(a) Initially, the thick, black bond is unscreened, but as (b) the atom enters
the region of influence (light gray ellipses) the bond weakens. (c) The atom
has moved into the close vicinity of the bond (dark gray region), effectively
disabling it while creating two new bonds (red lines). The Baskes screening
functions [376] are defined the aspect ratio of the light gray region
($C_{\text{max}}$) and the dark gray ($C_{\text{min}}$), defining a measure of
bond screen that is independent of the absolute lengths of the bonds in the
system.

The Baskes screening functions [376] were applied to empirical bond-order
potentials independently by Pastewka et al. [378, 25, 401] and Kumagai et al.
[402]. Both groups emphasized that the screened potentials significantly
improved the properties of amorphous carbon modeled with REBO2 or Tersoff-type
potentials. In addition, the screening functions served to overcome the issue
with dissociation of a bond under external stress discussed in Refs. [377,
378, 25, 401]. This enabled modeling of fracture in crystalline and amorphous
carbon systems [403]. Perriot et al. [404] presented slightly different
screening concept requiring the REBO potential to be refitted.

Screening functions can be rationalized as originating from nonorthogonality
in a tight-binding framework. This nonorthogonality leads to an environment-
dependence of the bond-integrals, when the nonorthogonal tight-binding is
“coarse-grained” to an equivalent orthogonal tight-binding model. Nguyen-Manh,
Pettifor and Vitek [405, 406] showed, that the theory of the bond-order
expansion, briefly touched upon in Sect. 5.8, can be used to derive screening
functions from a nonorthogonal tight-binding model. This first-principles
construction lends additional support to Baskes’ screening functions and other
screening approaches, such as empirical environment-dependence introduced in
the context of orthogonal tight-binding shortly after Baskes work [407, 408,
409, 410].

##  10 Summary and perspectives

Eugene Wigner allegedly said: “ _It is nice to know that the computer
understands the problem. But I would like to understand it too._ ” One main
motivation for writing this review was to assist people with similar ambitions
as Wigner. To this end, we summarized our understanding of what properties in
condensed-matter systems can be induced by the functional form of the
potentials used for their description. In our endeavor, we felt compelled to
create much own data and new figures with the purpose to create insight and to
convey trends and differences between potential classes rather than to produce
numbers for a specific system. When doing so, we did our best to embed
anything written into a historical context, which is summed up in Fig. 19.

Figure 19: Selected highlights of the development of interaction potentials.

Yet many times, we could not find appropriate references or quotes, which we
are certain do exist. As one of numerous examples, we found many papers
computing shear and bulk moduli of metals or vacancy-defect and cohesive
energies, but always missed the argument why their respective ratios are
correlated and how they relate to the ratio of melting and boiling
temperature. We expect our discussion to have satisfied Wigner’s desire for
understanding the correlation between these ratios. Despite certainly having
missed well known studies, we did find some old works, which may have been
underappreciated, such as Slater’s paper on the interaction between helium
atoms [84]. As mentioned earlier, Slater derived the exponential repulsion
between atoms with closed valence shell, which promoted Born and Mayer [85] as
well as Buckingham [7] to use this or slightly modified forms for the
repulsion in the potentials now carrying their names. However, Slater also
derived the dispersive coefficient for helium to within 15% accuracy, two
years before London [56] generalized the results to other closed-valence
shells.

Despite the length of this article, we could only scratch the surface of the
large field of interatomic potentials. Many central aspects were not touched
upon. Most importantly, we barely discussed how to adjust parameters, in
particular the pros and cons to fit to experimental or to in-silico data.
There are good reasons to follow the main-stream opinion that the quality of a
potential increases the less empirical the data on which the potential is
parameterized. Computer-generated reference data is much more versatile than
that provided from experiments. Forces on individual atoms can be used for
characteristic bonding situations or rare but important configurations like a
transition state occurring during a chemical reaction or a collective phase
transformation [411]. Moreover, in-silico data does not contain quantum
effects, which frequently need to be accounted for when comparing computer-
generated data to experiments. Describing how to do that properly would have
required us to outline how to approximately subtract the nuclear quantum
effects from experimental data or how to incorporate quantum fluctuations into
the simulations, e.g., through path-integral techniques, or, by encoding their
effect into effective temperature-dependent, many-body potentials, which would
have been beyond the scope of this review.

Thus, there is scarcely any argument to gauge parameters on experiments, if
there was not the small but important detail that experimental data is by and
large more accurate than density-functional theory, which cannot be deemed
exact, as long as the exact functional has not been identified. It could be
argued that we base one theory or potentials with uncontrolled approximations
on another one with uncontrolled approximations, which has trouble to predict
two dislike molecules or clusters separated by a large distance to each
acquire an integer charge [412, 413]. When dismissing empiricism as
fundamentally problematic, one may also keep in mind that one of the greatest
theoretical achievements in chemistry, arguably in all of science, was the
construction of the periodic table by Mendeleev. He even predicted the
existence of unknown elements including some of their physical and chemical
properties with an accuracy that people using potentials or even DFT might
have a hard time to match if they did not know what they had to predict, or,
rather postdict. Moreover, the amount of data that Mendeleev could build on
was noticeably less than what is required in machine learning.

