Indian Journal of Chemistry VoI.39A, Jan-March 2000, pp. 32-39

A density functional calculation of Ar++(3s23p3nl) satellite states Vikas, Amlan K Roy & B M Deb'!

Theoretical Chemistry Group,Department of Chemistry Panjab University, Chandigarh 1 60 014, India

Received 2 October 1999; 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 spec­tra. These satellite lines are due to valence electron ex­citation, concurrent with the ejection of photoelectrons (shake-up) I . Since it occurs mainly through electron-elec­tron correlation, the phenomenon provides a way to study the dynamics of many-electron processes. The satellite states converge to the double ionization potential, where­upon two photoelectrons proceed from the doubly charged core (shake-off) . The correlation of the shake­up lines with the actual processes and the transitions in­volved provide precise information about the energy lev­els of the ion. Shake-up and shake-off also lead to de­tailed information about relaxation processes in atoms.

The experimental study of "satellite states" of a dication, e.g. Ar++ requires high efficiency and resolu­tion of threshold spectroscopy. Recently, allied to the selectivity of electron-electron coincidence techniques, threshold photoelectron coincidence (TPEsCO) spectros­copy has been developed to investigate the dication states of noble gases2-4. In the present work, we discuss satel­lite states of the argon dication, namely, Ar++(3s23p3nl). 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 con­figuration-interaction (CI)5.6 and Green's function? ap-

t 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 calcu­lated for the first time.

In a recent work9, our group has successfully com­puted 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 includ­ing autoionizing and satellite states 10- 1 7. In the present work, the same method within a single determinantal approach investigates Ar++(3s23p3nl) satellite states, us­ing single determinantal energies along with Slater's sum rule 1 8 . Other workers 19-22 had also employed such an ap­proach to calculate excited-state and excitation energies .

Since an independent-particle picture cannot describe correlation states, a many-electron description includ­ing 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 en­ergy functionals, viz. , Wigner (W we) and Lee-Yang-Parr (W LW) ' for these satellite states; and (iii) to test the effi­cacy of the present single-determinantal method, incor­porating electron correlation, in investigating such intri­cate many-electron processes. To achieve this, 3p-ns, 3p­np (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 com­puted.

VIKAS et al. : DENSITY FUNCfIONAL CALCULATION OF Ar++ 33

Section II of this paper describes the methodology and Section III discusses the results .

II. The Method of Calculation

The methodology involves a density-functional-based formalism9-1 7 in which a nonrelativistic Kohn·-Sham-type differential equation23 is solved numerically in a cen­tral-field approximation (atomic units employed),

. . . ( 1 )

in order to obtain the self-consistent set of orbitals { ¢ i } . vjr) is the Hartree electrostatic potential, including elec­tron-nuclear attraction and interelectronic repulsion,

v (r) = - Z + f p (r'L d r' <., r I r - r' l . . . (2)

W xc(r), the total exchange-correlation potential, is partitioned as

. . . (3)

In the work-function formalism of Harbola et aF4,25, 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. ,

where

_ f p x

( r, r' ) (r - r') cx (r) - 3 I r - r' l

. . . (4)

d r' . . . (5)

and I denotes the path of integration . The two correla­tion potentials employed are :

(i) The local parametrized Wigner-type functionaF6 [ a + bp -1/3 1 Wwc (r) = - ( )2 a + cp ·11 3 . . . (6)

where a = 9.8 1 , b = 28.583, and c = 2 1 .437.

(ii) The closed-shell, nonlocal functional of Lee et aL. 27

wLVP (r� � -a(F;p + F, ) - abCF psn (G; p + { G , )

-��; p Iv p i ' + G; (3 IV p 12 + 2p V 2 p ) +4G I V' P ] -;� PG; P IVp 1 2 + G; (SIVp I' + 6p V 2 p ) + 4G , v' p ]

where

a = 0.0491 8, b = 0. 1 32, C = 0.2533, d = 0.349 , CF = (311 0) {31T.2)2/3

. . . (7)

FI'and Gil are the first derivative, respectively, with re­spect to p ; G/' is the second derivative.

The total energy is the sum of kinetic (T), electro­static ( V) and exchange-correlation ( Vxc) energies:

T = - ! "Jf (r) V21/! (r) d r 2 � I I

i

v = -zfp (r) d r +!Jfp (r)p (r') d r d r' " r 2 I r - r1

v =!ffP (r)Px (r, r') d d " x I 1 r r ,

2 r - r

y(r,r,) = " f(r')l/! (r) � , ,

v = -f p (r) d r we (9.8 1 + 2 1 .437p -I I J )

\.iyp=-af I -//3 [p+bp-213 l+dp .

