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Quantum mechanical perturbation theory in terms of characteristic functions 1977

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QUANTUM MECHANICAL PERTURBATION THEORY IN TERMS OF CHARACTERISTIC FUNCTIONS by MARTIN GODFREY LUIS GOMBERG B.A., U n v e r s i t y o f E s s e x , 1973 A THESIS SUBMITTED IN PARTIAL FULFILLMENT THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in THE FACULTY OF GRADUATE STUDIES (Department o f C h e m i s t r y ) Me a c c e p t t h i s t h e s i s as c o n f o r m i n g t o t h e r e q u i r e d s t a n d a r d THE UNIVERSITY OF BRITISH COLUMBIA A u g u s t , 1977 Copyright © 197? by M.G.L. Gomberg In presenting th is thes is in pa r t i a l fu l f i lment of the requirements f o r an advanced degree at the Un ivers i ty of B r i t i s h Columbia, I agree that the L ibrary sha l l make i t f r ee ly ava i l ab le f o r reference and study. I further agree that permission for extensive copying of t h i s t h e s i s for scho lar ly purposes may be granted by the Head of my Department or by h is representa t ives . I t is understood that copying or p u b l i c a t i o n of th is thes is for f i n a n c i a l gain shal l not be allowed without my writ ten permission. Department of Chem/sfry The Univers i ty of B r i t i s h Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Date flyytfih '9^7 A quantum mechanical perturbation theory, for f i n i t e dimensional cases, based not on the perturbed Hamiltonian operator i t s e l f , but on the c h a r a c t e r i s t i c function i s developed. & perturbation hierarchy in terms of derivatives of the c h a r a c t e r i s t i c function i s constructed. From t h i s hierarchy, perturbation ser i e s for i n d i v i d u a l eigenvalues are found. Various cases of degeneracy and degeneracy l i f t e d i n various orders are examined i n d e t a i l . This perturbation theory for i n d i v i d u a l eigenvalues i s generalized. Perturbation theory i s developed for a set of eigenvalues considered together. Here the perturbation s e r i e s are for the c o e f f i c i e n t s of a 'reduced c h a r a c t e r i s t i c function* for t h i s set of eigenvalues. These perturbation s e r i e s are found by a contour i n t e g r a l method and by an algebraic method. The expressions for the i n d i v i d u a l eigenvalues and t h e i r generalization, the expressions for the reduced c h a r a c t e r i s t i c function, both of which are i n terms of derivatives of the (f u l l ) c h a r a c t e r i s t i c function, correspond, respectively, to the f a m i l i a r matrix element expressions i n Hayleigh-Schroedinger, and Van Vleck perturbation theories. Some i l l u s t r a t i o n s and applications of the c h a r a c t e r i s t i c function perturbation formulae are given. General expressions are found, to second order, for the perturbed Hiickel 7f-molecular o r b i t a l energy l e v e l s , of any perturbed even-membered r i n g of carbon atoms. The f a m i l i a r i i i Rayleigh-Schroedinger perturbation foraalae are rederived from th e i r corresponding c h a r a c t e r i s t i c function expressions. The rel a t i o n s h i p between energy derivatives and physical properties i s discussed v i t h p a r t i c u l a r reference to simple spin systeas. Expressions for the dipole and guadrupole spin p o l a r i z a t i o n s and for spin p o l a r i z a b i l i t i e s i n siaple spin systeas are found froa the c h a r a c t e r i s t i c functions of the spin systeas. These properties are useful i n connection with weak hyperfine coupling, and for predicting the i n t e n s i t y of peaks occuring i n p o l y c r y s t a l l i n e spectra. i v TABLE OF CONTENTS Page Abstract . i i Table of contents i v L i s t of tables , .. v i L i s t of figures v i i Acknowledgements v i i i Chapter 1 Introduction 1 1.1 The ideas ..................................... 2 1.2 Thesis content 9 Chapter 2 Perturbation theory for single eigenvalues . 12 2.1 The method 13 2.2 Orientation 15 2.3 The hierarchy of perturbation eguations ....... 18 2.3.1 Comments ............................... 19 2.4 An al t e r n a t i v e notation 22 2.5 Special cases a r i s i n g i n perturbation theory .. 29 2.5.1 Non-degenerate eigenvalues ............. 29 2.5.2 2-fold degeneracy l i f t e d in f i r s t order . 32 2.5.3 2-fold degeneracy not l i f t e d i n f i r s t order .. 34 2.5.4 3-fold degeneracy and degeneracy p a r t i a l l y l i f t e d 37 2.5.5 q-fold degeneracy ...................... 40 2.6 Summary , 42 Chapter 3 The reduced c h a r a c t e r i s t i c function ........ 46 3.1 The reduced c h a r a c t e r i s t i c function ........... 47 3.2 The eigenvalue 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 r 54 V 3.3.1 Non-degenerate eigenvalues 56 3.3.2 2-fold degeneracy 58 3.3.3 3-fold degeneracy 68 3.3.4 Near-degeneracy 71 3.4 Contour i n t e g r a l method: H—rr ................ 73 3.4.1 I l l u s t r a t i o n : 2-fold degeneracy 76 3.5 An algebriac method: f - * - r 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 applications 94 4.1 A perturbation c a l c u l a t i o n for even-membered rings of carbon atoms 95 4.2 Derivation of the Hayleigh-Schroedinger perturbation formulae from the c h a r a c t e r i s t i c function 104 Chapter 5 Physical properties as energy d e r i v a t i v e s : Application to simple spin systems ......... 111 5.1 Physical properties as energy derivatives 112 5.2 Some properties or simple spin systems as energy derivatives 116 5.3 Applications 127 5.3.1 A spin system weakly coupled to some nuclei ............................ 127 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 ................................ 129 Bibliography 134 Appendix A Conditions for a n a l y t i c i t y of eigenvalues of an Bermitian operator .................. 136 Appendix B More than one perturbation parameter ...... 138 Appendix C Calculation of ^aHr/3x, 3xa 140 v i LIST OF TABLES Table I The f i r s t eight equations of the perturbation hierarchy 26-28 Table II The non-zero terms of the equations of the perturbation hierarchy for some sp e c i a l cases 43-45 Table III 2-fold degeneracy: The eigenvalue power sums 63-65 Table IV 2-fold degeneracy: The c o e f f i c i e n t s of the reduced c h a r a c t e r i s t i c function 66 Table V 2-fold degeneracy: The discriminant of the reduced c h a r a c t e r i s t i c function ................. 67 Table VI 3-fold degeneracy: The eigenvalue power sums ............... 69 Table VII 3-fold degeneracy: The c o e f f i c i e n t s of the reduced c h a r a c t e r i s t i c function • 70 Table VIII Algebraic r e l a t i o n s between the 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 Table IX The c h a r a c t e r i s t i c functions f o r systems of spin 1, 3/2, and 2 ................... 117 Table X Expressions for the spin 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 tensor f o r the p-th (non-degenerate) state of systems with spin 1 and 3/2 122 v i i L I S T OF FIGOBBS Figure 1 The variation of the c h a r a c t e r i s t i c function (equation 1.6) with 2 , for two fixe d values of /\ ....................... Figure 2 i n example of the variation of a c h a r a c t e r i s t i c function with z for two fixe d values of }\ 15 Figure 3 Guide to the perturbation hierarchy ( i ) : Hon-degenerate eigenvalues , 31 Figure H Guide to the perturbation hierarchy ( i i ) : 2- f o l d degeneracy 36 Figure 5 Guide to the perturbation hierarchy ( i i i ) : 3- f o l d degeneracy 39 Figure 6 The contour r£L , t^+t,x» a n d i n t a e complex z-plane 72 Figure 7 Some examples of the 'best' contour ..... 88 v i i i ACKNOWLEDGEMENTS I wish to thank Dr. J.A.R. Coope for suggesting t h i s research t o p i c , for guidance in both the research and thesis preparation, and for always being a v a i l a b l e for discussion. I aa also indebted to ay fellow graduate students, i n part i c u l a r David Sabo and E r i c Turner, from whom I have learned much. Por f i n a n c i a l support during my two year stay I would l i k e to thank both the Chemistry and Physics departments for teaching assistantships and the University of B r i t i s h Columbia for summer scholarship money. Finally, I would l i k e to thank Kathryn L e s l i e f o r her help i n typing t h i s t h e s i s . 1 CHAPTER 1. INTRODUCTION. This t h e s i s develops a quantum mechanical perturbation theory based on the c h a r a c t e r i s t i c function, <1.1, f C * , a ) = d e M z - H ( » J ; of a perturbed, f i n i t e dimensional Hermitian operator H ( A ) . The one or more perturbation parameters are represented by A. This approach i s i n contrast to the usual perturbation theory which i s based on the operator i t s e l f or i t s matrix elements. A perturbation theory based on the c h a r a c t e r i s t i c function -Pfe,^) amounts to a study of the behaviour of the roots of the polynomial equation (1.2) -TYZ,*) = .. + a„(n) = o as the c o e f f i c i e n t s ex, i  = ±,...,n vary with A. It i s assumed that H ( ^ ) , and thus the a;(a), are a n a l y t i c functions of A i n some neighbourhood of |AI=0. 2 1.1 The ideas. An equation that aodels a physical system can r a r e l y be solved exactly. Quantum mechanical perturbation theory i s motivated by t h i s f a c t . Rayleigh (1894) i n the context of •vibrations of a s t r i n g with small inhomogeneities" and Schroedinger (1926) i n the context of his new wave mechanics developed what i s now known as Rayleigh-Schroedinger perturbation theory. In t h i s , an Hermitian operator of the form i s considered. (Here, and in the rest of the t h e s i s , only time-independent operators are considered.) The eigenvalues E;(6) and eigenvectors lEj(o)> of the unperturbed Hermitian operator H° are assumed to be known. The Hermitian operator V provides the perturbation, with A a small r e a l parameter. The eigenvalues E;(A) and eigenvectors |E,-(a)> of the perturbed operator H(Xj are found as power series i n A . For example, i f Ep(o) i s a non-degenerate unperturbed eigenvalue, then the perturbed eigenvalue £p(A) and eigenvector l^pCX)) are given, to f i r s t order, by (1 . 3 ) Ep(l) = Epip) + A Mpp•+• OCX) and where Mip *<E,\ (o) \ V / Ep(o)>. Rayleigh-Schroedinger perturbation theory s t a r t s with an operator and expressions are obtained i n terms of matrix elements of this operator. Perturbation theory i n terms of c h a r a c t e r i s t i c functions 3 s t a r t s instead with the c h a r a c t e r i s t i c function of the operator. An example of the c h a r a c t e r i s t i c function approach. for an i l l u s t r a t i o n of the c h a r a c t e r i s t i c function approach consider the 7 T-aolecular o r b i t a l s i n the cyclopropenyl s y s t e i i n which one carbon atom has been perturbed in some way. The Hiickel matrix, i n units of /3 r e l a t i v e to « as a zero of energy, i s (1.5) H(%) = 3 I I l o i I i o The c h a r a c t e r i s t i c function of t h i s operator i s (1.6) = Z'- 5z-2. -»- 2 and the unperturbed eigenvalues are (1.7) E,(o) = * a , Ea(o) = e 5(o) = "1. The perturbed eigenvalues are the roots of the c h a r a c t e r i s t i c equation (1.8) $U,V> = o (see Figure 1). Eguation (1.8) defines Z as an i m p l i c i t function of A . The f i r s t order perturbed eigenvalue can therefore be determined by f i r s t order i m p l i c i t d i f f e r e n t i a t i o n , using the standard formula (1.9) # = - h where the subscripts l a b e l the p a r t i a l d e r i v a t i v e s . For the non-degenerate eigenvalue Ei fa) we have (1.10) so that (1.11) 4E> da (o) =  4MI d^/s-e.(o) * = o z,C\) = E.0>) + 2. -h I. z=e,(o) = 2 + ±% + O(^). 3 <90* a) In the usual Rayleigb-Schroedinger theory t h i s saae r e s u l t obtained by f i r s t finding the unperturbed eigenvector |£,(o)> - and then evaluating the matrix element i n equation (1.3): 5 # ( o ) = <E,(o)|V|E.(0)> = ± . In general, the c h a r a c t e r i s t i c equation defines 2 as an i m p l i c i t function of A. The problem i s to convert t h i s i m p l i c i t information to an e x p l i c i t functional dependence, 2 = Perturbation theXory gives t h i s dependence, though not i n closed form, as a power series i n A. The c o e f f i c i e n t s of t h i s s e r i e s , namely the derivatives, can in p r i n c i p l e be obtained by higher order i m p l i c i t d i f f e r e n t i a t i o n . Returning to the example, i t may be observed that eguation (1.9) cannot be used, as i t stands, for the i n i t i a l l y degenerate eigenvalues EAC\) and £3 (3) , since both £ a n < J H vanish when evaluated at z = E x ( o ) - ~ l t 7i~o, The perturbation theory for i n d i v i d u a l eigenvalues i n degenerate cases, which depends on the order at which degeneracy i s l i f t e d , i s developed i n d e t a i l i n Chapter 2. The c h a r a c t e r i s t i c function approach does not y i e l d the perturbed eigenvectors d i r e c t l y . Indeed, the unperturbed eigenvectors, which form the basis i n which H° i s diagonal, need not be known, and t h i s can be an advantage i n c e r t a i n c a l c u l a t i o n s . The determinant in eguation ( 1 .1 ) i s invariant to basis change: 6 (1.12) £ = d e H z - H I = del-1 l / ( s - H ) ( J | = dehl*-U+HU\ ? where U i s a unitary matrix. However, information contained i n the eigenvectors can be found by taking derivatives of the energy with respect to appropriate perturbation parameters. In p a r t i c u l a r , i f the perturbation parameters are the matrix elements of the operator, themselves, the derivatives y i e l d the matrix elements of the density matrix. Kato (1949) developed a quantum mechanical perturbation theory i n terms of resolvents and contour i n t e g r a l s . Coulson (1940) had already used resolvents and contour i n t e g r a l s i n a quantum mechanical context. The resolvent was not i d e n t i f i e d as such i n Coulson*s work and was expressed not i n terms of operators but i n terms of c h a r a c t e r i s t i c functions. Resolvents and contour i n t e g r a l s provide the mathematical base for part of Chapter 3, and allow the r e l a t i o n s h i p between operator and c h a r a c t e r i s t i c function approaches to be exhibited i n a p a r t i c u l a r l y clear way. In t h e i r perturbation theory of conjugated systems, Coulson and Longuet-Higgins (1947) give f i r s t and second order formulae i n terms of the c h a r a c t e r i s t i c function. Their Hamiltonian H referred to the electrons i n the rr-system of a conjugated molecule, and t h e i r perturbation parameters were the matrix elements H r sof t h i s Hamiltonian i n the atomic o r b i t a l basis (Huckel parameters). They related f i r s t derivatives of the energy, with respect to Hrs, to the elements of the density matrix P s r, and second derivatives were re l a t e d to f i r s t derivatives of the density matrix, i . e . , p o l a r i z a b i l i t i e s . 