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Accurate calculations of bound- and quasibound-state energies of some three -body systems, and cross sections for photodecay

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Accurate calculations of bound- and quasibound-state energies of some three -body systems, and cross sections for photodecay

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Content NOTE TO USERS This reproduction is the best copy available. ® UMI Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ACCURATE CALCULATIONS OF BOUND- AND QUASIBOUND-STATE ENERGIES OF SOME THREE-BODY SYSTEMS, AND CROSS SECTIONS FOR PHOTODECAY by Tieniu Li 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 (PHYSICS) May 2005 Copyright 2005 Tieniu Li Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. UMI Number: 3180479 INFORMATION TO USERS The quality of this reproduction is dependent upon the quality of the copy submitted. Broken or indistinct print, colored or poor quality illustrations and photographs, print bleed-through, substandard margins, and improper alignment can adversely affect reproduction. In the unlikely event that the author did not send a complete manuscript and there are missing pages, these will be noted. Also, if unauthorized copyright material had to be removed, a note will indicate the deletion. ® UMI UMI Microform 3180479 Copyright 2005 by ProQuest Information and Learning Company. All rights reserved. This microform edition is protected against unauthorized copying under Title 17, United States Code. ProQuest Information and Learning Company 300 North Zeeb Road P.O. Box 1346 Ann Arbor, Ml 48106-1346 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. D ed ica tio n This dissertation is dedicated to my wife, Lan Hao, my son, Zheyu (Joey) Li and my family. ii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A ck n ow led gem en ts I would like to thank my advisor, Professor Robin Shakeshaft, for his guidance and advice through the years of my study at University of the Southern California. He is the greatest m entor for my research and life. I want to thank the faculty of the D epartm ent of Physics and Astronomy at University of Southern California, especially Professor Maxim OP Shanii, Professor V italy Kresin, Professor Stephan Haas and Professor Joseph Kune of the D epartm ent of Aerospace Engineering, for being on my guidance and dissertation committee. I also w ant to thank the staffs in the D epartm ent of Physics and Astronomy especially Betty, Lisa and M ary Beth, for their supports to my study. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C on ten ts D ed ica tio n ii A ck n ow led gem en ts iii L ist O f Tables v List O f F igu res vii A b stract ix 1 In trod u ction 1 2 T h e background th e o ry o f th e tw o-electron sy stem s 12 2.1 Atomic H am iltonian in inter-particle c o o rd in a te s ............................................................ 12 2.2 The interaction between Atoms and the field ................................................................... 17 2.3 Ionization R ate and Branching R a tio .................................................................................... 20 2.4 Projection onto open channels................................................................................................. 23 2.5 Perim etric c o o rd in a te s............................................................................................................... 28 2.6 Random P erturbation M ethod .............................................................................................. 30 3 R esu lts and d iscu ssion 34 3.1 The ground-state energy of H e liu m ....................................................................................... 34 3.2 the ground-state energy of H _ and Ps~ ............................................................................. 37 3.3 One photon ionization rate ..................................................................................................... 37 3.3.1 the case of H e liu m ......................................................................................................... 37 3.3.2 the case of H~ ............................................................................................................... 41 3.4 two photon ionization rate of H e liu m .................................................................................... 43 3.5 One photon ionization branch r a t e ......................................................................................... 50 3.5.1 Direct implem entation w ith complex b a s is ............................................................. 50 3.5.2 Projection with real b a s i s ........................................................................................... 53 3.6 Electron distributions of H ~ , He and P s ~ ......................................................................... 57 3.7 The Bound states vs. Z re la tio n s h ip .................................................................................... 67 4 A u toion ization reson an ces o f th e p ositron iu m n egative ion 77 4.1 Complex-Rotation M ethod ..................................................................................................... 77 4.2 Hamiltonian of Ps~ w ith L = 0 .............................................................................................. 81 4.3 Result and D is c u s s io n .............................................................................................................. 82 B ibliograp hy 91 iv Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. List O f Tables 2.1 The eigenvalues of overlap m a tric e s......................................................................................... 31 3.1 The ground state of Helium com puted by the basses w ith k\ = 2.3a( 7 1 and k - 2 = 1.9a7x 35 3.2 The ground-state energies of H~ and P s ~ ........................................................................... 37 3.3 The total one-photon ionization cross section of Helium below double ionization threshold ....................................................................................................................................... 39 3.4 The total ionization cross section of Helium above double ionization threshold . . 40 3.5 Helium 12 and 13 Autoionization Resonances....................................................................... 44 3.6 H” ionization rate below the shape re so n a n c e .................................................................... 44 3.7 Helium energies of I 1# , 21P and 3lD s t a t e s ........................................................................ 47 3.8 Oscillator strengths of Helium: l x5 — > 2 'P and 2l P — > 3l D ................................... 48 3.9 Helium two-photon ionization rates below the ionization threshold................................ 48 3.10 Different path contributions of two-photon ionization rate of Helium below the ionization threshold ................................................................................................................... 50 3.11 The n = 1 eigenenergies of a Z = 2 etherial system with a complex b a s is ............... 53 3.12 The n = 2 eigenenergies of a Z = 2 etherial system with a complex b a s is ............... 54 3.13 The example of the etherial eigenenergies (n = 1, Z = 1) w ith a real b a s i s ............ 55 3.14 The symmetric components of 15 wavefunction of the Z = 1 etherial system in the mixed b a s i s .................................................................................................................................... 56 3.15 The example of eigenenergies of P H P .................................................................................. 56 3.16 The example of eigenenergies of Q HQ in the range of the shape resonance E r = — 0.124a.«. and T = 0.0013a.u.................................................................................................... 57 v Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 3.17 The critical value of Z where the bound state starts to d is a p p e a r................................ 68 4.1 The 1S doubly excited states of P s- with one-length scale m e t h o d ............................ 83 4.2 The 3S doubly excited states of Ps~ with one-length scale m e t h o d ............................ 84 4.3 The 1Sf resonances of P s- : a comparison between two different m e th o d s .................. 85 4.4 The 3S resonances of P s- : a comparison between two different m e th o d s .................. 85 4.5 Convergence of the 1S doubly excited states of P s- with one-length scale m ethod . 88 4.6 Convergence of the x5 doubly excited states of P s- with two-length scales m ethod 89 4.7 Convergence of the 3S doubly excited states of P s- with one-length scale m ethod . 92 4.8 Convergence of the 3S doubly excited states of P s- with two-length scales m ethod 93 4.9 An illustration of the sensitivity of eigenenergies of the rotated H am iltonian H (rel6) of P s- to the value of the angle 6............................................................................................. 94 4.10 The rotation angle dependence of the 1S and 3S doubly excited states of P s- with one-length scale m ethod............................................................................................................... 95 4.11 The rotation angle dependence of the l S and ' S doubly excited states of P s- with two-length scale m ethod............................................................................................................... 96 vi Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. L ist O f F igu res 2.1 Coordinates of 2-electron s y s t e m ............................................................................................. 13 3.1 The total ionization cross section of He close to N = 2 lim it............................................ 41 3.2 The total ionization cross section of He close to IV = 3 lim it............................................ 42 3.3 A doubly excited Rydberg Resonance of Helium at E = 71.235eP ................................ 43 3.4 H~ photoionization cross section below 2 1 P shape resonance ...................................... 45 3.5 H~ photoionization cross section in shape r e s o n a n c e ....................................................... 46 3.6 The diagrams of two photon absorption pro cesses.............................................................. 47 3.7 Two-photon ionization rate of Helium below the ionization th re s h o ld ......................... 49 3.8 Ionization process of H eliu m ...................................................................................................... 51 3.9 P artial cross sections for photodetachm ent of H“ in the region of the shape reso nance E r = — 0.124 a.u................................................................................................................... 58 3.10 Total cross section for the photodetachm ent of H _ in the region of a shape resonance 59 3.11 The density of the probability of finding one electron at r i ............................................ 62 3.12 The density of the probability of finding one electron at r\ compared to the distri bution of the electron of Hydrogen-like atom 63 3.13 the contour plot of \^> \2dV of H e ................................................................. 64 3.14 the contour plot of \^\2dV of H ~ ................................................................. 65 3.15 the contour plot of |W|2dV of P s" .............................................................. 66 3.16 E (Z ) on the complex energy plane of H ~ — like system: Z = 0.91 and Z = 0.915 . 69 3.17 E (Z ) of H “ -like systems on the complex— E p l a n e .......................................................... 71 vii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 3.18 T h e low est e ig en en erg y o f H — like sy ste m w ith ato m ic n u m b e r Z o n th e co m p lex energy plane..................................................................................................................................... 72 3.19 The lowest eigenenergy of Ps~ — like system with atomic number Z on the complex energy plane..................................................................................................................................... 73 3.20 The ground-state probability distribution when one electron is fixed at its peak position. The left top one corresponds to Z — 0.92; the left bottom one corresponds to H “ ;the right top one corresponds to He; and the right bottom one corresponds to Li+ w ith Z = 3. The horizontal axis is scaled to r — ► Z r .......................................... 74 3.21 The ground-state probability distribution when one electron is fixed at its peak position. The top one corresponds to Z = 0.92 and the bottom one corresponds to H “ ....................................................................................................................................................... 75 3.22 The ground-state probability distribution when one electron is fixed at its peak position. The top one corresponds to He and the bottom one corresponds to Li+ with Z = 3........................................................................................................................................ 76 4.1 Spectrum transform under complex coordinate ratation ................................................ 80 4.2 P s" autoionization resonances ( 2s2s and 2.s.3.s ) on complex energy plane................. 90 viii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A b stract In this dissertation I use the perim etric coordinates and Sturm ian-type basis, with two scale pa rameters, to study the three-body systems, especially H - , P s- and He. I propose two m ethods to deal with the numerical linear dependence which appears as the size of the basis is increased. I study the one- and two-photon ionization rates for H~ and He. I also study the S wave autoion ization resonances of P s- and report very accurate results. I present a new m ethod to calculate the branching rate of H where a projection operator is used. The projection operator is constructed through a real basis. I also study the properties of the electron distribution in the ground states of three-body systems. The relationship between the existence of a bound state and the nuclear charge Z is presented too. My results for the critical value of Z for bindings agree with the results, where available, of other methods. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C h apter 1 In trod u ction It is well known th at the three-body system has no analytic solution in classical mechanics. As its counterpart in quantum mechanics, the three-body Coulomb system, such as He, H _ and P s", also has no closed analytic form since its Hamiltonian, H ( n , f 2, f 3) = V? - — V* - — V? - + p - ^ 7 , (1-1) m i m 2 m 3 |r x — r 3| |r2 - r 3| \ri - r2\ is not separable due to the presence of the repulsive interaction of two electrons. The Ritz variational m ethod and perturbation schemes have to be used to study the quantum three-body problem. He, H~ and Ps~ are the simplest three-body systems. And the research on them is not only of purely methodological interest; while such research provides an im portant test for the various approximation methods, it is also of practical interest — for example H _ is of great im portance for the opacity of the Solar atmosphere. Historically an accurate treatm ent of He and H _ has been of great interest since the birth of quantum mechanics. It is natural to look for a quantum treatm ent for He after the success of the explanation of the spectrum of H in the frame of ’ ’old quantum mechanics” . Unfortunately the ’ ’old quantum theory” failed completely on such a ’ ’simple” system— for sure we now know it is not simple at all. 1 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. By using a product of Hydrogen wavefunctions of rq and r 2 with reduced atom ic num ber Z , which is 'J' = e~(z ~a' >rie~t 'Z~C T ' ,r2, the Ritz variational m ethod yields the following well-known result for the energy of the ground-state (Bethe and Salpeter, 1957, pp:146-148): Eground = - { Z - ^ ) 2- (1-2) The value of E g = — 2.8477a.u. for He, obtained by putting Z — 2 in Eq. ( 1.2 ), is not bad comparing to the more accurate value of E g = — 2.9035a.u.. However, putting Z = 1 in Eq. ( 1.2 ) implies th a t II- dissociates into H + e~ . The value of E g = — ,473a.u. of H _ gives a negative ionization potential J = — Eg — Z 2/2 where —Z 2/ 2 is the ground-state energy of H. This means the simple independent particle approxim ation has no bound state for H ~. Now it is well known—from experiments (NIST-Database) and theoretical studies (Drake, 1988; Frolov and Yeremin, 1989) — th a t H~ has one and only one bound state w ith E g = — 0.5277510 • ■ •. It is interesting th a t when Chanderasekhar (1944) investigated a trial function of the form $ = (e- « n - 6r2 + + ^ ^ 3) where r 3 = |Fi — r 2| is the distance between the two electrons, he found the variational m ethod gave a value J = 0.0135a.u. even when the r \2 term was om itted w ith c = 0, a = 1.039 and b = 0.283. The corresponding ground-state energy Eg = — 0.5135 is poor but it indeed establishes a bound state and suggests th a t one of the two electrons of H~ is almost in the ground-state of the H atom and another electron is very slightly tied to the H atom, roughly ’ ’located” at 1/6 « 3.5a0—here ao is the Bohr radius. Chanderasekhar (1944) found a value for the ground- state energy of Eg = - 0 .52509c/,.tt. with a = 1.075, 6 = 0.478 and c = 0.312 which is better th an the 3 param eters Hylleraas (1928) value for H _ . On the other side, the simple independent Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. particle approxim ation gives a reasonable ground-state energy of He. This fact means both of the two electrons of helium are tightly bounded to the nucleus. The earliest wave mechanical calculation on the He atom was carried by Kellner (1927). Then Hylleraas published a series of papers (Hylleraas, 1928, 1929) which used two elliptic coordinates plus the interparticle distance s = ri + n 2, t = T\ - r2, u = r-i for