# Hubbard-U band-structure methods

R C Albers1, N E Christensen2, and A Svane2 1Theoretical Division, Los Alamos
National Laboratory, Los Alamos, NM 87545, USA  2Department of Physics and
Astronomy, Aarhus University, Denmark<EMAIL_ADDRESS>
###### Abstract

The last decade has seen a large increase in the number of electronic-
structure calculations that involve adding a Hubbard term to the local density
approximation band-structure Hamiltonian. The Hubbard term is then solved
either at the mean-field level or with sophisticated many-body techniques such
as dynamical mean field theory. We review the physics underlying these
approaches and discuss their strengths and weaknesses in terms of the larger
issues of electronic structure that they involve. In particular, we argue that
the common assumptions made to justify such calculations are inconsistent with
what the calculations actually do. Although many of these calculations are
often treated as essentially first-principles calculations, in fact, we argue
that they should be viewed from an entirely different point of view, viz., as
phenomenological many-body corrections to band-structure theory.
Alternatively, they may also be considered to be just a more complex Hubbard
model than the simple one- or few-band models traditionally used in many-body
theories of solids.

###### pacs:

71.10.-w,71.15.-m,71.27.+a,71.28.+d

††: J. Phys.: Condens. Matter

##  1 Introduction

Since the very early days of quantum mechanics, the exact electronic
Hamiltonian, written in terms of the kinetic energy and electrostatic
interactions between the electrons and with the nucleus, has been known. In
non-relativistic second-quantized form, for example, this can be written as

$\displaystyle\hat{H}$ $\displaystyle=$ $\displaystyle\sum_{\sigma}\int
d\bm{r}\psi^{\dagger}_{\sigma}(\bm{r})[\frac{-\hbar^{2}}{2m}\nabla^{2}+V_{N}(\bm{r})-\mu]\psi_{\sigma}(\bm{r})$
(1) $\displaystyle+$ $\displaystyle\frac{1}{2}\sum_{\sigma\sigma^{\prime}}\int
d\bm{r}d\bm{r^{\prime}}\psi^{\dagger}_{\sigma}(\bm{r})\psi^{\dagger}_{\sigma^{\prime}}(\bm{r^{\prime}})V_{C}(\bm{r}-\bm{r^{\prime}})\psi_{\sigma^{\prime}}(\bm{r^{\prime}})\psi_{\sigma}(\bm{r}),$

where $V_{N}(\bm{r})$ is the Coulomb interaction between the electrons and the
nuclei, $V_{C}(\bm{r}-\bm{r^{\prime}})$ is the Coulomb interaction
($e^{2}/|\bm{r}-\bm{r^{\prime}}|$) between the electrons,
$\psi^{\dagger}_{\sigma}(\bm{r})$ and $\psi_{\sigma}(\bm{r})$ are creation and
destruction operators for an electron at $\bm{r}$ with spin $\sigma$, and
$\mu$ is the chemical potential. Since this Hamiltonian has strong
interactions between all the electrons in a solid, it is fundamentally a many-
body problem by nature, and is too intractable to be capable of exact
solution. Density functional theory and its application to calculations of the
electronic band structure (BS) of materials[1, 2, 3] has had a profound impact
on Condensed Matter Physics. With fast computers, excellent numerical
algorithms, and good basis sets for expanding the resulting one-particle
equations, it has been possible to reliably predict many properties of
materials from first-principles, such as the correct ground-state crystal
structure and good values for the internal atomic positions and lattice
parameters by minimizing the total energy of the band structure.

Nonetheless, despite the excellent success of the local density approximation
(LDA) band-structure (LDA-BS) theory in this respect (note that by the term
LDA we also include the gradient corrected or GGA versions of this theory),
there have been many well known examples of its failures as well, such as band
gaps in semiconductors that are about 50–80% smaller than the experimentally
measured values and an incorrect description of many aspects of the electronic
structure of strongly correlated electron systems such as mixed-valence,
transition-metal oxides, heavy fermions, and high-temperature superconductors.
These failures can usually either be attributed to the use of a local
potential to represent exchange, which is actually non-local, or more
importantly to not adequately treating the many-body electronic correlations.
Most involve experiments aimed at some type of spectroscopic or excited-state
properties of the electronic quasiparticles such as photoemission or the
presence or absence of band gaps.

To remedy these problems, many new techniques have been developed,
particularly with respect to strongly correlated electron systems. These
methods usually add a model Hubbard-Hamiltonian term to the band-structure
Hamiltonian, and hence have the generic form:

$\displaystyle\hat{H}$ $\displaystyle=$
$\displaystyle\sum_{\bm{k},ilm_{l}\sigma,i^{\prime}l^{\prime}m_{l^{\prime}}\sigma^{\prime}}[H^{0}(\bm{k})]_{ilm_{l}\sigma,i^{\prime}l^{\prime}m_{l^{\prime}}\sigma^{\prime}}\hat{c}_{\bm{k}ilm_{l}\sigma}^{\dagger}\hat{c}_{\bm{k}i^{\prime}l^{\prime}m_{l^{\prime}}\sigma^{\prime}}$
(2) $\displaystyle+$ $\displaystyle\frac{1}{2}\sum_{i,m\sigma\neq
m^{\prime}\sigma^{\prime}}U_{im\sigma,im^{\prime}\sigma^{\prime}}\hat{n}_{im\sigma}\hat{n}_{im^{\prime}\sigma^{\prime}}+\hat{V}_{DC}.$

The first term is an LDA one-electron band-structure Hamiltonian summed over
Bloch vectors $\bm{k}$ and with orbitals at lattice sites $i$, orbital
momentum $l$, azimuthal quantum number $m_{l}$, and spin $\sigma$. The second
term is a Hubbard-$U$ term that only applies to the $f$ orbitals, which we
will use as a prototype for the strongly correlated orbitals (in many of the
materials mentioned above, these orbitals are actually $d$ orbitals). To
simplify the discussion we use an orthogonalized form of the Hamiltonian that
ignores any potential overlap integrals between the orbitals. The term
$V_{DC}$ is usually called the “double-counting” correction term (for a fairly
complete discussion of this term in the literature, see Ref. [4], and
references therein). We will refer to this Hamiltonian as a Hubbard-$U$ band-
structure (HU-BS) Hamiltonian. Different methods solve this Hubbard-$U$ term
to various degrees of sophistication.

The techniques involving the HU-BS Hamiltonian have been considered by many to
be a revolutionary new devlopment in electronic structure theory, especially
for strongly correlated electronic systems. A recent review article by Held on
dynamical mean-field theory (DMFT)[5], for example, ends with a claim that is
commonly echoed in many places, viz., that “the advances in electronic
structure calculations through DMFT put our ability to predict physical
quantities of such strongly correlated materials onto a similar level as
conventional electronic structure calculations for weakly correlated
materials—at last.”

Given such optimism in the field, we feel that it is timely to review these
new HU-BS methods from the perspective of basic electronic structure theory.
Because the mathematical details of the methods have already been heavily
reviewed many times recently, we will focus this review primarily on more
fundamental aspects. Given the history of electronic structure methods and
what we know about the underlying theory, what is the role and usefullness of
HU-BS approaches? How predictive are they in practice? How much can we trust
the results of such theories and how optimistic can we be that they represent
the revolutionary breakthrough ascribed to them? In addition, how seriously
should we view the development of even more sophisticated methods based on
this approach, especially in light of considerations involving the underlying
foundations of the starting Hamiltonian upon which these sophisticated
mathematical tools are employed?

