Epstein, S. T.

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SUM RULES FOR VARIATIONAL WAVE FUNCTIONS* 
by 
S a u l  T. Eps t e in  
T h e o r e t i c a l  Chemistry I n s t i t u t e  and Physics Department 
Univer s i t  y of Wiscons i n ,  Mad i s  on, Wis c ons i n  
ABSTRACT 
Condit ions are given under which a v a r i a t i o n a l l y  determined 
frequency dependent p o l a r i z a b i l i t y  w i l l  s a t i s f y  the  o s c i l l a t o r  
s t r e n g t h  sum r u l e .  
* This  r e s e a r c h  was supported by the  Na t iona l  Aeronaut ics  
and Space Admin i s t r a t ion  Grant NsG-275-62. 
. 
I n  t h e i r  v a r i a t i o n a l  c a l c u l a t i o n s  of t he  frequency dependent 
L 
d ipo le  p o l a r i z a b i l i t y  of helium, Chan and Dalgarno found t h a t  
t h e i r  approximate o s c i  1 l a t o r  s t r e n g t h s  a u t  ora  t i c a  1 lp sa t  i s  f ied the  
o s c i l l a t o r  s t r e n g t h  sum r u l e  whatever t h e  s i z e  of t he  basis s e t  
they used. I n  t h e  f i r s t  p a r t  of t h i s  note  we e x h i b i t  the under ly ing  
r eason  fo r  t h i s  r e s u l t ,  and i n  the  second p a r t  we w i l l  g ene ra l i ze  
2 
t h e  r e s u l t  t o  s i t u a t i o n s  involving non l i n e a r  v a r i a t i o n a l  parameters .  
We w i l l  use t h e  n o t a t i o n  of re ference  1 throughcut ,  except  t h a t  
i n s t ead  of cons ide r ing  the  dipole  p o l a r i z a b i l i t y  we w i l l  consider  
an a r b i t r a r y  p o l a r i z a b i l i t y .  Hence we r e p l a c e  t h e i r  11 by V . 
Expansion of t he  t r i a l  func t ion  x ( W  ) i n  terms of a f ixed  
3 b a s i s  s e t  i s  r e a d i l y  seen 
p o l a r i z a b i l i t y  of a system descr ibed by a Hamiltonian H under a 
p e r t u r b a t i o n  V where t h e  bar denotes  p r o j e c t i o n  onto the  b a s i s  
s e t  p l u s  . Thus, by t h e  usua l  argument the  sum of the  
approximate osc i 1 l a t o r  s t r eng ths  w i  11 s a t  i s  f y  
t o  be equ iva len t  t o  f i n d i n g  t h e  exac t  
- 
- 
4 
0 
where t h e  square b racke t  denotes a corrunutator. 
t h a t  i f  \6 % 
t he  work r epor t ed  i n  re ference  1 )  then 
We w i l l  now show 
i s  a member o f  t he  b a s i s  s e t  (as  was the  case  i n  
1 
. . .  
2 
which of course then  e x p l a i n s  the r e s u l t  found i n  r e f e r e n c e  1. 
6 The prl)of i s  simple w e  merely look a t  the two s o r t s  of  
terms involved. F i r s t  consider  T, [%, SGG+V-l 
Since v%o i s  i n  t h e  b a s i s  s e t  we have \d .%.o= v$.o whence 
xi= c+o,vGv $ 0 ,  which i n  t u r n ,  for  t he  same reason,  
equa l s  c%, V M\d VI?) . F i n a l l y  consider ILz (\ko,V k H  +o) 
This  can be w r i t t e n  s u c c e s s i v e l y  a s  E, e*@,% %7) = 
Eo c40 ,Vv  t a l  5 c h i  vvq  +o) . Equation (1) then c l e a r l y  f 
follows from t h e s e  r e s u l t s .  
c-- 
7 
We now t u r n  t o  a more g e n e r a l  s i t u a t i o n .  I f  one uses non- 
l i n e a r  v a r i a t i o n a l  parameters then t h e  whole concept of an 
o s c i l l a t o r  s t r e n g t h  w i l l  probably be meaningless (it  w i l l  depend 
on CI) ). However one may s t i l l  ask,  does one have 
We w i l l  now show t h a t  i f  one g e t s  t he  frequency dependence c o r r e c t  
then the c o e f f i c i e n t  w i l l  be c o r r e c t  provided (as  was the case  above) k&k 
i s  an  allowed v a r i a t i o n .  
B 
I n  accord wi th  thwaassurnptiori Let us a s s u m  t h a t  ;wc  t r i a l  
funct ions can be w r i c t e n  as  
