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Hindawi Publishing CorporationAdvances in Physical ChemistryVolume 2011, Article ID 541375, 14 pagesdoi:10.1155/2011/541375
Research Article
Invariant Graphical Method for Electron-Atom ScatteringCoupled-Channel Equations
J. B. Wang1 and A. T. Stelbovics2
1 School of Physics, The University of Western Australia, WA 6009, Australia2 Institute of Theoretical Physics, Curtin University of Technology, WA 6102, Australia
Correspondence should be addressed to J. B. Wang, [email protected]
Received 9 September 2010; Revised 28 December 2010; Accepted 13 January 2011
Academic Editor: Wybren Buma
Copyright © 2011 J. B. Wang and A. T. Stelbovics. This is an open access article distributed under the Creative CommonsAttribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work isproperly cited.
We present application examples of a graphical method for the efficient construction of potential matrix elements in quantumphysics or quantum chemistry. The simplicity and power of this method are illustrated through several examples. In particular, acomplete set of potential matrix elements for electron-lithium scattering are derived for the first time using this method, whichremoves the frozen core approximation adopted by previous studies. This method can be readily adapted to study other many-bodyquantum systems.
1. Introduction
The fundamental problem encountered by a theoreticianworking in the field of quantum physics or quantumchemistry is to evaluate the so-called observables 〈ψ|Ω|ψ〉 tocertain precision. Here, ψ is the wave function of the systemunder investigation, and Ω is the operator representing thephysical quantity one needs to evaluate. This seeminglyeasy task encounters many obstacles, one of which isthe complexity of the physical structure of a compositesystem. For example, there are several electrons and nucleiinteracting with each other in an atomic or molecular system.Even though the exact form of such interactions is wellknown, it is exceedingly difficult to obtain accurate solutionsfrom the corresponding Schrodinger or Dirac equation.
However, most physical systems have symmetry proper-ties that influence their dynamics and structures in a wayindependent of the detailed interactions. A good exampleis the motion of a particle in a central field. The symmetryof the interaction leads to the conservation of angularmomentum that is independent of the detailed form ofthe interaction potential V(r). Angular momentum theorydeals with rotational symmetries of a composite system.In classical physics, angular momentum treatments arestraightforward following the simple manipulation rules of
vectors. In the quantum world, not only the magnitudeof angular momenta is quantized, but also their spatialorientations.
Angular momentum calculations in quantum theoryprove to be most laborious, especially when many-bodysystems or complicated operators are involved. Furthermore,the final expressions are often incomprehensible even forexperienced researchers in the field. Unless one derives theequations himself by a painstakingly tedious procedure, onecannot really tell where the phases and the n- j symbols comefrom. It is also quickly apparent, once one begins to derivethe complete reduced potential matrix-elements for systemsbeyond four electrons, that the expressions become rapidlyintractable to algebraic derivation by conventional methods.The problems are simply due to the vast number of Clebsch-Gordan coefficients that comprise the expressions, each ofwhich is summed over several angular momentum labels. Inaddition, there are many pages of practical formulas that oneoften needs in order to perform these calculations, such asthose listed in the appendixes of Brink and Satchler [1].
In view of these difficulties, we started to search for amethod that is transparent, free of the incomprehensiblephases, makes minimal use of formulas, and leads easilyto a correct final expression. We will demonstrate in thispaper that the invariant graphical method initiated by Danos
2 Advances in Physical Chemistry
i i
b
c
d
f h
ea
g
(a)
i ib
c
d
f h
(b)
Figure 1: (a) Basic recoupling transformation graph; (b) recou-pling of three tensors.
and Fano [2] meets all the above criteria. We will thenapply this method to derive the reduced matrix elementsof selected electron-atom interaction potentials, which arenot available mainly due to the difficulties mentioned above.This method can be readily adapted to study other many-body systems such as two-photon double ionization of atoms[3, 4], two-nucleon knockout reactions [5], spin networks[6], and entangled N spin-1/2 qubits [7, 8].
A central point to this method is the concept that mostphysical quantities are invariant, that is, they are independentof the coordinate system one chooses for their description.This is the case for energies, cross-sections, and transitionprobabilities. In other words, these quantities are scalarquantities and have zero angular momentum. Consequently,the tensors they may contain would couple to a total ofzero angular momentum. We will demonstrate, in the nextsection, that angular momentum calculations become muchsimpler once we take this invariance into account. Atomicunits are used throughout this paper.
2. Elements of the Invariant Graphical Method
In the invariant graphical method, there is only one basicgraph (Figure 1(a)) for angular momentum coupling andrecoupling. The essential elements are the horizontal lines,each representing a tensor with a given angular momentumlabeled above the line. Here, tensors are used to represent allquantities including wave functions, operators, amplitudes,and so forth. In this way, a scalar quantity is a tensor ofrank zero. A vector is a tensor of rank one and so on.As an example, the state with angular momentum L canbe represented by a tensor Ψ of the rank L. This graphprovides the basic transformation associated with fourangular momenta to rearrange them into a different couplingscheme. The transformation is represented by a recouplingbox, mathematically associated with a square 9- j symbolsuch that
[[A[a]B[b]
][e][C[c]D[d]
][ f ]][i]
=∑gh
⎡⎢⎢⎢⎣a b e
c d f
g h i
⎤⎥⎥⎥⎦[[A[a]C[c]
][g][B[b]D[d]
][h]][i]
,
(1)
where [A[J]B[K]][I]M = ∑
mJmK(JKmJmK |IM)A[J]
mJB[K]mK is rep-
resented by K
JI
. The square 9- j symbol is related to theWigner 9- j coefficients by
⎢⎢⎢⎢⎣a b ec d fg h i
⎥⎥⎥⎥⎦ = e f g h
⎧⎪⎨⎪⎩a b ec d fg h i
⎫⎪⎬⎪⎭, (2)
where a = √2a + 1.This basic graph is the key building block of this method.
All possible recoupling transformation can be performedbased upon this graph. For example, the recoupling of threeangular momenta can be derived using the graph illustratedin Figure 1(b) as
[B[b][C[c]D[d]
][ f ]][i]
=∑h
⎢⎢⎢⎢⎣0 b bc d fc h i
⎥⎥⎥⎥⎦[C[c][B[b]D[d]
][h]][i]
.
(3)
In this method, there is no need to use 6- j symbols, whichcarry extra phases and normalization constants that can leadto tedious book-keeping and are prone to errors. There isalso no need to remember the many sum rules, which arerequired to eliminate unnecessary summation indexes inother conventional methods.
