Physica Scripta. Vol. 61, 690^695, 2000
Doubly-Excited States of Beryllium-Like Ions
W. J. Pong and Y. K. Ho
Institute of Atomic and Molecular Sciences, Academia Sinica, P.O. Box 23-166, Taipei, Taiwan, 10764, Republic of China
Received November 29, 1999; accepted in revised form January 24, 2000
pacs ref: 31.20.Di, 32.80.Dz
Abstract
The resonance energies and widths for doubly-excited (1s23d2, 3p4f, 3d4d) 1Ge
states of beryllium-like ions are determined by calculating the density of
resonance states using the stabilization method. A model potential is used
to represent the interaction between the outer electrons and the core electrons.
Products of Slater Orbitals are used to represent the two-electron wave
functions. We present results for some lower-lying doubly-excited states for
Z ˆ 5 10. Comparisons are made with other calculations in the literature.
1. Introduction
In recent years, considerable experimental activities have
been carried out to investigate double-electron transfer processes in collisions involving a highly ionized ion and an
atom (or a molecule) [1^6]. Studies on A…Z 2†‡ (1s2) ‡He,
H2 have been carried out, where A ˆ C; N; O, and Ne. In
order to interpret the experimentally observed electron
spectra, accurate theoretical energy levels and autoionization rates (related to the resonance widths) are needed
for the doubly excited states A…Z 4†‡ (1s2nlnl0 † of the Be-like
systems.
In a recent related theoretical development, Mandelshtam
et al. [7] have proposed a procedure to calculate resonance
energy Er and width G for a resonance by calculating the
density of resonance states using the stabilization method.
The use of the stabilization method to investigate atomic
resonances has a long history (Taylor [8], Hazi and Taylor
[9]). The method has been used to estimate resonance positions for di¡erent atomic systems. In the latest
development, Mandelshtam et al. [7] applied the procedure
to a model problem with successful results. The method
was also applied by Muller et al. [10] to calculate doubly
excited S-wave resonances in He, and by Bachau for triplyexcited states in N4+ ion [11]. Recently, we have applied
the stabilization method to calculate the resonance energies
and widths for doubly excited states in H^ (Ref. [12]),
and for 1De states in four-electron Be-like ions [13], and
in Mg [14].
Here. we apply the stabilization method to investigate
the (1s2 ln0 l 0 †) 1Ge resonance states of the four-electron
beryllium-like B+, C2+, N3+, O4+, F5+, and Ne6+ ions below
the N ˆ 3 thresholds of the three-electron systems. We concentrate on the 1Ge states in the present work due to the fact
that the widths for the lower lying 1Ge states are quite broad,
and that they are predominately produced in ion-H2
collisions [1^3]. Accurate resonance parameters (positions
and widths) for such states are called for. Here, we use a
model potential to represent the interaction between the core
electrons and the outer electrons. Products of Slater orbitals
are used to represent the doubly-excited two-electron wave
functions. Basis sets of M ˆ 935 terms are used for these
Physica Scripta 61
systems to calculate the 1s3ln0 l 0 1Ge states. The use of the
products of Slater orbitals should be able to provide accurate
results for such angular momentum states. Usually, for
lower angular momentum states (L 1), highly correlated
wave functions such as the Hylleraas-type would be more
appropriated. The use of extensive Hylleraas functions to
construct the stabilization plots would require a major computational e¡ort, and is outside the scope of our present
investigation. The computational procedure demonstrated
in this work should be applicable for other angular
momentum states and for other types of wave functions.
