University of Southern California Dissertations and Theses

Green'S Function Technique In Atomic And Molecular Physics To Special Consideration Of Electron Scattering Problems In The Generalized Random Phase Approximation (Grpa)

(USC Thesis Other)

Green'S Function Technique In Atomic And Molecular Physics To Special Consideration Of Electron Scattering Problems In The Generalized Random Phase Approximation (Grpa)

PDF

Download

Open document

Flip pages

Download a page range

Download transcript

Copy asset link

Request this asset

Transcript (if available)

Content | CSANAK, Gyorgy, 1941- I GREEN’S FUNCTION TECHNIQUE IN ATOMIC AND MOLECULAR V PHYSICS TO SPECIAL CONSIDERATION OF ELECTRON } SCATTERING PROBLEMS IN THE GENERALIZED RANDOM I PHASE APPROXIMATION (GRPA). | University of Southern California, Ph.D., 1971 Chemistry, physical ,v i University Microfilms, A X ER O X Com pany, Ann Arbor, Michigan THIS DISSERTATION HAS BEEN MICROFILMED EXACTLY AS RECEIVED GREEN'S FUNCTION TECHNIQUE IN ATOMIC AND MOLECULAR PHYSICS TO SPECIAL CONSIDERATION OF ELECTRON SCATTERING PROBLEMS IN THE GENERALIZED RANDOM PHASE APPROXIMATION: (GRPA), by Gyorgy Csanak A Dissertation Presented to the FACULTY OF THE GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree DOCTOR OF PHILOSOPHY ( Chemistry ) A u g u s t 1971 UNIVERSITY OF SOUTHERN CALIFORNIA T H E G R A D U A T E S C H O O L U N IV E R S IT Y PA R K L O S A N G E L E S , C A L IF O R N IA 9 0 0 0 7 This dissertation, w ritten by GYORGY CSAHAIt under the direction of hAs.... D issertation C o m m ittee, and a p p ro ved by all its m em bers, has been presented to and accepted by T h e G radu ate School, in partial fulfillm ent of require ments of the degree of D O C T O R O F P F I I L O S O P H Y f ' T r t a ^ f o Dean DISSERTATION COMMITTEE 2. PLEASE NOTE: Some Pages have indistinct print. Filmed as received. UNIVERSITY MICROFILMS DEDICATION To my friends, with whom I used to walk ACKNOWLEDGEMENTS I would like to thank Professor Howard S. Taylor who made possible my study and research in the United States under unusual circumstances, overcoming numerous administrative and other diffi culties . It is also a great pleasure to express my appreciation to the University of Southern California for providing excellent facilities for my studies. The Chairman of the Department of Chemistry, Professor David A. Dows and the Dean of the Graduate School, C. Mayo, paid special attention to my graduate studies; I thank them. I would also like to thank Professor Robert Void, who helped me to overcome administrative difficulties. I would like finally to thank Miss Carol Snyder for her patient typing and proofreading of this extremely difficult manuscript. iii TABLE OF CONTENTS Page DEDICATION..................................................... ii ACKNOWLEDGEMENTS ............................................... iii INTRODUCTORY N O T E .............................................. vi Chapter I. INTRODUCTION ............................................ 1 II. MANY-PARTICLE GREEN's FUNCTIONS AND PHYSICAL QUANTITIES ............................................... 4 A. The One-Particle Green's Function and Its Relationship to Physical Quantities ............... 4 B. The Two-Particle Green's Function ................. 14 III. COUPLED SYSTEM OF EQUATIONS FOR GREEN'S FUNCTIONS. THE METHOD OF FUNCTIONAL DIFFERENTIATION. THE DYSON EQUATION. THE BETHE-SALPETER EQUATION.................. 21 A. A System of Equations for the Many-Particle Green's Function .................................... 21 B. The Dyson Equation and the Self-Energy............. 22 C. Hierarchies for Green's Functions and Related Quantities; The Method of Functional Differentiation .................................... 25 D. The Bethe-Salpeter Equation. Another Hierarchy of Equations......................................... 29 IV. SCATTERING.............................................. 35 V. NON-PERTURBATIVE APPROXIMATION METHOD ................. 50 VI. GENERALIZED THOMAS-FERMI THEORY FOR SCATTERING AND THE SHAM-KOHN MODEL (LOCAL DENSITY APPROXIMATION) . . . 64 A. Introduction......................................... 64 B. The Thomas-Fermi Theory and Quantum Corrections . . 65 C. The Effective Interaction, A New Hierarchy of Equations for the Many-Body Problem............... 71 iv D. Baraff's Quasiclassical Theory ..................... 76 E. The Sham-Kohn Model (The Local Density Approximation)........................... 82 VII. SUMMARY................................................. 86 APPENDIX A. The Angular Momentum Analysis of the Dyson Equation......................................... 87 APPENDIX B. The Angular Momentum Analysis of the GRPA Equation......................................... 95 REFERENCES......................................... 100 v INTRODUCTORY NOTE This thesis was written to serve simultaneously as an intro duction to the field of many-body physics and as a presentation of new results. It was aimed at the audience of quantum chemists with no training in this field. As such, the new material and the hopefully simplified explanations of known results are intermingled in the written presentation. The following is a list of what I am presenting as new work with the pages upon which it is found. 1. The S-matrix formula for electron-atom (or molecule) inelastic scattering (pp. 40-41). 2. The observation of the fact that one can simultaneously calculate electron-atom and electron-ion scattering cross sections in the GRPA scheme and the derivation of the relevant formulas (pp. 42-49). 3. A rigorous derivation of the GRPA method through the Bethe- Salpeter equation, which makes the physical meaning more trans parent for the iterated quantities (pp. 52-53). 4. The reduction of the GRPA equation to eigenvalue form (p. 59). 5. The angular momentum analysis of the Feynman-Dyson and Bethe- Salpeter amplitude for atomic systems (pp. 87-99). vi Chapter I INTRODUCTION One of the classical approaches to the description of the scattering of electrons on atoms or molecules is the optical model.^ In this model the scattering is visualized as the propagation, refrac tion and absorption of the electron-wave by the target-medium, similar to that of a light wave. There are several mathematical formalisms to 2-4 which the previously described physical picture can be attributed. In the present work, the many-body or field-theoretical formalism will 4 5 be used. The real or renormalized Green's function will be defined and the optical potential will be identified with the self-energy of the electron in an inhomogeneous electron system (the electrons of the target). The formulation of the optical model in terms of Green's functions is particularly useful for low energies because the exchange effect is automatically included. The present work is an outgrowth and continuation of the work of B. Schneider, H. S. Taylor and R. Yaris. They propose a method for the self-consistent calculation of the frequency-dependent polariza- bility (linear response function) and electron-atom scattering cross section (one-particle Green's function). The Green's function formal- 4 ism has been used for the optical potential and with the aid of the Schwinger variational principle^ an approximation is postulated in the course of which a pair of equations is obtained for the one-particle 1 Green's function and the linear response function. Using the Hartree- Fock approximation for the one-particle Green's function, the random- * phase approximation (RPA) is obtained for the linear response function. A self-consistent solution of the pair of equations is postulated starting from the Hartree-Fock approximation for the Green's function and proceeding iteratively. The coupled, self-consistent equations are called generalized random phase approximation (GRPA). The present work is concerned with the possibilities of calculating several differ ent types of cross sections in the course of the procedure described above. The linear response function in the RPA can be used for the calculation of electron-ion scattering cross sections in the framework of the Chew-Low^ formalism. g The Lehmann - Symanzik- Zimmermann reduction formalism was developed for atomic problems and applied to the calculation of the electron-atom or molecule elastic scattering cross section. The same method was used to obtain electron-ion scattering cross section in the GRPA scheme (generalized Chew-Low theory). An inelastic scattering S-matrix formula was derived for the electron-atom or molecule scattering problem. This formula forms the basis of an optical model for inelastic scattering. References to RPA are given in Ref. 6. The present work proves that the second equation in the GRPA system can be cast into a form identical to the RPA eigenvalue problem. 9 10 The second part of this work deals with the quasiclassical ’ and the local density approximation^ for the optical potential. These approximation schemes were defined and used extensively for bound-state calculations. The Green's function formalism of the optical model makes possible the use of these schemes in scattering calculations. In the appendices the angular momentum analysis of the Dyson equation and of the Bethe-Salpeter equation is given. This will be useful in the solution of the GRPA system. Chapters II-V are the condensed form of a paper being published in Advances in Atomic and Molecular Physics (edited by D. R. Bates and Immanuel Estermann) under the joint authorship of Gy. Csanak, H. S. Taylor, and R. Yaris. Chapter II MANY-PARTICLE GREEN'S FUNCTIONS AND PHYSICAL QUANTITIES12"14 A. The One-Particle Green's Function and Its Relationship to Physical --------. --U I I I I pj ' ~ .. ‘ *----- Quantitxes-*»> • LJ The n particle Green's function is defined with the following formula: Gn(l,2,...,n;l/,2/,...n/) = = (i)"n <Y|T[Hl)...'Kn) *t(n,)...i|r+(l/)] |Yq> (2.1) Here i = r.t., where r. means spatial and spin coordinates, t. is the IX x x time coordinate, and |^ ) is the ground-state wave-vector of the N particle system; i|f(i) is called the field operator in the Heisenberg representation. The symbol T is the time ordering operator which, when applied to a product of operators, arranges them in chronological order of their time arguments with a multiplicative factor of ± 1 depending on whether the chronological order is an even or odd permu tation of the original order. The one-particle Green's function can be written in the form: G1(l,l/) = (i)"1 {©(tj-t') <i|r(l) t V 7)) - - ©(tjj-tp d a ' ) HI))} (2-2) * where © is the Heaviside/unit step/function. Where it will not cause confusion we shall usually use angular 4 G^(l,l7) describes the propagation of a particle in an N- particle ground-state. We can define the following related functions G>(1,1/) = i <f(l) i | r +(l')> (2.3a) G<(1J1/) = "T ( i j f t(l/) t(l)> (2.3b) These are called correlation functions. G^ depends only on t = ti“ti and can be written as G1(l,l/) = (i)"1 SOrXiK^) e"i(H_Eo)T /(r^)) - ®(-T)<t+(r1) ei(H_Eo)T s = G^r^r'sT) (2.4) where i|i(r) is ttie field operator in the Schrodinger representation. Similar, but more complicated, formulas can be written for higher Gn> It will now be demonstrated how a knowledge of G^ (or G )enables one to calculate the one-particle density matrix of the system (and thus all expectation values of one-particle operators over the ground- state Y ) and also the ground-state energy. brackets to denote an expectation value with respect to |Y ); i.e., <0 ) = < Y |o |y >. ° p o 1 p 1 o The density operator, pQ^(r), whose expectation value is the "4 # m density at position r, is given in the Schrodinger representation as: n v * ‘H <2-5> i=l and in the second quantized form (Heisenberg representation) as PQp( r t ) = ^(rt) ilf(rt) (2.6) Therefore, the density at position r and time t is p(rt) = M m (l|/^(rt/) f(rt)) (2.7) t-t /-*-0 Since in taking the limit, t' approaches t from above (-0 means t' is * always larger than t), we can insert T and write p(rt) = Aim i)/^(rt') i]/(rt)]) = t-t/—*-0 = Aim C-iG(r,r;T)} (2.8) T—*-0 From Eq. (2.7) one could alternatively have written p(rt) = iG<(rt, rt) (2.9) < —» _> ** and we see that G (rt,rt) gives the density. In practice G(t ) * Where it will not cause confusion, we shall drop the subscript 1 on G-^, r ^, t^ • Sometimes we waive writing out all coordinates. will be determined as its time Fourier transform G(iu) where: 00 G(iu) = J" dT G (t) ( 2. 10) for any complex tu and the inverse transform is CO (2. 11) Similarly, we can define G<(od) , G>(cu) . Then Eq. (2.8) simply means that if G(cu) is known, then ip(r) is determined by integrating this function over a semicircular contour going along the real u) axis from _co to 4°° by closing counter-clockwise in the upper half plane (u.h.p.). After fixing the path of integration, we can take the limit behind the integral sign and a unit factor will be obtained instead of the expo nential. The density is now known if G (t) or G ( t ) or G(cu) or the poles and residues of G(u)) are known in the u.h.p. In the practical problems which we are concerned with in this work, the residues and poles of G(oj) shall be solved for and used to obtain the density. In other cases we have to calculate the integral directly. This is accomplished most easily by rotating the contour counterclockwise by tt/2. One can prove that the rotated contour encloses the same poles as the original one. On the infinite semicircle in the left half plane, the form of G(uo) will be obvious and the integral can be per formed analytically. On the part along the imaginary axis, one can integrate numerically (there are no singularities along the imaginary axis) moving symmetrically (up and down) from the real axis until G(tu) reaches its large uu asymptotic form, at which point analytic integration again can be used. It will now be shown that the same residues and poles deter mine the ground-state energy also. The ground-state energy can be written in terms of the one- 13 particle Green's function as: Eq = 2 f dr ^im [It " ih^^] GO^r'jT T“*-0 —* / —* r -*r 00 = " ^TT J dtu f dr ^ + G(r,r';(u) (2.12) I T “* 1 T _co Again, the limit T-*-0 simply means that one must close the contour in the ( j o plane in the u.h.p. and therefore Eq is determined by the same poles and residues as p. Note that both Eq and p depend only on G<(t). From the interpretation of the one-particle Green's function, it is obvious that G^ will be related to the elastic scattering cross- 14 15 section. ’ Though the formula for the cross-section is simple, the explanation is not straightforward, and we should refer to the adia batic principle and the formulation of field theory in the Heisenberg i5,16 representation. The elastic scattering cross section in the Green's function 4 formalism was first given by Bell and Squires and by Namiki. One can formally simplify the calculation using the so-called in-out formalism in which adiabatic decoupling is applied to the operators. Let us denote by 'Krt) the electron field operator and by ) the ground-state of the target."