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Perspective: the chemical information in Electron Density

Julia Contreras-Garc´ıa, a, , Weitao Yang b

a UPMC Univ Paris 06, CNRS, UMR 7616, Laboratoire de Chimie Th´eorique, case courrier 137,

4 place Jussieu, F-75005, Paris, France and

b Department of Chemistry, Duke University, Durham, North Carolina 27708, United States

(Dated: today)

Abstract

In this Perspective we review the chemical information encoded in electron density, as different ingredi- ents used in semilocal functionals. This information is usually looked at from the functional point of view, the exchange density or the enhancement factor are dissected in terms of the reduced density gradient. But what parts of the molecule do they correspond to? We will look at these quantities in real space, aiming to understand the electronic structure information they encode and providing an insight very much used in the Quantum Chemical Topology (QCT). Generalized Gradient Approximations (GGAs) add information on the presence of chemical interactions, whereas meta-GGAs are able to differentiate between bonding types. By merging these two worlds, we will show new insight into the failures of semilocal functionals along three main errors: fractional charges, fractional spins and non-covalent interactions. We will build on simple models. We will analyze delocalization error in hydrogen chains, showing the ability of QCT to reveal the delocalization error introduced by semilocal functionals. Then we will show how the analysis of localization can help understand fractional spin error in alkali atoms, and how it can be used to correct it. Finally, we will show that the poor description of GGAs of isodesmic reactions in alkanes is due to 1,3-interactions.

PACS numbers:

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I. INTRODUCTION: DFT AND BONDING

The last decades have been characterized by a constantly growing interest in the development of exchange-correlation (xc) functionals, E xc = E x + E c , within the Kohn-Sham density functional theory (DFT) framework. This has lead to a very large number of functionals being proposed.

As a classification scheme, the Jacob ladder was developped, classifying functionals in terms of their explicit ingredients of electron density. The first three rungs are occupied by the so-called semilocal approximations which consist of a single integral for E xc

E xc [ ρ ] =

Z ρ (r) ε xc [ ρ (r)]dr, (1)

and where ε xc , the exchange-correlation energy density per electron per unit volume, is a function of the following functions : 1

• 1 st rung: the electron density, ρ , in the local density approximation (LDA),

• 2 nd rung: ρ and its first derivative, ∇ ρ , in the generalized gradient approximation (GGA), and

• 3 rd rung: ρ , ∇ ρ and the kinetic-energy density and/or ∇ 2 ρ in the meta-GGA approximation.

Each of these steps of the ladder includes chemical information in terms of 3D functions which is also being used in the analysis of chemical bond, within the Quantum Chemical Topology (QCT) framework. 2 QCT analyzes the shape of these functions in real space in order to decode the bonding pattern. The most documented example is the electron density. In an approach known as Quantum Theory of Atoms In Molecule (QTAIM), the shape (critical points, zero-flux gradi- ent surfaces) of the electron density is used to define chemical concepts such as bonds or atoms. 3,4 However, many other functions used in QCT derive from other information used in semilocal Den- sity Functinal Approximations (DFAs). The Electron Localization Function 5–7 and the Localized Orbital Locator 8,9 are some representative examples. Indeed, ingredients in DFAs are probably the biggest source of chemically meaningful functions used in QCT. However, this connection is not usually acknowledged, not even exploited. DFAs provide functions which are studied in depth within the QCT framework. However, the chemical insight obtained is rarely employed in DFA development.

In this paper, we will focus on analyzing the similarities between both fields of knowledge, pay-

ing special attention to the use of topological information as a source of insight into DFA behavior.

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Indeed, this information can be used to better understand the content and the failures of current E xc . With this aim in mind, the paper will be divided into two main sections: (1) the chemical content of the functions used in semilocal DFAs and the progressive inclusion of chemical infor- mation along Jacob’s ladder, and (2) the use of chemical content in the understanding of DFAs errors, which can be used to understand DFA, or even to help choosing and improving functionals.

