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Quantum mechanical perturbation theory in terms of characteristic functions Gomberg, Martin Godfrey Luis 1977-12-31

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QUANTUM MECHANICAL PERTURBATION THEORY IN TERMS OF CHARACTERISTIC FUNCTIONS by MARTIN GODFREY LUIS GOMBERG B.A., Unversity of Essex, 1973 A THESIS SUBMITTED IN PARTIAL FULFILLMENT THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in THE FACULTY OF GRADUATE STUDIES (Department of Chemistry) Me accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA August, 1977 Copyright © 197? by M.G.L. Gomberg In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of Chem/sfry The University of British Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Date flyytfih '9^7 A quantum mechanical perturbation theory, for finite dimensional cases, based not on the perturbed Hamiltonian operator itself, but on the characteristic function is developed. & perturbation hierarchy in terms of derivatives of the characteristic function is constructed. From this hierarchy, perturbation series for individual eigenvalues are found. Various cases of degeneracy and degeneracy lifted in various orders are examined in detail. This perturbation theory for individual eigenvalues is generalized. Perturbation theory is developed for a set of eigenvalues considered together. Here the perturbation series are for the coefficients of a 'reduced characteristic function* for this set of eigenvalues. These perturbation series are found by a contour integral method and by an algebraic method. The expressions for the individual eigenvalues and their generalization, the expressions for the reduced characteristic function, both of which are in terms of derivatives of the (full) characteristic function, correspond, respectively, to the familiar matrix element expressions in Hayleigh-Schroedinger, and Van Vleck perturbation theories. Some illustrations and applications of the characteristic function perturbation formulae are given. General expressions are found, to second order, for the perturbed Hiickel 7f-molecular orbital energy levels, of any perturbed even-membered ring of carbon atoms. The familiar iii Rayleigh-Schroedinger perturbation foraalae are rederived from their corresponding characteristic function expressions. The relationship between energy derivatives and physical properties is discussed vith particular reference to simple spin systeas. Expressions for the dipole and guadrupole spin polarizations and for spin polarizabilities in siaple spin systeas are found froa the characteristic functions of the spin systeas. These properties are useful in connection with weak hyperfine coupling, and for predicting the intensity of peaks occuring in polycrystalline spectra. iv TABLE OF CONTENTS Page Abstract . ii Table of contents iv List of tables , .. vi List of figures viAcknowledgements viiChapter 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 5 2.3 The hierarchy of perturbation eguations ....... 18 2.3.1 Comments ............................... 19 2.4 An alternative notation 22 2.5 Special cases arising in perturbation theory .. 29 2.5.1 Non-degenerate eigenvalues ............. 29 2.5.2 2-fold degeneracy lifted in first order . 32 2.5.3 2-fold degeneracy not lifted in first order .. 34 2.5.4 3-fold degeneracy and degeneracy partially lifted 37 2.5.5 q-fold degeneracy ...................... 40 2.6 Summary , 42 Chapter 3 The reduced characteristic function ........ 46 3.1 The reduced characteristic function ........... 47 3.2 The eigenvalue pover sums as contour integrals . 51 3.3 Contour integral 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 63.3.4 Near-degeneracy 71 3.4 Contour integral method: H—rr ................ 73 3.4.1 Illustration: 2-fold degeneracy 76 3.5 An algebriac method: f-*-r 78 3.6 Convergence radii 86 3.7 Summary 93 Chapter 4 Tuo illustrative applications 94 4.1 A perturbation calculation for even-membered rings of carbon atoms 95 4.2 Derivation of the Hayleigh-Schroedinger perturbation formulae from the characteristic function 104 Chapter 5 Physical properties as energy derivatives: 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 Singularities in polycrystalline spectra ................................ 129 Bibliography 134 Appendix A Conditions for analyticity of eigenvalues of an Bermitian operator .................. 136 Appendix B More than one perturbation parameter ...... 138 Appendix C Calculation of ^aHr/3x, 3xa 140 vi LIST OF TABLES Table I The first eight equations of the perturbation hierarchy 26-28 Table II The non-zero terms of the equations of the perturbation hierarchy for some special cases 43-45 Table III 2-fold degeneracy: The eigenvalue power sums 63-65 Table IV 2-fold degeneracy: The coefficients of the reduced characteristic function 66 Table V 2-fold degeneracy: The discriminant of the reduced characteristic function ................. 67 Table VI 3-fold degeneracy: The eigenvalue power sums ............... 69 Table VII 3-fold degeneracy: The coefficients of the reduced characteristic function • 70 Table VIII Algebraic relations between the coefficients of the reduced and full characteristic function 84-85 Table IX The characteristic functions for systems of spin 1, 3/2, and 2 ................... 117 Table X Expressions for the spin polarization vector and polarizability tensor for the p-th (non-degenerate) state of systems with spin 1 and 3/2 122 vii LIST OF FIGOBBS Figure 1 The variation of the characteristic function (equation 1.6) with 2, for two fixed values of /\ ....................... Figure 2 in example of the variation of a characteristic function with z for two fixed values of }\ 15 Figure 3 Guide to the perturbation hierarchy (i): Hon-degenerate eigenvalues , 31 Figure H Guide to the perturbation hierarchy (ii): 2- fold degeneracy 36 Figure 5 Guide to the perturbation hierarchy (iii): 3- fold degeneracy 39 Figure 6 The contour r£L , t^+t,x» and in tae complex z-plane 72 Figure 7 Some examples of the 'best' contour ..... 88 viii ACKNOWLEDGEMENTS I wish to thank Dr. J.A.R. Coope for suggesting this research topic, for guidance in both the research and thesis preparation, and for always being available for discussion. I aa also indebted to ay fellow graduate students, in particular David Sabo and Eric Turner, from whom I have learned much. Por financial support during my two year stay I would like to thank both the Chemistry and Physics departments for teaching assistantships and the University of British Columbia for summer scholarship money. Finally, I would like to thank Kathryn Leslie for her help in typing this thesis. 1 CHAPTER 1. INTRODUCTION. This thesis develops a quantum mechanical perturbation theory based on the characteristic function, <1.1, fC*,a)= deMz-H(»J; of a perturbed, finite dimensional Hermitian operator H(A). The one or more perturbation parameters are represented by A. This approach is in contrast to the usual perturbation theory which is based on the operator itself or its matrix elements. A perturbation theory based on the characteristic function -Pfe,^) amounts to a study of the behaviour of the roots of the polynomial equation (1.2) -TYZ,*) = .. + a„(n) = o as the coefficients ex, ii = ±,...,n vary with A. It is assumed that H(^), and thus the a;(a), are analytic functions of A in some neighbourhood of |AI=0. 2 1.1 The ideas. An equation that aodels a physical system can rarely be solved exactly. Quantum mechanical perturbation theory is motivated by this fact. Rayleigh (1894) in the context of •vibrations of a string with small inhomogeneities" and Schroedinger (1926) in the context of his new wave mechanics developed what is now known as Rayleigh-Schroedinger perturbation theory. In this, an Hermitian operator of the form is considered. (Here, and in the rest of the thesis, 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 real parameter. The eigenvalues E;(A) and eigenvectors |E,-(a)> of the perturbed operator H(Xj are found as power series in A. For example, if Ep(o) is a non-degenerate unperturbed eigenvalue, then the perturbed eigenvalue £p(A) and eigenvector l^pCX)) are given, to first order, by (1.3) Ep(l) = Epip) + A Mpp•+• OCX) and where Mip *<E,\ (o) \ V / Ep(o)>. Rayleigh-Schroedinger perturbation theory starts with an operator and expressions are obtained in terms of matrix elements of this operator. Perturbation theory in terms of characteristic functions 3 starts instead with the characteristic function of the operator. An example of the characteristic function approach. for an illustration of the characteristic function approach consider the 7T-aolecular orbitals in the cyclopropenyl systei in which one carbon atom has been perturbed in some way. The Hiickel matrix, in units of /3 relative to « as a zero of energy, is (1.5) H(%) = 3 I I l o i I i o The characteristic function of this operator is (1.6) = Z'- 5z-2. -»- 2 and the unperturbed eigenvalues are (1.7) E,(o) = *a , Ea(o) = e5(o) = "1. The perturbed eigenvalues are the roots of the characteristic equation (1.8) $U,V> = o (see Figure 1). Eguation (1.8) defines Z as an implicit function of A. The first order perturbed eigenvalue can therefore be determined by first order implicit differentiation, using the standard formula (1.9) # = - h where the subscripts label the partial derivatives. 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 this saae result obtained by first finding the unperturbed eigenvector |£,(o)> -and then evaluating the matrix element in equation (1.3): 5 #(o) = <E,(o)|V|E.(0)> = ± . In general, the characteristic equation defines 2 as an implicit function of A. The problem is to convert this implicit information to an explicit functional dependence, 2 = Perturbation theXory gives this dependence, though not in closed form, as a power series in A. The coefficients of this series, namely the derivatives, can in principle be obtained by higher order implicit differentiation. Returning to the example, it may be observed that eguation (1.9) cannot be used, as it stands, for the initially degenerate eigenvalues EAC\) and £3(3), since both £ an<J H vanish when evaluated at z = Ex(o)-~lt 7i~o, The perturbation theory for individual eigenvalues in degenerate cases, which depends on the order at which degeneracy is lifted, is developed in detail in Chapter 2. The characteristic function approach does not yield the perturbed eigenvectors directly. Indeed, the unperturbed eigenvectors, which form the basis in which H° is diagonal, need not be known, and this can be an advantage in certain calculations. The determinant in eguation (1.1) is invariant to basis change: 6 (1.12) £ = deHz-HI = del-1 l/(s-H)(J| = dehl*-U+HU\ ? where U is a unitary matrix. However, information contained in the eigenvectors can be found by taking derivatives of the energy with respect to appropriate perturbation parameters. In particular, if the perturbation parameters are the matrix elements of the operator, themselves, the derivatives yield the matrix elements of the density matrix. Kato (1949) developed a quantum mechanical perturbation theory in terms of resolvents and contour integrals. Coulson (1940) had already used resolvents and contour integrals in a quantum mechanical context. The resolvent was not identified as such in Coulson*s work and was expressed not in terms of operators but in terms of characteristic functions. Resolvents and contour integrals provide the mathematical base for part of Chapter 3, and allow the relationship between operator and characteristic function approaches to be exhibited in a particularly clear way. In their perturbation theory of conjugated systems, Coulson and Longuet-Higgins (1947) give first and second order formulae in terms of the characteristic function. Their Hamiltonian H referred to the electrons in the rr-system of a conjugated molecule, and their perturbation parameters were the matrix elements Hrsof this Hamiltonian in the atomic orbital basis (Huckel parameters). They related first derivatives of the energy, with respect to Hrs, to the elements of the density matrix Psr, and second derivatives were related to first derivatives of the density matrix, i.e., polarizabilities. 