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Quantum mechanical perturbation theory in terms of characteristic functions Gomberg, Martin Godfrey Luis 1977

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QUANTUM IN  MECHANICAL  TERMS  PERTURBATION  OF C H A R A C T E R I S T I C  THEORY  FUNCTIONS  by  MARTIN B.A.,  A THESIS THE  GODFREY  Unversity  LUIS  GOMBERG  o f Essex,  1973  SUBMITTED I N P A R T I A L  FULFILLMENT  REQUIREMENTS  DEGREE  MASTER  FOR T H E OF  OF  SCIENCE  in THE  F A C U L T Y OF GRADUATE (Department  Me a c c e p t t h i s to  THE  o f Chemistry)  thesis  the required  UNIVERSITY  197?  as conforming standard  OF B R I T I S H  August,  Copyright ©  STUDIES  COLUMBIA  1977  by M.G.L. Gomberg  In p r e s e n t i n g t h i s  thesis  an advanced degree at the  agree  fulfilment  the U n i v e r s i t y o f B r i t i s h  L i b r a r y s h a l l make i t  I further  in p a r t i a l  freely  available  o f the r e q u i r e m e n t s f o r Columbia,  I agree  that  f o r r e f e r e n c e and s t u d y .  t h a t p e r m i s s i o n for e x t e n s i v e copying o f t h i s  thesis  f o r s c h o l a r l y purposes may be granted by the Head o f my Department o r by h i s r e p r e s e n t a t i v e s . of  this  thesis for  I t i s understood t h a t c o p y i n g o r p u b l i c a t i o n  financial  gain s h a l l not  written permission.  Department The  of  Chem/sfry  U n i v e r s i t y of B r i t i s h Columbia  2075 Wesbrook P l a c e Vancouver, Canada V6T 1W5  Date  flyytfih  '9^7  be allowed without my  A  quantum  dimensional  mechanical  cases,  operator i t s e l f ,  is  perturbation  based  not  on  the  theory, perturbed  for  finite  Hamiltonian  but on the c h a r a c t e r i s t i c f u n c t i o n  developed. & p e r t u r b a t i o n h i e r a r c h y i n terms of  characteristic perturbation Various  function series  cases  of  is  for  derivatives  of  the  c o n s t r u c t e d . From t h i s h i e r a r c h y ,  individual  eigenvalues  are  found.  degeneracy and degeneracy l i f t e d i n v a r i o u s  orders are examined i n d e t a i l . T h i s p e r t u r b a t i o n theory generalized.  Perturbation  for  theory  individual is  developed  e i g e n v a l u e s c o n s i d e r e d t o g e t h e r . Here the a r e f o r the c o e f f i c i e n t s of a 'reduced for found The  this  set  of  eigenvalues.  by a contour expressions  eigenvalues  perturbation  both  These p e r t u r b a t i o n s e r i e s are  the  individual  eigenvalues  of  which  are  function,  correspond,  the  matrix  element  familiar  Hayleigh-Schroedinger,  are  found,  7f-molecular even-membered  to  ring  their  characteristic  respectively, expressions  a p p l i c a t i o n s of the  formulae  second  orbital  and  to in  and Van Vleck p e r t u r b a t i o n t h e o r i e s .  Some i l l u s t r a t i o n s and perturbation  method.  i n terms of d e r i v a t i v e s of the  (full) characteristic  function  series  c h a r a c t e r i s t i c function*  g e n e r a l i z a t i o n , the e x p r e s s i o n s f o r the reduced function,  f o r a s e t of  i n t e g r a l method and by an a l g e b r a i c for  is  order, energy of  characteristic  are g i v e n . General for  the  levels,  carbon  expressions  perturbed of  atoms.  Hiickel  any  perturbed  The  familiar  iii Rayleigh-Schroedinger  perturbation  t h e i r corresponding c h a r a c t e r i s t i c  f o r a a l a e are r e d e r i v e d function  r e l a t i o n s h i p between energy d e r i v a t i v e s and is  discussed  Expressions and  v i t h p a r t i c u l a r reference  f o r the d i p o l e and  expressions. physical  spin  The  properties  to simple s p i n  guadrupole  from  systeas.  polarizations  f o r s p i n p o l a r i z a b i l i t i e s i n s i a p l e s p i n s y s t e a s are found  f r o a the c h a r a c t e r i s t i c f u n c t i o n s of the  spin  systeas.  properties  with  weak  coupling,  are and  useful  in  connection  These  hyperfine  f o r p r e d i c t i n g the i n t e n s i t y o f peaks o c c u r i n g  polycrystalline  spectra.  in  iv TABLE OF CONTENTS Page Abstract  .  Table of c o n t e n t s  iv  L i s t of tables  ,  ..  vi  L i s t of figures  v i i  Acknowledgements Chapter 1  viii  Introduction  1.1 The i d e a s  1  .....................................  1.2 T h e s i s content Chapter 2  i i  2 9  Perturbation  theory f o r s i n g l e e i g e n v a l u e s .  12  2.1 The method  13  2.2 O r i e n t a t i o n  15  2.3 The h i e r a r c h y 2.3.1  of p e r t u r b a t i o n  e g u a t i o n s .......  18  Comments ...............................  19  2.4 An a l t e r n a t i v e n o t a t i o n  22  2.5 S p e c i a l cases a r i s i n g i n p e r t u r b a t i o n 2.5.1  theory ..  29  Non-degenerate e i g e n v a l u e s .............  29  2.5.2 2 - f o l d degeneracy l i f t e d 2.5.3  in first  2 - f o l d degeneracy not l i f t e d order ..  order .  32  in first 34  2.5.4 3 - f o l d degeneracy and degeneracy partially lifted  37  2.5.5  40  2.6 Summary Chapter 3  q - f o l d degeneracy ...................... ,  42  The reduced c h a r a c t e r i s t i c f u n c t i o n ........  3.1 The reduced c h a r a c t e r i s t i c f u n c t i o n  ...........  46 47  3.2 The e i g e n v a l u e pover sums as contour i n t e g r a l s .  51  3.3 Contour i n t e g r a l method: -P  54  r  V  3.3.1  Non-degenerate eigenvalues  56  3.3.2 2 - f o l d degeneracy  58  3.3.3 3 - f o l d degeneracy  68  3.3.4  71  Near-degeneracy  3.4 Contour i n t e g r a l method: H—rr  ................  3.4.1 I l l u s t r a t i o n : 2 - f o l d degeneracy 3.5 An a l g e b r i a c  method: f - * - r  73 76 78  3.6 Convergence r a d i i  86  3.7 Summary  93  Chapter 4  Tuo i l l u s t r a t i v e a p p l i c a t i o n s  4.1 A p e r t u r b a t i o n c a l c u l a t i o n f o r even-membered r i n g s of carbon atoms 4.2 D e r i v a t i o n o f the Hayleigh-Schroedinger p e r t u r b a t i o n formulae from the c h a r a c t e r i s t i c function Chapter 5  Physical  properties  95  104  as energy d e r i v a t i v e s :  A p p l i c a t i o n t o simple s p i n systems ......... 5.1 P h y s i c a l p r o p e r t i e s  94  as energy d e r i v a t i v e s  111 112  5.2 Some p r o p e r t i e s o r simple s p i n systems as energy d e r i v a t i v e s  116  5.3 A p p l i c a t i o n s  127  5.3.1 A s p i n system weakly coupled to some n u c l e i ............................ 5.3.2 S i n g u l a r i t i e s i n p o l y c r y s t a l l i n e spectra  ................................  Bibliography Appendix A  127 129 134  Conditions f o r a n a l y t i c i t y of eigenvalues of an Bermitian operator ..................  136  Appendix B  More than one p e r t u r b a t i o n  138  Appendix C  C a l c u l a t i o n o f ^ H /3x, 3x a  r  a  parameter ......  140  vi LIST OF TABLES  Table I Table I I  Table I I I Table IV  Table V  Table VI T a b l e VII  The f i r s t e i g h t equations of t h e perturbation hierarchy  26-28  The non-zero terms o f the e q u a t i o n s of t h e p e r t u r b a t i o n h i e r a r c h y f o r some s p e c i a l cases  43-45  2 - f o l d degeneracy: The e i g e n v a l u e power sums  63-65  2 - f o l d degeneracy: The c o e f f i c i e n t s o f the reduced c h a r a c t e r i s t i c function  66  2 - f o l d degeneracy: The d i s c r i m i n a n t o f t h e reduced c h a r a c t e r i s t i c f u n c t i o n .................  67  3 - f o l d degeneracy: The e i g e n v a l u e power sums ...............  69  3 - f o l d degeneracy: The c o e f f i c i e n t s o f t h e reduced c h a r a c t e r i s t i c function  70  •  Table V I I I  A l g e b r a i c r e l a t i o n s between t h e c o e f f i c i e n t s of the reduced and f u l l c h a r a c t e r i s t i c function 84-85  T a b l e IX  The c h a r a c t e r i s t i c f u n c t i o n s f o r systems of s p i n 1, 3/2, and 2 ...................  117  E x p r e s s i o n s f o r the s p i n p o l a r i z a t i o n v e c t o r and p o l a r i z a b i l i t y t e n s o r f o r the p-th (non-degenerate) s t a t e of systems with s p i n 1 and 3/2  122  Table X  vii LIST  OF  FIGOBBS  Figure 1  The v a r i a t i o n of the c h a r a c t e r i s t i c f u n c t i o n (equation 1.6) with 2 , f o r two f i x e d values o f /\ .......................  Figure 2  i n example of the v a r i a t i o n of a c h a r a c t e r i s t i c f u n c t i o n with z f o r two f i x e d values o f }\  15  Guide to the p e r t u r b a t i o n h i e r a r c h y ( i ) : Hon-degenerate eigenvalues ,  31  Guide t o the p e r t u r b a t i o n h i e r a r c h y ( i i ) : 2- f o l d degeneracy  36  Guide t o the p e r t u r b a t i o n h i e r a r c h y 3- f o l d degeneracy  39  Figure 3 Figure H Figure 5 Figure 6 Figure 7  The contour r£ , ^+t,x» complex z - p l a n e t  a  n  d  i  n  (iii): t  a  e  L  Some examples of the 'best' contour  72 .....  88  viii ACKNOWLEDGEMENTS I wish t o thank research  topic,  Dr. J.A.R.  Coope  f o r suggesting  this  f o r guidance i n both t h e r e s e a r c h and t h e s i s  p r e p a r a t i o n , and f o r always being a v a i l a b l e f o r d i s c u s s i o n . I aa a l s o indebted  to  ay  fellow  graduate  students,  in  p a r t i c u l a r David Sabo and E r i c Turner, from whom I have l e a r n e d much. Por to  thank  teaching  f i n a n c i a l support both  during my two year s t a y I would  t h e Chemistry  and  Physics  like  departments f o r  a s s i s t a n t s h i p s and the U n i v e r s i t y o f B r i t i s h  Columbia  f o r summer s c h o l a r s h i p money. Finally,  I would l i k e t o thank Kathryn L e s l i e f o r her help  in typing t h i s t h e s i s .  1  CHAPTER  1.  INTRODUCTION.  T h i s t h e s i s develops  a  quantum  theory based on t h e c h a r a c t e r i s t i c  fC*,a)=  <1.1, of  mechanical  perturbation  function,  deMz-H(»J  ;  a perturbed, f i n i t e dimensional H e r m i t i a n o p e r a t o r  or more p e r t u r b a t i o n parameters a r e r e p r e s e n t e d by A. T h i s  one  approach is  H ( A ) . The  based  i s i n c o n t r a s t t o t h e u s u a l p e r t u r b a t i o n theory on  the  operator  itself  o r i t s matrix elements. A  p e r t u r b a t i o n theory based on the c h a r a c t e r i s t i c f u n c t i o n amounts  to  a  study  of  the  which  behaviour  of  -Pfe,^)  t h e r o o t s o f the  polynomial e q u a t i o n (1.2)  as t h e c o e f f i c i e n t s that  .. + a„(n)  -TYZ,*) =  ex,  ii = ±,...,n  H ( ^ ) , and thus the a ; ( a ) ,  some neighbourhood o f |AI=0.  =  o  vary with A. I t  is  assumed  a r e a n a l y t i c f u n c t i o n s of A i n  2  1.1  The i d e a s . An equation t h a t aodels a p h y s i c a l system  solved  exactly.  motivated  Quantum  by t h i s f a c t .  •vibrations Schroedinger developed  of  a  mechanical Rayleigh  string  can  rarely  be  perturbation  theory  is  context  of  (1894)  with  in  small  inhomogeneities"  (1926) i n the context o f h i s what  is  now  known  the  new  as  wave  and  mechanics  Rayleigh-Schroedinger  p e r t u r b a t i o n t h e o r y . In t h i s , an Hermitian o p e r a t o r of the form  i s considered.  (Here, and  time-independent E (6) ;  and  operator  in  operators  the are  e i g e n v e c t o r s lEj(o)> H°  rest  of  the  considered.) of  the  The  and  eigenvectors  operator H(Xj are found Ep(o)  is  a  perturbed e i g e n v a l u e  |E,-(a)>  as power s e r i e s i n A .  non-degenerate  unperturbed  Hermitian  Hermitian operator V  are assumed to be known. The  E;(A)  only  eigenvalues  unperturbed  p r o v i d e s the p e r t u r b a t i o n , with A a s m a l l r e a l eigenvalues  thesis,  parameter. of For  the  perturbed  example,  eigenvalue,  The  if  then  the  £p(A) and e i g e n v e c t o r l^pCX)) are g i v e n ,  to  f i r s t o r d e r , by (1.3)  E (l) p  = E ip) p  + A Mpp•+• OCX)  and  where M  ip  *<E,\ (o) \ V / Ep(o)>.  Rayleigh-Schroedinger  perturbation  theory s t a r t s with an o p e r a t o r and e x p r e s s i o n s a r e o b t a i n e d terms of matrix elements of t h i s Perturbation  theory  in  operator.  i n terms of c h a r a c t e r i s t i c f u n c t i o n s  3  starts  instead  with  the  characteristic  function  of  the  operator.  An example of the c h a r a c t e r i s t i c for  an  illustration  f u n c t i o n approach.  of  the  approach c o n s i d e r the 7 T - a o l e c u l a r s y s t e i i n which one carbon  characteristic  function  o r b i t a l s i n the c y c l o p r o p e n y l  atom has been p e r t u r b e d i n some  The Hiickel matrix, i n u n i t s of /3 r e l a t i v e t o «  as  a  zero  way. of  energy, i s  3  I  I  l o i I i o  (1.5)  H(%) =  The c h a r a c t e r i s t i c  f u n c t i o n o f t h i s operator i s  (1.6)  = Z'-  and the unperturbed  eigenvalues are  -»- 2  E,(o) = * a , E (o) = e (o) = " 1 .  (1.7) The  5z-2.  5  a  perturbed  e i g e n v a l u e s are the r o o t s of the c h a r a c t e r i s t i c  equation  $U,V> = o  (1.8) (see  1).  Figure  function therefore  of  A. be  (1.8)  Eguation The  first  defines  order  determined  by  (1.9)  where  #  the  subscripts  = -  label  non-degenerate e i g e n v a l u e  perturbed first  d i f f e r e n t i a t i o n , u s i n g the standard  Z  as  an  implicit  eigenvalue order  can  implicit  formula  h  the p a r t i a l d e r i v a t i v e s .  Eifa)we have  For the  (1.10)  4E>  4MI  ==  da(o)  d^/s-e.(o) *= o  so t h a t (1.11)  z,C\)  = E.0>) +  =  2  2.  + ±% 3  -h I. z=e,(o)  +  <90* ) a  O(^).  In the u s u a l Rayleigb-Schroedinger t h e o r y t h i s saae obtained by f i r s t  finding  |£,(o)>  the unperturbed  result  eigenvector  -  and then e v a l u a t i n g the matrix element i n e q u a t i o n  (1.3):  5  = <E,(o)|V|E.(0)>  #(o)  In  general,  the  to  Perturbation  an  ±  .  c h a r a c t e r i s t i c equation d e f i n e s 2 as an  i m p l i c i t f u n c t i o n of A. The information  =  problem i s t o c o n v e r t t h i s  explicit  theXory  functional  gives  this  implicit  dependence, 2 =  dependence,  though  c l o s e d form, as a power s e r i e s i n A. The c o e f f i c i e n t s s e r i e s , namely the  can  in  not i n of  this  derivatives,  principle  be  obtained  by  higher  order  implicit  differentiation. Returning t o the example, i t may (1.9)  cannot  degenerate  be  used,  as  it  E C\)  eigenvalues  stands,  and  A  be observed for  £3(3),  at z = E ( o ) - ~ l  t h a t eguation  the  initially  s i n c e both £  evaluated  theory  individual  e i g e n v a l u e s i n degenerate c a s e s , which  the  at  for  depends developed The perturbed  on  order  which  i n d e t a i l i n Chapter characteristic eigenvectors  eigenvectors,  which  need not be known, and c a l c u l a t i o n s . The b a s i s change:  directly. the  t h i s can  determinant  degeneracy  is  perturbation  lifted,  is  approach does not y i e l d  the  2.  function  form  t  The  H  vanish when  x  7i~o,  a n < J  Indeed,  the  unperturbed  b a s i s i n which H° i s d i a g o n a l , be  an  in eguation  advantage (1.1)  in  certain  i s i n v a r i a n t to  6  (1.12)  £  where U  =  d e H z - H I  =  del-1  l / ( s - H ) ( J | = dehl*-U+HU\  ?  i s a u n i t a r y matrix. However, i n f o r m a t i o n c o n t a i n e d i n  the e i g e n v e c t o r s can be found energy  by  taking  derivatives  of  the  with r e s p e c t t o a p p r o p r i a t e p e r t u r b a t i o n parameters.  particular, i f  the  perturbation  parameters  elements of the o p e r a t o r , themselves,  are  the  In  matrix  the d e r i v a t i v e s y i e l d  the  matrix elements of the d e n s i t y matrix. Kato  (1949)  developed  a quantum mechanical p e r t u r b a t i o n  theory i n terms of r e s o l v e n t s and (1940)  had  integrals.  c o n t e x t . The r e s o l v e n t was  such i n Coulson*s  work  and  was  expressed  not i d e n t i f i e d not  3, and allow  characteristic  their  Coulson  and  formulae  conjugated  relationship approaches  perturbation  terms H  to  be  theory  of  referred  molecule,  of  (Huckel  the energy, matrix P ,  the  characteristic  and in  a  and second function.  order Their  and t h e i r p e r t u r b a t i o n parameters were the  parameters).  second the  systems,  t o the e l e c t r o n s i n the rr-system o f a  atomic  orbital  d e r i v a t i v e s of  t o the elements of the  derivatives  density  the  They r e l a t e d f i r s t rs  d e r i v a t i v e s of  exhibited  (1947) g i v e f i r s t  with r e s p e c t t o H , and  operator  conjugated  r s  s r  of  base f o r part of  between  matrix elements H o f t h i s Hamiltonian i n basis  terms  way.  Longuet-Higgins  in  Hamiltonian  the  function  particularly clear In  in  as  but i n terms of c h a r a c t e r i s t i c f u n c t i o n s . Resolvents  and contour i n t e g r a l s p r o v i d e the mathematical Chapter  Coulson  a l r e a d y used r e s o l v e n t s and contour i n t e g r a l s i n a  quantum mechanical  operators  contour  matrix,  were i.e.,  related  density to  first  polarizabilities.  