(1)
where Y is the antisymmetric N-electron function which yields r(r), minimizes
, and is an eigenstate of some Hamiltonian of the form [3]
(1)
where Y is the antisymmetric N-electron function which yields r(r), minimizes , and is an eigenstate of some Hamiltonian of the form [3]
(2)
where
(3)
(4)
and
(5)
_______________________
This chapter is, verbatim, the paper: Shubin Liu and Robert G. Parr, "Expansion of the correlation energy density functional and its kinetic component in terms of homogeneous functionals", Phys. Rev. A, 53, 2211(1996).
where l is the coupling constant that measures the interaction strength between electrons of the system. The correlation energy density functional Ec[r] then can be defined as [4]
(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)
Here, r is any specific N and n representable electron density.
Some time ago, Levy and Perdew [4] derived two relations for Ec[r] and Tc[r],
(10)
and
(11)
Eliminating the integral term between these equations, one obtains [5]
(12)
Consequently, using Eq. (8), we have [5,6]
(13)
If one takes the functional derivative of Eq. (12) with respect to r(r), multiplies by r(r)r×Ñ, and then integrates over all space, there results
(14)
After Eq. (11) is employed to eliminate the integral on the right-hand side, this becomes
(15)
Notice that Eqs. (11) and (15) respectively involve only Ec[r] and Tc[r]. These formulas all are exact identities.
3.2 First-order approximations for Ec[r] and Tc[r]
Following Levy and Perdew [4], we first suppose that Ec[r] can be expanded in a Taylor series in l up to the first order, i.e.,
(16)
Here E'c is by definition the l-independent derivative 
. However, 
[4], so that
(17)
and Eq. (11) becomes
(18)
as was already given by Levy and Perdew [4]. Inserting Eq. (17) into Eq. (12), one finds
(19)
Combination of Eqs. (15) and (19) gives
(20)
Eqs.(18) and (20) mean that in the first-order approximation Ec[r] and Tc[r] are homogeneous of degree zero in coordinate scaling (see Appendix for discussion).
Consider now the case that Ec[r] and Tc[r] are local functionals, which means (see Appendix) that
(21)
and
(22)
The locality condition then implies [7]
(23)
and
(24)
By partially integrating Eqs. (18) and (20), and using above relations, one arrives at the conclusion that Ec[r] and Tc[r] are homogeneous of degree one with respect to density scaling, i.e.,
(25)
and
(26)
See the Appendix for discussion of homogeneity with respect to coordinate scaling and homogeneity with respect to density scaling. These are different concepts.
As shown in the Appendix, the only way a functional X[r] can be homogeneous of degree one in density scaling and homogeneous of degree zero in coordinate scaling is for it to be of the form X[r] = CN[r] = CN. The first-order assumption of Eq. (16) would therefore imply that both Ec and Tc would be proportional to N. Indeed, actual data demonstrate that these proportionalities hold roughly. This is shown in Figures 1 and 2, in which Ec and Tc values are plotted against N for the atoms He through Ar. The data for Ec were obtained from the very recent optimized-effective-potential (OEP) calculation by Grabo and Gross [8]. The data for Tc were taken from Morrison and Zhao [9]. It is clear that the homogeneities of Eqs. (19)-(20) and (25)-(26) are approximately satisfied.
One sees from the figures that there certainly is an approximate proportionality between Ec and N and also between Tc and N, though, not surprisingly, the proportionality assumption is better satisfied if one replaces N with N-1. Best fits give Ec = -0.0377N and Ec = -0.040(N-1), with regression coefficients 0.957 and 0.984, respectively; Tc = 0.0257N and Tc = 0.028(N-1), with regression coefficient 0.956 and 0.974, respectively.
Test of validity of first-order approximation
We may expand Ec[r] to second order and use known data to check whether the second-order term is smaller than the first-order term. In place of Eq. (16), write
(27)
Given l=1, Eq. (27) becomes
(28)
Inserting Eq. (27) into Eq. (12), one obtains
(29)
With l=1, Eq. (29) becomes
(30)
Employing reference values of Ec [8] and Tc [9], from Eqs. (28) and (30) one can estimate magnitudes of E'c and E"c. Table 1 shows the results for the atoms from H to Ar. The magnitude of E"c is roughly 20% of that of E'c, which shows that Eq.(16) gives reasonably good approximations for both Ec[r] and Tc[r]. Note that the only approximation here is that Ec[r] is quadratic in l; no locality assumption has been made.