^ It can be shown that as t-+F°° with simultaneous adiabatic decoupling of the electron-atom interaction, the field operator will converge to an asymptotic form which obeys free equation of motion.^ Therefore, in M m i|f(itt) = iim i | ; OUt:(rt) = M m ^ ree(rt) (2.13) where i | i ln(rt) and i|f°ut(rt) is the asymptotic form of ijr(rt) in the distant past and distant future. The adiabatic decoupling considered here refers to the electron-atom interaction. This process can be handled in a nontrivial manner in the field theoretic formalism (A. Klein and A. Klein and C. Zemach^) . ^ n(rt) and \ | r ° Ut(rt) can be expanded in terms of propagating plane waves cp^(rt) of momentum k: in in ♦OUt(?t) = Z ^ a S ut cp£(rt) (2.14) m This defines a£Ut* Now, the scattering state corresponding to an / incoming) , _ 7* , . < . . > electron of momentum k can be written as (outgoing f in tout I |^> = ai°ut|Yo> (2.15) 10 and from Eqs. (2.13) and (2.14) in ai°Ut = Aim ai(t) (2.16) where a^(t) is the expansion coefficient appearing in the expansion of Krt) : K^t) = 2^1 ak(t) c Pk^rt) (2.17) k Equation (2.15) says that the scattering functions |Y^) are those that connect adiabatically, when the electron-atom interaction is turned off infinitesimally slowly to the state in which a free electron moving "in" toward or "out" from the target is created in the field of the ground state. The scattering matrix can be written as sk'k ’ " - M m Or |aC ,(t') al(c) |f ) - . o 1 k t—*-oo t'-M® = i Aim J dr' dr cp-^r't') G^r't^rt) cp-*(rt) (2.18) t~»-oo t ,—*a Equation (2.18) shows clearly how elastic scattering is related to the one-particle Green's function, G. A knowledge of G can be shown to give the natural orbitals and the ionization potentials of the bound electrons of the target. A slight generalization of Eq. (2.8) shows 11 that G gives the density matrix, and the natural orbitals are those functions which diagonalize this matrix. The density matrix can be written as p(r,r') = < i | r +(r) i|r(*')> = = - i Aim G(r,r ;T) = = - i G<(1+,1/) (2.19) The density matrix is diagonal in the natural orbitals: X^(r) and it can be expressed as: p(r,r') = ^ n. ^(r) X*(r') (2.20) i where n^ are the occupation numbers. The Xj_(ir) 1 s are orthonormal as p(r,r ) is a Hermetian matrix. The Green's function method is clearly the way to calculate the density matrix and the natural orbitals 18 directly. This way was followed by Reinhardt and Doll. To further illuminate the relationship between the natural orbitals, the ionizations energies, and the density matrix, it is well to write what is known as the spectral representation of the one- particle Green's function. Starting with Eq. (2.2) and inserting a complet set of N+l and N-l particle state vectors gives: 12 n m ,r: ' * < yn_1 U (i) | yn> (2. 21) The superscript now denotes the number of particles. Introducing the Feynman-Dyson amplitudes with the definition: fn(1) = <Yc J K l ) l'Fn+1 > = __. i . ,_N+1 _N. . = <Y I K O |YN+1> e” n “V H 3 o 1 1 n ./t.N+1 _ - C \ -i(E -E ) t. = f (r..) e n o 1 n 1 (2.22a) g ^ D = <lN“1|tO?1)|lN> ei(En “Eo^ tl = n i o . /r,N-l Jtf. „ . i(E -E ) 11 = gn(rx) e n ° 1 (2.22b) Equation (2.21) becomes G(l,l') = -i |®(t) ^ fn(^ fn(?/) 6 N N-l n/ \ ^ *t-'\ -i(E -E ) T - ®(-T) > Sm(r) gm(r ) e ° m ^ m m m . ,„N+1 Nn -x(E -Eq) T (2.23) 13 From (2.19) and (2.22b) follows: p(r,r7) = g (r') g* m m (r) (2.24) Usually the g's are not linearly independent, and they should not be identified with the natural orbitals. The only thing that can generally be proven about f's and g's is the completeness of the total set. f (r) f (r7) + V gm(r) g(r') = 6(r-r7) (2.25) n n m m n m 19 • • 20 Goscinski and Lindner have shown how to use Lowdin's method of canonical orthogonalization to transform the g's into the X's* The ionization energies and the electron attachment energies to various states of the N-l and N+l particle system clearly appear in Eq. (2.23). To see how the ionization energies are poles of G(iu), 13 Eq. (2.23) is Fourier transformed providing the spectral representa tion of G(oj) : G(r,r/;cu) = = Aim Tt—+0 * f (r) f (r7) n 'cu-(EN+1-EN) + iT| n n o + — ♦ W — * . g»(r) em(r > - m m o m (2.26) Clearly, in the limit, the poles of the second term are at the ioniza tion energies which can take on both discrete and continuous values. One discrete value occurs for each bound state of the N-l particle 14 system. For a neutral N electron system, the physical poles of the second term appear in the second quadrant of the complex plane and fall on a line oj = ill, Tj> 0; i.e., infinitesimally above the real axis. They are discrete for small negative real cu and merge into a branch cut for a larger negative to. The physical poles of the first term of Eq. (2.26) aK ll lie in the lower half plane (l.h.p.) along a line to = - iT), T|>0. If there exist bound states of the negative ion, dis crete poles appear in the third quadrant. A cut always appears along this line in the fourth quadrant. 12 13 B. The Two-Particle Green's Function 5 In G2 four times appear and different specific time orderings yield different information. To illustrate this, several time order ings shall be considered. Case I. Set > t2,t2 ^or arkitrary order of t^ and t'. Then for this case G2(l,2;l/,2/)1 = -OlfJ||T[*(l> *+(l')] T[<|r(2) +t(2/)]|^> = = - ^ V l ' l ' ) V 2^') (2.27) n Here an N-particle-state closure has been inserted and the hole- particle Bethe Salpeter amplitudes have been defined as ^(1,1') = <^|T[»jr(l) /(I')] I*”) (2.28a) 15 (2.28b) It is well to note that it was the specific time ordering that resulted in two hole-particle amplitude products . Other time orderings will give other pairings, such as hole-hole and particle-particle. Only for the hole-particle pairings do the intermediate states have to be N-particle states. It is well to write out the right-hand side of Eq. (28a) in detail. If this is done after some algebra, the follow ing result is obtained: i ,_N _N, / (2.29) where Using Eq. (2.27), this gives G2(l,2;l/2/)1 00 (2.31) n=0 Here the following changes have been introduced: T 2(tl+tl) " 2(t2+t2) Ti ti"ti (l 1,2) 2V 1 1 2 2 2 16 Equation (2.31) has three relative times, the exponential argument of one of them being simultaneously completely factorable from the matrix elements and being an excitation energy. The only other time ordering giving this property is the other hole-particle case, t2,t2 > tl,tl" Case II. Set t2,t' > tl,tl* 00 G2(l,2;l/,2/)11 = T X ^ ) (2.32) n=0 Other time orderings do not have factorable exponentials in which PN _N E -E appear. o n G2(l,2;l/,2/) = G2(l,2;l/,2/)1 ® (t - \\^ | - j | t 2 | ) + + G2(l,2;l/,2/)11 ® ( - t - y|t 1| - 2 lT2l) + other orderings = = G2 ^(l,2;l/,2/) + other orderings (2.33) We have defined the hole-particle Green's function: G2 ^(1,2; lf ,2f) as the first two terms in Eq. (2.33). Fourier transforming the variable t and using Eqs. (2.31), (2.32), and (2.33), one obtains G2(ti»t2»( J «) = G2(T1’T2’U))1 + G2(t1»t2»U)^11 + + other terms = G*1 ^(t^,t2,0)) + other terms (2.34) where 17 G2^T1,T2,U^ = I V * \.(V ;2’V ‘ i „iS 2-, ^ ( e V ) + n o X e and o ✓ v XX 2^T1’T2,CU^ 1 ,. V 1 \i(rl,rl,Tl) \.(r2-r2>T2> ari-(E“-EH) - ill n n o -|[mt<E“-E“)][|r1| +|t2|] X e N Let us define a new variable u) which runs through all E - n n N N ^o~^n va- * - ues • Correspondingly Sgn(cun) = 1 and Sgn(iun) = ■ define: V ' l ’V V 1£ “n > 0 and V ri>riiTi> * ^ * n < V ?r Ti> if “n < 0 X^r^r';^) if o>n > 0 Xn(rl’rl;Tl) Xn(r1,r';T1) if ^ < 0 Let us define also (2.35a) (2.35b) •E^ and all o ■1. Let us (2.36a) (2.36b) 18 Xn(l>l/) = e“iu)nt/ Xn(r1>r';T1) Xn(l,l/) = e1(Unt where t ' = • With this notation one can write: g£"p(1,2;1',2') = = - 5 ] x n(l,l') Xn(2,2') 0[Sgn(a)n) T - ^ItJ - ||x2|] (2.38) (2.37) tn n and G2"P(t1,t2,( u) = X^(rn ,r';T.,) X^(r2,r2;T2) Sgn(con) t X 7 1 -M-O “ ” wn ‘ i T 1 ‘ (D n f Sgn(in )(<jd-u)_)(|t1| + |t„|) X n n l Z (2>3g) G\ P(t^,t2,uj) has poles at ± (E^-E^J) . The "other terms" in Eq. (2.34) do not have poles at these excitation energies. The information contained in G2 P is clear, namely its poles give the target excitation (or deexcitation) energies. As interest ing is that the residue of G2 P at the nth pole gives Xn(r,r^;T) which is from Eq. (2.30) in the limit t-*-0 the matrix element i|i^(r/) i[r(r) |^) • This is a case which shall be obtained in the RPA and GRPA schemes. This residue then allows the calculation of all 19 one-particle transition matrix elements between |y ) and 17^). If 1 o n 0 = E 0 is a one-particle operator, then in second quantized P “ P form 0^ = j ' J ’ dr dr' i | r ^ (r') 0p(r) \Ji(r) 6(r-r') Therefore <yn |o l^) = o p 1 n ■// dr dr' Op(r)(^Uf(r') *(*> l^> 5^ ' ) where the integral over the delta function is taken only after the operator is applied. In particular, from Eq. (2.6) (Y^|\j;^(r)\|f(r) |Y^> the diagonal element in the position representation is the transition j,N|_ ,_.x |,„N (2.5) to get .Mi I N\ density: (Yq |pop(r)|^n). Moreover, if one Fourier-transforms Eq. N N \ s r,-' e/~* \ ikr. \ ikr. = 2^j J dr (r-r ^ e 1 = e 1 i=l i=l It is seen that the Fourier transform with respect to r of the element N ('l'^|i|f^(r) i]f(r) |^> is just (Y^lE ed^ri|Y^) the generalized oscillator 21 1=1 strength. Therefore, the pole and residues of the hole-particle part of the two-particle Green's function, in the form in which they are ob tained in the RPA and in the GRPA, shall yield the excitation energies, the generalized oscillator strengths, and enable one to calculate all one-particle transition matrix elements. To calculate expectation values of n-particle operators (n>l) other than the Hamiltonian in the ground or in excited states, takes a knowledge of higher Green's functions and the rest of GThese are more difficult to calculate in the sense that the GRPA will not yield this information. Chapter III COUPLED SYSTEM OF EQUATIONS FOR GREEN'S FUNCTIONS„ THE METHOD OF FUNCTIONAL DIFFERENTIATION. THE DYSON EQUATION. THE BETHE-SALPETER EQUATION. 12 13 A. A System of Equations for the Many-Particle Green's Function ’ In this chapter various useful and equivalent forms of the equations of motion for G^ and shall be given. The field operator i|f(rt) obeys the Heisenberg equation of motion: i ^ Krt) = [*(rt),H] 01) A useful shorthand notation is obtained using the definitions 1 2 rlfcl 2 2 r2fc2 V(l-2) = VCr^-r^) 6(t1~t2) and dl s dr^ dt^ d2 = dr^ dt^ where is the interaction potential. Substituting H into Eq. (3.1) in second quantized form, one obtains [i - h(l)J Kl) 2 V(l-2) v | ; +<2> i | r (2) Kl) (3.2) where h(l) is the one-particle part of the Hamiltonian. Using the definition of G^(l,l/), Eq. (2.2) and Eq. (3.2), one obtains 21 22 -/d2 V(l-2)<T[*+(2) K2) K D 'I'V')] (3.3) The matrix element on the right-hand side can be related to G2> if a positive infinitesimal time e is added to t2 in i | r (2) and the limit e-*0 is inserted inside the integral sign. Now that there are two -4 —+ - { “ - J - different indices r2t2 2t2 H ^ ' anc* Permutat^ - ons time order- ing under T can be made as discussed in the previous chapter. These permutations are made to give the ordering as in G^, with the result Eq. (3.3) becomes [i - h<l>] Gl(l,l') + + ± J d2 V(l-2) G2(l,2;l/,2+) = 6(1-1') (3.4) + + where 2 implies that the limit 2 -* 2 (e-*0) must be taken before integration. This equation is the first one in the coupled hierarchy relating G ^ to G^ The rest of the hierarchy can be derived by analogous methods. This coupled hierarchy is the new form of the Schrodinger equation in Green's function theory. B. The Dyson Equation and the Self-Energy . Introducing [G°] ^ with the definition: [G^d,!')]"1 - h(l)] 6(1-1') (3.5) 23 Equation (3.4) can be written in a condensed form (considering the variables as matrix indices) [G®]"1 Gx + iVG2 = I (3.6) where I is the unit matrix. Operating from the right on Eq. (3.6) with G^ gives [G°]_1 + iVG^”1 = G"1 (3.7) a form which lends itself to the introduction of a new function, £, called the self-energy, the optical potential, or the one-particle effective potential, defined as £ = - i V G ^ 1 (3.8) or more explicitly: E(l,l') = - i J ' dZ dl' V(l-2) G2(12,2'2+) G_1(2/,l/) (3.9) Using E, Eq. (3.6) obtains the form £i - h(l)J G1(l,l/) -y* d2 £(1,2) G1(2,l/) = 6(1-1') (3.10) or G1 = G° + G° E Gj^ (3.10') Equations (3.10) and (3.10') are respectively the differential and integral forms of the equation of motion for G^ and are called the 24 Dyson equation. In this equation £ plays the role of an effective potential which is generally energy dependent, complex and nonlocal. If £ is a local, energy independent potential, reduces to the Green's function used in mathematical physics. Equation (3.10) is a one-particle equation which, if £ is known, is equivalent to the Schrodinger equation. The effective one-particle potential has folded in it all the effects of the rest of the system; as such it is com plex, nonlocal and energy-dependent. Thus, an exact one-particle picture has been achieved at the price of a complicated potential. A detailed study of £ will show it to be real when cu is set to an energy value below the first inelastic threshold of the system. Above 22 the threshold £ is complex. The complex part can be shown to represent the absorption of incident particles by the system. This describes the inelastic processes. Since such phenomenological potentials play a key role in optics, they are called optical poten tials . Equation (3.10) can be written also in the form (after Fourier transform for the time) Substituting the spectral representation of G(r,r/;e) into Eq. (3.10"), Eq. (2.26) and taking the equality of the residues of the poles on [e-hCrp] G(r^,r^; s) - (3.10") 25 both sides in Eq. (3.10"), one obtains the Dyson equation for the Feynman-Dyson amplitude: [e-hCrp] - J d?2 EOrpr^e^ fn(r2) = 0 (3.11) This equation will be derived in Chapter IV with another method and in integral form. Equation (3.11) is an effective one-particle Schrodinger equation for the Feynman-Dyson amplitude (similar equation holds for g ). Therefore, f and g can be considered as an effective °n n °n particle or hole wavefunction. C. Hierarchies for Green's Functions and Related Quantities; The Method of Functional Differentiation Now that is known given E, equations for finding G2 or E must be developed. In the spirit that the equations for G^,G2,..., or G^,E,... will be solved simultaneously, it will be assumed that G^ is known and an equation for G2 or E shall be the object of this sub section. Clearly, the equation for G2 or £ will involve a knowledge of Gg or its equivalent. As in the case of G^, physical insight and incisive approximation will be facilitated by writing the equation for G2 in terms of a closed equation with an effective (two-particle in the media) potential, called H. Several sets of equivalent functional variables are in the process of being defined; viz., G^jG^G^j ...; G^j62>H,...; G^jSjH,...; G^,2,6E/6u,•••, etc. The various hierarchies are all formally equivalent, in the sense that no new physical content will be contained in any of the alternative forms to the G^jG^G^,... 