(1)
where
(2)

(3)
and
(4)
____________________

• This chapter is, verbatim, the paper: Shubin Liu, "Local density approximation, hierarchy of equations, functional expansion, and adiabatic connection in current-density functional theory", Phys. Rev. A, 54, 1328(1996).
  

(7)
and the exchange energy functional Ex is homogeneous of degree one in coordinate scaling [1],

(8)
Consequently, one obtains [1]

(9)
and
(10)
For Ec and Tc, Erhard and Gross [1] derived,

(11)
Eqs. (9)-(11) are generalizations of Levy-Perdew formulas [12], in which j(r) = 0 is implied.
When gauge invariance is required for each of the above functionals, a new variable has to be introduced. Eqs. (9)-(11) then become

(12)

(13)
and
(14)

6.2 Local density approximations for Ts[r, v] and Ex[r, v]
Following Vignale and Rasolt [2], let us suppose that for an almost uniform electron gas Ts[r, v] and Ex[r, v] can be approximated by the variable separation

(15)
and
(16)
respectively. If one inserts Eq. (15) into Eq. (12) , there follow

(17)
and
(18)
Inserting Eq. (16) into Eq. (13), one obtains

(19)
and
(20)
Eqs. (17) and (19) are the conventional Levy-Perdew virial relations [12] for Ts[r] and Ex[r], respectively.
Now suppose all quantities are local, which means that the energy density of each of them is only a function of the density, i.e., for any local Q[r],

(21)
It has been shown that for the local functional Q[r] [13],

(22)
Partial integration of Eq. (22) and insertion of it into Eqs. (17)-(21), respectively, gives

(23)

(24)

(25)
and
(26)
which mean that Ts[r] and Ex[r] are homogeneous of degree 5/3 and 4/3 with respect to the density, respectively, and Ts[v] and Ex[v] are homogeneous of degree 5/2 and 2 with respect to v(r), respectively. According to the uniqueness theorem of local functionals [14], the only functional homogeneous of degree k in density is and the same argument applies to homogeneous functionals in v(r). Thus we must have

(27)

(28)
where C1-C4 are constants to be determined. In the homogeneous electron gas case, C1 = CTF = 2.8712 and C3 = Cx = 0.7386 [15]. Eq. (28) was first proposed by Vignale and Rasolt [2], who gave

(29)
where kF is the Fermi momentum, cL is the diamagnetic susceptibility, and 0L stands for the diamagnetic susceptibility of the non interacting gas.

6.3 Hierarchy of equations
Using Eqs. (12)-(14) and following Kugler [16] and Nagy [17], one derives hierarchies of equations for Ts[r, v], Ex[r, v], Ec[r, v], and Tc[r, v], respectively. A general functional expansion scheme, which is a generalization of that of Parr, Liu, Kugler and Nagy [13], is obtained.

6.3.1 Hierarchy of equations for Ts[r, v]
Functional differentiation of Eq. (12) with respect to r(r) leads to

(30)
and functional differentiation of Eq. (12) with respect to v(r) leads to

(31)
Again, taking functional differentiation of Eq. (30) with respect to r(r) and v(r), respectively, one obtains

(32)
and
(33)
and taking functional differentiation of Eq. (31) with respect to r(r) and v(r), respectively, leads to

(34)
and

(35)
Similarly, one can obtains higher order terms. Rewriting the virial relation Eq. (12) with the aid of the hierarchy of Eqs. (30)-(35) and so on, there results

(36)

6.3.2 Hierarchy of equations for Ex[r, v]
Equation 13 serves as the zeroth equation of the hierarchy for Ex[r, v], we get the first equations by taking functional differentiation of Eq. (13) with respect to r(r) and v(r), respectively,

(37)

(38)
Starting with Eqs. (37) and (38), one is able to obtain higher order equations as done for Ts[r, v]. Rewriting Eq. (13) with the aid of Eqs. (37) and (38) and higher orders, we find

(39)
6.3.3 Hierarchy of equations for Ec[r, v] and Tc[r, v]
Following a similar procedure, beginning with Eq. (14), one ends up with

(40)
Thus
(41)
and
(42)
where "Const" stands for a constant independent of r(r) and v(r). Eqs. (36), (39), (41) and (42) are the generalization of the functional expansion scheme by Parr, Liu, Kugler and Nagy [13], in which any well-behaved functional Q[r] may be expanded as

(43)
When one set v(r) = 0, Eqs. (36), (39), (41) and (42) reduces to Eq. (43).

