density functional calculations on low-lying singly excited states of open-shell atoms

13 November 1998

Ž .Chemical Physics Letters 296 1998 530–536

Density functional calculations on low-lying singly excited statesof open-shell atoms

Ranbir Singh 1, Amlan K. Roy, B.M. Deb ) ,2

Theoretical Chemistry Group, Department of Chemistry, Panjab UniÕersity, Chandigarh-160014, India

Received 8 June 1998; in final form 15 September 1998

Abstract

By employing a simple density-functional approach, low-lying singly excited states of several open-shell atoms, viz. B,C, O, F, Na, Mg, Al, Si, P and Cl, have been calculated and compared with experimental results. The work-function-basedexchange potential has been used in a nonrelativistic, single-determinantal framework. The effects of two different

Ž .correlation energy functionals local Wigner and nonlocal Lee–Yang–Parr on excitation energies and excited-state energieshave been studied. While the exchange-only results show good agreement with numerical Hartree–Fock results, thecorrelation energy functionals do not show any significant improvements in excitation energies over the exchange-onlyresults, although the excited-state energies are improved significantly. q 1998 Elsevier Science B.V. All rights reserved.

1. Introduction

Ž .Density functional theory DFT has achievedimpressive successes in explaining the electronicstructure and properties of many-electron systems intheir ground state. However, the calculation of ex-cited-state energies and densities has remained abottleneck. Recently, a simple DFT-based approachhas been employed for dealing with atomic singly,

) Corresponding author. Fax: q91-0172-541409; telex: 395-7464 RSIC IN; e-mail: [email protected]

1 Present Address: Department of Physics, Brooklyn College ofthe City University of New York, Brooklyn, NY 11210, USA.

2 Also from the Jawaharlal Nehru Centre for Advanced Scien-tific Research, Bangalore-560064, India.

doubly, triply, valence- and core-excited states ofwatoms, including autoionizing and satellite states 1–

x7 . Excited-state energies, radial densities, excitationŽ .energies 8–2000 eV and other energy differences

Ž .0.03–23.5 eV were satisfactorily reproduced withina single-determinantal framework.

Ž .The objectives of this paper are: i to calculateseveral low-lying singly excited states of open-shell

Ž .atoms B, C, O, F, Na, Mg, Al, Si, P and Cl throughŽ .the present single-determinantal approach; ii to

study the effects of two different correlation energyŽ .functionals, viz. Wigner W and Lee–Yang–ParrWC

Ž . ŽW , both of which are designed for ground-stateLYP. Ž .calculations , on these excited states; and iii to find

the efficacy of the present approach, using sphericaldensities, especially in the light of large differencesbetween calculated DFT and experimental excitation

w xenergies, as reported by Nagy and Andrejkovics 8 .

0009-2614r98r$ - see front matter q 1998 Elsevier Science B.V. All rights reserved.Ž .PII: S0009-2614 98 01031-8

( )R. Singh et al.rChemical Physics Letters 296 1998 530–536 531

2. Method of calculation

The work-function-based nonvariational exchangew xpotential is 9,10

r ™™ ™ ™W r sy e r d l 1Ž . Ž . Ž .Hx x`

™ ™Ž .where e r is the electric field due to the Fermi-holex™™XŽ .charge distribution r r,r and l denotes the pathx

of integration. The two correlation potentials em-ployed are:

Ž .i The local parametrized Wigner-type functionalw x11

y1r3aqbr™W r sy 2Ž . Ž .WC 2y1r3aqcrŽ .

where as9.81, bs28.583 and cs21.437.Ž .ii The closed-shell, nonlocal functional of Lee etw xal. 12

™ X 5r3W r sya F rqF yabC rŽ . Ž .LYP 1 1 F

=8 ab

2X Y < <G rq G y G r =r1 1 1ž /3 42X 2 2< <qG 3=r q2 r= r q4G = rŽ .1 1

ab2 2Y X< < < <y 3G r =r qG 5=rŽ1 172

2 2q6r= r q4G = r 3Ž .. 1

Ž . Ž y 1r 3 .y 1 Ž .where F r s 1 q d r ; G r s1 1Ž . y5r3 Ž y1r3.F r r exp ycr , as0.04918, bs0.132,1