7 However, they were interested i n derivatives of the t o t a l energy of the system. Thus the i r c a l c u l a t i o n s involved sums over a l l occupied o r b i t a l s and, in p a r t i c u l a r , sums over degenerate o r b i t a l s . Pukui et. a l . (1959) tr e a t s i n g l e non-degenerate o r b i t a l s but s t i l l sum over degenerate o r b i t a l s . A perturbation theory based on c h a r a c t e r i s t i c functions presupposes that these functions are known. (In an operator approach the matrix elements of an operator are assumed to be known.) For some classes of systems of chemical i n t e r e s t , the c h a r a c t e r i s t i c functions can i n fact be obtained. Two such classes are c e r t a i n conjugated systems and simple spin systems. Rutherford (1945,1951) organized the r e s u l t s of e a r l i e r workers giving general formulae for some *continuant determinants* or c h a r a c t e r i s t i c functions which occur i n simple Huckel type cal c u l a t i o n s . Hallion and Rigby (1976), using graph-theoretical methods, have obtained c h a r a c t e r i s t i c functions f o r an a r b i t r a r i l y weighted graph. This corresponds to having the c h a r a c t e r i s t i c function of a Huckel matrix f o r a r b i t r a r y <x> and /Srs. Coope (1966) wrote down some c h a r a c t e r i s t i c functions for simple spin systems. Here the c o e f f i c i e n t s of the ch a r a c t e r i s t i c function are scalars and are simpler i n structure than the matrix elements of the Hamiltonian of the spin system, which are in general t e n s o r i a l functions of angles. The development of a perturbation theory i n terms of c h a r a c t e r i s t i c functions i s motivated primarily by the i n t e r e s t of generalizing the work started by Coulson and Longuet-Higgins, f i r s t l y to apply to a r b i t r a r y perturbation 8 parameters, secondly to apply to any order of perturbation, and t h i r d l y , and i n p a r t i c u l a r , to include the treatment of degenerate eigenvalues. I t i s also 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, and by the fact that the c h a r a c t e r i s t i c function approach makes no e x p l i c i t use of wave functions, which can be tedious to calc u l a t e . 9 1.2 Thesis content. In Chapter 2 a perturbation theory for single eigenvalues i s developed by extending the i m p l i c i t function approach used in the example i n section 1.1. A perturbation hierarchy i s AmE derived from which a l l the energy derivatives TTfap(°) c a n D e a A found. P a r t i c u l a r cases of degeneracy l i f t e d i n various orders are considered. This approach i s the c h a r a c t e r i s t i c function analogue of Rayleigh-Schroedinger perturbation theory, and the Hayleigh-Schroedinger formulae can be obtained from th e i r corresponding c h a r a c t e r i s t i c function formulae, as shown i n Chapter 4., In Chapter 3, perturbation theory for single eigenvalues i s generalized. , Instead of si n g l e eigenvalues, a reduced c h a r a c t e r i s t i c equation for a set of eigenvalues i s considered, and a perturbation theory for the c o e f f i c i e n t s of t h i s reduced c h a r a c t e r i s t i c equation i s developed..These c o e f f i c i e n t s are computed by two methods; by contour integration and by an algebraic method. ,<The 'reduced c h a r a c t e r i s t i c equation* can be regarded simply as the c h a r a c t e r i s t i c equation of an e f f e c t i v e operator H, as introduced by Van Vleck (1929). In Van Vleck*s perturbation theory an e f f e c t i v e operator H i s constructed so that i t s eigenvalues are a subset of the eigenvalues of the perturbed operator H. The e f f e c t i v e operator H i s then found correct to some order i n A . , The roots of the '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 the perturbed eigenvalues of H, correct to the same order i n A. The Van Vleck approach i s represented by the path 1 0 H • H In chapter 3 however, the path H H i i •P » r i s taken, ., The c o e f f i c i e n t s of the reduced c h a r a c t e r i s t i c function r are found not i n terms of matrix elements of an operator but from the ( f u l l ) c h a r a c t e r i s t i c function JBy contour integration they are determined i n terms of derivatives of the c h a r a c t e r i s t i c function. . In addition, an algebraic method i s used to r e l a t e the c o e f f i c i e n t s of i ~ 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 also shown that the path H H £ r can be taken, without mention of an e f f e c t i v e operator H. Here the c o e f f i c i e n t s of r are found i n terms of matrix elements of H. In Chapter 4 the perturbation formulae derived 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 function, are applied to two i l l u s t r a t i v e examples. The f i r s t example i s a perturbation c a l c u l a t i o n , within the context of Hiickel molecular o r b i t a l theory, for an even-membered r i n g of carbon atoms. In the second example i t i s shown that the f a m i l i a r Bayleigh-Schroedinger perturbation formulae can be obtained from the corresponding c h a r a c t e r i s t i c function expressions. In Chapter 5 the rel a t i o n s h i p between energy derivatives and properties i s discussed with p a r t i c u l a r reference to simple spin systems. , Properties such as the spin polarization vector and the spin p o l a r i z a b i l i t y tensor are found using the c h a r a c t e r i s t i c function of the spin system and two examples of t h e i r use are given. 12 CHAPTER 2. PERTDSBATION THEORY FOB SINGLE EIGENVALUES.- Introduction. . In t h i s chapter a perturbation theory f o r single eigenvalues, i n terms of c h a r a c t e r i s t i c functions, i s developed. The theory i s r e s t r i c t e d to eigenvalues which are analy t i c functions of the perturbation A in some neighbourhood of IM=om For a f i n i t e dimensional Hermitian operator, i t s e l f an anal y t i c function of A i n some neighbourhood of |A/ = 0, the eigenvalues w i l l be an a l y t i c i f either 1. there i s only one perturbation parameter, or 2 . the unperturbed eigenvalue i s non-degenerate (see Appendix A). ,, 13 2.1 The method. • Consider an n-diaensional Hermitian operator Hft) which i s an an a l y t i c function of one perturbation parameter A. The c h a r a c t e r i s t i c function f(z>A) i s given by (2.1) $&>V s del- lz-H(A)J , which has the f a c t o r i z a t i o n (2.2) H*,V) = 7T(z-e .-0O), where the E, (A) are the eigenvalues of H(A) and are r e a l valued ana l y t i c functions of A ., The values of H for 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 are the eigenvalues of W(A). , The Taylor seri e s f o r any par t i c u l a r eigenvalue £p(3}, expanded about A-4 i s (2.U) E^A) = E p(o) + A ̂ EP(O) fe>(o) + ... . I t i s the object of t h i s chapter to determine the derivatives AJrT(o)t rr> = 1,2,: • The zeroth derivative or unperturbed eigenvalue £p{o) i s assumed to be known. , The idea behind the approach i s to regard the 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 defining Z as a function of A, for ̂  i n some neighbourhood of the unperturbed eigenvalue ^(o), and for A i n some neighbourhood of »This function wil be q-valued i f the unperturbed eigenvalue i s q-fold degenerate. For example, consider the 2-fold degenerate case, P(z,3)=ia-A% where e-ft>̂ = ea(o) =o. 14 A hierarchy of equations for the i m p l i c i t derivatives ^3 can ATT be obtained. Since d"2 I _ 4mE A-o an expression for 9L_^ obtained from the hierarchy, when evaluated at Z^ep(o) and ^ o , w i l l y i e l d the required derivative <±SP(O). Thus the terms i n the Taylor s e r i e s expansion (equation (2.4)) can be found and the perturbed eigenvalue E"P(A) obtained, correct to any order in A. The r e s u l t s obtained for a non-degenerate unperturbed eigenvalue with one perturbation parameter can be generalized to the case of more than one perturbation parameter. Degenerate eigenvalues, on the other hand, are guaranteed to be ana l y t i c only i f there i s one perturbation parameter. Thus the results obtained for degenerate unperturbed eigenvalues cannot be generalized to the case of more than one perturbation parameter. , 15 2.2 Orientation. , In c h a r a c t e r i s t i c function language quantum mechanical perturbation theory becomes the study of the way i n which the roots of a polynomial change as i t s c o e f f i c i e n t s are varied a l i t t l e , i . e . , as A varies (see Figure 2). Eatt) E5C*) Figure 2., An 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 function with 2 f o r two fixed values o f A . ? 16 Since the c h a r a c t e r i s t i c equation defines z as a function of /\, the t o t a l d i f f e r e n t i a l with respect to A i s given by» d_ = a. + d ? 9 . 3A dA 9z ' Taking the t o t a l 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 respect to A y i e l d s (2.5, ^ + f z S J | = O , where the subscripts denote p a r t i a l derivatives. Figure 2 i l l u s t r a t e s two t y p i c a l cases. The unperturbed eigenvalue E,(o) i s non-degenerate, whereas B.z(p)-Es(o) are 2-fold degenerate eigenvalues. For the non-degenerate eigenvalue, froa equation (2.5), we have For the degenerate eigenvalues, E,(o) and £3(0), the s i t u a t i o n i s quite d i f f e r e n t , since ^(E a (o),o)-0 (which i s the condition that £a(o) be at l e a s t 2-fold degenerate), and -PafeitP^Oj-o2. Osinq L'Hospital's rule a quadratic for i s obtained, lSee, for example, Goursat(1904), volume I, page 41, where the expression i s given not i n operator form but as d f = 3 f . 3 f . d_z 9* 32 dA - 2 I n general, for q—fold deqeneracy, the c h a r a c t e r i s t i c function has the f a c t o r i z a t i o n Thus a l l aixed derivatives of order less than the order of the degeneracy vanish, when evaluated at z^EpCo), ~X^ot i . e . . 17 (2.6) The two roots of t h i s quadratic, when a l l derivatives of -P are evaluated at z=£^(°)# A=o, are the derivatives ^ ( o ) and d§3(o). Equation (2.6) originated from the c h a r a c t e r i s t i c equation. The c h a r a c t e r i s t i c equation i s known to be s a t i s f i e d when z. i s equal to the unperturbed eigenvalue, £4(0), and when A=c. Equation (2.6) i s also s a t i s f i e d f o r these values of 2 and A. Thus i t i s only the derivatives of Ej(X) and E^CA) with respect to A, evaluated at 7\-of that can be found.. Indeed, the derivatives jr^ffa) with }\?o could only be found i f Epfa), /\*o was O A already known. .This i s not usually the case. 18 2.3 The hierarchy of perturbation equations. To obtain expressions for <LSP{O) , successive t o t a l derivatives of the c h a r a c t e r i s t i c eguation (eguation (2.3)), with respect to A, are taken: ,2.7, i f - -O 1 d ^ c/AA (2.10, d i f - > " i fcpfef.J&f=0. ^ - » - i > ' , + - a y J + . . . + ' 7 y 0 - n (<* = v, + ...+vn) (See comment 1, below, for explanation of summation i n eguation (2.10).) This set of equations w i l l be referred to as the perturbation hierarchy, or hierarchy for brevity. Equation (2.10) i s the n—th equation of the hierarchy. 19 2.3.1 Comments.7 1., The summation in equation (2.10) i s taken over a l l i n t e g r a l values of /Szo, Z;>09 ; = 1,~.yn, with the r e s t r i c t i o n that 2. The combinatorial c o e f f i c i e n t * ——= — i s the number of ways n objects (the A*s) can be partitioned into sets: /3 i n the f i r s t set, 1 i n the next Xt sets, 2 in the next ^ sets,...,n i n the l a s t )fn sets, where the order within each set and the order of the 'next ^ s e t s 1 i s unimportant. The t o t a l number of sets i s 1 + Jf, + ... +- y n = 1 +• <L . 3. Dimensionally we have and Since * = X,+...+in and p-+i*v+a)tn+...ntin^, i t follows that and each term i n the sum i s dimensionally correct., »See, for example, Abramowitz and Segun (1965), page 823. 20 4., Eguation (2.10) can be derived i n the following way*: the t o t a l d i f f e r e n t i a l dA 29i dA 9?; i s such that JA d5\k+l For example, & • k Ik •* - (Ir+a & {*$) W ( f & fc • The term ^ ^ J r - i s the r e s u l t of i acting on dz i n the product IV 1z da da 4?<L, the term (4^tl^-T i s the re s u l t of part of i , 2 . , d * 9* Ida/ UzJ * d* d3 3z- acting on =L i n the same product ^? 2 - , and so on. Cl e a r l y , ?L 3z da 3z clA" w i l l r e s u l t i n a sum of terms of the form Each «_? xs the r e s u l t of Q_ acting on ? £ , the 2 # i s the d a k da da"-' da"*' r e s u l t of d- acting on d z , and so on, down to being the d * d a k ' a \ d a * r e s u l t of 4- having acted on 4J£. Since t h i s 4* o r i g i n a l l y d * da dA iEquation (2.10) i s given, without proof, by Krasnosel'skii e t . al..(1969), page 329. In t h i s study equation (2.10) was f i r s t obtained i n the notation of section 2.4. 21 appears i n the product £_ t the power to which 2_ i s raised, to 3 z <*, i s given by ot — tf, •+- ... + Yn . In a l l , d_ operates n times. It has operated l%i-Wi+-nXn times to to produce M^^.The number of times ' l e f t over* w i l l fee the power to which iL must be raised, i . e . . To establish the combinatorial c o e f f i c i e n t , suppose each A i s l a b e l l e d , i . e . , — j > d_ d- ... d_ Each "\lt i n w i l l appear once i n each term of the sum. A l l permutations of the 'A; occur except that ^ j l r j ^ j f ; j ( c l l f ) ° r A.A a A3» ^ o r example, w i l l appear only once, not 3! times. Hence the order of the "next ^ sets,' as well as the order within each set, i s unimportant. On removing the l a b e l s , the combinatorial c o e f f i c i e n t i n eguation (2.10) i s obtained.. 22 2.4 An alternative notation. r I t i s possible to group the terms occurring in each equation of the hierarchy i n a way which brings out a structure which w i l l be made use of l a t e r . The terms i n the second equation (equation (2 . 8 ) ) can be grouped into a quadratic l i k e part, terms i n the t h i r d equation (equation (2.9)) can be grouped into a cubic l i k e part, and so on. The following notation i s used, (2.11) ^ = ILir)^^^^ > r-o where I'i] i s the binomial c o e f f i c i e n t , i n t h i s L R V " > (n-r-)iri notation T^P has the following property, rCn) c(n+>) r-Cn-i) a < 2 ' 1 2 > 3 * 2 P = ° d f a ' as can e a s i l y be shown.„ In t h i s notation the f i r s t equations of the hierarchy can be written 23 (2.15) To write the next equations i n a systematic way a new combinatorial c o e f f i c i e n t ( J w i l l be introduced, representing the number of ways •*-dk.