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Many body perturbation calculations of photoionization
Computer Physics Communications 17 (1979) 99—111 © North-Holland Publishing Company
MANY BODY PERTURBATION CALCULATIONS OF PHOTOIONIZATION H.P. KELLY Department ofPhysics, University of Virginia, Charlottesville, VA 22901, USA
The application of many body perturbation theory to the calculation of atomic photoionization cross sections is reviewed. The choice of appropriate potential for the single-particle states is discussed and results are presented for several atoms including resonance structure. In addition to single photoionization, the process of double photoionization is considered and is found to be significant.
1. Introduction The many body perturbation theory (MBPT) of Brueckner [1] and Goldstone [2] has been very useful in many areas of physics [3,4] including atomic physics [5]. As in most perturbation theory formalisms, one requires appropriate complete sets of single particle states and different methods to obtain these states must be developed for different physical problems. Frequently, it is necessary to use numerical
potentials. The Hamiltonian for atoms of atomic number Z is N
N
i=1
i
~II~ T, +
H=
(1)
~1
where T1 =
V~/2 Z/r~ (2) Atomic units are used throughout this paper. In a first approximation, ~u~1is replaced by ~ V~, where V~is a Hermitian potential which may be arbitrarily chosen. The approximate resulting Hamil-
methods to obtain the appropriate single particle states and matrix elements involving these states. In this paper, MBPT is applied to the calculation of atomic photoionization cross sections, and results are reviewed for such atoms as Zn, Ba, Cd and Cl. The process of double photoionization is also considered for which the absorption of one photon results in the ejection of two electrons. Double photoionization results for neon and argon are discussed. Important reviews of atomic photoionization have been given by Fano and Cooper [6] and in the book edited by
—
.
tonian is N
H0
=
~
v~)
(T, +
,
(3)
~=
and the resulting states 1a satisfy H0’t~ 0 = ~ The states 4~are determinants containing N singleparticle solutions ~ of
Wuileumier [7]. A description of the very successful results using R-matrix theory may be found in articles by Burke and Taylor [8] and by LeDourneuf [9].
(7’ + V) ~n = nctln (4) One can systematically include correlation effects by using perturbation theory with .
N
2. Many body perturbation theory
H’
N
=
VI.
—
i
2.1. Linked cluster expressions for energy and wave functions
(5)
i=1
According to the results of Brueckner [1] and Goldst one [2],
Consider N electrons interacting with two-body 2 /r potentials v~= e 11 in addition to the one-body
=
~ L
99
(
1 —
H0 H’) ~
(6)
H.P. Kelly / Many body perturbation calculations
100 and
t0~=(I
L~E0=E —E 0lH’I’I’~>, (7) where indicates that only “linked” terms [1,2] are included. Terms in the perturbation expansions of eqs. (6) and (7) may be represented by diagrams [1—4]. 0
0
~L
for large n. Inserting (11) into (10), we find 2 , Imc~(w)(2/k)I(kIzjp>l
(14)
where k = (2~+ 2w)”2
(15)
.
The photoionization cross section u(w) is (in
atomic units) [Ill 2.2. Frequency-dependent polarizability and the photoionization cross section
u(w)
Cox~idera perturbing electric field E = F~cos wt with the accompanying time-dependent perturbation
21) z
1.
P = ct(w)E,
(9)
where ct(w) is the frequency-dependent polarizability which may be evaluated by MBPT [10]. The lowest-order contribution to a(w) from the
orbital cb~occupied in ~I’~is given by (kjzlp>12( _______1 \e~— Ek
+—
W
—
1
~,
Ek + ~
/
(10)
where the sum over k includes all excited states (i.e., those not occupied in uum.
both bound and contin-
~
In eq. (10) if Ip), 1k) and are exact eigenstates and eigenvalues respectively ofH and z represents ~ z~,then eq. (10) gives cr(w) exactly. Since e~ may we add a small imaginary part ii~ andwuse thevanish, formula ~,
5
+
lim (er—
Ek + W +
recalling the normalization of eq. (12). From eq. (14),
we obtain the familiar results [6]. (17)
It can be shown that the many-body diagrams for Im a(w) may be factored to give the exact (‘J!k X ~ z~ I ‘I’d> times its complex conjugate times correla-
The induced dipole moment P is given by
—
(16)
,
(8)
i=1
—21)k
(8irw/ck) I (kIzlp>~2
u(w)=(4irw/c)Imcs(w).
N
V(r, t) =Fcos wt
=
tions due to normalization diagrams
[51.Effects of
normalization diagrams have been less than 5% for all cases studied to date. Diagrams contributing to (~‘PkI~ z~IW0)in which there is a p ~ k transition are shown in fig. 1. The solid dot represents matrix elements of z and the cross represents interaction with —V. The other dashed lines represent Coulomb interactions. The diagrams of fig. I are read from bottom to top corresponding to right to left in the mathematical expressions. In the diagrams, Coulomb interactions (i.e.,
with H’) below the heavy dot correspond to correlations in the initial state I ‘I’~)and Coulomb inter-
actions above the heavy dot Exchange corresponddiagrams to correlaare tions in the final state not explicitly shown in fig. 1 but should be included. 1’k>.
‘
Diagram 1(a) is the lowest order diagram and repre-
ii sents the matrix element (kjzjp). In the next order
1)’
of perturbation theory, diagrams 1(b), (c), (d) and
=P(e~ w)~ i~(e~ Ek + w), (11) where P represents principal value integration. In the summation continuum are included by (2/ir) f~dk,~which assumes states continuum states are —
+
—
—
(e) occur and also exchange diagrams corresponding
to (b), (c) and (e). Diagram 1(b) represents correlations in given by
4’k
and is
‘
normalized according to
~(kqIvIpk’)(k’IzIq>(eq Rk—~cos[kr+6l+(q/k)ln2kr—~(l—1)ir]/r (12) as r where V(r) q/r. Bound states are included by finite summation and use of the extrapolation —~
KnIzIp)12
-+
—~
C/n3
(13)
~k’
+
w)’.
(18)
k
The exchange of 1(b) is obtained by replacing
(kqlvlpk’) by —(qkl vlpk’). The expression for diagram 1(c), which represents
H.P. Kelly I Many body perturbation calculations
/
P~k
\.L_. (b)
q
(a)
the corresponding velocity cross sections, including correlations tends to bring them closer.and However, agreement of length and velocity cross sections is only a necessary but not sufficient condition to indicate accuracy of the solutions.
k
0
101
———-
(c)
q~k
3. Potential V for single-particle states e
Although the choice of potential V is, in principle, arbitrary, it may have a drastic effect upon the con-
q
I!
q
k
~
k
__.UI’ (g)
(h)
Fig. 1. Diagrams contributing to <‘ikI ~ z~’I~&. The solid dot represents a matrix element of z and the cross represents interaction with — V. Other broken lines represent Coulomb interactions. Exchange diagrams are also included.
vergence properties of the perturbation expansion. It is usually desirable to choose V so that we have Hartree—Fock(HF) orbitals occupied in 4~.However, as discussed previously [14—18],the potential for HF ground state orbitals is not unique as is readily seen by considering the potential [17,1 8] V= R
+
(1
—
Q) ~7(l Q),
(22)
—
where &2 is an arbitrary Hermitian operator and R is a ground state correlations is 21)(qlzlk’> (kk’IuI pq>(ep + 6q
—
6k
—
6k’).
(19)
potential which gives the N Hartree—Fock orbitals occupied in I~.The operator N
21)
k
The sum of diagrams (d) and (e) and the exchange of (e) is
‘~
N
(iIRIj) w)’
.
(23)
where states In> are occupied in I~. For example, R is often written [1—4]
~ (—(kI V]k’>+(kqlvlqk’>) 6k’ +
In)(nI,
(20)
X (k’IzIp> (e~ Diagrams l(f)—(i) represent some of the diagrams which are second order in H’. We note that all the diagrams of fig. 1 except for 1(i) are included in the random phase approximation with exchange [12,13]. In calculations it is customary to evaluate both “length” (z) and “velocity” (dfdz) forms of the matrix elements. When “k> and I ‘I’o> are exact eigenstates of H, they are related by
21)
((inluljn)—
1)).
n1
—
N
‘I’ )
fI
I
Z~
— —
E
(
—
~ — Ef)
N
< \/~, f ~ ~d
~ 0/.
~21
In most cases studied so far, the lowest-order (Hartree—Fock) length cross sections are greater than
Since (1 Q)In) = 0 when In> is occupied in Vin) = RIn>. For an “excited” orbital, however, VIk> = Rik> + (1 Q)~2Ik>.We may choose 12 to accelerate the convergence of the perturbation expansion. In those cases where the excited states are of a symmetry different from that of any orbitals occupied in ~ the potential may be chosen arbitrarily and one does not need eq. (22) to ensure ortho—
~
—
gonality to occupied orbitals. As an example of the single-particle potential, consider argon 3p6 lg• Dipole absorption of a photon leads to final many-particle states 3p5kd ‘P and 3p5ks ‘P. If these states are represented by singleparticle states, we obtain an appropriate potential for the kd and ks orbitals by setting (3p5k ‘PIH’I3p5k ‘P>
0
,
(25)
H.P. Kelly / Many body perturbation calculations
102
or 5k’PI ~u~I3p~k
‘1~.
(26)
(kIVIk’)= (3p / For ks, eq. (22) is needed so that the ks orbitals are
orthogonal to hs, 2s and 3s. The Vof eq. (26) results
in the cancellation of fig. 1 diagrams (d), (e) and also (b) when q belongs to the same subshell asp [12,19].
