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\begin{document}

\title{The Devil in the Details: A Tutorial Review on Some Undervalued Aspects
of Density Functional Theory Calculations}



\author[1]{Pierpaolo Morgante}%
\author[1,2]{Roberto Peverati}%
\affil[1]{Florida Institute of Technology}%
\affil[2]{IJQC Interactive Papers}%


\vspace{-1em}



  
  \date{March 14, 2025}


\begingroup
\let\center\flushleft
\let\endcenter\endflushleft
\maketitle
\endgroup





\selectlanguage{english}
\begin{abstract}
Density functional theory, or DFT, has become ubiquitous for chemical
applications in research and in education. The exact functional at the
foundation of DFT is unfortunately unknown, and issues arise when
choosing an approximation for a specific application. With this tutorial
review, we tackle the selection problem and many related ones, such as
the choices of a basis set and of an integration grid, that are often
overlooked by occasional practitioners and by more experienced users as
well. We offer a practical approach in the form of a commented notebook
containing 12 experiments that can be run on a simple computer in just a
few hours. We propose this review as a primary source for those who are
willing to include DFT in their everyday research or teaching activities
in a way that reflects the research advances of the field in the last
couple of decades%
\end{abstract}%



\sloppy




\textbf{Introduction}

This Tutorial Review provides a concise introduction to some practical
aspects of running Kohn--Sham density functional theory (KS-DFT)
calculations. It is intended as a guide for non-experts, students,
occasional practitioners, or anyone that wants to start its own
adventures in the ``zoo'' of DFT approximations. It is organized in
three parts. In the first part, we answer a few questions to establish a
necessary (but far from sufficient) foundation. In the second part, we
concentrate on the methodological components and we introduce the
discussion on the choice of the functional and basis set. In the third
part, we deal with some technical issues that are important for the
reliability and reproducibility of calculations. The purpose of this
tutorial is to draw awareness to the details, rather than being a
comprehensive guide for all DFT-related answers. To this extent, it
represents a complementary tutorial to other that have appeared in this
series, in particular to the foundational essay by K. Burke and L. O.
Wagner entitled ``DFT in a Nutshell''.\textsuperscript{1} Because of its
general audience, the technical details of the reported issues have been
simplified to some degree. To make up for it, we provide a broad body of
references where the interested readers can find the important
technicalities of each discussed topic. This review is also accompanied
by a notebook of 12 calculation experiences, which is formatted for
students of an advanced undergraduate or graduate course, but it is
challenging enough to be interesting for more experienced users as well.

\textbf{Foundations}

\textbf{Foundation 1: What is Kohn--Sham DFT?}

Short answer: Kohn-Sham density functional theory (KS-DFT) is the
theoretical framework at the foundation of the majority of modern,
practical applications of density functional theory in chemistry.

Long(er) answer: DFT is an exact theory based on the idea that the
electron density can be used as the basic quantity to calculate all
components of the electronic structure of a system, including its total
electronic energy. A first attempt at using the density as a basic
variable dates back to the 1920s, when Thomas, Fermi, and later Dirac
proposed the first density
functional.\textsuperscript{2--5}Unfortunately, the Thomas--Fermi--Dirac
functional has very limited practical utility for chemistry because it
predicts molecules to be less stable than their separated constituent
atoms.\textsuperscript{6} In 1964, Hohenberg and Kohn established the
existence of a universal, exact energy functional of the
density.\textsuperscript{7} One year later, Kohn and Sham introduced a
useful framework to circumvent the difficulties of calculating the
kinetic energy of the electrons as a functional of the electron density
by means of a basis set expansion.\textsuperscript{8} Their KS-DFT
framework uses orbitals to describe a fictitious system of
non-interacting electrons with the same density as the real, interacting
one, and reorganizes all the differences between the two systems,
together with all the other effects that are non-trivial to calculate,
into the unknown\emph{exchange--correlation} (\emph{xc} ) functional.

For further details on the mathematical foundations we refer the
interested readers to ``DFT in a Nutshell'',\textsuperscript{1} as well
as to some of the main textbooks in the
field,\textsuperscript{9--15}while for research challenges we point them
to recent reviews\textsuperscript{16--23} and references therein.

