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N
37
The difference between E0 and %EÅ‚i is due to particle-particle interactions. The ad-
i
ditional terms on the right-hand side of (73) give mathematical meaning to the common
statement that the whole is more than the sum of its parts. If E0 can be written approx-
N
imately as %EÅ‚) (where the %EÅ‚) are not the same as the KS eigenvalues %EÅ‚i) the system
i
can be described in terms of N weakly interacting quasiparticles, each with energy %EÅ‚).
Fermi-liquid theory in metals and effective-mass theory in semiconductors are examples
of this type of approach.
35
0
A
AKS
HOMO (N 1)
xc
I
LUMO (N)
KS
HOMO (N)
Figure 2: Schematic description of some important Kohn-Sham eigenvalues
relative to the vacuum level, denoted by 0, and their relation to observables.
See main text for explanations.
the HOMO and LUMO become the top of the valence band and the bottom
of the conduction band, respectively, whereas in metals they are both iden-
tical to the Fermi level. The vertical lines indicate the Kohn-Sham (single-
particle) gap "KS, the fundamental (many-body) gap ", the derivative dis-
continuity of the xc functional, "xc, the ionization energy of the interacting
N-electron system I(N) = -%EÅ‚N (N) (which is also the ionization energy of
the Kohn-Sham system IKS(N)), the electron affinity of the interacting N-
electron system A(N) = -%EÅ‚N+1(N + 1) and the Kohn-Sham electron affinity
AKS(N) = -%EÅ‚N+1(N).
Given the auxiliary nature of the other Kohn-Sham eigenvalues, it comes
as a great (and welcome) surprise that in many situations (typically char-
acterized by the presence of fermionic quasiparticles and absence of strong
correlations) the Kohn-Sham eigenvalues %EÅ‚i do, empirically, provide a rea-
sonable first approximation to the actual energy levels of extended systems.
This approximation is behind most band-structure calculations in solid-state
physics, and often gives results that agree well with experimental photoemis-
sion and inverse photoemission data [65], but much research remains to be
done before it is clear to what extent such conclusions can be generalized,
and how situations in which the KS eigenvalues are good starting points for
approximating the true excitation spectrum are to be characterized micro-
36
scopically [66, 67].38
Most band-structure calculations in solid-state physics are actually calcu-
lations of the KS eigenvalues %EÅ‚i.39 This simplification has proved enormously
successful, but when one uses it one must be aware of the fact that one is
taking the auxiliary single-body equation (71) literally as an approximation
to the many-body Schrödinger equation. DFT, practiced in this mode, is not
a rigorous many-body theory anymore, but a mean-field theory (albeit one
with a very sophisticated mean field vs(r)).
The energy gap obtained in such band-structure calculations is the one
called HOMO-LUMO gap in molecular calculations, i.e., the difference be-
tween the energies of the highest occupied and the lowest unoccupied single-
particle states. Neglect of the derivative discontinuity "xc, defined in Eq. (65),
by standard local and semilocal xc functionals leads to an underestimate of
the gap (the so-called  band-gap problem ), which is most severe in transition-
metal oxides and other strongly correlated systems. Self-interaction correc-
tions provide a partial remedy for this problem [71, 72, 73, 74].
4.2.3 Hartree, Hartree-Fock and Dyson equations
A partial justification for the interpretation of the KS eigenvalues as start-
ing point for approximations to quasi-particle energies, common in band-
structure calculations, can be given by comparing the KS equation with
other self-consistent equations of many-body physics. Among the simplest
such equations are the Hartree equation
h2"2
¯
- + v(r) + vH(r) ÆH(r) = %EÅ‚HÆH(r), (76)
i i i
2m
and the Hartree-Fock (HF) equation
h2"2 ³(r, r2 )
¯
- + v(r) + vH(r) ÆHF (r) - q2 d3r2 ÆHF (r2 ) = %EÅ‚HF ÆHF (r),
i i i i
2m |r - r2 |
(77)
where ³(r, r2 ) is the density matrix of Eq. (46). It is a fact known as Koop-
man s theorem [49] that the HF eigenvalues %EÅ‚HF can be interpreted as unre-
i
laxed electron-removal energies (i.e., ionization energies of the i th electron,
38
Several more rigorous approaches to excited states in DFT, which do not require the
KS eigenvalues to have physical meaning, are mentioned in Sec. 6.
39
A computationally more expensive, but more reliable, alternative is provided by the
GW approximation [68, 69, 70].
37
neglecting reorganization of the remaining electrons after removal). As men-
tioned above, in DFT only the highest occupied eigenvalue corresponds to
an ionization energy, but unlike in HF this energy includes relaxation effects.
The KS equation (71) includes both exchange and correlation via the
multiplicative operator vxc. Both exchange and correlation are normally
approximated in DFT,40 whereas HF accounts for exchange exactly, through
the integral operator containing ³(r, r2 ), but neglects correlation completely.
In practise DFT results are typically at least as good as HF ones and often
comparable to much more sophisticated correlated methods  and the KS
equations are much easier to solve than the HF equations.41
All three single-particle equations, Hartree, Hartree-Fock and Kohn-Sham
can also be interpreted as approximations to Dyson s equation (38), which
can be rewritten as [48]
h2"2
¯
- + v(r) Èk(r) + d3r2 £(r, r2 , Ek)Èk(r2 ) = EkÈk(r), (78)
2m
where £ is the irreducible self energy introduced in Eq. (38). The Ek ap-
pearing in this equation are the true (quasi-)electron addition and removal
energies of the many-body system. Needless to say, it is much more compli-
cated to solve this equation than the HF or KS equations. It is also much
harder to find useful approximations for £ than for vxc.42 Obviously, the KS
equation employs a local, energy-independent potential vs in place of the non-
local, energy-dependent operator £. Whenever this is a good approximation,
the %EÅ‚i are also a good approximation to the Ek.
The interpretation of the KS equation (71) as an approximation to Eq. (78)
is suggestive and useful, but certainly not necessary for DFT to work: if the
KS equations are only used to obtain the density, and all other observables,
such as total energies, are calculated from this density, then the KS equa-
tions in themselves are not an approximation at all, but simply a very useful [ Pobierz całość w formacie PDF ]

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