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

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Autoionization References 399

There are many other areas in which autoionization

can play an important role, such as satellite line forma-

tion [25.42], inner-shell ionization [25.43], to mention

only a few (Chapt. 62 and [25.44]). In addition, sig-

niﬁcant application of the phenomena associated with

autoionization to diagnostics of astrophysical [25.45]

and fusion [25.46] plasmas, for example, shows that au-

toionization has considerable applied utility.

References

25.1 H. Feshbach: Ann. Phys. (N. Y.) 19, 287 (1962)

25.2 For a more complete review of much of this mater-

ial: A. Temkin, A. K. Bhatia: Autoionization, Recent

Developments and Applications, ed. by A. Temkin

(Plenum, New York, 1985), pp. 1 ff, 35 ff

25.3 H. S. Taylor: Adv. Chem. Phys. 18, 91 (1970)

25.4 A. Berk, A. Temkin: Phys. Rev. A 32, 3196 (1985)

25.5 A. Temkin, A. K. Bhatia: Phys. Rev. A 31, 1259 (1985)

25.6 C. L. Pekeris: Phys. Rev. 112, 1649 (1958)

25.7 B. R. Junker, C. L. Huang: Phys. Rev. A 18,313(1978)

25.8 B. F. Davis, K. Chung: Phys. Rev. A 29,313,2437

(1978)

25.9 M. Bylicki: J. Phys. B 24, 413 (1991)

25.10 J. Brunt, G. King, F. Read: J. Phys. B 10, 433 (1977)

25.11 Y. Hahn, T. F. O’Malley, L. Spruch: Phys. Rev. 128,

932 (1962)

25.12 Y. Hahn: Ann. Phys. (N. Y.) 58, 137 (1960)

25.13 T. F. O’Malley, S. Geltman: Phys. Rev. A 137,1344

(1965)

25.14 A. Temkin, A. K. Bhatia, J. N. Bardsley: Phys. Rev. A

7, 1633 (1972)

25.15 K. T. Chung: Phys. Rev. A 20, 1743 (1979)

25.16 K. T. Chung, B. F. Davis: Hole-projection method

for calculating Feshbach resonances and inner

shell vacancies. In: Autoionization, Recent De-

velopments and Applications, ed. by A. Temkin

(Plenum, New York 1985) p. 73

25.17 K. T. Chung, B. F. Davis: Phys. Rev. A 26, 3278 (1982)

25.18 Y.-K. Ho, A. K. Bhatia, A. Temkin: Phys. Rev. A 15,

1432 (1977)

25.19 U. Fano: Phys. Rev. 124, 1866 (1960)

25.20 A. K. Bhatia, A. Temkin: Phys. Rev. A 11, 2018 (1975)

25.21 R. P. Madden, K. Codling: Astrophs. J. 141,364

(1965)

25.22 H. Morgan, D. Ederer: Phys. Rev. A 29, 1901 (1984)

25.23 A. K. Bhatia, P. G. Burke, A. Temkin: Phys. Rev. A 8,

21 (1973)

25.24 A. K. Bhatia, A. Temkin: Phys. Rev. A 29, 1895 (1984)

25.25 K. Smith: The Calculation of Atomic Collision Pro-

cesses (Wiley-Interscience, New York 1971) p. 47

25.26 R. J. Drachman: Phys. Rev. A 15, 1432 (1977)

25.27 E. B. Balslev, J. W. Combes: Comm. Math. Phys. 22,

280 (1971)

25.28 T. Scholz, P. Scott, P. G. Burke: J. Phys. B 21, L139

(1988)

25.29 A. K. Bhatia, Y. K. Ho: Phys. Rev. A 41, 504 (1990)

25.30 J. Callaway: Phys. Rev. A 26, 199 (1982)

25.31 A. K. Bhatia, A. Temkin: Rev. Mod. Phys. 36,1050

(1964)

25.32 A. K. Bhatia, Y. K. Ho: Phys. Rev. A 42, 1119 (1990)

25.33 C.-Y. Hu, A. K. Bhatia: Muon Catalyzed Fusion 5/6,

439 (1990/1991)

25.34 I. A. Ivanov, Y. K. Ho: Phys. Rev. A 68, 033410

(2003)

25.35 F. Martin, O. Mo, A. Riera, M. Yan

ez: Europhys.