7 However, they were interested in derivatives of the total energy of the system. Thus their calculations involved sums over all occupied orbitals and, in particular, sums over degenerate orbitals. Pukui et. al. (1959) treat single non-degenerate orbitals but still sum over degenerate orbitals. A perturbation theory based on characteristic 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 interest, the characteristic functions can in fact be obtained. Two such classes are certain conjugated systems and simple spin systems. Rutherford (1945,1951) organized the results of earlier workers giving general formulae for some *continuant determinants* or characteristic functions which occur in simple Huckel type calculations. Hallion and Rigby (1976), using graph-theoretical methods, have obtained characteristic functions for an arbitrarily weighted graph. This corresponds to having the characteristic function of a Huckel matrix for arbitrary <x> and /Srs. Coope (1966) wrote down some characteristic functions for simple spin systems. Here the coefficients of the characteristic function are scalars and are simpler in structure than the matrix elements of the Hamiltonian of the spin system, which are in general tensorial functions of angles. The development of a perturbation theory in terms of characteristic functions is motivated primarily by the interest of generalizing the work started by Coulson and Longuet-Higgins, firstly to apply to arbitrary perturbation 8 parameters, secondly to apply to any order of perturbation, and thirdly, and in particular, to include the treatment of degenerate eigenvalues. It is also motivated in part by the availability of various characteristic functions, and by the fact that the characteristic function approach makes no explicit use of wave functions, which can be tedious to calculate. 9 1.2 Thesis content. In Chapter 2 a perturbation theory for single eigenvalues is developed by extending the implicit function approach used in the example in section 1.1. A perturbation hierarchy is AmE derived from which all the energy derivatives TTfap(°) can De a A found. Particular cases of degeneracy lifted in various orders are considered. This approach is the characteristic function analogue of Rayleigh-Schroedinger perturbation theory, and the Hayleigh-Schroedinger formulae can be obtained from their corresponding characteristic function formulae, as shown in Chapter 4., In Chapter 3, perturbation theory for single eigenvalues is generalized. , Instead of single eigenvalues, a reduced characteristic equation for a set of eigenvalues is considered, and a perturbation theory for the coefficients of this reduced characteristic equation is developed..These coefficients are computed by two methods; by contour integration and by an algebraic method. ,<The 'reduced characteristic equation* can be regarded simply as the characteristic equation of an effective operator H, as introduced by Van Vleck (1929). In Van Vleck*s perturbation theory an effective operator H is constructed so that its eigenvalues are a subset of the eigenvalues of the perturbed operator H. The effective operator H is then found correct to some order in A., The roots of the 'reduced characteristic equation* r(z,A) - deHz-Hl = O are the perturbed eigenvalues of H, correct to the same order in A. The Van Vleck approach is represented by the path 10 H • H In chapter 3 however, the path H H i i •P » r is taken, ., The coefficients of the reduced characteristic function r are found not in terms of matrix elements of an operator but from the (full) characteristic function JBy contour integration they are determined in terms of derivatives of the characteristic function. . In addition, an algebraic method is used to relate the coefficients of i~ to the coefficients of -P directly. It is also shown that the path H H £ r can be taken, without mention of an effective operator H. Here the coefficients of r are found in terms of matrix elements of H. In Chapter 4 the perturbation formulae derived in Chapters 2 and 3, in terms of the characteristic function, are applied to two illustrative examples. The first example is a perturbation calculation, within the context of Hiickel molecular orbital theory, for an even-membered ring of carbon atoms. In the second example it is shown that the familiar Bayleigh-Schroedinger perturbation formulae can be obtained from the corresponding characteristic function expressions. In Chapter 5 the relationship between energy derivatives and properties is discussed with particular reference to simple spin systems. , Properties such as the spin polarization vector and the spin polarizability tensor are found using the characteristic function of the spin system and two examples of their use are given. 12 CHAPTER 2. PERTDSBATION THEORY FOB SINGLE EIGENVALUES.-Introduction. . In this chapter a perturbation theory for single eigenvalues, in terms of characteristic functions, is developed. The theory is restricted to eigenvalues which are analytic functions of the perturbation A in some neighbourhood of IM=om For a finite dimensional Hermitian operator, itself an analytic function of A in some neighbourhood of |A/ = 0, the eigenvalues will be analytic if either 1. there is only one perturbation parameter, or 2. the unperturbed eigenvalue is non-degenerate (see Appendix A). ,, 13 2.1 The method. • Consider an n-diaensional Hermitian operator Hft) which is an analytic function of one perturbation parameter A. The characteristic function f(z>A) is given by (2.1) $&>V s del- lz-H(A)J , which has the factorization (2.2) H*,V) = 7T(z-e.-0O), where the E, (A) are the eigenvalues of H(A) and are real valued analytic functions of A., The values of H for which the characteristic equation (2.3) f (Z,A) = O is satisfied are the eigenvalues of W(A). , The Taylor series for any particular eigenvalue £p(3}, expanded about A-4 is (2.U) E^A) = Ep(o) + A ^EP(O) fe>(o) + ... . It is the object of this chapter to determine the derivatives AJrT(o)t rr> = 1,2,: • The zeroth derivative or unperturbed eigenvalue £p{o) is assumed to be known. , The idea behind the approach is to regard the characteristic equation (equation (2.3)) as implicitly defining Z as a function of A, for ^ in some neighbourhood of the unperturbed eigenvalue ^(o), and for A in some neighbourhood of »This function wil be q-valued if the unperturbed eigenvalue is 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 implicit 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, will yield the required derivative <±SP(O). Thus the terms in the Taylor series expansion (equation (2.4)) can be found and the perturbed eigenvalue E"P(A) obtained, correct to any order in A. The results 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 analytic only if there is 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 characteristic function language quantum mechanical perturbation theory becomes the study of the way in which the roots of a polynomial change as its coefficients are varied a little, i.e., as A varies (see Figure 2). Eatt) E5C*) Figure 2., An example of the variation of a characteristic  function with 2 for two fixed values ofA. ? 16 Since the characteristic equation defines z as a function of /\, the total differential with respect to A is given by» d_ = a. + d? 9. 3A dA 9z ' Taking the total differential of the characteristic equation with respect to A yields (2.5, ^ + fz SJ| = O , where the subscripts denote partial derivatives. Figure 2 illustrates two typical cases. The unperturbed eigenvalue E,(o) is 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 situation is quite different, since ^(Ea(o),o)-0 (which is the condition that £a(o) be at least 2-fold degenerate), and -PafeitP^Oj-o2. Osinq L'Hospital's rule a quadratic for is obtained, lSee, for example, Goursat(1904), volume I, page 41, where the expression is given not in operator form but as df = 3f . 3f. d_z 9* 32 dA -2In general, for q—fold deqeneracy, the characteristic function has the factorization Thus all 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 this quadratic, when all derivatives of -P are evaluated at z=£^(°)# A=o, are the derivatives ^(o) and d§3(o). Equation (2.6) originated from the characteristic equation. The characteristic equation is known to be satisfied when z. is equal to the unperturbed eigenvalue, £4(0), and when A=c. Equation (2.6) is also satisfied for these values of 2 and A. Thus it is 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 if Epfa), /\*o was OA already known. .This is not usually the case. 18 2.3 The hierarchy of perturbation equations. To obtain expressions for <LSP{O) , successive total derivatives of the characteristic eguation (eguation (2.3)), with respect to A, are taken: ,2.7, if - -O 1 d^ c/AA (2.10, dif - > "i fcpfef.J&f=0. ^-»-i>',+-ayJ+...+'7y0-n (<* = v, + ...+vn) (See comment 1, below, for explanation of summation in eguation (2.10).) This set of equations will be referred to as the perturbation hierarchy, or hierarchy for brevity. Equation (2.10) is the n—th equation of the hierarchy. 19 2.3.1 Comments.7 1., The summation in equation (2.10) is taken over all integral values of /Szo, Z;>09 ; = 1,~.yn, with the restriction that 2. The combinatorial coefficient* ——= — is the number of ways n objects (the A*s) can be partitioned into sets: /3 in the first set, 1 in the next Xt sets, 2 in the next ^ sets,...,n in the last )fn sets, where the order within each set and the order of the 'next ^ sets1 is unimportant. The total number of sets is 1 + Jf, + ... +- yn = 1 +• <L . 3. Dimensionally we have and Since * = X,+...+in and p-+i*v+a)tn+...ntin^, it follows that and each term in the sum is dimensionally correct., »See, for example, Abramowitz and Segun (1965), page 823. 20 4., Eguation (2.10) can be derived in the following way*: the total differential dA 29i dA 9?; is such that JA d5\k+l For example, & • k Ik •* - (Ir+a & {*$) W(f& fc • The term ^^Jr- is the result of i acting on dz in the product IV 1z da da 4?<L, the term (4^tl^-T is the result of part of i, 2. , d* 9* Ida/ UzJ * d* d3 3z-acting on =L in the same product ^? 2- , and so on. Clearly, ?L 3z da 3z clA" will result in a sum of terms of the form Each «_? xs the result of Q_ acting on ? £ , the 2 # is the dak da da"-' da"*' result of d- acting on d z , and so on, down to being the d* dak'a \ da* result of 4- having acted on 4J£. Since this 4* originally d* da dA iEquation (2.10) is given, without proof, by Krasnosel'skii et. al..(1969), page 329. In this study equation (2.10) was first obtained in the notation of section 2.4. 21 appears in the product £_ t the power to which 2_ is raised, to 3z <*, is given by ot — tf, •+- ... + Yn . In all, d_ operates n times. It has operated l%i-Wi+-nXn times to to produce M^^.The number of times 'left over* will fee the power to which iL must be raised, i.e.. To establish the combinatorial coefficient, suppose each A is labelled, i.e., —j> d_ d- ... d_ Each "\lt i n will appear once in each term of the sum. All permutations of the 'A; occur except that ^jlrj^jf; j (cllf ) °r A.Aa A3» ^or example, will appear only once, not 3! times. Hence the order of the "next ^ sets,' as well as the order within each set, is unimportant. On removing the labels, the combinatorial coefficient in eguation (2.10) is obtained.. 22 2.4 An alternative notation. r It is possible to group the terms occurring in each equation of the hierarchy in a way which brings out a structure which will be made use of later. The terms in the second equation (equation (2.8)) can be grouped into a quadratic like part, terms in the third equation (equation (2.9)) can be grouped into a cubic like part, and so on. The following notation is used, (2.11) ^ = ILir)^^^^ > r-o where I'i] is the binomial coefficient , in this LRV"> (n-r-)iri notation T^P has the following property, rCn) c(n+>) r-Cn-i) a <2'12> 3* 2P = ° dfa ' as can easily be shown.„ In this notation the first equations of the hierarchy can be written 23 (2.15) To write the next equations in a systematic way a new combinatorial coefficient ( J will be introduced, representing the number of ways •*-dk.