II. THE CHEMICAL CONTENT IN LOCAL FUNCTIONALS

A. 1

st

rung: the electron density

The earliest and simplest spin-density functional for the exchange-correlation energy was the local density approximation (LDA): 10

E xc LDA =

Z ρ (r) ε HEG [ ρ ]dr (2)

where ε HEG = ε HEG [ ρ ] is the exchange-correlation energy per particle of a Homogeneous Electron Gas (HEG) of density ρ (r).

Just as the electron density is the key property of any electronic system from the energy point of view, 10 it is also so from the bonding viewpoint. The main chemical properties of the system may be quantitatively calculated from ρ (r). The analysis of the electron density topology provides valuable information on atomic and bonding property, stability and chemical reactivity.

The shape of the electron density can be understood from the analysis of atomic behavior and its superposition. 11 For an atom, A, the value of the density at the nucleus position, r A , can be related to its nuclear charge, Z A :

r lim → r

A

δ

δ r + 2Z A

ρ ¯ (r) = 0, (3)

where ¯ ρ (r) is the spherical average of ρ (r). When going away from the nucleus, an exponential decay is observed, so that at long distances, the asymptotic decay is related with the first ionization potential, I:

ρ (r) ∝ e 2

2I | r | (4)

Finally, very far away from the nucleus, the electron density falls to zero:

ρ (r → ∞) = 0 (5)

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FIG. 1: a) Electron density on a plane that contains the LiH molecule (left). b) Electron density on the plane perpendicular to the LiH bond that crosses the BCP (right).

When a molecule is formed, the general shape of the electron density is dictated by the position of the nuclei (which is described in the external potential). The density shows cusps at the nuclei positions (related to their Z). Fig. 1a highlights the position of Li and H atoms in the LiH molecule as cusps of the electron density.

As can be seen from Fig. 1a, all the main features of the electron density are already dictated by the position of atoms Li and H. The relaxation (SCF) cycle leads to a perturbation of this atomic densities. This feature is commonly used in crystallorgraphic studies, where the initial non-relaxed density is used as a first approximation to fit structure factors.

The existence of bonding interactions can also be derived directly from the electron density.

First order saddle points of the electron density (also known as bond critical points - BCPs) appear in between bonded atoms. 3,4 At the BCPs, the electron density is minimal along the bonding line, and maximal across the perpendicular plane (Fig. 1b).

The electron density cusps enable identifying the position, R A , and nature of the atoms in the system ,Z A . Zero flux gradient surfaces can be used to define non-overlapping atomic regions and their properties (e.g. charge, volume). Finally, we can derive the bonds present in the system from the position of the first order saddle points.

B. 2

nd

rung: the reduced density gradient

The semilocal information on the electron density was used to introduce the electron density

gradient by a second-order gradient expansion (GEA) 12 and generalized gradient approximation

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(GGA), 13–15

E xc GGA = E x LDA + ∑ Z F(s) ρ 4/3 (r)dr (6)

where F (s) is the enhancement factor, a function of s(r) for a given spin with

s(r) = 1 C s

| ∇ ∇ ∇ ρ (r) |

ρ (r) 4/3 , (7)

C s = 2(3 π 2 ) 1/3 and the 4/3 exponent of the density ensures that s(r) is a dimensionless quantity.

The electron density gradient is at the heart of the QTAIM approach, since it becomes zero at points where interactions are expected. More specifically, as we will see below, properties of s(r) have been investigated in depth in the process of developing increasingly accurate functionals due to its deep relationship with the chemical region of the molecule (shells, bonds). 16–21 Indeed, the reduced density gradient can be related to local density inhomogeneities:

• It assumes large values in the exponentially-decaying density tails far from the nuclei, where the density denominator approaches zero more rapidly than the gradient numerator.