7  However,  they  were  interested  energy o f the system. over  a l l occupied  degenerate  Thus  their  orbitals  orbitals.  in  Pukui  perturbation  presupposes approach  theory  calculations  and,  in  et. a l .  non-degenerate o r b i t a l s but s t i l l A  derivatives  involved  particular, (1959)  treat  single orbitals.  based on c h a r a c t e r i s t i c f u n c t i o n s known.  (In an  characteristic  functions  can  in  fact  interest,  systems.  (1945,1951) o r g a n i z e d t h e r e s u l t s of e a r l i e r  g i v i n g g e n e r a l formulae f o r some *continuant characteristic  functions  which  c a l c u l a t i o n s . H a l l i o n and Rigby have  arbitrarily  obtained  weighted  occur  in  workers  determinants* simple Huckel  or type  (1976), u s i n g g r a p h - t h e o r e t i c a l  characteristic  graph.  the  be o b t a i n e d . Two such  c l a s s e s a r e c e r t a i n conjugated systems and simple s p i n  methods,  operator  matrix elements o f an o p e r a t o r a r e assumed t o be  known.) For some c l a s s e s o f systems of c h e m i c a l  Rutherford  sums  sums over  sum over degenerate  t h a t these f u n c t i o n s a r e  the  o f the t o t a l  This  functions  corresponds  for  an  t o having the  c h a r a c t e r i s t i c f u n c t i o n o f a Huckel matrix f o r a r b i t r a r y <x> and /S . Coope (1966) wrote down some c h a r a c t e r i s t i c f u n c t i o n s f o r rs  simple  spin  systems.  characteristic structure  function  than  s p i n system,  the  which  Here  the  coefficients  are s c a l a r s  and  are  of  simpler  the in  matrix elements o f the Hamiltonian o f the are  in  general  tensorial  functions of  angles. The  development  of  a  perturbation  theory  i n terms o f  c h a r a c t e r i s t i c f u n c t i o n s i s motivated p r i m a r i l y by t h e i n t e r e s t of  generalizing  Longuet-Higgins,  the  work  firstly  to  started apply  by  Coulson  and  to arbitrary perturbation  8 parameters, s e c o n d l y t o apply t o any order o f p e r t u r b a t i o n , and thirdly,  and  degenerate  in  particular,  eigenvalues.  It  to  include  the  treatment  i s a l s o motivated i n part by the  a v a i l a b i l i t y of various c h a r a c t e r i s t i c functions, fact  that  e x p l i c i t use calculate.  the of  characteristic wave  of  functions,  function which  and  approach can  be  by  makes tedious  the no to  9  1.2 T h e s i s c o n t e n t . In  Chapter  2 a p e r t u r b a t i o n theory f o r s i n g l e  i s developed by extending t h e i m p l i c i t f u n c t i o n in  the  example  d e r i v e d from found.  in  section  approach  used  1.1. A p e r t u r b a t i o n h i e r a r c h y i s  which a l l the energy  A mE  derivatives  TTfa (°)  a  p  c  a  n  D  e  A  P a r t i c u l a r cases o f degeneracy l i f t e d i n v a r i o u s o r d e r s  are c o n s i d e r e d . T h i s approach i s t h e analogue  corresponding Chapter  characteristic  function  o f Rayleigh-Schroedinger p e r t u r b a t i o n theory, and the  Hayleigh-Schroedinger  formulae  characteristic  can  be  function  obtained  from  their  formulae,  as shown i n  3, p e r t u r b a t i o n theory f o r s i n g l e  eigenvalues  4.,  In Chapter is  eigenvalues  g e n e r a l i z e d . , Instead  of  single  eigenvalues,  a  reduced  c h a r a c t e r i s t i c equation f o r a s e t o f e i g e n v a l u e s i s c o n s i d e r e d , and a p e r t u r b a t i o n theory f o r t h e c o e f f i c i e n t s o f t h i s characteristic  equation  computed by two methods; algebraic  is  developed..These c o e f f i c i e n t s a r e  by  contour  method. ,<The 'reduced  integration  and  H,  by  effective  as i n t r o d u c e d by Van Vleck (1929). In Van Vleck*s  p e r t u r b a t i o n t h e o r y an e f f e c t i v e o p e r a t o r H i s c o n s t r u c t e d that  i t s eigenvalues  are  to  some  order  so  a subset o f the e i g e n v a l u e s o f the  perturbed o p e r a t o r H. The e f f e c t i v e operator H correct  an  c h a r a c t e r i s t i c equation* can be  regarded simply as t h e c h a r a c t e r i s t i c equation o f an operator  reduced  i n A.,  The  roots  i s then  of  the  found  'reduced  c h a r a c t e r i s t i c equation*  r(z,A) - d e H z - H l = O are t h e perturbed e i g e n v a l u e s of H, c o r r e c t t o the in  same  A. The Van Vleck approach i s r e p r e s e n t e d by the path  order  10  H  • H  In c h a p t e r 3 however, the path H  H  i i  •P is  taken, ., The  function  »  r  coefficients  of  the  r are found not i n terms  operator  but  from  the  contour i n t e g r a t i o n  (full)  matrix  characteristic elements  characteristic  method  is  to  function. . In relate  of  an  c h a r a c t e r i s t i c function  they are determined i n terms of  of t h e  used  of  reduced  the  addition,  coefficients  JBy  derivatives  an  algebraic i~  of  to  the  c o e f f i c i e n t s of -P d i r e c t l y . I t i s a l s o shown that the path H  H  £  r  can be taken, without mention the  of an e f f e c t i v e o p e r a t o r H.  Here  c o e f f i c i e n t s of r are found i n terms of matrix elements o f  H. In Chapter 4 the p e r t u r b a t i o n formulae d e r i v e d i n Chapters 2 and 3, i n terms of the c h a r a c t e r i s t i c f u n c t i o n , to  two  illustrative  perturbation molecular  examples.  calculation,  orbital  within  The the  first  are  applied  example  context  of  is  a  Hiickel  t h e o r y , f o r an even-membered r i n g of carbon  atoms. In the second example i t Bayleigh-Schroedinger  is  perturbation  shown  that  formulae  can  the  familiar  be o b t a i n e d  from the corresponding  c h a r a c t e r i s t i c function expressions.  In Chapter 5 the r e l a t i o n s h i p between  energy  derivatives  and p r o p e r t i e s i s d i s c u s s e d with p a r t i c u l a r r e f e r e n c e t o  simple  spin  vector  and  systems. , P r o p e r t i e s such as the s p i n p o l a r i z a t i o n the  spin  characteristic  polarizability  tensor  are  found  f u n c t i o n o f the s p i n system and two  t h e i r use are g i v e n .  using  the  examples of  12 CHAPTER 2.  PERTDSBATION THEORY FOB SINGLE EIGENVALUES.-  Introduction. . In  this  eigenvalues,  chapter in  a  terms  perturbation of  characteristic  developed. The theory i s r e s t r i c t e d to analytic of  IM=o  m  analytic  theory  for  functions, i s  eigenvalues  or  which  are  f u n c t i o n s o f t h e p e r t u r b a t i o n A i n some neighbourhood For a f i n i t e  dimensional Hermitian o p e r a t o r , i t s e l f an  function of A i n  some  neighbourhood  o f |A/ = 0,  e i g e n v a l u e s w i l l be a n a l y t i c i f e i t h e r 1.  single  t h e r e i s o n l y one p e r t u r b a t i o n parameter,  2 . t h e unperturbed  (see Appendix A). ,,  e i g e n v a l u e i s non-degenerate  the  13 2.1 The method. • Consider  an n - d i a e n s i o n a l Hermitian o p e r a t o r Hft) which i s  an a n a l y t i c f u n c t i o n  of  characteristic function  perturbation  parameter  A.  The  f(z>A) i s given by  $&>V  (2.1)  one  s  del- l z - H ( A ) J ,  which has the f a c t o r i z a t i o n  (2.2)  H*,V)  where  = 7T(z-e.-0O),  the E, (A) a r e the eigenvalues o f H(A) and are r e a l valued of A . , The  analytic functions  values  of H  f o r which  the  c h a r a c t e r i s t i c equation (2.3)  f (Z,A) = O  i s s a t i s f i e d a r e t h e e i g e n v a l u e s o f W(A). , The  Taylor  expanded about  series  f o r any  fe>(o)  p  + ... .  I t i s the o b j e c t o f t h i s chapter t o determine AJrT(o)t  > =  rr  1  eigenvalue £p(3},  A-4 i s  E^A) = E (o) + A ^EP(O)  (2.U)  particular  the  ,2,: • The z e r o t h d e r i v a t i v e or unperturbed  derivatives eigenvalue  £ {o) i s assumed t o be known. , p  The  idea  behind  the  approach  i s to  regard  c h a r a c t e r i s t i c equation (equation (2.3)) as i m p l i c i t l y Z as a f u n c t i o n o f A,  for ^  in  some  neighbourhood  defining of the  e i g e n v a l u e ^(o), and f o r A i n some neighbourhood  unperturbed  the  of  »This f u n c t i o n w i l be q-valued i f t h e unperturbed eigenvalue i s q-fold degenerate. F o r example, c o n s i d e r the 2 - f o l d degenerate case, P(z,3)=i -A% where e-ft>^= e (o) =o. a  a  14 A h i e r a r c h y o f equations be o b t a i n e d .  f o r the i m p l i c i t d e r i v a t i v e s ^3 ATT  can  Since  d"2  I  _  4E m  A-o  an  f o r 9L_^  expression  obtained  from  the  hierarchy,  e v a l u a t e d at Z^e (o) and ^ o , w i l l y i e l d the r e q u i r e d  derivative  p  <±SP(O).  Thus the terms i n t h e T a y l o r s e r i e s expansion  ( 2 . 4 ) ) can be found and t h e perturbed eigenvalue  when  (equation  E" (A) o b t a i n e d , P  c o r r e c t t o any order i n A. The  results  obtained  for  a  non-degenerate  e i g e n v a l u e with one p e r t u r b a t i o n parameter can  be  unperturbed generalized  to t h e case o f more than one p e r t u r b a t i o n parameter. Degenerate eigenvalues,  on  the other hand, a r e guaranteed t o be a n a l y t i c  o n l y i f there i s one p e r t u r b a t i o n parameter. Thus obtained  f o r degenerate  generalized parameter. ,  to  the  case  unperturbed of  more  eigenvalues than  one  the  results  cannot  be  perturbation  15  2.2  Orientation. , In  characteristic  function  language  p e r t u r b a t i o n theory becomes the study of the roots  of  quantum mechanical way  in  which  a polynomial change as i t s c o e f f i c i e n t s are  l i t t l e , i . e . , as A v a r i e s  (see F i g u r e  An example o f the function  varied  2).  Eatt)  F i g u r e 2.,  the  E C*) 5  v a r i a t i o n of a c h a r a c t e r i s t i c  with 2 f o r two  f i x e d values o f A .  ?  a  16  Since the c h a r a c t e r i s t i c equation d e f i n e s z as a f u n c t i o n of /\, the t o t a l d i f f e r e n t i a l with r e s p e c t to A i s g i v e n by»  d_  a.  =  d? 9. dA 9z '  +  3A  Taking  the  total  d i f f e r e n t i a l of the c h a r a c t e r i s t i c  equation  with r e s p e c t t o A y i e l d s  (2.5,  ^  + f  z  = O ,  S J |  where t h e s u b s c r i p t s denote p a r t i a l  derivatives.  F i g u r e 2 i l l u s t r a t e s two t y p i c a l eigenvalue  cases.  E,(o) i s non-degenerate, whereas  The  unperturbed  B. (p)-E (o)  are 2-fold  s  z  degenerate e i g e n v a l u e s . For the non-degenerate e i g e n v a l u e , equation  froa  ( 2 . 5 ) , we have  For t h e degenerate e i g e n v a l u e s , E,(o) and £3(0), t h e s i t u a t i o n i s q u i t e d i f f e r e n t , s i n c e ^ ( E ( o ) , o ) - 0 (which a  i s the condition  t h a t £a(o) be a t l e a s t 2 - f o l d degenerate), and -PafeitP^Oj-o . Osinq 2  L'Hospital's rule a quadratic for  l  s  obtained,  S e e , f o r example, Goursat(1904), volume I , page 41, e x p r e s s i o n i s given not i n o p e r a t o r form but as df  2  i  =  3f  where  the  . 3 f . d_z  9* 32 dA In g e n e r a l , f o r q — f o l d deqeneracy, the c h a r a c t e r i s t i c f u n c t i o n has the f a c t o r i z a t i o n  Thus a l l a i x e d d e r i v a t i v e s of order l e s s than the order of the degeneracy v a n i s h , when e v a l u a t e d at z^EpCo), ~X^o i . e . . t  17  (2.6)  The two r o o t s o f t h i s q u a d r a t i c , when a l l d e r i v a t i v e s of -P a r e e v a l u a t e d a t z=£^(°)# A=o, a r e t h e d e r i v a t i v e s ^ ( o ) and d§3(o). Equation  (2.6)  originated  from  the  characteristic  e q u a t i o n . The c h a r a c t e r i s t i c equation i s known t o be when  z.  i s equal t o the unperturbed  A=c. Equation  e i g e n v a l u e , £ 4 ( 0 ) , and when  (2.6) i s a l s o s a t i s f i e d f o r these v a l u e s o f 2 and  A. Thus i t i s only the d e r i v a t i v e s o f Ej(X) and E^CA) with to  A,  satisfied  evaluated  at  7\-o  f  that  can  be  respect  found.. Indeed, the  d e r i v a t i v e s jr^ffa) with }\?o c o u l d o n l y be found i f Epfa), /\*o O A  a l r e a d y known. .This i s not u s u a l l y t h e case.  was  18  2.3  The  h i e r a r c h y of p e r t u r b a t i o n  To  obtain  expressions  equations.  for  d e r i v a t i v e s of the c h a r a c t e r i s t i c with r e s p e c t t o A, are  ,2.7,  i  f  eguation  ,  successive  (eguation  total  (2.3)),  taken:  -O  -  1  (2.10, d i f -  <LSP{O)  d  ^  >  ^ - » - i > ' , + - a y  c/AA  fcpfef.J&f=0.  " i  J  + . . . + ' 7 y  - n  0  (<* = v, + ...+v ) n  (See comment 1, below, f o r e x p l a n a t i o n o f summation i n eguation (2.10).)  This  set  of  perturbation hierarchy, (2.10) i s the n—th  equations or  equation  will  hierarchy o f the  be r e f e r r e d to as for  brevity.  hierarchy.  the  Equation  19 2.3.1  Comments.  7  1., The summation values  of /Szo,  i n equation Z;>0  9  (2.10) i s taken  ; = 1,~. n,  with  y  2. The c o m b i n a t o r i a l c o e f f i c i e n t *  is  the  number  the  ——=  o f ways n o b j e c t s  ^  sets,...,n i n the l a s t  3. D i m e n s i o n a l l y  X  t  sets, 2 in  n  ^  sets  1  is  unimportant.  n  we have  and  Since * = X,+...+i  n  and p-+i* +a)t +...nti ^, i t f o l l o w s t h a t v  n  the  )f s e t s , where t h e order w i t h i n  1 + Jf, + ... +- y = 1 +• <L .  number o f s e t s i s  that  (the A*s) can be p a r t i t i o n e d  each s e t and t h e o r d e r of t h e 'next The t o t a l  restriction  —  i n t o s e t s : /3 i n t h e f i r s t s e t , 1 i n the next next  over a l l i n t e g r a l  n  and each term i n t h e sum i s d i m e n s i o n a l l y c o r r e c t . ,  »See, f o r example, Abramowitz and Segun  (1965), page 823.  20 4., Eguation total  (2.10)  can be d e r i v e d i n t h e f o l l o w i n g way*: the  differential  dA  d A 9?;  29i  i s such t h a t  JA  d5\k+l  For example,  - (Ir+a & {*$) W ( f &fc•  & • k Ik •* ^^JrIV 1z  The term  4?<L, d * 9*  the term  i s t h e r e s u l t of i  da  I(4^tl^-T da/ UzJ  a c t i n g on dz i n the product da  i s t h e r e s u l t o f part of i  *  a c t i n g on =L i n t h e same product 3z  ,  d*  2 .  d3 3z-  ,  ^? 2 - , and so on. C l e a r l y , ?L clA"  da 3z  w i l l r e s u l t i n a sum o f terms o f t h e form  Each «_? da  xs t h e r e s u l t o f Q_ a c t i n g on ? £ , t h e 2 da  k  r e s u l t o f d- a c t i n g on d z d  result  *  da ' k  o f 4d  *  having  da"-'  , and so on, down to  acted  da*  on 4J£.  S i n c e t h i s 4*  da  i s the  being  \  a  #  da"*'  dA  the  originally  iEquation (2.10) i s g i v e n , without proof, by K r a s n o s e l ' s k i i e t . a l . . ( 1 9 6 9 ) , page 329. In t h i s study equation (2.10) was f i r s t obtained i n the n o t a t i o n o f s e c t i o n 2.4.  21  appears i n the product  £_  t  the power to which 2_  is  raised,  3z  to  <*, i s given by ot — tf, •+- ... + Y In  .  n  a l l , d_ o p e r a t e s n times. I t has operated  l%i-W +-nX i  times  n  to  to  M ^ ^ . T h e number of times ' l e f t  produce  power t o which iL  fee the  over*  will  must be r a i s e d , i . e . .  To e s t a b l i s h the c o m b i n a t o r i a l c o e f f i c i e n t , suppose each A is labelled, i.e., —j> Each  i  "\  lt  n  permutations A.A A » a  Hence  3  the  ^  of o r  d_  d-  ...  d_  w i l l appear once i n each term of the sum. A l l the  'A;  occur except t h a t  ^ j l r j ^ j f ;  example, w i l l appear only once, not  order  of  the "next ^  j  3!  (cllf  )  °  r  times.  s e t s , ' as w e l l as the order  w i t h i n each s e t , i s unimportant. On removing  the  labels,  c o m b i n a t o r i a l c o e f f i c i e n t i n e g u a t i o n (2.10) i s o b t a i n e d . .  the  22 2.4 An a l t e r n a t i v e n o t a t i o n . r  It  i s possible  equation  to  group  the  terms  occurring  i n each  o f the h i e r a r c h y i n a way which b r i n g s out a s t r u c t u r e  which w i l l be made use equation  (equation  p a r t , terms i n the  of  later.  The  terms  in  the  second  ( 2 . 8 ) ) can be grouped i n t o a q u a d r a t i c third  equation  (equation  (2.9))  can  like be  grouped i n t o a c u b i c l i k e p a r t , and so on. The f o l l o w i n g n o t a t i o n i s used,  ^ = ILir)^^^^ >  (2.11)  r-o  where  I'i] i s the binomial coefficient V"> n o t a t i o n T^P has the f o l l o w i n g p r o p e r t y ,  ,in  this  equations o f the h i e r a r c h y  can  L R  r  <2  '  12>  Cn)  3 *2 P  c  (n+>)  =  °  r-Cn-i)  (n-r-)iri  a  d f a'  as can e a s i l y be shown.„ In t h i s n o t a t i o n the f i r s t be w r i t t e n  23  (2.15)  To  write  t h e next  combinatorial representing  equations  coefficient the number  ( of  p a r t i t i o n e d i n t o k s e t s , with the order  in a J ways  systematic  way  will  introduced,  be  •*-d .