Appraisal of first order perturbation iteration fast converging metho

Hirschfelder, J. O.

Huntington, J. H.

Robinson, P. D.

English

NASA Technical Reports Server

1 
AN ~ I S A L  OF THE FOPM FAST-CONVERGING PERTURBATION BETHOD* 
Peter  D. Robinson? John H. Huntington, 
and Joseph 0. Hirschfelder  
Universi ty  of Wisconsin Theore t ica l  Chemistry 
I n s t i t u t e ,  Madison, Wisconsin 
ABSTRACT 
The p r a c t i c a l i t y  of t h e  Fir s t -Order -Per t u r  bat  ion- I  ter a t i on- 
Method (FOPM) is appraised. After t he  f i r s t  i t e r a t i o n ,  t he  
expec ta t ion  value of the energy is given by a non-analytic funct ion 
of the per tu rba t ion  parameter 3 . The asymptotic expansion of 
t h i s  func t ion  gives  the energy accura te ly  up t o  O( A'). 
convergence of t he  asymptotic expansions is  discussed.  
examples are  considered: a perturbed ground s t a t e  hydrogen atom, 
The 
Two 
and a perturbed ground state l inear  harmonic o s c i l l a t o r .  
t '  * This research  was  supported by National Aeronautics and Space 
Admini s tr a t  ion  c$. an t N s  G- 2 75 -6 2. 
+ R e s e n t  address:  Dept. of Mathematics, Univers i ty  of Sussex, 
Falmer, Brighton, Sussex, England. 
https://ntrs.nasa.gov/search.jsp?R=19660023884 2020-03-16T21:00:23+00:00Z
. 
The F i r s t  -Order -Perturbation- Iteration-Me thod (FOPIM) f o r  
192 s o lv ing  t ime - independent quantum mec hani  ca 1 pro b lems 
c a r e f u l  s c ru t iny .  After t h e  n-th i t e r a t i o n  t h e  expec ta t ion  value 
of t h e  energy i s  given by a complicated non-analyt ic  func t ion  of 
t h e  pe r tu rba t ion  parameter 8 which may be expanded i n  an asymptotic 
r equi r  e s 
series in h a c c ~ r s t e  UP to  terns of the order of x r a i ~ d  to 
t h e  2*' power. Thus, FOPIM requ i re s  the s o l u t i o n  of fewer 
d i f f e r e n t i a l  equations t o  obtain high p r e c i s i o n  than would be 
r equ i r ed  by other procedures. However, t h e  method is not easy t o  
apply. 
soon become i n t r a c t a b l e ,  and numerical techniques are requi red  for  
f u r t h e r  progress.  
The i n t e g r a t i o n s  and the d i f  f e r e n t i a 1  equations involved 
The d i f f i c u l t i e s  are i l l u s t r a t e d  by two r a t h e r  
t r i v i a l  examples: (1). A ground s ta te  hydrogen atom perturbed by 
a small a d d i t i o n a l  charge a t  the nucleus, and (2). A ground state 
l i n e a r  harmonic o s c i l l a t o r  perturbed by an add i t iona l  quadra t ic  
p o t e n t i a l .  
t he  r e l e v a n t  theory and a l s o  consider the question of t he  convergence 
of t h e  i t e r a t e d  pe r tu rba t ion  s e r i e s .  
Before proceeding with t h e  examples, we  b r i e f l y  r e c a l l  
11. T!IEORY 
W e  start wi th  the  usua l  Bayleigh-Schrgdinger pe r tu rba t ion  
theory for non-degenerate s t a t e s .  L e t  ho be t h e  unperturbed 
Hamiltonian which has 
e igenfunct ion  for  t he  s t a t e  under cons idera t ion .  
Hamiltonian i s  
6, and % as i t s  eigenvalue and normalized 
The perturbed 
H 
hO 
+ 
0 
A V  
1 
2 
We seek the perturbed energy 
9 (A) . If E and are a n a l y t i c a l  func t ions  of , 
E( h ) and t h e  per turbed e igenfunct ion  
The f i r s t  three co r rec t ions  t o  the  energy can be expressed2 i n  
terms of vo , “yo , and yo (1) > 
Here vo(’) s a t i s f i e s  t he  equat ion  
In &he usual s i t u a t i o n  where ho c o n s i s t s  of a k i n e t i c  energy 
i t  i s  convenient operator  T p lus  a p o t e n t i a l  energy func t ion  u 0 ’  