2. The stabilization method
The spectral density of states has two parts: rP …E† and rQ …E†
where P and Q refer to the open (P) and closed (Q) spaces,
respective, in the Feshbach projection formalism. rP …E† is
a smooth function of energy E, and rQ …E† is a result of
complex poles of the Green's function. The spectral density
of a resonance pole is given by
"
X
1
Im
p
…E
k
rQ …E† ˆ
#
1
;
Ek † ‡ iGk =2
…1†
where Ek iGk =2 is the kth complex pole of the Green's
function. Mandelshtam et al. [7] have shown that rQ …E)
can be obtained by
Q
r …E† ˆ
Z
1
L2
L1
L2
L1
rL …E†dL;
…2†
where
rL …E†
X
d…Ej …L†
E†;
…3†
j
and L is the size of the box in which the Hamiltonian is
diagonalizied. For an atomic system, we can replace the
``hard wall'' by a ``soft wall'' (Muller et al. [8]), and Eq.
(2) can be replaced by the following
rQ …E† ˆ
Z
1
amax
amin
amax
amin
ra …E†da;
…4†
where a is a non-linear parameter in the wave function (to be
discussed below). Eq. (3) becomes
ra …E† ˆ
X
d…Ej …a†
E†:
…5†
j
# Physica Scripta 2000
Doubly-Excited States of Beryllium-Like Ions
Using the equation
1
Z
df d…f f …x††g…x†dx ˆ g…x† ;
dx f …x†ˆf
where
…6†
Eq.(2) can be evaluated as
XdEj …a† 1
1
Q
:
r …E† ˆ
amax amin j da Ej …a†ˆE
…7†
For an isolated resonance rQ …E† can be derived from Eq. (1)
(Bowman [15]), with
rQ …E†  p
1
…Er
G=2
:
E†2 ‡ G2 =4
…8†
rQ …E† can be calculated from the stabilization graph using
Eq. (7), and the resonance parameters (Er and G) can be
obtained by ¢tting rQ …E† to Eq. (8).
3. The Hamiltonian and the model potential
We use a model potential to represent the interaction
between the inner core electrons with the outside valence
electrons. Recently Bachau et al. [16] have proposed a
generalization of the Feshbach method to study the
1s2 nln0 l 0 doubly excited states of Be~like systems [17]. The
e¡ect of the ls2 core is represented by a model potential
Vm , with (energy in Rydberg units)
Vm …r† ˆ
4
r
4
…1 ‡ br†e
r
2b r
;
…9†
where b is the e¡ective charge felt by the two electrons
described by the 1s orbitals. In the present work, the
Z-dependent b is given as b ˆ 0.6845Z^0.3944 [16]. The parameter was determined by ¢tting the exact energy of the
lowest 1s2 2s 2S state of the corresponding three-electron ion.
The use of the model potential would lead to an estimated [18]
error of about 0.0004 Ry (more bound) for the 2s 2S state. For
the N ˆ 3 states, the estimated uncertainties are within
0.0032 Ry, 0.0053 Ry, and 0.0004 Ry, for the 3s 2S, 3p
2
P, and 3d 2D states, respectively. For two-electron systems,
the estimated di¡erences in energy are about doubled.
The Hamiltonian for the four-electron atomic systems is
given by
HM ˆ H o …1† ‡ H o …2† ‡ Vm …1† ‡ Vm …2† ‡
691
2
;
r12
Zai …r† ˆ rnai exp… xai r†:
…13†
In Eq. (12), A is the antisymmetrising operator, S is a
two-particle spin eigenfunction and the Z are individual
Slater orbitals. Y is an eigenfunction of the total angular
momentum L,
XX
…1; 2† ˆ
C…la ; lb ; L; Mla ; mlb ; M†Yla ;mla …1†Ylb ;mlb …2†;
YlLM
a lb
mla mlb
…14†
with C denoting the Clebsch^Gordan coe¤cients.
For 1Ge states, we use an expansion length of M ˆ 935 to
construct the stabilization plots. The basis sets are constructed by using Slater orbitals of 11s-type, 11p-type,
10d-type, 9f-type, 8g-type, 7h-type, 6i-type, 5k-type,
41-type, 3m-type, 2n-type, and lo-type. To establish the
``soft wall,'' we multiply the exponents of the Slater orbitals
(Eq. (13)) by a scaling factor a. By changing a, the di¡useness
of the wave function is changed, thereby changing the range
of the potential in which the wave function expands.