26 set of coupled equations. However, since the set of coupled equations must be truncated by an approximation at some stage, different forms of the hierarchy lead naturally to different approximations and hence to different truncation procedures. 12 13 Perturbation expressions can be derived ’ which give G£ in terms of V and G^. The perturbation expansion for G^ involves and G^. The two closed equations for G^ and G^ and the expansion for G^ again give a self-consistent perturbation theory of higher order. Of course, closing the set of equations at G^ will be much more work than closing at GOnly experience will show how high in the coupled equations the truncations must be made. Fortunately, since anything above G^ will be inordinately difficult to calculate, diagrammatic 27 analysis and experience tends to indicate that G^ will not be needed for most atomic and molecular problems. The completion of the G^ truncation perturbative strategy does not require the derivation of the closed equation for G2 in terras of H since it has already been indicated how the equation for G^ in terms of G^ is derived. The closed equation from the perturbative point of view is only an alter nate equation in the sense that Eq. (3.10) is an alternate to Eq. (3.4). The real reason that this closed alternate form will be de rived is to introduce the method of functional differentiation and the expression of G2 in terms of functional derivatives of lower order quantities. This method will lead to non-perturbative approximations to be discussed later. The fundamental idea of the functional 27 5 13 derivative method of Schwinger ’ is to introduce into the problem an arbitrary nonlocal time dependent potential UCljl7) that is turned on slowly (adiabatically) at t = -“ and off at t = + 00. This potential "probes1 1 the system, and the physical quantities ,£, etc., are calculated in the limit U(l,l/) -» 0. The idea is exactly that of studying generalized response properties of the system. The method is quite physical in that all experiments actually probe the system and measure its response in one of its several forms, e.g., absorp tion coefficient, dielectric constant. It shall be shown that G^ (for a given G1) can be replaced by a knowledge of 6G,/6u| ; i.e., the lu -o variation of G^ with respect to a small probe potential in the limit that the potential goes to zero. What will result is a sequence of coupled equations relating S (or G^) to 6E/6U; 6£/6u to second deri vatives, etc. The advantages of the new hierarchy over the original one are both physical and formal. Physical arguments will allow one to make approximations to variations that were not evident when G^jG^,..., etc., were used. In the situation that a small arbitrary external nonlocal two time dependent potential U(2/,2) is turned on at t = - “ and off at t = + oo; G1(l,l/ ;U) = 7<*0(U) |T[yi) ^ ( l /)]|to(U)> (3.12) where the symbol U reminds us that the new Hamiltonian has U(2/,2) added to the one of the previous chapter. Since the defining equation 28 was for an arbitrary Hamiltonian, the derivation still holds and the equation can be written [G1(U)]_1 = [G°]_1 - U - E We can consider U as a functional variable and we can form the func tional derivative of a functional of U. 5 13 The Schwinger equation ’ which will be of fundamental importance in the following has the form: fiGjU.l'jU) 6U(2',2) = - G2(l,2;l/,2/) + G 1(1,1/) 6^2,2') (3.13) u=o This functional relation replaces G„ by 6G./6UI |U=0 For a given G^, the derivation of the hierarchy for E in terms of 62/6U can be easily carried through. The following equation can be obtained: 6G (1,2') _ _ - G l(l,4) Gl(5,2') + r 62(2,1') +J d2 dl' G1(l,2) G1(l',2/) (3.14) which on using Schwinger1s equation gives: - G2(l,5;2'4) +6^1,2') G1(5,4) = 0^1,4) 0^5,2') + y*d2 dl' G1(l,2) G1(l',2/) (3.15) 29 We can not substitute 6E/6u for in the definition of E. In this definition G2(l,2;l/,2+) , the so-called "three point" Green's function appears, thus in the hierarchy for I! in terms of 6E/6U only local per turbing potential [u(2,2/) s U(2) 6(2-2')] need be used. Using the expression for G^ in the definition of E, one obtains the relation: 2(1,1') = - i6(l-l') y*d2 V(l-2) G1(2,2+) + + iV(l-l') G1(l,l'+) + + x j & 2 V(l-2) 6^1,3) (3.16) Equation (3.16) is the first equation of a new hierarchy which for a given G^, replaces the coupled equations for G^ in terms of G^, G^ in terms of G^, etc., with exceptions relating E to 6E/6U, 6E/6U to second variations, etc. Higher equations in the hierarchy are ob tained by functional differentiation of Eq. (3.16). D. The Bethe-Salpeter Equation. Another Hierarchy of Equations. For purposes of physical insight and ease of derivation of the exact cross sections, a third form of the hierarchy will now be derived. This third form, which is equivalent to the previous two forms, stresses the "optical" potentials. Closed equations shall be given for each Green's function. The unknown part of each equation will be an effective potential that requires a knowledge of higher 30 Green's functions or equivalently higher functional derivatives of E. The first equation of the hierarchy is again (3.10) whose effective potential nature has already been discussed. The second equation is obtained by replacing by a more convenient functional variable called the generalized linear response function and defined as: SGjU, l';U) R(l,2;l',2') = 6U(27,2) (3.17) U=0 R(l,2;l7,27) is the coefficient of the linear term in the expansion of G^(1,1/;U) in a power series in U(2,27). For small U, it is the most important term describing the effect U has on the system. The usual linear response is a special case of Eq. (3.17) and is (recall- /+ ing that l7 = r7,tj+e) 60 <l,l'+;U) E(l,2;l'+,2+) i r----- 6U(2 ,2) 6U(2) (3.18) U=0 Therefore, Eq. (3.18) gives the linear term in the expansion of the density matrix in a local potential. Comparing Eqs. (3.13) and (3.17) it is clear that for a given G^, R and G^ give equivalent information; i.e., R(l,2;l',2/) = - G2(l,2;l7,2') +0^1,1') 0^2,2') (3.19) To obtain a closed equation for R, Eq. (3.14) is combined with 31 Eq. (3.17) and 6E/6U is replaced, using .^-(2,3), = f 6Z(2,3) !gl(6)7) = 6U(4,5) J a° 6G1(6,7) 6U(4,5) = / d 6 d7 H(2,7;3,6) R(6,5;7,4) (3.20) In the last step Eq. (3.17) has been used and the new definition s<2-7;3-6> - <3-21> has been introduced. The result is, after some changes of dummy integration variables, the Bethe-Salpeter equation, R(l,2;17,2 ') = G1(l,2) G1(2,l/) + + J d3 d37 d4 d47 G]L(1,3) G . ^ 7,!7) E(3,4;37,47) X X R(47,2;4,2') (3.22) Equation (3.22) is a closed equation for R (or G^) , which when com pared to Eq. (3.10') shows that it is an integral equation for R describing the motion of two "Dyson" or "dressed" particles. H re quires a knowledge of 6E/6G, which can be shown to be equivalent to a knowledge of G^ or higher functional derivatives of E with respect to U. Similar closed equations for higher Green's functions can be derived for dressed particles in the media. Equation (3.22), called the Bethe-Salpeter equation, and the other effective-potential equa tions, form our last hierarchy. 32 Solution of Eq. (3.22) is facilitated by deriving an equation for the spectral amplitude which has fewer variables. The property of Eq. (3.22) that and t^ are parametric is very useful for such a derivation. Changing the parametric time variable to T2 = t2 " t2 C 2^2 + t2') dt2 dt2 = dT2 dt2 Equation (3.22) can be written as R^l'jr^r'jt2,^) = Ro(l,l/;r2,r2;t2,T2) + + / d 3 d3' d4 d4' R0(l.S 'il'.S ) HO,4,3'.*') X X R(4/,4;r2,r2;t2,T2) (3.23) where R0(l,2 ;l' ,2 ') = G]L(1,2/) G1(2,l/). With Fourier transform for t2 and using the spectral repre sentation for R one obtains: Xn(l,l/) = y*d3 d3' d4 d47 Rq(1,3 '; 1' ,3) X X H O ^ j S V ) Xn(4/,4) (3.24) Equation (3.24) is the desired equation for the Bethe-Salpeter ampli- tude. It has been assumed that R has no poles at tu . This is o r n clearly not true for scattering states and hence Eq. (3.24) is a 33 bound-state equation. However, for atomic and molecular problems one can span the relevant region of space with a discrete basis set. Within such a basis set, the assumption holds and Eq. (3.24) can be used for the X's necessary to evaluate R. Equation (3.24), the Bethe-Salpeter amplitude equation for bound states, is clearly a closed equation for the two particle, hole- particle amplitude. Since Rq depends on and not on G°, the hole and particle move in Dyson orbitals and interact via H. H is the effective potential for hole-particle interactions of dressed parti cles and contains only true two-particle interactions with the single particle average effects removed. For continuous states the following equation can be obtained: Xn(l,l/) = Jd2 dl' G(l,2/) G(2,1') Xn(2,27) + + y*d3 d3/ d4 d4' G(l,3) G(3',l') S(3,4;3',4') X X Xn(4',4) (3.25) which is the Bethe-Salpeter amplitude equation for continuum states; i.e., unbound hole-particle states. This equation has an inhomo- geneous term on the right-hand side. In Chapter IV we will elaborate this inhomogeneous term. In summary, three equivalent forms of the hierarchy of equa tions have been derived in this chapter: (i) that for G^ related to G£, to G^, etc. 34 (ii) that for related to E, E to 6E, 6E to 6^E and (6E)^s etc. (iii) that for given by a closed equation with a potential 2 which requires a knowledge of R (or G^) and a closed equation for R in terms of a potential H, which depends on Gg or higher variations in S. Chapter IV SCATTERING4 The formulas and equations for scattering are here further developed using the exact relations of the previous chapter. First, the case of elastic scattering is considered. The pur pose is to give a simple prescription for calculating the T matrix when 2 is given in exact, or approximate, form. The S matrix has already been given in Eq. (2.18) in terms of G. It is here assumed that 2 has already been solved for exactly, or, on a finite basis set, approximately. To do this, consider the one-particle amplitude that corresponds to the boundary conditions of an "in" or "out" elastic scattering experiment; i.e., to one related to Ifi. This amplitude is, from Eq. (2.22a), ^(rt) = <Y0(N)U(?t)|4(N+l)> (4.1) Using Eq. (2.18) and inserting T, since the limit already specifies the ordering, gives f|(rt) = Aim (YjiKrt) = t —00 = M m i f dr' G^rt.r't') cp^(r't') (4.2) t *-oo J where cp^ is a free particle function. 35 36 Putting Eq. (4.2) into Eq. (2.18) gives for the "+" case: = M m /'dr7 cp£,(l') fjd') (4.3) t * 0 0 J Therefore a knowledge of f^(rt) gives Substituting Eq. (3.101) into Eq. (4.2) and using Eq. (4.2) again, the expression for f£ becomes fi(rt) = M m | /dr7 G (rt,r7t7) cpr’d't') + k t [J ° k + y ’ dr1dt1dr2dt2 Go(rt,?ltl) ^(rpl’^ V X X G1( r2t2>r 7 t / ) c p ^ (r7 t 7) | = c p £ (rt) + + y*dr1dt1dr2dt2 Go(rt,rltl) E ^ . r p p f ^ t g ) (4.4) It is now necessary to Fourier transform Eq. (4.4). To do this, time homogeneity is invoked to give £(1,1 ) = £(r,r ;t-t ) and the fact that fi(rt) can be written as fi(rt) = e ^ q 1 " fi(r) where e-* = q q q q eS+1 - E^ and f“*(r) = (^U(^) l ^ ’N+^) to give from Eq. (4.4) f J ( r ) = cp~(r) + + Aim /*G (r,r. ;e-*+iT|) £(r.. ,r ;e-*) fi(r ) (4.4') H-h-o J ° l q l z q q 2 where the iT) is added, because since t -* + °°, t is positive, which requires that the inverse transform of GQ(e) be done by closing the 37 contour in the lower half plane. The "if]" guarantees, by the residue theorem, that non-zero solutions only occur when the contour is closed properly and that Eq. (4.4') has the proper boundary conditions. Eqs. (4.4) and (4.4') are the integral forms of the Dyson equation for the Feynman-Dyson amplitude mentioned in Chapter III. (They can be called also generalized Lippman-Schwinger equations.) Hence, if E is solved for in any way, the solution of the elastic scattering problem is exactly the same as a nonlocal potential problem, for which many methods are known. The determinental method 28 is especially appropriate. In deriving Eq. (4.4) the integral with out E was replaced by cp£, the plane wave. To see the correctness of this procedure, note that Here is the unperturbed ground state and \ [ f Q(rt) is the field particle, one-particle amplitude which, by the Dyson equation, since E is here zero, is an incoming plane wave. It is now convenient to have a T matrix form and to convert from time variables to energy variables. To this end, if Eq. (4.4) t -*-eo = <i0H 0(rt)|$i> operator for the unperturbed system. t y 0(rt) ^-s t*ie free 38 is substituted into Eq. (4.3) and the analog for Eq. (2.23) is used for G , one obtains o Go(rt,?'t') = «P£(rt) cpj(r7t7) for t > t7 s The equation for becomes Sk7k = ' eim ' ldx' + j / d*' ' pE(;'t ' + L T f & ' <PI<rV) X Kk s X J *dr dt dr7 7 dt7 7 cp^(rt) E(rt,r77t7/) fi(r77t77) j Using the conservation of energy and momentum this gives sk7k = 6<er e2') j6® ' + + \ J "dr 7dt7dr7 7 dt7 7 cp^,(r7t7) £(?7t 7,r77t77) X X f±(r77t77) J (4.5a) Now defining T£/£ by the standard relation sk7k = 6(er eE 7) 6Sk7 + Tk7k Tk7k = i fd l d1' v (1) 2<1»1/) fk(1,) (4-5b) To express the T matrix in terms of f^(r), Eq. (4.5b) is written as 39 = 5^ " e^ \ f ^ d*' x s(r3r'; er*) fi(r') (4.6) pq pqij p p q Equation (4.6) is what shall be used in practice, since £(e^) &nd f^*(r ) is what is usually available when time-independent methods (which are easier to work with and commonly used) are used. The next equation to be derived is for inelastic scattering. Analogous to the derivation of Eq. (2.15), the expression for the S matrix element for a scattering process that excites the target from state o to state n with electron initial and final momentum p and q respectively is s _ _ = (y-- 2 , N+1 U +-AN+1) = op,nq op 1 nq = Aim (^la^t') &i(t) |¥^) = ° P q n t-*-CO = Aim Jdv dr' cp^(r/t/) Xn(r/t/,rt) cp^(rt) (4.7) *-CO It is evident that the two-particle, hole-particle amplitude is needed here, which requires a knowledge of S (or G^). An alternative viewpoint comes from a comparison of the definitions of and which shows that ^ is reasonably considered an "off-diagonal one- particle Green's function" and should be related to off-diagonal optical potentials and responses. If the bound state Bethe-Salpeter equation for hole-particle amplitudes, Eq. (3.24), is substituted into 40 Eq. (4.7), another form, which is possibly more useful for making approximation, is obtained S - - = / dl d2 d3 d4 & * (1) X °P»nq ,2 / P I 6 X E(l,4;2,3) ^(3,4) f±(2) (4.8) where the f's are special solutions of the ground-state Dyson equation defined by fi(2) = M m i f dr' cp-(2') G(2,2') q t^-cc J 2 q f-*(l) = Jlim i C dr' G(l'l) cp2(l') ^ t f —t c o J 9 (4.9) The time independent form of Eq. (4.8) showing the proper energy conservation is obtained by substituting Eqs. (4.4') and (2.34) into Eq. (4.8), with the changes of variables T = *=3 " *4 t = 2(t3 + t4)j ^ = tl -fc2 S = -^(t- + t„) a = s - t; a) = E - E Z J L z n n o to obtain S -* = -7T / d a d t dp, dT d r1d r _ d r „ d r . f-» ( r . ) f i ( r „ ) X op,nq /I J ^ 1 2 3 4 p r qv 2' v i(e-*-e-*)a i(e-*-e-*-cu )t i(e-W-e-0(j,/2 . X e p q' e p q n' e v p q/K" s ^ r ^ r ^ T a ) 41 Integration over t gives the energy conserving 6-functions. Integra- integration over T after substituting in the Fourier integral repre sentations of H(t) and x(T) gives finally If E is known, then the inelastic electron scattering cross section can be evaluated from Eq. (4.10). The equation for inelastic scattering of an electron off a target, since it only uses electron amplitudes for a ground state target[see Eq. (4.9)], has a somewhat strange appearance. The more usual description of inelastic scattering as in Eq. (4.7) has the electron leaving the target with the target in its excited state. It is just this necessity of describing the system in terms of its excited state wavefunction that the Bethe-Salpeter amplitude equation does away with. That is the excited state wavefunction is considered to have been created adiabatically from an excitation of the ground state system, this process will be described in detail below for a somewhat different case. If one recalls the derivation of the Bethe- Salpeter amplitude equation in Chapter III, where one starts with an equation for R in terms of an XX product, one sees that this exactly analogous to the process of obtaining excited state information such tion over a and p . Fourier transforms E ( ( j, , t , ct) to S(e-*+e-*,T, e-*-e-0 P 4 P 4 (4.10) as oscillator strengths by looking at the poles and residues of a ground state property, the frequency dependent polarizability. There, one obtains inelastic photon scattering information, the absorption oscillator strength, by using an elastic process, the polarizability, whose defining equation involves only ground state quantities. To complete the scattering picture, formulas for are needed when u>n is greater than the ionization potential. Here ^ is a con tinuum function and R has poles (or rather a cut at (« ) so that the o n bound state Bethe-Salpeter equation no longer holds. Special con tinuum are needed if exact R or are to be calculated and the discrete basis approximation not made. A second reason for calculat ing the continuum is that it obviously contains information about the ionization continuum of the target. From this information, it should be possible to obtain information about scattering from the ion. To do this, it is necessary to study the effect of adiabatic decoupling on the electrons of the target atom itself. The ground state and some specific states of the ion can be described in the independent particle model as a hole in the ground state of the atom. where |$q) is the ground state in this approximation and m is a quantum number referring to a ground state occupied orbital. In the following the Hartree-Fock independent particle model is used and creates a Hartree-Fock hole. Also the field operator is 43 *<?> - <%•*■& n The Heisenberg operator is expressed as \|r(rt) = an(fc) n HF -* HF -* where cp (r) and cp (rt) are the Hartree-Fock orbitals for the time n n independent and a freely propagating time dependent case, respectively. an(t) is neither a pure Heisenberg not interaction operator, but is a mixed representation. It is defined by the equation for an^ i)r(rt) which are related by the usual Heisenberg transform. The symbol "~n on the a^ is simply to distinguish it from that for free (neutral targets) or Coulomb (charged target) waves. Now starting from the un coupled state by adiabatically turning on of the correlation potential, an exact ion state can be reached: |Y ) = aln |Y > 1 m m 1 o where ~in . ~ . a = Him a (t) £ — ♦..00 This |Ym> will be used as an ionic target state in the formalism used previously for electron-atom scattering. In this case, the scattering matrix assumes the form: 44 s_ -4 = <y" -*|'i'+ -*> pm15qm2 m] L p 1 n^q' = Aim /dr dr'dr' Gd't'.r^.rjt ,r't') X 1 2 » < W T ? * 4 ■ < * 4 — 4 X <pj(rltl) f>mi(r2t2) ^ (r2t2) (4 n) •4 >4 HF HF where cp-*(rt) , cp-*(rt) are Coulomb waves, whereas cp (rt), cp (rt) are p q m2 Hartree-Fock one-particle stationary states (m^ and m2 are Hartree- Fock quantum numbers). The change here from the electron-atom scatter- ■ 4 “4 ing case is that the indices p and q refer to Coulomb-wave numbers and — 4 -— 4 cp-*(rt) and cp-*(rt) are Coulomb functions. The use of Coulomb functions Tp is conditioned by the long-range electron-ion interaction. The scattering matrix formula can be simplified by defining the Bethe- Salpeter amplitude referring to the state x±raa , 2) = Cro |T[t(i) /(2 )]|Y ± m > = Aim <Y I tC K 1) 1^(2)] a (t) a^(t') |Yo> = t '-*-00 ^ Aim(-l) f dr dr' G2(l,rt,2,r't') cp-* (r't') cpHF*(rt) —.-co J q m t '+e-*-° (4.12) Using this expression in the scattering matrix formula, it becomes S-* -* Aim /dr/dr' cp-*(r,'t') cpHF(r't') X (r't/,r't') (4.13) pm1,qm2 1 2 Tpv 1 V 2 2 1 1’ 2 2' 45 From the Bethe-Salpeter equation, an inhomogeneous (nonlinear) integral equation can be derived for the yi ^(1,2) Bethe-Salpeter amplitude. Inserting Eq. (3.19) and (3.22) into Eq. (4.13) and using the following identities M m f dr dr' 6(1,1') tp-(l') 9^(1) t-*-oo J q m t /+e-*-co Aim <a (t) a+(t')> = (t 1^ ) = 0 t— oo m o' qxm and Aim i f dr{ 6(1,1') cp-(l') = f±(l) • ( * +„co * / H n / HF* * dx± 6(1,2) cp“ (1) = - gm(2) the following equation is obtained for the Bethe-Salpeter amplitude: x± (1,2) = fi(l) g*< q m q m + J d3 d3' d4 d4' Rq(14,23) E(34',43') ^ (3V) (5.14) where f^(l) is the solution of the Dyson equation with incoming Coulomb wave of wave vector q boundary condition and gm(2) is the solution of the Dyson equation with Hartree-Fock boundary condition S„<2> - <p"F(2> + / G HF<2,3) Zcorr(3,4> gn<4> (4.15) 46 where Gl,„(2,3) is the Hartree-Fock one-particle Green's function and xir ^corr = ^ " ^HF wllere ^hf is the Hartree-Fock potential. This equation follows from the following form of the Dyson equation. G = G + G X G HF HF corr Now, the substitution of from Eq. (4.14) to the scattering matrix expression given by Eq. (4.13) gives 2> + +J2 Y L J im / d?i ^ P P x -*/ / t--«> J v v p m 1 X M m f / F(r't') g*/(r't') dr' X m1 2 2 m 22 2 X y*d3 d3' d4 d4' f^ (3 ) gm/(4) H(34, ,43/) X^m ( 3 V ) (4.16) This formula can be simplified, noticing that M m fdnt' c A r V ) f-*,(r't') = S-~, (4.17) t J p P PP The same expression occurred previously in the electron-atom scatter- ing formula, however p and p here refer to Coulombic functions. Similar expressions can be defined for the "hole scattering" 47 Aim f dr' cpHF(r't') g ,(r't') = 1> , (4.18) m m^,m Finally the following formula is obtained: S-* -* = S-+-* S + / / S-*-*t S / T-♦/ t (4.19) pm1,qm2 pq m ^ pp m-jin p m ,qm2 p m ' where T-/ / - = /*d3 d3' d4 <14' f£?(3) g ,(4') X p m ,qm2 J p n X E(34',43') £ m (3',4') (4.20) 2 The first term in Eq. (4.19) describes the independent scattering of the particle and the hole; the second term expresses the interference of these scatterings. The energy-dependent form of the T matrix in Eq. (4.19) is T-*/ t -* = 6(eX + e' - (v* ) X p m ,qm2 p m qm2 X/ d ? f ?'(?l> V (?2> X X H ( r ^ , ^ 5 -cm/) X ^ , ? ' ) (4.21) It is this form of the equation for the electron-ion T matrix which should prove most useful when combined with the perturbation expansion for 3. The formula for the scattering matrix Eq. (4.19) can be made 48 more symmetric. The equation for the Dyson orbital f^(l): 4 (1> “ 5Pg(l) + / G0(l,2) 2(2,3) f|(3) d2 d3 (4.22) can be solved in two steps. In the first step we construct the HF Hartree-Fock orbital cp£ (1) with outgoing Coulomb wave boundary condition and then solve the Dyson equation with 2 = 2 - 2,„„ with J H corr iiF HF cpj* as inhomogeneous term. Then can be written as = f m / d? 1 ‘ P f + tl K° + M m yT' f dr' cp|F(r't') cpS(r't') X tl-°° ? X cpSF*(2) 2 (2,3) fi(3) d2 d3 (4.23) r corr q In the eigenphase representation the Hartree-Fock scattering matrix is diagonal. Therefore '1 and consequently M m /*cf§F(l') qWl') dv[ = S(p) 6- - (4.24) £ KO J ^ 9 * where S-* -* = S(p) 6-*-* + / , S(r) 6-*-* T-*-* = pq Pq “ rp rq = S(p) 6— . + S(p) T-* = S(p) S— • (4.25) pq pq pq T_^ = f cp5F*(2) 2 (2,3) f±(3) d2 d3 (4.26) Pq J P corr qv ' v ' The substitution of the form of into Eq. (4.29) gives S—• —• = S(p) S - * - * s’ + pm1,qm2 r/ pq E E + 7 j 7 j s(P) s-~, s / t-*/_ / - ♦ */ i p m pp mm p m ,qm2 Chapter V NON-PERTURBATIVE APPROXIMATION METHOD13,6521,29(GRPA) Now that the formal equations are developed, an "equation de coupling procedure" is needed to truncate any of the three equivalent sets of coupled equations of Chapter III. This approximation will also simplify the scattering formulas involving H. The idea behind the approximation is to guess a functional form for E. (The reason that the G^, G^, G^j ... hierarchy was replaced by the two other equivalent ones which were stressed was to make 2 more visible.) A "functional form" means a specified dependence of E on G^, for unspecified G^. Since this guess defines the model and will be used to generate the physics of the problem, it must be physically well motivated. Once this is done the hierarchy can be closed in any one of an infinite number of ways, each of which gives a higher order self-consistent set of equations, viz.: If E^(G^) is put into the Dyson equation, the system is truncated to the closed equation for G^ requiring no other information. Since this equation is nonlinear, it will be solved iteratively (or self-consistently). If, on the other hand, E E^(G^) is used in 6E « 6E^(g^) with 6n£ 0, for n ^ 2, a set of two coupled equations are obtained; namely, the Dyson equation for G^ in terms of E, a formula (not "equation") for E in terms of G^, V and R, and a closed equation for R in terms of G^. These two coupled equations 50 51 must be solved iteratively and hence self-consistently. If the approximation is made for higher functional derivatives, i.e., 6n£/6u n , n=0, 1 is unspecified, but 6^£ py 6^£^(G^) with 6n£/6Un = 0, n=3,4, ... larger and higher self-consistent sets are defined. Hence, it is seen how higher and different types of self-consistencies can be developed. It is hoped (and moreover physically reasonable) that higher self-consistencies are better, since the model £^ is being used only to calculate smaller and smaller variations of £ rather than 2 itself. The problem now is to choose the form of S^(G^). From the dis cussion above, especially with reference to the single equation truncation, the choice of as Hartree-Fock functional form is suggested. Now ^(1,1'jGp = 2hf(M ' ) = - i&Cl-l7) f d2 V(l-2) 0^22+) + + iV(l-l') G1(l,l/+) (5.1) because if G- is taken as Gu_ (i.e., the Dyson orbitals and energies 1 Hr in the spectral representation of G^ are taken as Hartree-Fock orbi tals and energies) becomes the Hartree-Fock potential, since Gi ^ ®ihf* ^ aPProximati°n is made in the Dyson equation, the total theory reduces to the Hartree-Fock model which is unsatisfactory for anything except the ground state density and expectation values of one-particle operators. The next step and the contribution contained in Ref. 6 is to do the second step, i.e., leave £ along, but use 52 5 2 - 6 ^ (5.2) This defines the GRPA scheme. With this, using Eq. (3.21), the approximate form of H is obtained 6S (33') H(34;3'4') = ^ ' ■ ' * « — — ----- = Ha(34,3'4') = 5G1(4'4) 6g1(4'4) A = i6(3-4') 6(3/+-4) V(3-3') - - i6(3-3') 6(4-4'+) V(3-4') (5.3) Equation (5.3) is the first order term in the perturbation expansion of H as a function of G; this is the GRPA form of H. Now inserting Eq. (5.3) into Eq. (3.24) gives Xn(l,l') = - i f62 62' Ro(12/l'2) V(2-2') Xn(2'2/+) + + 1 f 62 62' Rq(12/1/2) V(2-2 ') Xn(22/+) (5.4) Using the 6(t2~t') hidden in the definition of V(2-2') shows that on the right-hand side of Eq. (5.4), the first and second terms contain respectively X (r_,r„,t ;-0) and X (r„,r„,t ;-0). Hence, in this n z z n 2 / approximation X (1,1') is obtained from a knowledge of XR(r^jr^;t^,-0) -4 — * » 2 and G^. ^n(r2,r2’t 0^ta^ne<^ the closed equation obtained from Eq. (5.4) by choosing t-[=t^, so that (t^t') - (t '“t^t1',T1=-0) 53 - 1 • / dV ?2dt2 Ro(?lt/’?2t2’?lt/+’?2t2) X X V ( | r 2 - ^ | ) Xn ( ? ^ t 2 -0 ) + + i J'dr2dr'dt2 R ^ r ^ ' ,r2t2 ,r.Jt/+,r2t2) V(r2-r2) Xn(r2r2t2-0) (5.5) where from Eqs. (2.36a), (2.29), and (2.30) -i<u t ‘ /,,,NI ■ t Xn(ri?(t'-°) = -e_iV (r') K r p l Y ”), u>n > 0 -e""iuJnt' O ^ U V ' ) I , iun < 0 -iiu t/ „ ,-*/-* \ = -e n Xn(riri) One obtains: = ^ / d?2d?2 / de |G< * l V > G(?2?r e" V X X V ( r 2 - ? p Xn ( ? ' ? 2 ) - G (? 1? 2 e) 0 ( ^ , 0 - ^ ) X (5.6) X V^r2_r2^ Xn^r2’r2^ If these results are used, the definitions made that Vkl(?3] (5.7) I I > to k _ ♦ ¥ r2> V(?2-r3) sj(? 2) = f-*, N-» = 0; Re e-» > 0 n q q = g-*3 N r = 1> Re e-* < 0 (5.8) n q q is the integral form of the GRPA equation (N-/-N-) cp-*(r ) 9?'(r{) X (r,'r,) = - 7 . — 3----3--3— ---3---— X 54 * X/ d?4 Xn(V 4 ) + (N^-MH) cpS'Q:') + e-*- e—»/ -a) -*-*/ q q n qq X J dr3d?4 cpS(r3) V(r3-?4) cp^,(r4) Xn(?4,r3) (5.9) The (N-*-N-*/) factor assumes that only hole-particle X's are solved r ; 6 for. A closed set of equations has finally been achieved. Equation (5.9) can be solved by standard non-Hermetian matrix diagonalization techniques. If the Dyson orbitals and energies are known, the know- ledge of X (r,,r„) and cu values enables the complete construction of n 1 2 n only a special case of the general R(l,2,1',2'); namely, from Eq. (2.39) the hole-particle R^^r^j^jr'jr^jO^O*, e) ; i.e., R(12,l'V+> = Rh!’ (?i;2,;';',0+,0+,S : ) - 1 V ' SSn(°0 = T > , ■ ■ ■ A g / ^ -- (5.10) i e-cu +xl] Sgn(iu ) u > * > n This and are all that is needed to calculate E in this approxima tion consistent with Eq. (5.2). One obtains: 55 Z(l,l') = ^pC1*1') - y*d2 d3 V(l-2) G(13) V(l'-3) R(321/+2+) + + G(1,1')J d2 d3 V(l-2) V(l'-3) R(323+2+) (5.11) Noting the 6(t^-t^) in VCl^-S) indicates that the part of the general R needed in Eq. (5.11) is R(t^jt^jt^t^) = R(0+,0+Jt/-t^) which is the Fourier transform of Eq. (5.10). In Fourier space Eq. (5.10) is: ‘ V V l ) - ^ / d?2d?3dz' V< V ?2> * Vm 4-4- . —♦ —». —* —♦ T /. X R (r3r2r3r2’0 ° Z > V(r3“rl) G(r!rJz-z > + " L a "♦ / “4 —♦ >4 <4 <4 4- 4- + 277 7 dr2dr3dz V(rr r2) RA(r3r2rlr2° ° z) X X V(r3~r') G(r1r3,z-z7) (5.12) The final set of self-consistent approximate equations are then: 1) Eq. (4.41) — the Dyson equation 2) Eq. (5.9) — called the generalized RPA equation 3) The formula in Eq. (5.12) after Eq. (5.10) is substituted in. The method of solution is then to start in Eq. (4.4') with and solve for the Hartree-Fock orbitals and energies which are the cp and e . Tn n These are then introduced into Eq. (5.9) which now becomes the I 56 RPA equation (since the cp's are *s) . The X's and (l^'s are found and combined with the 9n's and en's to form a new E. The procedure is repeated until self-consistency is achieved. On higher iterates Eq. (5.9) is no longer the RPA equation, but the GRPA equation. Eq. (5.9) need be solved for > 0 only as seen from Eq. (5.6). Once conver gence is reached and G is known, Eq. (2.12) gives Eq, and Eq. (2.19) gives the one-particle density matrix from which one-particle aver ages and natural orbitals can be found. As previously noted, the are the excitation energies and the give the generalized oscillator strengths. Of course R ^ is the linear response (not the general response) from a generalization of the RPA or coupled time-dependent Hartree-Fock (which are just different names for the same equation) which is consistent with the one-particle orbital equation. 