3. Applications
Coupled-channel equations have been used extensively in thedescription of electron-atom scattering. Their applicationhas been particularly successful in the so-called convergentclose-coupling approach (CCC) implemented by Bray andStelbovics [9] and subsequently extended to other atomswith one or two valence electrons outside an inert core[10, 11, 13]. To perform such calculations, one requires thereduced matrix elements of the coupled-channel potentials,which we denote by 〈L′‖V‖L〉 with L representing thecomplete set of quantum numbers specifying a particularconfiguration of the system under study. For electronscattering from an N-electron target, the coupled-channelreduced matrix elements have the following structure
⟨L′‖V‖L⟩
=⟨L′∥∥∥∥∥−
N
r0+
N∑i=1
1r0i
−
⎛⎜⎜⎜⎝−
N∑i=0
N
ri+
N∑i, j=0i> j
1ri j− E
⎞⎟⎟⎟⎠
N∑i=1
P0i
∥∥∥∥∥L⟩
,
(4)
which includes both the direct and exchange terms. Here, Pis the permutation operator, E is the system energy, subscript0 denotes the incoming electron, and subscripts 1-N denotethe target electrons. Using the explicitly antisymmetrized
Advances in Physical Chemistry 3
Y[ℓa]Y[ℓb][ℓ]
Y[λ]Y[λ][0
]Y[ℓa]Y[ℓb][ℓ]
λ λ
λℓbℓb
ℓa
ℓa
ℓbℓb
ℓa
ℓaλ
l
l
l
22
2
λ λ 0λ λ 00 0 0
ℓa ℓb ℓℓa ℓb ℓλ λ 0
ℓ
ℓl/ℓ
[λℓaℓa]
[λℓbℓb]
Figure 2: Recoupling graph for (7).
0 1/2 1/21/2 1/2 s1/2 s′ S
1/21/2
1/2
1/21/2
1/2
1/21/2
1/2
1/21/2
1/2s′s
S S
⟨σ ′0(σ ′1σ ′2)
σ0(σ1σ2)⟩ ⟨
σ ′0(σ ′1σ ′2)σ1(σ0σ2)
⟩
1/S1/S
Figure 3: Recoupling graph of the spin tensors for (11)–(15).
λλ
λ
λ
λ λ 0λ λ 00 0 0
ℓ0 ℓ Lℓ0 ℓ Lλ λ 0
[λℓ0ℓ0
]
[λℓ1ℓ1
]L
L
ℓ
ℓ1ℓ1
ℓ2
ℓ1
ℓ1
ℓ2
ℓ2δℓ2ℓ2
ℓ0
ℓ0
ℓ0
ℓ0
ℓ
1/L
Y[ℓ0]
(�r0)Y[ℓ1]
(�r1)Y[ℓ2]
(�r2)[ℓ][L]
Y[λ]
(�r0)Y[λ]
(�r1)[0
]Y[ℓ0
](�r0)
Y[ℓ1
](�r1)Y
[ℓ2
](�r2)
[ℓ][L]
ℓ1 ℓ2 ℓℓ1 ℓ2 ℓλ 0 λ
Figure 4: Recoupling graph of the orbit tensors for (11).
4 Advances in Physical Chemistry
λλ
λ
λ
λ λ 0λ λ 00 0 0
ℓ0 ℓ Lℓ1 q Lλ λ 0
0 ℓ0 ℓ0
ℓ1 ℓ2 ℓℓ1 q L
[λℓ0ℓ1
]
[λℓ1ℓ0
]L
L
ℓ
ℓ1
ℓ1
ℓ2
ℓ1
ℓ1
ℓ2
ℓ2δℓ2ℓ2
ℓ0ℓ0
ℓ0
ℓ0
q
ℓ
1/L
Y[ℓ0]
(�r0)Y[ℓ1]
(�r1)Y[ℓ2]
(�r2)[ℓ][L]
Y[λ]
(�r0)Y[λ]
(�r1)[0
]Y[ℓ0
](�r1)
Y[ℓ1
](�r0)Y
[ℓ2
](�r2)
[ℓ][L]
ℓ1 ℓ2 ℓℓ0 ℓ2 qλ 0 λ
Figure 5: Recoupling graph of the orbit tensors for (12).
target states, the reduced matrix elements can be simplifiedto⟨L′‖V‖L⟩
=⟨L′∥∥∥∥N(− 1r0
+1r01
)
−(
1r01
+N − 1r02
+N − 1r12
+(N − 1)(N − 2)
2r23−N
(E +
N
r0
))P01
∥∥∥∥L$.
(5)
Each of these terms will be derived explicitly for e-He ande-Li scattering in Sections 3.2 and 3.3.
By employing the invariant graphical method, we willderive a complete set of direct and exchange potential matrixelements for electron-atom scattering, which are beyond thefrozen core approximation adopted by Wu et al. [12], Fursaand Bray [13], Bray et al. [10, 16], and Zhang et al. [14].
3.1. Two-Electron System. To illustrate the use of the invari-ant graphical method, we first consider the direct Coulombmatrix elements for a two-electron system, that is,〈(�′a�′b)�‖1/r12‖(�a�b)�〉, which was also discussed in detail inDanos and Fano [2]. The complete coupling and recouplinggraph is shown in Figure 2. As described earlier, each linein the figure represents a tensor with a given angularmomentum indicated by the label above the line. The joiningof two lines into one line represents coupling of their angularmomenta. Recoupling into a different tensor set is indicatedby a transformation box and needs to be carried out untileach component of the graph is expressible in terms of single-particle matrix elements (e.g., the end boxes [λ|�′a|�a] and
[λ|�′b|�b] in this case). The invariance of such a triple productprovides a means of eliminating unnecessary intermediateindexes. For example, it demands that the coupling of �′aand �a must give rise to a tensor with angular momentumλ instead of an arbitrary index. It also helps in identifyingthe most direct and economical recoupling scheme. We firstexpand the two-body Coulomb potential into its multipolecomponents
1r12
=∑λ
Rλ(r1r2)[Y [λ]Y [λ]
][0], (6)
where Rλ(r1r2) = (4π/λ2)(rλ</rλ+1> ). The corresponding final
expression can then be directly read off from Figure 2 as
⟨(�′a�
′b
)�∥∥∥∥ 1r12
∥∥∥∥(�a�b)�$
=∑λ
Rλ(r1r2)i�′a+�′b−�a−�b
×⟨[Y [�′a]Y [�′b]
][�]∥∥∥∥[Y [λ]Y [λ]
][0]∥∥∥∥[Y [�a]Y [�b]
][�]$
=∑λ
Rλ(r1r2)i�′a+�′b−�a−�b 1
�
×
⎡⎢⎢⎢⎣�′a �′b �
�a �b �
λ λ 0
⎤⎥⎥⎥⎦
⎡⎢⎢⎢⎣λ λ 0
λ λ 0
0 0 0
⎤⎥⎥⎥⎦[λ∣∣�′a∣∣�a]
[λ∣∣∣�′b∣∣∣�b]
,
(7)
where Y [L] = (−i)LYL and YL is the regular sphericalharmonic tensor. This definition is introduced to avoid
Advances in Physical Chemistry 5
λλ
λ
λ
λ λ 0λ λ 00 0 0
ℓ0 ℓ Lℓ1 q Lλ λ 0
0 ℓ0 ℓ0
ℓ1 ℓ2 ℓℓ1 q L
[λℓ0ℓ1
]
[λℓ2ℓ2
]
L
L
qℓ
ℓ1
ℓ1
ℓ2 ℓ2
ℓ1
ℓ2
ℓ2
ℓ0δℓ1ℓ0
ℓ0ℓ0
ℓ0
ℓ
1/L
Y[ℓ0]
(�r0)Y[ℓ1]
(�r1)Y[ℓ2]
(�r2)[ℓ][L]
Y[λ]
(�r0)Y[λ]
(�r2)[0
]Y[ℓ0
](�r1)
Y[ℓ1
](�r0)Y
[ℓ2
](�r2)
[ℓ][L]
ℓ1 ℓ2 ℓℓ0 ℓ2 qλ 0 λ
Figure 6: Recoupling graph of the orbit tensors for (13).
