density functional calculation of complex atomic spectra

THEO CHEM

Journal of Molecular Structure (Theochem) 361 (1996) 33-39

Density functional calculation of complex atomic spectra

Ranbir Singh, B.M. Deb*

Theoretical Chemistry Group, Department of Chemistry, Panjab Uziversity. Chandigarh 160014. India

Received 6 March 1995; accepted 20 March 1995

Abstract

A density functional formalism, based on the Harbola-Sahni approach, is suggested for calculating the term energies associated with open-shell atomic configurations. The accuracy of the method is demonstrated for carbon and silicon electronic configurations corresponding to single- and double-electron excitations. The exchange-only energy values for multiplets associated with the ground-state configuration have also been calculated, the maximum error being 0.03% compared to the Hartree-Fock results. The error in the single- and double-electron excitation energies is within 4.5% compared to the experimental results.

1. Introduction

In spite of its remarkable successes in providing

lucid explanations and insights for various ground- state electronic properties of atoms, molecules and solids over the last three decades, density functional theory (DFT) has not acquired the status of an independent and complete density-based quantum mechanical formalism. The major reason for this has been its inability to satisfactorily deal with the excited states of many-electron systems. As a consequence, the two most important areas of chemical physics, namely spectroscopy and molecular reaction dynamics, have remained out- side its scope.

Nevertheless, the formulation of a general excited-state DFT has been a frontline area in theoretical research for two decades. A number of

* Corresponding author. Fax: 91-172-541409. Telex: 395-7464 RSIC IN. Also at Jawaharlal Nehru Centre for Advanced Scien- tific Research, Bangalore 560064, India.

density-based formalisms have been proposed, although with limited success. The most promising line of approach is based on Slater’s [l] transition- state theory. A major thrust in this direction was given by Theophilou [2] who put Slater’s theory on a firm footing by rigorously deriving fractional occupation numbers for the orbitals. Gross et al. [3,4] further generalized and extended this approach to derive a Rayleigh-Ritz variational principle. Their approach led to an exact expression relating excitation energies to the Kohn-Sham eigenvalues. However, because of certain approxi- mations required for numerical implementation, encouraging results could not be obtained. Another complication of this approach is that calculation of all the M - 1 lower eigenstates is necessary in order to determine the Mth state. Therefore, the calculation of high-lying excited states would be computationally very difficult. The accuracy of the results would also be unsatis- factory for higher states.

Following MacDonald [5], Valone and Capitani

0166-1280/96/$15.00 0 1996 Elsevier Science B.V. All rights reserved SSDI 0166-1280(95)04299-7

34 R. Singh. B.M. Deb/Journal of Molecular Structure (Theochem) 361 (1996) 33-39

[6] proposed a variational principle based on density to calculate the energy of a given excited state. However, this formalism also had implementation difficulties as it required quadratures invol- ving fi2. Another problem with this method was that their functional, unlike the Hohenberg-Kohn functional, did not have universal properties. Later, Lieb [7] argued that a universal density- functional procedure does not exist which can yield an individual excited state.

Recently, Koga [8] calculated with reasonable accuracy the 2 ‘S state of He using the local scaling transformation method [9]. This method, however, may not be regarded as a genuine first principles method as it only improves a given wavefunction $+, and does not provide a new unknown wavefunction. Since it requires multivariate nonlinear opti- mization of the adjustable parameters on which the trial wavefunction depends, the whole process becomes very tedious as the numbers of parameters and electrons increase.

This paper proposes to calculate singly and double excited-state energies and densities by a combination of DFT and the conventional wavefunction-based quantum mechanics. This approach has already yielded encouraging results for atoms [lo-121. We solve a Kohn-Sham-type equation with the nonvariational local exchange potential of Harbola and Sahni [13] in order to obtain the radial function for a given electronic configuration of an atom. Using this radial function, we then employ Slater’s [ 141 diagonal sum rule to obtain the energies of multiplets associated with that electronic configuration.

The method is outlined in Section 2. Section 3 discusses the results, followed by concluding remarks in Section 4.

2. The method of calculation

According to the Harbola-Sahni (HS) approach [ 151, the exchange potential within the central-field model for open-shell atoms is calculated by spherically averaging the radial component of the electric field due to the Fermi hole. This makes the result- ing potential spherically symmetric.

