Variational methods and degenerate perturbation theor

Epstein, S. T.

English

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I L. 
VARXTYONAL WFHODS AN3 2EEGE;NEitiTE PERTLWBAT33N TBEORV 
by 
* 
S a u l  T .  Eps t e in  
The o re  t i c s  B CE.er5s t r y  Ins ti Lute &xd Depsr tmen t ~f Phys ics  
U n i v e r s i t y  of Wiseonein, Madison, ‘Fdisconsin 
Aba t r a c  t 
&-7 - I373 
Var Lat i o n a l  metkigds r e l a t e d  t o  degenera te  p e r t u r b a t i o n  
theory  a r e  d i sccs sed  i n  a generwl way throu.gh second order  i n  
t h e  energy and f k s t  o rder  in t he  wave func t ion .  
- - - - -  
* This  r e s e a r c h  Kqlc‘ soap~or t ed  by the  fa l lowing  g r a n t :  
Na t iona l  A e r m s u t i c s  arid Space Adminis t ra t ion  Grant NsG-275-62. 
.. 
. 
I .  I n t r o d u c t i o n  
V a r i a t i o n a l  methods have been much used and s tud ied  as a 
means of approximately c a l c u l a t i n g  t h e  e f f e c t s ’  of a p e r t u r b a t i o n  
on a non-degenerate e igenvalue  . I n  t h i s  no te  w e  wish t o  make a 1 
s t a r t  on ex tending  the  d i s c u s s i o n  t o  t h e  degenera te  case  by t r y i n g  
t o  g e t  an  idea  of t he  types  of problems which w i l l  be encountered.  
S p e c i f i c a l l y  w e  w i l l  cons ide r  s i t u a t i o n s  i n  which t h e  degeneracy 
i s  broken i n  e i t h e r  f i r s t  o r  second o rde r .  
It i s  a prime c h a r a c t e r i s t i c  of degenera te  p e r t u r b a t i o n  
theo ry  t h a t  a w r i o r i  w e  do not  know t h e  c o r r e c t  z e r o  o rde r  wave 
func t ion .  Thus t h e  appropr i a t e  t o o l s  t o  use a r e  t h e  v a r i a t i o n a l  
p r i n c i p l e s  given on page 299 of r e f e r e n c e  1 which a r i s e  from d i r e c t  
expansion of t h e  v a r i a t i o n  p r i n c i p l e  f o r  t h e  t o t a l  energy, and 
which a l low one t o  v a r y  t h e  zero  order  func t ion .  
be t h e  z e r o  order  Hamiltonian, a, Let 2 0  , t h e  pe r tu rba t , i on  
, e t c .  v a r i a t i o n a l  t r i a l  func t ions  which a r e  t o  
approximate t h e  z e r o  order  wave func t ion ,  f i r s t  o rde r  c o r r e c t i o n ,  e t c .  
3i‘7 , J‘” e t c .  by Now d e f i n e  3 , C0-l 
1 

From t h e  d i s c u s s i o n  i n  r e f e r e n c e  1 i t  then  fo l lows  t h a t  t h e  
a r e  t o  be msde s t s t l o n a r y  wi th  r e s p e c t  t o  v a r i a t i o n s  of t h e  , “y 
ca 
t h e  c o n s t a n t s  E be ing  held f i x e d .  The l a t t e r  a r e  then  t o  
be determined from r’’zo , and provlde  v a r i a t i o n a l  approximations 
t o  t h e  z e r o  o rde r  energy, f i r s t  o rde r  energy c o r r e c t i o n ,  e t c .  
Now a l though w e  d o n ‘ t  know t h e  c o r r e c t  $b’ w e  w i l l  assume 
t h a t  w e  d o  kncw a complete s e t  o f  degenera te  e igen func t ions  of 80 
belonging  t o  t h e  z e r o  order  e igenvalue  i n  ques t ion .  Denoting 
t h e s e  e igen func t ions  by y, where o(= ‘ 1 2 7  - - - - %  , D 
be ing  t h e  e x t e n t  of t h e  degeneracy, and the  common e igenva lue  
E - LO3 
by , t hen  t h e  only  6( w e  w i l l  cons ide r  w i l l  be of t h e  
form 
a 
where t h e  ca- a r e  cons t an t s  t o  be determined.  
