Energy states of helium isoelectronic series investigated by 50-term variational wave function with two nonlinear parameter

Machacek, M.

Sanders, F.

Scherr, C. W.

English

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L 1 
Finite-Mass 
Milos Machacek , * 
Helium ' A t o m s  11. 
Frank Sanders,  and Char l e s  W. Scher r  
The Z3P State' 
Phys ics  Department GPO $ 
The Unive r s i ty  of Texas CFSTl PRICE(S) $ 
/. 0 Aust in ,  Texas 78712 Hard copy (HC) 
q v ? r *  Microfiche (M F) 
ABSTRACT , !* 
ff 653 July 65 
3 
I The 2 P s ta tes  of t h e  helium i s o e l e c t r o n i c  series are i n v e s t i -  " 1 
g a t e d  v i a  50-term v a r i a t i o n a l  wavefunctions c o n t a i n i n g  two n o n l i n e a r  
parameters .  The Hamil tonian,  s p i n  free and n o n - r e l a t i v i s t i c ,  is 
w r i t t e n  d i r e c t l y  i n  center-of-mass c o o r d i n a t e s  so  t h a t  no adiabat ic  
approximation is r e q u i r e d .  The e x p e c t a t i o n  v a l u e s  of the Hamil tonian 
and of moments of t he  i n t e r p a r t i c l e  s e p a r a t i o n s  are r e p o r t e d .  I s o t o p e  
s h i f t s  are found,  and, as noted i n  a p re l imina ry  repor t ,  are i n  
e x c e l l e n t  agreement w i t h  experiment.  I n  order t o  compare t h e  r e s u l t s  
w i t h  p rev ious  t h e o r e t i c a l . r e s u l t s  t h e  systems were s i m i l a r l y  s t u d i e d  
i n  t h e  i n f i n i t e  n u c l e a r  m a s s  l i m i t  (adiabatic approximation) .  Tile 
e n e r g i e s  so o b t a i n e d  are t h e  deepes t  so f a r  o b t a i n e d  by a direct  
c a l c u l a t i o n  w i t h  a v a r i a t i o n a l  wavefunction. The e x p e c t a t i o n  v a h u e s  
of t he  o p e r a t o r s  were s u b j e c t e d  t o  a d i f f e r e n c i n g  p rocess  t o  o b t a i n  
estimates of t h e  p e r t u r b a t i o n  expansion c o e f f i c i e n t s  f o r  them, and 
t h e  r e s u l t s  are i n  good agreement w i t h  t he  d i r e c t l y  c a l c u l a t e d  
r e s u l t s  of Knight and Sche r r .  &&a 
'This research w a s  suppor ted  by a g r a n t  from t h e  Na t iona l  Aeronaut i l  .i 
and Space Admin i s t r a t ion .  
P r e s e n t  address: 80 Burley S t r e e t ,  Danver, Massachuset ts .  * 
https://ntrs.nasa.gov/search.jsp?R=19650019825 2020-03-17T02:00:24+00:00Z
I 
I .  
-2- 
I .  INTRODUCTION 
I n  paper I t h e  2'P states  of t h e  helium atom i s o - e l e c t r o n i c  
series were treated by a v a r i a t i o n a l  procedure i n  which t h e  
Hamiltonian e x p l i c i t l y  conta ined  t h e  correct mass dependence 
( t r u e  mass c a l c u l a t i o n ) ! '  
i n  t h e  i n f i n i t e  nuc lea r  mass approximation ( a d i a b a t i c  approxi- 
mat ion) .  The t r u e  mass c a l c u l a t i o n  r e s u l t s  were used  t o  s t u d y  
t h e  i s o t o p e - s h i f t s  i n  these systems,  t h u s  provid ing  a direct  
l i n k  w i t h  experiment .  As a f u r t h e r  i n d i c a t i o n  o f  t h e  re l iab i -  
l i t y  of  t h e  c a l c u l a t i o n s ,  t h e  i n f i n i t e  mass c a l c u l a t i o n  r e s u l t s  