! i Vp (r)j' I 2 tW = 8 p er) - s V p (r)

. . . (8)

. . . (9)

. . . ( 1 0)

. . . C I I )

. . . ( 1 2)

. . . ( 1 3)

. . . ( 1 4)

The orbitals { ¢ (r) } are used to construct various I . . determinants which in tum can be employed to calcu-late 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 W X<r), given by Eq.(4), is path-inde­pendent (irrotational) only for spherically symmetric systems. However, results discussed in the next section indicate that, for many of the states considered, the rota­tional component of W ir) is unlikely to be signifi­cant2S.29. 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. 1 7 . In general, if E(O) and E(M) denote the energies of the determinants and multiplets respectively, constructed from the orbit­als obtained from the self-consistent numerical solution of the radial equation, associated with a given configu­ration, the E(M) is calculated following Slater's diago­nal sum rule as

. . . ( 1 5)

It has been emphasized that the work-function for­malism is not based on a variational principle, but is derived from a physical interpretation associated with the Fermi-Coulomb hole-charge distribution of the in­teracting 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 equa­tion 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 orthogo­nalities 14· 16 in order to prevent variational collapse to the ground state. A detailed interpretation of the electron­interaction energy functional and its functional deriva­tive (potential) in terms of two fields (one field accounts ·

for Pauli and Coulomb correlations while the other ac­counts 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 rig­orous31 . Furthermore, although OFT guarantees the ex­istence 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 in­terpretation for the local ground-state potrytial leads to a possible argument for the existence of ilocal excited­state potential that incorporates all correlation ef­fects24.30.3 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 ex­periment results, are given in Table I . Excitation ener­gies are calculated relative to the Ar+(3s23p4) 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 configura­tions, most of the configurations (3p3np and 3p3nd) form more than one series (states labelled with P forms two series and those labelled with i3 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 i llustrated below.

The 3p34p configuration gives rise to 43 determinants, with the equations for 30 state (employing LYP for cor­relation here),

3F = ( J + I -O' } ') = - 524.9824 a.u. . . . ( 1 6)

3F + 3D + 3D = ( 1 ' 1-0'0+) + ( J + I -- I ' I +) + ( 1 '0'0- 1 +) ( 1 7 . . . ) = - 524.985 1 + (- 524.9370) + (- 524.9525) a.u.

After subtracting 3F and averaging, 30 = - 524.946 1 H .U. ( 28.99 1 3-eV with respect to Ar++3p4CP) main line). Here the left- and right- hand sides in Eqs ( 1 6),( 1 7) de­note 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 CZO)3p34pCO) and CZP)3p34pCO) states. Hence in Table 1 , states labeled with P and i3 are averaged. Experimental results are avail­able from threshold photoelectron coincidence spectros-copy3 12,'4 I . ,. - , n some cases, expenmental results are not available for the complete series; however, they are re­portee! 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 ei­ther a triplet or a quintet state. In the present work, about fo:rty states are reported for the first time. No experi­mental data seem to exist for these. All the states re­ported 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( lF) LYP shows best agreement w i thin 0 .005 e V for 3p3np states and 3p34dCO) LYP within 0.024 eV for 3p3nd states. But, in case of 3p34sCSS) , 3p34sCS) , 3p33d( lO), 3p33dCF), 3p35dCF), 3p33d( lP), 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++(3s23p3nl) relative to the main Ar++(3s23p4) 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 n -E (a.u.l Relative energy (eV) Expt. (eV) LYP WC LYP WC

3p3_ns 3D

4 525. 1 1 50 525. 1 1 29 24.3952 24.4850 , - 24.38532' 5 524.7728 524.78 1 9 33.707 1 33 .4921 33 .75502',33.75h 6 524.635 1 524.6465 37.4541 37. 1 766 37.65h

5S 4 525.24 14 525.2393 20.9557 2 1 .0455 2 1 .620 1 8' 5 524.8933 524.9026 30.4281 30.2077 3 1 .08527a,3 1 .05h 6 524.7542 524.7659 34.2 1 32 33.9275 35.05h

'D 4 525.0947 525.0926 24.9448 25.0374 24.7675 1 ' 5 524.7673 524.7764 33 .8567 33.64 1 8 6 524.6327 524.6441 37.5 1 94 37.24 19

