EXPLORATION OF THE TWO-ELECTRON CONTINUUMPROBLEM USING R-MATRIX METHODSbyKURT WAYNE MEYER
B. S., University of Nebraska, 1988
M. S., George Washington University, 1991
M. S., University of Colorado, 1994

A thesis submitted to the
Faculty of the Graduate School of the
University of Colorado in partial ful llmentof the requirements for the degree ofDoctor of Philosophy
Department of Physics
1997This thesis for the Doctor of Philosophy degree byKurt Wayne Meyer
has been approved for the
Department of
PhysicsbyChris H. Greene

John Cooper

Date
The nal copy of this thesis has been examined by the signators, and we nd that both thecontent and the form meet acceptable presentation standards of scholarly work in theabove mentioned discipline.

iii
Meyer, Kurt Wayne (Ph. D., Physics)
Exploration of the Two-electron Continuum Problem Using R-Matrix Methods
Thesis directed by Professor Chris H. Greene
If we ever hope to describe complicated physical systems in nature, it is imperative
to rst understand much simpler systems. In this work, we explore the fundamental problemof two electron escape processes, which is itself a particular case of the more general threebody problem in nonrelativistic quantum mechanics. A large body of experimental andtheoretical work performed in the last 80 years provides strong evidence that this theory
adequately describes most processes in atomic systems. Our study is also important in thesense that the equation which describes our system of three particles is one of the simplest
nonseparable partial di erential equations in quantum physics. For any n-particle system(with n  3) interacting through position or velocity dependent forces (including classicalsystems), a nonseparable partial di erential equation is required to describe the system andallow for exchange of energy and momentum between the particles.
In this work, we adopt a method (the eigenchannel R-matrix method) with a provencapability to accurately describe single electron escape. We make a major extension of thismethod to treat escape of two electrons here. We focus on the process of double photoejectionof helium and H . A single photon is absorbed by the target, transferring enough energy forboth electrons to escape. However, one observes that single electron escape occurs most of
the time, while double electron escape happens much less frequently. Since the interactionof the electromagnetic eld with the atom is described by a single particle operator, double
electron escape can only occur due to interactions between the two electrons. One of the
major goals of this work is a deeper understanding of the role of interactions between thetwo electrons in the resulting single or double escape process.
Our initial calculations of helium double photoionization cross sections were in fairagreement with then existing experimental values. However, these calculations also exhibitediv
an internal inconsistency caused by inaccurate wavefunctions that were used to describe theinitial and nal states of our system. A subsequent application of the eigenchannel R-matrix
approach to a simpli ed model of electron-hydrogen (and electron-He+) scattering nevertheless indicated that this method could describe two electron escape processes accurately. To
overcome the inconsistency in our original calculations of helium double photoionization, we
next implemented the eigenchannel R-matrix method using a nite element basis set. Application of this approach to two electron photoejection in helium and H reduced the internal
inconsistency by more than an order of magnitude. The newest helium calculations are ingood agreement with the most recent theoretical and experimental studies. Furthermore,calculations of partial cross sections for the single escape processes h + He ! He+ (n) +e agree with recent experimental values for n = 2 6.

DEDICATION
Not in the victory, but the action:
not the goal but the game:in the deed the glory.
-Hartley Burr Alexander

I would like to dedicate this work to all of the men and women of the past andpresent who have endured and continue to endure hardships, and who have made and continue to make sacri ces for the sake of creating a better world for the rest of us.

vi

ACKNOWLEDGEMENTS
I would like to acknowledge the National Science Foundation for supporting thisresearch, and also the National Center for Supercomputer Applications for use of their computer resources.
On a more personal note, I would like to acknowledge all those who have had apositive in uence on my life. This includes the many classmates and teachers with whom I
have had contact over the past 26 years of my formal education. I would also like to thankmy current advisor Chris Greene and past advisors for giving me the opportunity to workon interesting problems and the chance to develop my skills and abilities. In addition, I
acknowledge several members in the atomic physics community with whom I have had thepleasure of working, especially Igor Bray, James Samson, and Janine Shertzer.
Some of my most productive collaborations have come from within the group. I
gratefully acknowledge the support of the group over the years, including postdocs Francis
Robicheaux, Je Stephens, Gregory Miecznik, John Bohn, and Hugo van der Hart, and fellow
graduate students Bob Wood, William Clark, Brett Esry, Jim Burke, and Brian Granger. I
would also like to thank Chela Kunasz and Anne Hammond for providing computer supportand helping me out on many occasions.
Throughout my life I have been blessed with the support and encouragement of avast network of aunts and uncles and roughly 80 cousins. I would like to especially mention the more immediate members of my family, including my brothers Je and Kevin andsister-in-law Madeline, who have provided me with much helpful advice, encouragement,
comraderie, and fun over the years, and my nieces Marie and Megan, and nephew Michael
who have been constant reminders of the joys of life. Finally, I would like to thank myviiparents for providing my daily source of inspiration. For the often dicult learning experiences on the farm to their constant examples of professionalism, dedication, compassion,and charity, I would like to especially thank my parents for teaching me the most importantlessons of life.

CONTENTS
CHAPTER
1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

2 TWO ELECTRON ESCAPE: PUTTING THE PROBLEM IN PERSPECTIVE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9

2.1 Preliminary Theory and Initial Assumptions . . . . . . . . . . . . . . .

9

2.2 Simpli ed Descriptions of Two Electron Escape . . . . . . . . . . . . .

12

2.3 Discrete Methods for Treating the Double Continuum . . . . . . . . . .

14

2.4 Dependence of Double Escape on Projectile Energy . . . . . . . . . . .

21

2.5 Experimental Progress in Measuring Double Photoionization Cross Sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

28

2.6 Observables and Targets as Probes of Electron-Electron Correlations .

29

3 INITIAL APPROACH: EIGENCHANNEL R-MATRIX CALCULATION WITH
A HYDROGENIC BASIS . . . . . . . . . . . . . . . . . . . . . . . . . . . .

32

3.1 Double Photoionization of Helium . . . . . . . . . . . . . . . . . . . . .

32

3.2 Simpli ed Model of Electron Scattering . . . . . . . . . . . . . . . . . .

40

3.2.1 Discussion of the Temkin-Poet Model . . . . . . . . . . . . . . .

42

3.2.2 R-matrix Application to Electron Scattering . . . . . . . . . . .

43

3.2.3 Averaging Techniques . . . . . . . . . . . . . . . . . . . . . . . .

46

3.2.4 Frame Transformation . . . . . . . . . . . . . . . . . . . . . . .

50

3.2.5 Application to e-H and e-He+ Scattering . . . . . . . . . . . . .

57

4 FINITE ELEMENTS: A NEW APPROACH . . . . . . . . . . . . . . . . . .

65

4.1 Method of Finite Elements . . . . . . . . . . . . . . . . . . . . . . . . .

65

4.2 Double Photoejection Cross Sections of He and H

73...........

ix
4.3 Other Observables Associated with Two Electron Escape . . . . . . . .

86

5 CONCLUDING REMARKS . . . . . . . . . . . . . . . . . . . . . . . . . . .

97

BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

100

APPENDIXA EQUIVALENCE OF DIPOLE OPERATOR FORMS IN A FINITE VOLUME 108
B DETAILS OF OUR FINITE ELEMENT IMPLEMENTATION . . . . . . .

115

x

FIGURESFIGURE
1

R-Matrix Con guration Space . . . . . . . . . . . . . . . . . . . . . . . . .

192Discrete Spectrum of Target Eigenstates . . . . . . . . . . . . . . . . . . .

34

3

Gailitis Averaging Technique . . . . . . . . . . . . . . . . . . . . . . . . . .

38

4

He Double Photoionization Cross Section . . . . . . . . . . . . . . . . . . .

41

5

He Double to Single Photoionization Ratio . . . . . . . . . . . . . . . . . .

41

6Energy Dependence of Box Size . . . . . . . . . . . . . . . . . . . . . . . .

47

7Illustration of Box Averaging . . . . . . . . . . . . . . . . . . . . . . . . . .

47

8Transformation Factor for Frame Transformation . . . . . . . . . . . . . .

54

9

Frame Transformation and Convolution Method . . . . . . . . . . . . . . .

56

10 e-H Singlet Elastic and Inelastic 1s 2s Cross Sections . . . . . . . . . . .

59

11 e-He+ Singlet Elastic and Inelastic 1s 2s Cross Sections . . . . . . . . .

60

12 e-H Singlet Ionization Cross Section . . . . . . . . . . . . . . . . . . . . . .

61

13 e-He+ Singlet Ionization Cross Section . . . . . . . . . . . . . . . . . . . .

61

14 Hermite Polynomials Adopted for Finite Element Basis . . . . . . . . . . .

68

15 He and H Radial Ground State Wavefunctions . . . . . . . . . . . . . . .

76

16 He Double to Single Photoionization Ratio . . . . . . . . . . . . . . . . . .

78

17 H Double to Single Photodetachment Ratio . . . . . . . . . . . . . . . . .

78

18 Helium Double Photoionization Cross Section Dependence on Angular Congurations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

83

19 H Double Photodetachment Cross Section Dependence on Angular Congurations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

83

20 Scaling Laws for He Isoelectronic Sequence . . . . . . . . . . . . . . . . . .

85

xi
21 Helium Partial Cross Sections . . . . . . . . . . . . . . . . . . . . . . . . .

90

22 H Partial Cross sections for n=1 and n=2 . . . . . . . . . . . . . . . . .

91

23 H Partial Cross Sections for n=3-6 . . . . . . . . . . . . . . . . . . . . .

92

24 Scaling of Partial Cross Sections with n . . . . . . . . . . . . . . . . . . . .

93

25 Dipole Operator Forms: Square Well Example . . . . . . . . . . . . . . . .

109

xii

TABLESTABLE
1Calculated Ground State Energies of He and H . . . . . . . . . . . . . . .

77

2

Dependence of Ground State Energies on Angular Con gurations . . . . .

77

3

Maximum and Asymptotic Ratios for He Isoelectronic Sequence . . . . . .

87

CHAPTER 1
INTRODUCTION
In this work, we attempt to describe a special case of the three-body problem in thecontext of nonrelativistic quantum mechanics. The general three-body problem remains oneof the longstanding unsolved problems in physics. The n-body problem was rst outlined
precisely by Newton [1]: For n particles (point masses), each of known mass interactingwith each other only through a gravitational force, given the positions and velocities of theparticles at any one time, the problem is to calculate the positions and velocities of theparticles at any other time. The motion in the n-body problem is described by a set of
3n second-order di erential equations. Therefore, 6n constants of integration are required
to describe the complete motion of the system. However, only 10 integrals of motion (asdetermined by Euler [1]) are known to exist: six constants from the center of mass motion,
three constants from conservation of angular momentum, and one constant from conservationof energy. Using the set of 10 integrals of motion, the set of equations describing the motion of
the n-body system can be reduced to 6n 10. By replacing time with another coordinate andusing the \elimination of the nodes" as introduced by Jacobi [1], the order of the equationsof motion can be reduced to 6n 12. Therefore, no known solution exists for the generaln-body problem for n > 2. However, particular solutions (solutions for a special set of initialconditions) of the three-body problem are known.
One of the most well known particular solutions of the classical three-body problem was rst given by Lagrange in 1772 [1]. He showed that for special cases the three-bodycon guration will retain its geometrical form (e.g., three points on the vertices of an equilateral triangle, or three colinear points), although the shape may rotate or change in size

2[2]. Another special case of the three-body problem called the circular restricted three-bodyproblem has been extensively studied by Poincare. For this particular case, two massive
particles move in circles about the common center of mass and attract (but are not attracted to) a third particle of negligible mass. This case is immediately applicable to an
Earth-Moon-spacecraft system. For the case in which the mass of the third body is small
but nonnegligible, the starting point of the problem is to describe the two massive bodies with a two-body solution, then treat the change in motion of this system due to thethird body with a perturbative approach. The problem of n point charges interacting only
through electrostatic forces can be mapped onto the general n-body problem as proposed
by Newton, since the classical form of electrostatic forces is essentially equivalent to thatof gravitational forces; the only di erence is that electrostatic forces can be both attractive
and repulsive. Although the classical three-body problem has received a great amount ofattention from astronomers and mathematicians, it remains unsolved three centuries after it
was rst proposed.
Attempts were made early in this century to describe the three-body problem inatomic and molecular systems using classical and semiclassical ideas. Bohr succeeded inderiving the energy spectrum of hydrogen in 1913; however, his attempt to nd a reasonableestimate for the ground state energy of helium was a dismal failure [3]. Another system of
early interest was the H+2 molecule. The approximate classical analog of this problem was
rst solved by Euler [4]. He treated the problem of two stationary masses attracting a third
light mass. By introducing elliptical coordinates, the system can be separated into threesecond order di erential equations which have elliptic functions as solutions. Sommerfeld
later developed a semiclassical theory by generalizing the early ideas of Bohr. He gave
Wolfgang Pauli the task of applying these semiclassical rules to describe H+2 for his Ph.D.
thesis. Pauli concluded (wrongly) that H+2 could only be metastable. Despite the recent
resurgence in semiclassical methods for treatment of quantum systems, a proper semiclassical3
treatment of helium has remained elusive [5]. These early failures of semiclassical methods toaccurately describe atomic and molecular systems were extremely important in motivating
the development of a new quantum theory.
With the introduction of nonrelativistic quantum mechanics by Schrodinger [6]and Heisenberg [7], the concepts of classical trajectories and orbits all but disappeared;
however, notions of energy and angular momentum remained. One of the rst issues whicharose shortly after quantum mechanics was introduced was the application of this theory
to systems of particles [8]. This \new" quantum theory was quickly adopted to address
the three-body problem for the cases of the atom, the nucleus, and the molecule [9]. The
molecular case is di erent from the atomic and nuclear cases, as it is a system composed of
two heavy particles and one light particle. Thus, this system can be considered a special
case of the more general three-body problem. For the molecular three-body problem (e.g.,
H+2 ), the rapid motion of the electron dominates over the slow motion of the heavy nuclei.
This di erence in relative speeds allows an approximate separation (using prolate spheroidal
coordinates, see Refs. [10] and [11]) of the system into two parts: 1) electronic motion about
two xed centers, and 2) slow nuclear motion in a time averaged eld of 1), as rst shown
by Born and Oppenheimer [12].
For atomic and nuclear systems, the description of the three-body problem is more
complicated than that for the molecular case. For these systems, all interparticle forces
are comparable. Also, there is no single fast particle to be identi ed; this precludes thedescription of motion in a time averaged eld. The molecular system can be transformed
into the atomic or nuclear system by gradually increasing the mass of the light particle. Upon
performing this transformation, two new features of the modi ed system become apparent.
First, the molecular potential no longer acts between the two nuclei, but rather between
one heavy particle and the center of mass of the heavy and light particle. Second, as themass of the light particle is increased, the velocities of the two parts of the system become

4comparable to the internuclear motion. Using these types of ideas, Wheeler [9] addressed
nuclear structure from a molecular viewpoint, and, in addition calculated the binding energyof 3 H.
One problem of early interest in understanding the nature of nuclear forces was
the scattering of slow neutrons by deuterons. Early attempts to describe these types ofnuclear collisions involving more than two particles [13, 14, 15, 16] used an approximate
total wavefunction of the form =Xii fi , where i is a product of the internal wavefunc-

tions of the two colliding nuclei, and fi is the wavefunction of relative motion. The sum isperformed over all pairs of nuclei whose formation is energetically possible. Although thisform of the wavefunction is accurate when the two nuclei are widely separated, it does notaccurately represent the wavefunction when the two nuclei are interacting with each other.
Motz and Schwinger [17] performed a detailed calculation of neutron-deuteron scattering
using a variational approach. Kohn [18] later developed a variational approach similar to
the Rayleigh-Ritz method and applied it to calculate neutron-protron and neutron-deuteronscattering cross sections.
Another early test of Schrodinger quantum mechanics in addressing three-bodysystems was an accurate description of the energy levels of helium. Earlier attempts todescribe the structure of helium (and heavier atoms) adopted a central eld approximation[19, 20, 21]. The basic idea of the central eld approximation is that each electron moves inan e ective spherically symmetric potential V (r) representing the interaction of the nucleus
and the other electrons. Adopting this approximation with the self-consistent eld approach,the early attempts were able to give energy values in fairly good agreement with theoretical values. Hylleraas [22] adopted the Rayleigh-Ritz variational method and performed atransformation of the independent electron coordinates r1 and r2 (s = r1 + r2 , t = r1 r2 ,and u = r12  j~r1 ~r2 j) to obtain a very accurate description of the helium ground state.
Today, the structure of the helium atom is considered essentially \solved." For example, the5theoretical and experimental values of the helium ionization energy agree to one part in 107[11].
In contrast to the bound state structure of helium, our e ort to understand thehelium continuum (the continuum refers to states in which one or both electrons possessenough kinetic energy to escape to in nity) is still a work in progress. Much progress has been
made in describing single electron escape processes in atomic physics [23]. Although the crosssection pro les of these processes often exhibit a rich and complex set of features, re ecting
a diverse range of interactions, theoretical methods exist to reliably reproduce these electron
escape cross sections. Scattering processes in which the target absorbs enough energy for
two electrons to escape are much more dicult to describe. For single escape, if the energyof the incoming projectile is known, then by energy conservation, the energy of the escaping
electron at asymptotic distances is constrained to be the initial target state energy plus the
projectile energy less the energy of the remaining fragment. However, for double escape, there
are no longer any constraints on the energy distribution between the two escaping electrons(other than the indistinguishable particle requirement that this distribution is symmetric).
This added diculty of not being able to analytically express the asymptotic wavefunction
for double escape has been a major barrier in past attempts to properly describe this process.
In this work, we attempt to accurately describe the two electron continuum wavefunction. Theoretical e orts to accurately describe double electron escape have had limited
success until recently. These e orts have been enhanced by the availability of ecient modern day computer resources. An \exact" numerical treatment of two escaping electrons out
to in nity remains impractical. However, suitable approximations allow an accurate description of many details in these processes. Double electron escape processes can occur inboth electron scattering and photoabsorption. In this work, the focus is primarily on double
photoejection of H and He.
The rst quantum mechanical treatment of two electron systems can be traced back6
to e orts to describe the energy spectrum of helium. It therefore seems natural to revisit thehelium system in addressing the double continuum problem. The Schrodinger equation for
helium is one of the simplest nonseparable equations which occurs in atomic physics. As such,
this system cannot be described by a separable analytical expression in independent particle
coordinates. Helium is an ideal system for studying the e ects of electron correlations, sinceno core electrons are present to modify the electron-nucleus Coulomb interaction. Double
electron escape by photoejection is easier to model than escape by electron scattering, due tothe absence of long-range forces between the incoming projectile and the target state. The
experimental community has also expended much e ort in measuring helium photoionization
processes. For these reasons, the helium double photoionization process represents an ideal
test case for theoretically describing the details of two electron escape.
Past studies of helium have been motivated by e orts to understand the importanceof electron-electron correlations in various processes. The importance of these correlations
was experimentally probed by Madden and Codling [24]. They observed the autoionizationof doubly excited states of helium after photoabsorption in the early 1960's. A few years
later, experimental studies of helium photoabsorption were performed at photon energies
above the double escape threshold. Carlson [25] rst measured the ratio of double to single
photoionization cross sections in helium in the photon energy range of 100-600 eV. Later
experiments gave results that were inconsistent with the earlier experimental results of Carlson and with each other [26]. Various theoretical attempts to calculate the branching ratio
also gave a broad distribution of values [26]. Twenty- ve years after the original experimentof Carlson, a quantitatively accurate description of the fundamental two electron process ofhelium double photoionization in the intermediate energy range (100-600 eV) remained elusive. It was the lack of agreement among then-existing experimental and theoretical values ofhelium double photoionization cross sections in this energy range which strongly motivatedthe subject of this dissertation.

7
In the early 1990's, extensions of close coupling methods which treated the continuum with a discretized representation showed promise in describing two electron escapeprocesses [27]. Most of these methods were developed to describe electron scattering processes. Previous calculations of helium double photoionization used perturbative approaches
such as many-body perturbation theory, the Born approximation, or distorted wave Born
approximation. One might expect these methods to work well at high energies. For these energies the importance of interactions between electrons as they escape is small in comparison
to their large kinetic energies, thereby allowing the 1=r12 term to be treated perturbatively.
However, at lower energies one would expect the 1=r12 term to signi cantly in uence twoelectron escape. Therefore, electron correlations cannot be accurately treated perturbatively
in this energy range.
In this work, we extend the eigenchannel R-matrix approach to describe two electron
photoejection processes. This method treats electron-electron interactions nonperturbatively
in the region near the target nucleus (the reaction volume), where these e ects are assumed to
be important. Outside the reaction volume, the wavefunction is approximated by a separable
solution: the outer electron is subject to only a Coulomb potential of charge Z 1 (Z being
the charge of the nucleus), assuming the inner electron eciently screens the nucleus. One
expects this approximation to break down at energies just above threshold, but it should bevalid for energies at least a few eV above threshold.
In Chapter 2 of this work, we attempt to put the double continuum problem into
perspective. Early theoretical attempts to describe two electron escape are discussed, in
addition to more recent alternatives to the eigenchannel R-matrix approach. We next discuss the dependence of two electron escape on projectile energy, experimental progress inmeasuring double photoionization processes, and other aspects of the double escape process.
Chapter 3 is devoted to our initial R-matrix treatment of helium double photoionization [28]. The eigenchannel R-matrix method and its extension to describing two electron

8
escape are discussed in detail. Double photoionization results are presented for energies of
80-280 eV, and the discrepancy between calculations using di erent forms of the dipole operator is addressed. An application of our approach to a simpli ed model of electron scatteringis also included [29]. This chapter concludes with a discussion of the problems that arise
when treating the continuum with a discretized set of states, and outlines systematic techniques used to overcome these problems. In particular, a frame transformation technique isintroduced as a procedure for distinguishing between single and double escape cross sections.
In Chapter 4, we combine a nite element basis set with the eigenchannel R-matrix
method to describe double photoejection of helium and H [30]. This local nite element
basis set is adopted to replace the global hydrogenic basis set used in our initial calculations.
Application of this new approach greatly reduces discrepancies present in results obtainedusing our original method. We also use this nite element method to address other detailsof two electron escape [31]. A discussion of the similarities and di erences between heliumand H targets is included.

