Indian Joual of Chemistry VoI.39A, Jan-March 20, pp. 32-39 A density functional calculation of Ar++(3s 2 3p 3 nl) satellite states Vikas, Amlan K Roy & B M Deb'! Theoretical Chemistry Group,Department of Chemistry Panjab University, Chandigarh 1 60 0 14, India Received 2 October 1 999; accepted 15 November 1999 The correlation states (satelli tes) of atoms, particularly of charged ions, are diff icult to compute accurately. In this work, we present and discuss the results of Ar(3s23p3nl) satellite states, calculated by employing a simple density-functional formal ism 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 tus out to be better About for ty 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 gases 2 -4. In the present work, we discuss satel- lite states of the argon dication, namely, Ar + +(3s 2 3p 3 nl). 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 ? 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 calcu- lated for the first time. In a recent work 9 , 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++(3s 2 3p 3 nl) satellite states, us- ing single determinantal energies along with Slater's sum rule 18. 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 (W ) and Lee-Yang-Parr (W L W) ' for these satellite states; and (iii) to test the ef fi- 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++(3s 2 3p 3 nl), origi nating from simultaneous ionization and excitation of 3p valence electrons have been com- puted.
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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 + 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
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 . . . )
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
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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