J. Phys. B: At. Mol. Phys. 18 (1985) 3481-3486. Printed in Great Britain
Doubly excited states of helium atoms between the N = 2 and
N = 3 He+ thresholds
Y K Ho and J Callaway
Department of Physics and Astronomy, Louisiana State University, Baton Rouge, Louisiana
70803, USA
Received 8 February 1985, in final form 30 April 1985
Abstract. Some doubly excited autoionising states of helium atoms converging on the
N = 3 He" threshold are calculated by use of a method of complex coordinates. Hylleraastype wavefunctions are used for L = 0 and L = 1 resonances with products of Slater-orbitaltype wavefunctions with expansion lengths up to 319 terms for , 5 3 2 resonances. States
of parities (-1)= and (-l)L+' and with angular momenta of L S 4 are calculated. Resonance
energies and autoionisation widths are reported.
1. Introduction
Studies of doubly excited resonances in helium atoms are of continuous interest from
both the experimental and theoretical viewpoints. Recent experimental interest
includes observations of optical transitions for doubly excited states in beam-foil
spectra (Bruch et a1 1984, Cederquist et a1 1983). In the experiment of Bruch et a1
(1984) some transitions involving doubly excited states of extremely narrow width
below the N = 3 He+ threshold were observed. In the beam-foil experment of Cederquist et a1 (1983), the autoionisation width of the 2s2p3P0 state was measured by
observing different transitions involving the 2s2p 3P0 state. In addition, they also
observed some transitions that were believed to involve the 3p3d3D0 state. On the
theoretical side there are several studies of resonances between the N = 2 and N = 3 He'
thresholds. These studies are close-coupling calculations (Burke and Taylor 1969,
Wakid and Callaway 1980), different variants of the Feshbach projection formalism
(Chung 1972, Oberoi 1972, Herrick and Sinanoglu 1975, Lipsky et a1 1977, Chung and
Davis 1980) and a method of complex coordinates (Ho 1979). In studies of resonance
phenomena, the parameters that are of interest are resonance energy positions and
resonance widths (related to the inverse of the autoionisation lifetimes of the doubly
excited states). Other theoretical studies of doubly excited resonances are the investigations of the dynamics of these resonances, as well as the symmetry of the Hamiltonians
for two-electron systems. The theoretical studies of such approaches include investigations of the doubly excited states by use of a group theoretical method (Herrick and
Sinanoglu 1975, Kellman and Herrick 1980) and by use of hyperspherical coordinates
(Lin 1983).
The doubly excited resonances of He can be produced in a formation-type experiment (i.e. in e--He' scattering) or in a production-type experiment (i.e. in photon-atom
or ion-atom scattering). Once the doubly excited resonant states are formed, the decay
0022-3700/85/ 173481 + 06$02.25
0 1985 The Institute
of Physics
3481
Y K Ho and J Callaway
3482
of such states will go through one of the following routes. They may autoionise (by
ejecting an electron) to one of the He+ ion states, or they may cascade by radiative
transitions to one of the lower (singly or doubly) excited states of He. The radiative
transition and autoionisation processes compete with each other. For resonances with
large autoionisation widths (hence short lifetimes) the autoionisation process is the
dominant factor for low-Z ions. For resonances with narrow or extremely narrow
widths, the radiative decay rates are comparable with, or even dominate, the autoionisation rates. It was such doubly excited states with narrow widths that were observed
in the recent beam-foil experiments (Bruch et a1 1984).
Because of the recent experimental and the continuous theoretical interest, we now
carry out an investigation of the resonance states in helium atoms between the N = 2
and N = 3 He+ thresholds. A method which is able to calculate both resonance positions
and total widths with accurate results is the method of complex coordinates (see reviews
by Ho 1983, Junker 1982 and Reinhardt 1982). The advantage of using this method
is that resonance parameters can be obtained by using bound-state-type wavefunctions
and no asymptotic wavefunctions are necessary. Such an advantage becomes apparent
when we are calculating a resonance in which many channels are open. the calculation
of the resonance position and total width for a many channel resonance is as straightforward as that for an elastic resonance. The use of this method has been very successful
for calculations of resonances for L = 0 and L = 1 states of two-electron systems (Ho
1979, 1982). We now extend this method to L z 2 states for helium atoms.
2. Wavefunctions and calculations
The wavefunctions used in this work are of the Hylleraas type for the L = 0 and L = 1
resonances,
W=
Ckmn
exp[-a
(rl+
r2)IrL( r; Y
Y +yoo(
~ 1
*
yLo(2) r2”C + L
yoo(2) yLo( 11)
(1)
+ +
where ( k n m ) s w, with w a positive integer, and products of Slater orbitals for
resonances with L 2 2.
where
qa,(r) =
exp(-&rr).