Lett. 4, 799 (1987)

25.36 L. Lipsky, M. J. Conneely: Phys. Rev. A 14, 2193 (1976)

25.37 H. Feshbach: Ann. Phys. (N. Y.) 43,410(1967)

25.38 D. C. Grifﬁn, M. S. Pindzola, F. Robicheaux,

T. W. Gorczyca, N. R. Badnell: Phys. Rev. Lett. 72,

3491 (1994)

25.39 V. A. Mandelshtam, T. R. Ravuri, H. S. Taylor:

J. Chem. Phys. 101, 8792 (1994)

25.40 C.-D. Lin (Ed.): Classiﬁcations and properties of

doubly excited states of atoms. In: Review of Fun-

damental Processes and Applications of Atoms and

Ions (World Scientiﬁc, Singapore 1993) p. 24

25.41 D. R. Herrick, O. Sinanoglu: Phys. A 11, 97 (1975)

25.42 F. Bely-Debau, A. H. Gabriel, S. Valonte: Mon. Not.

Roy. Astron. Soc. 186, 305 (1979)

25.43 C. J. Powell: Inner shell ionization cross sections.

In: Electron Impact Ionization, ed. by T. D. Mark,

G. H. Dunn (Springer, Berlin, Heidelberg 1985)

Chap. 6

25.44 R. D. Cowan: The Theory of Atomic Structure and

Spectra (Univ. of California Press, Berkeley 1981)

25.45 G. Doschek: Diagnostics of solar and astrophys-

ical plasmas dependent on autoionization. In:

Autoionization, Recent Developments and Appli-

cations, ed. by A. Temkin, A. K. Bhatia (Plenum,

New York 1985) p. 171, op. cit. in [25.2]

25.46 M. Finkenthal: Atomic processes responsible for

XUV emisssion. In: AIP Conference Proceedings #

206; Atomic Processes in Plasmas, 1990, ed. by Y.-

K. Kim, R. E. Elton (American Institute of Physics,

New York 1990) p. 95

Part B 25

401

Green’s Functi

26. Green’s Functions of Field Theory

The discussion in this chapter is restricted

to Green’s function techniques as applied to

problems in atomic physics, speciﬁcally to the

calculation of higher order (correlation, Breit, as

well as radiative) corrections to energy levels,

and also of transition amplitudes for radiative

transitions of atoms which are gauge invariant (GI)

at every level of approximation.

Green’s function techniques were ﬁrst applied

to many-electron atoms in 1971 as speciﬁc instances

of the use of ﬁeld theory techniques in many-

particle problems [26.1, 2]. They initially provided

alternative derivations of known approximations

such as the Hartree–Fock (HF) approximation

26.1 The Two-Point Green’s Function ........... 402

26.2 The Four-Point Green’s Function .......... 405

26.3 Radiative Transitions ........................... 406

26.4 Radiative Corrections ........................... 408

References .................................................. 411

and random phase approximation (RPA). The start-

ing point for the derivations was a nonrelativistic

(NR) ﬁeld-theoretical effective Hamiltonian for the

system, which involved the nucleus–electron po-

tential and only Coulomb interactions between

electrons.

It subsequently became apparent that it was possible

to formulate the problem more generally, with the

full quantum electrodynamic (QED) Hamiltonian as

a starting point [26.3, 4]. Thus, the theory contains

both relativity, and virtual and real transverse photons,

as well as the Coulomb interaction between the elec-

trons. The relativistic (R) approximate equations, such

as the Dirac–Fock (DF) and relativistic random phase

approximation (RRPA) are the natural outcomes of the

formalism, and the NR results can be viewed as ap-

proximations to these R cases. Moreover, the Green’s

function approach (GFA) now provides a means of car-

rying out programs involving systematic approximations

of successively higher and higher accuracy. The GFA

provides a framework which allows one to make cor-

rections to results obtained in the DF approximation or

in the coupled cluster approximation (CCA)andmany-

body perturbation theory (MBPT), including magnetic

(Breit type) interactions [26.5, 6]. It ensures that there

is neither double counting nor omission of contribu-

tions. For radiative transitions, the formalism allows for

a systematic treatment of such effects which is gauge

independent that at any given level of approximation

[26.7–10]. (It should be noted that there is a subtle differ-

ence between gauge invariance and gauge independence.

The ﬁrst refers to the transition amplitude and the second

to quantities which are directly observable experimen-

tally.) Finally, the GFA is numerically implementable.

Renormalization can be carried out for radiative cor-

rections, resulting in ﬁnite and calculable expressions

[26.11–13]. The integro-differential equations (i. e., the

Dyson equations) needed to calculate energies or tran-

sition amplitudes in nontrivial approximations are also

soluble [26.8, 9, 14].

In quantum ﬁeld theory, Green’s functions are de-

ﬁned in terms of vacuum expectation values of products

of ﬁeld operators. While this restriction can be relaxed,

expectation values must still be taken for a nonde-

generate state. As a practical matter, this requirement

ultimately restricts one to consider atoms with electron

numbers associated only with closed shells or sub-

shells, and those with closed shells or subshells plus or

minus one or two electrons. The corresponding Green’s

functions considered here are the two- and four-point

functions for energy levels, and the three- and ﬁve-point

functions for transition amplitudes (leading to oscillator

strengths). The restriction in electron numbers is clearly

not a severe one. It allows one to cover many atomic

species.

