Jinhua_Xi_PhD_Thesis_Phototetachment_of_Negative_Atomic_Ions

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PHOTODETACHMENT OF NEGATIVE ATOMIC IONS: A STUDY OF THE
He~, Li~, AND Be" SYSTEMS
By
Jinhua Xi
Dissertation
Submitted to the Faculty of the
Graduate School of Vanderbilt University
in partial fulfillment of the requirements
for the degree of
DOCTOR OF PHILOSOPHY
in
Physics
May 2000
Nashville, Tennessee
Approved: Date:
3,rl<Je)<y
tJ OAAcJ^ ■ itij.v s A s I 1 0 0 n
3 h X B t o
/ ; /
'Z I . Ob-d i
/

UMI Number 9970085
Copyright 2000 by
Xi, Jinhua
All rights reserved.
___ ®
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ACKNOWLEDGMENTS
I thank my advisor, Dr. Froese Fischer, for her invaluable advice and guidance on
my research, in both theoretical atomic physics and computational science, and for
coaching me through the steps to completion of this degree program.
I really appreciate the help from Dr. Albridge, and Dean Reed in helping me
resolving all the details regarding my application for both a master degree in Com
puter Science and a Ph.D degree in Physics. I thank my other outstanding committee
members, Dr. Ernst, Dr. Kephart, and Dr. Umar, who provided advice and support
when I was in need.
I would like to express my appreciation to other faculty and staff and gradu
ate students in the Physics Department and the Computer Science Department. I
appreciate their help to me during this degree program.
Thanks should also go to the Physics Department for providing me the opportunity
to continue my graduate study on a part-time basis. I also appreciate the help from
the Computer Science Department, who maintained my computer account in these
years. I thank Gallagher Financial Systems, Inc., my employer, for providing me
assistance relating to my graduate study.
I greatly appreciate the support and encouragement of my family (my wife and
my sons) through this long lasting effort. Because I can only work on my research
and dissertation during my spare time, it is impossible to get things moving through
to the current stage without their support.
This work was supported by the Division of Chemical Sciences, Office of Basic
Energy Sciences, Office of Science, U.S. Department of Energy.
iii

TABLE OF CONTENTS
Page
ACKNOWLEDGMENTS ............................................................................................. iii
LIST OF F IG U R E S ....................................................................................................... vi
LIST OF T A B L E S ........................................................................................................... xi
Chapter
I. INTRODUCTION ............................................................................................. 1
Background of S tu d y ................................................................................ 1
The Nature of Negative Atomic I o n s .................................................. 2
The Photodetachment Process............................................................... 4
Review of Experimental and Theoretical M ethods........................... 5
A Brief Review of Negative Atomic I o n s ........................................... 8
II. THEORETICAL A P P R O A C H ...................................................................... 12
Introduction................................................................................................. 12
Atomic State Wave Function ............................................................... 13
The Independent-Particle M odel.............................................. 13
The Central-Field M o d el............................................................ 15
The Multi-Configuration A p p ro a c h ........................................ 17
The B-Spline B a sis................................................................................... 17
Solution of One Electron System Using Spline-Galerkin Approach 20
The Continuum State Wave F unction.................................................. 22
The Spline-Galerkin Method and the Interaction M a tr ix ............. 24
The Orthogonality R equirem ents......................................................... 26
The Boundary C ondition......................................................................... 27
Multi-Channel Inverse Iteration A p p ro a c h ........................................ 28
Normalization of the Continuum State Wave F u n c tio n ................. 30
Photodetachment Cross Section............................................................ 35
Angular Distribution of Photoelectrons.............................................. 37
Program Packages ................................................................................... 40
III. PHOTODETACHMENT OF THE HE- (1S2S2P 4P °) SYSTEM . . . 42
Introduction................................................................................................. 42
Photodetachment Below the He(n = 4) T hreshold........................... 46
The Bound State Wave Functions and E n e rg ie s................. 46
The 4P States of He- ................................................................ 50
iv

The 45 States of He- ............................................................... 57
The 4D States of He- ............................................................... 58
Partial and Total Photodetachment Cross Sections and An
gular Distributions of the Photoelectrons.............................. 61
Resonance P a ra m e te rs............................................................... 66
Photodetachment in the Region of the Is DetachmentThreshold . 66
Bound State Orbital Set and Final State Configurations . . 68
The Photodetachment of the 2s, 2p Electrons, and the 2s2p24P
R esonance...................................................................................... 70
The Photodetachment of the Is E lectro n .............................. 72
The Total Cross Sections......................................................................... 82
Conclusion ................................................................................................ 82
IV. PHOTODETACHMENT OF THE LI“ (15*2S2 lS ) S Y S T E M .............. 84
Introduction................................................................................................ 84
The Continuum State Wave F unction.................................................. 87
Results and Discussion ......................................................................... 88
Conclusion ................................................................................................ 98
V. PHOTODETACHMENT OF THE BE" (1S22S2P24P ) SYSTEM . . . 99
In tro d u c tio n ............................................................................................. 99
The Continuum State Wave F u n c tio n .............................................. 101
Results and Discussion ......................................................................... 102
Conclusion ................................................................................................ I l l
BIBLIOGRAPHY .......................................................................................................... 112
v

LIST OF FIGURES
Figure
2.1. A distribution of P-splines £?Ii5(x) in the region x 6 [0,1] where the
horizontal coordinate labels the x value and the vertical one indicates
the value of the P-splines. The 14 5-splines of order 5 are defined by
the knot sequence ti = ••• = t5 = 0, t,- = tf_i + 0.1, for i = 6, 7, •••, 14
and £15 = ••• = tig = 1........................................................................................
3.1. Photodetachment cross section from He~( AP °) initial state to He- ( 4P )
final state in both length form and velocity form. The vertical dashed
lines indicate the positions of the target energies..........................................
3.2. The ls 2p2 AP resonance state of He- . The length form and velocity form
are in very good agreement and are indistinguishable on the scale of this
figure........................................................................................................................
3.3. The eigenphase shift in the area of the s2jr1AP resonance of He- . . .
3.4. The photodetachment cross section of He- (ls2s2p4P°) in the energy
region of the ls3p2 AP resonance of He- . The small resonance on the
right of the ls3p2 AP is the ls3p4p4P resonance..........................................
3.5. The eigenphase shift in the area of the 1s3jP AP resonance of He- . This
figure shows the change of 7r on the eigenphase shift at the energy region
of the resonance....................................................................................................
3.6. The resonance structure of the cross section of the 4P state of He- be
tween the He(n = 3) and He(n = 4) thresholds. The vertical dashed
lines indicate the energy position of the He target states...........................
3.7. The partial cross sections from the s2pkpAP and the s3pkpAP chan
nels with energy above the n = 3 threshold. The vertical dashed lines
indicate the energy position of the He target states. We can see that the
cross section of the s3pkp channel is much stronger than that of the
ls2pkp channel......................................................................................................
3.8. Photodetachment cross section in both length form and velocity form
from He- ( AP°) initial state to He- ( AS ) final state. The vertical dashed
lines indicate the positions of the target energies..........................................
3.9. Partial cross section in both length form and velocity form from the
s2sks AS channel. The vertical dashed lines indicate the positions of
the target energies................................................................................................
vi
Page
18
52
53
53
55
55
56
56
57
58

3.10. Partial cross section in both length form and velocity form from the
ls2pkp 4S channel. The vertical dashed lines indicate the positions of
the target energies................................................................................................. 59
3.11. A close up view of the cross sections from the ls2sfcs4S and ls2pkp4S
channels at the energy region of the Is3s4s 4S resonance. We see from
this figure that the contribution of the ls 2sA;s 4S channel and the s2pkp 4S
channel are comparable at the region of the Is3s4s 4S resonance. . . . 59
3.12. The eigenphase shift for the ls2sks 4S channel and the s2pkp 4P chan
nel at the energy region of the Is3s4s 4S resonance 60
3.13. Photodetachment cross section of He"( 4P°) to He~( 4D ) final state. The
vertical dashed lines indicate the positions of the target energies. . . . 61
3.14. Partial photodetachment cross section of He~( 4P°) via the He{ 3S ) chan
nel. Experimental data of Pegg et al [97] are also shown in the figure. 62
3.15. Partial photodetachment cross section of He"( 4P°) via He{ 3P ) channel.
Experimental data of Pegg et al [97] are also included................................ 62
3.16. Total cross section and the sum of the partial cross sections via the
He(n = 2) channels. The difference for energy higher than the He(n = 3)
threshold indicates the contribution from the H e(n > 3) channels. The
result is in good agreement with the Experimental data of Pegg et al [97]
in the limited energy region................................................................................ 63
3.17. Comparison of the present calculation with experimental data of Walter,
Seifert, and Peterson[77]. The experimental data is scaled by a factor of
| to get the agreement as shown in the figure................................................ 64
3.18. Asymmetry parameter /3 for the H e(2, 3S ) channel. Experimental data
of Thompson et al [98] in the limited energy region are also shown in the
figure......................................................................................................................... 65
3.19. Asymmetry parameter for the He{2, 3P ) channel and the experimental
data of Thompson et al [98]................................................................................ 65
3.20. Partial cross section of the He- ( ls2s2p4P°) photodetachment to the
final 4P state showing the 2s2pi 4P resonance. We can see that the
cross section for length form and velocity form agree very well, with a
difference of 0.03% at the peak of the resonance........................................... 71
vii

