L I B R I? it 1- ‘1 ‘ Michigan 3.! 1 “‘5 Unix-1131?; F GE" J" '5 1m 3M"3-fi“'~-" m This is to certify that the thesis entitled SCATTERING APPROACHES TO TWO DISCRETE STATE PROBLEMS presented by SUSAN JEAN NIEMCZYK has been accepted towards fulfillment of the requirements for PH.D. Jememflemicalihysics ll / \J 0 Major profeufl Date February 22, 1974 0-7639 BINDING BY nous & soNS' anux smnmv mc. ,mRARY BINDERS if“? ‘- 'nnnv "Inn-Ann ABSTRACT SCATTERING APPROACHES TO TWO DISCRETE STATE PROBLEMS By Susan Jean Niemczyk Two different approaches to many-body scattering problems are investigated: an S-matrix (scattering matrix) formalism is used in a model study of resonant vibrational excitation of diatomic mole- cules by lowbenergy electrons and a generalized Green function for- malism is used to discuss an exact approach to hydrogen atom and helium atom calculations. Both studies are of phenomena which are usually treated as bound-state phenomena but are considered here from unified scattering points of view. The vibrational excitation cross sections are interpreted as being due to shape resonances. No intermediate compound-state is considered. The molecule is replaced by a system of uncoupled spherically symmetric short-range attractive potentials, all of the same radius. The depth of each well is related to a vibrational state of the molecule. Inter- action between the molecule and the incident electron is permitted by another short-range potential. Perturbation theory is applied to the resulting set of distorted-wave scattering equations. Expressions are obtained for the wavefunctions and the corresponding S-matrix elements. These expressions are used to calculate the S-matrix elements through third order and therefore the elastic and the inelastic cross sections through third order and fourth order respectively for the scattering of a spinless particle by a set of superimposed interacting square wells. The calculated cross sections are compared qualitatively to those Susan Jean Niemczyk obtained experimentally for H N and 0 . The insight into the physi- 2’ 2 2 cal processes gained by using a perturbative approach is discussed. Formal expressions for the wavefunctions and the corresponding S-matrix elements for a multichannel short-range potential system are obtained through infinite order. The possible cross sections resulting from these expressions are discussed. The use of generalized Green functions in bound-state calculations is discussed. A general expression for the closed form of the partial- wave single-particle generalized Green function of at least any Jost function treatable potential is obtained in terms of Jost functions. This expression is used to obtain the generalized Green function for the hydrogen atom. The results of the attempted extension to three bodies for application to helium atom calculations is presented. SCATTERING APPROACHES TO TWO DISCRETE STATE PROBLEMS BY Susan Jean Niemczyk A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Program of Chemical Physics 1974 ACKNOWLEDGMENTS I wish to thank the Gulf Oil Corporation for awarding me its Fellowship in Chemical Physics. I also wish to thank the Atomic Energy Commission, the National Science Foundation and the Chemistry Department at Michigan State University for various assistantships. My thanks are due to Dr. Jack B. Kinsinger for his being in the chairman's office when needed. I am deeply grateful to my research director, Dr. George V. Nazaroff, for his consultation and his direction but mostly for his faith in me and his support of me. I wish to thank Mrs. Naomi Hack for her patience and her recipes. Finally I must thank my husband, Thomas, for his encouragement. ii TABLE OF CONTENTS Page LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . v LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . vi CHAPTER I. INTRODUCTION . . . . . . . . . . . . . . . . . . 1 II. A SIMPLE MODEL FOR RESONANT VIBRATIONAL EXCITAe TION OF DIATOMICS . . . . . . . . . . . . . . . . 3 A. Introduction . . . . . . . . . . . . . . . . . 3 B. Finite-order Perturbation Equations . . . . . . 8 1. Introduction . . . . . . . . . . . . . . . . 8 2. Fundamental Perturbation Equations . . . . . 8 3. Solutions . . . . . . . . . . . . . . . . . 18 C. Calculations . -.- . . . . . . . . . . . . . . 27 1. Introduction . . . . . . . . . . . . . . . . 27 2. Some S-matrix Elements Order by Order . . . 29 3. Some Cross Sections Order by Order . . . . . 46 4. Calculated Diatomic Cross Sections . . . . . 53 D. Infinite-order Perturbation Formalism . . . . . 62 1. Introduction . . . . . . . . . . . . . . . . 62 2. Single Resonance in a Single Channel . . . . 62 3. Resonances in a Multichannel System . . . . 66 iii 4. Discussion . . . . . . . . III. GENERALIZED GREEN FUNCTIONS . . A. Introduction . . . . . . . . B. Single-particle Generalized Green Functions 1. General Expression for a Generalized Green Function . . . . . . . . . 2. Generalized Green Function for a Coulomb Potential . . . . . . . . C. N-particle Generalized Green Functions 1. General Expressions for N—particle Genera- lized Green Functions . . 2. Two-particle Generalized Green Function for a Coulomb Potential . . . B IBL IOGMPHY O O O O O C O O O O O O O 0 APPENDIX A. EXCITATION OF SOME DIATOMICS APPENDIX B. APPENDIX C. APPENDIX D. TION TREATMENTS . . . . . . iv EVALUATION OF SOME PRODUCT INTEGRALS THE NORMALIZATION IMPLIED BY UNITARITY EXPERIMENTAL RESULTS OF RESONANT VIBRATIONAL DISCRETE STATE AND CONTINUUM STATE PERTURBAP Page 72 77 77 80 8O 87 96 96 105 107 109 115 122 124 Table LIST OF TABLES Maxims and minima of the partial-wave S-matrix elements and components given in Figures 2-8... Maxima and minima (in cm.2) of the partial- wave cross sections given in Figures 9-12...... Page 31 47 Figure LIST OF FIGURES Effective potential (- centrifugal potential + electrostatic potential + polarization potential) = centrifugal potential + arbitrary short-range potential (not drawn to scale)................... Zeroth-order S-matrix elements for the target initially in the ground state and in the third- exc1ted StateOOO0.0....0.0000COOOOOOIOIOCOOOOOOOO First-order S-matrix elements for the target initially in the ground state and finally in states £000.00...0.0.0....OOIOOOOOOOOOOOOOOOOOOOO First-order S-matrix elements for the target initially in the third-excited state and finally in states £00....OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO Various components of the second-order inelastic S-matrix elements for the target initially in the ground state and finally in the first-excited stateCOOOOIOOOOOO0..0....OOOOOCOOOOOOOOOOCCOOOOOO Various components of the second-order inelastic S-matrix elements for the target initially in the ground state and finally in the third-excited stateOOOOOOOOOII...OOOO...OOOOOOOOOOOOOOOOOOOOOOO Various components of the second-order elastic S-matrix element for the target initially in the ground stateIOOOOOOOOOOOOOOOOOOIIOOOOOOOOOOOOOOOO Various components of the second-order elastic S-matrix element for the target initially in the thitd’6XCited stateOOOOOOOOOOOOOOOOOOOOOOOOOOO... Zeroth-order (elastic) cross sections for the target in the ground state and in the third- exc1ted stateOOOOOI00.00.000.000...0.00.00.00.00. vi Page 32 36 38 40 41 43 45 48 Figure Page 10 Second-order inelastic cross sections for the target initially in the ground state and initially in the third-excited state......... 50 11 Second-order elastic cross sections for the target initially in the ground state and initially in the third-excited state......... 52 12 Third-order inelastic cross sections for the target initially in the ground state and initially in the third-excited state......... 54 13 Calculated partial-wave elastic and inelastic cross sections through third order for low- energy electron scattering from H2........... 57 14 Calculated partial-wave inelastic cross sections through third order for low-energy electron scattering from N2.................. 59 15 Contour of integration in the k-plane for single-particle Green functions.............. 84 16 Contours of integration in the k-plane for a coulomb potential-0....OOOOOOOOOOIOOOCOOOOOOO. 90 17 Analytic structure of E -p1ane (E -p1ane) for any two-particle Green nfunction.‘:’........... 101 A1 Experimental differential cross sections (at 60°) for low-energy electron scattering from gaseous O . F = .5 meV. (F. Linder and H. Schmidt, Z. Naturforsch. 265, 1617 (1971).)..................................... 111 A2 Experimental differential cross sections (inelastic at 20°, elastic as indicated) for low-energy electron scattering from gaseous N2. P = .15 eV. (H. Ehrhardt and K. Will- mann, Z. Physik 294, 462 (1967).)............ 112 A3 Experimental total cross sections for low- energy electron scattering from gaseous H . P = 2 to 4 eV. (H. Ehrhardt, L.Langhans, F. Linder and H. S. Taylor, Phys. Rev. 113, 222 (1968).)................................. 