Ž 2 .2r3 Xcs0.2533, ds0.349, C s3 3p r10. F andF 1

GX are the first derivatives respectively, with respect1

to r; GY is the second derivative.1

With these potentials, a nonrelativistic Kohn–Sham-type differential equation is solved numeri-cally in the central-field approximation,

1™ ™ ™ ™2y = qW r qW r f r se f rŽ . Ž . Ž . Ž .es xc i i i2

4Ž .in order to obtain the self-consistent set of orbitals

™� 4 Ž .f . W r is the Hartree electrostatic potential,i es

including electron-nuclear attraction and interelec-™Ž .tronic repulsion; W r is the exchange-correlationxc

potential, W qW , where W is either W orx c c WC

W .LYP

Ž .The total energy is the sum of kinetic T , electro-Ž . Ž .static V and exchange-correlation V energies:es xc

1™ ™ ™) 2Tsy f r = f r d r , 5Ž . Ž . Ž .ÝH i i2 i

™ ™ ™Xr r 1 r r r rŽ . Ž . Ž .

X™ ™ ™V syz d rq d r d r ,H HHes X™ ™r 2 < <ryr6Ž .

™ ™™X1 r r r r ,rŽ . Ž .x X™ ™V s d r d r ;HHx X™ ™2 < <ryr2X™™

g r ,rŽ .X™™

r r ,r sy , 7Ž .Ž .x ™2 r rŽ .V sV qV ; V sV or V , 8Ž .xc x c c WC LYP

™r rŽ .

™V sy d r , 9Ž .HWC y1r39.81q21.437r

1y2r3 5r3V sya rqbr C rHLYP Fy1r3 ½1qd r

1 12y2 t q t q = rw W 5ž /9 18

= ™y1r3exp ycr d r , 10Ž .Ž .2™1 =r r 1Ž .

™2t s y = r r . 11Ž . Ž .W ™8 8r rŽ .™� Ž .4The orbitals f r are used to construct variousi

determinants which in turn can be employed tocalculate the various multiplets associated with aparticular electronic configuration. The use of Slater’s

w xdiagonal sum rule 13 for calculating the multipletsw xhas been described earlier 1–7 . This procedure was

w xalso employed by others 14–19 .

3. Results and discussion

Table 1 reports the ground-state, excited-state andexcitation energies of the atoms studied, using W -x

only, W qW and W qW , comparing themx WC x LYPŽ . w xwith numerical Hartree–Fock HF 20 , other DFT

w x w x8 and experimental 23 results. For each atom,calculations were performed for an excited statecorresponding to an electronic configuration where

( )R. Singh et al.rChemical Physics Letters 296 1998 530–536532

Table 1Nonrelativistic total energies and excitation energies from the ground state, in a.u., of open-shell atomic states. ‘‘Exact’’ ground-stateenergies are given in parentheses in column 4. For every state, the top number in columns 4 and 6 is the W -only result whereas the middlex

number and the third number refer to W qW and W qW respectively. Except in column 4, all numbers in parentheses denotex WC x LYP

percentage deviations with respect to experimental results. In column 7, CA:VWN denotes Ceperley–Alder: Vosko–Wilk–Nusair

Ž .Atom state Total energy yE Excitation energya a c dHF Present HF Present CA:VWN Expt.