3.21. Partial Is detachment cross section of the He” ( Is2s2p 4P°) to the fi
nal 4P states. The first peak of the cross section is contributed by
the 2s2p(3P °)kp 4P channel. The peaks at 40 eV and 40.3 eV are
caused by the 2p2( 3P )ks 4P and 2p2( 3P )kd 4P channels. The large
peak at 43.352 eV indicates the 2s3p2 4P resonance. The peaks after
this resonance show the interaction between the 2s3p( 3P°)kp 4P and
the 2p3s(3P °)kp 4P channels.............................................................................. 73
3.22. The partial cross sections from the 2s3p( 3P°)kp 4P and the 2p3s( 3P°)kp 4P
channels. Strong interaction between these two channels is shown at en
ergy near 43.75 eV................................................................................................ 75
3.23. The contribution of the partial Is detachment cross section from the final
4S state. From this figure we cansee the narrow 2s3s4s 4S resonance. 75
3.24. The partial Is detachment cross section to the final 4D state. The first
two peaks in the figure are caused by the interaction of the 2s2p( 3P°)kp 4D
channel and the 2s2p( 3P °)kf 4D channel. The third peak is from the
2p2( 3P )kd 4D channel. The peak at 43.486 eV is a resonance caused by
the 2p3s( 3F°)3p 4D perturber state................................................................. 77
3.25. The partial cross sections from the 2s2pkp4D , 2s3skd4D , 2s3pkp4D ,
and 2p3skp 4D channels. Only the result from the length form are plot
ted in the figure. The difference between the velocity form and the length
form at the peak of each partial cross section is 5% for the 2s2pkp 4D
channel and the 2s3skd 4D channel and 9% for the 2s3pkp 4D channel
and the 2p3skp4D channel. The first peak at 43.135 eV is from the
2s3s( 3S )kd 4D channel. The large resonance at 43.486 eV is caused
by the 2p3s( 3P°)3p 4D perturber state. The cross sections for both the
2s2pkp 4D channel and the 2s3skd 4D channel have a maximum value of
about 20 Mb at the resonance position. The peak at 43.81 eV shows the
channel interaction between the 2s3pkp 4D channel and the 2p3skp 4D
channel when the cross section from the 2p3skp 4D channel reaches its
maximum................................................................................................................. 78
3.26. The average eigenphase shift for the 2s2pkp 4D channel and the 2s3skd 4D
channel at the resonance region of the 2p3s( 3P°)3p 4D perturber state.
We can see from the figure that the average eigenphase shift has a change
of 7r at 43.486 eV................................................................................................... 80
3.27. The total cross section from the Is detachment channels. The large peak
at 39 eV is mainly from the 2s2pkp 4D detachment channel. Below 43 eV
we can see the 2s3s4s 4S resonance. Detailed structures are also shown at
energy around 43.5 eV, including the 2s3p2 4P and the 2p3s( 3P°)3p 4D
resonances................................................................................................................ 81
viii

3.28.
4.1.
4.2.
4.3.
4.4.
4.5.
4.6.
4.7.
4.8.
5.1.
5.2.
5.3.
The total cross section from the He_ (ls2s2p4P°) initial state. The re
sults of Kim, Zhou, and Manson are also shown in the figure.................... 83
Total cross section in length form and velocity form for the photode
tachment of Li- from 1s 22s 2 lS initial state. The verticaldashed lines
indicate the energy positions of the Li target states..................................... 90
Partial cross section in length form and velocity from the 2skp channel.
The vertical dashed lines indicate the energy positions of the Li target
states........................................................................................................................ 95
The partial cross sections in length form and velocity from the 2pks
channel. The vertical dashed lines indicate the energy positions of the
Li target states...................................................................................................... 95
The partial cross sections from the 2pkd channel. The verticaldashed
lines indicate the energy positions of the Li target states................. 96
The partial cross section from the 3skp channel. The vertical dashed
lines indicate the energy positions of the Li target states................. 96
The partial and total cross sections in the energy regionbetween the
Li(ls23s 2S ) and the Li(ls23p 2P°) thresholds. The partial cross sections
show a small dip at about 4.2 eV before the resonance. The vertical
dashed lines indicate the energies of the H e(ls23s 2S ) and H e(ls23p2P°)
target states............................................................................................................ 97
Partial cross sections in the vicinity of the resonance below the L i(ls23p 2P°)
threshold................................................................................................................. 97
The total cross section and eigenphase shift in the region of the res
onance. From this figure we can see that the eigenphase shift has a
change of 7r at the resonance.............................................................................. 98
The partial photodetachment cross section from the B e~(4P ) initial
state to the Be- ( 4P°) final state via the ks channel. We can see that
the cross section reaches a maximum at an energy very close to the
ls22s2p3P ° threshold. A resonance isshown at about 3.94 eV.......................104
The partied cross section to the Be- ( 4P°) final state via the kd channel
showing a smooth increase near threshold and a resonance at about 3.94
eV.............................................................................................................................. 105
A close up view of the resonance structure for the partial cross sections in
both the length form and the velocity form via the ks and kd channels to
the Be- ( 4P°) final state. The vertical dashed line indicates the threshold
energy of the Be(ls22s3s 3S ) state. We can see that the position of the
resonance is well below the threshold energy. ............................................ 106
IX

5.4. The photodetachment cross section in length form and velocity form to
the Be“ ( 4P°) final state via the Be(ls22s2p 3P°)ks -+- kd channel com
pared with the sum of the cross sections via other excited channels. We
can see that the contribution from these excited channels is sm all but not
negligible. The vertical dashed lines indicate the threshold energies of the
Be(ls22s3s 3S ), Be(ls22s3p 3P°), Be(ls22p2 3P ), and Be(ls22s3d 3D ) states,
respectively. ...................................................................................................... 107
5.5. The photodetachment cross section to the Be“ ( 4D°) final state, the first
peak comes from the ls22s2p( 4P°)kd 4D° channel and the second peak
is caused by the ls22s3p( 4P°)kd 4D° channel. The vertical dashed lines
indicate the threshold energies of the Be(ls22s3p3P°), B e(ls22p2 3P ),
and B e(ls22s3d3D ) states, respectively. ...................................................... 108
5.6. Total photodetachment cross section for the Be- (ls22s2p2 4P ) state.
Experimental values of Bae and Peterson[146] and Pegg et al [155] are
also shown in the figure. The vertical dashed lines in the figure in
dicate the threshold energies of the Be(ls22s3s 3S ), B e(ls22s3p3P°),
Be(ls22p2 3P ), and Be(ls22s3d3£ ))states, respectively. ....................... 109
5.7. The angular asymmetry parameter ft for the B e(ls22s2p 3P °) target
state. Experimental value of Pegg et al [155] is included for compari
son............................................................................................................................ 110
x

LIST OF TABLES
Table Page
3.1. Energies ( E’tar9et ) of the triplet states of He obtained from the present
MCHF calculation are compared with accurate theoretical or experimen
tal values (Eacc)...................................................................................................... 47
3.2. The channel wave functions for the 4P , 4S , and 4D final states. The
corresponding target states and their energies (in eV) relative to the
initial He- (ls2s2p4P°) state are also listed.................................................... 49
3.3. The discrete energy levels of the 4P state of He- , obtained from bound
state Cl calculation, and the corresponding energies relative to the initial
state. The energy values within the curly brackets ( {...} ) come from
the discretization of the corresponding continuum channels and thus the
corresponding states can not form resonances when continuum channels
are included in the final state wave function................................................... 50
3.4. The resonance parameters for the ls2p2 4P shape resonance are deter
mined by fitting the total cross section to Eq.(2.88) in the energy region
from 1.22 eV to 1.28 eV. Threshold energy was fixed in the fitting process.
Other theoretical and experimental results are also listed for comparison. 67
3.5. The positions and widths of the Feshbach resonances below the n = 3
and n = 4 thresholds............................................................................................. 67
3.6. Energies of target states and their relative values in eV to the initial state.
The energy of the initial state is -2.1780499 a.u. (1 a.u. = 27.2076134
eV is used to convert a.u. to e V .) ................................................................... 69
3.7. Channel wave functions for the final states. The target energies relative
to the initial state are also included. The energy of the initial state is
-2.1780499 a.u. (1 a.u. = 27.2076134 eV is used to convert a.u. to eV.) 70
3.8. Position and width of the 2S2P2 4P resonance compared with those from
other theoretical calculations............................................................................. 72
3.9. Resonance position (Er) and width (r), and other parameters (a0, a, p,
and q) from the Fano-Cooper formula (Eq.(2.88) and Eq.(2.89)) for the
resonance states identified in this paper. The energy ranges (eV) of the
resonances are also listed. The parameters are determined by fitting the
Fano-Cooper formula to the cross section for both the length form and
the velocity form in the defined energy ranges............................................... 81
xi

4.1. Energies (in a.u.) of relevant target states and the photon energies (in
eV) required to reach these states from the initial state. The energy of
the initial state is -7.5002510 a.u. (1 a.u. = 27.2092 eV is used to convert
a.u. to e V . ) .......................................................................................................... 89
4.2. Available numerical data of experiment total cross sections for the pho
todetachment of Li~(ls22s2 lS ) and a comparison with other theoretical
results in length form (L) and velocity form (V).......................................... 91
5.1. Energies of the target states. Core excitation is not considered. Orbitals
are generated from an MCHF calculation of the initial state and target
states. The energy values AE relative to the initial state are also listed.
1 a.u. = 27.20967 eV is used in the conversion. The energy of the initial
state in this calculation is -14.529246 a.u........................................................ 103
xii

