Interchange theorems and variational principle in perturbation theor

Epstein, S. T.

English

NASA Technical Reports Server

. 
TWO NOTES ON PERTURBATION THEORY* 
I e INTERCHANGE THEOREMS AND THE VARIATION PRINCIPLE 
11. pv I N  A DIFFERENT NOTATION 
by 
Saul T .  Eps t e in  
T h e o r e t i c a l  Chernis t r y  I n s t i t u t e  and Phys ics  Department 
Un ive r s i ty  of Wisconsin, Madison, Wisconsin 
. 
* This  r e s e a r c h  was supported by the  Nat ional  Aeronautics and 
Space Adminis t ra t ion Grant N s G -  275-62.  
https://ntrs.nasa.gov/search.jsp?R=19680012734 2020-03-12T09:01:30+00:00Z
I. Interchange Theorems and t h e  Var i a t ion  P r i n c i p l e  
Interchange theorems have been proven and used i n  Rayleigh 
1 
Schroedinger p e r t u r b a t i o n  theory and i n  p e r t u r b a t i o n  theory wi th in  
t h e  Hartree-Fock approximation. I n  t h i s  no te  we wish t o  e x h i b i t  2 
t h e  common o r i g i n  of  t h e s e  theorems and a l s o  t o  i n d i c a t e  t h a t  such 
theorems can be expected t o  hold i n  o the r  s i t u a t i o n s  , by showing 3 
how interchange theorems a r i s e  out of t h e  v a r i a t i o n a l  p r i n c i p l e .  
We w i l l  con f ine  our a t t e n t i o n  t o  f i r s t  o rde r  p r o p e r t i e s .  4 
We consider  a Hamiltonian 
1"" H = H + ) , V  + 0 
L e t  be an opt imal  t r i a l  funct ion which w e  have der ived from 
the  v a r i a t i o n a l  p r i n c i p l e  : 
where, f o r  s i m p l i c i t y  of no ta t ion ,  we assume eve ry th ing  t o  be r e a l .  
We w i l l  suppose t h a t ,  i n  q u i t e  a gene ra l  way, 
N 
where 'p 
c o n s t a n t s )  which have been determined v a r i a t i o n a l l y  by f r e e  v a r i a t i o n  
s t a n d s  f o r  some funct ions ( p o s s i b l y  they  a r e  simply 
1 
2 
/v 
of t h e i r  func t iona l  form. We w i l l  f u r t h e r  assume t h a t  'p 
and can be expanded i n  a double power s e r i e s  i n  
'L 
I n s e r t i n g  these expansions and the  corresponding ones f o r  6.k-. 
(1) and ( 2 )  and equa t ing  t h e  c o e f f i c i e n t s  o f  each order  t o  zero we 
f ind  i n  p a r t i c u l a r  
i n t o  
5 
3 
and 
w 
Now we have assumed t h a t  i s  f r e e l y  v a r i a b l e ,  hence cons ider  
'L 
the  v a r i a t i o n  wi th  no o the r  p a r t  o f  
changing. From ( 4 )  and (5) t h i s  then y i e l d s  
4 
whence from ( 7 )  we f ind  
Similar  l y  ,FcO‘) # and a l l  o the r  v a r i a t i o n s  zero,  
y i e  I d s  
F i n a l l y  w e  no te  t h a t  from ( 6 )  
Combining (8) and (11) i t  then  fol lows from (9)  and (10) t h a t  
we have the  in t e rchange  theorem 
Q. E.D. 
. 
5 
11. PV i n  a D i f f e r e n t  Notation 
. 
I n  thLs n o t e  w e  wish t o  de r ive  one of t h e  e s s e n t i a l  r e s u l t s  i n  
3 
t h e  i n t e r e s t i n g  paper of Silverman and van Leuven , us ing  a 
n o t a t i o n  which i s  perhaps more f a m i l i a r  ( c e r t a i n l y  i t  i s  t o  u s ) .  
Namely we wish t o  d e r i v e  the  r e s u l t  which i s  summarized i n  t h e  
d i s c u s s i o n  fol lowing t h e i r  equat ion (24 ) .  
We cons ide r  now a s i n g l e  p e r t u r b a t i o n ,  however we w i l l  cont inue 
(L 
t o  allow the  t o  denote a se t  of func t ions ,  no t  n e c e s s a r i l y  
c o n s t a n t s  as i n  r e f e r e n c e  3 .  Then w e  have i n s t e a d  of ( 4 )  and (5) 
while  from (1) we d e r i v e  t h e  sequence of equat ions 
c 
. 
6 
e t c .  
Equation (14) then determines 
, e t c .  
, equa t ion  (15) 
We now remark t h a t  i n  the n ’ t h  order  - l\) determines .e 
equat ion,  y‘*’ , and hence y’ , occurs  e x p l i c i t l y  i n  only t w o  
terms 
and 
7 
I 0 
Now consider  any v a r i a t i o n  which i s  such t h a t  6'qLQ'=* . Then we 
no te  the  fo l lowing :  
rL 
( i )  C l e a r l y  (18) does n o t  c o n t r i b u t e  s i n c e  6+@''=0* 
a y " 7  
( i i )  The only p a r t  of which involves y*' w i l l  be 
which i s  j u s t  i n  the form of 6?'D' wi th  
.$'v?= &w? 
t o  ( 1 7 ) .  
, and hence, from (14) w i l l  make no c o n t r i b u t i o n  
Thus we s e e  t h a t  v a r i a t i o n s  wi th  %pb*)=o lead o n l y  t o  
c o n d i t i o n s  on t h e  lower o rde r func t ions ,  cond i t ions  which, s i r ice  t hese  
func t ions  have a l r e a d y  been determined, m u s t  t h e r e f o r e  be s a t i s f i e d  
a u t o m a t i c a l l y  i n  any c o n s i s t e n t  v a r i a t i o n a l  c a l c u l a t i o n  (one can 
of course a l s o  check t h i s  e x p l i c i t l y  bu t ,  a s  we have s a i d ,  i t  must  
be t r u e  i f  eve ry th ing  i s  t o  be c o n s i s t e n t ) .  Hence we conclude, i n  
agreement wi th  t h e  r e s u l t  given i n  r e f e r e n c e  3, t h a t  only 
need be v a r i e d  i n  order t o  determine the  . 
'Led) 
'W-) 
8 
Footnotes  and References  
1. A .  Dalgarno and A .  L. S tewar t ,  Proc.  Roy. S O C .  A238, 269 (1956);  
A247, 245 (1958).  
2 .  See M.  Cohen and A. I lalgarno, Proc.  Roy. SOC. A261, 565 (1961) 
and many subsequent papers .  
3.  For example i n  t h e  pv scheme of J .  N .  Silverman and J .  C .  van 
Leuven, Phys. Rev. 162, 1175 (1967) i f  extended t o  double 
p e r t u r b a t i o n s .  
4 .  For f u r t h e r  d e t a i l s  on n o t a t i o n ,  e t c .  see t h e  review by 
J .  0. H i r sch fe lde r ,  W .  B .  Brown and S.  T.  Eps t e in  i n  "Advances 
i n  Quantum Chemistry" (New York: Academic Press, (1964) Vol.  1, 
p .  354. 
5. See a l s o  K. M .  Sando and J .  0 .  H i r sch fe lde r ,  Proc.  Nat. Acad.  S c i .  
- 52, 434 (1964). 


Two notes on perturbation theory. 1 - Interchange theorems and the variation principle. 2 - PV in a different notation

Abstract

Similar works

Full text

Available Versions

NASA Technical Reports Server