CHAPTER 2
TWO ELECTRON ESCAPE: PUTTING THE PROBLEM IN PERSPECTIVE

2.1 Preliminary Theory and Initial Assumptions
The formalism for calculating photoionization cross sections is quite straightforward. The starting point is commonly referred to as Fermi's Golden rule [11]:dW = 2h jh f jHint: j i ij2 f (E )(Ef Ei );(1)

where dW is the transition rate, Hint: is the interaction Hamiltonian between the atom andeld, and f (E ) is the density of nal states. The full Hamiltonian of the system is given by

H = Hatom + Hint: + Hradiation ;(2)where the atomic Hamiltonian is

Hatom =

Xi~pi2 Ze2 + X e2 :2m j~ri ji>j j~ri ~rj j(3)

The classical Hamiltonian (in Gaussian units) describing an atom in an electromagnetic eld(Hatom + Hint: ) is given by:~
H = 2m p~i + eA(~cri ; t)
i
i
X

1

!2

+ V (~r1 ;~r2 ;    ;~rN );(4)

where V is the electrostatic potential energy of the atom. In this analysis, we neglect theinteraction between the nucleus and the radiation eld, and we neglect the spin interactionof the electron with the magnetic eld, normally expressed as:Xi

~ i  B~ (~ri ; t):

(5)

10
We also neglect reduced mass e ects, assuming an in nite mass nucleus is located at theorigin of our coordinate system. The interaction Hamiltonian is identi ed as:

Hint: =

N
X
i=1





e hp~  A~ (~r ; t) + A~ (~r ; t)  ~p i + e2 A~ (~r ; t) 2 :
i
i
i
i
2mc i
2mc2

(6)

For the weak eld regime considered here, the jA~ j2 term can be neglected. That is, we neglect
two-photon processes. Choosing the Coulomb gauge for A~ (r  A~ = 0), p~ and A~ commute.
Therefore, Hint: can be expressed as a single term. Introducing the vector potential A~ inthe formr2h ^ aei~k~ri + ay e i~k~ri  e i!t ;A~ (~ri ; t) = 2c!V(7)

where ~k and ~! are the wavenumber and angular frequency of the radiation, V is the volume,and ^ is the polarization vector. The electric dipole approximation can now be used toreplace ei~k~ri by unity, since for the photon energies considered in this work, the radiation
wavelength is much larger than atomic dimensions. The interaction Hamiltonian is thengiven byNXre 2c2 h (^  p~ ) a + ay  e i!t :Hint: = mc
i
!V
i=1

(8)

From the Golden rule (Eq. 1), the transition rate is given by

W1!2 =

Z

dW1!2

(9)
2

42 e2 h jX ^  p~ j i :
= m
i 1
2
2 !V
i

(10)

Dividing the transition rate by the incident ux (c=V ), and introducing the ne structure
constant = e2=hc, the cross section  is given by2

2 h h jX ^  p~ j i ;1!2 = 4m
i 1
2
2!
i

(11)

or in atomic units (e = h = m = 1),
2

2
X1!2 = 4! h 2 j ^  p~i j 1 i (a:u:):

i

(12)

11
Cross sections in this study will be given in units of barns (1 barn = 10 28 m2 ). Assuming
linearly polarized light (the case treated in this work) in the z-direction, and also assuming
the total orbital angular momentum of the initial state j 1 i is zero, the cross section is given
by

2

2
X
1!2 = 43! h 2 k @z@ k 1 i ;@ k i represents a reduced dipole matrix element.
1
@z
i
i

X

where h 2 k

(13)

i

i

The above form for the cross section is referred to as the velocity form. This equation has corresponding length and acceleration forms (see Appendix A). Using the identity

h j[~r; H ]j i = h j imh p~j i = (E
2

1

2

1

2

E1 )h 2 j~rj 1 i;(14)

we can obtain the cross section by evaluating the matrix element h 2 j~rj 1 i. Similarly, one
can use the identity:

h j[p~; H ]j i = (E212E1 )h 2 jp~j 1 i = h 2 j[p~; V ]j 1 i = ihh 2 j

For a potential of the form V = Z
is given byXiri j i:1(15)

e2 + X e2 , the matrix element h jX r j i
i 1
2ii ri i<j j~ri ~rj j

X

Ze2h 2 j~ri j i:
1
3
r
i i

X

(16)

There is no net contribution to the above expression from the gradient operator operatingon the Coulomb repulsion term, as ri and rj applied to this term give equal and opposite
contributions. If the wavefunctions j 1 i and j 2 i are exact, then all three forms (length,
velocity, and acceleration) should give identical results. In this work, we use all three forms
as a method for evaluating the accuracy of our wavefunctions. Since the length form is
proportional to ~r, it samples the large r behavior of the wavefunction. In contrast, theacceleration form is proportional to 1=r2 , and so samples the wavefunction's small r behavior.

12

2.2 Simpli ed Descriptions of Two Electron EscapeThe double ionization cross section for a two electron system can be expressed as
2
XZ E Z E
++ (E ) = (2J4+ 1)!
d d0 (h! ED:I:  0 )jMfi j2 ;
0

l

0

0

(17)

where (for linearly polarized light in the z-direction)
Z
Z
Mfi = d~r1 d~r2 f (~r1 ;~r2 )( @z@ + @z@ ) i (~r1 ;~r2 );
1

2

(18)

J0 is the total angular momentum of the initial state, ED:I: is the double ionization thresholdenergy, and  and 0 are the energies of the two escaping electrons. Information about theinitial and nal state is all that is required to calculate this cross section. Also, since theinitial state is spatially con ned to a small region, a description of the nal state wavefunction
for only relatively small values of j~r1 j and j~r2 j is required to evaluate Mfi . However, theasymptotically large r form is required to determine the short-range r behavior of the nal
state wavefunction. An accurate description of the ground state wavefunction (the Hylleraas
approach [22]) has been available for quite some time. The diculty which remains is an
accurate description of the double continuum nal state wavefunction.
The simplest treatment of two electron escape adopts the sudden approximation[25]. In this approximation, the \fast" electron is assumed to suddenly escape. The amplitude for the inner electron to escape subsequently is proportional to the projection ofa one-electron wavefunction representing the inner electron onto a continuum hydrogenic
wavefunction. This approximation was originally applied in the context of beta decay, where
the e ect of electron excitations was investigated for a sudden change in the charge of a
nucleus [32]. This approach is known to give reasonable results for atoms with relatively
large values of the nuclear charge Z (Z  25, see Ref. [32]).
More sophisticated methods have been applied to describe the probability of asecond electron escaping an atom due to the change in the potential from the escape ofthe rst electron; this is known as the \shake-o " mechanism [25]. These methods give13
roughly the correct order of magnitude for the double escape cross sections, but tend tounderpredict the correct values [25, 33]. Other mechanisms were proposed in an attempt
to describe two electron escape. Samson observed empirically that the ratio of double tototal ionization was approximately proportional to the cross section for electron impact
ionization of an ion [34]. This motivated his development of a classical model which describeddouble photoionization as a two step process (and therefore labeled as a \two-step one"
contribution in many body perturbation terminology): rst, the photon is absorbed by one
electron, and second, the excited electron scatters from the remaining electron as it leaves
the ion. This simple description was recently adopted in a semiclassical treatment of helium
double photoionization and found to give reasonable results [35]. Many body perturbation
treatments [36, 37] identi ed the contributions to double ionization as falling into three
categories: ground-state correlations, shake-o , and two-step one. These last two processes
are nal state correlation e ects. However, it was shown that the relative strengths of thesemechanisms depend on the form of the dipole interaction [38, 39]. Consequently, as seenby their dependence on the dipole form, these mechanisms can not be viewed as physicalprocesses; they are only useful for visualizing the process of double photoionization in aqualitative sense.
Early studies [40] illustrated the importance of including an accurate ground state
wavefunction in order to get accurate double ionization cross sections. The rst attempts
to calculate double photoionization normally adopted a nal state wavefunction represented
as a symmetrized product of uncorrelated Coulomb wavefunctions [40, 41]. Each of these
Coulomb functions were for a charge of Z = 2. That is, screening between the electrons was
ignored. Although these studies obtained results that were in fairly good agreement withexisting theory, a notable discrepancy appeared between calculations using various dipole
forms.
Further attempts to analytically represent the three-body Coulomb continuum nal

14state achieved mixed success. Using a product wavefunction consisting of three two-body
Coulomb wavefunctions (also known as the Brauner-Briggs-Klar(BBK) wavefunction [42]),
Maulbetsch and Briggs [43] were able to accurately reproduce the shape of electron angular
distributions in double photoionization, but predicted absolute cross sections that were ordersof magnitude too low. Recent attempts to address this normalization problem have beensomewhat successful [44].

2.3 Discrete Methods for Treating the Double Continuum
The greatest obstacle to the theoretical description of two electron escape derives
from an inadequate understanding of how to impose the proper boundary conditions at innity. For this reason, methods that aspire to directly represent the two-electron continuum
as a channel expansion of an uncountable in nity of states are infeasible. Approximate
perturbative methods which treat the double continuum in this manner (e.g., the Born
approximation) have been successful in calculating double continuum cross sections at suciently high energies (on the order of 1 keV [45]). However, alternate methods are required
to describe double escape processes at lower energies.
Methods that use a discrete representation of all possible excitation channels haveenjoyed tremendous success at low energies where only single escape is energetically allowed.
This includes, for instance, the R-matrix method, variants of the close-coupling method thatuse pseudostates, and the convergent close-coupling (CCC) approach. In this work, we showthat the eigenchannel R-matrix method is capable of describing a class of problems in the twoelectron continuum, excluding energies very close to the threshold energy for double escape.
Other coupled-channel methods have been similarly extended to treat the double continuum
in recent years, notably the intermediate energy R-matrix (IERM) method [46], the CCC
method [47], the R-matrix with pseudostates (RMPS) method [48], and the hypersphericalclose-coupling (HSCC) method [49]. Although many of these methods have been applied
primarily to describe electron scattering rather than photoejection, both processes involve15
the same nal state dynamics, and are thus closely related. It is important to explore thedouble escape energy region, in part because of its intrinsic theoretical interest. Yet anotherreason is that even processes involving single-electron escape could be adversely a ected if
the double continuum regions are linked in some sense, e.g., through constraints such as theoscillator strength sum rule (often referred to as the Thomas-Reiche-Kuhn sum rule [50]).
A couple of early e orts that demonstrated the promise of discrete methods in describing two electron escape included a pseudostate treatment of electron-hydrogen scatteringby Callaway and Oza [51], and a multichannel J-matrix calculation of double photoejectionof H by Broad and Reinhardt [52]. Both of these pioneering attempts were hindered by thelack of powerful computing resources which are presently available. The rst true demonstration of the capability of a discretized method to obtain accurate total double escapecross sections was shown in the CCC treatment of electron-hydrogen scattering by Bray andStelbovics [27]. They were able to obtain excellent agreement with experiment over the wide
energy range 4-500 eV above the ionization threshold. Complementing this e ort was thework of Kato and Watanabe [53], which used the hyperspherical close-coupling method to
calculate ionization cross sections of electron-hydrogen scattering. They were able to obtain
excellent agreement with experimental cross sections for energies as low as 0.1 eV above
the ionization threshold. These calculations are quite noteworthy, as e orts to theoretically describe this fundamental process accurately in the previously 50 years were largely
unsuccessful.
The CCC method employs a close-coupling representation to solve the momentumspace Lippmann-Schwinger equation that derives from the full three-body Schrodinger equation. A Sturmian basis set is chosen to diagonalize the target Hamiltonian in momentum
space. This choice of basis functions eliminates numerical problems associated with convergence and linear dependence. For a xed energy value, the basis set size is systematicallyincreased until the observable of interest converges. This discrete basis set is increased in16such a way that additional members extend to successively larger radii. Further details of
the CCC method can be found in Refs. [45] and [47].
In addition to calculating accurate two-electron continuum cross sections for electronhydrogen scattering, the CCC approach has produced reliable cross sections for electron-He+
ionization from threshold to 700 eV [54], for positron-hydrogen scattering [55], and more recently for electron-helium scattering in the frozen core approximation [56]. This approachalso determined accurate observables in electron scattering by Na, judging from its close
reproduction of spin-resolved measurements; furthermore, it should also be straightforward
to apply it to electron scattering by other targets with one valence electron (e.g., Li and K)[57]. In addition to obtaining bound state excitation and ionization cross sections, the CCC
formulation was used to calculate spin asymmetries and di erential, integral, and total crosssections. This approach has also been recently applied to calculate helium double photoionization cross sections [58]. A summary of applications of the CCC approach is reviewed inRef. [45]. Although this method has enjoyed great success in describing electron scattering processes, problems with this method in obtaining converged energy distributions haverecently surfaced [59]. The origin of these problems is still not fully understood.
The hyperspherical close-coupling (HSCC) approach has been used to describe twoelectron escape in both electron scattering [53, 60, 61] and photoabsorption processes [62].
This method replaces the independent particle radial coordinates r1 and r2 by a pair of colp

lective variables R and . The hyperradius R = r12 + r22 measures the \size" of the electron
system and the hyperangle = tan 1 (r2 =r1 ) describes the degree of electron-electron radial
correlations. Reexpressing the two electron atom Schrodinger equation in these coordinates,
one can apply the adiabatic approximation in the hyperradius (the only remaining in niterange variable). In the past, this hyperspherical coordinate method had been useful mainlyin the qualitative description of electron correlations [63, 64]. However, using the HSCC

17method, Tang et al. were able to obtain an accurate, high precision description of doubly excited resonances just below the double escape threshold by treating electron-electroninteractions accurately out to very large distances (R=170 a.u.) [65].
One of the drawbacks of the hyperspherical coordinate method is the slow convergence in calculating adiabatic potential curves when using the traditional basis set, namely
a set of functions of the hyperradius multiplied by adiabatic channel functions (functionsof the ve angular coordinates ; r^1 ; and r^2 , which are solutions of the adiabatic Hamiltonian for the parameter R). Tang et al. adopted an alternative numerical method, called
the diabatic-by-sector method [66], to solve the hyperspherical close-coupling equations. A
two-dimensional independent coordinate matching scheme was adopted [67] to match theinner region wavefunction to the asymptotic solution. Using this HSCC method, Kato and
Watanabe were able to reproduce the experimental ionization cross sections for electronhydrogen scattering down to energies as low as 0.1 a.u. [53]. They were also able to provide
numerical evidence of the 1.127 value of the Wannier exponent (de ned in Eqs. 19 and 20)
and the uniform energy distribution predicted by Wannier theory (for the near-threshold energy range) [61]. Calculations of helium double ionization cross sections were performed forphoton energies from threshold to 280 eV [62]. Very good agreement was achieved between
the length and acceleration forms of the calculation using a matching radius of R=30 a.u.
A third computational approach, R-matrix theory, was rst introduced by Wigner
and Eisenbud to study resonance reactions in nuclear physics [68]. The advantage of Rmatrix methods over CCC and HSCC type approaches is the fact that R-matrix methods
essentially describe the electron scattering/photoabsorption process independent of the projectile energy. Therefore, R-matrix methods are ideally suited to the study of resonancephenomena, where observables can be calculated over a ne energy mesh with little additional computational e ort.
In R-matrix theory, con guration space is partitioned into two (or more) regions

18
by a sphere of radius r=r0, as shown in Fig. 1. The most complicated physics is includedin the nite internal region (region I in Fig. 1), called the reaction volume. In the external
region, electron-electron correlation e ects are neglected, which is a good approximation forlarge enough values of r0 . It is the fact that exchange and correlation e ects are con ned
to a small volume that makes the R-matrix method appropriate even in the presence oflong-range Coulomb elds. Once the wavefunction is determined both inside and outside thereaction volume, the full solution is obtained by matching the logarithmic derivative of the
wavefunction across the surface of the reaction volume and solving the Schrodinger equationin the exterior region.
In this work, we adopt the eigenchannel R-matrix method. We use this method to
calculate variationally a particular set of Schrodinger solutions: each solution has a constantnormal logarithmic derivative across the surface of the reaction volume. Further details of
this method will be discussed in Chapter 3. Other variations of the R-matrix method canbe used to obtain identical results for two electron processes, provided an identical basis
set is used [69, 70]. Two alternative approaches of R-matrix theory that have recently been
applied to address double continuum processes include the intermediate energy R-matrix(IERM) method and the R-matrix with pseudostates (RMPS) method.
The major advancement of the IERM method over previous R-matrix methods
was the inclusion of continuum-continuum orbital couplings in the inner region [46]. Thatis, all electron exchange and correlation e ects between both continuum electrons and thecore are included in the reaction volume. In their approach, the internal region of con guration space is further divided into a number of subregions to treat inelastic e ects. The
systematic introduction of continuum pseudostates in the inner region leads to pseudoresonances at intermediate energies. A T-matrix energy averaging scheme [71], which involves
continuation to complex energies, is adopted to average out the ux contributions of thesepseudoresonances and therefore obtain smooth cross sections.

19

r2

II

IV

I

III

r0

r0

r1

Figure 1. Radial con guration space for two electrons used in R-matrix theory. Region I
represents the reaction volume, regions II and III represent single electron escape, and region
IV represents the double continuum. This gure is taken from Ref. [28].

20
The IERM method has been applied to calculate elastic, 1s 2s, and 1s 2p crosssections for electron-hydrogen scattering at intermediate energies [72, 73]. This method
has also been applied to calculate electron-hydrogen ionization cross sections for projectile
energies up to 50 eV [74]. These calculated values were slightly larger than the experimental
values of Shah, Elliot, and Gilbody [75] at higher energies. Recent ionization cross section
calculations for a simpli ed s-wave model of electron-hydrogen scattering (the Temkin-Poetmodel) in the energy range from threshold to 5 eV above threshold compared favorablywith similar CCC and RMPS calculations [76]. By going to a large reaction volume, they
were able to represent the energy region above the ionization threshold with a dense setof pseudostates. Converged results were obtained for a reaction volume radius of 150 a.u.
Preliminary IERM calculations for the full electron-hydrogen scattering problem using areaction volume radius of 63 a.u. appear promising; ionization cross sections for the 1 S epartial wave are consistent with the E 1:127 power law.
The R-matrix with pseudostates method (RMPS) recently developed by Bartschat
et al. [48] utilizes the Belfast R-matrix codes developed by Burke et al. [77]. The motivation

for using these existing computer codes, which have been developed over many years, isthe desire to extend this method to handle electron scattering processes in the low andintermediate energy range for a general N -electron atom or ion. In this method, both target
eigenstates and pseudostates are included in the basis set. The pseudostates consist of anite expansion of Sturmian orbitals. These two types of basis functions included in the
wavefunction expansion are not orthogonal. However, Bartschat et al. adopt the Schmidt
orthogonalization procedure to generate a new orthogonal basis set. Following an initial
application to electron-hydrogen scattering [78], the RMPS method has been applied to
study electron scattering by helium [79], beryllium [80], and boron [81]. Ionization crosssections for the simpli ed s-wave (Temkin-Poet) model of electron-hydrogen scattering havealso been calculated using this method [82]. An alternate version has recently been developed

21
by Gorczyca and Badnell to study photoionization-excitation processes in helium [83].

2.4 Dependence of Double Escape on Projectile Energy
An ideal scattering/photoejection theory should be capable of calculating reliablecross sections independent of the projectile energy. In practice, this ideal theory has not
yet been developed. To appreciate the limitations of current numerical methods to calculate
double escape cross sections over the entire energy range of an incoming projectile, it isuseful to understand the de ning features of two electron escape in each energy region. In
this work, our focus is on the intermediate energy region, as de ned below.
The minimum projectile energy required to produce double photoejection or electron impact ionization processes is called the double escape or ionization threshold (78.98 eVfor helium double photoionization, 14.35 eV for H double photodetachment). At energies
just above threshold, the double escape process is dicult to describe with discrete methods,
as interactions between the two escaping electrons are important out to very large distancesfrom the target. Also, the low speeds of near threshold particles translate into long times
over which correlations can develop. Much e ort has been expended in exploring threshold laws to describe the energy dependence of escape cross sections in the energy region
just above threshold. The primary theoretical motivation for studying threshold laws lies in
the understanding that the energy dependence is very sensitive to long range interactionsin the nal state. Therefore, threshold laws provide a diagnostic for characterizing longrange interactions (e.g., polarization and dipole forces), a subject of broad interest in atomic
physics.
One of the earliest works on this subject was contributed by Wigner [84], who
identi ed several key features of threshold laws. Since threshold laws are only dependent ona single parameter, namely the excess energy E of the nal state (for the case of inelastic
processes, as considered here), the threshold law is insensitive to the details of the initial
preparation of the state. This provides yet another close link between electron scattering

22
and photoabsorption. For example, the same threshold law which describes the electronhydrogen scattering ionization cross sections also describes double photodetachment of H .
Since threshold laws are controlled by long-range interactions, the threshold exponent is
independent of the physics within the reaction zone, the region in which the particles are
all close together. This enables one to obtain a threshold law without having to solve forthe full solution in the reaction zone, where much of the complicated physics occurs. An
additional explanation can be given as to why the region in which the outgoing particles havea large separation controls the threshold law. For large separations the potential interaction
is small, so small values of E are signi cant. However, when the particles are close together,large interaction energies are involved. Therefore, small variations in E will have little e ect
on particles of small separation.
For the three-body Coulomb continuum problem (e.g., electron impact ionizationof hydrogen or double photodetachment of H ), the Wigner threshold law is dependent
on the assumption one makes for the nal state [85]. If one represents the nal state as a
product of two one-electron wavefunctions, there are two limiting cases for modeling electron
correlations. If one makes the assumption that the slower electron completely screens thenucleus, the nal state can be described as a product of a Coulomb function and a planewave, leading to the threshold law  / E 3=2 . However, if the nal state is represented as a
product of two Coulomb functions, then the threshold law  / E results. The real thresholdlaw is expected to lie between these two cases, i.e.,  / E m , where 1  m  3=2.
The next major contribution to our understanding of threshold laws was provided
by Wannier [86]. Wannier's theory is a classical theory. Since for energies above the double
ionization threshold the electrons are characterized by small wavelengths at intermediate
distances, he felt a classical treatment was justi ed for addressing this problem. Wannier
recognized the importance of correlations in describing threshold laws and concluded thatany representation of the nal state as a product of one-electron wavefunctions would be23insucient for understanding the threshold behavior. He abandoned the independent radial
coordinates, replacing them with the hyperspherical coordinates R and .
A central concept of Wannier theory is the importance of dynamic screening. As
previously mentioned, the picture of static screening in which the nal state is represented
as a separable solution of the Schrodinger equation is insucient. Rather, a picture of thescreening changing as energy is partitioned between the two electrons is a more accurate view.
If one particle is slightly closer to the target nucleus than the other particle, the closer particle
will contribute more to screening of the outer particle. As a result, the outer particle gains
a greater share of the energy, leading to a further enhancement of the unequal screening of
the two escaping particles. Wannier reasoned that two electron escape processes would only
occur at low energies for tight energy correlations. He restricted double escape processes incon guration space to a small region neighboring the point 12 =  (12  cos 1 (^r1  r^2 )) and= =4. The point 12 =  describes the two electrons as moving in opposite directions fromthe nucleus, while = =4 corresponds to r1 = r2 , namely, the two electrons escaping suchthat they remain equal distances away from the nucleus. By analyzing classical trajectories
for L = 0 in the vicinity of the point (12 = ; = =4), Wannier arrived at the following
threshold law for double electron escape: / Em;
where

r

Z 9
m = 41 100
4Z 1

(19)
1:
4

(20)

For Z = 1, m = 1:056. As Z ! 1, m ! 1. Later quantum= 1:127, and for Z = 2, m mechanical treatments of the threshold law [87, 88] reproduced the results of Wannier. (Analternative dipole threshold law has been proposed by Temkin [89]: / [E=(ln E )2 ]f1 + C sin[ (R) ln E + ]g;p(21)

where = b 1=4, b is the dipole moment seen by the outer electron, and C and  are24constants. However, this threshold law is not generally accepted [90].)
One consequence of the Wannier picture is the prediction concerning the energy
distribution between the two escaping electrons. According to Wannier theory, all possible
partitions of E into E1 and E2 are equally likely near threshold [91]. This result has been
experimentally veri ed in electron impact ionization of helium [92] and double photoionization of helium [93]. Concerning experimental evidence for the Wannier threshold law, theseinclude experiments on electron impact ionization of hydrogen [75, 94], electron impact ionization of helium [92], double photodetachment of H [95], and double photoionization ofhelium [96]. It now appears possible that the Wannier threshold law can be explored by direct numerical calculations [61, 76, 97]. However, accurate calculations for this energy region
are not possible for the eigenchannel R-matrix method adopted here for reasons which will
be discussed in Chapter 3. One issue that often arises concerning discussions of thresholdlaws is the energy range over which threshold laws are valid. Although this question does
not have a clear answer, one can normally assume that a threshold law is valid up to 1-2 eVabove threshold.
The intermediate energy range can be characterized by the following two criteria:
1) the physical processes which occur in the reaction zone are important in describing thetwo electron escape process, and 2) the electron-electron interactions in the reaction zone
are large enough that they can no longer be treated perturbatively. The rst criterion
eliminates the region just above threshold, as the energy dependence of double escape crosssections depends solely on the longe range forces. The second criterion eliminates the highenergy regions, where perturbative methods (e.g., the Born approximation) are sucient for
accurately describing two electron escape processes. In properly describing these processes
for intermediate energies, an accurate nonperturbative method for treating the complicatedelectron-electron interactions in the reaction zone (reaction volume) is essential. This has
been the focus of recent numerical discrete methods, discussed in the previous section.