The potentials discussed in this review pertain mostly to situations, in which
bonds can be clearly classified as dispersive, metallic, covalent, or ionic.
For situations, where this simple categorization cannot be made, different
potential classes are combined in a mix-and-match fashion into compound
potentials. Prominent examples are the adaptive intermolecular REBO (AIREBO)
[252] (combining Brenner’s potential with nonbonded Lennard-Jones
interaction), the Streitz-Mintmire potential [414] (combining EAM with charge
transfer), the charge-optimized many-body potential (COMB) [415, 416, 417]
(combining Tersoff’s potential with charge transfer), a merger of REBO with
split-charge equilibration [333, 418], as well as early combinations of
Keating-type with charge-transfer potentials potentials [419, 420]. Of course,
the widely-used ReaxFF potential [421, 12], which merges a bond-order approach
(different in nature than the approaches discussed in this review) with non-
bonded interactions and charge transfer, must also be mentioned.

While compound approaches can be extremely powerful, many of them simply add
different energy terms. This can be problematic even for seemingly simple
alloys or intermetallics formed by elements of large electronegativity
difference. Put simply, negatively charged atoms grow in size while positively
charged atoms shrink. This symmetry breaking between negative and positive
charge is not reflected when simply adding charge equilibration to a (post)
Ducastelle potential. Yet, it is supposedly responsible for why the negatively
charged atoms in intermetallics have the tendency to close pack while positive
atoms occupy interstitial positions, as it happens, for example, for Al2Au,
also called the purple plague: Au atoms form an fcc lattice while Al atoms
assume interstitial positions. Although promising steps toward true compound
potentials have been taken [422], e.g., by augmenting or reducing the valence
density of a neutral atom with a term proportional to its partial charge,
systematically merged potential remain a dream.

Novel paths that are taken with machine learning potentials seem extremely
promising. However, a puzzling question is why machine-learned potentials
outperform parameterized potentials. The claim that they are parameter free or
free of functional constraints is not entirely justified. Many of the local
descriptors are suspiciously close to what is used in potentials, as indeed
they are often “physics-inspired” [352]. However, the big advantage of MLPs is
that they do not make strong assumptions like pair-wise additive repulsion,
which might be one of the most important sources of error in classical
interatomic potentials.

A show-stopping problem central to all potentials is the curse of
dimensionality. Fitting multi-species (or alloy) potential requires a number
of pair-parameters that scales asymptotically as $N_{\textrm{s}}^{2}$ with the
number of atomic species $N_{\textrm{s}}$. The scaling becomes even less
favorable if we need specific parameters for triplets ($\propto
N_{\textrm{s}}^{3}$), quadruplets ($\propto N_{\textrm{s}}^{4}$) and so on,
quickly becoming intractable for a large number of species. The compression of
chemical fingerprints has recently been proposed to circumvent the curse of
dimensionality for MLPs [423]. Using explicit functional forms, it can be
possible to circumnavigate the curse of dimensionality with combining rules.
However, they are only available for few interactions types and may be plagued
with poor transferability.

As a final note, we would like to point out that despite the fact that (with
the exception of bare Coulomb interaction) all potentials discussed here are
local, chemistry can be quite non-local. By non-locality we do not mean the
range of the bare interaction, such as the range of the bond-integrals in a
tight-binding formulation. We mean the non-locality intrinsic in the
diagonalization of the quantum mechanical Hamiltonian. In hydrocarbon
chemistry, the non-locality manifests itself for example in bond conjugation
and in metals through an algebraic decay of the density matrix [424], while in
group 15–17 in the periodic table, it is reflected in the Peierls deformation
causing elemental crystals to reduce from the simple cubic to less symmetric
structures [425]. As another example, carbon chains – also called carbynes –
can exist in a polyynic form of alternating single and triple bonds or a
cumulenic form of repeating double bonds [426, 427, 428]. Which form is chosen
depends on whether the chain is odd or even numbered and how it is terminated.
This crucially affects how they interact with their environment, for example
with oxygen [429, 430]. Such non-local effects even manifest in bulk
materials: Force-locality tests on amorphous carbon by Deringer and Csányi
showed that chemistry in low-density, graphite-like amorphous carbon is much
longer ranged than in denser more diamond-like carbon [355]. Approaches for
incorporating true quantum non-locality into potentials currently do not
appear to exist. Modeling it appears to require new classes of potential,
e.g., the ability of an EAM or MEAM potential to make atoms adjust their
donating charge density in response to the environment in a fashion that
allows for multistability.

We hope that this review was successful in highlighting the incredible
achievements throughout the last century in understanding the bonding of
matter, and molding these insights into simple analytical expressions. The
wide availability of high-accuracy electronic structure calculations and
advances in statistical modeling have moved the field into exciting new
directions. We would also like to add that the wide availability of present-
day interatomic potentials in the form of open-source software, ideally
embedded in a standard database [13] or a standard code [310, 311], is
accelerating quick adoption of potentials into practice — not to mention the
savings in students’ lifetimes, by not having to dissect which of the $50$
parameters just manually copied from printed publication XYZ is missing a $0$
in print. (Yes, we are thinking about our own PhD theses.) Of course,
significant challenges remain, both for traditional fixed-form as well as
machine-learned interatomic potentials, of which we believe the curse of
dimensionality and the coupling of electron transfer and Coulomb interaction
to the electronic bond as most crucial.