Starting from relativistic QED, the electron ﬁeld op-

erator ψ(r, t) written in the Heisenberg picture, satisﬁes

equal-time anticommutation relations



ψ(r, t), ψ(r



, t)





†

(r, t), ψ

†



, t)



= 0 , (26.1)



ψ(r, t), ψ

†



, t)



= δ

(r−r



). (26.2)

(Spinor labels are suppressed.)

Part B 26

402 Part B Atoms

The time-translation operator, acting on any Heisen-

berg operator O(r, t) ={ψ(r, t), ψ

†

(r, t), j

(r, t),...},

O(r, t) = e

iH t

O(r, 0) e

−iH t

, (26.3)

where H is the full QED Hamiltonian.

Notations and Deﬁnitions

For brevity, plainface numbers are used to denote both

a coordinate and time, while boldface numbers denote

a coordinate vector alone. For example,

1 ≡ (1, t

) ≡ (r

, t

1 ≡ d

, (26.4)

1 ≡ d

For radiative transitions, M



) denotes the tran-

sition amplitude for the emission of a single pho-

ton of energy k

for an N



-electron atom from

an initial energy E



to a ﬁnal energy E



where



) ≡ (2π)





− E



−k





(26.5)

The notation on the right-hand side of (26.5)issome-

what redundant, since k

is taken to be k

= E



−



The current density operators at t = 0aregiven

respectively by

(r, 0)















†

(1, 0), u

(12;r)ψ(2, 0)



for R



2ψ

†

(1, 0)u

(12;r)ψ(2, 0),

for NR

(26.6)

with

(12;r)











eα

(1−r)δ

(2−r), for R





2im



(∇

−∇

)δ

(1−r)δ

(2−r)



for NR ,

(26.7)

where α

= (α, 1), and the components of α are the

usual Dirac matrices.

The corresponding charge operators are deﬁned as

Q =















†

(1, 0), ψ(1, 0)



, for R



1 ψ

†

(1, 0)ψ(1, 0), for NR .

(26.8)

The transition amplitude for a photon of polarization

four-vector 

(k),momentumk,energyk

=|k|),

and photon attachment point r is



) =



ik·r

√



(r;k

). (26.9)

In terms of the current density operator, M



(r;k

) can

be written as



(r;k

) =









(k) j

(r, 0)







≡







(r, 0)







(26.10)

where







is a state of leptonic charge number N



, with



corresponding to an atom of N



electrons, with total

energy E



.Thetermlepton charge is used to refer to

the charge of electrons and positrons.

In the dipole approximation, e

ik·r

≈1, 

contains

the quantity



(k)u

(12;r) ≡ λ

(r)δ(1−r)δ(2−r), (26.11)

where, in the radiation gauge,

(r) =



e(k) · α , velocity form

iek

(k) · r , length form .

(26.12)

26.1 The Two-Point Green’s Function

The two-point Green’s function, or one-body propagator

[26.3, 4], for a system of lepton charge Ne is deﬁned as

the expectation value of a time-ordered product

(1, 1



) ≡−i







ψ(1)ψ

†



)







(26.13)

where





is the ground state of leptonic charge num-

ber N, with N corresponding to an atom with electron

number N in a ﬁlled shell or subshell (N = 2, 4, 8,...).

The relative time (t

−t



) Fourier transform

of (26.13) yields the spectral representation of

Part B 26.1

Green’s Functions of Field Theory 26.1 The Two-Point Green’s Function 403

(1, 1













ω −ε

+iη











ω −ε

+iη

,η= 0

(26.14)

where

(1, 1



) ≡1|G



, u

(r) ≡







, etc. ,

(26.15)

and where the symbol





denotes a summation over dis-

crete and integration over continuous states. The two

terms in (26.14) are obtained by counting each time-

ordering separately in (26.13), introducing a complete

set of intermediate states and using the time transla-

tion operator (26.3). The functions u

(r) and v

(r) are

deﬁned by

(r) ≡





ψ(r, 0)



N+1







ψ(r, t)



N+1



= e

i(E

−E

N+1

(r), (26.16)

and

(r) ≡



N−1



ψ(r, 0)







N−1



ψ(r, t)





= e

i(E

N−1

−E

(r). (26.17)

Here,





is the ground state of an atom of lepton charge

number N (corresponding to a nondegenerate state –

a closed shell or subshell) of energy E



N+1



atomic state of total energy E

N+1

and leptonic charge

number N +1, and



N−1



is the state of energy E

N−1

and leptonic charge number N −1. These several states

satisfy the eigenvalue equations





= E







N+1



= E

N+1



N+1





N−1



= E

N−1



N−1



(26.18)

The energy parameters ε

and ε

are deﬁned by

≡ E

N+1

− E

, (26.19)

≡ E

− E

N−1

. (26.20)

Equations (26.19)and(26.20) are generalizations of

Koopmans’s theorem [26.15] (see Sect. 21.4.1). In the

DF approximation, the state



N+1



can be thought of as

N effective particles (electrons) making up the core,

plus one valence electron, with energy label j.The

atom can also be an isoelectronic ion with nuclear

charge number Z.Inthis(DF) approximation, the

state



N−1



is one of two possible types. It can have

N independent electrons making up the core, with one

of the core electrons missing, or equivalently the N elec-

tron core with one electron–hole, with energy label ζ .

