Partial wave expansion in spheroidal coordinates for diatomic molecule

Brown, W. B.

Hake, R. B.

English

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https://ntrs.nasa.gov/search.jsp?R=19670002903 2020-03-16T16:55:49+00:00Z
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i 
A PARTIAL WAVE EXPANSION IN SPHEROIDAL COORDINATES 
FOR DIATOMIC MOLECULES* 
R. B. Hake and W. Byers Brown 
* This research was supported by the following grant: 
Aeronautics and Space Administration NsG-275-62. 
National 
In t roduc t ion  
' ?  
The method of p a r t i a l  wave expansicr, f s s  been adopted a number 
of  times i n  t h e  t h e o r e t i e z l  invesLiga t ion  of t he  s t r u c t u r e  and 
p r o p e r t i e s  of atoms and one- and two-electron diacornic molecules . 
The method c o n s i s t s  of w r l r i n g  the wave func t ion  of c5e system i n  
t h e  form ( f x  a o n c - e l x t r x  s y t e ~ )  
1 
where y(u3 Q,+) a r e  the s u r f a c e  harmonics. This  has  t h e  
advantage t h a t  i t  a f f o r d s  a s t r a igh t fo rward  nethod of success ive  
approximation t o  t h e  t o t a l  vava func t ion .  In a p p l i c a t i o n s  of t h i s  
method t o  diatomic systems the expansion (I) has been made about 
some p o i n t  i n  the  molecular framework, such a s  the  c e n t r e  of charge.  
One might expect  t h a t ,  f o r  d i a r o m i c  niolecules a more r a p i d l y  
convergent p a r t i a l  wave expansion could be obtained by employing 
p r o l a t e  sphe ro ida l  coord ina tes ,  which provide a n a t u r a l  coord ina te  
system. 
For t h e  diatomic system AB i n  which the  n u c l e i  A and B 
a r e  separa ted  by a d i s t a n c e  R a s  shown i n  Figure 1, 
I 
C' 
2 
w e  def ine  
and #; 
i n t e rnuc lea r  a x i s .  
i s  the angular  coord ina te  of t h e  po in t  about  t h e  
Appl ica t ion  t o  the  N-electron diatomic molecule 
A space component of t h e  t o t a l  wave func t ion  f o r  an  N-electron 
diatomic system may be w r i t t e n  a s  
_-- and 2 N where the summation i s  over a l l  v a l u e s  of - 
--- m i . e .  \ W L I ~  4; permiss ib le  va lues  of ml, m2 
N '  
and 
a l s o  we must have 
1 , 
3 
N 
where 
and 
A l s o  
The expansion (2) i s  formally exac t ;  however, i n  a l l  p r a c t i c a l  
a p p l i c a t i o n s  the  series must be truncated. The number of terms 
r e t a i n e d  i n  t h e  t runca ted  expansion w i l l  l e t e rmine  t h e  degree of 
approximat ion. 
If t he  nuc lea r  charges i n  the diatomic system a r e  Z and Z b  a 
t h e  Hamiltonian may be w r i t t e n  (using p r o l a t e  sphe ro ida l  coord ina tes )  
where U- = Z + Zb : A = Z b  - Z 
a a 
and 
# . '  i 
and 
where 5 (4) is t h e  lesser and 2. (bl the g r e a t e r  of 
We now wish t o  t a k e  t h e  wave equat ion  
and 
mult ip ly  through by t h e  product  
, 
4 
and i n t e g r a t e  over t h e  ' angu la r '  variables and C/J If t h i s  is 
5 
done f o r  every product  of t he  form (5) then  a set of  equat ions  
I n  the  fol lowing we s h a l l  f ind i t  convenient t o  in t roduce  the 
r o t  a t  i on 
Write the  t o t a l  Hamiltonian as  
a 
(c.f. equat ion  ( 3 j j  
., . 
c '  
6 
Multiply ( 6 )  by the product (5)  and integrate over the variables 
and for a l l  of the electrons,  making use of integrals  
2 given by Livingston , special  cases of which are given i n  the 
appendix. 
Thi s  gives 
7 
I 
t 
where 
O( = hi/2 i 
= 2 ki P i  
i = 2 k l + l  
o( i =(hi - i)/2 
Xi even (31 = 2ki + 1 
y = 2ki 
1 odd 
ti) 
W e  have 
Consider one of the '1-6; terms multiplied by the product (5). 
.' 
8 
I n t e g r a t i n g  over t h e  
and j gives  ze ro  f o r  a l l  terms b u t  i k  = )&k ', mk = p k  
( k  = \ A , * . .  N n& i ,  j ) I n t e g r a t i n g  over t he  coord ina te s  
, (# coord ina te s  of a l l  e l e c t r o n s  bu t  
3; J d ;  and 7; J + i  gives 
where M + m i - ) A i - O  
and the C ' s  are t h e  Clebsch-Gordon c o e f f i c i e n t s :  see, for  
3 
example, Rose . 
9 
Also 
. 
? 
where M +pi - ” j  = 0 
Using t h e s e  r e s u l t s  t he  t o t a l  ‘ITij term i s :  
.’ 
10 
iii) 
t h e  Y’S and g ives  simply 
Mul t ip ly ing  the  R.H.S. of ( 4 )  by (5) and i n t e g r a t i n g  Over 