1/L
λλ
λ
λ
λ λ 0λ λ 00 0 0
[λℓ1ℓ0
]
[λℓ2ℓ2
]L
L
ℓ
ℓ1
ℓ2 ℓ2
ℓ1
ℓ1
ℓ2
ℓ2
ℓ1δℓ0ℓ1
ℓ0
ℓ0
ℓ0
ℓ
ℓ
ℓ0 ℓ Lℓ1 ℓ L0 0 0
0 ℓ0 ℓ0
ℓ1 ℓ2 ℓℓ1 ℓ L
Y[ℓ0]
(�r0)Y[ℓ1]
(�r1)Y[ℓ2]
(�r2)[ℓ][L]
Y[λ]
(�r1)Y[λ]
(�r2)[0
]Y[ℓ0
](�r1)
Y[ℓ1
](�r0)Y
[ℓ2
](�r2)
[ℓ][L]
ℓ1 ℓ2 ℓℓ0 ℓ2 ℓλ λ 0
Figure 7: Recoupling graph of the orbit tensors for (14).
unnecessary phases entering the coupling and recouplingscheme.
Each square 9- j symbol in (7) corresponds to a recou-pling box in Figure 2. The end-box represents an invariantmatrix element containing three spherical harmonic tensorswritten as [�|k| j], which equals
[�|k| j] = i�+k+ j �k j√
4π
⎛⎝� k j
0 0 0
⎞⎠. (8)
For numerical computation purposes, (7) can be treated asa final expression. Nevertheless, further simplification can be
made considering the many zeros in the 9- j symbols. Thisleads to the well-known result (see, e.g, [15])
⟨(�′a�
′b
)�∥∥∥∥ 1r12
∥∥∥∥(�a�b)�$
=∑λ
Rλ(r1r2)(−1)λ+� �′a�′b�a�b
×⎛⎝�′a λ �a
0 0 0
⎞⎠⎛⎝�
′b λ �b
0 0 0
⎞⎠{�′a �′b ��b �a λ
}.
(9)
6 Advances in Physical Chemistry
0 ℓ0 ℓ0
ℓ1 ℓ2 ℓℓ1 ℓ L
L
L
Liℓ0+ℓ1+ℓ2−ℓ0−ℓ1−ℓ2
ℓ
ℓ1
ℓ2
ℓ1
ℓ2
ℓ0
ℓ0
ℓ
1/L
Y[ℓ0]
(�r0)Y[ℓ1]
(�r1)Y[ℓ2]
(�r2)[ℓ][L]
Y[ℓ0
](�r1)
Y[ℓ1
](�r0)Y
[ℓ2
](�r2)
[ℓ][L]
Figure 8: Recoupling graph of the orbit tensors for (15).
0 1/2 1/21/2 p s1/2 s′ S
1/21/21/2
1/2
1/21/21/2
1/2
1/21/21/2
1/2
1/21/21/2
1/2s′s
p
S S
⟨σ ′0(σ ′1σ ′2σ ′3)
σ0(σ1σ2σ3)⟩ ⟨
σ ′0(σ ′1σ ′2σ ′3)σ1(σ0σ2σ3)
⟩
1/S1/S
Figure 9: Recoupling graph of the spin tensors for (17)–(22).
3.2. Electron-He Scattering (A Three-Electron System). As asecond more complicated example, we will derive themomentum-space direct and exchange matrix elements fora three-electron system, such as electron scattering from ahelium atom. The general configuration of the system canbe represented by |L〉 ≡ |k(�0(�1�2)�)L; (σ0(σ1σ2)s)S〉, wherek, �0, and σ0 are, respectively, the linear, orbital angular andspin momentum of the incoming electron, �1 and �2 are theangular momenta of the target electrons, σ1 and σ2 are thespin momenta of the target electrons, � and s are the totalorbit angular and spin momentum of the target atom, and Land S are the total orbit angular and spin momentum of theentire system. Note that the linear radial integral part of the
direct and exchange elements is straightforward to work outand has the general form of⟨n′2�
′2,n2�2
⟩⟨k′�′0, k�0;n′1�
′1,n1�1
⟩
≡⟨ϕn′2�′2 | ϕn2�2
⟩
×∫u�′0 (k′r0)u�0 (kr0)ϕn′�′1 (r1)ϕn�1 (r1)
rλ<rλ+1>
dr0dr1,
(10)
where ϕn� are the target atomic orbitals with quantum num-bers n and �, and u� is the �th partial wave of the projectileelectron. The coupling and recoupling graph for the direct
Advances in Physical Chemistry 7
λ
λ
λ
λ
λ λ 0λ 0 λ0 λ λ
ℓ0 ℓ Lℓ0 ℓ Lλ λ 0
ℓ1 ℓ23 ℓℓ1 ℓ23 ℓλ 0 λ
ℓ2 ℓ3 ℓ23ℓ2 ℓ3 ℓ23
0 0 0
[λℓ0ℓ0
]
[λℓ1ℓ1
]
L
L
ℓℓ1
ℓ1
ℓ2
ℓ1
ℓ1
ℓ2
ℓ2δℓ2ℓ2
ℓ3δℓ3ℓ3
ℓ0
ℓ0
ℓ0
ℓ0
ℓ3
ℓ3
ℓ23
ℓ23
ℓ
1/L
Y[ℓ0]
(�r0)Y[ℓ1]
(�r1)Y[ℓ2]
(�r2)Y[ℓ3]
(�r3)[ℓ23
][ℓ][L]Y[λ]
(�r0)Y[λ]
(�r1)[0
]
×Y[ℓ0
](�r0)
Y[ℓ1
](�r1)
Y[ℓ2
](�r2)Y
[ℓ3
](�r3)
[ℓ23
][ℓ][L]
Figure 10: Recoupling graph of the orbit tensors for (17).
