Exact Exchange versus Correlation
It is possible to partition the xc energy into a part that is, in principle, exactly known and a remaining energy contribution. Using the Slater determinant of the noninteracting KS system introduced in Eq. (2.10), the exact-exchange (EXX) energy is defined via
Exex[{ϕiσ[nσ]}] = hΦminn |W|ˆ Φminn i −EH[n]
= −1 2
Nσ i,
∑
j=1 σ=↑,↓Z Z ϕiσ∗(r)ϕjσ(r)ϕiσ(r0)ϕ∗jσ(r0)
|r−r0| d3rd3r0. (2.21) This energy contribution strictly follows from the Pauli exclusion principle. It resembles the Fock exchange integral evaluated with KS orbitals.
The remaining part ofExc is referred to as correlation energy
Ec[{nσ}] = F[{nσ}]−Tni[n]−EH[n]−Exex[{ϕiσ[nσ]}] (2.22)
= hΨ0|Tˆ|Ψ0i − hΦminn |Tˆ|Φminn i+hΨ0|Wˆ|Ψ0i − hΦminn |Wˆ|Φminn i. (2.23) Thus, correlation summarizes all contributions of the electronic kinetic and interaction energy that cannot be described by a single Slater determinant of KS orbitals but rather require knowledge of the many-body ground-state wavefunctionΨ0. Other than this formal correspondence, no explicit expression is in general known for the exact correlation energy in contrast to the exchange part.
In practice, usually both the exchange and correlation energy are approximated. In this case, the exact distinctions of exchange and correlation of Eq. (2.21) and Eq. (2.23) do not apply. However, it is convention to distinguish between exchange and correlation even though neither is described exactly, i.e.,Exc[{nσ}] =Ex[{nσ}] +Ec[{nσ}]andvxc[{nσ}](r) =vx[{nσ}](r) +vc[{nσ}](r)for the approximate xc energy and potential.
Uniform Coordinate Scaling
One direct approach to characterize the xc energy is given via the uniform coordinate scaling of the electron density defined as [LP85]
nγ(r) =γ3n(γr). (2.24)
2.4 Exact Properties of the Exchange-Correlation Functional
The density scales such thatnγ(r)always reproduces the correctN according to Eq. (2.4). While γ<1 stretches and expands the density,γ>1 compresses it.
For the Hartree and noninteracting kinetic energy terms of Eq. (2.12) evaluating the correct scaling behavior is straightforward. Based on Eq. (2.13) and Eq. (2.14), one finds
EH[nγ] = γEH[n] and (2.25)
Tni[nγ] = γ2Tni[n]. (2.26)
However, for the exact xc energy the situation is more complicated. The exchange part, whose expression in Eq. (2.21) is similar in its basic structure toEH[n], scales as
Exex[nγ] =γExex[n]. (2.27) For the correlation part, on the other hand, no straightforward scaling rule exists. Instead,
Ec[nγ] =γ2Ec1/γ[n], (2.28)
where Ec1/γ[n] denotes the correlation in a system with reduced electron interaction ˆW →Wˆ/γ [PK03]. The scaling ofEccan further be expressed by the inequality [LP85]
Ec[nγ]
Ec[n] <γ for γ>1. (2.29)
Note that Eq. (2.27) is not only fulfilled by EXX, but holds for other approximate exchange energy functionalsExas well. In fact, in Ref. [Lev91] it is argued that the scaling rule of Eq. (2.27) defines the exchange part of anyExc, whereas the part with no simple scaling rule should be declared as correlation according to [KK08]
Ex[n] = γ→∞lim Exc[nγ]/γ
and (2.30)
Ec[n] = Exc[n]−γ→∞lim Exc[nγ]/γ
. (2.31)
The uniform coordinate scaling further provides a rule to distinguish functionals that treat exchange 100% exactly in contrast to functionals that only partially include EXX (see Sec. 3 for an introduction to such functionals) [PS01]. In Ref. [PSTS08], functionals that fulfill
γ→∞lim Exc[nγ]
Exex[nγ]=1 (2.32)
are termed to have full EXX. Consequently, said functionals automatically satisfy all constraints on their exchange part. It is reasonable to assume that the more exact constraints a functional fulfills, the better it performs for different physical situations [PSTS08, KPB99]. In this light, Eq. (2.32) provides a desirable aim for the construction of approximations toExc.
