% \section{ANALYSIS OF CHEMICAL BONDING}
% \label{sec:bonding}

%------------
%\subsection{Introduction}
%\label{sec:bond-intro}
%------------

%
As discussed in Sec.~\ref{sec:alt-loc}, there is a long tradition
in the chemistry literature of using localized molecular orbitals
\cite{boys-rmp60,foster-rmp60a,foster-rmp60b,edmiston-rmp63,boys-66}
as an appealing and intuitive avenue for investigating the nature
of chemical bonding in molecular systems.  The maximally-localized
Wannier functions (MLWFs) provide the natural generalization of this
concept to the
case of extended or solid-state systems. Since MLWFs are uniquely
defined for the case of insulators and semiconductors, they are
particularly suited to discuss hybridization, covalency, and
ionicity both in crystalline and disordered systems. In addtion,
in the supercell approximation they can be used to describe any
disordered bulk, amorphous, or liquid system \cite{payne-rmp92},
providing a compact description of electronic states in terms of
their Wannier centers, their coordination with other atoms, and
the spatial distribution and symmetry of the MLWFs. As such, they
are often very useful for extracting chemical trends and for allowing
for statistics to be gathered on the nature of bonds (e.g covalent
bonds vs.\ lone pairs), be it in the presence of structural complexity,
as is the case of an amorphous solid, or following the intrinsic
dynamics of a liquid or an unfolding chemical reaction. They also
share the same strengths and weaknesses alluded to in section
\ref{sec:alt-loc}, whereby different localization criteria
can provide qualitatively different representations of chemical
bonds. This arbitrariness seems less common in extended system, and
often some of the most chemically meaningul information comes from
the statistics of bonds as obtained in large-scale simulations,
or in long first-principles molecular dynamics runs.  Finally, localized
orbitals can embody the chemical concept of transferable functional
groups, and thus be used to construct a good approximation for
the electronic-structure of complex systems starting from the
orbitals for the different fragments \cite{hierse-prb94,benoit-prl01,lee-prl05},
as will be discussed in Section \ref{sec:basis}.

%------------
\subsection{Crystalline solids}
\label{sec:bondcrystals}

One of the most notable, albeit qualitative, characteristics of MLWFs
is their ability to highlight the chemical signatures of different band manifolds.
This was realized early on, as is apparent from Fig. \ref{fig:si-gaas}, showing the
isosurfaces for one of the 4 MLWFs in crystalline silicon and gallium arsenide, respectively.
These are obtained from the closed manifold of four valence
bands, yielding four equivalent MLWFs that map into one another under
the space-group symmetry operations of the crystal.\footnote{It should
  be noted that the constructure procedure does not necessarily lead to MLWFs that
  respect the space-group symmetry.  If desired, symmetries can be enforced by
  imposing co-diagonalization of appropriate operators \cite{posternak-prb02} or by using
  projection methods.\cite{ku-prl02,qian-prb08}}
It is clearly apparent that these MLWFs represent the intuitive chemical
concept of a covalent bond, with each MLWF representing the bonding
orbital created by the constructive interference of two atomic $sp^3$
orbitals centered on neighboring atoms. In addition, it can be seen that
in gallium arsenide this covalent bond and its WFC are shifted towards
the more electronegative arsenic atom.  This has been explored further
by \textcite{abu-farsakh-prb07}, to introduce a formal definition of
electronegativity, or rather of a bond-ionicity scale, based on the
deviation of WFCs from their geometrical bond centers.  It is worth
mentioning that these qualitative features of Wannier functions tend
to be robust, and often independent of the details of the method used
to obtained them - maximally localized or not.  For example, similar
results are obtained for covalent conductors
whether one makes use of symmetry considerations
\cite{satpathy-pss88,smirnov-prb01,smirnov-ijqc02,usvyat-ijqc04},
finite-support regions in linear scaling methods \cite{fernandez-prb97,skylaris-prb02},
or projection approaches \cite{stephan-prb00}, or even if the MLWF
algorithm is applied within Hartree-Fock \cite{zicovich-wilson-jcp01}
rather than density-functional theory.

