The self-interaction correction to the local spin density model: Effect on atomic momentum space properties

Volume 120, number 1

CHEMICAL

27 September

PHYSICS LETTERS

THE SELF-INTERACT~~N CORRECTION TO THE LOCAL EFFECT ON ATOMIC MOMENTUM SPACE PROPERTIES Shridhar

R. GADRE

Depar:menr 01 Chcmisrry.

1985

SPIN DENSIT-Y MODEL:

and Subhas CHAKRAVORTY Uniuersi~y of Pwna.

Pune 41 I 007. India

Received 6 June 1985

Perdew and Zunger showed rhal the exact energy densily funclional for the ground stale is s~%lly self-inleraclion-free. However, the local spin densily (LSD) approximalion lacks this self-interaction correclion (SIC). Perdew and Zunger studied Ihe effect on coordinale-space properties of incorporatin, 0 the SIC. In the present study we examine the effecl of SIC on values. eleclron momenhxn densities and the momemum space properties, viz the electron momentum distribulion. (p”) Complon profiles for atoms He LOAr. A remarkable improvement is seen in all the momenlum space properties of the SIC LSD model over the LSD model when compared IO their near Harlree-Fock counrerparls.

1. Introduction

p(r) =

The Xa theory has become extremely popular among chemists and physicists for the study of atomic and molecular properties. This theory is basically a Kohn-Sham scheme, in the density functional sense. Its spin-polarized version is also known as the local spin density functional (LSD) model. The one-electron equations within the spin-polarized Xol theory are given by

and

[fl+V=(r)+

y.?“(r)1~,o(r)=~,#,(l(r)~

(1)

llcm

lu,

(r)l’,

P, @) = z ‘larJ lu, Q

(5)

(r> I2

(6)

is the election density with spin quantum number u. In this approximation, the total interelectronic energy (V-1 is given as a sum of direct Coulomb and exchange contributions: =

(v,)+E,,,

(7)

where the exchange-correlation

where & =+v;-z/rl,

(2)

E,,

= cjV~(r)p,(r) 0

the Coulomb potential

(3) and the exchange-correlation exchange parameter a is V?(f)

C au

=‘-3a

($7~)‘!~ pii

potential, (r).

with a Slater

(4)

Here the one-electron orbitals are u_. with occupation numbers n,, a and u being quantum numbers. The-electron density b(r) is defined as

0 009-2614/85/S (North-Holland

03.30 0 Elsevier Science Publishers B.V. Physics Publishing Division)

energy

dr.

(8)

Recently, Perdew and Zunger [l] demonstrated that there is an exact cancellation of the Coulomb and exchange-correlation terms for a completely filled spin orbital. They defined a self-interaction (SIC) version of the LSD approximation as ’

[ f1 + vc (r) + V?(r) =&a

%u (r),

+ v,sbc (r)] I.4pi (r>

(9)

where 101

Volume

120, number 1

CHEhIICAL

k(r)=-~;;~;;; dr’+30:(+n)“3 PO

PHYSICS

l~,,(r)l~‘~. (10)

This

approach

led to remarkable

improvement

of

the coordinate space properties of atoms including (rn) values, electron densities, binding energies, etc. There have been attempts [2- 41 to study the Compton profdes of atoms using the spin non-polarized Xo density functional model, but thus far no systematic study of the momentum space properties using the LSD and the SIC-LSD models has been reported. Thus, it was felt worthwhile to embark upon a detailed study of momentum-space properties, including electron momentum distributions, various (p”) values and Compton profiles of atoms He to Ar within the framework of the LSD and SIC-LSD models.