Finally, to be clear about the focus of this review, we note that we will
ignore electron-phonon and other vibrational aspects of electronic structure
in this paper, as well as pairing and superconductivity, and will only
consider the case where all of the atoms are at static positions within the
unit cell of a periodic solid.

##  2 Failures of Band Structure

Before turning to specific aspects of the HU-BS methods, it is useful to begin
by reviewing the failures of conventional LDA-BS methods that motivate the
search for improvements. By understanding what has gone wrong, we will gain
insight into what the HU-BS methods are attempting to achieve.

A brief catalog of typical failures include: (1) BS predictions of metallic
materials that are experimentally known to be insulating (e.g., CoO and FeO),
(2) absence of magnetism for materials that are magnetic (e.g., for many
undoped high-temperature superconducting oxides) and vice versa (such as Pu),
(3) band gaps that are much too small compared with experiment (e.g., for many
semiconductors), (4) electronic specific heats that are drastically too small
(e.g., for heavy fermion materials), (5) missing peaks at the Fermi energy
(e.g., Kondo-like peaks), and (6) missing satellite spectra (e.g., as occurs
in Ni). More examples can no doubt be found, but this list suffices for our
purposes.

When examining this list, it becomes clear that many of the problems listed
have to do with the spectral properties of the electronic structure of a
material. However, as explained carefully in the early classic papers[1, 2] on
LDA, this type of theory is designed to minimize the total ground-state energy
of the electrons in a material as a functional of the spatial distribution of
the number density of electrons. Thus, the eigenvalues of the Kohn-Sham
equations[2] were never supposed to represent the actual quasiparticle
spectrum of electrons. Nonetheless, because the eigenvalues often, in fact,
are a reasonably good representation of the spectral properties measured in
experiments, this identification is commonly made in practice. Hence, although
everyone admits that this has no justification, most attempts to improve BS
theory are actually attempts to make corrections to the eigenvalue spectra to
bring it into agreement with various spectroscopies that probe the
quasiparticle properties of the materials, such as optical and photoemission
spectra. Even the metal versus insulator problem involves this issue, since
this distinction depends upon knowing the quasiparticle spectral distribution
as a function of energy.

From this very basic point of view, one could strongly question why band-
structure theory should be used for any spectral property, since there is no
formal justification for such an application! So, in this respect, a correct
starting point for a HU-BS description should actually begin with an
explanation of what spectral features an LDA-BS description can be expected to
accurately predict, and what many-body modifications need to be made to
improve this description. If LDA bands are to be used for the quasiparticle
description of the non-$f$ electrons in this approach, it is important that
this part of the theory should be placed on a firmer foundation. As far as we
know, this has never been done in any satisfactory way.

In order to examine this question, the best approach is probably to consider
the GW approximation[6, 7]. Such a theory is developed in a Green’s function
formalism, which is necessary in order to calculate spectral properties. The
one-shot GW approximation can be written as an RPA-like correction to any one-
particle Hamiltonian, such as, for example, an LDA band-structure Hamiltonian.
GW theory has a formal derivation, and it is very clear what physics it
includes and what it does not include.

In this type of approach, one could ignore the original derivation of LDA
theory, and simply treat the Kohn-Sham equations as an approximate one-
electron representation of the electronic Hamiltonian. The Green’s function
for this Hamiltonian can be used to calculate spectral properties, and, if
desired, to lowest-order in the screened Coulomb interaction, these results
can then be approximately corrected to provide a better many-body theory of
the electronic structure. Framed in this way, one can ask if better one-
electron Hamiltonians than LDA would provide a better starting point for
spectral properties. In fact, for example, ideas based on GW theory for such
an improved Hamiltonian has been proposed by van Schilfgaarde, Kotani, and
coworkers[8]. Of course, better many-body corrections would then need to be
added to any such one-electron approach.

Historically, electronic-structure methods have forked into two paths. The
beginnings of this division were seen even 40 years ago[7]: “on the one hand,
we have had the enormous wealth of energy band calculations which have had
tremendous success in explaining the properties of specific solids, but in
which the connection with first principles is not always apparent. On the
other hand, we have seen the spectacular progress of many-body theory applied
to the solid state, which has given a number of new results, although often of
a rather general and formal nature, such as to provide the justification and a
formal basis for a one-electron theory.” In today’s perspective, a very large
effort has gone into improving the first-principles local-density
approximations in order to provide the best possible one-electron theory of
electronic structure, with the main focus on the accuracy of the total energy
functional. The advantage of this “fork” is that such theories are usually
first-principles (i.e., parameter free) and provide a detailed calculation of
specific wave functions and their spatial distribution with respect to the
actual crystal structure of the material, and also include the atomic number
and core electrons of the relevant atoms. It is also usually possible to find
the optimal atomic locations by energy minimizations. The weakness of these
types of theories is their poor treatment of the many-body and quasiparticle
aspects of the electron-electron interaction. On the other hand, approaches in
the second fork have attempted to focus on this many-body character, albeit in
the form of simplified model Hamiltonians such as the Hubbard or Anderson
Hamiltonians, which can then be solved by a variety of sophisticated many-body
techniques involving various levels of approximation.

More recently, these two “forks” have been merged into unified approaches,
e.g., LDA+U (see, for example, Ref. [9] and references therein) or dynamical
mean-field theory, DMFT (see, for example, Refs. [10, 11, 5] and references
therein), that are believed to include the best aspects of both types of
approaches. These are the HU-BS methods mentioned above. In order for theory
to provide a proper guidance or context for various experiments on different
types of materials, it is essential to retain the details about the types of
atoms, their orbital character, and the atomic locations of the atoms in the
unit cell. Otherwise, the calculations often become generic and less useful.
This is included in the BS part of the theory. On the other hand, many
materials clearly exhibit important many-body effects that must be treated
with more sophistication than LDA-BS methods. This is treated by many-body
methods applied to the model Hubbard term in the theory. Because these HU-BS
theories have been “built” on the BS Hamiltonian, in the literature and at
scientific conferences, one commonly finds that many of these calculations
have been de facto considered as quasi first-principles methods. It is the
purpose of this paper to counter this prevailing assumption and to provide a
proper context for these new techniques.

As mentioned above, the types of electronic-structure calculations that we
will consider, in particular, are all based upon adding an additional
Hubbard-U term (or one of its variants) to the band-structure Hamiltonian and
then solving the resulting many-body problem to some level of treatment. When
examining such approaches, the critical question to ask is what such “hybrid”
approaches mean, or how one should understand them. How first principles are
such approaches and do they provide an adequate treatment of the electronic
structure? This type of discussion rarely occurs in the literature, but yet is
crucial if the field is to properly advance.

##  3 What is a good band structure; what does a band structure measure?

Since HU-BS methods are designed to correct band-structure calculations and to
make them more realistic (i.e., to have better agreement with experiment), it
is useful to review what we may mean by a good band structure or what a band
structure actually measures. In this regard, we can begin by reminding
ourselves what goes into a band-structure and what quantities result. The
input to a band structure are the atomic positions and types of atoms (e.g.,
Cu or Si) within a unit cell. The band-structure method then involves
generating a one-electron Hamiltonian and calculating the electronic wave
functions and energy eigenvalues. It also provides a total energy, and number
density (or charge density) and spin density as a function of position. From
the energy eigenvalues the density of states can be calculated.