2 The converged E^ can be put in Eq. (4.4') with e = q /2 set to i f c the desired scattering energy to give an equation for the f^ of Eq. (4.1). These scattering orbitals, with the boundary conditions given in Eq. (4.4/), can be solved for by any and all methods used for solv- 28 _i ing potential scattering problems. When solving for t the finite basis set used in solving Eqs. (4.4') and (5.9) is no longer used. Equation (4.5b) with E^ substituted E gives the approximate T matrix for elastic scattering. An approximate expression for inelastic scattering is obtained by substituting Eq. (5.3) into Eq. (4.8) which, after Fourier transforming, gives 57 S-* -» = - 6(e -e -a) np,oq p q n d^ld^2 X The approximate equation for inelastic scattering, (5.13), is expected to be less accurate than the other approximate equations given in this chapter. The reason for this is that the truncation approximation, Eq. (5.2), is closer in the hierarchy to the desired quantity than in the case of quantities depending on G^. The phil osophy of this approximation scheme is that a first-order approxima tion to H when integrated over gives a moderately good R (or X^), which when integrated over gives a good G^. In the case of inelastic scattering, however, one is not integrating over R (thus averaging out its deficiencies), but using the Xn directly (and hence exposing them in their nakedness), so it would not be as surprising to see the approximation show deficiencies. If that proves to be the case, it will be necessary to either use higher order self-consistent sets of 2 2 coupled equations (e.g., 6 E « 6 E^ ) or use higher order truncations of the Bethe-Salpeter equation. Clearly, a unification of the time-dependent Hartree-Fock theory with the Hartree-Fock theory has been achieved, with the advan tage of overall self-consistency. The result is to give a theory that calculates with one basic approximation Eq, p(l7,l), Xn, u)n, and the linear response. Hopefully, if the scheme is performed on a 58 finite basis set, the calculations will be tractable. , In solving these equations, the non-Hermetian matrix diagonalization, the energy dependence of E, and the linear dependence of the set of Dyson orbi tals, are all new, but hopefully tractable problems. The discrete basis set should not be any restriction since the cp's and repre sent phenomena that are localized in a small region of space. The successful computation, using such basis, or orbitals and frequency dependent moments for atoms and molecules is well documented. The topic of the use of a discrete basis also brings up a point of caution. Equation (5.9) comes from Eq. (3.24) and is valid for bound type functions only; i e., functions that go to zero as r^ or r^ go to infinity. In a finite basis this is true of all functions obtained. If a method of solution is used which includes continuum functions, the continuum hole-particle amplitude equation (4.14), will have to be used for the continuum amplitudes. After Eq. (5.2) is sub stituted in and the usual Fourier transform (the changes are essen tially the same as for the bound state problem) taken an equation is derived that is exactly as Eq. (5.9) except that n -• p^m and an inhomo- geneous term appears on the right-hand side which is simply the product outgoing wave particle and hole Dyson states with particle index of incoming standing p^ and hole index m, respectively. For the purposes of calculating R, the normalization of the Dyson orbitals does not matter since they differ by a phase factor which in turn causes the X to have a differ- ent phase factor which cancels in the produce XX appearing in Eq. (5.11). 59 A convenient matrix form of X is easily derivable even though the cp's are linearly dependent. Simply noting that Eq. (5.9) can be written as V ?Pi> " S xr ' <5-14> qq with x» = — 3 — 3. qq q q j /dr V-~,(?) X (?'?) f - n j J qq nv J "dr dr' qA(r) V(r-r') cfr*(r) Xn(r'r) (5.15) Substituting Eq. (5.14) into Eq. (5.15) gives the desired matrix form (e-*-e-*/-to ) X^/-* = q q n q q = (Nq"Nq')^^ (qp'Mq'p) - <qp'|v|pq'> (5.16) P P PP where (ab|v|cd) = J "dr dr' cfM(r) cpg(r') V(r-r') cp^(r) cp^(r') (5.17) Similarly, the matrix form of R^P is Rhp<;1;2^ 20+0+z) ■ qq with <u -z-ie Sgn(m ) n n q q P P (5.18b) V *0 Now that the basic approximate equations are exposed, a further discussion of the physics of the approximation is in order. Up to now the approximation has been introduced by appealing to the attractive ness of having higher order self-consistent theories which unify the Hartree-Fock and the coupled time dependent Hartree-Fock approxima tions. Also the point has been stressed that even if S « is not a very good approximation, 6E * = « 61!^ may be a much less damaging one. Further justification for expecting that the approximation will give good results comes from the experience that coupled time dependent Hartree-Fock equation gives in actual calculation good frequency 30 dependent responses, that the RPA gives reasonable excitation ener- 31 gies and that schemes which (i) use the RPA, (ii) solve for the ground state RPA wavefunction, (iii) use the latter in place of cp^ in Eq. (4.22) to solve for new orbitals, and (iv) use the orbitals in Eq. (5.10) to get new and get generally improved agreement with 32 experiment with only one iteration. Moreover, it has been shown that the types of correlation effects required in R, i.e., hole- 27 particle effects are all included in our iterated R. Note that the final linear response here is "nonlinear" in the sense that the zero'th order model has been greatly refined upon iteration. Perhaps the most persuasive argument for the approximation comes from the physical model of elastic scattering that is implied. Here one starts with the Hartree-Fock virtual continuum orbital as the scattering orbital. This is called the static exchange approximation. The target electrons are also taken as in Hartree-Fock orbitals with Hartree-Fock exclusion correlations. The scattered or "test electron" in the virtual orbital now causes the target to respond. This response is calculated in the coupled time dependent Hartree-Fock (or RPA) approximation. The response, which in this approximation is much better than perturbation theory applied to the Hartree-Fock Hamil tonian and which contains correlation and exchange effects and depends on the energy and position of the test particle, is then coupled with the Hartree-Fock orbitals which "drive" the response to give a new effective potential. This potential is then used to calculate new target and scattering orbitals. Everything is then iterated until the scattering orbital, the target orbital, and the response are all self- consistent. Since the electrons are all indistinguishable, the correlations in the target are as well represented as the ones between the test particle and the target; this must be reasonably well done since as said above even the RPA gives good responses. To obtain the generalized RPA equation for an electron scatter ing from an ion, in terms of quantities calculated for the neutral particle, Eq. (5.3) is substituted into Eq. (4.20) which, after inte grating over the delta functions, gives: 62 tgrpa ^ qlml’q2m2 i f dl d2 f*+(l) g* (2) V(l-2) xi (1,2+) J q ^ m ^ q2m2 - i /'dl d2 fj+(l) g* (1) V(l-2) xi (22+) (5.19) J 1 1 2 2 This can further be approximated by its first iterate the RPA. Then where X's are the first iterate to Eq. (5.10). Also making the above substitution in Eq. (4.30) reduces the S and s’ matrices to their 6- function terms only. For the elastic RPA case, they become the unit case the first term in Eq. (4.30) vanishes, and in the second term there is only one non-zero term in the summation. We now remind the reader that the inelastic cross-section formula refer for only states that can be achieved by adiabatic coupling from a particle-hole in dependent particle state. For helium, there is only one particle-hole state, i.e., the lsq state and therefore for electron-helium scatter ing this formalism is not able to describe inelastic process at all. £ is replaced by £^ in the generalized RPA Eq. (4.9), f^(^) by c Pr,(r)„„ and g (r) by cp (r)T,_. These substitutions reduce Eq. (5.19) Tk HF m ' , m HF - i n to , . matrix and the summations over p and m drop out. For the inelastic 2 In Be one can think of two hole-particle states, i.e., (Is) 2sq and 2 (Is)(2s) q/. As such one can study electron scattering on either of these states or inelastic between them. (Though this inelastic process is not very interesting physically.) We note also, that even in the case where the formula can be applied for an inelastic process, the approximation is probably poor, because the RPA is a refinement of the HF and the HF is not able to describe inelastic processes. The formula so obtained is well known as the RPA elastic scattering 33 30 formula. Dalgarno and Jamieson have used this equation by solving for Xn using the RPA equation for X with G taken as G ^ and with Hartree-Fock hole-particle boundary conditions. They solved this RPA scattering elastic formula for the p-wave phase shift for the elastic scattering of an electron off a helium positive ion. Their agreement 34 with the close coupled results of Burke and McVicar was quite good. Although obvious, it should be mentioned that in molecular systems, noting the facts that: (1) for each geometry excitation energies are calculated self-consistently and directly, and (2) Eq. (2.12) can be used to get the consistent absolute energy of the ground state (hope fully the errors in this formula will not be very sensitive to the geometry changes), implies that all potential surfaces of the molecule can be calculated self-consistently in one calculation. Chapter VI GENERALIZED THOMS-FERMI THEORY FOR SCATTERING AND THE SHAM-KOHN MODEL (LOCAL DENSITY APPROXIMTION) A. Introduction After having described how the Green's function technique can be applied in the theory of electron-atom or electron-molecule scatter ing, a quasiclassical or statistical approximation will be postulated that can be useful in relating the Green's function formalism to pre vious theories and also introducing more physical insight into the formalism. The accuracy of this approximation cannot be compared with the rigorous quantum mechanical calculation; however, it might serve as a first approximation to it. In the potential scattering approximation the electron density distribution of the atom plays a central role and therefore the quantum-orbitals for the atomic electrons are not so important. In many cases, especially for large atoms (Z £ 20), the Thomas-Fermi potential is a good approximation to the Hartree potential and gives a result not very much different from the Hartree or Hartree-Fock 35 approximation. Some of the earliest calculations for the scatter ing of electrons on heavy atoms have been done using the Thomas-Fermi 36 potential. Henneberg used the WKB approximation for the scattering problem with the Thomas-Fermi potential and applied the method for 64 65 Hg, Kr, and Ar atoms. For large energies (E > 100 eV) the calculation properly described the angular distribution of the scattered electrons. 37 Similar calculations have been done by Massey and Mohr and by 3 Bullard and Massey. By the inclusion of quantum corrections to the Thomas-Fermi potential better agreement is expected with the Hartree- Fock result. 39 However, for the inclusion of polarization effects, a scheme should be considered that includes correlation among the electrons. The major theorem which will be used is that the Thomas-Fermi theory can be considered as the quasiclassical limit of the Hartree-Fock theory. Considering a more general scheme that includes correlation (e.g., RPA or GRPA) the quasiclassical limit of some low order correc tion will also include correlation effects. In the following a simplified version of the GRPA will be con sidered (by neglecting the second and higher order exchange effects appearing in the GRPA but retaining self-consistency). This scheme was postulated by Baraff^ and the quasiclassical approximation has been worked out for this case by him. The quasiclassical approxima tion up to second order will be considered and the possible numerical calculation will be described. 35 B. The Thomas-Fermi Theory and Quantum Corrections 35 In the Thomas-Fermi atom model the electron-gas is treated as a locally homogeneous system and the Fermi-Dirac statistics is 66 applied. The Thomas-Fermi theory in the usual form refers to the calculation of the ground state energy and its physical meaning for scattering problems is not clear. The first step in the direction 40 toward a more general Thomas-Fermi theory was made by Dirac who proved that the Thomas-Fermi equation (with an exchange correction) can be obtained by considering the quasiclassical limit to the Hartree-Fock equations. The same conclusion has been obtained by 41 Fenyes who proved that the zeroth order WKB approximation to the Hartree equation provides the Thomas-Fermi equation. The next step 42 was made by Theis introducing the phase-space representation of 43 physical quantities, following the earlier work of Wigner. Theis showed that using an expansion for ft, the zeroth order approximation to the Hartree-Fock equation provides the Thomas-Fermi equation. (Exchange correction is not included.) 44 Kompaneets and Pavlovskii used essentially this method (WKB approximation to the Hartree-Fock) of obtaining quantum corrections to the Thomas-Fermi equation. They have established that both the 40 exchange (or Dirac ) and the kinetic energy inhomogeneity (or 45 Weizsacker ) corrections are of the same order. Finally, Baraff and 9 Borowitz used Schwinger's Green's function technique and the power series expansion for f t to the derivation of quantum corrections to the Thomas-Fermi equation. They have obtained identical results for the second order correction with Kompaneets and Pavlovskii. 