J . Phys. Chem. 1988, 92, 3060-3061
3060
Derivation of a Local Formula for Electron-Electron Repulsion Energyt Robert G . Parr Department of Chemistry, University of North Carolina, Chapel Hill,North Carolina 2751 4 (Received: July 22, 1987)
By an interparticle-coordinate expansion and Gaussian resummation of the second-order density matrix, an assumption of local behavior, and imposition of a normalization condition, an approximate formula is derived for the total electron-electron repulsion energy; namely, Ve,[p]= C(N - 1)2/3Sp4/3(F) d? (C = = 0 . 7 9 3 7 ) , where p ( F ) is the electron density and N is the number of electrons. This is an improvement of a formula earlier used, but never itself deduced, in a "local density functional theory" (Parr, R. G.; Gadre, S . R.; Bartolotti, L. J. Proc. Nutl. Acud. Sci. U.S.A. 1979, 76, 2522).
Introduction The electron-electron repulsion energy of an atomic or molecular system is given exactly by the formula
where p2 is the diagonal element of the spinless second-order density matrix
Vee[p] = B."R/31p4/3(7)d7, B = 0.7544
(8)
This formula was introduced in an ad hoc manner invoking dimensional arguments. Here a derivation will be provided. Theory Introduce the interparticle coordinates 7 = (7, SO that
+ F2)/2,
s' = 7, - F2
(9)
p2(71r72)= p2(7++"/2,+3/2).Equations 1 and 3 then become
and which has the normalization
N ( N - 1) 2 =
1
JPz(7172) d72
(3)
where r2(7,s)is the spherical average
For a ground state, it is known from density functional theory' that V,, is a functional of the electron density (4) and one may write Vee[pl. The total repulsion energy usually is broken up into two pieces where
The idea is to make a good approximation to r2which preserves the important normalization condition of (1 1). Now r2(7,s)to good approximation possesses a formal Taylor expansion without a linear term
Here /3*(7) is a function of 7 that for the present purposes can be regarded as an unknown function that is yet to be determined. A Gaussian resummation of (13) now suggests itself5s6
and - K [ p ] is the rest. In Hartree-Fock theory K [ p ] is the exchange energy, but in general it contains correlation effects; no simple exact formula exists for K [ p ]. A famous approximation to K[p] going back to Dirac is the local formula
r2(7,s)
N
p2(7,7)e-S2/2b2(i)
(14)
With this approximation, (10) and (1 1) become Vee = ~TJ-PZ(~,F) P2(3 dr'
(15)
N ( N - 1) = 2 ( 2 ~ ) ~ / ~ 1 p ~ (p23/2(?) 7 , ? ) d7
(16)
and Elsewhere2 it has been shown how (7), and improvements on it, can be derived by approximate resummation of an expansion of the Hartree-Fock first-order density matrix followed by a localization assumption and renormalization. In the present paper a similar technique is used to obtain a similar approximation to the whole V,[p]. After all, it is the whole V, that is of interest, and in (5) there is considerable cancellation between J and K (the orbital self-repulsion terms, which indeed are most of K ) . Furthermore, such a study is pertinent because of the existence in the l i t e r a t ~ r e of ~ . a~ "local density functional theory" employing the formula 'Research aided by grants from the National Science Foundation and the National Institutes of Health to the University of North Carolina.
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These equations could themselves be numerically tested. Without doing that, one may proceed further by making the assumptions that p2(7,F) and @2(3 both are local functions of p: (1) Parr, R. G. Annu. Reu. Phys. Chem. 1983, 34, 631. (2) Lee, C.; Parr, R. G. Phys. Rev. A 1987, 35, 2377. (3) Parr, R. G.; Gadre, S. R.; Bartolotti, L. J. Proc. Nut/. Acud. Sci. U.S.A. 1979, 76, 2522. (4) Gadre, S. R.; Bartolotti, L. J.; Handy, N. C. J. Chem. Phys. 1980, 72, 1034. (5) Berkowitz, M. Chem. Phys. Lett. 1986, 129, 486. See also Meyer, J.; Bartel, J.; Brack, M.; Quentin, P.; Aicher, S. Phys. Lett. B 1986, 172, 122. (6) Other approximate summations are possible. For example, the "trigonometric"summation of ref 2 leads to the same form as the present (23), but with the value 0.7386 for the constant.
0 1 9 8 8 American Chemical Society
J . Phys. Chem. 1988, 92, 3061-3063 P2V9r3
= PAP(r3),
PL?)
= P2(P(?)