CHAPTER I
INTRODUCTION
Background of Study
The influence of negative ions on the nature of physical phenomena has been un
derstood in great detail in the past decades. This applies to absorption in stellar,
including solar, atmospheres, to electric discharge and breakdown phenomena in elec
tronegative gases and to the terrestrial ionosphere. Negative ions have also been used
for several decades in a variety of accelerator applicationsfl, 2, 3, 4]. Negative ion
based neutral beam injection has been considered as one of the candidates for heat
ing and/or current drive on the magnetic fusion device[5], which is one of the most
important options for the energy supply in the future. Because of the importance
of negative ion beams, there has been an international symposium held every three
years on the Production and Neutralization of Negative Ions and Beams since 1977.
Because of the important applications of negative atomic ions, understanding their
structure and dynamics has been an important task in atomic physics.
Negative atomic ions are excellent prototypes for the study of atomic structure
and dynamics. It is well known that negative ions are extremely sensitive to electron
correlation effects. Because of the diffuse characteristic of the wave function and
weak coupling among atomic state configurations, negative ions provide a good test
for theoretical computational models. Its weak coupling makes theoretical calculation
much more difficult compared to those for neutral atoms or positively charged atomic
ions. Because of their unique properties, the negative ions have been the targets of
1

experimental and theoretical investigation.
The Nature of Negative Atomic Ions
Negative ions constitute a special type of atomic system. Since they are formed by
adding an extra electron to neutral atoms, the electron affinities are usually very small.
Apart from the halogens, the alkali metals are the only main-group elements that can
form stable closed-shell negative atomic ions. Other negative ions are usually formed
in the form of metastable states. The structure of atomic negative ions is usually
associated with the characteristics of short-range potentials [6]. Consequently, the
properties of negative ions are governed by electron correlation. For a neutral atom
or positively charged atomic ion, the outermost electron sees a positively charged
ion core formed by the inner electrons and the atomic nuclei. Thus motion of the
outermost electron is dominated by the long-range Coulomb potential, which leads to
an infinite number of Rydberg resonances. Although the effect of electron correlation
is important in the case of doubly excited states of neutral atoms or positive ions, the
independent particle mode gives a reasonable zero-order approximation. In the case
of negative ions, however, the inner electrons effectively screen the nuclear charge, and
the outermost electron experiences an interaction from the atomic core via correlation
effects with inner electrons. The consequence is that negative ions only possess one or
a few bound and quasi-bound states, but no infinite series of such states as known for
neutral atoms or positive ions. There are exceptions, however, when the inner atomic
core can form a permanent electric dipole moment, as for the case of H- . In this case,
the outermost electron experiences an asymptotic long-range dipole tail potential.
However, long series of resonances are observed in negative ions such as H” and
2

He~ (see, for example, Table I and Table II in Ref.[6]). In order to interpret this
phenomena, Fano[7] proposed a mechanism based on the idea of a Wannier ridge[8, 9].
Under the Wannier theory, two electrons move on the opposite sides of the residual
ion and at large distance from the ion. This configuration leads to a long-ranged
Coulomb potential which support a Rydberg-like series of resonances near thresholds.
The Wannier-ridge mechanism has been applied successfully to the interpretation of
the Rydberg-like resonances.
Similar to the resonances of neutral atoms or positive ions, resonances in negative
ions can be classified into two groups, Feshbach resonances and shape resonances.
A shape resonance is formed when an incident electron tunnels through a potential
barrier of the target atom, remains confined within the barrier for the lifetime of
the resonance, and then tunnels out again. The potential barrier can be formed by
the repulsive centrifugal potential and the attractive atomic mean field. In contrast,
a Feshbach resonance involves the process of the capture of an electron to form a
negative ion in it’s excited states, and the release of the electron when it re-acquires
enough energy to escape. But these two types of resonances cannot be strictly dis
tinguishable in real atomic systems. Usually a resonance exhibits a characteristic of
a mix of these two types. For example, although an atomic mean field may yield a
potential barrier to support a shape resonance, the resonance energy and width may
depend sensitively on correlated interaction of the electron and the target. On the
other hand, a Feshbach resonance may derive from a simple potential barrier struc
ture provided by an excited state of the target, and would be manifested as a shape
resonance in an experiment on electron scattering by that excited state.
3

The Photodetachment Process
The absorption of photons by atoms and ions is a fundamental process of nature
and an important route of study in atomic physics and other areas[10]. A great deal
of work has been done in the past decades in this area for neutral atoms and positive
ionsfll, 12]. In comparison with the photoionization of neutral atoms, relatively few
photodetachment cross sections for negative ions have been determined. There is
a definite deficiency of information available on the photodetachments of negative
atomic ions.
The absorption of radiation by a negative ion resulting in the destruction of the ion
to form a neutral atomic system, constitutes one of the most recent, and potentially
the most important, photoabsorption process to be investigated. This process, known
as the photodetachment process, can be represented as
hu + A ~ — >A - f-e” . (1.1)
There are a number of reasons for pursuing photodetachment investigations. First,
photodetachment measurements provide a uniquely precise method for determining
binding energies of atomic negative ions by recording the onset of photoelectron or
residual atom production in the photodetachment process. Second, photodetachment
provides a source of continuous opacity in the visible and infrared spectra of hot gases
and stellar atmospheres[29]. Reliable estimates of the photodetachm ent cross sections
are required, for example, in the interpretation of the properties of low temperature
plasmas and in the field of upper-atmosphere physics. Third, photodetachment cross
sections between specific initial and final states are of indirect importance, since they
are frequently the most accessible route to the cross sections for the reverse process
4

between the same states, namely the radiative attachment of an electron to a neutral
atom.
W ith the advent of lasers and the availability of negative ion beams, there has
been a huge surge in the number of exciting experiments being performed on photode
tachment of negative ions. Photodetachment cross sections and angular distributions
can be measured directly via photodetachment spectroscopy technology. This makes
comparison of experimental and theoretical values possible not only on the resonance
parameters such as position and width, but also on the absolute values of the detach
ment cross sections as a function of photon energy. Theoretically, negative atomic
ions present a considerable challenge since the inclusion of correlation is crucial for
even a basic qualitative description and the coupling between the initial photon and
the target electrons is generally weak. Therefore, the photodetachment process stands
out as a particularly sensitive probe of properties of negative ions.
Review of Experimental and Theoretical Methods
The principal means of experimental study of negative ions are electron-atom
collision, laser photodetachment spectroscopy, and negative-ion-atom collision. De
tailed description of these techniques can be found from the article by Schulz[13] and
Buckman and Clark[6] and the references cited therein.
Electron-impact spectroscopy has provided most of the information on atomic
negative-ion resonances, with resolution up to 10 meV. Negative-ion resonances formed
by electron impact are generally studied by detecting reaction products in one or more
of the decay channels energetically available to the compound state. Such studies in
volve the measurement of either the total or differential scattering cross section for
5

the reaction products as a function of the incident electron energy.
Photodetachment spectroscopy technology was developed in the 1970s as a result
of advances in both laser and negative-ion beam technology. This approach has been
one of the most important approaches in recent years in the experimental study of
negative atomic ions, including electron affinity, cross sections, angular distribution,
and other properties.
Various theoretical approaches have been developed for resonance calculation.
These approaches can be classified into three categories according to Nicolaides[14]:
- computation of the wave functions of continuum states, from which resonance
parameters are determined via the analysis of scattering matrices or eigenphase
shifts. Methods in this category include /2-matrix, close-coupling, random-
phase, many-body perturbation, etc. Much of the early theoretical work, par
ticularly on H~ resonances, was done in the close-coupling approximation (see,
for example, the review published by Schulz[13] ). The /2-matrix method is
among the most widely used methods for computation of atomic collision phe
nomena. It is based on enclosing the N + 1 electron system within a sphere
of radius ro and imposing a fixed logarithmic derivative boundary condition
on the wave functions on its surface. Only one electron is allowed to move
in the region outside the sphere. Though the method can be applied to the
calculation of any atomic system, the treatment for negative ions is different.
One respect in which /2-matrix calculations of negative-ion resonances differ
from those of neutral atoms and positive ions is that the magnitude of r0 can
be significantly larger in order to accommodate diffuse excited states of the
target. This increases the size of the calculation significantly. Recently, the

finite basis set approach with B-splines has come to play an important role
in this category. There are several implementations of this approach, among
them, the spline-Galerkin method has been shown to be very successful and
will be discussed in detail later. Since the wave functions of the continuum
states are calculated, methods in this category are capable, in principle, of
describing all observable phenomena, such as differential angular distribution
and partial excitation cross sections, from which properties of any possible
resonances such as position and width can be determined.
- direct calculation of complex energies of decaying states, by treating the reso
nance as an eigenfunction of a non-Hermitian system. This approach casts
the problem into a form where only square-integrable wave functions are
determined, and which therefore leads immediately to spectroscopic assign
ments analogous to those applied to bound states. The most widely used
such approach during the past decades includes the Feshbach projection and
complex-coordinate rotation methods. The basic idea of the Feshbach projec
tion operator technique is to treat a resonance as a ’’discrete state embedded
in a continuum,” by deriving an equation of motion for the component of the
wave function that is orthogonal to the continuum. The method, originally
introduced by Feshbach in nuclear physics[15], has been adapted to the many-
electron atomic calculations[16, 17] and has provided very accurate results for
two-electron systems[18, 19] and some three electron systems[20]. The idea of
complex-coordinate rotation is[21]: For a general atomic system, if all particle
coordinates are scaled by a factor et0, the transformed Hamiltonian H{0) may
7

support discrete eigen functions with complex energies e e~t4>, provided that
6 > 0 / 2. By diagonalizing H{9) in an appropriate basis of many-electron wave
functions composed of square-integrable orbitals, the resonance position and
width can be determined. Numerous calculations of negative-ion resonances
by complex-coordinate rotation method have been performed[22, 23].
- computations that treat resonances like ordinary bound states and neglect
interactions with the continuum. In this approach, the effects of continuum
channel interactions are ignored. Such calculations represent resonance states
by diagonalizing the Hamiltonian in a finite basis of iV-electron configuration
state functions constructed from square-integrable orbitals. A projection oper
ator is applied to the basis set to remove any continuum component. Despite
its unsure theoretical foundations, this method has been quite successful in
determining resonance energies.
A Brief Review of Negative Atomic Ions
H~ is the simplest system in the family of the negative atomic systems. So it
is used as a prototype for understanding the physics of such weakly bound systems.
Though it is a member of the helium isoelectronic series, it’s behavior is quite different
from other helium-like systems. Since the nuclear charge is smaller than the number
of electrons, the electron-electron interaction is as strong as the nuclear-electron in
teraction, and therefore the motion of the electrons is highly correlated, especially for
continuum states. Unlike most of the other negative atomic ions, the nature of the
continuum state of H- is governed by the long-range interaction between one electron
8