67 As we have mentioned, the Hartree-Fock theory, and therefore its quasiclassical approximation also, does not contain correlation. In a second paper, Baraff^ started out from a scheme that includes correla tion. The scheme he has used is essentially the GRPA scheme without the exchange terms in the GRPA equations. Baraff^ developed the quasiclassical approximation to this scheme, using a power-series expans ion in ft. In the following the Planck constant, ft, will be considered as a parameter. Therefore, the f t = 1 unit will not be used, as has been done in earlier chapters. The fundamental equation for G^ and G^ has the form (the defi nition of Gn has not been changed) of Eq. (3.4), [ i f t ^ / d t p - H ^ r p ] G -^r -jt ^r' t') + = ft6(rx-rp 6(tx-t') (6.1) The Hartree-Fock approximation is defined by the formula: G2(12,l/2/) » Gjd.l') G1(2,2/) - 0^1,2') 6^2,1') (6.2) Neglecting relativistic effects, the Green's function would not depend on the spin and the spin-summation can be executed. The following (Hartree-Fock) equation is obtained: 68 [iMd/dtp - HqC^)] + + iy*dr2 VO^-rp [2 ,r2t+) Gprp.r't') - “ V V l ’V t ) Gl(?2tr ?lt]L^ = = ftS^-rp eCt^-t') (6.3) where integration refers only to space coordinate (Baraff and Boro- 9 witz ). Define the k(l,2) function by the formula /k(l,2) d2 G1(2j1,)=A6(1-1/) (6.4) (Here the number means space and time coordinates.) In the Hartree-Fock approximation M r p ^ r p p = [ift(9/dtp - ^ ( r p + + 2i J dr' VOrj-r') G^r'tpr't*)] 6(tj-t2) S ^ - r p - - i V(rr r2) G1(?1t1;?2t+) 6^ - t p (6.5) This is, except for the first two terms, the optical potential in the Hartree-Fock approximation: ^ ( r p ^ r p p = - 2i f dr' V(r1-r') G1(r/t;r't+) 6(t1-t2) fiC^-i-p + 69 + i VCr^-rp G^r-jt^r^*) 6(t]L-t2) (6.6) and k(l,2) = [±ft<a/at1> - Ho(r1>] 6(1-2) - E(l,2) (6.7) The phase-space representation of k is defined by the formula, (this 43 transformation was originally introduced by Wigner ) k(R,p,aj) = y*d(*V^2) d(ti"t2^ k^ l tl’^2t2^ X X exp | - i [pO^-rp - a>(t1-t2)] /ft (6.8) where S = 2(?1^ 2) (6*9) Similar transformation provides G(R,p,(o), ^(R>Pjto). The quasi- classical approximation is obtained by using an expansion for f t as a parameter. The neglect of higher order terms in f t means the neglect of the powers of the operator ft 2- -i- n 3R Bp i.e., the variation of the function within the phase cell. This can be expressed also that in classical mechanics there is no uncertainty relation, the Fermi cell is of zero volume; therefore the "change of In the following we drop the index 1 in G^. 70 the function’ 1 within the Fermi cell is exactly zero. If k and Gf are slowly varying functions within the Fermi cell, then the lower order approximation is good. This is the usual quasiclassical assumption. 9 Baraff and Borowitz prove that the zeroth order approximation pro vides the Thomas-Fermi equation, i.e., Gq(RjPjU)) = f t 3[ko(R,p,io)] 1 = f t 3[uj-e] 1 (6 .10) where E(r,p) = p /2m + $Q(r) (6. 11) and §o(R) = - Ze^/R + 2 / v (I -r) nQ(r) dr (6.12) They have obtained for $Q(R) the Thomas-Fermi equation: V2§ (R) = - -^3 3tt* 12m [p--$o(R)]} 3/2 (6.13) 2 3-1 3 where n (R) = ( 6tt h ) Pr,(R) is the electron density in the Thomas- o F Fermi model. The first order correction is zero. For the second order correction to the optical potential is obtained: s2(r,p) = - k2(R,p) = *2Cr) + ^ 3 Ttfl 2 2 PF-P 2p An PF“P PF+P (6.14) ! ■ - _ 71 where p^ is the local Fermi momentum, defined as pp(R) - 2m[n-§o(R)] 1/2 (6.15) and $2^) obeys the following equation: - v 2 * + ^ [ 2m(ll,-}o)]1/2 i2 » nn me2[2m(ij,-l )] Unfi' [«2#o + ( ^ ) _ (6.16) 44 This equation was also derived by Kompaneets and Pavlovskii. They have solved it numerically. This equation contains exchange and kinetic energy inhomogeneity effects, i.e., Dirac and Weizsacker correction. (However, this latter one has a factor of 1/9 to the 45 correction originally introduced by Weizsacker. ) The second term in is the well known exchange-hole. C. The Effective Interaction, a New Hierarchy of Equations for the Many-Body Problem In Chapter III.C several hierarchies of equations have been postulated for the solution of the many-body problem. The most important was for the purposes of the present work the hierarchy that included the quantities G^, Z, S, etc. The RPA and the GRPA have been formulated in terms of these quantities. In the GRPA, 3 was approxi mated by a simple expression containing V [Eq. (5.3)], and then G and 72 E was solved self-consistently. In the following, a new hierarchy 46 will be postulated. This hierarchy has been used by Hubbard and essentially defined with the variational technique by Martin and Schwinger.^ This is called also the shielded potential approxima- 13 tion. The reason for the postulation of this formalism is the fact that the exchange term in the RPA [the second term in Eq. (5.4)] cannot be handled easily in many-body theory. This new scheme will be used in scattering theory for similar reasons. (In this new formalism the exchange term is a higher order term than the appro priate direct term. We suspect that this corresponds to the real situation.) This is one of the simplest formalisms that can be formu lated with the Green's function technique and includes correlation effects. Later, analyzing the quasiclassical limit of this scheme, it will be pointed out how the previous limitation could have been released. The new hierarchy can be postulated by introducing the effec- * tive external one-particle potential: V(l) = U(l) - i.J V( 1-3) G(3,3+) d3 (6.17) where U(l) is the external potential. r 47 We follow here Hedin. The Dyson equation has the form [ift(S/at1) - H(l) - V(l)] G(l,2) - - / M(l,3) G(3,2) d3 = ft6(1-2) where M(l,2) is defined as M(l,2) = - i f V(l-3) G(l,4) d3 d4 where k is defined by the formula: y*G( 1,2) k(2,3) d2 = *6(1-3) The effective (or screened) interaction W(l,2) is defined as; W(l,2) = / v ( 1-3) d3 6U(3) The vertex function is: r (12;3) = - = 6(1-2) 6(1-3) + Then M(l,2) can be expressed as: M(l,2) = iy*W(l,3) G(l,4) T(4,2;3) d3 d4 In the lowest order approximation r(l,2;3) is substituted by 6(1-2) 6(1-3) r (o )(l,2;3) = 6(1-2) 6(1-3) (6.18) (6.19) ( 6. 20) (6. 21) (6. 22) (6.23) (6.24) 74 Then follows M(1)(l,2) = i G(l,2) W(l,2) (6.25) and W(l,2) obeys the equation: W(1)(l,2) = V(l-2) - i /VC1.3) G(3,4) G(4,3) W(1)(l,2) d3 d4 (6.26) in this approximation. Using the form Eq. (6.26) for M, it is solved self-consistently for G and W. Baraff^ in his quasiclassical approximation starts from this scheme. A comparison between the GRPA and the scheme just mentioned can be made by relating the second order terms in the perturbation series for the optical potential. In the GRPA the perturbation series for 2(1,1') can be obtained by substituting the perturbation series for R(12,l'2'). Taking the zeroth order term for R, R°(12, 1 '2') = G(12') G(2l') (6.27) the second order perturbation form is obtained for E: 2(1,1') = 2^ ( 1,1') - - y*d2 d3 V(l-2) G(l,3) V(l'-3) G(32+) G(21/+) + (6.28) 75 The second order term can be represented diagrammatically as 2 > 2. By solving the integral equation for R(12,l/2/), similar terms are summed up to infinite order (direct and exchange "bubble" diagrams). The perturbation series for the optical potential in the scheme defined in this chapter (Hubbard, Martin-Schwinger) can be obtained from the perturbation expansion of the screened interaction W(l-2) = V(l-2) - - \ f V(l-3) G(3,4) G(4,3) V(4,2) d3 d4 + ... (6.29) The second order correction to in this scheme is + G(l,l') J d2 d3 V(1-2) V(l'-3) G(3,2) G(2,3) (6.30) only one of the previous two, represented diagrammatically by: 76 \ By solving the integral equation for the effective interaction similar diagrams are summed up to infinite order (all direct "bubble" dia grams) . Therefore, this latter formalism lacks certain exchange terms (except the first order) that are included into the GRPA. However, the effect of the exchange term is usually smaller than the direct term. In many-body theory the second order exchange term gives a con tribution of a constant to the energy; it does not depend on the density. Therefore, it is considered to depend on the derivative of the density, which is neglected in the present scheme.^ However, the second order exchange term can be brought into the formalism by modi- 46 fying the effective interaction, as Hubbard did, or adding to the optical potential as an extra term, essentially the method applied by 48 Gell-Mann and Brueckner. D. Baraff's Quasiclassical Theory Baraff"^ used the approximation described above for the optical potential; i.e., he takes the form: £(1,2) = - i 6(1-2)/d3 V(l-3) G1(3,3+) + + i W(l,2) G1(l,2+) (6.31) 77 where the effective (screened) interaction W is defined by the equation: W(l,2) = V(l-2) ~ J f d3 d4 V(l-3) G(3,4) G(4,3) W(4,2) (6.32) It is worth stressing that this scheme is self-consistent and for the quasiclassical approximation the self-consistency is a highly desir able feature. Baraff^ considers the quasiclassical approximation and solves 9 the problem with a method similar to that which Baraff and Borowitz have used for the Hartree-Fock case, k(R,p,tu) has the form: k(R,p,ui) = tu - p^/2m + Ze^/R - _ 2 f n£l_e^ d r _ J |R-r| - i(2nft)"4 y ' dp duo W(R,p-p' ,cu-u/) X X 'GCRjp' ,u/) exp (iu/0+) (6.33) G and k are expanded as previously, and it is assumed that W can be expanded as CO W(R,p,tu) ? hi W^(R,p,cu) (6.34) j=0 The zeroth order approximation here is also the Thomas-Fermi equation; i.e., 78 G0(R,P,cu) = *"3(u)-E)-1 (6.35a) E =' E(R,p) = p2/2m + $q(R) (6.35b) where $Q(R) is the Thomas-Fermi potential. The first order correction is zero in this case also. The second order correction can be written as = " " X(R>P>u>) (6.36) where $2^) °beys bhe following inhomogeneous differential equation: ,2 2 Vz $ (R) - =2|- p (R) $2(R) = Sy-g p (R) J(R) + TTft TT h 2 me 12nfi3pF(R) (6-37> J(R) can be expressed as J(R) = 4t t2* {Es-P2/2m} ( 6 .3 8 ) where Ec = p2/2m + E (pj + E (p_) (6.39) S F ex F corr F the total electron self-energy on the Fermi surface. E is a function O of R now. Baraff^ generalized the formalism by proposing of the sub stitution for E the expression that is appropriate at the density D present at the point R. This step, however, breaks down the consis tency of the formalism. (In the previous model, Eg is supposed to be 79 calculated with the model described previously (Hubbard, Martin- Schwinger) which is a good approximation only at high densities.) Baraff's equation was solved using for E both a low density and a o i 4-9 high density approximate form by Viswanathan and Narahari Achar , by P. Venkatarangan"*^ and by Tietz and Krzeminski.^ The second part of X0*»P>U))» defined by the formula: x W2(R,P-P>V) Go(R,p/,0)/) eiu) 0 (6.40) Here 4ne2/p2 W2(R,p,cu) --- = ~ ^ Z ---------- <6*41> l+2Qo(R,p,u)) e(R,p,w) the energy-dependent dielectric function has been introduced which, in this case, has the form: e(Pj(Ju) = 1 + C(R)/x2 f(x,y) (6.42a) where F and C(R) = 2me2/nftp„ (6„42b) r x = p /p f (6.42c) y = tu/(p2/2m) (6.42d) f(x,y) = 1 + 2x 80 s [L - i (*+ £)* * -* 52 e(po)) is also called the Lindhard dynamic dielectric constant. Now, the following observation is important for the calculation of X(r jPjUJ)* The R-dependence comes into X only via p_(R) and $ (R) . r O The value of x CRjP*01) is exactly equal to the value of the self-energy in the homogeneous system if the density is defined by the Thomas- Fermi value at R, i.e., by its local value, and p = p„(R) is substi- r r tuted and ou - $Q(R) for the energy, i.e., X(R,p,oj) = ^hcraitp^-^CR)] (6.43) where is the optical potential in the same approximation (i.e., the Hubbard or Martin-Schwinger) for the homogeneous system. This approximation can be called the local density approxima tion, i.e., the physical meaning of this correction can be given as the appropriate self-energy, calculated for a homogeneous system from point to point, using the density given by the Thomas-Fermi theory, and the energy variable is taken as the total energy minus the appro priate Thomas-Fermi potential energy. This observation will lead to another, quantum-mechanical model, that will be discussed later. Con- sequently, xCR’P*^) can he calculated from the numerical results of 81 calculations for homogeneous system with the same method. The most 53 extensive calculation with this method has been done by Lundqvist 54 using Quinn and Ferrel's computational technique. Therefore, the numerical calculation of the second order correction looks feasible. By solving numerically the Dyson equation with the optical potential, we can obtain the appropriate Feynman- Dyson amplitude and then formula (4.6) provides the elastic scattering T matrix. Summarizing the previous steps, the following operational pro cedure can be given: 1. Solve the Thomas-Fermi equation for $q(R). (This has been done numerically; also approximate analytical forms are available."^) 2. Construct the Fermi momentum function p (R). F 3. Substitute this value of p = p„(R) into the dielectric F i ? function. 4. With the dielectric function given, substitute e = cju-$o (R) for the energy-variable in ^ lom(P>e)* (Use the table given by Lundqvistf*^) The quantity calculated is called XCRjPj^) • 5. Obtain the result for from Ref. 49, 50 or 51. 6. Construct the quantity: E(R"p,Sder>“ * o « + + 7. Calculate the Fourier transform 82 (2nd order) , /> (2nd order) \ ^ lr2U)) = -----3 / ‘ ^ R»P>U)) e1R 1 r2 dp (2rrft) J 8. Solve the scattering problem (i.e., the Dyson equation) Eq. (4.4') and use the solution in formula (4.6) for the scattering matrix. E. The Sham-Kohn Model' * ' ' * ' (The Local Density Approximation) Baraff's^ expression for the optical potential (up to second order in ft) could be explained on the basis of a simple physical model. This model has been developed on a more general basis also by Sham and Kohn.^ These authors write the optical potential in the following form: £(r,r' ;cu) = cp(r) 6(r-r/) + M[r,r';ui-cp(ro)] (6.44) where rQ = |(r+r') (6.45) and cp(r) is the electrostatic potential of the target. [The Hartree ■ 4 — or the Thomas-Fermi potential is an approximation to cp(r).J They prove the following important theorems: (a) E(r,r/, u j ) is a short-range kernel with a range of order |r-r^| ~ Xp = 2ir/kp where Ak^ is the Fermi momentum corresponding to the local density. (b) The functional form of M depends only on the density n(s) in the vicinity of r and r in the sense of 83 s-?o| ~ Max(XF,XTF) (\ is the Thomas-Fermi screening length.) ir On the basis of these theorems, they propose a model for inho- mogeneous electron system with the assumption: statistical theory where cp(r) was the Thomas-Fermi potential, and this relation has been observed for the second order term. However, in the Sham-Kohn model, M^[r,a),n(rQ)] is calculated in an approximation which is suitable at the density n(rQ) and not necessarily with the high density (Hubbard, Martin-Schwinger) approximation. This means that Baraff's idea of calculating Ec is valid also for the calculation b The Sham-Kohn model physically looks to be a better approxima tion than the second order quasiclassical approximation, although it is not self-consistent. Sham and Kohn^ also propose an approximation for the scatter ing problem with this optical potential. First, the relation for the static potential is used: (6.46) where is the optical potential for the homogeneous system at density n(rQ). This is exactly the mathematical form of our observation in of X(R.P.UJ) • | i = cp(r) + nhCn(r)] (6.47) 84 where ^n(n) is the chemical potential for the homogeneous system at density n. (This equation is fulfilled in the quasiclassical approxi- mation when cp(r) is the Thomas-Fermi potential.) With this assumption they obtain: M[r,r';E-cp(rQ)] « Mj^Cr-r' ;E-|i-Hxh[n(ro)] ;n(rQ)} (6.48) Finally, they propose to solve the scattering problem with the WKB approximation, i.e., from the equation | P2 + p,E-|J/+p,h[n(r)] ;n(r)} = = E - p . + |i^[n(r)] (6.49) one obtains p as a function of r, i.e., p = p(r) and substituting this into a local potential is obtained U(r,E) = l^fp^) ,E-|i4jj,h[n(r)] ;n(r)3 (6.50) This potential is simple; however, the WKB assumption (6.49) is a strong restriction for electron-atom or molecule scattering. There fore, the use of the original form is preferable.'