Indian Journal of Chemistry
VoI.39A, Jan-March 2000, pp. 32-39
A density functional calculation of Ar++(3s23p 3nl) satellite states
Vikas, Amlan K Roy & B M Deb'!
Theoretical Chemistry Group,Department of Chemistry
Panjab University, Chandigarh 1 60 01 4, India
Received 2 October 1 999; accepted 15 November 1999
The correlation states (satellites) of atoms, particularly of charged ions, are difficult to compute accurately. In this work, we present
and discuss the results of Ar(3s23p3nl) satellite states, calculated by employing a simple density-functional formalism within a single
determinantal approach along with Slater's sum rule. A Kohn-Sham-type differential equation is solved numerically by employing the
work-function-based potential of Harbola and Sahni for exchange while for correlation, the effects of two different correlation energy
functionals (local Wigner and nonlocal Lee-Yang-Parr) have been studied. In some cases, Lee- Yang-Parr functional gives better results,
while for others Wigner functional turns out to be better. About forty states are reported for the first time.
I. Introduction
Under high resolution, satellite lines are observed
around the main core electron line in photoelectron spec­
tra. These satellite lines are due to valence electron ex­
citation, concurrent with the ejection of photoelectrons
(shake-up) I . Since it occurs mainly through electron-elec­
tron correlation, the phenomenon provides a way to study
the dynamics of many-electron processes. The satellite
states converge to the double ionization potential, where­
upon two photoelectrons proceed from the doubly
charged core (shake-off). The correlation of the shake­
up lines with the actual processes and the transitions in­
volved provide precise information about the energy lev­
els of the ion. Shake-up and shake-off also lead to de­
tailed information about relaxation processes in atoms.
The experimental study of "satellite states" of a
dication, e.g. Ar++ requires high efficiency and resolu­
tion of threshold spectroscopy. Recently, allied to the
selectivity of electron-electron coincidence techniques,
threshold photoelectron coincidence (TPEsCO) spectros­
copy has been developed to investigate the dication states
of noble gases2-4. In the present work, we discuss satel­
lite states of the argon dication, namely, Ar++(3s2 3p3nl).
In this process, two electrons are ejected and a third one
is promoted to an unoccupied orbital, converging to the
triple ionization potential of Ar. From a theoretical point
of view, the calculations of satellite states employ con­
figuration-interaction (CI)5.6 and Green ' s function? apt Also
at Jawaharlal Nehru Centre for Advanced Scientific
Research, Bangalore 560 064, India.
proaches; these have been well studied for unications8 .
In the present work, dication states of argon are calcu­
lated for the first time.
In a recent work9 , our group has successfully com­
puted various satellites in the neon atom and the overall
agreement with experiment was satisfactory. The simple
density-functional methodology employed in this work
has been employed earlier in our laboratory to calculate
various excited states of atomic systems, viz., single,
double, triple, low-lying and inner excited states includ­
ing autoionizing and satellite states 10- 17 . In the present
work, the same method within a single determinantal
approach investigates Ar++(3s2 3p3nl) satellite states, us­
ing single determinantal energies along with Slater's sum
rule 1 8 . Other workers 1 9-22 had also employed such an ap­
proach to calculate excited-state and excitation energies.
Since an independent-particle picture cannot describe
correlation states, a many-electron description includ­
ing correlation effects is essential. The objectives of this
paper are (i) to calculate satellite states of argon dication ;
(ii) to study the effects of two different correlation en­
ergy functionals, viz. , Wigner (Wwe) and Lee-Yang-Parr
(WLW) ' for these satellite states; and (iii) to test the effi­
cacy of the present single-determinantal method, incor­
porating electron correlation, in investigating such intri­
cate many-electron processes. To achieve this, 3p-ns, 3p­
np (n=4,5 ,6) and 3p-nd (n=3,4,5) satellite states of
Ar++(3s23p3nl), originating from simultaneous ionization
and excitation of 3p valence electrons have been com­
puted.
VIKAS et al. : DENSITY FUNCfIONAL CALCULATION OF Ar++
Section II of this paper describes the methodology
and Section III discusses the results.