Also, higher dsder diagrams such as (0 are cancelled by V. Using this potential the first-order corrections in H’ come exclusively from diagram (c) and its exchange which account for ground state correlations, This choice of potential is that appropriate for Hartree—Fock continuum states obtained by b (3pSkiPIHI3pSk ‘P> = 0 , (27)
near threshold by an order of magnitude from the experimental of Marr and Austin [24] as shown in fig. 2. Theredata are resonances from 3d np excita—*
tions and the lowest-order diagrams contributing to these is fig. 1(b) with p = 4s, q = 3d, k’ = np. Higher-
order diagrams may be summed exactly geometrically as indicated in fig. 3 to give effective widths to the resonances. An indication of the non-resonance correlations is given by the curves NRV (velocity) and NRL (length) in fig. 2. These were calculated by inclusion of only those diagrams which do not contribute to resonances. Diagrams contributing to resonances contain single
3d tip excitations alone in one or more denominators. The lowest-order curves are noticeably reduced, -~
and proved very useful in MBPT calculations of u(w) for argon [20]. I
~
6
4. Single photoionization cross sections HFL x
5k—
Many-body perturbation theory has been applied to the calculation of many properties such as correlation energies, polarizabilities, oscillator strengths, fine structure and hyperfine structure [21,22]. In this paper results are presented only for photoionization
HFV
-
-
cross sections. In summing over single-particle states as in eqs. (18)—(20), we must ensure that our set of
single-particle states is effectively complete. Generally, this is done with approximately ten bound excited states and use of eq. (13) to account for the remainder and with approximately thirty continuum states. For the outer shells of atoms, a maximum value k = 30.0 usually suffices. The initial spacing in k is usually 0.1 and then becomes coarser after 1.0. Sums over the continuum are done numerically.
4.1. Zinc As an example of the application of MBPT to photoionization cross sections, we consider the (4~)2 subshell MBPT calculations [23]. The final state in this case is 4s kp ip and the potential is chosen as in section 3 so that in diagram 1(b) we only take q to
represent inner shells. In this calculation only correlations among 4s and 3d electrons were included. It was discovered that there are very large correlation effects with the lowest-order cross section differing
r b
2
r
~--~
vor
-
VCF [____~_~
-
100
95 PHOTON
---
-
-
05
00
ENERGY (eVI
Fig. 2. Cross section 0(w) for photoionization of the 4s sub-
shell of ZnI near threshold. The curves HFL and HFV represent Hartree—Fock length and velocity cross sections using frozen-core (FC) orbitals. Curves NRV and NRL represent correlated cross sections omitting contributions from resonance diagrams. Curves VCF and VCI represent correlated velocity cross sections with FC and IC (ionic-core) orbitals respectively. The circles are experimental results by Marr and
Austin 1241.
H.P. Kelly /Many body perturbation calculations
4s
kp
4s
kp
-—
--
3d
mp
3d
np
--
--
4s
kp
4s
kp
-~
3d
94s2np’P,,
one must consider resonance states 3d
(b)
kp
3d
——
3d
summed geometrically. The resulting expression [23] is equivalent to that derived previously by Fano [25] for the case of one continuum and many interacting resonances. In the segment shown in diagram 3(c), one sums over all excitations mp and this accounts for the interaction between resonances mp and m’p. For Zn, the resonance structure due to 3d np excitations shows effects of spin—orbit splitting and -*
T’p
(a) 4s
103
~ —-
mp
4s
kp ~
I
3d
4S ~
P
3D,, 3P, which are sufficiently close to interact. The intermediate-coupling energies and mixing coefficients for these states were calculated by the method of Wilson [26]. The resulting calculated cross section is shown in fig. 4 along with the experimental results of Marr and Austin [24]. The absorption window between the two large resonances is obtained in calculations only when interactions between resonances are included.
(d)
(c)
Fig. 3. Diagrams and diagram segments associated with reso-
nances. The horizontal line indicates that the denominator should be treated according to —iir~(D). The denominator for diagram (d) includes both PD1 and —iira(D).
4.2. The 4d’°subshell of barium The 4dio subshell of BaI was calculated [27] to .
.
compare with the interesting experimental results of
particularly in the length case. Including both resonance and non-resonance diagrams, the curve VCF of fig. 3 was obtained with the length curve being 10% lower. The VCF curve was calculated using kp orbitals of the configuration 3d’04s kp in which occupied orbitals of ZnI were used to calculate a frozen core (FC) potential. The velocity curve labeled VCI used occupied orbitals of Zn~in the potential for kp. All diagrams of fig. 1 involving 4s and 2d electrons were
included except for 1(i). Large correlation effects came from 3d
—*
kf and 4s
—*
4p excitations. Diagram
1(f) with p 4s, q = 3d, r = 4s was found to be quite significant, reducing diagram 1(a) by approximately 20%. This does not indicate a slow convergence of the
Connerade and Mansfield [28] and of Rabe, Radler and Wolff [29]. A previous calculation by Wendin [30] had predicted a collective resonance for this cross section, and it was of interest to test this prediction by a perturbation calculation. The excited states were calculated in a VN~4d9kip potential using orbitals of Bal. Both 4d kp and 4d kf transitions were calculated, and most of the cross section was observed to come from 4d kf. The experimental 4d removal energy (99.28 eV) of Connerade and Mansfield [28] was used in the calculations. The spin— orbit splitting between 2D 2D 512 and 312 cores was included, and the cross sections for these cores was assigned from the non-relativistic result according to statistical weights. Results of the calculations are -+
-~
—*
compared with experiment in fig. 5, where the exper-
perturbation expansion, however, since this is the lowest order in which the large 4s 4p dipole matrix
imental curve was normalized to give an oscillator strength of ten. The dashed curve includes the lowest-
element occurs in final state correlations,
order diagram of fig. 1(a) and diagram 1(c) (and
The lowest-order resonance diagram of fig. 1(b) is modified by higher-order diagrams as shown in fig. 3(a) and (b). The horizontal line indicates that the denominator is treated according to —ix6(D) as in eq. (11). The section of diagrams 3(a) and 3(b) which is mdicated in 3(c) is repeated in higher orders and may be
exchanges) with p, q = 4d which represents ground state correlations in the 4dio subshell. The remaining first-order (in H’) diagrams (1(b), (d), (e)) cancel because of the choice of potential. There are contributions due to diagram 1(b) where q refers to other subshells but these were estimated to be small. The
—~
H.P. Kelly /Many body perturbation calculations
104 60,
I
I
I
150-
,~,
4030
-
I
-
-
-
/~ ii
120II 0 —
-
L/
-
94s24p resonance region. Solid line, correlated velocity cross section; dash—dot line Fig. 4. Cross section of ZnI o(w) 11.0in the 3d 11.2 114PHOTON 11.6 ENERGY 11.8 (eV) 12.0 122 124 correlated length cross section; dashed line, experimental data of Marr and Austin [241.
lowest-order (Hartree—Fock) calculations yielded
velocity results into close agreement. The calculated
length and velocity curves which bracketed the normalized experimental curve. Inclusion of the ground state correlations of fig. 1(c) brought the length and
curve of fig. 5 is an average of the final length and velocity results. The solid curve in fig. 5 is obtained from the random phase approximation with I
I
I
I
80
-
70
-
60
-
/
N
/
E50 040
‘S
I,‘/I
//
\
f
b 30 2010-
\..
“
-
•
//
-..--.‘..
N....
-
‘1
//‘ “I
95
‘..
100
---.---_.~..
I
105
110
115
120
-
I
I
I
125
130
135
140
145
PHOTON ENERGY (eV) 2D 2Dresults for °4dof Bal: Fig. 5. Comparison of calculated and experimental , average oflength and velocity form RPAE results including effect of splitting of 512 and 312 cores; , experimental data of Connerade and Mansfield [28] normalized to give an oscillator strength of 10; — — —, average of length and velocity ground state correlation results including spin— orbit spitting of cores.
H.P. Kelly /Many body perturbation calculations exchange (RPAE) and includes diagrams such as (g) and (h) in fig. 1 and also higher-order diagrams of this type. It is interesting that the low-order Hartree-Fock calculation with ground state correlation gives such good results, possibly indicating significant cancellation among higher-order diagrams. 4.3. The 4d1°subshell of cadmium Calculations recently carried out [31] on the 4d’° subshell of cadmium help us to understand the poor agreement of RPAE results with experiment in the case of Ba. The excited kf and kp states were first calculated in a VA’_i Hartree-Fock potential 4d9ktP. The Hartree—Fock length cross section was approximately twice that calculated by the velocity operator. However, including the ground state correlation of fig. 1(c) brought the curves into close agreement. The geometric mean of the resulting length and velocity cross sections is labelled GMU (geometric mean unrelaxed) in fig. 6. We have here adopted the suggestion of Hansen [32] of using the geometric mean of the length and velocity results. The potential for the excited kf states in this case used the unrelaxed orbitals of Cdl. The second order RPAE diagrams of fig. 1(g) and (h) were then included and the resulting curve labelled GMS (geometric mean second order) is plotted in fig. 6 and is higher than the curve GMU. Other second-order diagrams which represent relaxation effects as shown in fig. 1(i) were then
is labelled GMR (geometric mean relaxed) in fig. 6. It is interesting to note that the average of the GMS and
GMR curves is close to GMU, thereby implying considerable cancellation between the second-order RPAE effects and relaxation effects. In fig. 7 comparison is made with the recent experimental results of Codling et al. [33] and with earlier results by Marr and Austin [34] and Cairns et al. [35]. Fig. 7 also contains a Hartree—Slater calculation by McGuire [36]. The importance of the accurate treatment of exchange effects is very clear. The discrepancy of the MBPT results with experiment near maximum is not understood, and investigation of higher-order diagrams and possible relativistic effects would be interesting. Beyond 100 eV, calculated curves are below experiment due to omission of the 4p-subshell cross section. The correlation-corrected dipole matrix elements were also used [31] to calculate the asymmetry parameter $3 of the photoelectron angular distribution. 4.4. Chlorine The neutral chlorine atom 3s23p5 2P represents an interesting open-shell atom to test computational methods. Because of its proximity to argon in which correlation effects are significant, one might expect correlations to be important for Cl. However, a recent
/
approximated by calculating the excited kf states in a V”~44d9kfip potential using orbitals of Cd~4d95s2.