*''*" However for atomic systems, especially for scattering problems outside the atom the electron density is very small. Therefore X is large (\ -• °° F F if r m) and the local density approximation cannot be good. There fore, for scattering problems we have included other terms. Sham and Kohn'*'^ also give a formula for the correction term. They write where n^(s) = n(s) - n(rQ). In the low-density region, n(r) « 0 and the second term describes the long-range polarization effects. The first term is responsible for the short-range polarization. The further develop ment of the theory in this direction looks promising. Another possibility is of using the Sham-Kohn potential for the variation of E, i.e., the 6E -* 6E approximation. This is an SK alternative to the 6E -* which defined the GRPA scheme. Chapter VII SUMMARY The Green's function formalism was presented with its possible applications to atomic and molecular problems. It was shown how the GRPA formalism can be used for the calcu lation of electron-atom, electron-ion elastic and inelastic cross sections. The electron-atom elastic cross section is expected to be fairly good. The electron-atom inelastic result with GRPA is probably less accurate. For the calculation of the electron-ion cross sections, one needs to solve the GRPA equation in the continuum region. After this has been done, the electron-ion elastic cross section is expected to be fairly good. It has been proved that the GRPA equation can be reduced to a form identical to the RPA eigenvalue problem. The Hartree-Fock orbi tals are substituted with the Feynman-Dyson amplitudes. The spurious solution which comes from the overcompleteness of the Feynman-Dyson amplitude set must be eliminated. The classical approximation and the local density approximation was considered. The classical approximation was interpreted as a local density approximation with the Thomas-Fermi potential and density. The local density approximation describes only the short-range correlation; therefore, long-range effects must be included by other terms. 86 I Appendix A THE ANGULAR MOMENTUM ANALYSIS OF THE DYSON EQUATION The time-independent Feynman-Dyson orbital is defined by the formula: where n, L, M quantum numbers refer to the energy, angular momentum magnitude and projection, i.e., H|Y T„ > = E |Y > (A.2a) 1 nLMh n 1 nLMo' ' ■ L(L+1) I ’W <A‘2b> Lz ^ TnLI-fa) = M^ n U f a ^ < A - 2 c ) and o' is the notation for additional quantum numbers. In the follow ing it will be shown that the angular dependence of the Dyson orbital fnT.My(r) : ' ' s t*ie weH known spherical harmonics; i.e., f _ (r) = f (r) Y (0 ,cp) nLMo' nLo< LM T The Dyson orbital for a general state |Y) will be denoted by f(r); i.e., f(r) = <Yo|*(r>|Y> (A.3) Let us define the operator %z acting on f(r) by the formula: 87 88 2 f(r) = <Y |Kr) L |Y> (A.4) The commutator of ijf(r) and L can be obtained from field theory: z where &z is the one-particle angular momentum projection operator acting on \)i(r) . Using (A.5) in (A.4) the following expression is obtained: 2 f(?) = (Y | ['lf(r) ,L ] |Y) + z o z + <Y |l +(?)|y> = o 1 z T = <Yo|jlz ii(r)|Y) + <Yo|lz «?)|y> (A.6) Let us assume that the ground-state of the system is an S state; i.e., . 2 i L |Y ) = 0 and consequently L Y > = 0 z o (A. 7) Therefore, 1 f(r) = i <Yn|*(r)|y> = J & f(r) (A.8) z o' This is true for any Feynman-Dyson orbital, and so 2 - SL (A.9) z z A This means that j I is the usual one-particle angular momentum pro- z jection operator. Let us apply this result to f (r): J nLMy K fnLMc/r^ = Lz ^nLMaf^ (A.10) However, because of (A.2c) this can be written as: J L f TM (r) = M<Y \Ur)\V T« > = M f (*> (A.11) z nLMcv o,TV ' • nLMcy nUMcr ' ' This means that - * - s an eigenfunction of the operator with eigenvalue M. In spheroidal system of coordinates, the operator is repre sented as: and from (A.11) one obtains: fnLIfc/r^ fnLMc/r’0^ 6 ^ (A. 13) a simple angular dependence for *■2 -* Let us define the I operator for f(r) by the formula: i2 f(?) = <YQ|*(r) L2|Y> (A.14) 2 L can be written as L2 = -|(L+L_ + L_L+) + L2 (A. 15) where 90 From these definitions follows: i . ; - and (A.17) L! = L+ For these operators, the following general identities are true: [i|r(r),L 1 = ZJf(v) + + (A.18) [i|f(r),L_] = Z_ty(r) where j l+ and Z is the appropriate one-particle operator, operating on i|r(r) here. The commutation relations for L+, L and Lz are [ \ ' V ‘ L+ [L ,L ] - - L (A.19) Z “ “ [L+,L_] = 2Lz From these commutation relations follows that L+ |^nTM^) is an eigen- 2 state of Lz with eigenvalue M + 1 and of L with eigen value L(L+1) (but not properly normalized). Let us calculate: <YoU(r)L+L_|Y> (A.20) Using the |$) = L |Y) notation, this is equal to: 91 <Y U(r) L.|§> (A.21) From Eq. (A.18) follows: <Yo|nr)L+|§> = <Yo|[Kr),L+] j$> + + <Yo|L+iK?)U> (A.22) The conjugate of (Yo |l+ is L |Y ) = 0, because L_ lowers M, but M=0 for |Yq). Consequently, <YoU(?)L+ |§) = <Yo|f+K?)l*> = = f+<YoU(r)|$> = J & +0roU<r)L_|’ r> (A.23) With similar argument for L , we obtain finally: <Yo|Kr)L+L_|Y> = A+£_<YoU(r)|Y> = ! L +SL_ f(?) (A.24) Quite analogously for L L+ the result is obtained: I1 f(r) = <YoU(^)L2|Y) = = <Y0I ^ ( L+L- + L_L+) + = l\v>+l_: + Z J +) + A2] f(?) = JS2f(r) (A.25) o where S L is the usual one-particular angular momentum magnitude operator. 92 Because Eq. (A.25) is valid for any f(r), the identity is obtained a2 2 r s jr (a.26) Applying this theorem to > one obtains: ■ L(L+1) <A -27> From Eqs. (A.11) and (A.27) follows that the angular dependence of fnu^(r) is the usual spherical harmonics; i.e., fnLMc/r^ = fnLffc/r^ YLM^ 0,Cf^ (A.28) This is our main result. Let us apply this expression in the Dyson equation: fnLMc/r^ = C p nLMb'^r^ + + / Go(¥ l ;V S(V 2eq) fnLlfa(?2> d?l d?2 (A*29) (q = nLMar) Substituting Eq. (A.28) into Eq. (A.29) and considering: CPnL*fa(?) = Cpn(r) YLM(n) (A‘30) we obtain: 93 W r) Ym<® ' ‘ p »(t) Yi»<n) + + . . V (r> YL'M,(n) V (rl> YL'm'(D] e /- e n'L'M' n q X fnLMa(r2) Y^CO,) d ^ dr2 (q = nLMoO * Multiplying both sides with Y— (fi) and integrating for Q: liM • fnLtoCr) 6ll ■ ‘P„(r) 6ll 6w + /-V-> V <r) V (rl> * V ^ ^ ( 1 n' n q X W V W / V rldrldnir2dt2dn2 Define: <P'(r) c Pn#(rlJ --- = G (r,r ;e ) T 6n/_eq ° 1 q and f w S^rlr2’0q^ YLM^n2^ dni dn2 = S U4,LM(rlr2' ) then from Eq. (A.32) we obtain, if L / L or M ^ M, Tz~ = 0 lM,LM X (A031) (A.32) (A.33) (A.34) (A.35) 94 the non-diagonal elements are zero. If L = L and M = M, then the Dyson equation is obtained for the radial part: W (r) = CPn (r) + / Go(r’rl;eq) SLM,LM(rl’r2) X X fnLto(r2) rldrlr2dr2 (A'36) Appendix B THE ANGULAR MOMENTUM ANALYSIS OF THE GRPA EQUATION The time-independent Bethe-Salpeter amplitude is defined by the formula for co > 0: n Xn(??l) = <^0l*+(?/)*(?l)I V and the GRPA eigenvalue equation is given by: V ' l V ■ / d'2d?2 V ' l V ' l ?2 X )X V< V ?2 ' > V ?2?2> " - Jdx2 d?' X V<?2_72) X„(?2'2) <B'2> here: V ^ l V ^ r V = ^ f G&l*2e) G(*2?le" V dS (B<3) Let us assume that an atomic system is given and the |^ ) is 2 an eigenvector of L and L , as well as of H; i.e., z n^nLMcf'* = EJ YnLMo''> l2I W = l(l+1) I W <b -4> l | y _„ > = m | y TVf ) z 1 nIMv 1 nLMa? and a is the notation for additional quantum numbers. In the follow ing the angular momentum analysis of the Bethe-Salpeter amplitude will be given and the radial form of the GRPA equation will be derived. 95 96 Let us define the operator X XnL*fc/rlrP <*CI* ^rl ^ rP Lz lYnLMb^ ^B'5^ It can be shown in a similar way, as in Appendix A, that K = \ (1/) + xz(1) (B-6) A i.e., $ , is the two-variable angular momentum projection operator. a2 Let us define X with the formula J2 <b-7> It can be proved that X2 = [X(l') + X(l)]2 (B.8) A2 i.e., X is the two-particle angular-momentum magnitude square operator. However, because of Eq. (B.4), we obtain: [Xz(l ^ + ^z^1^ XnLMc/rlrP = M XnLlfc/rlrP (B.9a) and [1(1') + J(l)]2 - L(L+1) <B.9b) Denoting by t^ie eigenfunction of the X(l'^X(lj\ X X [X(l') + X(l)]2 and X (1') + X (1) operators, X (r' r..) can be z z n.LMcy J L 1 expressed as: XnUt/rlrl^ = i S XX'X1(rlrl) YX.'X W ^ l rl^ (B.10) „ 1 « 1 ^ 1A1 97 Here given by the well known formula: T “r Yj&'j&-LM(rlrl* ] ^ ( V , 1A1mJ il£lLM* Yw /(rj) (r^ 11 / 1 1 1 1 mimi (j&^m^j&jm^|j&^j£jLM) is the usual Clebsch-Gordan coefficient. • 4 " 4 * “ *♦ f j R (r,r. ,rnr0 ,uu ) can be expanded in the form: o 1 2 1 2 n __ v ^ V V ^ i ^ i V .. V v z ^ i v V = Z - r Z * 7~f — x A'A LM rlr2,rlr2 x Yje'ALM^riri^ x x Y£/ALM(r2r2) Substituting Eq. (B.10) and (B.12) into (B.2), one obtains: lrl> ArA1 ,nLM x Viui<;i;i> x ><y'YA/^LM(r2r2) PX*r2r2* Yje'^2LM(r2r2) dr2dr2 (B.ll) (B.12) The angular part can be integrated. One obtains, using the Wigner- Eckart theorem: / i t A . A v A » A A »A A » A YA'jem^r2r2^ F), 2 C2 > YA'^2LM^r2r2 2 2 " 6u. 6* CB-14) / it A A .A A • >A >A « A A » YA'2m( 'r2r2-1 2r2 Y«'l!2LM<r21 : 2> 2 r2 " 6L L 6* «>.15) Here the c and d coefficients can be expressed with the 3j and 6j X5 A A symbols. Multiplying Eq. (B.13) by LM^rlrP an<* inteSrat:i- n8 for r£ and r^, one obtains: 99 nLM . , . E E * X ^ i 2 f o ^rlr2r2rl^ * f r2dt2rZiri ~ U 7------ V V 2 > XS (r2r2) ri 1 1 ^ RA,,i,(rlr2r2rl) ' E /d t 2r22dr2 — 7 ^ r* r . A2A2 rlrl Vl/r2r2^ ( r ' r ' ) X d C A j A ^ ^ j L ) z 2 2> (B.16) This is the radial form of the GRPA. REFERENCES The optical model was first postulated in particle scattering by H. A. Bethe, Phys. Rev. 57_, 1125 (1940) and used by S. Fernbach, R. Serber and T. B. Taylor, Phys. Rev. 75, 1352 (1949). For a review, see A. L. Fetter and K. M. Watson in Advances in Theo retical Physics, edited by K. A. Brueckner (Academic, New York, 1965) Vol. 1, pp. 115-194. The first mathematical postulation of the optical model was given by N. C. Francis and K. M. Watson, Phys. Rev. 92_, 291 (1953). See also Fetter and Watson.'*' An important formulation of the optical model was given by H. Feshbach, Ann. Phys. (N.Y.) 5, 357 (1958), ibid. 19, 287 (1962). See also Fetter and Watson.'*’ J. S. Bell and E. J. Squires, Phys. Rev. Letters 3, 96 (1959) and M. Namiki, Progr. Theoret. Phys. (Kyoto) 23, 629 (1960) have given the many-body (or field theoretic) interpretation of the optical potential. This method was followed in the paper by Gy. Csanak, H. S. Taylor and R. Yaris, Phys. Rev. A3, 1322 (1971). See also Fetter and Watson.'*" The renormalized Green's function was first defined in field theory by J. Schwinger, Proc. Natl. Acad. Sci. U.S. 37_, 452 (1951), ibid. 37, 455 (1951) and used in many-body theory by P. C. Martin and J. Schwinger, Phys. Rev. 115, 1342 (1959). 101 6. B. Schneider, H. S. Taylor and R. Yaris, Phys. Rev. 1A, 855 (1970). 7. G. F. Chew and F. E. Low, Phys. Rev. 101, 1570 (1956). 8. H. Lehmann, K. Symanzik, and W. Zimmerman, Nuovo Cim. JL, 205 (1955). 9. G. A. Baraff and S. Borowitz, Phys. Rev. 121, 1704 (1961). 10. G. A. Baraff, Phys. Rev. 132, 2287 (1961). 11. L. J. Sham and W. Kohn, Phys. Rev. 145, 561 (1966). 12. For the definition and application of the Green's function, see any book on field theory; e.g., S. S. Schweber, An Introduction to Relativistic Quantum Field Theory, Row Peterson and Co., Illinois (1961). 13. For the application of Green's functions in many-body physics see e.g., A. A. Abrikosov, L. P. Gorkov, I. E. Dzyaloshinski, Methods of Quantum Field Theory in Statistical Physics, Prentice- Hall, New Jersey (1963); L. P. Kadanoff and G. Baym, Quantum Statistical Mechanics, Behjamin, New York (1962); A. L. Fetter and J. D. Walecka, Quantum Theory of Many-Particle Systems, McGraw-Hill, New York (1971). 14. For the expression of the scattering matrix, see M. Namiki, Progr. Theoret. Phys. j23, 629 (1960); T. Kato, T. Kobayashi and M. Namiki, Supplement of the Progr. Theoret. Phys. JJ5, 3 (1960); J. S. Bell and E. J. Squires, Phys. Rev. Letters _3, 96 (1959). 102 15. P. Roman, Advanced Quantum Theory, Addison-Wesley, Reading (1965), pp. 308-370. 16. A. Klein, Progr. Theoret. Phys. JL4, 580 (1956); A. Klein and C. Zemach, Phys. Rev. 108, 126 (1957). 17. P.-O. Lowdin, Phys. Rev. 97, 1474 (1955). 18. W. P. Reinhardt and J. D. Doll, J. Chem. Phys. 5£, 2769 (1969). 19. 0. Goscinski and P. Lindner, J. Math. Phys. _11, 1313 (1970). 20. P.-O. Lowdin, Advan. Phys. 5^, 1 (1956). 21. B. Schneider, Phys. Rev. A2, 1873 (1970). 22. N. F. Mott and H. S. W. Massey, The Theory of Atomic Collisions, Clarendon, Oxford (1965). 23. A. J. Layzer, Phys. Rev. 129, 897 (1963). 24. P. M. Morse and H. Feshbach, Methods of Theoretical Physics, McGraw-Hill, New York (1953), Part I, pp. 884-886. 25. J. H. Wilkinson, The Algebraic Eigenvalue Problem, Clarendon, Oxford (1965). 26. M. L. Goldberger and K. M. Watson, Collision Theory, Wiley, New York (1964). 27. H. P. Kelly, in Advances in Theoretical Physics, edited by K. A. Brueckner (Academic, New York, 1968), Vol. 2, pp. 75-169. 28. W. P. Reinhardt and A. Szabo, Phys. Rev. Al, 1162 (1970). 29. G. Baym and L. P. Kadanoff, Phys. Rev. 124, 287 (1961). 30. A. Dalgarno and G. A. Victor, Proc. Roy. Soc. A291, 291 (1966); 103 M. J. Jamieson, Ph.D. Thesis, The Queen's University of Belfast, Belfast (1969). 31. T. H. Dunning and V. McKoy, J. Chem. Phys. 47, 1735 (1967). 32. H. Gutfreund and W„ A. Little, Phys. Rev. 183, 68 (1969); D. J. Rowe, Rev. Mod. Phys. 40, 153 (1968). 33. K. Dietrich and K. Hara, Nucl. Phys. Alll, 392 (1968); R. H. Lemmer and M. Veneroni, Phys. Rev. 170, 883 (1968). 34. P. Burke and P. P. McVicar, Proc. Phys. Soc. 86, 989 (1965). 35. For the Thomas-Fermi theory see e.g., P. Gombas, Die Statistische Theorie des Atoms und ihre Anwendungen, Springer, Wien (1949). 36. W. Henneberg, Zs. f. Phys. J53, 555 (1933); Naturwiss 2£, 561 (1932). See also Ref. 35, p. 252. 37. Massey and Mohr, Nature 130, 276 (1932). See also Ref. 22, p. 578. 38. Bullard and Massey, Proc. Camb. Phil. Soc. 26, 556 (1930). See also Ref. 22, p. 460. 39. For the importance of polarization effects, see Ref. 22, p. 575. 40. P. A. M. Dirac, Proc. Cambridge Phil. Soc. 26, 376 (1930). 41. I. Fenyes, Zeit. f. Physik, 125, 336 (1944). 42. W. R. Theis, Zeit. f. Physik, 142, 503 (1955). 43. E. Wigner, Phys. Rev. 40, 749 (1932). 44. A. S. Kompaneets and E. S. Pavlovskii, Soviet Phys. JETP 31(4), 328 (1957). 104 45. C. F. Weizsacker, Zeit. f. Physik, 9<3, 431 (1934). 46. J. Hubbard, Proc. Roy. Soc. A240, 539 (1957); ibid. A243, 336 (1958). 47. L. Hedin, Phys. Rev. 139, A796 (1965). 48. M. Gell-Mann and K. Brueckner, Phys. Rev. 106, 364 (1957). 49. K. S. Viswanathan and B. N. Narahari Achar, Can. J. Phys. 42, 2332 (1964). 50. P. Venkatarangan, Can. J. Phys. 43, 1157 (1965). 51. T. Tietz and S. Krzeminski, Acta. Phys. Hung. 2J_, 161 (1969). 52. J. Lindhard, Dan. Math. Phys. Medd. 28, No. 8 (1954). 53. B. I. Lundqvist, Physik Kond. Materie 1 _, 117 (1968). 54. J. J. Quinn and R. A. Ferrel, Phys. Rev. 112, 812 (1958).