The total energy as a function of unit cell dimensions and atomic positions is
very useful since changes in the total energy can provide forces on atoms that
can be used in molecular dynamics programs, for example, and can provide
energy differences between different crystal structures. A good band structure
could be defined in terms of how well it calculates this total energy.
However, this is not the main focus of this article. We are more concerned
with quasiparticles, spectral properties, and the energy distribution of
electrons. These come from the energy eigenvalues or dispersion relations
(energy eigenvalues as a function of the k-vector in the Brillouin zone).

To answer the question of what a good band structure is and how we can
experimentally confirm such a band structure, one has to first ask first what
are the fundamental intrinsic mathematical formulations of the electronic
structure of a solid and secondly how various experimental spectroscopies are
related to this formulation. About 40 years ago, Hedin and Lundqvist wrote a
very significant review article[7] that very clearly delineated the answer to
these questions.

With respect to the first question, the answer surely must be that fundamental
theoretical functions that must be calculated are the one-particle and, more
generally, $n$-particle Green’s functions. The one-particle Green’s function,
for example, provides information on the energy needed to add or remove one
electron from the solid as well as the energy dependent spectral density,
which can be written in terms of the imaginary part of the Green’s function.
The two-particle Green’s function arises in a simple and direct calculation of
the total energy (although this can be reformulated in terms of the one-
particle Green’s function as well), as well as the dielectric and other
response and correlation functions. Many of these are needed to evaluate
neutron scattering and magnetic response functions, for example, such as
magnetic susceptibilities or superconducting pairing. The value of the Green’s
function approach is that it can incorporate simple approximations like one-
electron approaches and yet can be generalized to contain the full many-body
physics of the electronic structure, also including, for example, plasmons or
other collective excitations.

When the one-electron band structure is put into a Green’s function form, the
results are very simple. The imaginary part of the Green’s function is just a
sum over delta functions at the energies of the different eigenvalues. Because
the band-structure is a one-electron theory each electron acts independently
and excitations involve only differences between the various energy
eigenvalues with no correction effects. Because the quasiparticle spectra are
just a series of delta functions, the lifetime of each quasiparticle is
infinite (there is no broadening of the spectral function by lifetime
effects). Also, because of the independent particle approximation, there are
no collective excitations.

For this reason, the predicted spectrum of the band-structure is simply a
series of sharp quasiparticle excitations (band energies as a function of
$\mathbf{k}$). Corrections to this spectrum could come in two possible forms:
(1) single-electron corrections that would shift the energy eigenvalues as a
function of $\mathbf{k}$, and (2) addition of a frequency (and
$\mathbf{k}$-dependent) self-energy that could also shift the effective
quasiparticle energies (through the real part of the self energy evaluated at
the quasiparticle energy) as well as provide quasiparticle lifetimes (through
the imaginary part of the self energy evaluated at the quasiparticle energy),
and other-excited state effects such as satellite features at other energies.
As we will see below, mean-field Hubbard model theories are examples of the
first type of correction, and dynamic many-body theor es like DMFT the latter
type.

To understand how “good” this band structure is requires experimental
verification of the spectral properties or other ways of evaluating the
Green’s function that band-structure predicts. This is far from an easy task
and in general involves correcting raw experimental data for a variety of
matrix elements, and other surface and experimental effects (for example,
secondary electrons and experimental resolution, etc.). These issues are
discussed later in this article (see Section 8). Here, it should only be noted
that many of these correction effects are often not carefully taken into
account and our knowledge of the “experimental” spectral functions are
probably not very good for most materials.

Finally, since HU-BS methods only correct the “strongly correlated” orbitals
and leave the other (usually $s$, $p$, and some $d$) orbitals unchanged, the
question of how well conventional band-structure theory applies to the
spectroscopic properties of these more extended orbitals is actually very
important and should be studied much more systematically than has been done up
to now.

##  4 The Hubbard term

There are several important features about the HU-BS Hamiltonian that must be
emphasized. First, the band-structure part of the Hamiltonian identifies
specific orbitals and various hybridizations that provide a realistic
description of the underlying electronic structure and take into account the
correct underlying crystal structure. In addition, such calculations are
first-principles and involve no adjustable parameters. Secondly, the Hubbard
term requires knowing the occupation numbers of the “correlated” $f$ orbitals
(as mentioned above, we use the convention that $f$ orbitals will be the
correlated orbitals in this paper).

It should be pointed out that this “hybrid” Hamiltonian has no derivation. It
is written down based on an intuitive understanding of the electronic
structure. The two terms are simply added together with no formal
justification. The connection between the two terms comes through the $f$
occupation numbers in the Hubbard term, which are assumed to be the same
orbitals as the $f$ orbitals of the underlying band-structure (the first term
of the Hamiltonian). Hence the many-body treatment, which will only be applied
to the second or Hubbard term involves a projection of the Bloch states onto
the $f$ orbitals.

The main assumption made by theories that involve adding a Hubbard $U$ term is
that band-structure calculations treat the Coulomb repulsion between electrons
at the mean-field level and that a more sophisticated many-body treatment is
necessary to handle strong electronic correlation effects. Thus the Hubbard
$U$ term is reintroduced in a simplified way so that a proper many-body
treatment can be performed on this term. To avoid double counting, the mean-
field evaluation of this term is subtracted out in the belief that this
removes the same amount of Coulomb repulsion from the band-structure part of
the Hamiltonian. Hence, many-body effects are then included at some level of
sophistication while mean-field effects are cancelled out.

It is important to consider whether these assumptions make any sense. We
believe that in fact they are seriously flawed. For example, the Hubbard $U$
term is strongly screened and appears nowhere in the original Hamiltonian,
which directly treats the explicit unscreened Coulomb repulsion. In addition,
LDA calculations include a local exchange-correlation potential and hence
involve more than a mean-field (or Hartree) treatment of Coulomb repulsion. A
more straightforward approach would be to do a Hartree calculation of the
electronic structure and then add an unscreened Hubbard $U$ term upon which to
do the many-body treatment (including screening). This would be a disaster and
such a theoretical approach would lead to enormous errors in the electronic
structure.

One useful way to assess the validity of adding this term to the BS
Hamiltonian is to take the local limit of this theory. It is often asserted
that HU-BS methods become more exact as the correlated orbital becomes more
localized. An extreme version of localization is to consider the isolated
atom. For example, it should be possible to do both LDA+$U$ and DMFT for an
isolated atom. In addition, the constrained LDA methods for calculating the
effective $U$ should be very easy!

If this were to be done, the results would probably be very poor indeed.
Certainly the $U$ would not have the large screening of the solid and would
most likely revert back to the 20–30 eV characteristic of the unscreened
Coulomb integrals. In addition, most of the multiplet structure of the atom
would be missing, unless it involved only direct $f$-$f$ multiplets and hence
perhaps was specifically taken into account by the Hubbard-$U$ term.