There is a ﬁnite number N of such hole states, which

shall be labeled a. There are no other states in the NR

(HF) case. In the R (DF) case, the second group of

states



N−1



can also describe an atom with a core of

N electrons, plus one positron. Its continuum energy la-

bel, will be taken as ζ =

. This energy will appear with

anegativesignin(26.14).

The second step in leading to an explicit G

(1, 1



)

is the generation of a Dyson equation which it sat-

isﬁes. In both the DF and HF approximations, the

Dyson equation can be obtained through a succes-

sive series of steps [26.11]. Working in the Coulomb

gauge (see Sect. 27.2), begin with the Coulomb in-

teraction between electrons and neglect the exchange

of transverse virtual photons. The Dyson equation for

the resulting two-point function is then expanded as

a power series in the electron–electron (ee) interac-

tion

≡ V =

|x−y|

(26.21)

resulting in an inﬁnite set of Feynman diagrams

(α = e

/4π in rationalized mks units). As a next ap-

proximation, consider only diagrams involving single

Coulomb exchanges and their iterates (“ladders”). That

is, set aside for later consideration nonladder Feyn-

man diagrams of two or more Coulomb photons and

their iterates (e.g., two crossed Coulomb photon lines).

A summation of the inﬁnite set of these remaining terms

generates an equation for a propagator labeled G

ω,Coul

which contains Coulomb radiative corrections in its

kernel. Finally, modify the kernel by isolating these

radiative corrections (self energy and vacuum polar-

ization) by constructing a spectral representation of

“0”

which mimics that for the usual QED propa-

gator, which is a vacuum (rather than an N-lepton

ground state) expectation value. This requires shift-

ing the poles corresponding to core energies of the

atom from the upper to the lower complex ω plane.

The shifting of poles is accomplished by means of the

equation

x −iη

−

x +iη

= 2πiδ(x),

(26.22)

Part B 26.1

404 Part B Atoms

which, when used to shift the poles of core electrons in

the ω plane, gives

ω,Coul

= 2πi



a=1







δ(ω −ε

) +G

“0”

(26.23)

It is the ﬁrst term on the right-hand side of this equa-

tion which occurs in the kernel of (26.24) below and

generates the DF approximation.

Some of the terms set aside in the course of this se-

quence of approximations are reconsidered in Sect. 26.4

to obtain more accurate energy results. What remains

at the end of this sequence is an approximation to G

of (26.14), designated as G

ω,DF

, and which satisﬁes the

self-consistent Dyson equation

ω,DF

= g

ω,DF

, (26.24)

where



is the kernel deﬁned in (26.27). For the

R(DF) case,

−1

= ω −h

(r), h

(r) = α · p+mβ − Zα/r ,

(26.25)

and the corresponding h

,usedintheHF case, is

(r) =

−

Zα

(26.26)

The function g

in (26.25)istheRorNR Coulomb

Green’s function. It is the solution of the cor-

responding inhomogeneous c-number Schrödinger or

Dirac equation. It is a single-particle equation which

has had a long history of speciﬁc treatments (see

Chapt. 9 and [26.16–19]). It is the semiclassical limit

of the two-point propagator we consider here when

only the c-number nuclear Coulomb potential is

kept and all other (q-number) interactions are turned

off.

In the DF and HF approximations in (26.14), we re-

place ε

by e

> 0,





by |j (the valence energies and

states of an atom with N +1 electrons and a frozen re-

laxed core of N electrons), ε

by e

> 0, |u

 by |a,

for the N discrete electron core states, and ε

by e



< 0,

 by |

, for the continuum of negative energy states

(which do not appear in the HF approximation). The

kernel Σ

contains only core states |a,inboth the DF

and HF approximations, and is given by the sum over

states







= α



a=1



|x−y|



m|x



a|y



× (



x|n



y|a



−



x|a



y|n



)

≡



a=1





−





≡



a=1





(26.27)

for arbitrary states m and n. The ﬁrst term in brackets is

the Hartree term and the second is the electron exchange

term in the DF (HF) approximation. The DF (HF) equa-

tion is the homogeneous equation corresponding to the

inhomogeneous (26.24). Thus,



−h −Σ



|n=0,



n|n





= δ



. (26.28)

The states |n(valence, core, and negative energy states)

are orthonormal and complete. In the coordinate basis,

(26.28) takes the more familiar form

[

−h(x)