Linked assets

University of Southern California Dissertations and Theses

Conceptually similar

PDF

Application of many-body Green's function formalism to the elastic scattering and bound state properties of helium atom

PDF

Relaxation Oscillations In Stimulated Raman-Scattering And Brillouin-Scattering

PDF

Brillouin Scattering In Molecular Crystals: Sym-Trichlorobenzene

PDF

A Tamm-Dancoff Approximation Applied To The Green'S Function Formalism For The Electron Gas

PDF

Raman-Scattering In Superfluid Helium

PDF

Electron-Scattering From The Singly Negative Atomic Hydrogen Ion And Helium Atom

PDF

Contributions To The Theory Of Raman-Scattering

PDF

Electric field induced spectra of carbon disulfide and its dipole moment in an excited state

PDF

Kramers-Kronig Dispersion Analysis Of Infrared Reflectance Bands And The Optical Properties Of Sodium-Chlorate

PDF

Numerical Solution Of The Dirac Equation By The Method Of Characteristics

PDF

The Angular Momentum Formalism In Classical Mechanics And The Classical Point-Particle

PDF

Raman scattering from localized vibrational modes in Gallium Phosphide and dielectric parameterization of Raman lineshapes of plasmon-phonon coupled modes

PDF

Structural Studies Of Selected Boron-Carbon Compounds

PDF

Utility Analysis And Laser Interactions In Gases

PDF

The Infrared Spectrum Of The Azide Ion In Potassium Azide And The Potassium Halides

PDF

Application Of The Method Of Stationary Phase To Potential Scattering Theory

PDF

The Calculation Of Resonant States Of Atoms And Molecules By Quasi-Stationary Methods

PDF

Study Of The Local Mode Of Calcium-Fluoride Doped With Negative Hydrogen Ion With Intense Carbon-Dioxide Laser Lines

PDF

Laser Saturable Resonators And Criteria For Their Bistable Operation

PDF

A Comparison Of The Physical Properties And Biological Activities Of Somecrystalline Phases Of Fluprednisolone

Asset Metadata

Creator Csanak, Gyorgy (author)

Core Title Green'S Function Technique In Atomic And Molecular Physics To Special Consideration Of Electron Scattering Problems In The Generalized Random Phase Approximation (Grpa)

Contributor Digitized by ProQuest (provenance)

Degree Doctor of Philosophy

Degree Program Chemistry

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag chemistry (physical chemistry),OAI-PMH Harvest

Language English

Advisor Taylor, Howard S. (committee chair), Marburger, John H. (committee member), Segal, Gerald A. (committee member), Thompson, Richard S. (committee member)

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c18-548570

Unique identifier UC11362301

Identifier 7206047.pdf (filename),usctheses-c18-548570 (legacy record id)

Legacy Identifier 7206047.pdf

Dmrecord 548570

Document Type Dissertation

Rights Csanak, Gyorgy

Type texts

Source University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

Access Conditions The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the au...

Repository Name University of Southern California Digital Library

Repository Location USC Digital Library, University of Southern California, University Park Campus, Los Angeles, California 90089, USA