±n o b j e c t s k  objects i n the i - t h set,  a  new  can be where  w i t h i n each s e t , and between s e t s c o n t a i n i n g the same  number o f o b j e c t s , i s unimportant. F o r example, we have  /  5  [3,3}  )=  _£L._L  a<3( '  With t h i s n o t a t i o n the next e g u a t i o n s o f t h e h i e r a r c h y become  (2.16)  24  is)  (2.17)  o  +-  /5\  dW  2  at  this  n^-th eguation  point  a  p a t t e r n i s beginning  t h e r e a r e terms  t o emerge. I n the  like  (n- f o f , *...-»-«<|,71  where <>a, i-X,.. k.t f  counting*).  <** + -^k^n t  and *tt — --- —  (to a v o i d  'double  I f by the symbol  n , •• • ol we  mean  K  t h e sum o f a l l terms above i n which t h e 4;*s s a t i s f y  the r e s t r i c t i o n s s t a t e d , then the n-th eguation  can be w r i t t e n  (2.18)  where m i s the l a r g e s t i n t e g e r such t h a t 6-th eguation  of the hierarchy i s  m ^ a . J o x  example, t h e  25  UJ  dA  £  fc  The n o t a t i o n i n t h i s s e c t i o n i s u s e f u l i n t h a t i t p r o v i d e s a compact  way  of  writing  any  particular  eguation  in  the  h i e r a r c h y . , In a d d i t i o n , the s t r u c t u r e o f each e q u a t i o n i s made c l e a r . Each e q u a t i o n i s simply implicit  d e r i v a t i v e s 4!Zs ,  a  polynomial  as equation  in  the  (2.19) i l l u s t a t e s .  f i r s t d e r i v a t i v e 4* i s embedded i n the symbol t  p  notation,  The  .  The f i r s t e i g h t equations o f the h i e r a r c h y are this  desired  given,  in  i n Table I . The combinatoral c o e f f i c i e n t s have  been r e p l a c e d by t h e i r numerical v a l u e s .  Table I  3.  3.  The  V  +  £'  f i r s t e i g h t equations of the  0)  o dA  a  4:  +  (3>  d*  ...+ A-.  a  — Z  o  d**  -*-  d*  a  -f  dA  3  =o  da  z  «da/  2  .. .-MO 4?<ft  d**  .+5  4" ^  d ^  a  + io £^W^\  wVwW  -  + -- -  27 Table I . continued. r  (t,)  r- ft** ,X-  -Fa  6 0  £  dV>  d^\ .d# W  d*£ + IStgMa|,  d?r jo* 2  da  -j- - -  5  =  o  dtf  7.  f ^ ^ U f - f t  \05  da*  iOS  da*/Ida/ 3  2  .  m  dA* is  dA  2.  &  5  d/f  *dr  35  =  (da/Ida*/ 3  o  ^ 3  Table I . continued  #)  ft)  r  840  dl*  * way  da>  z  ... + 70 K,  Cf^ff °  W)\dr)  £M die-  d^  z  —  dr  da* ... +  - a *o  ^ / ^ l 1^1+• - - • (dWld^J ......  +,  5  29 2.5 S p e c i a l cases a r i s i n g i n p e r t u r b a t i o n theory.Some s p e c i a l cases a r i s i n g i n p e r t u r b a t i o n theory w i l l now be c o n s i d e r e d : non-degenerate e i g e n v a l u e s ,  2-fold  and  3-fold  degeneracy l i f t e d i n v a r i o u s o r d e r s , and g — f o l d degeneracy. The equations o f the h i e r a r c h y s p e c i a l i z e i n each case.  2.5.1 Bon-degenerate e i g e n v a l u e s . If for  i s a non-degenerate e i g e n v a l u e , i . e . , E ( o ) ^ E,Co),  Ep(o)  i^p, then  P  ^  Z  ( E .  P  ( O ) , O ) ^ O .  In t h i s case a simple  rearrangement  of t h e equations o f t h e h i e r a r c h y l e a d s to e x p l i c i t for  a l l the d e r i v a t i v e s  <¥!!3p(o\r  down we have  (2.20,  (2.21,  z >  no)  expressions  m - l,..,n. .. .. . W r i t i n g these  30  s**i?]-nf*4i-f I Ik, n  (2.23)  n  where a l l terms on t h e r i g h t hand s i d e s of t h e s e equations evaluated  at  the unperturbed  \-0. (Eguation the  o b t a i n e d from derivatives  eigenvalue, i . e . , f o r 2. = £^b) and  (2.22) i s o b t a i n e d from equation  alternative  notation  eguation of  lover  of  are  (2.10) and, i n  s e c t i o n 2.4, eguation  (2.18).)  To  find  (2.23) i s  4!^P(o),  all  the  o r d e r must have a l r e a d y been c a l c u l a t e d  from t h e precede ding eguations i n the sequence. The way i n which the equations o f t h e h i e r a r c h y  a r e used,  f o r non-degenerate  e i g e n v a l u e s , i s r e p r e s e n t e d by t h e schema i n F i g u r e 3. In t h e case o f s e v e r a l p e r t u r b a t i o n parameters, the  perturbed  Taylor series,  (X -- \), >  >  e i g e n v a l u e E^ft) has a m u l t i p l e p e r t u r b a t i o n , o r expansion, t  (2.24)  X=  /  = M°>  + Y  L X  ff^)  +  >-->  Y  a  X2i l l r  ;i=->  (o) + ... , J  In t h i s case we have  ^P(o) = - f* / 4  (2.25)  (2.26)  :  ,  J hi  2 *Sr5 f  93,  (p) =  - [ 4 , + 4 * ^ + 4%  ^  4-4-' a«  3?]  /-P  where a l l terms on t h e r i g h t hand s i d e a r e e v a l u a t e d f o r z = and  3 =<V,The h i g h e r mixed d e r i v a t i v e s  ^Ef -(o) can be o b t a i n e d  31  F i g u r e 3.  Guide t o the p e r t u r b a t i o n h i e r a r c h y f i ] Hon—degenerate  eigenvalues.  X  dlE  Tda  <fe>(o)  In t h i s f i g u r e and i n F i g u r e s H and 5,  n  r e p r e s e n t s the n-th eguation o f t h e h i e r a r c h y .  by modifying  the  expression  f o r 4H§p(o).,  To  illustate  this  m o d i f i c a t i o n , c o n s i d e r the t h i r d , one parameter, d e r i v a t i v e ,  ,2.27, J 5 w  Each  term  ^'^g^^W^tygj&fe*  =  in  corresponding  A  a  and  this  e x p r e s s i o n c o n t a i n s A t h r e e times. I n the  expression f o r  A  3  each  _  A;,  — C o )  <=',^,.3  each  occurring  f o l l o w i n g m o d i f i c a t i o n s must be made:  term  will  contain  o n l y once. Thus, t h e  32  -IU  (2.2.8)  (2.29)  — ^  A  a a  3^d?_^ fell ^ +  £  1  3 i+  ,  -P  and so on. The c o m b i n a t o r i a l c o e f f i c i e n t , (2.29), i s the number of nays the  term  partial  4 ^ <Jz-.  As  \,  derivatives ^ z X , ^  a  n  n,  having  written  down  3  3  in  eguation  a r e permuted w i t h i n  i n s e c t i o n 2.3.1, the mixed ^2^3, only  d  g e n e r a l , then, t o o b t a i n ~? ^ ~o) Ep  the  Aj and A  discussed  3z  311  count  once.  f o r some p a r t i c u l a r  (  t h e e x p r e s s i o n f o r 4%p(p),  In  value o f  one simply  makes the a p p r o p r i a t e m o d i f i c a t i o n s . A l l t o t a l d e r i v a t i v e s d^Z? become  partial  derivatives  3  --  Zr  e x p r e s s i o n f o r 4-£p( )  i s rewritten  permutations  A  0  of  coefficient*  the —  a ;  within  /*' ^  and  each  by  listing  term  i n the  a l l allowed  each term. The c o m b i n a t o r i a l , i s j u s t the number  of  such  permutations. .,  2.5.2 2 - f o l d degeneracy l i f t e d If  E (o) i s P  a  in first  2-fold  degenerate  o.  In  h i e r a r c h y , eguation eguation  eigenvalue,  then -F (^pCo) ) - O  Epto)=Ep+,Co)?E,Co) f o r  $z.*(Ep(o),o)-*  order.  2  this  case  (2.7), i s  the  first  identically  (2.9), a q u a d r a t i c i n 43, g i v e s JA  j0  eguation zero., The  i.e , of  but the  second,  33  (2.30)  This  efl'fa) =  is  in  agreement  L*Hospital's rule distinct,  (eguation  t h e degeneracy  d§^'(o)w S u b s t i t u t i o n o f  4%o)  d%\t-o  e<f>> i  n  t  o  t  n  e  Jfea) -4*4 a  ±  remaining  with  (2.6)).  these  third  eguation  of  result  I f these  i s lifted two  eguations  y i e l d s a l l t h e d e r i v a t i v e s j~£Co) the  the  in  obtained two  first  values f o r  t h e h i e r a r c h y then  and i p 5 » + i ( 0 ) , m>I.  the h i e r a r c h y  are  order, i - e « #  different of  roots  using  For  gives  example,  the second  d e r i v a t i v e s as  or,  i n t h e n o t a t i o n o f s e c t i o n 2.4,  (2.32)  dr  6>)  3£  When the two d i f f e r e n t ^§p+'(b)»  values  f o r dl|z=E,£o)»  i . e . , dj&Ycrt  and  a r e s u b s t i t u t e d i n t o these e g u a t i o n s , the corresponding  second d e r i v a t i v e s sLff  and <L5~i(o), a r e o b t a i n e d . „  The q u a d r a t i c whose r o o t s a r e given by eguation  (2.30) i s  the  same  quadratic  that  Rayleigh-Schroedinger p e r t u r b a t i o n the  ( 2 - f o l d ) 'degenerate  2.5.3  first  condition  be  theory  when  obtained  diagonalizing  that  2-fold  in first  order.  degeneracy  i s not l i f t e d i n  order i s t h a t t b e two r o o t s given by eguation  equal.  This  will  be  in  block*.  2 - f o l d degeneracy not l i f t e d The  would  the  case  i f the  (2.30)  be  d i s c r i m i n a n t o f the  q u a d r a t i c v a n i s h e s , which i s the case when the Hessian o f i s z e r o , i . e . , when  (^J  (2.33)  If  -  t h i s i s t h e case,  (2.34)  &(o)  da  =  finding  -  dJFP+'fo)  da the  =O  -  &  then  K  (In Rayleigh-Schroedinger to  to)  p e r t u r b a t i o n theory t h i s  •degenerate  block* d i a g o n a l with the d i a g o n a l  elements equal.) I f degeneracy i s not l i f t e d then  the *~  can  third  equation  corresponds  of  first  order,  the h i e r a r c h y , eguation  (2.15),  a  d—3-, t u r n s out t o be  in  identically  zero,  be shown by t a k i n g the t o t a l d e r i v a t i v e o f equation  with r e s p e c t t o A and u s i n g equations  (2.33)  f o u r t h equation o f the h i e r a r c h y , equation  and  as  (2.33)  (2.34).  The  (2.16), reduces t o  35  (2.35)  The  3^.-  two  given  /d^j  +  4f ,  J i  2  a  V  +  =o  y i e l d 4lfp(o)  r o o t s of t h i s equation  and d*&>WoW They are  da*  by  (2.36,  ±  /  t  da'  *  W  A= o If  these  roots  second order and  are  different,  substitution  the h i e r a r c h y s i l l  the  degeneracy i s l i f t e d  i n the higher order eguations  then y i e l d the higher d e r i v a t i v e s ,  42Ff>(o)  in of and  da*"  If the  degeneracy i s not l i f t e d  sixth  eguation  i n the t h i r d  derivative  41?  .  In  this  <">4- are zero, i . e . , the s i x t h eguation  in  the  3  4^3 »  reduces t o  *  UT)W)  g u a d r a t i c i n ^f?-. ,In summary, the way da  sixth  3-  * da  a  case,  5  a l l the c o e f f i c i e n t s of the i m p l i c i t d e r i v a t i v e s  da  then  o f the h i e r a r c h y i s the r e g u i r e d q u a d r a t i c d*  eguation,  i n f i r s t or second order,  i n which the  equations  3  of the h i e r a r c h y are used  for  v a r i o u s o r d e r s , i s represented  2-fold  degeneracy,  lifted  by the schema i n F i g u r e  4.  in  37  2.5. 4 3 — f o l d degeneracy and degeneracy p a r t i a l l y - l i f t e d . -. Ep(°)  If  is  a  3-fold  Ef>C°)~ Ep-ni°)= p+ato)^E;to) f o r  =  £z  2  =  ^  =  when a l l d e r i v a t i v e s a r e e v a l u a t e d eguation  £\ = o a  at  3=o.  o f the h i e r a r c h y which i s not i d e n t i c a l l y  t h i r d , eguation  a cubic  eigenvalue, i . e . ,  /V/>, p+i, p+a, then  £  •Re- =  degenerate  The  first  zero i s the  ( 2 . 9 ) . T h i s eguation reduces t o v  in  . The t h r e e r o o t s of t h i s eguation, da e v a l u a t e d a t ^E 0>) and feo, a r e t h e f i r s t d e r i v a t i v e s  when  p  It first  can happen t h a t degeneracy i s only p a r t i a l l y l i f t e d i n  o r d e r . Consider the case i n which  da  d-x  </a  For t h e e i g e n v a l u e  E  substitution  dJfrfa)  of  pt  which into  h i e r a r c h y w i l l y i e l d the second of both dfpco) and 4^P(O) <^a c/A  i s no the  longer fourth  derivative  degenerate, eguation  4^3P(O)»  the  of t h e  Substitution  i n t o t h e f i f t h equation o f the hierarchy  a  will  yield  the  third  d e r i v a t i v e <L3p(p\,  and so on. Thus t h e  T a y l o r s e r i e s f o r E (X) i s o b t a i n e d . However, the p  situation  EfmCA) and 5>*a£l) i s q u i t e d i f f e r e n t . Since the f i r s t 4&>*'6>) and ^ ( o ) are  e q u a l , E>,<>) and E^,£A) a r e  degenerate  order.  to  first  +  Consequently,  for  derivatives still  2-fold  t o f i n d the second  38  d e r i v a t i v e s , a q u a d r a t i c equation must  be  s o l v e d . , The  fifth  equation i n t h e h i e r a r c h y i s the r e q u i r e d q u a d r a t i c i n dV-  In summary, t h e c a l c u l a t i o n f o r 3 - f o l d degeneracy i s given by the schema i n F i g u r e 5.  39 Figure 5 .  Guide t o the p e r t u r b a t i o n h i e r a r c h y  (iiil  3 — f o l d degeneracy. (P)~]  False  'True. 1  4-  dr  1  dr  6 da*  40  2« 5.5.  q—fold  ;  E (o) i s a q— f o l d degenerate  If  P  In t h i s  and  degeneracy.  eigenvalue,  then  case  consequently  the  first  non-vanishing  eguation  of t h e  h i e r a r c h y i s the q-th. T h i s f o l l o w s because c e r t a i n l y ^ ^ < % i f n<c^, so t h a t a l l terms i n t h e summation i n equation be  z e r o , when e v a l u a t e d a t z ~ f= Co), P  any  i-3.,...,n  %~O  will  I f n-<^, then °t+/? l i f <e  m  and <<->fi^% i f a l l  t  (2.10)  / = a,... /?. Hence f o r v  the non-zero terms i n the sum,  and t h e q-th equation i n t h e h i e r a r c h y reduces t o  The  r o o t s o f t h i s equation a r e t h e q f i r s t  derivatives,  which  may o r may not be d i s t i n c t . , On  writinq  equation  out  last  two terms of t h e summation i n  (2.37),  (2.38)  it  the  +  f ^  =0,  i%  can be seen t h a t i f the q - f o l d degeneracy i s not  first  order  e v a l u a t e d at  (i.e., z=^o),  i f eguation  in  (2.38) has e g u a l r o o t s when  3 ^ o ) , then t h e f i r s t  e i g e n v a l u e s i s merely  lifted  order s h i f t of  a l lg  41  dE''(o) =  (2.39)  « A  z=e {p)  f1  ;  A=o  z  £ since  i s the sum  o f the g r o o t s o f eguation  t h i s case a g-th o r d e r eguation i n t h e second is  required,  The  2q-th  quantity  derivatives  and t h i s i s the 2q-th e q u a t i o n of the h i e r a r c h y .  equation T ^/s_?A 2  (2.38). I n  >  is  the  first  to  contain  the  required  which occurs i n Co)  If of  degeneracy i s l i f t e d  each r o o t of equation  values  of  #  in first  o r d e r , then  (2.37), i . e . , each  into  the  higher  of  substitution  the  different  o r d e r eguations of the  h i e r a r c h y y i e l d s t h e c o r r e s p o n d i n g higher order d e r i v a t i v e s .  42 2.6 Summary. , I t has  been  characteristic  shown  equation  how  the t o t a l  of  a  derivatives  of the  perturbed o p e r a t o r , taken with  r e s p e c t t o a p e r t u r b a t i o n parameter, y i e l d e x p r e s s i o n s f o r t h e perturbed  eigenvalues  t o any order i n /\. As made c l e a r by t h e  n o t a t i o n o f s e c t i o n 2.4,  the  effect,  polynomial  a  sequence  of  perturbation  hierarchy  is,  in  equations i n t h e i m p l i c i t  d e r i v a t i v e s 4Cj^ , whose c o e f f i c i e n t s are t h e p a r t i a l  derivatives  A7T  of  t h e c h a r a c t e r i s t i c f u n c t i o n , with r e s p e c t t o z and a.  partial  derivatives  are evaluated  at z=£ 6>) P  These  and 7i-o and t h e  e q u a t i o n s s o l v e d t o g i v e t h e e i g e n v a l u e d e r i v a t i v e s 4£jh(p) Some m  important s p e c i f i c examples have been d i s c u s s e d i n s e c t i o n 2.5. For each s p e c i f i c specialize i.e., terms,  example,  the  eguations  of  the h i e r a r c h y  i n a p a r t i c u l a r way, when e v a l u a t e d a t z = ^ o ) ,  some terms  vanish.  i n the f i r s t  Table  II  lists  the  '\-o  t  non-vanishing  few eguations of t h e h i e r a r c h y , f o r the  s p e c i a l cases c o n s i d e r e d i n t h i s c h a p t e r .  43 Table II  The  non-zero terms of the e q u a t i o n s of the  p e r t u r b a t i o n h i e r a r c h y f o r some s p e c i a l c a s e s . A l l d e r i v a t i v e s of f are equations of  in  evaluated  a=o.  hand s i d e .  2 — f o l d degeneracy, l i f t e d  +  o  +  3  +i  Ose Table I,  in f i r s t  d2  dfl aa*  -  order.  o  a  . . . -4-  4- -Fj^ d ^  Ida*/ -  3  da  o  5  2 - f o l d degeneracy, l i f t e d  i o z  =  i n second  order^  O  Ida*/ TS  >  The  t h i s t a b l e have been obtained from the equations  Non-degenerate e i g e n v a l u e s .  2.  p  the h i e r a r c h y i n T a b l e I as i n d i c a t e d by the numbers on  left  3.  a t z. - E (o),  ,(5)  Ida'/  da  3  (dWldav  the  44  T a b l e I I . continued. 2 - f o l d degeneracy, l i f t e d  in third  t?° - O  2.  dA  2  2  1^  / U S  '  3 — f o l d degeneracy, l i f t e d  4.  Ida*]  Idav  z  10  3.  order.  i  w  &Ka*& z  =o  da  z  Id^J  2  *  d/l  5  O -  i n f i r s t order.  - O  (  ^  3  Table I I . c o n t i n u e d . 3 - f o l d degeneracy, p a r t i a l l y l i f t e d  order.  0>  r  3.  in f i r s t  = 0  + o z  For P C * ) . e  tt.  -r o ?  •+ b i i z  <L?  =  O  da*  For E p ^ ) a n d £*»«C\i  da* 3 - f o l d degeneracy, not l i f t e d  =  Ida^i in first  order.  O  da*  Ida  2  /s^Vd^f  (da*/  46  CHAPTER  THE  3.  REDUCED C H A R A C T E R I S T I C  FOBCTIOS. ,  Introduction.?  In  this  eigenvalues  chapter  but rather  generalization Chapter  attention on  but  characteristic  of  rather function.  series for  the  n o t on  eigenvalues.  of the s i n g l e eigenvalue  2. , P e r t u r b a t i o n  eigenvalue  sets  i s focused  are  individual This  perturbation obtained  coefficients  theory  not of  i s  a  for  a of  one  reduced  47  3.1 The reduced c h a r a c t e r i s t i c The term  'reduced c h a r a c t e r i s t i c f u n c t i o n ' w i l l be used t o  denote a p o l y n o m i a l subset, E  P )  function  £(z,3)  .. E }  w  (3.1)  function.,  r  p n  .  