The electric field according to the HS prescription is

Zx(i) = J p,(K w- 7’) di,

Ii- 713

The spherically averaged radial component of this field is

cx(r) = - -& J d 1 p,(i, i’) -- dr]T_ ,Idr”dRr (2)

where

where y(r’,7’) is the single-particle density matrix which has been spherically averaged over coordi- nates of the electrons of a given orbital-angular- momentum quantum number. The exchange potential is then the line integral

r WX(?) = - J &(J’) .di’ (4) cc

With this local exchange potential, we numerically solve a Kohn-Sham-type equation:

[-iv’ + U,,(T) + Wx(?)]4i(f+) = Cj$i(T) (5)

in order to obtain a self-consistent set of orbitals {#i(I)} for the given electronic configuration of the atom. The total energy of the atomic system is the sum of the following terms:

This sum is equivalent to the expectation value of the nonrelativistic hamiltonian taken with a Slater determinant of orbitals.

Now, starting from Eq. (6), we outline our procedure based on Slater’s diagonal sum rule [14] for calculating the term differences between various multiplets corresponding to an open-shell atomic

R. Singh, B.M. Deb/Journal of Molecular Structure (Theochem) 361 (1996) 33-39 35

configuration. It is based on the general theorem stating that the sum of the diagonal elements of a matrix is equal to the sum of its eigenvalues.

In the central-field model, the electronic wavefunctions are separable and may be written as

tinrm(3 = &l(r) Y,(R 4)4s) (7)

where R,,(r) is the radial component, Y,(I~,$) is the angular component and O(S) represents the spin part of the wavefunction.

The interaction between the electrons in the open shell of the atomic system is responsible for the multiplet splitting in a given open-shell electronic configuration. In this paper, we restrict ourselves to cases where spin-orbit coupling is neglected. The interactions are represented by electron-electron coulomb repulsion and the exchange term in Eq. (6). The coulomb repulsion is incorporated into V es, which also includes the electron-nuclear attraction. Therefore, denoting V,, as a sum

v,, = V,” + v,,

we deal only with the V,, part since Ven, which represents the electron-nuclear attraction, does not affect the term splittings.

Employing Eq. (7), the V, and V, terms can be expanded as follows:

(8)

where x’ combines space and spin variables. The electron density p(Z) can be written in terms of the occupied spin-orbitals as

= C &dr)Yh(Q)*~2(4 Eq. (8) then becomes

) dx dx’ dR dR’

(9)

(10)

Using the expansion

& = Jr,G,& Y~~,(n)Y,.,(n’)--&

where r< (r,) is smaller (larger) than ( 71 and 17’ 1,

V,, = :C J J R~,(r)R~,,,(r’)(T2(S)d2(S’)r2y’2

1” x & dr dr’ ds ds’

5

x J

Yh(R) Y,,,,(Q) Y;!,,,,,(R) dR

x Y;,,,(~‘)Y,,,,(~;~‘)Y,.,,I(R’)~~’ s

(11)

Using the orthogonality condition as well as the coupling rule [l&16] for the spherical harmonics and also integrating out the spin, we obtain

V,, = ~~~~~~,(r)Ri.,(i)~r2~2drdr r>

x C(II”I; mOm)C(11”1; 000)c(l’l”l’; m’Om’)

x C(Z’Z”1’; 000) (12)

where C are Clebsch-Gordan coefficients. Simi- larly, using the expansion for py(x’, 2’) as

where

y(x’,Z’) = E $Jf (x’)&(x”) i

and following the same procedure as for V,,, the expression for the exchange energy becomes

V, = - c (pairs with parallel spin)

x ss

R,,(r)R,,,,(r)R,,(r’)R,,,,(r’) &r2r” drdr

(21f 1) x ~ C*(ll”i’; m, m’ - m,m’)C*(11”1’; 000)

(21’ + 1)

(13)

Following Slater [16], a one-to-one connection between the functions Fk(nl, n’l’) and Gk(nl, n’l’)

36 R. Singh, B.M. Deb/Journal of Molecular Structure (Theochetn) 361 (1996) 33-39

immediately becomes clear. According to Slater

Fk(nl, n’l’) = e2(47r)2 s.I

&W&(~)

k

x & r2rt2 dr’ dr r >

energies of multiplets for any given open-shell configuration. However, in cases where a particular term occurs more than once in the configuration, this method would only give the sum of the energies of the like terms.