Thus the f i n a l  r e s u l t  i s  obtained by w r i t i n g  ( 7 )  + (8) = ( 9 )  
These r a t h e r  cumbersome equat ions  may be w r i t t e n  i n  a more 
compact form i f  w e  def ine 
Hence we ob ta in :  
11 
= E 
f o r  every  product  of t h e  form (5). 
The equat ions  (10) a r e  a s e t  of coupled d i f f e r e n t i a l  equat ions  
11 
which are  equiva len t  t o  t h e  Schrodinger equat ion  of t he  system, 
equa t ion  ( 4 ) .  Thus t h e  wave func t ion  and energy of t he  diaroniic 
system may be obtained by so lv ing  t h e  d i f f e r e n t i a l  equat ions  (10) 
f o r  t h e  func t ions  9 f 8 - f  ( f , .  . . %"* f3 
12 
Example of t h e  H2+ Ion  
Here w e  d i scuss  b r i e f l y  t h e  a p p l i c a t i o n  of t he  foregoing theory 
t o  t h e  s i m p l e  one-electron problem of t h e  H l  molecular ion,  t h e  
Schrodinger equat ion  of which i s  separable  i n  p r o l a t e  sphe ro ida l  
coord ina tes ,  and has been solved a c c u r a t e l y  by Bates, Ledsham and 
Stewart  and more r e c e n t l y  and more ex tens ive ly  by Peek . 
9 9  
4 5 
The wave func t ion  can be w r i t t e n  i n  the  form (see  (1)) 
1, vvf 
Mult ip ly ing  the  wave func t ion  by x'(q, +) and i n t e g r a t i n g  
gives ,  u s ing  the  r e s u l t  ( l o ) ,  and r e s t r i c t i n g  our i n t e r e s t  t o  t h e  
ground s t a t e  where m = 0 
13 
I *  
I 1  
This  set  of equat ions  r e p l a c e s  the  o r i g i n a l  Schrodinger equat ion,  
so  s o l u t i o n s  of t h e s e  eqdafions w i l l  determine the  wave func t ion  
and energy of t h e  system. The accuracy of t he  r e s u l t s  obtained i s  
determined by t h e  po in t  where the summation i s  t runcated.  
The exac t  ground s t a t e  wave func t ion  f o r  Hg+ may be w r i t t e n  a s  
where the  $(P,f) are cons t an t s  and 
Comparing (13) wi th  (11) w e  see t h a t  
Using (12) i t  i s  now poss ib l e  t o  derive a cond i t ion  which must be 
s a t i s f i e d  by t h e  f ' s  . 
The equat ion  fo r  z(4) given i n  r e fe rence  (3)  i s  
where C i s  a sepa ra t ion  constant .  This  equat ion i s  the  same as 
c 
14 
Hence 
So from (12) we have 
c 
c 
’ * .  
5 
and from t h i s  we ob ta in  
~ 
15 
Now 
6 
See, f o r  example, Jahnke and Emde . Writ ing 
and s u b s t i t u t i n g  i n t o  (15) we have 
As j 1, h 300 , so  for  t h i s  t o  be t r u e  f o r  v a l u e s  oE 
w e  r e q u i r e  
r .  
16 
The equat ion (15) may a l s o  be used t o  d e r i v e  a three- te rms  r e c u r s i o n  
r e l a t i o n  fo r  t he  f ' s .  
Using (15) w e  f i r s t  form t h e  sum 
1 7  
Then we Eind t h a t  evslng t h e  ~ecur rence  r e l a t i o n  for Q% 
s 
Thus we have 
w 
= 6  which gives  
It should a l s o  be noted here  t h a t  t h e  e s s e n t i a l  d i f f e r e n c e  
between the p a r t i a l  wave expansion method and t h e  ‘usua l ’  s e p a r a t i o n  
t reatment  a r i s e s  from the  f a c t  t h a t  i n  t h e  l a t t e r  approach t h e  
SchrGdinger wave i s  mul t ip l i ed  through by the  f a c t o r  (g’- T2) i n  
order  t o  f a c i l i t a t e  separa t ion .  
a term involv ing  ( i’- Tx)-’ which, when expanded, g ives  an i n f i n i t e  
s e r i e s .  
I f  t h i s  i s  not  done then w e  have 
19 
APPENDIX 
Some useful  i n t e g r a l s  for the  foregoing theory are 
= o  
L - 0  ! 9 d . 4  
* *  
3. 
20 
I - 
for  n + x  even 
= o  : fi + t  odd 
+I odd 
- o 6 + 1 even 
2 1  
R e  f e r  ence s 
. 
1. P. Luke, R. Meyerott and W. Clendenin, Phys. Rev. 85, 401 (1952); 
A. Temkin, J. Chem. Phys. 39, 161 (1963); A. Temkin and A .  K. 
Bhatia,  J. Chem. Phys. 42, 644 (1965); H. Rabinovich, J. Chem. 
Phys. 43, 3144 (1965). 
2.  P. M. Livingston,  J. Math. Phys. 325 (1966). 
3. M. E. Rose, "Elementary Theory of Angular Momentum" John Wiley (1957). 
4. D. R.  Bates,  K. Ledsham and A .  Le Stewart ,  P h i l .  Trans. Roy. SOC. 
A246, 215 (1953). 
J. M. 'Peek, Sandia Corporation Report SC-RR-65-77 (1966). 5. 
6. E. Jahnke and F. Emde, "Tables of Function", Dover (1945). 
t 


A Partial Wave Expansion in Spheroidal Coordinates for Diatomic Molecules

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