and exchange potential spin part is shown in Figure 3 andthose for the orbital part are shown in Figures 4, 5, 6, 7, and8. The corresponding final expressions can then be directlyread off from the figures as
⟨k′(�′0(�′1�
′2
)�′)L;(σ ′0(σ ′1σ
′2
)s′)S
×∥∥∥∥ 1r01
∥∥∥∥k(�0(�1�2)�)L; (σ0(σ1σ2)s)S$
=∑λ
⟨n′2�
′2,n2�2
⟩⟨k′�′0, k�0;n′1�
′1,n1�1
⟩
× ⟨σ ′0(σ ′1σ ′2) | σ0(σ1σ2)⟩i�′0+�′1+�′2−�0−�1−�2
×⟨[
Y [�′0](�r0)[Y [�′1](�r1)Y [�′2](�r2)
][�′]][L]
×∥∥∥∥[Y [λ](�r0)Y [λ](�r1)
][0]∥∥∥∥
×[Y [�0](�r0)
[Y [�1](�r1)Y [�2](�r2)
][�]][L]⟩
=∑λ
⟨n′2�
′2,n2�2
⟩⟨k′�′0, k�0;n′1�
′1,n1�1
⟩
× i�′0+�′1+�′2−�0−�1−�2�2
Lδ�′2�2
[λ∣∣�′0∣∣�0][λ∣∣�′1∣∣�1]
×
⎡⎢⎢⎢⎣�′0 �′ L
�0 � L
λ λ 0
⎤⎥⎥⎥⎦
⎡⎢⎢⎢⎣�′1 �′2 �′
�1 �2 �
λ 0 λ
⎤⎥⎥⎥⎦
⎡⎢⎢⎢⎣λ λ 0
λ λ 0
0 0 0
⎤⎥⎥⎥⎦,
(11)
⟨k′(�′0(�′1�
′2
)�′)L;(σ ′0(σ ′1σ
′2
)s′)S
×∥∥∥∥ 1r01
P01
∥∥∥∥k(�0(�1�2)�)L; (σ0(σ1σ2)s)S$
=∑λ
⟨n′2�
′2,n2�2
⟩⟨k′�′0,n1�1;n′1�
′1, k�0
⟩
× ⟨σ ′0(σ ′1σ ′2)|P01|σ0(σ1σ2)⟩i�′0+�′1+�′2−�0−�1−�2
×⟨[
Y [�′0](�r0)[Y [�′1](�r1)Y [�′2](�r2)
][�′]][L]
×∥∥∥∥[Y [λ](�r0)Y [λ](�r1)
][0]∥∥∥∥
×[Y [�0](�r1)
[Y [�1](�r0)Y [�2](�r2)
][�]][L]⟩
=∑λq
⟨n′2�
′2,n2�2
⟩
× ⟨k′�′0,n1�1;n′1�′1, k�0
⟩i�′0+�′1+�′2−�0−�1−�2
8 Advances in Physical Chemistry
λ λ
λ
λ
λ λ 0λ 0 λ0 λ λ
ℓ0 ℓ Lℓ1 q Lλ λ 0
ℓ1 ℓ23 ℓℓ0 ℓ23 qλ 0 λ
ℓ2 ℓ3 ℓ23ℓ2 ℓ3 ℓ23
0 0 0
0 ℓ0 ℓ0
ℓ1 ℓ23 ℓℓ1 q L
[λℓ0ℓ1
]
[λℓ1ℓ0
]L
L
qℓℓ1
ℓ1
ℓ2
ℓ1
ℓ1
ℓ2
ℓ2δℓ2ℓ2
ℓ3δℓ3ℓ3
ℓ0
ℓ0
ℓ0ℓ0
ℓ3
ℓ3
ℓ
ℓ23
ℓ23
1/L
Y[ℓ0]
(�r0)Y[ℓ1]
(�r1)Y[ℓ2]
(�r2)Y[ℓ3]
(�r3)[ℓ23
][ℓ][L]Y[λ]
(�r0)Y[λ]
(�r1)[0
]
×Y[ℓ0
](�r1)
Y[ℓ1
](�r0)
Y[ℓ2
](�r2)Y
[ℓ3
](�r3)
[ℓ23
][ℓ][L]
Figure 11: Recoupling graph of the orbit tensors for (18).
× �2
Lδ�′2�2
[λ∣∣�′0∣∣�1][λ∣∣�′1∣∣�0]
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣
012
12
12
12
s
12
s′ S
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦
×
⎡⎢⎢⎢⎣
0 �0 �0
�1 �2 �
�1 q L
⎤⎥⎥⎥⎦
⎡⎢⎢⎢⎣�′0 �′ L
�1 q L
λ λ 0
⎤⎥⎥⎥⎦
×
⎡⎢⎢⎢⎣�′1 �′2 �′
�0 �2 q
λ 0 λ
⎤⎥⎥⎥⎦
⎡⎢⎢⎢⎣λ λ 0
λ λ 0
0 0 0
⎤⎥⎥⎥⎦,
(12)
⟨k′(�′0(�′1�
′2
)�′)L;(σ ′0(σ ′1σ
′2
)s′)S
×∥∥∥∥ 1r02
P01
∥∥∥∥k(�0(�1�2)�)L; (σ0(σ1σ2)s)S$
=∑λ
⟨k�0,n′1�
′1
⟩⟨k′�′0,n1�1;n′2�
′2,n2�2
⟩
× ⟨σ ′0(σ ′1σ ′2)|P01|σ0(σ1σ2)⟩i�′0+�′1+�′2−�0−�1−�2
×⟨[
Y [�′0](�r0)[Y [�′1](�r1)Y [�′2](�r2)
][�′]][L]
×∥∥∥∥[Y [λ](�r0)Y [λ](�r2)
][0]∥∥∥∥
×[Y [�0](�r1)
[Y [�1](�r0)Y [�2](�r2)
][�]][L]⟩
=∑λq
⟨k�0,n′1�
′1
⟩
× ⟨k′�′0,n1�1;n′2�′2,n2�2
⟩i�′0+�′1+�′2−�0−�1−�2
× �0
Lδ�′1�0
[λ∣∣�′0∣∣�1][λ∣∣�′2∣∣�2]
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣
012
12
12
12
s
12
s′ S
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦
×
⎡⎢⎢⎢⎣
0 �0 �0
�1 �2 �
�1 q L
⎤⎥⎥⎥⎦
⎡⎢⎢⎢⎣�′0 �′ L
�1 q L
λ λ 0
⎤⎥⎥⎥⎦
×
⎡⎢⎢⎢⎣�′1 �′2 �′
�0 �2 q
0 λ λ
⎤⎥⎥⎥⎦
⎡⎢⎢⎢⎣λ λ 0
λ λ 0
0 0 0
⎤⎥⎥⎥⎦,
(13)
Advances in Physical Chemistry 9
λλ
λ
λ
λ λ 0λ 0 λ0 λ λ
ℓ0 ℓ Lℓ1 q Lλ λ 0
ℓ1 ℓ23 ℓℓ0 ℓ23 q0 λ λ
ℓ2 ℓ3 ℓ23ℓ2 ℓ3 ℓ23
λ 0 λ
0 ℓ0 ℓ0
ℓ1 ℓ23 ℓℓ1 q L
[λℓ0ℓ1
]
[λℓ1ℓ0
]L
L
qℓ
ℓ1
ℓ1ℓ2
ℓ2
ℓ1
ℓ2
ℓ2
ℓ0δℓ0ℓ1
ℓ3δℓ3ℓ3
ℓ0
ℓ0
ℓ0
ℓ3
ℓ3
ℓ23
ℓ23
1/L
ℓ
Y[ℓ0]
(�r0)Y[ℓ1]
(�r1)Y[ℓ2]
(�r2)Y[ℓ3]
(�r3)[ℓ23
][ℓ][L]Y[λ]
(�r0)Y[λ]
(�r2)[0
]
×Y[ℓ0
](�r1)
Y[ℓ1
](�r0)
Y[ℓ2
](�r2)Y
[ℓ3
](�r3)
[ℓ23
][ℓ][L]
Figure 12: Recoupling graph of the orbit tensors for (19).
