(2.3), (2.4), and (2.8) are solved iteratively. They form the backbone of the KS scheme, which is an in principle exact reformulation of the many-body problem in quantum mechanics. The major advantage of the KS scheme is that it provides a significant conceptual and practical simplification of the many-body problem that is traceable for up to hundreds or even thousands of electrons when applying appropriate xc-functional approximations.
2.2 Fundamental relations with respect to photoemission
The persistent drawback of KS DFT is that the ultimate xc functional remains unknown for systems of practical relevance. Despite relying on approximations for the xc functional, one can formulate fundamental relations which the ultimate xc functional and potential, respectively, must obey. Sat-isfying as many of these exact constraints as possible has become one of the promising philosophies in the development of xc approximations [PRT+05]. I will mainly present relations that have impact on the prediction of photoemission observables, for more details on exact constraints I recommend Ref. [PK03].
The first relation that I want to discuss has immediate implications on photoemission. It guarantees that the HOMO eigenvalue obtained with the ultimate xc functional equals the negative, relaxed, vertical ionization potential [PPLB82,LPS84,AvB85,PL97],
εHOMO(N) =E0(N)−E0(N−1) =−IP(N). (2.9) The relation follows from the asymptotic decay of the true electron densityn(r)∼exp(−2√
2IP r) of a finite system [AvB85]. As the KS density is governed by the least bound occupied KS orbital for largerwhich, in turn, decays with its eigenvalue|ϕHOMO(r)|2∼exp(−2√
−2εHOMOr) [KKGG98], the IP and the exact HOMO eigenvalue have to be identical. Relation (2.9) is often referred to as the IP theorem. In contrast to Koopmans’ theorem [Koo34], which is its counterpart in Hartree-Fock (HF), the IP theorem in DFT has the advantage of implementing the process of electronic relaxation. This is apparent from the total-energy difference of Eq. (2.9), whereE0(N) andE0(N−1)are self-consistent ground-state energies. Accordingly,E0(N−1)and the underlying N−1-electron density are fully adapted to the loss of one electron.
Since the IP theorem is rigorously valid only for the ultimate functional, HOMO eigenvalues from approximate functionals usually deviate from the exact IP. Whether approximate HOMO eigenvalues can still serve as reliable IP predictions will be part of the discussion within this chapter. To obtain an entire photoemission spectrum from DFT, also ionization energies of more strongly bound electrons have to be accessible. Unfortunately, DFT doesn’t offer any equivalently exact relation between IPs and eigenvalues that lie energetically beneath the HOMO. It has been demonstrated, though, that KS eigenvalues from accurateab initiodensities can be decent approx-imations to IPs of outer valence electrons. Reported deviations to experiment are on the order of 0.1 eV [CGB02]. Furthermore, KS eigenvalues are connected to quasiparticle energies by an expansion in which they form the leading contribution [CGB02,GBB03,KK10]. These relations are of paramount importance as they put the customary approach of predicting photoemission spectra with DFT, i.e., approximating IPs by DFT eigenvalues from self-consistent ground-state calculations [BC95,AMH+00,MKHM06,SB09,KK10,KSRAB12], on solid ground.
An accurate IP prediction is further related to the asymptotic decay of the exact xc potential with
−e2/r [LPS84, AvB85]. This fall-off is plausible considering a single electron far away from the system as it leaves behind a positively charged ion. The electron will experience the−e2/r
potential, which matches the leading contribution from a multipole expansion of the electrostatic Coulomb potential of a single charge. A wrong asymptotic behavior is known to cause HOMO eigenvalues to deviate from the IP [CS13,SKKK14].
Yet another constraint can be deduced from one of the most paradigmatic systems: a single electron bound to a single nucleus. In this system there should obviously be no electron-electron interaction.
