Electronic charge density for ground state of helium as function of nuclear charg

Epstein, J. H.

Epstein, S. T.

English

NASA Technical Reports Server

" 
ELECTRONIC CHARGE DENSITY I N  HELIUM I N  THE FIRST ORDER 
SHIELDING APPROXIMAT ION* 
by 
Jean H. Epstein 
Un ive r s i ty  of Wisconsin Theore t i c a  1 Chemistry I n s t i t u t e  
Ma d i son, W i s co ns i n  
and 
Saul  T. Epstein 
Theore t i ca l  Chemistry I n s t i t u t e  and Physics  Department 
Univer s i t  y of W i  sc oris i n ,  Mad i s  on, Wiscons i n  
ABSTRACT 
We have c a l c u l a t e d  the  e l e c t r o n i c  charge d e n s i t y  for  t he  
ground s t a t e  of helium us ing  the f i r s t  order  s h i e l d i n g  approximation. 
The r e s u l t s  a r e  d isappoin t ing .  
* This  r e s e a r c h  was supported by the fol lowing g r a n t :  
Nat iona l  Aeronaut ics  and Space Adminis t ra t ion Grant NsG-275-62. 
https://ntrs.nasa.gov/search.jsp?R=19660006960 2020-03-16T23:11:49+00:00Z
. 
I. I n t r o d u c t i o n  
C. Schwartz’ has  ca lcu la ted  the  f i r s t  order  c o r r e c t i o n  t o  the  
2 
e l e c t r o n i c  charge d e n s i t y  i n  the ground s t a t e  of Helium-like ions, 
due t o  t h e  i n t e r a c t i o n  between the  e l e c t r o n s .  H i s  r e s u l t  i n  atomic 
u n i t s  f o r  an i o n  of nuclear  charge Z i s  3 
where 
+ S  + 3 c 
Y 2 
and where 
I n  t h e  d e r i v a t i o n  of (1) only the  coulomb i n t e r a c t i o n  between 
t h e  e l e c t r o n s  was t r e a t e d  a s  a p e r t u r b a t i o n .  I n  the  f i r s t  order  
1 
. + 
2 
4 sh ie ld ing  approximation one has  a s  an a d d i t i o n a l  p e r t u r b a t i o n  the  
e f f e c t  of the d i f f e r e n c e  between the a c t u a l  nuc lear  charge 2 and 
an assumed nuc lear  charge 3 . 
i n  t he  expec ta t ion  va lues  of any opera tor  b/ 
e x p l i c i t l y  on 3 then i f  wi th  \5 = 2 one f i n d s  
I n  p a r t i c u l a r  i f  we a r e  i n t e r e s t e d  
which does no t  depend 
i t  is  easy t o  show t h a t  i n  t h e  f i r s t  order  s h i e l d i n g  approximation 
one w i l l  f ind  
4 
Following Dalgarno and Stewart  one then  determines \ 
f i r s t  order c o r r e c t i o n  vanishes :  
so  t h a t  t he  
d-5 
, For our problem where w =  1 6 6C2-2,) t$t%-?a--> 
and r being the  e l e c t r o n i c  coord ina te s ,  t h i s  means t h a t  we 1 1  -2 
w i l l  choose a d i f f e r e n t  va lue  of 3 f o r  each va lue  of r . 1 
3 
11. Calcula  t i ona  1 Procedure 
Combining (l), ( 4 )  and ( 5 )  one f i n d s  the  equat ion  
2-W-3 
which i s  t o  be solved t o  y i e l d  3 
given z . Since G i s  a l lun iversa l  function' '  of y~ 2 9 ~  , 
as  a func t ion  of 4 f o r  a 
t h e  procedure i s  q u i t e  simple.  For a given va lue  of y one e v a l u a t e s  
G . Then one uses  ( 6 )  t o  determine 2 , whence one can determine 
-t from 
+ =  
( 7 )  
Unfor tuna te ly  t h i e  procedure runs  i n t o  d i f f i c u l t y  because G 
Refer r ing  t o  F igure  I, which i s  a graph is  s i n g u l a r  a t  y ' =  3 '. 
of G ve r sus  y , w e  could confine our a t t e n t i o n  t o  t h e  " f i r s t  
branch" running from y=O t o  y=3 , y=O y i e l d i n g  r = O  and G& 
y i e l d i n g  't = 00 , However t h i s  seemed phys ica l ly  u n s a t i s f a c t o r y  