30
~20
r~i~i~i,i, GM5
:~:f 9
CADMIUM
-
4d—kp I
0
—
0
-
40
I
I
I
PHOTON ENERGY 160
~
200
/ ~
bi:
i;’
-~
I
I
/
2
I
/
The resulting curve including diagrams 1(a) and 1(c) 2~
105
240
e Fig. 6. Cadmium 4d —~ kf and 4d —~ kp cross sections. Curves shown are the geometric mean of length and velocity results. The 4d -+ kf curves are described in text.
50
100
—
150
200
250
PHOTON ENERGY (eV)
Fig. 7. Cadmium 4diO photoionization cross section. Experimental data points from Marr and Austin [34] x; Cairns et al. [351 0; Codling et al. [331 A. Chain curve is the Hartree— Slater calculation by McGuire [36]. Full curve is the geometric mean oflength and velocity results from the first-order MBPT calculation for the 4d —~kf plus 4d -# kp cross section [31]. Broken curve includes relaxation effects.
106
H.P. Kelly / Many body perturbation calculations
(RPAE) open-shell calculation random phase by Starace approximation and Armstrong with exchange [37] indicated that electron correlations are small. This work stimulated further calculations, and chlorine
+
—
+
— ~
now seems to be a standard system to test methods of calculating photoionization cross sections for openshell atoms. Three very recent calculations by different methods all find that correlation effects are important although there are differences among these calculations. Cherepkov and Chernysheva used the RPAE
+ ~
k P~\~Ø
q,r
—
p,q
[38]; Lamoureux and Combet-Farnoux [39] used the R-matrix methods [38]; and Brown et al. [40] used
MBPT with LS-coupled states which is essentially use of Rayleigh—Schradinger perturbation theory. The diagrams of MBPT are still used to indicate various processes but the states now represent LS-coupled
states. In the calculations of Brown et al. it was found that there are large contributions from diagrams such as fig. 1(b) in which p refers to a given ionic core of Cl~and q refers to a different core. For example,
p -÷ k in fig. 1(b) could refer to 3p5 2P -÷ 3p4 3Pkd 2D and q ~ k’ to 3p4 iDkld 2D or 3p4 1Sk”d 2D. In this case fig. 1(b) refers to a term in perturbation theory where H’ causes a transition from a single excitation of one core to a single excitation with a different ionic core. This type of interaction occurs in the next order indicated by the diagram of fig. 1(f). The diagonal contributions (with fixed core) are again eliminated by appropriate choice of the potential. Because of the slow convergence of these interactions involving different multiplets of 3p4, Brown et al. [40]
r\b
- ~--~
--
Fig. 8. Pictorial representation of coupled integral equations used to sum classes of diagrams as shown.
The MBPT lowest order (Hartree—Fock) length
result (not shown) is close to that of Starace and Armstrong, being 12% lower at the 1S edge and becoming closer at higher energies. The lowering of the length curve is due both to final state mixing among the core states and to correlations in the initial state. Near the 1S edge the MBPT results are in close
90 80 70
calculated these terms to all orders by solving the coupled integral equations indicated in fig. 8. States labelled p, q, r refer to the 3P, 1D, 1S multiplets of Cl~.In these calculations, the continuum was represented in each channel by 32 continuum states and 10 bound excited states. The higher bound states were included by a back extrapolation of the continuum into the bound region. The resulting 126 X 126 matrix was then inverted to solve for the correlated dipole matrix elements. In the case of 3p4 1D kd 2S, the dominant mixing is with 3s3p6 2S. The resulting cross section is shown in fig. 9 and includes both 3p —* k and 3s -÷ k cross sections. The only other calculation to include the inner shell 3s —~ kp excitations is that by Lamoureux and Cornbet-Farnoux [39].
=
40
3 20 10 15
20
25
30
35
40
45
PHOTON ENERGY leVI
Fig. 9. Photoionization cross section for ClI including 3p and 3s subshell cross sections [401. Solid (dashed) curve represents dipole-length (velocity) cross section. Solid circles, calculation by Starace and Armstrong [37]. Open circles, calculation by Lamoureux and Combet-Farnoux (length) [39]. Crosses, calculation by Conneely (length) [41]. Triangles, calculation by Cherepkov and Chernysheva 138].
H.P. Kelly /Many body perturbation calculations 3.5 3.0
I,’,,
107
Carter and Kelly used LS-coupled states which allowed
ii. .
—2 5 ~ a02.0
a determination of LS core-structure effects. In order to obtain the double photoionization cross section a~(w),one calculates the dipole length (or velocity) many-particle matrix elements
-~15s~
Z(,pq
-~
~I.O
k’k) = (‘hfI ~I~z1I’I’0>,
(28)
b
.5 fl~•
25
I
I
30 35 40 45 PHOTON ENERGY leVi
50
where I’I’~)is a many-particle state with orbitals p. q excited to k’, k. The corresponding velocity matrix element is obtained from eq. (21). Many-body diagrams for Z(pq kk’) are given in fig. 11 with (a)—(d) —~
Fig. 10. Total 3s photoionization cross section for Cli calculated by Brown et al. [40]. Upper curves are length (solid)
and velocity (dashed) including correlations. Lower curves are
being the lowest-order diagrams for this process and (e), (f), (g) are higher-order diagrams corresponding to effects found to be important in single photoioniza-
Hartree—Fock length (HFL) and velocity (HFV).
agreement with the length calculatidn of Conneely [41]. However, his velocity calculation (not shown) is almost a factor of two lower [41]. The coupled equations methods of this4 calculation forand the 1D, 3p4 ‘S, account 3s3p5 3P, resonances before the 3p 3s3p5 ‘P edges but they are not shown in fig. 9. The 3s -÷ kp cross section [40] is shown in fig. 10. It is seen that correlations with the 3p5 shell causes a qualitative change in the 3s cross section dust as for Ar [42,8].
PVkV~. (a)
V -.
(c)
Other MBPT calculations of interest include those onA[20],Na[43,44],K [44],C [45],C andF andA [46], Si~[47], [48]. and the isoelectronic sequences of Ne
(b)
\k~,~/
~
(d)
V
5. Double photoionization cross sections The process of double photoionization, where one photon ejects two electrons, is very interesting since it only occurs because of electron correlations. There has now been considerable experimental work on this
process for the rare gases [49—53]where it is found that double photoionization can contribute a significant fraction of the total cross section when it is energetically allowed. With the exception of helium, there have been very few calculations of this process and all (except for He) have used MBPT: Ne by Chang,
Ishihara and Poe [54] and by Chang and Poe [55]; Be by Winkler [56]; C by Carter and Kelly [57]; and ~ and A by Carter and Kelly [58]. The calculations by
k
k
—
- -
(e)
(f)
-..
(9) Fig. 11. Diagrams contributing to the matrix element Z(pq k’k). Full circles indicate matrix element of the dipole operator z; broken lines represent Coulomb interactions. Exchange diagrams are also included.
JLP. Kelly / Many body perturbation calculations
108
tion calculations. Exchange diagrams are understood to be included, With the normalization of eq. (12), 2
C
dk IZ~q k’k)I
0
(29)
where
k” Defining
Ep+Eq
,,
~
l’h’k’>=
[2(e~ + Eq
-~
~k2 + ~
I,
(31)
,
-Ek’Ek”
I
1k ),kk v]pq)
k” Ep + Eq =
,
,
)~kk vlpq)
DaE~~Z~
kmax
u~(w) = 16 ~
60]. The expression for this diagram is
—
(32)
Ek’
1/2
Da
(klzI’I’k’).
=
(33)
and kmax
=
[2(e~ +
+
~
(30)
1/2
For both neon and argon, Z(pq k’k) were calculated for cross sections in which an (np)2 pair is ejected, leaving the ion in the (np)4 3P, ‘D, or is -~
levels and also for cross sections in which an nsnp
pair is ejected leaving the ion nsnp 3P or 1P. The angular momentum of the outgoing k’k pair of elec-
Letting E = (E —
H
+ ~q
T’k’) =
0) I ‘
—
Ek’,
eq. (32) may be written
~II~ Ik” )(k’k” I v Ipq).
(34)
k
Applying closure, one obtains (E—Ho)I’I’k’)R ~‘F(r)
where
‘h’k’)R
(35)
is the radial part of I’1k’>, and
trons must couple with the orbital and spin angular
momentum of the ion to give a total ip state, and so there is an infinite number of types of outgoing electron pairs. For ejection of (3p)2 from argon, the outgoing continuum pairs which were included were kskp, kpkd, kskf, and kdkf. For ejection of 3s3p pairs, the calculated continuum pairs were k’pkp, k’dkd, and kskd. For neon, all these pairs were
included as well as k’sks, kpkf and k’fkf for 2s2p ejection. In sums over excited states, eight bound excited states were calculated and the continuum was approximated by 30 orbitals ranging from k = 0.062 5 to 24.0 au. Matrix elements Z(pq -÷ k’k) were deter-
dr’ Pk’(r)PP(r’)
0
X (r
T~)~~qfr)—
I~ln>R(lcnlvlpq),
n 0cc
(36)
where r< (r>.) is the lesser (greater) of r and r’. Results for u~(w)for argon are shown in fig. 12 27
~
24 21
mined using LS-coupled wave functions and perturba-
states indicated by the diagrams of fig. 1 l(a)—(d). tion theory with the configurations for intermediate The calculations were also checked by use of the usual MBPT formalism which does not distinguish subcross sections corresponding to the different multiplets of Ne++ or A++. In all cases the matrix element Z(pq -~k’k) was represented as a 10 X 10 matrix of k values ranging from 0.0625 to 5.0 au. In evaluating eq. (29), four point Lagrange interpolation was used. The diagrams of fig. 11 were evaluated by summation over the excited bound states and numerical integration over the continuum. The diagram of fig. 11(a) can be evaluated this way but requires care since the dipole matrix element is strongly peaked for k = k”. This diagram was also evaluated by the differential-equation or effective-operator technique [59,
f
F(r) =
-
00
-
18
‘~
I’,
1 12 0
b ~ ‘~
70
100
160
PHOTON
ENERGY
190
cv)
Fig. 12. Total o~~(w)for neutral argon. Full curves are lowest-order results for dipole-length (L) and dipole-velocity (V) matrix elements. Broken curves include second-order correlations [58]. Experimental data points are from Schmidt et al. [49], full circles; Wight and Van der Wiel [50], open circles; Samson and Haddad [51], open triangles; Carlson [53], full triangles. Arrow mark, respectively, excitation thresholds for 3s23p4(3P, 1D, 1S) and 3s3p5(3P, 1P) core levels.