±n objects can be partitioned i n t o k sets, with objects in the i - t h set, where the order within each set, and between sets containing the same number of objects, i s unimportant. For example, we have / 5 ) = _£L._L [3,3} a<3( ' With t h i s notation the next eguations of the hierarchy become (2.16) 24 (2.17) is) o +- /5\ 2 dW at t h i s point a pattern i s beginning to emerge. In the n^-th eguation there are terms l i k e (n- f o f , *...-»-«<|,71 where <>a, i-X,..fk.t <**t+ -^k^n and *tt — --- — (to avoid 'double counting*). I f by the symbol n , •• • olK we mean the sum of a l l terms above in which the 4;*s s a t i s f y the r e s t r i c t i o n s stated, then the n-th eguation can be written (2 .18) where m i s the largest integer such that m ^ a . J o x example, the 6-th eguation of the hierarchy i s 25 U J £ dAfc The notation i n t h i s section i s useful i n that i t provides a compact way of writing any p a r t i c u l a r eguation i n the hierarchy., In addition, the structure of each equation i s made cl e a r . Each equation i s simply a polynomial i n the desired i m p l i c i t derivatives 4!Zs , as equation (2.19) i l l u s t a t e s . The f i r s t derivative 4* i s embedded i n the symbol t p . The f i r s t eight equations of the hierarchy are given, i n t h i s notation, i n Table I. The combinatoral c o e f f i c i e n t s have been replaced by t h e i r numerical values. Table I The f i r s t eight equations of the 3. V + £' 0) dA a o 3 . 4:(3> d * a + ...+ Z d * * — o A-. d * a -*- d A 3 -f = o z da2 «daa/ .. .-MO 4?<ft + io £^W^\ + - - -d** wVwW .+5 4 " ^ - d ^ T a b l e I . c o n t i n u e d . 27 r(t,) r- ft** ,X- dV> d?r 6 0 -F£a d*£ + IStgM jo* W d̂ \ - j - - - a|,.d# 2 d a 5 dtf = o 7. f ^ ^ U f - f t d a * 2 dA* . is d A 5 d/f da*/Ida3/ m 2. & \05 iOS ^ 3 * d r 35 = o (da3/Ida*/ Table I. continued r#) ft) dl* * way 8 4 0 Cf̂ ff + ,°5 z da> W)\dr) ... + 70 K, £M die- z d^ — dr da* - a *o ̂ / ^ l 1^1+• - - • (dWld^J ...+ ...... 29 2.5 Special cases a r i s i n g i n perturbation theory.- Some s p e c i a l cases a r i s i n g i n perturbation theory w i l l now be considered: non-degenerate eigenvalues, 2-fold and 3-fold degeneracy l i f t e d i n various orders, and g—fold degeneracy. The equations of the hierarchy s p e c i a l i z e i n each case. 2.5.1 Bon-degenerate eigenvalues. I f Ep(o) i s a non-degenerate eigenvalue, i . e . , E P ( o ) ^ E,Co), f o r i^p, then ^ Z ( E . P ( O ) , O ) ^ O . In t h i s case a simple rearrangement of the equations of the hierarchy leads to e x p l i c i t expressions for a l l the derivatives <¥!!3p(o\r m - l,..,n. .. .. . Writing these down we have (2.20, z > (2.21, no) 30 (2.23) s**i?]-nf*4ni-fn I Ik, where a l l terms on the rig h t hand sides of these equations are evaluated at the unperturbed eigenvalue, i . e . , f o r 2. = £^b) and \-0. (Eguation (2.22) i s obtained from equation (2.10) and, i n the alt e r n a t i v e notation of section 2.4, eguation (2.23) i s obtained from eguation (2.18).) To f i n d 4!^P(o), a l l the derivatives of lover order must have already been calculated from the precede ding eguations i n the sequence. The way i n which the equations of the hierarchy are used, for non-degenerate eigenvalues, i s represented by the schema i n Figure 3. In the case of several perturbation parameters, /X = (X>-->\), the perturbed eigenvalue E^ft) has a multiple perturbation, or Taylor s e r i e s , expansion, t L a (2.24) = M°> + Y X ff^) + Y X2i l l r (o) + ... , >--> ;i=-> J In t h i s case we have (2.25) ^P(o) = - f*: / 4 , J hi (2.26) fSr (p) = - [ 4 , + 4 * ^ + 4% ^ 4 -4 - ' a« 3 ? ] /-P 2 * 5 93, where a l l terms on the rig h t hand side are evaluated f o r z = and 3 =<V,The higher mixed derivatives ^Ef -(o) can be obtained 31 Figure 3. Guide to the perturbation hierarchy fi] Hon—degenerate eigenvalues. X dlE T <fe>(o) da In t h i s figure and i n Figures H and 5, n represents the n-th eguation of the hierarchy. by modifying the expression for 4H§p(o)., To i l l u s t a t e t h i s modification, consider the t h i r d , one parameter, derivative, ,2.27, J 5 w = ^'^g^^W^tygj&fe* _ Each term i n t h i s expression contains A three times. In the corresponding expression f o r — C o ) each term w i l l contain Aa and A 3 each A;, <=',̂,.3 occurring only once. Thus, the following modifications must be made: 32 (2.2.8) -IU — ^ A a a (2.29) 3^d?_^ fell ̂  + £ 1 , 3 i +-P311 3z and so on. The combinatorial c o e f f i c i e n t , the 3 i n eguation (2.29), i s the number of nays \, Aj and A3 are permuted within the term 4 ^ <Jz-. As discussed i n section 2.3.1, the mixed p a r t i a l derivatives ^ z X , ^ a n d ^2^3, only count once. In general, then, to obtain ~?Ep^(~o) for some pa r t i c u l a r value of n, having written down the expression for 4%p(p), one simply makes the appropriate modifications. A l l t o t a l derivatives d^Z? become p a r t i a l derivatives 3 Zr-- and each term in the expression f o r 4-£p(0) i s rewritten by l i s t i n g a l l allowed permutations of the A a ; within each term. The combinatorial c o e f f i c i e n t * — /*' ̂  , i s just the number of such permutations. ., 2.5.2 2-fold degeneracy l i f t e d i n f i r s t order. I f E P(o) i s a 2-fold degenerate eigenvalue, i . e Epto)=Ep+,Co)?E,Co) for then -F 2(^pCo) j 0) - O , but $z.*(Ep(o),o)-* o. In t h i s case the f i r s t eguation of the hierarchy, eguation (2.7), i s i d e n t i c a l l y zero., The second, eguation (2.9), a quadratic i n 43, gives JA 33 (2.30) efl'fa) = ± Jfea)a-4*4 This i s i n agreement with the resu l t obtained using L*Hospital's rule (eguation (2.6)). I f these two roots are d i s t i n c t , the degeneracy i s l i f t e d in f i r s t order, i - e « # 4%o) d§ '̂(o)w Substitution of these two d i f f e r e n t values for d%\t-oe<f>> i n t o t n e remaining eguations of the hierarchy then yi e l d s a l l the derivatives j~£Co) and i p5»+i (0 ) , m>I. For example, the t h i r d eguation of the hierarchy gives the second derivatives as or, i n the notation of section 2.4, When the two d i f f e r e n t values for dl|z=E,£o)» i . e . , dj&Ycrt and §̂p+'(b)» are substituted i n t o these eguations, the corresponding second derivatives sLff and <L5~i(o), are obtained. „ The quadratic whose roots are given by eguation (2.30) i s (2.32) dr 6>) 3 £ the same quadratic that would be obtained i n Rayleigh-Schroedinger perturbation theory when diagonalizing the (2-fold) 'degenerate block*. 2.5.3 2-fold degeneracy not l i f t e d i n f i r s t order. The condition that 2-fold degeneracy i s not l i f t e d i n f i r s t order i s that tbe two roots given by eguation (2.30) be equal. This w i l l be the case i f the discriminant of the quadratic vanishes, which i s the case when the Hessian of i s zero, i . e . , when (2.33) ( ^ J - to) = O I f t h i s i s the case, then (2.34) &(o) = dJFP+'fo) - - & da da K (In Rayleigh-Schroedinger perturbation theory t h i s corresponds to f i n d i n g the •degenerate block* diagonal with the diagonal elements equal.) I f degeneracy i s not l i f t e d i n f i r s t order, then the t h i r d equation of the hierarchy, eguation (2.15), da *~ —3-, turns out to be i d e n t i c a l l y zero, as can be shown by taking the t o t a l derivative of equation (2.33) with respect to A and using equations (2.33) and (2.34). The fourth equation of the hierarchy, equation (2.16), reduces to 35 (2.35) 3^.- /d^j + 4 f 2 , J i a + V = o The two roots of t h i s equation y i e l d 4lfp(o) and d*&>WoW They are da* da' given by A = o (2.36, ± / t W * I f these roots are d i f f e r e n t , the degeneracy i s l i f t e d i n second order and substitution in the higher order eguations of the hierarchy s i l l then y i e l d the higher derivatives, 42Ff>(o) and da*" I f degeneracy i s not l i f t e d i n f i r s t or second order, then the s i x t h eguation of the hierarchy i s the reguired quadratic i n the t h i r d derivative 41? . In t h i s case, i n the s i x t h d * 5 eguation, 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 derivatives 4^3 » <">4- are zero, i . e . , the s i x t h eguation reduces to da 3 - * da3 * UT)W) a guadratic i n ^f?-. ,In summary, the way i n which the equations d a 3 of the hierarchy are used for 2-fold degeneracy, l i f t e d i n various orders, i s represented by the schema i n Figure 4.  37 2.5. 4 3—fold degeneracy and degeneracy p a r t i a l l y - l i f t e d . -. I f Ep(°) i s a 3-fold degenerate eigenvalue, i . e . , Ef>C°)~ Ep-ni°)= £p+ato)^E;to) f o r /V/>, p+i, p+a, then •Re- = = £z2 = ^ = £ \ a = o when a l l derivatives are evaluated at 3=o. The f i r s t eguation of the hierarchy which i s not i d e n t i c a l l y zero i s the t h i r d , eguation (2.9). vThis eguation reduces to evaluated at ^Ep0>) and feo, are the f i r s t derivatives a cubic i n . The three roots of t h i s eguation, when da I t can happen that degeneracy i s only p a r t i a l l y l i f t e d i n f i r s t order. Consider the case i n which da </a d-x For the eigenvalue Ept which i s no longer degenerate, the substitution of dJfrfa) i n t o the fourth eguation of the hierarchy w i l l y i e l d the second derivative 4^3P(O)» Substitution of both dfpco) and 4^P(O) into the f i f t h equation of the hierarchy <̂a c/Aa w i l l y i e l d the t h i r d derivative <L3p(p\, and so on. Thus the Taylor series for Ep(X) i s obtained. However, the s i t u a t i o n for EfmCA) and 5>*a£l) i s quite d i f f e r e n t . Since the f i r s t derivatives 4&>*'6>) and ^ ( o ) are equal, E>+,<>) and E^,£A) are s t i l l 2-fold degenerate to f i r s t order. Consequently, to f i n d the second 38 derivatives, a quadratic equation must be solved. , The f i f t h equation i n the hierarchy i s the required quadratic i n dV- In summary, the ca l c u l a t i o n for 3-fold degeneracy i s given by the schema i n Figure 5. 39 Figure 5 . Guide to the perturbation hierarchy ( i i i l 3—fold degeneracy. (P)~] 'True. 1 4- dr 1 dr False 6 da* 40 2« 5.5. q—fold degeneracy. ; I f EP(o) i s a q— f o l d degenerate eigenvalue, then In t h i s case and consequently the f i r s t non-vanishing eguation of the hierarchy i s the q-th. This follows because c e r t a i n l y ^^<% i f n<c^, so that a l l terms i n the summation in equation (2.10) w i l l be zero, when evaluated at z ~ f=PCo), %~Om I f n-<^, then °t+/?<el i f any i-3.,...,nt and <<->fi^% i f a l l / = a,... v/?. Hence for the non-zero terms i n the sum, and the q-th equation i n the hierarchy reduces t o The roots of t h i s equation are the q f i r s t derivatives, which may or may not be d i s t i n c t . , On writinq out the l a s t two terms of the summation i n equation (2.37), (2.38) + fi%^ = 0 , i t can be seen that i f the q-fold degeneracy i s not l i f t e d i n f i r s t order ( i . e . , i f eguation (2.38) has egual roots when evaluated at z = ^ o ) , 3^o), then the f i r s t order s h i f t of a l l g eigenvalues i s merely 41 (2.39) dE''(o) = « A fz1 z=e;{p) A=o £ since i s the sum of the g roots of eguation (2.38). In t h i s case a g-th order eguation i n the second derivatives i s required, and t h i s i s the 2q-th equation of the hierarchy. The 2q-th equation i s the f i r s t to contain the required quantity T 2 ^ / s _ ? A > which occurs i n Co) I f degeneracy i s l i f t e d i n f i r s t order, then substitution of each root of equation (2.37), i . e . , each of the d i f f e r e n t values of # into the higher order eguations of the hierarchy yi e l d s the corresponding higher order derivatives. 42 2.6 Summary. , I t has been shown how the t o t a l derivatives of the c h a r a c t e r i s t i c equation of a perturbed operator, taken with respect to a perturbation parameter, y i e l d expressions for the perturbed eigenvalues to any order i n /\. As made clear by the notation of section 2.4, the perturbation hierarchy i s , i n e f f e c t , a sequence of polynomial equations i n the i m p l i c i t derivatives 4Cj^ , whose c o e f f i c i e n t s are the p a r t i a l derivatives A7T of the c h a r a c t e r i s t i c function, with respect to z and a. These p a r t i a l derivatives are evaluated at z=£ P6>) and 7i-o and the equations solved to give the eigenvalue derivatives 4£jh(p)m Some important s p e c i f i c examples have been discussed i n section 2.5. For each s p e c i f i c example, the eguations of the hierarchy s p e c i a l i z e i n a pa r t i c u l a r way, when evaluated at z=^o), '\-ot i . e . , some terms vanish. Table II l i s t s the non-vanishing terms, i n the f i r s t few eguations of the hierarchy, for the sp e c i a l cases considered i n thi s chapter. 43 Table II The non-zero terms of the equations of the perturbation hierarchy f o r some s p e c i a l cases. A l l derivatives of f are evaluated at z. - Ep(o), a=o. The equations i n t h i s table have been obtained from the equations of the hierarchy i n Table I as indicated by the numbers on the l e f t hand side. Non-degenerate eigenvalues. Ose Table I, 2 — f o l d degeneracy, l i f t e d i n f i r s t order. 3 . + o + 3 + i d 2 - o dfla aa* Ida*/ . . . -4- 4- -Fj^ d 3 ^ - o da5 2 - f o l d degeneracy, l i f t e d i n second order^ 2 . i z o = O Ida*/ Ida'/ T S > , (5) da3 (dWldav 44 Table I I . continued. 2 - f o l d degeneracy, l i f t e d i n t h i r d order. 2. t?° - O 2 dA2 z Idav Ida*] 3 ' w - 10 1̂  / U S i ^ O 3—fold degeneracy, l i f t e d i n f i r s t order. 3 . ( - O 4 . &Ka*& = o z z da2 Id̂ J * d / l 5 Table I I . continued. 3 - f o l d degeneracy, p a r t i a l l y l i f t e d in f i r s t order. r 0> 3 . + z o = 0 For ePC*). tt. -r?o •+ b izi <L? = O da* For Ep^)and £*»«C\i da* Idâ i 3 - f o l d degeneracy, not l i f t e d i n f i r s t order. = O da* Ida2 /s^Vd^f (da*/ 46 CHAPTER 3. THE REDUCED CHARACTERISTIC FOBCTIOS. , I n t r o d u c t i o n . ? I n t h i s c h a p t e r a t t e n t i o n i s f o c u s e d n o t on i n d i v i d u a l e i g e n v a l u e s b u t r a t h e r on s e t s o f e i g e n v a l u e s . T h i s i s a g e n e r a l i z a t i o n o f t h e s i n g l e e i g e n v a l u e p e r t u r b a t i o n t h e o r y o f C h a p t e r 2. , P e r t u r b a t i o n s e r i e s a r e o b t a i n e d n o t f o r one e i g e n v a l u e b u t r a t h e r f o r t h e c o e f f i c i e n t s o f a r e d u c e d c h a r a c t e r i s t i c f u n c t i o n . 47 3.1 The reduced c h a r a c t e r i s t i c function., The term 'reduced c h a r a c t e r i s t i c function' w i l l be used to denote a polynomial Kz,A) of degree <̂  i n z, whose zeros are a subset, E P ) .. } E p n . l f of the zeros of the ( f u l l ) c h a r a c t e r i s t i c function £(z,3) w i . e . , a subset of the eigenvalues of H: (3.1) r ^ , A ) = 7T (z-e,-) The c o e f f i c i e n t s of 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 function the i n d i v i d u a l eigenvalues can be found as the roots of the 'reduced c h a r a c t e r i s t i c eguation* (3.2) r(^.,7i) = O - The set of eigenvalues w i l l always be taken to be a complete degenerate or nearly degenerate set (or any number of such s e t s ) . The dependence of the eigenvalues i s then i n general more complicated than that of the c o e f f i c i e n t s c,-. The c o e f f i c i e n t s of the reduced c h a r a c t e r i s t i c function, ct)....)cc^, are symmetric functions of the eigenvalues, E P ,  Ep+i-i> namely. 