Approximate ground state wave function used in variational principle for second order energ

Epstein, J. H.

Epstein, S. T.

English

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... 
c - -.*, 
SOME REMARKS ON 
PRINCIPLE FOR 
THE USE OF 
THE SECOND 
by 
THE VARIATIONAL 
ORDER ENERGY 
* 
Jean H. Epstein 
and 
Saul T. Epstein 
Theore t ica l  Chemistry I n s t i t u t e  and Physics Department 
Universi ty  of Wisconsin, Madison, Wisconsin 53715 
ABSTRACT 
It i s  pointed out t h a t  i f  one uses an approximate ground s t a t e  
wave funct ion i n  the  fami l ia r  va r i a t iona l  p r inc ip l e  f o r  t he  second- 
order  energy, that the  approximate energy need have no s p e c i a l  r e l a t i o n -  
. sh ip  t o  the  exac t  energy (it may be l a rge r  or smaller). Fur ther  it is 
shown t h a t  i n  some cases even with f u l l  use of the  v a r i a t i o n  pr inc ip le ,  
good accuracy of the 
sponding accuracy of 
ground s t a t e  wave funct ion does not  imply cor re-  
t he  approximate energy. 
pup' 
* 
This research was supported by the following grant:  National 
Aeronautics and Space Administration Grant NsG-275-62. 
i 
https://ntrs.nasa.gov/search.jsp?R=19660024130 2020-03-16T20:21:42+00:00Z
4. 
m 
a 
& 
- .  
* -  
a 
. 1 
I n  s i t u a t i o n s  where one wishes  t o  know a second-order energy correc-  
t i o n  E('), but  does n o t  have an accurate  so lu t ion  t o  the  zero-order 
problem, a f requent ly  used technique i s  t o  use  as an approximation t o  1 
the  quant i ty  J &ere 
N 
Here 6 i s  an approximate wave funct ion f o r  t he  zero-order problem (we 
HO 
will assume throughout t h a t  w e  are dea l ing  with t h e  ground state),  
the  zero-order Hamiltonian, V the per turba t ion ,  EO 5 (40 
- n d  'L 
H~I 4) , 
/ - (0 
.and * i s  t o  be determined 
w I 
v a r i a t i o n a l l y  from 4 3 = 0 .  
ry 1%) 
I f  we vary 3. f r e e l y  ( i n  p rac t i ce  of course one is r a r e l y  ab le  
@ td 
t o  do t h i s )  then one is l ed  to a d i f f e r e n t i a l  equat ion f o r  the  "exact"$ ; 
.. 
4 
We w i l l  denote the  so lu t ion  of t h i s  equat ion by 9"' and t he  value of 
t o  which i t  leads  by 3, One then r e a d i l y  f i n d s  success ive ly  t h a t  
where we have introduced t h e  eigen states of H : N p  (bnnc v 
0 
2 - 11) 
Further i f  we write a general  4 =\t)t\’+ Glt.“’ then 
one ‘has immediately t h a t  
From Eq. (2) and (3)-we then have t h e  well-known r e s u l t  t h a t  i f  
N %. 
bs dpo (and hence GoS E 0 ) then2 = EL’> , and E q .  (5) then 
expresses the w e l l  known f a c t  t h a t  under these same condi t ions an a r b i -  
-. 
4 
(2) 
& 
t r a r y  T’’ y i e l d s  a 3 which i s  an upper bound t o  J and hence t o  E . 
& 
I n  t h i s  note  we wish t o  d i scuss  the  r e l a t i o n s h i p s  among J, 3 and 
fb 
E ( 2 )  i n  the more r e a l i s t i c  case $o 4.0 . F i r s t ,  concerning t h e  
r e l a t i a n s h i p  between J and 5 i t  follows immediately from E q .  (5) t h a t  
cu 
137 J i n  general  i s  not a lower bound t o  J s ince  a poss ib l e  6’k i s  - 
c e r t a i n l y  an a r b i t r a r y  mul t ip l e  of bo . Thus i n  general  6 3 ‘-0 
simply y i e lds  a s t a t i o n a r y  po in t  althaugh, of course, i f ,  as i s  o f t e n  - (a) 
the  case i n  pract icelone i s  dea l ing  with a 9 
i t  is q u i t e  poss ib l e  t h a t  
of r e s t r i c t e d  v a r i a b i l i t y ,  
may i n  f a c t  appear t o  y i e l d  a minimum. 3 8 5 ‘0 
We w i l l  not attempt a general  discussion of t he  r e l a t i o n s h i p  between 
J and E ( 2 ) ,  b u t  i n s t ead  tu rn  t o  two simple examples. 