109
H.P. Kelly /Many body perturbation calculations
and compared with experimental results. These curves include 30 individual cross sections representing different LS-coupled terms and configurations as already described. The solid curves are the lowestorder results represented by the diagrams of fig. 1 1(a)—(d). Selected higher order terms as shown in fig. 11(f) and (g) were also calculated because the lower parts of (f) and (g) were found in previous dipoletomatrix work provide elements. important Thecorrelations effect of including in the p
-~
k”
these diagrams is shown by the broken curve in fig. 12. The experimental points were normalized with the total argon cross section data of West and Marr [61] and assuming that a3~can be neglected. In fig. 13 separate results for 3p2 and 3s3p cross sections are presented, and it is noted that 3s3p transitions account for more than 25% of the total ci~4(w). Contributions from different partial waves contributing to the 3p2 cross section are shown in fig. 14. Although the cross section is dominated by 3p2 -~ kpkd, the sum of other partial waves contributes considerably. it is reasonable that the remaining infinite sum of partial waves (e.g., 3p2 -~ kpkg, kf7cg, etc.; 3s3p kpkf, kdkg, etc.; and 3~2-~ kk’) might contribute an additional 10—15% to the cross section. Similar calculations for neon were carried out and resulted in the u~’cross section shown in fig. 15. The discrepancy with experiment of the neon u~maximum position and initial slope is more pronounced than in argon. For both argon and neon, continuum -~
8 6
12
,.~-
~
/
~~10
//
8 6
//
b~ 2
~d
-
kf
-~
40
70
100
130
160
190
220
250
PHOTON ENERGY 1Ev)
Fig. Partial-wave cross for four 3p2(solid -÷ k’k channels. 14. Results are plotted forsections both dipole-length curves) and dipole velocity (broken curves) matrix elements.
states were calculated in the Hartree—Fock V’ potential [5] which is asymptotically hr. One might expect a VN_2 potential [5] which is asymptotically 2/r would be more appropriate because of the plus two charge of the residual ion. However, the kinetic energy spectrum da~~16w IZ(pq -~ k’k)12 = — (37) dek c k’k —
-
30
0
A
£
~25 ~2O
•~ A
27
A
215
24
21
A A
7
18 -:~l5 012 b9
1
6
~__
~p2
:
10 5 /0101 A A I//1
1111
/
________________________________________________ 40
70
100 130 160 PHOTON ENERGY 1EV)
190
220
Fig. 13. Cross sections for total 3p2 —~k’k and 3s3p -+ k’k excitations in neutral aigon from [58]. Results are plotted for both dipole-length (solid curves) and dipole-velocity (broken curves) matrix elements.
250
210 I 260 PHOTON ENERGY 1EV)
310
Fig. 15. Total a++(w) for neutral neon. Lowest-order calculated results are shown for dipole-length (solid curve) and dipole-velocity (broken-curve) matrix elements [58]. Experimental data points from Schmidt et al. [49], full circles; Wight and Van der Wiel [50], open circles; Samson and Haddad [51], open triangles; Carlson [53], full triangles; Lightner et al. [52], open square. Arrows mark, respectively, excitation thresholds for 2s22p4(3P, 1D, 1S) and 2s2p5(3P, 1P) core levels.
110
H.P.
Kelly / Many body perturbation calculations 2 cases. The curve between the VN_l and VN labelled CP in fig. 7 is the 2p2 -~ kpkd velocity result
9 -
from the MBPT calculation by Chang and Poe [55] -
7
-
-
~:I
E~e,
w
0
J
:
which used VNI states. The discrepancy between the CP and VN~ curves in fig. 17 is not fully under-
\\\\
~
desirable total result to considerably investigate further above the experiment. effects ofItscreenis very
/~~A71___
ing, choice of potential and effects of higher order diagrams.
_______________________________ 50
0
100
150
stood [58]. While the CP curve initially shows better agreement with experiment, inclusion of other partialwave channels including 2s2p ~ k’k will push the
200
250
e(eVl
Fig. 16. Kinetic energy distributions for the neon 2p2 —° k’k (solid curve) and 2s2p —~ k’k (broken curve) excitations calculated at w = 278 cV [58]. Chain curve is the sum of both
6. Concluding remarks In the preceding sections it has been shown how
contributions. many-body perturbation theory can be applied to the calculation of many different aspects of photoabsorp-
is plotted for neon in fig. 16 and shows that in general the available kinetic energy is unequally distributed. The faster electron may therefore “feel” a residual charge of + 1, while the slower electron experiences a charge of approximately +2. In fig. 17 are plotted lowest-order velocity-form calculations for neon 2p2 kpkd excitations using single particle states calculated in both the VN_2 and VN~ potentials. Near threshold the slope of the VN_2 curve agrees -+
better with experiment but is too steep. Presumably the correct a~ curve for 2p2 —~kpkd is intermediate
tion including resonances and double photoionization. In addition, this theory has previously been applied to the calculation of many other atomic properties. A principal characteristic of this theory is that one has a well-defined procedure to calculate complete sets of single particle states including, of course, the continuum. These same sets of states may be used to calculate many different atomic properties. In this theory one is, in practice, usually able to calculate only two or three orders of perturbation. however,
it is sometimes possible to include particular interactions to all orders as was discussed in connection with resonances. Also, the potential may be chosen to
35
30L
facilitate convergence by including certain types of interactions. When there is a slow convergence, or divergence, of the perturbation expansion, it may be possible to sum the diverging terms to all orders by
25~ E20~
~
methods such as the coupled integral equations discussed in connection with chlorine. The MBPT methods are very appropriate for calculations in large
~1
atoms as well as small atoms, and calculations are also now underway using relativistic states [62].
51 0 60
110
160 210 260 PHOTON ENERGY 1EV)
310
2 —~kpkd transitions. Curves labelled andfor VN2 refer 2p to conFig. 17. Velocity-form crossVN~ sections the neon tinuum states calculated in these potentials [58]. Curve labelled CF is from V~~’1 calculation by Chang and Poe [551.
Acknowledgements I wish to thank my many collaborators particuport from the U.S. National Science Foundation is gratefully acknowledged.
larly E.R. Brown, S.L. Carter and A.W. Fliflet. Sup-
H.P. Kelly / Many body perturbation calculations
References [11 K.A. Brueckner,
Phys. Rev. 97 (1955) 1353; Phys. Rev. 100 (1955) 36; The many-body problem (John Wiley,
New York, 1959). [21 J. Goldstone, Proc. Roy. Soc. (London) A239 (1957) 267. [3] A.L. Fetter and J.D. Walecka, Quantum theory of manyparticle systems (McGraw-Hill, San Francisco, 1971). [4] N.H. March,W.H. Young and S. Sampanthar, The manybody problem in quantum mechanics (Cambridge University Press, London, 1967). [5] H.P. Kelly, Advan. Chem. Phys. 14 (1969) 129. [6] U. Fano and J.W. Cooper, Rev. Mod. Phys. 40 (1968) 441. [7] Photoionization and other probes of many-electron interactions, ed. F. Wuileumier (Plenum, New York, 1976). [8] PG. Burke and K.T. Taylor, J. Phys. B8 (1975) 2620. [9] M. LeDourneuf, Invited Talks of X ICPEAC, to be published. [10] H.P. Kelly, Advan. Theoret. Phys. II (Academic New York, 1968) 75, [11] H.A. Bethe and E.E. Salpeter, Quantum mechanics of one and two-electron atoms (Academic, New York, 1957) p. 306. [12] M.Ya. Amusia, NA. Cherepkov and L.V. Chernysheva, Zh. Eksp. Teor. Fiz. 60 (1971) 160 [Soy. Phys. JETP 33
(1971) 90J. [13] G. Wendin in Vacuum ultraviolet radiation physics, eds. E.E. Koch, R. Haensel and C. Kunz (Pergamon Vieweg Braunschweig, 1974) p. 225. [14] L.M. Franta, R.L. Mills, R.G. Newton and A.M. Sessler, Phys. Rev. Lett. 1(1958) 340. [15] B.A. Lippman, M.H. Mittleman and K.M. Watson, Phys. Rev. 116 (1959) 920. [16] R.T. Pu and E.S. Chang, Phys. Rev. 151 (1966) 31. [17] H.J. Silverstone and M.L. Yin, J. Chem. Phys. 49 (1968) 2026. [18] S. Huzinaga and C. Arnau, Phys. Rev. Al (1970) 1285. [19] T. Ishihara and R.T. Poe, Phys. Rev. A6 (1972) 111. [20] H.P. Kelly and R.L. Simons, Phys. Rev. Lett. 30 (1973) 529. [21] H.P. Kelly, in Atomic physics 2, eds. G.K. Woodgate and P.G.H. Sandars (Plenum, London, 1971) p. 227. [22] 1. Lindgren, in Atomic physics 4, eds. G. zu Putlitz,
111
[281 J.P.Connerade and M.W.D. Mansfield, Proc. Roy. Soc. A341 (1974) 267. [29] A. Rabe, K. Radler and H.-W. Wolff, in Vacuum ultraviolet radiation physics, eds. E.E. Koch, R. Haensel and C. Kunz (Pergamon Vieweg, Braunschweig, 1974) p. 247. [30] G. Wendin, Phys. Lett. 46A (1973) 119. [31] S.L. Carter and H.P. Kelly, J. Phys. Bil (1978) 2467. [32] A.E. Hansen, Molec. Phys. 13(1967)425.