This example is actually a very useful illustration of how dangerous it is to
assume that the HU-BS Hamiltonian is a good Hamiltonian to describe the
overall electronic structure of a system. It directly demonstrates how
strongly the HU-BS method truncates the original exact Hamiltonian and
therefore how severe this approximation is. Is it possible to really assume
that the screening of a solid can kill off so many aspects of the electronic
structure that such a simplified Hamiltonian as in the HU-BS method is
justified? Thus, it shows very clearly how drastically we have reduced the
actual complexity of the electronic structure when applying HU-BS methods.
Obviously, one must take the results from such theories with many misgivings.
In fact, as we argue elsewhere in this review, it only makes sense to give up
on the notion that these types of calculations are first principles in any
sense of this word, and that they can only reflect a convenient way to modify
the spectral weight of the band-structure predictions so as to better fit and
interpret experimental data.

##  5 Mean-Field Corrections

Mean-field corrections to the original band-structure through the use of the
Hubbard term (i.e., LDA+$U$; see, for example, Ref. [9] and references
therein) provides an important illustration of how the addition of model
Hamiltonian terms modifies and affects the original band structure. These
applications are especially simple in that they do not change the one-electron
character of the Hamiltonian and hence can be solved simply and accurately. In
effect, they are simply a slightly different band-structure than the LDA
starting point.

In a Hubbard framework, these modifications are written in terms of the
occupation operator of specific orbitals (the “strongly correlated” orbitals).
Hence they all have the same generic form:

$H_{MF}=\sum_{im\sigma}V_{im\sigma}\hat{n}_{im\sigma}$

where $V_{im\sigma}$ is a function of the occupation numbers of the $f$
orbitals on the same site $i$, and $\hat{n}_{im\sigma}$ is the number operator
for the $f$ orbital $im\sigma$. In the mean-field approximation this is
usually a linear function of the occupation numbers

$V_{im\sigma}=V_{im\sigma}^{0}+\sum_{m^{\prime}\sigma^{\prime}}U_{im\sigma,im^{\prime}\sigma^{\prime}}n_{im^{\prime}\sigma^{\prime}},$

where $V_{im\sigma}^{0}$ and $U_{im\sigma,im^{\prime}\sigma^{\prime}}$ are
numerical constants, and $n_{im^{\prime}\sigma^{\prime}}$ are the occupation
numbers of the $f$ orbitals (which have to be solved self-consistently in the
course of the calculation). This approach can, of course, also be generalized
for nearest-neighbor or more distant Coulomb-like interactions.

The first point to note about these relationships is that they depend on the
number operator of the correlated orbitals. Hence they are orbital-dependent
interactions and require a projection of the electronic states onto the number
density on these orbitals in order to specify the interaction. Thus they
depend specifically on the basis set that is used. Intuitively, they are meant
to be intra-atomic corrections, so that one prefers that these orbitals look
as atomic-like as possible. There are actually two different choices that can
be made in this regard. Since many BS methods involve muffin-tin basis sets
that have specific numerical wave functions for each type of orbital angular
momentum, one could project these occupation numbers onto an occupation number
for only these parts of the wavefunctions. As the wavefunctions in a solid
extend both into the interstitial region and other atomic spheres, such
occupation numbers would always be less than one when projected onto the
radial wave function of any specific sphere. Alternatively, one can view these
atom-centered basis functions to be Wannier functions centered on each site,
and to use maximally localized Wannier functions so that the portion of each
Wannier function has as much atomic-like character as possible on the relevant
atomic center. Each of these choices has some drawback. The first choice is
somewhat ill-determined since it involves the way the wavefunctions are
normalized, and the second choice puts parts of the Wannier function into the
interstitial region and onto the $s$, $p$, and $d$ orbitals of other atoms and
hence loses some of the intra-atomic character that is being corrected for.
Both choices depend on the size of the muffin-tin radii and how much of the
wavefunctions are localized within a given sphere. Either choice is only
likely to be somewhat satisfactory for very localized or atomic-like
wavefunctions and to become less well defined as the wave functions become
more diffuse. In practice, there is some interplay between the value of the
projection used and the parameters of the model Hamiltonian term that is used.
For example, if the method chosen for the projections leads to smaller
occupation numbers, one can correct for this by increasing the values used for
the mean-field (i.e., the Hubbard-$U$) parameters.

The first correction factor ($V_{im\sigma}^{0}$) is usually described as a
“double counting” term and simply shifts the bare energy level of that
specific $f$ orbital up or down in energy. It can be used to precisely place
the energy of any given orbital wherever it needs to be in order to achieve
good agreement with experiment. If it is independent of the z-projection of
the orbital ($m$), it simply shifts all the $f$ orbitals up or down. An
alternative way of viewing this correction is as a way of modifying the
occupation number of any orbital. By shifting their energy up or down, one
controls how much of the orbital is occupied. For example, this type of
interaction is identical to that which is used as the Lagrange parameter in
constrained calculations of the Hubbard $U$.

In the literature, different choices are made for this first correction factor
for different “flavors” of mean-field theories. Since the “double counting” is
actually an illusion (one is actually not adding and substracting terms from
the original starting Hamiltonian), the only satisfactory way to choose which
method one wants to employ (or perhaps to add even a different constant shift
of the orbitals) is to compare with experiment. Otherwise there is no
fundamental physics argument to choose one method in preference to the other.

The second term in mean-field theories (involving
$U_{im\sigma,im^{\prime}\sigma^{\prime}}$) is an orbital polarization term. It
causes different $f$ orbitals to shift in terms of their relative energies
depending on the specific occupations of each orbital and on the values of the
coefficients $U$ that are chosen (especially if they are positive or
negative). Given this functional form, any polarization that is desired in
order to fit experiment can be forced upon the band-structure solution if the
proper coefficients are chosen to do this.

In addition to these effects, the orbital dependence of these Hubbard terms
also makes it possible to include non-local exchange effects, since orbital-
dependent interactions can be used to represent a non-local function. For
example, in pseudopotential theories an $l$-dependent potential is often added
to represent the non-local character of the pseudopotentials. Since the
starting LDA potentials use a local exchange potential, the Hubbard terms can
be a way of correcting the LDA band structure for non-local exchange. In the
quasi-particle self-consistent method screened non-local exchange interactions
coming from the GW approximation are included as orbital-dependent potentials
to correct the one-electron band structure[8]. Similarly, it has long been
known that the LDA-BS method suffers from self-interaction errors, and the
LDA+U may be viewed as a method to remove self-interactions. In Hartree-Fock
theory, for example, the self-Coulomb and self-exchange interactions exactly
cancel, but once the exchange interaction is described in terms of a local
potential, this cancellation is no longer exact. The screening of the Hubbard
U parameters may then be justified by the fact that part of the self-exchange
is being removed by the local potential.

##  6 Dynamical Corrections

The mean-field treatments considered in the previous sections are basically
different variations on band-structure calculations. All are one-electron
theories and, at best, simply modify the disperson relations of the bands
(energy versus $\mathbf{k}$). However, as was well explained in the early
classic paper by Hedin and Lundqvist[7], the exact Green’s function for the
electronic structure contains a significant frequency dependence in its self
energy. Since a band-structure calculation is a static approximation for the
electronic structure, it has no frequency dependence, and completely misses
this structure. Hence, if one is going to correct band-structure theory in
order to provide a more realistic electronic structure, it is essential to
consider how to incorporate self-energy effects.