(ii) The closed-shell, nonlocal functional of Lee et aL. 27
wLVP ( r �
�
II. The Method of Calculation
The methodology involves a density-functional-based
formalism9- 17 in which a nonrelativistic Kohn·-Sham-type
differential equation23 is solved numerically in a cen­
tral-field approximation (atomic units employed),
... ( 1 )
in order to obtain the self-consistent set of orbitals { ¢ i } .
vjr) is the Hartree electrostatic potential, including elec­
tron-nuclear attraction and interelectronic repulsion,
v
<.,
(r) = - Z
r
+
f
p (r '
L d r'
I r r' l
-
... (2)
. .. (3 )
In the work-function formalism of Harbola et aF4,2 5 , the
exchange potential W x<r) is the work required to move
an electron against the electric field Ex(r) arising out of
its own Fermi-hole distribution, px (r,r,), viz. ,
-a(F;p + F, ) - abC F p s n (G; p + { G , )
-��; p Ivp i' + G; IVp 12 + 2p V 2 p ) + 4 G V' P ]
-;� P G; P IVp 1 2 + G; (SIVp I' + 6p V 2 p ) + 4G, v' p ]
(3
I
... (7)
where
a = 0.0491 8, b = 0. 1 32,
3
= (311 0) {31T.2)2/
C
,
= 0.2533, d = 0.3 49 C F
F I 'and Gil are the first derivative, respectively, with re­
spect to p ; G/' is the second derivative.
The total energy is the sum of kinetic (T), electro­
static ( V) and exchange-correlation ( Vxc) energies:
T
Wxc(r), the total exchange-correlation potential, is
partitioned as
33
J
= - ! " f (r) V 21/!I (r) d r
2�
f
i
v
r
= -z p ( ) d r + !
v
=!
"
x
2
. . . (8)
I
2
r
Jfp (I rr)p- r1(r') d r d r'
... (9)
ffP (rI)rP-xr(1r, r') d r d r" ,
y(r, r,) = " f (r')l/! (r)
�,
,
. . . (4)
where
cx ( r) -
_
f
p ( r, r ' ) (r - r')
3
r - r' l
x
d r'
I
... (5)
and I denotes the path of integration. The two correla­
tion potentials employed are:
...C I I )
vwe
=-
\.iyp=-a
[(
a + bp
a + cp
-1/3
·11 3
)2
1
where a = 9.8 1 , b = 28.583,
!
tW = 8
. .. (6)
and
c = 2 1 .437.
f
f
p (r )
dr
(9.8 1 + 2 1 .437p -I I J )
I
l+dp .
-//3
[p+bp-21
... ( 1 2 )
3
... ( 1 3)
(i) The local parametrized Wigner-type functionaF6
Wwc ( r) = -
... ( 1 0)
i Vp (r)j'
p er)
I
... ( 1 4)
- s V p ( r)
2
.
The orbitals { ¢ I(r). } are used to construct various
determinants which in tum can be employed to calculate the various multiplets associated with a particular
34
INDIAN J CHEM, SEC. A, JAN - MARCH 2000
electronic configuration. It should be pointed out that in
the present calculations spherical densities have been
employed since WX<r), given by Eq.(4), is path-inde­
pendent (irrotational) only for spherically symmetric
systems. However, results discussed in the next section
indicate that, for many of the states considered, the rota­
tional component of W ir) is unlikely to be signifi­
cant2S .2 9 . In the central-field
. approximation, W ir) is
defined by Eq.(4).
The use of S later ' s sum rule l s for calculating the
multiplets has been described earlier 9. 17 . In general, if
E(O) and E(M) denote the energies of the determinants
and multiplets respectively, constructed from the orbit­
als obtained from the self-consistent numerical solution
of the radial equation, associated with a given configu­
ration, the E(M) is calculated following Slater's diago­
nal sum rule as
... ( 1 5)
It has been emphasized that the work-function for­
malism is not based on a variational principle, but is
derived from a physical interpretation associated with
the Fermi-Coulomb hole-charge distribution of the in­
teracting fermion system. In the variationally derived
Hohenberg-Kohn-Sham OFT, all many-body interactions
are accounted for in the loca! multiplicative potential,
SExc[p]/Sp . Although the exact functional form for
Exc[p] remains unknown, good approximations to it are
available. However, although a Kohn-Sham-type equa­
tion is solved with the work-function potential in order
to obtain the energy and density of an excited state, one
need not ensure Hamiltonian and wave-function orthogo­
nalities 14·16 in order to prevent variational collapse to the
ground state. A detailed interpretation of the electron­
interaction energy functional and its functional deriva­
·
tive (potential) in terms of two fields (one field accounts
for Pauli and Coulomb correlations while the other ac­
counts for the correlation-kinetic contribution) has been
given by SahnPo . Also, with the approximate forms for
Exc[p), the bounds of the total energy are no longer rig­
orous3 1 . Furthermore, although OFT guarantees the ex­
istence of a local effective potential for the ground state,
a proof justifying the existence of such a potential for
excited states is still lacking. However, the physical in­
terpretation for the local ground-state potrytial leads to
a possible argument for the existence of ilocal excited­
state potential that incorporates all correlation ef­
fects 24.30.3 1 .