22B 2s 2p P 24.5290 24.528924.652024.6616

bŽ .24.65442 Ž .B 2s2p P 24.4507 24.4491 0.0783 0.0798 y39.2 0.209 0.1312

Ž . Ž . Ž .24.5722 y40.3 0.0798 y39.2 59.3Ž .24.5831 0.0785 y40.2

2 Ž .D 24.3119 24.3108 0.2171 0.2181 0.0 0.2181Ž . Ž .24.4324 y0.46 0.2196 0.69

Ž .24.4432 0.2184 0.142 Ž .S 24.2481 24.2347 0.2809 0.2942 1.59 0.2896

Ž . Ž .24.3553 y3.00 0.2967 2.45Ž .24.3661 0.2955 2.04

2 Ž .P 24.1790 24.1547 0.3500 0.3742 13.2 0.3305Ž . Ž .24.2752 5.90 0.3768 14.0

Ž .24.2863 0.3753 13.632 2C 2s 2p P 37.6886 37.6881

37.848437.8647

bŽ .37.84553 Ž .C 2s2p S 37.5992 37.5972 0.0894 0.0909 y40.9 0.302 0.1537

Ž . Ž . Ž .37.7579 y41.8 0.0905 y41.1 96.5fŽ .37.7753 0.0894 y41.8 0.1152

Ž .y25.03 fŽ .D 37.3944 37.3936 0.2942 0.2945 0.86 0.3087 0.2920

Ž . Ž . Ž .37.5525 0.75 0.2959 1.34 5.72Ž .37.5702 0.2945 0.86

3 fŽ .P 37.3370 37.3640 0.3516 0.3241 y5.48 0.3662 0.3429Ž . Ž . Ž .37.5229 2.54 0.3255 y5.07 6.79

Ž .37.5404 0.3243 y5.421 fŽ .D 37.1696 37.1638 0.5190 0.5243 17.6 0.5389 0.4460

Ž . Ž . Ž .37.3209 16.4 0.5275 18.3 20.8Ž .37.3390 0.5257 17.9

3 fŽ .S 37.1421 37.0764 0.5465 0.6117 26.9 0.5648 0.4821Ž . Ž . Ž .37.2337 13.4 0.6147 27.5 17.2

Ž .37.2522 0.6125 27.11 fŽ .P 37.1158 37.0742 0.5728 0.6139 12.4 0.5949 0.5462

Ž . Ž . Ž .37.2305 4.87 0.6179 13.1 8.92Ž .37.2484 0.6163 12.8

32 4O 2s 2p P 74.8094 74.808875.058375.0838

bŽ .75.06735 Ž .O 2s2p P 74.1839 74.1826 0.6255 0.6262 8.85 0.5753

Ž . Ž .74.4303 8.73 0.6280 9.16Ž .74.4574 0.6264 8.88

1 Ž .P 73.8720 73.8682 0.9374 0.9406 8.74 0.8650Ž . Ž .74.1145 8.37 0.9438 9.11


Ž .Table 1 continued


Ž .74.1422 0.9416 8.8622 5F 2s 2p P 99.4093 99.4136

99.714699.7418

bŽ .99.73426 Ž .F 2s2p S 98.5312 98.5357 0.8781 0.8779 14.3 0.7680

Ž . Ž .98.8346 14.3 0.8800 14.6Ž .98.8636 0.8782 14.4

2Na 3s S 161.8589 161.8669162.2634162.2856

bŽ .162.2572 Ž .Na 3p P 161.7864 161.7946 0.0725 0.0723 y6.47 0.0775 0.0773

Ž . Ž . Ž .162.1891 y6.21 0.0743 y3.88 0.26Ž .162.2058 0.0798 3.23

12Mg 3s S 199.6146 199.6281200.0658200.0882

bŽ .200.0593 Ž .Mg 3s3p P 199.5467 199.5593 0.0679 0.0688 30.9 0.0995 0.0996

Ž . Ž . Ž .199.9954 y31.8 0.0704 y29.3 y0.10Ž .200.0155 0.0727 y27.0

1 Ž .P 199.4711 199.4659 0.1435 0.1622 1.57 0.1597Ž . Ž .199.8874 y10.1 0.1784 11.7

Ž .199.9061 0.1821 14.022Al 3s 3p P 241.8767 241.8943

242.3733242.3954

bŽ .242.35642 Ž .Al 3s3p P 241.7909 241.8067 0.0858 0.0876 y33.7 0.1322

Ž . Ž .242.2848 y35.1 0.0885 y33.1Ž .242.3055 0.0899 y32.0

2 D 241.6917 241.7081 0.1850 0.1862242.1849 0.1884242.2051 0.1903

2 Ž .S 241.6453 241.6529 0.2314 0.2414 2.37 0.2358Ž . Ž .242.1290 y1.87 0.2443 3.60

Ž .242.1491 0.2463 4.4522 Ž .Al 3s3p P 241.6061 241.5949 0.2706 0.2994 16.0 0.2581