Figure 1 shows plots for F5+ (Z ˆ 9) energy eigenvalues
vs a: We use 261 points to cover the range a ˆ 1:5 to
a ˆ 2:8. The 21st to 37th eigenvalues in the energy range
10:5 Ry to 6.5 Ry are shown here. It is seen that an
eigenvalue near E ˆ 9:74 Ry exhibits stabilization
character. It represents the lowest 3d2 1Ge resonance.
Furthermore, there are two resonant states in the energy
range of 8.0 Ry to 7.5 Ry that belong to the 1Ge doubly
excited states with (1s23l4l 0 ) con¢gurations. To determine
the resonance parameters for the lowest 1Ge state from
the stabilization plot, we examine the 27th eigenvalue in
the interval a ˆ 1:6 to 1.8, as well as the 26th eigenvalue
…10†
where H o is a hydrogenic Hamiltonian of charge Z, with
Ho ˆ
2Z
;
r
r2
…11†
and Vm is the model potential described above.
4. Wave functions and calculations
We use products of Slater orbitals to represent the
two-electron functions. The products of Slater orbitals
are the following:
XX
FˆA
Cai bj Zaj …r1 †Zbj …r2 †YlLM
…1; 2†S…s1 ; s2 †;
…12†
a lb
la lb
ij
# Physica Scripta 2000
Fig. 1. Stabilization plot …E vs a† for the 1s2ln0 l 0 1Ge resonance eigenvalues for
Z ˆ 9.
Physica Scripta 61
692
W. J. Pong and Y. K. Ho
in the interval a ˆ 1:75 to 1.95. In doing such a procedure,
we investigate the stabilized eigenvalue for the lowest 1Ge(1)
resonance in a region called ``avoided crossing.'' We further
apply the following function to obtain the inverse of the
slope
rn ˆ
an‡1
En‡1
an
En
1
1
;
…15†
resonance 1Ge(2) state to the Lorenzian form (Eq. (16)).
The ¢t gives Er ˆ 7.85460 Ry and G ˆ 0:01160 Ry.
Similarly, the 1Ge(3) state is determined by the ¢t shown
in Fig. 4, giving resonance parameters as Er ˆ 7.70075
Ry and G ˆ 0:01203. These two resonance states have
approximate con¢gurations of 3p4f 1Ge and 3d4d 1Ge,
respectively.
where r is called the density of states. The calculated values
of r are then ¢tted to the following equation
rn …E† ˆ
…E
c…G=2†
‡ d;
Er †2 ‡ G2 =4
…16†
and with which the resonance position Er , and width G are
deduced. In Eq. (16), c and d are ¢tting constants with c
representing the overall normalization factor for the density
of resonance state, and d representing a small background.
They do not a¡ect the determination of Er and G.
In Fig. 2, the solid circles are the results of actual
calculation of rn using Eq. (15). The solid line is the ¢tted
curve using the Lorenzian form of Eq. (16). The ¢t gives
Er ˆ 9:74318 Ry and 0.04388 Ry. In general, we have also
performed ¢ts to other avoided-crossings in the stabilization
curve. We choose the best ¢t's (w2 gives a minimum value in
the non-linear squares ¢t) as our ¢nal results, and they
are summarized in Table I. We apply a similar procedure
to deduce the resonance parameters for the 1Ge(2) and
1 e
G (3) states. In Fig. 3, we show the ¢t of the density of
Fig. 3. Fit of the density of resonance states to the Lorenzian form of Eq. (1 6),
for Z ˆ 9, 1s23p4f 1Ge(2) state. The solid line is the ¢t.
Fig. 2. Fit of the density of resonance states (circles) to the Lorenzian form of
Eq. (16), for the Z ˆ 9, 1s23d2 1Ge(1) state. The solid line is the ¢t.