113 vii Figure A4 Page Experimental angular dependence of low- energy electron scattering from gaseous H2. (H. Ehrhardt, L. Langhans, F. Linder and H. S. Taylor, Phys. Rev. 113,.222 (1968)) 114 viii CHAPTER I INTRODUCTION Dealing with systems as complicated as molecules has tradition- ally been left to the chemists and so there has naturally been a tendency to treat molecules and molecule-like systems from a chemist's point of view. That viewpoint has been synonymous with a bound-state approach and bound-state techniques have been adopted and/or bent to suit the specific need of the system being investigated. But not all behavior exhibited by such systems is best described in terms of bound- state behavior. For example two colliding atoms or molecules often form a resonance, that is, a state that "thinks" for a short time that it is a bound state but then it "realizes" that it is not and so the two colliding particles separate. Typically this behavior has been treated as a bound state which decays. But those resonances most important in reactions are short-lived resonances which are least well-treated by a bound—state approach. A bound state is forever bound; it has no past or future other than being a bound state. A resonance is however only briefly apparently bound; it both starts and ends with its participants being free with respect to each other. To describe a resonance as a decaying bound state is to neglect one of its important aspects--its past. Most correctly such resonance states should be treated by a uni— fied approach which includes both scattering and discrete phenomena. In such a unified treatment scattering states, resonances and bound states are all considered on an equal footing. The physicists long 1 ago developed formalisms for such a unified treatment but these for- malisms have seldom been applied to systems other than those of high- energy physics. Obviously application of these formalisms might give insight into the more complicated systems of chemical physics, and it is this approach which we adopt in this thesis. We not only use one of these formalisms to investigate apparent resonant behavior, but we also use another one to investigate the possibility of advantageously studying bound-state behavior from such a vantage point. Although such formalisms are well-developed, in practice actual calculations on atomic and molecular systems are formidable. There- fore we attempt two different approaches. The first is a model study of resonant vibrational excitation of diatomic molecules by lowbenergy electrons. Rather than attempt elaborate and detailed calculations which give little insight into the physics of the situation we try to obtain qualitative agreement with experimental results by using a simple model which we hope has all the relevant physics of the elec- tron-molecule system. This work we present in Chapter II. The second is the initiation of an investigation of the possibility of doing bound-state calculations by a method which is different than usual bound-state techniques. This work we present in Chapter III. CHAPTER II A SIMPLE MODEL FOR RESONANT VIBRATIONAL EXCITATION 0F DIATOMICS A. Introduction Resonant vibrational excitation of diatomic molecules by low- - * energy electrons has been studied extensively experimentally1 7’ as well as theoreticallyS-lz’f. Fairly detailed descriptions of the states of the studies of resonant scattering from various diatomics can be found in the literature and so will not be repeated here. 'In this chapter we attempt to gain insight into the physical processes involved in resonant vibrational excitation by applying a perturbation treatment to a simple potential scattering model. The tendency has been to treat molecule-electron resonances as "compound states", that is, as bound states which were perturbed and caused to decay. But bound states are independent of how they were formed while a resonance, especially a short-lived one, "remembers" its past. Therefore resonances might be more naturally treated as a scattering phenomenon in which their entire history is automatically considered. So we approach the problem from a scattering viewpoint. As the incident electron approaches a target molecule the mole- cule is polarized by interaction with the electron. The electron in turn experiences an induced electric multipole field caused by the distorted molecule. For large distances r of the electron from the *Also see the references listed in 1-6. +Also see the references listed in 1-12. molecule this polarization potential is a net attractive potential. At small distances r the electron essentially experiences the same electrostatic potential which one of the outer molecular electrons experiences. Consequently it moves in a large orbit which is essen- tially that of the lowest unoccupied molecular orbital. The centrif- ugal repulsion combines with the attractions experienced by the elec- tron to form an effective barrier potential which traps the electron. Eventually the electron tunnels out of the barrier. Exact representation of the effective potential the electron experiences is extremely difficult. The form of the polarization potential customarily used* is valid only for the asymptotic region and calculations using this form can be shown to be very sensitive to the arbitrary cut-off radius imposed. Calculation of the electro- static potential becomes rather intractable for molecules larger than hydrogen. Also artifacts of the method of calculation are often dif- ficult if not impossible to separate meaningfully from the subtle interaction effects which actually cause a shape resonance. Insight into the fundamental aspects of the physics of the problem is thus obscured by calculational problems. Therefore we chose to study a model system which hOpefully would contain all the relevant physics of the problem but would not contain the calculational difficulties of a more exact study. We assume that the lack of spherical symmetry of the target mole- cule can be neglected and that the essential physics of the problem of a slow electron interacting with a nonexcitable molecule can be * _ The polarization potential is usually represented as -ar 4 where a is the polarizability. represented by a slow electron interacting with a potential such as that represented by l = O in Figure 1. That is, it is seen by super- imposing the polarization potential, the electrostatic potential and the centrifugal potentials that we obtain a series of wells for various angular momenta resembling those in Figure 1. Therefore we consider the effective potential as the superposition of an arbitrary yet-to—be—determined short-range potential and the appropriate centrif— ugal potential. Obviously the effect of the long-range polarization potential is merely to decrease the repulsive centrifugal tail for large r and so we can consider the effect of decreasing of the barrier by the polarization potential after we consider our proposed effective potential. Since the resonant behavior of an arbitrary short-range potential is essentially independent of its shape13 we can consider whatever short-range potential is most convenient to discuss. To allow for vibrational excitation of the target we assume a coupled set of such short-range wells with the depth of each well related to the vibrational energy of a specific vibrational state of the target. An arbitrary perturbing potential is added to allow coup— ling between the various wells, that is, between the vibrational states. For calculational ease we assume that all the coupled wells and the perturbing potential have the same radius r0. Also we assume that the perturbing potential is independent of r. To attempt to gain insight into the processes involved we use a perturbation approach. We assume14 that the resonances which we are considering are shape resonances (as opposed to Feshbach resonances). In Section B for a set of initially uncoupled distorted-wave scattering equations wide shape E ' -. v 7’ narrow shape (=0 resonance Figure 1. Effective potential (= centrifugal potential + electrostatic potential + polarization potential) = centrifu- gal potential + arbitrary short-range potential (not drawn to scale). we develop perturbation formulae for the interaction-region wave- functions. From these we obtain perturbation formulae for the cor- 15’16 elements and in turn from these we obtain responding S-matrix perturbation formulae for the corresponding cross sections. The unitarity and the symmetry of the perturbed S-matrix are discussed. We use a Green function technique to obtain usable expressions for the wavefunctions and the S-matrix elements through third order. In Section C we represent graphically some of the zeroth-order, the first- order and the second-order elements for a given set of parameters. Also we display some of the zeroth-order elastic, the second-order elastic, the second-order inelastic and the third-order inelastic cross sections associated with the same parameters. We then discuss the physical "pictures" corresponding to the various orders of the Sdmatrix elements and the cross sections. We attempt to imitate the experimental cross sectionsl’z’l7’18 b y calculating both the elastic cross sections through third order and the inelastic cross sections through fourth order for a coupled square-well system. In Section D we derive a formalism which allows summation of a perturbation series for our system through infinite order. Expressions for the wave- functions, their corresponding S-matrix elements and the associated resonance energies all through infinite order are obtained. B. Finite-order Perturbation Equations 1. Introduction In this section we perform a perturbation treatment of the scattering wavefunctions for a system of coupled not-too-singular short-range potentials. From the perturbation equations for the wave— functions we obtain expansions for the corresponding S-matrix elements. The implications of unitarity as they apply to the various orders of the S-matrix elements are discussed. Formulae for the various orders of the cross sections are developed and discussed. The zeroth-order interaction-region Green functions are used to obtain expressions for the wavefunctions and the corresponding S-matrix elements. 