III. Results and Discussion
For Ar++(3s23p3nl), the calculated non-relativistic en­
ergies and excitation energies of various satellites (3p3_
ns, 3p3_np, n = 4,5,6 and 3p3 _ nd, n = 3,4,5) employing
Wx + Wwc and Wx + WLyP ' comparing them with ex­
periment results, are given in Table I . Excitation ener­
gies are calculated relative to the Ar+(3s2 3p4) 3p main
line [the calculated energy is -526.0 1 1 5 a.u.(LYP) and 526.01 27 a.u. (Wigner)] . Except for 3p3ns configura­
tions, most of the configurations (3p3np and 3p3nd) form
more than one series (states labelled with P forms two
series and those labelled with i 3 form three series). The
present approach cannot separate the two or more series
given here; it can only obtain an average of the two, as
illustrated below.
The 3p34p configuration gives rise to 43 determinants,
with the equations for 3 0 state (employing LYP for cor­
relation here),
3F =
( J + I -O' } ') = -
3F + 3D + 3D
=
=
524.9824 a.u.
( 1 ' 1 -0'0+) + ( J + I -- I ' I +) + ( 1 '0'0- 1 +)
- 524.985 1 + (- 524.9370) + (- 524.9525) a.u.
. .. ( 1 6)
. .. ( 1 7)
After subtracting 3 F and averaging, 30 = - 524.946 1
4
H .U . ( 28.99 1 3-eV with respect to Ar++3p CP) main line).
Here the left- and right- hand sides in Eqs ( 1 6),( 1 7) de­
note E(M) and E(O) respectively; ( 1 ,0) denote the m{
values and (+,-) denote the ms values. Thus, the state
3p34pCO) is obtained here as a mean of the CZ O)3p34pCO)
and CZP)3p 34pCO) states. Hence in Table 1 , states labeled
with P and i3 are averaged. Experimental results are avail­
able from threshold photoelectron coincidence spectros.
copy 3 ,12
. ,'-4, In some cases, expenmental results are not
available for the complete series; however, they are re­
portee! bere and labelled as ire), i(i3), etc.
For the ionic states containing four open shells, the
first three unpaired electrons can be coupled to give ei­
ther a triplet or a quintet state. In the present work, about
fo:rty states are reported for the first time. No experi­
mental data seem to exist for these. All the states re­
ported here have been calculated for the first time. For
the 3p3ns states, the best agreement between the present
and experimental results is within 0.01 e V as shown by
3p34sCO) LYP. S imilarly, 3p34p( l F) LYP shows best
agreement w i thin 0 . 005 e V for 3p 3 np states and
3p34dCO) LYP within 0.024 eV for 3p3nd states. But, in
case of 3p34sCSS), 3p 34sCS ), 3p3 3d( l O), 3p 33dCF),
3p3 5dCF), 3p3 3d( l P), Wigner gives better agreement as
compared to LYP. Hence, both the local Wigner correla-
VIKAS et al. : DENSITY FUNCTIONAL CALCULATION OF Ar++
35
Table 1 - Nonrelativistic energies and relative energies of satellites in Ar ( 3s2 3p3nl) relative to the main
Ar++(3s2 3p4) 3p line (LYP,-526.0 1 1 5 a.u.; WC,-526.0 1 27 a.u.). I a.u. = 27. 2 1 1 65 eY.