In equation (2), A is the antisymmetrising operator, S is a two-particle spin eigenfunction and the 77 are individual Slater orbitals. Y is an eigenfunction of the total angular
momentum L,
yk,c;lb(l, 2, =
c
C(la, zb,L;mla, mlb, M)Y/a,m,~(1)Y/b,mrh(2)
mi, mih
with C the Clebsch-Gordan coefficients. It has been suggested (Ho and Callaway
1983) that the use of Slater orbitals to calculate resonances of high angular momenta
and associated with higher excitation thresholds can produce accurate results. In a
calculation of resonances in hydrogen negative ions, it was found that the use of
Slater-orbital-type-wavefunctions indeed produced results for some L z 2 states that
agreed well with an extensive 18-state close-coupling calculation.
Doubly excited states of helium
3483
In this work, quite extensive basis sets are used for the wavefunctions. For example,
for the IDe resonances we use a total of 9 s-type, 8 p-type, 7 d-, 6 f-, 4 g-, 3 h-, 2 i-,
2 k-, and 1 1-type orbitals. These orbitals would couple to a total of 281 terms for the
'De states. The actual numbers of terms for other resonant states differ slightly
depending on different angular momenta and parities. However, in any case, no less
than 220 terms are used in the present calculation. The most extensive case is the 3Ge
states for which 319 terms are used. By using such extensive basis sets, components
of open channels below the N = 3 He+ threshold are included in the wavefunctions
either explicitly or implicitly. This is the reason why we are able to calculate the total
widths for such resonances.
The theoretical aspects of the complex rotation method have been discussed in
previous publications (see Ho 1983, for example) and will not be repeated here. Instead
we only briefly describe the computational procedures. First, we use the stabilisation
method to obtain optimised wavefunctions with which complex-coordinate calculations
will then be carried out. The use of the stabilisation method as a first step for the
method of complex-coordinate rotation has been demonstrated in a recent review (Ho
1983). Once the stabilised wavefunctions for a particular resonance are obtained, a
straightforward complex rotation method is applied, and the so called 'rotational paths'
are examined. The final resonance parameters, both resonance positions and widths,
are then deduced from the conditions that a discrete complex eigenvalue was stabilised
with respect to the non-linear parameters in the wavefunctions (equations ( 1 ) and (2))
and with respect to 8, the so-called rotational angle of the complex transformation
I + r exp(i8).
3. Results and discussions
Table 1 shows results of doubly excited resonance of parities ( - l ) L and associated
with the N = 3 He+ threshold. It should be mentioned that some of the lower-lying
' S e and 'Poand 3P0states have already been published (Ho 1979, 1982). For completeness, we include them here. The 3Se resonances are new calculations using the method
of complex coordinates. Table 2 shows results of doubly excited resonances of parities
of (-l)LcL. For resonances where widths are not shown in tables 1 and 2, we estimate
the widths of such states to be less than 10-6Ryd. The resonance parameters are
believed to be quite accurate. Here, we mention other theoretical studies for the
resonances between the N = 2 and N = 3 He+ thresholds. These include the closecoupling calculations (Burke and Taylor 1969, Wakid and Callaway 1980), a stabilisation calculation (Callaway 1978) and different variants of Feshbach projection calculations (Chung 1972, Oberoi 1972, Herrick and Sinanoglu 1975, Lipsky et a1 1977, Chung
and Davis 1980). In the close-coupling calculations, both the resonance positions and
widths are calculated. In these calculations, however, only a limited number of
resonances are reported. In comparison, the resonances reported in the present work
are more extensive. For the calculations of Feshbach resonances mentioned above,
no Feshbach shifts (the interactions between the open and closed channels) are
included. Also, no widths are provided in the Feshbach calculations except for those
of Herrick and Sinanoglu (1975), who used a formula of the Fermi golden-rule type.
However, their open-channel scattering wavefunctions were approximated by Coulomb
wavefunctions. Such an approximation would lead to uncertainties in the widths.
Furthermore, the small number of expansion sets used to represent the closed part of
3484
Y K Ho
and J Callaway
Table 1. Autoionising energies and widths (in Ryd) of He doubly excited states associated
with the N = 3 He+ threshold. Parties of these states are (-l)L.