3p 4 525.0370 525.0348 26. 5 1 78 26.6 1 03 25.69362' 5 524.6940 524.7033 35.85 1 3 35.6309 35.4607 1 ',35.5Qh 6 524.556 1 524.5675 39.6038 39.3263 36.85h

Ip 4 525.0 167 525.0146 27.0701 27. 1 599 · 26. 1 6858' 5 524.6885 524.7569 36.00 10 34. 1 724 6 524.5537 524.565 1 39.669 1 39.39 1 6

3S 4 525 . 1984 525. 1 964 22. 1 258 22. 2 129 22.40 1 26' 5 524.8 1 95 524.8901 32.4363 30.5478 3 1 .3 1 547',3 1 . 35h 6 524.7474 524.7594 34.3982 34. 1 044 35.05h

3p1-np 5p

4 525. 1 05 1 525. 1 052 24.6646 24.6946 25.39238' 5 524.8420 524.8509 3 1 .8240 3 1 .6 145 6 524.7292 524.7407 34.8935 34.6 1 32

3F 4 524.9824 524.9824 28.0035 28.0362 28. 1 006 1 ' 5 524.7224 524.73 1 0 35.0785 34.8772 6 524.6 1 03 524.62 1 6 38 . 1 290 37.8541

3D 4 524.946 1 524.9461 28.99 1 3 29.0239 28.8695 1 " ;' 5 524.6838 524.6924 36. 1 289 35.9275 6 524.5683 524.5825 39.27 1 9 38 .9 1 8 1

'D 4 524.9 1 45 524.9 1 45 29.85 1 2 29.8838 29.78247'';' 5 524.6739 524.6824 36.3983 36. 1 997 6 524.5682 524.5780 39.2746 39.0406

IF 4 524.9763 524.9764 28. 1 695 28. 1 994 28. 1 7469' 5 524.7204 524.7290 35. 1 330 34.93 1 6 6 524.6094 524.6207 38 . 1 534 37.8786

3p 4 524.973 1 524.9732 28.2566 28.2865 28.28238" ;' 5 524.7285 524.7373 34.9 1 25 34.7057 6 524.6208 524.63 1 4 37.8432 37.5874

(contd . . . . . )

36

Satellite states

Ip

3S

I S

3pJ-nd 3G

IG

�D

�P

3F

IF

3D

ID

3p

Ip

INDIAN J CHEM, SEC. A, JAN - MARCH 2000

Table 1 - Nonrelativistic energies and relative energies of satellites in Ar++(3s23p3nl) relative to the main ArH(3s23p4) 3p line (LYP,-526.0 1 1 5 a.u.; WC,-526.0 l 27 a.u.) . 1 a.u. = 27.2 1 165 eV (contd . . . )

