(1)
where N is the total number of electrons and the integration extends over all space. Some examples of functionals of interest are the kinetic energy T[r], the electron-electron interaction energy Vee[r] and the exchange-correlation energy Exc[r]. The second purpose of this communication is to develop a number of particularly interesting consequence.
In Ref. [1], starting from the virial theorem, one of the present authors derived the following identity for the kinetic energy functional T0[r] of an inhomogeneous system of noninteracting fermions:

(2)

______________________

• This paper is, verbatim, the paper: Robert G. Parr, Shubin Liu, Alfred A. Kugler, and A. Nagy, "Some Identities in Density-Functional Theory", Phys. Rev. A, 52, 969(1995).

(3)

If the quantities T'0(x;[r]), T"0(x1, x2;[r]), ... are known, Eq. (2) can be regarded as a functional expansion of T0[r] in terms of the density r(x). In Ref. [1] it is also shown that for the homogeneous noninteracting Fermi gas in one and two dimensions, the series expansion (2) terminates after a few terms, thus giving the kinetic energy in terms of the first few functional derivatives.
Functional expansions such as Eq. (2) have been presented in several recent papers [2-4]. In Sec. 2 we establish that for any general, well-behaved functional F[r] an identity or functional expansion such as Eq. (2) holds. In Sec. 3 we develop a number of consequences. These results are of interest in density functional theory for two reasons. First, they may reveal interesting connections between quantities of physical interest. Second, they may permit the evaluation of some needed but unknown quantity in terms of other quantities. In particular, they may help in the evaluation of some quantity in terms of its functional derivatives.

2.2 General functional expansion
The identity in question is provided by the following:
Theorem 1. For any well-behaved functional F[r], up to a constant

(4)

Here "well-behaved" means that all of the indicated functional derivatives exist and that the indicated functional series converges, at least in a region near r. The value of the constant is zero if the region of convergence of the series includes r=0 and F[0]=0. However, there may be cases in which r=0 is not included in the region of convergence and for which the value of the constant is not zero. See below and the Appendix 1.
For a preliminary incomplete proof of Eq. (4), consider the expansion

(5)

where the successive functional derivatives are all evaluated at the density r1. Taking 1 = r and r2 = 0, Eq. (4) is immediately obtained. It is not necessary, however, to view Eq. (4) as an expansion about zero density, as is shown in the Appendix 1. All that is necessary is for the functional derivatives to exist and the series to converge, at the density of interest. One could say that F[r] must be "reasonably analytic" at r. The Appendix gives the proof of Eq. (4) for the case in which r=0 is not necessarily included in the region of convergence.
Comment 1. The significance of Eq. (4) from a physical point of view is that it provides an expression for the quantity F[r] in terms of the actual density r(x) and functional derivatives which can be related to kernels or correlation functions associated with the atomic or molecular system (see e.g. [1, 2]).
Comment 2. Eq. (4) is the functional analog of the following identity for functions that can be derived as follows. Suppose

(6)
Then it is readily verified upon similarly expanding f'(y), f"(y), ... that

(7)
This remark is due to Cedillo [8].
Comment 3. For functionals of physical interests, the region of convergence of Eq. (4) often includes r=0, with F(0)=0. This is the case for the functionals mentioned in the Introduction. In what follows, for convenience, we take the constant in Eq. (4) to be zero.
Comment 4. The validity of Eq. (4) is not dependent on whether or not the functional F[r] obeys any simple scaling relation. Thus, while a scaling property can be used to establish Eq. (4) in particular cases, the converse does not hold, i.e. the relation (4) does not imply the existence of a scaling property for F[r].
Comment 5. Equation (4) was used in Ref. [2] in connection with an expansion of what was there called the hardness functional.
Comment 6. An example where the expansion (4) terminates after a few terms is provided by the classical Coulomb interaction energy

(8)

In this case

(9)
(10)
(11)

and the right side of Eq. (4) becomes

(12)

Comment 7. An important example of the application of Eq. (4) for which the constant is not zero would be the description of a molecular electronic density as a perturbation of superposed atomic densities.

2.3 Some additional identities for functional expansion

2.3.1 Strictly Local Functionals
By a strictly local functional we here mean a functional that can be written as an integral of a function f of r(x), i.e.