l f  Kz,A) o f degree <^ i n z, whose zeros a r e a o f the z e r o s o f the ( f u l l )  characteristic  i . e . , a subset o f the e i g e n v a l u e s o f H:  ^ , A ) = 7T (z-e,-)  The c o e f f i c i e n t s o f r , qi,i=i...,<j, depend on A. Given the reduced c h a r a c t e r i s t i c f u n c t i o n the i n d i v i d u a l e i g e n v a l u e s can be found as t h e r o o t s o f t h e 'reduced c h a r a c t e r i s t i c  eguation*  r(^.,7i) = O -  (3.2)  The s e t o f e i g e n v a l u e s w i l l always be taken t o degenerate  o r nearly  s e t s ) . The  degenerate  be a  complete  s e t (or any number of such  dependence o f t h e e i g e n v a l u e s i s then  i n general  more complicated than t h a t o f the c o e f f i c i e n t s c,-. The  ct)....)cc^, E  P,  coefficients  a r e symmetric  p+i-i> namely.  E  o f the reduced c h a r a c t e r i s t i c functions  of  the  function,  eigenvalues,  48  (3.3)  = (-IP'TV  Co >  C  Accordingly,  L  (E;) ,  1  /=/»  21  -1  =  knowledge  of  E/  the  c o e f f i c i e n t s i s eguivalent to  knowledge o f the e i g e n v a l u e power sums  m - i , . ; ^ ,  d e f i n e d by  P+%-1 (3.4)  5  The two are r e l a t e d  {E;)  m  by Hewton^s* formulae:  s,  • •  (3.5)  In  \  =  m  +  c,  =  O  particular,  (3.6)  c, =-S  (  ,  Two methods w i l l be used t o compute the and  the  coefficients  reduced c h a r a c t e r i s t i c f u n c t i o n s f o r non-degenerate, 3-fold  initially  degenerate  method, contour i n t e g r a t i o n »See, f o r example, T u r n b a l l  eigenvalues.,, In  i s used to  determine  (1952), page 72.  of  2-fold  the f i r s t  perturbation  series  f o r the e i g e n v a l u e power sums, 5 . The second method i s m  a l g e b r a i c and y i e l d s p e r t u r b a t i o n s e r i e s f o r t h e  coefficients  c;, d i r e c t l y . The idea o f the reduced c h a r a c t e r i s t i c f u n c t i o n i s c l o s e l y related  to  a  f a c t o r i z a t i o n theorem o f B e i e r s t r a s s * . T h i s , i n  turn, i s closely related  t o t h e theorem  total  t h e subspace a r i s i n g from an i n i t i a l l y  projection  onto  degenerate e i g e n v a l u e i s a n a l y t i c of  |A/=<3. , The  out  o f the ( f u l l ) c h a r a c t e r i s t i c  (Kato, 1949)  i n A, i n  some  that  the  neighbourhood  reduced c h a r a c t e r i s t i c f u n c t i o n can be f a c t o r e d function:  (3.7)  The f a c t o r i z a t i o n theorem t e l l s degenerate  eigenvalues  us t h a t as  long  as  initially  a r e c o n s i d e r e d t o g e t h e r , i . e . , i f only  complete degenerate s e t s a r e i n c l u d e d , then t h e c o e f f i c i e n t s o f the c o r r e s p o n d i n g reduced c h a r a c t e r i s t i c f u n c t i o n a r e functions  of  neighbourhood  the p e r t u r b a t i o n parameters  analytic  (%,-.•, X^) /  o f \/\\~ O . Conseguently they have a T a y l o r  i n some  series,  or p e r t u r b a t i o n s e r i e s , o f the form  (3.8)  which i s convergent f o r a l l /\ i n some neighbourhood The perturbed e i g e n v a l u e s themselves do not, have  such  in  a s e r i e s expansion i f they are i n i t i a l l y  and i f t h e r e i s more example,  of  suppose  perturbation  than  for a  parameters,  *See, f o r example, Osgood  one 2-fold that  perturbation degenerate the  reduced  (1913), page 181.  \%\-0, general,  degenerate  parameter.  For  problem with two characteristic  50  function i s  (3.9) The  rUA) = Z -(X-K) a  coefficients  (trivially)  of t h i s reduced c h a r a c t e r i s t i c f u n c t i o n have  p e r t u r b a t i o n s e r i e s of the form  (3.8); the perturbed e i g e n v a l u e s , ±J A?-*- ^1 cannot  be  dealt  with  by  theory or i t s c h a r a c t e r i s t i c Chapter  given  in  eguation  , do not. Such cases  Bayleigh-Schroedinger p e r t u r b a t i o n function  analogue  developed  in  2. appendix A d i s c u s s e s the c o n d i t i o n s f o r a n a l y t i c i t y  o f eigenvalues.„  51 3. 2 The e i g e n v a l u e power sums as contour i n t e g r a l s . In  this  section  the e i g e n v a l u e  power  sums,  Sm, are  expressed i n terms o f contour i n t e g r a l s . , Since t h e c h a r a c t e r i s t i c f u n c t i o n has t h e f a c t o r i z a t i o n  i t follows  that  i-l  T h i s f u n c t i o n has a pole a t each eigenvalue, t o t h e degeneracy. I t f o l l o w s  S,*, =  (3.11)  with r e s i d u e  equal  that  <p 2T  ft  c/z; ,  r>l  where  is a  relevant  eigenvalues,  The in  contour i n t h e complex Z^-plane e n c l o s i n g the E  integrand  p  ,..,  E +^-t • , p  f  has no pole on the contour  for A  neighbourhood o f fAf = £>; i t i s an a n a l y t i c f u n c t i o n o f  some  A, i n t h i s neighbourhood./It f o l l o w s t h a t t h e s  m9  c/#  ,  and thus t h e  a r e a l s o a n a l y t i c f u n c t i o n s o f A, i n t h i s neighbourhood. Some  obtained  insight  into  the n a t u r e o f e q u a t i o n  by r e c o g n i z i n g t h e g u a n t i t y  resolvent,  G--(z-H)' , 1  t  as the t r a c e  For an Hermitian o p e r a t o r ,  i n the absence of e i g e n n i l p o t e n t s ,  (3.11) can be of t h e  or generally  52  G-  V- —  =  •  i  where f)  i s the  eigenvalue.,It  (3.12)  Eguation  Tr  projection follows  Cr  (3.11)  =  onto  the  subs pace  of  the  i-th  that  - J  V  4  =  can t h e r e f o r e  m  a l s o be w r i t t e n i n the o p e r a t o r  form  S  (3.13)  m  JJ-S-ZT  =  am  I f the t r a c e i n eguation eguation  (3.11)  is  G-  (3.13) i s  obtained,  f u n c t i o n can be found i n terms function.  This  .  JfA  taken  of  the  found  This i s discussed Equation  (full)  characteristic  a n a l y s i s i s c a r r i e d out i n s e c t i o n 3.3. I f the 1  is  integration,  and the reduced c h a r a c t e r i s t i c  t r a c e i s taken a f t e r i n t e g r a t i o n . the function  before  in  reduced  characteristic  terms o f the o p e r a t o r matrix elements.  in section  (3.13) could  3.4.,  have  been  obtained  alternatively,  s t a r t i n g from the r e l a t i o n .  *Se see that the s , and thus the reduced c h a r a c t e r i s t i c f u n c t i o n r , are d e f i n e d even i f H i s i n f i n i t e d i m e n s i o n a l , i n which case the c h a r a c t e r i s t i c f u n c t i o n f i t s e l f i s not d e f i n e d . m  53  r-%  between the p r o j e c t i o n onto the a s s o c i a t e d eigenspaces, and  the  r e s o l v e n t . Eguation (3.13) then f o l l o w s from 5^  =  H"P .  Tr  The p o l e s of the i n t e g r a n d s i n equations are  located  (3.11) and  the e i g e n v a l u e s of H(2). f o r [M&o,  at  (3.13)  under the  i n f l u e n c e o f t h e p e r t u r b a t i o n , these e i g e n v a l u e s are not known. To compute t h e s e i n t e g r a l s , the i n t e g r a n d s are IA|=o.  The  poles  l o c a t e d at the  of  each  unperturbed  term  in  expanded  about  the expansions are then  eigenvalues  of  which  are  assumed t o be known. The expansions are i n t e g r a t e d term by term and, from the r e s i d u e s at the poles i n s i d e the contour Taylor Finally,  series the  expansion coefficients  of  S,,,, of  the  about  lAI = o  reduced  f u n c t i o n can be found u s i n g Hewton's r e l a t i o n s .  #  ^<^»  the  i s obtained. characteristic  54 3.3  Coptour i n t e g r a l method: f-»r. The  first  few  t e r n s of the T a y l o r s e r i e s expansions f o r c h a r a c t e r i s t i c f u n c t i o n r~  the c o e f f i c i e n t s o f t h e reduced now  be  determined,  derivatives  of  particular,  from  the  eigenvalues  of  contour  finite  f u n c t i o n f.  In  functions  for  characteristic 3-fold,  #  and  dimensional  will  i n t e g r a l s , i n terms o f  characteristic  reduced 2-fold  a  the  (full)  the  non-degenerate,  7  nearly  degenerate  Hermitian  operator  are  constructed., Expansion of the i n t e g r a n d i n eguation  (3.11) about  |A(=o  yields  (3.14)  For s i m p l i c i t y , i n the f o l l o w i n g , i t w i l l be assumed t h a t there i s o n l y one eguation  perturbation  parameter*., Then  the  integrand  in  (3.14) becomes  L -P  L -P  -p  a  t  J  r  ~r  -P* J  J  where -j- and i t s d e r i v a t i v e s are e v a l u a t e d at \ % - ° . The  general  term  in  the  expansion  of the i n t e g r a n d i s  simply  C o n v e r s i o n o f a one-parameter expression i n t o the corresponding many-parameter e x p r e s s i o n i s s t r a i g h t f o r w a r d , though t e d i o u s (see Appendix B).  55  2"  £  41  I t i s of i n t e r e s t t o note t h a t  al.  differentiating i n t e g r a t i n g by  &  =  eguation  _3i_  since  -P*  (3.14, with  parts, y i e l d s  respect  to  A  and  then  56  3.3.1  Non-degenerate e i g e n v a l u e s . The  Taylor  series  for  a  non-degenerate e i g e n v a l u e has  a l r e a d y been g i v e n i n s e c t i o n 2.5.1. I t can a l s o be d e r i v e d the  present  general  method,  and  this  calculation  illustrates  the  procedure.,  L e t the unperturbed Without  loss  of  eigenvalue  generality,  E-p(o) set  be  E (o)^o l  p  non-degenerate. The  c h a r a c t e r i s t i c f u n c t i o n , with the perturbed eigenvalue i t s z e r o , can be f a c t o r e d  out  of  the  (full) \A\-0,  f u n c t i o n , f o r A i n some neighbourhood o f (3.16)  fe,^)  where r ( £ C \ ) , ^ = O, p  -  r C z ^ - g C ? ^ )  =  (2r+-c,Vg(z,:\),  ^£pCh)/X)^o.  the T a y l o r s e r i e s expansion,  W -o. To determine  for  by  (3.17)  reduced E  p0\)  as  characteristic i n the manner  The d e r i v a t i v e s appearing i n given i n (3.15), a r e a l l e v a l u a t e d  them, t h e r e l a t i o n s  f (z,o) =  ^  and (3.18) are  f n(o,o) = 2  n^n-.(o,oj  ;  n ^ o  used. Eguations  (3.14),  (3.15), and (3.17) g i v e  *This simply means that the z e r o z.p o f the reduced c h a r a c t e r i s t i c f u n c t i o n i s the g u a n t i t y E Q-\) — E > ( O ) , i . e . , the perturbation of E p ( A ) . , The d e r i v a t i v e o f £Cz,a) e v a l u a t e d a t £=e co) , where E > ( P ) * o , i s t h e same as t h e d e r i v a t i v e o f f {%i-E W, A) e v a l u a t e d a t z = o . Thus, i n any a p p l i c a t i o n the a x i s s h i f t z - * z-«-£>(o) need not be c a r r i e d out; the d e r i v a t i v e s a r e simply e v a l u a t e d a t z = E ( P ) , i n s t e a d o f a t z=o. P  p  e  P  57  (3.19)  where: the contour  encloses  E (p) F  but no other  l  unperturbed  two terms i n the i n t e g r a n d , ik and 3 9 do n o t have p o l e s i n s i d e t h e contour f£i and do not c o n t r i b u t e  e i g e n v a l u e . , The f i r s t  r r to  t h e x n t e g r a l . , The  one.  t h i r d term, - z *a , has a pole o f order r  The r e s i d u e g i v e s  (3.?o)  since result 2.5.1,  E c\) =  +  p  g(o,o) = ^ ( ° , ° ) , a s p e c i a l case o f r e l a t i o n i s i n agreement  with  This  t h e r e s u l t obtained i n s e c t i o n  and i s simply t h e u s u a l formula  for  an  d e r i v a t i v e . . The h i g h e r o r d e r terms a r e found The  (3.18).,  implicit  first  i n a s i m i l a r way.  r e s u l t i s g e n e r a l i z a b l e t o many p e r t u r b a t i o n parameters, as  described i n section 2.5.1.  58  3.3.2 2 - f o l d  degeneracy.  Suppose the unperturbed 2-fold  degenerate  pair.  Ep(p>= t .,(o)=o. F o r  Without  in  pt  eigenvalues  some  E ( o ) =E +,(o) e  loss  form  p  of  generality,  neighbourhood  of  a  set  [W-O, the  c h a r a c t e r i s t i c f u n c t i o n has t h e f a c t o r i z a t i o n  -P(Z,S\) =  (3.21)  where r ( C ( ^ ^ = o ;  r  <^  r-C^A)  A)*0, i =p, P^-I.  (3.22)  =  Now  z: gfco) a  while (3.23)  (o,o)  which r e p l a c e eguations Equations  where  the  unperturbed  n ( n - i ) g n - i :' > o / jn  2  (3.17) and (3.18).  (3. 14) , (3.15), and (3.37) g i v e  contour  encloses  e i g e n v a l u e s . The f i r s t  £p(o> = Ep-n(p), but no other term i n t h e i n t e g r a n d has  a  simple pole a t z=o with r e s i d u e  2 = o = |A| This  vanishes  since  "Ep<p)  simple  pole  by  eguation  (3.23),  i s 2 - f o l d degenerate.  i . e . , because { - C o o ) - o , f  The f i r s t  a t z - o , with r e s i d u e <*-f^> Z ^ * *  p o l e , with r e s i d u e  To f i r s t  order term has a  n  d  a  third  order, t h e r e f o r e .  order  a  59  s, =  0.25)  For  and  has  residues  The  integrand  the  the  gives  algebra  S,  i s o f order  reduced  A  correct  (3.23),  (3.27)  c,  =•  (3.28)  c  =  Table 2  E  p f  +  To therefore  { p+tft E  leading  eguation  Evaluation of the  yields  a r e found  &C??)»  i n a s i m i l a r way, T a b l e I I I (page  Finally,  Hevton's  C, and c , t h e c o e f f i c i e n t s A  of the  To l e a d i n g o r d e r t h e s e a r e  A ^ f y /  /  I V (page 6 6 ) g i v e s them c o r r e c t  c -=i Bp .Sp*.,  for  a  to  characteristic function.  a  A.  becomes more c o m p l i c a t e d . S  and  of  o r d e r t e r m s have z e r o r e s i d u e s ,  h i g h e r o r d e r t e r m s i n s, and s  f o r m u l a e c a n be u s e d t o g i v e  A ~ (  i n  i n (3.15), using eguation  though  s  +  no p o l e , t h e f i r s t  the leading contribution  ocr).  a*k\  S,, t h e z e r o t h o r d e r t e r m  (3.14)  63)  -3  i s , in  general,  a t each order i n  to simpler  The e x p r e s s i o n than  that  for  A.,  order, t h e reduced  characteristic  function i s  60  (3.29)  r(2E^)  =  Z  a  -+- Q'X  ,  where the d e r i v a t i v e s of -f- are  evaluated  a t z = o,  /AJ=o. The t  r o o t s o f t h i s eguation g i v e t h e p e r t u r b a t i o n o f t h e e i g e n v a l u e s correct  to  the r e s u l t However,  f i r s t o r d e r , and these r o o t s a r e i n agreement with obtained  from  the  e i g e n v a l u e s £p(fa)  section  i s made about t h e  and  p  of  the  2.5.2., 2 i n two  analyticity  Secondly,  E +,(2)-  obtained f o r the c o e f f i c i e n t s function  in  t h e c a l c u l a t i o n d i f f e r s from t h a t i n Chapter  ways. F i r s t l y , no assumption the  hierarchy  the  reduced  of  formulae  characteristic  do depend on t h e order a t which degeneracy i s l i f t e d .  Formulae f o r  the  eigenvalues  considered  separately  depend on the order a t which degeneracy i s l i f t e d ,  do not  as was shown  i n s e c t i o n s 2.5.2 and 2.5.3. The E C\)  perturbed,  initially  2 - f o l d degenerate,  and Ep+,C\) a r e t h e r o o t s o f  P  the  reduced  eigenvalues  characteristic  e g u a t i o n , i n t h i s case a g u a d r a t i c , given by  E ; ( A ) =  (3.30)  Hith  no  loss  - £ L± •3.  J (C) ~  ;  a  =  p  ^  p  +  ,  A  o f g e n e r a l i t y , t h e unperturbed  e i g e n v a l u e s have  been s e t e g u a l t o z e r o . . F o r the Hermitian o p e r a t o r s c o n s i d e r e d , the  perturbed  eigenvalues  d i s c r i m i n a n t , (c,f — 4 - c  J #  are  real.,  Consequently  the  must be even i n A, t o l e a d i n g o r d e r ,  i . e., (3.31)  If,  (C) -4-0,  t h e r e f o r e , the A  a  =  QCtf")  t  term o f the d i s c r i m i n a n t  1,0.,...  .  i s zero,  then  61 A  the  3  term  must  also  he  zero., In  d i s c r i m i n a n t i s given c o r r e c t t o GOT). be l i f t e d i n f i r s t  second  (3.32)  ((4a) - 4* fa*)  /  z  agreement with s e c t i o n 2 . 5 . 3 . , I f  =  ^,  0  =  O ,  t h i s c o n d i t i o n h o l d s , then,  order,  Epfr)  (3.33)  and  not t o  o r d e r , t h a t i s , from Table ¥,  a  to f i r s t  For degeneracy  order t h e c o n d i t i o n i s t h a t t h e d i s c r i m i n a n t  v a n i s h through  in  Table V (page 67) the  the  first  -  = Ep+i(?)  3 &3 I  4 * U=o=/a/  derivatives  of  the i n i t i a l l y  degenerate  e i g e n v a l u e s with r e s p e c t t o A e x i s t , even i f A r e p r e s e n t s than  one  perturbation  degenerate  eigenvalues  eigenvalue t o f i r s t  parameter.  In  a r e behaving  a  sense,  like  a  order and thus have f i r s t  vanishing  <A  3  2-fold  non-degenerate  order d e r i v a t i v e s  with r e s p e c t t o A..Furthermore, i f c o n d i t i o n (3.32) the  the  more  holds, then  term o f t h e d i s c r i m i n a n t reduces t o ( i n the  n o t a t i o n o f s e c t i o n 2.4) t  (3.34)  2  .  e  T h i s was a l r e a d y shown t o be i d e n t i c a l l y z e r o i n s e c t i o n In t h i s case the f o u r t h order term i n t h e d i s c r i m i n a n t  2.5.3*, reduces  to  (3.35)  /»(o)»a  —  m in  agreement  with eguation  3 5  m  '  (2.35), f o r t h e case o f degeneracy  62 not l i f t e d  in first  Suppose second  order  order. i n equation c,  i n A.,Clearly  (3.30) i s  required  correct  to  must be known c o r r e c t t o second  order.  