±n objects can be partitioned into k sets, with objects in the i-th set, where the order within each set, and between sets containing the same number of objects, is unimportant. For example, we have / 5 ) = _£L._L [3,3} a<3( ' With this notation the next eguations of the hierarchy become (2.16) 24 (2.17) is) o +-/5\ 2 dW at this point a pattern is beginning to emerge. In the n^-th eguation there are terms like (n- f of, *...-»-«<|,71 where <>a, i-X,..fk.t <**t+ -^k^n and *tt — --- — (to avoid 'double counting*). If by the symbol n , •• • olK we mean the sum of all terms above in which the 4;*s satisfy the restrictions stated, then the n-th eguation can be written (2.18) where mis the largest integer such that m^a.Jox example, the 6-th eguation of the hierarchy is 25 UJ £ dAfc The notation in this section is useful in that it provides a compact way of writing any particular eguation in the hierarchy., In addition, the structure of each equation is made clear. Each equation is simply a polynomial in the desired implicit derivatives 4!Zs , as equation (2.19) illustates. The first derivative 4* is embedded in the symbol tp . The first eight equations of the hierarchy are given, in this notation, in Table I. The combinatoral coefficients have been replaced by their numerical values. Table I The first eight equations of the 3. V + £'0) dAa o 3. 4:(3> d*a + ...+ Z d** — o A-. d*a -*-dA3 -f = o z da2 «daa/ .. .-MO 4?<ft + io £^W^\ + - - -d** wVwW .+5 4" ^ -d^ Table I. continued. 27 r(t,) r- ft** ,X-dV> d?r 60 -F£a d*£ + IStgM jo* W d^\ -j- - -a|,.d# 2 da5 dtf = o 7. f^^Uf-ft da* 2 dA* . is dA5 d/f da*/Ida3/ m 2. & \05 iOS ^3 * dr 35 = o (da3/Ida*/ Table I. continued r#) ft) dl* * way 840 Cf^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 arising in perturbation theory.-Some special cases arising in perturbation theory will now be considered: non-degenerate eigenvalues, 2-fold and 3-fold degeneracy lifted in various orders, and g—fold degeneracy. The equations of the hierarchy specialize in each case. 2.5.1 Bon-degenerate eigenvalues. If Ep(o) is a non-degenerate eigenvalue, i.e., EP(o)^ E,Co), for i^p, then ^Z(E.P(O),O)^O. In this case a simple rearrangement of the equations of the hierarchy leads to explicit expressions for all 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 all terms on the right hand sides of these equations are evaluated at the unperturbed eigenvalue, i.e., for 2. = £^b) and \-0. (Eguation (2.22) is obtained from equation (2.10) and, in the alternative notation of section 2.4, eguation (2.23) is obtained from eguation (2.18).) To find 4!^P(o), all the derivatives of lover order must have already been calculated from the precede ding eguations in the sequence. The way in which the equations of the hierarchy are used, for non-degenerate eigenvalues, is represented by the schema in Figure 3. In the case of several perturbation parameters, /X = (X>-->\), the perturbed eigenvalue E^ft) has a multiple perturbation, or Taylor series, expansion, t L a (2.24) = M°> + Y X ff^) + Y X2i llr (o) + ... , >--> ;i=-> J In this 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 all terms on the right hand side are evaluated for 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 this figure and in Figures H and 5, n represents the n-th eguation of the hierarchy. by modifying the expression for 4H§p(o)., To illustate this modification, consider the third, one parameter, derivative, ,2.27, J5w= ^'^g^^W^tygj&fe* _ Each term in this expression contains A three times. In the corresponding expression for —Co) each term will contain Aa and A3 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, 3i +-P311 3z and so on. The combinatorial coefficient, the 3 in eguation (2.29), is the number of nays \, Aj and A3 are permuted within the term 4^ <Jz-. As discussed in section 2.3.1, the mixed partial derivatives ^zX,^ and ^2^3, only count once. In general, then, to obtain ~?Ep^(~o) for some particular value of n, having written down the expression for 4%p(p), one simply makes the appropriate modifications. All total derivatives d^Z? become partial derivatives 3 Zr-- and each term in the expression for 4-£p(0) is rewritten by listing all allowed permutations of the Aa; within each term. The combinatorial coefficient* — /*' ^ , is just the number of such permutations. ., 2.5.2 2-fold degeneracy lifted in first order. If EP(o) is a 2-fold degenerate eigenvalue, i.e Epto)=Ep+,Co)?E,Co) for then -F2(^pCo)j0) - O , but $z.*(Ep(o),o)-* o. In this case the first eguation of the hierarchy, eguation (2.7), is identically zero., The second, eguation (2.9), a quadratic in 43, gives JA 33 (2.30) efl'fa) = ± Jfea)a-4*4 This is in agreement with the result obtained using L*Hospital's rule (eguation (2.6)). If these two roots are distinct, the degeneracy is lifted in first order, i-e«# 4%o) d§^'(o)w Substitution of these two different values for d%\t-oe<f>> into tne remaining eguations of the hierarchy then yields all the derivatives j~£Co) and ip5»+i(0), m>I. For example, the third eguation of the hierarchy gives the second derivatives as or, in the notation of section 2.4, When the two different values for dl|z=E,£o)» i.e., dj&Ycrt and ^§p+'(b)» are substituted into these eguations, the corresponding second derivatives sLff and <L5~i(o), are obtained. „ The quadratic whose roots are given by eguation (2.30) is (2.32) dr 6>) 3£ the same quadratic that would be obtained in Rayleigh-Schroedinger perturbation theory when diagonalizing the (2-fold) 'degenerate block*. 2.5.3 2-fold degeneracy not lifted in first order. The condition that 2-fold degeneracy is not lifted in first order is that tbe two roots given by eguation (2.30) be equal. This will be the case if the discriminant of the quadratic vanishes, which is the case when the Hessian of is zero, i.e., when (2.33) (^J - to) = O If this is the case, then (2.34) &(o) = dJFP+'fo) - - & da da K (In Rayleigh-Schroedinger perturbation theory this corresponds to finding the •degenerate block* diagonal with the diagonal elements equal.) If degeneracy is not lifted in first order, then the third equation of the hierarchy, eguation (2.15), da *~ —3-, turns out to be identically zero, as can be shown by taking the total 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 + 4f2, Jia + V =o The two roots of this equation yield 4lfp(o) and d*&>WoW They are da* da' given by A = o (2.36, ±/tW * If these roots are different, the degeneracy is lifted in second order and substitution in the higher order eguations of the hierarchy sill then yield the higher derivatives, 42Ff>(o) and da*" If degeneracy is not lifted in first or second order, then the sixth eguation of the hierarchy is the reguired quadratic in the third derivative 41? . In this case, in the sixth d*5 eguation, all the coefficients of the implicit derivatives 4^3 » <">4- are zero, i.e., the sixth eguation reduces to da3-* da3 * UT)W) a guadratic in ^f?-. ,In summary, the way in which the equations da3 of the hierarchy are used for 2-fold degeneracy, lifted in various orders, is represented by the schema in Figure 4. 37 2.5. 4 3—fold degeneracy and degeneracy partially -lifted. -. If Ep(°) is a 3-fold degenerate eigenvalue, i.e., Ef>C°)~ Ep-ni°)= £p+ato)^E;to) for /V/>, p+i, p+a, then •Re- = = £z2 = ^ = £\a = o when all derivatives are evaluated at 3=o. The first eguation of the hierarchy which is not identically zero is the third, eguation (2.9).vThis eguation reduces to evaluated at ^Ep0>) and feo, are the first derivatives a cubic in . The three roots of this eguation, when da It can happen that degeneracy is only partially lifted in first order. Consider the case in which da </a d-x For the eigenvalue Ept which is no longer degenerate, the substitution of dJfrfa) into the fourth eguation of the hierarchy will yield the second derivative 4^3P(O)» Substitution of both dfpco) and 4^P(O) into the fifth equation of the hierarchy <^a c/Aa will yield the third derivative <L3p(p\, and so on. Thus the Taylor series for Ep(X) is obtained. However, the situation for EfmCA) and 5>*a£l) is quite different. Since the first derivatives 4&>*'6>) and ^(o) are equal, E>+,<>) and E^,£A) are still 2-fold degenerate to first order. Consequently, to find the second 38 derivatives, a quadratic equation must be solved. , The fifth equation in the hierarchy is the required quadratic in dV-In summary, the calculation for 3-fold degeneracy is given by the schema in Figure 5. 39 Figure 5. Guide to the perturbation hierarchy (iiil  3—fold degeneracy. (P)~] 'True. 1 4-dr 1 dr False 6 da* 40 2« 5.5. q—fold degeneracy.; If EP(o) is a q— fold degenerate eigenvalue, then In this case and consequently the first non-vanishing eguation of the hierarchy is the q-th. This follows because certainly ^^<% if n<c^, so that all terms in the summation in equation (2.10) will be zero, when evaluated at z ~ f=PCo), %~Om If n-<^, then °t+/?<el if any i-3.,...,nt and <<->fi^% if all / = a,... v/?. Hence for the non-zero terms in the sum, and the q-th equation in the hierarchy reduces to The roots of this equation are the q first derivatives, which may or may not be distinct. , On writinq out the last two terms of the summation in equation (2.37), (2.38) + fi%^ =0, it can be seen that if the q-fold degeneracy is not lifted in first order (i.e., if eguation (2.38) has egual roots when evaluated at z=^o), 3^o), then the first order shift of all g eigenvalues is merely 41 (2.39) dE''(o) = « A fz1 z=e;{p) A=o £ since is the sum of the g roots of eguation (2.38). In this case a g-th order eguation in the second derivatives is required, and this is the 2q-th equation of the hierarchy. The 2q-th equation is the first to contain the required quantity T2^/s_?A > which occurs in Co) If degeneracy is lifted in first order, then substitution of each root of equation (2.37), i.e., each of the different values of # into the higher order eguations of the hierarchy yields the corresponding higher order derivatives. 42 2.6 Summary. , It has been shown how the total derivatives of the characteristic equation of a perturbed operator, taken with respect to a perturbation parameter, yield expressions for the perturbed eigenvalues to any order in /\. As made clear by the notation of section 2.4, the perturbation hierarchy is, in effect, a sequence of polynomial equations in the implicit derivatives 4Cj^ , whose coefficients are the partial derivatives A7T of the characteristic function, with respect to z and a. These partial derivatives are evaluated at z=£P6>) and 7i-o and the equations solved to give the eigenvalue derivatives 4£jh(p)m Some important specific examples have been discussed in section 2.5. For each specific example, the eguations of the hierarchy specialize in a particular way, when evaluated at z=^o), '\-ot i.e., some terms vanish. Table II lists the non-vanishing terms, in the first few eguations of the hierarchy, for the special cases considered in this chapter. 43 Table II The non-zero terms of the equations of the perturbation hierarchy for some special cases. All derivatives of f are evaluated at z. - Ep(o), a=o. The equations in this table have been obtained from the equations of the hierarchy in Table I as indicated by the numbers on the left hand side. Non-degenerate eigenvalues. Ose Table I, 2—fold degeneracy, lifted in first order. 3. + o + 3 +i d2 - o dfla aa* Ida*/ ... -4- 4- -Fj^ d3^ - o da5 2-fold degeneracy, lifted in second order^ 2. izo = O Ida*/ Ida'/ TS> ,(5) da3 (dWldav 44 Table II. continued. 2-fold degeneracy, lifted in third order. 2. t?° - O 2 dA2 z Idav Ida*] 3 'w -10 1^ /US i ^ O 3—fold degeneracy, lifted in first order. 3. ( -O 4. &Ka*& = o z z da2 Id^J * d/l5 Table II. continued. 3-fold degeneracy, partially lifted in first order. r 0> 3. +zo =0 For ePC*). tt. -r?o •+ b izi <L? = O da* For Ep^)and £*»«C\i da* Ida^i 3-fold degeneracy, not lifted in first order. = O da* Ida2 /s^Vd^f (da*/ 46 CHAPTER 3. THE REDUCED CHARACTERISTIC FOBCTIOS. , Introduction.? In this chapter attention is focused not on individual eigenvalues but rather on sets of eigenvalues. This is a generalization of the single eigenvalue perturbation theory of Chapter 2. , Perturbation series are obtained not for one eigenvalue but rather for the coefficients of a reduced characteristic function. 