2.2 Kohn-Sham approach to DFT
2.2.1 Overview of the Kohn-Sham theory
While in DFT the full many-body problem is reduced to finding the ground state charge density of the system, the interactions are embodied in an intract-able electron-electron interaction term ˆVe ein Eq. (2.2). Kohn and Sham [13]
provided a route that allowed DFT to become the practical workhorse for ab-inito computations: The problem of the interacting many particle system is mapped onto a system of non-interacting particles subject to a common local potential, termed the Kohn-Sham potentialVKS(r), and with the charge density identical to the fully interacting many-body problem system.
For the non-interacting electrons the Levy-Lieb energy functional (Eq. 2.8) becomes
ELLKS[n] = inf
!n
D Tˆ+ ˆVKS E
, (2.28)
where the search of the infimum is limited to the ground states of non-interacting electrons, namely wave functions expressed through a Slater determinant composed of single particle states{ }
(r1,r2,r3, . . .rN) =
Conveniently, we rewrite Eq. (2.28) as ELLKS[n] =TS[n] +
ˆ
VKS(r)n(r) dr, (2.30) where the first term represents the density functional of the kinetic energy of non-interacting particles
TS[n] = inf
!n
D Tˆ E
. (2.31)
It follows that the Euler equation (Eq. 2.13) yields TS
n(r)=µKS VKS, (2.32)
whereµKS is the chemical potential of the non-interacting electrons.
The kinetic energy of the non-interacting electrons, together with the clas-sical Coulomb (Hartree) energy
EH[n] = 1 2
¨ n(r)n(r0)
|r0 r| drdr0, (2.33)
and the exchange and correlation density functional EXC[n] which embodies all the many-body interactions, constitute the Levy-Lieb functional:
FLL[n] =TS[n] +EH[n] +EXC[n]. (2.34) For a given density, we can (formally) define the XC term as
EXC[n] = (T[n] TS[n]) + (Ve e[n] EH[n]), (2.35) which is composed of two contributions (bracketed): (i) The di↵erence between the kinetic energy of interacting and non-interacting particles. (ii) The di↵erence between the energy of the electron-electron interaction energyVe e[n] and the Hartree energy functional. It should be noted that while the KS approach may merely appear as a transformation of the original problem of accounting for the many-body interactions, its strength lies in the fact that the XC energy is usually only a small contribution to the total energy and that it can be suitably approximated as we will see in Section 2.2.2.
Since the density of the non-interacting electrons in the Kohn-Sham system is identical to the physical system, we can combine the Euler equation (Eq. 2.13) with the result of Eq. (2.32), and obtain
µ= ELL
n(r)=µKS VKS(r) +VH(r) +VXC(r) +Vext(r). (2.36) Here, the Hartree potential term is given as
VH(r) = EH
n(r)=
ˆ n(r0)
|r r0|dr0, (2.37) and the exchange-correlation potential is
VXC(r) = EXC
n(r). (2.38)
From Eq. (2.36) we can determine the Kohn-Sham potential (up to a constant) as
VKS(r) =VH(r) +Vext(r) +VXC(r). (2.39) It follows that for each non-interacting particle we can write a Schr¨odinger-like
equation
1
2r2+VKS(r) n(r) ="KSn n(r), (2.40) and the total charge density is given as
n(r) =
NXocc
i
| i(r)|2. (2.41)
The sum is taken over all occupied KS eigenstates , i.e. the states with energy
"KSi lower or equal to the chemical potentialµin Eq. (2.36).