Fig. 4. Fit of the density of resonance states to the Lorenzian form of Eq. (16),
for the Z ˆ 9, 1s23d4d 1Ge(3) state. The solid line is the ¢t.
Table I. Comparisons of the 1s23ln0 l 0 1Ge states for Z ˆ 5 to Z ˆ 7:
Zˆ5
3d2 1Ge(l)
3p4f 1Ge(2)
3d4d 1Ge(3)
Physica Scripta 61
Er (Ry)
G(Ry)
Er (Ry)
G(Ry)
Er (Ry)
G(Ry)
Zˆ6
Zˆ7
Present
Bachau
et al. [17]
Present
Bachau
et al. [17]
Present
Bachau
et al. [17]
^1.54620
0.00987
^1.32764
0.00078
^1.28376
0.00403
^1.56000
0.02070
^1.32840
0.00094
^1.28960
0000613
^2.94461
0.02430
^2.42886
0.00483
^2.37069
0.00929
^2.94020
0.02888
^2.42680
0.00546
^2.36900
0.00954
^4.76456
0.03194
^3.88708
0.00775
^3.79861
0.01019
^4.76340
0.03526
^3.88420
0.00885
^3.79480
0.01066
# Physica Scripta 2000
Doubly-Excited States of Beryllium-Like Ions
We further de¢ne E ˆ
6. Results and Discussions
To represent the characteristics of the resonant states for the
whole isoelectronic sequence, we draw their energy variation
with the e¡ective nuclear charge (Z^2). We express the
energy of a doubly excited state as the sum of the energy
for the inner valence electron (with N ˆ 3) moving in a
Coulomb ¢eld of charge (Z^2), and the energy for the outer
valence electron moving in a Coulomb ¢eld of charge (Z^3),
with an e¡ective orbital quantum number N*. The doubly
excited energy E can therefore be expressed as
Eˆ
2†2
…Z
N2
…Z 3†2
:
…N †2
…17†
E ˆ fE ‡ ‰…Z
693
…N † 2 , and obtain
2†2 =N 2 Šg…Z
3† 2 :
…18†
Figure 5 shows a plot of E* vs (Z 2)^1. Our present results
which are obtained by using the density of states are compared with those of Bachau et al. [17]. These comparisons
are shown in Table I for Z ˆ 5 to Z ˆ 7 and in Table II
for Z ˆ 8 to Z ˆ 10. It is seen from Tables I and II and
Fig. 4 that for all the three 1Ge states with Z 6, our energies lie lower than those of Bachau et al. [17]. But for
Z ˆ 5, our results lie higher for all three 1Ge states.
Basically, Bachau et al. [17] employed a procedure to calculate the eigenvalue of QHQ, in the language of Feshbach
projection formalism. However, they have not calculated
the Feshbach shift, the interaction between the open~channel
components of the wave function PC, and the closedchannel components of the wave function QC. On the other
hand, we employ a procedure to calculate the resonant poles
directly, without dividing the wave functions into di¡erent
open and closed components. A comparison between our
results and those of Bachau et al. [17] indicates that the
Feshbach shifts would be negative for the three 1Ge states
with Z 6, and positive for Z ˆ 5. We should mention that
Bachau et al. [17] also employed a model potential to represent the interaction between the core and the outer valence
electrons.
Figures 6, 7 and 8 show, respectively, the resonance widths
for the 1Ge(1), 1Ge(2) and 1Ge(3) states for Z ˆ 5 to Z ˆ 10.
Shown in the ¢gures are also the results of Bachau et al. [17].
Fig. 6. G vs 1=…Z 2† for the 3d2 1Ge(1) state. Shown here are also results by
Bachau et al. [17], and by Vaeck and Hansen [18].
Fig. 5. Comparison of E* with results in Ref. [17].