and a neutral hydrogen atom due to a permanent electric dipole moment. Negative
hydrogen has a ground (Is2 lS ) state and an excited bound state, (2p2 3P )[24, 25].
Resonance states of negative hydrogen have been investigated in the past decades, a
list of these states is shown in Table I in Ref. [6].
Next to H- is the He” system. The concept of a rare gas anion goes against the
basic chemical concepts. When in their ground state, these atoms are closed shell
systems and thus have no readily available low energy orbitals in which to place an
extra electron. Thus, rare gas atoms like He are considered incapable of binding an
extra electron in their ground state[26]. However, when one considers excited states
of these neutral atoms, the closed-shell argument no longer holds and the possibility
of a rare gas anion bound state arises. The only two bound states of He" are excited
bound states, namely, the (ls2s2p4P°) state suggested by Hibby[27], and the (2p3 4S )
state predicted by Nicolaides and Beck[25]. Both states have been observed experi
mentally and their energies with respect to the (ls2s 3S ) state and the (2p22P ) state
of He, respectively, measured with high accuracy. Recently, Mercero et al studied the
stability of these bound excited states in an environment represented by the stati
cally screened Coulomb potential [28]. According to Mercero et a l , the stability of the
bound excited states of atomic negative ions is a subject of considerable importance
in many areas of physics, including electron scattering in atomic gases[6], and studies
on the opacity in stellar atmospheres [29]. The reason for the presence of a statically
screened Coulomb potential is that such screened potentials are important for many
areas of physics, such as plasmas, nuclear and elementary particle physics, atomic
physics, and solid state physics (see, for example, Mercero et al [28] and references
cited therein). The negative ion of helium provides one of the simplest structures
9

and has thus prompted many experimental and theoretical investigations over the
last 20 years. Only in the 1990’s, however, studies of excited states of He" involving
subshells with n > 3 began to appear. A list of resonance states of He- is shown in
Table II in Ref. [6].
Like all other alkali atoms, Li- negative ion has a stable ground state, (Is22s2 lS ).
Besides, it also has two excited states, namely, (ls2p35S°) and (ls2s2p25P ) . These
excited states were first reported by Bunge[30, 31]. There have been few studies of
the resonance states of Li- until recently when both experimental and theoretical
studies begin to appear.
The first beam of Be" was produced thirty years ago[32]. The lowest bound state
of Be- is (ls 22s2p24P ).
There are also many other negative ions being identified and their structures
are subject to theoretical and experimental investigation. One of the key aspect of
study for negative ions is to find the bound states and determine the electron affini
ties. Early works are mainly focused on this area. The early review of Hotop and
Lineberger[34] in 1975 summarized the progress on the investigation of the ground
states of negative atomic ions and their electron affinities. Ten years later, the authors
gave another comprehensive review on the same topic[35]. In Table 1 of Ref. [35] Ho
top and Lineberger listed all the negative ions, their ground states, and their electron
affinities. The authors also listed the metastable states of negative ions, including the
He- (ls2s2p4P°) state and the Be- (ls2ls2p2) state. In a nonrelativistic fixed-core
valence-shell configuration interaction calculation, Bunge, Galan, and Jauregui also
made a systematic search of possible excited stable and metastable states of negative
ions in the range H through Ca[36]. A more recent review of metastable excited states
10

is given by Nicolaides, Aspromallis, and Beck[37].
The interest in the study of resonance states of negative ions has increased consid
erably. Achievements have been made in the last decades, both experimentally and
theoretically on the study of resonance states of negative ions. Several reviews are
already available on these achievements[13, 6]. In the recent review of Buckman and
Clerk[6], the resonance states of all negative ions are studied in detail.
We have made a comprehensive study on the photodetachment of the He- , Li- ,
and Be- systems. In the following chapters, we illustrate the result of our study.
In chapter II we propose our theoretical approach for the photodetachment study of
negative ions. In chapter III through chapter V we illustrate our photodetachment
study on the He- , Li- , and Be- systems, respectively.
11

CHAPTER II
THEORETICAL APPROACH
Introduction
An efficient and accurate numerical method plays an important rule in the study
of complicated atomic systems. One of the fundamental tasks is to solve coupled
differential equations efficiently. Several approximation methods have been developed
over the past decades, including finite basis set approach. In recent years, the attempt
to construct finite basis sets with B-splines has achieved great success. This method
was shown to be an efficient method for representing both the bound and continuum
radial orbital wave functions of an atomic state wave function. It is used in a variety
of fields in theoretical calculations of atomic properties.
The jB-spline technique was employed by Johnson and et al [38, 39, 40] in the rela-
tivistic many-body perturbation calculation of atoms and by Froese Fischer et al [41,
42, 43, 44, 45] in solving radial orbital equations of continuum atomic states. The
5-spline basis was combined successfully with the Galerkin and the inverse iteration
approach to solve continuum state problems of two electron atomic systems with
only one open channel[43, 46]. The Galerkin approach was used for establishing the
interaction m atrix whereas inverse iteration was used for solving the system of lin
ear equations for the continuum states. Chang et al [47, 48] combined the 5-spline
basis with an L2 approach to study photoionizations of two electron systems. Xi
et al combined the 5-splines based with many-body perturbation theory[49] to in
vestigate hyperfine interaction[50] and electron correlational]. A two-dimensional
12

5-spline basis was also used by Xi et al to study the behavior of hydrogen atom in
super strong magnetic fields[52, 53]. This two-dimensional approach was later applied
to a hyperspherical coordinate system to study the hyperspherical potential curves of
three-electron atomic systems[54].
In this chapter, we will introduce the 5-spline basis into our calculation of the
continuum atomic state wave function.
In the following sections, we will explain briefly about the atomic state wave
functions, the idea of the 5-spline basis and it’s properties, the 5-spline expansion of
an electron radial wave function, the boundary conditions, and the Galerkin approach.
We will also discuss in detail the application of the 5-spline functions to atomic
of a many electron system, the independent particle model, and the Hartree-Fock
method. These topics have already been covered in great detail elsewhere[49, 55, 56].
Our coverage here is purely for convenience of description for the followed sections.
The Independent-Particle Model
For an iV-electron atom with nuclear charge Z, the nonrelativistic Hamiltonian
can be written as (in atomic units (a.u.)),
systems.
Atomic State Wave Function
In this section, we give a brief description about the atomic state wave function
(2.1)
13

where is the distance of electron i from the nucleus, and rtJ its distance from
electron j.
The idea of the independent-particle model is that we assume each electron moves
independently of the other electrons in an average field produced bythe nucleus and
the other electrons. This provides us an approximate descriptionof the atom and
also serves as the starting point for more accurate calculations. Let
H = H0 + Ves: (2.2)
where
tfo = £> o(*) (2-3)
i=l
is a sum of one-electron operators
ho{i) = - v 2i - - + u{fi) (2.4)
and
K , = - 5 > ( n ) + i ; ^ - - (2.5)
i i<j r'3
The approximate Hamiltonian H0 represents an average interaction with the nucleus
and the other electrons. Ves causes departures from this single-particle description.
This departure can be covered in a more accurate model such as in a perturbation
calculation or via multi-configuration approaches.
The wave functionof anN electron atomic system moving under an average po
tential can be written asa product of the wave functions of each electron. Considering
the requirement of the Pauli exclusion principle, the wave function of a many-electron
system must be antisymmetric with respect to interchange of any two of the elec
trons. This lead to the Slater determinant representation of the wave function of an
14

iV-electron atomic system under the assumption of the independent-particle model.
$
<Pa(l)<Pa(2)-<Pa(N)
<pb(l)<pb(2)...(pb(N)
(2.6)
¥Jn(l)V>n(2)...{fln(N)
where a, b,... stands for the set of quantum numbers of a single electron state, and
1, 2,... stands for the space and spin coordinates of electrons 1, 2,.... <pa(i) represents
the wave function of electron i, satisfies the Schrodinger equation
(2.7)
The Central-Field Model
In order to solve Eq.(2.7) we need to choose an appropriate coordinate system. A
natural choice is thespherical coordinate system. If we can assumethatthe average
potential u(fj) is spherically symmetric, which means it only dependson r,- and not
on the angular coordinates, then the single-electron Schrodinger equation (2.7) can
be re-written as
~ j + u(r)J <p = e<p, (2.8)
By expanding the wave function ip under the spherical coordinate,
¥>= — YUe,<t>)xm„ (2.9)
r
we get the following radial Schrodinger equation
LP(r) = cP(r), (2.10)
15

where
1 ( f 1(1 + 1) _ Z u(r)
2 dr2 2r 2 r r
(2.11)
For the wave function <p to be finite at the origin, it is necessary that P(r) —> 0 as
r —» 0. Also tp must be normalizable for bound states, which means that the integral
With the central-field approximation, the atomic state wave function (Eq.(2.6) can
be represented by configuration state functions with an LS term. Solving Eq.(2.10) for
each electron we get the wave function and energy of the atomic system. In order to
solve Eq.(2.10), however, we need to determine the average potential u(r). One of the
most commonly used average potential is the Hartree-Fock potential. It is obtained
by applying the variational principle together with the Brillouin theorem[57, 58] to
the expectation value of the total energy for an atomic state represented by the
configuration state function. Since the potential u(r) of one electron depends on the
wave function of other electrons, Eq.(2.10) is usually solved via a self-consistent-field
approach. Detailed description of the numerical methods for solving Eq.(2.10) is given
by Froese Fischer et al [55, 56]. In the next section, we will introduce a 5-spline basis
expansion approach.
should be finite. This condition leads to the boundary condition P(r) —►0 as r —>oc.
For a continuum state where e > 0, the radial wave function should be normalized to
a linear combination of the regular and irregular Coulomb functions,
Cc .(r) -> (2.12)
where k is the momentum of the free electron (k2 = 2e) and 5i is the phase shift.
16