[331 K. Codling, J.R. Hamley and J.B. West, J. Phys. Bli (1978) 1713. [34] G.V. Marr and J.M. Austin, Proc. Roy. Soc. A310 (1969) 137. [35] R.B. Cairns, H. Harrison and R.I. Schoen, J. Chem. Phys. 51(1969) 5440. [36] E.J. McGuire, Phys. Rev. 175 (1968) 20. [37] A.F. Starace and L. Armstrong, Jr., Phys. Rev. A13 (1976) 1850. [38] NA. Cherepkov and LV. Chernysheva, Phys. Lett. 60A (1977) 103. [39] M. Lamoureux and F. Combet-Farnoux, Abstrs. 5th VUV Conf. (Montpellier, France, 1977). [40] E.R. Brown, S.L. Carter and H.P. Kelly, Phys. Lett. 66A (1978) 290. [41] M.J. Conneely, Ph.D. Thesis (London University, 1969) unpublished. [421 M.Ya. Amusia, V.K. Ivanov, N.A. Cherepkov and L.V.
Chernysheva, Phys. Lett. 40A (1972) 361. [43] T.N. Chang, J. Phys. B8 (1975) 743.
[44] J.J. Chang and H.P. Kelly, Phys. Rev. A12 (1975) 92. [45] S.L. Carter and H.P. Kelly, Phys. Rev. A13 (1976) 1388.
[46] T. Ishihara and T.C. Foster, Phys. Rev. A9 (1974) 2350. [47] G.R. Daum and H.P. Kelly, Phys. Rev. A13 (1976) 715. [48] T.N. Chang, Phys. Rev. A16 (1977) 1171. [49] V. Schmidt, N. Sandner, H. Kuntzemüller, P. Dhez, F. Wullleumier and E. K~llne,Phys. Rev. Al3 (1976) 1748. [50] G.R. Wight and M.J. Van der Wiel, J. Phys. B9 (1976) 1319. [51] J.A.R. Samson and G.N. Haddad, Phys. Rev. Lett. 33 (1974) 875.
[52] G.S. Lightner, R.J. Van Brunt and W.D. Whitehead, Phys. Rev. A4 (1971) 602. [53] T.A. Carison, Phys. Rev. 156 (1967) 142. [54] TN. Chang, T. Ishihara and R.T. Poe, Phys. Rev. Lett. 27 (1971) 838. [55] T.N. Chang and R.T. Poe, Phys. Rev. A12 (1975) 1432. [56] P. Winkler, J. Phys. BlO (1977) L693.
E.W. Weber and A. Winnacker (Plenum, New York, 1975) p. 747. [23] A.W. Fliflet and H.P. Kelly, Phys. Rev. AlO (1974) 508. [24] G.V. Marr and J.M. Austin, J. Phys. B2 (1972) 168.
[57] S.L. Carter and H.P. Kelly, J. Phys. B9 (1976) 1887.
[25]U. Fano, Phys. Rev. 124 (1961) 1866. [261 M. Wilson, J. Phys. Bl (1966) 736. [27] A.W. Fliflet, R.L. Chase and H.P. Kelly, J. Phys. B7 (1974) L443.
Phys. Rev. All (1975) 758. [61]J.B. West and G.V. Marr, Proc. Roy. Soc. A349 (1976)
[58] S.L. Carter and H.P. Kelly, Phys. Rev. A16 (1977) 1525.
[59] R.M. Sternheimer, Phys. Rev. 115 (1959) 1198. [60] S. Garpman, I. Lindgren, J. Lindgren and J. Morrison,
[62] S.L. Carter and H.P. Kelly, to be published.
MANY BODY PERTURBATION CALCULATIONS OF PHOTOIONIZATION H.P. KELLY Department ofPhysics, University of Virginia, Charlottesville, VA 22901, USA
The application of many body perturbation theory to the calculation of atomic photoionization cross sections is reviewed. The choice of appropriate potential for the single-particle states is discussed and results are presented for several atoms including resonance structure. In addition to single photoionization, the process of double photoionization is considered and is found to be significant.
1. Introduction The many body perturbation theory (MBPT) of Brueckner [1] and Goldstone [2] has been very useful in many areas of physics [3,4] including atomic physics [5]. As in most perturbation theory formalisms, one requires appropriate complete sets of single particle states and different methods to obtain these states must be developed for different physical problems. Frequently, it is necessary to use numerical
potentials. The Hamiltonian for atoms of atomic number Z is N
N
i=1
i
~II~ T, +
H=
(1)
~1
where T1 =
V~/2 Z/r~ (2) Atomic units are used throughout this paper. In a first approximation, ~u~1is replaced by ~ V~, where V~is a Hermitian potential which may be arbitrarily chosen. The approximate resulting Hamil-
methods to obtain the appropriate single particle states and matrix elements involving these states. In this paper, MBPT is applied to the calculation of atomic photoionization cross sections, and results are reviewed for such atoms as Zn, Ba, Cd and Cl. The process of double photoionization is also considered for which the absorption of one photon results in the ejection of two electrons. Double photoionization results for neon and argon are discussed. Important reviews of atomic photoionization have been given by Fano and Cooper [6] and in the book edited by
—
.
tonian is N
H0
=
~
v~)
(T, +
,
(3)
~=
and the resulting states 1a satisfy H0’t~ 0 = ~ The states 4~are determinants containing N singleparticle solutions ~ of
Wuileumier [7]. A description of the very successful results using R-matrix theory may be found in articles by Burke and Taylor [8] and by LeDourneuf [9].
(7’ + V) ~n = nctln (4) One can systematically include correlation effects by using perturbation theory with .
N
2. Many body perturbation theory
H’
N
=
VI.
—
i
2.1. Linked cluster expressions for energy and wave functions
(5)
i=1
According to the results of Brueckner [1] and Goldst one [2],
Consider N electrons interacting with two-body 2 /r potentials v~= e 11 in addition to the one-body
=
~ L
99
(
1 —
H0 H’) ~
(6)
H.P. Kelly / Many body perturbation calculations
100 and
t0~=(I
L~E0=E —E 0lH’I’I’~>, (7) where indicates that only “linked” terms [1,2] are included. Terms in the perturbation expansions of eqs. (6) and (7) may be represented by diagrams [1—4]. 0
0
~L
for large n. Inserting (11) into (10), we find 2 , Imc~(w)(2/k)I(kIzjp>l
(14)
where k = (2~+ 2w)”2
(15)
.
The photoionization cross section u(w) is (in
atomic units) [Ill 2.2. Frequency-dependent polarizability and the photoionization cross section
u(w)
Cox~idera perturbing electric field E = F~cos wt with the accompanying time-dependent perturbation
21) z
1.
P = ct(w)E,
(9)
where ct(w) is the frequency-dependent polarizability which may be evaluated by MBPT [10]. The lowest-order contribution to a(w) from the
orbital cb~occupied in ~I’~is given by (kjzlp>12( _______1 \e~— Ek
+—
W
—
1
~,
Ek + ~
/
(10)
where the sum over k includes all excited states (i.e., those not occupied in uum.
both bound and contin-
~
In eq. (10) if Ip), 1k) and are exact eigenstates and eigenvalues respectively ofH and z represents ~ z~,then eq. (10) gives cr(w) exactly. Since e~ may we add a small imaginary part ii~ andwuse thevanish, formula ~,
5
+
lim (er—
Ek + W +
recalling the normalization of eq. (12). From eq. (14),
we obtain the familiar results [6]. (17)
It can be shown that the many-body diagrams for Im a(w) may be factored to give the exact (‘J!k X ~ z~ I ‘I’d> times its complex conjugate times correla-
The induced dipole moment P is given by
—
(16)
,
(8)
i=1
—21)k
(8irw/ck) I (kIzlp>~2
u(w)=(4irw/c)Imcs(w).
N
V(r, t) =Fcos wt
=
tions due to normalization diagrams
[51.Effects of
normalization diagrams have been less than 5% for all cases studied to date. Diagrams contributing to (~‘PkI~ z~IW0)in which there is a p ~ k transition are shown in fig. 1. The solid dot represents matrix elements of z and the cross represents interaction with —V. The other dashed lines represent Coulomb interactions. The diagrams of fig. I are read from bottom to top corresponding to right to left in the mathematical expressions. In the diagrams, Coulomb interactions (i.e.,
with H’) below the heavy dot correspond to correlations in the initial state I ‘I’~)and Coulomb inter-
actions above the heavy dot Exchange corresponddiagrams to correlaare tions in the final state not explicitly shown in fig. 1 but should be included. 1’k>.
‘
Diagram 1(a) is the lowest order diagram and repre-
ii sents the matrix element (kjzjp). In the next order
1)’
of perturbation theory, diagrams 1(b), (c), (d) and
=P(e~ w)~ i~(e~ Ek + w), (11) where P represents principal value integration. In the summation continuum are included by (2/ir) f~dk,~which assumes states continuum states are —
+
—
—
(e) occur and also exchange diagrams corresponding
to (b), (c) and (e). Diagram 1(b) represents correlations in given by
4’k
and is
‘
normalized according to
~(kqIvIpk’)(k’IzIq>(eq Rk—~cos[kr+6l+(q/k)ln2kr—~(l—1)ir]/r (12) as r where V(r) q/r. Bound states are included by finite summation and use of the extrapolation —~
KnIzIp)12
-+
—~
C/n3
(13)
~k’
+
w)’.
(18)
k
The exchange of 1(b) is obtained by replacing
(kqlvlpk’) by —(qkl vlpk’). The expression for diagram 1(c), which represents
H.P. Kelly I Many body perturbation calculations
/
P~k
\.L_. (b)
q
(a)
the corresponding velocity cross sections, including correlations tends to bring them closer.and However, agreement of length and velocity cross sections is only a necessary but not sufficient condition to indicate accuracy of the solutions.
k
0
101
———-
(c)
q~k
3. Potential V for single-particle states e
Although the choice of potential V is, in principle, arbitrary, it may have a drastic effect upon the con-
q
I!
q
k
~
k
__.UI’ (g)
(h)
Fig. 1. Diagrams contributing to <‘ikI ~ z~’I~&. The solid dot represents a matrix element of z and the cross represents interaction with — V. Other broken lines represent Coulomb interactions. Exchange diagrams are also included.
vergence properties of the perturbation expansion. It is usually desirable to choose V so that we have Hartree—Fock(HF) orbitals occupied in 4~.However, as discussed previously [14—18],the potential for HF ground state orbitals is not unique as is readily seen by considering the potential [17,1 8] V= R
+
(1
—
Q) ~7(l Q),
(22)
—
where &2 is an arbitrary Hermitian operator and R is a ground state correlations is 21)(qlzlk’> (kk’IuI pq>(ep + 6q
—
6k
—
6k’).