6.4 Adiabatic connection formulation of the Current-Density Functional Theory
Consider the Hohenberg-Kohn universal functional F[r, j] defined within an extended domain via the constrained-search [11] by the prescription

(44)
where Y is the antisymmetric N-electron wave function which yields r(r) and j(r), minimizes , and is an eigenstate of some Hamiltonian of the form

(45)
where
(46)
(47)
and
(48)
where l is the coupling constant that measures the interaction strength between electrons of the system, and u and A are the local scalar and vector external potential which keep the ground state density (r) and the paramagnetic-current-density j(r) of fixed for all l>0. The correlation energy density functional Ec[r, j] is defined as

(49)
which can be decomposed into two components, namely, the kinetic energy part Tc[r, j] and the potential energy part Vc[, j]:

(50)
with
(51)
and
(52)
The total energy of the N-electron system is then

(53)
where
(54)

The variational principle of CDFT ensures that [1]

(55)
and
(56)
Multiplying Eq. (55) by r(r) and Eq. (56) by j(r), integrating over all space, and inserting the results into Eq. (53), there follows

(57)
Following Erhard and Gross [1], one has the virial theorem

(58)

where Since Ts[r, j] and Ex[r, j] are l - independent, Eqs. (12) and (13) holding for any l, one obtains
(59)
and
(60)

Meanwhile, according to the Hellmann-Feynman theorem [12],

(61)
Differentiating Eq. (57) with respect to l and setting the result equal to Eq. (61) gives

(62)
By definition,

(63)
Combining Eqs. (60),(62) and (63) leads to

(64)

Equating Eqs. (62) and (63), one gets [18,14]

(65)
Taking functional derivatives with respect to the density and the paramagnetic-current-density, respectively, and eliminating quantities including Ec[r, j], one ends up with

(66)
Notice that Eqs. (64) and (66) include only Ec[r, j] and Tc[r, j], respectively. If the requirement of the gauge invariance is taken into consideration, Eqs. (64) and (66) then become

(67)
and

(68)

6.5 Expansions and the local density approximation for Ec[r, v] and Tc[r, v]
Now, let us look at the consequences of Eqs. (67) and (68). For simplicity, following Vignale and Rasolt, we take the approximation that Suppose that, up to the first order, Ec[r, v] can be expanded in a Taylor series in l as

(69)
where Ec=0[r, j] = 0 according to the definition of Eq. (50), and . Eq. (67) then becomes

(70)
and using Eqs. (65) and (69), Eq. (68) also reduces to Eq. (70). This means that A1[r] is homogeneous of degree zero in coordinate scaling [19], as early discovered by Levy and Perdew [12]. It is interesting to note that A1[v] does not show any homogeneity property in coordinate scaling.
Furthermore, suppose that A1[r] and A1[v] are local functionals of r(r) and v(r), respectively. Under the locality assumption, one has [13]

(71)
and
(72)
Combination of Eqs. (70)-(72) leads to

(73)
and
(74)
which means that A1[r] is homogeneous of degree one in the density r(r), as early discovered by Liu and Parr [14], and A1[v] is homogeneous of degree 3/2 in v(r). According to the uniqueness theorem of local and homogeneous functionals [14], the forms of A1[r] and A1[v] in Eqs. (73) and (74) are

(75)
and
(76)
respectively, where C1 and C'1 are constants to be determined.
In general, let us suppose that following Taylor series exists

(77)
and thus

(78)
Using Eq. (65), one has

(79)
and
(80)
Inserting Eqs. (77) and (78) into Eq. (67), for any l, there results

(81)
and inserting Eqs. (79) and (80) into Eq. (68), for any l, one has

(82)
The important consequence is that

(83)
which means that An[r, v] is homogeneous of degree (1-n) in coordinate scaling, i.e.,
(84)
If one supposes that

An[r, v] = An[r] + An[v], (85)
Eq. (83) reduces to

(86)
which gives the known formula [14]

(87)
which means that An[r] is homogeneous of degree (1-n) in coordinate scaling, and

(88)
Set l = 1, Eqs. (77) and (79) become

(89)

(90)
respectively, which show that both Ec and Tc can be expanded in terms of functionals An[r, v], n = 1, 2, 3, ..., each of which satisfies Eq. (83) exactly, and is homogeneous of degree (1-n) in coordinate scaling. If the two variables in An can be treated separately, i.e., Eq. (85) is valid, Eqs. (87) and (88) are then followed, showing that the density part is a homogenous functional of degree (1-n) in coordinate scaling, for the current-density part, however, no homogeneity property is observed.
Now if we suppose that both An[r] and An[v] are local, one has [13]
(91)
and
(92)
Combining Eq. (91) with Eq. (87), and Eq. (92) with Eq. (88), there follow