3 
a r e  v a r i a t i o n a l  parameters and w h e r e x  can be P where d and 
“anything“. For convenience bu t  without l o s s  of  g e n e r a l i t y  we w i l l  
however assume t h a t  
F u r t h e r ,  i n  accord with our 
=-o 
assumptions we w i l l  assume t h a t  a s  
b->& the  opt imal  t r i a l  funct ion takes  the form 
A +,g I - - -  L-= - w Lo4 
Indeed t h i s  i s  probably no t  r e a l l y  an assumption s::Tze the  a n a l y s i s  
below i m p l i e s  t h a t  for  consis tency,  t he  a l r e a d y  ass..in.ed form 05 t he  
t r i a l  f u n c t i o n  p l u s  t h e  v a r i a t i o n  p r i n c i p l e ,  p r e t t y  much impl i e s  ( 2 ) .  
The proof now proceeds a s  fol lows:  Ev iden t ly  &Vt.= T b  
i s  an allowed v a r i a t i o n  whence it fol lows from the  v a r i a t i o n  
p r  i n c i p  l e  
t h a t  t he  opt imal  value of t h e  p o l a r i z a b i l i t y  w i l l  be 
Suppose now t h a t  we vary O W \ %  Cd ,%en we have t h a t  
Then l e t t i n g  t&cpOa and us ing  ( 2 )  one f i n d s  from the  terms 
independent of Q t h a t  
4 
Thus from ( 4 )  and ( 5 )  we have t h a t  a s  U 4 9 1  
\ale w i l l  now show t h a t  A cc L $o Then (61, i n  conjunct ion 
w i t h  ( 3 ) ,  w i l l  y i e l d  the announced r e s u l t  s i n c e  
To show t h a t  AS -V#,, we no te  t h a t  from ( q )  i t  follows 
t h a t  
where 
can on ly  a r i s e  from X , i . e . ,  a s  W + 00) m u s t  go t o  a 
c v% ) AL’) ~0 , and f u r t h e r  i t  i s  c l e a r  t h a t  A L 
. Then we l e a r n  f rom r . Now l e t  us  vary k c o n s t a n t  times 
t h e  v a r i a t i o n a l  p r i n c i p l e  t h a t  
L3 
. 
which, i n  t h e  i n f i n i t e  f requency l i m i t ,  y i e l d s  
C A I ,  A l  = o  
or 
5 
which proves t h e  p o i n t .  
6 , 
Footnotes  and References 
1. Y .  M. Chan and p .  Dalgarno, Proc. Phys. SOC. ,  86, (777) (1965), 
2. Similar  conclusions have been reached by A .  Dalgarno and 
G. A .  Victor  (A. Dalgarno, p r i v a t e  communication). 
3. See f o r  example H. J .  Kolker and H. H. Michels, J .  Chem. Phys., 
43, 1027 (1965). 
4. We w i l l  assume t h a t  $o i s  exac t ,  t h a t  i s  t h a t  H . 6 ,  = I$; =E-$, 
However our r e s u l t s  s t i l l  apply t o  the  c a l c u l a t i o n s  i n  
reference-though t h e m  
SI- 
i s  n o t  t he  exac t  ground s t a t e  
e igen func t ion  f o r  He, t h e i r  method of c a l c u l a t i o n  i n  e f f e c t  
replaced t h e  exac t  helium Hamiltonian by another such t h a t  
h%o ‘1 $0 i s  s a t i s f i e d .  Since f u r t h e r  t h i s  
Hamiltonian involves  only a l o c a l  p o t e n t i a l ,  t h e  d i p o l e  
(53 
o s c i l l a t o r  s t r e n g t h  sum r u l e  i s  unchanged. 
where T i s  the k i n e t i c  energy ope ra to r .  Thus they  e f f e c t i v e l y  
r e p  laced the  helium Hamiltonian by the  Sternheimer Hamiltonian, 
R .  M. Sternheimer,  Phys. Rev. 96, 951 (1954). 
6 .  For a r e l a t e d  r e s u l t  s e e  R .  E .  Johnson e t  a l ,  J.  Chem. Phys. 
- 47, 1 2 7 1  (1967), foo tno te  8 .  
7 .  I n  connection with the  d i s c u s s i o n  i n  Footnotes  5 and 6 we may 
no te  t h a t  i f  t) q~ 3.  ED*^ t hen  one r e a d i l y  f i n d s  
t h a t  the l e f t  hand s i d e  of (1) equals  t he  r i g h t  hand s i d e  of ( 1 )  
where # q o  s G o  *v . Note t h a t  t he  discrepancy i s  independent 
of t h e  s i z e  of t h e  b a s i s  s e t .  
_ _ ~ ~ -  ~ 
. 


Sum rules for variational wave functions

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