were ana lysed  and a comparison made with some r e s u l t s  of p e r t u r -  
b a t i o n  theo ry .  The work s t a r t e d  i n  Paper I is completed here 
S t u d i e s  were also made of these s ta tes  
w i t h  a s i m i l a r  s e t  o f  c a l c u l a t i o n s  f o r  t h e  2'P s ta tes  of t h e  
same s y s t e m s .  
The comparison of t h e  i n f i n i t e  m a s s  c a l c u l a t i o n  r e s u l t s  
w i t h  t h e  r e s u l t s  of p e r t u r b a t i o n  theo ry  is achieved  through a 
well-known consequence of t h e  Schrgdinger p e r t u r b a t i o n  procedure:  
f o r  atomic systems t h e  t o t a l  wavefunction Y can be w r i t t e n  as an 
expansion i n  i n v e r s e  powers of t h e  n u c l e a r  charge Z .  Consequently,  
e x p e c t a t i o n  va lues  of any o p e r a t o r  Cl are o b t a i n e d  i n  a power series 
i n  2 - l :  
JdTYn'l' = (Cl) = C Za'n(L?)ne (1)  
where t h e  v a l u e  of a depends on t h e  n a t u r e  of t h e  o p e r a t o r  R. 
The Y is assumed t o  be real  and t o  be normalized. 
t h e  e x p e c t a t i o n  va lue  of t h e  HamiStonian is t h e  t o t a l  energy,  E ( Z ) ,  
I n  p a r t i c u l a r ,  
_ _  
a 
-3- 
(2)  
2 E(Z)  = Z C Z - n ~  n ,  
I t  is  u s u a l l y  e a s y  t o  compute t h e  lead c o e f f i c i e n t  i n  E q s .  (1) and 
(2) by an elementary i n t e g r a t i o n ,  and i n  a number of cases t h e  next  
few c o e f f i c i e n t s  have been numerical ly  approximated? 
i t  is known t h a t  
I n  a d d i t i o n ,  
3 
-m 
where 1: is t h e  e lec t ron-nucleus  s e p a r a t i o n ,  and u is t h e  e l e c t r o n -  
e l e c t r o n  s e p a r a t i o n .  Est imates  of t h e  l ead ing  c o e f f i c i e n t s  i n  
E q s .  (1)  and (2)  may be obtained from t h e  r e s u l t s  of a series of 
v a r i a t i o n a l  c a l c u l a t i o n s  v i a  a d i f f e r e n c i n g  technique  t h a t  r ega rds  
t h e  data as t r u n c a t e d  expansions of t h e  form of Eq.  (1). 
11. RESULTS 
The va lue  of t h e  lowest energy found f o r  each va lue  of Z 
cons idered  i n  d isp layed  i n  Table I f o r  both t h e  i n f i n i t e  mass 
c a l c u l a t i o n s  and f o r  t h e  more common o r  s table  i s o t o p e s  of t h e  
atoms. (These c a l c u l a t i o n s  do not  g i v e  a s tab le  2 P s t a t e  fo r  
H-.) From these d a t a  and from t h e  Rydberg va lues  i n  Table I1 
of paper  I ,  i s o t o p e  s h i f t s  a r e  c a l c u l a t e d  and p resen ted  i n  Table 
11. The p e r t u r b a t i o n  energy expansion c o e f f i c i e n t s  were recovered 
from EZ(Z).  
3 
For Z from 2 t o  10 i n c l u s i v e ,  where E2’(Z) is  de f ined  by 
E2(Z) - E(Z)  + (5/8)Z2 - (1481/6561)Z 
for the 2% systems, the results are p resen ted  i n  Table 111. 
/ . .  
-4- 
Total e x p e c t a t i o n  v a l u e s  of t h e  moments of t h e  i n t e r p a r t i -  
c le  c o o r d i n a t e s  are also computed. As is Paper  I ,  these d a t a  
are ana lysed  v i a  t h e  d i f f e r e n c i n g  technique  i n  two ways :  (1)  
d i r e c t  a n a l y s i s  of t h e  basic d a t a , ( 2 )  a n a l y s i s  o f  t h e  data a f t e r  
removal of t h e  exactly-known zero o r d e r  c o e f f i c i e n t .  The first 
way g i v e s  r e s u l t s  s u b s t a n t i a l l y  s imilar  t o  t h e  r e s u l t s  of t h e  
a n a l y s i s  of  t h e  2 P d a t a ,  and they are n o t  r e p o r t e d .  The r e s u l t s  
of t h e  second way are p resen ted  i n  Table  I V .  I n  a d d i t i o n ,  a t h i r d  
a n a l y s i s  is performed: 
a f t e r  removal of t h e  exactly-known zero-order  c o e f f i c i e n t  and 
of t h e  f i r s t - o r d e r  c o e f f i c i e n t ,  known v i a  Eqs. (3)  or ( 4 ) .  
The ( r - l ) l  = -1/201 is known e x a c t l y ,  wh i l e  t h e  (u - ’ )~  = 2c2 
is taken  from r e f e r e n c e  2,  where i t  has  been numer ica l ly  o b t a i n e d  
t o  more s i g n i f i c a n t  f i g u r e s  than are needed h e r e .  The r e s u l t s  of 
t h i s  t h i r d  a n a l y s i s  are p resen ted  i n  Table  V n o t  on ly  f o r  t h e  
2 P s ta tes  b u t  a lso f o r  t h e  2 P states.  
1 
(3)  a n a l y s i s  of  t h e  (r-’) and ( u - l )  d a t a  
3 1 
111. DISCUSSION 
P e k e r i s  S c h i f f  and Lifson3 employed a 220-term v a r i a t i o n a l  
wavefunction i n  a s t u d y  of  2P and 3P s ta tes  of t h e  helium atom. 
T h e i r  energy r e s u l t 4  f o r  t h e  i n f i n i t e  m a s s  helium 2 P s t a t e  is 
-2.1331633 a . u .  As i n  paper  I ,  t h e  deeper  energy of t h e  50-term 
wavefunction used  here is to  be a t t r i b u t e d  t o  t h e  f a c t  t h a t  Pe- 
k e r i s ,  S c h i f f  and L i f son  p r e s e l e c t e d  the i r  non- l inear  parameters .  
T h e i r  3 P energy v a l u e  is -2.0580715 a.u. , ,  and t h e  50-term v a l u e  
is -2.0563023 a . u .  
3 
3 
- 5- 
1 A s  i n  t h e  2 P c a l c u l a t i o n s ,  t h e  e x t r a p o l a t i o n  of t h e  non- 
l inear  parameters  t o  t h e  case of i n f i n i t e  2 is not  c l e a r c u t  
because of m u l t i p l e  minima. Approximately, t h e y  e x t r a p o l a t e  t o  
1.0182 and 0 . 6 2 2  and t h e  Peke r i s ,  Sch i f f  and Li fson  va lues  t o  Z 
1 and 0.52 cor respondingly ,  (The 2 P 50-term va lues  e x t r a p o l a t e d  t o  
1.0052 and 0.642.)  
The i s o t o p e  s h i f t s  f o r  Z equals  2 and 3 c a l c u l a t e d  i n  t h i s  
n o t e  have already been d iscussed  elsewhere! 
i s o e l e c t r o n i c  series, presented  i n  Table 11, can be roughly f i t t e d  
t o  an express ion  whose l ead ing  terms are 
The r e s u l t s  f o r  t h e  
4 2  S h i f t  (cm”) = 1.03 x 10 Z mp’l(1 - 1.502-I +...) 
where g-l is t h e  magnitude of the d i f f e r e n c e  of t h e  r e c i p r o c a l  
masses of t h e  two n u c l e i  involved and m is t h e  e l e c t r o n i c  mass. 
If t h e  s h i f t  is c a l c u l a t e d  from a s imple ,  p rope r ly  symmetrized, 
6 product  wavefunction of t h e  type cons idered  by Hughes and Ecka r t ,  
bu t  u s i n g  t h e  P e k e r i s  et  a1 p r e s c r i p t i o n  for t h e  o r b i t a l  exponents ,  
then  t h e  r e s u l t  is 
4 2  Z S h i f t  (cm”) - 1.00 x 10 ( 1  - 1.51 Z - l  +...). 
FOOTNOTES 
1. 
2. 
