Drake G.W.F. (editor) Handbook of Atomic, Molecular, and Optical Physics

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Part B 24

391

Autoionization

25. Autoionization

The phenomenon of autoionization, or more

particularly the autoionization state itself, is

treated for the most part in this chapter as

a bound state. The process is rigorously a part

of the scattering continuum (Chapt. 47), but, due

mostly to the work of Feshbach [25.1], a rigorous

formulation can be established whereby the main

element of the theory can be made into a bound

state problem with the scattering elements built

around it. The major constituent of both these

features is accomplished with projection operators.

A brief description of the above elements of the

theory, centered around projection operators, is

the aim of this chapter [25.2], although some

additional methods are discussed in Sect. 25.5.

Rydberg units are used unless otherwise noted.

25.1 Introduction ....................................... 391

25.1.1 Auger Effect .............................. 391

25.1.2 Autoionization, Autodetachment,

and Radiative Decay .................. 391

25.1.3 Formation, Scattering,

and Resonances........................ 391

25.2 The Projection Operator Formalism ....... 392

25.2.1 The Optical Potential ................. 392

25.2.2 Expansion of V

The QHQ Problem....................... 392

25.3 Forms of P and Q ................................. 393

25.3.1 The Feshbach Form.................... 393

25.3.2 Reduction for the N =1Target..... 394

25.3.3 Alternative Projection

and Projection-Like Operators .... 394

25.4 Width, Shift, and Shape Parameter ....... 394

25.4.1 Width and Shift ........................ 394

25.4.2 Shape Parameter....................... 395

25.4.3 Relation to Breit–Wigner

Parameters............................... 396

25.5 Other Calculational Methods ................. 396

25.5.1 Complex Rotation Method .......... 396

25.5.2 Pseudopotential Method ............ 397

25.6 Related Topics ..................................... 398

References .................................................. 399

25.1 Introduction

25.1.1 Auger Effect

Autoionization falls within the general class of phe-

nomena known as the Auger effect. In the Auger

effect an atomic system “seemingly” (Sect. 25.1.3) spon-

taneously decays into a partition of its constituent

parts.

25.1.2 Autoionization, Autodetachment,

and Radiative Decay

If the initial, composite system is neutral, or positively

charged, and its constituent decay particles are an elec-

tron and the residual ion, then the process is called

autoionization. If the original system is a negative ion,

so that the residual heavy particle system is neutral, then

the process is technically called autodetachment;forthe

most part, the physics and the mathematical treatment

are the same.

It is also possible, before electron emission takes

place, that the system will alternatively decay radia-

tively to an autoionization state of lower energy, or a true

bound state of the composite system. The latter process

is called radiative stabilization (which is a basic part of

dielectronic recombination (Chapt. 55)).

25.1.3 Formation, Scattering,

and Resonances

Autoionizing states are formed by scattering processes

and photoabsorption. These are the inverses of the auto-

ionization and photon emission processes by which

they can decay. In the scattering process, formation

of the autoionization state corresponds to a resonance

Part B 25

392 Part B Atoms

in the scattering cross section (Chapt. 47). Autoion-

ization is the process which corresponds to the decay

of the resonance. The decay of the resonance (au-

toionization) is then seen to be the last half of the

resonant scattering process. Strictly speaking there-

fore, the resonant or autoionization state, although it

may be long-lived, is not completely stationary, and

that is the reason that the word “seemingly” was used

to describe the Auger process. The compound system

can also be formed by absorption of photons im-

pinging on a bound state (usually the ground state)

of the compound system, in which case the aution-

ization state shows up as a “line” in the absorption

spectrum.

25.2 The Projection Operator Formalism

In the energy domain where the Schrödinger equation

(SE)

HΨ = EΨ

(25.1)

describes scattering, the wave function does not vanish

at inﬁnity, i. e.,

lim

→∞

Ψ =0 , (25.2)

where r

is the radial coordinate of ith electron.