(19)
potential which gives the N Hartree—Fock orbitals occupied in I~.The operator N
21)
k
The sum of diagrams (d) and (e) and the exchange of (e) is
‘~
N
(iIRIj) w)’
.
(23)
where states In> are occupied in I~. For example, R is often written [1—4]
~ (—(kI V]k’>+
In)(nI,
(20)
X (k’IzIp> (e~ Diagrams l(f)—(i) represent some of the diagrams which are second order in H’. We note that all the diagrams of fig. 1 except for 1(i) are included in the random phase approximation with exchange [12,13]. In calculations it is customary to evaluate both “length” (z) and “velocity” (dfdz) forms of the matrix elements. When “k> and I ‘I’o> are exact eigenstates of H, they are related by
21)
((inluljn)—
1)).
n1
—
N
‘I’ )
fI
I
Z~
— —
E
(
—
~ — Ef)
N
< \/~, f ~ ~d
~ 0/.
~21
In most cases studied so far, the lowest-order (Hartree—Fock) length cross sections are greater than
Since (1 Q)In) = 0 when In> is occupied in Vin) = RIn>. For an “excited” orbital, however, VIk> = Rik> + (1 Q)~2Ik>.We may choose 12 to accelerate the convergence of the perturbation expansion. In those cases where the excited states are of a symmetry different from that of any orbitals occupied in ~ the potential may be chosen arbitrarily and one does not need eq. (22) to ensure ortho—
~
—
gonality to occupied orbitals. As an example of the single-particle potential, consider argon 3p6 lg• Dipole absorption of a photon leads to final many-particle states 3p5kd ‘P and 3p5ks ‘P. If these states are represented by singleparticle states, we obtain an appropriate potential for the kd and ks orbitals by setting (3p5k ‘PIH’I3p5k ‘P>
0
,
(25)
H.P. Kelly / Many body perturbation calculations
102
or 5k’PI ~u~I3p~k
‘1~.
(26)
(kIVIk’)= (3p / For ks, eq. (22) is needed so that the ks orbitals are
orthogonal to hs, 2s and 3s. The Vof eq. (26) results
in the cancellation of fig. 1 diagrams (d), (e) and also (b) when q belongs to the same subshell asp [12,19].
Also, higher dsder diagrams such as (0 are cancelled by V. Using this potential the first-order corrections in H’ come exclusively from diagram (c) and its exchange which account for ground state correlations, This choice of potential is that appropriate for Hartree—Fock continuum states obtained by b (3pSkiPIHI3pSk ‘P> = 0 , (27)
near threshold by an order of magnitude from the experimental of Marr and Austin [24] as shown in fig. 2. Theredata are resonances from 3d np excita—*
tions and the lowest-order diagrams contributing to these is fig. 1(b) with p = 4s, q = 3d, k’ = np. Higher-
order diagrams may be summed exactly geometrically as indicated in fig. 3 to give effective widths to the resonances. An indication of the non-resonance correlations is given by the curves NRV (velocity) and NRL (length) in fig. 2. These were calculated by inclusion of only those diagrams which do not contribute to resonances. Diagrams contributing to resonances contain single
3d tip excitations alone in one or more denominators. The lowest-order curves are noticeably reduced, -~
and proved very useful in MBPT calculations of u(w) for argon [20]. I
~
6
4. Single photoionization cross sections HFL x
5k—
Many-body perturbation theory has been applied to the calculation of many properties such as correlation energies, polarizabilities, oscillator strengths, fine structure and hyperfine structure [21,22]. In this paper results are presented only for photoionization
HFV
-
-
cross sections. In summing over single-particle states as in eqs. (18)—(20), we must ensure that our set of
single-particle states is effectively complete. Generally, this is done with approximately ten bound excited states and use of eq. (13) to account for the remainder and with approximately thirty continuum states. For the outer shells of atoms, a maximum value k = 30.0 usually suffices. The initial spacing in k is usually 0.1 and then becomes coarser after 1.0. Sums over the continuum are done numerically.
4.1. Zinc As an example of the application of MBPT to photoionization cross sections, we consider the (4~)2 subshell MBPT calculations [23]. The final state in this case is 4s kp ip and the potential is chosen as in section 3 so that in diagram 1(b) we only take q to
represent inner shells. In this calculation only correlations among 4s and 3d electrons were included. It was discovered that there are very large correlation effects with the lowest-order cross section differing
r b
2
r
~--~
vor
-
VCF [____~_~
-
100
95 PHOTON
---
-
-
05
00
ENERGY (eVI
Fig. 2. Cross section 0(w) for photoionization of the 4s sub-
shell of ZnI near threshold. The curves HFL and HFV represent Hartree—Fock length and velocity cross sections using frozen-core (FC) orbitals. Curves NRV and NRL represent correlated cross sections omitting contributions from resonance diagrams. Curves VCF and VCI represent correlated velocity cross sections with FC and IC (ionic-core) orbitals respectively. The circles are experimental results by Marr and
Austin 1241.
H.P. Kelly /Many body perturbation calculations
4s
kp
4s
kp
-—
--
3d
mp
3d
np
--
--
4s
kp
4s
kp
-~
3d
94s2np’P,,
one must consider resonance states 3d
(b)
kp
3d
——
3d
summed geometrically. The resulting expression [23] is equivalent to that derived previously by Fano [25] for the case of one continuum and many interacting resonances. In the segment shown in diagram 3(c), one sums over all excitations mp and this accounts for the interaction between resonances mp and m’p. For Zn, the resonance structure due to 3d np excitations shows effects of spin—orbit splitting and -*
T’p
(a) 4s
103
~ —-
mp
4s
kp ~
I
3d
4S ~
P
3D,, 3P, which are sufficiently close to interact. The intermediate-coupling energies and mixing coefficients for these states were calculated by the method of Wilson [26]. The resulting calculated cross section is shown in fig. 4 along with the experimental results of Marr and Austin [24]. The absorption window between the two large resonances is obtained in calculations only when interactions between resonances are included.
(d)
(c)
Fig. 3. Diagrams and diagram segments associated with reso-
nances. The horizontal line indicates that the denominator should be treated according to —iir~(D). The denominator for diagram (d) includes both PD1 and —iira(D).
4.2. The 4d’°subshell of barium The 4dio subshell of BaI was calculated [27] to .
.
compare with the interesting experimental results of
particularly in the length case. Including both resonance and non-resonance diagrams, the curve VCF of fig. 3 was obtained with the length curve being 10% lower. The VCF curve was calculated using kp orbitals of the configuration 3d’04s kp in which occupied orbitals of ZnI were used to calculate a frozen core (FC) potential. The velocity curve labeled VCI used occupied orbitals of Zn~in the potential for kp. All diagrams of fig. 1 involving 4s and 2d electrons were
included except for 1(i). Large correlation effects came from 3d
—*
kf and 4s
—*
4p excitations. Diagram
1(f) with p 4s, q = 3d, r = 4s was found to be quite significant, reducing diagram 1(a) by approximately 20%. This does not indicate a slow convergence of the
Connerade and Mansfield [28] and of Rabe, Radler and Wolff [29]. A previous calculation by Wendin [30] had predicted a collective resonance for this cross section, and it was of interest to test this prediction by a perturbation calculation. The excited states were calculated in a VN~4d9kip potential using orbitals of Bal. Both 4d kp and 4d kf transitions were calculated, and most of the cross section was observed to come from 4d kf. The experimental 4d removal energy (99.28 eV) of Connerade and Mansfield [28] was used in the calculations. The spin— orbit splitting between 2D 2D 512 and 312 cores was included, and the cross sections for these cores was assigned from the non-relativistic result according to statistical weights. Results of the calculations are -+
-~
—*
compared with experiment in fig. 5, where the exper-
perturbation expansion, however, since this is the lowest order in which the large 4s 4p dipole matrix
imental curve was normalized to give an oscillator strength of ten. The dashed curve includes the lowest-
element occurs in final state correlations,
order diagram of fig. 1(a) and diagram 1(c) (and
The lowest-order resonance diagram of fig. 1(b) is modified by higher-order diagrams as shown in fig. 3(a) and (b). The horizontal line indicates that the denominator is treated according to —ix6(D) as in eq. (11). The section of diagrams 3(a) and 3(b) which is mdicated in 3(c) is repeated in higher orders and may be
exchanges) with p, q = 4d which represents ground state correlations in the 4dio subshell. The remaining first-order (in H’) diagrams (1(b), (d), (e)) cancel because of the choice of potential. There are contributions due to diagram 1(b) where q refers to other subshells but these were estimated to be small. The
—~
H.P. Kelly /Many body perturbation calculations
104 60,
I
I
I
150-
,~,
4030
-
I
-
-
-
/~ ii
120II 0 —
-
L/
-
94s24p resonance region. Solid line, correlated velocity cross section; dash—dot line Fig. 4. Cross section of ZnI o(w) 11.0in the 3d 11.2 114PHOTON 11.6 ENERGY 11.8 (eV) 12.0 122 124 correlated length cross section; dashed line, experimental data of Marr and Austin [241.
lowest-order (Hartree—Fock) calculations yielded
velocity results into close agreement. The calculated
length and velocity curves which bracketed the normalized experimental curve. Inclusion of the ground state correlations of fig. 1(c) brought the length and
curve of fig. 5 is an average of the final length and velocity results. The solid curve in fig. 5 is obtained from the random phase approximation with I
I
I
I
80
-
70
-
60
-
/
N
/
E50 040
‘S
I,‘/I
//
\
f
b 30 2010-
\..