(13)
where f does not contain gradient terms or higher order derivatives of r(x). We note that for such a functional, the first few functional derivatives are

(14)
(15)
(16)

where f(n)(r) º dnf(r)/drn. Hence, substituting Eqs. (14)-(16) into Eq. (4), we obtain the functional expansion for functionals satisfying Eq. (13):

(17)
This type of functional includes the class of homogeneous functionals,

(18)
where the index k is a positive number. For such a functional

(19)
and the identity (17) is equivalent to

(20)
Eq. (20) is obviously correct since the quantity in square brackets is the binomial expansion for (1-1)k. As another example, the reader may easily verify the identity (17) for the functional

(21)
Theorem 2. For the functional defined by Eq. (13), the following identity holds:

(22)
Proof: Although a shorter proof is indicated below [Eqs. (33), (34)], we first follow a longer path to show the connection and consistency with the expansion (17). Consider the functional In["Symbol"] defined by

(23)
where n is a positive integer. Integration by parts leads to the identity

(24)
Multiplying both sides of Eq. (24) by (-1)(n-1)/(n-1)! and summing over n from 1 to ¥ we notice that on the left side all terms but the first cancel out, leaving only the term on the right of Eq. (22). The summation on the right side yields precisely the series (17). This establishes the identity (22) for the strictly local functional of Eq. (13). [Note that if the number of spatial dimensions is d rather than the factor 3, the factor 1/3 in Eqs. (22) and (24) must be replaced by 1/d.]
Comments: For some particular F[r] such that F and dF/dr(x) can be calculated, Eq. (22) can be used as a test of whether F[r] is strictly local. The identity (22) is a generalization of the one obtained in Ref. [8] for a spherically symmetric system.
Suppose now that the functional F[r], in addition to being strictly local, also satisfies a scaling relation of the form:

(25)
where r(x) = l3(lx) and l is a positive scale factor. It is well known [5-7] that in this case F[] obeys the identity

(26)
(27)
where Eq. (27) is obtained from Eq. (26) by integration by parts.
Theorem 3. A strictly local functional that obeys the scaling relation (25) is homogeneous of degree (1 + k/3), i.e. satisfies the relation

(28)
Proof. Eq. (28) follows directly from Eqs. (14), (22) and (27).
Comments: Eq. (28) is well known and forms the basis of the (strictly) local density approximation (LDA) in density functional theory [9,10]. As examples, for the noninteracting kinetic energy functional, T0[r], the scaling index k=2, whereas for the exchange (only) energy functional, Ex[r], k=1. Equation (28) shows then that T0[r] and Ex[r] are homogeneous functionals of degree 5/3 and 4/3 in r(x), respectively, i.e. the results of the Thomas-Fermi model and Dirac exchange model [11].

2.3.2 Inclusion of Gradient Terms
We now wish to extend the analysis to the case where the functional F[r] also depends on the gradient of the local density, i.e. we consider a functional of the form

(29)
where g is a function of x, r(x), and Ñr(x), but does not contain higher order gradient terms of (x). By allowing g to depend explicitly on the position variable x, the functional (29) also covers the case of the so-called weighted density approximation [cf. Eqs. (65)-(73) below].
Theorem 4. For the functional defined by Eq. (29), the following identity holds:

(30)
where

(31)
and summation over repeated indices is implied (here and in what follows).
Proof. For the functional (29), the functional derivative is given by

(32)
As is easily seen from an integration by parts, the functional defined by Eq. (29) can also be represented as

(33)
where ÑT indicates the total derivative with respect to x, i.e.

(34)
From Eqs. (29), (32) and (34) it follows that

(35)
Upon integration by parts

(36)
so that, altogether, the right side of (35) yields

(37)
Since the middle term in large parentheses of the this expression vanishes, we are left with the expression on the right side of Eq. (30). Q.E.D.
Corollary: In most cases of practical interest, the function g depends on the magnitude of the gradient, i.e. on

(38)
In this case, the functional derivative (32) becomes
(39)
and the identity (20) becomes

(40)
Proof. Equations. (39) and (40) follow from Eqs. (32) and (30), respectively, since in this case.

(41)
leading to Eq. (39), while

(42)
leading to Eq. (40).
Of particular interest is the case of spherical symmetry. In this case,

(43)
The functional derivative (39) becomes the simple expression

(44)
and the identity (40) becomes

(45)
The reader may verify Eq. (45) directly starting from the identity

(46)
where dg/dr donates the total derivative with respect to the variable r. Equation (46) assumes, of course, that for r¥ and 0, the function g is such that the limits both vanish
Comment. Equation (45) also obtains for the spherical average of the functional defined by Eq. (29). Thus, defining the function by

(47)
where the integration is over the solid angle, we find that the functional

(48)
satisfies Eq. (45).