because, as  one r e a d i l y  sees, t h i s  would mean t h a t  
would be decreas ing  only a s  +-3  as 3--> 00 . On t h e  o ther  
hand t h e  second branch c l e a r l y  y i e l d s  '5 3 cons tan t  a s  r 4 . 
* 
4 
Thus what w e  have chosen t o  do i s  t o  use the  f i r s t  branch f o r  small  
va lues  of r and the second branch f o r  l a rge  va lues  of r , br idg ing  
the  d i s c o n t i n u i t y  smoothly by eye. 
111. Resul t s  and Discussion 
The r e s u l t s  f o r f  (r) f o r  2 = Z are shown i n  F igure  11. The 
p o i n t s  a r e  obtained i n  the  way we have ind ica t ed  while  the  smooth 
, curve,  r ep resen t s  the  a n a l y t i c  i n t e r p o l a t i o n  formula 
The values  of t he  Ai and ki a r e  given i n  Table I. 
To g e t  some idea  of the accuracy of our r e s u l t s  we have computed 
the  averages of some powers of ? us ing  ( 9 )  and d e f i n i n g  
Th r e s u l t s ,  and t h e i r  comparison wi th  exac t  va lues  and wi th  valr 
der ived us ing  the  f i r s t - o r d e r  s h i e l d i n g  approximation are  given 
es 
i n  Table 11. Clea r ly  on a l l  counts :  t h e o r e t i c a l  ( t h e  s i n g u l a r i t y )  
and p r a c t i c a l  (agreement wi th  exac t  v a l u e s )  , t h e  r e s u l t s  are 
r a t h e r  d i sappoin t ing .  
7 
5 
Footnotes and References 
1. C. Schwartz, Annals of Phys. 6, 156 (1959). 
2. G. G. Hal l ,  L. L. Jones, and D. Rees, Proc. Roy. SOC. (London) 
A283, 194 (1965) have derived analogous r e s u l t s  for  t h e  
s t a t e  of He-like ions  and for  the  ground s t a t e  of L i - l i k e  ions .  
3. Schwartz 's  paper con ta ins  a misp r in t .  We have chosen t o  r e t a i n  
L& the  " '  on the  -t hand s i d e  of equat ion (25) and t o  d e l e t e  5 
the  "2" i n  equat ion ( 2 1 ) .  Schwartz (Table I) does t h e  r eve r se .  
I n c i d e n t a l l y ,  we have a l s o  reder ived  (1) us ing  double p e r t u r b a t i o n  
theory.  (See J. 0. Hi rschfe lder ,  W. Byers Brown, and S. T. 
Epstein,  Advances i n  Quantum Chemistry, ed i t ed  by P. 0. Lowdin 
(Academic P r e s s  Inc . ,  New York 1964) footnote  9a, page 291. 
11 
4 .  A. Dalgarno and A. L. Stewart ,  Proc. Roy. SOC. (London) A257, 
534 (1960). 
5.  Another procedure one might use t o  determine 3 i s  t o  r e q u i r e  
t h a t  %/bx=o (See W. A. Sanders and J. 0. Hi rschfe lder ,  
J. Chem. Phys. 42, 2904 (1965)). However, t h i s  involves  
s i n g u l a r i t i e s  a t  2 1 r = 3 2 6  so  w e  d i d  no t  pursue i t  
f u r t h e r .  
6 .  See Table I W. A. Sanders and J. 0 .  Hirschfe lder ,  Footnote  6 
and r e fe rences  given there .  
7 .  Unfortunately,  an  accurate  graph of 3 does not  seem t o  be 
a v a i l a b l e  i n  t h e  l i t e r a t u r e ,  and an o v e r a l l  comparison might show 
our approximation t o  be b e t t e r  than the  numbers i n  Table I1 suggest .  
For example, our !LO) i s  i d e n t i c a l  t o  t ha t  given by t he  usua l  
s h i e l d i n g  approximation and i s  q u i t e  accu ra t e  . 
w 
6 
TABLE I 
4 
-.44 
10.0 
c 
I I 
I I 
t 
t 
7 
This  
Appr ox. 
TABLE I1 
Exact 6 Shielding 6 
Appr ox. 
n = -1 I 
1 . 4 3  I 1.19 I 1.17  
1.01 I . 92  1 . 93  
c 
t 
4 
I 
, 
? 
9 


Electronic charge density in helium in the first order shielding approximation

Abstract

Similar works

Full text

Available Versions

NASA Technical Reports Server