++
Satellite states
3p3_ns
n
-E(a.u.l
LYP
3D
3p
Ip
3S
3p1-np
Relative energy (eV)
LYP
Expt. (eV)
WC
4
5
6
525. 1 1 50
524.7728
524.635 1
525. 1 1 29
524.78 1 9
524.6465
24.3952
33.707 1
37.4541
24.4850 , 33.4921
37. 1 766
24.38532'
33.75502',33.75h
37.65h
4
5
6
525.24 1 4
524.8933
524.7542
525.2393
524.9026
524.7659
20.9557
30.4281
34.21 32
2 1 .0455
30.2077
33.9275
2 1 .620 1 8'
3 1 .08527a,3 1 .05h
35.05h
4
5
6
525.0947
524.7673
524.6327
525.0926
524.7764
524.6441
24.9448
33.8567
37.5 1 94
25.0374
33.64 1 8
37.24 1 9
24.7675 1 '
4
5
6
525.0370
524.6940
524.556 1
525.0348
524.7033
524.5675
26. 5 1 78
35.85 1 3
39.6038
26.61 03
35.6309
39.3263
25.69362'
35.4607 1 ',35.5Qh
36.85h
4
5
6
525.0 1 67
524.6885
524.5537
525.0146
524.7569
524.565 1
27.0701
36.00 1 0
39.669 1
27. 1 599
34. 1 724
39.39 1 6
· 26. 1 6858'
4
5
6
525 . 1984
524.8 1 95
524.7474
525. 1 964
524.8901
524.7594
22. 1 258
32.4363
34.3982
22. 2 1 29
30.5478
34. 1 044
22.401 26'
3 1 .3 1 547',3 1 .35h
35.05h
4
5
6
525. 1 05 1
524.8420
524.7292
525. 1 052
524.8509
524.7407
24.6646
3 1 .8240
34.8935
24.6946
3 1 .6145
34.61 32
25.39238'
4
5
6
524.9824
524.7224
524.61 03
524.9824
524.73 1 0
524.621 6
28.0035
35.0785
38. 1 290
28.0362
34.8772
37.8541
28. 1 006 1 '
4
5
6
524.9461
524.6838
524.5683
524.9461
524.6924
524.5825
28.99 1 3
36. 1 289
39.27 1 9
29.0239
35.9275
38.9 1 8 1
28.8695 1 " ;'
4
5
6
524.9 1 45
524.6739
524.5682
524.91 45
524.6824
524.5780
29.85 1 2
36.3983
39.2746
29.8838
36. 1 997
39.0406
29.78247'';'
4
5
6
524.9763
524.7204
524.6094
524.9764
524.7290
524.6207
28. 1 695
35. 1 330
3 8 . 1 534
28. 1 994
34.93 1 6
37.8786
28. 1 7469'
4
5
6
524.973 1
524.7285
524.6208
524.9732
524.7373
524.63 1 4
28.2566
34.9 1 25
37.8432
28.2865
34.7057
37.5874
28.28238" ;'
5S
'D
WC
5p
3F
3D
'D
IF
3p
(contd ..... )
INDIAN J CHEM, SEC. A, JAN - MARCH 2000
36
Table 1 - Nonrelativistic energies and relative energies of satellites in Ar++(3s23p3nl) relative to the main
ArH(3s23p4) 3 p line (LYP,-526.01 1 5 a.u.; WC,-526.0 l 27 a.u.). 1 a.u. = 27.2 1 165 eV
(contd... )
Satellite states
n
Ip
3S
IS
3pJ-nd
3G
IG
�D
�P
3F
IF
3D
ID
3p
Ip
-E (a.u.)