Ž . Ž .242.0705 4.84 0.3028 17.3Ž .242.0904 0.3050 18.2

32 2Si 3s 3p P 288.8544 288.8746289.3977289.4211Ž .b289.374

53 eŽ .Si 3s3p S 288.7630 288.7813 0.0914 0.0933 y38.2 0.247 0.151Ž . Ž . Ž .289.3038 y39.5 0.0939 y37.8 63.6

Ž .289.3262 0.0949 y37.23 Ž .D 288.6201 288.6398 0.2343 0.2348 6.49 0.2205

Ž . Ž .289.1610 6.26 0.2367 7.35Ž .289.1832 0.2379 7.89

3P 288.5803 288.5988 0.2741 0.2758


Ž .Table 1 continued


289.1194 0.2783289.1416 0.2795

1D 288.4664 288.4796 0.3880 0.3950288.9990 0.3987289.0212 0.3999

3S 288.4493 288.4371 0.4051 0.4375288.9567 0.4410288.9790 0.4421

1P 288.4292 288.4374 0.4252 0.4372288.9570 0.4407288.9784 0.4427

42 3P 3s 3p S 340.7188 340.7405341.3103341.3352

bŽ .341.27244 Ž .P 3s3p P 340.4165 340.4379 0.3023 0.3026 11.5 0.2713

Ž . Ž .341.0057 11.4 0.3046 12.3Ž .341.0301 0.3051 12.5

24 Ž .P 3s3p D 340.2776 340.2997 0.4412 0.4408 33.7 0.3297Ž . Ž .340.8663 33.8 0.4440 34.7

Ž .340.8908 0.4444 34.82 Ž .S 340.2093 340.2267 0.5095 0.5138 54.6 0.3323

Ž . Ž .340.7930 53.3 0.5173 55.7Ž .340.8173 0.5179 55.9

2 Ž .P 340.1430 340.1528 0.5758 0.5877 90.0 0.3094Ž . Ž .340.7184 86.1 0.5919 91.3

Ž .340.7429 0.5923 91.422 5Cl 3s 3p P 459.4821 459.5061

460.1749460.2030

bŽ .460.19626Cl 3s3p S 458.9168 458.9407 0.5653 0.5654

459.6072 0.5677459.6355 0.5675

a w xNumerical Hartree–Fock results, taken from Ref. 20 .b w x w xGround-state results, taken from Ref. 21 for ZF10 and from Ref. 22 for Z)10 with Lamb shift correction.c w xOther DFT results, taken from Ref. 8 . See also footnote f below.d w x Ž 3 5 .Taken from Ref. 23 , except for Si 3s3p , S .e w xAs cited in Ref. 8 .f Ž . w xResults from 48-state calculation theory 2 , from Ref. 24 .

an electron in the outermost s orbital was excited tothe adjacent p orbital. For both ground and excitedstates considered here, the W -only results agreex

quite well with HF results, with deviations in therange of 0.0004–0.2%. Inclusion of correlation leadsto a significant improvement in the ground-state

w xresults, compared with the ‘‘exact’’ results 21,22 ,with deviations in the range of 0.002–0.01%. There-fore, one might assume that the excited-state ener-

gies would also improve significantly due to correla-tion, leading to improved estimates of the excitationenergies relative to the ground state. Interestingly,the second part of this assumption is not borne outby the results in Table 1, when compared with

Ž 3 3 .experimental results; except for C 2s2p , P , NaŽ 2 . Ž 3 . Ž 2 4 .3p, P , Mg 3s3p, P , Al 3s3p , P and SiŽ 3 5 .3s3p , S , the W -only excitation energies are bet-x

ter than those obtained by using correlation. In other


words, if only excitation energies are required then,due to a partial cancellation of errors, even the HFŽ .or exchange-only DFT results generally give betterestimates of excitation energies for these atoms.Furthermore, both the local Wigner correlation func-tional and the nonlocal LYP functional yieldground-state energies in good agreement with the‘‘exact’’ results.