Once conduction bands are included via the disentanglement procedure, results depend
on both the target dimensions of the disentangled manifold and on the states considered
in the procedure (e.g. the ``outer window'').  In this case, it becomes riskier to draw
conclusions from the qualitative features of the MLWFs.
Still, it is easy to see how MLWFs can make the connection between
atomic constituents and solid-state bands, representing a formal derivation of
``atoms in solids.'' That is, it can reveal the atomic-like orbitals
that conceptually lie behind
any tight-binding formulation, but that can now be obtained directly from first principles
according to a well-defined procedure. For crystalline silicon, the four-dimensional
manifold disentangled from the lowest part of the conduction bands gives rise
to four identical antibonding orbitals (see Fig.~\ref{fig:si-anti})
originating from the destructive interference of
two atomic $sp^3$ orbitals centered on neighboring atoms, to be contrasted with
the constructive interference shown in Fig.~\ref{fig:si-gaas} for the valence WFs.
In addition, an eight-dimensional manifold
disentangled from an energy window including both the valence bands and the lowest part of
the conductions bands gives rise to the atomic $sp^3$ tight-binding orbitals of crystalline
silicon (see Fig.~\ref{fig:si-sp3}). These can form the basis of
the construction of Hamiltonians for model systems
(e.g., strongly-correlated) or large-scale nanostructures, as will be discussed
in Chap. \ref{sec:basis}.

\begin{figure}
\begin{center}
\includegraphics[width=7.0cm]{figs/rmp_si_cond}
\end{center}
\caption{\label{fig:si-anti} (Color online) Four-dimensional manifold
(blue circles) disentangled from the full band manifold (black lines) for Si,
together with one of the four antibonding MLWFs that are obtained after
wannierization of this four-dimensional manifold.  (The other three are
equivalent under space-group symmetry operations.)  The frozen inner
window is also indicated.}
\end{figure}

\begin{figure}[b]
\begin{center}
\includegraphics[width=7.0cm]{figs/rmp_si_sp3}
\end{center}
\caption{\label{fig:si-sp3} (Color online)
Eight-dimensional manifold (blue circles)
disentangled from the full band manifold (black lines) for Si,
together with two of the eight atom-centered sp$^3$ MLWFs
that are obtained after wannierization of this manifold. (The other six are
equivalent under space-group symmetry operations.)  The frozen inner
window is also indicated.}
\end{figure}

These considerations also extend to more complex systems. The case
of ferroelectric perovskites was studied relatively early, as by
\textcite{marzari-acp98,baranek-prb01,evarestov-ssc03}, thanks to the
presence of well-separated manifolds of bands \cite{king-smith-prb94}.
A classic example is shown in Fig.~\ref{fig:bto3}, showing for BaTiO$_3$
the three MLWFs per oxygen derived from the nine oxygen $2p$ bands.  While
in the classical ionic picture of perovskites the B-cation (here Ti) is
completely ionized in a 4+ state, the covalent nature of the bond becomes
clearly apparent here, with the MLWFs showing a clear hybridization in
the form of mixed $p-d$ orbitals. Such hybridization is at the origin
of the ferroelectric instability, as argued by \textcite{posternak-prb94}.
The analysis of the MLWF building blocks can also extend to quite different crystal types.
For example, \textcite{cangiani-prb04} discussed the case of TiO$_2$
polytypes, where bonding MLWFs associated with the O $2s/2p$
valence manifold are seen to be similar in the rutile and anatase
form, with the third polytype (brookite) an average between the two
\cite{posternak-prb06}.
Applications to other complex systems can be found, e.g.,
for antiferromagnetic MnO \cite{posternak-prb02} or for
silver halides \cite{evarestov-ijqc04}.

\begin{figure}
\begin{center}
\includegraphics[width=7.0cm]{figs/Wannier_pano}
\end{center}
\caption{\label{fig:bto3} (Color online) The three maximally-localized
Wannier functions derived from the O $2p$ bands of BaTiO$_{3}$,
showing the hybridization with the nominally empty Ti $3d$ orbitals.
The left panel shows the
O$[2p_z]$--Ti$[3d_{z^2}]$ MLWF, the central panel adds the
O$[2p_y]$--Ti$[3d_{yz}]$ MLWF, and the right panel adds to these the
O$[2p_x]$--Ti$[3d_{xz}]$ MLWF.  Adapted from \textcite{marzari-acp98}.}
\end{figure}