Table II. Comparisons of the 1s23ln0 l 0 1Ge states for Z ˆ 8 to Z ˆ 10:
Zˆ8
3d2 1Ge(1)
3p4f 1Ge(2)
3d4d 1Ge(3)
Er (Ry)
G(Ry)
Er (Ry)
G(Ry)
Er (Ry)
G(Ry)
# Physica Scripta 2000
Zˆ9
Z ˆ 10
Present
Bachau
et al. [17]
Present
Bachau
et al. [17]
Present
Bachau
et al. [17]
^7.03649
0.03838
^5.69645
0.00972
^5.57596
0.01117
^7.03080
0.04013
^5.69320
0.01071
^5.57060
0.01184
^9.74318
0.04388
^7.85460
0.01160
^7.70075
0.01203
^9.74280
0.04390
^7.85160
0.01185
^7.69440
0.01296
^12.90187
0.04755
^10.36137
0.01256
^10.17306
0.01265
^12.89920
0.04688
^10.35840
0.01257
^10.16600
0.01395
Physica Scripta 61
694
W. J. Pong and Y. K. Ho
Fig. 7. G vs 1=…Z 2† for the 3p4f 1Ge(2) state. Shown here are also results by
Bachau et al. [17].
Fig. 8. G vs 1=…Z 2† for the 3d4d 1Ge(3) state. Shown here are also results by
Bachau et al. [17].
Table III. Comparisons of the autoionization widths (G in
Ry) for 1s23d2 1Ge(1) state of C2+ and N3+ ions.
C2+
N3+
G(Ry)
G(Ry)
Present
Vacek and Hansen
(1989) [18]
Bachau et al.
(1990) [17]
0.02430
0.03194
0.04397
0.04934
0.02888
0.03526
In general, our widths are narrower than those of Bachau et
al. [17] for all the three 1Ge states reported here. In particular
for Z ˆ 5, our results di¡er from those of Ref. [17] by about
20% for the 1Ge(3) state to a factor of two for the 1Ge(1)
state. Also from Fig. 5, it is seen that their results for the
1 e
G (1) state agree with ours better for high Z ions than
for low Z ions. Such results are not unexpected. In the calculations of resonance widths, Bachau et al. [17] used a static
exchange approximation for the scattering non-resonance
continuum. Such an approximation works better when Z
is large. In Fig. 7, the results by Bachau et al. [17] for
the 1Ge(3) state deviate more from our present values for
increasing Z once Z is larger than 6. This may re£ect the
di¡erent approximations used to represent the closed parts
(QC) of the wave functions. In Fig. 5 and Table III, we also
compare the resonance widths obtained by Vaeck and
Hansen [18] who employed Cowan's Hartree^Fock approximation code [19], together with the use of basis sets of N ˆ 3
con¢gurations. Their widths are considerable larger than
our present results. This may re£ect the simple approximation used for their wave functions.
In Table IV, we compare the resonance energies of the
1s2 ln0 l 0 1Ge(1), 1Ge(2) and 1Ge(3) states for N3+ ions with
the other earlier calculations. The resonance energies given
in van der Hart and Hansen [20] and in Chen and Lin [21]
were obtained by using the Truncated Diagonalization
Method together with the use of model potentials that are
very similar to ours. They used con¢gurations ranging from
the N ˆ 3 to the N ˆ 8 shell, but excluding those below
N ˆ 3. Their resonance energies hence represent the
un-shifted Feshbach QHQ eigenvalues, while our resonance
energies do include the Feshbach shifts, as the con¢gurations
below the N ˆ 3 shell are also used in our present calculation.
In general, the di¡erence between our results and those listed
in the table for a particular state is less than 0.008 Ry.
