L
(‘I IINI~SI~.
JOCilINAL 01: I’I IYSI(‘S
VOL.. 25, NO. I
SPRING 1987
Saddle-Point Variation Method for Autoionizing States of Atomic Systems
Kwong T. Chung ( $%j X $iR )
Department of Physics
North Carolina State University
Raleigh, N. C. 2769.5-8202
(Received 2 December 1986)
The application of Saddle-point variation method to the calculation of resonance energy and wave function is discussed. Examples are
given which show that this is a highly accurate method. To calculate
the lifetime of a resonance, the saddle-point method is combined with
the complex-rotation method. This gives a fast convergence method for
resonance energy and width.
I. INTRODUCTION
Doubly or multiply excited atomic systems play a very important role in the understanding of an atomic system. In most cases they are coupled to the continuum via Coulomb
interaction. Thus decaying to the continuum is a main mechanism for the de-excitation of
these atomic systems. Experimentally these multiply excited state appears as prominent
peaks in the electron spectroscopy. Therefore the study of these states is crucial in correctly
in.terpreting the-observed spectra.
Although these excited states are coupled to the continuum even in the non-relativistic
formulation, the coupling are usually very weak such that a square-integrable function may
give an excellent approximation to the wave function - a fact that was recognized as early
as 1933 by T. Y. Wu. [ 11. However, this coupling to the continuum present a very difficult
theoretical problem and not very much progress has been made to resolve this difficulty in
the next quarter of a century.
In the early 1960’s, with the advance of the computer technology, theoretical methods
have been developed to study these ‘resonances’ via scattering calculations [ 21. In the meantime, the analysis of Fano [3] and Feshbach [4] also placed the theoretical understanding
of these states on a firm mathematical basis. Feshbach, in particular, showed explicitly
how these states arise from the closed-channel segment of the scattering wave function, and
the name ‘Feshbach resonance’ or ‘closed-channel resonance’ has now been commonly
215
216
.
SADDLE-POINT
VARIATION METHOD FOR AUTOIONIZING STATES OF .4TOMIC SYSTEMS
adopted. Since then many theoretical methods have been developed. In this article, I will
discuss a method that is especially well suited for dealing with systems where one or more
core-electrons are vacated thus leaving the system in a highly excited state. It is called the
saddle-point technique [ 5 1.
‘In the saddle-point technique, the absence of the core is accounted for by building the
vacancy orbital directly into the wave function. The vacancy orbital wave function is
parametrised. The open-channel continua are eliminated by maximizing the total energy
with respect to the parameters in the vacancy orbital. Thus, the closed-channel resonances
are exposed as isolated discrete states, and the minimization procedure of the RayleighRitz variation method is restored.
Usually, an atomic system is characterized by its configuration and symmetry, which
is given by the good quantum numbers of the total wave function plus the principle and
angular quantum number of the individual electrons. However, many of the multiply excited
states are highly degenerate. For example, for the so called (2~2~)‘s resonance of helium,
the 2~2s configuration contributes 72 percent to the normalization whereas 2p2p contributes 28 percent. Hence the description (2~2~)‘s is only approximate. -However it is essentially correct in saying that it is the lowest i S resonance with two 1 s vacancies.
How do we build vacancy into a wave function ? Let us assume that J, (1, 2, . . ..n) is a
configuration interaction wave function with the proper angular and spin symmetries
of interest. This wave function is not antisymmetrized; hence, each particle in $ has a well
defined angular momentum. An antisymmetrized wave function with a vacancy can be
given by
$ = A [ I-- I
G,(rj)><@,(fj) II J/ (1, 2,. . 4
(1)
where A is an antisymmetrization operator. We are assuming that @,(rj) is the vacancy
orbital function for which electron j has the same symmetry and is therefore the only
particle that may fill the vanancy. Note that #0(r) does not contain any spin, the proiection
operator I $o(r) >< #o(r) 1 does not effect the s$n part of .+ (1, 2, . ..n).
Of course, the exact form of the vacancy orbital am is not available. However, a
theorem which is proved in Ref [5] can help us determine this function. In a variation
calculation, we adopt a certain wave function 4,,(r) with parameters q; the unprojected wave
function ti .contains linear parameters C and nonlinear parameters 0~. Thus the energy
expectation value be%omes
E (~,a,
s> =
<q/HI*>
<\k I*>
(2)
If we minimize this E with respect to the linear parameters C, a secular equation is obtained.
The roots of this secular equation will be a function of CY and q. For atomic systems, it is
well known that the energy of a discrete state should be a minimum with respect to (Y. The
KM'ONG T.
CHCNG
217
theorem in reference [ 51 suggests that the energy should be a maximum with respect to q.
Hence the energy of a multiply excited atomic state appear as a saddle point in this variation calculation.
In comparing with the existing theoretical methods for solving resonance problems,
the greatest advantage of the saddle-point method is the ease with which computations can
be carried out. Since only square integrable basis functions are used, the structure of the
secular equation is essentially the same as that of the ground state or singly excited states.