However, t h e order t o which  known  depends  lifted  i n f i r s t order,- then t h e square r o o t o f t h e d i s c r i m i n a n t  on  the  discriminant  must  be  when degeneracy i s l i f t e d . I f degeneracy i s  i s of the form  Jtf^+T^+GCV-) and If  the d i s c r i m i n a n t  = ACe  a  +  Ae )' 5  /a  -+-  0(A ) S  must be known c o r r e c t t o t h i r d order i n A . ,  deqeneracy i s not l i f t e d  in first  order,  then < the  square  r o o t o f the d i s c r i m i n a n t i s of t h e form  and t h e d i s c r i m i n a n t must be kown c o r r e c t t o f o u r t h order i n A*.  63  Table I I I .  2-fold The  8  1  degeneracy:  e i g e n v a l u e power  sums.  —  "  " "(tf ^  8  f c k rW a < W V * 4 4 » W f a S  (V)'  • f 4M 4* - H M * 4* * « +  4J»J +...  • •-4*a4» 4* * 4M 4*;t 4* * 4M 4a* 4a + -• ?+  ,6  5 2  Table I I I , continued.  16  ...+  7  W^a)- W ^ f *  © f t - ) .  %  «fka)*£»J  65 Table I I I , c o n t i n a e d .  3!  L  4-  ' SI % ^(t^ +  ----»- ^ 4 *  .... J .  * * * * *  ~  ^ H a fta * 3  ^  M  %  ^ (^n^n +...  %  ^  ^  h  *  -&A*(£*f ] + •  4a] • +  2-fold  degeneracy;  The c o e f f i c i e n t s o f the reduced c h a r a c t e r i s t i c function. -s,  21  I  3>  I E  ;  67  2-fold  Table V  degeneracy:  The d i s c r i m i n a n t o f t h e r e d a c e d c h a r a c t e r i s t i c function, (c,) -4-c a  5  -  L  *  a  ;  =  .  fz'  1  •• 5£ r&j'. +  5  (fa)  (£>)  < ^ 4 j ] . . . ,  +  s  1  +  J  68 3.3.3 3 - f o l d degeneracy.„ The  method  degenerate  of  calculation  case,except  i s similar  to  the  2-fold  t h a t here  {3.36)  r(^,A) =  (3.37)  £(z,o) = H  -1-C3  2 -*3  3  f  3 ( ^ , 0 ) ,  and (3.38)  ^n(o,o)  =  n(n-')(o-3)  As b e f o r e , the unperturbed  c,,  c  j9  s  lt  ,  z  degenerate  egual t o z e r o . Table 71 g i v e s gives  g n-3(o,o)  n > o -  e i g e n v a l u e s have been s e t  3^ and  S, 3  and c^, c o r r e c t t o t h i r d order. ,  and  Table  VII  69  3-fold  t a b l e VI.  degeneracy:  Tbe e i g e n v a l u e  ... + -  f j ^ 4"** 2  * 5 4*34*"J +  7 ^ { | ( 4 * / ^ * f  +  3  +  '-  4* W £**-'4»fei4»"f  -  4 * a (4M)* + ' (4M)* 4 M ] + • s  [ » (4*>  power sums.-  4a»*f€*4*  *  L  l  | ? [ -  a  3  J  +  J  te i  4*(£M)  4 3 ' +  (f^fi  ^ [3 4M4a'* f e  2  5 V  4^4M+ 4*4 ] 3  s  +  70  Table VII.,  3-fold The  degeneracy:  c o e f f i c i e n t s of the reduced  function  43  •2-' /  c  5 31  1  J  43 j  characteristic  71 3.3.4  Near-degeneracy. A  set  of  non-degenerate  . - - Ep+c^-i Co)  E C°), P  i  t  eigenvalues the  said  s  to  are much c l o s e r  spectrum.  constructed  :  &  reduced  with  a  unperturbed  be  t o one  nearly  degenerate  if  the  another than t o the r e s t of  characteristic  nearly  eigenvalues  degenerate  eguation  can  be  s e t of e i g e n v a l u e s as  r o o t s , an advantage i s t h a t the convergence r a d i i of the T a y l o r series  expansions  characteristic  for  the  function  coefficients  will,  this  reduced  i n g e n e r a l , be l a r g e r  convergence r a d i i o f  the  nearly  eigenvalues  degenerate  of  Taylor  series  than  the  for  the  separately.  The  expansions  considered  reasons f o r t h i s improvement w i l l be d i s c u s s e d i n s e c t i o n For  simplicity,  eigenvalues  EpC'X)  consider  2-fold  nearly  3.6.  degenerate  E ,C\), with a f i n i t e i s o l a t i o n d i s t a n c e  and  p+  ,i.e., (3.39)  The  j E (o) p  reduced  -  £„  (o) J  p  characteristic  d  > O  f u n c t i o n , with  i t s z e r o s , can be f a c t o r e d out  of  the  .  EpC*) and  (full)  f u n c t i o n . In t h i s case, however, the unperturbed  Ep+,fa) as  characteristic characteristic  f u n c t i o n i s g i v e n by £ fe,o) = ( 2.~ E p C o ) ) 0 - Ep+, to) g  (3.40) Thus both  the z=  E (o) P  Since  terms and  feP)  .  i n the i n t e g r a n d (3.15) have p o l e s l o c a t e d at z=  E  +i(o).  P  ,  72  Ep-,(o)  F i g u r e 6.  The contours  tt,t.  fpti,  i . and  fp,a  in  the complex z - p l a n e .  (3.41)  where t h e paths o f i n t e g r a t i o n  a r e shown  i n Figure  6, the  c o e f f i c i e n t s o f t h e reduced c h a r a c t e r i s t i c f u n c t i o n rteA)  (3.42)  can  be  constructed  =  by  non-degenerate e i g e n v a l u e s .  2  a  + C , Z + C  treating This  A  EpC*)  and E ^ f a )  i s accomplished  a  s  u s i n g the  non-degenerate formulae g i v e n i n s e c t i o n 2.5.1 i n t h e symmetric function  expressions  (3.3).  The  advantage  i s that the  convergence r a d i i o f t h e T a y l o r s e r i e s f o r c, and c2t in  this  »ay  #  obtained  a r e g e n e r a l l y g r e a t e r than t h e convergence r a d i i  of t h e T a y l o r s e r i e s f o r each separate e i g e n v a l u e .  73  3 . 4 Contour i n t e g r a l method: H-frr. Eguation provides  a  with the t r a c e taken  (3.13),  simple  route  to  after  expressions  integration,  for  c h a r a c t e r i s t i c f u n c t i o n i n terms o f operator  the  matrix  reduced elements. ..  The concept o f an e f f e c t i v e operator H i s not used. Suppose (3.43)  H  =  H ° +  -XM  .  The r e s o l v e n t can be developed about A=o i n the f o l l o w i n g  Gr  (3.44)  =  —L_  =  — —  z-H°  where Gr i s the unperturbed  • ' G-„  =  „ 2 -  I t e r a t i o n on eguation  z.-Hf-'M  resolvent  a  (3.45)  !  1  z-H  nay:  H  g i v e s the expansion  (3.44)  oo  For  perturbation  expressed  in  eigenvalue. the  = 2^G°^VGB^ *  Gr  (3.46)  terms  Let  unperturbed  degenerate, g-dimensional  then  of of  the the  P (<=>) eigenvalue  subspace  r e s o l v e n t S , e v a l u a t e d at  eigenvalue,  reduced  Gr  resolvent  must be  0  for  that  be the p r o j e c t i o n onto the subs pace of  p  fp(o)  p-th  is of  E (o).  if  p  the J  f  p  Ef>C°)r can be  l  i  p  projection  E (p) .. . E ^_ p  E (o) onto  (o).  written  The  s  the  g-fold whole reduced  74  5  provided  =  (l-P Co)).  ! Ep(b)-H°  p  the i n v e r s e operator  • (/-PpCo)),  i s d e f i n e d a p p r o p r i a t e l y . I t has  the s p e c t r a l r e s o l u t i o n  k - t h power o f S has t h e e x p r e s s i o n  The  (3.48)  and  i t i s convenient  negative  to  define  the  *zeroth  power*  as t h e  o f t h e p r o j e c t i o n , i . e . , by S°  (3.49)  =  - P (o) . p  Then t h e expansion of  G-  (3.50)  =  0  about  z = EpO>) can be w r i t t e n  L^- p( )  2-^^  £  o)  O  ,  and t h i s y i e l d s  s \i...\fs"" n  (3.5D  Q  0  ( V ^ = J J - . )  p  ,  +  -  +  ^ - ^ . ^  f o r the t e r n s i n the expansion Without Z Gm  t  loss  appearing  of  (3.46).  g e n e r a l i t y , put E ( 6 ) = O. ,The r e s i d u e o f P  i n eguation  e n c l o s i n g ?=o, but no other terms  w  y » - ^ - ^  (3.13),  with  a  contour  ^p,<^  p o l e s o f G-„, w i l l come from a l l t h e  i n t h e expansion o f Q- which c o n t a i n  (z-E ,Co))~*' ~ '^ m  f  h  . The  75 e i g e n v a l u e power sums are found t o be  (3.52)  » = ZI  (3.53)  SA  (3.54)  5 f n  S  °° '  Tr  B  C<N)  V  = TV ^  = T r ^ A" M  ,  C n >  where  8  (3.52a)  (3.53a)  Here  C  ( n >  = + y  S 'V.. V 5 k  x  k  " '  S V...V5 " k n  k ,  C n >  indicates a  sum over  a l l integer  k;>o, with t h e  restriction k,-h ...-+• k ^ j n  =• j .  The compactness o f these e x p r e s s i o n s i s a l i t t l e example, (3.55)  8^  deceptive..For  i s made up o f t h r e e terms,  6<a>=  S°\IS°\/S + ,  S VS'V50+ S'VS0V5° , 0  but o n l y those terms with matched ends c o n t r i b u t e t o the t r a c e . If  5 '  appears  a t one end, and 5  J  matched i f e i t h e r k,-=ro=kj  S  0  a t the o t h e r , the ends are  or k;*o*Jcj.  V S ' V 5  0  For example, the term  76 i s t h e o n l y term i n (3.55)  3.4.1 I l l u s t r a t i o n : 2 - f o l d Suppose i.e.*  £,(o) and  degeneracy.2-fold  E (o) are a a  degenerate  pair,  E, (o) = Ej(o). .in t h i s case P,(o)  and  with matched ends.  | t,(o>X E,(o)) -+-• I E^(c?)X E^Co)  =  J  one f i n d s that  where  (3.56)  For  Qpp,"'  -<E (o)lVS VS V.-\/S V|£ ,(q)> .  k  ,  i  k  p  simplicity,  p  assume  that t h e degenerate block  has already  been d i a g o n a l i z e d , i . e . * t h a t (3.57) and  V .  =  pp  S" « < E ( o ) | V | E ( o ) > , pp  p  p  t h a t , with no l o s s o f g e n e r a l i t y , S, = E,0) •+•  E,(o)=0. one f i n d s t h a t  E fa) A  and  = A ((v f (v ) ) a  +  M  A ((v„-y XQ; -<?; ))^... ?  a  M  +  u  1  a  v In t h i s n o t a t i o n not  lifted  discriminant  in  i t i s easy t o see that i f degeneracy  first  order,  i . e . , i f V,, =.  ,  then  is the  77 is  to l e a d i n g o r d e r . One  I f degeneracy ( V M - V ^ )  This  is  not  vanish a n d  lifted  has  in  first  order,  the  factors  &  e x p r e s s i o n f o r the  d i s c r i m i n a n t corresponds to that  given  i n Table y i n terms of the c h a r a c t e r i s t i c f u n c t i o n . • I n summary, with the  trace  integration,  the  function  are  found i n terms of  contrast  to  integration, (full) section  the  coefficients  results  where the  characteristic 3.3.  i n eguation of  the  matrix  obtained by  coefficients function  (3.13) taken  reduced  after  characteristic  elements., T h i s  is  t a k i n g the  before  are  found  derivatives,  trace in  terms  in  of  as d i s c u s s e d i n  78  3,5 ftn a l g e b r a i c method: f - ^ r . The  expressions  characteristic  7  f o r the coefficients  function,  given  of  i n Tables  the  IV  and  l e n g t h y . They s e r e t e d i o u s t o compute v i a contour and  sere  formulae. has  i n fact  coefficients derivatives  the  advantages, of r , of  the  integration,  indirectly,  v i a Newton's  o f these c o e f f i c i e n t s , which  i s given  the  i n this  section.,  c,-, a r e not found  (full)  VII, are  characteristic  The  i n terms  function  -P,  of but  a r e found d i r e c t l y i n terms o f t h e c o e f f i c i e n t s o f -P,  d;. The r e s u l t i n g  e x p r e s s i o n s a r e more compact  i n terms o f d e r i v a t i v e s . f o r t h e case o f 2 - f o l d Consider  (3.58)  eigenvalues.  those  degeneracy. f u n c t i o n o f an n-dimensional  H = W(a),  -P(2,A) = H ° + E,(o)=  than  The method w i l l be e x p l a i n e d i n d e t a i l  the c h a r a c t e r i s t i c  Hermitian o p e r a t o r  Suppose  only  & more d i r e c t d e r i v a t i o n  certain  instead  obtained  reduced  are  EjtCo)  Then  the  a  (  z  n  - ' H - . . . +  a  2-fold  a  n  .  degenerate  (full) characteristic  pair  of  f u n c t i o n has the  factorization  with  for  A  i n some  eigenvalues shift  neighbourhood  £#(o)  and  o f |A/=0,  The  unperturbed  E ( ° ) a r e s e t e g u a l t o z e r o by an a x i s 3  so t h a t , to l e a d i n g order,,  79 &(*)  (3.59)  E;M-=  (3.60)  EjCA) = OO)  Leibnitz* functions  = , j  /,a .  theorem* f o r t h e d i f f e r e n t i a t i o n o f a product o f  yields  <3.61,  =  r  +_^L_  ^  e  +  (  'i'  ^ T  I*"*-  T h i s e x p r e s s i o n may be rearranged, with a l l f u n c t i o n s e v a l u a t e d  at 2--o, but I'M^o,  t o give  To o b t a i n e x p r e s s i o n s f o r the c o e f f i c i e n t s first  c,  and  c  a  i tis  noted f o r ?=o, ['Afeo, t h a t  and  -Pza = The r a t i o s  <2g -f- 2 - c # g z -+- c a g a . z  and f / £ a y i e l d  (3.63)  C, =  (3.64)  c  a  =  / |+ af  j  C,Cjg-+-  / J + <Vg  z  +  C  a  g  z  a )  These  eguations,  linked  i t e r a t i v e e g u a t i o n s f o r c, and  *See, f o r example,  and Segun  g  ^  ..  t o g e t h e r with eguation  ftbramowitz  _ Q, C j a  c^,  (3.62), form a s e t o f since  the leading  (1965), page 12.  80 order e s t i m a t e s a r e  -PCo.a)  (3.65)  = OCc ) = a  • 4^(o,a) = Die)  =  O(^)  Ofr)  , t  where n i s the degree o f -f i n 2. To  leading  order,  eguations  (3.63)  and  (3.64)  give  substituted  back  directly  (3.66)  c, =  and  (3.67)  c  a  =  3 £(A*>  These f i r s t e s t i m a t e s f o r c, and into  the  right  c  s i d e o f eguations  a  may  be  (3.63) and (3.64). .Eguation  (3.62) i s used t o express 2 - d e r i v a t i v e s o f g i n Cj,  and  the z - d e r i v a t i v e s  estimate f o r c, and c  A  terms  of  of f . I n t h i s way a second  c,,  order  can be obtained, and t h i s process may be  continued t o higher o r d e r s . An e x p r e s s i o n f o r c, or c  correct  t o some order i n A i n c l u d e s some higher order terms., This  iterative  scheme  yields  expressions  f o r the  c o e f f i c i e n t s o f t h e reduced c h a r a c t e r i s t i c e g u a t i o n i n terms o f Z - d e r i v a t i v e s o f -P e v a l u a t e d a t f = o , given  in  section  3.3,  but a t YX\-O  previously  M  obtained  i n t e g r a t i o n , can be found by expanding ^ t C ° , A )  The  results  by  contour  about  YAI-OZ  81  ^(°A)  (3.68)  =  +^2;  ^t(P,o)  (0 )  -»-...  }0  .  I f t h e r e i s o n l y one p e r t u r b a t i o n parameter, then  (3.69)  c, =  ZSkCQAKom^  =  n  2[£ (o,o)+A^M  +  z  -P^ ( 0 , 0  Ofr)y[^(o )^OC\y\ j0  ocx),  ^  -P a(o,o) z  and, s i m i l a r l y ,  (3.70)  3  =  +  -fy  a  f a  }  (0,0)  2  i n agreement with eguations  ew)  (3.27) and  (3.28) of s e c t i o n 3.3.2.  However, more compact e x p r e s s i o n s r e s u l t by n o t i n g that \ t C o  (3.7D  y  \ )  =  fc-'  a-n-t: (*)  •  To l e a d i n g order t h e r e s u l t i s  (3.72)  C,  -  and  (3.73)  In t h i s way the function,  .  =  the  of the ( f u l l ) of t h e reduced  coefficients  of  the  reduced  characteristic  c - , can be r e l a t e d d i r e c t l y to the c o e f f i c i e n t s t  characteristic characteristic  f u n c t i o n , the a,-. The function f o r  other  coefficients degeneracies  82 can be found i n a s i m i l a r nay, Those f o r non-degenerate, and  3-fold  correct  to  degenerate OC?),  eigenvalues  OCX),  and  In the c a l c u l a t i o n that  E,(X)  both  being z e r o . , In function  of  6(7?),  f o r 2-fold  and  E^CX)  fact,  even  a  T a b l e VIII are s t i l l  degeneracy  i t  if £(2^)  is  operator,  was  E,(o) a  assumed E*(o)  and  characteristic  with  eigenvalues  a p p l i c a b l e , ft u s e f u l example i s  HC\)  fl  respectively.  Puiseux s e r i e s , the formulae  the c h a r a c t e r i s t i c f u n c t i o n  (3.7 )  g i v e n i n Table V I I I ,  were of order \,  non-Hermitian  e x p r e s s i b l e i n convergent  are  2-fold  o f t h e non-Hermitian  given i n given  by  matrix  1 * X 1 1 o  =  OO  A  The c h a r a c t e r i s t i c e g u a t i o n i s (3.75)  -fe,A) =  with unperturbed  2  - 3 z  5  + 2 ( l - ^  a  O,  E,(o)=  E (o) A  = E C°) S  To c o n s t r u c t the reduced c h a r a c t e r i s t i c pair,  characteristic (3.77) with  f ^ o  roots  (3.76)  degenerate  ) +  a  E^  and  =  /.  eguation f o r the  E , the a x i s i s s h i f t e d ,  2-fold The  3  eguation becomes  £(x+/,*) =  X H- X 3  A  ^ X C ~ A - ^ > ~ ^  - O , -  (n=3)  a  (3.?8)  =  n  a  n-  a  -  1  , >  Q _, n  =  ' ,  =  ~(X+V) °-n-rr>  -  O  ,  From Table V I I I , the c o e f f i c i e n t s of the reduced function  f o r the p e r t u r b a t i o n of  and  E  s  are  m  >  3  .  characteristic  83  13.79)  c, = =  -IM*):-*-^*)*  +  A.]-*-  0(A  3 / a  )  o + ©Cx *), 5  In t h i s case t h e l e a d i n g order grouping i n t h e t a b l e i s (3.80)  c,  ^  C  =  2  Gfr«)\+.QMl+e(7J '*) +... s  :  !  OCX)  \+O0?«)i+OCtf)  : +  and (3.81)  E,-ft) =  X  ± A** +- O f t * ) 3  ^,  84 Table VIII.  A l g e b r a i c r e l a t i o n s between the c o e f f i c i e n t s of t h e reduced and f u l l c h a r a c t e r i s t i c functions.,  Terms  which  are  the  sase  leading  order  are  grouped  together. Non-degeneracy.  c,  =  Qn  ;  (ci )  +  a -s  a  n  n  +a(Qn) (an- f „ ( a f n  a  (a -.)  5  C,=  5  J  +  ...  degeneracy.  Qn-i  -f (Qn-j)*<Xn-3- <*n Qn-3 ' - f e i f S e d t  (a„- F;  (a„_ )^  (<w^  3  a  -+S(Qn-.r(ar,-s) _ 5 a o ^ - l f a n - S (<*n-a)7 (an-a)^ 5  -  n  (a„_,)^  n  2-fold  g_  3  J O  a  +  +^(Q'Hf  (a„-a)  13 Qn lan-«f Q n - S q n - ^ (Q f l - a )5  n l " - ' f ( Q / 7 - s ) + ( Q n f n - 5 - 5 ( a ) a n - 3 Q -^ ^ (Q _ )fc (a -a) (Qn-a) " Q  5  Q  a  n  3  n  2  ( Q r £ i 9 ^ f  n  ! +  OCX 5).  n  4  +.-• 5  85 Table VIII. continued.  :  3 - f o l d degeneracy.-  I  "  (QrJ_3)S  9^zL  c  3  = '  Bo  (CX ^y n  «  Qn-» _  +  "  (9 fa ) . 3  ' + <9£\ 3 )  86 3.6 Convergence  radii..  An advantage o f s e t t i n g up guantum mechanical p e r t u r b a t i o n theory  i n terms  of  r e s o l v e n t s and contour i n t e g r a l s i s t h a t  e s t i m a t e s of convergence r a d i i can be e a s i l y made, as shown Kato  by  (1949). Central  t o t h e c a l c u l a t i o n s i n t h i s c h a p t e r have been the  contour i n t e g r a l formulae.  (3.11)  -  Sm  -J—$  2L  M  &  ,  and, i n the o p e r a t o r form,  (3.13)  S  Jh.  =  m  AzTGr  d z .  