]

x|n−







= 0 ,

(26.29)

where













a=1



x|m







≡



a=1





≡



(x, y) d

yy|n . (26.30)

One can also generate equations corresponding to higher

approximations than DF using the same approach. For

example, one can obtain a Brueckner equation [26.20],



−h −Σ

−Σ

)



|n=0 , (26.31)

and the states now satisfy the orthonormality condition

lim

ω→e





n|x





y|n









(ω −e

)δ

(x−y) −







(ω) −Σ

)







ω −e



= δ



, (26.32)

where the energy-dependent kernel Σ

) arises from

irreducible Feynman diagrams involving two Coulomb

photons. The kernel is given by





)









a,i, j

1/2

−e











Part B 26.1

Green’s Functions of Field Theory 26.2 The Four-Point Green’s Function 405





a,b,i

1/2

−e











(26.33)





involves summations over core states (a, b),and

summation and integration over valence states (i, j).

In perturbation theory, the lowest order contribution of

) is an ee correlation term.

26.2 The Four-Point Green’s Function

The four-point Green’s function [26.4, 8, 11], or two-

body propagator, is deﬁned as

(12, 1



)

≡−







ψ(1)ψ

†



)ψ(2)ψ

†



)







(26.34)

In order to avoid unnecessary complication, only sim-

ple ladders of V

≡ V are considered. There are

4! possible time orderings. Of these, there are four

groups of four with t

, t



> t

, t



, t



< t

, t



, t

> t



, t



,andt

, t

< t



, t



, corresponding to the

particle–hole (PH/HP) and the two-particle/two-hole

(2P/2H) cases for the ﬁrst eight and last eight time

orderings, respectively. For each of these cases, intro-

duce a total time and relative time variable deﬁned

T =



), T



t = t

−t



, t



= t

−t



, (26.35)

for the ﬁrst eight cases (PH/HP)and

T =

), T



t = t

−t

, t



= t



−t



, (26.36)

for the last eight cases (2P/2H). For a particular set

of eight time orderings, a time translation with respect

to the relevant c.m. time t or t



, followed by a sepa-

rate Fourier transformation for each case with respect

to T −T



(with integration variable dΩ), yields contri-

butions with poles in the separately deﬁned Ω-planes

(iα)

= E

(iα)

− E

, for PH (26.37a)

−ω

(iα)

= E

− E

(iα)

, for HP (26.37b)

(ij)

= E

N+2

(ij)

− E

, for 2P (26.38a)

(ab)

= E

− E

N−2

(ab)

, for 2H . (26.38b)

Equations (26.37a,b) parallel (26.19)and(26.20)as

generalizations of Koopmans’ theorem. The spectral

representation is of a form similar to (26.14), with wave

functions corresponding to (26.16)and(26.17)givenby

(ia)

(11



) ≡





(ia)

(t)







−iω

(ia)







ψ(1)ψ

†



)





(ia)



(26.39)

for the PH/HP case and

(ij)

(12) ≡



(ij)

(t)e

−iω

(ij)





[

ψ(1)ψ(2)

]



N+2

(ij)



(26.40)

(ab)



) ≡







(ab)



)



−iω

(ab)





N−2

(ab)





ψ(1



)ψ(2



)







(26.41)

for the 2P and 2H cases, respectively. The antisymmetry

under exchange follows from the deﬁnitions (26.40)and

(26.41):

(τ)

(12) =−ζ

(τ)

(21), ζ

(τ)

= ϕ

(ij)

,γ

(ab)

, (26.42)

where it is understood that the suppressed spinor in-

dices are interchanged as well as the coordinate and

time variables.

The three amplitudes deﬁned above satisfy Bethe–

Salpeter (BS) equations. The PH/HP case is the analog

of the positronium atom and the 2P case is analogous

to He. For the case of Coulomb ladder exchanges, to

which we have restricted ourselves, these BS equations

can be reduced to simpler ones, with the relative time

set equal to zero (the Salpeter equation [26.21]). The

corresponding BS wave functions in the DF (HF)basis

(rather than the coordinate basis), are for PH/HP,





(ia)





≡





m|11|χ

(ia)

(0)|1



1



|n ,

(26.43)

with m =k and n =c,orm =cand n = k,andfor2P/2H,





(τ)



≡



2m|1n|2





(τ)

(0)



(26.44)

where ζ

(τ)

= ϕ

(ij)

,γ

(ab)

. The states |i, |j, |a, |b, |m,

and |n label one-particle DF (HF) eigenkets and

Part B 26.2

406 Part B Atoms

, e

, and e

, the corresponding eigen-

values.