This means the procedure for calcu&ing the energy of a resonance is essentially the same as
calculating that of the ground state. With the maximum-minimum principle inherent in the
method, the computation becomes very straight forward. Unlike other methods for resonances in which fictitious solution may arise all soluti,ons of the Saddle-point method below
a corresponding excitation threshold are true Feshbach re,sonances. These advantages have
enabled us to obtain a large amount of theoretical data for the purpose of comparing with
existing experiments. The calculated result are found to be very accurate.*They have been
helpful in explaining some of the unresolved questions in past experiments [6], as well as
correcting some misinterpretations 171 in the literature. In the case where absolute calibration is difficult in an experiment, it is also used to make calibration [ 81. A typical
comparison between the saddle-point .result with that of the experiment for some lithiumlike atomic systems is given in Table I.
TABLE I. Comparisons of Auger energies calculated from saddle-point method and experimental results. (in eV>
State
nuclear charge
ls2s2sZS
Exp.
Theo.
ls(2s2p)3PZP
Theo.
Exp.
ls(2s2p)‘PZP
Theo.
Exp.
1 s2p2pz D
Theo.
Exp.
4a
96.14
96.1 20.1
100.58
100.58tO.l
102.79
102.79+-0.1
104.24 104.25kO.l
sa
154.85
154.94kO.l
161.26
161.24kO.l
164.12
164.12kO.l
166.36 166.38kO.l
6a
227.17
227.23kO.3
235.55
235.44kO.2
238.78
238.86k0.2
242.08 242.15+0.2
gb
412.62
412.7 kO.1
424.99
425.0 kO.1
429.71
429.6 +O.l
434.38 434.4 -10.1
a. See R. Bruch, et al. Phys. Rev. A31, 3 10 (1985) and the references therein.
b. R. Bruch, et al. Phys. Rev. A (1987, to be published).
Perhaps the most distinct advantage of the saddle point method over the other theories
is for those atomic systems where both inner core(ls) electrons are vacated. In this case,
the system is highly excited and there are an infinite numbers of open channels. For systems
218
SADD LE-POINT VARIATION METHOD FOR AUTOIONIZING STATES OF ATOMIC SYSTEMS
of more than two electron, computations become very involved or impractical for many
theoretical methods. However, it is still very straight forward for the saddle-point method.
The highly accurate results obtained by this method have helped to identify some very
interesting experiments in the literature [8,9].
One obvious limitation of the saddle-point method is that it only gives the energy
position of the resonance but not the width of the state nor the shifted energy caused by
the interaction of the discrete state with the continuum. To resolve this difficulty, one
may incorporate the saddle-point method with the well known complex-rotation method
[ 101. Let us assume
H (r, , r2 ,-. . . r,)=H(R,)= H(RN.RN)
where RN represents the set of radial corrdinates rl r2. . . rN, and RN represents the angular
coordinates collectively. The complex-rotation method suggests that one can calculate the
energy of a resonance with variation method using square integrable basis function. That is
provided that one rotate the coordinate of the Hamiltonian into the complex plane
i0
H = H(R,e ,RN).
The main difficulty of the complex-rotation method is convergence and numerical stability.
In. the Ritz-Rayleigh variation Jnethod, the solutions to the secular equation are upper
bounds to the corresponding eigenvalues of the Hamiltonian. These upper bounds approach
the eigenvalues monotonically when the basis function space is expanded. However in Eq.
(4), only stationary points are searched. In this case, increase the number of terms in a
variational wave function does not necessarily give a better energy, nor does a solution with
a lower energy always imply that it is a better solution.
It is well known that the speed of convergence in a variation calculation depends on
the quality bf the trial wave function. If the trial function can approximate the true eigenfunction closely and it is also sufficiently flexible, then the convergence is expected to be
fast. Hence, it is important to have a good idea of the solution in choosing the trial function.
This is very difficult with the rotated complex Hamiltonian. A poor choice of trial function,
on the other hand, may cause serious numerical instability.
If one has a good approximate trial function to the real Hamiltonian in the real space,
one could recover the convergence by rotating the coordinates of this trial function with
the Hamiltonian. However, this would be equivalent to the calculation in the real space.
The energy eigenvalues so obtained will be ail real; i.e., no width will come out of this
calculatibn.
RU’ONG T. CHUNG
219
The width of a resonance is the result of coupling between the closed- and openchannel segments of the resonant wave function. For narrow resonances, this coupling is
very weak. The resonant wave function can then be approximated by the closed-channel
component alone. The resonance energy position is largely determined by this component
which can be accurately obtained by using the saddle-point method. Based on these considerations a saddle-point complex-rotation method can be constructed [ 1 1 I.