47 3.1 The reduced characteristic function., The term 'reduced characteristic function' will be used to denote a polynomial Kz,A) of degree <^ in z, whose zeros are a subset, EP) .. } Epn.lf of the zeros of the (full) characteristic function £(z,3) w i.e., a subset of the eigenvalues of H: (3.1) r^,A) = 7T (z-e,-) The coefficients of r, qi,i=i...,<j, depend on A. Given the reduced characteristic function the individual eigenvalues can be found as the roots of the 'reduced characteristic eguation* (3.2) r(^.,7i) = O -The set of eigenvalues will always be taken to be a complete degenerate or nearly degenerate set (or any number of such sets). The dependence of the eigenvalues is then in general more complicated than that of the coefficients c,-. The coefficients of the reduced characteristic function, ct)....)cc^, are symmetric functions of the eigenvalues, EP, Ep+i-i> namely. 48 (3.3) Co = (-IP'TV1 (E;) , > /=/» CL = -1 21 E/ Accordingly, knowledge of the coefficients is eguivalent to knowledge of the eigenvalue power sums m-i,.;^, defined by P+%-1 (3.4) 5m = \ {E;)m The two are related by Hewton^s* formulae: (3.5) • • s, + c, = O In particular, (3.6) c, =-S( , Two methods will be used to compute the coefficients of the reduced characteristic functions for non-degenerate, 2-fold and 3-fold initially degenerate eigenvalues.,, In the first method, contour integration is used to determine perturbation »See, for example, Turnball (1952), page 72. series for the eigenvalue power sums, 5m. The second method is algebraic and yields perturbation series for the coefficients c;, directly. The idea of the reduced characteristic function is closely related to a factorization theorem of Beierstrass*. This, in turn, is closely related to the theorem (Kato, 1949) that the total projection onto the subspace arising from an initially degenerate eigenvalue is analytic in A, in some neighbourhood of |A/=<3. , The reduced characteristic function can be factored out of the (full) characteristic function: The factorization theorem tells us that as long as initially degenerate eigenvalues are considered together, i.e., if only complete degenerate sets are included, then the coefficients of the corresponding reduced characteristic function are analytic functions of the perturbation parameters (%,-.•,/X^) in some neighbourhood of \/\\~ O. Conseguently they have a Taylor series, or perturbation series, of the form which is convergent for all /\ in some neighbourhood of \%\-0, The perturbed eigenvalues themselves do not, in general, have such a series expansion if they are initially degenerate and if there is more than one perturbation parameter. For example, suppose for a 2-fold degenerate problem with two perturbation parameters, that the reduced characteristic *See, for example, Osgood (1913), page 181. (3.7) (3.8) 50 function is (3.9) rUA) = Za -(X-K) -The coefficients of this reduced characteristic function have (trivially) perturbation series of the form given in eguation (3.8); the perturbed eigenvalues, ±J A?-*- ^1 , do not. Such cases cannot be dealt with by Bayleigh-Schroedinger perturbation theory or its characteristic function analogue developed in Chapter 2. appendix A discusses the conditions for analyticity of eigenvalues.„ 51 3. 2 The eigenvalue power sums as contour integrals. In this section the eigenvalue power sums, Sm, are expressed in terms of contour integrals., Since the characteristic function has the factorization it follows that i-l This function has a pole at each eigenvalue, with residue equal to the degeneracy. It follows that (3.11) S,*, = <p 2T ft c/z; , r>l where is a contour in the complex Z^-plane enclosing the relevant eigenvalues, Ep,.., Ep+^-t • ,f The integrand has no pole on the contour , for A in some neighbourhood of fAf = £>; it is an analytic function of A, in this neighbourhood./It follows that the sm9 and thus the c/# are also analytic functions of A, in this 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 in the absence of eigennilpotents, 52 G- = V- — • i where f) is 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 in the operator form (3.13) Sm = JJ-S-ZT G- . am JfA If the trace in eguation (3.13) is taken before integration, eguation (3.11) is obtained, and the reduced characteristic function can be found in terms of the (full) characteristic function. This analysis is carried out in section 3.3. If the trace is taken after integration1. the reduced characteristic function is found in terms of the operator matrix elements. This is discussed in section 3.4., Equation (3.13) could have been obtained alternatively, starting from the relation. *Se see that the sm, and thus the reduced characteristic function r, are defined even if H is infinite dimensional, in which case the characteristic function f itself is not defined. 53 r-% between the projection onto the associated eigenspaces, and the resolvent. Eguation (3.13) then follows from 5^ = Tr H"P . The poles of the integrands in equations (3.11) and (3.13) are located at the eigenvalues of H(2). for [M&o, under the influence of the perturbation, these eigenvalues are not known. To compute these integrals, 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 series expansion of S,,,, about lAI = o# is obtained. Finally, the coefficients of the reduced characteristic function can be found using Hewton's relations. 54 3.3 Coptour integral method: f-»r. 7 The first few terns of the Taylor series expansions for the coefficients of the reduced characteristic function r~ will now be determined, from the contour integrals, in terms of derivatives of the (full) characteristic function f. In particular, the reduced characteristic functions for non-degenerate, 2-fold# 3-fold, and nearly degenerate eigenvalues of a finite dimensional Hermitian operator are constructed., Expansion of the integrand in eguation (3.11) about |A(=o yields (3.14) For simplicity, in the following, it will be assumed that there is only one perturbation parameter*., Then the integrand in eguation (3.14) becomes L -P L -P -pa J t r ~r -P* J J where -j- and its derivatives are evaluated at \%-°. The general term in the expansion of the integrand is simply Conversion of a one-parameter expression into the corresponding many-parameter expression is straightforward, though tedious (see Appendix B). 55 2" £ 41 It is of interest to note that since al. & = _3i_ -P* differentiating 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 in section 2.5.1. It can also be derived by the present method, and this calculation illustrates the general procedure., Let the unperturbed eigenvalue E-p(o) be non-degenerate. Without loss of generality, set Ep(o)^ol- The reduced characteristic function, with the perturbed eigenvalue Ep0\) as its zero, can be factored out of the (full) characteristic function, for A in some neighbourhood of \A\-0, in the manner (3.16) fe,^) - rCz^-gC?^) = (2r+-c,Vg(z,:\), where r(£pC\),^= O, ^£pCh)/X)^o. The derivatives appearing in the Taylor series expansion, given in (3.15), are all evaluated for W -o. To determine them, the relations (3.17) f (z,o) = ^ and (3.18) f2n(o,o) = n^n-.(o,oj ;n^o are used. Eguations (3.14), (3.15), and (3.17) give *This simply means that the zero z.p of the reduced characteristic function is the guantity EPQ-\) — E>(O) , i.e., the perturbation of Ep(A)., The derivative of £Cz,a) evaluated at £=epco) , where E>(P) * o , is the same as the derivative of f {%i-EeW, A) evaluated at z=o. Thus, in any application the axis shift z-* z-«-£>(o) need not be carried out; the derivatives are simply evaluated at z = EP(P), instead of at z=o. 57 (3.19) where: the contour encloses EF(p)l but no other unperturbed eigenvalue., The first two terms in 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 third term, -rz *a , has a pole of order one. The residue gives (3.?o) Epc\) = + since g(o,o) = ^(°,°), a special case of relation (3.18)., This result is in agreement with the result obtained in section 2.5.1, and is simply the usual formula for an implicit first derivative.. The higher order terms are found in a similar way. The result is generalizable to many perturbation parameters, as described in section 2.5.1. 58 3.3.2 2-fold degeneracy. Suppose the unperturbed eigenvalues Ee(o) =Ep+,(o) form a 2-fold degenerate pair. Without loss of generality, set Ep(p>= tpt.,(o)=o. For in some neighbourhood of [W-O, the characteristic function has the factorization (3.21) -P(Z,S\) = r-C^A) where r(C;(^^=or <^ A)*0, i =p, P^-I. Now (3.22) = z:agfco) while (3.23) (o,o) n (n-i)g2n-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 first 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 {-Cofo)-o , since "Ep<p) is 2-fold degenerate. The first order term has a simple pole at z-o , with residue <*-f^> Z^** and a third order pole, with residue To first order, therefore. 59 0.25) s, = -3 a*k\ + ocr). For S,, the zeroth order term in the integrand of eguation (3.14) has no pole, the first order terms have zero residues, and the leading contribution is of order A. Evaluation of the residues in (3.15), using eguation (3.23), yields The higher order terms in s, and sa are found in a similar way, though the algebra becomes more complicated. Table III (page 63) gives S, and SA correct to &C??)» Finally, Hevton's formulae can be used to give C, and cA, the coefficients of the reduced characteristic function. To leading order these are (3.27) c, =• A^fy / (3.28) ca = / Table IV (page 66) gives them correct to The expression for c2-=i Bp .Sp*., is, in general, simpler than that for sA~(Epf + {Ep+tft at each order in A., To leading order, the reduced characteristic function is therefore 60 (3.29) r(2E^) = Za -+- Q'X , where the derivatives of -f- are evaluated at z = o, /AJ=o. t The roots of this eguation give the perturbation of the eigenvalues correct to first order, and these roots are in agreement with the result obtained from the hierarchy in section 2.5.2., However, the calculation differs from that in Chapter 2 in two ways. Firstly, no assumption is made about the analyticity of the eigenvalues £p(fa) and Ep+,(2)- Secondly, the formulae obtained for the coefficients of the reduced characteristic function do depend on the order at which degeneracy is lifted. Formulae for the eigenvalues considered separately do not depend on the order at which degeneracy is lifted, as was shown in sections 2.5.2 and 2.5.3. The perturbed, initially 2-fold degenerate, eigenvalues EPC\) and Ep+,C\) are the roots of the reduced characteristic eguation, in this 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 real., Consequently the discriminant, (c,f — 4-cJ# must be even in A, to leading order, i. e., (3.31) (C)a-4-0, = QCtf") t 1,0.,... . If, therefore, the A term of the discriminant is zero, then 61 the A3 term must also he zero., In Table V (page 67) the discriminant is given correct to GOT). For degeneracy not to be lifted in first order the condition is that the discriminant vanish through second order, that is, from Table ¥, (3.32) ((4a)a - 4* fa*) /z=0^, = O , in agreement with section 2.5.3.,If this condition holds, then, to first order, (3.33) Epfr) = Ep+i(?) - 3 &3 I 4* U=o=/a/ and the first derivatives of the initially degenerate eigenvalues with respect to A exist, even if A represents more than one perturbation parameter. In a sense, the 2-fold degenerate eigenvalues are behaving like a non-degenerate eigenvalue to first order and thus have first order derivatives with respect to A..Furthermore, if condition (3.32) holds, then the vanishing <A3 term of the discriminant reduces to (in the notation of section 2.4) (3.34) t2e . This was already shown to be identically zero in section 2.5.3*, In this case the fourth order term in the discriminant reduces to (3.35) /»(o)»a — 3 m 5 m ' in agreement with eguation (2.35), for the case of degeneracy 62 not lifted in first order. Suppose in equation (3.30) is required correct to second order in A.,Clearly c, must be known correct to second order. However, the order to which the discriminant must be known depends on when degeneracy is lifted. If degeneracy is lifted in first order,- then the square root of the discriminant is of the form Jtf^+T^+GCV-) = A Cea+ Ae5)'/a -+- 0(AS) and the discriminant must be known correct to third order in A., If deqeneracy is not lifted in first order, then < the square root of the discriminant is of the form and the discriminant must be kown correct to fourth order in A*. 