In the PH/HP case, the BS equation is

−(ω

(ia)

−e

)





(ia)



















(ia)















(ia)







(26.45)

(ω

(ia)

−e

)





(ia)









j,b











(ia)















(ia)







In the 2P/2H case, the coupled pairs of BS equations

are

−(ω

(τ)

−e

)



cd|

(τ)







m>n









(τ)



(26.46)

(ω

(τ)

−e



)



k|

(τ)







m>n











(τ)



with m and n in these equations labeling either both

core, or both valence states.

The wave functions also satisfy the additional con-

ditions:





(ia)









(ia)







= 0 ,

PH/HP (no 2P/2H terms)

(26.47)





(ij)







(ab)



= 0 ,

2P/2H (no PH/HP terms)

(26.48)

The single indices a, b, c, d refer to core and i, j, k,

to valence DF (HF) states. The BS wave functions satisfy

the orthonormality conditions



k,c





(ia)







(ia)











(ia)







(ia)







= δ

(ia)(ia)



, (26.49)

for the PH/HP case, and





c>d



(τ)







(τ



)



−



k>



(τ)



k



k



(τ



)





= δ

(τ)(τ



)

, (26.50)

where +(−) corresponds to the 2H(2P) case and

(τ) = (ab) or (ij ).

The PH/HP case in the GFA, involving Coulomb

ladders for the ee interaction, is just the R and NR

RPA [26.2]. The labels c, d should also refer to anti-

particles, but the contributions of the integrals from these

terms to (26.45)and(26.46) are negligible.

26.3 Radiative Transitions

For the majority of applications, one begins with the

function Γ

(12;3), the reducible three-point vertex:

(12;3) =−







ψ(1) j

(3)ψ

†

(2)







(26.51)

The usual strategy is followed. The spectral representa-

tion serves to identify the functions of ultimate interest.

(Energies are not relevant in this case.) One then gen-

erates Dyson equations in the chosen approximation by

summing an inﬁnite series of perturbation terms.

There are 3! time orderings in (26.51). As with

the four-point function, not all of them are subse-

quently useful. The two useful cases are t

> t

and t

> t

. To obtain a spectral representation

for these two cases, one ﬁrst carries out a time trans-

lation of t

, using the operator exp(iH t

) of (26.3),

so that t

→ τ

= t

−t

, t

→ τ

= t

−t

.Onethen

introduces complete sets of intermediate states. The

functions deﬁned in connection with the two-point

function in (26.16)and(26.17) will now appear, as

well as the radiative transition amplitude, deﬁned

in (26.10). If one next carries out a separate time

translation of τ

and τ

and Fourier-transforms the

resulting expressions, one obtains (with 3 replaced

by r)

(r;ω

)





j

M

N+1

j

(r)u



(ω

−ε

+iη)(ω

−ε



+iη)





ζχ

M

N−1

ζχ

(r)v

(ω

−ε

−iη)(ω

−ε

−iη)

+··· .

(26.52)

Part B 26.3

Green’s Functions of Field Theory 26.3 Radiative Transitions 407

We next deﬁne the three-point irreducible electron

vertex Λ

from the reducible vertex Γ

(the electron

“legs” in Γ

are missing in Λ

) as





(r;ω

)





≡

















(r;ω

)















(26.53)

and pick out the residues of the ω

and ω

poles at

speciﬁc energies ε

and ε

. [Λ

(r;ω

) has no such

poles.] With the equivalent of the DF(HF) approximation

to the Dyson equations (26.57) below, the corresponding

kets form an orthonormal set, and scalar products are

calculated with respect to m| and |n. The second term

on the right-hand side of (26.52) refers to hole states and

is of less interest than the ﬁrst term describing 1P–1P

transitions. From the ﬁrst term, the transition matrix

element in terms of Λ

N+1

(r;k

) =





(r;e

)





(26.54)

Generation of a Dyson Equation

The approximation of only Coulomb “ladder” ee inter-

actions in the two-point Green’s function produces an

inﬁnite set of Feynman diagrams to which one end of

a single transverse photon line is attached in all possi-

ble ways. A resummation of these diagrams generates

the G

ω,Coul

functions, which are approximated by the

DF (HF) propagators written in their spectral form. The

scalar products leading to (26.54) can then be taken.

A further simpliﬁcation results because of the

Coulomb ladder approximation, which is the same as

in the four-point Green’s function case: the relative time

can be set equal to zero, which corresponds to inte-

grating over the relative frequency, ω = ω

−ω

. With

the total frequency deﬁned as Ω =

(ω

+ω

),andthe

deﬁnitions

(r,Ω)=

2π



dωΛ



r;ω +

Ω, ω −

Ω







(Ω)







ik·r

√





(r,Ω)





Ω

= e

−e

= k

, (26.55)

with a relation similar to (26.55) for the quantity λ

(r)

deﬁned in (26.11), i. e.,



λ(k

)





ik·r

√

m|λ

(r)|n , (26.56)

the matrix elements in the DF (HF) basis are [26.14,20]





)





=m|λ(k

)|n







−e









)





−e

−k









)





(26.57)

As discussed in Sect. 26.1, the label a should include not

just hole states but also negative energy states, which

have been neglected in (26.57). Note also that only

PH or HP matrix elements appear on the right-hand

side of (26.57). Therefore, a closed set of inhomoge-

neous linear algebraic equations for b|

)| and

|

)|bresults from setting m = b, n = or m =,

n = b, respectively, in (26.57). These equations can be

solved numerically, and the resulting





)





and





)





substituted in (26.57) to obtain the ﬁnal

result

N+1

) ≡





)





(26.58)

An integro-differential equation form, which pro-

vides the option of choosing alternative numerical

techniques [26.14], is obtained from (26.57), after

some rearrangement and passage to a coordinate basis.