Lastly, note that the high-density limit in Eq. (2.32) does not describe a merely theoretical limiting case. It represents, e.g., the physical situation of an atom with fixed electron numberNand core chargeZ→∞. Such a system becomes increasingly hydrogenic, withTni dominatingEHand Exas follows from Eqs. (2.25), (2.26), and (2.27) [PK03].
Exchange-Correlation Hole
Based on the concept of the xc hole ¯nxc(r0,r)[GJL76, FNM03], one can in general express the xc energy via [BP95, PK03]
Exc[n] =1 2
Z Z n(r)n¯xc(r0,r)
|r−r0| d3rd3r0. (2.33) Illustratively, the xc hole describes the reduction in the probability of finding an electron atr0given that there is one atr[Cap06]. Therefore, it obeys the sum rule
Z
¯
nxc(r0,r)d3r0=−1, (2.34) i.e., the electron atris considered to be taken out of the system [PK03]. Notably, the exact xc hole has a cusp atr0→r[Kim73, PK03], and ¯nxc(r0,r)is obtained via the coupling-constant integration (see Sec.3). The exact xc energy can further be expressed by [GJL76, FNM03]
Exc[n] =N 2
Z ∞
0 4πu2hn¯xc(u)i
u du, (2.35)
withu=r−r0. Therefore, the xc energy is only defined by the spherically averaged xc hole hn¯xc(u)i= 1
N Z
d3r n(r)
Z n¯xc(r−u,r)
4π dΩu. (2.36)
Significance of the Highest Occupied KS Eigenvalue
For the exactExcthere exists a simple correspondence between the negative highest occupied (ho) KS eigenvalue and the first vertical ionization potential (IP) defined asI(N) =E0(N−1)−E0(N) for a finite system withN electrons and ground-state energy E0(N). Labeled IP theorem in the following, this relation reads [Jan78, PPLB82, LPS84, AvB85, PL97]
−εho(N) =−εN(N) =I(N). (2.37)
This relation is strongly connected to the asymptotic decay of the electron density in finite systems.
Since the KS orbitals fall off exponentially with their decay determined by their corresponding KS eigenvalue according toϕiσ(r) −→
|r|→∞exp(−√−2εiσ· |r|)[KKGG98], the density is dominated by a single KS orbital in the asymptotic limit and thusn(r) −→
|r|→∞exp(−2√
−2εho· |r|)[AvB85].
Extending the IP theorem to a system with N+1 electrons naturally provides the electron affinity (EA), which is defined asA(N) =I(N+1) =E0(N)−E0(N+1), thus yielding
−εho(N+1) =−εN+1(N+1) =A(N). (2.38) Note that the determination ofA(N) of the N electron system requires knowledge of the ho KS eigenvalue of theN+1 electron system, i.e., the anion.
It is important to emphasize that Eqs. (2.37) and (2.38) only provide a strict physical meaning for the corresponding ho eigenvalue. For all other KS eigenvalues, however, no rigorous correspon-dence to electron removal energies can be derived mathematically. For a more detailed discussion of this matter, I refer the reader to Sec. 4.7 of this thesis.
2.4 Exact Properties of the Exchange-Correlation Functional
Derivative Discontinuity
The energy difference ∆g=I(N)−A(N), commonly referred to as fundamental gap, is of direct physical relevance as it gives, e.g., the band gap of semiconductors [BGvM13]. However, based on the correspondence to occupied eigenvalues defined above, it follows that calculating∆grequires self-consistent solutions of the KS equations for theN and N+1 electron systems. One might therefore ask if there is a way to express the fundamental gap in terms of quantities related to theN electron system only.
In 1982, Perdew et al. provided an answer to this question. Based on a statistical mixture between two integer states, they expanded the realm of KS DFT to noninteger particle numbers, i.e., N =N0+ω with N0 ∈N and w∈[0,1[ [PPLB82]. Importantly, it can be shown that the ground-state energy varies linearly with the fractional electron number between adjacent integer points (cf. Sec. 2.5 and 5.1 for a more detailed discussion of this behavior).