% \dvm{Thanks, the caption of Fig.~\ref{fig:bto3} is better now.  I made slight
% further changes.}

%------------
\subsection{Complex and amorphous phases}
%------------

Once the electronic ground state has been decomposed into well-localized
orbitals,
it becomes possible and meaningful to study their spatial
distribution or the distribution of their centers of charge
(the WFCs).  \textcite{silvestrelli-ssc98} were the first to argue
that the WFCs can be a powerful tool for
understanding bonding in low-symmetry cases, representing an insightful
and an economical mapping of the continuous electronic degrees of freedom into
a set of classical descriptors, i.e., the center positions and spreads
of the WFs.

The benefits of this approach become apparent when
studying the properties of disordered systems.
In amorphous solids the
analysis of the microscopic properties is usually based on the coordination
number, i.e., on the number of atoms lying inside a
sphere of a chosen radius $r_c$ centered on the selected atom
($r_c$ is typically inferred
from the first minimum in the pair correlation function).
This purely geometrical analysis is
completely blind to the actual electronic charge distribution,
which ought to be important in any description of chemical bonding.
An analysis of the full charge distribution and bonding in terms of the
Wannier functions, as for example in Fig.~\ref{fig:a-si} for the distorted
tetrahedral network of amorphous silicon,
would be rather complex, albeit useful to characterize
the most common defects \cite{fornari-cms01}.

Instead, just the knowledge of the positions of the WFCs and
their spreads can capture most of
the chemistry in the system and can identify the defects present.
In this approach, the WFCs are treated
as a second species of ``classical particles'' (representing electrons),
and the amorphous solid is treated as a statistical assembly of the two kinds of
particles (ions and WFCs).
Pair-correlation functions can thus be constructed for ions and classical electrons,
leading to the definition of novel bonding criteria based on the locations
of the WFCs.  For the case of amorphous silicon, for example,
the existence of a bond between two ions can be defined by their sharing
a common WFC within a distance that is smaller than
the first minimum of the silicon-WFC pair correlation function.
Following this definition, one can provide a more meaningful definition
of atomic coordination number,
argue for the presence (or absence) of bonds
in defective configurations, and propose specific electronic signatures for
identifying different defects \cite{silvestrelli-ssc98}.

\begin{figure}
\begin{center}
\includegraphics[width=8.2cm]{figs/boron}
\end{center}
\caption{\label{fig:boron} (Color online) Charge densities for
the MLWFs in $\beta$-rombohedral boron. Red isosurfaces correspond
to electron-deficient bonds; blue correspond to fully
occupied bonds.  From \textcite{ogitsu-jacs09}.}
\end{figure}

The ability of Wannier functions to capture the electronic structure of
complex materials has also been demonstrated in the study of
boron allotropes. Boron is almost unique among the elements
in having at least four major crystalline phases -- all stable or
metastable at room temperature and with
complex unit cells of up to 320 atoms -- together with an amorhpous phase.
In their study of $\beta$-rhombohedral boron, \textcite{ogitsu-jacs09}
were able to identify and study the relation between two-center and
three-center bonds and boron vacancies, identifying
the most electron-deficient bonds as the most chemically active.
Examples are shown in Fig.~\ref{fig:boron}.
\textcite{tang-prb10} were also able to study the evolution
of 2D boron sheets as they were made more compact (from hexagonal to
triangular), and showed that the in-plane bonding pattern
of the hexagonal system was preserved, with only minor changes in the
shape and position of the MLWFs.


\begin{figure}
\begin{center}
\includegraphics[width=8.7cm]{figs/nano1}
\end{center}
\caption{\label{fig:nano1} (Color online)
Collapse and amorphization of a Si cluster under
pressure: increasing to 25 GPa (a) and then to 35 GPa (b), and then
back to 5 GPa (c).  Small red ``atoms'' are the Wannier centers.
From \textcite{martonak-cms01}.}
\end{figure}