(17)
Dimensional arguments on (1 5 ) and (16) then immediately yield
= A ~ P ~P /Z (~P ) ,@ 2 3 / 2 ( ~ ) = A ~ P
P ~ P &) ( P )
(18)
TABLE I: Test of Local Formulas for Electron-Electron Repulsion Eneres (au)
atom
Be (4) Ne (10) Ar (18) Kr (36) Xe (54)
Hartree-Fock" 1.03 4.49 54.0 201.4 1078 2701
Rn (86)
8244
(Z)
He (2)
or, equivalently p2
= A3p2,
p 2 = A4p-'t3
(19)
Here A I ,A2, A,, and A4 are constants. But in the closed-shell Hartree-Fock case, exactly 1 1 (20) P2(j;?) = iP(3 P(r3, A3 =
4
With this value for A,, ( 1 5 ) , (16), and (19) give
V,, = . 1 r A ~ l p ~ / di: ~(r3
(21)
and 1 N ( N - 1) = - ( ~ T A ~ ) ~ / or ' N nA4 = 2-'/3(N- l y 3 2
(22)
Insertion of (22) in (21) gives, finally
V, = 2-'l3(N- 1 ) 2 / 3 1 p 4 / 3 ( di: ?) = 0.7937(N - 1)2131p4/3(7')di:
(23)
3061
old formulab 1.43 5.95
new formula'
52.3 195.5 987
51.3 198.0
0.95 5.17
2489
1019 2586
7423
7749
"Values from Clementi, E.; Roetti, C. A t . Data Nucl. Data Tables 1974, 14, 177. *Equation 8 of text and ref 2. CEquation23 of text. This is the desired approximate formula for Vee.6
Numerical Values and Discussion The reduction of (23) to (8) is trivial for large N, there is only a 5% difference in the values of the constant. Equation 8 already being known to give reasonably good numerical prediction^,^ ( 2 3 ) will as well. Some values are given in Table I. Note that (23) is good for H and He, while (8) fails for them. For high N , (23) remains superior. The interest in (23) is not so much in it as a tool for direct numerical prediction, of course, but as a component in densityfunctional-theory models of Thomas-Fermi or X a type.
Electronic-Structure Methods for Heavy-Atom Moleculest Russell M. Pitzer* Department of Chemistry, Ohio State University, Columbus, Ohio 4321 0
and Nicholas W. Winter Lawrence Livermore National Laboratory, Livermore. California 94550 (Received: August 17, 1987)
Methods are derived to simplify and expand the scope of ab initio electronic-structure calculations using relativistic core potentials. The spin-orbit operator obtained at the same level of approximation is expressed in a simpler form to facilitate matrix-element computation. Double-group results are used, when sufficient spatial symmetry is present, both to block the Hamiltonian matrix and to make it real, even though the wave functions are necessarily complex.
Introduction Electronic-structure calculations on molecules containing heavy atoms have proven to be done most effectively using relativistically derived core potentials and spin-orbit Other approaches using perturbation theory and all-electron relativistic methods have so far only been successful for smaller molecule^.^ We have reformulated many of the steps needed to carry out relativistic core-potential and spin-orbit calculations in order to make them more practical for polyatomic molecules. Computer programs have been modified extensively to make use of these methods in molecules of various sizes. The steps treated in this paper are ( 1 ) modifying the form of the spin-orbit operator obtained with relativistic core potentials, (2) using double-group-adapted linear combinations of Slater determinants to block diagonalize the Hamiltonian matrix, and (3) deriving a method, given sufficient spatial symmetry, to make all the Hamiltonian matrix elements real. Further work to develop methods of evaluating atomic-orbital integrals of core potentials This work was supported by the National Science Foundation under Grant CHE-8312286, by the Division of Materials Science of the Office of Basic Energy Sciences, Department of Energy, and by the Lawrence Livermore National Laboratory under Contract W-7405-Eng-48.
0022-3654/88/2092-3061$01.50/0
and spin-orbit operators will be given elsewhereS6
Form of Spin-Orbit Operator Relativistic core potentials (derived from Dirac-Fock atomic calculations) depend on both the I (orbital) and j (total) angular momentum magnitude quantum numbers of the atomic shells they r e p r e ~ e n t .When ~ used, therefore, they must be multiplied by projection operators that operate on both space and spin functions:
The summation limits in this and subsequent summations are determined by standard properties of angular momentum operators: ( I ) Pitzer, K. S . Int. J . Quantum Chem. 1984, 25, 131. (2) Krauss, M.; Stevens, W. J. Annu. Rev. Phys. Chem. 1984, 35, 357. (3) Christiansen, P. A,; Ermler, W. C.; Pitzer, K. S. Annu. Reu. Phys. Chem. 1985, 36, 407. (4) Balasubramanian. K.: Pitzer. K. S . Adu. Chem. Phvs. 1987. 67. 287. (Si Ermler, W. C.; Ross, R. B.; Christiansen, P. A. Adu: Quantum Chem. 1988, 19, 139. ( 6 ) Pitzer, R. M.; Winter, N. W., paper in preparation.
0 1988 American Chemical Society