The Multi-Configuration Approach
The independent-particle model can only give an approximate description of the
wave function of an many-electron atomic system. This description is far from ad
equate for an accurate calculation of such systems. For strongly correlated systems
such as negative atomic ions, the independent model will completely fail. In spite of
these limitations, the wave functions from the independent model are a good start
ing point for more accurate treatment. The independent particle model gives us an
infinite set of atomic state wave functions, which can be used as basis to form a
multi-configuration wave function
* (7 LS) = J > $ ( a f£S ) (2.13)
i
where $(ctiLS) is the atomic state wave function defined in Eq.(2.6) under the LS
coupling. Eq.(2.13) has become the starting point for the accurate study of atomic
systems.
The B-Spline Basis
The B-splines are piecewise polynomials defined on a given knot sequence {£,, i =
1, 2,...}, with the following properties:
- Bi(t) = 0, £ $? [£,-, ti+fc]
- Bi{t) > 0, £e [£,-,£,-+*■]
- £ , £,(£) = M 6 [£i ,£at]
- Bi is K —2 times differentiable
17

0.0
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0 0.1 0.2 0.4 0.6 0.6 0.7 0.80.3 0.0
Figure 2.1: A distribution of 5-splines B,-i5(:e) in the region x € [0,1] where the
horizontal coordinate labels the x value and the vertical one indicates the value of
the B-splines. The 14 5-splines of order 5 are defined by the knot sequence t t =
■•• = t5 = 0, t{ = ti-i + 0.1, for i = 6,7, •••, 14 and t i5 = ■■• = tl9 = 1.
The 5-splines of order K on the r axis are defined as
1 , fj ^ t <C £ t+ l
BiAr)
B iM r)
= {
0, otherwise,
-----------------------------+ t+K_ — Bi+itK-i(r).
H + K —I H 1-i+K W+l
(2.14)
(2.15)
A sample example of a set of 5-splines is shown in Figure 2.1. A detailed illustration
of the properties of the 5-splines can be found in the book by deBoor [59].
The selection of the knot sequence is very important to the efficiency of the 5 -
spline approach. The basic rules of setting knot sequences for atomic problems can be
found in the literature[41, 46]. Generally, in the inner region of the atom, the atomic
orbital wave functions scale with respect to the nuclear charge Z , so we should set the
knot sequence based on t = Zr, where r is the radial coordinate. For the continuum
18

state problem, the wave function oscillates in the asymptotic region, so the spline
functions should be uniformly distributed in this region. Based on these considera
tions, the knot sequence of a set of N splines defined on the r axis are selected as
follows:
U = 0 fo r i = 1 ,..., K
ti+ = U + h fo r i = K , ..., K + m
ti-t-L = £ » (l "b h ) f 01“ 1 ^ ^t+1 h m c
tj+ 1 = t i "f" h m a x f OT t i <C Z r p u u J
set N + 1 = m ai(i)
tff+i = tN+1 fo r i = 2 ,..., K
where h is the base step size, obtained from h = 2-m,m is an integer, hmax is
the maximum step size and Tmax the maximum radial radius of the wave function,
specified by the user.
The following principles are followed in selecting these parameters for a continuum
state problem,
- The order of the spline basis plays an important role for the accuracy of the
spline representation of the atomic wave functions. Numerical experiments
show that a proper selection is K — 8.
- rYjmx must be larger than the radius of all bound state orbitals, and must
be large enough so that the minimum continuum energy of interest can be
studied. Our experience indicates that this can be determined from the relation
kTmax > (3tt ~ 57t), with k2f 2 being the energy of the electron in a continuum
19

channel, in atomic units. For the multi-open-channel case, this should be the
energy of the electron in the highest open channel (with smallest k value).
- hmax cannot be too large, with a proper K value (K = 8, for example), a
suitable choice for hmfir is that the biggest step size A = r N-+i — satisfies
the relation ^ A < 7r/3, with k corresponding to that of the electron in
the lowest channel, i.e., we should put at least 3 knots within a half period of
oscillation of the continuum orbital wave function in the asymptotic region.
From this analysis we see that it is not appropriate to study a large energy region
with many open channels by using the same knot sequence because the channel with
low target energy requires a small /W r value whereas the channel with high target
energy requires a large rmnT value, which can lead to an interaction matrix of very
large dimension.
Solution of One Electron System Using Spline-Galerkin Approach
One of the core problems we need to deal with is the radial wave function of
the ejected electron. In this section, we outline the basic idea of solving the radial
Schrodinger equation using the B-spline and Galerkin method. W ith this, we can
easily extend the method to a multi-configuration atomic system.
To solve Eq. (2.10) we expand the radial wave function P(r) in terms of fl-splines,
P(r) « ^ a j B j i r ) , (2.16)
i= i
The boundary condition at r = 0 is imposed by setting ai = 0. The boundary
condition at r = Tmax cannot be pre-determined since it depends on the energy of the
electron.
20

By inserting Eq. (2.16) into Eq.(2.10) we get
AT
y^(L —e)djBj(r) = res(r) ss 0 (2.17)
j=2
where res(r) is the residual.
The Galerkin condition[60j requires that this residual be orthogonal to the solution
space. This is achieved by choosing a set of test functions, ri? that spans the solution
space, and setting up the equations
< Tj|res > = 0,for i = 1, 2, , ..., N (2.18)
A natural choice for those test functions are the B-splines, then, we get,
N
'Eiflij - e B ^ a j = 0, for i = 2, 3,..., N (2.19)
i= 2
where
Hij = < BiLBj > (2.20)
Bij = < BiBj > (2.21)
Eq.(2.19) represents a generalized eigenvalue problem. Solving this equation we get
the expansion coefficients a,- and thus get the solution of the radial wave function of
the electron.
In the next sections, we will extend this method to handle general multi-configuration
atomic state wave functions, and we will discuss in detail the solution of the eigen
value problem using an inverse iteration approach. The reason for choosing inverse
iteration approach will also be discussed later.
21

The Continuum State Wave Function
The multi-configuration wave function (Eq.(2.13)) of an N electron atomic system
can be extended easily to represent a system of N + 1 electron in it’s continuum
state. The wave function of a continuum state of a negative ion can be defined as
a combination of the wave functions of the perturber states which are the bound
configuration wave functions of the N + 1 system in the desired L S term, and the
wave functions of open or closed channels. Channel wave functions are formed by
coupling the multi-configuration wave function | 77 > of a target state of the N-
electron system with the single-particle wave function of the detached electron, with
the condition that the coupled result forms an antisymmetric wave function of the
N + 1 electron system with desired L S term. So the continuum state wave function
can be written as
Mp Mc ___
(TLS) = £ c(i)<t>{aiLS) + £ |( 7 > | ^ >)L S > (2-22)
i=I i=l
where the first part represents the summation of perturber states of interest, the
second term is a summation of the channel states of interest, each channel state
being represented in terms of the coupled state of the target state | 77 > and the
channel orbital | n,Z, >. Each target state | 71 > is a multi-configuration state
of the target atomic system. Both the bound perturber states 0(a,L 5) and the
target states | 77 > are defined in terms of the fixed orbitals obtained from a multi
configuration Hartree-Fock (MCHF) calculation. Orthogonality is required among
all target state wave functions and all configuration state wave functions (including
perturber-perturber, perturber-channel, and channel-channel). We also require that
all channel orbital wave functions be orthogonal to all fixed orbital wave functions.
22

The orthogonality is not required among channel orbital wave functions.
The wave function in Eq. (2.22) must satisfy the Schrodinger equation
H * = E * (2.23)
where H is the hamiltonian of the system defined in Eq.(2.1) and E is the total energy
of the system.
For a continuum state system, the energy E can be expressed as the sum of the
target energies Eti and the corresponding electron energy associated with this target,
i.e.
E = E ti + kf/2 (2.24)
To solve the Schrodinger equation for the wave function we approximate the wave
function with a set of atomic configuration state wave functions as shown in Eq.(2.22),
and we further expand the radial channel orbital wave function P (r) of the atomic
state wave function in terms of a B-spline basis set, B,(r), as
P M = (2.25)
j
then the wave function in Eq.(2.22) can be written as
mp Mc N
W(jLS) = J T Ci<f>(aiLS) + J E f ljC O k ij > (2-26)
i=l i=lj=l
where rij > is the i-th channel function with the radial orbital function substituted
by the j-th B-spline basis function, so that
n j >= 1(77* I ruh(Bj) >)LS > (2.27)
23

The Spline-Galerkin Method and the Interaction Matrix
With the wave function expanded in terms of the B-spline basis as in Eq.(2.26),
we expect that under a good approximation the residual (res) of the system is
res = (H - E)*(~fLS) « 0 (2.28)
The Galerkin condition requires that the residual be orthogonal to the solution space
for a set of test functions. This is achieved by choosing a set of test functions, r,-,
that spans the solution space and requiring
< rAres > = 0 (2.29)
The test functions in the current case are
<p(atiLS),i = 1,..., Mp
Applying the Galerkin condition we get the following generalized eigenvalue problem
Here C is the solution vector
(H - ES)C = 0
c
a (l)
C = a(2)
(2.30)
(2.31)
a(M c)
where c is the column vector of coefficients of the expansions defining the perturbers
in Eq.(2.22) and a(i), i = 1 ,2 ,..., M c is the column vector of B-spline coefficients
for channel i.
24