The basic idea of the projection operator formal-

ism [25.1] is to deﬁne projection operators P and Q

which separate Ψ into scattering-like (PΨ ) and quadrat-

ically integrable (QΨ )parts:

Ψ = PΨ +QΨ.

(25.3)

Implicit in (25.3)are

completeness: P +Q =1 ,

idempotency: P

= P , Q

= Q ,

orthogonality: PQ = 0 ,

(25.4)

and the asymptotic properties

lim

→∞







PΨ = lim

→∞

QΨ = 0

(25.5)

25.2.1 The Optical Potential

Straightforward manipulation of (25.1)and(25.3) leads

to an important relation between QΨ and PΨ :

QΨ = (E −QHQ)

−1

QHPΨ. (25.6)

From (25.6) a basic equation for PΨ (which, by

virtue of (25.5), contains all the scattering information)

emerges:

(PHP +V

−E)PΨ = 0 . (25.7)

The most signiﬁcant part of (25.7) is the optical potential

given by

= PHQ(E − QHQ)

−1

QHP . (25.8)

is a nonlocal potential; the most incisive way to give

it meaning is to the deﬁne the QHQproblem.

25.2.2 Expansion of V

:TheQHQ Problem

The following eigenvalue problem constitutes the heart

of the projection operator formalism:

QHQΦ

= E

. (25.9)

For calculational purposes, it is best to recast (25.9)in

the variational form



ΦQHQΦ

ΦQΦ



= 0 .

(25.10)

This equation may yield a discrete plus a contin-

uous spectrum in an energy domain where the SE

has only a continuous spectrum. Moreover, if the

Q operator is appropriately chosen, then the dis-

crete eigenvalues E

are close to the desired class

of many-body resonances, called variously Fesh-

bach resonances, core-excited [25.3], or doubly-excited

states. In terms of these solutions, V

has the

expansion





PHQ|Φ

(E −E

)

−1

Φ

|QHP . (25.11)

Part B 25.2

Autoionization 25.3 Forms of P and Q 393

25.3 Forms of P and Q

25.3.1 The Feshbach Form

Projection operators are not unique. Feshbach [25.1]has

sketched a derivation of a “robust” projection operator

for the general N-electron target system. Robust means

that QΦ is devoid of open channels. The complete ex-

pression for P, including inelastic channels has been

derived in [25.5]

P =

N+1



i=1





(i)







(i)





) · ψ



(i)



) · ψ



(i)



−1



(25.12)

where the prime on the second summation means that

terms with λ

=1 are to be omitted, r

denotes the radial

coordinate of electron i,andr

(i)

stands for the angular

and spin coordinates of electron i plus all coordinates

of the remaining N electrons. Thus r

(i)

indicates the

totality of all coordinates of the (N +1) electrons except

the radial coordinate of the ith electron r

. ψ(r

(i)

) is

the vector channel wave function in which the angular

momentum and spin of the electron i are coupled to

the target in state ν (ν = 0, ... ,ν

max

). A component of

a vector labels the inelastic channel, and dot products

represent sums over channels. The Q operator is then

made explicit by completeness: Q = 1 − P,(25.4).

The α-indexed quantities in (25.12) arise from ex-

change; they are the eigensolutions of the integral

eigenvalue problem

) = λ

K(r

) · v

)

. (25.13)

Here, K(r

) is a matrix with components

µν

) ∝ N





(i)





( j)



(ij)

, (25.14)

and r

(ij)

indicates that all variables, except r

and r

are

integrated over. In the inelastic regime, therefore, the

,λ

] are not associated with speciﬁc channels, but

rather with the totality of open channels. This means

that every component of v

is associated with all inelas-

tic channels [25.5]. The v

are orthogonal, and can be

normalized so that v

· v

=δ

αβ

.Theλ

obey several

sum rules [25.4], of which the most useful is



α=1

(λ

)

−1

max



ν=0

K

νν

(r|r)

, (25.15)

where n

is the number of eigenvalues of (25.13). The

can be accurately calculated by use of a variational

principle [25.4]. A test of (25.15)forthelowestHe

−

(

resonance, using Hylleraas functions to construct Q in

the evaluation of QHQ, is shown in Table 25.1 [25.4],

and results for the resonance position are compared in

Table 25.2.