“
-
•
//
-..--.‘..
N....
-
‘1
//‘ “I
95
‘..
100
---.---_.~..
I
105
110
115
120
-
I
I
I
125
130
135
140
145
PHOTON ENERGY (eV) 2D 2Dresults for °4dof Bal: Fig. 5. Comparison of calculated and experimental , average oflength and velocity form RPAE results including effect of splitting of 512 and 312 cores; , experimental data of Connerade and Mansfield [28] normalized to give an oscillator strength of 10; — — —, average of length and velocity ground state correlation results including spin— orbit spitting of cores.
H.P. Kelly /Many body perturbation calculations exchange (RPAE) and includes diagrams such as (g) and (h) in fig. 1 and also higher-order diagrams of this type. It is interesting that the low-order Hartree-Fock calculation with ground state correlation gives such good results, possibly indicating significant cancellation among higher-order diagrams. 4.3. The 4d1°subshell of cadmium Calculations recently carried out [31] on the 4d’° subshell of cadmium help us to understand the poor agreement of RPAE results with experiment in the case of Ba. The excited kf and kp states were first calculated in a VA’_i Hartree-Fock potential 4d9ktP. The Hartree—Fock length cross section was approximately twice that calculated by the velocity operator. However, including the ground state correlation of fig. 1(c) brought the curves into close agreement. The geometric mean of the resulting length and velocity cross sections is labelled GMU (geometric mean unrelaxed) in fig. 6. We have here adopted the suggestion of Hansen [32] of using the geometric mean of the length and velocity results. The potential for the excited kf states in this case used the unrelaxed orbitals of Cdl. The second order RPAE diagrams of fig. 1(g) and (h) were then included and the resulting curve labelled GMS (geometric mean second order) is plotted in fig. 6 and is higher than the curve GMU. Other second-order diagrams which represent relaxation effects as shown in fig. 1(i) were then
is labelled GMR (geometric mean relaxed) in fig. 6. It is interesting to note that the average of the GMS and
GMR curves is close to GMU, thereby implying considerable cancellation between the second-order RPAE effects and relaxation effects. In fig. 7 comparison is made with the recent experimental results of Codling et al. [33] and with earlier results by Marr and Austin [34] and Cairns et al. [35]. Fig. 7 also contains a Hartree—Slater calculation by McGuire [36]. The importance of the accurate treatment of exchange effects is very clear. The discrepancy of the MBPT results with experiment near maximum is not understood, and investigation of higher-order diagrams and possible relativistic effects would be interesting. Beyond 100 eV, calculated curves are below experiment due to omission of the 4p-subshell cross section. The correlation-corrected dipole matrix elements were also used [31] to calculate the asymmetry parameter $3 of the photoelectron angular distribution. 4.4. Chlorine The neutral chlorine atom 3s23p5 2P represents an interesting open-shell atom to test computational methods. Because of its proximity to argon in which correlation effects are significant, one might expect correlations to be important for Cl. However, a recent
/
approximated by calculating the excited kf states in a V”~44d9kfip potential using orbitals of Cd~4d95s2.
30
~20
r~i~i~i,i, GM5
:~:f 9
CADMIUM
-
4d—kp I
0
—
0
-
40
I
I
I
PHOTON ENERGY 160
~
200
/ ~
bi:
i;’
-~
I
I
/
2
I
/
The resulting curve including diagrams 1(a) and 1(c) 2~
105
240
e Fig. 6. Cadmium 4d —~ kf and 4d —~ kp cross sections. Curves shown are the geometric mean of length and velocity results. The 4d -+ kf curves are described in text.
50
100
—
150
200
250
PHOTON ENERGY (eV)
Fig. 7. Cadmium 4diO photoionization cross section. Experimental data points from Marr and Austin [34] x; Cairns et al. [351 0; Codling et al. [331 A. Chain curve is the Hartree— Slater calculation by McGuire [36]. Full curve is the geometric mean oflength and velocity results from the first-order MBPT calculation for the 4d —~kf plus 4d -# kp cross section [31]. Broken curve includes relaxation effects.
106
H.P. Kelly / Many body perturbation calculations
(RPAE) open-shell calculation random phase by Starace approximation and Armstrong with exchange [37] indicated that electron correlations are small. This work stimulated further calculations, and chlorine
+
—
+
— ~
now seems to be a standard system to test methods of calculating photoionization cross sections for openshell atoms. Three very recent calculations by different methods all find that correlation effects are important although there are differences among these calculations. Cherepkov and Chernysheva used the RPAE
+ ~
k P~\~Ø
q,r
—
p,q
[38]; Lamoureux and Combet-Farnoux [39] used the R-matrix methods [38]; and Brown et al. [40] used
MBPT with LS-coupled states which is essentially use of Rayleigh—Schradinger perturbation theory. The diagrams of MBPT are still used to indicate various processes but the states now represent LS-coupled
states. In the calculations of Brown et al. it was found that there are large contributions from diagrams such as fig. 1(b) in which p refers to a given ionic core of Cl~and q refers to a different core. For example,
p -÷ k in fig. 1(b) could refer to 3p5 2P -÷ 3p4 3Pkd 2D and q ~ k’ to 3p4 iDkld 2D or 3p4 1Sk”d 2D. In this case fig. 1(b) refers to a term in perturbation theory where H’ causes a transition from a single excitation of one core to a single excitation with a different ionic core. This type of interaction occurs in the next order indicated by the diagram of fig. 1(f). The diagonal contributions (with fixed core) are again eliminated by appropriate choice of the potential. Because of the slow convergence of these interactions involving different multiplets of 3p4, Brown et al. [40]
r\b
- ~--~
--
Fig. 8. Pictorial representation of coupled integral equations used to sum classes of diagrams as shown.
The MBPT lowest order (Hartree—Fock) length
result (not shown) is close to that of Starace and Armstrong, being 12% lower at the 1S edge and becoming closer at higher energies. The lowering of the length curve is due both to final state mixing among the core states and to correlations in the initial state. Near the 1S edge the MBPT results are in close
90 80 70
calculated these terms to all orders by solving the coupled integral equations indicated in fig. 8. States labelled p, q, r refer to the 3P, 1D, 1S multiplets of Cl~.In these calculations, the continuum was represented in each channel by 32 continuum states and 10 bound excited states. The higher bound states were included by a back extrapolation of the continuum into the bound region. The resulting 126 X 126 matrix was then inverted to solve for the correlated dipole matrix elements. In the case of 3p4 1D kd 2S, the dominant mixing is with 3s3p6 2S. The resulting cross section is shown in fig. 9 and includes both 3p —* k and 3s -÷ k cross sections. The only other calculation to include the inner shell 3s —~ kp excitations is that by Lamoureux and Cornbet-Farnoux [39].
=
40
3 20 10 15
20
25
30
35
40
45
PHOTON ENERGY leVI
Fig. 9. Photoionization cross section for ClI including 3p and 3s subshell cross sections [401. Solid (dashed) curve represents dipole-length (velocity) cross section. Solid circles, calculation by Starace and Armstrong [37]. Open circles, calculation by Lamoureux and Combet-Farnoux (length) [39]. Crosses, calculation by Conneely (length) [41]. Triangles, calculation by Cherepkov and Chernysheva 138].
H.P. Kelly /Many body perturbation calculations 3.5 3.0
I,’,,
107
Carter and Kelly used LS-coupled states which allowed
ii. .
—2 5 ~ a02.0
a determination of LS core-structure effects. In order to obtain the double photoionization cross section a~(w),one calculates the dipole length (or velocity) many-particle matrix elements
-~15s~
Z(,pq
-~
~I.O
k’k) = (‘hfI ~I~z1I’I’0>,
(28)
b
.5 fl~•
25
I
I
30 35 40 45 PHOTON ENERGY leVi
50
where I’I’~)is a many-particle state with orbitals p. q excited to k’, k. The corresponding velocity matrix element is obtained from eq. (21). Many-body diagrams for Z(pq kk’) are given in fig. 11 with (a)—(d) —~
Fig. 10. Total 3s photoionization cross section for Cli calculated by Brown et al. [40]. Upper curves are length (solid)
and velocity (dashed) including correlations. Lower curves are
being the lowest-order diagrams for this process and (e), (f), (g) are higher-order diagrams corresponding to effects found to be important in single photoioniza-
Hartree—Fock length (HFL) and velocity (HFV).
agreement with the length calculatidn of Conneely [41]. However, his velocity calculation (not shown) is almost a factor of two lower [41]. The coupled equations methods of this4 calculation forand the 1D, 3p4 ‘S, account 3s3p5 3P, resonances before the 3p 3s3p5 ‘P edges but they are not shown in fig. 9. The 3s -÷ kp cross section [40] is shown in fig. 10. It is seen that correlations with the 3p5 shell causes a qualitative change in the 3s cross section dust as for Ar [42,8].
PVkV~. (a)
V -.
(c)
Other MBPT calculations of interest include those onA[20],Na[43,44],K [44],C [45],C andF andA [46], Si~[47], [48]. and the isoelectronic sequences of Ne
(b)
\k~,~/
~
(d)
V
5. Double photoionization cross sections The process of double photoionization, where one photon ejects two electrons, is very interesting since it only occurs because of electron correlations. There has now been considerable experimental work on this
process for the rare gases [49—53]where it is found that double photoionization can contribute a significant fraction of the total cross section when it is energetically allowed. With the exception of helium, there have been very few calculations of this process and all (except for He) have used MBPT: Ne by Chang,
Ishihara and Poe [54] and by Chang and Poe [55]; Be by Winkler [56]; C by Carter and Kelly [57]; and ~ and A by Carter and Kelly [58]. The calculations by
k
k
—
- -
(e)
(f)
-..
(9) Fig. 11. Diagrams contributing to the matrix element Z(pq k’k). Full circles indicate matrix element of the dipole operator z; broken lines represent Coulomb interactions. Exchange diagrams are also included.