3. 
4. 
5. 
6. 
M. Machacek, F. Sanders,  and C. W. Sche r r ,  i n  p r e s s ,  
R. E. Knight and C. W. Scherr ,  Rev. Mod. Phys. - 35, 431(1963). 
C. L. Peker i s ,  B. S c h i f f ,  and H. L i fson ,  Phys. Rev. 126, 1057 
(1962). 
P e k e r i s ,  Sch i f f  and Li fson  also presen t  e x t r a p o l a t e d  r e s u l t s  
which should  b e $ e r y  accu ra t e .  These  are, f o r  t h e  2 P s ta te ,  
3 -2.1331641 a . u .  and f o r  the  3 P s t a t e ,  -2.0580812 a . u .  
M. Machacek and C. W .  Scher r ,  J .  Chem. Phys. 39, 3151(1963). 
D. S. Hughes and C. E c k a r t ,  Phys. Rev. - 36, 694(1930). 
-
3 
-
TABLE I 
THE ENERGIES IN REDUCED ATOMIC UNITS' 
--- 
I so tope  (m/~)xl~4 E ( t r u e  mass) E ( i n f i n i t e  mass) 
3 
4 
H e  
H e  
L i 6  
7 
9 
L i  
Be 
BIO 
B1l 
C12 
Cl3 
NI4 
N15 
OI6 
0l7 
0l8 
F1' 
N e  
N e  
20 
21 
1.8192 
1.3706 
0.9126 
0.7821 
0.6089 
0.5480 
0.4984 
0.4573 
0.4220 
0.3919 
0.36585 
0.3431 
0.3228 
0.3049 
0.2888 
0.2745 
0.2614 
-2.13317527 1 
-2.13317237 ) 
-5.02774288 
-5.02773889 
-9.17501548 
-14.5732063 
-14.5732000 
-21.2218005- 
J -21.2217935- 
-29.1206127 1 
'r 
-29.1206053 J 
-38.2695549 
-38.2695471 
-38.2695401 
-48.6685726 
-60.3176636 
-60.3176553 
-2.13316352 
-5.02771503 
-9.17497248 
-14.5731370 
-21.2217100 
-29.1205010 
-38.2694219 
-48.6684265- 
-60.3174880 
N e  22 0.2495 -60.3176476 I 
Reduced atomic energy u n i t s  i n  u n i t s  of pe4112. L e t  m be t h e  mass of a 
t h e  electron and M be t h e  mass of t h e  nucleus,  then t h e  reduced m a s s ,  
I.r = mM/(M+m). 
t h e  familar i n f i n i t e  m a s s  Rydberg. 
For t h e  i n f i n i t e  mass s y s t e m s  ~.r = m, and t h e  Rydberg is 
TABLE 11 
ISOTOPE SHIFTS 
~~ ~~ ~-~ 
-1 a - Systems S h i f t  (cm ) 
3 H e  ; 
Li6; 
C12; 
N14. 9 
0 l 6  J 
016. 9 
BIO 9 
Ne2' j 
Ne2'; 
4 H e  
L i 7  
B l l  
C l 3  
0 l 7  
0l8 
N 1 5  
21 N e  
Ne22 
0 .675 
0.633 
0.88 
0.96 
1.00 
1 .07  
2 . 0 0  
1.13  
2.14 
f 0.005 
f 0,006 
* 0 .05  
f 0.06 
f 0.06 
0.07 
f 0 .07  
f 0.08 
0.08  
The error est imate is a 
1 data of the  P states. 
the same as  w a s  made for the  corresponding 
TABLE I11 
THE sn I N  ATOMIC UNITS 
b n Recovered herea Knight and Scherr 
2 
3 
4 
-0,072997 
-0.01661 
-0.01013 
-0.072992603 
-0.016558519 
5 -0.0065- 
The Z = 2 and 3 values a 
d i g i t  is to be regarded 
bData from reference 2. 
are not included 
as unreliable.  
i n  the analysis .  The l a s t  
, + 
TABLE IV 
RECOVZRY~OF THE <n>, 
-1 r 
r 
2 r 
-1 U 
U 
2 U 
510 
1314 
33/2 
1481/6561 
70615/13122 
204153 /59049 
-0.11251 
1.969 
23 e99 
-0 . 14669 
3 e477 
48.17 
0 OOk09 
1.681 
24.77 
3.197 
4-8 '67 
-0.06620 
a The Z - 2 and 3 values  are not  included i n  the ana lys i s .  - 
' 8  
* ' b  
TABLE V 
-1 A REFINED RECOVERYa OF AND .(u )n 
IP r 5/8 17 0 5/ 13 122 - 0 .  00050 0 01604 
lP U -' 1705/6561 -0. 31404246b 0.0787 0.0131 
-1 
3P r -1 5/8 1481/13122 0. 00066 -0 .  01451 
3P U 148 1/ 6 56 1 -0, 145985206b -0.0490 -0.0518 -1 
aThe Z = 2 and 3 values are not included i n  the analysis. 
bData from reference 2. 


Finite-mass helium atoms. ii- the 23p state plus

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