2. Fundamental Perturbation Equations The Hamiltonian for the electron-molecule system can be written as H = H + KE + V (B.1) m e i 9 where Hm is the Hamiltonian for thezmolecule, KEe is the kinetic energy operator for the incident electron and V is the operator for i the interaction between the incident electron and the molecule. As was mentioned in the discussion of our model system we replace the 1 Assuming that all the target molecules are originally in state actual interaction V1 with some arbitrary short-range potential V a, where a represents collectively all quantum numbers applying to a molecule, the wavefunction for the system can be written as v = E xbwba(r). (B 2) where wba(;) is the scattering wavefunction for the electron at a 9 distance r from the center of the molecule and for which the molecule is in state a before the interaction and is in state b after the inter- action and xb(R) is the molecular wavefunction for state b with inter- nuclear coordinate R, that is, (Hm - ab)xb = O. (8.3) We assume that the electronic state of the molecule remains the same throughout the interaction; this is consistent with our assumption that no "compound-state" need be formulated. Also we assume that exchange can be ignored; thus exchange scattering is not allowed to occur. Further we assume that the nonsphericity of the molecule can be neglected; this means that we are assuming that the molecular wave- function has no angular dependence, that is, it is a function only of R, the distance between the two nuclei. Therefore we can perform a partial-wave expansion of the total wavefunction so that we can write V = x (R) w (r)Y (e.¢). (3.4) g b £20 lba 2m where Yzm(6,¢) is a spherical harmonic for angular momentum 2 and ¢£ba(r) is the radial wavefunction for the motion of the incident particle in which the molecule is in state a before the interaction and is in state b after the interaction. The Schrodinger equation for the electron-molecule system can consequently be written as (am + KEe + vi — E) E xb220¢2baY2m = o. (5.5) 10 we now perform a perturbation treatment on this system. For the zeroth-order Hamiltonian H(o) we choose H(o) = H +’KE + V (3.6) m e and for the perturbing potential we choose 3(1) = AV' (3.7) where v; 2 v + AV' (B.8) 1 potential is sufficiently small to allow a perturbation treatment. represents an arbitrary division of V such that the perturbing Since we have chosen that the electronic state of the molecule remain unchanged throughout the interaction, it is the scattering wavefunction which is perturbed. Therefore we can write 2 0° n(n) wlba(r) - f A ¢£ba , (3.9) n=0 where o::: is the nth—order wavefunction for the incident particle for scattering in which the target is in state a before the inter- action and is in state b after the interaction. A is the perturbation parameter. Multiplying on the left of (3.5) by Y:m(6,¢)xg(R) and integrating over the angular scattering coordinates and the molecular coordinate we obtain '2 ' (-Vl + V - Ebw’zba _ —E Vbcwlca ’ (3'10) 11 where VL2 s r‘2{(a/ar)[r2(a/ar)] - 2(3+1)}. (3.11) Eb E E - eb, (3-12) vbc s (XbIlV Ixc>(1-5bc). (3.13) Throughout the remainder of this section except where otherwise indi- cated we shall suppress the explicit expression of the subscript R on the partialdwave wavefunctions and all related functions. Substituting (8.9) into (3.10) we obtain for the interaction region '2 (O) _ (-V2 + V - Ea)¢aa - O, (8.14) (-v£ '2+ v-Eb)¢(1) = -vga¢§g) bia. (3.15) (“V 2+ va)¢(“) = - X Vgc¢§:'1) n=2,3,4,°°° (3.16) cib Similarly for the region outside of which the interaction occurs (- VE- E a)¢(0)= 0, (3.17) z-Eb)¢(n)=0 n-l,2,3,°°° (3.18) Obviously we can write the boundary conditions to which these wavefunctions are subject as [r¢(:)]|rao (3.19) e-i(kbr-£w/2) qbba r+m 1(2kb r) (GOnGbae S(n)ei(kbr-1L1r/2) seba ): (3.20) 12 3‘“) (int) | 3‘“) (ext) | (3. 21) 2:0 4'0 [(d/dr)¢(n)(1nt)] Ir=ro= [(d/dr)¢(n) (extu |,C._.r0 , (3. 22) where k: E Eb’ int denotes inside the interaction region of radius to and ext denotes outside the interaction region. The asymptotic condi— tion (3.20) defines the nth-order partial-wave S-matrix element for which the target is in state a before the interaction and is in state b after the interaction. The deltafunctions in (3.20) indicate that particles -l/2 b * factor ensures unitarity of the S—matrix therein defined . Similarly are incident only in the zeroth order and only in channel a. The k l -i(kbr-2w/2) 1(kbr-2w/2) 1/2 — 3ba Wink.) r) (3 e -s ba bae ) (3.23) defines the total partial-wave S-matrix element for which the target is initially in state a and finally in state b. Again the k;1/2 factor ensures unitarity of the S-matrix. By comparing (3.9), (3.20) and (3.23) it is evident that s x “3(“). (3.24) a ba = E b n: 0 Since both the zeroth-order problem and the total problem must be considered to be complete problems, both appropriate partial-wave S-matrices must be unitary. That is, |S§g)|2 = 1 (3.25) and Z Isba = 1. (3.26) * See Appendix C. 13 By expanding Sba in terms of the nth-order S-matrix elements and sub- tracting (3.25) from (3.26) we obtain n-huS (n)* SIEm) (0)* (0) ____ g [mZO nEons b8 )- sba Sba ] o. (3.27) Because of the presence of the perturbation parameter A, (3.27) is actually a series of conditions which must hold because of unitarity. That is (0)*S(1) 2 Re(Sa Saa)-= 0 (1) 2 (0)* (2) _ é ISba I + 2Re(saa Saa ) _ o (1) S(2) (0)* S(3) _ E 2Re(sba sba ) + 2Re (Sa 38 ) — 0, 2 |s(2)|2 + Z 2Re(S(l)*S(:)) + 23e(s(0)*s(4)) - 0 b be b ba ba aa aa _ ... (3.28) Unitarity of the S-matrix is the mathematical expression of the conservation of probability flux. Therefore unitarity applies only to open channels and so the explicit summations in all the unitarity con- straints are only over open channels. But the nth-order S-matrix ele- ments themselves for n 2 2 contain summations over all possible inter- mediate states including those both in open channels and in closed channels. If we define a component of an S-matrix element as the part of that element which is due to a specific series of intermediates taken in a certain order then we can consider the S-matrix element as the separable sum of all its possible components. Therefore we can divide any element into a part containing all those components in which all the intermediate states belong to open channels and another part 14 containing all those components in which at least one of the interme- diate states belongs to a closed channel. Then we can write each of (3.28) as two equations: one with all products involving only open- channel contributions and another with all products involving any closed-channel contributions. For example in the second-order con- straint the only closed-channel contribution comes from the second- order elastic S—matrix element. So we can write (0)* (2) Re[Saa Saa (closed)] - 0, (3.29) where closed indicates that only components in the second-order elastic elements containing closed-channel intermediate states are considered. Similarly an open-channel constraint can be written. These and similar separations of the unitarity constraints can be useful for checking calculations. More general expressions of unitarity than (3.25) and (B.26) are (0)* 8(0). g Sbc Sba Gca’ (3'30) * E Sbcsba= aca’ (3'31) which when combined give more general unitarity constraints than those given in (3.28). But those stated in (3.28) are sufficient for checking our calculations and so we will not further consider (3.30) and (8.31). As an aside it can be noted that the S~matrix can be shown straight- forwardly to be symmetric order by order provided both the zeroth-order Sdmatrix and the complete S-matrix are symmetric. The proof depends on the time-reversal invariance of both the zeroth-order and the com- plete wavefunctions from which the following relations can be obtained: 15 * (0) <0) _ g scb Sba 6ca’ (3'32) * g Scbsba - 6ca° (3.33) Replacing the S~matrix elements of (3.31) and (3.33) with their per- turbation expansions, combining the results with (3.25) and (3.32), and comparing term by term the desired symmetry is seen. This order by order symmetry can serve as another useful check of calculations. The elastic differential cross section we can write as _ -1 -2 m ' * 088(6) - 4 k £20 £'§0(2£+l)(22 +1)(Szaal)(S£.aa1)P£[cos(6)]P2,[cos(6)] (3.34) where we now explicitly denote the angular momentum dependence 1 of each S-matrix element. Similarly the inelastic differential cross section we can write as -1 -2 E oba(6) - 4 k X (22+l)(2£'+1)S£basz.baP2[cos(6)]P2,[cos(6)]. I '3 2 ° 2 0 (3.35) In either the elastic or the inelastic cross sections interference between various partial waves will be important only if the amplitudes for different partial waves are of the same order of magnitude. For resonant behavior such interference can usually be ignored; therefore we can consider the simpler total cross sections 11’ o = 2n I o(6)sin9d0. (3-36) 0 16 That is, -2 Q 2 ’2 u 2 a . "k, 2 (22+1)|1-82aa| = nka (2£'+1)|1-S,na I as 2‘0 a , (3.37) |2 (3.38) -2 m 2 O In “k8 Z (22+1)|S£ba 2"bal 9 -2 2 wk (23"+1)|s ba 2:0 a where the approximations are valid only if just one partial wave is close to resonance and in such cases 2" denotes the angular momentum nearest resonance. In the resonant behavior we are considering, the electron is almost bound to the molecule and consequently moves about the molecule in what is essentially the lowest unoccupied orbital. Because this orbital is large, a single-center expansion of the molec- ular orbital provides a good approximation to the orbital. The lowest angular momentum allowed in the expansion determines k", the dominant partial wave. Expanding the S-matrix elements in terms of their perturbation expansions and applying the unitarity constraints previously derived we obtain the various orders of the total elastic and the total inelas- tic cross sections: (0) -2 (0) 2 O '3 Tl’ka E (22+1)|1'Szaal 9 (0) abs = 0’ o<1) . 