LYP
WC
Relative energy (eV)
Expt. (eV)
LYP
WC
4
5
6
524.9297
524.6874
524.5793
524.9298
524.6965
524.5908
29.4376
36.0309
38.9725
29.4675
35.81 60
38.6922
28.85549';'
4
5
6
524.8704
524.6282
524.5214
524.8704
524.6369
524.5304
3 1 .05 1 2
37.64 1 9
40.548 1
3 1 .0839
37.4378
40.3358
29.65629,·i(i')
4
5
6
524.7634
524.5576
524.4590
524.7633
524.5663
524.4725
33.9629
39.5630
42.246 1
33.9982
39.3589
4 1 .9 1 1 4
3 1 .62935"
3
4
5
525.2398
524.79 1 8
524.6382
525.2383
524.7969
524.647 1
20.9992
33. 1 900
37.3698
2 1 .0727
33.0839
37. 1 602
2 1 .3490"
33.2 1 49b
3
4
5
525.2 1 55
524.7890
524.6372
525.2 140
524.7942
524.6461
21 .6605
33.2662
37.3970
2 1 .7339
33. 1 574
37. 1 874
2 1 .77976'
3
4
5
525. 1 804
524.7458
524.61 74
525. 1791
524.75 1 0
524.6263
22.61 56
34.44 1 8
37.9358
22.6836
34.3329
37.7262
1 7.9667 1 "
30.50578'
35.05b
3
4
5
525. 1 36 1
524.76 1 1
524.592 1
525. 1 694
524.7663
524.60 1 1
23.821 1
34.0254
38.6242
22.9476
33.9 1 66
38.4 1 20
25.39238"
3
4
5
525.243 1
524.7770
524.6356
525.2398
524.7822
524.6445
20.9094
33.5928
37.4405
2 1 .03 1 9
33.4839
37.23 1 0
2 1 .6879';'
34.00558',i" 33.90b.;'
37.25b,i(;')
3
4
5
525.0460
524.7332
524.6 1 56
525.0628
524.7385
524.6244
26.2728
34.7847
37.9847
25.8483
34.673 1
37.7779
24.83626·,iIi')
3
4
5
525.2 1 57
524.8224
524.7194
525. 2 1 57
524.8543
524.7284
21 .6550
32.3574
36. 1 602
2 1 .6877
3 1 .5220
34.9479
22.97 1 94';'
33.94506'/
37.28002',;' liJ)
3
4
5
525.0390
524.8373
524.6755
524.9864
524.8287
524.6938
26.4633
3 1 .95 1 9
36.3548
27.9273
32. 2 1 86
35.8894
22.2583",i1i')
3
4
5
525.222 1
524.8261
524.6024
525.2089
524.8225
524.6864
2 1 .4809
32.2567
38.3439
2 1 .8727
32.3873
36.0908
25.oo344'} Ii')
34.3 1 928',;' 1i'),34.40"·;' Ii'l
38. 1 8b,;' (;-')
3
4
5
524.99 1 7
524.7420
524.6282
524.9956
524.7452
524.5945
27.7504
34.5452
37.64 1 9
27.6770
34.4908
38.59 1 6
27.265 1 8a.i(i')
(contd . ....)
VIKAS et al. : DENSITY FUNCTIONAL CALCULATION OF Ar++
37
Table 1 - Nonrelativistic energies and relative energies of satellites in Ar+(3s23p3nl) relative to the main
Ar++(3s23p4) 3p line (LYP,-526.0 1 1 5 a.u.; WC,-526.0127 a.u.). I a.u. = 27.21 1 65 eY.
(contd ... )
Satellite states
n
LYP
-E (a.u.)
WC
Relative ene�y' (eV)
WC
LYP
Expt. (eV)
5S
3
4
5
525.8523
525.2800
525 . 1 267
525.8 1 70
525.285 1
525. 1 358
4.3321
19.9053
24.0769
5.3253
1 9.7992
23.861 9
3
4
5
524.9 1 7 1
524.6427
524.5267
524.9074
524.64 1 9
524.5386
29.7804
37.2473
40.4039
30.0770
37.301 7
40. 1 1 27
3
4
5
524.863 1
524.6325
524.4600
524.883 1
524.5896
524.4470
3 1 .2499
37.5249
42.21 89
30.7383
38.7249
42.6053
3S
25.38308··i(i'J
33.73230··i(i'J,33.75h,i(i'J
IS
"Ref [33-34].
"Ref [3].
i single series; P two series; i3 three series.
i(i2) out of two series, results of only one are available.
i(i3) out of three series, results of only one are available.
PW) out of three series, results of only two are available.
tion functional and the nonlocal LYP functional yield
good excitation energies in agreement with experimen­
tal results. For most of the remaining states, agreement
was observed to be between 0.005- 1 eY. Therefore, the
results for 40 new states may be useful for future inves­
tigation. Avaldi et at. 3 were unable to assign peaks at
3 1 .65, 34.45, 36.35, 38.25, 38.85, 39.95 eY. It is sug­
gested here that these peaks may be assigned as follows :
(i) peak at 3 1 .65 eV can be assigned to both 3p34p(5P)
and 3p34p(5 S); (ii) peak at 34.45 eV to 3p36p( 5P) ; (iii)
peak at 36.35 eV to 3p15pe O); (iv) peak at 38.25 eV to
3p3 6p e F) and 3p3 6p( ' F) ; (v) peak at 3 8 . 8 5 eV to
3p36p( l P), and (vi) peak at 39.95 eV to 3p36se p).