The GW approximation, as its name suggests, automatically generates a self-
energy that is proportional to a Green’s function times a screened Coulomb
energy. Although this self energy is the lowest order term in an expansion in
the screened Coulomb energy, it still incorporates some important features
that more sophisticated treatments will need to include. For example, for
weakly correlated systems it maintains the quasiparticle structure inherent in
the band structure. Hence the spectral function often has a strong peak at the
quasiparticle energy. The energy of this peak can be considered to represent
the corrected band-dispersion relations and the width provides a lifetime for
the quasiparticle. It can also correct the size of band-gaps in semiconductors
and open gaps in systems that otherwise would be metallic within LDA band-
structure theory (although sometimes this requires using LDA+U or other
theories to first create a gapped electronic structure as the starting point
for a one-shot GW calculation). Because it incorporates a non-local screened
exchange term, it can also provide the type of corrections that traditionally
have come from Hartree-Fock-like theories, for example, such as are often
added by LDA+U approaches. Hence it can account for some of the modifications
discussed in the previous section on mean-field approaches.

Although simpler many-body approaches can be incorporated into HU-BS
approaches[12, 13, 14], the state-of-the-art methods are now almost
exclusively DMFT. This involves a non-perturbative many-body solution of the
Hubbard term that is performed by mapping the original problem onto a single-
impurity Anderson model (SIAM) and solving the SIAM as exactly as possible. It
requires a projection onto the strongly correlated $f$ orbitals. The method
produces a self-energy coming from the $f$ orbitals only. These can generate
satellite spectra (lower Hubbard bands) as well as Kondo-like peaks at the
Fermi energy and large specific heat enhancements. They will also provide an
electronic lifetime for states that have a significant $f$ character. However,
these lifetimes only come from the $f$-electron self energy, while the other
$s$, $p$, and $d$ electrons have no self-energy or lifetimes and are the
original starting band-structure dispersions. Such theories are useful if
electronic correlations dominate the “interesting” parts of the electronic
structure. The Hubbard-$U$ parameter must either be estimated from constrained
Hubbard-$U$ calculations or fit to experiment. Although much success has been
claimed for these types of theories, the experimental verification is often
qualitative. The critical assumption of the single-site DMFT is that the self
energy of the correlated electron states is $\mathbf{k}$ independent.

One very serious issue with DMFT approaches is the “solver” for the SIAM
equation in the theory. At the present time, many different solvers are used.
Most SIAM solvers, whether from iterative perturbation theory or non-crossing
approaches, etc., have large uncertainties in the correctness of the many-body
solutions they provide. Only the quantum Monte Carlo and numerical
renormalization group solvers are exact. However, even for these, despite new
algorithmic advances such as continuous time quantum Monte Carlo techniques,
there is considerable uncertainty about the quality of their results. For
example, the quantum Monte Carlo methods requires an analytic continuation
from the imaginary frequencies that are calculated by the method to the real
frequencies needed for physical properties. Even with new and sophisticated
techniques for accomplishing this analytic continuation such as those
involving maximum entropy, the real frequency results are very sensitive to
small changes in the imaginary frequency results leading to concerns of large
errors. Also, the Monte Carlo solvers are most accurate at high temperatures
and become increasingly untrustworthy at low temperatures where most of the
interesting correlation physics lies. Overall, from the point of view of the
SIAM solvers, at this time one has to strongly question the accuracy of most
DMFT calculations. In particular, it is clear that different solvers will give
different results, as emphasized at the beginning of Sec. 5 of Held’s DMFT
review.[5]

In general many-body theories must be added to band-structure methods if the
correct electronic structure is to be produced. However, self-energy effects
need to be generalized to correct the non-correlated orbitals as well as the
correlated orbitals. All of the quasiparticles (except those exactly at the
Fermi surface) have finite lifetimes and are likely to require corrections to
their dispersion relations relative to the LDA starting point. In addition,
plasmon, lower Hubbard band, and other non-quasi-particle features will in
general be present in the electronic structure. Such effects are not included
in the LDA band structures.

##  7 Is the HU-BS approach a real electronic structure method?

At this point, based on the previous discussion, it is useful to summarize our
review of the content of the electronic structure implicit in the HU-BS
methods. The most striking comment that can be made on this method is the
starting Hamiltonian itself, Eq. (2). Compared with the exact Hamiltonian, Eq.
(1), it is clear that such a drastic simplification has been made that the
credibility of the HU-BS Hamiltonian cannot be taken at face value but must be
carefully assessed. Exactly what has been done?

From the form of Eq. (2) and the fact that the Hubbard term is a model term
whereas the first term is an attempt at a first-principles description of the
electronic structure, it is reasonable to interpret this Hamiltonian in terms
of its most fundamental part, the band-structure Hamiltonian, and a correction
term, the Hubbard term. In addition it is commonly assumed that the double-
counting term is just the same term but treated in the same mean-field way as
the local-density approximation, and thus one is essentially adding or
subtracting the same effect in order to do a more exact treatment of the most
difficult part of the physics. However, this is not really credible. Neither
the Hubbard term nor the double-counting term exist in the original
Hamiltonian. They simply represent an “ansatz” that has been inserted by hand.
Thus, they can only be sensibly understood as a “correction” to the band-
structure Hamiltonian. These terms are a means by which to include additional
many-body physics that was left out when the rather drastic approximations
needed to formulate the LDA Hamiltonian were made. Hence they make it possible
to build in new features such as satellite peaks and to adjust the
quasiparticle spectra of the band structure.

To approach Hubbard $U$ theories in this spirit provides new flexibility and
should make it easier to resolve certain controversies that often arise, such
as which LDA+U theory is the best approximation. For example, once it is
realized that double-counting is not an issue, one can focus more on what
electronic-structure effects have been left out of the band-structure approach
and what “model” terms could best correct for these effects with a better
many-body treatment. In fact, one could question, for example, whether other
expressions that are different from the Hubbard $U$ term would lead to better
corrections or whether one should instead add corrections to the self-energy
of the electronic Green’s function instead of adding additional terms to the
Hamiltonian.

An important consideration is whether the HU-BS approach can actually work.
How do we know what physics is left out, and why do we believe that the model
term is the right correction factor? Finally, can we actually do the many-body
physics in a sufficiently correct way to believe that we have significantly
improved our understanding of the electronic structure? A bad treatment might
actually lead us to a worse description. Also, an important aspect of this
approach is that we need to include parameters in the theory in order to mask
our ignorance of the real many-body microscopic theory that we are at present
unable to successfully attack. What are the implications of being forced into
a parameterized approach?