Gk(rzl, n’l’) = e2(47r)2 ST

R~r(~)R~ntc’(r)R,c(i)R,~r~(i) 3. Results and discussion

k

x & r2rt2 dr’ dr 6

The accuracy of the above formalism can be judged from the results in Table 2 which reports the total energy values for all three multiplets, i.e. 3P, ‘D and ‘S, from the two equivalent p electrons in the ground electronic configurations of carbon and silicon. In each case, the error is within 0.03% compared with the Hartree-Fock (HF) results [ 181. For the carbon and silicon ‘S terms, the error is 0.3% and 0.01% respectively, while that for the 3p and ‘D terms is 0.01% and 0.006% respectively. The (‘S - ‘D)/(‘D - 3P) ratio has been shown to be 3/2 for np2 configurations [17]. In our cal- culations, this ratio turns out to be 1.500 for carbon but 1.912 for silicon.

The ak and bk terms have been replaced by the Clebsch-Gordan coefficients in the present work. These functions (Fk, Gk) have been extensively employed for calculating the term energies within the single-determinant framework.

As an illustration, consider the two equivalent p electrons of carbon. Corresponding to this configuration, there would be 15 different determinants [ 171 according to the scheme shown in Table 1.

Now, applying the diagonal sum rule, the energy of the ‘D term would be the same as that evaluated from the single determinant corresponding to C m,(M,) = 2 and C m,(M,) = 0. Similarly, for C ml = 1 and Cm, = 1, i.e. for the 3P state, there is again only one wavefunction. For Cm1 = 0 and Cm, = 0 there are three wavefunctions, the eigenvalues of which when added give the sum of the energies of the 3P, ‘D, and ‘S terms. Hence, the energy of the ‘S term can be found by subtracting from this sum the energies of the 3P and ‘D terms. This method will in general give the

Table 1 Scheme of the 15 determinants for two equivalent p electrons of carbon

1 0 -1

2 (lfl-) 1 (l’Oi-) (1+0-)(1-o+) (l-o-) 0 (l+ - If) (‘I- l-)(1- - I:)(o+o-) (l- - I-)

-1 (o+ - l+) V/J;- - l ) (o- - 1-) -2

The +, - superscripts denote the spin of the electron; the numbers are ml quantum numbers associated with each of the electrons.

The excellent agreement of our results with the HF results can be understood from Figs. 1 and 2, in which the spherically averaged radial densities calculated for 3P, ‘D and ‘S states of C and Si from Clement-Roetti wavefunctions [18] as well as the radial density calculated from the radial function obtained using Eqs. (4) and (5) are plotted. The four plots drawn in the same Figure are virtually

Table 2 Term energies in Rydbergs of the multiplets associated with the ground-state configuration of carbon and silicon. Numbers in parentheses denote percentage errors compared with the Har- tree-Fock results

Term symbol

3P

‘D

‘S

-E

Present work Hartree-Fock

C Si C Si

15.3615 517.6763 75.3772 577.7086 (0.01) (0.006) 15.2522 577.5976 75.2627 577.6301 (0.01) (0.006) 75.0792 577.4471 75.0992 577.5171 (0.03) (0.01)

R. Singh. B.M. Deb/Journal of Molecular Structure (Theochetn) 361 (1996) 33-39 31

6.50

5.85

5.20

z 3.90

-: 3.25 Y

d 2.60

0.65 11 0 I- O

r ta.u.)

Fig. 1. Radial density plot of the ‘P, ‘D, ‘S states of C using Fig. 2. Radial density plot of the 3P, ‘D, ‘S states of Si using wavefunctions from Ref. [ 181 and the spherically averaged radial wavefunctions from Ref. [ 181 and the spherically averaged radial density calculated from the radial function obtained from Eq. density calculated from the radial function obtained from Eq. (5) for the ground (. .2p*) configuration of C. (5) for the ground (. .3p*) configuration of Si.

indistinguishable in the region close to the nucleus, i.e.uptor=0.6a.u.forcarbonandr= l.Oa.u.for silicon. In the intermediate region we notice small deviations among the four plots; in the asymptotic region the four curves again almost overlap on one another.