Table 25.1 Test of sum rule (25.15)forthelowest

−

(1s2s

S) autodetachment state [25.4]. Projection

operators are based on a 4 term Hylleraas φ

and the vari-

ational form of v

given below. Values of other constants

are given in [25.4]

∝ (1 +C

) exp(−γ

−γ

)

+(r

↔r

∝



(α)



exp(−γ



(α)



exp(−γ

r),

and the variational eigenvalues are [25.4]

= 1.009 453 λ

= 232.8540

= 80 101.08 λ

= 4 817 341

Summation Value (Ry

−1

)



α=1

(λ

)

−1

0.994 9425

K(r|r)

0.994 9514

Table 25.2 Comparison of methods for calculating the en-

ergy of the lowest He

−

(1s2s

S) autodetachment state.

The QHQ results are denoted (

E − E

)

Quasi

for the quasi-

projection method and (E − E

)

Complete

for the complete

projection method [25.4]. The entries labeled “Other re-

sults” give the full resonant energy. Units are eV

Target (

− E

)

Quasi

(E − E

)

Complete

Closed shell 19.366 19.593

Open shell 19.385 19.666

|1s1s



|+2p

19.388 19.615

4 term Hylleraas 19.381 19.496

10 term Hylleraas 19.379 19.504

Other results 19.402

,19.376

,19.367

,19.367 ±0.007

Values obtained using R

∞

= 13.605 698 eV and

=−79.0151 eV [25.6]

Complex rotation method; Junker and Huang [25.7]

Hole-projection complex-rotation; Davis and Chung [25.8]

Hermitian-representation complex-rotation; [25.9]

Experiment; Brunt et al. [25.10]

Part B 25.3

394 Part B Atoms

25.3.2 Reduction for the N = 1Target

Explicit rigorous P and Q operators of the above type

are only possible for N = 1 (i. e. hydrogenic) targets.

In that case, spin can be easily eliminated by using spa-

tially symmetric or antisymmetric wave functions. In the

elastic region, P and Q reduce to [25.11, 12]

P = P

+ P

− P

; Q =1 − P

− P

+ P

(25.16)

Here the P

= φ(r

)φ(r

) are purely spatial projectors.

Forms for the inelastic continuum are easily general-

ized [25.13].

There have been many calculations of QHQ for

two-electron systems (N =1), starting with fundamental

work of O’Malley and Geltman [25.13]. A small sample

isgiveninTable25.3. They are given for their historical

importance, demonstrating for the ﬁrst time the con-

vergence of eigenvalues to well deﬁned values in the

continuum. All eigenvalues (below the n = 2 threshold,

in this case) correspond to resonances.

Table 25.3 Energies E

of the He(2s2p

) autoionization

states below He

(n = 2) threshold from the variational

calculations of O’Malley and Geltman [25.13]. Units are

Ry. N is the number of terms in the trial function

N s = 1 s = 2 s = 3

9 −1.377 08 −1.178 92 −1.097 16

15 −1.380 44 −1.183 12 −1.104 32

20 −1.382 16 −1.183 48 −1.108 28

25 −1.383 16 −1.190 00 −1.111 88

25.3.3 Alternative Projection

and Projection-Like Operators

Two alternative methods based on the idea of projec-

tion are available: quasi-projectors and hole projection

operators.

Quasi-projectors [25.14] relax the condition of idem-

potency, but still maintain a discrete spectrum, which is

in a one-to-one correspondence with resonances, with

a predeterminable number of exceptions [25.2].