25
One sensitive probe of the strength of electron-electron interactions is the ratio R
of double-and-single photoejection cross sections (R  2e =1e ). Double photoejection could
not occur in the absence of electron-electron interactions, at least in rst-order perturbation
theory for the photon-atom interaction. Since the maximum value of this ratio as well as the
maximum value of the double escape cross section occurs in the intermediate energy range(e.g., see Ref. [52]), escape processes in this energy range are a subject of great interest.
Previous studies of helium double photoionization for these energies will be discussed in
Chapter 3. One issue which arises for processes in this energy range is the size of thereaction volume required to obtain accurate results. Other than the fact that the required
reaction volume radius grows as the inverse of the energy excess from threshold [61], there
is no clear theoretical value for the required reaction volume radius. In practice, the size ofthe reaction volume is increased until converged results are obtained.
At high enough (but nonrelativistic) projectile energies perturbative methods canbe used to calculate accurate double escape cross sections. Perturbative treatments of electron correlations rely on the assumption that one of the ionized electrons leaves the target
area relatively rapidly. For this situation, in which the electron kinetic energies overwhelm
the electron-electron interaction potential, the nal state correlation e ects are minimal. The
assumption that the escaping electrons have very di erent speeds is approximately valid forphoton energies much larger than the ionization potential of the target. The well known
Born approximation does not formally include exchange or short-range distorting potentials.
This method normally gives accurate results for projectile energies above 1 keV. Speci c
application of the Born approximation to two electron escape include the helium double
photoionization calculations of Brown [41] and the electron-hydrogen ionization cross section calculations of Peach, whose values are reported in Ref. [27]. The distorted wave Born
approximation (DWBA) improves on the Born approximation by including a short-range
potential; that is, plane wave solutions in the Born approximation are replaced by distorted

26
waves. Further improvements can be attained by including higher order approximations,such as the second-order distorted-wave Born approximation (DW2BA) [98], which unlike
the DWBA is able to include the e ects of dipole polarization. Many body perturbation
theory (MBPT) approaches have also been applied to calculate double photoionization crosssections in helium [33, 36, 37, 99]. The di erences between these various many body calculations are due to di erences in the choice of basis sets and in the methods used to estimate
higher order corrections. Although these methods were applied for photons in the intermediate energy range, it is unclear as to how electron-electron interactions can be accurately
treated perturbatively here.
One issue of great interest is the asymptotic (nonrelativistic) energy limit of theratio of double and single photoejection cross sections. It is known [38] that the energy
dependence of both single and double photoejection cross sections goes as E 7=2 for high(but nonrelativistic) energies (the decrease in the cross section with energy is actually slower
than E 7=2 for relativistic energies [100]), and thus the ratio R approaches a constant.
Dalgarno and Stewart [101] showed that the value of this constant is solely dependent on theinitial state wavefunction. Dalgarno and Sadeghpour calculated the ratio values of 0.0150
for H , 0.0167 for He, and 0.0087 for Li+ [38, 102].
A complete description of double photoejection at high energies is complicated by
the importance of the atomic Compton e ect [103], namely, h +He! h 0 +He2+ + e +e . Compton processes result from the presence of the jA~ j2 term (two photon process) in
Eq. 6, which we have previously ignored. The Compton e ect will become important inphotoejection when the energy increase of the electron(s) (the energy E = (p)2 =2m corresponding to the net momentum transferred (p) to the atom by the photons) is greater thanthe electron escape threshold. For helium, Compton processes are nonnegligable at photon
energies greater than roughly 2.5 keV for single photoionization and 4.5 keV for double
photoionization. These processes dominate at higher energies, since the single and double

27
photoionization cross sections fall o as E 7=2 while the Compton scattering is roughly
independent of energy. A new parameter of interest is then the asymptotic energy limitof the ratio RC of helium double-and-single ionization by Compton scattering. There wassome disagreement between theories in calculating this value. Amusia and Mikhailov [104]calculated a value RC =0.0168, while three other studies [105, 106, 107] calculated a valueof RC =0.008. Bergstrom, Hino, and Macek calculated a value of RC =0.016 at 20 keV, but
concluded that their value had not yet reached an asymptotic limit [108] (their values of

RC were still decreasing with energy). One recent helium ionization experiment by 57-keV
X rays obtained a value RC =0.0125 [109] while a di erent experiment with 58-keV X rays
measured a value which supports the smaller value of RC (0.008) [110].
In summary, it is dicult for a single numerical method to accurately predict twoelectron escape cross sections over the full range of projectile energies. The role of electron
correlations in double escape is highly dependent on the excess energy of the electrons, andso is dicult to describe. Discrete methods are primarily limited to the intermediate energy
range. Limitations in the size of the reaction volume hinder an accurate treatment of longrange correlations, which is required for describing near threshold behavior. Limitations inthe size of the basis set hinder an accurate representation of escaping electrons with large
excess energy, as the corresponding wavefunctions have small de Broglie wavelengths. The
general pro le of the double photoejection cross section is characterized by a zero value atthreshold, a nearly linear increase for the rst few eV above threshold, reaching a peak atintermediate energies before decreasing to zero at high energies, with an energy dependenceof E 7=2 . This pro le re ects the physical role of correlations. Double escape just above
threshold is limited to a small region of phase space, as there is little excess energy available
to partition between two escaping electrons. Likewise, double escape events at high energies
are suppressed, as the energetic electrons spend little time near each other, and therefore havelittle time to interact. The pro le of the ratio of double to single photoejection is similar to28
the double photoejection cross section, except that the peak is broader and located farther
from threshold, and the values fall o much more slowly with energy before eventuallyreaching a nonzero constant value. It is part of the challenge of discrete numerical methods
to calculate the location and value of the peak ratio.

2.5 Experimental Progress in Measuring Double Photoionization Cross Sections
Much progress has been made in developing experimental methods to observe twoelectron escape processes excited by photoabsorption. In Carlson's original experiment [25],double photoionization events were produced by using low energy x-rays emitted by an xray tube. Modern double photoionization experiments are performed at synchrotron lightfacilities, which provide a high intensity, high energy source of radiation [111]. In typicalexperiments, the synchrotron radiation is focussed on a small jet of gas. The jet then drifts
through an externally applied electric eld. A time-of- ight spectrometer is used to distinguish between the singly- and doubly- ionized target residuals [112] (since doubly charged
ions experience a greater acceleration in the eld than singly charged ions, these ions willreach the detector rst). A COLTRIMS (cold target recoil ion momentum spectroscopy)
technique has also been very useful in distinguishing between Compton scattering and photoabsorption processes. For Compton processes, the momentum and energy transferred by
the photon correspond to the energy and momentum of the emitted electron(s), whereas inphotoabsorption the recoiling target ion must compensate for the momentum of the escaping
electron(s). By using this technique to distinguish between events with small and large recoil
ion momenta, Spielberger et al. [113] were able to separate photoabsorption and Compton
scattering contributions.
Double photoionization experiments are dicult to perform, since these experiments usually involve the measurement of very small cross sections (< 10 24 m2 for helium).

29
The cross sections eventually decrease with increasing photon energy, compounding the problem. Consequently, even small sources of error can result in a signi cant distortion of themeasured ratio of double-and-single photoionization [112]. One possible source of error in
helium double photoionization experiments is the creation of spurious helium ions due tostray light or electrons. Since photoionization cross sections are much larger at energies
just above threshold than at higher energies, even a small amount of stray light can create
problems [112]. Another possible source of error is a di erence in detection eciencies of
He+ and He++ with the time-of- ight spectrometer and the channel plate, as the doubly andsingly charged ions will reach the detector at di erent speeds. Experimental measurements
are also very sensitive to contaminants in the helium gas, because the photoionization crosssections for these contaminants are often much larger than that for helium. Careful analysisof these possible sources of error has allowed recent measurements of the ratio of doubleand-single photoionization to be performed with considerably improved accuracy, comparedwith earlier measurements. In addition, modern experiments have been performed over amuch wider range of photon energies (e.g., see Ref. [111]).

2.6 Observables and Targets as Probes of Electron-Electron Correlations
Although much of the current investigation has focussed on calculating a single
process (photoabsorption) for a single target (helium ground state) for a single observable(ratio of double to single photoionization cross sections), the work presented here has amuch broader context. The close relationship between electron scattering and photoejection
has already been portrayed in previous sections. The dependence of two electron escapeon the projectile energy has also been discussed. Further information about two electron
escape processes can be obtained by calculating various observables. These include not only
single, double and total photoionization cross sections, but also partial cross sections (single
photoejection plus excitation of the remaining electron), energy distributions, and angular
distributions. Exploration of these other observables help to provide a more complete picture

30of two electron escape. A wealth of experimental data of observables for electron scattering
exists; I will focus primarily on photoabsorption processes here.
A paper summarizing past theoretical and experimental determinations of helium
partial cross sections has recently been published [114]. We will compare our calculated
values to the experimental results in this paper in Chapter 4. In addition, experimental
values exist for the angular distribution of helium photoelectron satellites He+ nl for n=2-7
over a large photon energy range [115]. Wehlitz et al. [93] have measured electron energy
distributions in helium double photoionization for excess energies of 5-41 eV. Calculations ofenergy distributions for helium double photoionization include those of Hino et al. [39] using
many body perturbation theory, and of Proulx and Shakeshaft [116] over the range of photon
energies 89-140 eV. Pont and Shakeshaft explored the dependence of singly di erential (inenergy) cross sections on excess energy for the equal energy sharing case (E1 = E2 = E=2)[117]. Theoretical studies of helium double photoionization angular distributions performedby Maulbetsch et al. [118] and by Pont and Shakeshaft [119] agree fairly well with the experimental values of Schwarzkopf et al. [120]. In principle, our eigenchannel R-matrix method
can be used to calculate all of these observables, although the procedure for calculating energy distributions is not necessarily straightforward. The accuracy of our method for thedescription of the double escape observables like angular distributions remains uncertain.
Further information about two electron escape can also be learned by probing different target states. Other than the isolated work of Teng and Shakeshaft [121], who exploreddouble photoionization of the He(1s2s3S ) metastable state, helium double photoionization
studies have focused entirely on the He target ground state. Advances in experimental
methods may allow measurements of double photoionization cross sections for the initialstate He(1s2s3 S ) and other metastable states to be undertaken in the future.
Although this work has concentrated primarily on double photoejection of H and
He, heavier elements in the helium isoelectronic sequence (Li+ ,Be++ , C4+ , O6+ ,   ) can also

31
be explored. Kornberg and Miraglia [122] have used an approximate analytical nal state
wavefunction to show that the ratio of double-and-single photoejection scales as a simple
function of the nuclear charge Z . Further discussion of this topic is included in Chapter 4.
The study of other heliumlike atom target states (Be, Mg, Ca, Sr, Ba, Ra) provide further
opportunities for exploration of two electron escape. Accurate treatments of the heavier
atoms (Ba, Ra) will require the inclusion of relativistic e ects. Further complications arise
at relatively high photon energies (50 eV above threshold), where excitations of the innershell core occur. Experimental double ionization studies of several gases (He, O, N, Ar, andNe) have been performed [123, 124]. Proper theoretical descriptions of these processes (otherthan helium) are dicult however, as these targets contain more than two electrons in theouter shell. In general, for both the case of theory and experiment, the study of two electron
escape processes for various target species has been quite limited for photoabsorption incomparison to electron scattering.

CHAPTER 3
INITIAL APPROACH: EIGENCHANNEL R-MATRIX CALCULATION WITH A
HYDROGENIC BASIS

3.1 Double Photoionization of Helium
The eigenchannel R-matrix method for single photoionization is extended here todescribe a class of two electron escape processes. We apply this approach to calculate thedouble photoionization of helium by single-photon absorption, for photon energies in therange 80-280 eV [28]. Calculations are performed using the length (Eq. 14), velocity (Eq.
13), and acceleration (Eq. 16) forms of the dipole operator; only the velocity and acceleration
results are presented here. Pseudoresonances in the double continuum are eliminated by
performing a Gailitis average and by averaging the nal spectrum over the size of the Rmatrix box (Region I of Fig. 1). The ratio of double and single photoionization cross sections,a key parameter for characterizing electron correlations, is compared to existing theoreticaland experimental values.
Methods that use a discrete representation of all possible excitation channels haveenjoyed tremendous success at low energies where only single escape is energetically allowed[23]. Earlier theoretical attempts (before 1994) to calculate double photoionization relied on
perturbative treatments of electron-electron interactions, such as many body perturbation
theory. In this study, theoretical calculations of double photoionization cross sections are
performed with a nonperturbative approach. While this approach also faces diculties whenapplied to the escape of two electrons, it is informative to see results from a completely
di erent formulation.
The implementation of our eigenchannel R-matrix approach starts from numerical33(closed-type) solutions of the one-electron radial Schrodinger equation for He+ , calculated
inside the reaction volume subject to the boundary condition that the radial wavefunction
vanishes at the boundary r = r0 . This generates a set of (closed-type) solutions having a
discrete energy spectrum (see Fig. 2). In essence, we are limiting the Hilbert space to theset of eigenstates of our target state Hamiltonian in a shrunken eigenspace (0 < r < r0 ).
Virtually all methods today utilize some such set of shrunken eigenstates (see section 2.3).
The lowest energy states (with principal quantum numbers n  3) are very similar to theordinary physical bound states of He+ , as they vanish at large distances anyway. For high nvalues, the spacing between energy levels is roughly proportional to n2 , characteristic of theenergy spectrum for a square well potential. These solutions are next used to construct a twoelectron basis set. The two electron trial wavefunction at each nal state continuum energy

E is written as a linear combination of these basis functions, that now describes a continuum
energy state. A variational method described elsewhere [23] is used to solve for the optimalcoecients in the basis set expansion of the trial wavefunction. We neglect exchange andcorrelation e ects outside of the reaction volume.
If only closed-type one-electron orbitals were used to construct two-electron trial
functions, no electrons would be able to escape from the reaction volume, since the wavefunction would be exactly zero everywhere on its surface. Accordingly, we also introduce\open-type" orbitals as in Ref. [125], de ned as eigenfunctions of the He+ Hamiltonian with
nonzero values at r = r0 . These open-type orbitals are typically chosen to have nonvanishing
derivatives at r = r0 . In this work, the open orbitals were determined by solving for eigenstates at speci ed energies; namely, the energies midway between the levels of the energy
spectrum for the closed-type eigenstates. Each \channel" in our problem is represented by
one closed-type state of He+ for the \inner" electron, multiplied by each member of the complete set of closed-type basis functions for a given outer electron partial wave, and by two
open-type basis functions for that partial wave as well. In this study, twenty-two closed-type

34

TARGETCONTINUUM
STATES ε

_
_n=4
_n=3
n=2

n=3
n=2

_
n=1
SHRUNKEN
TARGET STATES

n=1
PHYSICALTARGET STATES

Figure 2. Comparison of the energy spectra of the eigenstates of the physical target state
Hamiltonian with the eigenstates of the target state Hamiltonian in a shrunken con guration
space. The physical target states consist of a Rydberg series and an in nite set of continuum
states of energy , whereas only discrete states appear for the spectrum in the shrunken
space. Note the good correspondence in energies for the lowest few states, as these physicalstates t within the shrunken con guration space. This gure is taken from Ref. [31].

35basis functions and two open basis functions are included for each channel.
In addressing two electron escape, one might suspect that we need to include products of two open-type orbitals. However, this is not the case. Inclusion of such product typebasis functions would require a two-particle matching scheme with a wavefunction representing the motion of the two electrons outside of the reaction volume. The determination of an
exact analytical expression for the three-body Coulomb continuum wavefunction remains an
unsolved problem [44]. By selecting product basis functions in which the radial coordinate
of at least one electron is zero on the surface of the reaction volume, we are able to employ
the same one electron matching scheme used for describing single electron escape, provided
we represent the wavefunction outside the reaction volume as a separable expression. In thisouter region, wavefunctions are approximated by a linear combination of Coulomb functions
with unit charge for the outer electron, multiplied by an He+ eigenstate for the inner electron, followed by antisymmetrization. The channels included in the present photoionization
calculations are nsp, nps, npd, and ndp. That is, we include two partial waves for the
initial state and three partial waves for the nal state. A numerical test showed that f -waves
have a relatively minor e ect on both the single- and double-photoionization cross sections.
Approximately one thousand total two-electron con gurations are used to describe the nal
states reached in single and double photoionization.
Previous applications of eigenchannel R-matrix methods have treated the escapeof a single electron from the reaction volume (regions II and III in Fig. 1). The boundary
conditions imposed by these calculations do not allow for the possibility of direct electron
escape into the double continuum region of con guration space (region IV in Fig. 1). The
eigenchannel R-matrix method is extended in this study to obtain double photoionization
cross sections for helium, using a point of view similar to that applied by Robicheaux etal. [126] to the double continuum of H + e + e . We achieve this by reinterpreting themeaning of the remaining He+ bound state electron in our single photoionization treatment,

36
despite the fact that we actually impose the same (single-escape) boundary conditions. The
basic idea is to recognize that if the complete set of He+ closed-type eigenfunctions in the
R-matrix box are used to represent the \inner" electron in a close-coupling expansion, thenthose eigenfunctions that lie at positive energies (relative to the double escape threshold)must physically represent a (discretized) continuum state of He+ . Therefore, we interpret
all ux escaping in such channels with positive energy thresholds as ux that contributes
to double photoionization. Similarly, all ux escaping in channels at negative energies is interpreted as contributing to single photoionization, even for those negative energy He+ box
eigenstates which do not coincide with a physical He+ eigenstate. Physically, this re ects
our assumption that most of the photon's energy is absorbed by one of the electrons, leaving
the He+ ion in either a negative energy bound state or a discretized positive energy continuum state. Samson [34] has achieved some success using a model based on this qualitative
picture. We will introduce a frame transformation in the next section as an improvementfor distinguishing between the contributions to single and double photoionization.
One well-known [71] consequence of using a discrete spectrum of positive energy
He+ \ionization thresholds" is the appearance of pseudoresonances in the double ionization
cross section. These pseudoresonances are an artifact of the arti cial boundary conditions
imposed on the nite reaction volume; they have no real physical meaning. It is desired
to average out the e ects of these pseudoresonances in the double continuum to obtain asmooth cross section. This can be accomplished by implementing Gailitis [127, 128] andbox averaging techniques (the \box" is another term for the reaction volume). The Gailitis
technique is useful for averaging over resonances near channel thresholds. The technique
is easily implemented in the multichannel quantum defect theory (MQDT) portion of ourcalculation: we simply treat a weakly closed ionization channel (with Ei > E ) as though it
is open when solving the MQDT equations for the photoionization cross section. (Althoughclosed channels are energetically forbidden at larger radial distances, they are included in our

37calculation, as these channels still participate in the short-range physics.) In this analysis,
Gailitis averaging is performed from 2 eV below each He+ threshold all the way up to
threshold. This automatically removes all or most of the Rydberg resonances converging tothat particular ionization threshold (see Fig. 3). In order to average over box size, the crosssections are calculated for ve di erent box sizes (reaction volume radii) in the range 10-12
atomic units, after which they are simply averaged to obtain the nal cross sections reported
here. These averaging techniques will be discussed further in the next section.
Calculations of double photoionization cross sections have been performed using thelength, velocity, and acceleration dipole forms. The length form calculations give unreasonably high values of double photoionization cross sections for our R-matrix approach, while
the results obtained using the velocity and acceleration forms are stable, consistent with eachother, and in reasonable agreement with existing calculations. The length form weights thelarge-r part of the wavefunction the most, which is precisely where our unphysical boundary
conditions probably cause our variational solutions to be the least accurate. Consequently,
the large errors in the length form calculation might have been expected. However, single
ionization calculations using all three dipole forms gave nearly identical results. We only
report our velocity and acceleration results here. The averaged double photoionization crosssections obtained with these forms are shown in Fig. 4. There is a discrepancy of roughly
20% between the two calculated curves. The e ects of pseudoresonances are barely visible
in the small irregularities of the cross section pro les. These irregularities could presumably
be eliminated by averaging over more box sizes, or by applying a convolution technique, as
is discussed in the next section.
Considerable e ort has been expended, both experimentally [34, 96, 129, 130, 131,
132, 133] and theoretically [33, 36, 37, 39, 116] to determine the ratio R of the single and double photoionization cross sections for helium in the intermediate photon energy range. Our
calculated values for the ratio R are compared in Fig. 5 to several existing experimental and

38

7.0
Gailitis Averaging
No Gailitis Averaging++

σ (kb)

6.5

6.0
9d threshold

5.5
9p threshold
9s threshold
10d threshold
5.0
120

125

130
Photon Energy (eV)

135

140

Figure 3. Helium double photoionization cross sections are shown for the case of Gailitis averaging (heavy line) and no Gailitis averaging (light line). In this analysis, Gailitis averaging
is performed from 2 eV below each channel threshold up to threshold. The results shown inthis gure were calculated with the velocity form of the dipole operator, and a box size of
12 a.u. was used.

39
theoretical values. The curve generated by Samson's simple classical model agrees well withthe available experimental data, even though this model does not account for the symmetryof the nal state reached in the photoionization process. The MBPT calculations of Carter
and Kelly [36] and of Hino et al. [39] represent upper and lower limits on the theoreticalvalues shown in Fig.5. These perturbative treatments of the electron-electron interaction
are expected to have some shortcomings in this relatively low photon energy range. Our
eigenchannel R-matrix results agree more closely with the results of Hino et al., although
the ordering of the velocity and acceleration form curves is reversed in these two sets ofcalculations. Pan and Kelly [134] have used another MBPT approach to calculate a maximum R value of almost 5%, which is somewhat larger than the maximum value obtained inboth of our calculations. Proulx and Shakeshaft's [116] cross section calculations for photon
energies in the range 80-140 eV agree well with our velocity form calculations. They also
used their method to obtain values of the angular asymmetry parameter that are in good
agreement with experimental measurements, indicating that their adopted analytical form
for the nal state wavefunction may have some merit. Our results are slightly smaller thanthe experimental values shown in Fig. 5. This discrepancy is comparable to our theoretical\error bar," which we take to be the di erence between our velocity and acceleration form
calculations.
It may not be obvious why the present approach of imposing arti cial boundary
conditions at r = r0 should accurately represent the ux leading to double-ionization. In
particular, we initially neglect the wavefunction in region IV of Fig. 1. However, our reinterpretation of positive energy partial cross sections allows the inner electron to eventuallyescape to in nity through region IV. By describing nonperturbatively the electron correlationand exchange e ects within the reaction volume (region I), we attempt to accurately describe
the initial outward propagation of the electron pair that is induced by the photon absorption.
The electrons are mutually de ected as they escape to in nity beyond the reaction zone, and40this de ection is not properly described by our approach, but we nevertheless expect theamount of ux leading to double escape will not di er widely from our calculated values.
However, as we do not accurately describe the physics of the two electron escape beyond
the reaction volume, we cannot expect our current approach to give a realistic description ofother details of the double photoionization process such as the photoelectron distributions
in energy and angle. However, an improved version of the approach discussed in Chapter 4
may be able to accurately provide at least some details of two electron escape processes.
In summary, a new approach was developed to calculate double photoionization
cross sections of helium. We extended the R-matrix method by reinterpreting the meaningof discrete He+ states that have positive energy eigenvalues. Gailitis and box averaging
techniques were used to obtain smooth cross sections, minimizing the e ects of the arti cialpseudoresonances. Although a discrepancy in the ratio of double and single photoionization
exists between our calculations and experimental results, this discrepancy was comparable
to the uncertainty in our results. We will return to this problem in Chapter 4.