⟨k′(�′0(�′1�
′2
)�′)L;(σ ′0(σ ′1σ
′2
)s′)S
×∥∥∥∥ 1r12
P01
∥∥∥∥k(�0(�1�2)�)L; (σ0(σ1σ2)s)S$
=∑λ
⟨k′�′0,n1�1
⟩⟨n′�′1, k�0;n′2�
′2,n2�2
⟩
× ⟨σ ′0(σ ′1σ ′2)|P01|σ0(σ1σ2)⟩i�′0+�′1+�′2−�0−�1−�2
×⟨[
Y [�′0](�r0)[Y [�′1](�r1)Y [�′2](�r2)
][�′]][L]
×∥∥∥∥[Y [λ](�r1)Y [λ](�r2)
][0]∥∥∥∥
×[Y [�0](�r1)
[Y [�1](�r0)Y [�2](�r2)
][�]][L]⟩
=∑λ
⟨k′�′0,n1�1
⟩
× ⟨n′�′1, k�0;n′2�′2,n2�2
⟩i�′0+�′1+�′2−�0−�1−�2
× �1
Lδ�1�
′0
[λ∣∣�′1∣∣�0][λ∣∣�′2∣∣�2]
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣
012
12
12
12
s
12
s′ S
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦
×
⎡⎢⎢⎢⎣
0 �0 �0
�1 �2 �
�1 �′ L
⎤⎥⎥⎥⎦
⎡⎢⎢⎢⎣�′0 �′ L
�1 �′ L
0 0 0
⎤⎥⎥⎥⎦
×
⎡⎢⎢⎢⎣�′1 �′2 �′
�0 �2 �′
λ λ 0
⎤⎥⎥⎥⎦
⎡⎢⎢⎢⎣λ λ 0
λ λ 0
0 0 0
⎤⎥⎥⎥⎦,
(14)
⟨k′(�′0(�′1�
′2
)�′)L;(σ ′0(σ ′1σ
′2
)s′)S
×‖EP01‖k(�0(�1�2)�)L; (σ0(σ1σ2)s)S〉
= E⟨ϕn′�′1 | u�0 (k)
⟩⟨u�′0 (k′) | ϕn�1
⟩⟨ϕn′�′2 | ϕn�2
⟩
× i�′0+�′1+�′2−�0−�1−�2⟨σ ′0(σ ′1σ
′2
)|P01|σ0(σ1σ2)⟩
×⟨[
Y [�′0](�r0)[Y [�′1](�r1)Y [�′2](�r2)
][�′]][L]
×[Y [�0](�r1)
[Y [�1](�r0)Y [�2](�r2)
][�]][L]⟩
= E⟨ϕn′�′1 | u�0 (k)
⟩⟨u�′0 (k′) | ϕn�1
⟩⟨ϕn′�′2 | ϕn�2
⟩
10 Advances in Physical Chemistry
λ
λ
λ
λ
λ λ 0λ λ 00 0 0
ℓ0 ℓ Lℓ1 q L0 0 0
ℓ1 ℓ23 ℓℓ0 ℓ23 qλ λ 0
ℓ2 ℓ3 ℓ23ℓ2 ℓ3 ℓ23
λ 0 λ
0 ℓ0 ℓ0
ℓ1 ℓ23 ℓℓ1 q L
[λℓ2ℓ2
]
[λℓ1ℓ0
]
L
L
qℓℓ1
ℓ2
ℓ2
ℓ1 ℓ1ℓ2
ℓ2
ℓ1δℓ1ℓ0
ℓ3δℓ3ℓ3
ℓ0
ℓ0
ℓ0
ℓ3
ℓ3
ℓ23
ℓ23
1/L
ℓ
Y[ℓ0]
(�r0)Y[ℓ1]
(�r1)Y[ℓ2]
(�r2)Y[ℓ3]
(�r3)[ℓ23
][ℓ][L]Y[λ]
(�r1)Y[λ]
(�r2)[0
]
×Y[ℓ0
](�r1)
Y[ℓ1
](�r0)
Y[ℓ2
](�r2)Y
[ℓ3
](�r3)
[ℓ23
][ℓ][L]
Figure 13: Recoupling graph of the orbit tensors for (20).
λλ
λ
λ
λ λ 0λ λ 00 0 0
ℓ0 ℓ Lℓ1 q L0 0 0
ℓ1 ℓ23 ℓℓ0 ℓ23 q0 0 0
ℓ2 ℓ3 ℓ23ℓ2 ℓ3 ℓ23
λ λ 0
0 ℓ0 ℓ0
ℓ1 ℓ23 ℓℓ1 q L
[λℓ2ℓ2
]
[λℓ3ℓ3
]
L
L
qℓℓ1
ℓ2
ℓ2
ℓ1ℓ2
ℓ2ℓ1δℓ1ℓ0
ℓ0δℓ0ℓ1
ℓ0
ℓ0
ℓ3
ℓ3ℓ3 ℓ3
ℓ23
ℓ23
ℓ
1/L
Y[ℓ0]
(�r0)Y[ℓ1]
(�r1)Y[ℓ2]
(�r2)Y[ℓ3]
(�r3)[ℓ23
][ℓ][L]Y[λ]
(�r2)Y[λ]
(�r3)[0
]
×Y[ℓ0
](�r1)
Y[ℓa]
(�r0) Y[ℓb]
(�r2)Y[ℓc]
(�r3)[ℓ23
][ℓ][L]
Figure 14: Recoupling graph of the orbit tensors for (21).