0(1) = 0, aa ba * (2) -2 (0) (2) (2) a . wka E (22+1)[ 2 Re(s£aas£aa ) - 2 Re(S£aa)], 17 2 -2 oéa)- nka é (22+1)|s(1)|2, * 0:2) . wk"2 2 (22+1)[ 2 Re(S§g)S§:; ) - 2 Re(S§::)], m 2 0(3). -2 (1)S (2) Che ”ka E (22+1)[2 Re(S.Q.bas 2ba )] ’ ... (3.39) Near a dominant resonance only one partial wave contributes and for eadh order we can make the approximations made in (3.37) and (3.38). The zeroth-order problem is complete, that is, all the flux which enters the problem in zeroth-order is accounted for at the end of the zeroth-order scattering. Therefore for higher-order interactions flux must be removed from the zeroth-order problem and must be redistributed to other orders. The higher-order cross sections are therefore correc- tions to the zeroth-order cross sections. In a given order if flux is put into a channel we can expect a positive correction cross section, but if flux is removed from a channel we can expect a negative correc- tion cross section. It should be noted that by utilizing the unitarity constraints we can obtain the perturbational equivalents of the optical theorem. By considering total aa + X Oba (B°40) bfa we obtain (0) _ -2 ” _ (0) ototal _ nka Z (22+1)(2 2 Reszaa), <1) ,_ 0’ 0total 18 (2) —2 ” <2) Ototal - nka £§O(22+1)(-2 Reszaa), (3) - -2 ” _ (3) ototal nka 220(22+1)( 2 Reszaa), ... (3.41) 3. Solutions We now obtain the actual wavefunctions and S-matrix elements for our model. We use the zeroth-order interaction-region Green func- tions to generate the first few orders of the interaction-region wave- functions and then we use (3.21) to obtain the corresponding S-matrix elements. For orders greater than one we can obtain the scattering solutions for the interaction region of a short-range potential system by using the apprOpriate Green functions of the zeroth-order equations. The apprOpriate Green function for each vibrational state is the inter- action-region standing-wave Green function of the proper wavenumber which obeys the boundary conditions (3.21) and (3.22). That is, the Green function gb for the zeroth-order problem in which the molecule is in state b is gb = wl(Kbr<)w2(Kbr>)r—2{W[w1(Kbr)pw2(Kbr)]}-1, (3.42) where w1 and w2 are two linearly-independent solutions of the zeroth- order interaction-region equation for channel b such that w1(0) = 0, (3.43) lwg/w2(Kbr>llr=ro = [h+') + Yby(Kbr>)]r-2Y;1{W[j(Kbr).y(Kbr)]}_} (B.46) where _ 4‘! t + Yb = [1(Kbr0)h. (kbr0)-j (Kbro)h (kbr0)] (3.47) ' + +' —1 °[y (Kbr0)h (kbro) - y(Kbro)h (kbro)] . j and y denote the regular and the irregular spherical Bessel functions for the angular momentum under consideration. For {Vbc} independent of r we can express the wavefunctions for our model in terms of these Green functions as ¢ba ba 8b¢aa dr, (1) _ v' [to (0)r2 o (2) _ . r0 (1) 2 ¢ba cgbvch gb¢ca r dr' 0 ba b ca r ¢(3) - E v' I 0g ¢(2)r2dr. c b °'° (3.48) For any not-too-singular short—range potential we can write 20 r 0 2 _ 2 2 -1 Jo gbw1(Kar)r dr - (Kb Ka) [w1(Kar) + w1(Kbr) Cl(Ka’Kb)] bfia, r 0 w (K r)r2dr = (2K2)—1[- '(K r) - (K r)/2 + (K )C (K )] 0 8a 1 a a rwl a V1 a wl ar 2 a ’ ro ' 3 _ 2_ 2 ‘1 v 2 2- 2 -1 J0 gbw1(Kar)r dr - (Kb Ka) [rw1(Kar) + 2Ka(Kb Ka) w1(Kar) + wl(Kbr)C3(Ka.Kb)] bi‘a. to w'(K r)r3dr = --2 r0 w (K r)r2d +-1 r w (K r)C (K K ) O 8a 1 a 2 0 8a 1 a r 2 0 1 a 4 a’ a ’ (3.49) where a prime denotes differentiation with respect to r. 2 ' +_ _ I +-l C1(Ka’Kb) ' [wlahb w1 httnwlbh+ b wlbhb] : . + +' 2 + v + C2(Ka) _ [(wlaha + wlaha )/2 + roKawlaha + rowlaha - £(£+1)r01w la h+ a1][w aah - w1a hzll v +' H C3(Ka,Kb)= — {rowla hb +-w1a hb - rowlahb h'+ v +-1 ’ 2K2 a(Kb Ka>l ahb ’ w 1ahb]}[wlbfi b ' wlbhb] b*a’ + +' v +-1 C4 (Ka Kb) _ w 1ahb [wlbhb - wlbhb] , (3.50) where wla E wl(Kar0)’ v = v wla " w1(Kar)|r=r 21 + = + ha - h (karo), ! i + = + ha _ h (kar)|r=r0. (3.51) As is shown in Appendix B the integrals (3.49) involved in obtaining all the wavefunctions of order higher than zero can be done in a gen- eral form for any not-too-singular potential because the method of integration depends only on nonspecific properties of the Schrodinger equation. For our model with the target initially in state a and b # a we obtain where (1) aa ¢ (1) ¢ba ¢(2) 88 (2) ¢ba =0, Méi) “1(Kbr) + Bb aw1(Kar)’ = {{(B(l)/B(°))[3(1C)wl(xar) + B;Cwl(Kcr)] c¥a + v;c3;a(2K:)-1[—rwi(Kar) - w1(Kar)/2 + w1(Kar)C2(Ka)]}, c b,aVb + Z (3 (1)3' /B(0))W1(Kcr) + g VLCB;3(K:-K:)‘lwl(xar)’ c¥b,a ca be c b,a (3.52) (0) _ - -1 -1 3c 2 hC(wlc) (~21mc)(fC-dc-imc) , ' _ ' 2 2 -1 (0) Bbc : Vbc(Kb-Kc) Bc ’ 22 38:) 5 -Bbcwlc(wlb)-1(fc-db-imb)-l(fb-db-imb)-l’ (3'53) with c1C 2 Re(h:'/h:), mc :- Im(h:'/h:). (3.54) We then obtain the various orders of each S-matrix element by solving (3.21) using the expressions obtained in (3.52) for the inter- action-region wavefunctions. (Obviously we can obtain the same S-matrix elements as those obtained by the method just used by merely expressing the nth order of the interaction-region solution for channel b as the sum of an arbi- trary constant times the regular solution of the zeroth-order equation for channel b and the appropriate particular solution of the nth-order equation for channel b, applying (3.19)-(3.22) and solving for the arbitrary constant and the corresponding S-matrix element.) We now present our results for the S-matrix elements through third order. Because we are interested in the development of resonance structures in the cross sections as the perturbation treatment pro- gresses, we express the various orders of the S-matrix elements in notation which makes any resonant behavior obvious. Using (3.54) we can write the zeroth-order elastic S—matrix element as 23 21: _ 5(0) . e aa(f -d +im )(f -d -1m ) 1 aa a a a a a a 215as = e [l + Zima/(fa-da—ima)], (8.55) where 21g _ — + e ba : -ha/hb. (B.56) Near resonance .. _ (0) fa-da — B(E Era ), ma 2 -BFa/2, (3.57) where B is a constant, Big) is the resonance energy for channel a, and Pa is the half-width of that resonance. Therefore near resonance 215 (0) ~ aa _ . _ (0) -l Saa -'3 [1 1Fa(E Era +iFa/2)] . (3.58) Obviously the first term of the second factor varies only slowly with incident energy; it causes hard-sphere or potential scattering. The magnitude of the second term changes rapidly near the resonance energy; it causes the resonant or nondirect scattering. In order to enable us to compare the behavior of the higher-order Sdmatrix elements with the zeroth-order results we use (8.52) and (B.56) to express our higher-order elements as (1) SILaa 8 0’ (1) 1/2 t 2_ 2 -1 _ 3m - (k.a/kb) vba (Kb K3) 21ma<£b fa) .(fb-db-imb)_1(fa-da-ima)-lexp(21gba)’ 24 s(2)_ v v 2_ 2 -2 _ _ _ szaa- Z vacvcama Kc) 21ma(fa dc 1mc)(fa fc) cfa ’(f -d -im )'2(f -d -im )‘1exp(2ig ) a a a c c c aa - E v' v (KZ— K2)’1(2K2)'121m I ac ca c a a a a c a (f —d -im )‘2ex (215 ) a a a p aa ’ S(2)_1/2 2 -1 2_ -1 2_ 2 -1 SEba I"(ks/kb) CE; bvbcvca(Kc Ka) (Kb Ki) (Kb Ks) 21m -[ 2(f -£ )(f -d -im )-K2(f —f )(f -d -im ) a Kb c a b c c a c b a c c —K2(f -f )(f -d -im )]-(f -d -1 )’1 c b a c c c b b mb . "1 "1 21 ) °(fa-da-1ma) (fc-dc-imc) exp( gba , S(3)_ 2_ 2 -1 2_ 2 -2 2_ 2 -2 Slaa- E ac 2 Vcevea(Kc Ke) (Ks Ke) (Ka Kc) cfa e#c,a 4 21ma {Ka(fc-fe)(fa-de-ime)(fa-dC-imc) + K“(f -f )(f -d -im )(f —d -im ) e C a a C C e e e 4 + Kc(fa-fe)(fa-de-ime)(fc dc-imc) KK2 I(:-C(fa -f )(fa -f ee)(d +ime -d cc-im) 0: 9:30 (07:53 K(fa -f ca)[(f -d e-im ee)(f -d c-im c) + (fa -dc -imC )(fe -de -im e)] 25 2 2 KaKc(fa-fe)[(fa-dc imc)(fc-de-1me) + (fa-de-ime)(fc-dc-imc)]} -2 -1 -1 _ - _ - - - 1 (f8 :18 ima) (fc dc imc) (re de ime) exp<2 £33). 3 1 2 . . 2 2 -1 2 2 -1 2 2 -1 Sébl I (Ra/kb) I {Z vcevea(Kb-Kc) (Kb-Ks) (Kc-Ks) cib,a -(Ki-K:>'1fiw mdoauoom mmouo o>m3IHowuumn can no A~.au Gav oawofia vow usfixdzn .N manna 48 zero order W i=0 intensity (orb. units) |.5 539v collision energy Figure 9. Zeroth—order (elastic) cross sections for the target in the ground state and in the third excited state. 49 (2) fa only one process, the direct transition from channel a to channel f. (1) fa ' peaks, one at the location of the zorp of channel a and another at The second-order inelastic cross section 0 also represents Therefore it is readily interpreted in terms of S We expect two the location of the zorp of channel f. Fbrther we expect the peak occurring at the higher incident energy to have the greater amplitude. As f and a separate with reSpect to energy we expect the overall mag- nitude of o(: ) to decrease. The behavior of GE: ) for two different initial states of the target can be seen in Figure 10. (2) The second-order elastic cross section.o% is composed of two contributions: -2 Re(S(2)) and 2 Re(S;g)S ::)* ). The sum of these two terms can be regarded as representing interference between zeroth- order elastic scattering (given by (l-Sig))) and second-order elastic scattering (given by 8(2)). According to this interpretation -2 Re(S::)) represents interference between a completely unscattered zeroth-order wave in channel a and the second-order elastically scattered wave in channel a. Therefore this term represents the removal from the elastic channel of flux scattered in zeroth order but not scattered after second-order processes have been considered and the addition to the elastic channel of flux scattered after second- order processes have been considered but not scattered in zeroth order. We expect the ith component of -2 Re(S::)) to produce a resonance peak at the energy of the zorp of channel 1. Further we expect the ith component to produce some type of interference structure of exagger- ated magnitude at the energy of the zorp in channel a. From the (2) already discussed behavior of 8a we expect as 1 increases the inter— mediate—state peak will increase in magnitude relative to the cross section(orb. units) 50 second order inelastic 30 L: lam ll O-r3 3-2 K N 34 0-.4 \ A A - L5 1 3.30V L5 3.36V Figure 10. Second-order inelastic cross sections for the target initially in the ground state and initially in the third-excited state. 