(1)
where Y is the antisymmetric N-electron function which yields r(r), minimizes , and is an eigenstate of some Hamiltonian of the form [5]

(2)
where
(3)

(4)

________________________

• This chapter is, verbatim, the paper: Shubin Liu, "Expansion of the pair distribution and the second-order density matrix in terms of homogeneous functionals", Phys. Rev. A, in press.
  

(6)
which can be decomposed into two components, namely, the kinetic energy part Tc[r] and the potential energy part Vc[r]:

(7)
with
(8)
and
(9)
Starting from the exact relations for Ec[r] [5], Tc[r] [3], and between Ec[r] and Tc[r] [6], respectively,

(10)

(11)
and
(12)
and assuming that Ec[r] can be expanded in terms a Taylor series in l,

(13)
one obtains [1-3] that the functionals An[r] are homogeneous of degree (1-n) in coordinate scaling with n = 1, 2, 3, ...; that is,

(14)
Görling and Levy [1] have given the explicit formula for A1[r] in terms of the Kohn-Sham single-determinant wave function. It follows [1,3] that Tc[r], the kinetic component of Ec[r], can be expanded in terms of a Taylor series of homogeneous functionals in the form of

(15)
All above equations are exact, except Eqs. (14) and (15), whose validity depends on that of the Taylor series Eq. (13).
If one replaces the density r(r) with r in Eqs. (13) and (15), upon using Eq. (14), one finds

(16)
and
(17)
Notice that the validity of Eqs. (16) and (17) is not in general guaranteed since the adiabatic connection formulation requires that the ground state density r remain unchanged. It was first proved by Levy and Perdew [5] and then confirmed by Levy, Yang and Parr [7] that the ground state wavefunction yielding r minimizes only , but not appearing in Eq. (1). Therefore, Eqs. (16) and (17) may generally be invalid. For example, Eq. (16) does not satisfy the Levy low limit case for coordinate scaling: It was shown by Levy [8] that all expansions involving the correlation energy, the pair distribution functions, etc., are invalid for the density when g is small enough. Fortunately, in most cases Eqs. (16) and (17) provide a good approximation for Ec[r] and Tc[r], respectively, and have been demonstrated to satisfy most of the relations derived by Levy and Perdew [3]. In what follows, we assume that they are valid.
If one further assumes that a particular An[r] is local, it has been proved [3] that it has the unique form , where C is a constant. If all the An[r] are local, then for atoms and molecules Ec[r] and Tc[r] can be expressed approximately by . Locality is only approximate, but formulas of this form are remarkably accurate.
With Exc[r] expanded in terms of homogeneous functionals An[r], using the known homogeneity properties [5] of the kinetic and exchange energy density functionals, one now is able to expand the universal density functional F[r]:

(18)
(19)
and
(20)
Thus
(21)
and
(22)
In this paper we shall extend this expansion scheme to the pair distribution function gxc([r]; r, r'). We will find that it can be expanded in a similar way.

5.2 Expansion of gc([r]; r, r')
It is known that [4]

(23)
where
(24)
and gc([r]; r1, r2) is the pair distribution function for a system with a Coulomb repulsion of strength le2 and the external potential n(r) chosen to keep the density equal to the exact density. Levy [8] has proved two identities for the pair distribution function:

(25)
and
(26)
It is also known that [8]

(27)
where the potential energy part VcSymbol"[r] of the correlation energy density functional Ec[r] is defined in Eq. (7). According to its definition [8],

(28)
in which only gc([r]; r1, r2) is l-dependent. Meanwhile, from Eq. (7), upon using Eqs. (7), (13) and (15), one has

(29)
This form is consistent with the result of Levy, March and Handy [9], who suggested that Vc[r] must be linear in l for small enough l. Combination of Eq. (29) with Eq. (28) shows that gc([r]; r1, r2) can be expanded in a Taylor series in terms of some functionals an([r]; r1, r2) with n = 1, 2, 3, ... . in the form

(30)
So
(31)
Since An[r] is homogeneous of degree (1-n) in coordinate scaling, from Eq. (31), it is required that an([r]; r1, r2) be homogeneous of degree (-n) in coordinate scaling, i.e.,