n -E (a.u.) LYP WC

4 524.9297 524.9298 5 524.6874 524.6965 6 524.5793 524.5908

4 524.8704 524.8704 5 524.6282 524.6369 6 524.5214 524.5304

4 524.7634 524.7633 5 524.5576 524.5663 6 524.4590 524.4725

3 525.2398 525.2383 4 524.79 1 8 524.7969 5 524.6382 524.647 1

3 525.2 1 55 525.2 140 4 524.7890 524.7942 5 524.6372 524.6461

3 525 . 1 804 525 . 1791 4 524.7458 524.75 1 0 5 524.6 1 74 524.6263

3 525. 1 36 1 525. 1 694 4 524.76 1 1 524.7663 5 524.592 1 524.60 1 1

3 525.243 1 525.2398 4 524.7770 524.7822 5 524.6356 524.6445

3 525 .0460 525.0628 4 524.7332 524.7385 5 524.6 1 56 524.6244

3 525 .2 1 57 525. 2 1 57 4 524.8224 524.8543 5 524.7 194 524.7284

3 525.0390 524.9864 4 524.8373 524.8287 5 524.6755 524.6938

3 525.222 1 525.2089 4 524.8261 524.8225 5 524.6024 524.6864

3 524.99 1 7 524.9956 4 524.7420 524.7452 5 524.6282 524.5945

Relative energy (eV) LYP WC

29.4376 29.4675 36.0309 35.8 1 60 38.9725 38.6922

3 1 .05 1 2 3 1 .0839 37.64 1 9 37.4378 40.548 1 40.3358

33 .9629 33.9982 39.5630 39.3589 42.246 1 4 1 .9 1 1 4

20.9992 2 1 .0727 33. 1 900 33 .0839 37.3698 37. 1 602

2 1 .6605 2 1 .7339 33.2662 33. 1 574 37.3970 37. 1 874

22.6 1 56 22.6836 34.44 1 8 34.3329 37.9358 37.7262

23.82 1 1 22.9476 34.0254 33.9 1 66 38.6242 38.4 1 20

20.9094 2 1 .03 1 9 33.5928 33.4839 37.4405 37.23 1 0

26.2728 25.8483 34.7847 34.673 1 37.9847 37.7779

2 1 .6550 2 1 .6877 32.3574 3 1 .5220 36. 1 602 34.9479

26.4633 27.9273 3 1 .95 1 9 32. 2 1 86 36.3548 35.8894

2 1 .4809 2 1 .8727 32.2567 32.3873 38.3439 36.0908

27.7504 27.6770 34.5452 34.4908 37.64 19 38.59 1 6

Expt. (eV)

28.85549';'

29.65629,·i(i')

3 1 .62935"

2 1 .3490" 33.2 1 49b

2 1 .77976'

1 7.9667 1 " 30.50578' 35.05b

25.39238"

2 1 .6879';' 34.00558',i" 33.90b.;' 37.25b,i(;')

24.83626·,iIi')

22.97 1 94';' 33.94506'/ 37.28002',;' liJ)

22.2583",i1i')

25.oo344'} Ii') 34.3 1 928',;' 1i'),34.40"·;' Ii'l 38. 1 8b,;' (;-')

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. = 27.2 1 1 65 eY. (contd . . . )

Satellite states n -E (a.u.) Relative ene�y' (eV) Expt. (eV)

5S

3S

"Ref [33-34]. "Ref [3].

IS

3 4 5

3 4 5

3 4 5

LYP WC

525.8523 525.8 1 70 525.2800 525.285 1 525 . 1 267 525. 1 358

524.9 1 7 1 524.9074 524.6427 524.64 1 9 524.5267 524.5386

524.863 1 524.883 1 524.6325 524.5896 524.4600 524.4470

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.

LYP

4.3321 19 .9053 24.0769

29.7804 37.2473 40.4039

3 1 .2499 37.5249 42.2 1 89

tion functional and the nonlocal LYP functional yield good excitation energies in agreement with experimen­tal 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 inves­tigation. Avaldi et at. 3 were unable to assign peaks at 3 1 .65, 34.45, 36.35, 38 .25, 38.85, 39.95 e Y. It is sug­gested 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(5S) ; (ii) peak at 34.45 eV to 3p36p(5P); (ii i) peak at 36.35 eV to 3p15peO); ( iv) peak at 38 .25 eV to 3p36peF) and 3p36p( 'F) ; (v) peak at 38 . 85 eV to 3p36p( lP), and (vi) peak at 39.95 eV to 3p36sep).

The worst agreement (error 1 .43-4.72 e V) is observed for the s tate s , 3p36s(3P) , 3p34p(3S , I S ) , and 3p3nd(50,30, IO,5P,3P,3S) . However, i t 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 high­lights the difficulties involved in computation of corre­lation 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

WC

5.3253 1 9.7992 23.86 1 9

30.0770 25.38308··i(i'J 37.30 1 7 33.73230··i(i'J,33 .75h,i(i'J

40. 1 1 27

30.7383 38.7249 42.6053

approach in dealing with correlation states ( i.e. , not rep­resenting 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 (ba­sis-set-independent) calculations apparently not includ­ing 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 mix­ing of "doubles" and "triples" for their proper descrip­tion.

From the above arguments, i t appears that calcula­tions of atomic multiplets within a single-determinan­tal OFT framework may sometimes lead to large errors. However, there are variational methods within OFT, which have been employed with a certain degree of suc­cess. While the variational method employed by Nagy35 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-Fock­Slater 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 e V of experimental results. Krieger et al. 38-40 and Nagt' have presented a method for con­structing 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-depen­dent density functional approach by Petersilka et al. 44

gave comparative results. One may also refer to the time­dependent response theory45-47, which calculates the lin­ear response of the system to a time-dependent pertur­bation and determines the position of any discrete ex­cited state (see ref. 1 7 for a review on density functional approaches to exc ited state s ) . Furthermore, multireference coupled cluster methods48.50 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 single­determinantal 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[5l .

Acknowledgement We thank the CSIR, New Delhi and the lawaharlal

Nehru Centre for Advanced Scientific Research, Ban­galore for financial support.

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