\textbf{Foundation 2: How many functionals exist?}

KS-DFT is based on the existence of the \emph{exact
exchange-correlation} \emph{functional} ,\textsuperscript{7} i.e. a
mathematical object that returns a number (the \emph{xc} energy) given
the electron density of the system (a three-dimensional function) as
input. Unfortunately, Kohn and Sham did not give any indication on the
form of the exact functional, and to this day, the exact functional
itself is unknown and essentially unknowable.\textsuperscript{21,24} The
practical use of KS-DFT relies on the development of \emph{xc}functional
\emph{approximations} to the exact functional (simply called functionals
from now on).\textsuperscript{16,19,21,22,25,26} More than 300
approximations have been proposed to date,\textsuperscript{27} with the
total number growing substantially every year. According to our own
estimate---shown in \textbf{Figure 1} ---the functionals development
machine is at full production strength, having produced at least 10 new
functionals in each of the last 15 years. If the current trend holds for
the next decades, we will reach a total of 500 functionals right around
2030. At present, there is no apparent sign of slow down.

As the name suggest, an \emph{xc} functional is usually composed of an
exchange part and a correlation part. Some of the functionals have been
developed by optimizing for a specific property, for example, the
BB1K\textsuperscript{28} and BMK\textsuperscript{29} functionals have
been developed for kinetics, while the PBEsol
functional\textsuperscript{30} was introduced for solid state
calculations. Other functionals, like
B3LYP,\textsuperscript{31--34}PBE,\textsuperscript{35}
MN15,\textsuperscript{36} or \selectlanguage{greek}ω\selectlanguage{english}B97M-V,\textsuperscript{37} have been
developed to be applicable to a broader range of problems, at least in
chemistry. In 2001, Perdew and Schmidt introduced the Jacob's Ladder of
density functional approximations,\textsuperscript{38} as an attempt to
classify the functionals according to the ``ingredients'' used in their
definitions. The main idea behind Jacob's Ladder is that functionals
belonging to a higher rung yield more accurate results than functionals
from lower rungs, at the expense of a more complex functional form (and
to some degree, a slightly increased computational cost). The Jacob's
Ladder is composed of five rungs. The first rung corresponds to the
local spin density approximation, or LSDA. These functionals only depend
on the density of the system. On the second rung, we find functionals
that use the density and its gradient, called generalized gradient
approximations (GGAs). The PBE and\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image6/image6}

\end{center}
\end{figure}

\textbf{Figure 1} Estimate on the total number of exchange--correlation
functional approximations created since 1980. The assessment was
obtained by cross-referencing our own functional databases with those
released in the LibXC software library.

PBEsol functionals mentioned above belong to this second rung.
Functionals on the third rung depend on the density, its gradient, and
the Laplacian of the density (the second derivative), or equivalently on
the kinetic energy density \selectlanguage{greek}τ. \selectlanguage{english}These functionals are called\emph{meta}
-generalized gradient approximations (\emph{m} GGAs). Famous\emph{m} GGA
functionals are TPSS\textsuperscript{39} and SCAN\textsuperscript{40} of
Perdew and coworkers, M06-L,\textsuperscript{41}
M11-L,\textsuperscript{42} and MN15-L\textsuperscript{43} of Truhlar and
coworkers, and B97M-V\textsuperscript{44} of Mardirossian and
Head-Gordon. All functionals on the first three rungs depend only on
local quantities and are therefore also called ``local functionals''.
The notation semilocal is sometimes used in the physics literature for
differentiating rung-2 and rung-3 functionals from rung-1 ones. We
prefer to avoid it, since it is mathematically misleading (rung-2 and
rung-3 functionals still depends on variables that are mathematically
``local'', such as the gradient and/or the Laplacian of the density).
The additional ingredient on the fourth rung is a certain percentage of
non-local exact (``Hartree-Fock-like'') exchange. The B3LYP
approximation mentioned above is a hybrid-GGA, while the MN15 and
\selectlanguage{greek}ω\selectlanguage{english}B97M-V functionals are hybrid\emph{m} GGAs. The last rung includes
contributions from the unoccupied orbitals, in either
double-hybrid\textsuperscript{45} (or doubly hybrid\textsuperscript{46})
functionals, or functionals based on RPA\textsuperscript{47--51} or
other advanced techniques.\textsuperscript{52--56} Functionals on both
rung-4 and rung-5 are by definition non-local.