Deﬁning m

) from λ(k

) in analogy with the deﬁni-

tion of M

N+1

) from

) in (26.58), one has

N+1

) = m

) +



(a|λ(k

)|A

−



+A

|λ(k

)|a), (26.59)

where

[

h(r) ∓k

−e

]















=−





r|jj|





|a ,

(26.60)

j|v

|a=







, j|v

−

|a=







(26.61)

j|V

|a=











∓





(26.62)

The three-point Green’s function, as can be seen

from this summary, describes transition amplitudes

Part B 26.3

408 Part B Atoms

for radiative transitions between two valence states of

atoms with closed shells (subshells) plus one electron

(the 1P case). In the Coulomb ladder approximation,

closely related to the DF equation, the photon vertex





(r;ω

)





of (26.53), is nonlocal in space,

rather than the local vertex



(r)



of (26.11)

[as follows from the factor of δ

(1−r)δ

(2−r)]. The

presence of these additional nonlocal contributions to

N+1

) is made apparent in (26.59). The m

)

term is the contribution of the local vertex. Of course, if

one knew the exact N +1 electron wave functions and

energies, only the local vertex would enter and a GI re-

sult would be obtained. However, the length and velocity

versions of (26.12) are equal and GI even in the approx-

imation just discussed [26.3, 7]. Gauge invariance is an

essential constraint on radiative transition amplitudes.

It has been proven for general gauges not only in the

present approximation, but also in the ones discussed

below [26.8, 10], all of them arising in the GFA.Since

the effective potential in the DF (or HF) approximation

is nonlocal, the effective current must also be nonlocal

in order to maintain the GI.

The somewhat more complicated Dyson equa-

tion satisﬁed by the nonlocal vertex corresponding

to the Brueckner approximation has also been gener-

ated [26.22], and put in a numerically implementable

form.

An alternative approximation [26.3, 7]fortransi-

tion matrix elements, proposed earlier [26.2] than the

one just discussed, is based on the RPA [our PH/HP

case, with wave function

( fa)

the solution of (26.45)].

Reference to (26.6), (26.7), (26.10)and(26.11)gives

immediately

(r;ω

( fa)

) =−λ

(r)





( fa)





(26.63)

where k

= k =ω

( fa)

and N corresponds to atoms with

a closed shell or subshell of electrons.

Aside from being a different approximation to tran-

sition amplitudes than (26.54), (26.63) covers a different

set of cases than does (26.54), since the corresponding

initial states i in (26.54) are restricted to core states a

in (26.63). Thus, for example, the case N = 2in(26.54)

can describe transitions between any two valence states

of Li. The corresponding case for (26.63)isN = 4, but

only transitions to a higher level, originating in the 1s or

2s level of Li, can be described by the formalism.

Finally, we shall discuss radiative transitions [26.8]

for 2P atoms (closed shell/subshell plus two valence

electrons). The general case involves a ﬁve-point

nonlocal vertex. However, in the ladder approxima-

tion, this reduces to transition matrix elements which

contain combinations of DF,2PBS wave func-

tions, and three-point functions. The ﬁnal expressions

are

N+2

(r;k

)

( fp)



(r, k

)

−∆H

V −V

(r, k

)

−∆H



(iq)

(26.64)

The expressions appearing in this equation are, in more

detail, in the DF basis:



(r, k

)

−∆H



≡





(r, k

)





−e

(26.65)

where m, n, q, s label DF states (P or H),





≡|qs≡

|q|s;(26.65) serves to deﬁne ∆ H, as a difference of

two DF Hamiltonians, the argument r of

Λ refers to the

electron which emits the photon, and





(τ)

















(τ)







k











k



(τ)



(26.66)

In (26.66) we used the fact that, in the DF basis, |

(τ)



only has 2P or 2H components (26.46, 26.48).

26.4 Radiative Corrections

This section summarizes radiative corrections for 1P/1H

atoms, starting from the two-point Green’s function. The

Dyson equations are generated and solved perturbatively

for the energy to order α

m (α

a.u.). The perturbation

theory starts from the DF solution as the zero-order one.