48 (3.3) Co = (-IP'TV1 (E;) , > /= /» CL = -1 21 E/ Accordingly, knowledge of the c o e f f i c i e n t s i s eguivalent to knowledge of the eigenvalue power sums m - i , . ; ^ , defined by P+%-1 (3.4) 5 m = \ {E;)m The two are related by Hewton^s* formulae: (3.5) • • s, + c, = O In p a r t i c u l a r , (3.6) c, =-S( , Two methods w i l l be used to compute the c o e f f i c i e n t s of the reduced c h a r a c t e r i s t i c functions for non-degenerate, 2-fold and 3-fold i n i t i a l l y degenerate eigenvalues.,, In the f i r s t method, contour integration i s used to determine perturbation »See, for example, Turnball (1952), page 72. series for the eigenvalue power sums, 5 m . The second method i s algebraic and yi e l d s perturbation series for the c o e f f i c i e n t s c;, d i r e c t l y . The idea of the reduced c h a r a c t e r i s t i c function i s cl o s e l y related to a f a c t o r i z a t i o n theorem of Beierstrass*. This, i n turn, i s cl o s e l y r e l a t e d to the theorem (Kato, 1949) that the t o t a l projection onto the subspace a r i s i n g from an i n i t i a l l y degenerate eigenvalue i s an a l y t i c i n A, in some neighbourhood of |A/=<3. , The reduced c h a r a c t e r i s t i c function can be factored out of the ( f u l l ) c h a r a c t e r i s t i c function: The f a c t o r i z a t i o n theorem t e l l s us that as long as i n i t i a l l y degenerate eigenvalues are considered together, i . e . , i f only complete degenerate sets are included, then the c o e f f i c i e n t s of the corresponding reduced c h a r a c t e r i s t i c function are an a l y t i c functions of the perturbation parameters (%,-.•,/X^) i n some neighbourhood of \/\\~ O. Conseguently they have a Taylor series, or perturbation s e r i e s , of the form which i s convergent f o r a l l /\ i n some neighbourhood of \%\-0, The perturbed eigenvalues themselves do not, i n general, have such a serie s expansion i f they are i n i t i a l l y degenerate and i f there i s more than one perturbation parameter. For example, suppose f o r a 2-fold degenerate problem with two perturbation parameters, that the reduced c h a r a c t e r i s t i c *See, f o r example, Osgood (1913), page 181. (3.7) (3 . 8 ) 50 function i s (3.9) rUA) = Z a -(X-K) - The c o e f f i c i e n t s of t h i s reduced c h a r a c t e r i s t i c function have ( t r i v i a l l y ) perturbation s e r i e s of the form given i n eguation (3.8); the perturbed eigenvalues, ±J A?-*- 1̂ , do not. Such cases cannot be dealt with by Bayleigh-Schroedinger perturbation theory or i t s c h a r a c t e r i s t i c function analogue developed i n Chapter 2. appendix A discusses the conditions for a n a l y t i c i t y of eigenvalues.„ 51 3. 2 The eigenvalue power sums as contour i n t e g r a l s . In t h i s section the eigenvalue power sums, Sm, are expressed i n terms of contour i n t e g r a l s . , Since the c h a r a c t e r i s t i c function has the f a c t o r i z a t i o n i t follows that i-l This function has a pole at each eigenvalue, with residue equal to the degeneracy. I t follows that (3.11) S,*, = <p 2T ft c/z; , r>l where i s a contour i n the complex Z^-plane enclosing the relevant eigenvalues, E p , . . , Ep+^-t • ,f The integrand has no pole on the contour , for A i n some neighbourhood of fAf = £>; i t i s an analytic function of A, i n t h i s neighbourhood./It follows that the sm9 and thus the c / # are also analytic functions of A, i n t h i s neighbourhood. Some insight into the nature of equation (3.11) can be obtained by recognizing the guantity as the trace of the resolvent, G--(z-H)'1, t For an Hermitian operator, or generally i n the absence of eigennilpotents, 52 G- = V- — • i where f) i s the projection onto the subs pace of the i-th eigenvalue.,It follows that (3.12) Tr Cr = V - J = 4 m Eguation (3.11) can therefore also be written i n the operator form (3.13) Sm = JJ-S-ZT G- . am JfA I f the trace i n eguation (3.13) i s taken before integration, eguation (3.11) i s obtained, and the reduced c h a r a c t e r i s t i c function can be found i n terms of the ( f u l l ) c h a r a c t e r i s t i c function. This analysis i s car r i e d out i n section 3.3. I f the trace i s taken aft e r i n t e g r a t i o n 1 . the reduced c h a r a c t e r i s t i c function i s found i n terms of the operator matrix elements. This i s discussed in section 3.4., Equation (3.13) could have been obtained a l t e r n a t i v e l y , s t a r t i n g from the r e l a t i o n . *Se see that the sm, and thus the reduced c h a r a c t e r i s t i c function r , are defined even i f H i s i n f i n i t e dimensional, i n which case the c h a r a c t e r i s t i c function f i t s e l f i s not defined. 53 r-% between the projection onto the associated eigenspaces, and the resolvent. Eguation (3.13) then follows from 5 ^ = T r H " P . The poles of the integrands i n equations (3.11) and (3.13) are located at the eigenvalues of H(2). f o r [M&o, under the influence of the perturbation, these eigenvalues are not known. To compute these i n t e g r a l s , the integrands are expanded about IA|=o. The poles of each term in the expansions are then located at the unperturbed eigenvalues of which are assumed to be known. The expansions are integrated term by term and, from the residues at the poles inside the contour ^<^» the Taylor s e r i e s expansion of S,,,, about lAI = o # i s obtained. F i n a l l y , the c o e f f i c i e n t s of the reduced c h a r a c t e r i s t i c function can be found using Hewton's r e l a t i o n s . 54 3.3 Coptour i n t e g r a l method: f-»r. 7 The f i r s t few terns of the Taylor series expansions for the c o e f f i c i e n t s of the reduced c h a r a c t e r i s t i c function r~ w i l l now be determined, from the contour i n t e g r a l s , i n terms of derivatives of the ( f u l l ) c h a r a c t e r i s t i c function f. In p a r t i c u l a r , the reduced c h a r a c t e r i s t i c functions for non-degenerate, 2 - f o l d # 3-fold, and nearly degenerate eigenvalues of a f i n i t e dimensional Hermitian operator are constructed., Expansion of the integrand i n eguation (3.11) about |A(=o y i e l d s (3.14) For s i m p l i c i t y , i n the following, i t w i l l be assumed that there i s only one perturbation parameter*., Then the integrand i n eguation (3.14) becomes L -P L -P -pa J t r ~r -P* J J where -j- and i t s derivatives are evaluated at \ % - ° . The general term i n the expansion of the integrand i s simply Conversion of a one-parameter expression i n t o the corresponding many-parameter expression i s straightforward, though tedious (see Appendix B). 55 2 " £ 41 I t i s of i n t e r e s t to note that since al. & = _3i_ -P* d i f f e r e n t i a t i n g eguation (3.14, with respect to A and then integrating by parts, yields 56 3.3.1 Non-degenerate eigenvalues. The Taylor series for a non-degenerate eigenvalue has already been given i n section 2.5.1. I t can also be derived by the present method, and t h i s c a l c u l a t i o n i l l u s t r a t e s the general procedure., Let the unperturbed eigenvalue E-p(o) be non-degenerate. Without l o s s of generality, set Ep(o)^ol- The reduced c h a r a c t e r i s t i c function, with the perturbed eigenvalue Ep0\) as i t s zero, can be factored out of the ( f u l l ) c h a r a c t e r i s t i c function, for A i n some neighbourhood of \A\-0, i n the manner (3.16) fe,^) - r C z ^ - g C ? ^ ) = ( 2 r + - c , V g ( z , : \ ) , where r ( £ p C \ ) , ^ = O, ^£pCh)/X)^o. The derivatives appearing i n the Taylor ser i e s expansion, given i n (3.15), are a l l evaluated for W -o. To determine them, the rel a t i o n s (3.17) f (z,o) = ^ and (3.18) f 2n(o,o) = n ^ n - . ( o , o j ; n ^ o are used. Eguations (3.14), (3.15), and (3.17) give *This simply means that the zero z.p of the reduced c h a r a c t e r i s t i c function i s the guantity EPQ-\) — E > ( O ) , i . e . , the perturbation of E p(A)., The derivative of £Cz,a) evaluated at £=e pco) , where E > ( P ) * o , i s the same as the derivative of f {%i-EeW, A) evaluated at z=o. Thus, i n any application the axis s h i f t z-* z-«-£>(o) need not be carr i e d out; the derivatives are simply evaluated at z = E P ( P ) , instead of at z=o. 57 (3.19) where: the contour encloses E F ( p ) l but no other unperturbed eigenvalue., The f i r s t two terms i n the integrand, ik and 3 9 do not have poles inside the contour f£i and do not contribute r r to the xntegral. , The t h i r d term, - rz *a , has a pole of order one. The residue gives (3.?o) Epc\) = + since g(o,o) = ^ ( ° , ° ) , a sp e c i a l case of r e l a t i o n (3 . 1 8 ) . , This r e s u l t i s i n agreement with the r e s u l t obtained i n section 2.5 . 1 , and i s simply the usual formula for an i m p l i c i t f i r s t d e r i v a t i v e . . The higher order terms are found i n a s i m i l a r way. The r e s u l t i s generalizable to many perturbation parameters, as described i n section 2 .5.1. 58 3.3.2 2-fold degeneracy. Suppose the unperturbed eigenvalues E e ( o ) = E p + , ( o ) form a 2-fold degenerate pair. Without loss of generality, set Ep(p>= tpt.,(o)=o. For i n some neighbourhood of [W-O, the c h a r a c t e r i s t i c function has the f a c t o r i z a t i o n (3.21) -P(Z,S\) = r-C^A) where r ( C ; ( ^ ^ = o r <̂  A)*0, i =p, P^-I. Now (3.22) = z : a g f c o ) while (3.23) (o,o) n ( n - i ) g 2 n - i :'j n > o / which replace eguations (3.17) and (3.18). Equations (3. 14) , (3.15), and (3.37) give where the contour encloses £p(o> = Ep-n(p), but no other unperturbed eigenvalues. The f i r s t term in the integrand has a simple pole at z=o with residue 2 = o = |A| This vanishes by eguation (3.23), i . e . , because { - C o f o ) - o , since "Ep<p) i s 2-fold degenerate. The f i r s t order term has a simple pole at z-o , with residue <*-f̂ > Z^ * * a n d a t h i r d order pole, with residue To f i r s t order, therefore. 59 0.25) s, = - 3 a*k\ + ocr). For S,, the z e r o t h order term i n the in t e g r a n d of eguation (3.14) has no p o l e , the f i r s t order terms have ze r o r e s i d u e s , and the l e a d i n g c o n t r i b u t i o n i s of order A. E v a l u a t i o n of the r e s i d u e s i n (3.15), using eguation (3.23), y i e l d s The higher order terms i n s, and s a are found i n a s i m i l a r way, though the a l g e b r a becomes more complicated. Table I I I (page 6 3 ) g i v e s S, and SA c o r r e c t t o &C??)» F i n a l l y , Hevton's formulae can be used t o g i v e C, and c A, the c o e f f i c i e n t s of the reduced c h a r a c t e r i s t i c f u n c t i o n . To l e a d i n g order t h e s e are ( 3 . 2 7 ) c , =• A ^ f y / ( 3 . 2 8 ) c a = / T a b l e IV (page 6 6 ) g i v e s them c o r r e c t t o The e x p r e s s i o n f o r c2-=i Bp .Sp*., i s , i n g e n e r a l , s i m p l e r than t h a t f o r s A ~ ( E p f + {Ep+tft a t each order i n A . , To l e a d i n g order, 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 t h e r e f o r e 60 (3.29) r ( 2 E ^ ) = Z a -+- Q'X , where the derivatives of -f- are evaluated at z = o, /AJ=o. t The roots of t h i s eguation give the perturbation of the eigenvalues correct to f i r s t order, and these roots are i n agreement with the r e s u l t obtained from the hierarchy i n section 2.5.2., However, the ca l c u l a t i o n d i f f e r s from that i n Chapter 2 i n two ways. F i r s t l y , no assumption i s made about the a n a l y t i c i t y of the eigenvalues £p(fa) and Ep+,(2)- Secondly, the formulae obtained f o r the c o e f f i c i e n t s of the reduced c h a r a c t e r i s t i c function do depend on the order at which degeneracy i s l i f t e d . Formulae for the eigenvalues considered separately do not depend on the order at which degeneracy i s l i f t e d , as was shown in sections 2.5.2 and 2.5.3. The perturbed, i n i t i a l l y 2-fold degenerate, eigenvalues EPC\) and Ep+,C\) are the roots of the reduced c h a r a c t e r i s t i c eguation, i n t h i s case a guadratic, given by (3.30) E ; ( A ) = - £ L ± J ( C ) a ~ ; = p ^ p + , •3. A Hith no loss of generality, the unperturbed eigenvalues have been set egual to zero..For the Hermitian operators considered, the perturbed eigenvalues are r e a l . , Consequently the discriminant, (c,f — 4 - c J # must be even in A, to leading order, i . e., (3.31) ( C ) a - 4 - 0 , = QCtf") t 1,0.,... . I f , therefore, the A term of the discriminant i s zero, then 61 the A 3 term must also he zero., In Table V (page 67) the discriminant i s given correct to GOT). For degeneracy not to be l i f t e d i n f i r s t order the condition i s that the discriminant vanish through second order, that i s , from Table ¥, (3.32) ((4a)a - 4* fa*) / z = 0 ^ , = O , i n agreement with section 2.5. 3 . ,If t h i s condition holds, then, to f i r s t order, (3.33) Epfr) = Ep+i(?) - 3 &3 I 4 * U=o=/a/ and the f i r s t derivatives of the i n i t i a l l y degenerate eigenvalues with respect to A exist, even i f A represents more than one perturbation parameter. In a sense, the 2-fold degenerate eigenvalues are behaving l i k e a non-degenerate eigenvalue to f i r s t order and thus have f i r s t order derivatives with respect to A..Furthermore, i f condition (3.32) holds, then the vanishing <A3 term of the discriminant reduces to (in the notation of section 2.4) (3.34) t 2 e . This was already shown to be i d e n t i c a l l y zero i n section 2.5 . 3 * , In t h i s case the fourth order term i n the discriminant reduces to (3.35) /»(o)»a — 3 m 5 m ' i n agreement with eguation (2.35), for the case of degeneracy 62 not l i f t e d i n f i r s t order. Suppose i n equation (3.30) i s required correct to second order i n A.,Clearly c, must be known correct to second order. However, the order to which the discriminant must be known depends on when degeneracy i s l i f t e d . I f degeneracy i s l i f t e d i n f i r s t order,- then the square root of the discriminant i s of the form Jtf^+T^+GCV-) = A C e a + Ae 5 ) ' / a -+- 0(AS) and the discriminant must be known correct to t h i r d order i n A . , I f deqeneracy i s not l i f t e d i n f i r s t order, then < the square root of the discriminant i s of the form and the discriminant must be kown correct to fourth order i n A*. 63 T a b l e I I I . 2 - f o l d degeneracy: The e i g e n v a l u e power sums. 8 1 " — " "(t f ̂  8 f c k r W a < W V * 4 4 » W f a S ^ (V)' • + f 4M 4* - H M * 4* * « 4J»J +... • •-+ ?4*a4» 4* * ,64M 4*;t 4* * 5 2 4M 4a* 4a +- • T a b l e I I I , c o n t i n u e d . 