63 Table III. 2-fold degeneracy: The eigenvalue power sums. 81" — " "(tf ^ 8fckr Wa< WV*44» WfaS^ (V)' • + f 4M 4* -HM* 4* * « 4J»J +... • •-+ ?4*a4» 4* * ,64M 4*;t 4* * 52 4M 4a* 4a +- • Table III, continued.7 16 W^a)-% W^f* «fka)*£»J ...+ ©ft-). 65 Table III, continaed. 3! L 4-' + SI % ^(t^ **** *~M^%^^h* 4a]+• ----»- ^ 4* ^H3a fta * % -&A*(£*f ] + • .... J. ^ (^n^n +... 2-fold degeneracy; The coefficients of the reduced characteristic function.; -s, 21 I 3> I E 67 Table V 2-fold degeneracy: The discriminant of theredaced characteristic  function, ; (c,)a-4-ca = . 5 - L * 1 fz' (fa) (£>)s J ••+5£5r&j'.+ <^41j,]+... 68 3.3.3 3-fold degeneracy.„ The method of calculation is similar to the 2-fold degenerate case,except that here {3.36) r(^,A) = 23-*- -1-C3 f (3.37) £(z,o) = H3 3(^,0), and (3.38) ^n(o,o) = n(n-')(o-3) gzn-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 third order. , 69 table VI. 3-fold degeneracy: Tbe eigenvalue power sums.-... + f2j^ 4"** * 5 4*3 4*"J + - 7^{|(4*/^*f 4* W £**-'4»fei4»"f -+ 4*a (4M)* + 's(4M)* 4M ] + • [ » 4a»*f€*4* 4*(£M)3 J + (4*> L * J 3'- l te (f^fi + |?[-ia43'+ ^2[3fe4M4a'*5V4^4M+34*4s]+ 70 Table VII., 3-fold degeneracy: The coefficients of the reduced characteristic  function •2-' / 43 J c5 31 1 43 j 71 3.3.4 Near-degeneracy. : A set of non-degenerate unperturbed eigenvalues EPC°), . - -t Ep+c^-i Co) is said to be nearly degenerate if the eigenvalues are much closer to one another than to the rest of the spectrum. & reduced characteristic eguation can be constructed with a nearly degenerate set of eigenvalues as roots, an advantage is that the convergence radii of the Taylor series expansions for the coefficients of this reduced characteristic function will, in general, be larger than the convergence radii of the Taylor series expansions for the nearly degenerate eigenvalues considered separately. The reasons for this improvement will be discussed in section 3.6. For simplicity, consider 2-fold nearly degenerate eigenvalues EpC'X) and Ep+,C\), with a finite isolation distance ,i.e., (3.39) j Ep(o) - £p„ (o) J d > O . The reduced characteristic function, with EpC*) and Ep+,fa) as its zeros, can be factored out of the (full) characteristic function. In this case, however, the unperturbed characteristic function is given by (3.40) £ fe,o) = ( 2.~ EpCo))0- Ep+, to) g feP) . Thus the terms in 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. fpti, i. and fp,a in  the complex z-plane. (3.41) where the paths of integration are shown in Figure 6, the coefficients of the reduced characteristic function (3.42) rteA) = 2a+C,Z+CA can be constructed by treating EpC*) and E^fa) as non-degenerate eigenvalues. This is accomplished using the non-degenerate formulae given in section 2.5.1 in the symmetric function expressions (3.3). The advantage is that the convergence radii of the Taylor series for c, and c2t obtained in this »ay# are generally greater than the convergence radii of the Taylor series for each separate eigenvalue. 73 3.4 Contour integral method: H-frr. Eguation (3.13), with the trace taken after integration, provides a simple route to expressions for the reduced characteristic function in terms of operator matrix elements. .. The concept of an effective operator H is not used. Suppose (3.43) H = H°+ -XM . The resolvent can be developed about A=o in the following nay: (3.44) Gr = —L_ = —1— ! z-H z-H° z.-Hf-'M where Gra is the unperturbed resolvent (3.45) • ' G-„ = „ 2 - H Iteration on eguation (3.44) gives the expansion oo (3.46) Gr = 2^G°^VGB^ * For perturbation of the p-th eigenvalue, Gr0 must be expressed in terms of the reduced resolvent for that eigenvalue. Let Pp(<=>) be the projection onto the subs pace of the unperturbed eigenvalue Ep(o). if Ep(o) is g-fold degenerate, then fp(o) is 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-PpCo)). ! • (/-PpCo)), Ep(b)-H° provided the inverse operator is defined appropriately. It has the spectral resolution The k-th power of S has the expression (3.48) and it is convenient to define the *zeroth power* as the negative of the projection, i.e., by (3.49) S° = - Pp(o) . Then the expansion of about z = EpO>) can be written (3.50) G-0 = 2-^^ L^-£p(o)) O , and this yields (3.5D Q0(V^=JJ-.)p,+-+^-^.^wy»-^-^ sn\i...\fs""', for the terns in the expansion (3.46). Without loss of generality, put EP(6)= O. ,The residue of ZmG-t appearing in eguation (3.13), with a contour ^p,<^ enclosing ?=o, but no other poles of G-„, will come from all the terms in 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) 5fn = Tr^ A" MCn> , where (3.52a) 8(n> = +y x Sk'V.. V5k"' (3.53a) CCn> Sk,V...V5kn" Here indicates a sum over all integer k;>o, with the restriction k,-h ...-+• kn^j =• j . The compactness of these expressions is a little deceptive..For example, 8^ is made up of three terms, (3.55) 6<a>= S°\IS°\/S,+ S0VS'V50+ S'VS0V5° , but only those terms with matched ends contribute to the trace. If 5 ' appears at one end, and 5 J at the other, the ends are matched if either k,-=ro=kj or k;*o*Jcj. For example, the term S0VS'V50 76 is the only term in (3.55) with matched ends. 3.4.1 Illustration:2-fold degeneracy.-Suppose £,(o) and Ea(o) are a 2-fold degenerate pair, i.e.* E, (o) = Ej(o). .in this case P,(o) = | t,(o>X E,(o)) -+-• I E^(c?)X E^Co) J and one finds that where (3.56) Qpp,"'k -<Ep(o)lVS,VSiV.-\/SkV|£p,(q)> . For simplicity, assume that the degenerate block has already been diagonalized, i.e.* that (3.57) Vpp. = S"pp« <Ep(o)|V|Ep(o)> , and that, with no loss of generality, E,(o)=0. one finds that S, = E,0) •+• EAfa) and = Aa((vMf+(vM)a)+ A?((v„-yuXQ;1-<?;a))^... v In this notation it is easy to see that if degeneracy is not lifted in first order, i.e., if V,, =. , then the discriminant 77 is to leading order. One has If degeneracy is not lifted in first order, the factors (VM-V^) vanish and& This expression for the discriminant corresponds to that given in Table y in terms of the characteristic function. • In summary, with the trace in eguation (3.13) taken after integration, the coefficients of the reduced characteristic function are found in terms of matrix elements., This is in contrast to the results obtained by taking the trace before integration, where the coefficients are found in terms of (full) characteristic function derivatives, as discussed in section 3.3. 78 3,5 ftn algebraic method: f-^r. 7 The expressions for the coefficients of the reduced characteristic function, given in Tables IV and VII, are lengthy. They sere tedious to compute via contour integration, and sere in fact obtained only indirectly, via Newton's formulae. & more direct derivation of these coefficients, which has certain advantages, is given in this section., The coefficients of r, the c,-, are not found in terms of derivatives of the (full) characteristic function -P, but instead are found directly in terms of the coefficients of -P, the d;. The resulting expressions are more compact than those in terms of derivatives. The method will be explained in detail for the case of 2-fold degeneracy. Consider the characteristic function of an n-dimensional Hermitian operator H = W(a), (3.58) -P(2,A) = H° + a(zn-'H-...+ an . Suppose E,(o)= EjtCo) are a 2-fold degenerate pair of eigenvalues. Then the (full) characteristic function has the factorization with for A in some neighbourhood of |A/=0, The unperturbed eigenvalues £#(o) and E3(°) are set egual to zero by an axis shift so that, to leading order,, 79 (3.59) E;M-= &(*) = (3.60) EjCA) = OO) , j /,a . Leibnitz* theorem* for the differentiation of a product of functions yields <3.61, = +_^L_ re ^ +(^T 'i' I*"*-This expression may be rearranged, with all functions evaluated at 2--o, but I'M^o, to give To obtain expressions for the coefficients c, and ca it is first noted for ?=o, ['Afeo, that and -Pza = <2g -f- 2-c#gz -+- ca gza . The ratios and f/£a yield (3.63) C, = / |+ C,Cjg-+- j _ Q, Cjga ^ (3.64) ca = af / J + <Vgz +Cagza) .. These eguations, together with eguation (3.62), form a set of linked iterative eguations for c, and c^, since the leading *See, for 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 is the degree of -f in 2. To leading order, eguations (3.63) and (3.64) give directly (3.66) c, = and (3.67) ca = 3 £(A*> These first estimates for c, and ca may be substituted back into the right side of eguations (3.63) and (3.64). .Eguation (3.62) is used to express 2-derivatives of g in terms of c,, Cj, and the z-derivatives of f. In this way a second order estimate for c, and cA can be obtained, and this process may be continued to higher orders. An expression for c, or c correct  to some order in A includes some higher order terms., This iterative scheme yields expressions for the coefficients of the reduced characteristic eguation in terms of Z-derivatives of -P evaluated at f =o, but at YX\-OM The results given in section 3.3, previously obtained by contour integration, can be found by expanding ^tC°,A) about YAI-OZ 81 (3.68) ^(°A) = ^t(P,o) +^2; (0}0) -»-... . If there is 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, similarly, (3.70) = 3a -fy + ew) } f2a (0,0) in agreement with eguations (3.27) and (3.28) of section 3.3.2. However, more compact expressions result by noting that (3.7D \tCoy\) = fc-' a-n-t: (*) • To leading order the result is (3.72) C, -and (3.73) = . In this way the coefficients of the reduced characteristic function, the ct- , can be related directly to the coefficients of the (full) characteristic function, the a,-. The coefficients of the reduced characteristic function for other degeneracies 82 can be found in a similar nay, Those for non-degenerate, 2-fold and 3-fold degenerate eigenvalues are given in Table VIII, correct to OC?), OCX), and 6(7?), respectively. In the calculation for 2-fold degeneracy it was assumed that both E,(X) and E^CX) were of order \, E,(o) and E*(o) being zero. , In fact, even if £(2^) is a characteristic function of a non-Hermitian operator, with eigenvalues expressible in convergent Puiseux series, the formulae given in Table VIII are still applicable, ft useful example is given by the characteristic function of the non-Hermitian matrix (3.7fl) HC\) = 1 * X 1 1 o A OO The characteristic eguation is (3.75) -fe,A) = 25-3za+2(l-^a) + f^o with unperturbed roots (3.76) E,(o)= O, EA(o) = ESC°) = /. To construct the reduced characteristic eguation for the 2-fold degenerate pair, E^ and E3, the axis is shifted, The characteristic eguation becomes (3.77) £(x+/,*) = X3H- XA^XC~A-^>~^ - O , -with (n=3) (3.?8) an = , Qn_, = ~(X+V) an-a - 1 > = ' , °-n-rr> - O , m > 3 . From Table VIII, the coefficients of the reduced characteristic function for the perturbation of and Es are 83 13.79) c, = -IM*):-*-^*)* + A.]-*- 0(A3/a) = o + ©Cx5*), In this case the leading order grouping in the table is (3.80) c, ^ Gfr«)\+.QMl+e(7Js'*):!+... C2 = OCX) \+O0?«)i+OCtf) : + and (3.81) E,-ft) = X ± A** +- Oft3*) ^ , 84 Table VIII. Algebraic relations between the coefficients-of the reduced and full characteristic-functions., Terms which are the sase leading order are grouped together. Non-degeneracy. +a(Qn)3(an-af „(anf gn_5 J+ ... (an-.)5 (a„_,)^ c, = Qn ; + (cin)a an-s 2-fold degeneracy. -f (Qn-j)*<Xn-3- <*n Qn-3 ' -feifSedt +^(Q'Hf +.-• (a„_a)^ (a„-3F; (<w^ (a„-a)5 -+S(Qn-.r(ar,-s)5_ 5 ao^-lfan-S + 13 Qn lan-«f Qn-Sqn-^ (<*n-a)7 (an-a)^ (Qfl-a)5 -JO anlQ"-'f(Q/7-s)5+ (QnfQn-5 - 5(an)aan-3 Qn-^ ^ (Qn_2)fc (an-a)3 (Qn-a)4" (Qr£i9^f ! + OCX5). C,= Qn-i 85 Table VIII. continued. :  3-fold degeneracy.-I " " (QrJ_3)S (CXn^y 9^zL « Qn-» _ ' + <9£\3) c3 = ' Bo + (9 fa3) . 