(1)
where F[r] is a universal functional of r(r), which can be decomposed into three ingredients, i.e., the kinetic energy functional Ts[r], the classical electron-electron repulsion energy functional J[r], and the nonclassical exchange-correlation energy functional Exc[r],

. (2)
From (ii), one obtains the Euler-Lagrange equation

(3)
where m, the Lagrange multiplier, is the chemical potential of the system.
Since explicit dependence of E[r] or F[r] on r(r) is so far unfortunately unknown, one has to resort to approximations. The first and most widely-used implementation of DFT is the so-called Kohn-Sham scheme [8],

(4)
where

(5)
and the local Kohn-Sham potential uks(r) consists of the external potential, the Coulomb potential, and the exchange-correlation potential,

. (6)
with the Coulomb potential

(7)
and the exchange-correlation potential

(8)
A large amount of effort [1-6] has been devoted to finding approximate energy density functionals for the kinetic energy Ts[r], the exchange-only energy Ex[r], and the exchange-correlation energy Exc[r]. The first model is the local density approximation (LDA) proposed by Thomas and Fermi more than fifty years ago, in which Ts[r] and Ex[r] are homogeneous functionals of degree 5/3 and 4/3 in r(r), respectively. The theoretical expression of LDA is as follows. For any well-behaved functional F[r], LDA assumes that it takes the form

(9)
where f is the energy density. This means that f depends on the density only, which is the case for a system which is the homogeneous electron gas. It has been recently proved that for inhomogeneous systems the above homogeneity of Ts[r] and Ex[r] still remains unchanged, i.e., 5/3 and 4/3, respectively, in LDA [9].
LDA, however, is too crude an approximation to be quantitatively used for atoms and molecules. A deviation of around 15% is typical. An immediate improvement for F[r] in Eq. (9) would be

(10)
where the energy density f is a function of both r and r(r). This kind of approaches are called the weighted-density approximation (WDA) [9-10]. It is generally believed, however, that although it gives a much better result than LDA in classical fluid simulation, it behaves poorly in treating atoms and molecules [11]. In the further development, the gradient expansion approximation (GEA) and the generalized gradient approximation (GGA) and their Padé approximate summations [1-6] were employed. In these formulations, F[r] is assumed to take the form

(11)
Compared with Eq. (9), difference is immediately seen. GEA/GGA includes the influence of the gradient of density, which is nonlocal in nature, and thus more flexibility is expected in simulating the ground state energies. Emphasis has been given to the exchange-only functional Ex[r] and exchange-correlation energy functional Exc[r] in recent years. Good results have been observed, especially in energetics, geometry prediction, rotation-vibrational spectroscopy, etc.. A relative error of less than 5% is usually observed. See two recent reviews [5,6] and references therein.
On the other hand, energy functionals based on the GEA/GGA/Padé approximation perform much less satisfactorily in functional derivatives [12] as well as in some other kinds of applications, such as in describing London dispersion forces [13], spin resonance phenomena [14], transition states of some reactions [15, 16], structures of some molecules [17, 18], and so on. The main failures of the above functionals are believed to lie in three aspects: (i) failure to handle the correct asymptotic behavior [12,19] of their functional derivatives as well as their behavior near the nuclear cusp, (ii) failure to reproduce the local behavior of corresponding exact energy densities [20], and (iii) failure to handle the regions where a large gradient of density is presented [21].
This is where our present research starts from. It is the purpose of this work to go beyond the GEA/GGA/Padé approximations and develop new formulas for new kinds of the energy density functionals.
As the first step, a general formula of functional expansion for any well-behaved functional is established and rigorously proved in Chapter 2 [9]. One finds that for above F[r], if ones knows its nth-order functional derivatives with respect to the density, it can be expressed as

(12)
where C stands for a constant. As a case of much interest, we considered the situation where the energy density of F[r] is a function of three variables: r, r(r), and Ñr(r), i.e.,

(13)
This is a general case in which both GEA/GGA and WDA were taken into account. We find that the following identity holds [9]

(14)
where

(15)
and summation over repeated indices is implied. If one assumes locality of F[r], one obtains a compact identity,

(16)
which may be used as a definition and criterion to test the locality of any functional, as shown in Ref. [22].
Equation (12) may be further developed into a general form, i.e., expanded in terms of powers of density r [23],

 (17)
where a may be any number. If a=1, Eq. (17) reduces to Eq. (12). Eq. (17) may help us understand the homogeneity property of energy density functionals.
Another generalization is made in the current-density functional theory in Chapter 6 [24], in which F[r, v], any two-variable well-behaved functional, is expanded as