3.2 Simpli ed Model of Electron Scattering
In order to better test the accuracy of wave functions obtained with our previous
approach to helium double photoionization, we applied the eigenchannel R-matrix method
to describe electron impact ionization processes, where two electron escape arises from thearrival of an energetic electron rather than a photon. A simpli ed model of electron scattering, called the Temkin-Poet model, was adopted for two reasons: to simplify the numericalcalculations, and to allow a comparison with other highly accurate numerical results obtainedwith the convergent close-coupling (CCC) approach. Since the CCC method has been successfully applied to many di erent systems (see section 2.3), comparison of this method with
our R-matrix calculations provided a more conclusive test of the accuracy of our methodthan was possible for the case of helium double photoionization. As illustrated in Fig. 5,
there was no consensus in 1994 as to the proper values of the ratio of double and single

41

10.0

8.0

Velocity Form
Acceleration Form++σ (kb)

6.0

4.0

2.0

0.0
80.0

180.0
Photon Energy (eV)280.0

Figure 4. The averaged double photoionization cross section calculated for helium is shown
as a function of photon energy. The solid curve represents calculations performed withthe velocity dipole form, while the dotted curve represents calculations performed with theacceleration dipole form. This gure is taken from Ref. [28].
6.0

++

+

σ /σ (%)

4.0

2.0

0.0
80.0

Samson et al. [34]Schmidt et al. [96, 129]
Wight and Van der Wiel [130]Holland et al. [131]Lablanquie et al. [132]Samson et al. [34] (Classical Model)
Carter and Kelly [36]Hino et al. [39] (Length/Velocity Form)
Hino et al. [39](Acceleration Form)
Present Work [28] (Velocity Form)
Present Work [28] (Acceleration Form)

180.0
Photon Energy (eV)280.0

Figure 5. The ratio of double and single photoionization cross sections of helium is shown
versus the photon energy (status as of 1994). The individual symbols represent experimental
measurements, while the lines represent theoretical calculations. A plot of this ratio for more
recent experimental and theoretical values is given in Fig. 16. This gure is taken from Ref.
[28].

42
photoionization for helium in the intermediate photon energy range. Consequently, we wereunable to clearly evaluate the accuracy of our method when applied to this particular case of
two electron escape. A direct evaluation of our previous results was further complicated by
the discrepancy between calculations using the various forms for the dipole operator. Since
the nal state dynamics in electron impact ionization are very similar to those in helium double photoionization, an application of our method to describe electron scattering provided
an equally challenging but more informative test.

3.2.1 Discussion of the Temkin-Poet Model Ionization of hydrogen byelectron impact represents one of the simplest two electron escape processes. Accurate
calculations of ionization cross sections for this process have only recently been achieved[27]. Earlier attempts to address this problem included the adoption of a simpli ed model ofelectron scattering proposed by Temkin in which all non-zero orbital angular momenta areneglected [135]. That is, the hydrogen atom is treated as though it only has s states, and thetotal angular momentum of the electron-hydrogen system is set to zero. This model was later
studied in much greater detail by Poet [136, 137]. We will denote this model as the TemkinPoet (TP) model. Although the application of this oversimpli ed model to the real e-H
system is quite limited, it does provide a nontrivial test of the accuracy of double continuum
wave functions. As an example of its limited applicability, the TP model generates incorrect
behavior in the ionization cross section near threshold. A simpli ed model that provides
a more realistic description of threshold behavior was introduced by Peterkop. Peterkop's
model [87], which replaces 1=r12 by 1=(r1 + r2 ), does generate the correct threshold law(Wannier exponent) for double escape [88]. However, we do not consider the more realisticPeterkop model in this study, as accurate results are only currently available for comparison
with the TP model.
The Schrodinger equation for the TP model can be solved \exactly". Its replacement of the electron-electron interaction term by 1/r> (the rst term in the usual multipole

43
expansion of 1=r12 ) leads to a separable partial di erential equation. However, the boundary
conditions are not separable. Poet used an interpolation technique to obtain bound state excitation cross sections. Calculations of the ionization cross section for the TP model is a nontrivial exercise, as the presence of nonseparable boundary conditions requires numerical solution of an integral equation. Callaway and Oza [138] obtained a rough estimate of the ionization cross section for this model by combining information about the total cross section (givenby the optical theorem) with calculated excitation cross sections for the lowest three boundstates. Since accurate bound state excitation cross sections are known for the TP model,
this model is often used to test other methods [53, 60, 76, 82, 126, 139, 140, 141, 142, 143].

3.2.2 R-matrix Application to Electron Scattering We now apply our Rmatrix method to address electron scattering, and compare our results with those obtainedusing the CCC approach [29]. The CCC method has been previously applied to the TPmodel for e-H scattering [142]. Extensive data has been tabulated for this case in Ref.
[144]. In addition to addressing e-H scattering in the TP model, we also investigate e-He+scattering for this simpli ed model. To the best of our knowledge, results for the TP modelof e-He+ scattering have not previously been reported. Yet the TP model should be evensimpler to apply to e-He+ scattering than to e-H scattering, as the dominant electron-electron
correlations are more spatially con ned for this case. Nonetheless, the e-He+ model provides
a nontrivial test case for describing two electron escape. Also, the e ect of pseudoresonances
are more prominent in the e-He+ system due to the presence of the long-range Coulomb
attraction.
To explicitly describe the matching scheme adopted, a particular solution of thespatial part of the two-electron wavefunction can be expressed as
X

=A

i

i (r< )Fi (r> );

(22)

44
with a corresponding normal derivative@ = AX  (r )F (r ):i < i >
@r>
i
0

(23)

Here represents an index that distinguishes di erent independent solutions at the specied energy E , A is the antisymmetrization operator, i is a channel index, and r< and r>
represent the radial distances of the \inner" and \outer" electron from the nucleus. (Notethat the antisymmetry operator acts on the total wavefunction, with the form of the spatial wavefunction after antisymmetrization given by Eq. 81. For the S = 0 singlet caseconsidered here, the spatial wavefunction is symmetric.) Analogous to the case of double
photoionization of helium, the spatial part of the wavefunction is represented as a linear
combination of two-electron orbitals:
X

=A

nn

Cnn ns (r1 )n s (r2 ):
0

0

(24)

0

In our formulation, one summation (over n) includes only closed-type orbitals (representing
the \inner" electron), while the other summation (over n0 ) includes two open-type orbitals in
addition to the same number of closed-type orbitals. That is, our two-electron basis consistsof con gurations that are products of the closed-closed or closed-open type. Therefore, the\inner" electron orbital can always be factored out, reducing Eq. 24 to the form of Eq. 22.
Consequently, the matching of the two-electron wave function across the reaction volume is
reduced to a matching of the \outer" electron radial wave function only. Antisymmetrizationof our wavefunction (see Fig. 1) allows the matching to be imposed either at r1 = r0 as afunction of r2 (r1 representing the \outer" electron), or at r2 = r0 as a function of r1 (r2representing the \outer" electron). The values of Fi (r> ) and its derivative are numericallydetermined inside the reaction volume. Outside the reaction volume, the exact form ofFi (r> ) depends on the nature of the scattering target, or more precisely, on the eld of thee ective charge that the \outer" electron experiences as it leaves the reaction volume. The
general asymptotic form for the spatial part of the energy-normalized wave function with an45electron in incoming channel i0 is given by
i (r< ) p 1 fi+ (r> )Sii
2vi
i

X

i =A
0





0

fi (r> )ii ;

(25)

0

where fi are the outgoing and incoming wave solutions for the radial Schrodinger equation ofpthe outer electron in channel i. The factor 1= 2vi , where vi is the speed of the outermost
electron, is required to ensure that jSii (E )j2 represents the ratio of the outgoing radial0

probability ux in channel i to incoming radial probability ux in channel i0 [23]. For e-He+scattering at energies suciently high, the \outer" electron can be approximated by a linear
combination of unit charge Coulomb functions, as in helium double photoionization:
N
X

i =A
0

i=1

i (r< )[fi (r> )ii

0

gi (r> )Kii ]; r> > r0 ;

(26)

0

where fi (r> ) and gi (r> ) are the regular and irregular Coulomb functions, and Kii is the0

reaction matrix. For the case of e-H scattering in the TP model, however, the \outer"electron does not experience a Coulomb charge. Therefore, the radial solution in each channel
is represented by a linear combination of spherical Bessel functions. The coecients of thislinear combination (and therefore the scattering matrix) can then be obtained by matching
the wave functions and normal derivatives at the surface of the reaction volume,
There are several other di erences in e-H and e-He+ scattering. The main di erence is the deeper Coulombic core of the He+ target. Consequently, the maximum e-He+
ionization cross section is more than a factor of ten smaller than the e-H ionization crosssection. Since the hydrogen bound wave functions extend roughly twice as far in space as
the He+ bound wave functions, the reaction volume used for e-H scattering is roughly twiceas large as that used for e-He+ scattering. Consequently, for a given r0 and an identicalbasis set size, convergence of the basis set begins to deteriorate at a lower energy for e-H
scattering. However, the maximum cross section for e-H scattering occurs at a lower energy.
Due to the absence of long-range interactions in the TP model of e-H scattering, multichannel quantum defect theory (MQDT) is not needed, as weakly-closed channels generate no46
Rydberg manifolds. For the actual case of e-H scattering (in contrast to the TP model),a long-range dipole interaction between the outgoing electron and the degenerate hydrogen(bound state) core is also present. MQDT can be implemented to treat this case and obtainbound state excitation cross sections eciently [145]. The existence of a long-range Coulomb
potential between the outgoing electron and the core motivates our use of MQDT for thecase of e-He+ scattering. The physical scattering matrix Sij can be obtained from Eqs. 17-22
in Ref. [146]. The cross section is given by

j!i = k2 2S 4+ 1 jSij ij j2 ;j(27)

where the indices j and i refer to the incoming and outgoing electron channel states, kj =p2(E Ej ) denotes the asymptotic wave number of the incoming electron, and S denotes

the total spin of the system. (Note that we have ignored the Coulomb phase shift here forthe case of e-He+ elastic scattering. This was done for the present purposes of comparison.)

3.2.3 Averaging Techniques In order to calculate physical cross sections using the R-matrix approach, a method must be adopted to express the physical scatteringmatrix, with incoming and outgoing electron channels corresponding to the true physical hydrogenic eigenstates, in terms of our calculated \box" scattering matrix, which has incoming
and outgoing electron channels corresponding to the \box" hydrogenic eigenstates (sometimes referred to as \pseudostates"). Fig. 6 shows the energies of the lowest closed-type box
states of He+ as functions of box size. These discrete states (which represent the \inner"electron) can have both positive and negative energies, since the radial wave functions areforced to be zero at the surface of the reaction volume instead of at in nity. An energy
spectrum of these box states for a single box size is shown in Fig. 2. Imposing boundary
conditions at the box radius leads to box state energies that are dependent on the box size(the size of the reaction volume), as shown in Fig. 6.
In the previous application to helium photoionization, the double escape cross sections calculated with the eigenchannel R-matrix approach were obtained simply by summing

47
10s

5.0

9s
Energy (atomic units)

3.0

8s
7s
6s

1.0
5s
4s
3s
2s

Box
Averaging
Region

-1.0
1s
-3.0

8.0

10.0

12.0
14.0
16.0
Reaction Volume Radius r0 (atomic units)

18.0

Figure 6. Energy dependence of He+ box states as a function of box size (size of the reaction
volume). This gure is taken from Ref. [29].
2.5

2.5

2.5

+

+

He

No Box Averaging(Single Box Size 15 a.u.)

Average Over 5 Box Sizes(14,14.5,15,15.5,16 a.u.)

1.5

1.0

0.5

2

2

2

-3

Ionization Cross Section (units of πa0 x10 )

2.0

-3

Ionization Cross Section (units of πa0 x10 )

2.0

-3

Ionization Cross Section (units of πa0 x10 )He

Average Over 3 Box Sizes(14, 15, 16 a.u.)

2.0

1.5

1.0

0.5

(a)
0.0

+

He

50

100
150
200
Electron Energy (eV)

1.5

1.0

0.5

(b)
0.0

50

100
150
200
Electron Energy (eV)

(c)
0.0

50

100
150
200
Electron Energy (eV)Figure 7. Singlet ionization cross sections for the Temkin-Poet model of electron-He+ scattering using (a) a single box size, (b) an average over three box sizes, and (c) an averageover ve box sizes. The frame transformation was also used to obtain these results. This
gure is taken from Ref. [29].

48
over all discrete states with a positive energy. That is, we considered a box channel withan energy just above threshold as contributing entirely to ionization and a channel with anenergy just below threshold as contributing entirely to bound state excitation. However,each box channel actually possesses both bound and continuum properties. Since the energies of the box states depend on the size of the reaction volume, the number of ionization
channels included changes discontinuously with box size. However, errors associated with
our discretization of the continuum should cancel out if the nal spectra are averaged overa proper range of box sizes.
To implement box averaging, cross sections are generated with several di erent box
sizes, then simply averaged to obtain the nal cross section. This averaging procedure helps
to eliminate arti cial pseudoresonances that would otherwise appear in the continuum, andproduces a much smoother and more realistic ionization cross section. The pseudoresonancesthat appear in the ionization spectrum have no physical signi cance. Rather, they aresimply an artifact of our discretized continuum. These pseudoresonances are essentially due
to probability ux entering the series of Rydberg states attached to each box state in thecontinuum, occurring near the energies of these box states (see Fig. 3). The range of box
sizes over which to average is normally chosen such that a particular box state for the smallest
box size has approximately the same energy as the next higher state (next higher principal
quantum number) for the largest box size (see Fig. 6). This required range r of box sizes
actually depends on the energy range, as higher states will have a larger energy spacing@E=@n for a xed box size (see Figs. 2 and 6). The range r also depends on the averagebox size for which the calculations are performed. For example, with larger box sizes, therate of the energy shift with box size @E=@r0 decreases more slowly for any particular state(see Fig. 6). Since the higher energy levels can be approximated by those of a particle in a
one-dimensional square well, the box averaging range r will be roughly proportional to theinverse of the square root of the energy of the state being targeted for the averaging. This

49
result follows from performing a rst order Taylor expansion of the energy:@E
E (n1 ; r1 ) = E (n0 ; r0 ) + (n1 n0 ) @E
@n n0 + (r1 r0 ) @r r0 ;(28)

where E (n0 ; r0 ) is the energy of the box state with principal quantum number n0 and box size

r0 , and E (n1 ; r1 ) is the energy of the box state with principal quantum number n1 = n0 + 1
and box size r1 . Applying the criterion of equal energies and solving for r = r1 r0 gives
the condition for the required range of box sizes:
r '



@r0
@E





@E
@n0 :

(29)

For a one-dimensional box of width r0 , E (n0 ; r0 ) = C (n20 =r02 ) is the energy of the nth state,
where C is a constant of proportionality. Applying Eq. 29 for this case, r is found to be
proportional to the inverse square root of the targeted energy E . By including a sucientnumber of box sizes and averaging over the proper range, a smooth and reliable cross section
can be obtained.
Other competing factors must be considered to determine the optimal set of box
sizes over which to average. First of all, the box needs to be large enough to include signi cantelectron-electron correlations. In practice, this can be tested by varying the box size andobserving the stability and convergence of the cross sections. At the same time, convergencewith respect to basis set size will deteriorate as the box size is increased. Consequently, thebox size must be kept small enough to give accurate cross sections over the desired projectile
energy range. Again, this can in practice be easily tested by varying the basis set size untilconvergence is achieved. The energy range we have targeted here to obtain the most accurate
cross sections is near the maximum of the ionization cross section. The range r of box
sizes was then determined by selecting a box state with this energy and applying the criteria
used to obtain Eq. 29. The e ect of box averaging is illustrated in Fig. 7. The ionization
cross sections are shown for a single box size, an average over three box sizes, and an averageover ve box sizes. The e ect of the pseudoresonances is hardly noticeable for the average

50over ve box sizes. By averaging over a larger number of box sizes, an even smoother crosssection can be obtained.
For e-He+ scattering, a Gailitis averaging technique is also performed in order toremove the in nite number of Rydberg pseudoresonances near each ionization threshold (seeFig. 3). This technique was used previously to calculate helium double photoionization crosssections. The Rydberg series of pseudoresonances can be interpreted as a set of standing
waves. Each such wave is a superposition of outgoing ux and incoming ux that has beenre ected by a Coulomb barrier in a closed channel far beyond the surface of the reaction
volume. The main idea of Gailitis averaging is to treat the closed channels (channels forwhich Ei > E ) lying just below threshold (for our calculations, this procedure was applied
to channels from 2 eV below threshold up to threshold) as being open in the multichannel
quantum defect theory (MQDT) part of the calculation. As a result, in this energy region
there is no probability ux re ected (unphysically) inward. Therefore, the Rydberg series ofpseudoresonances is eliminated.
In addition to eliminating pseudoresonances, the box averaging technique roughly
averages out errors associated with treating box states as contributing entirely to ionizationor to bound state excitation. However, we will now develop an improvement which gives
the ionization cross sections to reasonably good accuracy even before any box averaging is
performed. This is accomplished with a frame transformation, which is described below.

3.2.4 Frame Transformation A projection technique which relates pseudostate excitation cross sections to the ionization cross section was originally introduced
by Gallaher [147], and later adopted by Callaway and Oza [51], and by Bray and Stelbovics[27]. Let jni represent a physical hydrogenic bound state and jni represent a box hydrogenic
state. Gallaher adopted the following formula to calculate the ionization cross section:

I (E ) =

X

n



1

X

n



jhnjnij n (E );
2

(30)

51
where n are box state excitation cross sections and I is the desired \true" ionization crosssection. The quantity in large parentheses in the above formula can be interpreted as thedi erence between the total cross section and the bound state excitation contributions tothe cross section. Alternatively, this formula can be interpreted as the fraction of the n-thbox state which is composed of true continuum wave functions.
The frame transformation method presented here is somewhat similar to Gallaher's method. However, we directly transform the box state scattering matrix amplitudes
instead of the box state excitation cross sections. By transforming amplitudes rather thanprobabilities, our approach is closer in spirit to frame transformation methods, which havebeen successfully applied to many problems [146, 148, 149]. We rst introduce an analog ofGallaher's technique, and then proceed to develop an alternate method to calculate ionization cross sections. We start by de ning a relationship between the physical (bound-bound)phys and the box scattering matrix S box :scattering matrix Snn
nn
00phys =
Snn
0

X

nn

box hn0 jn0 i:
hnjniSnn
0

0

(31)

Here n and n0 denote physical bound states of the target, whereas n and n0 represent theclosed-type box states representing the \inner" electron in the scattering process.
We will only investigate electron scattering from the ground state (n0 = 1) of thetarget here. Since the radial wavefunctions of the lowest two physical states (the 1s and 2s
states) t inside our reaction volume, the scattering matrix elements of interest for calculating
the desired bound state excitation cross sections are given by

Snphys=1XnhnjniSnbox :1(32)

Here we have used the fact that the closed-type box states form a complete orthonormal
basis since they satisfy the same Schrodinger equation and boundary conditions. The elastic
scattering cross section and the inelastic cross section 1s!2s can be calculated directly from
Eq. 27, as the matrix elements S11 and S21 are nearly identical to S11 and S21 . Ionization

52
cross sections and excitation cross sections for higher bound states can be calculated byusing the frame transformation technique that is described below. Although our physicalS -matrix is not exactly unitary (the largest error in unitarity that we have observed is about0.2%), deviations from unitarity do not signi cantly a ect our results. Assuming unitarity,
the ionization cross section is given by1X
2
I = k2 2S 4+ 1 1
jS1phys
n j
1
n=1


1 nX
max
X
2
S
+
1

box
2
j hnjniS1n j := k2 4 1
1
n=1 n=1
(33)

Obviously, this in nite sum over n can not be explicitly evaluated numerically, but rather
must be truncated at some nite n. Numerical tests have shown that the sum over nconverges slowly, thus requiring a relatively large number of terms to approach a converged
answer.
Rather than pursuing the approach based on Eq. 33, we develop an alternative
expression for the ionization cross section by introducing the physical hydrogenic continuum
states (Coulomb functions), which are energy normalized over the in nite radial domain
0  r < 1. We extend the frame transformation (Eq. 31) to relate continuum elements ofthe physical scattering matrix to our box scattering matrix elements:

Sphys=1XnhjniSnbox ;1(34)

where ji represents a physical energy-normalized continuum state at energy  for the innerhydrogenic electron. The ionization cross section is then given by
Z E
2I = k2 2S 4+ 1 d Sphys
1
0
1
2
Z E
X
2
S
+
1

box
d hjniSn1 := k2 40
1
n(35)

The transformation factor hjni determines the fraction of the box state n which will contribute to the ionization cross section at the continuum energy . A plot of jhjnij2 as a53function of the inner electron continuum energy  is shown in Fig. 8 for n=5 and n=9 (then=9 curve has been multiplied by a factor of two so it can be seen on the same graph). These
curves were generated for e-He+ scattering with a box size of r0 = 16:5 atomic units. The
pro le of the plot is similar to the familiar function [sin(x)=x]2 . In the limit of an in nite
reaction volume, the transformation elements approach Dirac delta functions, in which casethe physical and box cross sections become identical. For the size of the reaction volume
shown, the n = 5 state has an energy just above the ionization threshold. The maximumof the frame transformation factor occurs at the box state energy of n. In our previous
formulation (and application to helium photoionization), this channel would be interpreted
as contributing entirely to ionization. However, from Fig. 8, one observes that a signi cantportion of the ux in this channel (nearly half) will contribute to the excitation cross sectionsof high bound states. The zeros of the transformation factor occur precisely at the energiesof the other box states, owing to their orthogonality. Apart from a normalization factor, theclosed-type box states are proportional (for r < r0 ) to the physical continuum states, but
only at those discrete energies for which the continuum wave function happens to vanish at
the surface of the reaction volume.
In practice, the eciency of our method for calculating the frame transformation
matrix is greatly enhanced by employing the identity

hun jui = W2((u ; un))jr0 :n(36)Here un represents the box state radial wave function multiplied by r, u represents thecontinuum hydrogenic radial wave function multiplied by r, W (u; un )jr0 is the radial Wronskian evaluated at r = r0 ,  is the energy of the continuum state, and n is the energy ofthe box state. This identity is easily derived by manipulating the box and continuum stateradial Schrodinger equations. The greatest changes observed in our cross sections when theframe transformation technique is implemented (in comparison to our previous method) occur for box sizes in which the energy of one box state is very near threshold. For this case,

54

_
_n=5
n=9 (x 2)
6.00_ 2|<ε|n>|

4.00

2.00

0.00
0.00

0.50
1.00
1.50
Continuum Wave Energy ε (atomic units)

2.00

Figure 8. Plot of the square of the transformation factor used in the projection technique
as a function of the continuum wave energy. The dashed line corresponds to a box state ofn = 5, while the solid line corresponds to a box state of n = 9. The n = 9 state has been
multiplied by a factor of 2 so it can be seen more easily on the same graph. This gure istaken from Ref. [29].