a one-dimensional harmonic o s c i l l a t o r  
I n  each H desc r ibes  
0 
. These problems I n  example I , V = X  while i n  example 11, v = xs 
are r e a d i l y  solved exac t ly  and y i e ld ,  respect ively,  E ( 2 )  = - \/czarrOal 
and -% /( \bwLw3J 
\ 
3 
& -  
I - -  
N 
For each example we have evaluated J from equation (4) f o r  
4 given by 
The r e s u l t s  of the ca lcu la t ion  are shown i n  f igu res  I and 11. Let 
=3 
QS first &%.scuss exaiiiple I. 
- no t  3-71 EL%' f o r  t he  range of o( shown (see footnote  4). For o ther  
va lues  of O( 
smaller than E ( 2 ) .  
sh ip  between J 
&re the imporcam r e s u l t  is t h a t  J 4 
between 21 one can ge t  J values a r b i t r a r i l y  l a r g e r  o r  
When coupled with our e a r l i e r  remarks on the  r e l a t i o n -  
hr 
and J we have t h e  obvious moral-anything can happen- 
and we w i l l  not belabor the  point  fur ther .  
Now we tu rn  to  example 11. Here we f ind  J r /  E(2)  but  i n  a curious 
way, namely even f o r  d=Q , x+FL*' , and i n  f a c t  f o r  d = o  we 
have the  l a r g e s t  devia t ion  from E ( 2 ) .  
havior  is  not  hard t o  find; it i s  i n  t h e m a 0  term i n  Eq. (4), which 
from Eq. (3) can a l s o  be written as c &, CV- r"") hoX %, q'") , 
For o(=o t h i s  term appears as the indeterminant form OlO 
i f  one starts with d=o i n i t i a l l y  then, as i s  w e l l  known, Eq. (2) does 
not  determine cb, ,(") 
o the r  hand $ , c V - ~ ' ' ~ & >  0 whence 010 is t o  be read as 0 , 
and w e  have the  f a m i l i a r  r e s u l t  for  E(2) .  
t h a t  i f  one starts with Or*0 then (90, *'") is determined and 
The source of t h i s  pecu l i a r  be- 
. However 
, any f i n i t e  value i s  allowed, while on the  
On t he  o ther  hand one sees 
i n  f a c t  i s  propor t iona l  t o  'Id- 
t h a t  o!O i s  t o  be read as a c e r t a i n  f i n i t e  number. Hence 
discrepancy. c 
f w d  
so t h a t  i n  t h e  l imi t  d 3 0  
The moral here  then is, i n  problems f o r  which 
g c 9  
a r e  not i d e n t i c a l l y  zero (they w e r e  
we f i nd  
the 
4&J 
4 
5 i d e n t i c a l l y  zero i n  example I and i n  most p o l a r i z a b i l i t y  problems) , 
* 
t h a t  even i f  d?ii i s  very near ly  equal t o  4 , ? need not  be 
(2) very c lose  t o  E . 
We are indebted t o  D r .  Margaret Benston fo r  s eve ra l  he lp fu l  
discuss  ions. 
. 
1 . 
f 
2 
X 
i 
2 
2igilru 2: v = -33- 
. 
I I I I I .--- . ~ -  - .  _ _  . , 
3 ,  t 0.2 C 
Footnotes and References 
1. See f o r  example T. P. D a s  and R. Bersohn, Phys. Rev., 115, 897 (1959) 
and M, Karplus and H. 3. Kolker, J. Chem, Phys., 38. 1203 (1963). 
The latter authors also discuss  var ious a l t e r n a t i v e  procedures. 
2. We discuss  the  indeterminate Ma0 term i n  Eq. (4) a t  t h e  end of 
t h i s  note. 
3. This is presumably w h a t  D a s  and Bersohn (reference 1) have i n  mind 
i n  the  discussion following t h e i r  Eq. (13) (which is  i d e n t i c a l  t o  
our Eq. (5)). 
N 
4. I n  order t h a t  b o  have no nodes and go monotonically t o  zero as 1x1 
increases ,  e must be r e s t r i c t e d  t o  0 * 4 f e (with the  
usua l  d e f i n i t i o n  of t he  bin ). 
5. I f  one redef ines  E"'' as g'"a c4a2V40) , which a l s o  makes 
, then one g e t s  proper behavior a t  d=O and 
However t h i s  i s  not a p r a c t i c a l  way 
(Q- .e'") = o  
a graph l i k e  t h a t  of f i g u r e  I. 
out s ince  by hypothesis w e  don ' t  know +o . 


Some Remarks on the Use of the Variational Principle for the Second Order Energy

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