## Acknowledgement(s)

We thank Gábor Csányi, Volker L. Deringer, Christian Elsässer, Peter Gumbsch,
Judith Harrison, James Kermode, Pekka Koskinen, Gianpietro Moras, Michael
Moseler, Matous Mrovec, Toon Verstraelen and Michael Walter for many
enlightening discussions over the years. We are further indebted to James
Kermode for comments on the machine learning section of the manuscript as well
as Joshua Weißenfels and Jan Grießer for proofreading and commenting on the
full manuscript. We used gpaw [431] for all DFT calculations shown here that
are not obviously taken from third sources.

## Disclosure statement

No potential conflict of interest was reported by the author(s).

## Nomenclature/Notation

### Abbreviations

ACE | atomic cluster expansion
---|---
ATM | Axilrod-Teller-Muto interaction potential
DFT | density-functional theory
EAM | embedded-atom method
EOS | equation of state
GAP | Gaussian approximation potential
LJ | Lennard-Jones
MEAM | modified EAM
ML | machine learning
MLP | machine-learned potential
QEq | charge equilibration
REBO | reactive empirical bond-order potential
REBO2 | second generation REBO
SQE | split-charge equilibration
SW | Stillinger-Weber
TB, TB$n$M | tight-binding, TB $n$-th order moment expansion

### Symbols

$\alpha$, $\alpha^{\prime}$ | polarizability in SI and atomic units
---|---
$\alpha_{\text{M}}$ | Madelung constant
$\delta_{\alpha\beta}$ | Kronecker delta
$\epsilon$ | Lennard-Jones energy parameter
$\varepsilon_{0}$ | vacuum permittivity
$\varepsilon_{\textrm{r}}$ | dielectric constant
$\varepsilon_{\alpha\beta}$ | element of the Eulerian strain tensor
$\eta_{\alpha\beta}$ | element of the Lagrangian strain tensor
$\rho$ | charge density, number density
$\sigma$ | length scale parameter
$\sigma_{\alpha\beta}$ | element of the Cauchy stress tensor

$A$ | electron affinity
---|---
$B_{n}$ |  $n$-th order virial coefficient
$C_{\alpha\beta\gamma\delta}$ | element of elastic tensor
$C_{ij}$ | element of elastic tensor in Voigt notation
$C_{n}$ | dispersion coefficient of order $n$
$\hat{H}$ | Hamilton operator
$H_{i\alpha j\beta}$ | Hamiltonian integral between orbital $\alpha$ on atom $i$ and orbital $\beta$ on atom $j$
$I$ | ionization energy
$B$ | bulk modulus
$N$ | particle number
$Q_{i}$ | charge of atom $i$
$S_{ij}$ | square of the distance between atoms $i$ and $j$
$U$ | interaction energy
$U_{0}$ | dimer/molecular binding energy
$U_{\text{pa}}$ | potential energy per atom
$U_{\text{pa}}^{\text{eq}}$, $U_{\text{coh}}$ | equilibrium potential energy per atom (cohesive energy)
$U_{\text{pb}}$ | potential energy per bond
$U_{\text{pb}}^{\text{eq}}$ | equilibrium potential energy per bond
$U_{1}$, $U_{1}^{(i)}$ | single-body interaction energy
$U_{2}$, $U_{3}$ | pair and triplet interaction energy
$V$ | volume
$Z_{0}$ | coordination number
$Z_{s}$ | number of atoms in $s$’th nearest-neighbor shell

$a_{n}$ | distance between an atom with a $(n+1)$-nearest neighbor
---|---
$a_{0}^{\textrm{eq}}$ | equilibrium bond length
$a_{\textrm{B}}$ | Bohr radius
e | elementary charge
$f_{\text{c}}$ | cutoff function
$k_{\textrm{B}}T$ | thermal energy
$m$ | mass
$n(\varepsilon)$ | density of states
$n_{i\alpha}(\varepsilon)$ | local density of states of orbital $i\alpha$
$\hat{p}_{i}$ | momentum operator
$\mathbf{p}_{i}$, $p_{i}$ | dipole moment of species $i$ and its magnitude
$p$ | pressure
$\mathbf{q}_{i}$ | descriptor of the environment of atom $i$
$q_{ij}$ | bond charge donated from atom $i$ to atom $j$
$r$, $r_{ij}$ | (pair) distance
$r_{0}$ | equilibrium distance in a diatomic molecule
$r_{\textrm{c}}$ | cutoff radius
$\nu_{\textrm{s}}^{\alpha\beta}$, $\nu_{\textrm{s}}^{\alpha\beta\gamma\delta}$ | second- and fourth-rank shell tensor for the $s$’th nearest-neighbor shell
$v_{\text{pa}}$ | volume per atom

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