× (−1)�′0+�′1+�′2−�0−�1−�2
×
⎡⎢⎢⎢⎢⎢⎢⎢⎣
012
12
12
12
s
12
s′ S
⎤⎥⎥⎥⎥⎥⎥⎥⎦
⎡⎢⎢⎢⎣
0 �0 �0
�1 �2 �
�1 �′ L
⎤⎥⎥⎥⎦.
(15)
Minor simplification of (11)–(15) gives rise to the sameexpressions presented in [11], where the conventional alge-braic approach was used. However, we emphasize that theinvariant graphical method is so much simpler with thederivation completed by drawing the diagrams.
3.3. Electron-Li Scattering (A Four-Electron System). Deriva-tions for these momentum-space potentials have been
Advances in Physical Chemistry 11
0 ℓ0 ℓ0
ℓ1 ℓ23 ℓℓ1 q L
L
L
q
Liℓ0+ℓ1+ℓ2+ℓ3−ℓ0−ℓ1−ℓ2−ℓ3
ℓℓ1
ℓ2
ℓ1
ℓ2
ℓ0
ℓ0
ℓ3
ℓ3
ℓ23
ℓ23
ℓ
1/L
Y[ℓ0]
(�r0)Y[ℓ1]
(�r1)Y[ℓ2]
(�r2)Y[ℓ3]
(�r3)[ℓ23
][ℓ][L]Y[ℓ0
](�r1)
Y[ℓa]
(�r0)Y[ℓb]
(�r2)Y[ℓc]
(�r3)[ℓ23
][ℓ][L]
Figure 15: Recoupling graph of the orbit tensors for (22).
previously given for e-H [9], for e-Li [11], for e-Na [16], andfor e-He [16], all using the conventional algebraic approach.The frozen core approximation was adopted for e-Li and e-Na in these studies, which could be the main reason for thesmall discrepancies between the theoretical calculations andexperiments [10]. With the enormous success of quantumscattering theories in describing scattering from one- andtwo-electron targets, one is naturally seeking to performcalculations with more complex systems and free of approx-imations. However, such an extension has been proven to beextremely tedious and in some cases even intractable. In thefollowing, we derive a complete set of direct and exchangematrix elements for electron-Lithium scattering (previouslynot available) using the invariant graphical method. In thisway, we are able to remove the frozen core approximationadopted by Bray et al. [10, 16] and Zhang et al. [14]. Again,the linear radial integral part of the direct and exchangeelements is straightforward to work out and has the generalform of
⟨n′2�
′2,n2�2
⟩⟨n′3�
′3,n3�3
⟩⟨k′�′0, k�0;n′1�
′1,n1�1
⟩
≡⟨ϕn′2�′2 | ϕn2�2
⟩⟨ϕn′3�′3 | ϕn3�3
⟩
×∫u�′0 (k′r0)u�0 (kr0)ϕn′�′1 (r1)ϕn�1 (r1)
rλ<rλ+1>
dr0dr1.
(16)
The corresponding coupling and recoupling graphs areshown in Figures 9, 10, 11, 12, 13, 14, and 15, and the resultsare as the following:⟨�k′�′0
(�′1�
′2�′3
)�′‖V01‖�k�0(�1�2�3)�
⟩
=∑λ
⟨n′2�
′2,n2�2
⟩⟨n′3�
′3,n3�3
⟩
× ⟨k′�′0, k�0;n′1�′1,n1�1
⟩i�′0+�′1+�′2+�′3−�0−�1−�2−�3
× ⟨σ ′0(σ ′1(σ ′2σ ′3)) | σ0(σ1(σ2σ3))⟩
×⟨⎡⎣Y [�′0](�r0
)
×[Y [�′1](�r1)
[Y [�′2](�r2
)Y [�′3](�r3
)][�′23]][�′]
][L]
×∥∥∥∥[Y [λ](�r0)Y [λ](�r1)
][0]∥∥∥∥
×⎡⎣Y [�0](�r0
)
×[Y [�1](�r1)
[Y [�2](�r2
)Y [�3](�r3
)][�23]][�]][L]⟩
=∑λ
⟨n′2�
′2,n2�2
⟩⟨n′3�
′3,n3�3
⟩
12 Advances in Physical Chemistry
× ⟨k′�′0, k�0;n′1�′1,n1�1
⟩i�′0+�′1+�′2+�′3−�0−�1−�2−�3
× �2δ�2�′2�3δ�3�
′3
L
[λ∣∣�′0∣∣�0][λ∣∣�′1∣∣�1]⎡⎢⎢⎢⎣�′0 �′ L
�0 � L
λ λ 0
⎤⎥⎥⎥⎦
×
⎡⎢⎢⎢⎣�′1 �′23 �′
�1 �23 �
λ 0 λ
⎤⎥⎥⎥⎦
⎡⎢⎢⎢⎣λ λ 0
λ 0 λ
0 λ λ
⎤⎥⎥⎥⎦
⎡⎢⎢⎢⎣�′2 �′3 �′23
�2 �3 �23
0 0 0
⎤⎥⎥⎥⎦,
(17)
⟨�k′�′0
(�′1�
′2�′3
)�′‖V01P01‖�k�0(�1�2�3)�
⟩
=∑λ
⟨n′2�
′2,n2�2
⟩⟨n′3�
′3,n3�3
⟩
× ⟨k′�′0,n1�1; k�0,n′1�′1
⟩i�′0+�′1+�′2+�′3−�0−�1−�2−�3
× ⟨σ ′0(σ ′1(σ ′2σ ′3))|P01|σ0(σ1(σ2σ3))⟩
×⟨⎡⎣Y [�′0](�r0
)
×[Y [�′1](�r1)
[Y [�′2](�r2)Y [�′3](�r3)
][�′23]][�′]
][L]
×∥∥∥∥[Y [λ](�r0)Y [λ](�r1)
][0]∥∥∥∥
×⎡⎣Y [�0](�r1
)
×[Y [�1](�r0)
[Y [�2](�r2)Y [�3](�r3)
][�23]][�]][L]⟩
=∑λqp
⟨n′2�
′2,n2�2
⟩⟨n′3�
′3,n3�3
⟩
× ⟨k′�′0,n1�1; k�0,n′1�′1
⟩i�′0+�′1+�′2+�′3−�0−�1−�2−�3
× �2δ�2�′2�3δ�3�
′3
L
[λ∣∣�′0∣∣�1][λ∣∣�′1∣∣�0]