(18)
An immediate application [25] of Eq. (18) involves expanding the total energy E[r] in four different ensembles [26,27], resulting in four new expressions of the total energy in terms of chemical potential, chemical hardness (or softness), Fukui function, response function, and so on. Maximum Hardness Principle (MHP) [28-31] may be regarded as an immediate consequence of these formulas.
The second topic we are concerned with in this dissertation is looking for new functional forms. The correlation energy density functional Ec[r] and its kinetic component Tc[r] are the first examples we have investigated in Chapter 3. Based on the Levy-Perdew's relation for Ec[r] [32],

(19)
where l is the so-called coupling constant [32], we find [33],

(20)
Using Savin's relation between Ec[r] and Tc[r] [34]

(21)
if a Taylor series

(22)
is assumed, we arrive at the conclusion [33] that both Ec[r] and Tc[r] can be expanded in terms of homogeneous functionals in coordinate scaling. This is because we can prove that An[r] appearing in Eq. (22) is indeed homogeneous of degree (1-n) in coordinate scaling, i.e.,

(23)
It is interesting to note that both Ec[r] and Tc[r] can be determined simultaneously by the formulas

(24)
and
(25)
respectively, if An[r] are known. Equations (24) and (25) may be regarded as a new kind of functional expansion if one notices homogeneity of An[r], Eq. (23).
If the locality assumption of An[r] is employed, using Appendix 2, we obtain a new local form of the correlation energy density functional [33]

(26)
where a, b, and c are constants to be determined. It has been applied to atoms, ions, and simple molecules in Chapter 4 [35]. We find that among five local forms of Ec[r], Eq. (26) with fitted coefficients best reproduces experimental (or exact theoretical) results, and in some cases, it is even comparable with Lee-Yang-Parr form, a widely-used nonlocal formula for Ec[r]. In addition, this kind of functional expansion, Eq. (24), is also utilized to expand the pair distribution function and the second-order density matrix in Chapter 5 [36], and applied to the current-density functional theory in Chapter 6 [24].
The last chapter, Chapter 7, of this thesis deals with the general cusp problem of any strongly decaying properties reported in present GEA/GGA/Padé formulas. Four rigorous formulas for evaluation of these properties at any nuclear cusp are given [19]. Applications are made for the electronic density and its derivative for the first 18 neutral atoms. Also, the formulas are applied to evaluate cusp values of the exchange-only potential Vx(r) and the exchange-correlation potential Vxc(r). Consistent results are obtained and regularities are observed.
Other works done by us in the last couple of years but chosen not to be contained in this dissertation are: (i) Expanding the exchange-only energy density functional Ex[r] which is believed to be more nonlocal than the correlation energy density functional Ec[r], in terms of homogeneous functionals in density scaling [23]; (ii) Modeling Ex[r] from a theoretical basis by explicitly including its first-order contribution [37], which is absent in GEA/GGA/Padé methods; (iii) Comparing various models in the kinetic energy density functional Ts[r] [38]; (iv) Proving the uniqueness and asymptotic behavior of local kinetic energy density [39]; (v) Providing a new definition for the local temperature of a electronic system [40]; and (vi) Study on the exchange-correlation potential of excited states for some atomic systems [41]; (vii) Giving a new proof for the Maximum Hardness Principle (MHP) and new formulas for total energy expansion in terms of chemical potential, hardness, softness, Fukui function, and so on [25]; (viii) Starting from a new series, called Laurent series, with the same spirit of Ref. [33], a new formula was obtained for both Ec and Tc [42]; and (ix) a Padé approximation for Eq. (26) was obtained recently [43], whose simplest form is nothing but an old form due to Wigner. This formulation more or less justifies the origin of the Wigner form, which was first apparently obtained purely from empirical evidence.
We conclude by noticing that DFT is a young but much fast developing branch of theoretical chemistry. Although it does not yet provide a systematic way to improve its results (including approximate functionals themselves), we are quite optimistic that it offers great help for tackling the many-body problem. I believe that we are finally on the right track! This by no means implies that conventional ab initio methods, such as HF, CI, MCSCF, MBPT, CC, etc., are incorrect. I mean that in DFT one may have an alternative approach which can be much superior to others in understanding and describing the phenomena in nature. One evidence is perhaps convincing to support this point: Order N scaling approaches so far available in the literature to handle very large molecular systems all involve versions of DFT format.
Finally, I would like to dedicate this work to the many teachers who have influenced my life in one way or another.
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