2. Electron

momentum

LElTERS

27 September

where R, I(r) is the radial part of U, (r), & 1(p) is the radial part of its corresponding orbital in momentum space andir(pr) is the spherical Bessel function. The y(p) can be readily evaluated from the a,&) by using the relation

From the y(p) or.l(p) one =rr readily evaluate the spherically averaged Compton profile, J(q). With the advent of y-ray and synchrotron radiation very accurate measurements of Compton profiles (for a review, see ref. [8] ) of atoms and molecules and solids have become possible. There are various tabulations of theoretical Compton profiles in the literature [9,10]. There are also various calculations of atomic Compton profiles for the spin-unpolarized Xa model for atoms reported in the literature [2-41. The Compton profile, J(q) is obtained through the equation

distributions

In recent years several theoretical studies of electron momentum distributions have been reported *. There has also been an exhaustive survey [6] on the nonmonotonicity of the spherically averaged electron momentum density, r(p), and shell structure attributes of its radial momentum density, 1(p) = 4rrp*r@) using the highly accurate near-Hartree-Fock (NHF) wavefunctions of Clementi and Roetti [7] by the present authors. In order to obtain y(p) it is necessary to obtain @e@ l , p2, ___ , p,), the Fourier-Dirac transform of the wavefunction \k(rl, r2,___, rn) given by

(14)

where q is the projection of the electron momentum before collision upon the direction of the momentum transfer_ Recently Cadre et al. [ 1 l] reported a comprehensive compilation of various (~“9 values for atoms and ions using the wavefunctions of ref. [7] _From the 7(p) or I@) one can readily evaluate (~“1 employing the relation (p”)=

47rjy(~)$‘+~ 0

X exp(-i

1985

Cpi-r-)drldr2...dr,. i

(11)

I@)pR

dp.

(15)

0

One can also evaluate various (pn) values from tbeoretical or experimental Compton profiles using the equation [ 121 (p”)=2(n+l)~J(q)q”dq

For a central field problem this reduces to

dp=J

(16)

0

@” I(~) = (2/~)“*~R,l(r)i,(pr)

.’

dr>

(12)

0

* An excellent review of the calculations of atomic electron momentum densities is presnted by Mcnddsohn and Smith

[51102

for 0 G n < 4. The physical significance of various (pn) values was explicitly pointed out by Epstein [13] _ The (p”J value is simply the normalization constant_ Epstein speculated that the (p) value might be correlated with the shielding in nuclear magnetic resonance spectroscopy_ The electronic energy is numerically equal to $ (p2) by the virlal theorem. Another observable directly available from the Compton profile is J(O), which is equal to 4 (p-l)_

i ! 1 1 I 1

Volume

110. number

3. Results

CHEMICAL

1

and discussion

The y(p) for atoms employing spin-polarized Xa wavefunctions was evaluated via eqs. (12) and (13). The integrands in eq. (12) are highly oscillatory in nature, rendering normal methods of quadrature useless. Recently Sawant and Kanhere [4] computed the Compton profdes of the atoms neon and argon within the self-interaction corrected spin non-polarized Xa theory. However the relative and absolute errors involved in their calculations were not available for comparison. The Filon method 1141 for oscillatory integrands was found suitable by the authors for their calculations. The r-mesh employed for integration was the standard 441-point Herman-Skillman mesh with the default interval size inx of 0.0025. The Kohn-Sham value of a = f was employed for all atoms in the present study. Of the fist row atoms, the momentum densities are non-monotonic for 0, F and Ne within both the LSD and LSD-SIC frameworks. The atoms He to N, however, show a strictly monotonic y(p) for both models. These results are in agreement with the results obtained [6] for the NHF wavefunctions of ref. [7] _ In the second-row atoms, non-monoticity appears for the atoms Si to Ar, while the atoms Na, Mg and Al exhibit a strictly monotonically decreasing y@) using both LSD and SIC-LSD functionals. The above results are again exactly parallel to our NHF results [6] _Thus the earlier observation [6] that the non-monotonicity in 7@) occurs for atoms with their outer p shells occupied by two or more electrons supported by this study. Another interesting property is the “shell structure”, i-e_ the maxima in the radial momentum density, I@). Table 1 Compton areinau

27 September

PHYSICS_LETTERS

It is noteworthy that the first-row atoms Li and Be show two maxima or shells in their I(p), but for B td Ne there is only one maximum in the J(p) for both-the SIC-LSD and LSD frameworks. The second-row atoms Na to Ar all have two maxima in their I(p) for both models. The shell structure characteristics of I(p) within both the LSD and SIC-ISD models seem to agree very well with their NHF counterparts. The Compton profiles and the (p” > values were evaluated from the y@) via eqs. (14) and (15) using a standard integration subroutine CUBINT [I 5]_ This subroutine integrates a function tabulated at unequal intervals subsequent to a cubic spline interpolation. The (p”) value, the normalization constant, was found to be accurate to 5 significant digits. The errors in (p2) evaluated by a direct comparison with the kinetic energies obtained by both the LSD and SIC--LSD models were found to be less than 1 X 10m3 percent. Thus any Cp”> value reported in this study is correct to 5 significant places. The J(q) for atoms He to Ar within the LSD and the SIC-LSD frameworks have been computed. However only a few representative Compton profiles (those of Be, 0 and Ar) are presented in table 1. A substantial improvement in the J(0) value results on incorporation of the SIC correction (as seen from table 1). The SIC-corrected profiles are broader (have greater half-widths) compared to their LSD counterparts. Thus it may be seen that the SIC-corrected J(q) are in better overall agreement with the NHF values. The results for (pe2) and (p-l) are presented in table2. There is a considerable improvement in the values obtained using the SIC-LSD model. Both the (p-‘) and (p-l) values are higher in the LSD model than in SIC-

profdcs of beryllium. oxygen and argon atoms employing the LSD, SIC-LSD

4

0 0.1 0.5 1.0 5.0 10.0

J(q)

beryllium

J(q)

1985

and the Hartrce-Fock

J(9)

o.uygen

schemes. All values

=eon

LSD

SIC-LSD

HF”)

LSD

SIC-LSD

HFa)

LSD

SIC-LSD

HF3)

3.229 3.033 1.027 0.417 0.0207 0.000931

3.160 2.978 1.032 0.417 0.0214 0.000946

3.16 2.98 1.03 0.409 0.0206 0.00091

2.863 2.845 2.400 1.445 0.0911 0.0121

2.802 2.785 2.372 1.462 0.0921 0.0123

2.70 2.77 2.38 1.48 0.0903 0.012

5.126 5.101 4.366 2.636 0.360 0.0765

5.093 5.069 4.357 2.642 0.362 0.0769

5.06 5.04 4.35 2.66 0.359 0.075

al Ref. [lo].

103

CHEMICAL

Volume120,numb&l _ Table 2 (p’) and (pi’)

are in

PHYSICS

LEl-iERS

27.Septemb&l?85

-.

., values for atoms helium through argon. calculated in LSD, SIC-LSD

and the Hartree-Fqck

t

.,...

schemes. All values

au

(p-9

tp% LSD 4.6988

He

28.129 26.677 17.233 12.584 9.8479 8.0867 6.8111 5.8838 32.479 37.507 28.243 22.766 19.220 16.932 15.028 13.512

Li Be Bo C N 0 F Ne Na MI3 Al Si P S Cl Ar

SIC-LSD 410923 26.720 25.408 16.500 12.094 9.483 1 7.7482 6.5315 5.6474 3 1.258 36.357 27.599 22.36 1 18.927 16.660 14.802 13.320

-.

NHF=) 22706

4.0893 26.556 25.291 16.262 11.758 9.1019 7.5015 6.3460 5.4196 32.337 36.958 T7.848 22.350 18.727 16.611 14.614 13.107

5.3246 6.4583 6.1400 5.9203 5-7734 5.7270 5.664 1 5.6126 8.7757 10.353 10.387 10.315 10.254 10.290 10.275 10.x1

21410

2 140

5.1888. 6.3025 6.0001 5.7934 5.6569 5.6569 5.5478 5.5025 8.6416 10.226 10.278 10.228 10.181 10.216 10.206 10.187

5.185 6.318 5.979 5.754 5.597 5.552 5.503 5.455 8.711 10.294 10.298 10.216 10.135 10.214 10.156 10.12.9

a) Ref. [ll].

Table 3 (p) and (p*) in au

values for atoms helium through argon, calculated io LSD, SIC-LSD

(P) LSD

and the Harrree-Fock

NHFa)

LSD

2.7990 4.9056 7.4342 10.649 14.462 18.863 23.721

Mg Al Si P S

7.3192 10.508 14.302 18.677 23.499 28.920 34.920 40.479 46.290 52.481 59.034 65.926 73.082

2.7990 4.9101 7.4447 10.668 14.494 18.901 23.758 29.212 35.245 40.796 46.606 52.803 59.361 66.259 73.425

Cl AI

80.583 88.423

80.936 88.783

5.4773 14.387 28.447 48.127 74.224 107.42 147.98 196.95 254.98 321.29 396.49 480.7 1 574.36 677.77 791.03 914.55 1049.0

NC Na

a) Ref. Ill].

104

2.6997

4.7996

All values are

(P2) SIC-LSD

He Li Be B C r: 0 F

schemer

29.166 35.197 40.732 46.530 52724 59.280 66.184 73.301 80.848 88.700

SIC-LSD 5.7233 14.868 29.155 49.097 75.489 109.01 149.92 i99.27 257.72 324.43 400.05 484.68 587.75 68259 796.29 920.37 1055.2

NHFa) 5.7235 14.865 29.146 49.058 75.376 108.80 149.62 198.82 257.09 323.71 399.22 483.74 577.69 68 1.43 794.86 918.93 1053.6 . .

Vdhrrne 120,,number

1

..

.CHEMICAL

PHYSICS LE’l’&RS

LSD and NIJF. The values.@redicted by SIC-LSD are slightly overestimatedwhen compared to NHF results. In table. 3 the results for (p) and (p2) are given. Again the results of the SIC--LSD model are closer to the NHF values than those predicted by the LSD model. In conclusion, addition of the self-interaction correction to the local spin density approximation gives better estimates of atomic momentum space properties.

Acknowledgement Support through a research grant from the Indian National Science Academy, New Delhi, is gratefully acknowledged.

References [l] J.P. Perdew and A. Zunger, Phys. Rev. El23 (1981) 5048. [2] JR. Sabin and S.B. Dickey, J. Phys. B8 (1976) 2593. [3] S.R. SingIt and V-H. Smith Jr.. 2. Phys. 255 (1972) 83; R.N. Euwema and G-T_ Surratk J. Phys C7 (1974) 3655.

27 September.1985

141_ V-R. Sawant and D.G. Kanhere. . J_ Phvs. _ - B17 (1984)

3003. (51 L. Mendelsohn and V.H. Smith Jr.. in: Compton scattering, ed. B.G. Wdhams (McGraw-Hill, New York, 1977) p_ 103. [6] S.R. Gadre, S. Chakravorty and R.K. Patliak, J. Chem. Phys. 78 (1983) 4581. [7 ] E. Clementi and C. Roetti. At Data NucL Data Tables 14 (1974) 177. [S] B.G. WiBiarns, Phyb Scripta 15 (1977) 69, and references therein. [ 9) RJ. Weiss. A Harvey and WC. Phillips Phi!_ Mag 17 (1963) 241. [ 101 F. Biggs, L.B. Mendclsohn and J-8. Mann, At Data Nucl. Data Tables 16 (1975) 201. [ 1 l] S.R Cadre, S-P. Gejji and S. Ch.akravorty, At Data Nucl. Data Tables 28 (1983) 477. [ 121 R. Benesch and V.H. Smith Jr.. in: Wave mechanics the fnst fifty years, cds W.C. Price, S.S. Chissick and T. Ravensdak (Buttenworths, London, 1973) pp_ 357--377[ 13 ] 1-R. Epstein, Phys. Rev. A8 (1973) 160. [14] C.E. Froberg, Introduction to numerical analysis (Addison-Wesley, Reading, 1970) p_ 204. [ 151 P.J. Davis and P. RabinowiE, Methods of numerical integration (Academic Press New Y ork, 1975) p. 366.

105