-- i s  e x a c t .  F u r t h e r  w e  c l e t i r l y  have t h a t  no t  on ly  i s  
co> 
b u t ,  i f  6 7 ’  i s  any v a r i a t i o n  of  cf- t hen  a l s o  
3 
4 
Prom t h e s e  obse rva t ions  i t  then  f o l l o w s  t h a t  w e  may d i spense  
, and y’) by at07 - t?) , F 9 3 ’  wi th  , and r e p l a c e  3 
t h e  simpler express ions  
Afte r  t nese  p re l l i r i imr i e s  ; ~ e  nocJ t x r n  t o  a d e t a i l e d  d i s c u s s i o n  
encounter  va-ri3us carLplexEties  a l l  of whEch x i 1 1  be recognized 
a s  t h e  ans iogs ,  d i t h i n  cur var la r r iona l  tippraach, of w e l l  known 
compl ica t ions  i n  t h e  exac t  sc lm over sttirres nethod. 
IT.. The f i r s t  order energy c x r e c t i o n  
S ince  we have exhausted rrhe conten t  
I f  we i n s e r t  ( 6 )  i n t o  (11) and 
make 6 Jt’’=D the? we f ind2  t h e  s e t  
p 
I 
we now t u r n  t o  
vary the  C ,  s o  a s  t o  
of homogeneous equat ions  
- C\)  
t o  determine the  c, and the % . These o f  course a r e  
3 
p r e c i s e l y  t h e  equat ions  of the e x a c t  theory  . I n  p a r t i c u l a r  then  
the  -‘‘’ E w i l l  be exac t ,  hence we w i l l  d r o p  the  t i l d e .  
We w i l l  now go on t o  f n v e s t i g s t e  two  extreme c a s e s :  
Case I - a l l  
s a r e  uniquely def ined (and e x a c t ) .  
E ( l )  d i f f e r e n t ,  and hence the  corresponding 
-co-l , 
-a> 
Case I1 - a l l  e i d e n t i c a l .  
5 
I n t e r z e d i a t e  s i t ~ a t i o n s  carL be handled by analogous methods. 
6 
-4Q ?ti., 
111. and f o r  Case I. 
The previous  c o n s i d e r a t i o n s  e v i d e n t l y  exhaus t  t h e  c o n t e n t  of  
Denoting a p a r t i c u l a r  
s iQ3 p 3 /  
and . We now t u r n  t o  T‘O’ 1 
to? tl) norinnlized s o l u t i o n  of ( 9 )  by *i! , E, , and deno t ing  the  --- 
‘“io) ‘L t q  
corresponding  by q* w e  now cons ider  
This  now has  the form o f  the  f a m i l i a r  I Iy l le raas  v a r i a t i o n  p r i n c i p l e  
f o r  t he  second order  energy,  hence we j u s t  proceed as usua l  and 
1 
&C\) cz-) ’ 
vary  +- ; t he  c o n d i t i o n s  6T-  z o  J ” 0  
0- 
. Fur the r  - -7 - 0) t hen  y i e l d i n g  an  o p t  itnal c.- and an  op t ima l  E 
one can show by iiie s t anda rd  z r g c m e n t  ( r e f e r e n c e  1: page 2 7 8 )  
E cz) 
be an  upper bound t o  the  exac t  0 .  - L‘)
However, no te  t h a t  w e  do n o t  comple te ly  de te rmine  0- i n  
t h i s  way. Namely, no t  vn ly  d o  w e  have t h e  u s u a l  innocuous ambigui ty  
t h a t  < &, , 7 may be chosen a r b i t r a r i l y ,  bu t  w e  a l s o  
don’ t  deteriuine <.4’’i, f o r  b+k  . One sees t h e  most 
e a s i l y  l~g no t ing  t h a t  one can r e p l a c e  cs( D- 
“C\) 
+.-+ 
c o n s t a n t s ,  ant] get tile same va lue  f o r  3 o- 
rd, .- c\> 
t‘) - L\-) 
i n  (14) b y  
7 +2’ F!l&- , where t h e  F~c- a r e  a r b i t r a r y  
b =) I cz’, , s i n c e ,  a s  a 
7 
3 
consequence o f  (9) 
cL k\) 
To determine e') ye) then  w e  m u s t  e v i d e n t l y  go 
L e t  u s  then w r i t e  c d  beyond Tk 
I 
'L C\) u;) 
where yGL has been determined from 5, and i s  t o  be 
uniquely def ined  by t h e  requirement th t i t  i t  i s  or thogonal  t o  
a l l  . A l s o  w e  have chosen t h e  convenient  normal iza t ion  
'0 so t h a t  t h e  prime means no t e r m  b =& 
/ 4 *at?', 43- '" -2 
occurs  i n  t h e  sum. 
terms which e x p l i c i t l y  involve t h e  Fbb a r e  
I f  we now i n s e r t  t h i s  i n t o  Tz '  , chen t h e  
where w e  have omitted terms vrhich v a n i s h  i d e n t i c a l l y .  I f  now w e  
we f ind  
8 
. 
a r e  normalized and 
'ow 
B u t  f r o m  (15) - -  remember t h e  
o f  c o u r s e  a r e  or thogonal  - -  
and hence the  e x p l i c i t  s o l u t i o n  
T h u s  yb has n o w  been determined.  
I-' - ;,;- 
IV . E f o r  ca se  1 1 .  
For case  11 
so t h a t  t h ~ '  (1,- arc. as ycit completely a r b i t r a r y .  S i n c e  w e  
liavt' c'vidclnt l y  c ~ s l i a u s t c ~ t l  t l i c l  c o n t e n t  o f  -i" 3 
I n s c r t  i n g  ( h )  i n t ( 1  ( 1 0 )  ant1 v a i - y i n g  onp F i n d s  
/ I 
+:r> 
w e  t u r n  t o  ~ 
r *  
9 I 
which, i n  t h e  r o c a t i o n  of t h e  F r € V i O U S  s e c t i o n ,  can, from (17) )  
be w r i t t e n  a s  
- tz) 
So  h e r e  we have a se t  of r e l a t i o n s  between t h e  CL , G and 
I 
i n  7‘”’ t o  f i n d  -is.) * 
vL 
. To g e t  o t h e r s  we v a r y  - (33 
” L  
I n  g e n e r a l  t h e  s e t  of equat ions  (18) and ( 1 9 )  provides  u s  
w i t h  a non- l inear  coupled problem which must be solved i n  some 
s e l f - c o n s i s t e n t  way. ( I f ,  however, one r e s t r i c t s  onese l f  t o  
l i n e a r  v a r i a t i o n a l  parameters  and a f i x e d  b a s i s  s e t  t h e  problem 
becomes a simple l i n e a r  one ) .  One might, f o r  example, guess 
a se t  of L ‘ S  normalized s o  tharr c e ~ l L  = . One would 
t h e n  s o l v e  ( 1 6 )  f o r  + L  . Then w i t h  t h i s  Y L  , (15) 
w o i i l d  p rovide  u s  w i t h  a n  approximate 
s e t  of e,., . 
4 
c - t q  “IC.\ - 1%) 
E and a new normalized 
\ 
One would then i t e r a t e  u n t i l  a c o n s i s t e n t  s o l u t i o n  
i s  obta ined .  Having obtained one s o l u t i o n  one would t h e n  go on 
and look  f o r  another .  
. 
then it dces riot  ;&zm obv;ciJs c h a t  one cannot  ge t  m o r e  Ltian . 
I f  t h i s  ‘ s  i n J E t a  t b c  C ~ ~ P ; ‘  :kc_--, opc hc , , l d  h e  faced i i i t h  the problem 
. Wr i t i ng ry. i.h‘! ‘ will he  of  no h e l u  so w e  mu3L t u r n  L O  
11 
where we have aga in  dropped terms which vanish  i d e n t i c a l l y .  I n  
“cy 
a d d i t i o n  qLl i s  a l s o  invcslved i n  the  t e r m  
* 
I f  now we vary  FbQ we f l n d  
v z r 7  ?t 
=L whi le  i f  we vary  we f ind  
t 
and we m u s t  now so lve  t h i s  coupled s e t  t o  determine t h e  p&+ 
(and i n  a d d i t i o n  ~ G L ) .  
l i n e a r  v a r i a t i o n a l  parameters t he  s o l u t i o n  i s  s imple,  However, 
/Lc‘c) 
I f  one uses  a f ixed  b a s i s  s e t  and 
4 
s i n c e  i t  does not  seem poss ib l e  t o  give an e x p l i c i t  s o l u t i o n  
i n  gene ra l ,  we leave the  d i scuss ion  a t  t h i s  p o i n t .  
V I .  Discuss ion  
It i s  c l e a r  from t h i s  pre l iminary  survey t h a t  a v a r i a t i o n a l  
approach t o  degenera te  p e r t u r b a t i o n  problems w i l l  involve  one 
i n  p o t e n t i a l l y  q u i t e  complicated problems o f  so lv ing  coupled 
f u n c t i o n a l  equat ior ,s  i n  a s e l f  c o n s i s t e n t  way once the  degeneracy 
p e r s i s t s  beyond f i r s t  o r d e r .  A l s o  we have noted t h a t  t h e r e  may 
be d i f f i c u l t i e s  i n  i n t e r p r e t i n g  t h e  s o l u t i o n s .  