86 3.6 Convergence radii.. An advantage of setting up guantum mechanical perturbation theory in terms of resolvents and contour integrals is that estimates of convergence radii can be easily made, as shown by Kato (1949). Central to the calculations in this chapter have been the contour integral formulae. (3.11) Sm - -J— $ 2LM & , and, in the operator form, (3.13) Sm = Jh. AzTGr dz. r-l These integrals 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 analyticity of s^-s^fr) depends on the analyticity of the integrand, for z on ^<^, and for A in some neighbourhood of MI=o. The convergence radius for the power series expansion of Sm 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*^ is 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 in 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) HAVGolUl. By Schwarz^s inequality, IJAVG-oll £ ftlJIVJI.IIGbll , and consequently a sufficient condition for convergence is (3.84) (A/J/V/MlG-oll C I . If the sup norm is used, then the spectral resolution of the unperturbed resolvent implies that (3.85) \\Qx0\\ = [A(z)]~"' , where A(z) is 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 this equation is obtained when the contour keeps Ate) as large as possible for 2. on fp^.,The smallest value that A(z) attains for z. on t^^ 0 establishes the bound on | A/ . , Let be a circle enclosinq ^Just one eigenvalue E,(%) and centered on E.^o). Suppose the nearest unperturbed eigenvalue to E,(p) is E^(o) and that their isolation distance is d,)SL .Let the radius of the circle f"^, be The value of A(z} is dl>3/a. everywhere on this circle. This is the largest value A(z) can take for z on the real axis, between E,(p\ and EjpS (see Figure 7). Thus f^, is a •best* contour, fi lower bound for the convergence radius of the power 88 series for the perturbed eigenvalue E,(A) is thus d,^ /\\ t proportional to the isolation distance. The same is true for If and E2C\) are nearly degenerate eigenvalues, and are treated together, a reduced characteristic function with roots EC\) and E^(A) can be constucted. , The integration contour will be some tf^ (not necessarily a circle, see Figure 7)*, The smallest on is (3.87) daj3 /a ( > . A lower bound for the convergence radii of the expansions of the coefficients of the reduced characteristic equation is thus greater than that for the expansions of the nearly degenerate eigenvalues when considered separately. Kato (1949) points out that in general these lower bounds 89 cannot be improved on..This is shown by the following simple example: (3.88) H° = — I o o / 0 / 1 o Here the eigenvalues of H = H*vaV are whose power series, ±[\ + i? + -l , are convergent for |AJ Eguation (3.86) , with I and IIVII=/, gives the same bound.. It is not possible, in general, to obtain an explicit bound, such as that given by eguation (3.86), from eguation (3.11), since A is usually embedded in a complicated way in the characteristic function . However, an implicit bound is obtained by writing (3.89) f GE.,*) = -PC^o) - A vfe/\) and expanding the integrand of eguation (3.11) according to (3.90) Z." I* •F This series converges for (3.91) which yields the implicit bound 90 (3.92) W < j fU,o> For the example of equation (3.88), the characteristic function is (3.93) fCz,*) - Ha- l-A*. The best circular contour l~^tl around the unperturbed eigenvalue "/ is the circle of unit radius, given by z = -| + c'e , o±9^ 2-n. The implicit bound (3.68) becomes (3.94) m < J^j = j o±e+Qrr , from which, as before, it is found that m < i . In general, however, equation (3.92) is not particularly useful, although it does, in principle, provide a means for the calculation of a bound. To actually use equation (3.92), the minimum value that |^^'^~ | attains, for z on some contour tp^, is required for a range of values of 3. If I V(Z,^)( min. on ^) ^ for all \ such that o^lJ\)<6 where B is real and positive, and 91 / V(Z,C\) I min. (z on r£j%) y for \%\ \ S, then 6 is the bound on YXl • .• It is worth emphasizing that if an expansion of the integrand of (3.11) or (3.13) converges for all z. on some contour, ^ say, then this implies that EP(\) does indeed lie within the contour. Consider (3.95) EPM = & * Y v th p,t The p-th term in the summation can be expanded about H= Ep(p) ^ = ' (EP(A)~<zP(o)\n which expansion converges if (3.96) ) EP(A)-EP(O)1 < \^-EpCo)\ . The above inequality implies that Z is further from EpO>) than EPCA) is; thus EP(A) lies within . An alternative perspective on the convergence properties can be illustrated 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 analytic in ^ , 92 lie on the circle of convergence of E, and EA in the 3-plane*. Thns the expansions of E, and Fa about \AI-0 are convergent for /A/ < \±t) - I . It is for 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 in the complex A-plane would imply exact knowledge of convergence radii. From this point of view, the convergence radii are larger for the coefficients of a reduced characteristic function than for individual eigenvalues, because the degeneracies in the complex plane between the eigenvalues associated with the same reduced characteristic function are not relevant to the convergence of the coefficients of the latter, but do limit the convergence of the eigenvalues. *See, for example, Harkness (1898), page 178. 1 However, for a=/ the eigenvalue e, is still the only eigenvalue within the contour r?t, in the complex z-plane. Thus, while the convergence of an expansion of £, implies that e, lies within r*h, , the converse is not true. 93 3.7 Summary. A reduced characteristic function has been introduced. By taking the trace in eguation (3.13) before integration, the coefficients of the reduced characteristic function are found in terns of derivatives of the (full) characteristic function, taken with respect to z and A. In contrast, by taking the trace in eguation (3.13) after integration, the coefficients are found in terms of operator matrix elements. An algebraic method is developed which relates the perturbed coefficients of the reduced characteristic function directly to those of the (full) characteristic function.y Finally, convergence radii are estimated in terms of both operators and characteristic functions. 94 CHAPTER 4., TWO ILLUSTRATIVE APPIICATIOSS. Introduction., In this chapter : the results of Chapters 2 and 3 are applied to two illustrative examples. In the first example an even-membered ring of n carbon atoms, one of which is perturbed in some way, is considered in the context of Hiickel molecular orbital theory. Expressions for the energies of the perturbed Tf-molecular orbitals are found to second order and these general results are applied to the benzene ring, where n—6. In the second example the usual Bayleigh-Schroedinger perturbation formulae in terms of matrix elements are obtained from the characteristic function. The operator and characteristic function formulations of perturbation theory can be related, formally, through the contour integral expression for the eigenvalue power sums (eguation 3.13). The distinction there was the order in which trace and integral operations are performed.. In section 4.2, however, the interrelation is considered from a different point of view. It is shown how the well-known perturbation formulae in terms of matrix elements can be derived from the characteristic function expressions. An application of quite different character, namely to spin systems, is considered in the following chapter. 95 4. 1 A perturbation calculation for even-membered rings  of carbon atoms.T The Hiickel matrix for the 7Y-molecular orbitals of the perturbed n-atom ring, in units of the interaction parameter/3, relative to the carbon atom parameter <* as the zero of energy, is the nxn matrix rMo.... oi 1 0 1 o . o O 1 O 1 O - -(4.1) o 1 o o O 1 O 1 The characteristic function can be written, in the notation used by Coulson (1938), as (4.2) -Pfe,*) = Mn-APn_, , where M„ is the characteristic function for an unperturbed n-fold ring (4.3) and where (4.4) while P0_, function for =2. co.5 & t is the characteristic (n-/)-dimensional linear chain: (4.5) f^-i = 5inn&/sm9 . From eguation (4.3) the unperturbed eigenvalues are seen to be (4.6) Em(o)=Zcos**?r , m-o, m, g . There are two non-^degenerate unperturbed eigenvalues, (4.7) E„(o) = , En/a(o) = and (j-i) pairs of 2-fold degenerate unperturbed eigenvalues, (4.8) EwCo) = E_„(o) y m~+±,: an 96 The perturbation calculation 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 first few derivatives of PCz,^) , evaluated at \-Ot are accordingly as follows: (M.,o, fa = = , (4.11) ta _ n F Sinne-cos0_ n cosn^l - -nL , (4.12) -IU = 3co5£.P_a . fisinngr|-n*] = , (4.13) "^aAb ~ ° > 2 . The non-degenerate eigenvalues.r From Table I (or eguation 2.20) the first derivatives of the non-degenerate eigenvalues E0{X) and En/2(x) are (4.14) Jf.(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, in this case, must be evaluated by taking the appropriate limits. For E0(A) , the limit 8-+0 yields (4.16) 4^ = na (4.17) 4* = na(na-±)/6 , •z and thus (4.18) &>/<>) = J_ (oa-l) Similarly, the limit yields (4.19) dl§n/a(o)= ~_L (na-/) daa 6na In summary, the two non-degenerate eigenvalues have the perturbation expansions (4.20) E; (X) = E,(o)+ % 4E>(o) - £ dV,(o) + ... dA 21 d3a = ±2. -h A ± X(n*-1) + , . . , > = O (+), n (_) n /in1 • . a 98 The 2-fold degenerate eigenvalues. To find expressions for the perturbed, initially 2-fold degenerate eigenvalues, the approaches of both Chapter 2 and Chapter 3 vill be used. First the calculation is done by treating each eigenvalue separately, using the perturbation hierarchy of Chapter 2. . For comparison, the calculation is repeated, but instead of considering the eigenvalues separately, a reduced characteristic function for each pair of degenerate eigenvalues is constructed, using the results of Chapter 3., From Table II (or eguation 2.30), the first 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 is unchanged in first order., These unchanged eigenvalues vill be labeled in the following by the negative integers , -(§-/•). The other member of each pair has a first order shift that is twice the shift of the non-degenerate eigenvalues. These shifted eigenvalues will be labeled by the positive integers *•>,. The second derivatives, from Table II (or eguation 2.31), 99 are given by 2«5n(oJ A^O Substitution of the first derivatives (4.22) into this equation, together with the expressions (4.11)-(4.13) for the partial derivatives, yields (4.24) 4lEm(o)^ O ..-(g-l) and <4.25) —*(o) = A_ 4 da2 3n* ^a X cos0 n Accordingly, to second order, the initially 2-fold degenerate eigenvalues become (1.26) EmC\) = leasing! + 6CV) , m= -/, ..,-(§->) j <».27> r„m aasaar + a + t - Oft5),. 20 sin2' m = +1,...,+(%-!) Since all second and higher ^-derivatives of -f vanish, i.e., (4.28) f^^b =. O , b > X, 100 it can be seen, by inspection of the perturbation hierarchy, that the eigenvalues which were unchanged in first order are in fact unchanged in all higher orders, i.e., (a.29) Em(n) =. O , n>\, m=-*j--,-(%-l) . Reduced characteristic equation calculation.-Instead of treating the eigenvalues separately, a reduced characteristic eguation, (4.30) Ha+ C,Z +-Ca = for the initially 2-fold degenerate pairs of eigenvalues can be constructed. The non-zero terns in the expressions for ; the coefficients c, and ca in Table I? are l taJ -2U5 4* (4*/ 4* 4^J (4.32) C, = O -f- BW), where all derivatives are evaluated at the unperturbed eigenvalue, i.e., for 6=aJ2Jy, m=±l,..,±|0_i), a=Q. On substituting the partial derivatives (4.11)-(4.13), the reduced characteristic equation is found, to second order, to be The roots of this equation. 101 (4.34) z. = O = 2 _ 2a cos «2n yield the perturbations of the initially 2-fold degenerate s eigenvalues, in agreement with the expressions (4.26) and (4.27) previously obtained for the eigenvalues individually., Application to benzene. ; For the benzene ring, n=6 and eguations (4.6) and (4.7) give the unperturbed eigenvalues (4.35) E0(o) = +X , = , E3 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 radii of these perturbation series. In this casel|VII = l {see eguation 4.1), the isolation distance of all the eigenvalues is 1, and the bound becomes (4.37) W < VX . 