\textbf{Foundation 3: What is covered in this review?}

For the rest of this tutorial review, we will use research articles
published in the past decade to extract some simple representative
calculation that focus on different undervalued aspects of DFT. We
collected all calculations in the form of a series of experiments in a
laboratory notebook provided in the Supporting Information (SI). We
invite the readers to try all experiments on their own and to compare
their answers with the ones provided as a key. All calculations can be
performed on desktop computers or even on a laptop, and they should take
at maximum just a few hours. Mirroring the structure of this essay, we
collected the experiments into three groups, the first dealing with the
foundations of the theory, where we mainly test the knowledge of the
acronyms used in the computational chemistry community. The second group
deals with the choice of the method (where method refers to the
combination of a functional and a basis set), and the third deals with
the technical issues. The three groups have a total of 12 activities,
plus an additional final self-assessment test/questionnaire. The
experiments and their take-home messages are summarized in \textbf{Table
1} .

Given the vastness of the topic, it was necessary for us to make some
choices on the material to either include or exclude. For example, we
have decided to concentrate exclusively on molecular calculations in
gas-phase based on Gaussian basis functions and Kohn--Sham DFT, because
of their importance in chemistry and related fields of molecular
science. We will not consider orbital-free DFT methods,
projector-augmented wave and related methods, and several flavors of
plane-waves and periodic boundary conditions-based KS-DFT calculations,
despite their importance in the solid state and material science
community. For some of those topic, we refer the reader to recent
reviews\textsuperscript{57--67}

Even on the chemical side, we will not discuss some of the important
corollary aspects of molecular calculations, such as solvation models,
calculation of partial atomic charges, use and interpretation of
molecular orbitals, calculations with effective core potentials, as well
as calculations of associated properties (such as prediction of spectra,
polarizabilities, magnetic properties, etc). We left these topics out
because they are not strictly bound to KS-DFT itself, since they equally
apply to most of quantum chemical calculation methods, including
wave-function-theory--based ones. Each of these subjects deserves a
thorough treatment, for which we refer to specific field studies and
reviews, such as Baik et al.\textsuperscript{68} for solvation models.
Finally, for investigating the state-of-the-art in DFT research
development and application, we refer the readers to the recent reviews
and perspectives.\textsuperscript{16,19,22,25,26,36,43,69}

\textbf{Methodological Components.}

\textbf{Component 1: Which functional should I use?}

Given the sheer amount of options available, a key issue in modern
KS-DFT research is the choice of an approximate \emph{xc} functional.
This is not an easy task, as the accuracy and reliability of the results
is dependent on the functional chosen to describe the chemical system.
If one chooses a functional that is not suitable for a given chemical
application, the results will not be reliable (in accordance with the
old computer science adage: garbage in, garbage out). This does not mean
that DFT fails as a whole, but rather that the specific approximation is
inadequate for the chemical problem under investigation. In this
context, we may think of the functional (and not of DFT) as a tool
needed to perform a specific task. If the tool is not adequate, we
should choose a different tool, or a different task. If we need to put a
nail in the wall, we will pick a hammer, regardless of the number of
screwdrivers we have at our disposal. As Kathrin Hopmann recently
advised: ``Do not use DFT for mechanisms it cannot
handle''.\textsuperscript{70} Many topics at the center of the last
decades of DFT research, such as dispersion
corrections,\textsuperscript{71--76} broadly applicable \emph{meta} -GGA
functionals,\textsuperscript{36,37,40,43,44} and functionals for strong
correlation,\textsuperscript{36,42,69,77--81} have been introduced and
studied to provide better and more reliable approximations. Several of
these new advances have been facilitated by the introduction of large
databases of accurate data of broad chemical
interest.\textsuperscript{25,26,36,43,69,82} The use of databases
outside the theoretical chemistry community is encouraged as they are
the primary source for suggestions when choosing the \emph{xc}functional
for a specific application. It is good practice to compare the specific
reaction under investigation with a similar one in a computational
database before choosing a functional. In addition, databases can also
give an estimate of the expected accuracy of the results over a broad
spectrum of chemical properties. The reactions we will analyze in this
tutorial come from three important collections of chemical data, namely
MGCDB84,\textsuperscript{26}GMTKN55,\textsuperscript{25} and Minnesota
2015.\textsuperscript{36,43,69}