Before delving into such matters, however, it is useful to examine more
closely the Hubbard-$U$ term again. If $U$ is treated as a matrix and allowed
to depend on the $m$ projection of the $f$ orbitals, this term is exactly of
the same form as the original Coulomb integrals for a fixed basis of $f$
orbitals. In this sense it has the same physics as the Coulomb term for a
fixed (or frozen) atomic basis, although the basis functions are limited to
one type ($f$ orbitals only) and these functions are extremely limited in
scope (a minimal $f$ basis). If used for an atomic calculation, such a limited
basis would give very poor results for treating the electron-electron Coulomb
terms. So, why should we expect an accurate treatment in the solid? We believe
that, in fact, this term does not provide an accurate treatment. The $U$
matrix that is used in the HU-BS approximation is heavily screened. What is
actually going on is that the many-body treatment of the Hubbard-$U$ term is
being used on something that looks like the original Coulomb term. Hence, the
form of the results (the types of peaks and excitations in the Green’s
function) has a frequency dependence and quasiparticle spectra similar to what
a real Coulomb term would generate. By scaling down the Hubbard $U$ one
reduces the strength of this effect while retaining the same functional form
(freqency or spectral dependence). Thus, if the original electronic structure
is missing peaks or features, this is a way of reintroducing them. At the same
time one has a tuning parameter that can be used to fit the peaks in an
experimental spectra. Thus, such a theory provides realistic spectra that can
be fit to experiment, and with which to correct the LDA band structure for
these types of missing features. Because it is not the electronic structure
calculation of any actual electronic-structure Hamiltonian but of a model or
pseudo-Hamiltonian, the accuracy of the results really doesn’t matter. As long
as the right types of peaks or other features that are seen in experiment are
present in the many-body results, the Hubbard-$U$ parameter can be scaled up
or down to fit the experimental peaks or features. Essentially, the HU-BS
approach is just a model solution of a Coulomb-like term, with the final
results scaled and then mixed with some LDA band structure.

In practice, as discussed above, mean-field treatments of the Hubbard-$U$ term
are used to add in Hartree-Fock like atomistic structure into the one-particle
spectra. These can essentially orbitally polarize the correlated $f$ shell of
electrons. They can also account for SIC-like corrections. For dynamical
theories like DMFT, two effects are commonly introduced: (1) an additional
peak in the photoemission (the lower Hubbard-$U$ band), and a narrowing of the
correlated quasi-particle bands. These are all that are required to fit the
experimental data. Besides the Hubbard-$U$ parameter itself, additional
parameters such as the Hubbard-$J$, etc, can be added if the single $U$
parameter is too crude to fit the experimental data. Hence, since there are
plenty of available parameters and such limited data set with which to fit to,
the HU-BS approach is almost certain to be in good agreement with experiment.

This argument could be turned upside down, of course. Is there any
experimental data that show the essential correctness of the HU-BS approach
other than being a simple fitting procedure? We have been unable to find any
such examples, which leads us to the conclusion that HU-BS approaches are
simply ways to add in some crude many-body effects that are left out of the
original band-structure calculations. Similar questions could be framed in
another way. For example, are there any surprises from these types of
calculations that could not have been guessed from the model calculations
alone? How much physics does the band-structure piece of the Hamiltonian add
that is not included in the Hubbard-U term? What new physics has really come
from the merging of these two Hamiltonians? Is there any rigorous confirmation
of DMFT or any other HU-BS approach that goes beyond a fit to some
experimental data?

The ideal approach would, of course, be to start from the exact microscopic
theory of electronic structure and then to make various approximations that
then lead to different levels of sophistication in the solution. This is
similar to the line of theories starting with Hartree and Hartree-Fock
solutions, through various flavors of local-density approximations, and up
through GW-type theories. However, at this point our abilities to calculate
true many-body effects from first principles appears to have hit a dead end,
in the sense that it is unclear how to go further with a tractable theory.

This is, in fact, the driving force for developing Hubbard-$U$-like
approaches. Modern many-body theory has heavily focused on solving simplified
electronic-structure Hamiltonians based on a simple nearest-neighbor tight-
binding treatment of the Hubbard Hamiltonian, often based on a single orbital
per unit cell. By simplifying the electronic structure, it was possible to
focus on the complex mathematical manipulations that are necessary to treat
the many-body aspects of the theory. The price that was paid for this approach
was to lose the connection to the specific types of orbitals, atoms, and their
geometries possessed by real materials. Hence one ended up with “spherical
cow” approximations to the electronic structure of materials that could not
well describe Fermi surface or photoemission details of materials of interest.
To include these material-dependent properties the LDA band-structure was then
added back into the various approaches, with the treatment of the Hubbard $U$
term projected onto the most localized or strongly correlated orbitals. Since
the model calculations were viewed as simplifications of the real electronic
structure, this lead to the conclusion that one had to add and subtract terms
from the band-structure Hamiltonian, leading to the notion of double-counting,
etc.

We believe that the correct many-body treatment of the microscopic electronic
Hamiltonian is still too difficult for current levels of theory. Hence some
simplifications of the many-body effects will require the introduction of
model terms or expression that parameterize corrections of the first-principle
theory. What these additional terms do in practice is to push spectral weight
of the electronic structure away from that calculated by the original band-
structure theory. For example, in some materials, remnants of the original
atomic structure show up as satellite features (often described as lower
Hubbard bands) below the conventional valence band structure or as additional
peaks in the density of states at the Fermi energy (often described in terms
of Kondo effects in many theories). These effects cannot naturally arise in
the one-electron-type approach of band-structure theory. Since they cannot be
calculated from first-principles, one has to add in parameters to the theory
to force the electronic-structure theory to agree with available experimental
data. One can then question how best to correct the original band-structure
theory to force this agreement, and what understanding such a theory provides
about real electronic structure that an exact theory would predict. There are
also questions about the robustness of such corrections. For example, is each
correction materials specific, or can trends be determined for classes of
materials that continuously tune these parameters. Also, is there enough
physics in the “correction terms” to allow one to understand the correct
mechanisms controlling the functionality and many-body properties described by
such theories?

##  8 Experimental Verification

The key to making progress is good experimental data. Since we cannot trust
the current level of theory to accurately predict materials properties,
especially when part of the electronic structure depends upon unknown
parameters, experimental data is necessary to guide theory.

The chief obstacle with respect to experimental data is that most experiments
do not directly measure the fundamental mathematical properties of the
electronic structure, viz., the various Green’s functions and spectral
densities of states. If we need completely different electronic-structure
theories for each specific type of spectroscopy, we will only attain many
random bits of information that do not form a coherent whole. To be useful
there has to be a common ground where all the experimental data converge. This
common ground has to be a fundamental property of the electronic structure and
it must be amenable to theoretical techniques. Hence it makes sense to focus
on spectral and other fundamental properties of the electronic structure.

From this point of view, one has to ask what each type of spectroscopy
measures and how each one can shed light on the various Green’s functions or
their spectral representations. This will depend upon a very clear theoretical
understanding of all of the physical processes that are involved in each
spectroscopy and how to account for these in order to pull out of the raw data
the fundamental information about spectral functions. At the present time,
very little emphasis has been placed on this. Most interpretations of
experimental data are very simple minded and have not changed much in the last
30 years. Spectrometers and the physical and electronic equipment used in the
various techniques has undergone enormous improvements, but this is not the
case for the fundamental theory needed to interpret the data. This is a
situation that desperately cries out for improvement. The best way to advance
our understanding of strongly correlated electron materials, for example, is
to improve our understanding of what each spectroscopy accurately measures
about their properties. This can only be achieved if we have a proper
understanding of the fundamental physics of each spectroscopic method.