Small values of s(r) occur throughout for the HEG, close to bonding regions, due to the pres- ence of BCPs, and in Lewis pairs, due to its relationship with the One Electron Potential. 22 The effect of these inhomogeneities on the reduced density gradient is especially easy to visualize when s(r) is plotted versus the density (Fig. 2a). Assuming a Slater-type behavior, it is easy to show that graphs of s(r) versus ρ (r) assume the form s(r) = a ρ (r) 1/3 , where a is the Slater exponent. When there is overlap between atomic orbitals, s(r) goes to zero and a spike appears in the s( ρ ) diagram.

It can be shown that the information introduced by the reduced density gradient is provides a scaled approximation to the kinetic energy density: 22

t bose (r) = t w (r) t HEG (r) = 5

6 s(r) 2 (8)

where t w is the von Weizs¨acker kinetic energy density, resulting from assuming a Bosonic behavior (all electrons in the same orbital), and t HEG is the HEG value (the Thomas-Fermi kinetic energy density):

t w (r) = 1 8

(∇ ρ (r)) 2

ρ (r) (9)

t HEG (r) = 3

10 (3 π 2 ) 2/3 ρ (r) 5/3 (10)

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Examples are shown in Fig. 2. Fig. 2a shows s( ρ ) for water and the water dimer where the the inhomogeneity from the monomer to the dimer due to the presence of non-covalent interactions (hydrogen bond) is highlighted at small densities. 23,24 Fig. 2b shows the ability of s to reveal electron pairing in real space in N 2 . 19,25,26 Nitrogen-nitrogen bond and N lone pairs appear as isosurfaces of s.

a)

b)

FIG. 2: (a) s(ρ) for water and water dimer. Peaks for covalent and non-covalent interactions are labelled.

(b) Chemical features (bond, lone pairs) of N

2

as revealed by s(r) =0.33. Cores are not shown due to the grid size.

. ,

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In real space approaches, the reduced density gradient is most commonly used under the topo- logical approach for revealing non-covalent interactions due to the difficulty to identify them by other means. For example, the functions in section II C 2 only reveal localized electrons, thus they do not reveal non-covalent features. Furthermore, these interactions are also more difficult to identify from the geometry alone. Whereas covalent radii are very well defined, non-covalent interactions cover a much wider range of radii and angles. Moreover, in many cases they are not localized, so many atoms contribute to the same interaction. The reduced density gradient enables to cast these situations in a very intuitive way.

Fig. 3a-b highlights the ability of the reduced density gradient to provide intuitive insight into localized vs. delocalized interactions non-covalent interactions in benzene crystal. Whereas lo- calized CH-C interactions appear as a small atom-to-atom surface, delocalized CH- π interactions highlight the interaction of the hydrogen atom with the whole neighboring π system. The T-shape interaction is highlighted in Fig. 3b. Localization indexes are not able to reveal these features, only the electron density topology (section II A) is able to reveal non covalent interactions, but always as atom-to-atom (localized) visualizations (Fig. 3c).

FIG. 3: a) s(r) for the intermolecular interactions in benzene crystal: localized CH-C interaction (small isosurface) and delocalized CH-π interaction (bigger surface). b) lateral view of the T-shape CH-π interac- tion. c) bond paths (QTAIM). Reprinted with permission from Chem. Eur. J. 18, 15523. Copyright 2012 American Chemical Society.

Thus, using all the ingredients of the second rung ( ρ , ∇ ρ and s) provides very rich information

on the chemical interactions of the system, including non-covalent regions. All of them appear as

peaks of the reduced density gradient (s → 0).

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C. 3

rd

rung

Functionals in the third rung of Jacob’s ladder, known as meta-GGAs, include information on the Laplacian and/or kinetic energy density of the calculated Kohn-Sham orbitals. We have seen that the first rung includes information of atoms, the second rung includes information of interactions. What is the information conveyed by the Laplacian and the kinetic energy density?

These are functions that highlight electron pairing. We have seen that the reduced density gradient is also able to reveal electron pairing, however, the 3D functions in the 3 rd rung make a distinction, they are focused only on electron localization, they do not reveal non-covalent interactions. We will see in the following section how this can be related to the improved performance of meta- GGAs.