Hole projection operators have proven to be a more

practical and effective approach [25.15]. The method

uses one-particle (say, hydrogenic) orbitals, φ

(q;r),

to build holes via projectors, [1 −



φ

], op-

erating on the (N +1)-particle wave function. The

Rayleigh–Ritz functional is minimized with respect to

the parameters in Φ, but it is maximized with respect

to q, the nonlinear parameter in all the φ

. In a model

case, this minimax procedure has been shown [25.15]

to optimize the eigenvalues to describe resonance ener-

gies; many calculations since then [25.16] have veriﬁed

the minimax criteria in many-electron systems. More re-

cently, the technique has been combined with complex

rotation, so as to enable calculation of other resonant

quantities [25.17]. Remarkably accurate results have

been obtained.

Finally, hole projectors are ideally suited for inner-

shell vacancy states of many-electron systems if high

accuracy is required [25.2]. Effectively, this amounts to

a reliable method for optimizing parameters of a hole

orbital to be used in an Auger transition integral for ﬁll-

ing such a vacancy, although that method has apparently

never been used (Chapt. 62).

25.4 Width, Shift, and Shape Parameter

25.4.1 Width and Shift

Here one requires PΨ as well as QΨ . The former is

obtained from a “nonresonant continuum,” deﬁned as

the scattering solution of



PHP+V

(nr)

−E



PΨ

(nr)

= 0 , (25.17)

where the nonresonant potential,

(nr)

= V

− PHQΦ

(E −E

)

−1

Φ

QHP ,

(25.18)

excludes the resonant state s from the optical potential.

In terms of PΨ

(nr)

, whose phase shift, η

,issmooth

in the vicinity of E ≈ E

, a solution of the complete

problem, (25.7), can be constructed [25.13] with a phase

shift η

+η

, where the additional phase shift

= arctan



Γ/2

+∆

) −E



(25.19)

exhibits typical resonant behavior (0 <η

<π).

From (25.19) it is clear that the “true” position of the

resonance is

(E) = E

+∆

(E). (25.20)

The width Γ

and shift ∆

are given by [25.13]:

(E) = 2k|Ψ

(nr)

(E) PHQ Φ

|

, (25.21)

∆

(E) =Φ

QHPG

(E) PHQΦ

 , (25.22)

Part B 25.4

Autoionization 25.4 Width, Shift, and Shape Parameter 395

where k is the scattering momentum (i. e. k

= E −E

and G

is the Green’s function associated with (25.17);

can be simpliﬁed from the form given in [25.13]

[(2.28) of ﬁrst article of [25.2]].

Equation (25.20) is an implicit equation for the

energy at which the resonance occurs. It can be

solved graphically [25.18], and that energy deﬁnes

the Feshbach resonant energy E

, which differs (very

slightly) from the Breit–Wigner energy (Sect. 25.4.3 and

Fig. 25.1).

25.4.2 Shape Parameter

The shape of an isolated radiative transition between

an autoionization state and some other state can be de-

scribed by Γ , E, and an additional parameter q

,often

called the shape parameter [25.19]. The ratio of tran-

sition probabality (in, say, absorption from an initial

state labeled i) through the resonant state to its non-

resonant value, parametrized in its Fano form [25.19]

on the left-hand side of (25.23), can be equated to

its meaning in Feshbach terms on the right-hand side

of (25.23):

)

1 +e

|PΨ + QΨ |T|i|

|PΨ

(nr)

+QΨ

(nr)

|T|i|

, (25.23)

where e

is the scaled energy

= (E − E

)/(Γ

/2) (25.24)

and T is a radiative transition operator (25.25).

To analyze (25.23) in the Feshbach formalism, the

key point [25.13] is to recognize that the bras on the

right-hand side must include P as well as Q parts of

the respective wave functions, as is indicated in (25.23).

That is because the T operator is a perturbation and

not part of the dynamical problem. With T in length

form

T ∝

N+1



j=1

, (25.25)

the rhs of (25.23) can be calculated by noting that PΨ

and PΨ

(nr)

can be, in principle, determined from (25.7)

and (25.17). QΨ

is then derived from PΨ

using (25.6).