While one could discuss many different types of spectroscopy here, probably
the most heavily used spectroscopies that most directly measure electronic
spectral densities involve interactions of photons with matter, such as
optical spectroscopy and photoelectron spectroscopy. In each case, the
fundamental process involves the absorption of a photon by exciting an
electron from an occupied state of the solid to an unoccupied state. It is
useful to do at least a brief exploratory discussion of how these types of
experiments can be related to the fundamental electronic stucture. Here we
will only discuss photoelectron spectoscopy as a prototype for the types of
discussion that need to be more generally employed.

The simplest theories of photoemission (see, for example, the recent review
paper on the cuprate superconductors, Ref. [15], that cites many earlier
review papers, as well as the standard book on the subject, Ref. [16]) use the
three-step model developed in the early days by Spicer and coworkers. This
treats the photoemission process as involving: (1) an electronic excitation of
the system by a photoelectron, (2) the transport of the photoelectron to the
surface, and (3) the escape of the photoelectron through the surface to the
vacuum where it is detected. Even this very simplified model already hints at
how complicated photoelectron spectroscopy really is. Not only excitation
processes need to be described, but electron transport and detection as well.
Also, the surface clearly must play an important role.

If we just focus on the first process, the excitation of the electrons by the
photon, even this is not simply related to the spectral function of the
desired one-particle Green’s function. Clearly, this must involve the
transition from an occupied electronic level to an unoccupied. In a band
picture, one would calculate this from the Golden Rule, and this would involve
occupied electrons of a given band index and $\mathbf{k}$ being excited to a
higher lying band with some matrix element squared (which would also, of
course, have selection rules). This involves a convolution of occupied and
excited states. From this, how is the spectral function $A(\mathbf{k},\omega)$
to be determined?

In practice, the cuprate review article[15] (see their Eq. (12)) recommends
using the sudden approximation and the formula for the observed electron
intensity:

$I(\mathbf{k},\omega)=I_{0}(\mathbf{k},\nu,\mathbf{A})f(\omega)A(\mathbf{k},\omega)$

where $\mathbf{k}=\mathbf{k}_{\parallel}$ is the in-plane momentum, $\omega$
is the electron energy with respect to the Fermi level,
$I_{0}(\mathbf{k},\nu,\mathbf{A})$ is proportional to a squared one-electron
matrix element and therefore depends on the electron momentum as well as the
energy and polarization of the incoming photon, and $f(\omega)$ is the Fermi
function. The function $A(\mathbf{k},\omega)$ is the electron spectral
function.

With even this additional level of simplification, analysis is still not
completely simple. In most experimental papers, it appears as if the angle-
resolved experiments simply track peaks in the observed energy distribution
curves as a function of angle and energy. These peak energies are then plotted
relative to an estimated Fermi energy to produce an “experimental” band
structure. However, what about the matrix elements
$I_{0}(\mathbf{k},\nu,\mathbf{A})$. If these are strongly k or energy
dependent, they could certainly drastically distort apparent band positions.
One also has to question how valid the sudden approximation is (see, for
example, the classic discussion in Ref. [17]). The chief argument for such a
simplified analysis of photoemission is that the results appear somewhat
similar to one-electron band-structure calculations!

In the above formula, one has to question what happened to the unoccupied
states that the photoelectron is excited to? Hüfner’s book suggests using
free-electron band-structure expressions for accounting for this quantity,
which would involve a more complex analysis than given by the above formula.
However, electronic band-structure calculations suggest that there is
significant band-structure effects that strongly distort even fairly high
energy unoccupied electrons away from their free-electron energies. This is
likely to be the case for the relatively low energies used in the UV
photoelectron range, which has the highest precision. What are these
corrections and how much do they change the determination of
$A(\mathbf{k},\omega)$?

Finally, one should consider the effects of the surface and a large number of
other physical processes such as secondary electrons that complicate the
experiment (see, for example, the Hüfner book). Even 35 years ago, this
complexity was recognized as important for understanding the comparison
between band theory and photoemission experiments[18, 19]. However, today,
most of this complexity seems to have been swept under the rug, with simple
peak evaluations trusted as reliable estimates of the spectral functions! It
is certainly incumbent upon the experimentalists to correct their data as
carefully as possible in order to provide the best experimentally determined
spectral function as possible.

To us, one of the most problematical aspects of photoemission is its high
surface sensitive. Besides possible effects in shifting the peak positions in
which quasiparticle energies are based or on the appearance of new surface
electronic-structure peaks, another example of problems with surface sensitive
spectroscopies is the danger of artifical narrowing of strongly correlated
electron bands. Tight-binding theory suggests that the order of magnitude of
the band width is proportional to the number of near neighbors times the
nearest-neighbor hopping matrix element (see, for example, the analysis in
Ref. [18]). At a surface, there are fewer nearest neighbors and bands should
narrow. This is actually observed in LDA-like calculations of surfaces. See,
for example, Ref. [20]. Correlation effects may artificially enhance such
effects, leading to a significant narrowing of bands that is purely a surface
effect. In the HU-BS methods, band narrowing can be caused by increasing the
value of the Hubbard $U$. If this is fit to photoemission that is really
measuring the surface band width, significant errors may be introduced and
misleading conclusions drawn. Perhaps many strongly correlated materials are
far less correlated than they appear, and the band narrowing observed in
experiment is just a measure of enhanced “surface” electronic structure? It is
unknown how surfaces may modify satellite, Kondo, and other many-body
features. Again, it is possible that they greatly enhance such effects.

While it is clear that these types of experiments desperately need a good
theoretical underpinning to aid in the interpretation of the data, on the
other hand, the very fact that such experiments are surface sensitive makes it
very difficult to develop the precise theory needed to interpret them.
Surfaces introduce changes that depend on how they are prepared and are a much
less intrinsic property of a material than bulk electronic structure. Often
the presence of oxygen, hydrogen, or other impurities can significantly modify
the nature of the surface. In addition, there is the possibility of
preferential segregation of bulk impurities to surfaces, particularly if some
heat treatment or annealing has been performed. For a good theory to be
developed, it is necessary to have a very precise knowledge of all of the
atomic positions and types of atoms at a surface, before attempting to account
for the excitation process of the photon, and the transport of the resulting
photoelectron to the surface, emission through the surface, and its
collection. This involves very complex physics, and is quite difficult.
However, until we have a better understanding of the theory of photoemission,
and how the nature of these types of measurements affect the resulting
electronic properties that are measured, it will always be somewhat dangerous
to rely upon such experimental data to tune the parameters of a strongly
correlated material.

In addition to surface sensitivity, lifetime effects can also be problematic,
and may limit precise measurements to energy regions around the Fermi energy.
Usually the lifetime of an occupied electron state increases rapidly as its
energy moves farther below the Fermi energy. In photoemission, this rapidly
washes out the experimentally determined dispersion relations and it becomes
difficult to know what the quasiparticle energies are deep (or even
moderately) below the Fermi energy. Since the electronic lifetimes are an
intrinisic bulk effect, this effect cannot be reduced in any type of
spectroscopy. This can, for example, make it difficult to measure shifts
between the bottom of the $s$ band and the position of the bottom of the $f$
band in actinides, which might be useful to know if one wants to understand
how nonlocal exchange potentials shift localized electronic states relative to
delocalized.