×
⎡⎢⎢⎢⎢⎢⎢⎢⎣
012
12
12
p s
12
s′ S
⎤⎥⎥⎥⎥⎥⎥⎥⎦
⎡⎢⎢⎢⎣
0 �0 �0
�1 �23 �
�1 q L
⎤⎥⎥⎥⎦
⎡⎢⎢⎢⎣�′0 �′ L
�1 q L
λ λ 0
⎤⎥⎥⎥⎦
×
⎡⎢⎢⎢⎣�′1 �′23 �′
�0 �23 q
λ 0 λ
⎤⎥⎥⎥⎦
⎡⎢⎢⎢⎣λ λ 0
λ 0 λ
0 λ λ
⎤⎥⎥⎥⎦
⎡⎢⎢⎢⎣�′2 �′3 �′23
�2 �3 �23
0 0 0
⎤⎥⎥⎥⎦,
(18)
⟨�k′�′0
(�′1�
′2�′3
)�′‖V02P01‖�k�0(�1�2�3)�
⟩
=∑λ
⟨k�0,n1�′1
⟩⟨n′3�
′3,n3�3
⟩
× ⟨k�′0,n′1�′1;n′2�
′2,n2�2
⟩i�′0+�′1+�′2+�′3−�0−�1−�2−�3
× ⟨σ ′0(σ ′1(σ ′1σ ′3))|P01|σ0(σ1(σ2σ3))⟩
×⟨⎡⎣Y [�′0](�r0
)
×[Y [�′1](�r1)
[Y [�′2](�r2)Y [�′3](�r3)
][�′23]][�′]
][L]
×∥∥∥∥[Y [λ](�r0)Y [λ](�r2)
][0]∥∥∥∥
×⎡⎣Y [�0](�r1
)
×[Y [�1](�r0)
[Y [�2](�r2)Y [�3](�r3)
][�23]][�]][L]⟩
=∑λqp
⟨k�0,n1�
′1
⟩⟨n′3�
′3,n3�3
⟩
× ⟨k�′0,n′1�′1;n′2�
′2,n2�2
⟩i�′0+�′1+�′2+�′3−�0−�1−�2−�3
× �0δ�0�′1�3δ�3�
′3
L
[λ∣∣�′0∣∣�1][λ∣∣�′2∣∣�2]
×
⎡⎢⎢⎢⎢⎢⎢⎣
012
12
12
p s
12
s′ S
⎤⎥⎥⎥⎥⎥⎥⎦
⎡⎢⎢⎢⎣
0 �0 �0
�1 �23 �
�1 q L
⎤⎥⎥⎥⎦
⎡⎢⎢⎢⎣�′0 �′ L
�1 q L
λ λ 0
⎤⎥⎥⎥⎦
×
⎡⎢⎢⎢⎣�′1 �′23 �′
�0 �23 q
0 λ λ
⎤⎥⎥⎥⎦
⎡⎢⎢⎢⎣λ λ 0
λ 0 λ
0 λ λ
⎤⎥⎥⎥⎦
⎡⎢⎢⎢⎣�′2 �′3 �′23
�2 �3 �23
λ 0 λ
⎤⎥⎥⎥⎦,
(19)
⟨�k′�′0
(�′1�
′2�′3
)�′‖V12P01‖�k�0(�1�2�3)�
⟩
=∑λ
⟨k′�′0,n1�1
⟩⟨n′3�
′3,n3�3
⟩
× ⟨n′1�′1, k�0;n′2�′2,n2�2
⟩i�′0+�′1+�′2+�′3−�0−�1−�2−�3
× ⟨σ ′0(σ ′1(σ ′2σ ′3))|P01|σ0(σ1(σ2σ3))⟩
×⟨⎡⎣Y [�′0](�r0
)
×[Y [�′1](�r1)
[Y [�′2](�r2)Y [�′3](�r3)
][�′23]][�′]
][L]
Advances in Physical Chemistry 13
×∥∥∥∥[Y [λ](�r1)Y [λ](�r2)
][0]∥∥∥∥
×⎡⎣Y [�0](�r1
)
×[Y [�1](�r0)
[Y [�2](�r2)Y [�3](�r3)
][�23]][�]][L]⟩
=∑λqp
⟨k′�′0,n1�1
⟩⟨n′3�
′3,n3�3
⟩
× ⟨n′1�′1, k�0;n′2�′2,n2�2
⟩i�′0+�′1+�′2+�′3−�0−�1−�2−�3
× �1δ�1�′0�3δ�3�
′3
L
[λ∣∣�′1∣∣�0][λ∣∣�′2∣∣�2]
×
⎡⎢⎢⎢⎢⎢⎢⎢⎣
012
12
12
p s
12
s′ S
⎤⎥⎥⎥⎥⎥⎥⎥⎦
⎡⎢⎢⎢⎣
0 �0 �0
�1 �23 �
�1 q L
⎤⎥⎥⎥⎦
⎡⎢⎢⎢⎣�′0 �′ L
�1 q L
0 0 0
⎤⎥⎥⎥⎦
×
⎡⎢⎢⎢⎣�′1 �′23 �′
�0 �23 q
λ λ 0
⎤⎥⎥⎥⎦
⎡⎢⎢⎢⎣λ λ 0
λ λ 0
0 0 0
⎤⎥⎥⎥⎦
⎡⎢⎢⎢⎣�′2 �′3 �′23
�2 �3 �23
λ 0 λ
⎤⎥⎥⎥⎦,
(20)
⟨�k′�′0
(�′1�
′2�′3
)�′‖V23P01‖�k�0(�1�2�3)�
⟩
=∑λ
∑λqp
⟨k′�′0,n1�1
⟩⟨k�0,n′1�
′1
⟩
× ⟨n′2�′2,n2�2;n′3�′3,n3�3
⟩i�′0+�′1+�′2+�′3−�0−�1−�2−�3
× ⟨σ ′0(σ ′1(σ ′2σ ′3))|P01|σ0(σ1(σ2σ3))⟩
×⟨⎡⎣Y [�′0](�r0
)
×[Y [�′1](�r1)
[Y [�′2](�r2)Y [�′3](�r3)
][�′23]][�′]
][L]
×∥∥∥∥[Y [λ](�r2)Y [λ](�r3)
][0]∥∥∥∥
×⎡⎣Y [�0](�r1
)
×[Y [�a](�r0)
[Y [�b](�r2)Y [�c](�r3)
][�23]][�]][L]⟩
=∑λqp
⟨k′�′0,n1�1
⟩⟨k�0,n′1�
′1
⟩
× ⟨n′2�′2,n2�2;n′3�′3,n3�3
⟩i�′0+�′1+�′2+�′3−�0−�1−�2−�3
× �1δ�1�′0�0δ�0�
′1
L
[λ∣∣�′2∣∣�2][λ∣∣�′3∣∣�3]
×
⎡⎢⎢⎢⎢⎢⎢⎢⎣
012
12
12
p s
12
s′ S
⎤⎥⎥⎥⎥⎥⎥⎥⎦
⎡⎢⎢⎢⎣
0 �0 �0
�1 �23 �
�1 q L
⎤⎥⎥⎥⎦
⎡⎢⎢⎢⎣�′0 �′ L
�1 q L
0 0 0
⎤⎥⎥⎥⎦
×
⎡⎢⎢⎢⎣�′1 �′23 �′
�0 �23 q
0 0 0
⎤⎥⎥⎥⎦
⎡⎢⎢⎢⎣�′2 �′3 �′23
�2 �3 �23
λ λ 0
⎤⎥⎥⎥⎦
⎡⎢⎢⎢⎣λ λ 0
λ λ 0
0 0 0
⎤⎥⎥⎥⎦,
(21)
⟨�k′�′0
(�′1�
′2�′3
)�′‖EP01‖�k�0(�1�2�3)�
⟩
= E⟨ϕn′1�′1 | u�0 (k)
⟩⟨u�′0 (k′) | ϕn1�1
⟩
×⟨ϕn′2�′2 | ϕn2�2
⟩
×⟨ϕn′3�′3 | ϕn3�3
⟩i�′0+�′1+�′2+�′3−�0−�1−�2−�3
× ⟨σ ′0(σ ′1(σ ′2σ ′3))|P01|σ0(σ1(σ2σ3))⟩
×⟨⎡⎣Y [�′0](�r0
)
×[Y [�′1](�r1)
[Y [�′2](�r2)Y [�′3](�r3)
][�′23]][�′]
][L]
×⎡⎣Y [�0](�r1
)
×[Y [�a](�r0)
[Y [�b](�r2)Y [�c](�r3)
][�23]][�]][L]⟩
= (−1)�′0+�′1+�′2+�′3−�0−�1−�2−�3E
⟨ϕn′1�′1 | u�0 (k)
⟩
×⟨u�′0 (k′) | ϕn1�1
⟩⟨ϕn′2�′2 | ϕn2�2
⟩⟨ϕn′3�′3 | ϕn3�3
⟩
×∑pq
⎡⎢⎢⎢⎢⎢⎢⎢⎣
012
12
12
p s
12
s′ S
⎤⎥⎥⎥⎥⎥⎥⎥⎦
⎡⎢⎢⎢⎣
0 �0 �0
�1 �23 �
�1 q L
⎤⎥⎥⎥⎦.
(22)
4. Conclusions
The invariant graphical method developed by Danos andFano [2] is very powerful. The complex manipulation ofangular momenta is reduced to the drawing of a compactgraph, from which the final expressions can be readilyread off. The procedure is transparent and simple to apply.It avoids writing out the intermediate expansions, yields