The increase in error from 0.01 to 0.03% for carbon and from 0.006 to 0.01% for silicon occurs because we have used the same radial function (calculated from Eq. (15) using the spherically averaged exchange potential of Eq. (4)), whereas Clementi and Roetti calculated the complete eigen- function for each of the multiplets. As mentioned above, for carbon the (‘S - ‘D)/(‘D - 3P) ratio is 1.5 as expected theoretically, thus indicating that although the absolute energy from Eq. (6) is slightly higher than the HF energy, the excitation energies would match those calculated from HF results. In the case of silicon, the (‘D - 3P) differ- ence is 0.787 Ry from our results and 0.785 Ry as calculated from HF results. However, because of the higher error in the energy of the ‘S term, the (‘S - ‘D)/(‘D - 3P) ratio is 1.912 instead of 1.5; the corresponding ratio from HF results is 1.43.

(3D,1 D,’ P,3 P) of the six multiplets possible for the ls22s22p3p configuration. The agreement of our exchange-only results with the experimental values [19] is satisfactory, with an error range of 2-4.6%. Note that although our exchange potential is nonvariational, the ordering of the energy levels is the same as that found experimentally. Similar results are again observed with the excitation energy for the ‘P term when a single

Table 3 One-electron excitation energies in Rydbergs for carbon compared with experimental results from Ref. [15]. Numbers in parentheses denote percentage errors

Valence-shell configuration and term symbol

AE

Present work Experimental

Table 3 gives the single excitation energies for carbon. The first four values correspond to four

2p3p 3D 0.6073 (4.3) 0.6351 2p3p ‘D 0.6484 (2.0) 0.6617 2p3p 3P 0.6337 (2.5) 0.6502 2p3p ‘P 0.6021 (4.0) 0.6276 2~3s ‘P 0.5512 (2.4) 0.5648 2~4s ‘P 0.6840 (4.2) 0.7139 2~5s ‘P 0.7298 (4.5) 0.7644 2~6s ‘P 0.7513 (4.6) 0.7875 2~7s ‘P 0.7632 (4.6) 0.8000

0 0.7 1.4 2.1 2.8 3.5 4.2 4.9 5.6 6.3 ; r (au)

38 R. Singh. B.M. Deb/Journai of Motecuiar Structure (Theochem) 361 (1996) 33-39

V.“”

F 5.85 -

5.20 -.

4.55 --

c 3.90 --

; 3.25 .-

’ 2.60 -

1.95 -

1.30 -

0.65 -

0; 0 0.90 i .a0 2.70 3.60 4.50

(a) r (a.u.1

0.10

0.09 -

0.08 -

0.07 -

c 0.06 -

co.05 - e * 0.04 -

0.03 -

0.02 -

0.01 -

OO I I I I

9 18 27 36 45 54 63 72 81

(b) r (a.u.1

1

Fig. 3. Radial density plot for the ls’2s22p7s configuration of C: Fig. 4. Radial density plot for the ls22s22p63s25p2 configuration (a) for r = 0 to r = 4.5a.u.; (b) for r = 4.5 to r = 90.0a.u. of Si: (a) for r = 0 to r = 3.0a.u.; (b) for r = 3.0 to r = 30.0a.u.

electron is excited to an s shell, i.e. for configura- multiplets calculated after exciting the two p elec- tions of the type ls22s22pns (n = 3,. . . ,7), the error trons in carbon and silicon to higher p shells, i.e.

range is 2.4-4.6%. Figs. 3(a) and 3(b) show the for configurations of type 1 s22s2np2 for carbon and radial density plot for the 1 s22s22p7s configuration ls22s22p63s2np2 for silicon, the highest value of n of the carbon arom. An added advantage of this being 5 in each case. The radial density plot for the method is that it is not necessary to orthogonalize ls22s22p63s25p2 configuration of Si is given in Figs. the excited state being calculated with respect to all 4(a) and 4(b). The radial density plots for carbon as the lower states of the same space and spin symme- well as silicon have been split into two parts (a and try. Table 4 reports energy values for each of the b) to clearly show the maxima in each case.

16.0 ,

14.4 --

12.8 .-

11.2 -

- = 9.6

y" 8.0 -

' 6.4 -

4.8 -

3.2 -

1.6 -

0 III III I I 0 0.3 0.6 0.9 1.2 1.5 1.8 2.1 2.4 2.7 3.0

(a)

0.35

0.31

0.28

0.25

f 0.21

co.17

4 0.14 I-

0.11

0.07 ,_

0.04 I-

C '0

(b)

r fa.u.1

3 6 9 12 15 18 21 24 27 3 r (a.u.)

R. Singh, B.M. Deb/Journal of Molecular Structure (Theochem) 361 (1996) 33-39 39

Table 4 Acknowledgement Energies in Rydbergs of a few doubly-excited states of carbon and silicon

C Si

We thank the Department of Atomic Energy and the Council for Scientific and Industrial Research for financial support.

State -E

2s23pz ‘P 73.6362 2s24p2 ‘P 73.2481 2s25p2 ‘P 73.0828 2s23p2 ‘D 73.5940 2s24p2 ID 73.2264 2s25p2 ‘D 73.0691 2s23pZ ‘s 73.5307 2s24p2 ‘S 73.1935 2s25p2 ‘S 73.0487

State -E

3s24pl ‘P 576.6099 3s25pz ‘P 576.3274 3s24p2 ‘D 576.5768 3s’5p2 ‘D 576.3094 3sz4pz ‘s 576.5271 3sz5pz ‘s 576.2823

References

[1] J.C. Slater, The Self-Consistent Fields for Molecules and Solids, McGraw-Hill, New York, 1974.

[2] A.K. Theophilou, in N.H. March and B.M. Deb (Eds.), The Single-Particle Density in Physics and Chemistry, Aca- demic Press, London, 1987.

[3] E.K.U. Gross, L.N. Oliveira and W. Kohn, Phys. Rev. A, 37 (1988) 2805.

[4] E.K.U. Gross. L.N. Oliveira and W. Kohn, Phys. Rev. A, 37 (1988) 2821.

4. Conclusion

Thus, we have a general prescription for the atomic excited states within DFT which allows us to calculate the energies of individual excited states. The nagging problem of orthogonalization has been avoided. The present results are almost as accurate as HF results. However, since we have restricted ourselves to single-determinant represen- tations of the excited states, the accuracy for some of the states does suffer a bit, as exemplified by the ‘S state of the carbon and silicon ground-state configurations. Further, the method at present does not yield the electron density of the states sepa- rately because the spherically averaged potential is used to calculate the radial function. Since Eq. (5) is of the Sturm-Liouville type, extending this formalism to a multideterminantal formalism should be possible and would remove the above-mentioned drawbacks of the method in its present form.

[5] J.K.L. MacDonald, Phys. Rev., 46 (1934) 828. [6] SM. Valone and J.F. Capitani, Phys. Rev. A, 23 (1981)

2127. [7] E.H. Lieb, in R.M. Drieizler and J. da Providencia (Eds.),

Density Functional Methods in Physics, Plenum, New York, 1986.

[8] T. Koga, J. Chem. Phys., 95 (1991) 4306. [9] E.S. Kryachko and E.V. Ludena, Energy Density Func-

tional Theory of Many-Electron Systems, Kluwer Aca- demic, Dordrecht, 1990.

[lo] R. Singh and B.D. Deb, Proc. Indian Acad. Sci. (Chem. Sci.), 106 (1994) 1321.

[1 I] R. Singh and B.M. Deb, J. Chem. Phys. in press. [12] K.D. Sen, Chem. Phys. Lett., 188 (1992) 510. [13] M.K. Harbola and V. Sahni, Phys. Rev. A, 45 (1992) 1434. 1141 J.C. Slater, Quantum Theory of Atomic Structure, Vol. II,

McGraw-Hill, New York, 1960. [ 151 M.K. Harbola and V. Sahni, Phys. Rev. A, 45 (1992) 1434. [16] J.C. Slater, Phys. Rev., 34 (1929) 1293. [17] E.U. Condon and G.H. Shortley, The Theory of Atomic

Spectra, Cambridge University Press, 1970. [18] E. Clementi and C. Roetti, At. Data Nucl. Data Tables, 14

(1974) 177. [19] C.E. Moore, Atomic Energy Levels, Vol. I, United States

National Bureau of Standards, Washington, DC, 1949.

density functional calculation of complex atomic spectra

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