1. The Laplacian

We have seen that within the QTAIM approach, the position of nuclei and bonds are associated with critical points of the density. Within this theory, electron pairing (atomic shells, lone pairs) is recovered from the Laplacian, ∇ 2 ρ (r), which is one of the optional pieces of information added to the 3 rd rung functionals.

Assuming the electron density is given by ρ (r) = Ae −ζ r , the Laplacian is given by:

2 ρ (r) = Ne −ζr

ζ 2 2 ζ r

(11) The main limiting behaviors are as follows:

• It tends to − ∞ at the nucleus

• Since the electron density falls exponentially to zero at long distances, so does the Laplacian.

• In between these two asymptotes, the Laplacian in an isolated atom shows maxima and min- ima with spherical symmetry. These recover the number of shells. This chemical description in atomic shells is also maintained in ionic compounds.

The meaning of the Laplacian can also be understood from its relationship with the electron

density gradient through the divergence theorem. According to the divergence theorem, 27 the sign

of the Laplacian of a scalar function indicates whether a net flux of the gradient of the scalar

is entering (negative sign) or leaving (positive sign) an infinitesimal volume centered on a given

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a) b)

FIG. 4: − ∇

2

ρ (r) maps across the plane that contains the nuclei a) MgO (Mg on the left, O on the right). b) C

2

.

point. Hence the sign of the Laplacian of the density informs us on whether the the electron density is concentrating/compressing or diluting/expanding at the point. This yields to the Laplacian as a descriptor of charge depletion(accumulation) for a positive(negative) Laplacian. 28 Indeed, the local expression of the virial theorem: 3,29

1

4 ∇ 2 ρ (r) = 2t(r) + V (r), (12)

where

t(r) = 1 2 ∑

i

(∇ φ i (r) · ∇ φ i (r)) (13) is the positive definite kinetic energy density calculated from the molecular orbitals φ i , and V (r) is the potential energy density. Since t(r) is positive everywhere and V (r) is negative in most chemical cases regions (e.g. it is asymptotically positive in dianions), the theorem states that the sign of ∇ 2 ρ (r) determines which energy contribution, potential or kinetic, is in local excess relative to their average virial ratio of minus two. Most generally, a negative Laplacian reveals that the potential energy is in local excess (shared-electron situation), while a positive Laplacian denotes that the kinetic energy is locally prevailing (closed-shell situation).

Fig. 4a highlights the ability of the Laplacian to reveal the shells in MgO: two for Mg 2+ (K and L) and same for O 2 . In covalent compounds, the valence becomes united (see C 2 in Fig. 4b).

This different picture enables to distinguish bonding types in terms of the sign of the Laplacian. 28 It can be seen that in the first case (MgO) this leads to ∇ 2 ρ > 0 in the interatomic region, whereas

2 ρ < 0 for C 2 , as expected from their respective bonding type.

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2. Kinetic energy density

Due to the ability of the first order density matrix to reveal non local effects, kinetic energy densities are a good option to introduce extra bonding information and non-local information.

Since the kinetic energy density is not uniquely defined, the positive definite option, t, is usually used in bonding analyses, since it enables an easier interpretation.

3. LOL

The Localized Orbital Locator (LOL), v, 8 was introduced by Schmider and Becke as an intuitive measure of the relative speed of electrons as the kinetic energy density, t, scaled by the HEG value:

v(r) = t HEG (r)

t(r) , (14)

The scaling enables eliminating of the ρ 5/3 dependence of the kinetic energy density which would hide valence characteristics due to the core greater densities. 30,31

Since v(r) is bounded by zero from below, but has no upper boundary, a Lorentzian mapping is usually used, so that the index, LOL runs from 0 to 1:

LOL(r) = 1

1 + v(r) (15)

Its name is Localized Orbitals Locator because its 3D shape is determined by the ability of a point r to be described by a localized orbital:

At the positions of the stationary points of localized orbitals, v is driven to small values (LOL → 0).

In regions dominated by the overlap of localized orbitals, v attains large values (LOL → 1).