14 Advances in Physical Chemistry
immediately the selection rules for intermediate angularmomenta, and helps in finding the most direct and economi-cal intermediate recoupling. It also requires a minimal use ofrules in comparison with pages of formulas involved in otherconventional methods.
Acknowledgments
J. B. Wang would like to thank H. Y. Wu for cross-checking the phases in (11) using the conventional algebraicmethod. The authors also acknowledge support from TheUniversity of Western Australia, Murdoch University, andCurtin University of Technology.
References
[1] D. M. Brink and G. R. Satchler, Angular Momentum, OxfordUniversity Press, Oxford, UK, 1993.
[2] M. Danos and U. Fano, “Graphical analysis of angularmomentum for collision products,” Physics Report, vol. 304,no. 4, pp. 155–227, 1998.
[3] A. S. Kheifets, “Sequential two-photon double ionization ofnoble gas atoms,” Journal of Physics B, vol. 40, no. 22, pp. F313–F318, 2007.
[4] A. S. Kheifets, “Photoelectron angular correlation pattern insequential two-photon double ionization of neon,” Journal ofPhysics B, vol. 42, no. 13, Article ID 134016, 2009.
[5] E. C. Simpson, J. A. Tostevin, D. Bazin, and A. Gade, “Lon-gitudinal momentum distributions of the reaction residuesfollowing fast two-nucleon knockout reactions,” PhysicalReview C, vol. 79, no. 6, Article ID 064621, 2009.
[6] V. Aquilanti, A. C. P. Bitencourt, C. Da, A. Marzuoli, andM. Ragni, “Combinatorics of angular momentum recouplingtheory: spin networks, their asymptotics and applications,”Theoretical Chemistry Accounts, vol. 123, no. 3-4, pp. 237–247,2009.
[7] J. Suzuki, G. N. M. Tabia, and B. G. Englert, “Symmetricconstruction of reference-frame-free qudits,” Physical ReviewA, vol. 78, no. 5, Article ID 052328, 2008.
[8] A. Maser, U. Schilling, T. Bastin, E. Solano, C. Thiel, and J. VonZanthier, “Generation of total angular momentum eigenstatesin remote qubits,” Physical Review A, vol. 79, no. 3, Article ID033833, 2009.
[9] I. Bray and A. T. Stelbovics, “Convergent close-couplingcalculations of electron-hydrogen scattering,” Physical ReviewA, vol. 46, no. 11, pp. 6995–7011, 1992.
[10] I. Bray, J. Beck, and C. Plottke, “Spin-resolved electron-impactionization of lithium,” Journal of Physics B, vol. 32, no. 17, pp.4309–4320, 1999.
[11] D. V. Fursa and I. Bray, “Calculation of electron-heliumscattering,” Physical Review A, vol. 52, no. 2, pp. 1279–1297,1995.
[12] H. Wu, I. Bray, D. V. Fursa, and A. T. Stelbovics, “Convergentclose-coupling calculations of positron-helium scattering atintermediate to high energies,” Journal of Physics B, vol. 37, no.6, pp. 1165–1172, 2004.
[13] D. V. Fursa and I. Bray, “Electron-impact ionization of thehelium metastable 2 S state,” Journal of Physics B, vol. 36, no.8, pp. 1663–1671, 2003.
[14] X. Zhang, C. T. Whelan, and H. R. J. Walters, “Electron impactionization of lithium-spin asymmetry of the triple differential
cross section,” Journal of Physics B, vol. 25, pp. L457–L462,1992.
[15] R. D. Cowan, The Theory of Atomic Structure and Spectra,University of California Press, Berkeley, Calif, USA, 1981.
[16] I. Bray, “Convergent close-coupling method for the calculationof electron scattering on hydrogenlike targets,” Physical ReviewA, vol. 49, no. 2, pp. 1066–1082, 1994.
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