7 16 W^a)-% W ^ f * «fka)*£»J ...+ © f t - ) . 65 Table I I I , continaed. 3! L 4 - ' + SI % ^(t^ * * * * * ~ M ^ % ^ ^ h * 4 a ] + • ----»- ^ 4 * ^ H 3 a fta * % -&A*(£*f ] + • .... J. ^ (^n^n +... 2-fold degeneracy; The c o e f f i c i e n t s of the reduced c h a r a c t e r i s t i c function. ; -s, 21 I 3> I E 67 Table V 2-fold degeneracy: The discriminant of theredaced c h a r a c t e r i s t i c function, ; ( c , ) a - 4 - c a = . 5 - L * 1 fz' (fa) (£>)s J • • + 5 £ 5 r & j ' . + < ^ 4 1 j , ] + . . . 68 3.3.3 3-fold degeneracy.„ The method of c a l c u l a t i o n i s s i m i l a r to the 2-fold degenerate case,except that here {3.36) r ( ^ , A ) = 23-*- -1-C3 f (3.37) £(z,o) = H 3 3 ( ^ , 0 ) , and (3.38) ^n(o,o) = n ( n - ' ) ( o - 3 ) g z n - 3 ( o , o ) , n > o - As before, the unperturbed degenerate eigenvalues have been set egual to zero. Table 71 gives slt 3^ and S3, and Table VII gives c , , cj9 and c^, correct to t h i r d order. , 69 table VI. 3-fold degeneracy: Tbe eigenvalue power sums.- ... + f 2 j ^ 4 " * * * 5 4 * 3 4 * " J + - 7 ^ { | ( 4 * / ^ * f 4 * W £ * * - ' 4 » f e i 4 » " f - + 4 * a (4M)* + ' s(4M)* 4 M ] + • [ » 4 a » * f € * 4 * 4 * ( £ M ) 3 J + (4*> L * J 3'- l te ( f ^ f i + | ? [ - i a 4 3 ' + ^ 2 [ 3 f e 4 M 4 a ' * 5 V 4 ^ 4 M + 3 4 * 4 s ] + 7 0 T a b l e VII., 3 - f o l d degeneracy: The c o e f f i c i e n t s of the reduced c h a r a c t e r i s t i c f u n c t i o n •2-' / 4 3 J c 5 31 1 4 3 j 71 3 . 3 . 4 Near-degeneracy. : A set of non-degenerate unperturbed eigenvalues EPC°), . - -t Ep+c^-i Co) i s said to be nearly degenerate i f the eigenvalues are much closer to one another than to the rest of the spectrum. & reduced c h a r a c t e r i s t i c eguation can be constructed with a nearly degenerate set of eigenvalues as roots, an advantage i s that the convergence r a d i i of the Taylor serie s expansions f o r the c o e f f i c i e n t s of t h i s reduced c h a r a c t e r i s t i c function w i l l , i n general, be larger than the convergence r a d i i of the Taylor s e r i e s expansions for the nearly degenerate eigenvalues considered separately. The reasons f o r t h i s improvement w i l l be discussed i n section 3 . 6 . For s i m p l i c i t y , consider 2-fold nearly degenerate eigenvalues EpC'X) and Ep+,C\), with a f i n i t e i s o l a t i o n distance , i . e . , ( 3 . 3 9 ) j Ep(o) - £p„ (o) J d > O . The reduced c h a r a c t e r i s t i c function, with EpC*) and Ep+,fa) as i t s zeros, can be factored out of the ( f u l l ) c h a r a c t e r i s t i c function. In t h i s case, however, the unperturbed c h a r a c t e r i s t i c function i s given by ( 3 . 4 0 ) £ fe,o) = ( 2.~ EpCo ) )0 - Ep+, to) g feP) . Thus the terms i n the integrand (3.15) have poles located at both z= EP(o) and z = EP+i(o). , Since 72 Ep-,(o) Figure 6. The contours tt,t. f p t i , i . and f p , a i n the complex z-plane. (3.41) where the paths of integration are shown i n Figure 6, the c o e f f i c i e n t s of the reduced c h a r a c t e r i s t i c function (3.42) rteA) = 2 a + C , Z + C A can be constructed by treating EpC*) and E ^ f a ) a s non-degenerate eigenvalues. This i s accomplished using the non-degenerate formulae given in section 2.5.1 i n the symmetric function expressions (3 .3 ) . The advantage i s that the convergence r a d i i of the Taylor series for c, and c2t obtained i n t h i s »ay # are generally greater than the convergence r a d i i of the Taylor series for each separate eigenvalue. 7 3 3 . 4 Contour i n t e g r a l method: H-frr. Eguation ( 3 . 1 3 ) , with the trace taken aft e r integration, provides a simple route to expressions f o r the reduced c h a r a c t e r i s t i c function i n terms of operator matrix elements. . The concept of an e f f e c t i v e operator H i s not used. Suppose ( 3 . 4 3 ) H = H ° + -XM . The resolvent can be developed about A=o i n the following nay: ( 3 . 4 4 ) Gr = —L_ = — 1 — ! z - H z-H° z.-Hf-'M where Gra i s the unperturbed resolvent ( 3 . 4 5 ) • ' G-„ = „ 2 - H I t e r a t i o n on eguation ( 3 . 4 4 ) gives the expansion oo ( 3 . 4 6 ) Gr = 2 ^ G ° ^ V G B ^ * For perturbation of the p-th eigenvalue, Gr0 must be expressed i n terms of the reduced resolvent f o r that eigenvalue. Let Pp(<=>) be the projection onto the subs pace of the unperturbed eigenvalue E p ( o ) . i f E p(o) i s g-fold degenerate, then fp(o) i s the projection onto the whole g-dimensional subspace of Ep(p)J.. .f Ep^_l ( o ) . The reduced resolvent S , evaluated at Ef>C°)r can be written 74 5 = ( l - P p C o ) ) . ! • ( / - P p C o ) ) , E p ( b ) - H ° provided the inverse operator i s defined appropriately. I t has the spectral resolution The k-th power of S has the expression (3.48) and i t i s convenient to define the *zeroth power* as the negative of the projection, i . e . , by (3.49) S ° = - P p ( o ) . Then the expansion of about z = EpO>) can be written (3.50) G-0 = 2-^^ L^-£p(o)) O , and t h i s y i e l d s (3 . 5D Q 0 ( V ^ = J J - . ) p , + - + ^ - ^ . ^ w y » - ^ - ^ sn\i...\fs""', f o r the terns i n the expansion (3.46). Without l o s s of generality, put E P(6)= O. ,The residue of ZmG-t appearing i n eguation (3.13), with a contour p̂,<̂ enclosing ?=o, but no other poles of G-„, w i l l come from a l l the terms i n the expansion of Q- which contain (z-Ef,Co))~*'m~h'^ . The eigenvalue power sums are found to be 75 (3.52)  S» = TrZI B°° ' (3.53) SA = TV ̂  V C< N ) (3.54) 5 f n = T r ^ A" M C n > , where (3.52a) 8 ( n > = + y x Sk'V.. V 5 k " ' (3.53a) C C n > S k ,V . . . V 5 k n " Here indicates a sum over a l l integer k;>o, with the r e s t r i c t i o n k,-h ...-+• k n^j =• j . The compactness of these expressions i s a l i t t l e deceptive..For example, 8^ i s made up of three terms, (3.55) 6<a>= S°\IS°\/S,+ S 0 V S ' V 5 0 + S ' V S 0 V 5 ° , but only those terms with matched ends contribute to the trace. I f 5 ' appears at one end, and 5 J at the other, the ends are matched i f either k,-=ro=kj or k ; * o * J c j . For example, the term S 0 V S ' V 5 0 76 i s the only term i n (3.55) with matched ends. 3.4.1 I l l u s t r a t i o n : 2 - f o l d degeneracy.- Suppose £,(o) and Ea(o) are a 2-fold degenerate pair, i . e . * E, (o) = Ej(o). .in t h i s case P,(o) = | t,(o>X E,(o)) -+-• I Ê (c?)X E^Co) J and one finds that where (3.56) Q p p , " ' k - < E p ( o ) l V S , V S i V . - \ / S k V | £ p , ( q ) > . For s i m p l i c i t y , assume that the degenerate block has already been diagonalized, i . e . * that (3.57) Vpp. = S"pp« <E p(o ) |V|E p(o)> , and that, with no loss of generality, E,(o)=0. one finds that S, = E,0) •+• EAfa) and = A a ( (v M f + (v M ) a ) + A?((v„-yuXQ;1-<?;a))^... v In t h i s notation i t i s easy to see that i f degeneracy i s not l i f t e d i n f i r s t order, i . e . , i f V,, =. , then the discriminant 77 i s to leading order. One has I f degeneracy i s not l i f t e d i n f i r s t order, the factors ( V M - V ^ ) vanish and & This expression for the discriminant 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 function. • In summary, with the trace i n eguation (3.13) taken a f t e r integration, the c o e f f i c i e n t s of the reduced c h a r a c t e r i s t i c function are found i n terms of matrix elements., This i s i n contrast to the r e s u l t s obtained by taking the trace before integration, where the c o e f f i c i e n t s are found i n terms of ( f u l l ) c h a r a c t e r i s t i c function derivatives, as discussed i n section 3.3. 78 3,5 ftn algebraic method: f - ^ r . 7 The expressions for the c o e f f i c i e n t s of the reduced c h a r a c t e r i s t i c function, given i n Tables IV and VII, are lengthy. They sere tedious to compute via contour integration, and sere i n f a c t obtained only i n d i r e c t l y , v i a Newton's formulae. & more d i r e c t derivation of these c o e f f i c i e n t s , which has c e r t a i n advantages, i s given i n t h i s section., The c o e f f i c i e n t s of r , the c,-, are not found i n terms of derivatives of the ( f u l l ) c h a r a c t e r i s t i c function -P, but instead are found d i r e c t l y i n terms of the c o e f f i c i e n t s of -P, the d;. The re s u l t i n g expressions are more compact than those i n terms of derivatives. The method w i l l be explained i n d e t a i l for the case of 2-fold degeneracy. Consider the c h a r a c t e r i s t i c function of an n-dimensional Hermitian operator H = W(a), (3.58) - P ( 2 , A ) = H ° + a ( z n - ' H - . . . + a n . Suppose E,(o)= EjtCo) are a 2-fold degenerate pair of eigenvalues. Then the ( f u l l ) c h a r a c t e r i s t i c function has the fa c t o r i z a t i o n with for A i n some neighbourhood of |A/=0, The unperturbed eigenvalues £#(o) and E 3 ( ° ) are set egual to zero by an axis s h i f t so that, to leading order,, 79 (3.59) E;M-= &(*) = (3.60) EjCA) = OO) , j /,a . Leibnitz* theorem* f o r the d i f f e r e n t i a t i o n of a product of functions y i e l d s <3.61, = +_^L_ r e ^ + ( ^ T ' i ' I*"*- This expression may be rearranged, with a l l functions evaluated at 2--o, but I'M^o, to give To obtain expressions for the c o e f f i c i e n t s c, and c a i t i s f i r s t noted f o r ?=o, ['Afeo, that and -Pza = <2g -f- 2 - c # g z -+- c a g za . The r a t i o s and f/£a y i e l d (3.63) C, = / |+ C , C j g - + - j _ Q, C j g a ^ (3.64) c a = a f / J + <Vgz + C a g z a ) .. These eguations, together with eguation (3.62), form a set of linked i t e r a t i v e eguations f o r c, and c ^ , since the leading *See, f o r example, ftbramowitz and Segun (1965), page 12. 80 order estimates are (3.65) -PCo.a) = OCca) = O(^) , • 4^(o,a) = Die) = Ofr) t where n i s the degree of -f i n 2. To leading order, eguations (3.63) and (3.64) give d i r e c t l y (3.66) c, = and (3.67) c a = 3 £ ( A * > These f i r s t estimates for c, and c a may be substituted back into the r i g h t side of eguations (3.63) and (3.64). .Eguation (3.62) i s used to express 2-derivatives of g i n terms of c,, C j , and the z - d e r i v a t i v e s of f . In t h i s way a second order estimate for c, and c A can be obtained, and t h i s process may be continued to higher orders. An expression for c, or c correct to some order i n A includes some higher order terms., This i t e r a t i v e scheme y i e l d s expressions f o r the c o e f f i c i e n t s of the reduced c h a r a c t e r i s t i c eguation i n terms of Z-derivatives of -P evaluated at f = o , but at YX\-OM The results given i n section 3.3, previously obtained by contour integration, can be found by expanding ^ t C ° , A ) about YAI-OZ 81 (3.68) ^(°A) = ^t(P,o) +^2; (0}0) -»-... . I f there i s only one perturbation parameter, then (3.69) c, = ZSkCQAKom^ 2[£z(o,o)+A^M + Ofr)y[^(oj0)^OC\y\ = n -P^ ( 0 , 0 ^ ocx), -Pza(o,o) and, s i m i l a r l y , (3.70) = 3 a -fy + ew) } f 2 a (0,0) i n agreement with eguations (3.27) and (3.28) of section 3.3.2. However, more compact expressions r e s u l t by noting that (3.7D \ t C o y \ ) = fc-' a-n-t: (*) • To leading order the r e s u l t i s (3.72) C , - and (3.73) = . In t h i s way the c o e f f i c i e n t s of the reduced c h a r a c t e r i s t i c function, the ct- , can be related d i r e c t l y to the c o e f f i c i e n t s of the ( f u l l ) c h a r a c t e r i s t i c function, the a,-. The c o e f f i c i e n t s of the reduced c h a r a c t e r i s t i c function for other degeneracies 82 can be found i n a s i m i l a r nay, Those f o r non-degenerate, 2-fold and 3-fold degenerate eigenvalues are given i n Table VIII, correct to OC?), OCX), and 6(7?), respectively. In the c a l c u l a t i o n for 2-fold degeneracy i t was assumed that both E,(X) and E^CX) were of order \, E,(o) and E*(o) being zero. , In f a c t , even i f £ ( 2 ^ ) i s a c h a r a c t e r i s t i c function of a non-Hermitian operator, with eigenvalues expressible i n convergent Puiseux s e r i e s , the formulae given i n Table VIII are s t i l l applicable, ft useful example i s given by the c h a r a c t e r i s t i c function of the non-Hermitian matrix (3.7fl) HC\) = 1 * X 1 1 o A OO The c h a r a c t e r i s t i c eguation i s (3.75) -fe,A) = 2 5 - 3 z a + 2 ( l - ^ a ) + f ^ o with unperturbed roots (3.76) E,(o)= O, EA(o) = ESC°) = /. To construct the reduced c h a r a c t e r i s t i c eguation for the 2-fold degenerate pair, E^ and E 3, the axis i s s h i f t e d , The c h a r a c t e r i s t i c eguation becomes (3.77) £(x+/,*) = X 3 H - X A ^ X C ~ A - ^ > ~ ^ - O , - with (n=3) (3.?8) an = , Qn_, = ~(X+V) a n - a - 1 > = ' , °-n-rr> - O , m > 3 . From Table VIII, the c o e f f i c i e n t s of the reduced c h a r a c t e r i s t i c function for the perturbation of and Es are 83 13.79) c, = -IM*):-*-^*)* + A.]-*- 0(A 3 / a) = o + ©Cx 5 * ) , In t h i s case the leading order grouping i n the table i s ( 3 .80 ) c, ^ Gfr«)\+.QMl+e(7Js'*):!+... C2 = OCX) \+O0?«)i+OCtf) : + and ( 3 . 8 1 ) E,-ft) = X ± A** +- O f t 3 * ) ^ , 84 Table VIII. Algebraic r e l a t i o n s between the 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 - functions., Terms which are the sase leading order are grouped together. Non-degeneracy. +a(Qn)3(an-af „ ( a n f gn_5 J + . . . (a n-.) 5 (a„_,)^ c, = Qn ; + ( c i n ) a a n - s 2-fold degeneracy. -f (Qn-j)*<Xn-3- <*n Qn-3 ' - f e i f S e d t + ^ ( Q ' H f +.-• (a„_a)^ (a„- 3F; (<w^ (a„-a) 5 -+S(Qn-.r(ar,-s)5_ 5 a o ^ - l f a n - S + 13 Qn lan-«f Qn - S q n - ^ (<*n-a)7 (a n - a ) ^ (Q f l- a)5 - J O anl Q"-'f(Q / 7-s) 5+ ( Q n f Q n - 5 - 5 ( a n ) a a n - 3 Q n-^ ^ (Q n_ 2)fc ( a n - a ) 3 ( Q n - a ) 4 " ( Q r £ i 9 ^ f ! + OCX 5). C,= Qn-i 85 Table VIII. continued. : 3-fold degeneracy.- I " " ( Q r J _ 3 ) S (CXn^y 9^zL « Qn-» _ ' + <9£\3) c 3 = ' B o + (9 fa3) . 86 3.6 Convergence radii.. An advantage of setting up guantum mechanical perturbation theory i n terms of resolvents and contour i n t e g r a l s i s that estimates of convergence r a d i i can be e a s i l y made, as shown by Kato (1949). Central to the calculations i n t h i s chapter have been the contour i n t e g r a l formulae. (3.11) Sm - - J — $ 2 L M & , and, i n the operator form, (3.13) S m = Jh. AzTGr d z . r-l These in t e g r a l s are independent of the shape of the contour tp,c^t depending only on the residues of the poles of the integrand that f£><t encloses., The a n a l y t i c i t y of s^-s^fr) depends on the a n a l y t i c i t y of the integrand, for z on ^<^, and for A i n some neighbourhood of MI=o. The convergence radius fo r the power series expansion of S m about (31 » o can be estimated from the convergence radius for the expansion of the integrand, with z on JejLa., This estimate of the convergence radius depends on how tp*^ i s drawn. The best lover bound for the convergence radius, i . e . , the largest lower bound, reguires the 'best* contour Vp)C^* Suppose H = HtV3\/. Prom eguation (3.46) the integrand i n eguation (3.13) can be expanded about IAI= O as follows. 