102 Algebraic method. •• The expansions (4.36) can also be obtained by using the algebraic relations given in Table VIII. The characteristic function for the fT-molecular orbitals of benzene, in which one carbon atom is perturbed, is (4.38) #2,A) = Hfe-25A - b^ •+• 4-z-3A + 9z*~ 3z3 - 4- . On shifting the axis by putting 2-X+l, this becomes (4.39) £(x+l,3)* X6+X5M)^^-^)^3(-4--t^+xaHa+aA)+4.xA, i.e., E±i(°) has been shifted to the value zero. Table VIII shows how the coefficients of the reduced characteristic eguation (4.40) z2+ C,Z + Ca = 0, for the initially 2-fold degenerate pair E±,io), can be computed from the coefficients of powers of X in eguation (4.39), i.e., from the coefficients Table VIII therefore gives (4.42) c, ^ : + (Q5f a3 _ a±*z (9(?f) = -2 - 2f 4- <9ft3) 3 54-and 103 (4. u3) Co =- ^ i-f- as °-3 -+ eCx) Thus, to second order the reduced characteristic eguation for the perturbation of is (4.44) Z2- -i- -| -2? ) = 0, which yields (4.45) = * / correct to second order, and in agreement with the expressions (4.36) for E-,(A) and £+,fa) obtained earlier. 104 4.2 Derivation of the Bayleigfa-Schroedinqer perturbation  formulae from the characteristic function. Starting from the characteristic function expressions, the well-known perturbation formulae in terms of matrix elements can be obtained. , These are the usual Bayleigh-Schroedinger formulae, except that the series obtained in this way are correctly ordered, corresponding to the series of Kato (1966). The question of •ordering* in the degenerate case is not made clear, for example, in the usual discussions in quantum mechanics textbooks., The Bayleigh-Schroedinger matrix expressions will first be recalled. If E,(o) is a non-degenerate eigenvalue of H° and if (4.H6) Hfo) = H° + AV, then the perturbed eigenvalue E,fa) is given by (<1.»7) E.C0 = E,(o) H- ^V,, + X V +... ..^\Y ^V.SVSI _v„V- V,rVra (In the energy denominators E; = Ei(.°X), If c,(o) is o^-fold degenerate, say, then this formula can still be used provided degeneracy is lifted in first order. The usual prescription is to diagonalize the degenerate block, and redefine the zeroth order energies appearing in the denominator -r -105 as . new patting where E,(o), i-\t..}c^ are the initially q^-fold degenerate set. Op to third order any term in a summation with r or s egual to will be multiplied by the matrix element VrS , r,s^.<^. since the degenerate block has been diagonalized, these elements are zero. This is eguivalent to restricting the summations in eguation (4.47) so as to exclude the indices However, the fourth order terms in (4.47) are only apparently so in this degenerate case. , They do include summation indices and these give rise to terms which are actually third order in A. The terms in eguation (4.47) must accordingly be regrouped, to obtain a correct A-ordering. For a 2-fold degenerate level, when degeneracy is lifted in first order (\f„ VzX ) , this prescription leads to (4.48) EXA) = E,(o) H- ^N/„ * XY V"-Vr, fly V.rX-sVsi _V„V Vr. (In the energy denominators, E, = E;(o).) The perturbed eigenvalue EaQX) is given by a similar expression. The last summation of the X term in eguation (4.48) comes from the first summation of the A term in 106 equation (4.47). This reordering is necessary since the tero V,r Vrs VstVfc. can have s^z without X* or N/j, appearing in the numerator. In this case the denominator contains E.(O)+ *v„ ~EA(O)- -Ay,a = Oi (V,rV„) and the numerator is Vlf-VriVafcVfc, . Thus part of the fourth order term in equation (4.47) becomes third order in equation (4.48). These formulae can be obtained from the characteristic function approach, given an explicit expression for the characteristic function in terms of matrix elements.. In terms of the matrix representation of \J, in the basis in which H° is diagonal, the characteristic function, JTC^A) = dcl-Jz- w°-^v|, can be written jH-E,(o)-}V() W,a .... W,n (4.49) ^Cz,^) = dct Z-E„(O)-}V, nn The determinant may be expanded to give 107 (4.50) -P(Z,A) = rite-EjCo)-^,) - ty V/r5Vsr7T (E_E(-(b)-^V(j)-f-.. ^sr^tu^t TY(*E-E;(o)-M{i)+.. ...+ <9(*5). All indices in these summations are distinct and no double counting occurs. / For example, in the A* term V^V^, appears only once; ^,V(a is not counted, as well. The terms written out explicitly are evidently sufficient to specify any mixed derivative of the characteristic function, evaluated at an unperturbed eigenvalue and at A-O, up to fourth order in ^, i.e., all derivatives ^1-^(^(0)^0) with /S±4. In particular the first and second order derivatives are the following: 108 (4.51) 4^(0)^= TT (Ep(o)-E,(o>) , (4.52) £(EP(0\0) = - V TT i^p(P)-Et(o)) , (4.53) £aM>/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) is non-degenerate, the perturbation hierarchy in Table I leads to the expressions (2.20) and (2.21) for the first and second derivatives., They, along with eguations (4.51)- (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 first derivative: On substitution of expressions (4.53)-(4.55) for the partial derivatives, this becomes (4.58) dE:<(o) = V»+^a ± J (Vy-Mttf-t-4-^,1%! ^A a. a. ' As pointed out in section 2.5.2, this is simply the solution to the guadratic obtained when diagonalizing the degenerate block. If it is assumed that this block has been diagonalized, i.e., that MJ^O^VJ,, 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 lifted in first order (V„ ?tVia ) , expressions for the second derivatives 4?Jg'(p) and 5-ia(o) are obtained from Table II (or eguation 2.31), namely 110 3=o On substituting the partial derivatives obtained from eguation (4.50), this becomes (4.62) This expression is in agreement with the A* term of eguation (4.48), with the summation correctly restricted, i.e.* r* />*. Higher derivatives are obtained similarly, though the details will be omitted here. The expression found for the third derivative, ^7 ~^'{o) , is precisely the A term of eguation (4.48). Since it yields the Taylor series for E|(A) directly, the characteristic function approach automatically ensures that the ordering is correct. 1.11 CHAPTER 5. PHYSICAL PROPERTIES AS ENERGY DEHIVATIVES;-APPLICATION TO SIMPLE SPIN SYSTEMS.T Introduction., In this chapter the determination of various physical properties of a system from energy derivatives is considered. These energy derivatives can be found either from matrix element expressions or from derivatives of the characteristic function. Properties of simple spin systems such as the spin polarization vector and the spin polarizability tensor ^<1> are found using the characteristic 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 is to determine weak hyperfine interactions and one application of the spin polarizability tensor is to determine the intensities of the singularities in polycrystalline 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, Hrs, yields the matrix element of the density matrix, ifArs , in the same basis as the Hamiltonian. This can easily be seen from the expression (5.1) Ep = TrHPp where Pp is the projection onto the subspace of the non^degenerate eigenvalue Ep. The stationary property of eigenvalues (variation theorem) implies that Tr H = O It follows that (5.2) = {pr)sr , 3 Hr5 Thus the physical properties of the system, in 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 first 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 particularly with respect to matrix elements of the 11.3 Hamiltonian, but rather with respect to any perturbation parameters appearing, explicitly, in the operator representation of the Hamiltonian. The first derivatives then represent corresponding first order properties and the higher derivatives, polarizations of these., Consider a Hamiltonian of the form (5.3) H = H°+^VW+ a^v60, where V<0 and Vta> are the operators for two physical properties. If Ep is a non-degenerate eigenvalue of H, and |p> is the corresponding eigenvector, the expectation values of these properties are related to the corresponding energy derivatives by the Hellman-Feynmann theorem, (just the first order Bayleigh-Schroedinger perturbation expression), (5.4) < V(i)>p = <PJVWIP> = ^ , ' = -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 for 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 first derivatives of the physical properties 4yc'^p . The derivative of ^V*'*)p with respect to AJ is 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 in terms of characteristic functions are considered here, which, from equations (2.25) and (2.26), are the following: (5.7) <vc/)> = =25* Here, to maintain generality, it is assumed that the eigenvalue Sp=EpC\) is known for some value of A, say which is not necessarily zero. , The derivatives of •£ in equations (5.7) and (5.8) are evaluated for z=^»(£) and Equation (5.7) , for 115 example, will be written in the following as it being understood that the derivatives are evaluated for 116 5.2 Some properties of simple spin systems  as energy derivatives. The Hamiltonian j£€ for a simple spin system in a uniform magnetic field H is conventionally written (5.9) H + 3>(S£-£S*)+EtSi-SJ) . Following the notation used by Coope (1966), this can be written more compactly as (5.10) Si = i-S +• Bi^sf* where k=cj(^H), where is the irreducible second rank tensor operator, (5.H) [§f% = i (S^S* and where ID is the irreducible zero-field splitting tensor. In the principal axis system of ID, one has (5.12) H^2=|I> , J>XX-JDVV = 3E. It is 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): © = Tr |E?3, A number of characteristic functions for the Hamiltonian (5.10) were given by Coope (1966) using this notation., Those for systems of spin 1r 3/2, and 2, are listed in Table IX. „. 117 Table IX. The characteristic functions for systems of spin t. 3/2. and 2. U = h.S + J:[S](2) Spin 1. Spin 3/2. Spin 2. £ = z5 -+- aa23-»- 03^+ a5 a* = -(si?* ym3) , o3 = -(y&s + zi bjb-b) a5 6(3^3-/?-B>'L)(S1I>*-ZV?) 118 The Hamiltonian in eguation (5.10) is of the same general form as the Hamiltonian given in equation (5.3) except that the parameters X, and Ax are replaced by sets of 3 and 5 independent parameters, respectively, in ij and B> . The first derivatives of the energy of the p-th (non-degenerate) state yield,' the following first order properties of the p-th state: Spin polarization vector, <£>p = <pl5 )p>. (5.14) <-S> = 9EP P 2h, Magnetic moment <^x\ , 3H Quadrupole polarization tensor ([Sj00^! (5.16) <W\ = 2 ID It is worth remarking that the density matrix itself of a spin system can be written in terms of such first order properties (Fano, 1957). For a spin 1 system, for example, the density matrix /=" is given by (Coope, unpublished) /> = iif^).S + <[S7(a5>:[sf} . This example of the way in which the density matrix can be related to first order properties, and thus to derivatives of the energy, is complementary; to the representation of the type (5.2), due to Coulson and Longuet-Higgins (1947). 1-1.9 As discussed in section 5.1, the second order properties, the polarizabilities, can be related both to second derivatives of the energy and to first derivatives of first order properties. In particular, here, one has (5.17) £>l§f = 9<S>P (-5.18) = d<A>p = 3<[s]ca)^ 2hdJ£ 3 ID 2h (5.19) tlr = 3<E^>p In a similar way, derivatives with respect to other coupling constants in the Hamiltonian represent other spin properties.. In particular, derivatives with respect to the hyperfine tensor fl„ in a hyperfine coupling term (5.20) s . gn In , n represent the tensor coupling of S to Inz (5.21) <5r(ln)s>p = p 3(«o)rs If 6„ is isotropic (6„=<yJ) # then (5.22) • Io>p = 2J> These properties of the spin system can be calculated 120 using the characteristic function expressions (5.7) and (5.8). The spin polarization vector is given by (5.23) <S> = d§P = _ 4 and the spin polarizability tensor, 3<€>P , is given by (5.24) 3<^>P = 2?J=P Here <£>p<s> , ^<^§>pr etc. are dyadics. For the spin 1 system, the relevant partial derivatives, of the characteristic function given in Table IX, are 4 = 6z = _ 2Hk - 2 E> fa, ** - Bur, where U is the unit tensor, where lb]w= bb-ib*^, and where \ From eguation (5.23) the spin polarization vector for a spin 1 system is accordingly given by 121 (5.25, , <§> = 2t*y + £]'h j while, from eguation (5.24,, the polarizability tensor 9<£>p t dh for a spin 1 system, is (5.26, ?<£>P = 4zy^]^(^^>^<s>py-^<s>p<s>p 5t 3z2 _ (h*^ ^fl^) (when evaluated at z = EP ). Similarly, the quadrupole polarization tensor ^[Sj6*^, is given by (5.27, <[Sr\ = * fl> + KT~ [ST . The corresponding expressions for the spin polarization vector and polarizability tensor for spin 1 and spin 3/2 systems are given in Table X. 122 Table I. Expressions for the spin polarization vector and polarizability 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~ 3za-(h%-&Es,) Spin 3/2. 