51 initial-state structure; also we expect the overall magnitude of a given component's contribution to -2 Re(S§:)) to be inversely related to the closeness of the energies of a and i. 2 Re(S§g)Si:) ) repre- sents interference between the total outgoing zeroth-order wave in channel a and the second-order elastically scattered wave in channel a. By the second-order unitarity constraint we can see that this term equals - Z IS§:)I2, which is proportional to the total second-order inelasticbcross section. Therefore this term represents flux which is removed from the elastic channel and is redistributed to the inelastic channels. Obviously this contribution to the cross sec- tions must be everywhere negative. The behavior of some components of the second-order elastic cross sections for two different initial states of the target can be seen in Figure 11. represents inter- The third—order inelastic cross section oéz) ference between first-order inelastic scattering and second-order inelastic scattering. The unitarity constraints provide no simpli- fication for the interpretation as they did for the second-order elastic cross section. In third order flux can be removed not only from the elastic channel but also from the inelastic channels. Obviously for a given inelastic cross section 0:2) flux can be removed in any relative quantity only at the locations of the zorps in chan- nels f and a. In general we expect flux to be added at the locations of the zorps of intermediate channels. As in all inelastic cross (3) f sections we expect the overall magnitude of o a to decrease as f and a separate with respect to energy; further we expect the ith components' (3) fa respect to energy. We note that interference with a component for an contribution to o to likewise decrease as i, f and a separate with cross section(orb. units) 52 second order elastic M 3—2 J a, or 1;. .. V I I T r r I 1 LS 3.36V |.5 3.3eV Figure 11. Second-order elastic cross sections for the target initially in the ground state and initially in the third- excited state. 53 intermediate state of greater energy than f and/or a causes flux removal at the locations of the zorps of channels f and/or a; simi- larly interference with a component for an intermediate of lower energy than f and/or a causes deposition of flux at the locations of the zorps of channels f and/or a. The behavior of some components of the third-order inelastic cross sections for two different initial states of the target can be seen in Figure 12. 4. Calculated Diatomic Cross Sections In this subsection we present and discuss the results of our calculations through third order for the cross sections of low-energy electron scattering by H and N . Our results for O are analogous 2 2 2 to those for N2, except that the peaks for O are much narrower, and 2 so we do not repeat those cross sections here. We compare and con- trast our results with the experimental cross sections. lbr refer- ence some of the relevant experimental cross sections are given in Appendix A. -12 The H2 cross sections are given for V’- 5.49 x 10 ergs, - -1 r0 = 3.16 x 10 8 cm., V! = 5.49 x 10 3ergs and 2 - l. The NZ cross sections are given for V’= 4.4 x 10.11 ergs, r0 - 2.09 x 10”8 cm., V’ . 4.4 x 10-13 ergs and 2 = 2. Rar a given angular momentum the combination of the well radius ro and the initial well depth \Iwhich produces cross sections with peaks of the appropriate half-widths is highly specific. That is, only one combination of radius and well depth can be found to produce a given cross section. Upon comparing the parameters for which we cross section(orb. units) ‘2 i i: 9 54 third order inelastic ().. .J U 0-03 3~4 \ J \ L 3-IO 3‘2 I I I I I |.5 3.30V I5 I 3.39V Figure 12. Third-order inelastic cross sections for the target initially in the ground state and initially in the third-excited state. 55 obtained the best N cross sections with those obtained in doing much 19,20 2 more exact calculations our model does not appear to provide very good results. However when we add the effect of the polarization of the target molecule to our results, we see that our parameters actu- ally are in agreement with the previously obtained ones. Polarization reduces the barrier width and so decreases the resonance lifetimes and increases the resonance half-widths. Therefore to obtain the same cross section as before polarization was considered,the well must be deepened and its width must be decreased in a manner which keeps (Vt§)1/2 approximately constant. The previously obtained parameters can be obtained from ours by such variation of r0 and V. Ibr the parameters used to obtain the cross sections in Figure 13 our results are in close qualitative agreement with the experimental cross sections for H2. However as can be seen in Figure 14 our best attempt to imitate the experimental N2 cross sections is not as much in agreement with the experimental results as was our H attempt. 2 In our calculated N2 cross sections some peaks are missing or are too small compared with experimental cross sections. Also our calculated cross sections do not display exactly the same final—state dependence as the experimental ones do. Our calculations correctly show that the dominant structure of the cross sections shifts to higher energies as the final state increases but do not show that the resonance half-width apparently increases as the final state increases. Further the spacing between the peaks in our calculated elastic cross section does not change as significantly from the zeroth-order spacing as it does in the experimental cross section. We obtained only a very slight shift of any of the peaks. 56 Figure 13. Calculated partial-wave elastic and inelastic cross sections through third order for low-energy electron scattering from H2. cross-section (orb. units) 57 O-IO xIOO Q3 aoaV Figure 13. 58 Figure 14. Calculated partial-wave inelastic cross sections through third order for low-energy electron scattering from N2. 59 O-rl ’ J a e 2 4 O 3 # 0 0-4 AHH t W.) Amzc: .Qccv c0_._.ommImmo._o 3.3 eV l.5 Figure 14. 60 For the parameters we used for Hz the cross sections rapidly decrease in magnitude as the order of perturbation increases. But for the parameters we used for N the cross sections decrease only 2 slightly in magnitude as the perturbation order increases, so that many orders of perturbation seem to be significant. For the param- eters we used for 02 the cross sections decrease even less rapidly in magnitude as the perturbation order increases than for N2 so that even more orders of perturbation seem to be significant. These results are in agreement with what we should expect if we consider the life— times of the various resonances. The Hz-electron resonance is very short lived (IO-lssec.) and therefore only low orders of interactions should be required to account for its behavior. The Nz-electron l4 resonance is longer lived (10- sec.) and so higher-order inter- actions should be important. The Oz-electron resonance is even longer lived (10-12 sec.) and so even higher-order interactions should be sig- nificant. It should be noted that it is precisely the peaks which are absent from or too small in our N results (also our 0 results) which 2 2 would tend to be emphasized by higher orders of perturbation. Although higher-order calculations were indicated for N2 and for 02 by these considerations,our calculations could not be extended due to a lack of sufficient computer core (CDC 6500). The final-state dependence present in our cross sections is somewhat surprising. Because we took all the coupling elements {vba} to be equal, it is obvious that the usually proposed Franck- Condon factor effect21 used to account for the increasing importance of the higher energy peaks as the final state increases is only part of the total explanation of the final-state dependence of the cross 61 sections. The general dependence of low-energy cross sections on the incident energy is another very important factor in final-state dependence. The change in resonance width in the experimental N cross 2 sections as the final state increases is probably caused merely by the distortion of the resonance peaks due to overlapping. This effect might appear in our cross sections if the missing peaks were present. The difference in the spacing of the elastic cross section peaks and the zeroth-order peaks might also appear in higher-order calculations. Also it should be noted that the change in spacing may be related to the unequal spacing of the ground-state vibrational levels and our calculations considered only equal spacing. In summary we can say that our results seem to produce the correct qualitative results for H However higher orders of pertur- 2. bation are required to determine whether indeed our model is suffi- cient to account for all the observed phenomena for N2 and for 02. Despite the calculational shortcomings of our cross sections we have gained valuable insight into the physical processes involved in low-energy resonant vibrational excitation of diatomic molecules by using our perturbation approach. In the next section we use an infinite-order perturbation technique to study our model to help elucidate the results which might be expected if higher-order cal- culations were performed. 