The worst agreement (error 1 .43-4.72 eV) is observed
for the s tate s , 3p 3 6 s ( 3 P ) , 3p 34p( 3 S , I S ) , and
3p3nd( 50,30, I O, 5P,3P, 3S). However, it may be noted that
for Ne satellites9, the deviations of calculated results from
the experiment were in the range of 0.3-3.9%. This high­
lights the difficulties involved in computation of corre­
lation states, using a single-determinantal approach, even
though it includes correlation. One might argue that such
failyres are due to (i) inherent "weaknesses" of OFT in
dealing with excited states and hence correlation st<!.t�s,
(ii) the limitations of the present single-determinantal
approach in dealing with correlation states (i.e. , not rep­
resenting a correlation state as a linear combination of a
fairly large number of wavefunctions of the same space
and spin symmetry), (iii) the present fully numerical (ba­
sis-set-independent) calculations apparently not includ­
ing continuum functions, and (iv) the nonuniversality of
Wigner and LYP functionals with regard to all states.
One might also feel that the present discrepancies might
be due to the assumption of spherical symmetry in cal­
�ulating Wir). However, this is not fully supported by
the present results. Such large discrepancies between the
calculated and experimental energies can arise due to
the inability of the present single-determinantal approach
to describe electron correlation satisfactorily in these
correlation states which might require significant mix­
ing of "doubles" and "triples" for their proper descrip­
tion.
From the above arguments, it appears that calcula­
tions of atomic multiplets within a single-determinan­
tal OFT framework may sometimes lead to large errors.
However, there are variational methods within OFT,
which have been employed with a certain degree of suc­
cess. While the variational method employed by Nagy3 5
gave occasional large errors in calculating single excita-
38
INDIAN J CHEM, SEC. A, JAN - MARCH 2000
tion energies, Ziegler et al. 20.36 used the Hartree-Fock­
Slater method for several lowest-state calculations. The
latter method has also been utilized by von Barth37 , within
a local density approximation, for singlet and triplet states
with results within 1 eV of experimental results. Krieger
et al. 38-40 and Nagt' have presented a method for con­
structing an accurate spin-polarized exchange-only KS
potential (KLI) based on the analysis of an optimized
effective potential (OEP) integral equation42 .43 The KLI
potential, a functional of KS orbitals, yields results for
total energies, single-particle expectation values, spin
densities, etc. with good success. Also, a time-depen­
dent density functional approach by Petersilka et al. 44
gave comparative results. One may also refer to the time­
dependent response theory45 -47, which calculates the lin­
ear response of the system to a time-dependent pertur­
bation and determines the position of any discrete ex­
cited state (see ref. 1 7 for a review on density functional
approaches to exc ited state s ) . Furthermore,
multireference coupled cluster methods4 8.50 may prove
to be fruitful to understand correlation in the present
satellite states. It may, however, be noted that most of
the above methods do not yield the kind of accuracy for
such an extensive range of states that the present method
has been able to achieve.
IV Conclusion
Considering the difficulties associated with comput­
ing the energies and electron densities of the argon
correlation states described in this work, it is gratifying
to note that for a number of such states the present single­
determinantal approach leads to an agreement within 0. 1
eV between the calculated and experimental results. For
the other states, the agreement worsens, as discussed in
Section III, mainly because due accouht of electron
correlation could not be taken for such states. It may
also be noted that while correlation energy functionals,
such as local Wigner and nonlocal energy functionals,
such as LYP, which were designed for the ground state,
have been quite successful with a large number of atomic
excited states of various types, a systematic approach to
predict the nature of excited states where such functionals
may or may not work is necessary. In particular, the LYP
functional does not give the uniform gas limit correctly
and therefore one may have to adopt a more accurate
correlation functional such as that of Perdew et a[5 l .
Acknowledgement
We thank the CSIR, New Delhi and the lawaharlal
Nehru Centre for Advanced Scientific Research, Ban­
galore for financial support.
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