Besides its application to the study of
disordered networks \cite{meregalli-ssc01,lim-ms02,fitzhenry-jpcm03}, the
above analysis can also be effectively employed
to elucidate the chemical and
electronic properties accompanying structural transformations.
In work on silicon nanoclusters under pressure
\cite{martonak-prl00,molteni-jcp01,martonak-cms01},
the location of the WFCs was monitored during compressive loading (up
to 35 GPa) and unloading.  Some resulting configurations are shown in
Fig.~\ref{fig:nano1}. The analysis of the
``bond angles'' formed by two WFCs and their common Si atom shows
considerable departure from the tetrahedral rule at the transition
pressure.  The MLWFs also become significantly more
delocalized at that pressure,
hinting at a metallization transition similar to the one that occurs in
Si in going from the diamond to the $\beta$-tin structure.

%------------
\subsection{Defects}
%------------

\begin{figure}[b]
\begin{center}
\includegraphics[width=4.5cm]{figs/fourfold}
\end{center}
\caption{\label{fig:fourfold} (Color online) The fourfold coordinated
defect in Si. Si atoms are in green, vacancies in black, and the centers
of the MLWFs in blue. Adapted from \textcite{goedecker-prl02}.}
\end{figure}

Interestingly, the MLWFs analysis can also point to structural
defects that do not otherwise exhibit any significant electronic signature.
\textcite{goedecker-prl02} have predicted
-- entirely from first-principles -- the existence of a new
fourfold-coordinated defect that is stable inside the
Si lattice (see Fig.~\ref{fig:fourfold}).
This defect had not been considered before, but displays by far the lowest
formation energy -- at the DFT
level -- among all native defects in silicon. Inspection of the relevant
``defective'' MLWFs
reveals that their spreads actually remain very close to those
typical of crystalline silicon,
and that the WFCs remain equally shared
between the atoms in a typical covalent arrangement.
These considerations suggest that the electronic
configuration is locally almost indistinguishable from that of
the perfect lattice, making this defect difficult to detect with standard
electronic probes.
Moreover, a low activation energy is required for the self-annihilation
of this defect; this consideration, in combination with the
``stealth'' electronic signature, hints at the reason why such
a defect could have
eluded experimental discovery
despite the fact that Si is one of the best studied materials in the
history of technology.

For the case of the silicon vacancy, MLWFs have
been studied for all the charge states by
\cite{corsetti-prb11}, validating the canonical Watkins model
\cite{watkins-prl74}.  This work also demonstrated
the importance of including the occupied defect levels in the gap 
when constructing the relevant WFs, which are shown in the first
two panels of Fig.~\ref{fig:corsetti}.
%
% \dvm{Thanks, this is better now.  I made very slight further
% changes.}
%
For the doubly charged split-vacancy configuration, 
the ionic relaxation is such that one of the nearest neighbors of the
vacancy site moves halfway towards the vacancy, relocating to the
center of an octahedral cage of silicon atoms. This gives rise to
six defect WFs, each corresponding to a bond between $sp^3d^2$ hybrids on
the central atom and dangling $sp^3$ orbitals on the neighbors,
as shown in the last panel of Fig.~\ref{fig:corsetti}.

\begin{figure}
\begin{center}
\includegraphics[width=7.0cm]{figs/corsetti}
\end{center}
\caption{\label{fig:corsetti} (Color online) Contour-surface plots of 
the MLWFs most strongly associated with a silicon vacancy in bulk
silicon, for different charge states of the vacancy (from left to
right: neutral unrelaxed, neutral relaxed, and doubly negative
relaxed). 
Adapted from \textcite{corsetti-prb11}.}
\end{figure}