87 oo (3.82) = ^(WG-S . n - o This series converges for (3.83) H A V G o l U l . By Schwarz^s inequality, IJAVG-oll £ ftlJIVJI.IIGbll , and consequently a s u f f i c i e n t condition f o r convergence i s (3.84) (A/J/V/MlG-oll C I . I f the sup norm i s used, then the sp e c t r a l resolution of the unperturbed resolvent implies that (3.85) \\Qx0\\ = [A(z)]~"' , where A(z) i s the distance of z from the nearest unperturbed eigenvalue. Rearrangement of equation (3.84) then gives (3.86) /A/< ! » //vii .//G-Jj / / v « The best bound given by t h i s equation i s obtained when the contour keeps Ate) as large as possible f o r 2. on f p ^ . ,The smallest value that A(z) attains for z. on t^^ 0 establishes the bound on | A/ . , Let be a c i r c l e enclosinq ^Just one eigenvalue E,(%) and centered on E.^o). Suppose the nearest unperturbed eigenvalue to E,(p) i s E^(o) and that t h e i r i s o l a t i o n distance i s d,)SL . L e t the radius of the c i r c l e f"^, be The value of A(z} i s dl>3/a. everywhere on t h i s c i r c l e . This i s the largest value A(z) can take for z on the r e a l axis, between E,(p\ and EjpS (see Figure 7). Thus f^, i s a •best* contour, fi lower bound for the convergence radius of the power 88 series f o r the perturbed eigenvalue E,(A) i s thus d,^ / \ \ t proportional to the i s o l a t i o n distance. The same i s true for If and E2C\) are nearly degenerate eigenvalues, and are treated together, a reduced c h a r a c t e r i s t i c function with roots EC\) and E^(A) can be constucted. , The integration contour w i l l be some tf^ (not necessarily a c i r c l e , see Figure 7)*, The smallest on i s (3.87) d a j 3 / a ( > . A lower bound f o r the convergence r a d i i of the expansions of the c o e f f i c i e n t s of the reduced c h a r a c t e r i s t i c equation i s thus greater than that for the expansions of the nearly degenerate eigenvalues when considered separately. Kato (1949) points out that i n general these lower bounds 8 9 cannot be improved on..This i s shown by the following simple example: ( 3 . 8 8 ) H ° = — I o o / 0 / 1 o Here the e i g e n v a l u e s of H = H*vaV are whose power s e r i e s , ±[\ + i? + -l , are convergent f o r |AJ Eguation ( 3 . 8 6 ) , with I and IIVII=/, gives the same bound.. I t i s not possible, i n general, to obtain an e x p l i c i t bound, such as that given by eguation ( 3 . 8 6 ) , from eguation ( 3 . 1 1 ) , since A i s usually embedded in a complicated way i n the c h a r a c t e r i s t i c function . However, an i m p l i c i t bound i s obtained by writing ( 3 . 8 9 ) f GE.,*) = -PC^o) - A v f e / \ ) and expanding the integrand of eguation ( 3 . 1 1 ) according to ( 3 . 9 0 ) Z." I * •F This series converges for ( 3 . 9 1 ) which yields the i m p l i c i t bound 90 (3.92) W < j fU,o> For the example of equation (3.88), the c h a r a c t e r i s t i c function i s (3.93) fCz,*) - H a - l - A * . The best c i r c u l a r contour l~^tl around the unperturbed eigenvalue "/ i s the c i r c l e of unit radius, given by z = - | + c'e , o ± 9 ^ 2-n. The i m p l i c i t bound (3.68) becomes (3.94) m < J^j = j o±e+Qrr , from which, as before, i t i s found that m < i . In general, however, equation (3.92) i s not p a r t i c u l a r l y useful, although i t does, i n p r i n c i p l e , provide a means for the c a l c u l a t i o n of a bound. To actually use equation (3.92), the minimum value that |^^'^~ | attains, for z on some contour tp^, i s required for a range of values of 3. I f I V(Z,^)( min. on ^ ) ^ for a l l \ such that o^lJ\)<6 where B i s r e a l and p o s i t i v e , and 91 / V ( Z , C \ ) I min. ( z on r£j%) y f o r \%\ \ S , then 6 i s the bound on YXl • .• I t i s worth emphasizing that i f an expansion of the integrand of (3.11) or (3.13) converges for a l l z. on some contour, ^ say, then t h i s implies that EP(\) does indeed l i e within the contour. Consider (3.95) EPM = & * Y v t h p,t The p-th term i n the summation can be expanded about H= Ep(p) ^ = ' (EP(A)~<zP(o)\n which expansion converges i f (3.96) ) EP(A)-EP(O)1 < \^-EpCo)\ . The above inequality implies that Z i s further from EpO>) than EPCA) i s ; thus E P ( A ) l i e s within . An a l t e r n a t i v e perspective on the convergence properties can be i l l u s t r a t e d by returning to the example (3.88), for which the eigenvalues were In the complex ^-plane, E, and Ez become egual when A=i/. These singular points, about which E, and Eu are not a n a l y t i c i n ^ , 92 l i e on the c i r c l e of convergence of E, and E A i n the 3-plane*. Thns the expansions of E, and F a about \AI-0 are convergent f o r /A/ < \±t) - I . I t i s f o r 7i = ± \ that a contour can no longer separate E, and E3t since they have become egual*. In general the only non-analytic points occur at such degeneracies. Thus, knowledge of the degeneracy points i n the complex A-plane would imply exact knowledge of convergence r a d i i . From t h i s point of view, the convergence r a d i i are large r f o r the c o e f f i c i e n t s of a reduced c h a r a c t e r i s t i c function than for i n d i v i d u a l eigenvalues, because the degeneracies i n the complex plane between the eigenvalues associated with the same reduced c h a r a c t e r i s t i c function are not relevant to the convergence of the c o e f f i c i e n t s of the l a t t e r , but do l i m i t the convergence of the eigenvalues. *See, for example, Harkness (1898), page 178. 1 However, for a=/ the eigenvalue e, i s s t i l l the only eigenvalue within the contour r?t, i n the complex z-plane. Thus, while the convergence of an expansion of £, implies that e, l i e s within r*h, , the converse i s not true. 93 3.7 Summary. A reduced c h a r a c t e r i s t i c function has been introduced. By taking the trace i n eguation (3.13) before integration, the c o e f f i c i e n t s of the reduced c h a r a c t e r i s t i c function are found i n terns of derivatives of the ( f u l l ) c h a r a c t e r i s t i c function, taken with respect to z and A . In contrast, by taking the trace i n eguation (3.13) after integration, the c o e f f i c i e n t s are found i n terms of operator matrix elements. An algebraic method i s developed which rel a t e s the perturbed c o e f f i c i e n t s of the reduced c h a r a c t e r i s t i c function d i r e c t l y to those of the ( f u l l ) c h a r a c t e r i s t i c f u n c t i o n . y F i n a l l y , convergence r a d i i are estimated i n terms of both operators and c h a r a c t e r i s t i c functions. 94 CHAPTER 4., TWO ILLUSTRATIVE APPIICATIOSS. Introduction., In t h i s chapter : the r e s u l t s of Chapters 2 and 3 are applied to two i l l u s t r a t i v e examples. In the f i r s t example an even-membered ring of n carbon atoms, one of which i s perturbed i n some way, i s considered i n the context of Hiickel molecular o r b i t a l theory. Expressions f o r the energies of the perturbed Tf-molecular o r b i t a l s are found to second order and these general r e s u l t s are applied to the benzene r i n g , where n—6. In the second example the usual Bayleigh-Schroedinger perturbation formulae i n terms of matrix elements are obtained from the c h a r a c t e r i s t i c function. The operator and c h a r a c t e r i s t i c function formulations of perturbation theory can be r e l a t e d , formally, through the contour i n t e g r a l expression f o r the eigenvalue power sums (eguation 3.13). The d i s t i n c t i o n there was the order i n which trace and i n t e g r a l operations are performed.. In section 4.2, however, the i n t e r r e l a t i o n i s considered from a d i f f e r e n t point of view. I t i s shown how the well-known perturbation formulae i n terms of matrix elements can be derived from the c h a r a c t e r i s t i c function expressions. An application of quite d i f f e r e n t character, namely to spin systems, i s considered in the following chapter. 95 4. 1 A perturbation c a l c u l a t i o n for even-membered rings of carbon atoms. T The Hiickel matrix f o r the 7Y-molecular o r b i t a l s of the perturbed n-atom r i n g , i n units of the int e r a c t i o n parameter/3, r e l a t i v e to the carbon atom parameter <* as the zero of energy, i s the nxn matrix r M o . . . . o i 1 0 1 o . o O 1 O 1 O - - (4.1) o 1 o o O 1 O 1 The c h a r a c t e r i s t i c function can be written, i n the notation used by Coulson (1938), as (4.2) -Pfe,*) = M n - A P n _ , , where M„ i s the c h a r a c t e r i s t i c function for an unperturbed n-fold ring (4.3) and where (4.4) w h i l e P0_, function f o r = 2 . co.5 & t i s the c h a r a c t e r i s t i c (n-/)-dimensional l i n e a r chain: (4.5) f^-i = 5inn&/sm9 . From eguation (4.3) the unperturbed eigenvalues are seen to be (4.6) E m ( o ) = Z c o s * * ? r , m - o , m , g . There are two non-^degenerate unperturbed eigenvalues, (4.7) E„(o) = , E n / a(o) = and ( j - i ) pairs of 2-fold degenerate unperturbed eigenvalues, (4.8) E w C o ) = E_„(o) y m~+±,: an 96 The perturbation c a l c u l a t i o n reguires z.- and W-derivatives of -P(^,^)., according to eguation (4.4) the ^.-derivatives are related to the ©-derivatives by 0 . 9 ) 9. = de 3_ _ -I 9_ 3£ d£ 29 Xc,\n& 29 The f i r s t few derivatives of PCz,^) , evaluated at \-Ot are accordingly as follows: ( M . , o , f a = = , (4.11) t a _ n F Sinne-cos0_ n cosn^l - - n L , (4.12) -IU = 3co5£.P_a . fisinngr|-n*] = , (4.13) " ^ a A b ~ ° > 2 . The non-degenerate eigenvalues. r From Table I (or eguation 2.20) the f i r s t derivatives of the non-degenerate eigenvalues E0{X) and En/2(x) are (4.14) J f . ( o ) = _ & I = — i = o Q . n } 'a. From Table I (or eguation 2.21) the non-degenerate formulae for the second derivatives are 97 which, i n t h i s case, must be evaluated by taking the appropriate l i m i t s . For E0(A) , the l i m i t 8-+0 y i e l d s (4.16) 4^ = n a (4.17) 4* = n a ( n a - ± ) / 6 , •z and thus (4.18) &>/<>) = J _ ( o a - l ) S i m i l a r l y , the l i m i t y i e l d s (4.19) dl§n/a(o)= ~_L ( n a - / ) daa 6n a In summary, the two non-degenerate eigenvalues have the perturbation expansions (4.20) E; (X) = E,(o)+ % 4E>(o) - £ dV,(o) + ... dA 21 d3 a = ±2. -h A ± X(n*-1) + , . . , > = O (+), n (_) n / i n 1 • . a 98 The 2-fold degenerate eigenvalues. To find expressions for the perturbed, i n i t i a l l y 2-fold degenerate eigenvalues, the approaches of both Chapter 2 and Chapter 3 v i l l be used. F i r s t the c a l c u l a t i o n i s done by tre a t i n g each eigenvalue separately, using the perturbation hierarchy of Chapter 2. . For comparison, the c a l c u l a t i o n i s repeated, but instead of considering the eigenvalues separately, a reduced c h a r a c t e r i s t i c function f o r each pair of degenerate eigenvalues i s constructed, using the re s u l t s of Chapter 3., From Table I I (or eguation 2.30), the f i r s t derivatives of the 2-fold degenerate eigenvalues are (4.21) jE»(o) = ± Jp&F-t&frl ,m = t±, ,±{%-i) . Since $^-Ot these reduce to (4.22) i&n = O , ( o ) n One eigenvalue of each pair i s unchanged i n f i r s t order., These unchanged eigenvalues v i l l be labeled in the following by the negative integers , -(§-/•). The other member of each pair has a f i r s t order s h i f t that i s twice the s h i f t of the non-degenerate eigenvalues. These s h i f t e d eigenvalues w i l l be labeled by the positive integers *•>,. The second derivatives, from Table I I (or eguation 2.31), 99 are given by 2«5n(oJ A^O Substitution of the f i r s t derivatives (4.22) into t h i s equation, together with the expressions (4.11)-(4.13) for the p a r t i a l derivatives, y i e l d s (4.24) 4lEm(o)^ O ..-(g-l) and <4.25) —*(o) = A _ 4 d a 2 3 n * ^ a X cos0 n Accordingly, to second order, the i n i t i a l l y 2-fold degenerate eigenvalues become (1.26) EmC\) = leasing! + 6CV) , m= -/, ..,-(§->) j <».27> r „ m aasaar + a + t - O f t 5 ) , . 2 0 sin2' m = +1,...,+(%-!) Since a l l second and higher ^-d e r i v a t i v e s of -f vanish, i . e . , (4.28) f^^b =. O , b > X, 100 i t can be seen, by inspection of the perturbation hierarchy, that the eigenvalues which were unchanged in f i r s t order are i n fact unchanged i n a l l higher orders, i . e . , (a.29) Em(n) =. O , n > \ , m=-*j--,-(%-l) . Reduced c h a r a c t e r i s t i c equation c a l c u l a t i o n . - Instead of treating the eigenvalues separately, a reduced c h a r a c t e r i s t i c eguation, (4.30) H a + C , Z + - C a = f o r the i n i t i a l l y 2-fold degenerate pairs of eigenvalues can be constructed. The non-zero terns i n the expressions f o r ; the c o e f f i c i e n t s c, and c a i n Table I? are l t a J - 2 U 5 4 * ( 4 * / 4 * 4 ^ J (4.32) C, = O -f- BW), where a l l derivatives are evaluated at the unperturbed eigenvalue, i . e . , for 6=aJ2Jy, m=±l,..,±|0_i), a=Q. On substituting the p a r t i a l derivatives (4.11)-(4.13), the reduced c h a r a c t e r i s t i c equation i s found, to second order, to be The roots of t h i s equation. 101 (4.34) z. = O = 2 _ 2 a c o s «2n y i e l d the perturbations of the i n i t i a l l y 2-fold degenerate s eigenvalues, i n agreement with the expressions (4.26) and (4.27) previously obtained for the eigenvalues i n d i v i d u a l l y . , Application to benzene. ; For the benzene r i n g , n=6 and eguations (4.6) and (4.7) give the unperturbed eigenvalues (4.35) E 0 (o) = +X , = , E 3 Co) - -X . Eguations (4.20), (4.26), and (4.27) become (4.36) E „ W - ^ - I , - i + » - X ^ ecx*) , Eguation (3.86) gives a lower bound for the convergence r a d i i of these perturbation 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 distance of a l l the eigenvalues i s 1, and the bound becomes (4.37) W < VX . 102 Algebraic method. • The expansions (4.36) can also be obtained by using the algebraic r e l a t i o n s given i n Table VIII. The c h a r a c t e r i s t i c function for the fT-molecular o r b i t a l s of benzene, i n which one carbon atom i s perturbed, i s (4.38) #2 , A ) = H f e-2 5A - b^ •+• 4-z-3A + 9z*~ 3 z 3 - 4- . On s h i f t i n g the axis by putting 2-X+l, t h i s becomes (4.39) £(x+l,3)* X6+X5M)^^-^)^3(-4--t^+xaHa+aA)+4.xA, i . e . , E±i(°) has been s h i f t e d to the value zero. Table VIII shows how the c o e f f i c i e n t s of the reduced c h a r a c t e r i s t i c eguation (4.40) z 2 + C , Z + C a = 0 , for the i n i t i a l l y 2-fold degenerate pair E±,io), can be computed from the c o e f f i c i e n t s of powers of X i n eguation (4.39), i . e . , from the c o e f f i c i e n t s Table VIII therefore gives (4.42) c, ^ : + ( Q 5 f a 3 _ a±*z (9(?f) = -2 - 2f 4- <9ft3) 3 54- and 1 0 3 (4. u 3) Co =- ^ i - f - as °-3 - + eCx) Thus, to second order the reduced c h a r a c t e r i s t i c eguation for the perturbation of i s (4.44) Z 2- - i - - | -2? ) = 0 , which yie l d s (4.45) = * / correct to second order, and i n agreement with the expressions (4.36) for E-,(A) and £+,fa) obtained e a r l i e r . 