3.3 General expansions of Ec[r] and Tc[r] in terms of homogeneous functionals
Now let us make the general postulate that there exists a full Taylor series expansion of Ec in power of l,
(31)
Then
(32)
Inserting this into Eq. (12), one gets
(33)
so that
(34)
On insertion of Eqs. (31) - (32) into Eq. (11), and Eqs. (33) - (34) into Eq. (15), there result
(35)
and
(36)
for any l. The important consequence is that
(37)
This is to say, provided that the Taylor series of Eq. (31) exists and admits of the operations we have made on it, the n-th component of the Taylor series for Ec[r] and Tc[OL 114 \f "Symbol"] is a homogeneous functional of degree (1-n) in coordinate scaling; that is,*
(38)
where
(39)
In contrast, recall the fact that the kinetic energy density functional Ts[r] is homogeneous of degree two in coordinate scaling [4],
(40)
and that the exchange energy density functional Ex[r] is homogeneous of degree one in coordinate scaling [4],
(41)
With l = 1, Eq. (31) becomes
(42)
and Eq. (33) becomes
(43)
________________________
As confirmation of Eqs. (42) and (43), recall the relationship of Levy and Perdew [4],
(44)
which holds only at g = 1. Using Eqs. (38) and (42), one obtains
(45)
This expansion has been earlier generated by Levy [10] and by Görling and Levy [14].
Differentiating Eq. (45) with respect to g and setting g = 1, we obtain
(46)
One can readily confirm that Eqs. (42) and (43), together with the scaling property of An, satisfy various other identities of Ec[r] and its components proposed by Levy and Perdew [5,10].
3.3.1 Case of local functionals
If one assumes that An[r] is a local functional, that is,
(47)
then (see Appendix)
(48)
Using Eqs. (37), we will have
(49)
which means that, if it is local, An[r] is a homogeneous functional of degree (4-n)/3 in density scaling, with n = 1, 2, 3, ¼ Recall that for homogeneous systems under the locality approximation, Ts[r] is homogeneous of degree 5/3 in density scaling, and Ex[r] is homogeneous of degree 4/3. From the present derivation, we now know that Ec[r] and Tc[r] are combinations of functionals homogeneous in r of degrees: 1, 2/3, 1/3, 0, -1/3, ¼
Since the only form of a local functional X[r] that is homogeneous of degree k with respect to density scaling and homogeneous of degree m = 3k-3 (see Appendix) in coordinate scaling is C
(See Appendix for a detailed proof), where C is the constant to be determined, if one considers the Taylor expansion up to the third order, one would have, for both Ec and Tc,
(50)
If, however, one considers the expansion up to the fourth order, then one would have
(51)
where a, b, c and d are coefficients to be determined somehow. For atoms and molecules, n values greater than 4 are not be allowed, because exponential decay of the density then will cause divergence.
Figures 3 and 4 show the results for Ec and Tc, respectively, obtained using least squares to determine the coefficients in Eq. (50). The correlation coefficients for the two figures are 0.997 and 0.984, respectively. The results in Figures 3 and 4 show quite conclusively that Eq. (50) is a rather good approximation. Note in particular the decreasing values of the coefficients in the series. Notice also, however, that Eqs. (42) and (43), with all An local, very probably is not exact, because it has been previously demonstrated that Exc[r] is not completely local and for all the nonlocality to be contained in the exchange would be most unlikely[11]. Note also that the values of Ec[r] we have used here are not the exact Kohn-Sham values, and for larger atoms, for instance Ar, very accurate numerical calculations of Tc have not yet been achieved.
Since Eqs. (42) and (43) predict that Ec[r] and Tc[r] share the same expansion coefficients, in Figure 5, we report the results when they are fitted together in the form of Eq. (51) with one set of coefficients as predicted from Eqs. (42) and (43). The accuracy is remarkable, confirming the general validity of Eqs. (42) and (43) and a surprising accuracy of the local approximation.