A s  we  have mentioned t h e  s o l u t i o n s  t o  t h e s e  problems are  
s t r a i g h t  forward Lf one u s e s  l i n e a r  v a r i a t i o n a l  parameters  w i t h  
a f ixed  b a s i s  s e t  (indeed e x p l i c i r  use o f  t he  v a r i a t i o n  p r i n c i p l e  
i s  r e a l l y  supe r f luous  i n  t h i s  c a s e ) ,  bu t  how one handles  o the r  
pa rame t r i za t ions  w i  11 presumably vary  from case  t o  c a s e .  
I t  i s  a p l easu re  t o  acknowledge h e l p f u l  d i s c u s s l o n s  w i t h  
D r  . 110 j ing  K i m .  
b 
Y 
~ 
c 13 
. . . , I  Footnote’s and References ’ 
‘ A  
i 
1. 
2. 
3. 
4. 
5.  
For a r e c e n t  review s e e  J. 0. Hi r sch fe lde r ,  W .  Byers Brown, 
and S .  T .  Eps te in ,  Advan. Quantum Chem. I., 255-374 (1964).  
Note t h a t  w i t h  t h i s  s o r t  ui t r i a l  f u n c t i o n  6;‘’ ”1’1’ 
where i s  a small cons t an t ,  i s  a p o s s i b l e  v a r i a t i o n  whence 
-b) 
I =o . Simi la r  remarks w i l l  app ly  t o  6 J L * ” = ~  imp l i e s  p 
o t h e r  v a r i a t i o n s  which we w i l l  be u s i n g  la te r ,  s o  t h a t  we 
w i l l  never  need t o  use 9’20 e x p l i c i t l y .  
See f o r  example L. Paul ing and E. B. Wilson, Jr . ,  
I n t r o d u c t i o n  t o  Quantum Mechanics (McGraw H i l l  Book Co., 
New York, 1935, e s p e c i a l l y  s e c t i o n  24).  
With a f ixed  b a s i s  s e t  one i s  doing  p e r t u r b a t i o n  theo ry  - -  
e x a c t l y  f o r  t h e  ope ra to r  #otul  
and 8% p and where i s  the  p r o j e c t i o n  ope ra to r  
, where 
onto  t h e  b a s i s  se t .  I f  t hen  out: chooses the  b a s i s ,  as  one 
may always do, so a s  t o  d i a g o n a l i z e  
c 
$0 , t h e n  a l l  formulae 
look l i k e  t h e  e x a c t  sum over s ta tes  formulae except  t h e  sums 
are f i n i t e  (See a l s o  H. J.  Kolker and H. H. Michels,  J. Chem. 
Phys. 43, 1027 (1965)). 
I f  +o- i s  n o t  exac t  then  a l l  w e  can  prove i n  g e n e r a l  
col - C%> 
concern ing  t h e  r e l a t i o n  between the  E L  and t h e  exac t  - (13 
E, va lues  i s  t h a t  f o r  the ground s t a t e  
t o  the  lowest Elo 
i s  an  upper bound 
a s soc ia t ed  wi th  the  same symmetry 
14 4 
( see  foo tno te  11, page 300 of r e f e r e n c e  1) .  I f  one u s e s  l i n e a r  
v a r i a t i o n a l  parameters  then, of course ,  one can  s a y  more (J.K.L. 
Mac Donald, Phys. Rev. 43, 830 (1933). See a l s o  foo tno te  6) .  
6 .  This  o r t h o g o n a l i t y  assumption is  riot e s s e n t i a l  fo r  t h e  a n a l y s i s  
-197 
which fo l lows .  However, i f  l'.Lal i s  a p o s s i b l e  v a r i a t i o n  of 
td 
&bi and converse ly  then  it i s  e a s y  t o  prove from (18) and (19) 
(\c b3 "io') - L) 
t h a t  q- and +b w i l l  be or thogonal .  Thus given a set  *'a+ 
which d i d  pJ y i e l d  4 G ~ L  w e  can  always g e t  
o r t h o g o n a l i t y  by e n t e r i n g  t h e  v a r i a t i o n a l  p r i n c i p l e  a g a i n  w i t h  
t h e  new v a r i a t i o n a l  f u n c t i o n  f o r  'k )- : 
/I q 
- t.3 
where now the  e,, a r e  f ixed  and t h e  D, a r e  l i n e a r  
v a r i a t i o n a l  parameters  which we a r e  t o  de te rmine .  I n c i d e n t a l l y ,  
from MacDonald's theorem5 t h i s  w i l l  a l s o  ensure ,  a t  least  f o r  t h e  
ground s t a t e ,  t h a t  t h e  (new) E a  , ar ranged  i n  order  of & C !  
i n c r e a s i n g  va lue ,  w i l l  be upper bounds t o  t h e  s i m i l a r l y  ordered 
exac t  E, . c-7 
Y 


Variational methods and degenerate perturbation theory

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