-4
State
1
2
3
4
5
1
2
'se
3Se
0.574555
0.540 569
0.516 266
0.499918
0.497 996
0.000 06
0.000 094
0.000 040
0.000 005
0.000 022
'p"
0.671 25
0.571 90
0.565 65
0.542 45
0.535 285
0.514 86
0.503 20
0.501 55
0.496 57
0.491 1
0.014 0
0.000 056
0.002 9
0.006 2
0.000 02
0.000 03
0.001 0
0.002 4
3p0
0.70075
0.618 765
0.558 95
0.557 633
0.520 440
0.517 042
0.510 28
0.507 13
0.492 1
0.005 95
0.002 23
0.002 74
0.000 12
0.000 54
0.000 035
0.001 1
ID'
0.686 28
0.631 4
0.580 0
0.551 56
0.549 418
0.524 65
0.509 164
0.506 5
0.503 33
0.499 28
0.010 5
0.008 6
0.002 4
0.004 8
0.000 05
0.003 1
0.000 54
0.002 0
3De 0.650 65
5
6
7
8
9
10
5
6
7
8
9
10
'Fo
3
4
5
6
7
8
1
2
3
4
5
6
7
r
0.006 0
0.013 3
0.000 01
0.004 7
0.000 01
3
1
2
-Er
0.707 08
0.634 9
0.562 3
0.527 0
0.514 75
4
1
2
3
4
r
'G'
0.608 43
0.555 92
0.521 95
0.515 72
0.506 89
0.502 88
0.492 3
0.489 9
0.006 5
0.000 18
0.001 3
0.000 84
<1 x10-6
0.000 05
0.002 4
0.003 5
0.614 10
0.523 6
0.510 35
0.494 0
0.487 8
0.487 0
0.483 8
0.013 4
0.004 8
0.000 1
0.005 0
0.001 0
0.566 09
0.539 52
0.534 61
0.523 20
0.511 94
0.501 74
3Fo
0.66328
0.541 95
0.534 83
0.522 26
0.507 71
0.502 09
0.499 3
0.001 4
0.000 098
0.000 072
0.000 44
<1 x
<1x10-6
0.000 006
0.000 16
3Ge 0.539 135
0.518 29
the wavefunctions is another source of uncertainties in their calculations. Table 3
shows a comparison of the present 'De resonances with other theoretical calculations.
Comparisons for the S - and P-wave resonances with other calculations have been
discussed in a previous publication (Ho 1979).
On the experimental side, the present calculations are helpful for identifying optical
transitions of doubly excited states. Not only do the present results provide more
Doubly excited states of helium
3485
Table 2. Autoionising energies and widths (in Ryd) for doubly excited states of He
associated with the N = 3 He+ threshold. Parities of these states are (-l)'+'.
-E,
State
1
2
3
4
'pe
1
2
r
r
3Pe
0.670 5
0.582 30
0.542 8
0.008
0.000 05
0.004
0.000636
'Do
0.631 15
0.550 519
0.524 868
0.504 36
0.502 42
0.004 16
0.000 0384
0.001 38
0.537 07
0.521 91
0.497 73
0.000 001
0.000 09
3Fe
0.621 45
0.525 66
0.516 55
0.003 9
0.000 88
0.000 30
0.522 55
0.492 35
0.487 45
0.000 17
'Go
0.513 54
0.000 080
0.557 986
0.518 703
0.507 278
0.489 05
0.000088
'DO
0.656 46
0.539 48
1
2
3
'P
1
2
3
* G"
0.000 052
Table 3. Comparison of the 'De resonance energies, - E , (Ryd), and the width
given in brackets, obtained by various methods.
r (Ryd),
Close coupling
'De (1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
(9)
(10)
a
Complex
coordinates"
Feshbach
projectionb
0.686 28 (0.0105)
0.631 4
0.5800
0.551 56
0.549418
0.52465
0.509 164
0.5065
0.503 33
0.49928
0.689 6
0.630 12
0.581 16
0.553 16
0.549 52
0.524 08
0.509 16
0.507 24
0.503 28
0.499 04
3 cc + correlations'
9 ccd
0.684 8 (0.0113)
0.690 (0.0100)
Present calculations.
Oberoi (1972).
Burke and Taylor (1969) (3 atomic states plus 20 correlation terms).
Wakid and Callaway (1980) (6 atomic states and 3 pseudo-states).
accurate resonance energies, but they also provide more accurate values of the widths.
As a result, they are able to help to eliminate many transitions involving large autoionisation widths, and help to acheive more accurate identifications of the satellite lines in
high-resolution EUV beam-foil spectra.
Also in this work we report resonance parameters for the 3p3d 3D0
state. The width
for this state is 0.004 16 Ryd, and it is reported for the first time in the literature. This
3486
Y K Ho and J Callaway
result should also be of experimental interest. For example, Cederquist et a1 (1983)
have observed transitions at 1711 and 1821 A in the emission beam-foil spectrum of
helium. The two satellite lines were identified as 3p3p 3De-3p3d 3D0 and 3p3p 3Pe3p3d 3Do
transitions, respectively. In the experiments of Cederquist et a1 the authors
were able to extract the autoionisation widths for some doubly excited resonances.
The present theoretical values of the widths should be useful to experimentalists.
Finally, we mention an interesting finding of the present calculations. According
to the present results, the 3Fo(1) and 3F0(2)resonances have extremely narrow widths.
Such a finding contradicts those of Herrick and Sinanoglu (1975). It would therefore
be interesting to investigate experimentally the radiative decays of such resonances to
the lsnd 3D states of helium atoms.
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