To calculate the width of a resonance, the problem is first solved in the real space with
the saddle-point technique. It gives a highly accurate and relatively compact basis set
4j with optimised nonlinear parameters. We then add to this function the open-channel
segment. The total wave function is then given by
* (RN)= C Cj”j (RN)+A’
dik$i(RN_1)Uk (RN)
(6)
i, k
j
Here C and d are the linear parameters, $i are the open-channel target states, and A is the
antisymmetrization operator. The angular coupling in the second term of this equation is
suppressed. It is understood that if $ takes a set of good quantum number L, M, S, and
S, (IS coupling scheme), then the target state and the U, will couple in such a way that
the correct angular and spin quantum numbers will be obtained.
When H rotates through an angle 0, we adjust Eq. (6) as follows:
i0
*
WNe
,.Q~)=
CCj~j(R,eie,nN)+ACii(RN_leio,~N_l)U~
(RN)
i, k
j
(7)
With this $, the width and shift are calculated from the secular equation which is obtained
through Eq. (4). This method has been applied in many two-and three-electron systems.
It is found that in most cases the results are extremely stable. For instance, the real part of
the energy is stable beyond eight significant figures in many calculations, the imaginary
(width) part is stable to four or five significant figures. Only in the case where the width
is extremely small the stability of the imaginary part becomes much poorer. An example
of the energy and width calculation is shown in Table II.
TABLE II - Comparison of widths from saddle-point complex-rotation method and experimental results (in meV)
___..
__,
SYSTEM
RESONANCE
THEORY
EXPERIMENT
Hea
2s2p’ P
37.4
38*2
Hemb
1 s2s2s2 s
11.6
12,13, etc.
Lit
1 s2p2p2 D
11.0
10.4kO.26
.,
‘.
-
.’
..
-.--,
.
220
SADDLE- P OINT VARIATION METHOD FOR AUTOIONIZING
STATES OF ATOMIC SYSTEMS
Be +c
1 s(2s2p)3 PZ P
4.08
4.58?0.13 & 4.08+0.1 I
Be*
1 s2p2p2 D
27.6
30.3kl.l
a. See K.T. Chung and B.F. Davis Phys. Rev. A31, 1187 (2985) and the references therein.
b. See B.F. Davis and K.T. Chung Phys. Rev. A29, 1878 (1984) and the references therein.
c. See B.F. Davis and K.T. Chung Phys. Rev. A3 1, 30 17 (1985) and the references therein.
Accurate calculation for the energy position and width of resonance states is not only
important in itself, it is also important in the calculation of lifetimes for many metastable
states of the atomic systems. For example, the ls2~2p~Ps,~ 1,2 states of the Li-like system
is metastable against auto-ionization in the non-relativistic approximation. However, due
to Breit-Pauli interaction, they may decay to the ‘P continnum by directly couple to this
continuum or by the mixing of this quartet with the ls2s2p2 P,,, 3,2 resonances. This
coupling depends sensitively on the relative energy position of the ‘P and 4P states. We
found that these mixings are contributing a major part of lifetime for these 4P states. In
recent experiments [ 121, it is found that the lifetimes of the 1s2p2p4P,,, of NV and OVI
is much shorter than that of the ls2~2p~Ps,~ 1,2 components. This is caused by the coupling of the 4P,,2 with the 1 s2p2p2 D,,, . Hence, an accurate ‘D resonance energy and
width is crucial in obtaining the correct lifetime for this 4P state.
The application of this saddle-point method has been limited to the two-and threeelectron atomic systems thus far. This is mainly because here large amounts of experimental
data are available and accurate theorical calculations can be carried out to compare with
these data. We are also planning to apply this method to four electron systems in the near
future.
ACKNOWLEDGEMENT
This work is supported by the National Science Fobndation. Grant No. PHY-840 5 6 4 9
REFERENCE
1.
2.
3.
4.
5.
T.Y. Wu, Phys. Rev. 46,239 (1934); 66, 291(1944).
P.G. Burke and H.M. Schey, Phys. Rev. 126, 147( 1962)
U. Fano,Phys. Rev. 124. 1866(1961).
H. Feshbach, An. Phys. 5,357( 1958); 19, 287( 1962).
K.T. Chung, Phys. Rev. A20, 1743( 1979).
IiWONG T. CHUNG
221
6. K.T. Chung, Phys. Rev. A23, 2957( 1981); ibid 24, 1350( 1981).
7. K.T. Chung, Phys. Rev. A22. 1341( 1980); ibid 23, 1079( 1981).
8. K.T. Chung, Phys. Rev. A2.5, 1596( 1982’); B.F. Davis and K.T. Chung, J. Phys. Bl.5,
31 13(1982).
9. M. Rodbro, R. Bruch and P. Bisgaard, J. Phys. B12,2413( 1979).
10. E. Balslev and J. M. Combes, Commun. Math. Phys. 22, 280( 197 1).
1 1. K.T. Chung and B.F. Davis, Phys. Rev. A26, 3278( 1982).
12. A.E. Livingston, J.E. Hardis, L.J. Curtis, R.L. Brooks, and H.G. Berry, Phys. Rev.
A30, 2089( 1984).