62 D. Infinite-order Perturbation Formalism 1. Introduction In this section we use an infinite-order perturbative approach to consider perturbed resonances in systems of properly-behaved poten- tials. By a properly-behaved potential we mean any potential for which the Jost function322-24 can be analytically continued onto the unphysical sheet of the energy surface at least as far as the resonance energies. This group includes all not-too-singular short-range poten- tials. For an isolated perturbed resonance we derive formal expres- sions for the wavefunction in a single-channel system and for the wave— functions in a multichannel system- For s 861188 0f overlapping reso- nances we derive formal expressions for the wavefunctions in a multi- channel system. In all cases we obtain expressions for the resulting S-matrix elements. We make certain assumptions about some of the factors in our expressions for the S-matrix elements which are appro- priate for a system of coupled shape resonances at low energy. Then we use these assumptions to discuss the expected cross sections for low-energy electron resonant vibrational excitation of diatomic mole— cules. 2. Single Resonance in a Single Channel In terms of the zeroth-order Green function g, the zeroth- (0) order wavefunction ¢ and the perturbation V we can write13 the complete wavefunction for a single channel as w = X (gV>“¢(°)- n=0 (D.1) 63 Near a zeroth-order resonance the Green function can be written as) (0)) g = g + u rvi/(E— E (D.2) where urv: is the residue of the Green function at the resonance energy E<0) + , vr is adjoint to Ur, ur and Vr are both normalized to unity and g is orthogonal to the space of ur' If we define25 (1+GV) a Z (§V)n (v.3) n-O 25 and note that we can write 2 (gV)n = (1+GV) Z [w(1+GV)]n, (0.4) n=0 n-O where : + _(0) w _ urer/(E Er ), (D.5) it is readily seen that w = (1+év>¢(°) + (l+GV)w(1+GV)[1-w(l+GV)] 1¢ (0). (D.6) This can be rewritten as w.(1+év)¢(o) + (1+EV)ur(vr|v+v&v|¢(o))[E-EiO)-1’1, (D.8) instead of (D.6) and (D.7). Because ¢(O) is proportional to (E-E£0))-1 near resonance, the numerator in (D.8) is generally non- vanishing. From (D.8) we can obtain the S-matrix element as (0)+ s - 1 - 2n1<¢(0)"|v+vévl¢ )(E-EEO))[E-E:0)-(vrIV+VGVIur)]-l. (D.9) (0) From either (D.8) or (D.9) we can see that Er +(vrIV+VGVIur) is the resonance energy through infinite order for the resonance under consideration. It is conceivable that for certain values of the perturbation a conventional bound state could be produced from the resonance. In this case no resonance behavior is observed. We now prove that the value of the infinite-order resonance energy for the single-channel situation is as indicated by the formulae we have obtained for the wavefunction and the S-matrix element. For the well-enough behaved potential the only singularities of the Green function g on the unphysical energy sheet are virtual state poles, resonance poles, possibly some other poles and the branch point at the origin. Therefore we should be able to express the Green function on the unphysical sheet in terms of its residues at its vir- tual states, resonances and other poles on the second energy sheet and its behavior along the continuum. This is analogous to expressing the 65 the Green function on the physical sheet as an eigenfunction expansion. * We assume that we can write such an expression for g, that is, (0) n ). (13.10) g a X unv:/(E-E n where the summation is a summation over all poles on the unphysical sheet and is an integration along the continuum; also we assume that we can write a similar expansion for G, the complete Green function. Then obviously an the unphysical sheet the only poles of (vrlGlur) are located at the infinite-order resonance energies, the infinite— order virtual-state energies, et cetera. Therefore we seek to deter- mine the locations of these poles. From Lippmann-Schwinger formalism we can write (vrlGlur) = (vrlglur) + (vrIgVGIur), (D.1l) which can be rewritten as 0-1 ' (vrIGIur) I (E-E: )) [l + (vrIVIur)(vr|GIur) + g (vrIVIUm)KVmIG|ur)], where the prime indicates that the term with m-r is not included in the summation. Similarly we can write for pfr 0 -1 ' (vplclvp = (Ia-E; )> [(vp|v|ur> +E. (VPIVIuvamIGlurH- (v.13) Iterating the previous equation we find that (0) - (vaGIur)= (E-Ep ) 1 ... ~ ~ (vp'[V+VgV+VngV+...]|ur)(vr|G|ur). (D.14) * Obviously the same extensions for considering further singularities which apply to the physical sheet also apply to the unphysical sheet. 66 substituting this into our expression for (vrlclur) we obtain a _ (0) ’1 “' ( ‘ (vrIGIur) (E F.r ) [1 + (vrIV+VgV+...lur);vrIG|ur)]. (D.15) Therefore ( rl I r) [ r < rl I 17)] o ( .16) From the above expression the poles of G are obviously at E a E(0) r + (v |V+VGVIu ). (D.17) r r r Therefore our expressions for the wavefunction (8.8) and for the S-matrix element (D.9) for a single channel correctly indicate the exact resonance energy. 3. Resonances in a Multichannel System For a multichannel system such as we are predominantly con- cerned with in this chapter, (D.1)-(D.6) can be used directly provided (0) we use the following matrices: for the column vectors_! and 9_ the ith elements are given respectively by Vi = wia (D.18) and (0) = (0) ¢i — Gia¢aa ’ (D°19) where a denotes the initial state of the target and wia and ¢:g) are as defined earlier; for the column vector ur the ith element is given by (ur)i _ ui (D.20) if the ith channel is near resonance and otherwise it is zero; for the 67 column vector v the ith element is given by In (vr)i E v (B.21) if the ith channel is near resonance and otherwise it is zero; for any channel near resonance 0) _ ~ + _ ( 31 = 31 + uiVi/(E Eri ) (3.22) while for all other channels sc 5 8C (0.23) so that the ijth element of g.is given by : ~ + (0) gij _ 61j[gi + uiv1/(E Er1 )] (D.24) if the ith channel is near resonance and otherwise 81J 5 51381; (D.25) the ijth element of 9_is given by , + (0) wij uivivij/(E Eri ) (D.26) where .09. 5 ($91; (13.27) the ijth element of (1+GV) is given by (l + Gv)ij E 61j + (GV)ij (D.28) where (1+GV) is obtained from (D.3) using g_and y; and ~ 2 ~ . .2 gij 613g1 (D 9) 68 In place of (D.7) and (D.8) we obtain 1 = ($92“) + (1+GV)Q_(1+GV)[1_-I_n_(l+GV)] -’1i(0) (13.30) i -- (1:91) [I-g(_1__+51)]'1£(0). (13.31) For a single resonance 1 (D.30) and (D.3l) can be written as 1 - (1+GV)£(0) + (_1___+GV)u (v I(V+VGV)1L a¢| i?) o - _ [E‘E:1)-+ (0) she . -2ni { {I II (vi |¢ala )(E- E )/(E- E' i), (D.52) i k where Ilia):“ (v(3)|(v+v&v)|¢(°)+) (v.53) (0.49) and (D.50) are most appropriate not near a resonance in either channel a or channel b while (D.51) and (D.52) or the symmetric form of (D.52) are most appropriate near a resonance in either of these channels. For resonances in a multichannel system certain values of the per- turbation may cause the formation of either conventional bound states and/or bound states embedded in the continuum. The poles which become bound no longer contribute to the cross sections and the overlapping structure may be drastically altered from that which is indicated by our formulae. The infinite-order energies {Eri} can be determined by determinen- tal techniques or equivalently we can determine the zeros of each G1, the complete Green function for the ith channel. The expression obtained for the energies of each channel contains infinite-order resonance poles caused by all the zeroth-order resonances in every channel. It should be noted that to determine the infinite-order 72 energies all zeroth-order resonances in all the coupled channels must be considered, not just the ones we used to obtain expressions for the wavefunction with the praper resonant behavior near a certain energy. 4. Discussion In this section we use the formulae we have developed to con- sider low-energy electron resonant vibrational excitation of diatomic 2, N2 and 02. As we have already mentioned a single-center expansion of the molecules. Especially we consider scattering from H lowest unoccupied molecular orbital suffices to describe the behavior of the electron in resonance with a diatomic molecule. Therefore the Hamiltonian of the molecule is essentially rotationally invariant. ‘If a Hamiltonian is rotationally invariant then any of its resonance wavefunctions has a single value of the angular momentum. As is evi— dent from our expression for the S-matrix elements both the initial and the final states must be of that angular momentum for the partial- 0) i . the expressions we have obtained not only represent the total S-matrix wave cross section to exhibit resonance behavior near E: Therefore elements but also represent explicitly the partial-wave S-matrix ele- ments for the situation in which ui, ¢§g) and ¢ég) are all of the same partial wave. For the situation in which u1 and ¢(2; and/or ¢6g are of different partial waves the resonant portions of the S~matrix ele- ments vanish. The formalism we have developed for considering perturbed reso- nances does not provide much further information on the behavior of the resulting cross sections unless we are able to make estimates of (.1) 7-7 the behavior of {Iia}’ {¢ba}, {Rmm} and {II18 } for our system. 