\par\null

\textsuperscript{}\textbf{Table 1.} Structure of the laboratory notebook
and corresponding take-home message.\textsuperscript{}

\emph{Experiment:}

\emph{Experiment:}

\emph{Take-home message:}

\emph{Part 1: Foundations}

\emph{Part 1: Foundations}

0

Where can I seek more on DFT and functionals

Classify \emph{xc} functionals

\emph{Part 2: Methodological Components}

\emph{Part 2: Methodological Components}

1

Applying DFT to Organic Chemistry -- Part A

2

Applying DFT to Organic Chemistry -- Part B

Common traps in the selection of \emph{xc} functionals

3

Difficult Cases for DFT

4

Basis Set Incompleteness Error

5

Basis Set Superposition Error

Common traps in the selection of basis sets

6

Review of Modern Basis Sets

7

Why is the B3LYP/6-31G* Level of Theory So Successful?

Limitations of B3LYP/6-31G* and error cancellation

\emph{Part 3: Technical Issues}

\emph{Part 3: Technical Issues}

8

Integration Grids 1: The Argon Dimer

Integration grids

9

Integration Grids 2: The Case of But-2-yne

10

Stability Analysis and Transition Metals

Stability analysis

11

Comparison with Experimental Results

Required corrections to compare with experimental values

12

Final review

Setting up and running simple calculations, but knowing when to seek an
expert

Several simple reactions are used in the first three activities to show
that we cannot rely on chemical intuition alone to guide us in the
choice of a good \emph{xc} functional approximation. Every chemist is
familiar with the concept of conformational barriers, or\emph{cis}
-\emph{trans} isomerizations. These reactions are ``easy'' in the sense
that almost all the functionals that we tried (see SI) get quantitative
results. The reason of such great across-the-board performance is
because the electron densities of the molecules do not change much
throughout the reaction. Even if a functional provides quantitatively
inaccurate densities, it can still provide good results, as long as it
is consistent in the description of the (minimal) changes. This
compensation is called ``error cancellation''. In some cases, such as
these simple reactions, error cancellation works to our
advantage---making the results better than expected---but other times it
might work against us. Unfortunately, error cancellation is not
systematic, and there is no way to understand its behavior \emph{a
priori} . Looking at databases of accurate data can be diagnostic in
this sense, as it can help understanding whether some results are
affected or not.

The second group includes mainly Diels--Alder and isomerizations
reactions. The results for these reactions reveal that some functionals
might be quite inaccurate, even for simple organic reactions. In such
cases, the electron density of the products can be drastically different
than the density of the reactants. Therefore, error cancellation is not
as beneficial as for the molecules of the previous case, and results
must be evaluated carefully.\textsuperscript{83} Other problematic
reactions include molecules that are bound through non-covalent
interactions. As it has been repeatedly
shown,\textsuperscript{21,25,26,84--86} most functionals (especially
older ones) are not capable of describing van der Waals interactions. In
2012, the accurate description of non-covalent interactions was still
listed as a major challenge for DFT.\textsuperscript{87} Many experts in
the field\textsuperscript{25,26,88,89} have recently recommended to use
functionals in conjunction with some form of ``dispersion correction''
to substantially alleviate this issue. The two most popular corrections
are the -D of Grimme and coworkers,\textsuperscript{74,76,90,91} and the
VV10 functional of Vydrov and Van Voorhis.\textsuperscript{75} The -D
approach (in its four different flavors) is very popular, and it comes
at essentially no additional cost. While some functionals---such as the
Minnesota family---have been specifically designed to properly account
for the bonding in the van der Waals region, and therefore they do not
require additional corrections for the majority of cases, adding
dispersion correction to any other functional is in general always
strongly recommended,\textsuperscript{25,26,88,92} and we align with
this recommendation.\textsuperscript{a}