The interaction matrix (H —ES) has the following form:
h(PP) ~ E l M lp)T M2p)T — h(Mcp)T
h(lp) H ( U ) - E B H( 12) ... H{IMC)
h(2p) H(21) H(22) - E B ... 5(2M C) (2.32)
h(Mcp) H{MC1) 5 (M C2) ... H{MCMC) - E B
where I is an Mp x Mp unit matrix, B is the 5-spline overlap matrix of dimension
N x N,
By = < BiBj >, i j = 1,2,..., N (2.33)
h(pp) is an Mp x Afp matrix and comes from the perturber-perturber interaction.
h{piPj) = < <p(aiLSH<i>(aj L S >, i,j, = 1,2,..., Mp (2.34)
The h(np)'s are N x Mp matrices, representing the interaction between the n-th
channel and the perturber states.
Kmpj) = < TntiH<f>(ajLS >, (2.35)
where i = 1,2,..., N, j = 1,2,..., M P, and n = 1,2,..., Mc.
H(mn) are N x N matrices for channel-channel interaction.
) = < > (2.36)
where i,j = 1,2,..., N, m ,n = 1,2,..., Mc, and |rtJ > is defined in Eq.(2.27). It is the
z-th channel wave function with the radial orbital function substituted by the j-th
5-spline basis function.
The energy E of the system is determined by the energy of the photoelectron
relative to the first target state, as was shown in Eq.(2.24).
25

W ith the help of Racah’s algebra, the matrix elements can be expressed in terms
of radial Slater integrals[55], with the radial wave functions being replaced by the
B-spline basis functions.
The Orthogonality Requirements
Finally we need to deal with the problem of orthogonality requirements. In our
calculation, all bound orbitals are orthogonal with each other. This requirement is
automatically achieved in the MCHF calculation. But we also require that all channel
orbitals be orthogonal to bound orbitals. Starting withaone-electronSchrodinger
equation, if channelk is tobe orthogonal to orbital P (r), by introducing a Lagrangian
multiplier A and imposing the orthogonal condition, we can write
( H - E ) P k(r) + P (r) = 0 (2.37)
< P (r) |P * ( r)> = 0 (2.38)
where P*(r) and P(r) are the radial wave function of the channel k and the bound
orbited. Expanding the radial wave function in terms of B-splines,
PkiT) = E c.B .-tr) (2.39)
i
P(r) = £ 6 ;B ,(r ) (2.40)
j
then with Eq.(2.38) and Eq.(2.38) we get
(H - E B )c + B6A = 0 (2.41)
6‘Bc = 0 (2.42)
26

where b and c are the expansion coefficients in vector form. The above equations can
be re-written in m atrix form as
where Bb is a vector, it is zero everywhere except at the rows that corresponds to
channel k. 6 is the spline expansion coefficients of the orbital P(r). So one more
ity condition. We should note that Eq.(2.44) is no longer a generalized eigenvalue
equation, though it can be transformed to a generalized eigenvalue equation[61].
Up to now, we have set up the interaction m atrix and at this point, we are ready
to solve this system for the expansion coefficients.
orbital functions. First of all, the orbital wave function P(r) should be zero at r = 0.
Then, we must also require P(r = rmax) = 0 for closed channel orbitals. Whether
a channel is open or close depends on the energy of the system. From Eq.(2.24) we
learn that for a specified channel associated with target i, it is an open channel if
E —Eti > 0 a^d closed channel otherwise.
Now, we have already setup the interaction m atrix via the Galerkin approach and
applied the boundary condition. In the next section, we describe the inverse iteration
(2.43)
For a system of N -+- 1 electrons, the above equation can be extended as
(2.44)
equation and one scaler element (A ) are added to the equation for each orthogonal-
The Boundary Condition
It is very important to apply the appropriate boundary conditions to the channel
27

approach for solving continuum state wave function.
Multi-Channel Inverse Iteration Approach
We have already shown that the problem of solving the Schrodinger equation
(H - E)V = 0 (2.45)
can be converted approximately to the problem of solving a system of linear equation
AC = 0 (2.46)
where A is the matrix in Eq.(2.44) with appropriate boundary conditions applied.
We use the inverse iteration method to solve (Eq.(2.44). The advantage of inverse
iteration over eigenmatrix diagonalization is multifold. First, in order to use eigen-
matrix diagonolization method, we need to first convert Eq.(2.44)into aneigenvalue
equation, whereaswith inverse iteration, we only need to solvethelinear equation
with the given energy and the calculation is much more efficient. Second, with a given
5-spline setting, the eigenmatrix diagonalization method can only give a discrete set
of energies and the corresponding wave functions. In order to get enough energy
points to well describe the behavior near a threshold or a resonance, we need to ad
just the B spline setting (usually the cutoff radius) and re-generate the interaction
matrix. W ith inverse iteration, however, we can use the same interaction matrix to
determine the wave function at any given energy position.
The idea of inverse iteration can be understood as: When E is an eigenvalue of
H, the m atrix A will become singular as the accuracy increases. In an approximate
case, the energy E which satisfies Eq.(2.46) cannot be exactly equal to an eigenvalue
of the H which satisfies Eq.(2.45) but will be very close to one of the eigenvalues
28

of H which corresponds to the smallest eigenvalue of A. So for the continuum-state
problem with a given energy E which is an eigenvalue of H, the problem of solving
the system AC = 0 for the expansion coefficient C of the wave function turns out to
be a problem of finding the eigenvector corresponding to the smallest eigenvalue of A
(see references [41, 42, 43] for a detailed description). We should mention here that A
is energy dependent, so the smallest eigenvalue and the corresponding wave function
are specific to each given energy E. This is why we can use the same interaction
matrix to determine the wave function of the continuum state at any given energy.
An investigation by Brosolo, Decleva, and Lisini[62] showed that it is more stable
to solve A1A C = 0 instead of AC = 0. The inverse iteration approach was shown to
be very efficient in dealing with this problem [41, 46].
In the multi-open channel case, we need to determine several degenerate solutions
for each given energy E. The number of independent solutions to be determined
for each energy equals the number of open channels. In the approximate case, the
degeneracy is broken because the matrix A is not strictly singular. We can solve
them by finding the eigenvectors corresponding to the smallest eigenvalues of A. We
repeatedly perform inverse iteration for a solution and orthogonalize it to the ones
already obtained. Because the overlap m atrix S is not diagonal, the orthogonality
between solutions C ^ and C ^ should be carried out as
C(m)‘5 C (n) = Smn (2 4?)
On the other hand, however, the solutions obtained here are still not the physical
solutions. We will need to transform them to the K-matrix normalized form. It
has been demonstrated [63] that as long as the solutions obtained here are linearly
29

independent, the final physical solution should have the same form, independent of
the orthogonality schemes. Because of this, we can discard the overlap m atrix when
imposing the orthogonalization requirements among the solutions.
Normalization of the Continuum State Wave Function
A procedure combining the WKB method [64, 65] with the spline basis is used to
determine the normalization and phase shift of the continuum orbitals. To normalize
the open channel functions, we need to match the un-normalized channel functions
with Coulomb functions in the asymptotic region. The radial equation for an electron
under a Coulomb potential with effective nuclear charge Zeff is
y(r)" + w(r)y(r) = 0 (2.48)
where
w(r) = k2 + 2Zef f / r — 1(1 4- l ) / r 2 (2.49)
with k2/ 2 for the energy ( in a.u. ) of the electron and I for its orbital angular
momentum.
The radial function y(r) can be written as
y(r) = sin ^(r ) (2-50)
In the asymptotic region , C(r) satisfies the following equation,
C = (w(r) + C ^ A c -./2)i/2 (231)
A number of methods have been proposed to solve the above equation and determine
the phase function <j>(r) and normalization of the wave function [66, 67, 68]. Here we
30

use the Maple symbol manipulation package [69] and determine the iterates analyti
cally up to 4 iterations [70]. With sufficiently large r value, the iteration convergence
to an error of less than 10-9.
Having C(r) determined at a given large r value, we can determine the phase
function from Eq.(4), which gives
t a n * ( r ) = C / ( ^ + ^ ) (2.52)
where y'(r) and C(r) are the derivatives of y(r) and C(r) at the given r value, respec
tively.
The energy normalized regular and irregular Coulomb functions can be written as
F {r) = y ^ s i n ^ ( r ) (2.53)
G{r) = y^cos<?K r) (2.54)
These are then used to match the un-normalized radial channel function at the given
r value,
Pair) = Fi(r)Aij + Gi(r)Bij (2.55)
or
P = F A + GB (2.56)
in matrix form, where A and B are coefficient matrices, F and G are diagonal matrices
defined as
Fij = Fi(r)Sij (2.57)
Gij = Gi(r)6ij (2.58)
31

Multiplying Eq.(2.56) on the right by A~l , we get the K-matrix normalized
channel functions
P = F I + G K (2.59)
where
P = PA~l (2.60)
and K is the reaction matrix
K = B A ~ l. (2.61)
It should be mentioned that, care must be taken when calculating the K matrix.
When the phase shift of a certain channel approaches 7r/2, the matrix A~l will be
singular, this will lead to serious numerical errors, and cause pseudo resonances when
K is used. In this case, we should calculate K _1 = A B ~ l instead of calculating K.
The scattering matrix S is defined as [71]
e _ + lK ) fr, fio
S - ( l ^ l K ) (2'62)
The final state shouldbe subject to the boundarycondition of anincoming wave.
Then the wave function shouldbe normalized according to theS matrix,
P = F I + GS (2.63)
For the one open channel case, this differs from the K -matrix normalized wave
function by only a fixed constant, so we can simply use the latter as our normalized
wave function. For the multi-channel case, however, the Ff-matrix normalized wave
function no longer represents the physical situation and the S-m atrix normalization
should be used. The energy normalized channel wave function P subject to the final
32

physical boundary condition (Eq.(2.63) ) can be written as
P = PC (2.64)
where C is the normalization matrix,
Let 'Fy be the j-th solution of the un-normalized final state wave function, Wij the
weight coefficients of the i-th bound perturbers corresponding to We can rewrite
Eq.(2.22 ) as
_ mp __ Mc
= £ <Moi)Wv + Y . Wj (2-65)
i=l i=l
where ipjj isthej-th nn-normalized wave function of the i-thchannel state. From
Eq.(2.64), wecan get the normalized channel state wavefunction as
<p= tpC (2.66)
Similarly, we can get the normalized perturber weight, as
W = WC, (2.67)
or, the weight coefficient for the m-th perturber state in is
^ j m = E W im Cij (2.68)
i
Substituting Eq.(2.66) and Eq.(2.67) into Eq.(2.65), we get the j-th normalized
final state wave function as
* , = 2 ; * ^ (2.69)
t
Let dj be reduced dipole matrix elements between the initial state (&0) and the
j-th un-normalized finaJ state
dj = < ^ -|T |^ o > (2.70)
33