%------------
\subsection{Chemical interpretation}
%------------

It should be stressed that a ``chemical'' interpretation of the
MLWFs is most appropriate when they are formed from a unitary
transformation of the occupied subspace.  Whenever unoccupied
states are included, MLWFs are more properly understood as forming
a minimal tight-binding basis, and not necessarily as descriptors
of the bonding.  Nevertheless, these tight-binding states sometimes
conform to our chemical intuition.  For example, referring back
to  Fig.~\ref{fig:graph_pz}) we recall that the band structure of
graphene can be described accurately by disentangling the partially
occupied $\pi$ manifold from the higher free-electron parabolic
bands and the antibonding $sp^2$ bands.  One can then construct
either a minimal basis of one $p_z$ MLWF per carbon, if interested
only in the $\pi/\pi^\star$ manifold around the Fermi energy,
or a slightly larger set with an additional MLWF per covalent
bond, if interested in describing both the partially occupied
$\pi/\pi^\star$ and the fully occupied $\sigma$ manifolds.  In this
latter case, the bond-centered MLWFs come from the constructive
superposition of two $sp^2$ orbitals pointing towards each other
from adjacent carbons \cite{lee-prl05}.

On the contrary, as discussed in Sec.~\ref{sec:disentanglement} and
shown in Fig.~\ref{fig:cu_7},
a very good tight-binding basis for $3d$ metals such as Cu can be
constructed \cite{souza-prb01} with 5 atom-centered $d$-like orbitals and
two $s$-like orbitals in the interstitial positions.  Rather than
reflecting a ``true'' chemical entity,
these represent linear combinations of $sp^3$ orbitals
that interfere constructively at the interstitial sites, thus
providing the additional variational freedom needed to describe
the entire occupied space.
%
% \dvm{``where their lobes give rise to a constructive interference in the
% interstitial positions.''\\
% I thought this fragment is unlcear; not sure what to do with it.\\
% ``hence these $s$-like tight-binding states would
% survive atomic motion poorly, since the constructive interference would
% be rapidly destroyed''\\
% Do we know this, or is this just speculation?}
%
Somewhere in between, it is worth pointing out that
the atom-centered $sp^3$ orbitals typical of group-IV or III-V semiconductors,
that can be obtained in the diamond/zincblend structure by disentangling the
lowest 4 conduction bands, can have a major lobe pointing either to the
center of the bond or in the opposite direction \cite{lee-phd06,wahn-pssbssp06}.
% AAM: 20/12/2011 (post-submission)
%; both give rise to an equally good band structure, and indeed the
%``off-bond'' orbitals are, counter-intuitively, slightly more localized.
For a given spatial cutoff on the tight-binding Hamiltonian
constructed from these MLWFs, it is found that the former give a 
qualitatively much better description of the DFT band structure than the 
latter, despite the counter-intuitive result that the ``off-bond'' 
MLWFs are slightly more localized. 
The reason is that the ``on-bond'' MLWFs have a single dominant 
nearest neighbour interaction along a bond, whereas for the ``off-bond'' 
MLWFs there are a larger number of weaker nearest-neighbour interactions 
that are not directed along the bonds~\cite{corsetti-phd12}.

%------------
\subsection{MLWFs in first-principles molecular dynamics}
\label{sec:aimd}
%------------

The use of MLWFs to characterize electronic bonding in complex system
has been greatly aided by the implementation of efficient and robust
algorithms for maximal localization of the orbitals in the case of
large, and often disordered, supercells in which the Brillouin zone
can be sampled at a single point, usually the zone-center $\Gamma$ point
\cite{silvestrelli-ssc98,silvestrelli-prb99,berghold-prb00,bernasconi-jms01}.
Such efforts and the implementation in widely-available computer codes
have given rise to an extensive literature
dedicated to understanding and monitoring the nature of
bonding in complex and realistic systems under varying thermodynamical
conditions or during a chemical reaction. Such approaches are particularly
useful when combined with molecular-dynamics simulations, and
most applications have taken place within the framework first proposed
by Car and Parrinello \cite{car-prl85}. In fact, specialized algorithms have
been developed to perform on-the-fly Car-Parrinello molecular dynamics in a
Wannier representation \cite{sharma-ijqc03,iftimie-jcp04,wu-prb09}.