r-l  These i n t e g r a l s a r e independent o f t h e depending  tp,c^  t  integrand that  only f£  on  the residues  e n c l o s e s . , The  ><t  shape  of  of  the  t h e poles o f t h e  analyticity  of  depends on t h e a n a l y t i c i t y o f t h e i n t e g r a n d , f o r z on for  A  i n some neighbourhood o f  f o r t h e power estimated  series  expansion  s^-s^fr) ^<^, and  MI=o. The convergence r a d i u s of S  m  about  (31 » o  can  be  from t h e convergence r a d i u s f o r the expansion o f t h e  i n t e g r a n d , with z on JejLa., T h i s radius  contour  depends  on how  estimate  of  the  convergence  tp*^ i s drawn. The best l o v e r bound f o r  the convergence r a d i u s , i . e . , t h e l a r g e s t lower bound, r e g u i r e s the 'best* contour  Vp ^* )C  Suppose H = HV3\/. Prom e g u a t i o n t  eguation  (3.13) can be expanded  about  (3.46) IAI= O  the  integrand  as f o l l o w s .  in  87  (3.82)  oo  =  ^(WG-S  .  n-o  T h i s s e r i e s converges f o r  (3.83)  HAVGolUl.  By Schwarz^s i n e q u a l i t y ,  IJAVG-oll £ ftlJIVJI.IIGbll , and consequently  a s u f f i c i e n t c o n d i t i o n f o r convergence i s  (A/J/V/MlG-oll C I .  (3.84) If  t h e sup norm i s used, then t h e s p e c t r a l r e s o l u t i o n o f the  unperturbed r e s o l v e n t i m p l i e s t h a t (3.85)  \\Qx \\ = [A(z)]~"' , 0  where A(z) i s t h e d i s t a n c e o f z from  t h e nearest  eigenvalue.  (3.84) then g i v e s  Rearrangement o f equation  (3.86)  The  /A/<  best  bound  given  ! //vii .//G-Jj by  unperturbed  »  //v«  t h i s equation  i s obtained  when the  keeps Ate) as l a r g e as p o s s i b l e f o r 2. on f p ^ . , T h e  contour  s m a l l e s t value t h a t A(z) a t t a i n s f o r z. on t^^ e s t a b l i s h e s the 0  bound  on | A/ . , L e t  eigenvalue unperturbed isolation be  E,(%)  and centered  eigenvalue distance  be  a  circle on E.^o).  t o E,(p) i s  i s d,  )SL  enclosinq  ^Just one  Suppose the nearest  E^(o)  and that  their  . L e t the r a d i u s o f the c i r c l e f"^,  The v a l u e o f A(z} i s d /a. l>3  everywhere on t h i s  circle.  T h i s i s the l a r g e s t value A(z) can take f o r z on the r e a l a x i s , between  E,(p\ and EjpS (see F i g u r e 7). Thus f^,  is a  •best*  contour, fi lower bound f o r t h e convergence r a d i u s o f t h e power  88  s e r i e s f o r t h e perturbed  proportional  are  to  the  eigenvalue  t  i s o l a t i o n d i s t a n c e . The same i s t r u e f o r  If  and E2C\)  a r e n e a r l y degenerate e i g e n v a l u e s ,  and  treated  together,  a reduced c h a r a c t e r i s t i c f u n c t i o n  with  EC\)  roots  and  E^(A)  can  contour w i l l be some tf^ 7 ) * , The  smallest  lower  bound  a j 3  be  constucted. , The  integration  (not n e c e s s a r i l y a c i r c l e , see F i g u r e  on d  (3.87) A  E,(A) i s thus d,^ / \ \  / a  is ( >  .  f o r the convergence r a d i i o f t h e expansions o f  t h e c o e f f i c i e n t s o f t h e reduced c h a r a c t e r i s t i c equation g r e a t e r than t h a t f o r the expansions o f the eigenvalues Kato  when c o n s i d e r e d  nearly  i s thus  degenerate  separately.  (1949) p o i n t s out t h a t i n g e n e r a l these lower bounds  89  cannot  be improved on..This i s shown by  the  following  simple  example:  H°  (3.88)  =  — I  o  0  /  o  /  1  o  H e r e t h e e i g e n v a l u e s o f H = H*vaV  whose power s e r i e s ,  are  convergent  are  ±[\ + i? + -l ,  f o r |AJ  Eguation  (3.86)  , with  I and  IIVII=/, g i v e s the same bound.. I t i s not p o s s i b l e , i n as  that  general,  bound,  such  (3.11),  s i n c e A i s u s u a l l y embedded  to  given by eguation  characteristic function  obtain (3.86),  an  explicit  from  eguation  i n a complicated way i n the  . However, an i m p l i c i t  bound  obtained by w r i t i n g  f  (3.89)  and expanding  (3.90)  GE.,*) =  -PC^o) - A  v f e / \ )  the i n t e g r a n d o f e g u a t i o n  Z." I *  •F  T h i s s e r i e s converges f o r  (3.91)  which y i e l d s the i m p l i c i t  bound  (3.11)  according t o  is  90  W  (3.92)  <  j  fU,o>  For the example o f equation  (3.88), the c h a r a c t e r i s t i c f u n c t i o n  is  fCz,*)  (3.93)  The  best  circular  e i g e n v a l u e "/  i m p l i c i t bound  l-A*.  contour  l~^  a  - | + c'  ,  e  around  tl  the  unperturbed  o ± 9 ^  2-n.  (3.68) becomes  m < J^j  (3.94)  from  H -  i s the c i r c l e o f u n i t r a d i u s , g i v e n by  z = The  -  =  ,  o±e+Qrr  j  which, as b e f o r e , i t i s found t h a t  m <i . I n g e n e r a l , however, equation  (3.92) i s  not  particularly  u s e f u l , although i t does, i n p r i n c i p l e , provide a means f o r the calculation  of  a  bound. To a c t u a l l y use equation  minimum value t h a t |^^'^~ | tp^,  attains,  for  z  on  (3.92), the  some  contour  i s r e q u i r e d f o r a range of v a l u e s of 3. I f  I V(Z,^)( min. f o r a l l \ such t h a t o^lJ\)<6  on ^ )  ^  where B i s r e a l and p o s i t i v e ,  and  91  / V ( Z , C \ ) I min.  f o r \%\ \ It  S,  then 6 i s the bound on  is  worth  integrand  of  contour,  ^  emphasizing  (3.11) or  (3.13)  (z  r£ )  on  j%  y  YXl • .•  that  if  an  converges  expansion of the  for  a l l z.  E (\)  say, then t h i s i m p l i e s t h a t  on  some  does indeed l i e  P  w i t h i n the contour. Consider  EM  (3.95)  =  P  &  *  Y  t h  v  p,t  The p-th term i n the summation can be expanded about  =  '  (E (A)~<z (o)\ P  H=  E (p) ^ p  n  P  which expansion converges i f (3.96) The E CA) P  )  E (A)-E (O)1 P  \^-E Co)\  <  P  p  above i n e q u a l i t y i m p l i e s t h a t Z i s f u r t h e r from i s ; thus  E ( ) P  A  l i e s within  be  illustrated  by  EpO>)  than  .  An a l t e r n a t i v e p e r s p e c t i v e on the can  .  returning  convergence  properties  t o the example  (3.88), f o r  which the e i g e n v a l u e s were  In the complex ^ - p l a n e , E, and  E  z  become e g u a l when A=i/. These  s i n g u l a r p o i n t s , about which E, and E  u  are not a n a l y t i c  in ^ ,  92 lie  on the c i r c l e  of convergence  Thns t h e expansions o f E, and F /A/ < \±t) - I . I t i s f o r 7i = ± \ separate E, and E  In g e n e r a l the o n l y Thus,  complex A-plane From  larger  f o r the  function  this  than  degeneracies i n associated  with  that  \AI-0 a r e convergent f o r  a  contour  non-analytic  imply  point  exact  of  coefficients for the  complex  the  but do l i m i t  points  knowledge  view, of  individual  can  no  longer  occur  plane  of  reduced  convergence r a d i i are  characteristic  eigenvalues, between  such  p o i n t s i n the  t h e convergence a  at  the  because  the  eigenvalues  same reduced c h a r a c t e r i s t i c f u n c t i o n are  not r e l e v a n t t o t h e convergence latter,  i n the 3 - p l a n e * .  knowledge o f t h e degeneracy  would  radii.  about  A  s i n c e they have become e g u a l * .  3t  degeneracies.  a  o f E, and E  of  the convergence  the  coefficients  of the  of the eigenvalues.  *See, f o r example, Harkness (1898), page 178. However, f o r a=/ t h e e i g e n v a l u e e, i s s t i l l the only e i g e n v a l u e w i t h i n t h e contour r? , i n the complex z-plane. Thus, while the convergence o f an expansion o f £, i m p l i e s t h a t e, l i e s w i t h i n r* , , the converse i s not t r u e . 1  t  h  93 3.7  Summary. A  reduced c h a r a c t e r i s t i c f u n c t i o n  t a k i n g the t r a c e coefficients  of  i n eguation the  (3.13)  o f the  taken with r e s p e c t t o z  and  eguation  (3.13)  been i n t r o d u c e d .  before  integration,  reduced c h a r a c t e r i s t i c f u n c t i o n  i n t e r n s of d e r i v a t i v e s  in  has  after  A . In c o n t r a s t , by t a k i n g the integration,  i s developed which r e l a t e s the reduced c h a r a c t e r i s t i c f u n c t i o n  estimated functions.  in  function. terms  of  y  found  function,  perturbed  coefficients  convergence  operators  and  are  method of  d i r e c t l y t o those of the  Finally, both  trace  the c o e f f i c i e n t s  found i n terms of operator matrix elements. An a l g e b r a i c  characteristic  the  are  (full) characteristic  By  radii  the  (full) are  characteristic  94 CHAPTER  TWO  4.,  ILLUSTRATIVE APPIICATIOSS.  Introduction., In  this  chapter : t h e  a p p l i e d t o two  results  some  way,  Chapters  i l l u s t r a t i v e examples. In t h e f i r s t  even-membered r i n g of n carbon in  of  example  i s c o n s i d e r e d i n the context of Hiickel  orbitals  are  found  to  second  the  second  perturbation from  the  example  formulae  the  usual  molecular perturbed  order  and  g e n e r a l r e s u l t s are a p p l i e d t o the benzene r i n g , where In the  these  n—6.  Bayleigh-Schroedinger  i n terms of matrix elements are obtained  characteristic  function.  The  operator  c h a r a c t e r i s t i c f u n c t i o n f o r m u l a t i o n s of p e r t u r b a t i o n theory be  related,  f o r m a l l y , through  was  can  distinction  the order i n which t r a c e and i n t e g r a l o p e r a t i o n s are  performed.. In considered  and  the contour i n t e g r a l e x p r e s s i o n  f o r the e i g e n v a l u e power sums (eguation 3.13). The there  an  atoms, one of which i s perturbed  o r b i t a l theory. E x p r e s s i o n s f o r the energies of Tf-molecular  2 and 3 a r e  section  4.2,  however,  interrelation  is  from a d i f f e r e n t p o i n t o f view. I t i s shown how  the  well-known p e r t u r b a t i o n formulae  i n terms  the  of  matrix  elements  can be d e r i v e d from the c h a r a c t e r i s t i c f u n c t i o n e x p r e s s i o n s . An  application  of  quite  d i f f e r e n t c h a r a c t e r , namely t o  s p i n systems, i s c o n s i d e r e d i n the f o l l o w i n g c h a p t e r .  95 4. 1 A p e r t u r b a t i o n c a l c u l a t i o n f o r even-membered r i n g s of  carbon  atoms.  T  The Hiickel matrix f o r  the  7Y-molecular  orbitals  perturbed n-atom r i n g , i n u n i t s o f the i n t e r a c t i o n relative is  of  the  parameter/3,  t o the carbon atom parameter <* as the z e r o of energy,  the nxn  matrix r  Mo.... o i 1 0 1 o . O 1 O 1 O - -  o  (4.1)  o  o  O  1 o  1 O  The c h a r a c t e r i s t i c f u n c t i o n can be used by Coulson  -Pfe,*) = M„  n-fold  ring  is  written,  in  the  notation  (1938), as  (4.2) where  1  the  M -AP _, , n  n  characteristic  f u n c t i o n f o r an  unperturbed  (4.3) and where = 2 . co.5 &  (4.4) while  P _,  is  0  the  t  characteristic  function  for  an  (n-/)-dimensional l i n e a r c h a i n : (4.5)  f^-i  From eguation (  4.6)  E ( o ) = Z c o s * * ? r m  ,  m  non-^degenerate  (4.7)  (4.8)  5inn&/sm9  .  (4.3) t h e unperturbed e i g e n v a l u e s are seen to be  There are two  and ( j - i )  =  E„(o) =  - o ,  m ,  g .  unperturbed e i g e n v a l u e s ,  ,  E  n/a  (o) =  p a i r s o f 2 - f o l d degenerate unperturbed e i g e n v a l u e s , E  w  C o )  =  E_„(o)  y  m~+±,:  96  The of  W-derivatives  p e r t u r b a t i o n c a l c u l a t i o n r e g u i r e s z.- and  -P(^,^).,  according  t o eguation  (4.4)  the ^ . - d e r i v a t i v e s are  r e l a t e d t o the ©-derivatives by  9.  0.9)  The  3£ first  few  fa  (4.12)  t  a  at  \-O  cosn^l - - n L  ,  evaluated  =  F Sinne-cos0_  n  _  t  are  eigenvalues.r  non-degenerate Table  b  ~  °  >  I (or eguation  the non-degenerate e i g e n v a l u e s  Jf. ) ( o  n  n  The  a  ,  . fisinngr|- *]  a  "^ A  From T a b l e I  29  n  =  -IU = 3co5£.P_  (4.14)  Xc,\ &  PCz,^) ,  (4.13)  From  9_  -I  follows:  M.,o,  (4.11)  _  29  d e r i v a t i v e s of  a c c o r d i n g l y as  (  3_  de d£  =  E {X)  the f i r s t and  En (x) /2  d e r i v a t i v e s of are  =  _  =  — n  }  2.21)  the non-degenerate formulae f o r  (or eguation  the second d e r i v a t i v e s are  &  ,  2 .  2.20) 0  =  I  i =  o  Q . 'a.  97  which,  i n this  case,  must  be  evaluated  by  a p p r o p r i a t e l i m i t s . F o r E (A) , t h e l i m i t 8-+0 4^  =  n  (4.17)  4*  =  n (n -±)/6  and  •z  thus  (4.18)  &>/<>) =  S i m i l a r l y , the l i m i t  (4.19)  In  da  a  a  a  J_  ,  (o -l) a  yields  dl§n (o)= a /a  summary,  the  yields  0  (4.16)  taking  ~_L 6n  (n -/) a  a  t h e two non-degenerate e i g e n v a l u e s have the  p e r t u r b a t i o n expansions  (4.20)  E; (X)  = E,(o)+ % 4E>(o) - £ d  A  =  ±2.  -h A n  ± X(n*-1) / i n  21  dV,(o) + ... d3  a  + ,.. 1  , > = O (+), n (_) • . a  98  The  2 - f o l d degenerate To f i n d  degenerate Chapter  e x p r e s s i o n s f o r the eigenvalues,  3 v i l l be  treating  eigenvalues.  each  used.  but  2. . For  s e p a r a t e l y , a reduced degenerate Chapter  the  calculation  separately,  instead  initially  comparison, of  is  2 and  done  the  considering  calculation the  is  constructed,  is  eigenvalues  c h a r a c t e r i s t i c f u n c t i o n f o r each p a i r  eigenvalues  by  u s i n g the p e r t u r b a t i o n  of  using the r e s u l t s of  3.,  From T a b l e I I (or e g u a t i o n 2.30), the f i r s t the 2 - f o l d degenerate  jE»(o)  (4.21)  $^-O  Since  2-fold  approaches of both Chapter  First  eigenvalue  h i e r a r c h y o f Chapter repeated,  the  perturbed,  e i g e n v a l u e s are  =  i&n  (4.22)  Jp&F-t&frl  ±  t h e s e reduce  t  d e r i v a t i v e s of  =  ,m = t±, ,±{%-i) .  to  O  ,  ( o )  n One  e i g e n v a l u e of each p a i r i s unchanged i n f i r s t order., These  unchanged  eigenvalues  negative i n t e g e r s has  a  first  order  vill  be l a b e l e d i n the f o l l o w i n g by  , -(§-/•). The other member of shift  that  is  twice  the  each  The  pair  s h i f t of the  non-degenerate e i g e n v a l u e s . These s h i f t e d e i g e n v a l u e s l a b e l e d by t h e p o s i t i v e i n t e g e r s  the  will  be  *•>,.  second d e r i v a t i v e s , from T a b l e I I  (or eguation 2.31),  99 are given by  2«5n(oJ A^O  Substitution equation,  of  the f i r s t  together  derivatives  (4.22)  with the e x p r e s s i o n s  into  this  (4.11)-(4.13) f o r the  partial derivatives, yields  (4.24)  ..-(g-l)  O  4lEm(o)^  and  —*(o) da  <4.25)  2  = A_ 4 3n* ^  X  a  cos0 n  A c c o r d i n g l y , t o second o r d e r , t h e i n i t i a l l y  2-fold  degenerate  e i g e n v a l u e s become  (1.26)  E C\)  <».27>  r „ m  m  = leasing!  + 6CV) , m = -/, ..,-(§->)  aasaar + a + 2  t  0  a l l  second  and  f^^b  =. O  ,  5  sin ' +1,...,+(%-!)  higher ^ - d e r i v a t i v e s o f -f v a n i s h ,  i.e., (4.28)  Oft ),.  2  m =  Since  -  j  b > X,  100 it  can be seen, by i n s p e c t i o n of  the  perturbation  that the e i g e n v a l u e s which were unchanged i n f i r s t  hierarchy,  order a r e i n  f a c t unchanged i n a l l higher o r d e r s , i . e . ,  (a.29)  E  m(n)  =.  O  ,  n > \ ,  Reduced c h a r a c t e r i s t i c equation Instead  m=-*j--,-(%-l)  .  calculation.-  o f t r e a t i n g the eigenvalues s e p a r a t e l y , a reduced  c h a r a c t e r i s t i c eguation, H +  (4.30) for  C , Z +-C  a  the i n i t i a l l y  2 - f o l d degenerate  c o n s t r u c t e d . The non-zero t e r n s coefficients  c, and  l  (4.32)  where  C, =  all  O  =  a  p a i r s of e i g e n v a l u e s can be  in  the  expressions  c i n T a b l e I? a r e a  -2U  t J a  -f-  5  4*  (4*/  4 ^ J  4*  BW),  derivatives  are  evaluated  at  the  e i g e n v a l u e , i . e . , f o r 6= J2Jy, m=±l,..,±|0_i), a=Q. On a  the  partial  derivatives  (4.11)-(4.13),  c h a r a c t e r i s t i c e q u a t i o n i s found, t o second  The  f o r ; the  r o o t s of t h i s e q u a t i o n .  unperturbed substituting  the  order, t o be  reduced  101  z.  (4.34)  =  O  = 2  «2n  y i e l d the p e r t u r b a t i o n s  _  of  2  a  c o s  the  initially  2-fold  degenerate  s  e i g e n v a l u e s , i n agreement with the expressions (4.26) and p r e v i o u s l y o b t a i n e d f o r the e i g e n v a l u e s  (4.27)  individually.,  A p p l i c a t i o n t o benzene.  ;  For  the  benzene  g i v e the unperturbed  r i n g , n=6 and eguations  eigenvalues  E (o)  (4.35)  0  E Eguations  (4.20),  3  = X  ,  +  = Co) - -X  (4.26), and  E„W-^  (4.36)  -  , . (4.27) become  I  -i + » - X Eguation radii  of  ,  ^ ecx*) ,  (3.86) g i v e s a lower bound  these  (4.6) and (4.7)  perturbation  for  the  convergence  s e r i e s . In t h i s casel|VII = l  {see  eguation 4.1), the i s o l a t i o n d i s t a n c e of a l l the e i g e n v a l u e s i s 1, and the bound becomes (4.37)  W  <  VX .  102 A l g e b r a i c method. •• The expansions (4.36) can a l s o be obtained algebraic  relations  given  in  by  Table V I I I . The  using  characteristic  f u n c t i o n f o r t h e fT-molecular o r b i t a l s o f benzene, i n which carbon  the  one  atom i s perturbed, i s  # 2 , A ) = H -2 A - b^ 5  fe  (4.38)  •+•  4-z- A + 9z*~ 3 z 3 - 4- . 