then from Eq.(2.69) we get the normalized reduced dipole matrix elements for the
j-th state as
d ^ Y ^ d i C i j (2.71)
i
We have two equivalent methods for calculating the C matrix. It can be calculated
by means of the eigenvectors U and the eigenvalues tan S of the K m atrix (6 is called
diagonal phase shift), or K ~ l matrix
U~lK U = tan 5
C = A ~lQ
or
U~lK ~ lU = (tan6)-1
C = B ~ lQ
where
Qij = UikUjk cos(Sk)e lSk
or by means of A and B matrices [72, 73],
(2.72)
(2.73)
C = {A + iB) -i (2.74)
Let C = X + i Y , then, with Eq.(2.74) we have
(X + iY)(A + iB) = I (2.75)
or,
X A - Y B = /
X B + Y A = 0
(2.76)
(2.77)
If A 1 is not nearly singular, we have
-iK = B A
X = (A + K B )~ l
Y = - X K
(2.78)
(2.79)
(2.80)
34

otherwise,
K ~ l = AB~l (2.81)
Y = ~{B + K - lA)~l (2.82)
X = - Y K ~ X (2.83)
Photodetachment Cross Section
Photodetachment is a special type of photoionization where the extra electron is
detached from an negatively charged atomic system. The theory for photoionization
can be applied to photodetachment without change.
Considering that light produces transitions of an negative atomic ion to states of
its continuous spectrum, the photodetachment cross section for such a transition is
defined by[74]
a = P / I (2.84)
where P is the transition probability per unit time and I is is the quantum-mechanical
value of the incident light intensity expressed in photons per unit area per unit time.
Eq. (2.84) can be expressed in terms of transition operator, T, as
<r = 47r2a £ |< | T j tfo > |2 (2.85)
where a is the photodetachment cross section defined in atomic units (a.u.), a is
the fine-structure constant, E is the photon energy, and are the initial and
final state wave functions, respectively, represented in terms of the multi-configuration
and/or multi-channel states under L S coupling. By using the Wigner-Eckert theorem,
summing over the magnetic quantum numbers for the final state, and averaging over
35

those of the initial state, wecanget the cross section for eachfinal state in terms of
the reduced matrix elementsof the transition operator T£
4tt2
^ = 2 X ^ i a E (< II T II ^o(LoS) > |2 /(2L0 + 1) (2.86)
with A = 1 for a dipole transition. The dipole transition operator in length form and
velocity form are ( in a.u.)
n =
j
Ti, = £ § (2.87)
Under the non-relativistic limit, the total cross section is simply the sum of the
cross sections for all final states.
When a resonance appears, its position Er and width T can be determined from
the cross section. For a Feshbach resonance, these are determined by fitting the total
photodetachment cross section <r(E) by the many-channel Fano-Cooper formula [75]
<j(E) = a0[l + (E — Er)a]
with
(2.88)
e = 2{E —Er) / r, (2.89)
where a linear background cr0[l + (E — Er)a is assumed in theresonance region. <t0
is the background cross section at Er and a is a parameter.
For a shape resonance, the cross section can be well represented by the product
of the Wigner threshold law and the Breit-Wigner resonance formula [76, 77]
36

where I is the orbital angular momentum of the photoelectron associated with the
open channel which produces the shape resonance, is the partial cross section via
this channel, Eq is the corresponding threshold (target) energy. Shape resonances lie
energetically just above the threshold of the target state and occur when an electron
is temporarily trapped in a potential well arising from a combination of the repulsive
centrifugal force and the attractive short-range force due to the polarization of the
target state. If the cross sections via all other no-resonance channels can be regarded
as linear in the resonance region, then the resonance position and width can be
obtained by fitting the total cross section
(e - E0y +i/a '
<rtot(E) = o"o[l + a(E - £ r)] 1 + 6
(E - Ery + (r/2)2 (2.91)
Angular Distribution of Photoelectrons
In contrast to the total cross section, differential cross sections provide informa
tion about the angular distribution of the ejected particles, which provides data not
only on the amplitudes but also on their relative phases. At low energy (hv < 100
eV) photodetachment processes where the electric dipole approximation is valid, the
differential cross sections depend on only two dynamical parameters: the total cross
section a, which determines the intensity of the photoelectrons, and the asymmetry
parameter /3[78, 79], which determines the angular distribution of the photoelectrons.
In general, a and ft are energy dependent due to their dependence on the energy-
dependent transition amplitudes. Unlike a, however, /? is a dimensionless number
and is independent of energy when the scattering process under study has only a
single allowed final-state channel due to geometrical considerations. Because of this,
measurement of the deviation of /3 from a constant value for the dominant channel
37

provides a sensitive measure of the strength of the additional allowed channels.
The angular distribution asymmetry parameter of the photoelectron is defined
via the differential cross section as[78, 79]
^ = ^ [ 1 + /3 P 2(cos»)], (2.92)
where 6 is the angle between the photon polarization and the direction of the photo
electron and P2(cos0) represents the second order Legendre polynomial.
From Eq. (2.92) we can see that the angular distribution is determined completely
by the asymmetry parameter /3, which embodies all of the dynamical information
relevant to the angular distribution. The total cross section a determines the overall
intensity of the process. We also see that the requirement da/d£l be positive for all
values of 0 limits the magnitude of /3 to the range —1 < (3 < 2.
W ith the work of Fano and Dill[80, 81], the asymmetry parameter /3 can be rep
resented in terms of the following weighted average of the partial cross section <r{jt)
of all the angular momentum transfers j t:
0 = E j (2.93)
2Zjt a Jt)
where /3(jt) is the asymmetry parameter for angular momentum transfer j t, which is
defined by
jt = j-r ~ I = Lc —Lq, (2.94)
where LQ,L C and I are the orbital angular momentum of the initial state, the final
state target ( or core ) and the photoelectron, respectively. j y = 1 is the angular
momentum of the incident photon.
From Eq. (2.94) we have,
m a x (| L0 - Lc , I — 1 ) < j t < min(L0 + Lc,l + 1), (2.95)
38

The summation in Eq. (2.93) extends over all allowed values of j t. cr(jt) and
8{jt) are the partial cross section and asymmetry parameter for a given value of
jt■According to Fano and Dill[80, 81], the angular momentum transfer is named as
“parity favored” if 7r07rc = (—l)Jt and “parity unfavored” if = (—l)Jt+l, where 7r0
and 7rc are the parity of the initial state and target state, respectively. In the electric
dipole approximation, we have
7r<,7rc = (~ l),+I. (2.96)
With Eq. (2.96) we get
{ / ± 1, parity favored
(2.97)
I, parity unfavored
The cr(jt) and f3(jt) for each j t are determined from the reduced dipole matrix elements
as [80, 81, 82]
o /. (it + 1)l5'i(+i(it)l2 + (it - (it)I2 - 3 yjjttit + l )[s jt+Ut)Sft_y{jt) + c.c.]
Iav[Jt) ~ (2jt + l)(Sjt+l(jt)2 + Sjt. d j t ) 2)
» /« (* ) = p 2a E ^ ± { S i,+l(jt)(2 + S i, - l(j,)2] (2.98)
for parity favored j t, and
fiunfav(jt) = 1
&unfav(jt) = —7T2C*f?2£o _j_ ^I^Jt(it)|2 (2.99)
for parity unfavored j t.
Si(jt) is a reduced matrix defined as
SiUt) = e-ito-l* W ^ y /2 L + l ^ 1 L [ ( t f B||Tl ||tf0) (2.100)
1 Lq jt
39

where Si is the Coulomb phase shift of the radial channel wave function, L is the
angular momentum of the final state. In the present case of He- , the effective nuclear
charge acting on the photoelectron is zero, which results in Si = 0.
Program Packages
In this section, we give a brief description of the major program packages used in
our investigation of photodetachments of negative ions. Detailed description of these
programs can be found elsewhere[63].
The research group of Froese Fischer designed two program packages named
CHMAT and INVPHOTO[83]. These packages were used successfully in the pho
toionization calculation of two electron systems. There are limitations, however, in
these program packages: First, they can only handle two-electron systems, second,
they can only handle single configuration target state, and third, they can only handle
the case with one open channel.
Obviously, these packages do not meet our new requirement for photodetachment
of many-electron negative ions. So one of the primary programming tasks was to
modify these two program packages to handle multi-configuration target states and
multi-open channels. Extension is needed for the following aspects:
- For two electron systems, the target states are single configuration states. But
for many electron systems, to achieve acceptable accuracy, in general, we need
to build multi-configuration target states which are orthogonal to each other.
So we need to extend the CHMAT package to accept multi-configuration target
states.
40

- We need to deal with continuum states with multi-open channels instead of a
single open channel. We need to find a correct and efficient way to connect the
calculated continuum wave functions with the asymptotic physical solutions
and get the conversion matrix( a scalar quantity in single open channel case).
The old INVPHOTO package needs to be replaced by a new one.
- We need to extend the physics application package in INVPHOTO to deal with
the multi-open channel case and add more functionality such as the calculation
of the angular distribution which are a very important aspect in continuum
state problems.
We extended these program packages with the new functionality. Detailed description
of the design process and the usage of these packages and the supporting libraries
is available[63]. These packages have been proven to be a very powerful tool in the
photodetachment calculation of negative ions.
41