But QΨ

(nr)

excludes the sth term from the right-hand

side of (25.6):

QΨ

(nr)



n=s

Φ

QHPΨ

(nr)



E −E

. (25.26)

9.55744

9.5574

9.55736

9.55732

9.55728

0.04709

0.04707

0.04705

0.04703

0.04701

9.52 9.53 9.54 9.55 9.56 9.57 9.58 9.59

(eV)

1+E(k

) Γ (k

)

1+E

Fig. 25.1 Precision calculation of H

−

(

S) resonance. Solid curve:

E +1vs.k

,whereE = E +∆,(25.20). k

is that value of k

which E +1 = k

. Dashed curve is Γ vs. k

from (25.21), and

= Γ





. Curves are from calculations of [25.18]; results are

= 9.557 35 eV and Γ

= 0.047 0605 eV. Applying corrections,

(25.31) gives ﬁnally E

= E

+O(10

−6

), Γ

= 0.047 17 eV

With use of these relations in the rhs of (25.23)anex-

plicit formula for q

was derived in [25.2]; it is given

Φ

Q|T|i

k i|T|Ψ

(nr)

Ψ

(nr)

|PHQΦ



(25.27)

where

Q = Q +QHPG



n=s

QHPG

HQ|Φ

Φ

E −E

(25.28)

The Green’s function G

in (25.28) is the one associated

with (25.17). It can be expanded in terms of the eigenso-

lutions of (25.17), but its spectrum may have a discrete

as well as continuous part, in which case,



PΨ

(nr)

PΨ

(nr)

−E

℘



PΨ

(nr)



)PΨ

(nr)



)

√



−E



(25.29)

The sum over ν refers to the discrete part of the

spectrum of (25.17) (if there is one), and ℘ denotes

a principal value integral over the continuum solutions.

The latter are always assumed to be normalized as

Part B 25.4

396 Part B Atoms

Table 25.4 Comparison of high precision calculations with experiment for the resonance parameters of the He(

)

resonances below the n = 2 threshold. For photoabsorption, the appropriate Rydberg constant is R

= R

∞

/(1 +

m/M) [25.20]. The value used here is R

= 13.603 833 eV, and E

=−79.0151 eV [25.6]

Calculation

Experiment

Quantity Units s = 1 s = 2 s = 1 s = 2

Ry −1.385 7895 −1.194 182

∆ eV −0.007 13 0.000 6202

−E

eV 60.1444 62.7587 60.133 ±0.015

62.756 ±0.01

60.151 ±0.0103

Γ eV 0.0369

0.000 1165 0.038 ±0.004

0.038 ±0.002

−2.849

−4.606

−2.80 ±0.25

−2.55 ±0.16

Bhatia and Temkin [25.20], except as noted

Madden and Codling [25.21]

Morgan and Ederer [25.22]

Bhatia, Burke,andTemkin [25.23]

Bhatia and Temkin [25.24]

plane waves (not energy normalized) throughout this

chapter.

Equation (25.27) is a nontrivial example of what can

be done with the projection operator technique. Not only

does it allow very accurate calculations ([25.17,20]and

footnote e of Table 25.4), but it provides a theoretical

incisiveness which far exceeds all previous resonance

formalisms.

A formula for the resonant scattering cross section

can be derived which is of the same form as the left-

hand side of (25.23); however, in that case, the parameter

corresponding to q

is related to the nonresonant phase

shift [25.25], and has no quantitative relationship to the

above shape parameter (q

25.4.3 Relation to Breit–Wigner

Parameters

Inferring resonance parameters from experimental data

is generally done by ﬁtting to resonance formulae in

which the resonance parameters are assumed to be en-

ergy independent. A phase shift for example would be

inferred by assuming

η(E) = δ

(E) +arctan



−E



(25.30)

The relation between the above Breit–Wigner param-

eters and those of the Feshbach theory has been derived

in lowest order by Drachman [25.26]:

= E

−(1/4)Γ

(dΓ

/dE)

E=E

= Γ

(1 + d∆

/dE)

E=E

(E) = η

(E) −(1/2)dΓ

/dE . (25.31)

In only one precision calculation (for the lowest

resonance in electron–hydrogen scattering) have these

differences, thus far, been evaluated [25.18]. A précis is

given in Fig. 25.1.