9b 4-123 The use of the formulae in Table X can be illustrated for the spin 1 system. , In general, for a given h and JE>, the procedure is to find a root Ep of the characteristic function, numerically, and then to evaluate the expressions at this root. However, an analytic expression for the roots can be given in the particular case that h lies in a principal direction (the ^-direction) of ID, i.e., when ID - b = JPgz b and this case is convenient for illustration. In this case the characteristic function, for the spin 1 system, reduces to (5.28) f = Z3- (h**'***)* -£*i-h*) with roots (5.29) E0 =. - B>zz ^ E±i - £ 3>z^± \l ZDa-Zblh? + h*~ , which, after some algebra, can be written in the more familiar form (5.30) Eo = - #W , E±l = ±TJ>ZZ±J h^(^^lf , or, using the relations (5.12), in the form (5.31) E0 = -^3^ E±, = ^2> ±Jh*+Ea , It is useful to define a parameter (5.32) A - X^-E-,) which measures the separation of the two non-constant eigenvalues. In terms of this, the z-derivatives, evaluated at 124 the eigenvalues, are (5.33) ^|Eo = 2>a-A% (5.34) $z\£±i = 3A(A±2>), and the H-derivatives evaluated at the eigenvalues are (5.35) fb[ <5-«> fkltl - -a^J>±A) . From eguations (5.23) and (5.33)-(5.36) the spin polarization vectors are found to be (5.37) <S>±) = h\ h I E±/ _ ± Jb. and (5.38) <S>o = O By introducing the number /T)=O,±J 7 which becomes the ordinary magnetic quantum number at high fields, eguations (5.37) and (5.38) can be combined to give 125 (5.39) m m k m o} ± 1 Similarly, from equation (5.24), the spin polarizability tensors for the spin 1 system, with the effective field h along a principal direction of tt> , are found to be (5.40) and 91» (£+T>X*-3>) e-J> o o O -E-J> o o o o (5.41) 3 JL, ^(A±2>) E-I> O O O -£-£> o o o o These expressions can be combined into the single expression (5.42) 3<$>fn = (5ma-3)A-mJ> A(AA-DA; rE-r> O O O -E-D O O O O where (n-°,±-l. At large fields m becomes the usual magnetic quantum number, but these eguations hold for all field values. In a similar way the quadrupole polarization tensors are found, from equation (5.27), to be (5.43) (5.44) where % is the unit vector in 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 nuclei. Consider a strongly coupled spin system, with a Hamiltonian of the form (5.9), which is weakly coupled to a number of non-interacting nuclei. The Hamiltonian for the total system can be written rt rt ^ v.. The p-th level of the strongly coupled system with Hamiltonian is split into several nuclear sublevels. The effective eff nuclear spin Hamiltonian H giving this extra substructure is, to first order, • PP where It may be more convenient to compute ^§>p from the expressions in Table X than to compute i^>p as a matrix element. Second order contributions to the effective nuclear Hamiltonian are determined by the guantities 128 (5.46) <pl£lSXi\Slp> The contributions to the effective coupling between different nuclei can be computed from the real part of this expression, and hence from the polarizabilities ^LMF . ,Onfortunately the a Han effective guadrupole moment terms (self coupling) reguire the imaginary part of (5.46). Only the real part is given by the derivatives ii^p (c.f. eguation 5.5). 129 5.3.2. Singularities in polycrystalline spectra, > One use of the polarizability tensor is to determine the intensities of the singularities of polycrystalline spectra., Consider a spin system in a uniform magnetic field H =.(u,&,4). In an E.S.H experiment the freguency of incident radiation is usually fixed, the magnetic field being varied. Besonance occurs vhen {5.47) = Cm'(ti)- constant-. This can be regarded as an implicit function, though it is not usually known in closed form, giving the magnitude of the resonance field as a function of orientation: (5.48) Hr = Hr(0^) . In a polycrystalline sample all G and <f> are present at the same time., The intensity of the resonance signal is proportional to the number of molecules for which Hr(&j<P) actually eguals the magnitude of the applied field H. The main features of the spectrum are peaks of various sorts, where the resonance fields become stationary with respect to 0 and 4*, i.e., when (5.49) V±r = 3Jir = O . The intensity of these peaks is 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 3fMr 3X.3X, Here x, and Xz are orthogonal coordinates, proportional to arc lengths on a unit sphere. For example, if the stationary point {&o0 0o) lies at 0=90°, then possible coordinates are X,-e-009 *Z= The derivatives of Hr with respect to X, and Xa can be related to derivatives of the energy with respect to the components of ^. In Appendix C it is shown that, at a stationary point, the relation is (5.5D 3X; 3Xj H v;. J. v. = H 3;. where (5.52) and (5.53) V; = 3^ 3x; Here a =£J/H is the unit vector in the direction of H. The unit vectors y,, v,, and £ form an orthonormal set. , Finally, from eguation (5.47), the derivatives of ^ in eguation (5.52) are related to the energy derivatives by 131 (5.54) 9V - 3Em. _ 9Em 3h (5.55) - lEm Thus, the energy derivatives «L§m and ^l&n are related, by eguations (5.51)"(5.55), to second derivatives of the resonance field, 9a.Kr. , The Hessian of these second derivatives, in turn, ax, 3XJ determines the intensity of peaks occuring :i in polycrystalline spectra. To illustrate the use of these formulae consider again the spin 1 system in which U is in a principal direction of E>, (5.56) 5> b = fi^z h and suppose further that h> is also a principal direction of g, so that, in the principal axis system of 1t>, the g tensor has the form ~3KX 3xy o (5.57) g - hyx 9VV o Then this z-direction is a stationary direction..„From eguations (5.39) and (5.54) it is found that (5.58) ^b - AM ii where A0») = m'-/77, and from eguations (5.42) and (5.55), it is found that 132 (5.59) 9^ „ 2<sym, _ 3<5>r 9b 2b — ° A(m)) Z(m'+m)*-l> 0 -E_D 0 / A(Aa-J>a) o O O ifc-fefr)] in the principal axis system of ©.after some algebra it is found that J of eguation (5. $2) is given by (5.60) so that r E-Z> O O -3^aHag(,rr.'-<-fn)A-J> c. - O -E-fc o 3 + (5.61) {E-D O OJ O -E-l> ol o o o\ •i Two extreme cases are worth considering. ;lf ZD=0, so that the only anisotropy is in the g tensor, then only the second term contributes, and eguation (5.61) reduces to 133 (5.62) 1 3x In this case, from the Hessian (5.50), the peak intensity is proportional to (5.63) Oh the other hand, if the g tensor is isotropic, i.e., g =gt/# then only the first term in equation (5.61) contributes, andg disappears from this, so that (5.64) 2*Mr _ - H(5(/r7V/n)A,-2>) o -e-r> o o o o In this case, the peak intensity is proportional to (5.65) H(3(m V/n,A -J> (x»=»_ Eaj '/a These expressions are valid for all fields. 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 Phil. 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., 29, 74., Fukui, K., et. al., (1959), J. Chem. Phys., 3J, 287. Goursat, E. , 1904, 1916, A course in mathematical analysis (Ginn, Boston). , Harkness, J., and Horley, F., 1898, Introduction to the theory-of analytic functions (Hacmillan, London). Kato# T., 1949, Prog. Theoret. Phys. Kyoto, 4, 514. ,, Kato, T. , 1966, Perturbation theory for linear- onerators-(Springer-Verlag, Berlin), page 84, eguation (2.49). Kransnosel'skii, et. al., 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 details are given in this report. Their work has been submitted for publication but has not been seen.) Osgood, 8. F., 1913, Topics in the theory of functions of several complex variables, fladison Colloguium (Dover, New York, 1966, or Am. Math. Soc, 1914). Rayleigh, J.B.S.. 1894. Theory of sonndf 2nd. edition, volume I, page 115 (Hacmillan, London). Hellich, 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 Vleck, J.H., 1929, Phys. Hey., 33, 467. 136 APPENDIXA. CONDITIONS FOR ANALYTICITY OF EIGENVALUES OF AS HERMITIAN OPERATOR. The conditions for analyticity of the eigenvalues of an n-dimensional Hermitian operator H(^\) follow from the analyticity of the coefficients of the reduced characteristic function. -v A.1 One perturbation parameter. By assumption, the operator HO) is Hermitian for A real (in general it will not be Hermitian if A is complex with non-zero imaginary part) and thus, for A\ real, its eigenvalues are real., Suppose £P(o) is a g-fold degenerate eigenvalue. The reduced characteristic eguation, with the initially g-fold degenerate set Epfa), - -. Ep+^-i (3) as roots, is a polynomial eguation of degree in z, whose coefficients c, , i^i,...,^, are analytic functions of the single parameter A, in some neighbourhood of A=O. For such a polynomial, where Ep(°) is a root for A=O, the perturbed eigenvalue EPC\) can be written* as a power series in A (Puiseux series) convergent for small A. However, if EpC\) had fractional powers of A in its power series expansion, then, for/) real but negative, the perturbed eigenvalue Ep(A) would be complex. ,Since this is not the case, only integer powers of A can appear in the power series *See, for example, Goursat (1916), volume II, part 1, page 239. 137 expansion of Epfa)1. ,. Consequently EP(%) is an analytic function of A within some neighbourhood of /A/=o. ft.2 Non-degenerate eigenvalues.; If Ep(p) is a non-degenerate eigenvalue then the reduced characteristic eguation for EPC\) is simply rCz.'X) = z + c, ; where c, is an analytic function of X in some neighbourhood of l^lzzO, , Thus ELpC\), the root of the reduced characteristic equation, is also an analytic function of A in some neiqhbourhood of IXI=o. iRellich (1953), paqe 31. 138 APPENDIX B HOBE THAN ONE PBBTUBBATION PARAMETER. The results given in Tables III-VII--are valid for more than one perturbation parameter. Formally the following notation is used in the tables, r al =r —^ =2" Z—» a,!... ae/ and (B.2) V*** = + -If there is only one perturbation parameter the tables can be used directly. If there is more than one, then eguation (B.I) must be used. To illustrate, consider the A term of sa for 2-fold degeneracy (Table III) with A - ,It is given by 31 l i Using eguation (B.1) we find that 5 3 'J 140 APPENDIX C. CALCULATION Of 9aHr/d*, 2xj . . The resonance condition (eguation 5.47) (Cl) V(H) = V CH> X.jX*) = aonsta.nl-can be regarded as an implicit function of the magnitude of the resonance field This is just the kind of implicit function, considered in Chapter 2 except that 2 H f A, —x, f A* —e> Kz ^ It follows that (C2) =Jdy 3-W .2H.+ gV mQH QHl/zv 9x,3Xj ax;9H 2xy 3Xj9H 2K; 3H3H dX, 3XJ j/ 2H J where the derivatives are evaluated at the appropriate values of X., X: and H./Since ?a>ir, in eguation (5.50), is reguired at *> 3K,3XJ a stationary point, where (C3) 3Jir = 3 _ 0 3x, 3x. ' eguation (C.2) reduces to (C4) 9a^r = _ 3fV / 94> 9x,3Xj 3X;3XJ / 3H An orthonormal system a, y,, va can be defined by writing 141 (C.5) M = H u. (C.6) ^ = Hy; ; i-_i,a . dX; ' always assuming x,, and x% ace local cartesian coordinates. In this case we have <C.7) 2Ltl = - S;-- Hoc = _^ « . If, for the moment, the magnitude of H is denoted by x5, then, by the chain rule. (C.8) 22 = 3y . 3M i = and <c.9) ^ = - ^ 3# + ^ - 21* , i,a,3 3X,-3XJ ' 3x, 3Xj 3H 3x,3xj From relations (C. 5)-(C.7) eguations (C.8) and (C.9) can be written more explicitly (C.10) 2V 3x; = H ~ 3H (C.11) 3_y 3H - 3 V (C.12) 3X;9Xj Ha ( v, 9HdH ~- —-~ > ' > $ 3xj 3XJ (C. 13) 3x, 3H H (y; 3H9H 142 (C.14) 9^ = u. . , „ Using eguations (C.11) and (C.12) the expression (C.4) for the second derivative of the resonance field, at a stationary point, can be written as (C15) 5fHr =_(V (V;.PU> .V-) _ 5). H .£±L ]/u.dJ!f Since yf. yj == S}j , this can also be written in the form (c.16) a!±ir - H ( v;. J y.\ ax, Sxjj ~ ~ J / , where (C17) T = -f H* 3ff»? _ H ul/H.3u> m In any application it is simpler to express these derivatives, taken with respect to the applied field H, in terms of derivatives taken with respect to the effective field h. Since b = 3^b, it follows that,, (c-18' k = Oik'*** and thus, also that 143 CC.19) H. 3_ = h. 3_ = H3. = Ai 3_ . ay 3/n 3H Expression (C.17) can accordingly be written in the more convenient form (C20) o~ =-J>Haq.3l^ . a - h -SV.yl/h.dv 

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