CHAPTER III
PHOTODETACHMENT OF THE He" (ls2s2p4P°) SYSTEM
Introduction
It is well known that a He atom in its ground state cannot bind a third electron
to form a stable He- ion. A He" ion can exist only in a core-excited state which is
in the autodetachment continuum of the He atom. There are two groups of these
autodetaching states: states which interact with the continuum via an Coulomb elec
trostatic potential and states which interact with the continuum only via the weak
spin-orbital and spin-spin magnetic interactions. Autodetaching states of the first
group are short-lived and appear as resonances in the electron-scattering cross sec
tion. States in the second group are raetastable since the transition is spin forbidden
and autodetachment can only proceed at a lower rate.
Since the discovery of the metastable (Is2s2p 4P°) He- negative ion by Dopel[84]
and Hibby[27], The weakly bound He- negative ion has attracted considerable interest
in recent years.
The study of the He- electronic structure began with the study by Wu[85] who
suggested that the He- ion had a quartet state Is2s2p 4P°. Subsequently, Holoien
and Midtdal[86] showed theoretically that the He~(ls2s2p 4P°) state lies below its
parent He( 3S ) state. Since the autodetachment from the quartet He- ( 4P°) can only
occur through weak spin interactions, the lifetime of this ion is fairly large (of the
order of 10-5 s).
The binding energies of the 2s and 2p electrons in the He- ion were determined
42

by Brehm et al [87] to be 1.22 eV and 0.077 eV, respectively. The theoretical electron
affinity value calculated by Bunge and Bunge[88] was 77.51±0.04 meV. The accuracy
of this value was improved by a recent experimental and theoretical study of Kris-
tensen et al [89]. The reported theoretical value was 77.518±0.011 meV, in agreement
with their experimental result of 77.516±0.006 meV.
The investigation of the He- resonances has also been of great interest, both
experimentally and theoretically. Because of the weak coupling among electrons, the
oretical calculations cannot predict the resonance behavior correctly if correlation is
not properly included. The s2s2p4P° metastable state, with an extra electron bound
to the ls2s 3S state of He, can be studied via the photodetachment process. From
this study the quartet 4S , 4P , and AD excited states of He- can be investigated.
The first measurements of the He- ( 4P°) photodetachment cross section were
made by Compton, Alton, and Pegg [90] at selected photon energies in the ranges
1.77 - 2.75 eV, and Hodges, Coggiola, and Peterson [91], at energies in the range 0.12
- 4.0 eV. Though these experiments are incomplete, both experiments did indicate
that some resonance structure may exist in the He~ spectrum at an energy around
2.5-2.7 eV. The first theoretical calculation of the photodetachment cross section was
performed by Hazi and Reed [92] with energy from threshold up to 3 eV, using an
extensive configuration interaction (Cl) wave functions approach. This calculation
shows a large ls2p24P resonance immediately above the He(ls2p 3P°) threshold.
This resonance was later examined by a series of experiments [93, 76, 77] and theo
retical calculations[94, 95, 96]. Other experiments provided data on the partial cross
sections of the photodetachment and the angular distributions of the photoelectrons
[97, 98]. Though the position and width of this resonance have been determined to
43

a relatively high accuracy, the value of the maximum cross section at the resonance
is , however, still somewhat uncertain. The experimental value varies from 2400 Mb
to 8000 Mb. The recent experimental result by Walter, Seifert and Peterson[77] is
[5.8 ± 2.0] x 103 Mb.
Except for the ls2p2 4P shape resonance, a very weak ls3p2 4P Feshbach reso
nance was suggested by Hazi and Reed [92], with photon energy ranging between 2.4
and 3.3 eV. But this resonance did not appear in any other theoretical calculations
and experimental measurements until 1995, when Zhou, Robicheaux and Manson [99]
reported their calculation for a ls3p2 4P resonance at 3.06 eV. Until our recent report
on the photodetachment study of He- , no other resonance structure was reported and
theoretical and experimental works focused only on the low energy photodetachment
to the He(n = 2) (n is the principal quantum number for the outer electron) threshold.
In our investigation on the photodetachment study of He- [100], we calculated the
cross section and angular distribution of the Is2s2p 4P° state with energy from thresh
old to 4 eV, which covers the whole energy region up to the He(n = 4) threshold. We
employed an approach that uses a spline basis and multi-configuration Hartree-Fock
(MCHF) orbitals to calculate the interaction matrix and the wave functions of the
system. The l^ p 2 4P shape resonance was investigated in detail and excellent agree
ment with the experimental data was obtained. We also reported for the first time
other Feshbach resonances in these energy region, including the 1sZp* 4P resonance at
3.075 eV, the Is3p4p 4P resonance at 3.265 eV, the ls4p24P resonance at 3.811 eV,
the Is4p5p 4P resonance at 3.948 eV, and the Is3s4s 4S resonance at 2.959 eV. We
determined the width of these resonances to be 37.37,1.30, 38.19, 6.05, and 0.19 meV,
respectively. Two Cooper minima were also found in the ls2p( 3P°)kp 4P channel at
44

photon energies of 2.83 and 3.25 eV. This new result prompted the recent experiment
measurement of Klinkmuller et al [101, 102] and Kiyan et al [103], and the interest in
the photodetachment of He~ at high energy region. Klinkmuller et al measured the
photodetachment cross sections of He- using a collinear laser-ion beam apparatus in
the energy range 2.9-3.3 eV and successfully located the ls3p24P , Is3p4p 4P , and
Is3s4s 4S resonances at photon energies of 3.072, 3.264, and 2.959 eV, respectively,
in excellent agreement with our calculation. The width of the Is3s4s 4S resonance,
0.19(3) meV, also agrees with our calculation. The width of other resonances, how
ever, deviate from our predictions. New theoretical investigations have been reported,
trying to resolve these discrepancies and exploit other possible resonance structures at
higher energy regions. Bylicki[104] studied the 4P spectrum of the He- using complex
coordinate rotation method within an extensive basis sets of -correlated configura
tions. Chung[105] studied the effects of channel coupling and exchange interaction on
the formation of resonances in the He- system. Brandefelt and Lindroth[107] studied
the 4S resonances using a complex rotation with B-spline basis method. They found
four resonances with 4S symmetry below the H e(ls5s 3S ) threshold. Among them,
the position and width of the Is3s4s 4S resonance are in excellent agreement with
our prediction. Using R-matrix method, Ramsbottom and Bell[106] calculated the
cross sections with photon energy up to the He(n = 5) threshold and made a detailed
comparison with our results. Recently, Liu and Starace[108] made a comprehensive
study for the photodetachment of He~(ls2s2p4P°) within the energy region from the
He(n = 3) threshold to He(n = 5) threshold, using a eigenchannel R-matrix approach.
We also studied the cross section of the Is photodetachment from the He- (Is2s2p
45

4P°) state[109], with photon energy from threshold up to 44 eV. In the recent pub
lication of Kim, Zhou, and Manson[110], the photodetachment from the inner-Is
electron to certain selected channels was studied using the R-matrix method with
MCHF orbitals, where the energy covered the whole range from threshold to 100 eV.
In their paper, the large correlations are considered but fine correlation effects are
ignored, so small resonance structures near threshold did not appear. In our calcula
tion described later in this chapter, however, we will provide a more complete study
of the resonance structure and photodetachment property in the threshold region of
the Is detachment. We also report the result of the 2S2P2 4P Feshbach resonance.
The 2s2p24P state was first predicted by Chung[lll] using the saddle-point varia
tion method. Later the resonance position and width were investigated by Bylicld
and Nicolaides[112, 113], Chung[114], and Kim, Zhou, and Manson[110]. Recently,
Morishita and Lin[115] analyzed the resonance states of the He- system using the
hyperspherical adiabatic potential curves.
In the following sections, we will discuss in detail the photodetachment cross
sections and angular distributions from the He~(ls2s2p 4P°) state.
Photodetachment Below the He(n = 4) Threshold
The Bound State Wave Functions and Energies
The accuracy of the bound orbitals plays a fundamental rule in the accuracy of the
current calculation. Unlike the channel orbitals which are determined dynamically
via the channel coupling, the bound orbitals are fixed throughout the process in
determining the final state wave function. Because of this, we need to generate these
46

Table 3.1: Energies ( Etarget ) of the triplet states of He obtained from the present
MCHF calculation are compared with accurate theoretical or experimental values
(Eacc)-
state Etarget (a.U.) E ^ a . u . Y Error (meV)6
ls2s -2.1752238 -2.1752294 0.15
ls3s -2.0686873 -2.0686891 0.05
ls4s -2.0365108 -2.0365121 0.04
ls5s -2.0226185 -2.0226189 0.01
ls2p -2.1331574 -2.1331724 0.41
ls3p -2.0580500 -2.0580835 0.91
ls4p -2.0323025 -2.0323252 0.62
ls5p -2.0205496 -2.0205516 0.05
ls3d -2.0556362 -2.0556365 0.01
ls4d -2.0312887 -2.0312889 0.01
Isod -2.0200209 -2.0200211 0.01
ls 4 f -2.0312543 -2.0312535 -0.02
ls o f -2.0200026 -2.0200015 -0.03
a Energy values for 3S states are from Ref. [116], others are from Ref.[117]. 1 a.u. =
219444.12 cm -1 is used in converting the values from cm-1 to a.u. For consistency,
the energy of the ground state of He( 1S’), -2.9036697 a.u., is also taken from Ref.
[117], though a much more accurate value is provided in Ref. [116].
6 1 a.u. = 27.2076134 eV is used in the conversion.
orbitals with highest possible accuracy. Since we use the same orbital set to generate
the initial state wave function, the final perturber state wave function, and the target
state wave function, we need a special optimization algorithm to produce the best
average result.
In the following , we generate a set of MCHF orbitals using the MCHF atomic
structure program package [118]. The orbitals obtained span the orbital space for the
initial state, final state targets, and bound perturbers. The final state is represented as
a set of bound pseudo states, plus the channel states, which are formed by coupling
the channel orbitals to appropriate target states. In order to represent the wave
function correctly, it is crucial to have the targets as accurate as possible. As the
47

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