the two electron fixed nucleus problem—the He - like system, not include P s- whose "nuclear m ass” , the mass of e+ , is the same as of the electron. These coordinates are referred as Hylleraas coordinates. The trial functions are of the form <t>{s,t,u) = e~ts ^ 2 cnai,mSnt2lum , (1.5) l,m,n which is widely used and referred to as a Hylleraas basis for non-relativistic quantum three- body problems. The inclusion of term s in u efficiently incorporates the radial correlation of the two electrons. W ith the Hylleraas basis, Hylleraas (1928, 1929) got E g = — 2.9024a.u. w ith 3 param eters and got E g = -2.90324a.'u. with 6 parameters. Using a Hylleraas function w ith 18 param eters, Chanderaskhar et al. (1953) got E g = — 2.903715a.u.. K inoshita (1957) used a 38 param eter modified Hylleraas function, which included negative power of s, and got a value of E g = — 2.903723a.it.. W ith a 24 param eter Hylleraas function, Hylleraas and M idtdal (1956) obtained a value of E g = — 0.527725a.u. for H~. It is amazing th a t the Hylleraas-type basis led to very good results for the ground-state energy of He and H~ with the limited com puting power in 1950s. 3 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. In his pioneering work, Pekeris (1958) used the perim etric coordinates u = r 2 + r 12 - r 1, v = n + r i 2 - r 2, w = 2(r1 + r 2 - r 3) (1.6) and introduced a Sturm ian-type basis 4>{u, v , w) = e - * 1(n+t,+u,) Li(ku)Lm(kv)Ln(kw) (1.7) l,m,n into the three-body problem. Note th a t the perim etric coordinates were first used by Coolidge and Jam es (1937). Pekeris (1958) got E g = — 0.5277509744a.u. for the ground-state energy of H _ . And w ith a basis w ith 1078 term s of ( 1.7 ) Pekeris (1959) improved the ground-state energy of He to Eg = — 2.903724375a.w. The correct form of the trial function is one key to the success of a variational method. Intuitively it would seem to be im portant to choose a trial wave function which includes, as much as possible, the correct singularities, when the exact closed form solution is unavailable. B artlett (1937) pointed out th a t the exact solution of the Schrodinger equation for two-electron atoms includes term s in R ln R where R = \J r\ + r\. Fock (1954) used four-dimensional spherical coordinates to study the analytic form of the wave function of the He atom. Fock suggested the exact form for two-electron systems is [n-l] 9 = J 2 Rn^ E 0 ° S R )k^nk- (i-8) »= l,3/2,--- k=0 It is evident th a t there is a weak logarithmic singularity in the wavefunctions of a non-relativistic quantum three-body system. Following Fock’ s work, Schwartz (1956,1960) introduced a basis with 4 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. fractional powers. Using an 11 param eter fractional-power function, he found Eg = — 2.903704a.u., a result which differs by only 0.0007% from K inoshita (1959)’s 80 param eters solution E g = — 2.90372437a.u. Then Frankowski and Pekeris (1966) introduced the logarithmic term log(s) into the Hylleraas-type basis. They got the ground-state energy Eg = — 2.9037243770326a.n. for He and Eg = — 0.52775101635a.u. for H~ w ith 246 term s. The latest theoretical calculations follow these types of basis functions and there is a never ending search for more accurate ground-state energies for He and H~ (Freund et al., 1984; Frolov and Yeremin, 1989; Goldman, 1998; Korobov, 2000; Schwartz, 2004; Thakkar and Koga, 1994). W ith a high precision computing package, Schwartz (2004) reported the latest and the most accurate ground-state energy of He as Eg = — 2.90372437703411959831115924519440444a.«. (1.9) Incidentally Schwartz (2004) investigated the original Hylleraas basis, the negative powers basis, the fractional powers basis and the logarithmic basis. The bizarre outcome is th a t the original Hylleraas basis gave the above ground-state energy instead of those basses taking advantage of the true analytic structure ( singularities ) of the wave function. The P s- system is an extreme example of the three-body Coulomb system. Unlike other H e —like systems, all kinetic energies of the three particles have to be included in the Hamiltonian. The earliest calculations of the ground-state energy can be traced back to Wheeler (1946) and Hylleraas (1947). A binding energy E = 0.326eU was estim ated in those early days. O ther early calculations include the works of Cavaliere et al. (1975); Frost et al. (1964); Kolos et al. (1960). The latest calculations include (Bhatia and Ho, 1983; Frolov, 1993; Frolov and Yeremin, 1989; Ho, 1990, 1993). Ho (1993) reported a value Eg = — .52401014046571 Ry. for the ground-state energy of P s- . The doubly excited states of a He — like system, where neither of the two electrons is in the ground state, are above the single ionization threshold. They are embedded in the continuum 5 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. states of the H e+ ions or the H — like atoms. Therefore those doubly excited states are quasi bound states and will decay through em itting a photon or through dissociating to e~ + He~ ( e~ + H for H “ and e~ + Ps for P s” ). The latter decay m ethod is referred as autoionization. The earlier experimental and theoretical works on He and H” can be found in the reviews by Golden (1978) and Burke and Schey (1962). There are many m ethods which can be used to study the position and width of those autoionization resonance states (Bransden, 1983, chapter 7). The complex coordinate rotation m ethod, which now is the standard m ethod for studying the autoionization resonances—the quasi-bound states, was introduced by Balslev and Combes (1971); Simon (1973). The autoionization states of He, H” and P s” were extensively studied with the complex coordinate rotation m ethod ( Bachau (1984); Domke et al. (1996); Drake (1988); Ho (1982, 1991) for He; Drake (1988); Scrinzi and Piraux (1998) for H ” ; and Ho (1979, 1984, 1991, 1993); Ivanov and Ho (1999); Papp et al. (2002) for P s” ). The ionization rates of He and H ” have been well studied over the past decades— Chang and Fang (1995); Sanchez and M artn (1991) for the one-photon ionization of He; (Pont and Shakeshaft, 1995a) for the two-photon ionization below the ionization threshold of He; Ajm era and Chung (1975); Chang and Tang (1991) for one-photon ionization of H ” . The agreement between the length gauge and the velocity gauge is good but not perfect, especially for the ionization rate of H “ , among these works. In the early works, the discrepancy was about 20 — 30%. Chang and Tang (1991) reported a 1 — 2% discrepancy. This discrepancy should be eliminated if the calculation is accurate enough. The coordinates r i, r 2 and r$ = are referred to as interparticle coordinates. The most accurate three-body system calculations are based on the interparticle coordinates or a linear combination of them such as Hylleraas coordinates. The singularities of the Coulomb potentials appear a t r, = 0 and the basis functions can be chosen to well describe the singularities in the wavefunction (Frankowski and Pekeris, 1966; Schwartz, 2004, 1960). B ut the algebra involved in calculations with the interparticle coordinates is tedious for large angular momentum. The range 6 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. of r 3 is dependent on r\ and r 2— \r\ — r2\ < r 3 < r\ + r 2. Fortunately we can circumvent this difficulty w ith the perim etric coordinates ( 1.6 ). All u , v and w have the same range [0, oo). Here we use a Sturm ian-type basis, which follows the Pekeris (1958) scheme, and perim etric coordinates to study the three-body systems. The radial part of our basis functions are of the form <t>(u,v,w) = e~^(klU+k2V+kiW'> ^ 2 Li(kiu)Lm(k2v)Ln(k3w) + u + -> v. (1-10) l,m,n W ith the power of symbolic calculation of Mathematica® we are able to reduce all integrals needed in our calculation to manageable algebraic expressions even for the different length scales k\, k ,2 and k% . It is impossible to derive by hand the formulae to calculate the elements of the Ham iltonian of the three-body system—those formulae turn out to be too long, each more than 600 lines in Fortran code. There is an intrinsic disadvantage to th e basis functions of Eq. ( 1.10 ). W hen the electrons are far from each other, the correlation l / r 3 is small and the wave function of the excited states of the three-body system should be asym ptotically independent of r3. A Sturm ian-type basis such as ( 1.10 ) always incorporates r 3 in the term s of u and v. The asym ptotic independence of r 3 of the wavefunction relies on the cancelation of the coefficients of the basis functions, which seems wasteful and, moreover, may lead to round off error. By using two length scales, with fc3 equal to (ki + k2 ) / 2 to preserve the exponential decay of the wave function r\ and r 2 in the form e~k2ri~klT2, and by increasing the basis size, we can remedy the intrinsic disadvantage in our method. In C hap ter 2, we present a treatm ent for the two-electron fixed nucleus system, which is based on the decomposition the Hamiltonian by Euler-angles (B hatia and Temkin, 1964; Pont and Shakeshaft, 1995b). We discuss how to decompose the Hamiltonian of the three-body system w ith the Euler-angles and integrate the elements of the Hamiltonian and the overlap m atrices over the Euler-angles for L = 0,1 ,2 in S ection 2.1. Note th at the basis functions of ( 1.10 ) are not orthogonal. The dipole interaction between the bare atoms and the field is discussed in S ection 7 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2.2. The dipole interaction is restricted by the triangle rules for addition of angular m omentum. The total angular momentum can only differ by unity. Then the m atrix representation of the dipole operator is in co-diagonal blocks labeled by L' = L ± 1. S ec tio n 2.3 briefly introduces the formula used to calculate the one- and two-photon ionization rates. It also introduces a m ethod using a projection operator, th a t projects onto the subchannels of the corresponding etherial system, to calculate the partial rate for break up the three-body system. In S e c tio n 2.4 we explain the etherial system and the projection onto the subchannels of the etherial system. We also present a way to construct the projection operator which is used to calculate the partial ionization rate of H~ in S e c tio n 3.5.2. S e c tio n 2.5 discusses the details of th e perim etric coordinates and the Sturm ian-type basis th a t is used in our work. There should be no linear dependence when we choose the two length scales k\ and k2 properly—for example, when we choose them to be different irrational numbers. B ut in practical calculations, a numeric linear dependence arises when the size of the basis is very big and the difference between k\ and k2 is not large enough. The overlap m atrix S, whose elements are < 'Ihl'kj >, should be positive definite, but due to the linear dependence it can have zero eigenvalues, sometimes even negative eigenvalues. This introduces a difficulty to solve the generalized eigenvalue problem—H _\< pE > = ES\4>e >• A random perturbation to the overlap m atrix is introduced in S ectio n 2.6. We use the eigenvectors of the perturbed overlap m atrix as the basis functions. Then the new overlap m atrix is not singular any more. Accurate ground-state energies of the three-body system are obtained by using the Hylleraas coordinates or the perim etric coordinates. The Hylleraas functions and many modifications such as the inclusion of negative powers, fractional powers and logarithmic term s—are widely used. Schwartz (2004)’s investigation shows th a t the original Hylleraas basis, which is not tailored to the analytical singularities of the exact wavefunction, is the best to get the most accurate digits of the ground-state energy of He. This intrigued us to investigate the power of the Sturm ian-type basis in calculating the ground-state energies of three-body systems. Our results are reported in 8 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Section 3.1 and 3.2. O ur ground-states energies agree with the best variational results to at least the 10th digit. B ut our ground-state energies are still a little bit above those lowest values, which are the upper lim its of the true non-relativistic ground-state energies of the three-body systems He, H~ and P s- . The incorporation of two length scales does not help a lot especially for He. We only improve two more digits to Eg = — 2.9037243770a.u. compared to the value Eg = — 2.903724375a.u. of Pekeris (1958). O ur results of H~ and Ps~ are closer to the best known results. The one-photon ionization rates of He and H~ are extensively studied. We present our results of He in S ection 3.3.1 and the results of H“ in S ectio n 3.3.2. The total ionization rate of He is calculated by using one set of k\ and & 2 for the entire photon energy range from the ionization threshold to 300e V . The autoionization resonance states up to n = 3 threshold are identified on the ionization rate curve of He. There we also present the position and w idth of these resonances obtained by using the complex coordinate rotation method. Our results agree pretty well to the experimental values and other theoretical calculations. We calculated the total ionization rate of H _ . The agreement between the length gauge and the velocity gauge is about 0.1% or better over the entire energy range. It is better than any known preexisting calculations. The two-photon ionization rate below the threshold from the ground-state of He is presented in S ection 3.4. W hen the photon energy is lower than the threshold, there are two components in the two-photon ionization rate. One is the process S — > P — > S and another one is the process S — > P — » D. We present the contributions of these two components too. The contribution of S — ► P — > D is greater than the contribution of S — » P — > 5 outside the resonances of l P states. THis is expected since the photon absorption process prefers the high angular momentum configurations. We proposed a method, which uses the projection operators which project onto the subchannel of the etherial system, to study the partial ionization rates. In S ection 3.5.1 we discuss our efforts on a direct implem entation w ith complex basis which failed to calculate the correct partial rates. 9 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. W e also discuss the possible reason th a t leads to the failure. In S e c tio n 3.5.2 w e present the partial rates of H~ in the shape resonance E r = — 0.124a.u.. We introduced another m ethod to use two length scales w ithout the trouble of numerical linear dependence in this section. Taking advantage th a t the resolvent G (E ) = 1/{E — H ) can be represented by a real basis, we construct the projection operator on a real basis instead of a complex basis. O ur results agree fairly with the result of Sadeghpour et al. (1992). It is believed th a t one of the two electrons of H~ and P s- is only slightly bound to the core— the H atoms or the e+ e- systems. We investigate this phenom ena by calculating the probability distribution of one electron located at r i. The figures are plotted in Section 3.6. The figures confirm the idea. Only the electrons of He are tightly bound to the nucleus. H- has a much wider peak compared to the electron distribution of He. And the distribution of P s- has a very broad peak. It has been proved th a t IP has only one bound state (Nyden-Hill, 1977). It is also true for P s". In S ection 3 .7 we investigate an interesting question: when does the bound state of H ~ — like ion disappear as the atomic num ber Z is reduced? It turns out th at the bound state will disappear when Z < 0.9110 for H - and when Z < 0.9218 for P s- . This result also tells us th a t it is more difficult to form a negative ion as the nuclear mass becomes small. Ho (1979, 1984, 1991, 1993); Ivanov and Ho (1999) extensively studied the autoionization resonances of P s- . P app et al. (2002) reported their positions and widthes of the and 3S resonances of P s- . There is a discrepancy between the resonance widths of Papp et al. (2002) and Ho (1984). We investigate this discrepancy in C h ap ter 4. We briefly summarize the m ajor results of the complex coordinate rotation m ethod in S ectio n 4.1. In S ectio n 4.2 we introduce the Hamiltonian of the S states of Positronium negative ions. Compared to the Hamiltonians of H - and He, more mixed differential term s appears since we have to include the kinetic energy of the positron. Our results of the 1S and 3 S' autoionization resonances of P s- are presented in S ection 4.3. Most of our widths of the resonances agree with P app et al. (2002) except for two 10 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. resonances. But all values of the widths of the resonances are closer to the values of P app et al. (2002) than the values of Ho (1984). 