55
the frame transformation gives roughly an equal contribution to the bound state excitationand ionization cross sections for that particular near-threshold channel. An example of thissituation is shown in Fig. 9(a), where the ionization cross section for the TP model ofelectron-He+ is plotted with and without a frame transformation for box sizes of 16.5 and17 atomic units. For a box size of 16.5 atomic units, a box state lies just above threshold.
If no frame transformation is implemented, this channel would be treated as contributing
entirely to the ionization cross section, and as a result the ionization cross section would beoverestimated. For a box size of 17 atomic units, the box state now lies just below threshold.
Since this channel would be treated as contributing entirely to the bound state excitation
cross section, the ionization cross section would be underestimated. When the frame transformation is implemented, the proper ionization and bound state excitation contributions
are projected out, giving a nearly identical ionization cross section pro le for both box sizes.
Use of a frame transformation to calculate results which are independent of box size (alongwith a convolution technique to be described shortly) provides an ecient alternative tobox averaging for obtaining stable cross sections. This technique signi cantly reduces thenumber of computations required, since results can now be obtained by simply performing
an R-matrix calculation with a single box size.
An energy convolution technique has been introduced to improve the smoothnessof cross sections shown in Fig. 9(a). A typical ionization cross section pro le for a single
box size before the convolution approach is performed is illustrated in Fig. 7(a), with thelarge oscillations re ecting the presence of pseudoresonances. The convolution function used
to obtain the smooth pro les in Fig. 9(a) was a Gaussian function of variable width. We
allowed the width of the Gaussian to scale as dE=dn, since this factor describes the energy
spacing between pseudoresonances (for our approximation of a particle in a one-dimensionalpsquare well, dE=dn / E ). This approach forces a Gaussian of zero width to be used atthreshold, so that the convolved ionization cross section is ensured to vanish at threshold

56
0.0020+2

Ionization Cross Section (units of πa0 )He0.00150.00100.00050.000050.0

Box Size: 16.5 a.u.
No Frame Transformation
Frame Transformation
Box Size: 17 a.u.
No Frame Transformation
Frame Transformation

(a)

100.0
150.0
Incoming Electron Energy (eV)200.00.0020+2

Ionization Cross Section (units of πa0 )He0.00150.0010Box Averaging (r=14-16 a.u.)
Convolution (r=15 a.u.)

0.0005(b)0.0000
50.0

100.0
150.0
Incoming Electron Energy (eV)200.0

Figure 9. Illustration of the frame transformation and convolution methods to obtain reliablecross sections. Singlet ionization cross sections for the Temkin-Poet model of electron-He+
scattering (a) using box sizes of 16.5 and 17 a.u.: the dashed and long-dashed lines represent
the cross sections obtained without using a frame transformation, while the solid and dottedlines represent the cross sections obtained when a frame transformation is implemented, and(b) using an average of nine box sizes in the range of 14 to 16 atomic units, shown by thedotted line, and a convolution of a single box size of 15 a.u., shown by the solid line. Thesegures are taken from Ref. [29].

57(in contrast to a xed-width energy convolution). Fig. 9(b) shows a comparison of box
averaging of ionization cross sections over nine box sizes from 14 to 16 a.u., and a frametransformation and convolution applied to a single box size of 15 a.u. (at the midpointof the range of box averaging). The frame transformation/convolution approach generates
essentially the same magnitude of the ionization cross section as obtained by box averaging,but with a much smoother pro le. Furthermore, the number of required calculations is
reduced by a factor of nine (the number of box sizes). Although the convolution approachhas been useful in obtaining the results presented here, this technique should be used withprudence, since both physical and unphysical resonances will be broadened or else eliminatedaltogether. This is not a concern here, as no physical resonances occur above the threshold.
More generally, double continuum cross sections obtained with the R-matrix technique may
contain both physical and unphysical resonances. For example, double photoionization ofa heavier atom such as Ar is known to produce resonances embedded in double continua.
Box averaging should be used rather than the convolution approach for such cases in orderto ensure that only arti cial pseudoresonances are eliminated, with the \true" resonances
remaining una ected.

3.2.5 Application to e-H and e-He Scattering In the current study, up+

to 46 closed and two open orbitals per channel have been included in an R-matrix calculationof e-H scattering within the TP model. Fig. 10 shows a comparison of the singlet crosssections obtained by the CCC approach and by the eigenchannel R-matrix method for boththe elastic and inelastic 1s 2s cross sections. The two methods agree to within 3% for thesecross sections. The circles in the gure denote \exact" solutions of the TP model which were
calculated numerically [138, 141]. The energy range shown for the incoming electron is10-50 eV for the CCC approach, while results from the ionization threshold (13.60 eV) to
50 eV are shown for the R-matrix approach. A basis set of 30 Sturmian functions wasused to generate the CCC results. The R-matrix results were obtained by averaging over58
ve di erent box sizes from 24 to 28 atomic units. Although this averaging procedure did
not signi cantly change the results obtained for bound state excitation cross sections, it isrequired to eliminate arti cial pseudoresonances and to obtain a smooth ionization crosssection. The frame transformation technique described earlier is not required to generate
accurate elastic and 1s 2s inelastic cross sections, as the reaction volume is large enoughto contain the physical 1s and 2s wave functions of hydrogen.
Fig. 11 shows a comparison of the elastic and 1s 2s inelastic cross sections for thecase of singlet e-He+ scattering in the TP model. A basis set size of 30 Sturmian functions wasagain used to generate the CCC results. Up to 34 closed and two open orbitals per channel
were used to generate the R-matrix results. The R-matrix cross section was averaged overnine di erent box sizes ranging from 14 to 16 atomic units. Again, the two methods agree
to within 3% over most of the energy range.
Figs. 12 and 13 show the TP model singlet ionization cross sections for electron
scattering of both hydrogen and He+ targets. The same basis set sizes and averaging technique used to obtain the bound state excitation cross sections were adopted to obtain theionization cross sections. In addition, the frame transformation method, as given by Eq.
35, was used to generate the R-matrix results, and a convolution of the box averaged crosssection was performed to obtain a smooth pro le. The circles in Fig. 12 no longer represent
the \exact" solutions of the TP model, but rather a complicated interpolation calculated
by Callaway and Oza [138]. These interpolated values agree closely with the CCC results.
The agreement between CCC and the R-matrix approach is also fairly good, especially forincoming electron energies larger than 30 eV, where the agreement is better than 10%.
However, the R-matrix results are up to 70% smaller than those generated with the CCC
approach in the lowest energy range. We have less con dence in our R-matrix treatment at
lower energies, since our approximation of neglecting electron-electron correlations outside
the reaction volume is expected to break down in this energy range. The CCC approach is

59
0.50
H
References [138] and [141]CCC
R-Matrix

2

Elastic Cross Section (units of πa0 )

0.40

0.30

0.20

0.10

(a)
0.00
10.0

20.0

30.0
Incoming Electron Energy (eV)

40.0

50.0

0.050

2

Inelastic 1s -> 2s Cross Section (units of πa0 )H
References [138] and [141]CCC
R-Matrix

0.040

0.030

0.020

0.010

(b)
0.000
10.0

20.0

30.0
Incoming Electron Energy (eV)

40.0

50.0

Figure 10. Singlet elastic and inelastic 1s 2s cross sections for the Temkin-Poet modelof electron-hydrogen scattering. The values for the exact results, denoted by open circles,
are taken from references [138, 141]. The dashed line represents calculations performed with
the CCC approach, while the solid line represents calculations performed with the R-matrix
technique. These gures are taken from Ref. [29].

60
0.03
+

He

2

Elastic Cross Section (units of πa0 )CCC
R-Matrix

0.02

0.01

(a)
0.00
50.0

100.0
150.0
Incoming Electron Energy (eV)

200.0

0.005
+

2

Inelastic 1s -> 2s Cross Section (units of πa0 )He
CCC
R-Matrix

0.004

0.003

0.002

0.001

(b)
0.000
50.0

100.0
150.0
Incoming Electron Energy (eV)200.0

Figure 11. Singlet elastic and inelastic 1s 2s cross sections for the Temkin-Poet model ofelectron-He+ scattering. The dashed line represents calculations performed with the CCC
approach, while the solid line represents calculations performed with the R-matrix technique.
These gures are taken from Ref. [29].

61
0.030

2

Ionization Cross Section (units of πa0 )

H

0.020

Reference [138]CCC
R-Matrix

0.010

0.000
10.0

20.0

30.0
Incoming Electron Energy (eV)

40.0

50.0

Figure 12. Singlet ionization cross sections for the Temkin-Poet model of electron-hydrogenscattering. The interpolated values, as calculated by Callaway and Oza [138] are denotedby open circles. The dashed line represents calculations performed with the CCC approach,while the solid line represents calculations performed with the R-matrix technique. This
gure is taken from Ref. [29].
0.0020+2

Ionization Cross Section (units of πa0 )He0.00150.0010CCC
R-Matrix

0.00050.0000
50.0

100.0
150.0
Incoming Electron Energy (eV)200.0

Figure 13. Singlet ionization cross sections for the Temkin-Poet model of electron-He+
scattering. The dashed line represents calculations performed with the CCC approach, while
the solid line represents calculations performed with the R-matrix technique. This gure istaken from Ref. [29].

62also not expected to predict accurate ionization cross sections at a few eV above threshold.
Application of this method to the real case of e-H scattering from threshold to 500 eV gave
results which were up to 30% lower than the experimental values in the threshold region up
to 4 eV [27]. More accurate studies of the ionization cross sections for low energies havebeen discussed in sections 2.3 and 2.4.
Fig. 13 also shows fairly good agreement between the results calculated for e-He+
ionization in the TP model with the CCC and the R-matrix method over most of the energy
range. As with the case for e-H scattering, the R-matrix method signi cantly underestimatesthe cross section in the energy range just above threshold. Presumably, better agreement
could be achieved at lower energies if a larger reaction volume size was used for the Rmatrix method, as the electron-electron correlation e ects would then be included over amuch greater distance. Exploratory attempts to signi cantly increase the reaction volume
size indicated that this appears to be the case, although the use of a much larger basis set
size led to numerical instabilities which did not permit a smooth cross section to be obtained
over a very large energy range.
Although the TP model greatly simpli es the calculations required to obtain electron scattering cross sections, the R-matrix approach described here can be applied to more
complicated processes, such as the real case of e-H scattering. One reason for adopting the
TP model was to reduce the required amount of calculation. In this study, we required
roughly a couple of hours on a DEC 5000/240 workstation to obtain cross sections for asingle box size. To apply this R-matrix technique to real electron scattering processes, thetotal number of two-electron orbitals needs to be increased by roughly a factor of ve or
ten for each total orbital angular momentum of the system. Although the number of calculations required greatly increases due to a larger number of required orbital con gurations,
the R-matrix technique should be capable of generating accurate cross sections for this case.
Furthermore, a signi cant reduction in required calculations can be accomplished if reliable63
results are generated with a frame transformation/convolution approach for a single box size
rather than a box averaging technique.
By exploring a simpli ed model of electron scattering, many of the advantages
and limitations of both the CCC approach and the eigenchannel R-matrix technique havebecome apparent. One of the major problems encountered in extending the eigenchannel Rmatrix technique to double continuum processes involves our inability to obtain well-behaved
antisymmetric solutions outside the reaction volume in the double continuum region (region
IV of Fig. 1). This asymmetry can be seen most clearly in plots of the calculated partial
ionization cross section versus the \inner" electron energy . If antisymmetry was strictly
enforced, the partial cross section must be symmetric about the midpoint of the channel
energy, since there would be no way of distinguishing between the channel energy i of one
electron, and the energy E

i of the other electron, where E is the total energy of thesystem. However, this symmetry is not exhibited by our R-matrix or CCC results.
Neither method is able to provide accurate cross sections at low energies. However,
the CCC approach is able to predict better results near threshold. This re ects the limitationof our R-matrix approach to adequately treat electron-electron correlation e ects outside
the reaction volume. Approximations in our current approach also appear to limit accurate
calculations to total cross sections, whereas the CCC method has been used successfully togenerate accurate di erential cross sections. The limitations associated with basis set size
are more apparent for our R-matrix method, as the results shown here are for a smaller
energy range than has been previously reported in the CCC calculations.
A simpli ed model of electron scattering was adopted here in order to compare
ionization cross sections obtained with an eigenchannel R-matrix method and the CCC
approach. The singlet cross sections for e-H scattering have been explored for an energy range
from threshold up to 50 eV, while e-He+ singlet scattering cross sections have been calculated
for an energy range from threshold up to 200 eV. The results obtained by the two di erent

64
methods agree well over most of the energy range considered here. However, our R-matrix
calculations are found to underestimate the cross section for energies just above threshold.
This discrepancy at low energies indicates that our approximation to neglect the electronelectron correlations outside our reaction volume is not adequate for describing the twoelectron escape process for this energy range. Although the error associated with describing
electron impact ionization processes appears to be smaller than the error encountered in
describing helium double photoionization, the magnitude of the error associated with ourpresent approach may be too large to resolve the discrepancy between the experimental andtheoretical values of helium photoionization shown in Fig. 5. Nevertheless, the agreementwith the CCC approach gives us con dence that the eigenchannel R-matrix technique candescribe two-electron escape processes in the intermediate energy region, where the maximum
ionization cross section occurs, with an error of about 10-15%.

CHAPTER 4FINITE ELEMENTS: A NEW APPROACH
As has been shown in the previous chapter, an extension of the eigenchannel Rmatrix method achieved some success in treating double escape processes. Although thisapproach seems capable of describing two electron escape processes accurately, some problems arose in its application to helium double photoionization. A discrepancy between thevarious forms used in calculating dipole matrix elements indicated inaccuracies in the initial
and/or nal state wave functions. In particular, the discrepancy between the velocity andacceleration forms was roughly 20% (see Fig. 4), and the length form gave results which were
nearly an order of magnitude larger than the results obtained for other forms. Interestingly,
the convergent close-coupling method of Kheifets and Bray [58] gave unreasonable resultsin the length form, much like our R-matrix calculations. However, the hyperspherical closecoupling method of Tang and Shimamura [62] was able to obtain good agreement between
length and acceleration forms. In an e ort to reduce the discrepancy between the various
dipole forms, we have combined an eigenchannel R-matrix approach with a nite element
method to treat two electron photoejection.

4.1 Method of Finite Elements
The eigenchannel R-matrix method (Chapter 2) and its application to helium double photoionization and electron impact ionization (Chapter 3) have been previously discussed. The major di erence between the current approach and our previous application
involves our choice for the variational basis set. Instead of forming a two electron global
basis set consisting of products of one electron hydrogenic orbitals, we adopt a local niteelement basis set for the present study. Our expectation was that the nite element method

66(or other local basis set methods, e.g., B -splines) should be able to better represent theinitial and nal state wave functions, thereby reducing the discrepancy among the di erent
dipole forms. The suitability of a nite element approach for calculating accurate boundstate wave functions has been previously shown [150, 151]. This method has been used to
calculate accurate bound state excitation cross sections in elastic scattering [152]. A similar

B -spline local basis set has been combined recently with R-matrix theory to address electron
scattering of hydrogen [153]. In this work, we report the rst application of a nite element
basis set to two electron escape processes. The nite element set used in our approach consists of a set of six fth order Hermite interpolating polynomials that are shown in Fig. 14
and described elsewhere [152, 154].
In our original approach, we solved for the radial part of the one-electron hydrogenic
orbitals k (r) subject to \box" boundary conditions, that is, boundary conditions imposed at
the surface of the reaction volume. Orbitals that vanished at the outer radius of the R-matrix
box were referred to as \closed" orbitals, while \open" orbitals referred to orbitals whichwere nonzero at the box radius. In our more recent approach, we expand these hydrogenic
orbitals with a nite element basis set:k (r) =N X
6X
n=1 i=1

Cin ui (xn );(37)

where n denotes the sector number, and i denotes one of the six Hermite interpolating
polynomials. These six polynomials are de ned by enforcing a total of six \boundary"conditions at three points within a sector, namely, the polynomial value and its rst derivativeat the two endpoints and the center of the sector. Each of the six polynomials will have
only one nonzero boundary condition, with this boundary condition chosen to be unity (seeFig. 14). Also, ui (xn ) is de ned to be nonzero only in sector n, and xn is only de ned
over the interval [-1,1]. The transformation to the \physical sector" [ri ; ri+1 ] for a value r(ri < r < ri+1 ) is then given byr = ai xi + bi ;

(38)

67
where

ai = ri+12 ri ; bi = ri+12+ ri :(39)

Although there appears to be 6N independent coecients in Eq. 37, by enforcing boundary
conditions at the origin and the box radius r0 , and also by imposing the constraint of acontinuous wave function and rst derivative across each sector boundary, the number of
independent coecients is reduced to 4N .
As in our earlier R-matrix calculations, a two-electron wave function is formed bytaking products of one electron orbitals:(~r1 ;~r2 ) = A

N X
M X
6 X
6
X

n=1 m=1 i=1 j =1(^r1 ; r^2 );Cijnm ui (xn )uj (ym )YlLM
1 l2

(40)

where A is the antisymmetrization operator (see Appendix B for further details), andYlLM(^r ; r^ ) is a coupled spherical harmonic. Our two dimensional sectors are chosen to
1 l2
1

2

be rectangular, so that ui (xn ) and uj (ym ) are of similar form. Although there are 36 coecients for each sector, enforcing continuity of the wave function and rst derivatives acrosssector boundaries reduces the number of independent coecients to 16. To obtain the groundstate wave function, we impose the boundary condition that the wave function vanishes on
the surface of the reaction volume. This is easy to implement with a Hermite interpolating
polynomial basis set, as only one of the six polynomials (u5 (x), as denoted in Fig. 14) isnonzero on the outer boundary. Simply setting the corresponding coecient of this basis
function in the outermost sector to zero ensures that the wave function vanishes at the box
radius.
In order to express the nal state correctly, our expansion must include basis functions which are nonvanishing on the surface of the reaction volume in order to allow forelectron escape. We retain the idea in our original approach of also including closed-open
two electron con gurations consisting of a product of a closed and an open one-electron hydrogenic orbital. However, the open hydrogenic orbital is now replaced by a single Hermite

68

1.0

u1(x)
u2(x)
u3(x)
u4(x)
u5(x)
u6(x)

ui(x)

0.5

0.0

−0.5
−1.0

−0.5

0.0
x

0.5

1.0

Figure 14. The set of six Hermite interpolating polynomials adopted as the nite element
basis set in this study.

69interpolating polynomial in the outermost sector which is nonzero at the outer boundary(denoted by u5 (x)). In other words, we add basis functions of the form

A

M X
6
X
m=1 j =1

LMC5Nm
j u5 (xN )uj (ym )Yl1 l2 (^r1 ; r^2 )(41)

to our expansion in Eq. 40. Retaining the closed-open type of two-electron con gurations
allows us to use the same criterion as previously to distinguish between single and double
photoionization. That is, if the energy of our closed orbital representing the \inner" electron
is negative, then we consider it to remain bound (single ionization), whereas a positive
closed orbital energy is considered ionized (double ionization). This criterion along with a
box averaging technique was used to obtain single and double photoionization cross sections
in our original study. An improvement over this approach (the frame transformation, section
3.2.4) has been developed since then. In the context of photoabsorption, each closed orbital
will contribute to both the single and double ejection cross section. The contribution of eachclosed orbital to photoejection of the \inner" electron can be determined by projecting thewave function onto a continuum hydrogenic wave function. That is, analogous to Eq. 34 in
the case of electron scattering, for double photoejection processes the \box" dipole matrix

D( )box is related to the physical dipole matrix D( )phys by( )phys
Dl;g
=

X

nl

hljnliDnl;gbox ;

(42)

( )

( )boxrepresents the dipole matrix element for excitation of the inner electron from
where Dnl;gthe ground state \g" to the nal pseudostate nl, and l denotes the nal continuum statehydrogenic wavefunction. Similarly, single photoejection cross sections (with the \inner"electron remaining bound) are obtained by projecting the wave function onto a physicalbound state hydrogenic wave function. The double photoejection cross section is given (ina.u.) by2
 = 4

!

XZ E

l

0

d

X

nl

hljnliDnl;gbox
( )

2

;

(43)

70
( )boxwhere Dnl;grepresents the velocity form of the dipole matrix element for excitation of theinner electron from the ground state \g" to the nal pseudostate nl, is the ne-structure
constant, and ! is the photon energy in a.u. (compare to Eq. 12). This frame transformation
technique is used to obtain the results shown here.
Upon rst glance, it would appear that the nite element basis set has a disadvantage of requiring the solution of a larger number of expansion coecients than would
be required for the hydrogenic basis set used previously, ultimately limiting the number ofnite element sectors that could be used. However, since the local basis functions are only
de ned to be nonzero for a given sector, the Hamiltonian and overlap matrices are sparse.
Ecient numerical routines are implemented to take advantage of these sparse, symmetricmatrices. A Lanczos algorithm [155] is used to obtain the helium ground state wave functionand energy. This iterative numerical method allows one to obtain the lowest eigenvalues fora speci ed small number of eigenvectors without having to diagonalize the entire matrix.
The nal state is obtained by solving the usual generalized eigenvalue problem [23]C = bC;(44)= 2[ES H L];

(45)

where

S is the overlap matrix, H is the Hamiltonian matrix, L is the Bloch operator, and  is thesurface overlap matrix. The matrices in Eq. 44 are partitioned into closed and open basis
sets:
2
4

cc

co 3 2 C c 3

oc

oo

54

Co

2

b

5= 4

0

0

0 oo

32
54

Cc
Co

3
5

:

(46)

This leads to the reduced generalized eigenvalue problem:
oo C o = boo C o ;

(47)

71
where
oo = oooc ( cc ) 1 co :(48)

Once the open components C o of the solution eigenvector are obtained by solving Eq. 47,
the closed components are given by

C c = ( cc ) 1 co C o :(49)

The streamlined eigenchannel treatment [23] involves a transformation which diagonalizes
the Hamiltonian matrix. For a closed two-electron basis set which is orthogonal, the closed
portion of the overlap matrix is diagonal. Consequently, cc becomes diagonal after the
Hamiltonian is transformed, and so is trivial to invert. As this transformation of the Hamiltonian matrix only needs to be performed once, a great computational improvement is attained
over directly solving for the generalized eigenvalue problem at each desired energy.
For our purposes, application of this streamlined transformation is impractical owing to the large dimension of the Hamiltonian and overlap matrices. Rather than solve theoriginal generalized eigenvalue problem (Eq. 44), we would like to solve the reduced eigenvalue problem, given by Eq. 47. Note that oo involves the inverse of cc, which has large
dimension, but is sparse. Rather than directly calculating the inverse of this matrix, we use
a linear algebraic routine to solve the sparse system of equations
cc Qco = co :(50)

The oo matrix is then given byoo = oooc Qco :(51)

For our nite element application, the open-open part of the surface overlap matrix is diagonal, and in fact equal to the identity matrix. This reduces the generalized eigenvalue
problem given in Eq. 47 to the ordinary eigenvalue problem
oo C o = bC o :

(52)

72
The greatest computational e ort required to obtain results with a nite element
basis involves solving the system of equations in Eq. 50. Originally, a conjugate gradientmethod with a diagonal preconditioner was used to solve this system of equations. A generalized minimum residual method iterative method with an incomplete LU factorization(factorization of a matrix into the product of a lower triangular and an upper triangular matrix; see Ref [156] for further details) for preconditioning was later adopted from the sparselinear algebra package (SLAP) to perform this task more eciently. However, the amountof computational e ort required to solve this linear set of equations using these iterative
methods increases rapidly with energy. This is due in part to a larger number of channels
which are required to be included, and in part due to the need for a greater density of sectors(smaller sector sizes) in order to accurately represent the small wavelength behavior of the
wavefunction in some of the channels. Numerically, this translates into a larger number of iterations required to solve for each successive right hand side vector, and also a larger numberof right hand side vectors which require a solution. The number of right hand side vectors is
simply the total number of energetically allowed channels. Due to the large computationaltime which was required to calculate cross sections at high energies, practical applicationof this approach to helium was limited to energies no higher than 100 eV above the double
photoionization threshold.
A tremendous increase in the numerical eciency of our method for solving linear
systems of equations was achieved by adopting a complete factorization approach. Not onlyis the cc matrix in Eq. 50 a large sparse matrix, but for the appropriate numbering schemeof nite element nodes it also has a banded structure. LAPACK (Linear Algebra Package;
see Ref. [157]) routines for banded matrices are used to perform an LU factorization ofthis matrix. This leads to a numerical method whose time requirements are only weakly
dependent on the number of channels included (and thus more nearly energy-independent).