(32)
See proof in Appendix 3. Insert Eq. (30) into Eq. (24), there results

(33)
which means that the correlation part of the pair distribution function can be expanded in terms of homogeneous functionals, with respect to coordinate scaling, of degree -n, with n = 1, 2, 3, ××× . Notice that the constraint Eq. (26), proposed by Levy [8], is strictly satisfied if Eq. (34) is used to expand gc([r]; r1, r2), with use of the property of Eq. (32) for each an([r]; r1, r2).
Therefore, Ec[r] and Tc[r] become

(34)
and

(35)
respectively, where r12=|r1- r2|.
Notice that there are constraints that the exact gc([r]; r1, r2) must satisfy. The two well-known ones are the summation rule [10],

(36)
and the cusp condition [10]

(37)
where

(38)
Inserting Eq. (33) into Eqs. (36)-(37), one thus has

(39)
and

(40)

5.3 Expansion of the pair distribution function gxc([r]; r1, r2) and second-order density matrix r2(r1, r2)
In order to include the exchange part of the pair distribution function, one may use the formula [4]

(41)
with two conditions on gx([r]; r1, r2):

(42)
and
(43)
It is known that the exchange energy density functional Ex[r] is homogeneous of degree one in coordinate scaling [5], i.e.,
(44)
Applying the requirement Eq. (44) to Eq. (41), we see that gx([r]; r1, r2) is homogeneous of degree zero in coordinate scaling, that is,

(45)
Combining Eqs. (23) and (41), we find

(46)
where

(47)
with

(48)
Thus, the second-order density matrix can be expanded in the form

(49)

5.4 Further Approximation
If one introduces following recursive approximation,

(50)
where b([r]; r1, r2), according to Eq. (32), is homogeneous of degree -1 with respect to coordinate scaling, then gxc can be expressed in a closed form as

(51)

Simple choices for b([r]; r1, r2) would be ar12, or , etc., and the simplest choice of gx([r]; r1, r2) might be the form that applies to the homogeneous electron system [4]

(52)
with y = r12(3p2r)1/3, which is homogeneous of degree zero with respect to coordinate scaling. One could also choose gx([r]; r1, r2) from the weighted-density approximation (WDA), sometimes approximated by [11]

(53)
where rx(r1), the weighting parameter, is the radius of the exchange hole around an electron at r1.

5.5 Concluding Remarks
Equations (30)-(33), (46) and (49) are the main results of this work. In the previous papers [1-3,8], Ec[r] and Tc[r] were expanded in terms of An[r], which are homogeneous functionals of degree (1-n), n = 1, 2, 3, ... . In this paper, we have demonstrated that An[r] [i.e., Eq. (31)] can be expressed in terms of a homogeneous functional an([r]; r1, r2), with the several an in turn defining series expansions for the pair distribution function gxc([r]; r1, r2) and its correlation component gc([r]; r1, r2). All these formulas are exact.
As shown previously [3], the contributions of An[r] to become much smaller as n gets larger (the magnitude of the second term is about 20% of that of the first). One may thus expect that series of Ec[r] and Tc[r] of Eqs. (34) and (35) will converge fast. One may also expect similar behavior for homogeneous functions an([r]; r1, r2). In the pursuit of improved approximate exchange-correlation energy density functionals, one may focus on the first few terms of the expansions.
In the gradient expansion approximation (GEA) or generalized gradient approximation (GGA), Exc[r] is expressed as

(54)
where exc is a function of the density and its gradient. Although it gives much better results than LDA, one finds from actual calculations that Eq. (54) is still not good enough for accurately simulating Exc[r]. We believe that in the development of the next generation of approximate exchange-correlation energy density functionals, the double integral form of Exc[r], something like Eq. (34) or (46), may play an important role.
Finally, we conclude by noting that Eq. (30) answers Question #1 proposed by Professor W. Kohn at the end of the July 1994 Workshop on Density Functional Theory at the Institute for Theoretical Physics, University of California at Santa Barbara. This was "How does the pair distribution function depend upon the coupling strength l of the Coulomb interaction?" Eq. (30) gives the answer in a series expansion form. While we now know that the pair distribution function possesses some rigorous coordinate scaling properties (Eqs. (25) and (26)), the problem remaining is how to obtain the analytical or approximate form for an([r]; r1, r2), so that gc([r]; r1, r2), gc([r]; r1, r2), r2(r1, r2), and thus An[r], Exc[r] and so on, can be obtained explicitly.

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