The last group of reactions shows that seemingly simple molecules like
O\textsubscript{3} or CaO can be very challenging for many functionals.
In such situations, we suggest seeking experts' advice for guidance on
what to do. Before concluding this section, we want to underline one
more time that it is not the aim of this tutorial to provide answers on
how to solve all the issues that might arise in DFT calculations, nor it
is our intention to endorse the usage of newer functionals or to
discourage the usage of older approximations. We have shown (see SI for
numerical results) that for several simple cases 30- or
40-year-old\emph{xc} functionals---such as the popular B3LYP
approximation\textsuperscript{31--34}---are perfectly capable of
providing accurate answers. However, it is our purpose to show that
these older \emph{xc} functionals sometimes fail to provide reliable
results, and newer and more advanced methods are indeed available.
Moreover, for relatively unknown systems, we simply cannot predict in
advance whether older \emph{xc} functionals are going to be reliable or
not. In conclusion, the question used as title for this section, ``which
functional should I use?'', is probably the most difficult to
answer.\textsuperscript{93}

\emph{Our final suggestion:} Avoid underestimating the difficulty of
choosing a functional, since currently there is still no valid answer to
the question of which functional works best. Do not pick a functional
just because it is the most readily available and rely on recently
published studies and suggestions from experts.

\textbf{Component 2: Which basis set should I use?}

Choosing a good functional is not the only major component underlying a
successful DFT calculation. The complementary component is the choice of
a suitable basis set. The basis set issue is somehow even more general,
since it is present not only in DFT, but also in all
wave-function-theory--based methods as well, such as in Hartree-Fock
(HF), and in post-HF treatments. However, the relation between a DFT
calculation and its basis set presents unique aspects that are worth
highlighting in this context. Basis sets are usually collected in the
literature into ``families'', with those proposed by Pople and
coworkers,\textsuperscript{94--102} Dunning and
coworkers,\textsuperscript{103--106} Weigend and
Ahlrichs,\textsuperscript{107} and Jensen\textsuperscript{108--111}being
the most popular for electronic structure calculations. Navigating the
families of basis sets might be as intimidating as the choice of the
functional, especially for beginners. While in principle the issue of
basis set is solved by the simple axiom ``the larger, the better'', in
practice, choosing an appropriate basis set always means striking a
compromise between the accuracy of the results and the computational
cost of the calculations. Large molecules quickly become too expensive
with increasing basis set size, and small basis sets are often the only
option for practitioners who want to study large systems with limited
computational capabilities. At the core of the issue is the fact that
every basis set introduces at least two independent sources of error:
one is called basis set incompleteness error
(BSIE),\textsuperscript{112--114} and the other is called basis set
superposition error (BSSE).\textsuperscript{15,115} The magnitude of
both BSIE and BSSE is larger when a small basis set is used (e.g. with
double-\selectlanguage{greek}ζ \selectlanguage{english}basis sets), and diminishes with increasing basis set size. The
BSIE is defined as the difference between the results obtained with (or
extrapolated to) a complete basis set, and those obtained with a small
basis set. We discuss BSIE in experiment 4. The BSSE arises when the
calculated energy of a molecule is\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image7/image7}

\end{center}
\end{figure}

\textbf{Figure 2} General performance of the B3LYP and \selectlanguage{greek}ω\selectlanguage{english}B97M-V
functionals with different basis sets for some of the reactions used in
Experiments 1--3. All values are in kcal mol\textsuperscript{--1}

compared to the energy of two or more constituting fragments at infinite
separation, for example when a binding energy is calculated. The basis
set of the united system and those of the separated fragments are
intrinsically different, because the united system might benefit from an
artificial increase of the basis set size due to the simultaneous
presence of the basis functions of the monomers in the regions where
they overlap. As a consequence of this artificial local increase in
basis set size, the calculated binding energies can be fictitiously
overestimated. BSSE is thoroughly discussed in experiment 5. Finally, we
designed a review experiment on the choice of basis sets (experiment 6)
using proton affinities, electron affinities, and ionization potentials.
As part of this review experiment, we reiterate the general conclusion
that basis sets that are specifically optimized for DFT calculations are
superior to those optimized for wave function theory ones (not
necessarily in terms of accuracy, but mainly in terms of an improved
cost/accuracy ratio). These conclusions are in line with those found in
most of the recent basis sets studies of DFT
calculations.\textsuperscript{116--118} We need to point out, however,
that there is no universal consensus on them, and while a recent study
has shown that the Pople basis sets are not optimal for
DFT,\textsuperscript{116} other authors\textsuperscript{119} have also
pointed out that the Dunning's family still performs satisfactorily.