11 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C h ap ter 2 T h e background th eory o f th e tw o-electron sy stem s 2.1 A tom ic H am iltonian in inter-p article coord in ates The states of a quantum system define a H ilbert space, usually w ith infinite dimensions. Using Heisenberg’s m atrix representation, the Hamiltonian (H ) of a quantum system can be expressed by an infinite dimensional m atrix when a complete basis of the Hilbert space is used. For practical calculations, we always use a finite b a s is (^ , i = 1,2, ...,n ) in place of th a t infinite basis. This perm its the Ham iltonian to be approxim ated by a finite m atrix H_ is completely feasible. The reliability of this approxim ation can be tested by increasing the basis size and observing whether there is any change in the system ’s behavior. Then we can study the behavior of the system under a finite basis. The system ’ s eigenstates can be obtained by solving the generalized eigenvalue problem of H~x = E S lc , (2.1) where = < < j> i\H \< j> j > and S_i;j =< 4> i\4> j >■ The Green function G(E) = 1 /{ E — H ) can be approxim ated by G(E) = 1 /(ES_ - H_). We can apply this approximation to the two-electron system. W hen we use atomic coordinates (see figure 2.1), denoting r i and r 2 as the displacements of the electrons relative to the nuclei and 12 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 2.1: Coordinates of 2-electron system r 3 = r i — r 2 as the relative displacement of the two electrons, the Hamiltonian of this 2-electron system is: t f = _ I ( V ? + V l ) - - - - + — (2.2) 2 t \ r-i r3 v ' where Z is the nuclear charge and r 3 is dependent on r j and r2. Choosing the z axis as the polar axis, we can introduce two variables as: Ci = sin(0i)e1 '^1, (2.3) C 2 = sm(d2)el't‘2. (2.4) Recall th a t the angular momentum operators for a single particle are: L - i ^ , (2.5) ^ = +COt^ d e + sin2(< ? ) d4> 2^’ ^ i± = e±i^ ± - ^ + i c o t ( 9 ) - ^ ) . (2.7) 13 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Clearly is the eigenstate of lz w ith eigenvalue lz and acting on (} yields 0. Since 1 2 can be w ritten as tt = U++% + iz, (2.8) is also an eigenstate of I2 w ith eigenvalue 1(1+1). The total angular momenta of the two-electron system can be defined as : L = li T l2. (2-9) The angular m om enta lowering and arising operators are L ± = l ± , i + l±,2- ( 2-10) Using the properties of the single particle angular momentum operators, we have L+fCi1^ 2] = 0 and L z [ Ci1 C 22] = (h + Those facts give us th a t C 1 1 C 22 * s eigenfunction of L2 with eigenvalue L (L + 1) and the eigenfunction of L z w ith eigenvalue L = l\ +l2. The wavefunction 'I*LM of a 2-electron system w ith total angular momentum L(L + 1) and azimuthal angular momentum M = L can be expressed as V lm (n , r 2) = (r iCi)U(r2<2)'2 x /( t( 2( n , r 2, r 3), (2.11) ll+12— L where the functions l2(ri,r2,r 3) will be determ ined through the coupled differential equations of the radial part of the Hamiltonian of the system. By applying the lowering operator L_ to (2.11) we can get the eigenfunctions w ith total magnetic quantum number M < L. So the wavefunction of a singlet state, w ith M = 0, of the two-electron system can be w ritten as ^ n o ( r i,r 2) = ^ \(L )L ( n ^ ) 1 1 f o b ) 1 2 } x fiL 1:h( n , r 2,r3). (2.12) ll+l2=L 14 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. To determine / ^ j 2, we can apply the Hamiltonian operator H to the eigenstate (2.12). Since H commutes with L _, the action of H on a wavefunction can be expressed as H lL _(r1( 1y l (r2(2)h ]fiu h (ri,r2,r3) = £ [(L~)L(riCi)ll(r2C2)h ] x ha(l'1,l'2\h,l2)fiL ul2(ri,r2,rz) (2.13) I' 1 + 1' 2~ L where h0(^ , l2\h ih ) is the element of the tridiagonal (L + 1) x (L + 1) m atrix ha defined as j .. i i i i \ 1 , d2 2 d d2 2 d d 2 4 d h a i l u h l l u h ) = - 2 {d ^ + n d ^ + d^2 + 72 d ^ + 2 ^ f + ^ + (rj - r | + r%) d2 + (rf - r\ + r f ) d2 r i r 3 dr3dr3 r2r3 dr2dr3 z z | i a a h + i2 d r i r2 r3 r\ dr\ r2 dr2 r3 dr3 ’ for the diagonal elements and ha(h — l , h + l \ h , h ) = — i (2-15) r3 o r3 + 1^2 — 1 ^1 ,^) = — T, — , (2-16) f 3 or3 for the off-diagonal elements. We can solve the eigenstate problem to find a complete basis of the Hilbert space associated w ith the Hamiltonian of the system. B ut we can also choose a basis set for the associated Hilbert space and construct the approxim ate m atrix representation H_ of the 15 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. H am iltonian to solve the eigenenergies and eigenstates of the system. N oting th a t D_ - m atrix elements of H_ with respect to the individual term s in the expansion ( 2.12 ) are H, Y f dT(r ^ l ) l,1(r2^2 )l' 2fl\,l'2 {ri , r 2,r 3) l’1+l'2=LJ x (ri(i)ll(r2(2)h ha(l'1,l, 2\l1,l2)fi'u h (r1,r2,rz) p o o p o o p \ r i + r 2 \ / dri / dr2 / dr3 x n r 2r3 Jo Jo J\f'i— r2\ x (r i ,r2,r3) (m haji'^W uhfiuh (ri , r 2,r 3), where m is a (L + 1) x (L + 1) m atrix whose elements are r / p2t T p T T / > 2 7 T h + h ^ + h / da / siap d/3 / Jo Jo Jo Introducing r 12 r 2 i r 2 _ „ 2 rl + 2 r 3 we have, for example, if L = 1 ( m (L = !) = 3 r 2 r 2 | ( r f + r | - r | ) ^ ^ W i + rf-rf) rf while L = 2 = 2) = 15 r 4 2 |( 3 r f 2 - ^ 1 ) r 2r 12 l ( r 12 + 3 rl r l) r l r v l( 3 r f 2 - r ? r l ) /r»2„2 r l r 12 1 ' 12 Z/+ , the (2.17) (2.18) (2.19) (2.20) (2 .2 1 ) (2 .22) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ; for the case L = 0 m is ju st a constant: m(L = 0) = 1. (2.23) Decomposing the Hamiltonian of the two-electron with fixed nucleus by Euler angles was originally figured out by Hylleraas (1928). He applied it to the S states w ith L = 0. Breit (1930) generalized to the P states. Then B hatia and Temkin (1964) generalized the decomposition to any angular momentum L. The above discussion followed the derivation in (Pont and Shakeshaft, 1995b), which uses the Euler angle decomposition in a similar way of B hatia and Temkin (1964) and facilitates decomposing the Hamiltonian in term s of a Sturm ian-perim etric type basis. 2.2 T h e in teraction b etw een A tom s and th e field The dipole interaction is the simplest interaction between neutral atom s and fields. In the length gauge, the operator form of the dipole interaction of a two-electron atom is dlength T (2.24) where z\ and z2 are the z components of the electrons’ displacement respectively. In the term s of independent particle coordinates, z\ + z2 = ricos#i + r2cosO2- There is a com m utation relation for a spherical tensor: [L^.Tq1] = V 2N T IL 1 ? - 1. (2.25) Here we have Tq = Z \ + z2 = ricosOi + r2cos$2 , (2.26) 17 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. and T* = x t + iy1 + x2 + iy2 = risinOiez r t> + r2sin92el< t’ = r 1C i+ r-2C 2. (2.27) W ith the triangle rule for addition of angular momentum, we only need to consider the case L' = L ± 1. Since the m atrix elements of dipole operator L — > L + 1 and L — > L — 1 Eire related by symmetry, we will only derive the elements for the case L — > L + 1. The integral f v 4'i (f)T0 1L ^+1L£\E'L( f)d f is zero because the integrand is anti-symmetric of r. Then w ith ( 2.25 ) to ( 2.27 ), we have L++I(ricos0i + r2cos02) x {r2C 2)h = (riCi + r2(2)(L + (r2C 2)h fiL u h (ri,r2,r3) = (L + l)((2L )!)(nC i + (r2(2)h (2.28) The m atrix elements of the dipole interaction can be expressed as ih = J dT( t f £ ( ri>r2,r3)Y x (n cosOt + r2cos02) x $fli2(ri,r-2,r3) = ~ ) J 2@L + \) J dT^ ^ i ) (l(r2^ ) ^ /( f ,ii( r i>r 2^3) x (nC i + r 2C 2) x(riCi)h {r2C 2)h f,L lM {ru r2,r3) p o o p o o p \ r 1 + 7 - 2 ! = / dri dr2 dr3 x n r 2r 3/£ , , ( r i , r 2, r 3) x J o J o J \ r i - r 2 \ x fh,h(rh r2,r3), (2.29) 18 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. where d is a (L + 2) x (L + 1) m atrix whose elements are ^",(2 ;Ji,(2 = ~ \ j 2 ( 2 L + \ ) ^ l"'hSl^’h + Sl"’l' Sl2'h+l')- (2'3°) In the velocity gauge, the dipole operator can be expressed as - d d dvelocity = (2.31) Making the substitutions z — ► x — > ^ and y — > g^ifhe above derivations for the length gauge still holds. Then in velocity gauge the elements of the dipole interaction are V,,,. [i'2,hh = / d r ( ^ 1(r i - r 2, r 3) r x ( ^ - + ^ - ) x $ f ; (2(r1, r 2, r 3) = ~ y 2 ( 2 //+ 'i) J d T (r^iy'1 (r^2)l‘ 2 flil’ 2 {n,r2,r3 ) x (^ + ^ + i + ^ )[(riCl)'1(r2C2)'2/i^ (ri’r2’ r3)1- (2'32) Since rtQ = Xi + iyt, we have / 9 . d . . . d g (nti), d . 9 (q 7~ r (p -----1-*^— )(xi +tyi) dxi dyt d(riCi) dxt dyt = 0. (2.33) 19 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. We can move the term (ri£ i)(l (r2< f2)1 2 out of the differential operator and ( 2.32 ) becomes ¥-l\l'2,hh ~\ j 2(2L + 1) / d r(r i ^i)(,I(r2^)^/if,i'(n,r2,r3)(r-iCi)'1(r2C2);2 x . 8 8 8 8 , L , .. - J 2(2L + 1) / x 8rt 8 dri 8 dr2 8 dr2 8 L ( '8x1 8n 8Vl dn 8x2 dr2 dy2 dr2^fll’l^ ruT2'r3^ ~ ]J 2(2/. T l ) / dT^ 1^ ) (,1(r2^ ) ii/ ; f ^ ( r i>r 2 ,r3 )(riC i)i l (T -2C2)'2 x ((riCi)^ ^ r + (r2& 7 2^ r t ^ r i’r2’r3K - (2-34) Here we can replace by because the function ffc i2(ri,r2,r3) is only dependent of r,. We also use the facts th a t rf = xf + yf + zf and ^ = f 1. The partial derivatives related to 0 have no contribution. The integral in the velocity gauge can be rew ritten as p o o p o o p \ r i + r 2 \ / drx dr2 dr3 x rxr2r3f h v ( r i , r 2, r 3) x Jo Jo J \ r x - r 2 \ x fiL ui2(ri,r2,r3), (2.35) ^In-r2| where the elements of m atrix v are = ~ \ ] 2(2L+1)1)(^ ’ + (2-36) 2.3 Ion ization R a te and B ranching R atio We consider a bare atom interacting with a monochronic electromagnetic field through V (t) with frequency oj. The interaction can be decomposed as V(t ) = V+e~iu,t + W.eiwt, (2.37) 20 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. where V+ = Vl. If the system is initially in a state < / > o with energy E q, its response to the interaction of the field is determined by I4 > m >= Ga(EM + iv)V+\<pM-i >, (2.38) where M is the number of photons absorbed by the atom (Shakeshaft, 1999). The Ga(E) = ( E —Ha)*1 is the resolver for //„, the Hamiltonian of the bare atom, which describes the dynamics of the unperturbed atom. The photon ionization rate is Fm = — 2 /m < <}>M-i\V-\<i>M > ■ (2.39) In th e case M = 1, the response function of a system initially in the ground state \< p g > can be described by Ix > = Ga(Eg +Lj)V+\(f> g >, (2.40) where Eg is the system ’s ground energy. Then the single photon ionization rate is T = - 2 / m < < f > g\V-\x > — — 2 /m < (f> g|V— G(Eg 4- co)V+ \(f> g > . (2.41) The two photon ionization rate, when the photon energy is lower than the threshold for one-photon ionization, can be expressed as r<2> = - 2 I m < 4 > l\ V ^ \ 4 > 2> = - 2 / m < </>i|VT|G0(i?2 + > — — 2 /m < < f> g\V-Ga(E\ + irj)V-Ga{E2 + irj)V+Ga(E\ + irj)V+\<f>g > . (2.42) 21 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Considering the single ionization rate, we have i<x\(H-rf)\X > = i ( - < x\(E g + uJ - - H ) - ( E g +uJ - H ) i \ x >) = i ( - <x\v+\ 4 > 9> + <t9\vl\x» = —2 / m < < j> g\V-\x > = - 2 / m < 4 > g\V-G(Eg + uj)V+\4 > 9 > = T. (2.43) On the other hand, we have < X \ { H - H ') \ X > = i y V r x ( X*V2x - ( V 2x r x ) = ^ J n d S x ( x * { n - V ) x - x { n - Vx)*- (2.44) Here we applied Green’ s Theorem to convert the volume integral to a surface integral. Equation ( 2.44 ) means th a t only the asym ptotic form of x on a hyper-sphere of a very large radium contributes to the integral in the equation ( 2.43 ). W hen one of the two electron is liberated, the residual system will be in a bound state. Then for a two electron system, the asym ptotic form of X can be w ritten as X ( r i,r 2) — > 5Z </>n(ri)/n(r2 — ► oo) + F ( n — > oo,r2 — > oo) + ( n — > r2), (2.45) n where 4 > n(r) is the wave function of a bound state of the residual system such as H atom or H e+ ion. The partial ionization rate is defined as the rate for ionization rate when the residual system stays in a specified bound state. The total ionization rate can be expressed as r = ] T r n , (2.46) 22 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. where Fn corresponds to the term < / > n (r) in the asym ptotic form of x- A branching ratio is a ratio of two partial rates. To calculate the partial rate, we can project x onto different channels— term s in x ’s asym ptotic expansion. Let P be the projection operator to project onto a channel s; inserting it into equation ( 2.43 ), we get Ts = i < x \ P ( H - H ' ) P \ X > = 2 I m < x \ P { E - { H l + H2)P\X >, (2.47) where Hi is the Ham iltonian of the residual system with respect to the ith electron staying in the bound state of the residual system. Noting th a t P commute with Hi and the total Hamiltonian of the system is H = Hi + H 2 + I/V3, we finally get the partial rate as Ts = - 2I m [ < < t> g \ V - P \ X > + < X \ P } - J X > ■ (2.48) We will explain the details of Hi and how to construct projection operator P in next section. 2.4 P ro jectio n on to op en channels W hen one electron is photoejected from H~ or He, the three-particle system will be a bound H atom or He+ ion tied to a free electron through angular momentum coupling. The total angular momentum is L. The ejected electron — let us say, electron 2 — is far away from the nucleus while electron 1 is localized. We can ignore the interaction between electron 2 and the bound residual system. Consider the case where the ’ ’free” electron 2 has no kinetic energy but carries some angular momentum. Its behavior is the same as a noninteracting particle w ith infinite mass th a t is stationary but carries angular momentum. It is an etherial particle. Its Hamiltonian is a null operator defined by the limit 02 = — lim V | (2.49) 00 2 [A 23 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. W e refer to a sy ste m w h ich in clu d e s one e le ctro n b o u n d to a n u cleu s a n d a n o th e r ’’e le c tro n ” w h ich is n o n in te ra c tin g b u t h a s a n g u la r m o m e n tu m as a n ’ ’e th e ria l s y s te m ” . T h e H a m ilto n ia n of th e e th e ria l sy ste m is: 1 Z e 2 # ! = - — V ? + 0 2 (2.50) 2/j, n The etherial system has a spectrum very similar to the spectrum of a hydrogen-like ion with atomic number Z nuclear charges. B ut there is a big difference originating from the null operator 02 which acts on the state space of the etherial electron. The spectrum of the etherial system has infinite degeneracy because 02 has infinitely m any eigenstates all with 0 eigenvalues. Using the inter-particle coordinates and separating the angular momentum part, we can reduce the Hamiltonian of the etherial system to a bidiagonal (L + 1) x (L + 1) m atrix as: r r h : h ; h , h = _ h & _ , i i . , / , J L J L - 1 - ~ r 2 + r i ) 9 2 , 1 2 dr\ r\ dr\ dr 2 * •- a " _Z_ _ l ± _d_ _ h d r\ n dr I r3 dr3 ’ for the diagonal elements and H h - U 3+v,h,h = (2.52) 2 dr\ r\ dr\ dr$ r3 dr3 rir3 dridr3 Z h d h d (2.51) r3 dr3 h_d_ r3 dr3 ’ H h+i,h-iM 2 - (2.53) for off-diagonal elements. In the etherial system, the two electrons are not symmetric any more. W hen we exchange the two electrons, the Ham iltonian will change too. The Ham iltonian for the case where the electron 1 is etherial electron is: 1 7 ,p 2 ^ 2 = - ^ V | ----------+ 0J, (2.54) 2/i z r2 24 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. where Oj is defined similiarly as O 2. On the other hand, both of the electrons can be ejected from the true three-body system. We have to project the response function \x > = G(Eg + cj)V+\tf> g > onto two different configurations th at have different electrons as the etherial particle. Let Pi and P2 be the projection operators which project onto the configurations where the electrons 1 and 2, respectively, are in the bound state. The total projection operator is: P — P\ -\- P 2 — P 1P2 (2.55) The subtraction of P 1P2 removes th a t part of the subspace where both electrons are bound. We can construct these projection operators by diagonalizing the Ham iltonian of the etherial system. Let \En, 1 > be the eigenvector of //) with eigenvalue En; it can be decomposed to a symmetric p art and an antisymm etric part as: \En, 1 > = \En, 1; a > +\En, 1; a > , (2.56) where s denotes symmetric and a denotes antisymm etric. It is easy to show th at \En,2 > = \E„, 1;s > —\En, 1;a > (2.57) is the eigenvector of H 2 with the same eigenvalue En. Let |r i , r 2,*;s > and |r i , r 2, j ; a > be the symm etric and antisymm etric base kets for the Hilbert space associated w ith Hi and -ff2- Here the base kets satisfy \ri,r2,i;s >= \f2,ri,i\s > (2.58) and |f i , f 2,*;a > = — |f2,f i,* ;a > . (2.59) 25 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. And the Hamiltonians of the etherial systems also have the property H i{ fi,f2) = H2{f2,n ). (2.60) The m atrix respresentations of Hi and H2 have four types of elements. They are H k(i,i') = < f 1,r2:i;s\Hk\fi,f2,il;s >, = < r 1,r2,r,s\Hk\ri,r2lj';a> , Hk{j,i') = < r 1, f 2,j-,a\Hk\fi,r2,i'-,s>, Hk( j ,f ) = < r i , r 2,j-,a\Hk\fi,f2,j';a > . (2.61) Consider the equations from (2.58) to (2.60),We have H2(i,i') = < r 1,f2,j-s\H2(ri,f2)\fu f 2, j ’;s> = < r2,ri,j; s\H2(fi1f 2)\f2, f i , j > ; s > = < r2, n , j;s\Hi{r2,ri)\r2,ri,j';s > = < r‘ i , f 2,j;s\H1( fi,f2)lfu f 2,j';s > = Hi{i,i'). (2.62) In the penultim ate step, we ju st changed the integral variables name; th a t doesn’t change the integral’s value. Following the same procedure, we have H2( i,f) = H2{j,i') = H2(j,j') = Hj(j,j'). (2.63) 26 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Those relations ( 2.62 ) and ( 2.63 ) lead to the fact th a t when (2.56) is a eigenvector of Hi with eigenenergy E n, (2.57) is the eigenvector of H 2 with the same eigenvalue En. W ith the eigenvectors of Hk, the projection operator P\ can be expressed as and Ps as Pi = Ei < C Emax J 2 IEi > i< E, ill Ei<Em( E / \ \Ei,s>!< Ei,sh -\E i,s > 1< Et,a\i -\E i,a > i< Ei,s\i \Ei,a >!