Although the structure of Eq. (2.40) strongly resembles the Schr¨odinger equation, the eigenvalues "KSi cannot be interpreted as energies of the quasi-particles (holes or electrons) in the original (fully interacting) system. The
2.2. KOHN-SHAM APPROACH TO DFT
only exception is the eigenvalue of the highest occupied eigenstate"KSH which is (in principle) associated with the lowest energy needed to remove charge from the system, i.e. the lowest energy to create a hole. This is a consequence of the asymptotic behavior of the charge density. Katriel and Davidson [14] and Almbladh and von Barth [15] demonstrated that the asymptotic form of the bound wave function decays exponentially and leads to the charge density
rlim!1n(r) = exph 2rp
2Ii
, (2.42)
where I is the ionization potential (or equivalently the chemical potential µ , described in the previous section). At the same time, it is straightforward to show that the wavefunction of a state i (cf. Eq. (2.40)) decays asymptotically in an exponential way as and at large distancesrfrom the system the dominant contribution stems from the highest occupied eigenstate H. Since the density of the non-interacting system is identical to the real system, Eq. (2.43) implies that "KSH equals the ionization potential in Eq. (2.42) and we can write the so-called ionization po-tential theorem:
EGS(M) EGS(M 1) ="KSH . (2.44)
This finding has a very important consequence for practical calculations and their possible interpretation. While the ionization potential is predicted exactly in principle and is given directly by the negative of the KS eigenvalue"KSH , the energy of the first unoccupied eigenstate of the KS Hamiltonian "KSH+1 is not guaranteed to have any physical meaning, i.e. it cannot serve as estimate of the electron affinity. Indeed we will see below that further consideration is required in order to calculateA. This leads to a well knownband gap problem of DFT:
The energy di↵erence between the energies of the first unoccupied state and the last occupied state of non-interacting fermions of the KS system yields an eigenvalue gap
T ="KSH+1 "KSH , (2.45)
which is however distinct from the fundamental band gapEg (Eq. 2.20).
In a less stringent interpretation, the KS eigenvalues can be taken at least as an approximation to the quasiparticle energies [16]: With commonly used approximations to Vxc, Eq. (2.45) yields qualitatively correct descriptions in the vast majority of cases, i.e. T > 0 for most insulators and semiconduct-ors, though values are consistently smaller than the corresponding Eg (taken from experiments or higher order calculations) [17, 18]. Moreover, many-body perturbation theory employs the KS eigenvalues and eigenstates, assuming that they are approximations to the corresponding quasiparticle couterparts (see Section 2.4).
In order to gain additional insight into this issue, we investigate in more detail the piecewise linearity condition given by Eq. (2.19) and the related ex-pression for the fundamental band gap (Eq. 2.27). From Eq. (2.36) it follows that by evaluating the derivative of the total energy ( ELL/ n(r)) at the integer
point of electronsM from the sides of excess and deficient electronic charge, de-notedM+ andM respectively, we obtain
✓ TS
n(r)+VKS(r)
◆
M±
=µ±. (2.46)
Moreover from the previous discussion (Eq. (2.43) and Refs. [14, 15]) we know that µ ="KSH .
Following Eq. (2.27) we now require that the chemical potential changes discontinuously when an infinitesimal amount of charge is added to the system, i.e. theM+ side of the derivative. Such a discontinuity is trivially found in the kinetic energy term (Eq. 2.31) which explicitly contains sum over all occupied states (Eq. 2.3): By considering a system withM+ particles a new eigenstate ( H+1) needs to be occupied and hence it contributes to the total energy. If we now assume that the KS potential VKS does not change when the particle number crosses the integer point M (i.e. we are neglecting the interactions of the excess infinitesimal charge with the otherM particles) then
VKS(r) M = VKS(r) M+, (2.47)
The fundamental gap of non-interacting particles, which are considered in the KS approach, then naturally coincides with the eigenvalue gap T.
Due to the Hohenberg-Kohn theorems, the exact map between the ground state density and potential is guaranteed; the potential, however, is determined only up to a constant. In other words, an infinitesimal change in the charge density can be related to an infinitesimal change in the e↵ective potentialplus a constant term L [20–22]. Following Eq. (2.27), the fundamental band gap is then given as
Eg = ELL
n(r) M+
ELL
n(r) M = T + L, (2.49)
where the subscript on Ldenotes the fact that it is provided by a jump in the local KS potential, i.e. the assumption made in Eq. (2.47) is generally not valid.
By investigating the behavior of the potential terms appearing in Eq. (2.39) we see that bothVH andVextvary smoothly withn(r) and thus cannot contribute to L. The jump L, usually termed thederivative discontinuity, should thus be provided by the XC potential and we can write
L= EXC
n(r) M+
EXC
n(r) M . (2.50)
From this consideration it is clear that for systems where L is small, Eg
can be e↵ectively approximated by Eq. (2.45). However, the size of the deriv-ative discontinuity for a given system is unknown a priori. It should be noted that even insulators for which T = 0 exist: Highly correlated systems (Mott-Hubbard insulators), systems with local magnetic order (Mott-Heisenberg in-sulators) and charge transfer insulators are typical examples [23–25]. In these
2.2. KOHN-SHAM APPROACH TO DFT
cases the finite fundamental band gap arises from the electron-electron interac-tions and is given by L. The theoretical description of the electronic states in insulators remains one of the most challenging questions in electronic structure theory.