First applications
were to systems as diverse as high-pressure ice \cite{bernasconi-prl98},
doped fullerenes \cite{billas-jcp99}, adsorbed organic molecules
\cite{silvestrelli-prb00}, ionic
solids \cite{bernasconi-p02,posternak-prb02} and
the Ziegler-Natta polymerization \cite{boero-jacs00}.
This latter case is a paradigmatic
example of the chemical insight that can be gleaned by following
the WFCs in the course of an first-principles simulation. In the Ziegler-Natta
reaction we have an interconversion of a double carbon bond into
a single bond, and a characteristic agostic interaction between the C-H
bond and the activated metal center. Both become immediately visible once
the WFCs are monitored, greatly aiding the interpretation of the
complex chemical pathways.

Car-Parrinello molecular dynamics is particularly suited to the study of
liquid systems, and applications have been numerous in all areas of physical
chemistry.  Examples include the work of \textcite{
boero-jcp00,
raugei-jcp99,
sullivan-jpca99,
bako-jcp02, bernasconi-jcp04, blumberger-jacs04,
boero-jacs00, bucher-jpcb08, costanzo-jpcb08, dauria-jpca08,
faralli-jpcb06, heuft-jcp05,
ikeda-jcp05, jungwirth-jpca02,
kirchner-jcp04,
krekeler-jcp06, leung-jacs04, lightstone-c05,
lightstone-cpl01, odelius-jpca04, raugei-jcp02a,
salanne-jpcm08, schwegler-cpl01, suzuki-pccp08, tobias-jcp01, vanerp-jcp03,
vuilleumier-jcp01,
kreitmeir-jcp03, saharay-c04, todorova-jctc08}.

Water in particular has been studied extensively, both
at normal conditions \cite{grossman-jcp04,sit-prb07} and in
high- and low-pressure phases at high temperature
\cite{silvestrelli-prl99,silvestrelli-jcp99,
boero-potps00,boero-prl00,boero-jcp01,schwegler-prl01,romero-jcp01}
(a fast dissociation event from one of these simulations is shown in
Fig.~\ref{fig:hip_water}).  Behavior in the presence
of solvated ions \cite{lightstone-cpl01,schwegler-cpl01,tobias-jcp01,
raugei-jcp02a,bako-jcp02} or a hydrated electron
\cite{boero-prl03, boero-jpca07}, or at surfaces and interfaces
\cite{kudin-jacs08,kuo-s04,kuo-jpcb06,kuo-jpcc08,mundy-cr06,salvador-pccp03},
has also been studied.
Moreover, MLWFs have been used to calculate the electronic momentum
density that can be measured in Compton scattering \cite{romero-pssbr00}.
This work elucidated the relation between the anisotropy of the Compton profiles for
water and the nature of hydrogen bonding \cite{romero-jcp01}, and led to
the suggestion that the number of hydrogen bonds present can be directly
extracted from the Compton profiles \cite{sit-prb07}.

\begin{figure}
\begin{center}
\includegraphics[width=8.7cm]{figs/schwegler}
\end{center}
\caption{\label{fig:hip_water} (Color online) Snapshots of a rapid
water-molecule dissociation
under high-temperature (1390 K) and high-pressure (27 GPa) conditions;
one of the MLWFs in the proton-donor molecule is highlighted in blue, and
one of the MLWFs in the proton-acceptor molecule is highlighted in green.
From \textcite{schwegler-prl01}.}
\end{figure}

Even more complex biochemical systems have been investigated,
including wet DNA \cite{gervasio-prl02}, HIV-1 protease
\cite{piana-jacs01}, reverse transcriptase \cite{sulpizi-jpcb00},
phosphate groups (ATP, GTP and ribosomal units) in different environments
\cite{alber-jpcb99,minehardt-bj02,spiegel-jpcb03},
drug-DNA complexes \cite{spiegel-obc06}, and caspases and kinases
\cite{sulpizi-pfag03,sulpizi-jbc01}.  Extensive reviews of first-principles
quantum simulations and molecular dynamics, with discussions of MLWFs in
these contexts, have appeared reviews by \textcite{dovesi-rcc05, kirchner-prsopl07,
tuckerman-jpcb00, tuckerman-jpcm02, vuilleumier-incoll06,
tse-aropc02}, with \textcite{marx-book09} providing a very
comprehensive methodological overview.

Further applications of first-principles molecular dynamics oriented specifically
to extracting information about dipolar properties and dielectric
responses are discussed later in Sec.~\ref{sec:mol}.
