Approximation of expectation values by perturbation theor

Hirschfelder, J. O.

Sanders, W. A.

English

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. 
APPROXIMATION OF EXPECTATION VALUES* 
Joseph 0. Hirschfe lder  and W i l l i a m  A.  Sanders t 
Unive r s i ty  of Wisconsin Theore t i ca l  Chemistry I n s t i t u t e  
Madison, Wisconsin 
ABSTRACT 
The genera l  problem of c a l c u l a t i n g  expec ta t ion  va lues  for 
p r o p e r t i e s  o ther  t han  energy by t h e  use  of p e r t u r b a t i o n  theory , - -  
i s  considered.  
- - - - -  
* This  r e s e a r c h  w a s  supported i n  p a r t  by Nat iona l  Aeronaut ics  and 
One of t h e  Space Adminis t ra t ion  Grants  NsG-275-62 and NsG-512. 
au thor s  (JOH) wishes t o  thank t h e  Quantum Theory P r o j e c t ,  
Un ive r s i ty  of F lo r ida ,  fo r  t h e i r  suppor t .  
+ N a t i o n a l  Science Foundation Pos tdoc to ra l  Fellow, 1963-65. 
. 
1 It i s  w e l l  known t h a t  pe r tu rba t ion  theory  can be used i n  t h e  
quantum mechanical determinat ion of a n  expec ta t ion  va lue  <W> = (yl W Iy> 
of  an  opera tor  W. Here i s  an  e igenfunct ion  of t h e  Hamiltonian H. 
Usual ly  we  do not  know . Ins t ead  w e  are given an  approximate 
wave func t ion  % 
Ho 'yo = eo yo 
and 
Here yo(') i s  the  f i r s t  order func t ion  which i s  or thogonal  t o  
and s a t i s f i e s  the d i f f e r e n t i a l  equat ion  
which s a t i s f i e s  t h e  Schrgdinger equat ion  
. The d i f f e rence  H - Ho = V i s  t h e  p e r t u r b a t i o n  
can be expanded i n  t h e  series y = + q0(l) + ... . 
0 
The expec ta t ion  va lue  can  then be expressed as t h e  sum of  t h e  
c o n t r i b u t i o n s  from t h e  va r ious  o rde r s  of pe r tu rba t ion ,  (W> = Wo + W1+ 
Here Wo = (yo 1W lvo) and W1 r e p r e s e n t s  t he  f i r s t  c o r r e c t i o n  
f o r  t h e  badness of t h e  approximate wave func t ion ,  
... 
I f  W is a one-e lec t ron  operator ,  i t  i s  u s u a l l y  much easier t o  use 
t h e  Dalgarno Interchange Theorem 
equ iva len t  form 
W1 i n  t h e  mathematical ly  1 and express  
where x0(l) would be t h e  f i r s t  o rder  func t ion  i f  W r a t h e r  than  V 
2 
w e r e  the  p e r t u r b a t i o n  po ten t i a l ,  
x0(” be w r i t t e n  as t h e  product  2 Dalgarno and L e w i s  Suggested t h a t  
F vo where F i s  a 
Here the  summation i s  
Eq. ( 5 )  i s  f r equen t ly  
funct ion which sa:is€ies the  equat ion  
over the e l e c t r o n s .  For one-e lec t ron  W’s , 
separable  and F can e i t h e r  be determined 
e x a c t l y  or  else it can be s a t i s f a c t o r i l y  approximated. 
Largely on the  b a s i s  of i n t u i t i v e  arguments, Dalgarno and 
Stewar t3  suggested t h a t  Wo should be a good approximation t o  (W) 
provided t h a t  a parameter embedded i n  y o  i s  ad jus t ed  so  as t o  
make W = 0 A s i z e a b l e  number of expec ta t ion  va lues  have been 
es t imated  i n  t h i s  manner and fcund t o  be s u r p r i s i n g l y  accu ra t e .  
1 
4,5 
For the  ground s ta te  o f  two 