\emph{Our final suggestion:} Use particular care when choosing a basis
set, so to minimize BSIE and BSSE. Possibly prefer basis sets that have
been demonstrated to perform well for DFT, like the Jensen or Ahlrichs
families. The Ahlrichs' family is our own preferred
choice,\textsuperscript{b} since it has the additional advantage of
being defined for most of the periodic table (all elements up to Rn),
therefore avoiding issues that can arise when mixing basis sets
belonging to different families, most importantly when dealing with
organometallic complexes or heavy elements.\textsuperscript{120}

\textbf{Component 3: What is special about B3LYP/6-31G*?}

B3LYP\textsuperscript{31--34} is undoubtedly the most used and cited
functional,\textsuperscript{121} and it is the \emph{de facto} standard
for calculations involving organic molecules and even for teaching
purposes.\textsuperscript{122--126} B3LYP is the acronym for B88
exchange\textsuperscript{31} with Lee, Yang and Parr
correlation\textsuperscript{32} in the 3-parameter scheme of
Becke.\textsuperscript{33} B3LYP was proposed by Frisch et al. in 1994
for a study involving circular dichroism spectra of different
molecules,\textsuperscript{34} and it is based on a variation of the
B3PW91 functional of Becke.\textsuperscript{31,33,127,128} In B3LYP, the
PW91 correlation functional\textsuperscript{127,128} is replaced by the
LYP functional, but the remaining parts are unchanged (B88 exchange and
the 3-parameter scheme). The 6-31G* basis set is a double-\selectlanguage{greek}ζ \selectlanguage{english}basis set of
the Pople family,\textsuperscript{94--102} and it includes a very
manageable amount of basis functions. This method rose to popularity in
the mid 1990s, but since the mid 2000s many failures have been
documented in the specialized
literature.\textsuperscript{83,85,129--135} Unfortunately though,
B3LYP/6-31G* has become almost a synonym for DFT calculation itself in
much of the broader audience (non-specialistic)
literature.\textsuperscript{136--140} The educational literature is not
exempted, as this method is applied without any formal justification,
nor critical evaluation of the results even for reactions that are known
to be problematic.\textsuperscript{122--126,139,141} We specifically
designed experiment 7 to give first-hand experience on the limitations
of B3LYP/6-31G*. Selected results are reported in \textbf{Figure 2} ,
while full results are given in the SI. Instead of looking for systems
where B3LYP/6-31G* is successful, we encourage practitioners---as others
did before us\textsuperscript{131}---to move away from it and to seek
better methods for potentially problematic cases. As a guide, all other
hybrid GGA functionals are computationally as expensive as B3LYP (they
are on the same rung of Jacob's Ladder), while modern basis sets have
similar number of basis functions as 6-31G*
(def2-SV(P)\textsuperscript{107} has even less).

\textbf{Methodological Components: Final Remarks.}

The main message of this second section can be summarized as follows.

\begin{itemize}
\tightlist
\item
  Always follow recommendations from experts in the field when choosing
  any combination of functional and basis set.
\item
  Avoid picking a functional/basis set only because they are popular.
\item
  The best way to pick a functional is to compare the chemical problem
  under consideration with similar problems collected in databases of
  high-level computational data. Databases can also give an indication
  on the expected accuracy of the results (i.e. estimate the error bars
  associated with a calculation).
\item
  Basis sets are responsible for two types of errors -- BSIE and BSSE.
  Both errors can be alleviated by choosing a suitably large basis set,
  and in some specific situation by using appropriate techniques.
\item
  Do not use B3LYP/6-31G* blindly. It cannot be universally trusted.
\end{itemize}

\textbf{Technical Issues.}

The third part of this review (and notebook, see experiments 8--12 in
the SI) is dedicated to details of DFT calculations and accessory
techniques that are often overlooked. We cover integration grids,
stability analysis, and comparison with experimental data.

