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Recent Developments

and Applications

Page 136


We need not rigorously require 'lip and '¥Q to be orthogonal, in which
case this partitioning does not correspond to the rigorous projection in
Feshbach theory. We do not have to require this orthogonality, since we are
using the total wave function in the variational calculation. This partitioning
is also not unique, but merely represents a convenient way of viewing the
variational wave function so that physical insight can be incorporated into its
structure. For shape resonances, this partitioning may, in fact, be less useful,
although it can be used to partition different classes of configurations that
we might include in a variational wave function for a shape resonance, as
discussed later.

For Feshbach resonances, 'II Q contains closed-channel configurations as
well as polarization and correlation configurations. For the same reasons just
stated, these resonances can best be represented in terms of basis functions
with real nonlinear parameters. On the other hand, 'lip represents the open-
channel part, and as such, it will consist of an antisymmetrized product of
target function(s) and functions of the form in Eq. (64) or Eq. (65): i.e.,

'II = q>d" C A.TXreS
p L... v'+'v 11 (67)

where q> represents the projection of the appropriate spin and space group
symmetries, d is the antisymmetrizer, and ¢; is the target wave function for
the vth open channel. The real basis functions in ",~es for different channels
may be chosen to be the same or different functions. However, the complex
functions should certainly be different for different channels.

In the case of shape resonances, 'lip is basically constructed in the same
manner as previously discussed for Feshbach resonances. Correlation-type,
including polarization, configurations could be grouped into'll Q. For both
Feshbach and shape resonances, correlation between all open and closed
channels is automatically incorporated. Finally, the approximate resonant
energy Eres should be stabilized with respect to the parameters in ¢;, since the
presence of the additional electron will distort the target functions.

6.4. Field Ionization

Cerjan et al.(34) have used the numerical range* of

Hfie1d(O) = - [exp( - 2iO)/2]V2 + exp(iO)For (68)
and the fact that the Coulomb potential is compact relative to Hfie1d(O) to show
that the transformation in Eq. (1) can be used to uncover the Stark broadened

* The numerical range R(O)(34) of an operator 0 is defined as the set of all complex numbers that
might be obtained as a diagonal matrix element with respect to a normalized function Iu), i.e.,

R(O) == {z == <uIOlu)EC: <ulu) = I}

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124 B. R. JUNKER

and shifted atomic states. The theorem stating that the resonant wave
functions depend on rand e in only the combination r exp (iO) applies here also.
Thus, we should be able to compute the complex energies associated with the
Stark broadened and shifted atomic states by using the unrotated Hamil-
tonian directly with a square-integrable basis. For the nonrelativistic case of a
constant field directed along the z axis, for example, only M L remains an exact
quantum number that group theoretically characterizes the states. The
hydrogenic case is an exception where the fortuitous form of the Coulomb
potential permits complete separation of variables and, thus, the necessary
three quantum numbers that completely characterize each state. For an N-
electron atomic system, we could construct a variational wave function of the


'P = SIlL Lcp~(N - 1){ ~ C~jxr(N) + rL + 1j exp(iksjr)
L S j= 1

X {C~(ns+ l)rmj exp( - CXSjr) + C~(ns+ 2)rnj exp [ - (CXSj + I:Sjr)] YL} YL}

x exp (ikSjr) exp( - CXSjr) Yf ] (70)

where SII is as described above; C~j, C~(ns+ 1)' and C~(ns+2) are linear
variationally determined coefficients. The cp~(N - 1) represent wave
functions for the (N - 1) electrons, which are relatively stable with respect
to field ionization; the xr(N) are a set of basis functions describing the
ionized electron; and lj' mj, and nj are greater than or equal to zero. In writing
Eqs. (69) and (70), the assumption that only one particle is unstable with
respect to ionization is implicit, which is the case generally studied. The
integrals required for the preceding wave function are straightforward, where
as a Siegert-like wave function using basis functions representative of just
the case of a hydro genic system in a constant field would contain functions
with exponentials of fractional powers of r, cos X, and sin X. Such integrals
must be performed numerically. Finally, the effects of fields on resonances,
such as those studied by Broad and Reinhardt(35) and Wendoloski and
Reinhardt,(36) can be studied by a straightforward merger of Eqs. (66) and
(69) or (70).

These examples should indicate the general approach that could be
taken in constructing a variational wave function to study various resonant
phenomena. The fact that boundary conditions can be implicitly incorporated

Page 272


Resonant (cont.)
energy ('~), 15, 17,21-25

see also position
function, 13, 103, 104, 106-108, 121f
parameters, 2
part, 6
phase shift, 7, 24
position, 9, 103

see also energy
quantities, 6

see Shift

wave function, 103, 104, 106-108, 121f

see Width
Rotational harmonics, ('2ll functions), 19, 20
Rope resonance, 66

see Feshbach resonance
Rotational predissociation

see under Resonances
Rydberg constant,

of infinite mass (Rx), 32
of finite mass, (RM), 32

Sack, 50
Saddle-point technique, 76
Satellite lines, 199-20 I, 206
Scattering matrix

analytical models, 109f
pole, 103, 107

Schrodinger equation, I
Schulz resonance, 63

see He - [lS(2S)2 2S]
Secular equation, 75
Shape resonance, 88

see Resonance, Accordion resonance
Shift (a), 9, 10, II, 15, 16,25,27,28,29,

30, 31, 32, 33
see also Feshbach energy shift

Siegert boundary condition, 104, 106f, 125,
131, 132

Single particle projection operators, 77
Singularity, 5
Solar activity

active regions, 176, 177,220,225,233,

coronal transients, 177, 179
prominences, 177,241
sprays, 177

Solar activity (cont.)
surges, 177,225,241

see also Aares
Solar wind, 176
Spectrographs, 171, 189, 190

see Spectrographs

continuous, 5, 29
discrete, 5, 29
eigenspectrum, 12
spectral decomposition, 5

Spicules, 180
Spurious states, 55, 58, 59

eigenvalues, 56, 62
resonance, 57

Stabilization method
complex version, 143, 166-168
shape resonances, 156, 157

Stars, 182, 235-243
Stellar atmosphere

see Stars
Stellar flares, 237, 241
Sum rule, 40, 41
Supergranule network, 179
Supernova remnants, 182

Target (systems)
closed shell, 63
He (2'S), 64, 68
He (2'S), 64
one-electron, 35
open-shell, 47, 63



two- (or more) electron, 35, 47, 57
Three-body recombination

see Recombination
Three-electron system, 60-62

2S states, 58
see He-, Li

Tokomak spectra, 197,200,201,208,213
Transition probabilities

electric dipole, 193, 210
magnetic dipole, 210, 213, 231, 232
magnetic qudrupole, 245

Transition region, 178-180, 182, 187, 188,

Vacancy orbitals, 77

Page 273


Variational principle, 102f, 105

Width, 9, 10, II, 15, 16,25-28,30,31,

63, 64, 68, 69, 103

Width (cont.)
partial, 64, 68
wide, 51, 67, 68
see also Breit-Wigner resonance



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