This leads to a distribution of maxima and minima that reveals the atomic shell structure. 8,9 Morevoer, given its connection with localization, it has been shown to be able to reveal multi- center delocalization when averaged over atomic basins. 32

Fig. 5 (bottom) shows the evolution of LOL along the the N 2 bonding axis (red line). It is easy

to see that the function identifies the position of the cores (marked with vertical lines), the bonding

in between them and the lone pairs at each side.

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FIG. 5: Scaled reduced density gradient (green) and LOL (red) along internuclear (bottom) and resulting ELF (blue) (top) along N

2

interatomic axis. The zero is set at the BCP, atomic positions highlighted by dashed lines.

4. ELF

The two previously introduced kinetic energy density measures (LOL and s) constitute the chemical information used to construct another very well known topological index, the Electron Localization Function (ELF). 5 Its kernel, χ , according to Savin’s interpretation is given by: 7

χ (r) = t (r) − t w (r)

t HEG (r) = v(r) 1t bose (r) (16) Just like in the case of LOL, the kernel of ELF is re-scaled to provide a bounded function:

ELF (r) = 1

1 + χ (r) 2 (17)

Remember that t bose = 5/6s 2 . Due to the Pauli principle, electrons (Fermions) are faster on average than if they were Bosons. Hence t w is a lower limit for t, and t bose is a lower limit for v 1 . This has been highlighted in Fig. 5(bottom), where t bose (blue line) has been plotted along with LOL (red line). In other words, ELF has been developed to reveal Pauli behavior and its local effects on the kinetic energy.

What does this physical meaning lead to? Since Pauli repulsion leads to electron pairing and localization, the following features are obtained:

The upper limit ELF(r)=1 corresponds to perfect localization. In general, maxima of ELF(r)

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FIG. 6: ELF(r) localization domains for: N

2

( f =0.8).Monosynaptic and disynaptic basins are coloured in cyan and green respectively.

are identified with regions of high electron localization, such as cores, lone pairs, and cova- lent bondings.

• When ELF tends to zero, we are in a boundary region. ELF value at the saddle points has been shown 33,34 to be related to the delocalization between fragments via the overlap of the relevant orbitals involved in the interaction. This will be used in Section III A.

Fig. 6 highlights ELF’s ability to discern localized electrons in N 2 . Surfaces of ELF appear

at the N cores (not shown), the N ≡ N bond (in green) and the lone pairs (in cyan). It looks very

similar to the LOL (even to the s) picture. One may wonder what is the use of introducing a

more complex function to reveal chemical structure. The advantage of ELF is that it is rooted in

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the Pauli principle. Hence, the topology of ELF renders a quantitative Lewis picture of chemical systems. As an example, it has been possible to recover the Aufbau principle for shell filling from ELF integration of atomic shells along the periodic table. 35 This is not possible with LOL nor with s. Its use in meta-GGAs will be highlighted in the next section.

D. So what information is encoded in each rung?

The contributions of the different ingredients of semilocal functionals have of course received a lot of attention in the development of new DFAs. It is common practice to analyze the enhance- ment factor in terms of the reduced density gradient, F(s), as a way to understand its limiting behavior. 19–21,36 Although less traced, the real space perspective has been also proven to be a rich source of information in understanding the way from one rung to another one. Analyses of F (r) and s(r) 19,20,25,37 have lead to the understanding of the behavior of the GGA contribution in atoms and molecules, and even to the development of local hybrid functionals. An interesting example of such approach was introduced by Philipsen and Baerends, 37 who represented LDA and GGA contributions to the cohesive energy on spherical cells along the Cu lattice parameter as the dif- ference between the bulk and isolated atoms. This simple plot enables to see the contributions of the GGA term to the cohesion of the solid and to identify that it comes from the bonding region.

The analysis above (section II B) enables to understand this observation. Whereas the atomic core enhancement factor is constant from the atom to the bulk, important differences appear in s upon cohesion of the solid.

Maybe more interesting is the analysis of the difference between the 2 nd rung and the 3 rd rung.