5 approximate wave func t ions  
ope ra to r s  W, t he  va lue  of 
e l e c t r o n  atoms us ing  hydr ogenic 
and p o s i t i v e  d e f i n i t e  one-e lec t ron  
Wo + W i s  a lower bound t o  < W) 1 
and t h e  m a x i m u m  va lue  of W + W i s  obtained by s e t t i n g  w l = o .  @ I  
However, t h i s  behaviour is not gene ra l  as can be seen  from t h e  
fol lowing examples: 
Ca lcu la t e  <r> €or  t h e  ground sta. te of t h e  hydrogen atom. 
& 
F i r s t ,  u s ing  t h e  approximate wave func t ion  
w e  o b t a i n  <r>  = 2/oh and <r)l = ( 9 / 8  QI 2,  [ - ( 3 2 / 2 7 ) ]  . 
S e t t i n g  (x = 3 2 / 2 7  t o  make <‘>1 = 0 makes < r >  = 5 4 / 3 2  which i s  
yo = N r 2  ex?(-  r ) ,  
3 
l a r g e r  than  t h e  c o r r e c t  value < r >  = 4 8 / 3 2 .  Furthermore, t h e  
maximum va lue  of < r >  + (r > occurs  f o r  a = 6 4 / 7 5  and n o t  
a = 3 2 / 2 7 .  Simi lar ly ,  using %= II exp (- c[ r ) w e  o b t a i n  
= 0 f o r  = 5 / 4  , i n  which case  < r > o  = 2 . Furthermore, 
f o r  t h e  ground s ta te  of  t h e  h e l i u m  atom, t h e  o r b i t a l  %= N I exp(-IXr) 
l eads  t o  <rl> = 0 for  a = 1.9775 and correspondingly 
<r l )  
0.9293. 
= 1 . 2 6 4 2 ,  which i s  t o  be compared wi th  the  exact va lue  
w1 = 0 6 Recently,  Robinson has  shown t h a t  t h e  requirement  t h a t  
i s  equ iva len t  t o  s a t i s f y i n g  the h y p e r v i r i a l  r e l a t i o n  
where L i s  an ant i -Hermit ian opera tor  s a t i s f y i n g  t h e  cond i t ion  
There i s  no uniqueness i n  the  f u n c t i o n a l  form of L . 
assumed t o  be a f i r s t  order  d i f f e r e n t i a l  opera tor ,  
It might be 
7 
where g i s  the  product  of the metric scale f a c t o r s  of  t h e  
genera 1 i zed coord ina tes  
of a l l  of t h e  qk . With t h i s  form of L, t h e  h y p e r v i r i a l  
r e l a t i o n  Eq. ( 6 )  becomes 
and t h e  func t ions  f k  may be func t ions  
‘k 
5 
. 
4 
For a one-dimensional problerr, by v i r t u e  of Eq. (7), t he  func t ion  f 
i s  simply r e l a t e d  t o  t h e  Dalgarno func t ion  F, 5 
Tlius, i t  i s  no t  d i f f i c u l t  t o  ob ta in  the  r equ i r ed  h y p p r v i r i a l  
opera tor  L a s soc ia t ed  wi th  a p rope r ty  W 
7 As Epste in  and Hirschfelder  showed, t he  s a t i s f a c t i o n  of t h e  
y o  h y p e r v i r i a l  r e l a t i o n  Eq. ( 6 )  assures  that the  wave func t ion  
i s  e t i e rge t i c8 l ly  s t a b l e  wi th  r e spec t  t o  v a r i a t i o n s  of t h e  type 
Thus, i f  y o  s a t i s f i e s  Eq.  ( 6 ) ,  then  the l D w e s t  va lue  of 
is  given by A = 0 . Conversely, i f  yo does not  s a t i s f y  Eq. ( 6 ) ,  
t hen  t h e  func t ion  wi th  the energy optimized v a l u e  of A w i l l  
s a t i s f y  the  h y p e r v i r i a l  r e l a t i o n ,  Hence 
t h e  b e s t  approximation t o  < W >  when A is energy optimized. 
<+ 1 W )  4) should g ive  
I f  i t  i s  d i f f i c u i t  t o  deternrine t h e  func t ion  F corresponding 
t o  an  approximate wave funct ion % , perhaps one might no t  make 
a l a r g e  e r r o r  i n  us ing  a func t ion  which would be appropr i a t e  
0 
i 5 
f o r  a s impler  func t ion  vL That is, appraximate W1 by 
The func t ion  F changes only slightly as one goes from a crude 
approximate wave func t ion  t o  the exac t  func t ion ,  For example, 
u s ing  t h e  c o r r e c t  ground s t a t e  func t ion  fo r  the  hydrogen atom, 
, F = ( '5&a2-r z ) /2U.  I n  t h e  la t ter  --ar and f o r  = N r e  
two cases, t h e  va lue  of W i s  unchanged i f  Fo i s  used in s t ead  
of t h e  c o r r e c t  F, t h a t  i s  <% \ ( ~ - ~ . o ~ ~ ~  \vo > = 
1 
<N\(v-%e)F\II(,> . Thus us ing  Fo i n  p l a c e  of F l eads  
to the same optimum value  f o r  0% and < r) . 
A rougher approximation t o  F might be obtained i n  the  
fo l lowing  manner. A s  Lennard-Jones observed8" a f i r s t  order  
per turbed  wave func t ion  may be expressed in the  s p e c t r a l  form 
Here t h e  and t h e  6 are the  complete set  o f  e igenfunct ions  
and eigenvalues  of Ho 
make t h e  summation as s m a l l  as poss ib l e .  Neglect ing t h e  summation , 
j j 
The s ta te  "1" may be chosen so  as t o  
9 
To this approximation, 4 ( A ) might be replaced by 
$ ' ( A / )  = ( \ +  X'W)% . 
This is the basis for the well. known Eylleraas 10 or  Hasse 11 
approximation which leads to good values €or the polarizability of 
molecules. l2 Thus, might be approximated by <# I W I 4')  
where the value of 
value of H . 
< W > 
A' is adjusted so as t o  minimize the expectation 
. 
. *  
6 
7 
REFERENCES 
1. J. 0. Hirschfe lder ,  W. Byers Brown, and S .  T. Eps te in ,  "Advances 
i n  Quantum Chemistry, V o l .  I. (Academic Press, New York, 1964) 
p.  256-375. 
2. A. Dalgarno and J. T.  L e w i s ,  Proc. Roy. SOC. A233, 70 (1955). 
3. A. Dalgarno and A. L. Stewart ,  Proc.  Roy. SOC. A257,  534 (1960). 
4. A. Dalgarno, "Advances i n  Physics", 281 (1962)- 
5. W.  A .  Sanders and J. 0. Hi rschfe lder ,  J. Chem. Phys. 00. 0000 (1965). 
6. P. D. Robinson, Proc.  Roy. SOC. A283, 229 (1965). 
7. S. T. Eps t e in  and J. 0. Hi rschfe lder ,  Phys. Rev. 123, 1495 (1961). 
8. J. E. Lennard-Jones, Proc.  Roy. SOC. A129, 598 (1930). 
9. According t o  Ref. (5), for  t h e  ground s ta te  of t h e  helium atom 
w i t h  hydrogenic ze ro th  order wave func t ion ,  we  have e x a c t  
n 
express ions  f o r  F fo r  W = r . Thus, i n  t h e  proper u n i t s ,  1 
-1 , F = r - (3/2)  1 W = r  
2 
9 1 F = 3 - r  1 w =  r 
a 
-I 
Y 2 5  
3 - r  - 
11 F =  - 2 
1 2 1 w =  r 
I n  t h i s  case,  Eq. (15) would be a good approximation i f  W were 
equal  t o  
-1 2 
r l  . bu t  not  for  W equa l  e i t h e r  t o  or t o  r l  ' 
. 
. 
8 
10. 
11. 
12. 
E. Hylleraas ,  Z, Physik, 209 (1930). 
H. R .  Hasse, Proc. Cambridge P h i i .  
27, 66 (1931). 
J. 0. Hirschfe lder ,  C. F. C u r t l s s ,  
Theory of Gases and Liquids" (John 
1964), p.  942. 
Soc. 26, 542 (1930) ;  
and R.  
W i  l ey ,  
B. Bird, '%iolecular 
New York, 2nd P r i n t i n g ,  


Approximation of expectation values

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