In Table V, we compare the autoionization width for the
1s23d2 1Ge(1) state of the O4+ ion with other calculations
in the literature. The width shown in Ref. [22] was also
obtained by using the Feshbach projection formalism, with
an approximation for the scattering non-resonance continuum. The present work and those in Refs. [17] and [22]
all use similar model potentials. The di¡erences in widths
are mainly due to the di¡erent representations of the continuum functions. In our present work, the continuum
Table IV. Comparisons of the resonance energies for the 1s23ln0 l 0 1Ge(1), 1Ge(2) and 1Ge(3) states of
the N3+ ions.
3d2 1Ge(l)
3p4f 1Ge(2)
3d4d 1Ge(3)
Er (Ry)
Er (Ry)
Er (Ry)
Present
Van der Hart and
Hansen [20]
Bachau et al.
[17]
Chen and
Lin [21]
Vacek and
Hansen [18]
^4.76456
^3.88708
^3.79861
^4.7670
^3.8868
^3.7982
^4.7634
^3.8842
^3.7948
^4.756
^4.76
Table V. Comparisons of the autoionization widths for 1s23ln0 l 0 1Ge(1) and 1Ge(2) states of the
O4+ ions.
3d2 1Ge(1)
G(Ry)
Physica Scripta 61
Present
Vacek and Hansen
(1989) [18]
Bachau et al.
(1990) [17]
Lin
(1993) [22]
Nakamura et al.
(1994) [23]
0.03838
0.05321
0.04013
0.05324
0.06047
# Physica Scripta 2000
Doubly-Excited States of Beryllium-Like Ions
695
Table VI. Comparisons of the autoionization widths for the 1s23ln0 l 0 1Ge states of the Ne6+ ions.
Present
3d2 1Ge(1)
3p4f 1Ge(2)
3d4d 1Ge(3)
G(Ry)
G(Ry)
G(Ry)
0.04755
0.01256
0.01265
Vacek and Hansen
(1989) [18]
Bachau et al.
(1990) [17]
Boudjema et al.
(1993) [24]
Langeries et al.
(1994) [23]
0.0585
0.04688
0.01257
0.01395
0.0503
0.0289
0.00968
0.0503
0.0150
0.00987
functions are included implicitly by using the ``nearlycompleted'' sets of products of STO. Table V also shows
the result given by Nakamura et al. [23] who used a procedure
very similar to Vaeck and Hansen [18]. The di¡erence
between Vaeck and Hansen [18] and Nakamura et al. [23]
is that second order correlation e¡ects were included in
the wave function employed in the latter work [23]. In
general, the widths reported in Refs. [18, 22 ,23] are considerably larger than those of the present result and of Ref. [17].
Table VI shows a comparison of the widths for the three
doubly-excited 1Ge states in Ne6+ ion. Our widths agree
quite well with those of Bachau et al. [17] for such a high
Z system. The width for the 3d2 1Ge(1) state obtained by
Vacek and Hansen [18] has a considerably larger value
(0.0585 Ry) that ours (0.04735 Ry). Baudjema et al. [24]
and Langeries et al. [6] used the program SUPERSTRUCTURE [25] to calculate the bound state wave
functions. To calculate the autoionization rates, the former
authors used free waves and the latter used distorted waves
for the continuum wave functions. In general their widths
are larger than the present values for the 3d2 1Ge(1) and 3p4f
1 e
G (2) states, and smaller for the 3d4d 1Ge(3) state.
Furthermore, the value for the 1Ge(2) state obtained in
Ref [24] di¡ers from the rest shown in Table VI by a factor
of more than two.
In summary, this work presents a calculation of resonance
energy and width for the doubly-excited 3ln0 l 0 1Ge states of
B+, C3+ O4+, F5+ and Ne6+ by calculating the density of
resonance states using the stabilization method. The
newly-developed procedure which employs only L2-type
wave functions, together with the use of a model potential,
is demonstrated to have obtained accurate results for both
resonance energy and width for doubly-excited states in
four-electron atomic systems.
# Physica Scripta 2000
Acknowledgement
This work was supported in part by the National Science Council under grant
No. NSC 89-2112-M-001-013.
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