73 The matrix element aha represents coupling between state a and state b of the target molecule. In general for a given incident energy the magnitude of a should decrease as b and a separate from ba each other with respect to energy, while for a low incident energy and for a given a and b the magnitude of ¢ba should increase as the inci- dent energy increases. The matrix element Iia represents coupling between state a and zeroth-order resonance 1. In general for a given incident energy the magnitude of Iia should decrease as the zeroth- order energy in channel a and the zeroth-order energy in the channel in which 1 occurs separate from each other, while for low incident energy and for a given a and i the magnitude of Iia should increase as the incident energy increases. The matrix element Rmm represents a correction to the zeroth—order resonance energy for channel m. Coup- ling to closed channels should effectively deepen each well26 and thereby decrease the absolute magnitudes of both the real and the imaginary parts of each resonance energy. Because coupling is in gen- eral stronger between closer states the effect of coupling to closed channels should be greater on higher energy resonances than on lower energy resonances. The effect of coupling to open channels is not as easily generalized but should tend to counteract the effect of the coupling to closed channels, especially for low-energy resonances. J) ia should behave analogously to I The matrix elements 2 II 1 . a For a very wide barrier for which the resonances are very narrow and nonoverlapping, e.g. 02, we can see from (D.38) and (D.40) that peaks should occur in the same locations in all the inelastic cross sections. The overall magnitude of any inelastic cross section should depend upon the relative closeness of target states a and b; that is, 74 the closer a and b are with respect to energy the greater the magni- tude of the cross section a a should be. The magnitude of each peak b in each cross section depends on both the product Iiblia and the infinite-order resonance half-width P . In general for a given inci- i dent energy the amplitude of 1* should be of the same order of iina magnitude for at least all i such that Ii—bIIi-aI=Ib-a . For a given incident energy and for Ii-bl Ii-aI>Ib-a| , as Ii-bl Ii-al increases the amplitude of If should decrease. However as 1-increases so does iina the incident energy of importance and so for a given a and b the factor liina increases as 1 increases. For aw2n(r'> G£(r,r') =-; J dk 2 2 + Z 2 2 , (B.7) O k —k nfim k -k m m n where w:(k,r) is the delta-function normalized continuum function for wave number k and w£n(r) is the normalized bound-state function for wave number k.n which can be written as ¢ (k .r) ¢ (k .r) wine) = ‘1 n s ——"——‘—‘-—-. (3.8) [I ¢:(k .r)dr11/2 “ 0 n The integral in (3.7) can be rewritten as I =.g I” d k2+1 ¢£(k,r)f£(k,r') 1T (3.9) k (22+1):: 1 2 2 1+ f£+(k)(km-k )Zi by using (3.4), (3.5) and the analytic continuation of f£_(k,r). It is then readily seen that in the upper half k-plane the only singulari- ties of the integrand are simple poles at the bound states {kn} for n i m and either a double or a simple pole at the bound state km. Therefore the integral can be evaluated by contour integration using the contour shown in Figure 15. (Integration in the lower half plane T l' 10 \.l[ 84 k— plane Figure 15. Contour of integration in the k-plane for single- particle Green functions. 85 is in general discouraged by the possible presence there of virtual states, resonances and nonanalytic points in f2+(k,r).) The contributions to this integral from the contours along the inner and the outer semicircles can both be shown to vanish as [kl + O and as Ikl +'w, reSpectively. Therefore the integral can be written in terms of the contributions from the poles enclosed by the contour, that is, I: 2 g’ ), (3.10) (2ni 2 resk n n where resk denotes the residue of the integrand evaluated at k. The residue at any kn for n # m is 2+1 resk - kn :£(kn’r)f2+:kn’r') 2 2 2+ n 21 (21+1)!! [dkf£+(k)]kn(km-kn)’ (3.11) where the derivative evaluated at kn is l -2 d _ __~ k i [de ““011.“ — I (w [f (k. rm (k r)] -—-——-—---————--(2W),,Hkn =[{-C-;W[dk £+(k.r).f£+ (k r)] + c Hm (k r>.dk 1mm R -2 k 1 + ...}] (22+1)!! Ru 2 29+2 Nn a -4i(kn) ’ . (8.12) ((21+1) !!)2f£_(kn) Therefore for n # m “(k sr')¢£ (k or) res = 2 2 2 n (3.13) n 41(km-kM)N 86 and so w ()w*(') r r G:(r,r') = %(2ni Z resk + 21riresk ) + Z in 2 fin n#m n m n#m k -k m n = 4iresk . (3°14) m If m is not a bound state for i then f2+(k) does not have a zero at km and the pole of the integrand at km is a simple one. Then k“ (k )f (k ') m¢£ m’r £+ m’r res = - .£+l (3.15) km 41 (22+1)I:f£+(km) and the closed form of the partial-wave generalized Green function is 3 k ¢ (k ,r)f (k ,r') G?(r,r') = - i 1 m 1+ m . (3.16) i (22+1)!!fl+(km) But if m is a bound state for 2 then f£+(k) does have a zero at km and the pole of the integrand at km is a double one. Then the closed form of the partial-wave generalized Green function is 2+1 -2 k ¢ (k,r)f (k,r') Gm(r,r') = -21¥——--res[ 2 2+ 1 _ . (3.17) 2 (2£+l)!! f,+(k)(k:-k2) k—km (In general this residue is most readily evaluated by expanding the function about km in terms of 6, changing the derivative in the residue expression to be with respect to 6, and taking the limit as 6 + O, that is res[g(k)]k=k : lim ‘%E[(k-km)2g(k)] = lim-%E[62g(km+6)] ,(B.18) m k+k 6+0 m 87 where g(k) represents the function for which the residue is being determined.) Therefore the single-particle generalized Green function for r < r can be written as GmG'E') = Z i-z—ffillG‘EeJ'wlIcosexrr'fl, (3.19) i=0 where 6 is the angle between E and f7 and where G?(r,r') is given by (3.17) if m is a bound state for 2 and is given by (3.16) if m is not a bound state for 1. Similarly Gm(?,F') can be obtained for r > r'. The foregoing discussion is readily adapted to some central local potentials other than those discussed here simply by changing the boundary conditions (3.2) and (3.3) to the appropriate behavior and 36,37 carrying through the corresponding changes. Notably the Coulomb potential can be successfully treated this way. Also the results are kt directly applicable to central nonlocal potentials. 8 2. Generalized Green Function for a Coulomb Potential To illustrate the approach of the previous subsection we now obtain the generalized Green function for an attractive Coulomb poten- tial V(r) = —ar-1. We denote the scattering Coulomb wavefunctions as {¢£(k,r)}. 2+1 ikr e Using the substitution w£(k,r) = r g£(k,r) it is found that ** The Jost function of any nonlocal central potential is equal to the ratio of the Fredholm determinant of the integral equation of the scattering solution to the Fredholm determinant of the integral equation of the regular solution. The determinant of the regular solution may cause redundant zeros in fg+(k) bUt these cause no difficulty in obtaining the closed form of the Green function. 88 g£(k,r) obeys the confluent hypergeometric equation with g£(k,r) = ¢(£+1-i/k,22+2;-21kr) or w<2+1-1/k,22+2;-21kr) where o and T are the regular and the irregular solutions,respective1y. The regular solu- tion of the Coulomb radial Schrodinger equation is therefore 2+1 ikr e ¢2(k,r) = r ¢(£+l-i/k,2£+2;-21kr) (3.20) which behaves for small r like (3.3) while for large r -2-len/(2k) FC22+21e4 sin.(kr-k-11n2kr’ lfiz+62)° ¢2(k'r) r+m’ 2(2k) IP(£+1-i/k)T 2 (3.21) Because the Coulomb tail dominates at large r, f£+(k,r) can no longer obey (3.2). However by comparison with known Coulomb results f£+(k,r) can be taken such that lim f£+(k,r) = etikri(i/k)ln2kr. (B 22) r+m " Since lim w(2+1-1/k,22+2;-21kr) = (-2ikr)-£-l+i/k r7“ (3.23) then i“ _ - - .. f£+(k,r) = r£+le lkr W(2+l+i/k,2£+2;+21kr)(+Zik)£+1(-i) i/k. (3.24) Evaluating the wronskian W[f£+(k,r),¢£(k,r)] as r + m we obtain efl/(Zk) f9i(k) = r(2+1)/r(2+1$1/k). (3.25) Therefore the integration in (3.7) is complicated by the presence of a logarithmic branch point in f£+(k) at k = 0 but simplified by the absence of any poles in the k-plane other than simple ones at k = +(i/n). To perform the integration the integrand is divided into two parts with 89 respect to behavior for large Ikl. The contours are then taken as in Figure 16a, the upper and the lower contours being for those portions of the integrand which vanish asymptotically in the upper and the lower half k—planes,respective1y. Taking the sum of the two contours in Figure 16a for the partial integrands is equivalent to using the contour shown in Figure 16b for the entire integrand. By evaluating the residues at the poles surrounded in Figure 16b, the integral in (3.7) can thus be written as in (3.10). The results (3.11) through (3.14) follow. Therefore for the Coulomb potential for r‘< r: G:(r,r') can be written as G2(r,r') = i222+2(rr’)2+1(-l)£[(2£+1)!]-l k2£+2 k2 -1eik(r+r' ) res [k (ki- k) ¢(£+1-i/k,22+2;-Zikr) F(£+1—i/k)W(£+1—i/k,2£+2;-Zikr')] (3.26) k=iln where res denotes the second-order residue. To evaluate this residue at k = i/n we expand the integrand in terms of (i/n + 6), then differentiate the formal residue expression for a second-order pole and take the limit of the result as 6 approaches zero. Using the definition of ¢ and noting that n must be greater than we obtain [} . m-O where w(2) 5 F'(Z)/F(Z) is the psi function. 