104 4.2 D e r i v a t i o n o f t h e B a y l e i g f a - S c h r o e d i n q e r p e r t u r b a t i o n f o r m u l a e f r o m 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 t a r t i n g f r o m 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 x p r e s s i o n s , t h e w e l l - k n o w n p e r t u r b a t i o n f o r m u l a e i n t e r m s o f m a t r i x e l e m e n t s c a n b e o b t a i n e d . , T h e s e a r e t h e u s u a l B a y l e i g h - S c h r o e d i n g e r f o r m u l a e , e x c e p t t h a t t h e s e r i e s o b t a i n e d i n t h i s way a r e c o r r e c t l y o r d e r e d , c o r r e s p o n d i n g t o t h e s e r i e s o f K a t o ( 1 9 6 6 ) . T h e q u e s t i o n o f • o r d e r i n g * i n t h e d e g e n e r a t e c a s e i s n o t made c l e a r , f o r e x a m p l e , i n t h e u s u a l d i s c u s s i o n s i n q u a n t u m m e c h a n i c s t e x t b o o k s . , T h e B a y l e i g h - S c h r o e d i n g e r m a t r i x e x p r e s s i o n s w i l l f i r s t b e r e c a l l e d . I f E,(o) i s a n o n - d e g e n e r a t e e i g e n v a l u e o f H° a n d i f (4 .H6) H f o ) = H ° + A V , t h e n t h e p e r t u r b e d e i g e n v a l u e E , f a ) i s g i v e n b y (<1.»7) E.C0 = E,(o) H - ^V , , + X V +... ..^\Y ^V.SVSI _v„V- V,rVra ( I n t h e e n e r g y d e n o m i n a t o r s E; = Ei(.°X), I f c ,(o) i s o ^ - f o l d d e g e n e r a t e , s a y , t h e n t h i s f o r m u l a c a n s t i l l be u s e d p r o v i d e d d e g e n e r a c y i s l i f t e d i n f i r s t o r d e r . T h e u s u a l p r e s c r i p t i o n i s t o d i a g o n a l i z e t h e d e g e n e r a t e b l o c k , a n d r e d e f i n e t h e z e r o t h o r d e r e n e r g i e s a p p e a r i n g i n t h e d e n o m i n a t o r -r - 105 as . new patting where E,(o), i-\t..}c^ are the i n i t i a l l y q^-fold degenerate set. Op to t h i r d order any term in a summation with r or s egual to w i l l be multiplied by the matrix element V r S , r,ŝ .<̂ . since the degenerate block has been diagonalized, these elements are zero. This i s eguivalent to r e s t r i c t i n g the summations i n eguation (4.47) so as to exclude the indices However, the fourth order terms i n (4.47) are only apparently so i n t h i s degenerate case. , They do include summation indices and these give r i s e to terms which are actually t h i r d order i n A . The terms i n eguation (4.47) must accordingly be regrouped, to obtain a correct A-ordering. For a 2-fold degenerate l e v e l , when degeneracy i s l i f t e d i n f i r s t order (\f„ Vz X ) , t h i s prescription leads to ( 4 . 4 8 ) EXA) = E,(o) H- ̂ N / „ * X Y V"-Vr, f l y V.rX-sVsi _V„V Vr. (In the energy denominators, E, = E;(o).) The perturbed eigenvalue EaQX) i s given by a s i m i l a r expression. The l a s t summation of the X term i n eguation (4.48) comes from the f i r s t summation of the A term i n 106 equation (4.47). This reordering i s necessary since the tero V,r V r s VstVfc. can have s^z without X * or N/j, appearing in the numerator. In t h i s case the denominator contains E.(O)+ *v„ ~EA(O)- - A y , a = Oi ( V , r V „ ) and the numerator i s V l f - V r i V a f c V f c , . Thus part of the fourth order term i n equation (4.47) becomes t h i r d order i n equation (4.48). These formulae can be obtained from the c h a r a c t e r i s t i c function approach, given an e x p l i c i t expression f or the ch a r a c t e r i s t i c function i n terms of matrix elements.. In terms of the matrix representation of \J, i n the basis i n which H° i s diagonal, the c h a r a c t e r i s t i c function, JTC^A) = d c l - J z - w ° - ^ v | , can be written jH-E,(o)-}V() W , a . . . . W , n (4.49) Ĉz,̂) = d c t Z - E „ ( O ) - } V , nn The determinant may be expanded to give 107 (4.50) -P(Z,A) = rite-EjCo)-^,) - t y V/ r 5V s r7T (E_E (-(b)-^V ( j)-f-.. ^sr^tu^t TY(*E-E;(o)-M{i)+.. ...+ <9(*5). A l l indices i n these summations are d i s t i n c t and no double counting occurs. / For example, i n the A* term V^V^, appears only once; ^,V(a i s not counted, as well. The terms written out e x p l i c i t l y are evidently s u f f i c i e n t to specify any mixed derivative of the c h a r a c t e r i s t i c function, evaluated at an unperturbed eigenvalue and at A-O, up to fourth order i n ̂ , i . e . , a l l derivatives ^1-^(^(0)^0) with /S±4. In pa r t i c u l a r the f i r s t and second order derivatives are the following: 108 (4.51) 4^(0)^= TT (Ep(o)-E,(o>) , (4.52) £ (E P(0\0) = - V TT i^p(P)-Et(o)) , (4.53) £ a M > / 0 (<v(<>>-E;(°->), (4.54) ^fecov) ̂ - V ^ T\(EM-E^)-Y V, TT(^o)-£ / ( 03^ (4.55) £»M>V>> =2VpJ"^]T(M0^£;(0))-^Vp ;Xp 7T(F»-E,fe)). Bon-degenerate cases. If E,(o) i s non-degenerate, the perturbation hierarchy i n Table I leads to the expressions ( 2 . 2 0 ) and ( 2 . 2 1 ) f o r the f i r s t and second derivatives., They, along with eguations (4 . 5 1 ) - (4.55) t give (4.56) £L§»(oV = - 3=o and e.(o)-Er(o) 109 2-fold degenerate cases. s In the 2-fold degenerate case. Table II (or eguation 2.30) gives the following expression for the f i r s t d e r i v a t i v e : On substitution of expressions (4.53)-(4.55) for the p a r t i a l d e r i v a t i v e s , t h i s becomes (4.58) dE:<(o) = V»+^a ± J (Vy-Mttf-t-4-^,1%! ^A a. a. ' As pointed out i n section 2.5.2, t h i s i s simply the solution to the guadratic obtained when diagonalizing the degenerate block. I f i t i s assumed that t h i s block has been diagonalized, i . e . , that M J ^ O ^ V J , , then eguation (4.58) reduces simply to (4.59) dE.( D) = V„ dX ' and (4.60) dE*(o) = N/Ja In the case of 2-fold degeneracy l i f t e d i n f i r s t order (V„ ?tV i a ) , expressions for the second derivatives 4?Jg'(p) and 5-i a(o) are obtained from Table II (or eguation 2.31), namely 110 3=o On substituting the p a r t i a l derivatives obtained from eguation (4.50), t h i s becomes (4.62) This expression i s 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 . * r * />*. Higher derivatives are obtained s i m i l a r l y , though the d e t a i l s w i l l be omitted here. The expression found for the t h i r d derivative, ^7 ~^'{o) , i s precisely the A term of eguation (4.48). Since i t y i e l d s the Taylor s e r i e s f o r E|(A) d i r e c t l y , the c h a r a c t e r i s t i c function approach automatically ensures that the ordering i s correct. 1.11 CHAPTER 5. PHYSICAL PROPERTIES AS ENERGY DEHIVATIVES;- APPLICATION TO SIMPLE SPIN SYSTEMS.T Introduction., In t h i s chapter the determination of various physical properties of a system from energy derivatives i s considered. These energy derivatives can be found either from matrix element expressions or from derivatives of the c h a r a c t e r i s t i c function. Properties of simple spin systems such as the spin p o l a r i z a t i o n vector and the spin p o l a r i z a b i l i t y tensor <̂1> a r e found using the c h a r a c t e r i s t i c functions given by Coope (1966). The corresponding expressions in terms of matrix elements can be tedious to compute because they reguire prior determination of the eigenvectors. One application of the spin polarization vector i s to determine weak hyperfine interactions and one application of the spin p o l a r i z a b i l i t y tensor i s to determine the i n t e n s i t i e s of the 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. A discussion of these two applications concludes the chapter. 112 5 . 1 Physical properties as energy derivatives,~ The derivative of the energy, EF, with respect to the matrix element of the Hamiltonian, H r s, yields the matrix element of the density matrix, ifArs , i n the same basis as the Hamiltonian. This can e a s i l y be seen from the expression (5.1) Ep = T r H P p where Pp i s the projection onto the subspace of the non^degenerate eigenvalue Ep. The stationary property of eigenvalues (variation theorem) implies that T r H = O I t follows that (5.2) = {pr)sr , 3 H r 5 Thus the physical properties of the system, i n the state )p>, which can be found from the density matrix, can also be found by taking derivatives of the energy with respect to matrix elements of the Hamiltonian. The elements of the density matrix themselves can be regarded as f i r s t order properties. Their formulation as energy derivatives was pointed out by Coulson and Longuet-Higgins (1947). In the following, however, derivatives of the energy are taken, not p a r t i c u l a r l y with respect to matrix elements of the 11.3 Hamiltonian, but rather with respect to any perturbation parameters appearing, e x p l i c i t l y , i n the operator representation of the Hamiltonian. The f i r s t derivatives then represent corresponding f i r s t order properties and the higher derivatives, polarizations of these., Consider a Hamiltonian of the form ( 5 . 3 ) H = H ° + ^ V W + â v60, where V < 0 and V t a > are the operators for two physical properties. I f E p i s a non-degenerate eigenvalue of H , and |p> i s the corresponding eigenvector, the expectation values of these properties are related to the corresponding energy derivatives by the Hellman-Feynmann theorem, (just the f i r s t order Bayleigh-Schroedinger perturbation expression), ( 5 . 4 ) < V ( i ) > p = < P J V W I P > = ^ , ' = - a A, The second derivatives of the energy, given according to Bayleigh-Schroedinger perturbation theory by* »Eguation (5.5) can be obtained by taking the derivative of the expression f o r the energy, EP = jr A % . _j_dxr with respect to "X; and \. *r,'«Y», z-M 114 (5.5) 2Jy_)f> =Y <P)V(,))kXklV(i)/p> ̂ <p;V^)kXKl V ( 'V> ^ "V I— 4 represent f i r s t derivatives of the physical properties 4 y c ' ^ p . The derivative of ̂ V*'*)p with respect to AJ i s the same as the derivative of ̂ V^)p with respect to A,, i.e., (5.6, SO£>P =. £ £ r = 3<V(J,)P . The matrix element expressions (5.4) and (5.5) may be tedious to compute..The corresponding expressions i n terms of c h a r a c t e r i s t i c functions are considered here, which, from equations (2.25) and (2.26), are the following: (5.7) <v c / ) > = = 2 5 * Here, to maintain generality, i t i s assumed that the eigenvalue Sp=EpC\) i s known f o r some value of A, say which i s not necessarily zero. , The derivatives of •£ i n equations (5.7) and (5.8) are evaluated for z=^»(£) and Equation (5.7) , f o r 115 example, w i l l be written i n the following as i t being understood that the derivatives are evaluated f o r 116 5.2 Some properties of simple spin systems as energy derivatives. The Hamiltonian j£€ f o r a simple spin system i n a uniform magnetic f i e l d H i s conventionally written (5.9) H + 3>(S£-£S*)+EtSi-SJ) . Following the notation used by Coope (1966), t h i s can be written more compactly as (5.10) Si = i - S +• Bi^sf* where k = c j ( ^ H ) , where i s the i r r e d u c i b l e second rank tensor operator, (5.H) [§f% = i (S^S* and 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 axis system of ID, one has (5.12) H ^ 2 = | I > , J> X X-JD V V = 3 E . I t i s useful to make use also of the following two scalar parameters, eguivalent to X> and E, (5.13) S>2 = S>; ̂  = T> , D>3 = (V>W): © = T r |E? 3 , A number of c h a r a c t e r i s t i c functions for the Hamiltonian (5.10) were given by Coope (1966) using t h i s notation., Those for systems of spin 1 r 3/2, and 2, are 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 functions f o r systems of spin t. 3/2. and 2. U = h.S + J:[S](2) Spin 1. Spin 3/2. Spin 2. £ = z 5 -+- a a2 3-»- 0 3 ^ + a 5 a* = - ( s i ? * y m 3 ) , o 3 = - ( y & s + zi b j b - b ) a 5 6 ( 3 ^ 3 - / ? - B > ' L ) ( S 1 I > * - Z V ? ) 118 The Hamiltonian i n eguation (5.10) i s of the same general form as the Hamiltonian given in equation (5.3) except that the parameters X, and A x are replaced by sets of 3 and 5 independent parameters, respectively, i n ij and B> . The f i r s t derivatives of the energy of the p - t h (non-degenerate) state yield,' the following f i r s t order properties of the p-th state: Spin p o l a r i z a t i o n vector, <£>p = <pl5 )p>. (5.14) <-S> = 9EP P 2h, Magnetic moment <^x\ , 3H Quadrupole p o l a r i z a t i o n tensor ([Sj00^! (5.16) <W\ = 2 ID I t i s worth remarking that the density matrix i t s e l f of a spin system can be written i n terms of such f i r s t order properties (Fano, 1957). For a spin 1 system, f o r example, the density matrix /=" i s given by (Coope, unpublished) /> = i i f ^ ) . S + <[S7(a5>:[sf} . This example of the way i n which the density matrix can be related to f i r s t order properties, and thus to derivatives of the energy, i s complementary; to the representation of the type (5.2), due to Coulson and Longuet-Higgins (1947). 1-1.9 As d i s c u s s e d i n s e c t i o n 5.1, t h e s e c o n d o r d e r p r o p e r t i e s , t h e p o l a r i z a b i l i t i e s , c a n be r e l a t e d b o t h t o s e c o n d d e r i v a t i v e s o f t h e e n e r g y and t o f i r s t d e r i v a t i v e s o f f i r s t o r d e r p r o p e r t i e s . I n p a r t i c u l a r , h e r e , one h a s (5.17) £>l§f = 9<S>P (-5.18) = d<A>p = 3 < [ s ] c a ) ^ 2hdJ£ 3 ID 2h (5.19) t l r = 3 < E ^ > p I n a s i m i l a r way, d e r i v a t i v e s w i t h r e s p e c t t o o t h e r c o u p l i n g c o n s t a n t s i n t h e H a m i l t o n i a n r e p r e s e n t o t h e r s p i n p r o p e r t i e s . . I n p a r t i c u l a r , d e r i v a t i v e s w i t h r e s p e c t t o t h e h y p e r f i n e t e n s o r fl„ i n a h y p e r f i n e c o u p l i n g t e r m (5.20) s . g n I n , n r e p r e s e n t t h e t e n s o r c o u p l i n g o f S t o Inz (5.21) <5 r ( l n ) s > p = p 3(«o)rs I f 6 „ i s i s o t r o p i c (6„=<yJ) # t h e n (5.22) • Io>p = 2J> T h e s e p r o p e r t i e s o f t h e s p i n s y s t e m c a n be c a l c u l a t e d 120 using the c h a r a c t e r i s t i c function expressions (5.7) and (5.8). The spin p o l a r i z a t i o n vector i s given by (5.23) <S> = d§P = _ 4 and the spin p o l a r i z a b i l i t y tensor, 3<€>P , i s given by (5.24) 3<^>P = 2?J=P Here <£> p<s> , ^ < ^ § > p r etc. are dyadics. For the spin 1 system, the relevant p a r t i a l derivatives, of the c h a r a c t e r i s t i c function given i n Table IX, are 4 = 6 z = _ 2 H k - 2 E> fa, ** - B u r , where U i s the unit tensor, where lb]w= bb-ib*^, and where \ From eguation (5.23) the spin p o l a r i z a t i o n vector for a spin 1 system i s accordingly given by 121 (5.25, , <§> = 2t*y + £]'h j while, from eguation (5.24,, the p o l a r i z a b i l i t y tensor 9<£> p t dh f o r a spin 1 system, i s (5.26, ?