73
The time requirements for this approach now depend primarily on the number of nite element basis functions included. Applications of this method to helium double photoionization
have decreased the computational time required to obtain results by a factor of 30-50 overprevious iterative solvers. One drawback of the LAPACK banded factorization routine is
the greater memory requirements. Whereas previous routines could use sparse matrix storage routines which only stored nonzero elements (and their indices), all elements within the
maximum bandwidth of the banded matrix must be stored in a full matrix in order to use
the LAPACK routine. Application of this numerical routine to two-electron escape processes
using 18x18 nite element sectors has required memory usage of 750MB. Since this memory
requirement exceeds the current capabilities of our \in-house" computer resources, the useof a SGI (Silicon Graphics Incorporated) Power Challenge machine at the National Center
for Supercomputer Applications was necessary for performing the calculations presented in
the next section.

4.2 Double Photoejection Cross Sections of He and H
Although helium double photoionization processes have received a tremendous
amount of attention in the past, there has been a surprisingly great amount of disagreement concerning the maximum value of the ratio of double to single ionization, both fromexperiment and theory, at least in the region of intermediate photon energies. The maximum
value of this ratio occurs for photon energies in the range of 150-250 eV. It has been predicted
and measured to reach peak values anywhere from 0.030 to 0.055 before slowly falling to an
asymptotic value of 0.0167 at high energies [38, 102, 113, 158]. The discrepancies among different approaches to obtain the values for this parameter illustrate the diculties that havebeen encountered by theoretical attempts to describe the relatively infrequent occurrence of
two electron escape processes in this intermediate energy range.
In contrast to the case for helium, studies of one- and two-electron photoejection
from H at photon energies above the double escape threshold have been quite sparse. The

74
determination of H photodetachment cross sections has received a signi cant amount offocus for low photon energies [159], primarily due to its importance in astrophysics. The
threshold law for double photodetachment of H has been explored both theoretically [160]and experimentally [95], but the applicability of these studies are limited to just the rst one
or two eV above threshold. Although no experimental values of H photoejection at higher
energies are known to exist, it should be possible to perform these types of experimentsin the near future. To our knowledge, only one theoretical study has been undertaken toaddress photoejection of H at these energies. This sole theoretical study was performed by
Broad and Reinhardt over twenty years ago using a multichannel J-matrix technique with an
L2 basis set [52]. Total photodetachment cross sections were reported for photon energies in
the range of 1-69 eV. They claimed a convergence of their two electron photoejection crosssections to within 15% and quoted a value for this ratio of roughly 0.04 over this energy
range. Results presented in their paper indicated a maximum value for the ratio 2e =1e
which lies in the range of 0.055-0.085. Fluctuations in their calculated double photoejection
cross sections made it dicult to determine a precise value for the peak ratio. However,
we can ascertain an approximate \error bar" of 0:020 for these calculations of Broad andReinhardt.
Like helium, H is an ideal system for exploring two electron photoejection, due toits inherent simplicity. There are important di erences, however, between double photoionization of helium and double photodetachment of H . First, the ratio between the single
and double escape threshold energies is much smaller for H than for He, since H has onlya single bound state. Secondly, while a single electron escaping from helium experiences ascreened Coulomb potential (to rst approximation), this Coulomb potential is absent for asingle electron escaping from H . Due to the smaller nuclear charge of H in comparison tohelium, one expects electron interaction e ects to play an even greater role in the dynamics
of photoejection for this case. Fig. 15 compares the radial ground state wavefunction of

75
helium and H . The broader H wavefunction re ects its smaller nuclear charge.
We calculate the ratio of double to single photoejection cross sections of heliumand H for photon energies of 79-460 eV and 14.35-110 eV [30]. Roughly 180 hours ofsupercomputer time on a SGI Power Challenge machine was used to perform these crosssection calculations. The results presented here were obtained by using three partial waves
in the initial state and four partial waves in the nal state. This is in contrast to our earlier
study of helium photoionization, where two partial waves were used for the initial state andthree partial waves were used for the nal state. Box averaging was used to obtain the crosssection pro les presented here, and a frame transformation was implemented to distinguishbetween single and double photoejection contributions. A Gailitis averaging technique, as
outlined in Chapter 3, was used to obtain results for helium. Our calculated ground stateenergies of helium and H agree well with the \exact" nonrelativistic in nite mass values,
which are taken from Reference [161] (see Table 1). Note that the ground state energyof helium in our original calculation using a hydrogenic basis is much less accurate thanthe value obtained with a nite element basis. Part of this is improvement is due to theinclusion of an additional partial wave in our nite element calculations (see Table 2), butsome improvement is also due to our use of a local basis set.
Our results for the ratio of double to single photoionization for helium are compared
to recent theoretical and experimental values in Fig. 16. Experimental measurements of thisratio by Dorner et al. [26] and Levin et al. [158] over the intermediate energy range have
similar pro les, but di er in magnitude by 10-20%. Samson's values [162] in Fig. 16 haveactually been tted to a smooth curve, but these smoothed values di er very little from theoriginal experimental values (to within roughly a couple of percent). The error bar associatedwith Samson's measurements is approximately 5%, while Levin's experimental error bar iscomparable to the scatter in his reported values, as seen in Fig. 16. Recent theoretical
calculations of the ratio include our previous eigenchannel R-matrix calculations using a

76

Figure 15. Comparison of the He (top) and H (bottom) ground state radial wavefunctions.
These are plots of the square of the scaled radial wavefunction (jr1 r2 (r1 ; r2 )j2 ) for thepartial wave l1 = l2 = 0. The He wavefunction extends to approximately 4 a.u., while the
H wavefunction extends to about 10 a.u. This re ects the weaker binding energy of thetwo electrons to the nucleus for H .

77

Table 1. Comparison of ground state energies of helium and H , calculated with hydrogenic
and nite element basis sets. The corresponding ground state wavefunctions were used toobtain the helium and H double photoejection results in Chapters 3 and 4. The \exact"
nonrelativistic in nite mass values listed in the table were taken from Ref. [161]. Note thatthe ground state energy of helium in our original calculation using a hydrogenic basis is muchless accurate than the value obtained with a nite element basis. Part of this improvement
is due to the inclusion of an additional partial wave in our nite element calculations (seeTable 2), but some improvement is also due to the use of a local basis set.
He (a.u.) H (a.u.)\Exact"-2.9037 -0.52775
Finite Elements-2.9028 -0.52744
Hydrogenic Basis-2.8967 |||{

Table 2. Dependence of helium and H ground state energies on the number of angularcon gurations included. Finite element values for He were obtained using a reaction volumeof 10 a.u., while a reaction volume of 26 a.u. was used for the H values. The helium
calculations were performed with 12 sectors in each radial coordinate (a linear mesh of 4
sectors from 0-2 a.u. and of 8 sectors from 2-10 a.u.), while 16 sectors (a linear mesh of 4
sectors from 0-2 a.u. and of 12 sectors from 2-26 a.u.) in each radial coordinate were used
to obtain the values for H .
Num. of Ang. Con gs. He (a.u.) H (a.u.)
1-2.8790 -0.51448
2-2.9005 -0.52658
3-2.9028 -0.52744
4-2.9033 -0.52762

78
0.045
0.040
0.035

σ2e/σ1e

0.030
Present Results(v) [30]
Meyer and Greene(v) [28]Meyer and Greene(a) [28]Kheifets and Bray(v) [58]Kheifets and Bray(a) [58]Tang and Shimamura(l,a) [62]Levin et al. [158]
Doerner et al. [26]Samson [162]

0.025
0.020
0.015
0.010
0.005
0.000

100

150

200

250
300
Photon Energy (eV)

350

400

450

Figure 16. Comparison of the ratio of double to single photoionization for helium with othertheoretical and experimental values. Our results in a previous study are shown by the narrowsolid and dotted lines. This gure is taken from Ref. [30].

0.10
0.09
0.08
0.07

σ2e/σ1e

0.06
0.05
0.04
Present Results [30]
Present Results (Smoothed) [30]
Broad and Reinhardt(a) [51]Broad and Reinhardt(b) [51]Broad and Reinhardt(c) [51]

0.03
0.02
0.01
0.00

10

20

30

40

50
60
70
Photon Energy (eV)

80

90

100

110

Figure 17. Comparison of the ratio of double to single photodetachment for H with previous
calculations of Broad and Reinhardt [52]. The results of Broad and Reinhardt were obtainedusing the nal state basis sets: a) 10s; 10p; 6d; b) 7s; 10p; 6d; and c) 7s; 10p; 6d; 4f: This
gure is taken from Ref. [30].

79hydrogenic basis set [28], the convergent close-coupling calculations of Kheifets and Bray[58], and the hyperspherical close-coupling calculations of Tang and Shimamura [62]. Our
nite element results were obtained by box averaging over ve box sizes in the range 12-16
a.u. for photon energies of 80-280 eV, while a single box size of 10 a.u. was used to obtain the
results for 280-460 eV. A smaller reaction volume can be used at higher energies, as electronelectron interaction e ects are more tightly con ned near the nucleus. Our calculated valuesof the ratio are lower than most of the other theoretical and experimental values for the rst
few eV above threshold. This is not unexpected, as the near threshold region is where ourapproximation of unequal energy sharing is anticipated to fail. For higher photon energies,
our calculated results agree well with the experimental values of Samson and coworkers. Our
results are also consistent with the experimental values of Dorner, with the exception of his
highest energy measurement.
Fig. 16 shows only our velocity form calculations. Calculations performed using theacceleration form of the dipole operator are nearly identical (agreement to better than 1%)
to those of the velocity form over the entire energy range shown here. This is in contrast
to our previous results of helium photoionization (shown by the narrow solid and dottedlines in Fig. 16) using a hydrogenic basis set in which the discrepancy between the velocity
and acceleration forms was 20-25%. Furthermore, the length form calculations of double
ionization cross sections in this previous study were an order of magnitude larger than thecalculations in the other forms. For our nite element approach, calculations in the length
form agree closely with the other forms up to about 100 eV above the double escape threshold
before slowly diverging. At the highest energies shown in Fig. 16, double ionization crosssections calculated with the length form are about 10-15% higher than the values calculatedwith the other forms. Although the use of a nite element basis set requires a much greater
computational e ort in comparison to a hydrogenic basis set, this choice of basis set canbe systematically improved with less e ort. The resulting improved agreement among the80various dipole forms is dramatic, illustrating the usefulness of nite element techniques foraddressing such problems.
Our results for the double to single photoejection ratio in H are shown in Fig.
17. We averaged our calculation over ve box radii in the range 24-32 a.u. for photon
energies of 14.35-65 eV, and over two box radii of 24 and 26 a.u. for photon energies of 65-95
eV. The results for 95-110 eV were obtained with a single box radius of 24 a.u. A box ofroughly twice the size used in our helium calculations is necessary to accurately describe H
photodetachment, since hydrogen wavefunctions extend roughly twice as far as those of He+ .
We obtain a smooth pro le of our ratio by convolving our box averaged values (open circles)with a Gaussian function. The three curves of Broad and Reinhardt [52] in Fig. 17 wereobtained with di erent nal state basis sets. Our values for the H total photoabsorption
cross section agree well with those of Broad and Reinhardt. However, our maximum valueof the branching ratio for double photodetachment is signi cantly larger than their values.
The peak value of our ratio of 0.0940.006 lies in the photon energy range of 25-35 eV. We
believe that our new calculations are more accurate than the results of Broad and Reinhardt.
Evidence of this is suggested by the smaller amount of scatter among our results in Fig. 17.
However, our results are not seriously in disagreement with their calculations, since theircalculated values have correspondingly large \theoretical error bars". For this reason, thetwo calculations appear to be consistent, and the pioneering work of Ref. [52] is impressivein view of the limited computational resources available at that time.
It is interesting to note the di erences in two electron escape for helium and H .
To a rst approximation, one might expect the maximum in the ratio of double to single
photoejection to scale as a simple function of the nuclear charge [122]. However, H isquite di erent from other heliumlike atoms, and so would not be expected to obey any
simple scaling law. Our calculations indicate that the maximum value of the ratio for H of
0.0940.006 is about three times as large as the value for helium of 0.0370.001. Also, the81peak of the ratio occurs at an energy approximately six times smaller for H than for helium,
relative to the double escape threshold. This is a re ection of the small electron anity of
H , as a smaller required energy for removing one electron in H leaves more energy available
to remove the remaining electron. The maximum value of the double photoejection crosssection for H is observed to be about twenty times larger than that for helium, a re ectionof helium's deeper Coulombic well.
Another issue of interest is the predicted high energy limit of the branching ratio
between double and single photoejection. The value of this limit is predicted to be 0.0150 for
H and 0.0167 for He [38, 102]. However, our calculations show a much larger value of thepeak ratio for H than for helium. The high energy values for the ratio in H and helium

does not seem consistent with the idea that electron-electron correlations play a stronger rolefor smaller nuclear charge. One possible explanation for this apparent inconsistency is thefact that the n=2 partial cross section for H is comparable to the n=1 cross section [52],contributing more to the single detachment cross section, and therefore less to the double
detachment cross section [163].
One open question concerning double photoejection calculations is the number ofangular con gurations required to accurately describe the L = 0 initial and L = 1 nal state
wavefunctions, such that cross section values are converged. The dependence of groundstate energy on the number of angular con gurations is shown in Table 1. In our initial
calculations, we only included two angular con gurations (ss, pp) to describe the initialstate, whereas our latest calculations used three angular con gurations (ss, pp, dd). From
Table 1, we see that a signi cant improvement in the ground state energy is achieved byincreasing the number of angular con gurations from two to three, while the improvement
in energy is comparatively less when the number of angular con gurations is increased fromthree to four. This trend is similar for helium and H .
In contrast to the results found for the initial state, the contribution to the double

82
photoejection cross section for each nal state angular con guration is quite di erent for the
separate cases of helium and H . As seen in Fig. 18, the sp angular con guration gives
the majority of the contribution to the helium double photoejection cross section over thephoton energy range shown here, whereas the df con guration only gives about 10-15% of
the contribution to the cross section. In contrast, the pd con guration dominates the double
photoejection cross section of H in the energy range where the peak in the cross section
occurs (see Fig. 19). At higher photon energies, the df con guration contributes the largest
amount, whereas this con guration contributes the smallest amount for the case of helium.
Based on the results shown in Figs. 18 and 19, we have more con dence that our cross section
values for helium are converged with respect to the number of angular con gurations thanfor the case of H . These gures further illustrate the di erence in the double photoejection
dynamics for a target in which both electrons are \almost equally" bound (helium) and atarget in which there is one strongly bound and one weakly bound electron (H ).
Preliminary calculations of photoejection processes have been performed for otherions in the helium isoelectronic sequence (Li+ , Be++ , C4+ , and O6+ ) in an e ort to evaluate
the validity of scaling laws given in Ref. [122]. Although the adoption by Ref. [122] of asimple analytical approximation to the nal state wavefunction is probably not adequate todetermine accurate quantitative results, the derivation of approximate scaling laws is usefulfor understanding qualitative aspects of two electron escape. Ref. [122] uses two di erent
analytical wavefunctions: one is a product of three two-body continuum wavefunctions,while the other contains products of only two single-center Coulomb wavefunctions. They
determined that the threshold region (the region where the Wannier threshold law is valid)
extends to energies up to approximately E = (0:5eV )Z 2 , while the high energy region (theregion where the ratio of double to single photoejection is a constant asymptotic value)
begins at about E = (1:0keV )Z 2 . By taking the limit Z ! 1, the following scaling laws for

4.5

83

sp configuration
pd configuration
df configuration

4.0
3.5

σ2e (kb)

3.0
2.5
2.0
1.5
1.0
0.5
0.0

80

100

120

140

160
180
200
Photon Energy (eV)

220

240

260

280

Figure 18. Contributions to the double photoejection cross section of helium for each of thethree angular con gurations included to describe the nal state wavefunction. These values
were obtained from averaging over ve box sizes in the range 12-16 a.u. The con guration
with the lowest individual angular momentum states dominates, whereas the con guration
with highest individual angular momentum states contributes the least.
80

sp configuration
pd configuration
df configuration7060

σ2e (kb)

50

40

30

20

10

0

10

15

20

25

30

35
40
45
Photon Energy (eV)

50

55

60

65

Figure 19. Contributions to the double photoejection cross section of H for each of thethree angular con gurations included to describe the nal state. These values were obtained
from averaging over ve box sizes in the range 24-32 a.u. The pd con guration is observed
to give the dominant contribution at low energies, whereas the df con guration gives thelargest contribution at higher energies. This is in contrast to the case for helium (see Fig.
18).

84
single and double photoejection were determined:1e (Ef )  1Ze=1 (Ef =Z 2 )=Z 2 ;(53)

2e (Ef )  2Ze=1 (Ef =Z 2 )=Z 4 ;(54)

where Ef is the nal kinetic energy of the escaping electron(s).
Fig. 20 shows our calculated values of single and double photoejection cross sections
for six members of the helium isoelectronic sequence. Our results were scaled according to
the relations given in Eqs. 53 and 54. The box sizes used to obtain these results are given
in Table 3. Since the radial extent of hydrogenic orbitals scales as 1=Z , box sizes were alsochosen to scale as 1=Z . The results for H and helium have been obtained by box averaging;
therefore, the pro les for these curves are smoother than for the heavier ions. Consequently,
our results for the heavier ions should be interpreted as only providing approximate values, as
spurious e ects from pseudoresonances are still present. Evidence of this is apparent in Fig.
20(b), where a smaller relative energy mesh was chosen for C4+ calculations than for the otherheavier ions. However, fairly accurate ground state energies and good agreement between
the forms of the dipole operator give us con dence that our single box size calculations forthe heavier ions are accurate to about 10-15%.
As expected, the results for H deviate the most from the scaling laws given in
Eqs. 53 and 54. Although our single escape cross sections do not scale precisely as 1=Z 2,
the deviation from this scaling law is quite small. The 1=Z 4 scaling law for double escapeseems to be in even better agreement with our calculated values. All of the scaled double
photoejection peaks have the same approximate value (within 10-20%), and the peaks occur
at roughly the same energy relative to the double escape threshold when it is scaled by
1=Z 2. A simple \rule of thumb" implied by the results in Fig. 20 is that double escape crosssections for targets with charge Z in the helium isoelectronic sequence (with the exceptionof H ) will have a maximum value for photon energies (in eV) of approximately 7Z 2 + ED:I:,
where ED:I: is the double photoionization energy threshold of the target.

85

10.00
−

H
He
+
Li
++
Be
4+
C
6+
O

(a)

2

σ1eZ (Mb)

1.00

0.10

0.01

10

20

30

40

50
60
70
2
(Ephoton − ES.I.)/Z (eV)

80

90

100

−

0.14

120

(b)

H
He
+
Li
++
Be
4+
C
6+
O

0.12

110

0.08

4

σ2eZ (Mb)

0.10

0.06

0.04

0.02

0.00

0

10

20

30

40
50
60
2
(Ephoton − ED.I.)/Z (eV)

70

80

90

100

Figure 20. Scaling of six members of the helium isoelectronic sequence. Using the scalinglaws reported in Ref. [122], we have plotted (a) single photoejection cross sections scaledby Z 2 , and (b) double photoejection cross sections scaled by Z 4 . The energy relative to theappropriate threshold is scaled by 1=Z . The results for H and He were obtained by box
averaging, while the results for the heavier ions were obtained with a single box size.

86
Since 2e scales as 1=Z 4 and 1e scales as 1=Z 2, the ratio R would be expected toscale as 1=Z 2 . However, an ambiguity arises in the energy scaling, as Eq. 53 involves a scalingof the energy relative to the single escape threshold, while Eq. 54 scales the energy relative
to the double escape threshold. For this reason, scaled values of the ratio are not plotted
here. However, the maximum values of the ratio are listed in Table 3, and do appear toscale roughly as 1=Z 2 for Z  2. For the photon energy ranges considered here, the pro lesof R for Z  3 did not display a clear peak, but rather long plateau features. Further
investigation is required to more fully evaluate the relevance of the scaling laws proposed by
Kornberg and Miraglia [122].
The results shown here illustrate the usefulness of nite element methods in describing two electron photoejection processes. In particular, the good agreement achieved
between various forms of the dipole operator demonstrates the advantage of nite elements
over a hydrogenic basis set. Although only total photoejection cross sections have been
calculated with this method here, calculations of other observables will now be explored.
In particular, calculations of partial cross sections for the production of excited hydrogenic
bound states are performed in the next section. This will provide an alternate test of theimportance of electron-electron interactions. The frame transformation will once again playa central role in obtaining these results.

4.3 Other Observables Associated with Two Electron EscapeTo fully explore the usefulness of our method in describing two electron escapeprocesses, we now explore other details of the escape process for which electron-electroninteractions are important. Previously, our photoabsorption studies have focussed entirelyon either total single or double electron escape cross sections. One key issue that tests
the generality of our approach is single electron ejection from He or H accompanied byexcitation of the residual one-electron ion He+ (n) or atom H(n). We will refer to thesesingle escape processes as photoejection-induced excitation (PIE) processes.