3  On s h i f t i n g the a x i s by p u t t i n g 2-X+l, t h i s becomes (4.39)  £(x+l,3)*  i.e.,  E±i(°)  shows  how  X +X M)^^-^)^ (- -- ^ Ha+aA)+4.xA, 6  has the  5  been  3  shifted  coefficients  4  t  +xa  t o the value zero. Table VIII  of  the  reduced  characteristic  eguation z  (4.40) f o r the i n i t i a l l y from  2  +  C,Z + C  a  =  0,  2 - f o l d degenerate p a i r  E±,io),  the c o e f f i c i e n t s of powers o f X i n e g u a t i o n  from the c o e f f i c i e n t s  T a b l e VIII t h e r e f o r e g i v e s  (4.42)  c, ^  =  and  can be computed  : +  (Q f 5  -2  - 2f  3  54-  a  3 _  a±*z  4- <9ft ) 3  (9(?f)  (4.39), i . e . ,  103  (4. u )  Co  3  =-  i-f-  ^  s  °-3  a  -  +  eCx)  Thus, t o second order the reduced c h a r a c t e r i s t i c the p e r t u r b a t i o n o f  (4.44)  which  eguation  for  is  - | -2?  Z - -i2  )  =  0,  yields  (4.45)  correct  =  */  t o second order, and i n agreement with the e x p r e s s i o n s  (4.36) f o r  E-,(A) and £+,fa) o b t a i n e d e a r l i e r .  104 4.2  Derivation of  the  Bayleigfa-Schroedinqer  formulae  the  characteristic  from  Starting well-known can  be  perturbation  except  correctly  ordered,  question  clear,  for  mechanics The  that  of  the  •ordering*  example,  in  If  is  E,(o)  perturbed  E.C0  (<1.»7)  =  the  energy  If  c,(o)  s t i l l usual  be  i n the the  used  the  H° +  eigenvalue  E,(o)  H- ^V,,  the  in  series  this of  degenerate case  usual  elements  Bayleigh-Schroedinger  obtained to  matrix  the  way  Kato is  discussions  (1966).  not in  are  made  quantum  matrix expressions  +  eigenvalue  will of  first  H° a n d  be if  A V , E,fa)  X  is  given  by  V  +...  ^V.SVSI  V,rV a  _v„V-  r  -r -  d e n o m i n a t o r s E; = i(.°X), E  i  s  o^-fold  provided  prescription is  redefine  series  of  usual  a non-degenerate  ..^\Y  (In  the  expressions,  textbooks.,  Hfo) =  the  are  function  i n terms  corresponding  (4.H6) then  formulae  Bayleigh-Schroedinger  recalled.  function.  characteristic  o b t a i n e d . , These  formulae,  The  from t h e  perturbation  zeroth  degenerate, degeneracy to  order  say,  is  diagonalize energies  then  lifted the  this  formula  can  in first  order.  The  degenerate  block,  and  appearing  in  the  denominator  105  as . new  patting  where E,(o), Op  are the i n i t i a l l y q^-fold degenerate  i-\ .. c^ t  }  t o t h i r d order any term i n a summation with r or s e g u a l t o w i l l be m u l t i p l i e d by the matrix element V  the  degenerate  eguation  rS  , r,s^.<^.  eguivalent  to  restricting  the  summations  (4.47) so as to exclude the i n d i c e s  f o u r t h order terms i n (4.47) are o n l y degenerate  since  b l o c k has been d i a g o n a l i z e d , these elements are  zero. This i s  case. , They  do  terms i n e g u a t i o n  in  However, the  apparently  so  in  this  i n c l u d e summation i n d i c e s  and in  A.  (4.47) must a c c o r d i n g l y be regrouped,  to  these give r i s e t o terms which are a c t u a l l y t h i r d order The  set.  obtain a correct A-ordering. For a 2 - f o l d degenerate in f i r s t  order  (4.48)  EXA)  (\f„  The  perturbed  e x p r e s s i o n . The (4.48)  comes  "-Vr,  Vr.  E, = E;(o).)  eigenvalue  EaQX)  l a s t summation  of  the  V  V.rX-sVsi _V„V  denominators,  from  lifted  X  = E,(o) H- ^ N / „ * X Y  the energy  is  Vz ) , t h i s p r e s c r i p t i o n l e a d s t o  f l y  (In  l e v e l , when degeneracy  first  is the  given X  summation  by  term of  the  a  similar  in  eguation  A  term i n  106  equation  (4.47). T h i s r e o r d e r i n g i s necessary s i n c e the tero  V,r V  r s  VstVfc.  can have s^z without X * o r N/j, appearing i n the numerator. I n t h i s case the denominator c o n t a i n s  E.(O)+ *v„  ~E (O)-  - A y ,  A  =  a  Oi  (  V  ,  r  V  „  )  and the numerator i s  V -V V l f  Thus  part  r i  a f c  V ,  o f the f o u r t h order term i n equation  t h i r d order i n equation  approach,  (4.47) becomes  (4.48).  These formulae can be function  .  f c  obtained  given  from  an e x p l i c i t  the c h a r a c t e r i s t i c expression  for  c h a r a c t e r i s t i c f u n c t i o n i n terms o f matrix elements.. I n of  the  terms  the matrix r e p r e s e n t a t i o n o f \J, i n the b a s i s i n which H ° i s  d i a g o n a l , the c h a r a c t e r i s t i c  JTC^A) =  function,  d c l - J z -  w°-^v|,  can be w r i t t e n jH-E,(o)-}V  ()  (4.49)  W,  a  . . . .  W,  n  ^ C z , ^ )=  d c t  Z-E„(O)-}V,  nn  The determinant may be expanded t o g i v e  107  -P(Z,A) =  (4.50)  rite-EjCo)-^,)  - t  y  V/ V 7T sr  r5  ^sr^tu^t  (E_E -(b)-^V )-f-.. (j  (  TY(*E-E;(o)-M )+.. {i  ...+ <9(* ). 5  A l l i n d i c e s i n these summations counting only  o c c u r s . / For example,  once; ^,V The  (a  are  distinct  and  no  i n t h e A* term V^V^,  double appears  i s not counted, as w e l l .  terms w r i t t e n  out e x p l i c i t l y a r e e v i d e n t l y  sufficient  t o s p e c i f y any mixed d e r i v a t i v e o f t h e c h a r a c t e r i s t i c f u n c t i o n , e v a l u a t e d a t an unperturbed e i g e n v a l u e and a t A-O, up t o f o u r t h order  i n ^ , i . e . , a l l derivatives  p a r t i c u l a r the f i r s t following:  and  second  ^1-^(^(0)^0) order  with /S±4. I n  derivatives  a r e the  108  (4.51)  4^ ^=  (4.52)  £(E (0\0) = - V  (4.53)  (0)  TT (E (o)-E,(o>)  ,  p  TT i^p(P)-E (o))  ,  t  P  £aM>/0  (<v(<>>-E;(°->),  T\( M- ^)-Y E  E  (4.54)  ^fecov) ^  (4.55)  £»M>V>> =2V p J"^]T(M ^ ;( ))-^V p ; X p 7T(F»-E,fe)).  -  V  ^  0  V, TT(^o)-£ £  /(0  3^  0  Bon-degenerate c a s e s . If  E,(o) i s non-degenerate, t h e p e r t u r b a t i o n h i e r a r c h y i n  Table I leads t o the expressions first  and  second  (2.20)  d e r i v a t i v e s . , They,  ( 4 . 5 1 ) - (4.55) t g i v e  (4.56)  £L§»(oV  =  3=o  and  e.(o)-Er(o)  and  (2.21)  f o r the  along  with  eguations  109 2 - f o l d degenerate In  cases.  s  t h e 2 - f o l d degenerate  case. Table I I (or e g u a t i o n  g i v e s the f o l l o w i n g e x p r e s s i o n f o r the f i r s t  derivative:  On s u b s t i t u t i o n o f e x p r e s s i o n s (4.53)-(4.55)  f o r the  2.30)  partial  d e r i v a t i v e s , t h i s becomes (4.58)  =  dE:<(o)  V»+^a  ^A  ±  J  (Vy-Mttf-t-4-^,1%!  a.  a.  '  As p o i n t e d out i n s e c t i o n 2.5.2, t h i s i s simply the s o l u t i o n t o t h e g u a d r a t i c obtained when d i a g o n a l i z i n g t h e degenerate If  block.  i t i s assumed t h a t t h i s block has been d i a g o n a l i z e d , i . e . ,  that  MJ^O^VJ,,  (4.59)  then eguation  dE.( ) D  =  (4.58) reduces simply t o  V„  dX ' and  dE*(o)  (4.60)  In  t h e case  (V„ ? t V  ia  ),  of  =  2-fold  expressions  N/  Ja  degeneracy f o r t h e second  lifted  in  derivatives  first  order  4?Jg'(p) and  5 - i ( o ) a r e obtained from Table I I (or eguation 2.31), namely a  110  3=o  On s u b s t i t u t i n g t h e p a r t i a l d e r i v a t i v e s obtained from  eguation  (4.50), t h i s becomes  (4.62)  This  expression  is  i n agreement with the  A* term of eguation  (4.48), with the summation c o r r e c t l y r e s t r i c t e d , i . e . * Higher d e r i v a t i v e s details  will  be  are  omitted  t h i r d d e r i v a t i v e , ^7 ~^'{o) eguation  (4.48).  Since  obtained  similarly,  The e x p r e s s i o n found  ,  precisely  the  *  /  A  function  ensures t h a t the o r d e r i n g i s c o r r e c t .  approach  the  f o r the term  i t y i e l d s the T a y l o r s e r i e s f o r  d i r e c t l y , the c h a r a c t e r i s t i c  >*.  though  here. is  r  of E|(A)  automatically  1.11  CHAPTER 5.  PHYSICAL PROPERTIES AS ENERGY DEHIVATIVES;APPLICATION TO SIMPLE SPIN SYSTEMS.  T  Introduction., In  this  properties These  chapter  the  determination  of v a r i o u s  o f a system from energy d e r i v a t i v e s  energy  derivatives  can  be  found  is  considered.  either  element e x p r e s s i o n s o r from d e r i v a t i v e s of  the  physical  from matrix  characteristic  function. Properties  of  simple  spin  systems  such  as  the  p o l a r i z a t i o n v e c t o r and t h e s p i n p o l a r i z a b i l i t y t e n s o r ^<1> found using t h e c h a r a c t e r i s t i c f u n c t i o n s  g i v e n by Coope  The  c o r r e s p o n d i n g e x p r e s s i o n s i n terms o f matrix  be  t e d i o u s t o compute because they r e g u i r e  of the eigenvectors. vector  is  to  weak  hyperfine  intensities  spectra. chapter.  of  A discussion  the  singularities  r  e  (1966).  elements  spin  can  polarization  i n t e r a c t i o n s and one  a p p l i c a t i o n o f the s p i n p o l a r i z a b i l i t y t e n s o r i s the  a  p r i o r determination  One a p p l i c a t i o n o f the  determine  spin  in  of these two a p p l i c a t i o n s  to  determine  polycrystalline concludes  the  112 5 . 1 P h y s i c a l p r o p e r t i e s as energy d e r i v a t i v e s , ~ The  derivative  matrix element o f  of  the  the  EF,  energy,  Hamiltonian,  H , rs  ifArs ,  element of the d e n s i t y matrix,  with r e s p e c t  to the  yields  matrix  i n the same b a s i s as the  Hamiltonian. T h i s can e a s i l y be seen from t h e (5.1)  =  E  p  where  Pp  is  non^degenerate eigenvalues  the  The  =  H  the  subspace  stationary  property  taking  physical  =  { r)  sr  r 5  p r o p e r t i e s o f the system, i n the s t a t e )p>,  derivatives  of  themselves can be regarded as  and  as  energy  Longuet-Higgins  the  be  found  energy with r e s p e c t to matrix  first  order  matrix  properties.  d e r i v a t i v e s was pointed  Their  out by Coulson  (1947).  In the f o l l o w i n g , however, d e r i v a t i v e s of the taken,  of  ,  p  elements of t h e Hamiltonian. The elements of the d e n s i t y  formulation  the  O  which can be found from t h e d e n s i t y matrix, can a l s o by  of  that  3 H the  onto  ( v a r i a t i o n theorem) i m p l i e s t h a t  (5.2)  Thus  Ep.  expression  p  projection  eigenvalue  Tr  It follows  TrHP  the  energy  are  not p a r t i c u l a r l y with r e s p e c t t o matrix elements o f the  11.3  Hamiltonian,  but  rather  parameters  appearing,  with  respect  to  explicitly,  any  in  r e p r e s e n t a t i o n o f the Hamiltonian. The f i r s t represent  corresponding  first  perturbation the  operator  derivatives  then  order p r o p e r t i e s and the h i g h e r  d e r i v a t i v e s , p o l a r i z a t i o n s of these., Consider a Hamiltonian of t h e form  H = H° ^V  (5.3)  where V If  W  +  Ep  < 0  and V  is  t a >  by  the  60  a r e t h e o p e r a t o r s f o r two p h y s i c a l  the  expectation  of  i s the these  a r e r e l a t e d t o the corresponding energy d e r i v a t i v e s  Hellman-Feynmann  theorem,  (just  the  perturbation expression),  (5.4)  <  < V  ( i )  >  p  =  P  J V  W  I  P  >  =  ^  a  second  |p>  values  Bayleigh-Schroedinger  The  properties.  non-degenerate eigenvalue o f H , and  a  corresponding e i g e n v e c t o r , properties  a^v ,  +  derivatives  Bayleigh-Schroedinger  of  the  ,  A,  energy,  ' =  given  first  order  -  according t o  p e r t u r b a t i o n theory by*  »Eguation (5.5) can be obtained by t a k i n g t h e d e r i v a t i v e of the expression f o r the energy, E = jr A % . _j_dx with r e s p e c t t o "X; and \. * '«Y», z-M P  r  r,  114  (5.5)  2  Jy_)f>  =Y  <P)V )kXklV i /p> ^<p;V^)kXKl V ' V > (,)  ^  (  "V  )  (  4  I—  r e p r e s e n t f i r s t d e r i v a t i v e s o f the p h y s i c a l p r o p e r t i e s The  ^V*'*)  derivative of  d e r i v a t i v e o f ^V^)  p  =.  P  £  £  r  The matrix element e x p r e s s i o n s tedious  ' ^  p  .  i s the same a s t h e  3<V )  =  (J,  P  (5.4) and  .  (5.5) may be  t o compute..The corresponding e x p r e s s i o n s i n terms o f  characteristic equations  (5.7)  AJ  c  with r e s p e c t t o A,, i.e.,  p  SO£>  (5.6,  with r e s p e c t t o  4 y  functions  are considered  here,  which,  from  (2.25) and (2.26), a r e t h e f o l l o w i n g :  <v  c/)  >  =  = 2 5 *  Here, t o maintain g e n e r a l i t y , i t i s assumed t h a t the e i g e n v a l u e Sp=EpC\) i s known  f o r some  value  o f A, say  which i s not  n e c e s s a r i l y z e r o . , The d e r i v a t i v e s o f •£ i n equations (5.8)  are evaluated  f o r z=^»(£) and  Equation  (5.7)  and  (5.7) , f o r  115  example, w i l l be w r i t t e n i n the f o l l o w i n g  i t being  understood t h a t  the  derivatives  as  are  evaluated  for  116 5.2 Some p r o p e r t i e s o f simple  s p i n systems  as energy d e r i v a t i v e s . The  Hamiltonian  j£€ f o r a simple  magnetic f i e l d H i s c o n v e n t i o n a l l y (5.9)  written  + 3>(S£-£S*) EtSi-SJ)  H  Following  s p i n system i n a uniform  .  +  the n o t a t i o n  used  by  Coope  (1966),  this  can be  w r i t t e n more compactly as  where k = c j ( ^ H ) , tensor (5.H) and  = i - S +•  Si  (5.10)  Bi^sf*  where  i s the  i r r e d u c i b l e second rank  operator, [§f%  = i  (^* S  S  where ID i s the i r r e d u c i b l e z e r o - f i e l d s p l i t t i n g  tensor.  In  the p r i n c i p a l a x i s system o f ID, one has (5.12) It  i s useful  H^ =|I>  ,  2  to  make  J> -JD X X  use  V V  = 3 E .  a l s o of t h e f o l l o w i n g two s c a l a r  parameters, e g u i v a l e n t to X> and E, S> =  (5.13)  2  D>  S> ^  = T>  ;  ,  = (V>W) © = T r |E? 3 , :  3  A number o f c h a r a c t e r i s t i c f u n c t i o n s f o r the Hamiltonian were given by Coope systems o f s p i n 1  r  (1966)  using  this  (5.10)  n o t a t i o n . , Those f o r  3/2, and 2, a r e l i s t e d i n Table  IX. „.  117 Table IX.  The c h a r a c t e r i s t i c f u n c t i o n s f o r systems o f spin  t . 3/2. and 2.  U = h.S + J:[S]  (2)  Spin  1.  Spin  3/2.  Spin  2.  £  a* a  5  =  = z  - ( s i ? *  5  -+- a 2 -»- 0 3 ^ + 3  a  y m  3  )  ,  o  3  a  = - ( y & s + zi  6(3^3-/?-B>'L)(S1I>*-ZV?)  5  b j b - b )  118 The  Hamiltonian i n eguation (5.10) i s of t h e same g e n e r a l  form as the Hamiltonian g i v e n i n e q u a t i o n (5.3) except t h a t the parameters independent  X,  and  A  x  are  parameters,  replaced  by  respectively,  of  3  and  5  i n ij and B> . The f i r s t  d e r i v a t i v e s o f the energy o f t h e p - t h yield,' t h e f o l l o w i n g f i r s t  sets  (non-degenerate)  state  order p r o p e r t i e s of t h e p - t h s t a t e :  Spin p o l a r i z a t i o n v e c t o r , <£> = <pl5 )p>. p  (5.14)  <-S>  9E  =  P  2h,  P  Magnetic moment <^x\ ,  3H ([Sj ^!  Quadrupole p o l a r i z a t i o n t e n s o r  <W\  (5.16)  00  = 2 ID  It  i s worth remarking t h a t the d e n s i t y matrix i t s e l f  s p i n system can properties  be  written  in  terms  /> = i i f ^ ) . S T h i s example o f the way i n which  the  to  such  first  order  (Fano, 1957). For a s p i n 1 system, f o r example, the  d e n s i t y matrix /=" i s g i v e n by (Coope,  related  of  of a  first  unpublished)  + <[S7 >:[sf . (a5  the  }  density  matrix  can  be  order p r o p e r t i e s , and t h u s t o d e r i v a t i v e s o f  energy, i s complementary; t o the r e p r e s e n t a t i o n o f t h e  (5.2), due t o Coulson and Longuet-Higgins  (1947).  type  1-1.9  As  discussed  i n section  the  polarizabilities,  c a n be  of  the  to  energy  properties.  In  and  £>l§f  first  (-5.18) 2h £  coupling  a  t l r  =  similar  way,  constants  properties.. hyperfine  In  in  one  with  n  first  coupling  I  c a )  order  ^  respect  represent  derivatives  s .g  of  p  Hamiltonian  t e n s o r fl„ i n a h y p e r f i n e  (5.20)  derivatives  3<[s] 2h  derivatives  particular,  t o second  P  3<E^>  the  properties,  has  d<A>p = 3 ID  = dJ  In  both  order  derivatives  9<S>  =  the second  related  p a r t i c u l a r , here,  (5.17)  (5.19)  5.1,  with  to  other  other  spin  respect  t o the  term  ,  n  n  represent  the tensor  coupling  <5 (l ) >  (5.21)  r  n  s  of S  =  p  to  Iz n  p  3(«o)rs I f 6„ i s i s o t r o p i c  (5.22)  These  (6„=<yJ) #  •  properties  then  Io> = p  of the  spin  2J>  system  can  be  calculated  120 using The  the c h a r a c t e r i s t i c f u n c t i o n expressions  (5.7) and (5.8).  s p i n p o l a r i z a t i o n v e c t o r i s given by  (5.23)  <S>  d§P  =  _  =  4  and t h e s p i n p o l a r i z a b i l i t y t e n s o r , 3<€> , i P  s  given by  3<^>P = 2?J=P  (5.24)  Here < £ > < s > , p  ^ < ^ § >  p  etc.  r  are  dyadics.  For  the  spin  1  system, the r e l e v a n t p a r t i a l d e r i v a t i v e s , of t h e c h a r a c t e r i s t i c f u n c t i o n given i n T a b l e IX, are  4 =  =  6z  _ 2 k H  - 2 E>  fa,  **  - B  u  r  ,  where U i s the u n i t t e n s o r , where  lb]= bb-ib*^, w  and  From  where  eguation  \  (5.23) t h e s p i n p o l a r i z a t i o n v e c t o r f o r a s p i n 1  system i s a c c o r d i n g l y given by  121  (5.25, ,  <§>  w h i l e , from eguation  2t*y  =  + £]'h  j  (5.24,, the p o l a r i z a b i l i t y  tensor  9<£>  p t  dh  for a spin  (5.26,  1 system, i s  ?<£>P =  4 z y ^ ] ^ ( ^ ^ > ^ < s >  5t  (when  3  evaluated  z  at z = E 6  < [ S r \  The vector  ).  P  p o l a r i z a t i o n tensor ^ [ S j * ^ ,  (5.27,  _ (h*^  2  p  y - ^ < s >  < s >  p  ^fl^)  Similarly,  the  quadrupole  i s given by  =  * fl> + KT~  [ST  corresponding e x p r e s s i o n s f o r the and  p  polarizability  systems a r e g i v e n i n T a b l e X.  tensor  .  spin  f o r spin  1  polarization and s p i n  3/2  122 Table I .  Expressions f o r t h e s p i n p o l a r i z a t i o n vector and  p o l a r i z a b i l i t y t e n s o r f o r t h e p-tfa  (non-degenerate) s t a t e o f systems with spin  1 and 3/2.-  Spin 1.  9<£> 3  ~  Spin 3/2.  9b  4-  P  &  a.tey+&l+2tk<$> +<s> k)-(>z<z>p<s> P  P  3z -(h%-&Es,) a  P  123 The the  use o f t h e formulae i n Table X can be i l l u s t r a t e d  spin  1  system. , I n  general,  for  f o r a given h and JE>, the  procedure i s t o f i n d a r o o t Ep o f the c h a r a c t e r i s t i c  function,  n u m e r i c a l l y , and then t o e v a l u a t e t h e e x p r e s s i o n s a t t h i s r o o t . However,  an  a n a l y t i c expression  f o r t h e r o o t s can be given i n  the p a r t i c u l a r case t h a t h l i e s i n a p r i n c i p a l  direction  (the  ^ - d i r e c t i o n ) o f ID, i . e . , when ID - b and  =  JP  t h i s case i s convenient  b  gz  f o r i l l u s t r a t i o n . I n t h i s case the  c h a r a c t e r i s t i c f u n c t i o n , f o r t h e s p i n 1 system, reduces to (5.28)  with  f =  (h**'***)*  Z 3  -£*i-h*)  roots  (5.29)  E  =. - B>zz ^  0  -  E±i  £ 3> ^± z  \l Z D a - Z b l h ? + h*~ ,  which, a f t e r some a l g e b r a , can be w r i t t e n i n the more  familiar  form (5.30)  Eo  E  ±l  =  - # W ,  =  ±TJ> J  h ^ ( ^ ^ l f  ZZ±  o r , using the r e l a t i o n s (5.12), (5.31)  E  ,  i n t h e form  = -^ ^  0  3  = ^2>  E , ±  ±Jh*+E  a  ,  I t i s u s e f u l t o d e f i n e a parameter (5.32) which  A measures  eigenvalues.  -  X^-E-,)  the  separation  of  the  two  non-constant  In terms o f t h i s , t h e z - d e r i v a t i v e s , evaluated  at  124 the e i g e n v a l u e s , a r e  (5.33)  ^|  (5.34)  $z\  =  E o  2> -A% a  = 3A(A±2>),  £ ± i  and the H - d e r i v a t i v e s e v a l u a t e d a t t h e e i g e n v a l u e s a r e  fb[  (5.35)  < -«> 5  From  f k  eguations  l  t  -  l  (5.23)  - a ^ J > ± A )  .  and (5.33)-(5.36) the s p i n p o l a r i z a t i o n  v e c t o r s a r e found t o be  (5.37)  <S>  ±)  =  h\  h IE  _  ± Jb.  =  O  ± /  and  (5.38)  <S>  o  By i n t r o d u c i n g t h e number magnetic  quantum  number  /T)=O,±J7  which  becomes  the  a t high f i e l d s , eguations  (5.38) can be combined t o g i v e  ordinary (5.37) and  125  (5.39)  k  m  m  S i m i l a r l y , from equation tensors f o r the spin  o ±1  m  }  (5.24), t h e s p i n  polarizability  1 system, with the e f f e c t i v e f i e l d h along  a p r i n c i p a l d i r e c t i o n o ftt>, a r e found t o be e-J> (5.40) (£+T>X*-3>)  91»  O  o  o  -E-J> o  o  o  o  and O  E-I>  (5.41)  3 JL,  ^(A±2>)  O  O  - £ - £ >  o  o  o  o  These e x p r e s s i o n s can be combined i n t o the s i n g l e e x p r e s s i o n  (5.42)  3<$>fn = (5m -3)A-mJ> a  A(A -D ; A  where (n-°,±-l.  At  A  large  rE-r> O O O  O  -E-D O O  O  f i e l d s m becomes the usual magnetic  quantum number, but these e g u a t i o n s hold f o r a l l f i e l d In a s i m i l a r way the quadrupole found, from equation  polarization  ( 5 . 2 7 ) , t o be  (5.43)  (5.44)  where % i s the u n i t v e c t o r i n the z ^ d i r e c t i o n , and  values.  tensors are  '/3  o  O  o  O O  O  -*5  127  5.3  Applications.-  5.3.1  A s p i n system weakly coupled t o some n u c l e i . Consider  a  strongly  Hamiltonian o f the form  coupled  spin  (5.9), which i s  system,  weakly  coupled  with  a  to  a  number o f n o n - i n t e r a c t i n g n u c l e i . The Hamiltonian f o r t h e t o t a l system can be w r i t t e n  rt  rt  ^  v..  The  p-th l e v e l of the s t r o n g l y coupled system with is s p l i t into  several  nuclear  sublevels.  Hamiltonian  The  effective  eff  n u c l e a r s p i n Hamiltonian H to f i r s t  giving t h i s extra substructure i s ,  order,  • PP  where  It  may be more convenient t o compute ^ § >  i n T a b l e X than t o compute i^> Second  order  p  from the e x p r e s s i o n s  as a matrix element.  contributions  Hamiltonian are determined  p  to  the  by the g u a n t i t i e s  effective  nuclear  128  (5.46)  The  <pl£lSXi\Slp>  contributions  to the e f f e c t i v e c o u p l i n g between  n u c l e i can be computed from the r e a l p a r t of and  hence  from  imaginary  derivatives ii^p  of  (5.46).  reguire  Only the r e a l p a r t i s given by  ( c . f . eguation  5.5).  the  Han  guadrupole moment terms ( s e l f coupling) part  expression,  the p o l a r i z a b i l i t i e s ^LMF . ,Onfortunately  a effective  this  different  the the  129 5.3.2. S i n g u l a r i t i e s i n p o l y c r y s t a l l i n e s p e c t r a , > One use o f t h e p o l a r i z a b i l i t y t e n s o r the  intensities  of  the  singularities  i s t o determine of  polycrystalline  spectra., Consider  a  spin  system  H =.(u,&,4). In an E.S.H radiation  is  in  experiment  usually  a  uniform  the  magnetic  freguency  of  field  incident  f i x e d , t h e magnetic f i e l d being  varied.  Besonance occurs vhen {5.47)  m'(ti)-  C  =  constant-.  T h i s can be regarded as an i m p l i c i t usually  known  in  resonance f i e l d  closed  form,  as a f u n c t i o n  (5.48)  H  r  =  H  r  time., The  proportional  to  intensity the  ( 0 ^ )  number  of  resonance  molecules  of the a p p l i e d  signal  f o r which  is  H (&j<P) r  f i e l d H. The  main  of t h e spectrum a r e peaks of v a r i o u s s o r t s , where t h e with r e s p e c t  to 0  and  4*,  when  V±  (5.49)  The  o f the  .  the  resonance f i e l d s become s t a t i o n a r y i.e.,  t h e magnitude  a l l G and<f>a r e present a t the  of  a c t u a l l y e g u a l s t h e magnitude features  giving  though i t i s not  of o r i e n t a t i o n :  In a p o l y c r y s t a l l i n e sample same  function,  intensity  of  r  these  3Ji  =  r  = O  .  peaks i s i n v e r s e l y  mean c u r v a t u r e o f H , as measured by t h e r  Hessian o f second d e r i v a t i v e s  proportional  sguare  root  (Coope, 1969), namely.  to a  of the  130  3fMr 3X.3X,  (5.50)  Here  der  cv  proportional  to arc  l e n g t h s on a u n i t sphere. For example, i f t h e s t a t i o n a r y  point  {&o0  x, and Xz a r e orthogonal c o o r d i n a t e s ,  0o)  l i e s a t 0 = 9 0 ° , then p o s s i b l e c o o r d i n a t e s are X , - e - 0  0 9  *Z=  The related  d e r i v a t i v e s of H to  derivatives  components of ^ . stationary  In  with r e s p e c t  r  of  the  Appendix  to  energy  C  X, and with  i t i s shown  X  a  can  respect that,  be  t o the at  a  p o i n t , the r e l a t i o n i s  H  (5.5D  v;. J  . v. = H 3;.  3X; 3Xj  where  (5.52)  and  (5.53)  V;  Here a =£J/H vectors eguation  =  3^ 3x;  i s the unit vector  i n t h e d i r e c t i o n o f H . The u n i t  y,, v,, and £ form an orthonormal (5.47),  the derivatives of ^  r e l a t e d t o the energy d e r i v a t i v e s by  set. , F i n a l l y , i n eguation  from  (5.52) a r e  131  9V  -  3E . _  9E 3h  (5.55)  -  lEm  Thus, the energy d e r i v a t i v e s  «L§  (5.54)  m  m  m  ^l&n  and  are  related,  by  e g u a t i o n s (5.51)"(5.55), t o second d e r i v a t i v e s o f t h e resonance field,  9 .K . a  r  ax, 3XJ  , The Hessian o f these second d e r i v a t i v e s , i n turn,  determines the i n t e n s i t y o f peaks o c c u r i n g :i i n p o l y c r y s t a l l i n e spectra. To i l l u s t r a t e t h e use o f these formulae c o n s i d e r a g a i n the 1 system i n which U i s i n a p r i n c i p a l d i r e c t i o n o f E>,  spin  5> b  (5.56) and  =  fi^z  h  suppose f u r t h e r t h a t h> i s a l s o a p r i n c i p a l d i r e c t i o n o f g,  so t h a t , i n t h e p r i n c i p a l a x i s system o f 1t>, the g the  tensor  has  form  g  (5.57)  -  ~3KX  3xy  o  hyx  9  o  V V  Then t h i s z - d i r e c t i o n i s a s t a t i o n a r y direction..„From  eguations  (5.39) and (5.54) i t i s found t h a t  (5.58)  ^b  where A0») = m'-/77, found  that  and  AM  from  i i  eguations (5.42) and (5.55), i t i s  132  (5.59)  9^  „  2<sy , 9b  _  m  3  <5>  r  2b  —  A(m)) Z(m'+m)*-l>  / A(A -J> ) a  E  o  a  ifc-fefr)]  °  -_  0  D  0  O  O  i n t h e p r i n c i p a l a x i s system o f © . a f t e r some found t h a t J  o f eguation  (5. $2)  (5.60)  -3^ H g( rr.'- - n)A-J>  E-Z> a  ,  <  f  i t  is  i s given by  r  a  algebra  c. -  O  O  O  - E - f c o  3 +  so t h a t  {  E-D  (5.61)  Two  O  OJ  O  -E-l> ol  o  o o\ •i  extreme cases a r e worth c o n s i d e r i n g . ; l f ZD=0, so t h a t  the only a n i s o t r o p y  i s i n the g t e n s o r ,  term c o n t r i b u t e s , and eguation  then  only  (5.61) reduces t o  t h e second  133  1  (5.62)  In  3x  this  case,  from  the Hessian  (5.50), the peak i n t e n s i t y i s  proportional to  (5.63)  Oh the other hand, i f the g tensor i s then  only t h e f i r s t  term i n equation  isotropic,  (5.61) c o n t r i b u t e s ,  d i s a p p e a r s from t h i s , so t h a t  (5.64)  2*Mr  _  -  H(5(/r7V/n)A,-2>)  In t h i s c a s e , the peak i n t e n s i t y  (5.65)  o o  -e-r> o o o  i s proportional to  H(3(m V/n,A -J>  i . e . , g =gt/  (x»=»_ a j '/ E  These e x p r e s s i o n s are v a l i d f o r a l l f i e l d s .  a  #  andg  134 BIBLIOGRAPHY. Abramowitz, mathematical  H. and Segun, J.A. e d i t o r s , f u n c t i o n s (Dover, Hew Y o r k ) .  1965,  Handbook  of  Coope, J . A . 8 - # 1966, J . Chem. Phys., 44, 4431. , Coope, J.A.R., 1969, Chem..Phys..Letters, 3, 589. Coulson, C. A., 1940, Proc. Cambribge P h i l . Soc., 36, 201. Coulson, C. A. and Longuet-Higgins, Soc. ,. (London) , A 191, 39. Fano, 0., (1957),  H. C ,  1947, Proc. Roy.  Revs. Hodern Phys., 2 9 , 74.,  F u k u i , K., e t . a l . , (1959), J . Chem. Phys., 3 J , 287. Goursat, E. , 1904, 1916, (Ginn, Boston). ,  A  course  in  mathematical  analysis  Harkness, J . , and H o r l e y , F., 1898, I n t r o d u c t i o n t o the t h e o r y o f a n a l y t i c f u n c t i o n s (Hacmillan, London). Kato  #  T., 1949, Prog. Theoret. Phys. Kyoto, 4, 514. ,,  Kato, T. , 1966, P e r t u r b a t i o n theory f o r l i n e a r - onerators( S p r i n g e r - V e r l a g , B e r l i n ) , page 84, eguation (2.49). Kransnosel'skii, et.a l . , 1969, Approximatesolution o p e r a t o r equations ( S o l t e r s - N o o r d h o f f )  of-  M a l l i o n , R. B. and Bigby, M. J . , 1976, Oxford Dniversity T h e o r e t i c a l Chemistry Department Progress Report. 1975-1976. page 1. (No d e t a i l s a r e g i v e n i n t h i s r e p o r t . T h e i r work has been submitted f o r p u b l i c a t i o n but has not been seen.) Osgood, 8 . F., 1913, T o p i c s i n t h e t h e o r y of f u n c t i o n s o f s e v e r a l complex v a r i a b l e s , fladison Colloguium (Dover, New York, 1966, o r Am. Math. S o c , 1914). R a y l e i g h , J.B.S.. 1894. Theory o f sonnd I , page 115 ( H a c m i l l a n , London).  f  2nd. e d i t i o n ,  volume  Hellich, F., 1953, P e r t u r b a t i o n theory o f eigenvalue- problems.l e c t u r e s given a t New York U n i v e r s i t y (Gordon and Breach, New York, 1969). Rutherford, D.E., 1945, 1951, Proc.,Roy. Soc. (Edinburgh), 62, 229, 63, 232. Schroedinger,  E., 1926, Ann. P h y s i k . . 80. 62.  T u r n b a l l , H.B., 1952, Theory Edinburgh).  of  equations  (Oliver  and  Boyd,  Van  Vleck,  J.H.,  1929,  Phys.  Hey., 33,  467.  136 APPENDIXA.  CONDITIONS FOR  ANALYTICITY OF EIGENVALUES  OF AS HERMITIAN OPERATOR. The  conditions  n-dimensional analyticity  for  a n a l y t i c i t y o f the e i g e n v a l u e s of an  Hermitian of  H(^\)  operator  follow  the c o e f f i c i e n t s of the reduced  from  the  characteristic  function.  -v A.1  One  p e r t u r b a t i o n parameter.  By assumption, (in  general  it  the operator  will  non-zero imaginary  not  be  HO)  i s Hermitian f o r  A  real  Hermitian i f A i s complex with  part) and thus, f o r A\ r e a l , i t s  eigenvalues  are r e a l . , Suppose reduced  £ (o)  is  P  characteristic  degenerate  analytic  r o o t f o r A=O,  i n z,  functions  However, i f EpC\)  only  whose of  the  a s  E (A) p  integer  eigenvalue.  the  initially  roots,  i s a polynomial  coefficients single  c, ,  i^i,...,^,  parameter A,  E P C\)  i n some is  a  can be w r i t t e n * as  (Puiseux s e r i e s ) convergent powers  The  g-fold  For such a polynomial, where Ep(°)  had f r a c t i o n a l  expansion,  eigenvalue  with  the perturbed e i g e n v a l u e  power s e r i e s i n A  series  eguation,  degenerate  p  neighbourhood of A=O.  a  g-fold  s e t Epfa), - -. E +^-i (3)  e g u a t i o n o f degree are  a  of  A  f o r small A .  in  its  power  then, f o r / ) r e a l but n e g a t i v e , t h e perturbed  would be complex. ,Since t h i s i s not powers  of  *See, f o r example, Goursat  A  can  appear  in  the  case,  the power s e r i e s  (1916), volume I I , p a r t 1, page  239.  137 expansion  of  Epfa) .  ,. Consequently  1  E (%) P  f u n c t i o n o f A w i t h i n some neighbourhood  ft.2 Non-degenerate e i g e n v a l u e s . If  E (p) p  i s an  analytic  of /A/=o.  ;  i s a non-degenerate  e i g e n v a l u e then the reduced  c h a r a c t e r i s t i c e g u a t i o n f o r E C\) i s simply P  rCz.'X)  =  z  +  c,  ;  where c, i s an a n a l y t i c f u n c t i o n o f X i n some neighbourhood l^lzzO,  ,  Thus  equation,  is  neiqhbourhood  iRellich  ELpC\),  also of  the an IXI=o.  (1953), paqe 31.  root  analytic  of  the reduced  function  of  of  characteristic A  in  some  138  APPENDIX B  HOBE THAN ONE PBBTUBBATION The than  results  one  given  perturbation  in  PARAMETER.  Tables III-VII--are v a l i d f o r more  parameter.  Formally  the  following  n o t a t i o n i s used i n the t a b l e s ,  r  =r  al  Z—»  —^  =2"  a,!... a / e  and  V*** =  (B.2) If  there  +  -  i s only one p e r t u r b a t i o n parameter the t a b l e s can be  used d i r e c t l y . I f t h e r e i s more than one, then must  be  used.  To  2 - f o l d degeneracy  3 1  Using eguation  i l l u s t r a t e , c o n s i d e r the A  (Table I I I ) with  l  A -  eguation  (B.I)  term o f s f o r a  ,It i s given by  i  (B.1) we f i n d t h a t  5  3  'J  140 APPENDIX C.  CALCULATION Of 9 a Hr/d*, 2xj . . The  resonance c o n d i t i o n V(H)  (Cl)  =  (eguation 5.47)  V CH> X.jX*)  can be regarded as an i m p l i c i t resonance f i e l d  (C2)  H  A, — x ,  f  A* —e> K  f  z  ^  that  =  J  d  3-W .2H. 3 j 9 H 2K;  y  ax;9H 2xy  9x,3Xj  where  function,  i n Chapter 2 except t h a t  2 follows  f u n c t i o n of t h e magnitude of the  T h i s i s j u s t t h e kind o f i m p l i c i t  considered  It  aonsta.nl-  =  X  the d e r i v a t i v e s a r e evaluated  of X., X: and H./Since ? *>  a >  i , r  3K,3XJ  +  g V  QH  m  3H3H  QHl/zv  dX, 3XJ  at the appropriate  3Jir  eguation  (C4)  =  3  3x,  3x.  _  0  '  (C.2) reduces t o  9 ^r a  =  J  values  i n eguation (5.50), i s r e g u i r e d a t  a s t a t i o n a r y p o i n t , where  (C3)  j/ 2H  3fV  _  9x,3Xj  / 94>  3X;3XJ / 3 H  An orthonormal system a , y,, v can be defined by w r i t i n g a  141 M  (C.5)  ^ dX;  (C.6)  always this  assuming  case  we  x,, and  H  =  Hy  x  %  u.  ;  i-_i,a  ;  ace l o c a l  . '  cartesian  coordinates.  In  have  <C.7)  =  2Ltl  If,  =  f o r t h e moment,  by t h e c h a i n  - S;-- H o c  _ ^ « .  =  the magnitude  o f H i s denoted  by  x , 5  then,  rule.  22  (C.8)  =  3 y  . 3M  i =  and  <c.9)  From  ^ 3X,-3XJ  3  relations  written  more  - ^ ' ,  =  3 # 3Xj  x  (C.5)-(C.7)  2V 3x; 3_y  (C.11)  3H  H  (C.12)  a  3H  eguations  =  ,  i,a,  (C.9)  can  ~  9HdH  H (y; 3H9H  3  3x,3xj  (C.8)  and  H  ( v,  3X;9Xj  3x, 3 H  ^ - 21*  explicitly  (C.10)  ( C . 13)  +  3H 3 V  ~-  —-~ 3xj 3XJ  > '>$  be  142  (C.14)  9 ^  Using  eguations  u.  =  (C.11)  second d e r i v a t i v e of  .  ,  „  and (C.12) the e x p r e s s i o n  the  resonance  field,  at  (C.4)  a  f o r the  stationary  p o i n t , can be w r i t t e n as  5fHr  (C15)  =  _ ( V  (V;.PU>  .V-)  _ 5).  ]/u.dJ!f  H .£±L  yf. yj == S}j , t h i s can a l s o be w r i t t e n i n the form  Since  a!±i  r  (c.16)  -  H ( v;.  ~  ax, Sxjj  J y.\  ~  J/ ,  where  T  (C17)  In  =  any  - f  H *  3ff»?  application  _  H  it is  d e r i v a t i v e s , taken with r e s p e c t t o terms  -'  and  H  simpler the  m  to  applied  express field  these H,  of d e r i v a t i v e s taken with r e s p e c t t o the e f f e c t i v e  h. S i n c e b = 3^b,  (c 18  ul/ .3u>  i t f o l l o w s that,,  k  thus, a l s o t h a t  =  Oik'***  in field  143  CC.19)  H . 3_  = h. 3_  ay Expression  (C.17)  3/n  can  =  H 3 . 3H  accordingly  =  be  Ai 3_  .  written  in  convenient form  (C20)  o~ = - J > H  a  q . 3 l ^  . a -  h  -SV.yl/h.dv  the  more  

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