However, if one inserts the density of a one-electron systemn1(r)into the total-energy functional as it is partitioned in the KS scheme, the classical Hartree interaction will give a nonzero contribution,
EH[n1] =e2 2
Z Z n1(r)n1(r0)
|r−r0| d3rd3r0. (2.10) To correct the artificial Hartree self-interaction energy, the xc functional has to compensateEH[n1] exactly,
EH[n1] +Exc[n1,0] =0. (2.11) The bad news is that the most common xc approximations do not meet this requirement, giv-ing rise to one of the prominent deficiencies of DFT, namely the spurious self-interaction error (SIE) [PZ81]. Since the SIE definition of Eq. (2.11) is rigorously defined only in the one-electron limit, it is difficult to find a universal criterion for many-electron systems. Yet, in 1981 Perdew and Zunger extended the concept of one-electron SIE (OE-SIE) to the many-electron case in a straightforward way. Keeping up the interpretation of individual orbital densitiesni(r)as electrons, a system is declared to be free from OE-SIE if [PZ81]
∑
σ=↑,↓ Nσ i=1
∑
(EH[niσ] +Exc[niσ,0]) =0. (2.12) Concerning photoemission, the presence of the OE-SIE manifests in severe distortions of the energetic location of eigenvalues. As demonstrated in Ref. [KKMK09,KKMK10], the distinctly varying energy contributions to eigenvalues which arise from the OE-SIE are depending on the degree of localization of the corresponding orbitals. Whereas localized orbitals often suffer from a large OE-SIE, delocalized states are less affected. The eigenvalue spectra of PTCDA or NTCDA are prime examples since their outer valence electronic structures are composed both of localized σ orbitals and of rather delocalized π orbitals. Particularly in scenarios like these, functionals affected by the OE-SIE predict orbitals in a notably distorted energetic order [DMK+06,KKMK09, KKMK10,SK16b], [P0], [P1], [P3], and [P6].
In their seminal work on fractional particle numbers within DFT, Perdewet al.[PPLB82] provided the basis for further relations. One of them states that the total energy of a quantum-mechanical ensemble, which is designed to describe a statistical mixture of the pure N andN−1-electron ground states of a system, has to change linearly with respect to fractional removal (or addition) of an electron,
E(f) = (1−f)E0(N−1) +f E0(N). (2.13) Here, f specifies the fractional charge, which is confined between]0,1]for each linear total-energy segment [PPLB82]. Figure 2.1 illustrates the straight-line condition for total energies between N−1 andNas well asNandN+1 electrons. Whether piecewise linearity is satisfied has been studied intensively on the DFT and HF level from various perspectives [ZY98,RPC+06,MSCY06, CMSY08,MSCY08,TDT08,KK13,CAR+14,KSKK15,VESN+15,AZH+16,SK16b]. A
con-2.2 Fundamental relations with respect to photoemission
cave (or convex) deviation from the straight-line condition is often dubbed (de-)localization error [CMSY08]. The latter term originates from the fact that, e.g., convex deviations from linearity cause spurious delocalization of charges, which can be seen best in stretched H+2. Semilocal xc functional approximations yield a lower total energy compared to the piecewise linear total energy if both fragments are fractionally charged. Thus, the one electron erroneously favors a delocalization over both nuclei [CMSY08].
Further, the deviation from the straight line is often used as an alternative definition of the SIE in many-particle systems, termed many-electron SIE (ME-SIE). ME-SIE and the Perdew-Zunger definition of OE-SIE are not unrelated. Assuming that orbitals do not change when removing frac-tional charge from the HOMO (frozen-orbital approximation), only the occupation factor fHOMO
scales the total energy. While the kinetic energy and external energy are still changing linearly upon fHOMO, the Hartree energy does not. More precisely, the Hartree contribution to the one-electron self-interaction energy corresponding to the density of the HOMO,EH[nHOMO]as defined in Eq. (2.10), shows a quadratic dependence. Standard xc approximations are not able to compen-sate for the deviation from linearity. In fact, the OE-SIE often makes a major contribution to the total-energy behavior, giving rise to a close connection of both SIE definitions.