87

Table 3. Six members of the He isoelectronic sequence are listed along with the box sizes
used to obtain results for Fig. 20 and this table. Columns 2 and 3 are the experimental single
and double ejection thresholds, taken from Ref. [164]. The values in column 4 are based on
our numerical calculations. The asymptotic values for the rst three members listed weretaken from Refs. [38] and [102], while the last three values are based on the simple scaling
law 0:0087(3=Z )2 in Ref. [122].
Box Size(a.u.) S.E.(eV) D.E.(eV) Max. Ratio Asym. Ratio
H
24-32
0.755
14.360 0:094  0:006
0.0150He
12-16
24.58779.007 0:037  0:001
0.0167+Li
1075.638 198.083 0:020  0:003
0.0087Be++
7153.893 371.573 0:011  0:001
0.0049C4+
4.67
392.077 881.857 0:005  0:001
0.0022O6+
3.5
739.315 1610.035 0:003  0:001
0.0012

88
In contrast to the extensive current literature on the ratio of double to single photoejection in characterizing the strength of electron-electron interactions, the role of theseinteractions in PIE have been much less frequently studied. Wehlitz et al. [114] have recently
summarized previous experimental and theoretical attempts to determine PIE cross sectionsin helium. They also published their experimental values for the helium partial cross section
branching ratio (n)=(1s) (the ratio of the cross section for photoejection plus excitationof an n  2 state to the cross section for photoejection with the remaining electron in a
1s state) for a wide range of photon energies above the double escape threshold. Previous
studies were primarily limited to low n excited states (n  3) over a comparatively smallenergy range.
To calculate partial cross sections [31], we again adopt the frame transformation
method developed in the last chapter. We accomplish this by projecting the inner electron
pseudostates (calculated for a given box size r0 ) onto the in nite-space bound hydrogenic
target states. Analogous to Eq. 32 for electron scattering, the box dipole matrix elements

D( )box are related to the physical dipole matrix elements D( )phys by( )phys
Dnl;g
=

X

nl

hnljnliDnl;gbox ;
( )

(55)

where nl represent nal pseudostates corresponding to the inner electron, nl represent thenal bound target eigenstates, and g refers to the ground state. This frame transformation
technique has been used to obtain the results shown here.
Since the low n target states (n=1-3) t within our R-matrix box, the energies of theeigenstates n of the target state Hamiltonian in the shrunken con guration space correspondclosely to the energies of the physical target state for these n values (see Fig. 2). For thisreason, the frame transformation technique (Eq. 55) is not necessary for calculating accurate
PIE cross sections for these states. For excitation to higher lying n states, the physical target
states are represented by the set of pseudostates con ned to our R-matrix box. For these highn states, the electron has a high probability to be found outside our reaction volume. Since

89
this region is where we adopt a simpli ed assumption of electron-electron interactions, one
might suspect that method is incapable of describing excitations to high n states. That is,a plausible expectation is that only a method which accurately represents electron-electron
correlations out to large distances (large enough to contain the inner electron wavefunction)
should be able to accurately describe excitation to high n states. At the same time, wehave shown that our method can accurately describe helium double photoejection. Thosestudies employed a projection of pseudostates onto the physical continuum eigenstates which
extend out to in nity. This gives some indication that our approach may be able to describe
excitation to high-lying bound states that extend beyond our R-matrix box.
Our n =1s branching ratios are compared with the experimental values of Wehlitz
et al. [114] in Fig. 21. R-matrix results for photon energies from threshold to 200 eV above

threshold were obtained by averaging over ve box sizes in the range 12-16 a.u., while results
at higher energies were obtained with a single box size of 10 a.u. We can use a smaller
reaction volume at higher energies since electron correlations are more tightly con ned near
the nucleus. Gailitis averaging is again used to average cross sections for the energy range
below the pseudostate thresholds. Single photoejection with the remaining electron left in a
1s ground state accounts for approximately 90% of the total single ionization cross section inhelium. Our ratios for n=2 and n=3 agree well with the experimental values across an energy
range from the double escape threshold (78.98 eV) up to 500 eV, except for a small numberof our calculated values which lie slightly outside the experimental error bars. Surprisingly,
this good agreement also holds for n = 4 6, although these states do not t within our
R-matrix box. This result implies that long-range correlations between electrons have little
in uence on the electron energy distribution for PIE processes to high n states. In otherwords, the \decision" of the inner electron to be excited to a higher n state is made while
the electron is still near the nucleus.
Although the experiment of Wehlitz et al. was only able to measure cross section

90

0.08

n=2

0.07
0.06
0.05
0.015

n=3

σ(n)/σ(1s)

0.010
0.005
0.006

n=4

0.004
0.002
0.003

n=5

0.002
0.001

n=6

0.00150.00100.0005

100

200

300

400

Photon Energy (eV)

500

600

Figure 21. n =1s branching ratios of helium photoionization for n=2-6. The curves are
our calculated eigenchannel R-matrix values, while the circles are the experimental values of
Wehlitz et al. [114]. This gure is taken from Ref. [31].

91
2.5
Present Work
Broad and Reinhardt

σ(n=1) (Mb)

2.0

1.5

1.0

0.5

0.0

10

20

30

40

2.0

50
60
70
Photon Energy (eV)

80

90

100

110

100

110

Present Work
Broad and Reinhardt

σ(n=2) (Mb)

1.5

1.0

0.5

0.0

10

20

30

40

50
60
70
Photon Energy (eV)8090

Figure 22. Comparison of our n=1 and n=2 H partial cross sections with extrapolated
values taken from Fig. 1 of Ref. [52]. Note that unlike the case of photoejection-induced
excitation for helium (Fig. 21), where the remaining electron is left in a 1s state roughly
90% of the time, the n = 2 partial cross section for H is comparable in magnitude to then = 1 partial cross section.

92

0.15

n=3

0.10
0.05
0.00
0.06

n=4

σ(n)/σ(1s)

0.04
0.02
0.00
0.02

n=5

0.01

0.00
0.010

n=6
0.005

0.000

10

20

30

40

50

60

70

80

Photon Energy (eV)

90

100

110

Figure 23. n =1s branching ratios of H photodetachment for n=3-6. The irregularities
in the pro les for the rst 30 eV above threshold are a re ection of the denser energy meshused in this region.

93

n=1
n=2
n=3
n=4
n=5
n=6

0.6

3

n σ(n) (Mb)

0.5

(a)

0.4

0.3

0.2

0.1

0.0

80

100

120

140

160
180
200
Photon Energy (eV)

220

240

260

280

6
n=1
n=2
n=3
n=4
n=5
n=6

5

(b)

3

3

n σ(n) (Mb)

4

2

1

0

10

20

30

40

50
60
70
Photon Energy (eV)

80

90

100

110

Figure 24. Comparison of our (a) He and (b) H partial cross sections with the 1=n3 scaling
law, re ecting the properties of Rydberg electrons for large n values. This scaling law isobserved to be valid for n  3, and the agreement with the scaling law improves for larger
photon energies.

94
ratios, they obtained absolute cross sections indirectly by using total cross section measurements from Samson et al. [165]. Our n = 1 absolute cross sections are within the error barsof Wehlitz et al. over the entire energy range calculated. At high (but nonrelativistic) energies, the decrease in the single ionization cross section of heliumlike systems is described by
the power law E 7=2 [101, 166]. Our calculated partial cross sections decrease at a slightly
faster rate at high photon energies, indicating that the limiting E 7=2 energy dependence
has not yet been reached. Another noticeable feature in our partial cross section ratios is
the presence of a peak value at energies of several eV above threshold for n = 4 6. Thisfeature also appears in the experimental data for these n values. Peaks are observed in our
He+ (n) absolute cross sections for n = 5 and n = 6.
As seen in Figs. 23 and 24, partial cross sections calculated for H have features
similar to those found in helium. Analogous to the case for calculating total photoejection
cross sections, a reaction volume of roughly twice the radius used in our helium calculations
is necessary to accurately describe H PIE. For the results presented here, our calculations
were box averaged over ve box sizes in the range 24-32 a.u. for photon energies of 14.35-65
eV, and over two box sizes of 24 and 26 a.u. for photon energies of 65-95 eV. A single box sizeof 24 a.u. was used to obtain the results for photon energies of 95-110 eV. The occurrenceof irregular features in Fig. 23 only for the rst 30 eV above threshold re ects the fact that
we used a ner energy mesh over this range in comparison to calculations at higher energies.
Unlike the case for helium, the n = 2 partial cross section for H is roughly two-thirds aslarge as the n = 1 cross section. This is mainly a consequence of the comparatively large
overlap of the H ground state (see Fig. 15) with the n = 2 target eigenstates. As seen in Fig.
22, our n = 1 and n = 2 cross sections are consistent with the theoretical values extracted
from the paper of Broad and Reinhardt [52]. Our H total photoabsorption cross sections
also are found to agree well with the values presented by Broad and Reinhardt. Similar tothe case for helium, peaks in the n = 5 and n = 6 branching ratios and the absolute partial

95
cross sections occur at photon energies of several eV above the double escape threshold.
One topic of interest is the relationship between partial cross sections for di erent
values of n. Partial cross sections for helium are observed to scale as 1=n3 for n  3 (see Fig.
24(a)), a re ection of the 1=n3 frequency scale of Rydberg electrons to be located near the
nucleus. This 1=n3 scaling dependence improves for higher photon energies (greater than
100 eV above the double escape threshold). As seen in Fig. 24(b), the partial cross sectionsfor H also obey a 1=n3 scaling law for n  3.
Our present eigenchannel R-matrix approach does not allow for the exchange ofangular momentum and energy between the electrons in the region outside of the reaction
volume. Consequently, this approach can not hope to represent the details of a scattering
process that is sensitive to long range forces other than a screened Columb potential. Our
approximate treatment for the wavefunction outside of the reaction volume does not include
the e ect of long-range multipole interactions between the outer electron and the remaining
fragment. The e ect of including the long-range dipole interaction can be understood byadopting the form of this interaction as outlined by Gailitis and Damburg [167]. The dipole
interaction mixes states with the same speci ed principal quantum number but di erent
angular momentum values. Our method is capable of predicting the correct cross sections
for PIE to the n = 2 state, but is not expected to give proper amplitudes of the individual Gailitis-Damburg eigenstates (eigenstates of the dipole operator with basis elements

j2spi; j2psi; and j2pdi). A previous study [168] indicated that there are no major diculties in including the long-range dipole interaction in an eigenchannel R-matrix approach.
In conclusion, we have applied our method to obtain accurate partial cross sections
for helium and H photoabsorption. By adopting the frame transformation again, we wereable to obtain helium partial cross sections which agreed well with experimental data for nvalues as large as n = 6. This result is surprising, as it indicates that long-range correlations
are unimportant in determining the energy distribution of the remaining bound state electron

96
in PIE processes. These long-range correlations may be important in in uencing the angular
distributions of the remaining electron. Application of our approach to calculate angular
distributions has not yet been undertaken. The accurate description of PIE processes has
provided another demanding test of the ability of our eigenchannel R-matrix approach todescribe processes for which electron-electron interactions are important. PIE and double
escape are closely related processes. In fact, these processes are identical for the limitingcase of PIE with n ! 1. It is hoped that the results presented here will stimulate further
progress in the description of PIE processes, and therefore provide a deeper understandingof the role that electron correlations play in atomic processes.

CHAPTER 5
CONCLUDING REMARKSIn this work, we have extended the eigenchannel R-matrix method to address twoelectron escape processes. Although double escape is a much more complicated process thansingle escape, R-matrix methods are still able to describe this process accurately (provided
the excess escape energy of the two electrons is larger than a few eV). Our approach indicates that a proper description of electron-electron interactions is only required when theelectrons are relatively close (10-20 a.u.) to the nucleus in order to accurately treat double
escape. This perturbative method represents a vast improvement over nonperturbative methods, which were the only theoretical treatments of double photoejection in the intermediate
photon energy range at the time this study was initiated.
Although we have focussed primarily on double photoejection of helium and H ,this work has much broader applications. Our approach should be applicable to any quantum mechanical three-body system with two escaping particles, provided the solution forthe long-range interactions between any two particles is known, and the complicated shortrange physics between the particles can be described numerically. Therefore, this work may
have implications in several areas of physics, including chemical physics, condensed matter,
nuclear physics, and perhaps even elementary particle physics. Although both the classicaland quantum mechanical versions of the general three-body problem remain longstanding
unsolved problems in the realm of physics, we have shown that with the use of ecient modern computers and some well chosen approximations, one can accurately describe quantum
mechanically at least some of the details of two particle escape into the continuum.
We addressed two electron escape here for incident photons in the intermediate

98
energy range. This range is characterized by incoming energies high enough that the shortrange physics is important in in uencing the two electron escape process, yet low enough torender perturbative methods inadequate for the determination of double escape cross sections. By adopting a nite element basis set, we were able to address this energy range withan internally-consistent theoretical approach. Furthermore, most of the recent experimental
data is consistent with our theoretical results. Applications of our approach to calculate
partial cross sections of helium and H provided further insight into the nature of electronelectron interactions in electron escape. In particular, our results indicated that long-range
correlations have little importance in determining the energy distribution of the remaining
bound electron in single photoejection processes, at least in the intermediate photon energy
range.
Much progress has been made in the past 5 or so years in describing two electron
escape processes. These advances have been aided by rapid improvements in computational
resources. On the experimental side, the availability of modern synchrotron light sources withboth intense and high energy radiation beams, along with fast-gated coincidence electronics,
has allowed new experiments to be performed with unprecedented precision. Following therecent theoretical and experimental progress in this eld, attention has begun to turn tothe description of more complicated systems. Anticipated double photoejection experiments
for H and magnesium will provide additional tests of existing theoretical methods. An
accurate description of magnesium double photoionization will be further complicated by thepossibility of excitation of the core electrons at suciently high photon energies. Preliminary
calculations [169] have already been performed to address electron correlations in the threeelectron system of lithium. Double photoionization processes involving magnesium or lithium
represent a further level of complexity for large enough incident photon energies, as the
additional electron(s) in these systems can also be excited to higher states. A similarly
challenging problem involves a proper theoretical description of H2 double photoionization, a99
process that also results in dissociation of the molecule. It remains to be seen how successfully
discrete methods, of the type developed and extended in this dissertation, will be able todescribe these phenomena of still greater complexity.

BIBLIOGRAPHY[1] A.E. Roy, Orbital Motion (Adam Hilger, Bristol, 3rd edition, 1988).
[2] J.M.A. Danby, Fundamental of Celestial Mechanics (Willmann-Bell, Richmond, VA,
2nd edition, 1988).
[3] J.H. van Vleck, Philos. Mag. 44, 842 (1922).
[4] M.C. Gutzwiller, Chaos in Classical and Quantum Mechanics (Springer, New York,
1990).
[5] D. Wintgen, K. Richter, and G. Tanner, CHAOS 2, 19 (1992).
[6] E. Schrodinger, Ann. Physik 4, 79 (1925).
[7] W. Heisenberg, Z. Physik 43, 172 (1927).
[8] E.H. Kennard, Phys. Rev. 31, 876 (1928).
[9] J.A. Wheeler, Phys. Rev. 52, 1083 (1937).
[10] G. Arfken, Mathematical Methods for Physicists (Academic Press, New York, 2nd edition, 1970).
[11] B.H. Bransden and C.J. Joachain, Physics of Atoms and Molecules (John Wiley & Sons,
New York, 1983).
[12] M. Born and J.R. Oppenheimer, Ann. Physik 84, 457 (1927).
[13] G. Breit and E. Wigner, Phys. Rev. 49, 519 (1936).
[14] J.A. Wheeler, Phys. Rev. 52, 1107 (1937).
[15] W. Heisenberg, Z. Physik 120, 513 (1942).
[16] E.P. Wigner, Phys. Rev. 70, 15 and 606 (1946).
[17] L. Motz and J. Schwinger, Phys. Rev. 58, 26 (1940).
[18] W. Kohn, Phys. Rev. 74, 1763 (1948).
[19] D.R. Hartree, Proc. Camb. Phil. Soc. 24, 89 (1928).

101[20] J.C. Slater, Phys. Rev. 32, 339 and 349 (1928).
[21] V. Fock, Z. Physik 61, 126 (1930).
[22] E.A. Hylleraas, Z. Physik 54, 347 (1929).
[23] M. Aymar, C.H. Greene, and E. Luc-Koenig, Rev. Mod. Phys. 68, 1015 (1996).
[24] R.P. Madden and K.Codling, Phys. Rev. Lett. 10, 516 (1963).
[25] T.A. Carlson, Phys. Rev., 156, 142 (1967).
[26] R. Dorner, T. Vogt, V. Mergel, H. Khemliche, S. Kravis, C.L. Cocke, J. Ullrich, M.
Unverzagt, L. Spielberger, M. Damrau, O. Jagutzki, I. Ali, B. Weaver, K. Ullmann,
C.C. Hsu, M. Jung, E.P. Kanter, B. Sonntag, M.H. Prior, E. Rotenberg, J. Denlinger,
T. Warwick, S.T. Manson, and H. Schmidt-Bocking, Phys. Rev. Lett. 76, 2654 (1996).
[27] I. Bray and A.T. Stelbovics, Phys. Rev. Lett. 70, 746 (1993).
[28] K.W. Meyer and C.H. Greene, Phys. Rev. A 50, R3573 (1994).
[29] K.W. Meyer, C.H. Greene, and I. Bray, Phys. Rev. A 52, 1334 (1996).
[30] K.W. Meyer, C.H. Greene, and B.D. Esry, Phys. Rev. Lett. 78, 4902 (1997).
[31] K.W. Meyer, J.L. Bohn, C.H. Greene, and B.D. Esry, Phys. Rev. Lett. (submitted,
1997).
[32] J.S. Levinger, Phys. Rev. 90, 11 (1953).
[33] M.Y. Amusia, J. Phys. B 8, 1248 (1975).
[34] J.A.R. Samson, R.J. Bartlett, and Z.X. He, Phys. Rev. A 46, 7277 (1992).
[35] J.M. Rost, Phys. Rev. A 53, R640 (1996).
[36] S.L. Carter and H.P. Kelly, Phys. Rev. A 24, 170 (1981).
[37] T. Ishihara, K. Hino, and J.H. McGuire, Phys. Rev. A 44, R6980 (1991).
[38] A. Dalgarno and H. Sedeghpour, Phys. Rev. A 46, R3591 (1992).
[39] K. Hino, T. Ishihara, F. Shimizu, N. Toshima, and J.H. Mcguire, Phys. Rev. A 48, 1271
(1993).
[40] F.W. Byron and C.J. Joachain, Phys. Rev. 164, 1 (1967).
[41] R.L. Brown, Phys. Rev. A 1, 586 (1970).

102[42] M. Brauner, J.S. Briggs and H. Klar, J. Phys. B 22, 2265 (1989).
[43] F. Maulbetsch and J.S. Briggs, J. Phys. B 26, 1679 (1993).
[44] J. Berakdar and J.S. Briggs, Phys. Rev. Lett. 72, 3799 (1994).
[45] I. Bray and A.T. Stelbovics, Adv. At. Mol. Phys. 35, 209 (1995).
[46] P.G. Burke, C.J. Noble, and Penny Scott, Proc. R. Soc. London Ser. A 410, 289 (1987).
[47] I. Bray and A.T. Stelbovics, Phys. Rev. A 46, 6995 (1992).
[48] K. Bartschat, E.T. Hudson, M.P. Scott, P.G. Burke, and V.M. Burke, J. Phys. B 29,
115 (1996).
[49] J.Z. Tang, S. Watanabe, and M. Matsuzawa, Phys. Rev. A 46, 2437 (1992).
[50] H. Friedrich, in Theoretical Atomic Physics (Springer-Verlag, Berlin, 1990).
[51] J. Callaway and D.H. Oza, Phys. Lett. 72A, 207 (1979).
[52] J.T. Broad and W.P. Reinhardt, Phys. Rev. A 14, 2159 (1976).
[53] D. Kato and S. Watanabe, Phys. Rev. Lett. 74, 2443 (1995).
[54] I. Bray, I.E. McCarthy, J. Wigley, and A.T. Stelbovics, J. Phys. B 26, L831 (1993).
[55] I. Bray and A.T. Stelbovics, Phys. Rev. A 48, 4787 (1993); 49, R2224 (1994).
[56] I. Bray, D.V. Fursa, and I.E. McCarthy, J. Phys. B 27, L421 (1994); D.V. Fursa and I.
Bray, Phys Rev. A 52, 1279 (1995).
[57] I. Bray, Phys. Rev. A 49, 1066 (1994).
[58] A. Kheifets and I. Bray, Phys. Rev. A 54, R995 (1996).
[59] I. Bray, Phys. Rev. Lett. 78, 4721 (1997).
[60] S. Watanabe, Y. Hosoda, and D. Kato, J. Phys. B 26, L495 (1993).
[61] D. Kato and S. Watanabe, J. Phys. B 29, L779 (1996).
[62] J.-Z. Tang and I. Shimamura, Phys. Rev. A 52, R3413 (1995).
[63] U. Fano, Rep. Prog. Phys. 46, 97 (1983).
[64] C.D. Lin, Phys. Rev. 25, 76 (1982); 26, 2305 (1982); 27, 22 (1983).

103[65] J.-Z. Tang, S. Watanabe, M. Matsuzawa, and C.D. Lin, Phys. Rev. Lett. 69, 1633
(1992).
[66] J.C. Light and R.B. Walker, J. Chem. Phys. 65, 4272 (1976).
[67] B. Christensen-Dalsgaard, Phys. Rev. A 29, 2242 (1984).
[68] E.P. Wigner and L. Eisenbud, Phys. Rev. 72, 29 (1947).
[69] F. Robicheaux, Phys. Rev. A 43, 5946 (1991) (see Appendix); B.I. Schneider (unpublished).
[70] K. Bartschat and C.H. Greene, J. Phys. B 26, L109 (1993).
[71] P.G. Burke, K.A. Berrington, and C.V. Sukumar, J. Phys. B 14 289 (1981).
[72] M.P. Scott, T.T. Scholz, H.R.J. Walters, and P.G. Burke, J. Phys. B 22, 3055 (1989).
[73] T.T. Scholz, H.R.J. Walters, P.G. Burke, and M.P. Scott, J. Phys. B 24, 2097 (1991).
[74] T.T. Scholz, H.R.J. Walters, and P.G. Burke, J. Phys. B 23, L467 (1990).
[75] M.B. Shah, D.S. Elliot, and H.B. Gilbody, J. Phys. B 20, 3501 (1987).
[76] M.P. Scott, P.G. Burke, K. Bartschat, and I. Bray, J. Phys. B 30, L309 (1997).
[77] P.G. Burke, V.M. Burke, and K.M. Dunseath, J. Phys. B 27, 5341 (1994).
[78] K. Bartschat, E.T. Hudson, M.P. Scott, P.G. Burke, and V.M. Burke, J. Phys. B 29,2875 (1996).
[79] K. Bartschat, E.T. Hudson, M.P. Scott, P.G. Burke, and V.M. Burke, Phys. Rev. A 54,
R998 (1996).
[80] K. Bartschat, P.G. Burke, and M.P. Scott, J. Phys. B 29, L769 (1996).
[81] P.J. Marchalant, K. Bartschat, K.A. Berrington, and S. Nakazaki, J. Phys. B 30, L279
(1997).
[82] K. Bartschat and I. Bray, Phys. Rev. A 54, R1002 (1996).
[83] T.W. Gorczyca and N.R. Badnell, J. Phys. B (submitted, 1997).
[84] E.P. Wigner, Phys. Rev. 73, 1002 (1948).
[85] A.R.P. Rau, Comments At. Mol. Phys. 14, 285 (1984).
[86] G.H. Wannier, Phys. Rev. 90, 817 (1953).

104[87] R. Peterkop, J. Phys. B 4, 513 (1971).
[88] A.R.P. Rau, Phys. Rev. A 4, 207 (1971).
[89] A. Temkin, Phys. Rev. Lett. 49, 365 (1982).
[90] C.H. Greene and A.R.P. Rau, Phys. Rev. A 32, 1352 (1985).
[91] A.R.P. Rau, J. Phys. B 9, L283 (1976).
[92] S. Cvejanovic and F.H. Read, J. Phys. B 7, 1841 (1974).
[93] R. Wehlitz, F. Heiser, O. Hemmers, B. Langer, A. Menzel, and U. Becker, Phys. Rev.
Lett. 67, 3764 (1991).
[94] J.W. McGowan and E.M. Clarke, Phys. Rev. 167, 43 (1968).
[95] J.B. Donahue, P.A.M. Gram, M.V. Hynes, R.W. Hamm, C.A. Frost, H.C. Bryant, K.B.
Butter eld, D.A. Clarke, and W.W. Smith, Phys. Rev. Lett. 48, 1538 (1982).
[96] H. Kossmann, V. Schmidt, and T. Andersen, Phys. Rev. Lett. 60, 1266 (1988).
[97] S. Watanabe, D. Kato, and M. Matsuzawa, Comments At. Mol. Phys. 33, 95 (1996).
[98] D.H. Madison, I. Bray, and I.E. McCarthy, J. Phys. B 24, 3861 (1991).
[99] K. Hino, T. Ishihara, F. Shimizu, N. Toshima, and J.H. McGuire, Phys. Rev. A 48,
1271 (1993).
[100] R.H. Pratt, A. Ron, and H.K. Tseng, Rev. Mod. Phys. 45, 273 (1973).
[101] A. Dalgarno and A.L. Stewart, Proc. Phys. Soc. London 76, 49 (1960).
[102] H.R. Sadeghpour and A. Dalgarno, Phys. Rev. A 47, R2458 (1993) (See comment in
Ref. 15 of this paper).
[103] J.A.R. Samson, C.H. Greene, and R.J. Bartlett, Phys. Rev. Lett. 71, 201 (1993).
[104] M.Ya. Amusia and A.I. Mikhailov, J. Phys. B 28, 1723 (1995); Phys. Lett. A 199, 209(1995).
[105] L.R. Andersson and J. Burgdorfer, Phys. Rev. A 50, R2810 (1994).
[106] T. Suric, K. Pisk, B.A. Logan, and R.H. Pratt, Phys. Rev. Lett. 73, 790 (1994).
[107] M.A. Kornberg and J.E. Miraglia, Phys. Rev. A 53, R3709 (1996).
[108] P.M. Bergstrom, K. Hino, and J. Macek, Phys. Rev. A 51, 3044 (1995).