< Ei,a\i P 2 = E i < E max y~i 1 Ei > 2< Ei\2 Ei < E-m ax E \ \Ei,s >i<C Ei,s\i ^ 1^ Ei,o>\\ ^ \Eijd E u s h | > 1< Ei^d\\ y (2.64) (2.65) where Emax is a cut off energy. We only project onto those open channels with an energy lower than Emax. A projection operator satisfies P 2 = P. The total projection operator (2.55) can be constructed as = P\ + P2 — P 1P 2 = \ { P l + P 2 ) - \ { P l + P 2 ) 2 \Et, s > i< Ei, s|! = 3 E \ |Ei,a > i< Ei,a\i ^ ( ( E , < E c E |Ei, s Ei , sj (2 .66) (2.67) 27 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Here we use the fact that [P i,P 2] = 0. (2.68) 2.5 P erim etric coord in ates There is an inter-particle dependency among r% , r2 and r3. W hen we build up the matrices H_ and 5 , the integrations over these three variables cannot be done independently. To avoid this trouble, we can use different coordinates — perim etric coordinates. The perim etric coordinates were first tim e used by Coolidge and Jam es (1937). In a series of pioneer works, Pekeris (1958, 1959) applied the perim etric coordinates w ith a Sturm ian-type basis to calculate the S states of helium atoms, whose result of the ground-state energy of He stands best for a long time. The perim etric coordinates are defined as u = r2 + r3 - r i , (2.69) v = r i + r 3 - r 2, (2.70) w = 2(ri + r2 — r3). (2-71) Unlike r\, r2 and r3, the u, v and w vary independently of each other. All of them range from 0 to oo. T hat will simplify the integrations over r\, r2 and r3. Now the integrals will be calculated by /• o o r o c r \ r i + r 2 \ ^ /•o o r c c ro o / dri / dr2 / dr3x r ir2r3 ■ ■ ■ = — - / du dv dwx(u+v)(2u+w)(2v+w)... Jo Jo J in-r-ai 256 Jo Jo Jo (2.72) The perim etric coordinates are very useful to describe a system in which the two electrons are strongly correlated. There is special significance for the planes of u = 0 , v = 0 and w = 0. Two electrons are on a straight line which passes through the nuclei on those planes. W hen n o r * 28 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. equals 0, the two electrons are at the same side of the nuclei; when w = 0, they are on opposite sides of the nuclei. In principle, we can choose any complete function set as the basis of the H ilbert space to construct the m atrix representation of the Hamiltonian. The two-electron system ’s wave functions should be sym m etric or antisymm etric on switching the spatial coordinates of the two electrons. Switching r i and r 2 is equivalent to switching the angles (#, and < j> i) and u < -> v simultaneously. So f L(ri,r 2,ra) can be replaced by a function f L(u,v,w). A sturm ian-like function e~krLyl(2kr), where is a Laguerre polynomial, is used to construct a basis function. Here we use (Yang et al.. 1997): <l>imn(u,v,w) - e~Li(kiu)e ^ L m(k2v)e *~L„(k3w), (2.73) where Lp(x) is the pth order ordinary Laguerre polynomial and 1/fc, is a characteristic length scale of the two electron system. Plugging the definition ( eq. 2.73) into equation 2.12, the full symmetrized or anti-symmetrized basis functions are defined as: C (ri , r 2;± ) = & ( r i , r 2) ± </>%£ (r2, n ) , (2.74) where l\ + 13 = L and ^fmn2(r i - r 2) = j = [ ( L - ) L(ri(;i)h (r2(;2)h }4> imn(u,v,w). (2.75) We should note th a t the basis functions are not orthogonal to each other. The overlap m atrix S is not a unit m atrix. We can choose appropriate characteristic length scales ( 1/fcj ) for different systems. Actually we always choose k3 = (k\ + /c2) / 2 to make sure th a t the exponential part of tfiimn(u,v,w) has 29 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. the simple form e (*uri+ f e 2r2)_ The basis spans only a finite region of space, where the dynamics take place. The numerical results show us the use of different values for kj and & 2 gives us more power of calculation. Using ki = 1.4aO'1 and & 2 = 0.7aQ 1, a basis with 196 functions yield a ground energy of helium to the 6th digit, the same as the latest non-relativistic theoretical value. Using these length scales, we got good agreement on the oscillator strengths of S — > P and P —> D transitions of helium atom w ith other numerical results th at needed a much bigger basis size. 2.6 R an d om P ertu rb ation M eth od There should be no linear dependence when we use a basis defined by (2.74) w ith different values for fc,s. In practical calculations, a linear dependence problem arises when the to tal basis size is increased. This numerical linear dependence could be the result of round-off errors in the representation of real num bers in the computer. The real numbers are represented by 80 bits under the current IEEE standard. Using double precision, the reliable accuracy is only about 14 digits. W hen we calculate integrals numerically, the accumulated errors could introduce the linear dependence. We can check the eigenvalues of the overlap m atrix S_ to see this problem. The overlap m atrix should be positive-definite w ith the basis (2.74). W hen the basis size becomes large, the lowest eigenvalues of the overlap m atrix, whose elements are the inner-products of basis functions, are very close to zero ( See Table 2.1 ). Sometimes some of them are even less than zero . The problem is worse when the difference between A :,s is small. To avoid this numeric linear dependence problem, we introduce a random perturbation method. The random perturbation m ethod is implemented by the following procedure: Step 1, we choose the fcjS for the basis functions (2.74) and generate the overlap m atrix S_. 30 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Nodes = 3 Nodes = 5 0.000000000041656 0.000000000084500 0.000000000135705 0.000000000196221 0.000002675063950 0.000005407676959 0.000009372443615 0.000009490432868 0.000015218116349 0.000015994764079 0.000023075375114 0.000031224966903 0.002662769617628 0.006687821728046 0.000000000000000 0.000000000000000 0.000000000000000 0.000000000000000 0.000000000000000 0.000000000000001 0.000000000000004 0.000000000000008 0.000000000000012 0.000000000000012 0.000000000000019 0.000000000000021 0.000000000000026 0.000000000000031 Table 2.1: The eigenvalues of overlap matrices The overlap matrices S_ were constructed with k\ = 2.3«0 l. and = 1.9oq 1 Step 2, we add a very small random number to each diagonal element of S_u . Here we choose the m agnitude of the random numbers as 10-13— 10' 1 4 which is around the precision of double precision on 32-bit computers. Step 3, we diagonalize and get all the eigenvectors of the perturbed overlap m atrix S_' and normalize the eigenvectors to get a set of orthonorm al vectors |n! >s, which are in the form of: N I ri > = ^ c n/ji$ i ( r i , r 2), (2.76) t=i where 4>j(ri,r2) is defined in (2.74). Step 4, we construct the Ham iltonian H'[ with = < i'\H\j' > and the new overlap m atrix S ' with S( • = < i'\j' > . We can construct the m atrix representation of any other operator O on the new basis by O 'j =< i'\0\j' >. A question arises here—why | n' > s can be used as basis to the associated Hilbert space? The original basis (2.74) is nonorthogonal. We can diagonalize the overlap m atrix and use the eigenvectors as the orthogonal basis. We introduce only a very small random noise to 5, so 31 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. we expect th a t the eigenvectors of the perturbed overlap m atrix deviate only slightly from the eigenvectors of the unperturbed overlap m atrix and can be used as an approxim ately orthogonal basis as well. Actually we can prove the perturbed eigenvectors are very close to the original eigenvectors of the overlap m atrix. By the Wielandt-Hoffman theorem (Golub and Loan, 1996, pp:395-396), if A and A + E are n-by-n symmetric matrices, then the eigenvalues Aj(A) and \ { A + E) satisify where ||.£?||f is the Frobenius m atrix norm (Golub and Loan, 1996, pp:55). It is defined as In our random perturbation m ethod, the overlap m atrix is a real symmetric m atrix S_. The n (2.77) n n (2.78) This implies that (2.79) perturbation is a diagonal random m atrix R. Suppose x is the normalized eigenvector of S_ + R with eigenvalue A'; we have (S + R)x = X'x. (2.80) M ultiply 3? on both sides; we have xT (S_ + R)x = A' x 1^ Sx = A' — x? Rx xTSx ~ A' — max{Rii). (2.81) 32 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. From (2.79), we have A' ~ A ± ||£ || f = A d b (2.82) where A is an eigenvalue of S_. Following (2.81) and (2.82), we have xTSx ~ A ± ||fi||F — rnax(Ru). (2.83) Since our perturbation is in the m agnitude of 10 13 ~ 10 14, xT Sx is in the range of A ± s/n x 10~13 ~ 10-14. When the basis size is less than 3000,we have x Sx ~ A ± 6 x 10 (2.84) In fact, we relaxed the right hand side a lot. The real value of x 1 Sx should be in a much smaller range around A than above. The equation (2.84) means th a t * is very close to the eigenvector of the original overlap m atrix and can be used as an almost orthogonal basis. 33 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C h apter 3 R esu lts and discu ssion 3.1 T h e grou n d -state en ergy o f H elium Helium is one of the simplest 3-particle systems. The non-relativistic ground-state energy of helium has been com puted accurately in many theoretical works(Frankowski and Pekeris, 1966; Goldman, 1998; K inoshita, 1957; Schwartz, 2004). There are extensive calculational studies using variational m ethods w ith different trial functions. The perim etric-Sturm ian basis is very good for describing the low energy configurations of a strongly coupled system (Pekeris, 1958, 1959). We calculated the ground-state energy of helium to test our basis set. W hen we use two length scales k\ , k2 and put k$ = {k\ + k2)/2, the basis function ( 2.74 ) becomes («.«.«>) = ^ [ ( i - ) i (r1Ci)i l (»'2C 2)'2] e - ^ +^ ) - / 4Ln ((fc1 + k2)w/2) [e-(fclU)/2 L^ k lU) e - ^ v)/2Lm (k2v) + e - {klv)/2Ll(klV)e- ^ u^ 2Lm{k2u)}. (3.1) For the case of ground state, L = l\ = l2 = 0. The basis is normalized to (3.2) 34 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Basis Size Number of nodes ^ground 1 0 -2.684182892757 8 1 -2.901965521202 27 2 -2.903704128966 64 3 -2.903723206871 125 4 -2.903724180289 216 5 -2.903724320316 343 6 -2.903724355180 512 7 -2.903724365099 729 8 -2.903724371836 1000 9 -2.903724374118 1331 10 -2.903724375811 1728 11 -2.903724376367 2197 12 -2.903724376572 9261 20 -2.903724377009 Table 3.1: The ground state of Helium computed by the basses with k\ — 2.3a0 1 and /C 2 = 1.9a0 1 Notice th at the basis functions are not orthogonal to each other. We constructed the Hamiltonian m atrix = < tPi|//|T/ > of Helium and the overlap m atrix 5 ^ = < 'I',!'!'., >. Here we assigned an index to (u , v, w) in the order of the number of nodes included in all u, v, and w Laguerre polynomials. An inverse-iteration m ethod was applied to solve the generalized eigenvalue problem ( 2.2 ). We list the calculated ground energies of helium w ith different basis sizes in table 3.1. In most other variational m ethods the param eters were optimized to get the best convergence of the ground energy. In our calculation, we chose k\ = 2.3ag 1 and & 2 = l-9ag *. Unlike other basis sets, we did not optimize the length scales. W hen k$ = (ki + A ; 2)/2, the exponential part of perim etric-Sturm ian basis is proportion to e~(k2ri+kir2)/2. Notice th a t ki is proportion to 21 < ri > in our perim etric-Sturm ian basis. This facts suggests th a t ki and ki should be close to 2 since < r* > is around 0.9a0, which is calculated in a pre-run of our program. Although it is suitable to choose ki = ki when we only consider the ground state of helium, the two- length-scale should be better especially since we did not optimize the length scales. Intuitively, one of the ki should be assigned a value which is a little bigger than 2.2, where 2.2 is about twice 1/ < r, > , to describe the outer dynamic space and the other kt should be less than 2.2 35 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. to describe the inner dynamic space. We used the random perturbation m ethod to deal with the numeric linear dependence. This allowed us to include Laguerre polynomials with orders ranging from 0 to the same maximum order for all u, v, and w variables. As shown in table 3.1, the convergence of the ground state energy is very good. Our result agreed to the 10th digit w ith the best known non-relativistic Helium ground energy (Schwartz, 2004) when we use a basis with a size at 729, which includes ordinary Laguerre polynomials only up to the 8th order. O ur best result is Eg = — 2.90372437700970 obtained using a basis which includes up to 20th order ordinary Laguerre polynomials. This estim ate is slightly above the best calculated value E g = — 2.903724377034119 •• • (Schwartz, 2004). The two electrons of helium are symmetric in the ground state. Using a single length scale is also a good choice to calculate th e ground-state energy. We got Eg = — 2.90372437702361 w ith a basis with k = 2.3 of the size N = 10206. It is closer to the value of Schwartz (2004). We also calculated the energy of the 21S state of He. Using a basis w ith k\ = 1.73 and & 2 = 1.06 at the size N — 4096, which included the normal Laguerre polynomials up to the 16th order, we got E = — 2.145974045742. It is slightly higher than the extrapolated value E = — 2.145974046 in (Kono and H attori, 1986). We note here that Kono and H attori (1986) got a value E = — 2.1415974044 for the 21S state, which is higher than our result, before the extrapolation. Frankowski (1967) got a value as E = — 2.14159740457 with logarithmic term s in the basis functions. Our energy value is lower than his value. We can calculate eigenenergies and wavefunctions of excited states with our perim etric-Sturm ian basis. We did this for 21P and 31D states in order to check the oscillator strengths before com puting two-photon ionization rates for helium. In the com putation of two photon ionization rates, we used a basis set w ith ki = 1.51<2q 1 and ki = 0.61 and included 2744 perim etric-Sturm ian type functions, where the maximum order of Laguerre polynomials is 13. We got E = — 2.123843086597 for 2lP state; th a t agrees to the seventh decimal digit as the value E = — 2.123824471 • • • of (Kono and Hattori, 1986). Our value of the 31D energy is E = — 2.055620732653. It agrees to the tenth decimal digit to the value of E = — 2.05562073256 of (Sims and M artin, 1988). A Hylleraas type 36 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. basis was used in (Kono and H attori, 1986) and (Sims and M artin, 1988). We should notice that our values for 2lP and 31D energies were calculated w ith complex coordinates which might lose some digits of accuracy for bound state eigenenergies. The perfect agreement shows the power of using two length scales especially for the excited states 21P and 3lD. Notice again th a t we did not optimize the length scales here. 3.2 th e grou n d -state en ergy o f H~ and P s - H - and P s- are two extensively studied three-body systems. Their ground-state energies are of great interest to theoretical and experim ental researches. We present ground-state energies of H~ and P s" in table 3.2. Our results agrees to the referred results very well. Ref. H in a.u. P s in Ry. Scrinzi and Piraux (1998) -0.527 751 016 89 Drake (1988) -0.527 751 016 544 B hatia and Ho (1983) -0.524 010 130 Ho (1993) -0.524 010 140 465 Frolov and Yeremin (1989) -0.527 751 016 507 -0.524 010 140 464 present work -0.527 751 016 458 -0.524 010 140 456 Table 3.2: The ground-state energies of H and Ps 3.3 O ne p h oton ion ization rate 3.3.1 th e case o f H elium As discussed in previous section ( see section 2.3 ), the one photon ionization rate can be calculated from P = —21m < cj)g\V-\x > , (3.3) where \x > is the response function of the atom ic system responding to the external field, and V is the interaction between atoms and the field. For a linearly polarized light with the field 37 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. strength F and frequency u> , the dipole interaction can be expressed as V+ = V_ = F/2uid/dz in the velocity gauge and V+ = V_ = Fz/2 in the length gauge. The one photon ionization cross section is defined as the ratio of the ionization rate T to the energy flux of the incident light field I/uj, where I = c/(8itF2). The total cross section can be w ritten as 4tt d d = — < c t > g\Vv\x > (3.4) L u in the velocity gauge, and at = 167T < 4 > g\zi + z2\x > = 3 2 7 T < ^ |V i |x > (3.5) in the length gauge, where Vv and V ) are defined in ( 2.35 and 2.29, see section 2.2 ). The response function |x > = G(Eg + <J)V+\<jtg > is the solution of ({Eg + u)S - H )|x > = Vv\4> g > (3.6) for the velocity gauge and of {(Eg + u j ) S - H atom) \x > = V l\<t>g> (3.7) for the length gauge. All matrices H _atom, 5, V ) and Vv are built on the perim etric-Sturm ian type basis ( 3.1 ). We list the calculated total cross sections for photoionization of He from the ground state l 1 S under the double ionization threshold in table 3.3 and in table 3.4 for high photon energies above the double ionization threshold. We used a basis set w ith a size N = 2744, k\ = 1.51aQ 1 and 38 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Photon energy length gauge velocity gauge Tunan CC-CC Experim ent 24.6 7.393 303 7.393 484 7.403 7.40 25 7.212 531 7.212 526 7.219 7.21 27 6.408 174 6.408 180 6.412 6.40 29 5.703 677 5.703 682 5.707 5.70 31 5.090 363 5.090 367 5.085 5.10 33 4.557 541 4.557 544 4.563 4.57 35 4.094 476 4.094 478 4.097 4.09 37 3.691 307 3.691 309 3.689 3.68 39 3.339 384 3.339 386 3.336 3.32 41 3.031 300 3.031 301 3.027 3.01 43 2.760 801 2.760 802 2.759 2.72 45 2.522 662 2.522 663 2.523 2.48 47 2.312 553 2.312 555 2.313 2.28 49 2.126 954 2.126 956 2.128 2.10 51 1.963 126 1.963 129 1.963 1.94 53 1.819 270 1.819 272 1.819 1.77 55 1.695 235 1.695 237 1.694 1.67 57 1.596 210 1.596 213 1.594 1.61 59 1.575 068 1.575 071 1.582 1.56 Table 3.3: The total one-photon ionization cross section of Helium below double ionization thresh old fc2 = 0.61aQ 1 . The cross sections agree perfectly with the values of (Chang and Fang, 1995) for photon energies lower than double ionization threshold. They agree with the experim ental values of (Samson et al., 1994) for all photon energy ranges. O ur results also agree well w ith (Pont and Shakeshaft, 1995a) over the same photon energy range. We got excellent agreement between the length and velocity gauges. The cross sections from the length gauge and the velocity gauge agree up to the sixth or seventh digits, which is about 0.00001 — 0.0001%, for most photon energies except those right at the threshold. This is better than the best agreement of 0.1 — 0.2% in (Chang and Fang, 1995). The total cross sections of helium close to the N — 2 and N = 3 lim its are presented in figure 3.1 and figure 3.2. An extremely sharp resonance, which located at E = 71.235eV and belongs to N = 3 limit, was plotted alone in the figure 3.3. It is well known th a t the total cross section in the resonance range exhibits a Fano profile due to the interference between the direct photoionization and the autoionization after the creation of a doubly excited state embedded in the continuum (Fano, 1961; Fano and Cooper, 1965). Our figures showed Fano profiles in the resonance range. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. P hoton energy length gauge velocity gauge Experiment 100 0.40729 0.40728 0.393 110 0.31389 0.31391 0.306 120 0.25029 0.25032 0.244 130 0.20065 0.20068 0.196 140 0.16076 0.16078 0.160 150 0.13030 0.13032 0.131 160 0.10789 0.10790 0.108 170 0.09127 0.09128 0.0894 180 0.07832 0.07834 0.0760 190 0.06765 0.06767 0.0643 200 0.05848 0.05850 0.0550 210 0.05049 0.05052 0.0474 220 0.04358 0.04361 0.0409 230 0.03771 0.03773 0.0356 240 0.03279 0.03280 0.0315 250 0.02871 0.02873 0.0277 260 0.02534 0.02536 0.0245 270 0.02255 0.02257 0.0218 280 0.02023 0.02024 0.0194 Table 3.4: The total ionization cross section of Helium above double ionization threshold Due to the angular m om entum forbidden rules of the dipole interaction, those resonances belong to 1 P(, resonances. We got a total cross section of helium slightly bigger than the values of Sanchez and M artn (1991) in the N = 3 resonance range. For example, our total cross section at the peak of 3s3p resonance is 1.10M6 whereas Sanchez and M artn (1991) yielded 1.085Mb. Both values are bigger than the experim ental value 1.047M6 from Lindle et al. (1987). The peak position agrees very well w ith the experim ental and theoretical results of Bachau (1984), Domke et al. (1996) , Schulz (1996) and Lindle et al. (1987). Using the complex coordinate rotation m ethod ( see section 4.1 ), we calculated the autoioniza tion states embedded in the continuum which introduce the resonances. We presented the results in table 3.5. The autoionization states computed by the complex coordinate rotation m ethod confirmed the positions of the resonances on the total cross section curves ( the figures 3.1, 3.2 and 3.3 ). The results also agree very well w ith other theoretical results (Hamacher and Hinze, 1989; Ho, 1982, 1991; Oza, 1986; Zhou and Lin, 1993) . Domke et al. (1996) did a very high resolution study of 1Pq double excitation states. A mystery is th a t he did not get the resonance Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 3.0 2 .5 - X I E 2.0 - o c o o C D (/) C O C O e 1.5 - O 1.0 - < 0 + - « o I- 0 .5 - 0.0 58 60 62 64 66 Photon Energy (eV) Figure 3.1: The total ionization cross section of He close to IV = 2 limit. The solid line is the current work; the *s are experimental values from Samson et al. (1994). at E = 72.447 eV, which appeared in many theoretical calculations (Ho, 1991; Zhou and Lin, 1993). As shown in table 3.5 and figure 3.2, this resonance is wide enough and strong enough to be observed in Domke et al. (1996). We have to wait for the observation of this resonance in the further experimental works. 3.3.2 th e case o f H~ The photodetachm ent of the negative hydrogen ion is very im portant in astrophysics. Much theoretical and experimental research has been carried out (Ajmera and Chung, 1975; Chang and Tang, 1991; Smith and Burch, 1959). Using the same m ethod as in the previous section, the total cross section below shape resonance of II ~ was calculated. H _ appears ro be very similar to Helium atom. Not considering the difference of reduced mass, the only difference is th a t the 41 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1.12 2 1.0 5 - c o o a > c o u > < /> 2 0.98 - O 15 •4 — » o 0.91 - I — 70 T “ 71 ~I— 72 - r- 73 69 Photon Energy (eV) Figure 3.2: The total ionization cross section of He close to N = 3 limit. The solid line is the current work. The *s are the experimental values from Lindle et al. (1987) nuclear charge of H _ is one but Helium’ s nuclear charge is two. This difference has introduced much trouble in research on H “ . The coupling between the two electrons is the same as the nuclei-electron interaction in H~. The calculation for H _ is much more difficult then for Helium. The total cross section of Helium is converged at a very small basis w ith size N = 216. It needs a basis w ith size N = 1331 to get converged for H - . We present our results of the photodetachm ent cross section of H~ from the ground state in figure 3.4 when the photon energy is lower than the shape resonance E = — 0.124a.w. . It is evident th a t the current work agrees very well w ith Ajmera and Chung (1975) and Broad and R einhardt (1973). The total cross section for this shape resonance is presented in figure 3.5. The agreement between the length and velocity gauges is an im portant sign of how good the calculation is. The earlier works had very big discrepancies between the length gauge and the 42 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 0.9 9 - § 0.9 8 - o a > Vi Vi Vi 2 o £ 0.97 - £ 0 .9 6 - 71.12 71.19 71.26 Photon Energy (eV) Figure 3.3: A doubly excited Rydberg Resonance of Helium at E = 71.235eR velocity gauge. For example, it is about 20% in Geltm an (1962) and Doughty and McEachran (1966) and it is about 8% or better in Ajmera and Chung (1975). Chang and Tang (1991) reported an agreement in 1 ~ 2% percents or better for the entire energy range from 0.0 to 0.75 Ry. Here the agreement of the length gauge and the velocity gauge is about 0.1% or b etter for the entire photon energy range, which also includes the shape resonance at E = — 0.124a.u. . Here we put part of the result in the table 3.6. It is clear th a t the agreement of the length gauge and the velocity gauge is very good over the entire energy range. 3.4 tw o p h oton ionization rate o f H elium The interaction between the two electrons plays an essential role in the m ultiphoton ionization of helium. There are extensive studies on this topic (Parker et al, 1998). 43 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. E r r E l r 1 E* 12 60.145 0.0375 60.147 0.037 60.15 62.759 0.0001 62.758 < 0.0005 62.77 63.656 0.0081 63.658 0.010 63.66 64.118 < 0.0001 64.119 < 0.00005 64.135 0.00005 64.135 0.0003 64.14 64.465 0.0035 64.467 0.0035 64.47 64.657 0.000005 64.657 < 0.0001 13 69.872 0.1914 69.873 0.181 69.88 71.223 0.0009 71.23 < 0.005 71.24 71.308 0.0398 71.314 0.047 71.32 71.625 0.0789 71.623 0.082 71.63 71.999 0.00059 72.159 0.0141 72.160 0.023 72.181 0.0353 72.179 0.039 72.447 0.0077 72.45 Table 3.5: Helium 72 and 73 Autoionization Resonances. Er and T are calculated by complex coordinate rotation m ethod; The E l and T1 are experim ental d ata from Domke et al. (1996); E* are peaks identified from the total ionization cross section curves computed in this work ( see the figures 3.1 , 3.2 and 3.3 ). Photon energy (a.u.) Length Gauge (Mb) Velocity Gauge (Mb) 0.0400000 0.0800000 0.1600000 0.4000000 31.853870363919967 32.767524923472386 15.156577552078391 7.197976574137199 31.863242589960933 32.771934670802715 15.159656197010527 7.202747672956998 Table 3.6: H ionization rate below the shape resonance There are four kinds of diagrams involved in the two-photon absorption process ( see the figure 3.6 ). If we restrict the photon energy to be so low that a single photon is not able to ionize the helium atom , only Diagram 1 in figure 3.6 contributes to the two-photon ionization rate. The atom in the ground state absorbs one photon and is excited to some interm ediary state and thereafter it absorbs another photon to reach the continuum state, in which one electron is ejected and the atom becomes an ion. This fact simplifies the com putation of the two-photon ionization rate below threshold. Recall the two-photon rate is ( see section 2.42. ) r ( 2> = - 2 / m < 4 > g\V-Ga(Ei)V-Ga(E2)V+Ga(Ei)V+\4> g > . (3.8) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4 0 - (D 2 o - = 0.1 0.3 0.8 0.0 0.2 0.4 0.5 0.6 0.7 Photon Energy (Ryd) Figure 3.4: H~ photoionization cross section below 21 P shape resonance The solid line is current work of length gauge;the o is this work of velocity gauge;the • is from (Ajmera and Chung, 1975);and * is from (Broad and R einhardt, 1973). Initially the atom is in the 11SI ground state. Due to the forbidden rules of the dipole interaction, the only possible interm ediate state is a \ p state. This means that Ga(Ei) in the above equation is Ga{Ei) = 1 / { E \—H a{L = 1)). B oth the 1S and lD state can be reached from the interm ediate 1P state. So Ga(E2) can be G f {E2) = 1 /{E 2- H a(L = 0)) for a or G f ( £ 2) = 1 /{E 2- H a{L = 2)) for a l D final state. Let u > be the photon energy. The two-photon ionization rate will be T2 = -2 Im < 4 > g\V-G p (Eg + u,)V -G s a (Eg + 2u>)V+GZ(Eg +u})V+\4 > g > + -2 7 m < 4 > g\V-GZ(Eg +u>)V-G?(EB + 2u)V+Gp (Eg +uj)V+\c f> g > , (3.9) 45 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 0 .8 - C M O (0 K C o 0.6 - o 0) co (0 C O s 0.2 - 0.0 10.99 11.00 11.01 11.02 10.95 10.96 10.97 10.98 Photon Energy (eV) Figure 3.5: H~ photoionization cross section in shape resonance where u; is limited by Eg + u j < E g(H e+) = — 2.0a.u.\ E g + 2uj> E g(He+) = - 2 .0 a.u. (3.10) We need the Hamiltonian of L = 0 ,1 ,2 to calculate the two-photon ionization rate. As a check, we calculated the lowest eigenenergies for these three Hamiltonian matrices. We present them in table 3.7. All of them agreed fairly good w ith other works. 46 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 3.6: The diagrams of two photon absorption processes State Re(E) Im (E ) E ref 1*S -2.903 724 803 337 0.000 000 290 508 -2.903 724 3 7 7 ^ 21P -2.123 843 086 597 0.000 000 000 415 -2.123 843 430 31D -2.055 620 732 653 -0.000 000 000 200 -2.055 620 6 8 ^ Table 3.7: Helium energies of I 1 S'. 2l P and 31 states refer to (Schwartz, 2004); ^ refer to (Kono and H attori, 1986); ® refer to (Sims and M artin, 1988) The dipole matrices between S' — * P and P — > D are keys in the calculation of the two-photon absorption rate. One way to check the dipole operator is to calculate the oscillator strengths. Here we check the oscillator strengths from 1*S to 21P and from 21 P to 31D, which are used in this work, in table 3.8. The values of the length and the velocity gauges agreed to each other. All of them agree with the value of B. Yang (1997). We com puted the two-photon ionization rate below the threshold of helium. The result is listed and compared to Pont and Shakeshaft (1995a) in table 3.9. O ur results agreed fairly with the 47 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. transitions Gauge this work Yang P S — >2 l P length velocity 0.276 164 470 367 627 0.276 164 475 888 620 0.276 165 0.276 165 2X P — > 3lD length velocity 0.710 208 912 523 015 0.710 224 670 548 529 0.710 243 0.710 190 Table 3.8: Oscillator strengths of Helium: l 1 S' — > 21P and 21 P — > 31D values of Pont and Shakeshaft (1995a). As shown in ( 3.9 ), there are two components in the two- photon rate. One is through X S — S P — > 1 S; the other one is through 1S — A P — A D. We present the contributions through different paths in table 3.10. It is evident th a t the contribution of 1S — A P — A S is less than it of 1S — >1 P — > ' D except just in the resonances where there are real states — 2*P for E exces = 0.65 and 3 'P for E excesa = 0.80— right a t the interm ediate ’ ’virtual” states. Excess energy (a.u.) r / / 2 Velocity Gauge r / / 2 Length Gauge r / / 2 Ref[Proulx 1993] 0.15 0.235 146 0.235 173 0.231 70 0.30 0.145 983 0.146 001 0.143 57 0.45 0.094 444 0.094 471 0.092 46 0.60 0.102 099 0.102 086 0.098 39 0.75 0.027 367 0.027 335 0.026 84 Table 3.9: Helium two-photon ionization rates below the ionization threshold. We used a basis w ith k\ = 1.51 a( y 1.A ; 2 = 0.61 a,7 1, 0 = 20°, and including up to 13th order normal Laguerre polynomials for all u, v and w coordinates. The size of the basis is N = 2744. We plot the two-photon ionization rate vs. excess energy in the figure 3.7. Although the agreement of the length and the velocity gauges is not good in resonances, both of them show the correct resonances associated to the bound states corresponding to 21P and 3X P states of helium when the single photon carried an energy E = E?i P — Eg or E = E :ii P — Eg. 48 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 0 .5 0 .4 - 0 .3 - ^ 0.2 - 0 .1 - 0.0 - 0.0 0.6 0.8 0.2 0 .4 E xcess Energy ( a.u.) 0 .4 0 - 0 .3 5 - 0 .3 0 - 0 .2 5 - c o b 0 .2 0 - 0 .1 0 - 0 .0 5 - 0 .0 0 - 0.6 0.8 0.0 0.2 0.4 Excess Energy (a.u.) Figure 3.7: Two-photon ionization rate of Helium below the ionization threshold The top one is the total two-photon ionization rate of He below the single ionization threshold. The solid line in the top figure corresponds to the rate computed under the velocity gauge and the ▲ points correspond to the length gauge. The bottom one presents the contributions from the different paths, W here the A line corresponds to the S — ► P — > D component and the ▲ line corresponds to the S — ► P — > S component. 49 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Excess energy (a.u.) gauge LS - * 1 P — S 'S — P D total 0.15 length velocity 0.018 607 0.0185978 0.216 566 0.2165485 0.235 173 0.235146 0.30 length velocity 0.003 636 0.003 624 0.142 369 0.142 358 0.146 006 0.145 983 0.45 length velocity 0.000 018 0.000 002 0.094 454 0.094 442 0.094 471 0.094 444 0.60 length velocity 0.014 979 0.014 979 0.087 107 0.087 119 0.102 086 0.102 099 0.65* length velocity 1.314 431 1.317 408 0.632 348 0.635 127 1.946 779 1.952 535 0.75 length velocity 0.000 060 0.000 058 0.027 307 0.027 277 0.027 367 0.027 335 0.80* length velocity 0.068 735 0.070 605 0.002 444 0.004 019 0.0711787 0.074 624 Table 3.10: Different path contributions of two-photon ionization rate of Helium below the ion ization threshold The results are obtained by the same basis as table 3.9. The * indicates th at the photon energy is in the resonances. 3.5 O ne p h oton ion ization branch rate 3.5.1 D irect im p lem en tation w ith com p lex basis As discussed in the section 2.3 and 2.4, one of the electrons will be in a bound state of the residual system when another electron is ejected for photon energies below the threshold for double ionization. The rate for single ionization by one-photon absorption can be calculated by the formula ( 2.48 ) which is Ts = - 2 Im[< < f> g\V -P \x > + < X \P ^ \X > ■ The projection operator P can be constructed by the formula ( 2.66 ). We applied this m ethod to He atoms. We used the complex-rotated basis to compute the total ionization rate T = — 2Im[< < j > g\V-\x >• To construct the projection operator P, we used a mixed basis set which includes the symmetric basis + (3.11) 50 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. and the antisymm etric basis - C ( 4 h ) (3.12) where T 2) is defined in the equation 2.75. As discussed in section 2.4, due to the presence of the null 02 operator, the eigenstates of the etherial system are infinitely degenerate. Since we only use a finite basis, there is only a finite degeneracy for each eigenenergy of etherial system. We presented the eigenvalues of H x of the etherial system ( H e+ + e ) in tables 3.11 and 3.12. There are 686 symmetric basis and 686 antisym m etric basis included in the basis set. Figure 3.8: Ionization process of Helium The ground-state energy ( n = 1 ) of H e+ is E i = — 2.0 a.u. . The energy of 2 S and 2 P states ( n = 2 ) of IIe+ is E2 — — 0.5a.u.. The ground-state energy of He is E g ~ — 2.9073a.u.. Therefore the photon energy must be higher than E 2 — E g = 65eV to leave the bound electron of the etherial system in the n = 2 state ( see figure 3.8 ). We take a photon w ith energy E = 35eH to ionize the He atoms. The total cross section is a = 4.10m6. W ith this energy, the bound electron is in the 15 state of H e+. We expect to see the cross section for ionization to the n = 1 channel 51 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. to be the to tal cross section. We constructed the projection operator P which only projects onto the 15 subspace of the etherial system Hi ( 2.50 ). After we applied P to the response function and calculated the cross section for single ioniation by formula 3.11, the values are ju st garbage for both of gauges. After carefully studying the equation 3.11, we found the trouble came from the term < x \ P ^ \ x >• This term was not converged at all. It is necessary to take the complex conjugate for the bra < xl- Since we chose k{ for outgoing waves, the complex conjugate of |x > is a superposition of the incoming waves. A basis for the outgoing waves can not describe incoming waves well. This led to the divergence of the term < x \ P ^ \ x >■ If there is no interaction between the two electrons, the troublesome term will disappear from the equation 3.11. We did a calculation on a prototype system. In this system, we tu rn off the repulsive interaction between the two electrons. W hen we photo-ejected one electron, the other one can not see the field at all. We used a photon w ith E = 60eV to eject an electron from such a system. The total cross section is a = 2.4262M6, the cross section for ionization of hydrogen multiplied by two, as calculated by the known analytical formula. We used a basis w ith N — 343 for the ground state, N = 686 for the |x > and N = 1372 for the etherial system. We applied the operator Pi, which only projected onto the n = 1 subspace of the etherial system, to |x >• The first term of ( 3.11 ) yields o\ = 1.2132Mb for the length gauge and cy = 1.2168 for the velocity gauge. Since we only projected onto the ground state of H\ and the two electrons are interchangeable, it is the expected result ( 2<7i = a ). One interesting thing is th a t if we project onto all states of Hi, we have ai = a. And it m ust be right since Pi = JA \i >< i\ = I if we sum all states * > . At first glance, it looks a bizarre since the unperturbed electron must stay in the ground state. Those higher states ’ ’should” have no contribution. As we discussed in section 2.4, we can not distinguish Pi and P2 on their symm etric parts. W hen we apply Pi to |x > , only does the symm etric p art contributes since the full Hamiltonian is symmetric. Those higher states, especially the positive spectrum of Hi, includes the states th a t describe the other electron in the ground state. 52 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Re(E) a.u. Im(E) a.u. -2.01496145850679 -2.01401893021142 -2.00864973428571 -2.00780493342558 -2.00000060005763 -2.00000000106763 -2.00000000043792 -2.00000000016186 -2.00000000001067 -1.99999999999101 -1.99999999977835 -1.98814468837191 0.01139192300595 0.02214420297162 0.00512063179109 0.00412803529123 0.00000037427992 -0.00000000009523 -0.00000000127514 -0.00000000398457 -0.00000000020124 -0.00000000001208 -0.00000000189785 0.08133642327642 Table 3.11: The n = 1 eigenenergies of a Z — 2 etherial system with a complex basis 3.5.2 P r o jectio n w ith real basis The resolvent G(E) = 1 / (E — H) can b e c o n stru c te d from a real discrete basis representation of the Ham iltonian H (Shakeshaft, 2000, 2002; Shakeshaft and Piraux, 2000). The basis only needs to cover the space where the d y n a m ic s ta k e place. Therefore we need not worry about the asym ptotic boundary conditions imposed on the wavefunctions. The resolvent is analytic w ith respect to its dynam ic time-scale and can b e expressed as 1 00 1 G{E) = + i t ^ H ) ] T - I n (2Et^)L1 n_ 1(2t^H), (3.13) n=l where t$ = t0e is a complex unit of time, t0 is the time-scale during which the interaction takes place and Ljt_ 1(z) is an operator-valued associate Laguerre polynomial. The numerical coefficients In {2Et$) can be calculated through a recurrence relationship as n ln+i - 2(2 - z/2)In + n ln- i - 2 ( - l ) nz = 0. (3.14) By this way, both the resolvent G(E) and the projection operator P can be built on a real basis. After we calculate the resolvent and the projection operator, we can calculate the branch rate by the formula ( 2.48 ). 53 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Re(E) Im (E ) Re(E) Im (E ) -0.53000861722875 -0.01998516459430 -0.52113268687746 -0.01047941907456 -0.52074895024833 0.00144180936791 -0.50829973645241 -0.00152127487220 -0.50425860627346 0.00029138113277 -0.50421746144976 -0.00273953681369 -0.50411043692362 -0.00031475645941 -0.50356517121641 0.00021181809095 -0.50241365821142 0.00023096788765 -0.50060822395399 0.00047139053707 -0.50031142007380 0.00025957408040 -0.50026763553619 -0.00549566288570 -0.50022654293366 0.00023097924247 -0.50011274277267 0.00026814934154 -0.50006053992148 0.00017782396902 -0.50002007294446 0.00171659357142 -0.50000035181172 -0.00000280295710 -0.50000025551983 -0.00000015768978 -0.50000008826637 0.00000167886483 -0.50000000132180 0.00000001455455 -0.50000000008279 0.00000000021995 -0.49999905465977 -0.00000514973794 -0.49999999980913 -0.00000000007099 -0.49999999973802 -0.00000000547170 -0.49999995176144 0.00000002952000 -0.49999940918150 -0.00000010161055 -0.49999797185962 0.00001984883029 -0.49999715289723 0.00000310968252 -0.49999619495586 0.00001043021492 -0.49998509831113 -0.00001253144814 -0.49996507424555 0.00082988600160 -0.49995430169942 0.00006483732717 -0.49994003064459 0.00000147276330 -0.49993523730476 0.00061840818686 -0.49993348156665 0.00067935447131 -0.49989964392761 -0.00000866939251 -0.49983703269538 0.00000182802279 -0.49512825514697 0.00204976024404 -0.46219504774531 -0.00396979325900 -0.42999919907330 0.00122735976940 -0.32256046725304 -0.00912101278254 Table 3.12: The n = 2 eigenenergies of a Z = 2 etherial system w ith a complex basis We need a basis to describe well both the ground states and the excited states of the etherial system. This might require us to include different length scales in the basis functions. One way is to use two length scales and use the random perturbation to deal with the associated numerical linear dependence problem. Here we introduce another way to include two length scales. We introduce two groups of basis functions. One group w ith an optimized length scale to describe the ground states. A nother group will use another length scale to describe all other higher states. The procedure to construct the basis is: • 1. Construct H_ and S of the etherial system with a length scale k\ th a t is optimized for the ground states. We solve the ground states \4 > g,i >• We construct Pi = J T 1 4>gP >< Thereafter we construct and diagonalize P and select those eigenvectors of P with an eigenvalue very close to 1 as the basis to describe the ground channel of the etherial system. 54 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. • 2. R epeat the procedure 1 to construct P with a length scale k2. Diagonalize Q = 1 — P and choose those eigenvectors with a eigenvalue close to 1 as the basis to describe the excited energy channels of the etherial system. This m ethod is based on the fact th a t the ground state and the higher states of etherial system are of the quiet different spatial configurations. Although we use different length scales for them, they are still almost orthogonal to each other. Then the two groups of basis functions are not linear dependent. We list the ground channel energies of the Z = 1 etherial system in the table 3.13, where ki = 0.7(Zq 1 is optim ized for the ground channel and k2 — 0,2 a ^ is chosen for the higher channels. It is evident th a t k2 can not describe the ground channel well. And the mixed basis describes the ground channel very well. Actually the existence of the basis with k2 made contributions to improve the energies of the ground channel. li o < 1 k2 = 0.2 mixed basis -0.499996955542577 -0.499994862063548 -0.499989294880139 -0.498850719939367 -0.439112446578947 -0.429586976123187 -0.412587171804435 -0.362712899244077 -0.499997386846682 -0.499995435016121 -0.499989873710515 -0.498967316065049 Table 3.13: The example of the etherial eigenenergies (n = 1, Z = 1) with a real basis The wavefunction of the etherial system can be expressed as i* > = > + Y l b^ ’j >> (3-15) i j where |4>9,j >s, which describes the ground channel, are the eigenvectors of P with k\ and |$ , j >s, which describes the higher channels, are the eigenvectors of Q = 1 — P w ith k2. We present coefficients a, and bj of the symm etric p art of the four ground channel states in the mixed basis in the table 3.14. It is evident th a t most of components of the ground channel states are from the basis with k \ . The coefficients of the components of the basis w ith k2 are very small. 55 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. E -0.499997386846682 -0.499995435016121 -0.499989873710515 & i 0.015415997392058 0.656104810763908 0.261339955153956 -0.031462762187483 0.045176195043779 -0.263364682067521 0.645084319534984 -0.111649691842777 0.310392148546648 0.012416971495065 -0.124476410325587 -0.622895533174418 bj 0.000074443682388 0.000032456035676 -0.000092844503622 0.000263077250736 -0.000208203421013 0.000090135549683 -0.000036157117146 -0.000039743065018 -0.000077474822466 0.000037705507003 0.000293899775444 0.000048690796752 -0.000184170049382 -0.000074180194279 -0.000034650875920 -0.000090010804712 0.000055168335099 -0.000009566261717 -0.000004832804694 -0.000024208918457 0.000023616011509 0.000044527276924 -0.000017659559542 -0.000036512097514 -0.000195784936972 0.000135564246132 -0.000127538640282 Table 3.14: The sym m etric components of 15 wavefunction of the Z = 1 etherial system in the mixed basis It is very easy to generalize this m ethod to include more length scales if they are needed. For example, we can choose a kn optimized to describe the n th subchannel of the etherial system, where n is the principal quantum num ber of the bound electron of the etherial system. This m ethod can also be valuable to the whole system, not only for the etherial system. ~ E -0.049996319634483 -0.164140084186562 -0.251560868667170 -0.319249210841297 -0.371900664354740 -0.412711033410022 -0.443977766983010 -0.467351333173414 -0.483982898893576 -0.494661965028060 Table 3.15: The example of eigenenergies of P H P Here P projects onto the n = 1 subchannel of the etherial system. II is the full Hamiltonian, whose angular momentum is L = 1. We constructed the P which projects onto the n = 1 sub-channel of the etherial system. We present some eigenvalues of P H P in table 3.15 and QHQ in table 3.16 where Q = 1 — P. There are two Feschbach resonances in the eigenvalues of QHQ. There are 4 eigenvalues in the shape resonance with E r = — 0.124a.u.. Those eigenstates sim ulate this shape resonance. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. E -0.120540830804188 -0.120780339848436 -0.121737865473030 -0.122442790279272 -0.122561876965030 shape resonance -0.123561241302133 -0.123664671701951 -0.123956605106020 -0.124519176739792 -0.124754532546692 Feschbach resonance -0.125924928026966 Table 3.16: The example of eigenenergies of QHQ in the range of the shape resonance E r = — 0.124a.u. and T = 0.0013a.u. Here Q = 1 — P and P projects onto the n = 1 subchannel of the etherial system. H is the full Ham iltonian with L = 1 Using the projection operator P —projecting onto the n = 1 and n = 2 subchannels—and constructing the resolvent ( 3.13 ) we calculated the branch rates of the shape resonance at E r = — 0.124a.u. of H “ . The result is presented and compared w ith Sadeghpour et al. (1992) in figure 3.9. The shape of our curves is comparable to the shape of Sadeghpour et al. (1992)’s curves. The position of our peak is shifted a little bit to the right in comparison to Sadeghpour et al. (1992). We compare the total rate obtained from the real basis to the total rate obtained by our complex basis m ethod in figure 3.10. The values from the two m ethods agree very well. This work was published in (Li et al., 2003). The details are included in our paper. 3.6 E lectron d istrib u tion s o f H , H e and P s Let 'ltg(fl,r^) be the ground state wave function. The probability th a t an electron is located in [rio,7-io + 5] is proportional to p r i o + 6 p o o /*|ri+r2 | p[r0 € [r10,r 10 + 5}] ^ d n dr2 dr^rir2rz x \^ ( f i ,r 2)\2 (3.16) « /7 * io Jo J\r 1— 7 * 2! 57 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. o C O 0.8 M — o c . Z D 0.6 0.4 0.2 0.0 10.95 10.96 10.97 10.98 10.99 11.00 11.01 11.02 Energy(eV) Figure 3.9: P artial cross sections for photodetachm ent of H~ in the region of the shape resonance E r = — 0.124a.w. The broken line are the results from Sadeghpour et al. (1992) and the solid line from the projection by real basis. Upper (lower) curves are for the n = 2 ( n = 1 ) subchannel. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1.0 ‘ S 0.8 0.6 0.4 0.2 0.0 11.02 10.96 10.98 11.00 Energy ( eV) Figure 3.10: Total cross section for the photodetachm ent of H “ in the region of a shape resonance The solid line is the sum of the branch cross section with real basis, while the broken line was obtained using a complex basis w ith the formula of the total decay rate. We used two length scales k\ = 0.7a,g 1 and k2 = 0.2(Iq 1 with a rotation angle 6 = 20°. 59 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. From equation ( 2.69 ), we have 2v + w n = - r - 2 u + w T2 = 4 r 3 = (3.17) In perim etric coordinates, the ground state wave function can be expressed as N 'S>(u,v,w) = E Ci{(j>ii(kiu)(j)mi(k2v) 4- 4>ii{k\v)$mi{k2u))<j)ni{hw). (3.18) i- 1 The probability distribution along n is P [n e [»-io,no + < 5 ] = n o / / dvdw I du (2u + w)(u + v ) , T . .,2 x 1 p (3.19) We can integrate over the u coordinate since there is no constriction on u. Then we do numerical integration on the stripe n o < ^a±!L < ri0 4- §, We integrated on the v — w plane w ith grids. To get the probability density function along n , we should take the limit 5 — > 0. This reduced the two dimensional integral to a line integral on the v — w plane. The one-electron probability density function is defined as f i n 0] = n o J J dvdw J d u x ^2M + ^ ^ M + l;^ 4 '(u ,u ,w )|2. (3.20) 2 v + w = 2 r i o We present the probability densities in figure 3.11. The peak of the distribution of He is around n = 0.6a.u.; the peak of the distribution of H _ is around n = 1.4a.n..; and the peak of the distribution of P s_ is around n = 3.4a.it.. P s_ has a very wide peak. The w idth of the H~ 60 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. peak is much narrower than the w idth of the peak of P s~. B ut H - still has a much wider peak compared to He. These facts tell us the differences among the ground-states of He, H _ and P s_ . The electrons in the ground-state of H _ are loosely bound. The electrons in the ground-state of P s" are only very slightly bound since the probability for an electron to be far from the positron is significant. Only the electrons of He are tightly bound to the nucleus. The outer electron in P s" or H - polarizes the core positronium or hydrogen atom and is bound by the induced electric dipole. Then the outer electron has a great probability to show up in the area far from the core in Ps~ or H _ . But the two electrons are indistinguishable and their probability distributions along r\ and rz are mixed up. Therefore we can’t see two peaks in f{r\) in figure 3.11 corresponds to the inner and outer configurations. We compared the / (r) w ith the electron probability distribution of H — like atoms in figure 3.12, where the probability of H yd — like atom was multiplied by two. The electron in H yd — like atom is closer to the nuclei than f(r) in either He or H~. The electron distribution of H - spreads wider than the electron of H atom . This is the result th a t the outer electron of H _ is far from the core atom. There is no significant difference for the case of He in figure 3.12. We know the only stable configuration for the two-electron systems is th a t the two electrons sit on the same line but the opposite sides of the nucleus under classical electrodynamics. It is widely believed th a t the two electrons take a similar configuration under the framework of quantum mechanics. We present the contour plots of |$ |2dV, where dV = 7r/32(w + v)(2u + w)(2v + w) is the volume element, in figures 3.13, 3.14 and 3.15. One electron is located at the peak position given in figure 3.11. It is evident th a t in all cases, |4,|2dw reaches the maximum when 6 = n, which indicates th a t the two electrons sit on opposite sides of the nucleus. 61 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 0.8 - 0.6 - 0.4 - 0.2 - 0.0 0 2 4 6 8 10 12 14 r1 ( a.u.) Figure 3.11: The density of the probability of finding one electron at r\. The • line showed the distribution of H~, the A line showed for He, and the * line is with respect to Ps~. 62 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1.0- 0.8- 0.6- 0.4 - 0.2- 0.0- 12 4 6 8 10 0 2 r, (a.u.) Figure 3.12: The density of the probability of finding one electron at ri compared to the distrib ution of the electron of Hydrogen-like atom The open A and o computed by using the ground-state wave functions of the Hydrogen-like atom. 63 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 0.02500 0.05000 0.07500 Figure 3.13: the contour plot of |$|2dV of He Here r1 is fixed as ri = 0.6ao, where is the peak of / ( r j ) of He in figure 3.11 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 3.14: the contour plot of \^\2dV of H Here ri is fixed as r j = 1.4ao, where is the peak of /( r i) of H- in figure 3.11. 65 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 3.15: the contour plot of |iP|2dV of Ps Here r\ is fixed as r\ — 3.4ao, where is the peak of /( r i) of Ps- in figure 3.11. 66 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 3.7 T h e B ou n d sta tes vs. Z relation sh ip It is well known th a t the H~ only has one bound state (Nyden-Hill, 1977). An interesting question is: W hen does the bound state disappear for a H ” -like ion as the nucleus charge Z is reduced. If Z is small enough, for example Z - - - ■ 0. there is definitely no bound state. Auzinsh and Damburg (1999) estim ated Zc « 0.85355 for the cease of existence of a He — like atom . They used a very simple configuration of the two-electrons for the ground-state: Two electrons are located on a straight line and r 2 = — af). For the ground-state, they assign a = 1. Although this simple configuration gave close ground-state energies for large Z atoms, its ground-state energy is poor for the small Z ( it is — 0.5372a.u. for H ). So their estim ate of Z c is not reliable. From the ground states of the positronium negative ions and the H- ions, we know th a t one of the electrons is almost in the ground state of positronium or hydrogen and the other electron is very slightly tied to the core atom. Their bound state energies are only a little bit lower than the corresponding ground-state energies of the core atom . For H~, the ground-state energy is Eg = — 0.52775101a.w. and the ground-state energy of H atoms is E g = — 0.50a.m.; for P s- , it is Eg = — 0.262005a.it. and it is Eg = -0.25a.u. for the e+ e- system. So a useful criteria for the existence of a bound state of H - is th a t the lowest eigenenergy E (Z ) of H - is lower than the ground-state energy E = — [iZ2 j2 of the corresponding core. Then the critical value of Z , where the bound state begins to disappear, is the solution of f{Z ) = E ( Z ) + f i Z 2/ 2 = 0, (3.21) where fi = m em jv /(m e + m n ) is the reduced mass for the core system with mjv the mass of ’ ’nucleus” . Note th a t the Hamiltonian of a H - -like ion is H = T - Z- — ~ Z -— + ,^ 1 , (3.22) k i - r j v | |r2 -rAr| | n - r 2| 67 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. where T is the kinetic energy and is independent of Z. Then by Hellman-Feynman theorem (Feynman, 1935; Heilman, 1935), we have dE (Z ) d Z -'*'9*d z > -v g 1 1 ln-rjv| |r2-fW| - 2 < 4 > g \ \4 > g > ■ (3.23) Here we used the sym m etry associated with indistinguishability of the electrons. Combined with ( 3.21 ), we have / {Z) — — 2 < (f> g \ ^ \< j)g > . (3.24) Using the Newton-Raphson m ethod (Press et ai, 1992, Section 9.4), we can iterate to find the solution of the equation 3.21 as Z n+1 = Z n - f ( Z n) /f'( Z n) E{Zn) + n Z H 2 " fJ,Zn — 2 < (f>g\j~^~^\(j)g > ' ( - } In each step, we can solve the generalized eigenvalue problem 2.2 to get the energy E[Zn) and the wavefunction \< f> g > of the ’ ’ground” state. We present the ’ ’critical” values of Z for H _ and P s- basis size H- P s- 550 0.9111096086 0.9217995500 726 0.9110704655 0.9217988029 936 0.9110522732 0.9218012734 1183 0.9110399018 0.9218010141 1470 0.9110342172 0.9218018744 1800 0.9110303322 0.9218018258 2176 0.9110287020 0.9218021661 2601 0.9110275953 0.9218021643 Table 3.17: The critical value of Z where the bound state starts to disappear The coordinates were rotated by 6 — 15°. k = 0.7a,7 1 was used for H - and k = 0.225a7* for P s” For each basis size, we iterated 10 times to find the root with the Newton-Raphson method. 68 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. in table 3.17. The Zc converged to 0.9110 for a H ” -like ion and to 0.9218 for a Ps~-like system. Therefore when Z < Z c, there is no bound state for H ” -like ions. We can check the correctness of this statem ent by the complex-coordinate rotation method. We plot the eigenenergies of H” -like ions w ith different Z in the figure 3.17. It is evident th a t there are bound states for Z = 0.92, 0.93, and 1.0 and there is no bound state for Z = 0.90. In the case Z = 0.91 < Zc ( the left in the figure 3.16 ), the lowest eigenenergy, actually Re(Eg) = — 0.41378, is ju st at the right of the branch point E = —p Z 2/2 = — 0.41405 on the real axis. Actually it is a quasi-bound state ( see below). W hen Z = 0.915 > Zc ( the right in the figure 3.16 ), there is a bound state with Re(E) — — 0.4196 right at the left of the branch point E = — .418613 and not on the rotated continuum spectrum. Ze=0.915 0.00 -0.01 - -0.0 2 - -0.0 3 - -0.41 -0.40 •0.39 -0.38 -0.42 Ze=0.91 0. 00- -0.0 3 - -0.0 4 - -0.39 -0.40 Figure 3.16: E (Z ) on the complex energy plane of H ~ — like system: Z = 0.91 and Z — 0.915 The eigenenergies of H ” under the complex rotated coordinates. The horizontal axis is R e(E ) and the vertical axis is Im (E ). * — 0 = 15°;o — 9 = 30°. The • points in the figures are E = {—Z 2/ 2,0) which are the first branch point for II” like systems. We also presented the lowest eigenenergies of the II” system w ith nuclear charge Z on the complex energy plane in figure 3.18 and the same plot of P s” in figure 3.19. It is evident that the lowest eigenenergy starts to leave the real axis at Re(E) — —piZ2/ 2 rs — 0.41, W hich is with 69 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. respect to Z ~ 0.91 in figure 3.18. It is another evidence of the disappearance of the bound state. And the lowest eigenenergies sta rt to leave the real axis in figure 3.19 too. This tim e the curve leaves at Re(E) = — 0.2113, where Z « 0.92. We can make a conclusion th a t Z c, the solution of the equation 3.21, is the critical point: There is a bound state if Z > Zc; and there is no bound state if Z < Zc. 70 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Ze=0.90 0.00 -0 .2 0 - -0.40 -0.30 •0.25 .20 -0.15 •0.45 •0.35 -0. .05 Ze=0.91 .00 .05 .10 .15 .20 -0.45 -0.40 -0.35 •0.30 -0.20 -0.25 Z e = 0.92 -0.15 -0.35 -0.45 -0.40 •0.30 -0.25 -0.15 Z e=1.0 0.0 -0.1 -0.2 -0.6 -0.5 -0.4 -0.3 -0.1 - 0.2 -0.45 -0.40 •0.35 •0.25 -0.15 Figure 3.17: E (Z ) of H~-like systems on the complex— E plane The eigenenergies of H~ under the complex rotated coordinates. The horizontal axis is Re(E) and the vertical axis is Im (E ). * — 0 = 15° ;o — 0 = 30°. The • points in the figures are E = {—Z 2/ 2,0) which are the first branch point for II ~ like systems. 71 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 0.000 -0.002 - -0.004- -0.006- L U I -0.008- -0.010- -0.012- -0.014 -0.45 -0.40 -0.35 -0.30 -0.50 R efE j Figure 3.18: The lowest eigenenergy of H ~ — like system with atomic number Z on the complex energy plane. Here we rotate the coordinates w ith different angles: ▼ for 9 = 10°; A for 9 = 20°; V for 9 = 23°; • for 0 = 24°; A for 9 = 25°; ★ for 9 — 30°; and o for 0 = 40°. The 9th point from left corresponds to Z = 0.92, which is still a bound state. The 10th point from left corresponds to Z = 0.91, which is a quasi-bound state. 72 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 0 .000 - O ^ O -O ^ C K K K ^ -C b - 0.001 - -0.002 - 0 -0.003- -0.004- -0.005 - -0.006 -0.16 - 0.20 -0.18 -0.26 -0.24 - 0.22 Re(E) Figure 3.19: The lowest eigenenergy of Ps~ - like system with atomic num ber Z on the complex energy plane. Here we rotate the coordinates w ith different angles: ▼ for 9 = 10°; A for 9 = 20° ;★ for 9 = 30°; and o for 9 = 40°. The 8 th point from left corresponds to Z = 0.93, which is still a bound state. The 11th point from left corresponds to Z = 0.90, which is a quasi-bound state. 73 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 3.20: The ground-state probability distribution when one electron is fixed at its peak position. The left top one corresponds to Z = 0.92; the left bottom one corresponds to H - ;the right top one corresponds to He; and the right bottom one corresponds to Li+ w ith Z = 3. The horizontal axis is scaled to r — > Zr. 74 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 3.21: The ground-state probability distribution when one electron is fixed at its peak position. The top one corresponds to Z = 0.92 and the bottom one corresponds to H~. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 10 Figure 3.22: The ground-state probability distribution when one electron is fixed at its peak position. The top one corresponds to He and the bottom one corresponds to Li+ w ith Z = 3. 76 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C hapter 4 A u toion ization reson ances o f th e p ositroniu m n egative ion The positronium negative ion (e- ,e + ,e _ ) is one of the simplest three-body systems in which the only interactions are pure Coulombic. It is stable in the absence of positron annihilation (Hylleraas, 1947; Wheeler, 1946). This system was observed for the first time in the laboratory of (Mills, 1981, 1983). The structure of its autoionization resonances is very similar to the hydrogen negative ion system (e~,p,e~). There is a one-on-one correspondence between them (Ho, 1979). We can use the complex coordinate rotation m ethod to calculate the autoionization resonances. Here we will shortly introduce the complex coordinate rotation method (Balslev and Combes, 1971; Ho, 1983; Reinhardt, 1982; Simon, 1973). Then we will discuss the com putational results. 4.1 C om p lex-R otation M eth o d The complex coordinate rotation m ethod is widely used in studies of resonances in atomic and molecular physics. Those resonant phenom ena include the formation of quasi-bound state which has a lifetime long enough to be well recognized in the experimental and theoretical studies. At the tim e of formation (t = 0) the quasi-bounded state is the same as a localized bounded state in the space distribution. It decays as tim e evolves. The time evolvement of the resonance state can be described by ®reS(i) = e_ i^ ^ e S(0). (4.1) 77 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Unlike the stationary state, the energy of the resonance has to have an imaginary p art as Eres = E r - iT/2, (4.2) where E r is the energy and I is the lifetime of the resonance. This imaginary p art T /2 will force an exponential decay when tim e evolves. Unfortunately the Ham iltonian of a physical system is Herm itian so it has only the real eigenenergies. We can introduce complex coordinates into non-relativistic quantum mechanics to calculate those complex eigenenergies. We can rotate the real coordinate r to re 1 6 where 0 > 0. Then the H am iltonian H(r) will be changed to H (fel9). If all interactions are pure Coulomb forces, the Ham iltonian of the system is < « > * 1 i< j ■ > ' where e, is the charge of the particle i. After we apply the rotation of r) — > f)ete, the Hamiltonian becomes * (? .» , = -e-» E E («) i i<j 1 J ' The bound state »k(r) of H(r) includes an exponential decay p art e~cr. W hen we rotate the coordinate with 6 < 7r/2, e~cre'° still decays exponentially. The scattering states are linear combinations of e±lkr/r as r — ► oo. If k is changed to ke ~ l9 as r rotates, e±lkr will preserve the asym ptotic form. It is true since H {feie) = e~2ieT + e~i$V, (4.5) where T is the kinetic energy and V is the potential energy. W hen r — > oo, we have H {fei$) « e - 2ieT = {e-iek ) f / 2 m = [ k 'f/2 m . (4.6) 78 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Then the asym ptotic form of the scattering wavefunction is ei:tk're,l> /r e t6 = e±lfce t< > re 'e /re l9 = elkr I re1 0 at r — ► oo. Therefore the boundary conditions are preserved under the complex co ordinates rotation in both cases of bound states and scattering states. T hat suggests H(r) and H{reie) have the same real bound states, which means Eb0und{H{f)) = Ebound{H(fete)). B ut the continuum will rotate to the lower half of the complex energy plane and E cont — > Econte~2 % (> . Surprisingly the resonance states which are hidden in the spectrum of H (f) will become observable in the spectrum of H {fe‘6). The rigorous m athem atical theory of the above discussion was given by Combes (1971); Simon (1973) in the 1970s. More discussions can be found in Balslev and Combes (1971); Simon (1972); van W inter (1974) and Reed and Simon (1978, pp:51-60). Combes (1971) and Simon (1973) found th a t the Coulomb interaction is a special case of dilatation analytic potentials. Under a complex coordinate rotation, if it is restricted by 9 < 7r/2, the dilatation analytic potentials have some interesting properties. The bound states of H {fe ‘s) are independent of 9. Their eigenvalues are the same as those of the corresponding bound states of 77(f). W hen the system is ionized or dissociated, the scattering thresholds for different residual systems are also independent of 9. Each continuum series will be rotated to the lower half eigenenergy plane as arg(E — Ethreshoid) = — 29 since the continuum is determined by its kinetic energy instead of by the potentials. The spectrum of H {fel9) may contain some discrete complex eigenenergies which are located in the lower half complex energy plane. These eigenenergies are located at least above one of the rotated continuum series[see figure 4.1]. We can associate these eigenvalues to the resonance states. The resonance states will be independent of 9 when we vary 9 unless they are hidden by the continuum series. This gives us th e m ethod to study the resonance phenom ena in atomic and molecular physics. The position and the lifetime of the resonance will be obtained by solving the generalized eigenvalue problem (2.1) w ith a rotation of rz — > rte‘° in the basis function (2.12). 79 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ................ C om plex Rotation Figure 4.1: Spectrum transform under complex coordinate ratation upper p art is the spectrum of H(r). The lower p art shows the spectrum of H{reie). The x on the real axis represents the bound state. The • points are thresholds of continua. In the upper part, all continua are located on the real axis. They are rotated to lower half plane in the spectrum of H (re% e). The ® in the upper part shows a hidden resonance. It is exposed in the lower part. 80 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4.2 H am ilton ian o f P s w ith L = 0 The Ham iltonian of the positronium negative ion is different from the Hamiltonians of He and 11 ~ (2.2). The positron has the same mass as an electron. We need to include the kinetic energy of the positron e+ . Then the H am iltonian is defined by H (L = 0) = - ^ - V ? - ~ 1 -,- - 1 :,■ ■ ■ ■ ■ + ... 1 (4.7) 2m i 2m 2 2m p \rx - r :i| \r2 - r 3| \n - r 2| where , r 2 and t\3 are the spatial coordinates of electron 1 , electron 2 and positron respectively. All m asses m i , m 2 and m p are equal to one in Atomic units. We can introduce the inter particle coordinates as rtj = |f) — fj\. After removing the center-of-mass motion, the Hamiltonian ( 4.7 ) becomes (Ho, 1979) J _ J _ __________ 2 ________________________________—___— ) ~ m 1 + m 2 d rf2 + r 12 d r 12 m 2 + m p dr \ 3 + r 23 dr 23 _ ( J _ + ± ) ( ^ - + _ 1 (r? 3 -r2 23 + r? 2) d 2 m i m p d rl3 r 13 d r i3 m i 2 n 3r i 2 dr 13dr l2 1 (t* 2 3 ~ r~i3 + r?2) < 9 2 _ 1 (r?3 - r f 2 + r | 3) d2 m 2 2r 23r i 2 dr 2 3dri 2 m p 2 rX 3r 23 dr 13dr 23 ------ + — . (4.8) f 13 r 23 r i 2 In contrast to the Ham iltonians of H~ and He, there are cross term s in derivatives w ith respect to different interparticle coordinates. These cross term s arise from the fact th a t the center of mass of the three-body system does not sit at one of the comers of the triangle whose corners are the individual particles. In H _ and He, the mass of nuclei are huge compared to the mass of electrons; and the kinetic energy of the nucleus can be ignored in most of theoretical calculations. In the case of L = 0, the basis function ( defined by 2.12 and 2.74) is defined on the inter-particle coordinates. It can be used here to calculate the element of the P s- ’s Ham iltonian m atrix H _. 81 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4.3 R esu lt and D iscu ssion The autoionization states of the positronium negative ion were firstly calculated by Ho (1979) w ith the complex coordinate rotation m ethod. Ho studied the lowest resonance states with a Hylleraas-type wave function. In a series of publications on P s- (Bhatia and Ho, 1990; Ho, 1979, 1984; Ho and Bhatia, 1991, 1993; Ivanov and Ho, 1999), he studied the resonance structures extensively. Papp et al. (2002) proposed a new m ethod to treat the resonance states of the three- body system. The Faddeev-Merkuriev integral equations of the three-body Coulomb system were solved. Papp et al. (2002) used the Coulomb-Sturmian separable expansion technique and solved the homogeneous Faddeev-Merkuriev integral equations for complex energies. P app’s results of the L = 0 doubly excited resonance states 1S e and 3S e agreed with Ho’s early results for the lowly excited resonances. B ut there are big differences between their results for the more highly excited singlet resonances resonances and the first one of the triplet resonances. Using the perim etric-Sturm ian type basis function ( 2.12 ), we applied the complex-coordinate rotation m ethod to study the doubly excited resonances of the L = 0 series. We constructed H _ for P s- with the Hamiltonian (4.8). Then by solving the generalized eigenvalue problem of (2.1) we obtained the complex eigenenergies for the autoionization resonances. Due to the numerical linear dependence problem, we used two different m ethods to solve this generalized eigenvalue problem. In one m ethod we used one length scale ki = k 2 in the basis functions ( 2.73). W hen we only use one length scale, there is no numeric linear dependence. The disadvantage is th a t this single length scale has to describe the entire dynam ic spatial range. Another approach is to use the random perturbation m ethod to take care of the numerical linear dependence. We can choose two different length scales ki and k2 to describe the P s- system. Using two length scales has a big advantage in the description of the entire dynamic space. B ut its disadvantage is th a t it may lose some accuracy in the last few digits of double precision. 82 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. State 1 se Ho (1984) E real r x io -s (Papp et al., 2002) E r eal r x 10-3 This work E real T x 10-3 2 s 2 s 0.152 060 8 0.086 -0.151 9 0.085 -0.152 060 8 0.086 0 2s3s 0.127 30 0.02 -0.127 3 0.017 -0.127 298 0.017 34 3s3s 0.070 683 0.15 -0.070 7 0.15 -0.070 683 7 0.149 3s4s 0.05969 0.11 -0.059 68 0.11 -0.059 692 0.105 4s4s 0.040 45 0.24 -0.040 428 0.26 -0.040 427 8 0.260 A-pAp 0.035 0 0.3 -0.035 02 0.26 -0.035 020 7 0.263 4s5s 0.034 63 0.34 -0.034 62 0.32 -0.034 621 0.315 5s5s 0.0258 0.45 -0.026 06 0.21 -0.0260 618 0.211 5p5p 0.023 43 0.14 -0.0234 0.1 -0.0234 456 0.0869 Table 4.1: The 1S doubly excited states of P s" w ith one-length scale m ethod We used a single length k\ = k 2 = 0.135oq 1 to calculate the values in this table. The basis included norm al Laguerre polynomials up to 26th order for all u, v and w variables. The complex rotation angle is 0 = 22.5°. The size of the basis is N — 10206. *The unit of energy in this chapter is always in Ry. In the first method, where k\ = k2, we used 10206 basis functions to calculate the singlet l S e resonances of P s- . The fc’s were set to O.l'iaa^1. The basis was rotated by an angle 6 = 20.0°. The basis included Laguerre polynomials of order up to 26 for singlet resonances and up to 27 for the triplet resonances. By solving the complex rotated eigenvalue problem, we find the energy of the ground state of P s- to be Eground = — 0.524010418978 + 0.000000485419* Ry. The lowest variational ground energy of P s- is E ground = — 0.524010140465 Ry. (Frolov and Yeremin, 1989; Ho, 1993). Note th a t we lose the accuracy at the 7th digit since we used a com plex-rotated basis and the imaginary p art of Eground is nonzero at the 7th digit. The agreement between them ( up to the 6th digit ) means th a t the basis is well chosen to describe the P s- system. It also shows th a t the bound state is almost unchanged under complex rotation. Table 4.3 shows our results of the singlet 1S resonances w ith one length scale(/ci = k2) com pared to the results of Ho (1984) and Papp et al. (2002). It is clear th a t our results show near perfect agreement w ith the results of Papp et al. (2002). And for the 5s5s resonance, our work confirms the resonant width P = 0.00021 in P app et al. (2002) instead of the value T = 0.00045 in Ho (1984). As for the other discrepancy, the 5.s5.s 1S resonance, our work is closer to P app’s 83 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. State Ho (1984) E real r x 10“3 Papp et al. (2002) E real r x io - 3 This work E real P x 10“ 3 2s3s 0.127 06 0.01 -0.127 06 0.000 006 -0.127 074 0.000 006 3 3s4s 0.058 73 0.02 -0.058 73 0.000 37 -0.0587 413 0.000 375 4s5 s 0.034 15 0.02 -0.034 15 0.001 -0.034 202 0.001 42 Table 4.2: The 3S doubly excited states of P s- with one-length scale m ethod We used a single length fci = fc 2 = 0.135ag 1 to calculate the values in this table. The basis included norm al Laguerre polynomials up to 27th order for all u, v and w variables. The complex rotation angle is 0 = 22.5°. The size of the basis is N = 10584. The energy unit is Ry. results than Ho’s. It is curious that our estim ates of most of the resonance positions agree bet ter w ith Ho’s (Ho, 1984) th an with P app’s (Papp et al., 2002). For the highest resonance 5p5p discovered so far, this work finally converged to T = 0.00008689726 ( in Table 4.3 ). Considering the numerical stability, our resonance width can be rounded to T = 0.000087, which is a little bit narrower than the T of P app (Papp et al., 2002) and Ho (Ho, 1984). Table 4.3 shows our results for some triplet 3S resonances {k\ = fc 2) compared to Ho’s (Ho, 1984) and P ap p ’s (Papp et al., 2002) results. For the lowest triplet resonance 2s3s, Ho’s w idth is very big compared to P ap p ’ s result and our work. Ho (1984) did not give a very accurate width for the next triplet resonance 3s4s. Ho stated th a t his m ethod is not good for calculating very tiny widths. We present very good agreement on the w idths of the 2,s3.s and 3.s4s resonances w ith P app’s work. There is a big difference for the 4s5s resonance. O ur result P = 0.00000142 R y is almost one and a half tim es P app’s result P = 0.000001 Ry. Since no more results are available to make comparison, the difference has to be resolved in further works. We only confirm th a t the w idth of this triplet resonance should be very small instead of Ho’s P = 0.00002 R y . ( However we reproduced the small value of this width using the second m ethod — see below. ) In our second m ethod, we applied the random perturbation to the basis and then took the full power of two different length scales. The results for the singlet and the triplet states were computed in a basis w ith N = 4913 perim etric sturm ian type basis functions. The basis included Laguerre polynomials of order up to 16 for both singlet and triplet resonances. The length scales 84 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Ige single scale E r r 2 scales Er r 2 s 2 s 2s3s 3s3s 3s4s 4s4s 4p4p 4s5s 5s5s 5p5p -0.152060884679 -0.127298350648 -0.070683770720 -0.059692292237 -0.040427841795 0.035020714384 -0.034621870620 -0.026061860358 -0.023445642151 0.000086068981 0.000017357022 0.000149314094 0.000105423568 0.000260103704 0.000262943114 0.000315674225 0.000210593856 0.000086899726 -0.152060808190 -0.127298352711 -0.070683771475 -0.059692292373 -0.040427841644 -0.035020714249 -0.034621870632 -0.026061860366 -0.023445642136 0.000085954137 0.000017331569 0.000149316509 0.000105425558 0.000260103778 0.000262942967 0.000315674194 0.000210594103 0.000086899704 Table 4.3: The 1S resonances of P s- : a comparison between two different m ethods The two-length scales m ethod used a basis with fcl = .145oq 1 ,k2 = 0.095a( 7 1 and the coordinates are rotated by 0 = 20°. The basis also included the norm al Laguerre polynomials up to 16th order. The basis size is N = 4913. The one-length scale m ethod used a length scale k = .135ag 1 and a rotation of 9 = 22.5°. It included the normal Laguerre polynomials up to 26th order. Its size is IV = 10206. The energy unit is Ry. 3 se One scale E r r tw o scales Er r 2s3s 3s4s 4s5s -0.127074707408 0.000000006283 -0.058741373144 0.000000375804 -0.034202344849 0.000001424344 -0.127074707420 0.000000006271 -0.058741373144 0.000000375795 -0.034202344856 0.000001424366 Table 4.4: The 3 S' resonances of P s- : a comparison between two different m ethods The two-length scales m ethod used a basis w ith kl = .145ag 1,k2 = 0.095<2q 1 and the coordinates are rotated by 0 = 17.5°. The basis also included the normal Laguerre polynomials up to 16th order. The basis size is N = 4913. The one-length scale m ethod used a length scale k = .135<Zq 1 and a rotation of 0 = 22.5°. It included the normal Laguerre polynomials up to 27th order. Its size is N = 10584. The unit of energy is Ry. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. were set to kx = 0.145 a^ 1, and & 2 = 0.095 a ^ 1 The coordinate was rotated by 9 = 20° for the singlet resonances. It was rotated by 6 = 17.5° for the triplet resonances. Here we checked the ground energy of P s- again; we got E ground = — 0.524010624431 — 0.000000288212*, which is very close to the value in (Frolov and Yeremin, 1989; Ho, 1993). The agreement between the positions and widths of all resonances from two different m ethods is perfect. The differences are all less than one per thousand. For the 4 s 5s triplet resonance, the width is I' — 0.000001424366 Ry, which is same as the w idth of one length scale m ethod up to 9th digit. We checked the convergence of the positions and widths of the resonant states. A series of calculations were conducted on different basis sizes. We list the positions and widths of the singlet resonances 5s5.s and 5p5p in table 4.3 for the one-length-scale m ethod and in table 4.3 for the random perturbation two-length-scale m ethod. The other singlet resonances converged very quickly in both methods. For example th e singlet 2s3s resonance converged to the value E r — — 0.127298354518 and T = 0.000017355113 on a one length scale basis w ith a size of N = 1470 , which is very close to the final value listed in table 4.3. The convergence of the triplet resonances are listed in table 4.3 for the one-length-scale m ethod and in table 4.3 for the random perturbation two-length-scale method. Comparing the convergence speed of the two methods, it is evident th a t the two-length-scale method, where a random perturbation takes care of the numeric linear dependence, is much faster than the one-length-scale method. The two-length-scale method converged at the basis size of N = 2744 for singlet resonances. It converged very quickly in all triplet resonance states. This did not mean th a t the triplet state is easier to calculate. The reason th a t we need more basis functions for the singlet resonances is th a t 5s5s and 5p5p are higher energy configurations with more nodes. In both cases of singlet and triplet resonances, a much bigger basis was required for the one-length-scale m ethod. If we vary the length scales for each resonance, the basis size could be reduced significantly. On the other hand, the fact that all resonances can be calculated 86 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. by using the same length scales shows th a t the results are reliable. It shows the convergence is independent of special length scales. One interesting thing is th at the 3s4s state converged very quickly but the lowest one, the 2s3s resonance, is the slowest. We tried to vary the length scales in both m ethods—to see the changes of the convergence speed for the triplet resonances. A large value of k corresponding to a small length scale, which should favor the lowest excited state. B ut the 2s3s is always the slowest one and the 3s4s is always the fastest one. The problem can be related to the extremely narrow 2s3s width. It is very difficult to get th a t accuracy to the m agnitude of 10- 9 . As mentioned in the brief summ ary of the complex coordinate rotation method, the position and the w idth of a resonance state should be independent of the rotation angle 9 under the complex coordinate rotation (Balslev and Combes, 1971; Ho, 1982; Simon, 1973). A resonance state will disappear only when a continuum passes through it. The continuum spectrum will sweep along the rotation angle. B ut this is only true when an infinite size basis is used. W hen the basis is finite, a very small dependency between the eigenenergies of resonances and the rotation angle can show up. A really converged resonance state should be stable even if the basis has a finite size. The other eigenenergies, which belong to the continuum spectrum , are rotated as the angle changes. We list the eigenenergies of singlet states of P s- w ith different values of 9 in table 4.3. Except the two resonances, No. 20 and No. 21 in the table which are the 2s2s and 2s3s resonances, all eigenenergies changed significantly. The 2s2s and 2.s.3s singlet resonances are shown in figure 4.2. The circled point is the autoionization resonances which are hidden when we use a real coordinate. The * points are the eigenenergies for 9 = 15°. The • points correspond to 9 = 25°. From this figure we can see the rotation of the continua clearly. The angle dependence of the resonance states were shown in table 4.3 for the one length scale m ethod and in table 4.3 for the two length scale m ethod. For the single length scale method, we calculated the resonances for 9 = 15°, 25° and 35° on a basis with k\ — & 2 = 0.135<Iq 1 at the size 87 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reproduced w ith permission o f th e copyright owner. Further reproduction prohibited without permission. Nodes N 5s5s Ereal r 5p5p Ereal r 10 729 -0.025114959735 0.001739982192 -0.023333402812 -0.000182163886 11 936 -0.026275311195 0.000724565124 -0.023379279840 0.000106541091 12 1183 -0.026147892535 0.000199689197 -0.023430067433 0.000115069349 13 1470 -0.026067279117 0.000169516534 -0.023442239638 0.000094658506 14 1800 -0.026057697011 0.000205996468 -0.023444023729 0.000088969234 15 2176 -0.026061018267 0.000211826328 -0.023444999033 0.000089016767 16 2601 -0.026061855153 0.000210983144 -0.023445819997 0.000087917603 17 3078 -0.026061879052 0.000210627619 -0.023445806370 0.000087041411 18 3610 -0.026061860728 0.000210587188 -0.023445695479 0.000086861088 19 4200 -0.026061859062 0.000210589673 -0.023445650876 0.000086867812 20 4851 -0.026061860193 0.000210592944 -0.023445640972 0.000086888193 21 5566 -0.026061860376 0.000210594114 -0.023445640732 0.000086897303 22 6348 -0.026061860323 0.000210593984 -0.023445641586 0.000086899679 23 7200 -0.026061860465 0.000210594015 -0.023445642015 0.000086899930 24 8125 -0.026061860517 0.000210594192 -0.023445642135 0.000086899822 25 9126 -0.026061860411 0.000210594076 -0.023445642125 0.000086899754 26 10206 -0.026061860358 0.000210593856 -0.023445642151 0.000086899726 Table 4.5: Convergence of the 1S doubly excited states of Ps with one-length scale m ethod The basis used a length scale k\ = & 2 = 0.135<Iq *. The rotation angle is 6 = 22.5°. Reproduced w ith permission o f th e copyright owner. Further reproduction prohibited without permission. Nodes N 5s5s Ereal r 5p5p Ereal r 8 729 -0.026583748685 -0.001703426716 -0.023401853333 0.000045403893 9 1000 -0.025965684659 0.000086073926 -0.023439948593 0.000093995194 10 1331 -0.026066710493 0.000149335145 -0.023442283092 0.000086043209 11 1728 -0.026062695661 0.000211100056 -0.023446620585 0.000087348320 12 2197 -0.026061859067 0.000210570902 -0.023445607714 0.000086880969 13 2744 -0.026061860983 0.000210594149 -0.023445643425 0.000086899772 14 3375 -0.026061860191 0.000210593548 -0.023445641971 0.000086899677 15 4096 -0.026061860739 0.000210593699 -0.023445642139 0.000086899740 16 4913 -0.026061860366 0.000210594103 -0.023445642136 0.000086899704 Table 4.6: Convergence of the lS doubly excited states of Ps with two-length scales m ethod The basis used two length scales k\ = 0.145a7 1 and k2 — 0.095af 7 L . The rotation angle is 9 = 20°. 0.001 0.000 - 0.001 - - 0.002 - -0.003 - m -o.oo4 - s f -0.005- -0.006 - -0.007 - -0.008- -0.009 -0.155 -0.150 -0.145 -0.140 -0.135 -0.130 -0.125 -0.120 Re(E) Figure 4.2: Ps~ autoionization resonances ( 2.s2.s and 2.s.3.s ) on complex energy plane. The circled points are the 2s2.s ( E = — 0.15206Ry. and T = 0.0000857/fa/. )and the 2s3.s (E = — 0.127298/?//. and T = 0.0000173/?y.) resonances. Both of them are above the first continuum w ith a branch point located at E = -0.5Ry. which is not plotted in the graph. • points belong to 6i = 15° and * points belong to 62 = 25°. Note th a t we used different scales on different axes so th a t the angles are not shown exactly. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. of N = 5566. For the two length scales we calculated the resonances for 0 — 10°, 20° and 30° on a basis with k\ = 0.145cIq 1 and fc 2 = O.OQSa^1 at the size of N = 4913. As shown in the tables, the resonances are pretty stable over a wide range of the rotation angles. This fact means our results are well-converged and reliable. 91 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reproduced w ith permission o f th e copyright owner. Further reproduction prohibited without permission. Nodes N 2s3s Ereal T x 10“ 9 3s4s E real r x 10~7 4s5s f^real r x io ~ 6 Papp -0.12706 6 -0.05874 3.7 -0.0342 i 15 1920 -0.127074690638 62.087 -0.058741436387 4.29395 -0.034202145781 15.140022 16 2312 -0.127074706495 30.245 -0.058741393156 3.66612 -0.034203715811 5.281144 17 2754 -0.127074709100 14.366 -0.058741377258 3.66531 -0.034203114078 1.921535 18 3249 -0.127074708634 8.465 -0.058741373371 3.72170 -0.034202602507 1.253204 19 3800 -0.127074707975 6.568 -0.058741372886 3.74864 -0.034202397313 1.278197 20 4410 -0.127074707625 6.199 -0.058741373001 3.75657 -0.034202344974 1.365278 21 5082 -0.127074707469 6.188 -0.058741373099 3.75815 -0.034202339397 1.408586 23 6624 -0.127074707408 6.262 -0.058741373142 3.75811 -0.034202343889 1.424808 27 10584 -0.127074707408 6.283 -0.058741373144 3.75804 -0.034202344849 1.424344 Table 4.7: Convergence of the 3 S' doubly excited states of Ps with one-length scale m ethod The basses used a single length scale — 0.135aQ 1. The rotation angle is 0 = 20°. CO to Reproduced w ith permission o f th e copyright owner. Further reproduction prohibited without permission. Nodes N 2s3s E r T x 10"9 3s4s E r r x io~7 4s5s E r r x io ~ 6 Papp -0.12706 6 -0.05874 3.7 -0.0342 i 10 1331 -0.127074707210 7.507 -0.058741373619 3.75101 -0.034202322325 1.342830 11 1728 -0.127074707492 6.598 -0.058741373107 3.75471 -0.034202354915 1.416333 12 2197 -0.127074707431 6.165 -0.058741373161 3.75678 -0.034202345906 1.424009 13 2744 -0.127074707421 6.303 -0.058741373142 3.75794 -0.034202344919 1.423720 14 3373 -0.127074707439 6.299 -0.058741373193 3.75916 -0.034202344678 1.424490 15 4096 -0.127074707412 6.297 -0.058741373145 3.75805 -0.034202344842 1.424289 16 4913 -0.127074707420 6.271 -0.058741373144 3.75795 -0.034202344856 1.424366 Table 4.8: Convergence of the 3S doubly excited states of Ps w ith two-length scales m ethod The basses used two length scales k\ = 0.145ao 1 and k 2 = 0.095<Iq l . The rotation angle is 6 = 17.5°. 9 = 15° Re(E) Im (E ) 9 = 20° Re(E) Im {E) 19 -0.24332303392383 -0.15104248043252 -0.15291811924433 -0.40558246052203 20 -0.15206078386057 -0.00004286269284 -0.15206088703636 -0.00004302548159 21 -0.12729836214778 -0.00000863393891 -0.12729835205995 -0.00000867671839 22 -0.12520629985024 -0.00002143052490 -0.12523179751174 -0.00001230368277 23 -0.12476543915551 -0.00030989419119 -0.12497086992602 -0.00042516766730 24 -0.12453870697173 -0.00026247671991 -0.12465403046123 -0.00040363200106 25 -0.12402931131092 -0.00083046787023 -0.12449940080362 -0.00120059203044 26 -0.12390857993868 -0.00062020783261 -0.12418063317684 -0.00095416857501 27 -0.12302057019002 -0.00112289749110 -0.12377254144665 -0.00229653151793 28 -0.12294195779477 -0.00156098317894 -0.12351223940363 -0.00172866602261 Table 4.9: An illustration of the sensitivity of eigenenergies of the rotated Hamiltonian H (retff) of P s" to the value of the angle 9. Note eigenvalues 20 and 21 are insensitive since they are the autoionization resonances, 2s2s and 2s3s X S resonances respectively. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reproduced w ith permission o f th e copyright owner. Further reproduction prohibited without permission. 1s e 0 = 15° Er r 0 = 25° Er r 0 = 35° E r r 2 s 2 s 2s3s 3s3s 3s4s 4s4s 4p4p 4s5s 5s5s 5p5p -0.152060783861 0.000085725386 -0.127298362148 0.000017267878 -0.070683772273 0.000149332775 -0.059692288592 0.000105436498 -0.040427838838 0.000260102632 -0.035020713395 0.000262945128 -0.034621869431 0.000315672209 -0.026061850988 0.000210623483 -0.023445641406 0.000086901808 -0.152060887036 0.000086050963 -0.127298352060 0.000017353437 -0.070683770818 0.000149314932 -0.059692292245 0.000105424177 -0.040427841759 0.000260103658 -0.035020715684 0.000262943501 -0.034621870629 0.000315674737 -0.026061860413 0.000210594128 -0.023445638344 0.000086903242 -0.152060887051 .000086068925 -0.127298352218 0.000017357189 -0.070683770758 0.000149314693 -0.059692292384 0.000105424341 -0.040427841730 0.000260104075 -0.035020684008 0.000263090103 -0.034621860126 0.000315664018 -0.026061873064 0.000210652715 -0.023445694478 0.000086555608 2S e Er T Er r Er r 2s3s 3s4s 4s5s -0.127074707405 0.000000006293 -0.058741373144 0.000000375804 -0.034202344857 0.000001424325 -0.127074707405 0.000000006286 -0.058741373145 0.000000375803 -0.034202345053 0.000001424491 -0.127074707396 0.000000006178 -0.058741373193 0.000000375822 -0.034202342427 0.000001447982 Table 4.10: The rotation angle dependence of the 1S and 3S doubly excited states of Ps~ w ith one-length scale method. The length scale hi = & 2 = 0.135aQ 1 was used. The basis included the normal Laguerre polynomials up to the order of 21st for 1S states and of 25th for 35 states . The size of the basis is N = 5566 for singlet and N = 8450 for triplet states. to O i Reproduced w ith permission o f th e copyright owner. Further reproduction prohibited without permission. l S e e = io° E r r 6 = 20° E r r 6 = 30° Er r 2 s 2 s 2s3s 3s3s 3s4s 4s4s 4p4p 4s5s 5s5s 5p5p -0.152060893590 0.000086041037 -0.127298357381 0.000017357831 -0.070683769434 0.000149316742 -0.059692293952 0.000105422440 -0.040427840156 0.000260101754 -0.035020712311 0.000262964651 -0.034621869874 0.000315674376 -0.026061958426 0.000204848133 -0.023446949672 0.000087310022 -0.152060929576 0.000086113789 -0.127298351865 0.000017373897 -0.070683770964 0.000149311304 -0.059692291630 0.000105421804 -0.040427841950 0.000260104206 -0.035020714687 0.000262944133 -0.034621870646 0.000315674249 -0.026061860378 0.000210593825 -0.023445642114 0.000086899856 -0.152061957247 0.000089645000 -0.127298162299 0.000018232834 -0.070683845754 0.000149093087 -0.059692176557 0.000105458918 -0.040427916860 0.000259994489 -0.035020799409 0.000262831077 -0.034621918374 0.000315728848 -0.026061933581 0.000211148278 -0.023445650619 0.000087021346 3S e 2s3s 3s4s 4s5s -0.127074707465 0.000000006344 -0.058741372968 0.000000375721 -0.034202344668 0.000001424141 -0.127074707412 0.000000006278 -0.058741373144 0.000000375819 -0.034202344841 0.000001424336 -0.127074707403 0.000000006342 -0.058741373128 0.000000375761 -0.034202345148 0.000001424643 Table 4.11: The rotation angle dependence of the 1S and 3S doubly excited states of Ps~ with two-length scale method. The length scales k\ = 0.135og 1 and & 2 = 0.095ag 1 were used. The basis included the normal Laguerre polynomials up to the order of \Ath for 1S states and of 16th for 3S e states. The size of the basis is N = 3375 for singlet and N = 4913 for triplet states. co a > B ibliography Ajmera, M. P., and K. T. Chung, 1975, “Photodetachm ent of negative hydrogen ions,” Phys. Rev. A 12(2), 475. Auzinsh, M., and R. Damburg, 1999, “A simple formula for ground state energy of a two-electron atom ,” Latv. J. Phys. Tech. Sci. 5, 22, also see arXiv:quant-ph/990516. Bachau, H., 1984, “Position and w idths of autoionising states in the helium isoelectronic sequence above the n= 2 continuum,” J. Phys. B 17, 1771. Balslev, E., and J. M. Combes, 1971, Commun. M ath. Phys. 22, 280. B artlett, J. H., 1937, “The helium wave equation,” Phys. Rev. 51, 661. Bethe, H. A., and E. E. Salpeter, 1957, Quantum mechanics of one- and two-electron atoms (Springer-Verlag). Bhatia, A. K., and Y. K. 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Creator Li, Tieniu (author)

Core Title Accurate calculations of bound- and quasibound-state energies of some three -body systems, and cross sections for photodecay

School Graduate School

Degree Doctor of Philosophy

Degree Program Physics

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

Tag OAI-PMH Harvest,physics, atomic

Language English

Contributor Digitized by ProQuest (provenance)

Advisor Shakeshaft, Robin (committee chair), Hass, Stephen (committee member), Kresin, Vitaly V. (committee member), Kunc, Joseph (committee member), Shanii, Maxim Ol' (committee member)

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c16-391506

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