93 Using his expansion we obtain 1 n-R-l E§F1(£+l-i/k,2£+2;-Zikr')]k=(1/n)+6 = X Xm(r') m=0 2 n-l-l n-i-l an a X xm (r ) + in 26{ln(2r /n)[ X xIn (r )] m=0 m=0 . m m 22+1 n—l-l 21‘ "1|P+£-n||) _ I I _ __ _ 3: J: mélrm(r ) + mgo x m(r )[ (“+1'“)m m 22+l+m n-i-mrl - Z k‘1 _ Z {1 + Z {1]} k=1 k=1 k=l A(r') + nn26A(r') + in260(r'), where Y is Euler's constant and -1 Ym(r') (2r'/n)-mF(m)[(n-l)m(21+2)_m] Expanding the exponential terms we obtain [exp(-w/k)] = (-l)n(l-n26n) k=(i/n)+6 and {exp[ik(r+r')]}k=(i/n)+(5 Next we expand k22+2(k2+1/n2) 1 obtaining k224-2 2 2 -1 (k +1/n ) ]k=(i/n)+6 Noting that P(z)P(1—z) = n/sin(nz) = exp[-(r+r')/n][1+16(r+r')]. = (i/n)22'[12.-1-(3/4)+i(26n)”1 + Y (3. (3. (3. (3. ].(3. (3. 38) 39) 40) 41) .42) 43) 44) 94 it follows that [F(£+l+i/k)F(£+1—i/k)lk=(i/n)+5 = (£+n+in26)(2+n-l+in26)°°°(r+in26)F(r+in26)F(l-r-in26) (3.45) = (-1)n-2P(£+l+n)/[F(n—£)in26], (3.46) where r E n—R. The partial-wave generalized Green function for r < r' can then be written as 2£+1 -1 -2£-l 32(r,r') = -2 (rr')2+lF(2+1+n)[(22+1)!]—2[F(n-£)] n e'(r+r')/“([n‘1(22+3/2) - n-2(r+r')]A(r)A(r') + [n-13(r) + C(r)]A(r') - A(r)D(r')}. (3.47) The partial-wave generalized Green function for r > r' can be similarly obtained as 2£+l l -1 -2£-1 G2(r,r') = -2 (rr') +1r(2+1+n)[(22+1)I]‘2[r(n-2)] n e'(r+rv)/n{[n‘1(22+3/2) - n—2(r+r')]A(r')A(r) + [n‘13(r') + C(r')]A(r) - A(r')D(r)}. (3.48) For the simplest case in which n = 2+1 these reduce to G:+1(r,r') = 2 _ _ _ _ ' 2£+1 In 22 1e (r+r )ln (rr')2+1[(22+l)!] [-n“1(23+3/2) + n-2(r+r') - (2£+l)!fn2(2r>/n) + ln(2r/n) {(y)_mm} ] , (3.49) m=1 m=1 95 which for the ground state further reduces to G$(r,r') a 2(rr')exp[-(r+r')][- %-+ y + (r+r') - f(2r<) + 1n(2r>) - (2r>)‘1]. (3.50) This is the form obtained by Hameka29 with his f(x) being equal to our f10(x). The total generalized Green function for the hydrogen atom in the nth state can therefore be written as m R, - Gn(r.r') ' X igzgll'GE(r,r')P£(cos6)(rr') 1 , (3.51) i=0 where 2£+1 1 -1 -22-1 G2(r,r') = —2 (rr') +1I‘(9.+1+n)[(22+1)!]-2[I‘(n-£)] n e‘(r+r')’“{[n‘1(22+3/2) - n-2(r+r')]A(r)A(r') + [n'13(r<) + C(r<)]A(r>) - A(r<)D(r>)} n > 2 (3.52) and 2+ ———-' 2 1 1n 22 1e (r+r )/n G?/n) n §_£ (3.53) 96 C. N-particle Generalized Green Functions 1. General Expressions for N-particle Generalized Green Functions Interest has recently been shown in the Green functions for N* independent particles for performing atomic calculations.29’31’32 Especially the generalized Green function has been indicated for use in perturbation treatments of atoms.29 In such treatments an atom is regarded as a system of noninteracting electrons each bound by the attraction of the nucleus and perturbed by the interactions with the other electrons. For N identical noninteracting particles it has been demonstrated that the Green function can be constructed by merely iterating N single-particle Green functions and that the generalized Green function can be obtained from this iterated form by taking an appropriate limit of it.32 We extend this work by defining and discussing generalized N—particle Green functions other than the one previously considered.32 To acquaint the reader with the manipulations involved we briefly derive the iterated form of the N-particle Green function. We next present our extended defini- tion for generalized N—particle Green functions. As examples of the various Green functions we list expressions for all possible two- particle Green functions. Then we discuss the current status of using N—particle Green functions to perform atomic calculations. We indicate a method by which the closed form of the two-particle generalized Green function for both particles bound can perhaps be * N-particle here denotes what is usually considered to be (N+l)-parti- cle. We ignore (for counting purposes) the particle which is the source of the potential to which the N particles are subject. 97 obtained in general. The extension of this approach to N particles is indicated. For N identical noninteracting particles the unsymmetrized wavefunction of the system is a product of N single-particle wave- functions. The eigenfunction expansion form of the N-particle Green function can therefore be written straightforwardly. By com- paring the eigenfunction expansion form of the N-particle Green function G(A;l,l',2,2',°°°N,N') with the eigenfunction expansion form of the (N-l)-partic1e Green function G(A-En;2,2',°"N,N') it can readily be seen that C(A;l,1',2,2','°'N,N') = Slzwn(1)¢:(1')G(A-En;2.2'.'°'N.N') n a * V V U + an(l)wn(l )G(A-En;2,2 , N,N )dEn]. (C.1) 0 A and (A-En) are any complex numbers which are not eigenvalues of the N-particle and the (N-l)-partic1e Hamiltonians, respectively; wn(i) and wn(i) are discrete and continuum single-particle eigenfunc- tions for the particle located at f with single-particle eigenvalues 1 3n and En’ respectively; S is a symmetry operator which produces the desired symmetry in the expression on which it operates. By compar- ing (C.1) to the eigenfunction expansion of the single-particle Green function we can write G(A;1,l',2,---N,N') = (23i)-ISJG(En;l,1')G(A—En;2,2',°'°N,N')dEn, C (C.2) where C is a counterclockwise contour which surrounds all the eigen- values of the single-particle Green function and none of the 98 eigenvalues of the (N-l)-particle Green function. Clearly if any of the eigenvalues of the single-particle Green function coincide with any of the eigenvalues of the (N-l)-particle Green function then this contour can not be drawn. This is equivalent to saying that if A is an eigenvalue of the N-particle Green function then this contour can not be drawn. For a single particle the generalized Green function can be defined as n I I * G (r,r ) 11m [C(E;r.r ) - wnwn/(E-En)]- (C.3) 3+3 n Analogously for N identical noninteracting particles M of which are bound the generalized Green function can be defined as n ,n ’...n c 1 2 M(l,l',2,2',°°'N,N') it 2 lim SIan(1)wn(1')G(A-En;2,---N,N') 1+3 +3 +---3 +3 +---3 n n1 n2 nM nn+1 ”N + J¢n(l)W:(l')G(A-En;2,'°°N,N')dEn 0 * * * - Wn§1)¢n§1 )¢n§2)"’wn§”>¢ (M )¢n§fiil’°"¢n§N)?n§N’ (1-3 —3 ----3 -3 ----3 )"1]. (0.4) n1 n2 “M nn+1 ”N The previous definition 32 of a generalized N-particle Green function required all N particles to be bound. But the extension of the defini- tion to cover situations in which not all of the particles are bound may be useful for considering interaction between a system of M bound particles and (N-M) unbound particles. ,99 (C.4) can be rewritten as n ’n ’oosn c 1 2 M(l,l',2,°'°N,N') = lim (23i)-18JG(E ;l,l')G(A-E ;2,2',"’N,N')dE (C-5) 3 n n A+En+o 0 0E C' 1 nN where C' is a counterclockwise contour surrounding all of the eigen- values of the single-particle Green function and none of the eigen— values of the (N-l)-partic1e Green function except for those which coincide with eigenvalues of the single-particle Green function. As examples of all the various Green functions we now list expressions for the Green function and for the two generalized Green functions for two identical particles. According to (C.1) the two- particle Green function can be written as 6(1;1.1'.2.2') = 2‘1Ian(1)w:(1')c(1—3n;2,2') n + me(2)¢;(2')G(A-Em;l.l') + J¢n(1)w*(1')G(A-En;2,2')dEn m 0 n m * ' . I + [ww(2)ww(2 )G(A-Ew.l.l )dEw]. (C.6) 0 Therefore using (C.4) the generalized two-particle Green function for both particles bound is cab(1.1'.2.2') = 2‘1Iwa(1)w:(1')cb(2,2'> + wb(1>w;(1'>ca(2,2') * 1 1b ‘ V * U a ' + wa(2)wa(2 )2 (1.1 ) + wb(2)2b(2 )6 (1.1 ) + Xwn<1>w:(1')G(Ea+Eb-En;2.2') + Zwm(2)w;(2')Gyg1dr O 8 = a3yn(aa)jn(Ba) - 3! rzjn(8r)yn(ar)dr. (3.17) 0 Letting B = a + s, using the Taylor expansions (3.8) and (B.9), using the expression for the wronskian of jn(ar) and yn(ar), and taking the limits of both sides produces a I rayn(ar)j;(ar)dr = -a24-1a-1 + a32-1jn(aa)yn(aa) 0 ' (3/2){aZZ-1a-2[aj;(aa)yg(aa) + azajn(ar)yn(ar) - n(n+1)a-1jn(aa)yn(aa) + Z-IjA(aa)yn(aa) + 2-1jn(aa)yg(aa)] - (2n+1)4-1a-3}. (3°18) 120 a ' 3 I jn(Br)jn(ar)r dr 0 Letting fn(kr) - jn(ar) in (3.1) and then differentiating (3.1) with respect to r produces 20 v ' 2 22 ' [r jn(ar) + ern(ar)] + 2a rjn(ar) + [a r -n(n+1)]jn(ar) - 0. (3.19) Multiplying this equation by rjn(8r) and integrating leads to a . ' [rzjn(ar) + 2:1;(arnrjncsr) - I [rzjr'lmrfl unmr) + rjnwrndr O a a + 2oz! rzjn(ar)jn(8r)dr + a2! r33;(ar)jn(8r)dr 0 0 a - n(n+l)[ rj;(ar)jn(8r)dr = 0. (3.20) 0 ldultiplying (3.1) for jn(3r) by rj;(ar) and integrating by parts produces a . 2 . ' 2 a 3 n I rjn(ar)[r jn(Br)] dr + B I r jn(ar)jn(8r)dr 0 0 a - n(n+1)J rj;(ar)jn(8r)dr - o. (3.21) 0 Integrating the second integral in (3.20) by parts produces a2' ' ,v 2' v [r jn(ar)] [jn(8r) + r3n(8r)]dr - r jn(ar)[Jn(Br) + rjn(8r)] O a . - J rj;(ar)[r2j;(8r)]'dr. (3.22) o 121 Inserting (3.22) into (3.20), subtracting (3.21) from (3.20) and using (3.7) gives a n ' J r3j;dr = (82-02)-1{r33n(ar)jn(8r) + r21n 0 2 2 2 -1 - r3j; [jn(ar)j;(8r) - j;(ar)1n<3r)]}.|v Z (gPV)nI¢(O)). (3.2) n-O When a scattering state is perturbed its energy does not usually change and so we can write w = XO“¢‘°’. (3.3) n- ,(0) Because P has the same energy as both functions have the same normalization. From (3.3) it can be seen that in general each higher order of the wavefunction is not orthogonal to the zeroth-order 124 125 wavefunction; therefore eigenfunction expansions of the various orders of the wavefunction in terms of the zeroth-order solutions are not appropriate. H II H H T II III Ill. " I l | 3 03168 9924 3 129 rlmmmmmtmm