(93)
and
(94)
which mean that, under the locality and variable-separation assumptions, An[r] is homogeneous of degree (4-n)/3 in density scaling, and An[v] is homogeneous of degree (4-n)/2 in v(r), with n=1, 2, 3, .... Use the uniqueness theorem of local functionals [14], one ends up with
(95)
and
(96)
where a and b are constants to be determined. Consequently, Eqs. (89) and (90) become
(97)
and

(98)

where ai and bi are constants to be determined, and N is the number of electrons in the system. Notice that with only one set of ai and bi, Ec and Tc may be determined simultaneously. Eq. (97) and (98) are generalizations of our recent work on functional expansion of Ec[r] and Tc[r] [14].

6.6 Conclusions
In this paper, consequences of the virial theorem in the current-density-functional theory have been investigated. Local formulas for the kinetic, exchange and correlation energy functionals have been obtained separately under the variable separation assumption. The analysis shows that the kinetic energy density functional Ts[,v] is homogeneous of degree 5/3 in electronic density, which is the traditional Thomas-Fermi formula, and homogeneous of degree 5/2 in v(r), while the electronic part for Ex[r,v] is the conventional Dirac form (4/3 in r), and the v(r) part is a homogeneous functional of degree 2, as was first found by Vignale and Rasolt [2]. To obtain local forms for the correlation energy Ec[r,v] and its kinetic component Tc[r,v], the adiabatic connection formulation has been used. Eqs. (96) and (97) show that Ec and Tc are combinations of local functionals of degree 1, 2/3, 1/3, ..., in r(r), consistent with our recent result [14], and 3/2, 1, 1/2, ..., in v(r). Also, as shown in [14], Ec and Tc can be determined the same time in CDFT in terms of the series An[r, v], each being homogeneous of degree (1-n) in coordinate scaling. Also, starting from the virial relations, we have obtained hierarchies of equations for Ex, Ts, Ec and Tc, respectively.

References

[1]. S. Erhard and E.K.U. Gross, Phys. Rev. A 53, R5(1996).
[2].G. Vignale and M. Rasolt, Phys. Rev. Lett. 59, 2360(1987).
[3]. G. Vignale, in Density Functional Theory, Vol. 337 of NATO Advanced Srudy Institute, Series B: Physics, Edited by E.K.U. Gross and R.M. Dreizler (Plenum, New York, 1995), p.485.
[4]. M. Rasolt and G. Vignale, Phys. Rev. B 37, 10685(1988); M. Rasolt and G. Vignale, Phys. Rev. Lett. 65, 1498(1990).
[5]. S.M. Colwell, N.C. Handy, and A.M. Lee, Phys. Rev. A 53, 1316(1996); A.M. Lee, N.C. Handy, and S.M. Colwell, J. Chem. Phys. 103, 10095(1996).
[6]. S.M. Colwell and N.C. Handy, Chem. Phys. Lett. 217, 271(1994).
[7]. A.M. Lee, S.M. Colwell, and N.C. Handy, Chem. Phys. Lett. 229, 225(1994).
[8]. G. Vignale, Phys. Rev. B 47, 10105(1993).
[9]. M. Ferconi and G. Vignale, Phys. Rev. B 50, 14772(1994).
[10]. M. Ferconi M.R. Geller, and G. Vignale, Phys. Rev. B 52, 16357(1995).
[11]. M. Levy, Proc. Natl. Acad. Sci. USA 76, 6062(1979).
[12]. M. Levy and J.P. Perdew, Phys. Rev. A 32, 2010(1985).
[13]. R.G. Parr, S. Liu, A.A. Kugler, and A. Nagy, Phys. Rev. A 52, 969(1995).
[14]. S. Liu and R.G. Parr, Phys. Rev. A 53, xxxx(1996).
[15]. R.G. Parr and W. Yang, Density Functional Theory of Atoms and Molecules (Oxford Univ. Press, Oxford, 1989).
[16]. A.A. Kugler, Phys. Rev. A 41, 3489(1990).
[17]. A. Nagy, Phys. Rev. A 47, 2715(1993).
[18]. A. Savin, Phys. Rev. A. 52, 1805(1995).
[19]. S.K. Ghosh and R.G. Parr, J. Chem. Phys. 82, 3307(1985).