3.3.2 More general cases
If one supposes that the functionals of An[r] take form
(52)
Then the following identity holds (see Appendix):
(53)
Integrating Eq. (37) by parts, and using this identity, one thus obtains
(54)
In the GEA/GGA framework, in which An[r] is expressed as
(55)
Eq. (54) becomes
(56)
In the WDA approach, An[r] takes the form
(57)
Thus, Eq. (54) becomes
(58)
3.4 Concluding Remarks
In summary, in this paper, we have examined the consequences of two exact relations, Eqs. (12) and (15), for the correlation energy density functional Ec[r] and its kinetic component Tc[r]. Working from known identities and the assumption that Ec[r] can be expanded in a Taylor series in l, we have arrived at the conclusion that Ec[r] and Tc[r] are expressible as combinations of homogeneous functionals of different degrees in coordinate scaling, starting from 0, -1, -2, ..., (1-n), .... Furthermore, we have shown that when Ec and Tc are local functionals, for atoms and molecules, they are linear combinations of homogeneous functionals of the density of degrees 1, 2/3, 1/3 and 0. More specifically, Ec and Tc each has the form of 
where a, b, c and d are constants. Numerical results show the effectiveness of such series of local functionals.
It may be noted that some of the "allowed" components in these various series could vanish identically, or could be very small. Thus from Eq. (7) and the results we have obtained, the potential component Vc of the correlation energy density functional can be expressed as a combination of homogeneous functionals. Morrison and Parr [12] recently suggested that the principal component of Vc is approximately a local homogeneous functional of degree 1 in the density, and the principal component of Tc is approximately a local homogeneous functional of degree 0 in the density. These homogeneities accord with the results we here found in the present paper. A final interesting point is that terms
are missing from the expressions we have obtained for Ec and Tc. This is traceable to the presumption [4] that as l approaches zero, Ec goes to zero.
Table 1. Values of the Kohn-Sham correlation energy Ec, its kinetic component Tc, and expansion coefficients of the first- and second-order Taylor series of Ec and Tc for atoms from He to Ar. Atomic units.a
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a. Ec and Tc were from Ref. [8] and [9], respectively.
b. The value of Tc for Ar is rather suspicious as discussed in Ref. [9].
References
[1]. M. Levy, Proc. Nat. Acad. Sci. U.S.A. 76, 6062(1979).
[2]. M. Levy and J.P. Perdew, in Density Functional Methods in Physics, edited by
R.M. Dreizler and J. da Providencia (plenum, New York, 1985).
[3]. K.F. Freed and M. Levy, J. Chem. Phys. 77, 396(1982).
[4]. M. Levy and J.P. Perdew, Phys. Rev. A 32, 2010(1985).
[5]. A. Savin, Phys. Rev. A 52, 1805(1995).
[6]. M. Levy and A. Görling, Phys. Rev. A 52, 1808(1995).
[7]. R.G. Parr, S. Liu, A.A. Kugler and Á. Nagy, Phys. Rev. A 52, 969(1995).
[8]. T. Grabo and E.K.U. Gross, Chem. Phys. Lett. 240, 141(1995).
[9]. R.C. Morrison and Q. Zhao, Phys. Rev. A 51, 1980(1995).
[10]. M. Levy, Phys. Rev. A 43, 4637(1991).
[11]. Q. Zhao, R.C. Morrison and R.G. Parr, Phys. Rev. A 50, 2138(1994).
[12]. R.C. Morrison and R.G. Parr, Phys. Rev. A , submitted.
[13]. S.K. Ghosh and R.G. Parr, J. Chem. Phys. 82, 3307(1985).
[14]. A. Görling and M. Levy, Phys. Rev. B 47, 13105(1993).
Captions for figures
Figure 1. Correlation energy Ec of neutral atoms as function of number of electrons N. See text.
Figure 2. Kinetic energy contribution Tc to correlation energy of neutral atoms as function of number of electrons N. See text.
Figure 3. Curve of Ec vs. N. The fitted curve is:
.
The correlation coefficient is 0.997
Figure 4. Curve of Tc vs. N. The fitted curve is:
The correlation coefficient is 0.984.
Figure 5. Curves of Ec vs. N and Tc vs. N. The fitted curves are:
