, as seen below. To see the relativistic effects on the energy, we tabulate the relativistic a n d nonrelativistic total energies of the Fe a n d the Au atoms for various compressions P/Po in table 1 and make a comparison between them. Here p means the density of matter in a n arbitrary condition, while Po is the n o r m a l density (po= 7.85 g / c m 3 for Fe a n d 10.5 g / c m s for Au). The quantity P/Po is used in place of ro because it is widely used in the literature. The relation between the density p ( g / c m 3) a n d the corresponding atomic radius r0 ( c m ) may be described as
]xrgp = AW/NA,
(3.1)
where Aw a n d Ark are the atomic weight a n d Avogadro's n u m b e r , respectively. Reflecting the behavior of the electron distribution, a considerable lowering of the total energy is observed in the relativistic
19287.4 19282.5 19268.7 19068.7 18754.5 18080.6 17396.7 16047.2 12839.2
case. Note that even for the lighter atom the deviations of the relativistic energies from the nonrelativistic become quite remarkable as the atom is highly compressed. This clearly indicates the importance of the relativistic effects in the study of highly compressed atoms. Next, as another interesting quantity, we give a tabulation of pressure values at atomic surface in table 2. In the same m a n n e r as in the nonrelativistic case [ 9,10 ], the pressure can be calculated from the following formula: P=
1
dE
4~ro2 dro
=-[~(n)-tcan4/3-(V+#)n]r=,o.
(3.2)
Table 2 tells us that the effect of the relativistic correction on the pressure is less significant than on the energy. This may be understood from the smallness of the relativistic effect on the electron density near the atomic surface. Note, however, that the relativistic effect becomes quite significant as the compression increases. Therefore in the calculation of the pressure too, the inclusion of the relativistic correction becomes important for highly compressed atoms.
4. Conclusion I n the above we have observed some features of the relativistic T F D W model of compressed atoms. S u m m i n g up these observations, one may conclude 195
Volume 138, number 4,5
PHYSICS LETTERS A
26 June 1989
Table 2 The pressure values at the atomic surface for Fe and Au (in au). PR and PNR denote relativistic and nonrelativistic pressures, respectively, while PD means the percentage deviation, i.e. PD= 1 0 0 ( P R - - P N R ) / I P N R [ .
P/Po
1 5 10 50 100 200 300 500 1000
Fe (Z=26)
Au (Z=79)
PR
PNR
PD
PR
PNR
PD
2.3061E-3 3.6366E- 1 1.9728E+0 6.6231E+ 1 2.6669E+2 1.0198E + 3 2.1907E+3 5.6401E + 3 1.9803E + 4
2.3162E-3 3.6543E- 1 1.9828E+0 6.6669E+ 1 2.6881E+2 1.0300E + 3 2.2164E+3 5.7227E + 3 2.0210E + 4
-0.436 -0.484 -0.504 -0.657 -0.788 - 0.990 - 1.16 - 1.44 - 2.01
1.8780E- 3 4.6666E- 1 2.9005E+0 1.2851E+2 5.7190E+2 2.3847E + 3 5.3547E+ 3 1.4478E + 4 5.3691E + 4
1.9125E-3 4.7479E- 1 2.9541E+0 1.3142E+2 5.8644E+2 2.4544E + 3 5.5276E+ 3 1.5018E + 4 5.6564E + 4
- 1.70 - 1.71 - 1.81 -2.21 -2.47 - 2,84 -3.12 - 3.60 - 5.08
that e v e n for lighter a t o m s the relativistic c o r r e c t i o n c e r t a i n l y brings a b o u t a p p r e c i a b l e effects o n the electron density distribution and other important quantities o f a t o m s w h e n t h e y are highly c o m p r e s s e d . We c o n c l u d e this article by a d d i n g two r e m a r k s : ( 1 ) T h e e l e c t r o n d e n s i t y c h a n g e c a u s e d by the relativistic c o r r e c t i o n n a t u r a l l y gives rise to a change in the e f f e c t i v e p o t e n t i a l for electrons. T h e one-elect r o n energies for t h e e f f e c t i v e p o t e n t i a l are e x p e c t e d to e x h i b i t c o n s i d e r a b l e d e v i a t i o n s f r o m the n o n r e lativistic e v a l u a t i o n s . ( 2 ) A l t h o u g h we h a v e c o m p l e t e l y n e g l e c t e d the relativistic c o r r e c t i o n s to the g r a d i e n t a n d e x c h a n g e t e r m s here, t h e i r effects o f c o u r s e m u s t be e x a m i n e d in the n e x t step. R e c e n t l y a n e w d e v e l o p m e n t o f the relativistic T F D W m o d e l has b e e n a c h i e v e d by Engel a n d D r e i z l e r [ 11,12 ]. F r o m the theoretical p o i n t o f view, t h e i r m o d e l s e e m s to be q u i t e satisfactory, in that it takes t h e relativistic c o r r e c t i o n s to b o t h o f these two t e r m s i n t o a c c o u n t . It will be w o r t h w h i l e to e x a m i n e w h a t effects the relativistic c o r r e c t i o n s o f these two t e r m s give rise to. T h e s e two studies are n o w in progress a n d we h o p e that the result will be r e p o r t e d shortly.
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Acknowledgement T h e a u t h o r w o u l d like to express his cordial thanks to P r o f e s s o r Y. T o m i s h i m a for his k i n d interest in this work. T h a n k s are also due to P r o f e s s o r F u r u tani, w h o k i n d l y g a v e the a u t h o r an o p p o r t u n i t y to give a talk a b o u t this subject.
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