Viewed more broadly, excitation spectra can have also additional effects that
are unrelated to ground-state electronic structure, making it difficult to
know what the intrinsic electronic structure is in the absence of the specific
experimental probe used to measure the electronic structure. A well known
example of this is exciton effects in semiconductors. In an optical probe, the
incoming photon excites an occupied electron to an unoccupied state, creating
an electron-hole pair. The electron and hole repeatedly scatter off of each
other (this is usually calculated by a Bethe-Salpeter equation) and the
resulting excitation lies in the energy gap of the semiconductor. If not
accounted for, this would erroneously lead to a conclusion that the intrinsic
band gap is smaller than it actually is. Other issues are explicit excitation
processes that are different from ground-state electronic structure, such as
shake-up, shake-off, and other multiplet or other intra-atomic processes
involving electronic excitations that occur nearly simultaneoulsy with the
one-elecron process of interest.

Given these experimental difficulties, one can question how well we know the
experimental electronic structure, and whether the many-body corrections that
we are including by fitting to such data is really correct. There is thus a
real need to develop a better theoretical underpinning for the various
experimental techniques that are being used so that we can more reliably
interpret such data.

##  9 Summary

HU-BS methods involve adding a Hubbard model term to an LDA band-structure
Hamiltonian. The Hubbard model term is a static Coulomb interaction for frozen
orbitals with matrix elements that are scaled to fit experiment. A double-
counting term is actually just a way of preventing the average energy of the
correlated orbitals from being pushed too high in energy and should be
considered just another parameter of the theory that fits the correct average
occupation of the correlated orbitals. The LDA band-structure is a model for
the non-correlated orbitals. It replaces the small number of nearest-neighbor
hopping terms of traditional model Hamiltonians with the full complexity of
all of the relevant orbitals of the various atoms in the solid. However, it
suffers from the defect that LDA does not correctly treat the spectral
properties of these orbitals. In particular, non-local exchange and self-
interaction correction effects are improperly treated. The accuracy of the HU-
BS methods cannot be determined very well, because it is difficult to correct
any current spectroscopy sufficiently to accurately measure the intrinsic
spectral functions of the electrons in the solid. In practice, the HU-BS
methods add lower Hubbard band peaks and narrow the band-width of the
correlated states. The linear term can also be used to introduce Hartree-Fock
like structure to open band gaps and orbitally polarize the electrons. Because
these methods use parameters, they are fits to the experimentally observed
spectra (whether these are an accurate measurement of the actual spectral
functions or not) and are not first-principles methods. They should be viewed
as simply more elaborate model calculations to include more orbitals than
traditional Hubbard models, which often only have one or a very small number
of orbitals. Because the LDA term takes care of the non-correlated orbital
interactions, the number of fitting parameters of a traditional Hubbard model
is reduced for these extra orbitals at the price of the loss in accuracy
entailed by the LDA method.

Progress in the future has to involve two aspects. The first is better first-
principles starting points that incorporate more and more of the correct
physics. The better these are and the more physics they incorporate, the fewer
the corrections that need to be made to compare with experiments. Secondly,
better solutions to a variety of strongly interacting models are needed. What
does the frequency dependence of the exact self-energy for these various
models look like? Will they show any surprises, such as additional features in
their frequency dependence? If, for example, the exact theory simply shifts
the lower Hubbard sattelite away from the position of a less accurate theory,
this could be corrected for by simply modifying the strength of the Hubbard
$U$ parameter. What is particularly important here, and which has not been
carefully examined in the past, is the generic functional forms that the many-
body solutions involve. For example, the Hubbard Hamiltonian usually causes
lower and upper Hubbard-band satellite features. What is the functional form
of the frequency response of the self energy that causes such satellite
structure to appear in the spectral response of the many-body system? Could
one parameterize this in such a way as to compare different many-body
solutions and understand how to include model frequency dependent functional
forms in self-energy corrections to the starting band-structure solutions?
Could one model these in a way similar to what is done in Fermi liquid theory?
For example, if one knows that some quadratic or perhaps exponential frequency
dependence must show up in the exact theory, could one simply parameterize
this response to correct the band structure?

Finally, much more work must go into the interpretation of various
spectroscopies if these are to be accurately related to the bulk spectral
functions predicted by theory. Without good experimental bulk spectral
functions, it is impossible to tell how good or poor our current models of
electronic structure are. Without a true first-principles method, the
parameters of the HU-BS models must be fit to experiment. The resulting
electronic structure will heavily depend on the quality of the experimental
data that is fit to.

This work was carried out under the auspices of the National Nuclear Security
Administration of the U.S. Department of Energy at Los Alamos National
Laboratory under Contract No. DE-AC52-06NA25396, and the Center for
Theoretical Natural Science at Aarhus University.

## References

## References

  * [1] P. Hohenberg and W. Kohn.  Phys. Rev., 136:B864, 1965.
  * [2] W. Kohn and L. J. Sham.  Phys. Rev., 140:A1133, 1965.
  * [3] R. O. Jones and O. Gunnarsson.  Rev. Mod. Phys., 61:689, 1989.
  * [4] A. G. Petukhov, I. I. Mazin, L. Chioncel, and A. I. Lichtenstein.  Phys. Rev. B, 67:153106, 2003.
  * [5] K. Held.  Adv. in Physics, 56:829, 2007.
  * [6] L. Hedin.  Phys. Rev., 139:A796, 1965.
  * [7] Lars Hedin and Stig Lundqvist.  Effects of electron-electron and electron-phonon interactions on the one-electron states of solids.  In Frederick Seitz, David Turnbull, and Henry Ehrenreich, editors, Solid State Physics, volume 23, pages 1–181. Academic Press, New York, 1969\.
  * [8] Takao Kotani, Mark van Shilfgaarde, and Sergey V. Faleev.  Phys. Rev. B, 76:165106, 2007.
  * [9] V. I. Anisimov, F. Aryasetiawan, and A. I. Lichtenstein.  J. Phys.: Condens. Matter, 9:767, 1997.
  * [10] A. George, G. Kotliar, W. Krauth, and M. J. Rozenburg.  Rev. Mod. Phys., 68:13, 1996.
  * [11] G. Kotliar, S. Y. Savrasov, K. Haule, V. S. Oudovenko, O. Parcoullet, and C. A. Marianetti.  Rev. Mod. Phys., 78:865, 2006.
  * [12] M. M. Steiner, R. C. Albers, D. J. Scalapino, and L. J. Sham.  Phys. Rev. B, 43:1637, 1991.
  * [13] M. M. Steiner, R. C. Albers, and L. J. Sham.  Phys. Rev. B, 45:13272, 1992.
  * [14] M. M. Steiner, R. C. Albers, and L. J. Sham.  Phys. Rev. Lett., 72:2923, 1991.
  * [15] Andrea Damascelli, Zahid Hussain, and Zhi-Xun Shen.  Rev. Mod. Phys., 75:473, 2003.
  * [16] Stefan Hüffner.  Photoelectron Spectroscopy.  Springer, Berlin, second edition, 1996.
  * [17] Lars Hedin.  J. Phys.: Condens. Matter, 11:R489, 1999.
  * [18] N. Egede Christensen and B. Feuerbacher.  Phys. Rev. B, 10:2349, 1974.
  * [19] B. Feuerbacher and N. Egede Christensen.  Phys. Rev. B, 10:2373, 1974.
  * [20] O. Eriksson, R. C. Albers, and A. M. Boring.  Phys. Rev. Lett., 66:1350, 1991.