This linear dependence directly implies a surprising feature of the exactExc[n]: At the integerN0
the slope of the energy curve, and thus the chemical potentialµ(N) =∂E(N)∂N , exhibits discontinuous jumps [PPLB82]
µ(N) =
(−I(N0) =E(N0)−E(N0−1), N0−1<N<N0
−A(N0) =E(N0+1)−E(N0), N0<N<N0+1. (2.39) This quantity directly represents a discontinuity at integer electron numbers. Based on the Euler equation of Eq. (2.9) in combination with the KS energy partitioning of Eq. (2.12), it can be expressed as [PL83, SS83] The functional derivatives of the Hartree term and the external potential are continuous inN and thus do not appear here [PL83].
The fundamental gap is built up by two contributions. The first term contains the discontinuity of the noninteracting kinetic energy. In the literature, it is referred to as KS gap ∆KS and, using Eq. (2.10), it can be formulated as
∆KS=εN+1(N)−εN(N) =εlu(N)−εho(N). (2.42) Here,εlu(N)denotes the lowest unoccupied (lu) KS eigenvalue. The second contribution in Eq. (2.41) is the so-called derivative discontinuity of the xc potential∆xc, since
∆xc= lim The quantity∆xc marks a spatially independent energy contribution, and therefore represents the overall jump of the xc potential when traversing a point with integer particle number [SP08, GGS09, YCMS12, CC13, MSC14].
This jump can be understood as a direct manifestation of the principle of integer preference discussed in Ref. [Per90]. In order to ensure integer dissociation of, e.g., diatomic molecules
consisting of atoms with different electronegativity, a step-like structure appears in the exact xc potential (see Refs. [RPC+06, KAK09, MKK11] for details).
Asymptotic Behavior
Another exact constraint on the ultimateExc[n]is given by the long-range behavior of the xc energy density and potential for neutral finite systems. The former quantity, as introduced in Eq. (2.15), is asymptotically dominated by EXX and thus decays as [GJL79, vLB94]
exc(r)∼eexx (r) −→
|r|→∞− 1
2|r|. (2.44)
Similarly, the asymptotic behavior of the xc potential, as defined by the functional derivative in Eq. (2.18), is given via [LPS84, AP84, AvB85]
vxc(r)∼vexx (r) −→
|r|→∞− 1
|r|. (2.45)
This relation can be understood quite illustratively by considering a single electron far out in a finite neutral system. Leaving behindN−1 remaining electrons in the now ionized system, such an electron will effectively feel the Coulomb potential of a single positive charge in agreement with Eq. (2.45) [FNM03].
Size Consistency
Size consistency is a fundamental principle not only of DFT but electronic-structure theory in general [Per90]. It states that the energy of two systems A and B that are well separated by a large distance should equal the sum of the energies of the individual systems
E(A...B) =E(A) +E(B). (2.46)
For a detailed discussion of size consistency and the implications of its violation in the context of DFT, see, e.g., Refs. [KK08, Sav09, KKK13].
Homogeneous Electron Gas
The density of a homogeneous quantum gas or liquid of interacting electrons marks one of the oldest and most simple models in the theoretical description of condensed matter [Tho27, Fer27]. Notably, the foundations of DFT have their source in considerations regarding systems with a uniform density nhom(r) =nhom=const. [HK64, KS65].
For such a density, the xc energy density per particle (cf. Eq. (2.15)) becomes a direct function ofnhom. The exchange component can be derived analytically, yielding in the spin-unpolarized formulation [ED11]
nhom·ehomx (nhom) =−3 4
3 π
13
nhom43
. (2.47)
For the correlation part no such exact formulation is known. Yet, based on Quantum Monte Carlo calculations [CA80], very accurate and reliable approximate expressions ofehomc (nhom)were developed [VWN80, PZ81, WP92].
The importance of the homogeneous electron gas for DFT is twofold: First, it provides a limiting case that is physically relevant, e.g., for solids and extended systems with slowly varying