<£>P = 4 z y ^ ] ^ ( ^ ^ > ^ < s > p y - ^ < s > p < s > p 5 t 3 z 2 _ ( h * ^ ^ f l ^ ) (when evaluated at z = EP ). S i m i l a r l y , the quadrupole p o l a r i z a t i o n tensor ^ [ S j 6 * ^ , i s given by (5.27, < [ S r \ = * fl> + KT~ [ST . The corresponding expressions for the spin 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 tensor for spin 1 and spin 3/2 systems are given i n Table X. 122 Table I . Expressions f o r the spin polarization vector and p o l a r i z a b i l i t y tensor for the p-tfa (non-degenerate) state of systems with spin 1 and 3/2.- Spin 1. 9<£> P & a.tey+&l+2tk<$>P+<s>Pk)-(>z<z>p<s>P 3 ~ 3 z a - ( h % - & E s , ) Spin 3/2. 9 b 4- 123 The use of the formulae i n Table X can be i l l u s t r a t e d f o r the spin 1 system. , In general, for a given h and JE>, the procedure i s to f i n d a root Ep of the c h a r a c t e r i s t i c function, numerically, and then to evaluate the expressions at t h i s root. However, an ana l y t i c expression for the roots can be given i n the p a r t i c u l a r case that h l i e s i n a p r i n c i p a l d i r e c t i o n (the ^-direction) of ID, i . e . , when I D - b = JPgz b and t h i s case i s convenient for i l l u s t r a t i o n . In t h i s case the c h a r a c t e r i s t i c function, for the spin 1 system, reduces to (5.28) f = Z 3 - (h**'***)* -£*i-h*) with roots (5.29) E 0 =. - B>zz ^ E±i - £ 3>z^± \l Z D a - Z b l h ? + h*~ , which, a f t e r some algebra, can be written in the more f a m i l i a r form (5.30) Eo = - # W , E±l = ±TJ>ZZ±J h ^ ( ^ ^ l f , or, using the r e l a t i o n s (5.12), in the form (5.31) E 0 = - ^ 3 ^ E ± , = 2̂> ±Jh*+Ea , I t i s useful to define a parameter (5.32) A - X ^ - E - , ) which measures the separation of the two non-constant eigenvalues. In terms of t h i s , the z-de r i v a t i v e s , evaluated at 124 the eigenvalues, are (5.33) ^ | E o = 2 > a - A % (5.34) $ z \ £ ± i = 3 A ( A ± 2 > ) , and the H-derivatives evaluated at the eigenvalues are (5.35) fb[ <5-«> f k l t l - - a ^ J > ± A ) . From eguations (5.23) and (5.33)-(5.36) the spin p o l a r i z a t i o n vectors are found to be (5.37) <S> ± ) = h\ h I E ± / _ ± Jb. and (5.38) <S> o = O By introducing the number /T)=O,±J 7 which becomes the ordinary magnetic quantum number at high f i e l d s , eguations (5.37) and (5.38) can be combined to give 125 (5.39) m m k m o} ± 1 S i m i l a r l y , from equation (5.24), the spin p o l a r i z a b i l i t y tensors f o r the spin 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 of tt> , are found to be (5.40) and 91» (£+T>X*-3>) e-J> o o O -E-J> o o o o (5.41) 3 JL, ^ ( A ± 2 > ) E - I > O O O - £ - £ > o o o o These expressions can be combined into the sin g l e expression (5.42) 3<$>fn = (5m a-3)A-mJ> A(A A-D A; rE-r> O O O -E-D O O O O where (n-°,±-l. At large f i e l d s m becomes the usual magnetic quantum number, but these eguations hold for a l l f i e l d values. In a s i m i l a r way the quadrupole p o l a r i z a t i o n tensors are found, from equation (5 .27), to be (5.43) (5.44) where % i s the unit vector i n the z^direction, and '/3 o O O O o O -*5 127 5.3 Applications.- 5.3.1 A spin system weakly coupled to some n u c l e i . Consider a strongly coupled spin system, with a Hamiltonian of the form (5.9), which i s weakly coupled to a number of non-interacting n u c l e i . The Hamiltonian f o r the t o t a l system can be written rt rt ^ v.. The p-th l e v e l of the strongly coupled system with Hamiltonian i s s p l i t i n t o several nuclear sublevels. The e f f e c t i v e eff nuclear spin Hamiltonian H giving t h i s extra substructure i s , to f i r s t order, • PP where I t may be more convenient to compute ^§> p from the expressions i n Table X than to compute i^>p as a matrix element. Second order contributions to the e f f e c t i v e nuclear Hamiltonian are determined by the guantities 128 (5.46) <pl£lSXi\Slp> The contributions to the e f f e c t i v e coupling between d i f f e r e n t n u c lei can be computed from the r e a l part of t h i s expression, and hence from the p o l a r i z a b i l i t i e s ^LMF . ,Onfortunately the a H a n e f f e c t i v e guadrupole moment terms (self coupling) reguire the imaginary part of (5.46). Only the r e a l part i s given by the derivatives ii^p ( c . f . eguation 5.5). 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 spectra, > One use of the p o l a r i z a b i l i t y tensor i s to determine the i n t e n s i t i e s of the s i n g u l a r i t i e s of p o l y c r y s t a l l i n e spectra., Consider a spin system i n a uniform magnetic f i e l d H =.(u,&,4). In an E.S.H experiment the freguency of incident r a d i a t i o n i s usually f i x e d , the magnetic f i e l d being varied. Besonance occurs vhen {5.47) = Cm'(ti)- constant-. This can be regarded as an i m p l i c i t function, though i t i s not usually known i n closed form, giving the magnitude of the resonance f i e l d as a function of orientation: (5.48) H r = H r ( 0 ^ ) . In a p o l y c r y s t a l l i n e sample a l l G and <f> are present at the same time., The i n t e n s i t y of the resonance signal i s proportional to the number of molecules for which Hr(&j<P) actually eguals the magnitude of the applied f i e l d H. The main features of the spectrum are peaks of various s o r t s , where the resonance f i e l d s become stationary with respect to 0 and 4*, i . e . , when (5.49) V±r = 3Jir = O . The i n t e n s i t y of these peaks i s inversely proportional to a mean curvature of Hr, as measured by the sguare root of the Hessian of second derivatives (Coope, 1969), namely. 130 (5.50) cv der 3 f M r 3X.3X, Here x, and Xz are orthogonal coordinates, proportional to arc lengths on a unit sphere. For example, i f the stationary point {&o0 0o) l i e s at 0 = 9 0 ° , then possible coordinates are X,-e - 0 0 9 *Z= The derivatives of H r with respect to X, and X a can be related to derivatives of the energy with respect to the components of ^. In Appendix C i t i s shown that, at a stationary point, the r e l a t i o n i s (5 . 5 D 3X; 3Xj H v;. J . v. = H 3;. where (5.52) and (5.53) V; = 3 ^ 3x; Here a =£J/H i s the unit vector i n the d i r e c t i o n of H . The unit vectors y,, v,, and £ form an orthonormal set. , F i n a l l y , from eguation (5.47), the derivatives of ^ i n eguation (5.52) are related to the energy derivatives by 131 (5.54) 9 V - 3Em. _ 9Em 3 h (5.55) - lEm Thus, the energy derivatives «L§m and ^l&n are rel a t e d , by eguations (5.51)"(5.55), to second derivatives of the resonance f i e l d , 9 a .K r . , The Hessian of these second derivatives, in turn, ax, 3XJ determines the i n t e n s i t y of peaks occuring :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 the use of these formulae consider again the spin 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 of E>, (5.56) 5> b = fi^z h and suppose further that h> i s also a p r i n c i p a l d i r e c t i o n of g, so that, i n the p r i n c i p a l axis system of 1t>, the g tensor has the form ~ 3 K X 3xy o (5.57) g - hyx 9 V V o Then t h i s z-direction i s a stationary direction..„From eguations (5.39) and (5.54) i t i s found that (5.58) ^b - A M i i where A0») = m'-/77, and from eguations (5.42) and (5.55), i t i s found that 132 (5.59) 9^ „ 2<sym, _ 3 < 5 > r 9b 2b — ° A(m)) Z(m'+m)*-l> 0 -E_D 0 / A(A a-J> a) o O O ifc-fefr)] i n the p r i n c i p a l axis system of © . a f t e r some algebra i t i s found that J of eguation (5. $2) i s given by (5.60) so that r E-Z> O O - 3 ^ a H a g ( , r r . ' - < - f n ) A - J > c . - O - E - f c o 3 + (5.61) {E - D O O J O -E-l> ol o o o\ •i Two extreme cases are worth considering. ; l f ZD=0, so that the only anisotropy i s i n the g tensor, then only the second term contributes, and eguation (5.61) reduces to 133 (5.62) 1 3x In t h i s 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 i s o t r o p i c , i . e . , g =gt/# then only the f i r s t term i n equation (5.61) contributes, andg disappears from t h i s , so that (5.64) 2*Mr _ - H (5 ( /r7V/n)A , -2>) o -e-r> o o o o In t h i s case, the peak int e n s i t y i s proportional to (5.65) H(3(m V/n,A -J> (x»=»_ E a j '/a These expressions are v a l i d for a l l f i e l d s . 134 BIBLIOGRAPHY. Abramowitz, H. and Segun, J.A. editors, 1965, Handbook of mathematical functions (Dover, Hew York). 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, H. C , 1947, Proc. Roy. Soc. ,. (London) , A 191, 39. Fano, 0., (1957), Revs. Hodern Phys., 2 9 , 74., Fukui, K., et. a l . , (1959), J . Chem. Phys., 3 J , 287. Goursat, E. , 1904, 1916, A course i n mathematical analysis (Ginn, Boston). , Harkness, J . , and Horley, F., 1898, Introduction to the theory- of a n a l y t i c functions (Hacmillan, London). Kato # T., 1949, Prog. Theoret. Phys. Kyoto, 4, 514. , Kato, T. , 1966, Perturbation theory for linear- onerators- (Springer-Verlag, B e r l i n ) , page 84, eguation (2.49). Kransnosel'skii, et. a l . , 1969, Approximatesolution of- operator equations (Solters-Noordhoff) Mallion, R. B. and Bigby, M. J . , 1976, Oxford Dniversity Theoretical Chemistry Department Progress Report. 1975-1976. page 1. (No d e t a i l s are given i n t h i s report. Their work has been submitted for publication but has not been seen.) Osgood, 8 . F., 1913, Topics i n the theory of functions of several complex variables, fladison Colloguium (Dover, New York, 1966, or Am. Math. S o c , 1914). Rayleigh, J.B.S.. 1894. Theory of sonnd f 2nd. e d i t i o n , volume I, page 115 (Hacmillan, London). H e l l i c h , F., 1953, Perturbation theory of eigenvalue- problems.- lectures given at New York University (Gordon and Breach, New York, 1969). Rutherford, D.E., 1945, 1951, Proc.,Roy. Soc. (Edinburgh), 62, 229, 63, 232. Schroedinger, E., 1926, Ann. Physik.. 80. 62. Turnball, H.B., 1952, Theory of equations (Oliver and Boyd, Edinburgh). Van V l e c k , J.H., 1929, Phys. Hey., 33, 467. 136 APPENDIXA. CONDITIONS FOR ANALYTICITY OF EIGENVALUES OF AS HERMITIAN OPERATOR. The conditions for a n a l y t i c i t y of the eigenvalues of an n-dimensional Hermitian operator H(^\) follow from the a n a l y t i c i t y of the c o e f f i c i e n t s of the reduced c h a r a c t e r i s t i c function. -v A.1 One perturbation parameter. By assumption, the operator HO) i s Hermitian for A r e a l (in general i t w i l l not be Hermitian i f A i s complex with non-zero imaginary part) and thus, f o r A\ r e a l , i t s eigenvalues are r e a l . , Suppose £P(o) i s a g-fold degenerate eigenvalue. The reduced c h a r a c t e r i s t i c eguation, with the i n i t i a l l y g-fold degenerate set Epfa), - -. E p+^-i (3) a s roots, i s a polynomial eguation of degree i n z , whose c o e f f i c i e n t s c, , i ^ i , . . . , ^ , are analytic functions of the s i n g l e parameter A, in some neighbourhood of A=O. For such a polynomial, where Ep(°) i s a root for A=O, the perturbed eigenvalue E P C\) can be written* as a power serie s i n A (Puiseux series) convergent for small A . However, i f EpC\) had f r a c t i o n a l powers of A i n i t s power ser i e s expansion, then, for/) r e a l but negative, the perturbed eigenvalue Ep(A) would be complex. ,Since t h i s i s not the case, only integer powers of A can appear i n the power serie s *See, for example, Goursat (1916), volume I I , part 1, page 239. 137 expansion of Epfa)1. ,. Consequently EP(%) i s an analytic function of A within some neighbourhood of /A/=o. ft.2 Non-degenerate eigenvalues. ; I f Ep(p) i s a non-degenerate eigenvalue then the reduced c h a r a c t e r i s t i c eguation f o r EPC\) i s simply rCz.'X) = z + c, ; where c, i s an a n a l y t i c function of X i n some neighbourhood of l^lzzO, , Thus ELpC\), the root of the reduced c h a r a c t e r i s t i c equation, i s also an analytic function of A i n some neiqhbourhood of IXI=o. i R e l l i c h (1953), paqe 31. 138 APPENDIX B HOBE THAN ONE PBBTUBBATION PARAMETER. The re s u l t s given i n Tables III-VII--are v a l i d for more than one perturbation parameter. Formally the following notation i s used i n the tables, r al =r —̂  =2" Z — » a,!... a e / and (B.2) V*** = + - I f there i s only one perturbation parameter the tables can be used d i r e c t l y . I f there i s more than one, then eguation (B.I) must be used. To i l l u s t r a t e , consider the A term of s a for 2-fold degeneracy (Table III) with A - ,It i s given by 3 1 l i Using eguation (B.1) we f i n d that 5 3 ' J  140 APPENDIX C. CALCULATION Of 9aHr/d*, 2xj . . The resonance condition (eguation 5.47) ( C l ) V(H) = V CH> X.jX*) = aonsta.nl- can be regarded as an i m p l i c i t function of the magnitude of the resonance f i e l d This i s just the kind of i m p l i c i t function, considered i n Chapter 2 except that 2 H f A, — x , f A* —e> Kz ^ I t follows that (C2) = J d y 3-W .2H.+ g V mQH QHl/zv 9x,3Xj ax;9H 2xy 3 Xj9H 2K; 3H3H dX, 3XJ j/ 2H J where the derivatives are evaluated at the appropriate values of X., X: and H./Since ? a > i r , i n eguation (5.50), i s reguired at *> 3K,3XJ a stationary point, where (C3) 3J i r = 3 _ 0 3x, 3x. ' eguation (C.2) reduces to (C4) 9 a ^ r = _ 3 f V / 94> 9x,3Xj 3 X ; 3 X J / 3 H An orthonormal system a, y,, v a can be defined by writing 141 (C.5) M = H u . (C.6) ^ = H y ; ; i - _ i , a . dX; ' a l w a y s a s s u m i n g x , , a n d x % a c e l o c a l c a r t e s i a n c o o r d i n a t e s . I n t h i s c a s e we h a v e <C.7) 2Ltl = - S;-- Hoc = _ ^ « . I f , f o r t h e m o m e n t , t h e m a g n i t u d e o f H i s d e n o t e d b y x 5 , t h e n , b y t h e c h a i n r u l e . (C.8) 22 = 3 y . 3M i = a n d <c.9) ^ = - ^ 3 # + ^ - 21* , i,a, 3 3X,-3XJ ' 3 x , 3Xj 3 H 3x,3xj F r o m r e l a t i o n s ( C . 5 ) - ( C .7 ) e g u a t i o n s (C.8) a n d (C.9) c a n b e w r i t t e n m o r e e x p l i c i t l y ( C . 1 0 ) 2V 3 x ; = H ~ 3 H ( C . 1 1 ) 3 _ y 3 H - 3 V ( C . 1 2 ) 3X;9Xj H a ( v , 9HdH ~ - — - ~ > ' > $ 3xj 3XJ ( C . 13) 3x, 3 H H (y; 3 H 9 H 1 4 2 ( C . 1 4 ) 9 ^ = u. . , „ Using eguations ( C . 1 1 ) and (C.12) the expression ( C . 4 ) for the second derivative of the resonance f i e l d , at a stationary point, can be written as (C15) 5 fHr = _ ( V (V;.PU> .V-) _ 5). H .£±L ]/u.dJ!f Since yf. yj == S}j , t h i s can also be written i n the form ( c . 1 6 ) a!±ir - H ( v;. J y.\ ax, Sxjj ~ ~ J / , where ( C 1 7 ) T = - f H * 3f»? _ H ul/H.3u> m In any application i t i s simpler to express these derivatives, taken with respect to the applied f i e l d H , i n terms of derivatives taken with respect to the e f f e c t i v e f i e l d h. Since b = 3^b, i t follows that,, (c-18' k = Oik'*** and thus, also that 143 CC.19) H . 3_ = h. 3_ = H 3 . = Ai 3_ . ay 3/n 3H Expression (C.17) can accordingly be written i n the more convenient form ( C 2 0 ) o~ = - J > H a q . 3 l ^ . a - h -SV.yl/h.dv

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