Appendix 1. Derivation of Eq. (4) of Chapter 2

Suppose the functional F[r] is such that it allows a functional Taylor expansion about some density r1(x). In particular, this implies the existence of functional derivatives of F[r] of all orders in the function space surrounding the points (in function space) being considered. Define

(A1)
Then we have via the Taylor expansion about r(x)

(A2)
For brevity we set

(A3)

(A4)

etc. For the subsequent development it is important to note that the above functional derivatives are symmetric with respect to interchange of any pair of arguments. We rewrite (A2) as

(A5)

where the notation has been abbreviated. We also expand the quantities F'2, F"2, ... about r=r1(x). Thus

(A6)

(A7)
etc. Equivalently,

(A8)

(A9)
Using (A8), the term in (A5) can be rewritten as

(A10)

Substituting (A10) into (A5), the latter becomes

(A11)
where

(A12)

Using the symmetry of F"1 with respect to its arguments, the first term on the right-hand side of (A11) can be expressed as

(A13)

Using (A7), the right-hand side of (A13) can be written as

(A14)

With (A13) and (A14), (A11) becomes

(A15)
The pattern is now clear and may be continued in such a way that the quantity F1(n) becomes multiplied by r1r1"r1, from which is substracted the quantity F2(n) multiplied by r2r2¼r2. We thus arrive at the result

(A16)

where the notation (A12) is used. Apart from a possible constant term (see comment 3 in Sec II), (A16) is the same as Eq. (4) in the text.

Appendix 2. Scaling, homogeneity, and locality properties of density functionals

(I). Scaling and homogeneity
The homogeneity property of a density functional is related to a scaling process. In density functional theory, there are two types of scalings which are met. One is the density scaling, and the other is the coordinate scaling.
(i). Density Scaling. A functional is homogeneous of degree k in (or " with respect to ") density scaling if the functional Q[r] satisfies following condition.

(A1)
Another definition of the homogeneity with respect to density scaling is

(A2)
These two definitions are coincident with each other except when k=0. If k=0, Eq. (A1) implies Eq. (A2), but Eq. (A2) does not necessarily yield Eq. (A1). Instead, one then has

(A3)
which means that

(A4)
where X[r] is an arbitrary function or functional of "Symbol" such that

(A5)
This implies only that dX/dr is orthogonal to r, so that X[r] is not necessarily a constant. As an example, one may have

(A6)
(ii) Coordinate scaling. Homogeneity of a functional Q[r] of degree m in coordinate scaling is defined as

(A7)
where
(A8)
It has been proved [13] that, for any well-behaved Q[r] that obeys Eq. (A7),

(A9)
Notice that equivalence of Eqs. (A7) and (A9) also requires m¹0. If m = 0, it can be shown that Eq. (A7) implies Eq. (A9), but from Eq. (A9), one cannot derive Eq. (A7). Instead, one obtains

(A10)
which means that

(A11)
where Y[r] is an arbitrary functional such that

(A12)
Y[r] is not necessarily a constant. For instance, one might have

(A13)

(II). Locality
A energy density functional Q[r] is local if it can be expressed in the form

(A14)
where q is a function of the density r(r). It has been shown [7,11] that for any well-behaved local functional Q[r],

(A15)

(III). Relationship between homogeneity and locality
Theorem 1: If Q[r] is local, homogeneity in coordinate scaling is equivalent to homogeneity in density scaling; that is, either one implies the other.
Proof: suppose Q[r] is homogeneous of degree k in density scaling. Use Eqs. (A2) and (A15), it follows that Q[r] is homogeneous of degree 3k-3 in coordinate scaling. Conversely, if one suppose that Q[r] is homogeneous of degree m in coordinate scaling, then using Eqs. (A9) and (A15), it follows that Q[r] is homogeneous of degree (m+3)/3 in density scaling. In general, in the locality assumption, homogeneities of degree k in density scaling and degree m in coordinate scaling are related by

(A16)

Theorem 2: If Q[r] is local and homogeneous with respect to coordinate scaling and density scaling, it has the form

(A17)
where C is the constant to be determined. The homogeneity in density scaling is k; and the homogeneity in coordinate scaling is m = 3k-3.
Proof: If k ¹ 0, we have both Eq. (A2) and Eq. (A14). Taking functional derivative of Eq. (A2) with respect to the density, and since , one has

(A18)
The only solution of above equation for the well-behaved functional Q[r] is Eq. (A17).
If k = 0, however, according to Eq. (A16), m = -3. By using Eqs. (A9) and (A15), there results

(A19)
Using the same procedure as above, one has q((r)) = constant, which is a special case of k = 0 in Eq. (A17).
Comments: Note that the equivalence of the two kinds of scaling does not in general hold if Q[r] is not local. Notice also that it has been well-known that in the local density approximation (LDA) the kinetic energy density functional Ts[r] and the exchange-only density functional Ex[r] are homogenous of degree 5/3 and 4/3, respectively, with respect to density scaling. According to above theorem, the only forms of them are

(A20)
and

(A21)
respectively, where CF and CX are constants to be determined.

(IV) Functional expansions
It has been shown recently [7] that a well-behaved functional can be expanded in terms of its functional derivatives up to a constant. If Q[r] takes a form

(A22)
where q is a function of r, r(r), and Ñr(r), then [7]

(A23)
where
(A24)
Eq. (A23) includes two important special cases. One is the gradient expansion approximation (GEA) or the generalized gradient approximation (GGA), in which Q[r] takes the form