105[109] R. Wehlitz, R. Hentges, G. Prumper, A. Farhat, T. Buslaps, N. Berrah, J.C. Levin,
I.A. Sellin, and U. Becker, Phys. Rev. A 53, 3720 (1996).
[110] L. Spielberger, O. Jagutzki, B. Krassig, U. Meyer, Kh. Khayyat, V. Mergel, Th.
Tschentscher, Th. Buslaps, H. Brauning, R. Dorner, T. Vogt, M. Achler, J. Ullrich,
D.S. Gemmell, and H. Schmidt-Bocking, Phys. Rev. Lett. 76, 4685 (1996).
[111] V. Schmidt, Rep. Prog. Phys. 55, 1483 (1992).
[112] J.C. Levin, D.W. Lindle, N. Keller, R.D. Miller, Y. Azuma, N. Berrah Mansour, H.G.
Berry, and I.A. Sellin, Phys. Rev. Lett. 67, 968 (1991).
[113] L. Spielberger, O. Jagutzki, R. Doorner, J. Ullrich, U. Meyer, V. Mergel, M. Unverzagt,
M. Damrau, T. Vogt, I. Ali, Kh. Khayyat, D. Bahr, H.G. Schmidt, R. Frahm, and H.
Schmidt-Bocking, 74, 4615 (1995).
[114] R. Wehlitz, I.A. Sellin, O. Hemmers, S.B. Whit eld, P. Glans, H. Wang, D.W. Lindle,
B. Langer, N. Berrah, J. Viefhaus, and U. Becker, J. Phys. B 30, L51 (1997).
[115] R. Wehlitz, B. Langer, N. Berrah, S.B. Whit eld, J. Viefhaus, and U. Becker, J. Phys.
B 26, L783 (1993).
[116] D. Proulx and R. Shakeshaft, Phys. Rev. A 48, R875 (1993).
[117] M. Pont and R. Shakeshaft, J. Phys. B 28, L571 (1995).
[118] F. Maulbetsch, M. Pont, J.S. Briggs, and R. Shakeshaft, J. Phys. B 28, L341 (1995).
[119] M. Pont and R. Shakeshaft, Phys. Rev. A 51, R2676 (1995).
[120] O. Schwarzkopf, B. Krassig, J. Elmiger, and V. Schmidt, Phys. Rev. Lett. 70, 3008
(1993); O. Schwarzkopf, B. Krassig, V. Schmidt, F. Maulbetsch, and J.S. Briggs, J.
Phys. B 27, L347 (1994).
[121] Z.-J. Teng and R. Shakeshaft, Phys. Rev. A 49, 3597 (1994).
[122] M.A. Kornberg and J.E. Miraglia, Phys. Rev. A 49, 5120 (1994).
[123] J.A.R. Samson, E.-M. Lee, and Y. Chung, J. Electron Spectroscopy 66, 75 (1993).
[124] J.A.R. Samson and G.C. Angel, Phys. Rev. A 42, 5328 (1990).
[125] C.H. Greene and L. Kim, Phys. Rev. A 38, 5953 (1988).
[126] F. Robicheaux, R. Wood, and C.H. Greene, Phys. Rev. A 49, 1866 (1994).
[127] M. Gailitis, Zh. Eksp. Teor. Phys. 44, 1974 (1963) [Sov. Phys. JETP 17, 1328 (1963)].

106[128] M.J. Seaton, Rep. Prog. Phys. 46, 167 (1983).
[129] V. Schmidt, N. Sander, H. Kuntzmuller, P. Dhez, F. Wuilleumier, and E. Kallne, Phys.
Rev. A 13, 1748 (1976).
[130] G.R. Wight and M.J. Van der Wiel, J. Phys. B 9, 1319 (1976).
[131] D.M.P. Holland, K. Codling, J.B. West, and G.V. Marr, J. Phys. B 12, 2465 (1979).
[132] P. Lablanquie, K. Ito, P. Morin, I. Nenner, and J.H.D. Eland, Z. Phys. D 16, 77 (1990).
[133] R.J. Bartlett, P. Walsch, Z.X. He, Y. Chung, E-M. Lee, and J.A.R. Samson, Phys.
Rev. A 46, 5574 (1992).
[134] C. Pan and H.P. Kelly, J. Phys. B 28, 5001 (1995).
[135] A. Temkin, Phys. Rev. 126, 130 (1962).
[136] R. Poet, J. Phys. B 11, 3081 (1978).
[137] R. Poet, J. Phys. B 13, 2995 (1980).
[138] J. Callaway and D.H. Oza, Phys. Rev. A 29, 2416 (1984)[139] R. Poet, J. Phys. B 14, 91 (1981).
[140] D.H. Oza and J. Callaway, Phys. Rev. A 27, 2840 (1983).
[141] M. Le Dourneuf, J.M. Launay, and P.G. Burke, J. Phys. B 23, L559 (1990).
[142] I. Bray and A.T. Stelbovics, Phys. Rev. Lett. 69, 53 (1992).
[143] D.A. Konovalov and I.E. McCarthy, J. Phys. B 27, L407 (1994).
[144] I. Bray and A.T. Stelbovics, Atomic Data Nucl. Data Tables 58, 67 (1994).
[145] H.R. Sadeghpour, C.H. Greene, and M. Cavagnero, Phys. Rev. A 45, 1587 (1992).
[146] C.H. Greene and C. Jungen, Adv. At. Mol. Phys. 21, 51 (1985).
[147] D.F. Gallaher, J. Phys. B 7, 362 (1974).
[148] U. Fano, J. Opt. Soc. Am. 65, 979 (1975); U. Fano, Phys. Rev. A 24, 619 (1981).
[149] D.A. Harmin, Phys. Rev. A 26,2656 (1982).
[150] M. Braun, W. Schweizer, and H. Herold, Phys. Rev. A 48, 1916 (1993).

107[151] J. Ackermann and J. Shertzer, Phys. Rev. A 54, 365 (1996).
[152] J. Shertzer and J. Botero, Phys. Rev. A 49, 3673 (1994).
[153] H.W. van der Hart, J. Phys. B 30, 453 (1997).
[154] J.P. Burke, C.H. Greene, and B.D. Esry, Phys. Rev. A 54, 3225 (1996).
[155] M.T. Jones and M.L. Patrick, Applied Numerical Mathematics 12, 377 (1993).
[156] W.H. Press, B.P. Flannery, S.A. Teukolshy, and W.T. Vettering, Numerical Recipes(FORTRAN) (Cambridge University Press, Cambridge, 1989).
[157] E. Anderson, Z. Bai, C. Bischof, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum,
S. Hammarling, A. McKenney, S. Ostrouchov, and D. Sorensen, LAPACK Users Guide(SIAM, Philadelphia, 1992).
[158] J.C. Levin, G.B. Armen, and I.A. Sellin, Phys. Rev. Lett. 76, 1220 (1996).
[159] A.G. Abrashkevich and M. Shapiro, Phys. Rev. A 50, 1205 (1994), and references
within.
[160] J.F. McCann and D.S. Crothers, J. Phys. B 19, L399 (1986).
[161] G.W.F. Drake, in Atomic, Molecular, and Optical Physics Handbook, edited by G.W.F.
Drake and N.E. Hedgecock (AIP, New York, 1996).
[162] J.A.R. Samson (to be published) (1997).
[163] H.R. Sadeghpour (private communication).
[164] R.C. Weast, in Handbook of Chemistry and Physics (1st student edition), edited by
R.C. Weast (CRC Press, Inc., Boca Raton, Fl,1987).
[165] J.A.R. Samson, Z.X. He, L. Yin, and G.N. Haddad, J. Phys. B 27, 887 (1994).
[166] T. Aberg, Phys. Rev. A 2, 1726 (1970).
[167] M. Gailitis and R. Damburg, Proc. Phys. Soc. London 82, 192 (1963).
[168] H.R. Sadeghpour and M. Cavagnero, J. Phys. B 26, L271 (1993).
[169] X. Yang, C.G. Bao, and C.D. Lin, Phys. Rev. A 53, 3934 (1996).
[170] C. Bloch, Nucl. Phys. 4, 503 (1957).
[171] R.N. Zare, Angular Momentum (John Wiley and Sons, New York, 1988).

108

APPENDIX AEQUIVALENCE OF DIPOLE OPERATOR FORMS IN A FINITE VOLUMEIn an e ort to understand the nature of the discrepancy between calculations using
various forms of the dipole operator, we thoroughly explored the assumptions which weremade in transforming from one form of the cross section to another. For example, thetransformations given in Eqs. 14 and 15 of Chapter 2 have assumed that the Hamiltonian H^
is Hermitian. However, this is generally not true if the Hilbert space is restricted to a nite
volume. This observation will now be illustrated with a simple example.
Consider the nite volume example of a one-dimensional square well. Given thematrix element h 2 jx^j 1 i which appears in the length form of the cross section, and choosing
a closed (zero at outer boundary) wavefunction
at outer boundary) wavefunction

2

=

1

=

p

2= sin(x) and an open (nonzerop2= sin(x=2) (see Fig. 25), the matrix elementh jx^j i evaluates to -32/9+8/3. According to the relation given by Eq. 14 (substituting
2

1p^ with its di erential operator form ih dxd and using atomic units),dh jx^j i h E j dx jE i :
2

1

?
=

2

1

2

1

(56)

However, when the expression on the right hand side of this equation is evaluated, one
obtains the value -32/9, whereas the matrix element on the left hand side has the value-32/9+8/3. We will show that the proper relationship between the length and velocity
forms of the dipole operator is given bydh jx^j i = h E j dx jE i + [
2

1

2

1

2

1

2

1

+ 2 x^ ddx1 ddx2 x^ 1 ]jx0:2(E2 E1 )(57)

The only nonzero term in the second expression on the right hand side is the term involving
the derivative of1, which evaluates to 8/3. For this simple example, the transformation

109

ψ2

ψ1

0

x

π

Figure 25. Consider the transformation of the dipole matrix p
element from length to velocity
for wavefunctions con ned
to
a
nite
volume.
Let=2= sin(x) represent a closed
1p
wavefunction and 2 = 2= sin(x=2) represent an open wavefunction. For this simple
case, the Hamiltonian is not Hermitian, since unlike the case for an in nite Hilbert space,
the surface term (due to the nonzero derivative of 1 and the nonzero value of 2 on the outerboundary) no longer vanishes. Therefore, an additional term must be added in transformingfrom one dipole form to another for a nite volume Hilbert space.

110given by Eq. 56 is not correct; the additional term on the right hand side of Eq. 57 must be
added in order for the transformation to be valid.
As shown in the previous example, the Hamiltonian H^ is generally not Hermitian.
We can however introduce an operator ^ which is Hermitian in a nite volume by adding an
operator to the Hamiltonian; that is,

is Hermitian, where^  H^ + L^

(58)

@r
L^  21r (r r0 ) @r

(59)

is the Bloch operator [23, 170] in three dimensions (in this notation, the delta function
e ectively reduces the dimension of integration by one, leading to a \surface" term; theZ @ 1 dS , where @=@n is thegeneral integral form for the Bloch operator is given by 122
S @n
normal derivative). Therefore,H^ y = H^ + (L^ L^ y ):(60)

Using this identity to perform the transformation in Eq. 14, we obtain the expression

h j~^r j i = E i E h j^p~ j i + E 1 E h j(L^ L^y )~^r j i:
2

1

2

1

2

1

2

1

2

1

(61)

Note that the term involving the Bloch operator vanishes if the wavefunction and rst derivative of1

vanish on the surface, which are the boundary conditions normally adopted fordescribing a bound state wavefunction in an in nite Hilbert space. Eq. 57 was derived bysimply using the form of Eq. 61 with the one dimensional Bloch operator 12 (x x0 ) dxd .
The transformations between the length, velocity, and acceleration dipole operators
in one dimension can be summarized as follows:

dL = dV + CLV ; dV = dA + CV A ;(62)

dL  h 2 jx^j 1 i;

(63)

where

111

dV 

d
dx j 1 i = 1 h 2 jp^j 1 i ;

h j
2

E2 E1

(64)

i (E2 E1 )h j dV j i

h

CLV 
and

2

2

h

CV A 

dA  (E2 dxE )12 ;
1

1

(65)
i

+ 2 x ddx1 ddx2 x 1 jx0;
2(E2 E1 )d 1d 2
dx dx

(66)

i

2(V E1 ) 1 2 jx0:2(E2 E1 )2

Schrodinger's equation was used to eliminate a term in the second derivative of(67)
1

in theexpression for CV A . Note that the \surface" term connecting the velocity and acceleration
forms may be nonzero even if both wavefunctions vanish on the outer boundary.
We now develop the three-dimensional transformations and apply them to the test
case of photoionization of hydrogen. In three dimensions, the dipole matrix is given byh j^  r^j i. In the following derivation, we will make two assumptions: rst, the light is2

1

taken to be polarized in the z-direction (^ = r^ cos  ^ sin ), and second, the potential is
assumed to be a function of the radial coordinate only. For photoionization of hydrogen, theground state1

is in an s state ( 1 (~r) = '1 (r)Y00 (; )), which simpli es the form of thetransformations; however, we derive the more general case here. The commutator of interest
is now [H; r cos ]. The resulting angular integral can be factored out of the dipole matrix,
leaving a remaining integral in r. For convenience, we will introduce the scaled wavefunctionu(r), de ned by '(r) = u(r)=r. Note that when the three-dimensional form of the Blochoperator (Eq. 59) is evaluated in a matrix element involving the full wavefunctions, thematrix element is equivalent to the one formed by replacing the full wavefunctions by scaled
wavefunctions and the three-dimensional form of the Bloch operator by the one-dimensional
form. Applying the same approach used to obtain the one-dimensional results, the length
form of the dipole matrix is

dL  h 2 j^  ~r^ j 1 i = dLRAD FANG1

(68)

112
where

dLRAD  h'2 jr^j'1 i(69)

FANG1  hYl2 ;mj cos jYl1 ;m i:

(70)

and
The relationship between length and velocity forms is given by
6=0 F
dL = dlV1 RAD
ANG2 + [dV RAD + CLV ]FANG1 ;(71)

1
6=0  h'2 j r j'1 i ;dlV1 RAD(E E )

(72)

where

2

1

@ jY i;FANG2  hYl2 ;m j sin  @
l1 ;mh' j d j' idV RAD  (2E drE )1 ;
2
1
and





u u + u r du1 du2 ru j
CLV  2 1 22(Edr Edr) 1 r=r0 :
2
1

(73)
(74)
(75)

The superscript `l1 6= 0' denotes that the angular integral multiplying this term may only benonzero if l1 is nonzero. Note that for the case l1 = 0, the radial correction form expressedin terms of the scaled wavefunctions is identical to the form for the one-dimensional case.
The additional l1 6= 0 term arises from the additional component in the z-component of the@,angular momentum. Since the general form of the momentum is p~ = r^ @r@ + ^1r @@ + ^ r sin1  @
when dotted into the polarization vector, pz will have a term involving the derivative withrespect to , which can contribute for l1 6= 0. The transformation between the length andacceleration form is given by

where

dL = [dARAD + CLV + CV A ]FANG1 + CVl1A6=0 FANG2 ;(76)h' j dV j' idARAD  (E2 drE )12 ;

(77)

2

1

113
h

CV A 


 du
1
1
1
r2 u2 u1 + r u2 drand

CVl1A6=0 

r 2 u2 u11

iu2 ddr2 u21 r1 dudr2 u1 + dudr2 dudr1 jr=r0;2(E2 E1 )2
 du
 du 
1
1
2
r dr u1 + r u2 dr jr=r0 :1

2(E2 E1 )2

(78)

(79)

Note that the form of CV A is no longer the same as for the one-dimensional case, due tothe complications with the momentum operator operating on a scaled wavefunction. The
expression d2 u1 =dr2 appearing in this term can be replaced by 2[V (r) E1 +(l(l +1))=2r2]u1
using the Schrodinger equation.
We applied the above equations for the case of photoionization of the ground stateof hydrogen. For hydrogen, the potential is given by V (r) = 1=r, so dV=dr in the expression

dARAD is replaced by 1=r2 . For small enough box sizes (10 a.u.), the need for correction
terms to obtain equivalence of the di erent forms of the cross section was quite apparent.
Additional numerical tests were performed for the case l1 = 0 to verify that all of the terms
in Eqs. 75 and 78 are indeed correct. For our study of hydrogen photoionization, we foundthat the correction terms decreased roughly exponentially with increasing box size, re ecting
the exponential decay of our ground state wavefunction near the edge of the box.
This analysis was eventually extended to the case of helium double photoionization.
For this case, the Bloch operator becomes the sum of two operators of the form given in
Eq. 59, one for each of the independent particle radial coordinates. The equations for thetwo-electron system are very similar to the one-electron equations given above, so will not
be included here. One of the assumptions made in deriving the above equations was that thepotential was a central potential. However, for the case of helium there is additional angulardependence in the 1=r12 term. As explained in Section 2.1, this term does not contributeto the correction terms between various dipole forms. The correction terms were included
in our helium double photoionization computer code, but no signi cant di erence in ourcross section values were observed in comparison to our original values. Nonetheless, it is114
informative to understand the assumptions made in transforming from one dipole form toanother.

115

APPENDIX B
DETAILS OF OUR FINITE ELEMENT IMPLEMENTATION
In implementing our nite element approach, we had to overcome many diculties
encountered with imposing the antisymmetrization operator A, as it appears in Eqs. 40 and
41. Also, since our approach di ers from the approach outlined in Ref. [152], we will discussa few of the details of our implementation. In applying the antisymmetrization operator,
the general form of our spatial two-electron wavefunction is given by(~r1 ;~r2 ) =12(1 + n1 n2 l1 l2 )r2 ; r^1 )]:r1 ; r^2 ) + ( 1)S gl1 l2 (r2 ; r1 )YlLM[gl1 l2 (r1 ; r2 )YlLM
1 l2 (^
1 l2 (^
p(80)

We also require that the spatial permutation operator P12spatial acting on the wavefunction
satis es the following relation:

P12spatial j i = ( 1)S j i:(81)

In the applications considered in this work, we only consider singlet states (S=0), for whichcase the spacial wavefunction is symmetric. Using the identity [171]r ; r^ );r ; r^ ) = ( 1)l1 l2 L YlLM
YlLM
2 l1 (^
1 l2 (^
2

1

+

+

1

2

(82)

for the ground state of helium (L = S = 0, which implies l1 = l2 ), the coupled sphericalharmonic can be factored out of Eq. 81. Comparison of Eqs. 81 and 81 then implies

gl1 l2 (r1 ; r2 ) = gl1 l2 (r2 ; r1 ). That is, for the ground state of helium, antisymmetry of thetotal wavefunction requires that the two-electron radial wavefunction is symmetric about
the line r1 = r2 ; similarly, for triplet L = 0 states, the two-electron radial wavefunction is
antisymmetric about the line r1 = r2 .

116
As a consequence of the conditions imposed by the antisymmetry operator for the
ground state of helium, it is sucient to solve the Schrodinger equation in the half-planer1  r2 or r1  r2 . This is the approach adopted in Ref. [152]. By imposing the boundary
condition along the line r1 = r2 of a zero derivative of the radial wavefunction in the direction
perpendicular to this line, singlet wavefunctions are completely determined by solving in thehalf-plane. Similarly, triplet states are obtained by solving the Schrodinger equation in
the half-plane with the boundary conditions that the wavefunction vanishes along the line

r1 = r2 . Although Shertzer and Botero use a nite element grid of rectangular sectors, they
impose these additional boundary conditions in those sectors where the points r1 = r2 arelocated. A more straightforward approach would be to directly impose boundary conditions
along the line r1 = r2 using a nite element grid of triangular sectors; however, one would
then have to deal with the complications of using triangular sectors.
To determine the ground state wavefunction in our implementation, we solve theSchrodinger equation directly over both regions r1  r2 and r1  r2 without imposing any
boundary conditions along the line r1 = r2 . The only boundary conditions imposed are thatthe wavefunction vanish along the lines r1 = 0, r2 = 0, r1 = r0 , and r2 = r0 , where r0 is theradius of our reaction volume. A Lanczos iterative algorithm is used to partially diagonalize
our Hamiltonian and obtain the rst few lowest energy eigenvalues. The corresponding
eigenstates include both singlet and triplet states. Our method of solving for the groundstate in the full r1 r2 reaction volume space without imposing boundary conditions along
the line r1 = r2 is not as computationally ecient as the method used in Ref. [152].
However, since the determination of our initial state required very little computational timein comparison to the rest of our calculation, we chose to adopt a method based on simplicity
of implementation rather than numerical eciency.
For the L = 1 nal state wavefunction, we no longer have l1 = l2 , so the coupled
spherical harmonics cannot be factored out of Eq. 81. However, again using the identity

117
given in Eq. 82 and comparing the resulting expressions in Eqs. 81 and 81, we obtain thecondition gl1 l2 (r2 ; r1 ) = gl2 l1 (r1 ; r2 ). So, for example, if we determined the radial wavefunction g01 (r2 ; r1 ) in the region r1  r2 , then we have simultaneously determined the radial
wavefunction g10 (r1 ; r2 ) in the region r1  r2 . In the method implemented in Ref. [152],
separate matrices are formed for each ordering of angular con guration (e.g., one matrix
for jl1 = 0; l2 = 1i and one matrix for jl1 = 1; l2 = 0i), but only over half-plane regions.
For example, the matrix for the j01i angular con guration might be formed in the region

r1  r2 , while the j10i angular con guration would be formed in the corresponding region
r1  r2 . In our implementation, a single matrix contains the information for the wavefunctions gl1 l2 (r2 ; r1 ) and gl2 l1 (r1 ; r2 ) over the entire r1 r2 region inside our reaction volume.
In addition to di erences in our approach to antisymmetrization, the numbering
scheme for the ordering of our nodes was originally di erent than the approach adopted inRef. [152]. The term `nodes' refers to the points within each sector where the boundary
conditions of our Hermite interpolating polynomials (see Fig. 14) are enforced; in our case,there are nine nodes for each sector; four at the corners of each sector, four at the midpointsof the sides of each sector, and one at the center of each sector. Our original numbering
scheme resulted in matrices with large bandwidths, which were not suitable for factorization
methods. We eventually adopted the numbering scheme of Shertzer and Botero, and so wereable to make our computer code signi cantly more ecient.
Another important detail in our manipulation of nite element matrices was the
adoption of a sparse symmetric storage scheme. For the nal state wavefunction, the Hamiltonian is not Hermitian (see Appendix A), so is represented by a nonsymmetrical matrix.
However, the operator ^ introduced in Eq. 58 of Appendix A is Hermitian. To avoid nonsymmetric matrices, we always store the sum of the Hamiltonian and Bloch operators, which
is symmetric, rather than the nonsymmetric Hamiltonian and Bloch matrices separately.