\textbf{Issue 1: Should I care about the integration grid?}

When performing calculations using KS-DFT, the \emph{xc} energy is
calculated as an integral over the corresponding \emph{xc} energy
density (which in turn depends on the local quantities mentioned above).
The integration is usually performed on a three-dimensional real-space
grid obtained by partitioning the multi-center integral into atomic
contributions using a nuclear weight function. The nuclear partitioning
scheme of Becke\textsuperscript{142} is used by the vast majority of
software in conjunction with atomic grids that are further separated
into radial and angular parts. Specially constructed quadrature schemes
are usually used for the angular portion, such as the Lebedev
scheme.\textsuperscript{143--146} The radial part is generally treated
with the Euler-Maclaurin scheme.\textsuperscript{147} The
Lebedev--Euler-Maclaurin integration grids are represented by two
numbers---\emph{n} ,\emph{m} ---with \emph{n} denoting the number of
radial points and \emph{m} the number of angular points. For example,
the UltraFine grid in the Gaussian
program\textsuperscript{148}corresponds to a 99,590 grid, but many other
options are available. The quality of the quadrature grid follows the
same simple axiom discussed for basis set: ``the larger, the better''.
However, once again, software developers (and ultimately users) have to
strike a balance between accuracy and computational cost. Many recent
papers have focused on the importance of integration grids for molecular
calculations using DFT.\textsuperscript{149--153} The choice of the
integration grid can have a significant impact on the quality of
electronic energies, as shown for some members of the Minnesota family
of functionals,\textsuperscript{149--151,154} for which a grid of at
least 99 radial and 590 angular points is necessary to converge some
problematic cases (\textbf{Figure 3} shows the dissociation curve for
the argon dimer\textsuperscript{155} as calculated by the
MN15-L\textsuperscript{41} functional with the
def2-QZVPPD\textsuperscript{107} basis set and different integration
grids). Often such grids are not the default that is automatically
chosen by quantum chemistry programs, and the untrained users often
don't change the default setting. In some cases, avoiding the decision
on the integration grid (and hence relying on the default choice of the
selected software) can be as deleterious as choosing an inadequately
small basis set. Unfortunately, though, an untrained user is more blind
to this issue than the more common methodological ones discussed above.
The grid issue does not only affect electronic energies but might also
affect computed thermodynamic quantities such as entropies, Gibbs' free
energies and vibrational frequencies.\textsuperscript{152,156} The main
message of the experiment in this section is that the 99,590 integration
grid---corresponding to the UltraFine grid in the Gaussian program---is
just a minimum requirement that ensures convergence of molecular
calculations.\textsuperscript{149--152,156,157} When reporting computed
results, it is important to indicate the integration grid used to obtain
structures and energy values to make them reproducible.

\textbf{Issue 2: Do I Really Need to Perform Stability Analysis?}

The iterative self-consistent field (SCF) procedure is at the core of
each HF or KS-DFT calculation to optimize the coefficients of the atomic
basis set. The final coefficient obtained from a converged SCF
calculation correspond to a stationary point of the HF or KS energy
functional. For molecules containing only main-group elements, the
coefficients almost always correspond to (usually the lowest) minimum
energy solution, but in principle, there is no formal guarantee that
this behavior is universal.\textsuperscript{158} In fact, for some
problematic molecules---or for poorly chosen initial guesses---the SCF
procedure can finish on an excited state or on a maximum. In these
scenarios, the final wave function is said to be unstable, since it has
a higher energy than the corresponding stable one. In order to make sure
that the lowest-energy SCF solution is reached, we should check its
stability using a stability analysis algorithm, and eventually make sure
to remove the instability before reporting the final result. While in
principle every calculation---even on simple ``high-gap'' cases---can
erroneously converge to an unstable SCF solution, it becomes crucial to
perform stability analysis for systems that have several low lying
states, such as transition
metals.\textsuperscript{159,160}Unfortunately, some of the most popular
quantum chemistry software packages do not use the stability analysis
algorithm as a default procedure for every completed SCF calculation,
while a surprisingly large minority of software even lack such algorithm
altogether. This is a clear indication of the fact that even experienced
users often overlook the issue of the stability of the final solution.
Experiment 10 uses several low gap systems, some of them containing
transition metals, to remind the users of the importance of the
stability analysis.