2.3.3 Generalized gradient approximation
The forgoing section applies, in particular, to the so-called generalized gradient approximation (GGA) method, which has been applied in recent years in particular to the exchange (only) functional Ex[r] [12-16]. In the GGA (also referred to as the gradient expansion method up to the second order), functionals of interest are of the form

(49)
Note that here the dependence of g on x is only through r(x) and |Ñr(x)|; hence the last term on the right-hand side of Eq. (40) is absent and the latter becomes

(50)
Comments. In the absence of gradient terms in the function g, the right-hand side of Eq. (50) is zero, thus reproducing the identity (22) of the LDA. Also, Eq. (50) provides a criterion and test of the extent to which the GGA is, or is not, satisfied for a given functional F[YMBOL 114 \f "Symbol"].
Let us now assume that F[r] also satisfies the scaling or virial relations Eqs. (25)-(27). In this case, when these are combined with Eq. (50), we find the relation

(51a)
(51b)
where

(52a)
(52b)
Equation (51b) represents the generalization of Eq. (28) in the GGA. When the expression (52b) is substituted in Eq. (51b), we find that the latter can be reexpressed (after integration by parts) as

(53)
Equation (53) provides a simple and precise method to compare the relative contributions of (strictly) local and gradient terms to the value of the functional FGGA[r].
The two most prominent examples to which Eqs. (49)-(53) apply are the exchange-energy functional Ex[r] and the (noninteracting) kinetic-energy functional T0[r]. In the GGA, Ex[r] is written as

(54)
where ex denotes the exchange-energy density. This expression must satisfy Eq. (50). The so-called exchange potential ux(x; [r]) is the functional derivative

(55)
Ex[r] and ux(x; [r]) are related via the scaling relations with the index k=1; explicitly [5,6]

(56)
In the GGA, ux(x; [r]) must be evaluated via Eq. (52b) with substituted for g. We then find from Eqs. (51)-(53) the following equivalent relations for ExGGA[r]:

(57a)
(57b)
(57c)
The (noninteracting) kinetic-energy functional T0[r] obeys the scaling relations (25)-(27) with the index k=2; explicitly [5,6,1]

(58)
In the GGA, T0[r] is written as

(59)
where t0 denotes the kinetic-energy density. The expression (59) must satisfy the identity (50). Also

(60)
In this case, Eqs. (51)-(53) yield the following equivalent expressions for T0GGA[r]:

(61a)
(61b)
(61c)
The most frequently encountered functional T0[r] in the GGA is the Thomas-Fermi-Weizsäcker (TFW) functional (see, e.g., [1] and references therein) in which t0(r, |Ñr|) is taken as

(62)
where a and b are constants. For this functional one has, using Eq. (60),

(63)
and the reader may easily verify that Eqs. (61a) and (61b) are satisfied by TTFW[r].

2.3.4 Weighted-density approximation
In the weighted-density approximation (WDA), the functional F[r] is expressed as

(64)
i.e., there is an explicit dependence of g on the position variable x in addition to dependence on r(x). The WDA is obtained as a special case of Eq. (30); in this case the latter gives

(65)
where now

(66)
When combined with the scaling relation (27) we obtain the equivalent identities

(67a)
(67b)
In the case of spherical symmetry, Eqs. (67a) and (67b) become

(68a)
(68b)
Applying Eqs. (67) to the exchange energy (k=1) we have, in the WDA,

(69a)
(69b)
For the (noninteracting) kinetic energy (k=2), Eqs. (67) become

(70a)
(70b)
As the simplest example of ExWDA[r], consider the functional [3]

(71)
It is easily seen that this functional satisfies Eqs. (69a) and (69b). Moreover, the associated exchange potential is given by

(72)
and is independent of density. In fact, the expression (72) represents the exact exchange potential ux(x; [r]) in the asymptotic limit |x|-->¥ [17].

2.4 Concluding remarks
In this paper we have derived and discussed various identities relating to functionals of interest in density-functional theory and their expansion in terms of the density r(x). For the most general functional F[r] we have obtained an expansion Eq. (4), which expressed F[r] in terms of the densities at various points in space without invoking any "reference density" r0(x) about which F[r] is expanded. The cost for doing so lies in the (generally unknown) n-point functions or kernels F(n)(x1, ..., xn; [r]), which are the functional derivatives of F[r] evaluated at the actual density r(x).
We have shown the connection and consistency of the general functional expansion Eq. (4) to that obtained from Eq. (13) if F[r] is, or is approximated by, a strictly local functional (LDA). In this case, the functional expansion reduces to a very simple identity Eq. (17). Going beyond the LDA to include gradient terms in F[r], we have obtained a number of identities relating F[r] to dF[r]/dr(x). In particular, we have considered the generalized gradient approximation from a general point of view and have found a number of relations that must be satisfied regardless of the specific form chosen to express F[r].
These relations should serve to illuminate the general aspects underlying various calculations from the functional point of view.

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