' .  
t , 
. 
9 .  
3 
to let *yo = Fo 'yo where Fo i s  a func t ion  of the  coordinates .  
Then Eq. (7) reduces t o  
This is a d i f f e r e n t i a i  equation for 
be solved by quadrature i n  one-dimensional or separable  mul t i -  
Fo which can, i n  p r inc ip l e ,  
dtntensional problems. Thus, there is a f a i r l y  clear-cut way of 
der iv ing  9 and the  energy up t o  terms of t he  order of 'h . 4 
0 
The bas ic  i d e a  of FOP% is t o  i t e r a t e  t he  pe r tu rba t ion  procedure 
s t a r t i n g  with t h e  improved unperturbed eigenfunct ion 
where N is the normalization f ac to r  
+ 
Actually,  Eq. (8) only determines Fo up t o  an add i t ive  
constant  C If i s  the (asymptot ical ly)  c o r r e c t  f i r s t -  
order func t ion  fo r  which 
from 
then one may wr i t e  for  the f i r s t  i t e r a t e d  wave func t ion  
0 
< yo ,v:')> = 0 and Y/o(l) 'differs 
0 '  by 6 c 5 t h a t  i s ,  i n  a d i f f e r e n t  choice of C % 
Thus vi 
not  equiva len t  t o  a l t e r i n g  a m u l t i p l i c a t i v e  normalizat ion cons tan t .  
i s  no t  The energy expec ta t ion  va lue  
a f f ec t ed  by the  choice of Co up t o  terms of b( A ) . Furthermore, 
i f  v/ is  t h e  normalized func t ion  obtained a f t e r  t h e  f i r s t  
i t e r a t i o n ,  the  energy 
by the  choice of Co up t o  terms of O (  a*). Thus, for  our 
purposes, the  choice of  Co i s  immaterial and may be made (as i n  
our f i r s t  example) t o  s impl i fy  the  c a l c u l a t i o n s .  
v e r i f y  t h a t  h l \ Y l  = E 'yl where 
d i f f e r s  from 9, by terms of order A* . This  i s  
E, = < T ,  ff Y ,  
4 
2 
&2 = <% , ff v2> i s  not  a f f ec t ed  
It i s  easy t o  
and 
5 
Thus the perturbed Hamiltonian can be written i n  the form 
2 
H = hl + a  v1 
where 
2 The A vl( ) now plays the role of a perturbation. Corresponding to  
E q s .  (2) and (31, w e  have the expansions 
The new first-order wave function ( A )  m y  be expressed i n  
the form y l ( l )  ( A )  = F1( h ) v l (  3 
function sat i s fy ing  the equation 
where F l ( a )  i s  a 
& 
6 
(6a) 
Eq. (8a) only determines the  func t ion  
cons tan t  C1 . However, Eqs. (Sa) and (6a) are w r i t t e n  i n  such a 
form t h a t  the  values of 4 ,  (2) and 
va lue  of C1 . 
F1( ) t o  w i th in  an a d d i t i v e  
are independent of the  
I f  the expansion of t h e  energy, Eq.(2a), i s  terminated a f t e r  t he  
third-order  t e r m  
up t o  terms of the order of A . This  i s  the  same accuracy as could 
l6 el ( 3 )  ( 
8 
) , then E( 3 ) should be accura te  
be obtained from t h e  usua l  Rayleigh-Schr Gdinger pe r tu rba t ion  theory 
wi th  the  expansion of t he  wave funct ion,  Eq.(3), t runcated a f t e r  the  
t h i r d  order term y ( 3 )  . The advantage of FOPIM i s  t h a t  t o  
o b t a i n  t h i s  degree of accuracy i t  i s  only necessary t o  solve two 
d i f f e r e n t i a l  equat ions f o r  q0(l) and fo r  *,(')( A )  i n s t ead  of 
t h e  three  d i f f e r e n t i a l  equat ions fo r  9 , , and . 
7 
In  order t o  obtain higher accuracy, the FOPIM procedure can be 
repeated as  of ten as  required.  
I .  
111. CONVERGENCE. 
3 The theorems of Rellich,  &to, Titchmarsh, and others  enables 
one t o  discuos the convergence of the  Rayleigh-Schradinger energy 
s e r f r ~  Eq.(2) in te+m of the mttiatkai  properties of 
The convergence of the FOPIH energy s e r i e s  Eq.(Pa) is m o r e  d i f f i c u l t  
ho and v . 
0 
(1) hl and v as well as the E ,  , el , e tc . ,  I' t o  as8es8 since 
u e  functions of 'x . An addi t ional  d i f f i c u l t y  arises from 
h2 vl(  3 ) not being a rem.lar per turbat ion i n  the region of 
configuration apace for which f h F o l  1 . I n  t h i s  region, the 
T ~ y l t ~  series expamion i n  powers of A does not converge for the  
factor (1 I= aP0)-' Which occurs In vl( 'A ). No mat te r  how small 
the value? of 
I s  a bounded function (which it  seldom i r ) .  
h , there w t l l  alwaye be euch a region unless Fo 
Very l i t t l e  i s  known 
about the couvergence of 8 perturbation eerie- having such ccnuplica- 
t ione.  However, In tu i t i ve ly ,  one expects t h a t  a given number of 
FOP= energy terms should give a far bet te r  energy than the same 
.-mber of Rayleigh-Schr$dinger energy terms. 
~t t!i,? 
neighborhood of 
for 
the POPM series (2a) with the Rayleigh-Schrgdinger series (2) ae 
E,(') 1s are analyt ic  functions of x i n  some 
a =  0 ( i . e .  If they have convergent Taylor s e r i e s  
!h( less than some fixed A. ), it is  possible  t o  compare 
expansions i n  powers of h . From Eq. (121, it is clear that; Q, 
I s  analy t ic  i n  From Eqs. (4a), (91, and (24) we see tha t  el (1) . 
i 
2 is a l s o  ana ly t i c  s ince  i t  has the  form (ao + al h ) / (bo  + b l h  + b2 X ), 
The behavior of el (2) and 
following grgument shows, w e  cannot expect them t o  be a n a l y t i c  i n  
is  d i f f e r e n t .  Generally,  as the 
any neighborhood of A = 0 , no matter how small the  value of 3 . 
E q s .  (5a) and (6a) express  El (*) and as i n t e g r a l s  
involving vl - F1 v1 . The func t ion  F1 i s  found by 
so lv ing  Eq. (8a). 
real ,  T = - - 
(1) - 
I n  a simple one-dimensional case wi th  
d2/dx2 and F i s  the  i n t e g r a l  of 
yl assumed 
1 2 
Consideration of t h e  f a c t o r s  of (1 + AFo) occurr ing  i n  vl 
and y 
and (1 + AFo)-’ , and thus F1 involves ( 1  + AFo) - l  and 
log 1 1 + AFoI  . The logari thm i s  unpleasant as i t  cannot 
cance l  with any of t he  f a c t o r s  of (1 + A F o )  
t h e  i n t e g r a l s  of  E q s .  (5a) and (6a). 
and 
r eg ion  
i n  powers of 
i n d i c a t e  t h a t  dFl/dx has terms involving (1 + A F o ) - 2  1 
i n  the  numerators i n  
(2) These i n t e g r a t i o n s  for  el 
el  (3)  are taken over a l l  conf igura t ion  space inc luding  the  
1 XFo’{ > 1 i n  which t h e  Taylor series for  log \ 1 + A Fo\ 
does not  converge. It follows t h a t  e,(*) and 
themselves do not  have convergent Taylor series i n  powers 
Y , of i n  any neighborhood of h = 0 i f  F1 , and hence 
contains  log  I 1 + Fo 1 . One would s t i l l  expect E l ( 2 )  and 
t o  have asvmptotic expansions i n  ascending powers of  X,  
s i n c e  the smaller the  value of 
troublesome reg ion  where 1 h F o l  > 1 . But these  asymptotic 
h ,  the  less s i g n i f i c a n t  i s  the  
. I  
4 
0- . 
9 
expansions are not convergent f o r  any value of h . 
Thus, it i s  only i n  an asymptotic sense t h a t  one should expect 
t o  compare three or more terms i n  the  FOPIM series Eq.(2a) with the  
Rayleigh-Schrsdinger series Eq. (2). In t h a t  sense, n terms of 
Eq.(2a) should agree with 2x1 terms of Ed.(2). This is  evident 
because of the powers of 3 (rather than a ) which multiply the  2 
terms i n  (2a). Experience with asymptotic perturbation series has 
shovn t h a t  frequently only three or four t e r m s  are relevant before 
the  series must be corrected and terminated. On this account, the  
asser t ions  about agreement of t he  two series (2) and (2a) through 
high powers of 1 must be t en ta t ive  and questionable. 
Conparison of t he  two ser ies ,  Eqs. (3) and (3a), for t he  
perturbed eigenfunction q( h ) is s o l a e t i m e s  possible provided 
t h a t  the norms of the  two series are adjusted t o  be equal through 
the  power of h under consideration. With t h i s  proviso, it is  
evident that, for example, 2 ql( A 1 + h VI(') ( agrees 
is expandable i n  ascending powers of A, t h a t  i s  when I APo) < 1 . 
I V .  EXAMPLE: THE PERTURBED HYDROGEN ATOH. 
A ground-state hydrogen atom perturbed by the  addi t ion  of a 
charge -Ae a t  t he  nucleus is a simple example which can be used 
to  test our FOPM. 
Wigner , Trees, and Dalgarno t o  test per turba t ion  theories .  I n  
atomic uni t s ,  the  exact solutions are 
This example has previously been used by 
4 5 6 
We start  wi th  the unperturbed wave func t ion  
UI, = T-’ exp(- r )  
corresponding t o  
-1 and f = - f 1. 2 ‘. h = - 5 0  - r 
0 0 
The per turba t ion  p o t e n t i a l  and f i r s t  order energy are 
r: 1 
0 
-1 v = r  and 
0 
It is easy t o  so lve  E q . ( 8  for Fo giving 
= r exp( - 1: Fo = 1: and 
Here t h e  constant  of i n t e g r a t i o n  has been ad jus ted  S Q  as t o  
simplify the FOPIM c a l c u l a t i o n s  r a t h e r  than t o  make 
orthogonal t o  yo . From E q s . ( S )  and (6) w e  f ind  t h a t  
6 Indeed, a s  Dalgarno po in t s  out ,  a l l  of the  subsequent €IJ’ls 
are zero so t h a t  the f i r s t  t h ree  terms of t h e  Rayleigh-Schradinger 
energy series Eq. (2) s u f f i c e  t o  give the  exac t  energy of t h e  
perturbed system, Eq.(16). 
fo r tuna te .  
This f e a t u r e  of the example i s  
I n  accordance wi th  Eq. (91, t he  f i r s t - i t e r a t e d  FOPIM 
normalized "unperturbed" wave func t ion  is 
We note  t h a t  vl( 'x ) agrees wi th  the exac t  wave func t ion  *( A ) 
through the f i r s t - o r d e r  i n  7\ . From Eqs. (12) and (14), 
Using Eq. (4a), it follows tha t  
When I AI  < (7/12)' 4 0.764, t h e  expansion of Eq. (24) i n  
powers of is convergent giving 
So lu t ion  of Eq. (8a) y i e lds  
Here C1 is an a r b i t r a r y  constant  and 
vl i n  Eq. (5a) , we f ind  a f t e r  some F1 s e t t i n g  Yl (1) = 
90 
-X - in terms of the  exponent ia l  i n t e g r a l  Ei*( t )  = (e /x) dx  . 
I f  3. is negat ive so  t h a t  the integrand of I (  x )  has a 
s i n g u l a r i t y  a t  the  poin t  r = - l /?t  , then we must use the  
Cauchy p r i n c i p l e  value of Ei*(2/X ). The asymptotic expansion 
pf I(  'h ) i n  ascending powers of h i s  
13 
Expanding Eq. (28) with  t h e  use of Eq. ( 3 0 ) ,  w e  ob ta in  t h e  
asymptotic expansion 
Thus, from E q s .  (23), (25), and (31), it follows t h a t  
This illustrates the  asymptotic agreement of t he  f i r s t  t h r e e  terms 
of t h e  FOPIM series (2a) w i t h  t h e  f i r s t  t e r m s  of t he  Rayleigh- 
Schr8dinger series (2). Using Eq. (6a), we could evaluate E 1 ( 3 ) ( h )  
but  t h e  i n t e g r a t i o n s  would be very  lengthy. 
s i m i l a r  t o  (but more complicated than) t h a t  f o r  
new f e a t u r e s  would emerge. We would expect t o  f i n d  
W e  should ge t  a r e s u l t  
h )  and no 
The unnormalized f i r s t - o r d e r  co r rec t ed  f i r s t - i t e r a t e d  
eigenfunct ion is Yl + h2 yl(l) = (1  + X2 F1) yl . 
Here vl 
When 
is  given by Eq. (22) and F1 i s  given by Eq. (26 ) .  
1 a r  1 < 1 , t he  logarithm terms may be expanded so t h a t  
1 . 2  ( I - l - 3 h $ - 1 h Z )  
2 
c 
Here K = (1 + 3 A  + 3 . I f  C1 i s  chosen so  a s  t o  
make the asymptotic expansion of 
then the asymptotic expansion of vl + 1 ~ , ( l )  i s  propor t iona l  t o  
q 1 ( l )  orthogonal t o  % ’  
Thus t h e  agreement between the two s e r i e s  (3) and (3a) i s  a l s o  
asymptotic. 
I n  t h i s  example i t  is evident  from the  complicated na ture  of 
F1 thgl: a fu r the r  i t e r a t i o n  based on the  pe r tu rba t ion  
would be imprac t ica l .  4 8 (vl - e l ( ’ )  F1 (1 + ;h2 Flf-’ -. 1 
I 
15 
V. EXAMPLE: THE PERTURBED, LINEAR HARMONIC OSCILLATOB. 
I 2  
Another simple example i s  the linear harmonic osc i l l a tor  i n  its 
ground s tate ,  perturbed by addition of 
Using atomic units ,  and taking the unperturbed force constant t o  be 
iC t9  the force constant. 
unity 3 
7 
.. 
The exact energy and wave function of the perturbed system are 
Q 16 
In contrast to  the perturbed hydrogen example, l e t  us normalize 
< Y;': qe) = 0. yo( ' )  so 4s to be orthogonal t o  ye ; that i s  
Subject to t h i s  condition, solution of Eq.  (8) y ie lds  
or 
whence 
Now, the FOPIM equation i s  h,y  0 ,  J with 
17 
The new perturbation is 
. Tbe first integration of Eq. (8a) gives 
with 
c 
.- 
e 
For tunate l y  the 
does not appear 
an (2) are such that Jo = f n  f / i  erf(s) 
- 
i n  Eq. (48). Otherwise e *z [*e- %zdy dt 
1 '  
and related transcendentals would occur i n  F 
Solution of Eq. (8a) yields 
with 
b, 4 - A!- 
A = - 4  32 32 SI 2 
(51) i 
18 
The constant  C1 i s  chosen so as  t o  
funct ion i n  asymptotic agreement with Eq. 
t h a t  is, by the  condi t ion <y , y  >=O, it 1 
make 
(40) 
H e r  e 
t he  FOPIM wave 
through t h i r d  order ;  
and i n  eva lua t ion  of 
(31 the  energies  €:'and e ,  , one encounters i n t e g r a l s  which lead t o  
t ranscendental  funct ions of the  per turba t ion  parameter 3 . The 
i n t e g r a l s  a r e  of th ree  types : 
I 
1 @e- 
-@ 
I n  order t o  express the  d e f i n i t e  i n t e g r a l s  as power s e r i e s  i n  2 , 
one must expand the integrands by use of the  expansions 
and 
2 4 2  I b  
which converge for  X'G < \ O r  -x +# The 
Gaussian weight f ac to r  assures  c o r r e c t  values for  the  d e f i n i t e  
i n t eg ra l s  i n  the  l i m i t  of small per turba t ion .  
Thus 
6 & + - * e *  J 281 
44 f ,S20  
(54) 
The wave function obtained by the FOPM procedure is  
Expansion of  the logarithm gives 
19 
c 
1 
20 
Thus q2 i t 3  in  asymptotic agreement with through third order. 
Now, the approximate energy for the original  perturbed system Is 
. 
This agrees with E q .  (39) through seventh order but the eighth order 
7 
term is  an upper bound to the correct value. 
c 
21 
REFERENCES 
1. 
2. J.  0.  Hirschfelder, W. Byers Brown, and S. T. Epstein, Recent 
J. 0 .  Hirschfelder, J. Chem. Phys. 39, 2099 (1963). 
Advances i n  Quantum Chemistry, Vol. I (Academic Press, New York, 
1964), page 255. The FOPM procedure is discussed on page 307. 
3. T. W. Palmer, Univ. of Wisconsin Theoretical  Chemistry I n s t i t u t e  
Report WIS-TCI-62 (August, 1964); a l so  T. W. Palmer, reference 2, 
page 349, provide a review of research on the  convergence of 
perturbation series. 
4. E. P. Wigner, Phys. Rev. 94, 77 (1954). 
5. R. E. Treee, Phhs. Rev. 102, 1553 (1956). 
6. A. Dalgarno, Proc. Phys. SOC. (London) A69, 784 (1956). 
7. The eighth-order t e r m  was cafculated by means of Eqs. (11) and 
(12) of K. H. Sando and J. 0 .  Hirschfelder, Proc. N a t .  Acad. 
Sci.  52, 434 (1964). 


An Appraisal of FOPIM Fast-converging Perturbation Method

Abstract

Similar works

Full text

Available Versions

NASA Technical Reports Server