25.5 Other Calculational Methods

We now brieﬂy review two calculational methods used

for basic applications in autoionization of few body sys-

tems: (a) complex rotation and (b) a pseudopotential

method.

25.5.1 Complex Rotation Method

Complex rotation, which is based on a theorem of Bal-

slev and Combes [25.27], has been extensively applied

Part B 25.5

Autoionization 25.5 Other Calculational Methods 397

with great accuracy. Two additional basic systems to be

mentioned here are H

−

and Ps

−

(Ps = positronium). In

complex rotation the particle distances are multiplied by

a common phase factor

→r

iθ

. (25.32)

Under this replacement the Hamiltonian undergoes the

transformation

H → T e

−2iθ

+V e

−iθ

(25.33)

(only Coulomb interactions are assumed). A real varia-

tional wave function Φ is used (for the applications here,

they are of Hylleraas form, multiplied by rotational har-

monics of symmetric Euler angles of the desired angular

momentum, parity, and spin [25.31]). The functional

[E]=

Φ|H|Φ

Φ|Φ

Φ|T|Φe

−2iθ

+Φ|V|Φe

−iθ

Φ|Φ

(25.34)

is then evaluated. Minimization of (25.34) with respect

to the linear parameters, for a given value of θ, is car-

ried out in the usual way, but by virtue of the complex

dependence on rotation angle the matrix elements H

the matrix eigenvalue equation

det |H

(θ) −E∆

|=0 (25.35)

are complex. Thus the solution of (25.35)givesrise

to complex eigenvalues E

= E

(θ).Foragivenλ,

the optimum θ is the one for which E

(θ) is effec-

tively stationary as a function of θ [25.29]. Note that no

projection operators are used: the real part of E

corres-

ponds to the Breit–Wigner (i. e., experimental) position

of the resonance, and Im(E

) = Γ

/2, where Γ

cor-

responds to the Breit–Wigner width of the resonance.

These parameters thus include the full Feshbach values

plus corrections (25.31).

Using Hylleraas wave functions with up to 1230

terms in the complex rotation method, resonance

parameters have been obtained for resonance states

of H

−

below the n = 2 and 3 thresholds of H which

Table 25.5 Comparison of resonance parameters (in eV) obtained from different methods for calculating

states in H

−

Threshold Complex-cordinate R-matrix [25.28] Feshbach projection [25.20]

rotation [25.29] (25.31)

E Γ E Γ E

2 10.124 36 0.008 62 10.1252 0.008 81 10.1244 0.010

3 11.811 02 0.045 12 11.810 97

0.044 49

Close coupling (18-state), [25.30]

compare very well with those obtained using the

projection-operator, R-matrix and close-coupling meth-

ods. Results for the

states of H

−

are given in

the Table 25.5. Similar calculations for Ps

−

have been

carried out [25.32]. The complex rotation method has

been applied to the autoionization states of many

different systems including muonic systems [25.33],

as well as to study the combined effect of electric

ﬁeld and spin-orbit interaction on resonance param-

eters [25.34].

25.5.2 Pseudopotential Method

The second method that is included in this section is

done so for the reason that it represents a rather different

idea for the calculation of autoionization rather than be-

ing a more elaborate application of methodologies that

are already known, with results too numerous to be refer-

enced here. The method, described as a pseudopotential

approach, was introduced by Martin et al. [25.35]. An

effective Hamiltonian H

eff

is deﬁned as

eff

= H + MP, (25.36)

where M is a scalar parameter (i. e., a number), which

will be taken to be very large, multiplying the P operator,

(25.16). [Applications have thus far been restricted to

one-electron targets and resonances below n =3 excited

state.] In practice, one minimizes the expectation value

of H

eff

,i.e.,



Ψ

eff

|Ψ



Ψ



= 0 ,

(25.37)

using an arbitrary, quadratically integrable, variational

function Ψ

In order to understand the nature of the spectrum

that arises from this variation, we imagine Ψ

divided

into its P and Q space components:

= QΨ

+ PΨ

= Ψ

+Ψ

. (25.38)

Part B 25.5

398 Part B Atoms

The expectation value Ψ

eff

|Ψ

 is written in matrix

form as

Ψ

eff

|Ψ















(25.39)

The eigenvalue problem resulting from (25.37) reduces

to ﬁnding the eigenvalues of the determinant

det



H

−λ H



H

H

+M −λ



= 0 .

(25.40)

Note that only the bottom right component contains the

term M. As a result, in the limit of large M, the eigen-

values, which can readily be solved for from (25.39),

are

lim

M→large

λ =







M +H



H



(25.41)

The lower eigenvalue is the desired Feshbach resonant

energy E

=H

. The width is calculated from [with

our normalization, (25.21)]

Γ = 2k|Ψ

eff

|χ

|

(25.42)

Table 25.6 Resonance energies E

(Ry) and widths (eV) for

P states of He below n = 2 threshold (−1Ry)ofHe

State

Martin et al. [25.35]

Lipsky and Conneely [25.36] Bhatia and Temkin [25.20,24]

Position Width Position Width Position Width

1 −1.384 00 0.0382 −1.376 72 0.0341 −1.385 79 0.0363

2 −1.194 60 0.000 146 −1.193 12 0.000 131 −1.194 18 0.000 106

3 −1.127 52 0.000 860 −1.125 84 0.007 27 −1.127 72 0.0090

where χ

is the solution of the exchange approximation

−E)χ

= 0 (25.43)

It is emphasized that this method only calculates the

Feshbach energy; thus the shifts are not included. On

the other hand the method uses no projection operators

in calculating the matrix elements of H, and only the

matrix elements of P by itself occur. This is much easier

than a standard QHQcalculation (Sect. 25.2.2).

In practice, the matrix in (25.39) will expand to an

N × N matrix, where N is the number of linear param-

eters in Ψ

, and (if one uses a Hylleraas form of Ψ

for example) the matrix in (25.39) will not overtly

divide itself into the simple form of this heuristic ex-

position pictured in (25.39)or(25.40). Nevertheless,

the conclusion holds; in detail, the eigenvalue spec-

trum will span a range of values with those below the

threshold, appropriate to the P operator being used,

corresponding to real resonances, and the largest eigen-

value will approach the value of M used in the speciﬁc

calculation.

A sample of results for the He(

P) resonances below

the n = 2 threshold of He

taken from [25.35], with

some comparisons, is given in Table 25.6. Note that

the value of E

of the second resonance in the Martin

et al. [25.35] calculation is lower than the rigorous QHQ

calculation [25.20]. It is believed that this may be due to

the residual M dependence of H

eff

25.6 Related Topics

This chapter is necessarily of limited scope. Within the

projection operator formalism, overlapping resonance

theory [25.37] is discussed in Sect. 47.1.3. Recent cal-

culations [25.38] have shown that such effects, when

present, can induce signiﬁcant alteration of isolated

resonance results. Other prominent items not included

are stabilization methods [25.39] and hyperspherical

coordinate methods [25.40, 41]. The latter methods

have the appealing property of presenting energies as

a function of the hyperradius, R =







1/2

,which

look like potential energy curves of diatomic molecules

as a function of the intermolecular separation, which

is also usually denoted by R. The molecular struc-

ture analogy has also been used to uncover additional

(approximate) symmetries with corresponding quantum

labels [25.40,41]. They thus have a global character not

present in the foregoing methods. On a purely quanti-

tative level, however, they are not generally as accurate

as methods based on the projection operator or complex

rotation formalism.

Part B 25.6