However, the definitions of the OE-SIE and the ME-SIE are not mutually interchangeable. This can be exemplified by some distinct properties of both definitions. Freedom of the OE-SIE, for example, does not automatically imply a vanishing ME-SIE and vice versa [RPC+07,HKKK12, SK16b]. On the one hand, the ME-SIE provides a stringent condition that is naturally fulfilled for the exact functional. On the other hand, single-particle densities obtained with the ultimate functional do not necessarily force the OE-SIE condition in Eq. (2.12) to be zero [HK12b]. In order to satisfy the OE-SIE condition, single-particle ground-state densities have to be used to evaluate Eq. (2.12) since xc functionals are only defined rigorously for ground-state densities. Single-particle densities that are part of many-electron systems typically don’t meet this criterion. Yet, the compelling strength of the OE-SIE definition in the Perdew-Zunger sense is that it offers a intuitive and practical scheme to correct the OE-SIE [PZ81] while there is no obvious analogue for ME-SIE. Various studies point out that presently no xc expression is available that can globally restore (or intrinsically obey) the straight-line behavior without invoking system dependent parameters [MSCY06,VSP07,SK16b].
A consequence related to piecewise linearity is that the chemical potential
µ=
limf→0∂E/∂f|N−f =−IP(N)
limf→0∂E/∂f|N+f =−EA(N) (2.14) jumps discontinuously at integer particle numbers resulting in a kink of the total energy curve between two linear segments [PPLB82,PL83]. This behavior is illustrated in Fig 2.1. The jump arises because the electron ejection energyIP(N)and the electron affinityEA(N)of anN-electron system differ by the fundamental gap
∆f=IP(N)−EA(N). (2.15)
In DFT the jump of the chemical potential can be attributed to two contributions. The first one stems from the discontinuity of the noninteracting kinetic energy that is equal to the KS HOMO-LUMO gap,∆KS=εLUMO(N)-εHOMO(N). The second part is due to a constant jump of the xc
Figure 2.1:Left: Illustration of the straight-line condition. The black line corresponds to the linear total energy of the ultimate xc functional that is free from ME-SIE. The typical behavior of ME-SIE affected functionals (represented by the typical behavior of semilocal functionals) is sketched in blue.
Right: Illustration of Janak’s theorem and its relation to the IP.
potential when passing an integer electron number, which is the famous xc derivative discontinuity [PPLB82,PL83,SS83]. Critical for IP predictions from the KS HOMO eigenvalue in the spirit of the IP theorem in Eq. (2.9) is that common xc-functional approximations have an xc potential that is erroneously continuous at integer electron numbers. Instead of jumping, they yield an xc potential that averages over the xc derivative discontinuity which, in turn, results in a severe up-shift ofεHOMOwith respect to the true HOMO eigenvalue [PL83,TDT08].
The discussion of the constraints and relations has illustrated that most of them are closely con-nected. Even more insights into their relationship, especially regarding the IP prediction from approximate eigenvalues, is provided by Janak’s theorem [Jan78]. It states that the derivative of the total energy with respect to the occupation number has to be equal to the occupation-number-dependent eigenvalue,
∂E[n]
∂fiσ
=εiσ(fiσ). (2.16)
Remarkably, Janak’s theorem holds for any approximate density-dependent xc functional. A closer look at this relation is particularly interesting for the HOMO eigenvalue. For a total energy that is piecewise linear betweenNandN−1 electrons the HOMO eigenvalue needs to be constant and independent of the occupation number. If Eq. (2.16) is reformulated by integration over f between both adjacent integer charges,
IP=E0(N−1)−E0(N) =− Z 1
0
df εHOMO(f), (2.17)
the HOMO eigenvalue can be tied to the IP directly. In the exact case in which the HOMO eigenvalue is constant upon f one arrives at the IP theorem as defined in Eq. (2.9). However, once the total energy exhibits a curvature, the eigenvalues necessarily have to be a nontrivial function of f. As a consequence it is unlikely that an f-dependent HOMO eigenvalue will suffice the IP theorem. This immediately affects the accuracy of IP predictions from HOMO eigenvalues of