\textbf{Issue 3: Can I Directly Compare my Calculated Results with
Experimental Ones?}

Calculating the dissociation energy of a bond, while conceptually a
simple task, might results in several unexpected computational issues.
The calculated total atomization energies of small molecules, i.e. the
energy required to take all the atoms in a molecule apart from each
other,\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image8/image8}

\end{center}
\end{figure}

\textbf{Figure 3} Dissociation curve of the argon dimer calculated with
the MN15-L/def2-QZVPPD level of theory,\textsuperscript{41,107} using
four different integration grids: SG1; 75,302; 99,590; 175,974. The
oscillations observed in the curves calculated with the coarser grids
are unphysical artifacts that derive from an insufficient grid
choice.\textsuperscript{154} The energy is reported in kcal
mol\textsuperscript{-1}, while the distance is in \selectlanguage{ngerman}Å.

cannot be directly compared to the respective enthalpies of formation.
This is not a limitation of the computational methods themselves, but
rather a byproduct of the fact that the computational setup is different
than the experimental one. Several corrections that account for such
differences must be applied to the calculated data in order for a
meaningful comparison, as explained by Karton\textsuperscript{161} or by
Truhlar and coworkers.\textsuperscript{162,163} Experiment 11 is
designed to draw attention to these several corrections that are
required to make such comparison.

\textbf{Technical Issues: Final Remarks.}

The main message of this third section can be summarized as follows.

\begin{itemize}
\tightlist
\item
  Pay attention to which integration grid is used for DFT calculations.
  Override the default parameters of the quantum chemistry program and
  possibly choose at least a 99,590 (UltraFine) grid.
\item
  Always question the stability of the final solution, especially when
  dealing with potentially low-gap systems and transition metals.
\item
  Avoid the blind comparison of calculated values with experimental
  results. Particular care must be given to include accurate corrections
  to the former, before they can be compared with the latter.
\end{itemize}

\textbf{Conclusions.}

DFT is becoming the standard method for quantum chemical calculations in
both the teaching and research environments. We used recent research
articles, as well as modern computational research databases, to design
a notebook containing 12 experiment with the goal of answering a few
important questions about issues that might arise when running DFT
calculations. We also stressed the importance of different techniques
that are too often forgotten by students and occasional practitioners of
computational chemistry. Our overall goal is to raise awareness about
the struggle to obtain reliable and accurate results from DFT
calculations, and to fight the misconception that ``if you own a
computer, then you surely can run meaningful quantum chemical
calculations''.

\textbf{Keywords:} Density Functional Theory, DFT, Tutorial, Review,
Quantum Chemistry

Additional Supporting Information may be found in the online version of
this article.

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\textbf{AUTHORS BIOGRAPHIES}\selectlanguage{english}
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Pierpaolo (Pier) Morgante received his Master of Science degree from the
University of Turin, Italy, and a Bachelor of Arts in classical
saxophone from the Conservatory of Turin. He joined Dr. Roberto
Peverati's group in August 2017, where he started working on databases
of computational data, developing functional approximations to tackle
strongly correlated systems, and on applications of DFT to organic
chemistry.\selectlanguage{english}
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Roberto Peverati is a professor of theoretical and computational
chemistry at Florida Institute of Technology. He obtained a Ph.D. in
Chemistry from the University of Zurich, Switzerland, and worked as a
postdoc researcher at the University of Minnesota (Don Truhlar) and
University of California, Berkeley (Martin Head-Gordon). His research
focuses on density functional theory and database development, as well
as application to physical organic chemistry.

\textbf{GRAPHICAL ABSTRACT}\selectlanguage{english}
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Pierpaolo Morgante and Roberto Peverati

The Devil in the Details: A Tutorial Review on Some Undervalued Aspects
of Density Functional Theory Calculations

\emph{Text to accompany the graphical abstract (different than abstract
for main text)} : Max 75 words.

Density functional theory calculations have become ubiquitous in every
area of chemistry. This tutorial review focuses on different issues that
arise from the choice of the method, or from technical details that are
too often overlooked. It is mainly intended for non-experts in the field
or for occasional practitioners, but it can be a good first approach for
students wanting to start their own adventures in the ``zoo'' of
exchange-correlation functional approximations.

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