As done for the three-point function in Sect. 26.3,

the Dyson equation for radiative corrections involving

a single transverse photon is generated by expanding the

two-point propagator to all orders in the Coulomb ladder

approximation, inserting a transverse virtual photon in

all possible ways, and then resumming. The resulting in-

tegral equation [26.23] is quite complicated, containing

even the three-point vertex in its inhomogeneous term,

as well as a mass counter term to eliminate divergences.

Part B 26.4

Green’s Functions of Field Theory 26.4 Radiative Corrections 409

It is sufﬁcient for the present purposes to expand these

vertices through ﬁrst order matrix elements of V

≡ V,

but to exclude matrix elements of type





(where

a is the label for core states), since these are already

included in the DF approximation [see (26.27)].

Other radiative corrections are also generated, which

include Coulomb photons. Two of these corrections, in-

volving only such photons, have already been referred

to in the text between (26.21)and(26.23). Finally, to or-

der α

m, corrections involving two transverse photons

must be included.

Systematic application of the pole-shifting process

in (26.23) to the one- and two-transverse-photon and the

Coulomb-photon expressions yields two types of terms:

ﬁrst, photon exchange (of one transverse photon or of

two-photons of either kind, Coulomb or transverse) be-

tween core and valence states; and second, self-energy

and vacuum polarization contributions. Among the pho-

ton exchange terms, one can identify those contributions

included in other approaches (at least for a few special

cases, such as those arising from a single transverse pho-

ton interaction, and from two Coulomb interactions and

Coulomb–Breit interactions with positive energy inter-

mediate electron states). These are electron correlation

terms which are included as part of a MBPT calcu-

lation [26.5], or one which involves consideration of

an inﬁnite subset of MBPT terms [26.6]. They need

not be re-evaluated. The terms not covered by calcula-

tions of the type in [26.5, 6], which are of O(mα

involve retardation in Coulomb-transverse photon ex-

change, negative energy intermediate electron states

for two Coulomb and Coulomb-transverse photon ex-

changes, two transverse photon exchange, self-energy

and vacuum polarization terms, as well as anomalous

magnetic moment corrections.

After lengthy calculation, one obtains ﬁnal results

in ﬁnite analytic form, which are numerically imple-

mentable. The results given below are obtained after

further approximations. First, the remnants of the orig-

inal integral equation are solved iteratively. Second,

the self-energy terms are calculated in a joint expan-

sion [26.24–26]inα and αZ, and are thus only valid

for low Z in isoelectronic sequences. Finally, there are

characteristic logarithmic terms generated by low vir-

tual photon momenta in the self-energy contributions.

A standard approach is to scale these with a factor of

(αZ)

to obtain Bethe log (BL) terms as constants inde-

pendent of α, together with constant log (CL)termsof

the form ln(αZ)

.TheBL terms are independent of Z for

hydrogenic ions, and remain nearly so for other atoms.

BL and CL terms are associated with both nuclear and

ee Coulomb potentials. The ee CL terms arising from

core and valence self-energy terms are canceled exactly

by those coming from Coulomb and transverse photon

exchange. Thus, the numerous and rather complicated

corresponding ee BL terms, which also are individually

small compared to their associated CL terms, should

additionally almost cancel, and are therefore neglected.

One ﬁnally obtains (with all state labels denoting

principal and orbital quantum numbers as well as spin

indices: n ≡ (n,;m

))

∆E(n,)= Z|0|n|

F(n,), (26.67)

in a.u., where 0|n≡0|n, 0 is the s-state HF wave

function at the origin of coordinates, and serves to deﬁne

an effective (shielded) nuclear charge Z

n,eff

r|n,≡Ψ

n,

(r), |0|n|

≡

πn

n,eff

α)

(26.68)

(Just as in hydrogen, for which the ﬁnal expressions for

the radiative corrections require NR and not R wave

functions and energies, so in this case we use HF (NR)

and not DF (R) quantities.) F(n,)consists of a valence

or hole contribution F

v,h

, a core term F

core

(n,), and one

due to photon exchange between electrons F

(n,):

F(n,)= F

v,h

(n,)+F

core

(n,)+F

(n,),

(26.69)

v,h

(n,)=



N(Z)δ

0

+L(n) +U

v,h

(n)



(26.70)

core

(n,)=



N(Z)ρ(n) +L

core

(n)

+U(n) +U

core

(n)



, (26.71)

(n,)=



E(Z)K

(n) + K

(n)



(26.72)

with U

v,h

(n) and U

core

(n) being the spin-orbit terms,

N(Z) = ln

(Zα)

+ ZαC

, (26.73)

= 3π



1 +

128

−

ln 2



(26.74)

E(Z) =−

Zα

−

π,

(26.75)

L(n) =

i/4π

|0|n|



n,





p· L

(n)

−

(n) · p





n,



, (26.76)

(n) ≡ ln

2|e

− H|

(26.77)

Part B 26.4