JLP. Kelly / Many body perturbation calculations
108
tion calculations. Exchange diagrams are understood to be included, With the normalization of eq. (12), 2
C
dk IZ~q k’k)I
0
(29)
where
k” Defining
Ep+Eq
,,
~
l’h’k’>=
[2(e~ + Eq
-~
~k2 + ~
I,
(31)
,
-Ek’Ek”
I
1k ),kk v]pq)
k” Ep + Eq =
,
,
)~kk vlpq)
DaE~~Z~
kmax
u~(w) = 16 ~
60]. The expression for this diagram is
—
(32)
Ek’
1/2
Da
(klzI’I’k’).
=
(33)
and kmax
=
[2(e~ +
+
~
(30)
1/2
For both neon and argon, Z(pq k’k) were calculated for cross sections in which an (np)2 pair is ejected, leaving the ion in the (np)4 3P, ‘D, or is -~
levels and also for cross sections in which an nsnp
pair is ejected leaving the ion nsnp 3P or 1P. The angular momentum of the outgoing k’k pair of elec-
Letting E = (E —
H
+ ~q
T’k’) =
0) I ‘
—
Ek’,
eq. (32) may be written
~II~ Ik” )(k’k” I v Ipq).
(34)
k
Applying closure, one obtains (E—Ho)I’I’k’)R ~‘F(r)
where
‘h’k’)R
(35)
is the radial part of I’1k’>, and
trons must couple with the orbital and spin angular
momentum of the ion to give a total ip state, and so there is an infinite number of types of outgoing electron pairs. For ejection of (3p)2 from argon, the outgoing continuum pairs which were included were kskp, kpkd, kskf, and kdkf. For ejection of 3s3p pairs, the calculated continuum pairs were k’pkp, k’dkd, and kskd. For neon, all these pairs were
included as well as k’sks, kpkf and k’fkf for 2s2p ejection. In sums over excited states, eight bound excited states were calculated and the continuum was approximated by 30 orbitals ranging from k = 0.062 5 to 24.0 au. Matrix elements Z(pq -÷ k’k) were deter-
dr’ Pk’(r)PP(r’)
0
X (r
T~)~~qfr)—
I~ln>R(lcnlvlpq),
n 0cc
(36)
where r< (r>.) is the lesser (greater) of r and r’. Results for u~(w)for argon are shown in fig. 12 27
~
24 21
mined using LS-coupled wave functions and perturba-
states indicated by the diagrams of fig. 1 l(a)—(d). tion theory with the configurations for intermediate The calculations were also checked by use of the usual MBPT formalism which does not distinguish subcross sections corresponding to the different multiplets of Ne++ or A++. In all cases the matrix element Z(pq -~k’k) was represented as a 10 X 10 matrix of k values ranging from 0.0625 to 5.0 au. In evaluating eq. (29), four point Lagrange interpolation was used. The diagrams of fig. 11 were evaluated by summation over the excited bound states and numerical integration over the continuum. The diagram of fig. 11(a) can be evaluated this way but requires care since the dipole matrix element is strongly peaked for k = k”. This diagram was also evaluated by the differential-equation or effective-operator technique [59,
f
F(r) =
-
00
-
18
‘~
I’,
1 12 0
b ~ ‘~
70
100
160
PHOTON
ENERGY
190
cv)
Fig. 12. Total o~~(w)for neutral argon. Full curves are lowest-order results for dipole-length (L) and dipole-velocity (V) matrix elements. Broken curves include second-order correlations [58]. Experimental data points are from Schmidt et al. [49], full circles; Wight and Van der Wiel [50], open circles; Samson and Haddad [51], open triangles; Carlson [53], full triangles. Arrow mark, respectively, excitation thresholds for 3s23p4(3P, 1D, 1S) and 3s3p5(3P, 1P) core levels.
109
H.P. Kelly /Many body perturbation calculations
and compared with experimental results. These curves include 30 individual cross sections representing different LS-coupled terms and configurations as already described. The solid curves are the lowestorder results represented by the diagrams of fig. 1 1(a)—(d). Selected higher order terms as shown in fig. 11(f) and (g) were also calculated because the lower parts of (f) and (g) were found in previous dipoletomatrix work provide elements. important Thecorrelations effect of including in the p
-~
k”
these diagrams is shown by the broken curve in fig. 12. The experimental points were normalized with the total argon cross section data of West and Marr [61] and assuming that a3~can be neglected. In fig. 13 separate results for 3p2 and 3s3p cross sections are presented, and it is noted that 3s3p transitions account for more than 25% of the total ci~4(w). Contributions from different partial waves contributing to the 3p2 cross section are shown in fig. 14. Although the cross section is dominated by 3p2 -~ kpkd, the sum of other partial waves contributes considerably. it is reasonable that the remaining infinite sum of partial waves (e.g., 3p2 -~ kpkg, kf7cg, etc.; 3s3p kpkf, kdkg, etc.; and 3~2-~ kk’) might contribute an additional 10—15% to the cross section. Similar calculations for neon were carried out and resulted in the u~’cross section shown in fig. 15. The discrepancy with experiment of the neon u~maximum position and initial slope is more pronounced than in argon. For both argon and neon, continuum -~
8 6
12
,.~-
~
/
~~10
//
8 6
//
b~ 2
~d
-
kf
-~
40
70
100
130
160
190
220
250
PHOTON ENERGY 1Ev)
Fig. Partial-wave cross for four 3p2(solid -÷ k’k channels. 14. Results are plotted forsections both dipole-length curves) and dipole velocity (broken curves) matrix elements.
states were calculated in the Hartree—Fock V’ potential [5] which is asymptotically hr. One might expect a VN_2 potential [5] which is asymptotically 2/r would be more appropriate because of the plus two charge of the residual ion. However, the kinetic energy spectrum da~~16w IZ(pq -~ k’k)12 = — (37) dek c k’k —
-
30
0
A
£
~25 ~2O
•~ A
27
A
215
24
21
A A
7
18 -:~l5 012 b9
1
6
~__
~p2
:
10 5 /0101 A A I//1
1111
/
________________________________________________ 40
70
100 130 160 PHOTON ENERGY 1EV)
190
220
Fig. 13. Cross sections for total 3p2 —~k’k and 3s3p -+ k’k excitations in neutral aigon from [58]. Results are plotted for both dipole-length (solid curves) and dipole-velocity (broken curves) matrix elements.
250
210 I 260 PHOTON ENERGY 1EV)
310
Fig. 15. Total a++(w) for neutral neon. Lowest-order calculated results are shown for dipole-length (solid curve) and dipole-velocity (broken-curve) matrix elements [58]. Experimental data points from Schmidt et al. [49], full circles; Wight and Van der Wiel [50], open circles; Samson and Haddad [51], open triangles; Carlson [53], full triangles; Lightner et al. [52], open square. Arrows mark, respectively, excitation thresholds for 2s22p4(3P, 1D, 1S) and 2s2p5(3P, 1P) core levels.
110
H.P.
Kelly / Many body perturbation calculations 2 cases. The curve between the VN_l and VN labelled CP in fig. 7 is the 2p2 -~ kpkd velocity result
9 -
from the MBPT calculation by Chang and Poe [55] -
7
-
-
~:I
E~e,
w
0
J
:
which used VNI states. The discrepancy between the CP and VN~ curves in fig. 17 is not fully under-
\\\\
~
desirable total result to considerably investigate further above the experiment. effects ofItscreenis very
/~~A71___
ing, choice of potential and effects of higher order diagrams.
_______________________________ 50
0
100
150
stood [58]. While the CP curve initially shows better agreement with experiment, inclusion of other partialwave channels including 2s2p ~ k’k will push the
200
250
e(eVl
Fig. 16. Kinetic energy distributions for the neon 2p2 —° k’k (solid curve) and 2s2p —~ k’k (broken curve) excitations calculated at w = 278 cV [58]. Chain curve is the sum of both
6. Concluding remarks In the preceding sections it has been shown how
contributions. many-body perturbation theory can be applied to the calculation of many different aspects of photoabsorp-
is plotted for neon in fig. 16 and shows that in general the available kinetic energy is unequally distributed. The faster electron may therefore “feel” a residual charge of + 1, while the slower electron experiences a charge of approximately +2. In fig. 17 are plotted lowest-order velocity-form calculations for neon 2p2 kpkd excitations using single particle states calculated in both the VN_2 and VN~ potentials. Near threshold the slope of the VN_2 curve agrees -+
better with experiment but is too steep. Presumably the correct a~ curve for 2p2 —~kpkd is intermediate
tion including resonances and double photoionization. In addition, this theory has previously been applied to the calculation of many other atomic properties. A principal characteristic of this theory is that one has a well-defined procedure to calculate complete sets of single particle states including, of course, the continuum. These same sets of states may be used to calculate many different atomic properties. In this theory one is, in practice, usually able to calculate only two or three orders of perturbation. however,
it is sometimes possible to include particular interactions to all orders as was discussed in connection with resonances. Also, the potential may be chosen to
35
30L
facilitate convergence by including certain types of interactions. When there is a slow convergence, or divergence, of the perturbation expansion, it may be possible to sum the diverging terms to all orders by
25~ E20~
~
methods such as the coupled integral equations discussed in connection with chlorine. The MBPT methods are very appropriate for calculations in large
~1
atoms as well as small atoms, and calculations are also now underway using relativistic states [62].
51 0 60
110
160 210 260 PHOTON ENERGY 1EV)
310
2 —~kpkd transitions. Curves labelled andfor VN2 refer 2p to conFig. 17. Velocity-form crossVN~ sections the neon tinuum states calculated in these potentials [58]. Curve labelled CF is from V~~’1 calculation by Chang and Poe [551.
Acknowledgements I wish to thank my many collaborators particuport from the U.S. National Science Foundation is gratefully acknowledged.
larly E.R. Brown, S.L. Carter and A.W. Fliflet. Sup-
H.P. Kelly / Many body perturbation calculations
References [11 K.A. Brueckner,
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