Ž 2 4 .For the five states in Table 1, viz. B 2s2p , P ,Ž 3 5 . Ž 2 . Ž 3 .C 2s2p , S , Na 3p, P , Mg 3s3p, P and Si

Ž 3 5 . w x3s3p , S , other DFT results 8 show deviationsfrom experimental results in the range of 0.1–96.5%,compared with the present deviations of 3.2–41.8%.In general, wherever a comparison with experimentalresults could be made, the present calculated results

Ž .show large deviations as high as 91% making themunsuitable for use as a guideline for the experimen-talist. This is indeed puzzling in view of the earlier

w xsuccesses of the same approach 1–7 and the factthat even though open-shell atoms are being studied,the states investigated are singly excited andlow-lying. It is tempting to argue that such failures

Ž .are due to i inherent ‘‘weaknesses’’ of DFT inŽ .dealing with excited states, ii the limitations of the

present single determinantal approach in dealing withŽopen-shell systems i.e. not representing an excited

state as a linear combination of a fairly large numberof wavefunctions of the same space and spin symme-

. Ž . Žtry , iii the present fully numerical basis-set-inde-.pendent calculations apparently not including con-

Ž .tinuum functions, and iv the nonuniversality ofWigner and LYP ground-state correlation functionalswith regard to all states. However, these reasoningsdo not appear to be tenable, especially in view of therecent calculations of excited states of the carbon

w xatom by Dunseath et al. 24 . Their results on C3 Ž5 3 3 1 3 1 .2s2p S, D, P, D, S and P are included in

Table 1 for comparison. In perhaps the most elabo-rate calculations on these states reported so far, theseauthors employ an R-matrix method, with continuumorbitals and 48 states to represent each of the abovesix terms. Their results deviate from experiment by5.7–25.0%, compared with present deviations of0.9–41.8%. As yet, no viable explanation for suchlarge deviations has been forthcoming.

One might also feel that the present discrepanciesmight be due to the assumption of spherical symme-

™Ž .try in calculating W r . In other words, the rota-x

™Ž .tional component in W r , which was negligible inxw xall the excited states studied previously 1–7 , may

now play a significant role in the excited statesŽstudied in this paper note that, for Ne satellites, both

w xLDA and Becke 25 exchange potentials yieldedpoor excitation energies compared to the work-func-

w x.tion exchange potential 7 . However, this is notsupported by the present results in Table 1. For the

3 Ž5 .worst case with C 2s2p S , the exchange energy isin error by 0.002 a.u. whereas for the next worst caseŽ3 . Ž3 .S , the error is 0.066 a.u.; the best case D showsan error of 0.001 a.u. For all the other states in Table1, the error in exchange energy is from 0.001–0.041

4 Ž2 2 .a.u. Still, P 3s3p S, P , with errors in excitationenergy of 56 and 91%, respectively, show only 0.017and 0.009 a.u. errors, respectively, in exchange en-ergy. Clearly, exchange energy is not the reason forsuch large discrepancies between the calculated andexperimental excitation energies; these arise due tothe inability of the present single-determinantal ap-proach to describe electron correlation satisfactorilyin the present excited states which might requiresignificant mixing of ‘‘doubly-excited’’ determinantsfor their proper description. It would be of interest to

wapply multireference coupled-cluster methods 26–x28 as well as other density-functional approaches

Ž .such as the time-dependent TD optimized effectivew xpotential 29 and the TD density functional response

w xtheory 30–32 to the present excited states so thatthe nature of their electron correlation may be better

Ž .understood. In particular, the response theory RTapproach is a purely density-based method in con-trast to the present method which uses both density-based and wavefunction-based approaches to de-

Ž w xscribe the atomic excited states see also Ref. 33 fora combination of density functional and configura-

.tion interaction methods for molecular excited states .The RT approach calculates the linear responseŽ .first-order change in density of the system to a TD

Žperturbation, leading to the dynamic frequency-de-.pendent dipole polarizability whose poles and

residues yield the excitation energies and oscillatorstrengths, respectively. The method has been suc-cessfully applied to calculate molecular excitationenergies below 20 eV for N , CO, CH O, C H ,2 2 2 4

w x Ž w xpyridine and porphin 30–32 see Ref. 34 for areview on density functional approaches to excited

.states .


Acknowledgements

We thank the Council of Scientific and IndustrialResearch, New Delhi, University Grants Commis-sion. New Delhi, and the Jawaharlal Nehru Centrefor Advanced Scientific Research, Bangalore, forfinancial support.

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density functional calculations on low-lying singly excited states of open-shell atoms

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