Since Lewis pairing is already contained to a good approximation in the reduced density gradient, what is the advantage of looking at fermionic kinetic energy densities?

TABLE I: Values of kinetic energy descriptors in typical bonding types: t

bose

(Eq. 8), v

1

(Eq. 14), z = t

W

/t (Ref. 38), χ (Eq. 16)

GGA meta-GGA

Bonding type t

bose

v

1

z = t

W

/t χ

Covalent 0 5s

2

/3 1 0

Metallic 0 1 ≃ 0 ≃ 1

Non-covalent 0 1 ≃ 0 >> 1

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Table I) provides a summary of kinetic energy density relationships used in the construction of functionals. Bosonic contributions are able to identify all electronic structure features, but they all appear as poles of s, so that GGAs are not able to identify one feature from the others. Kinetic energy measures such as LOL or ELF are able to differenciate among bonding types.

Moreover, this approach can even be used to understand failures of a given functional. 38 Meta- GGAs based on either LOL and ELF (e.g. M06L 39 and MGGA MS2 40 ) are able to describe equi- librium non-bonding interactions, whereas functionals resorting to z = t w /t, such as revTPSS, 41 cannot. Looking at table I, it is easy to see that while ELF and LOL are able to differenciate between covalent, metallic and non-covalent bonding at the bonding critical points, z cannot.

Another example of the usefulness of plots in real space is the analysis of the instability in meta-GGAs. As we have seen, most of the reduced variables used in meta-GGAs show important structuring (critical points), some very steep, even divergent. When combined with small densities, spurious oscillations appear in real space that can be easily understood by plotting the functional along interatomic lines. 42

All this information can be used to take one further step. It can be used not just to understand failures, but also for improving functionals. Indeed, meta-GGAs were for a long time considered not to introduce great improvements over GGAs. The above analysis has finally helped enlarge the GGA vs meta-GGA distance. Favoring ELF information over LOL and z due to its chemical content and stability, Perdew and colaborators have recenlty proposed a new functional which makes meta-GGA systematically superior to GGA, able of describing many different bonding types including insulator to metallic transitions. 43,44

III. IDENTIFYING (AND CORRECTING) DFT ERRORS WITH CHEMISTRY DESCRIPTORS

Errors in DFT are difficult to dissect, largely due to the coexistence of various types of errors which affect the overall performance of DFT calculations.

One way to try to understand the different factors is to analyze these errors at the most ba-

sic level in very simple model atoms. These atomic models serve to separate the causes and thus

diagnose the failures independently. Following this approach, Yang and coworkers 45–48 have iden-

tified some basic errors which affect functionals in terms of fractional charges, fractional spins and

dispersion interactions. In order to understand what QCT can do for DFA development, we will

briefly describe each of these errors, analyze the 3D functions used to build them and exploit them

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to gain insight into the isolated errors. This insight will be further used to provide guidance on the choice of functionals, or even, on how to correct them.

A. Delocalization error and ELF

1. The error

Density functional approximations usually underestimate the barriers of chemical reactions, the band gaps of materials and charge transfer excitation energies. All these errors can be traced back to a common root, the delocalization error. 46,47 This error can be understood from the stretching of a very simple molecule, H + 2 . At the dissociation limit, this leads to two H atoms with half an electron each. The estimation of the energy of this system from available local functionals is too long (e.g. LDAs, GGAs, and even some hybrid functionals such as B3LYP). 15,49

This can be easily estimated from ensemble densities: 50 the energy of an atom with respect to the number of electrons is given by straight lines in between integer values. In other words, the energy of a H atom with half an electron, should be the average of an H atom with zero and one electrons. Approximate functionals, instead, lead to a convex behavior: they predict incorrect lower energies for fractional electrons (see Fig. 7). The consequence of this is straight forward:

Density Functional Approximations (DFAs) tend to over-delocalize electrons, to artificially spread them in the molecule. 46

With this simple model it is easy to understand the chemical nature of the above mentioned problems: transition states are systems where bonds are being broken and formed, they are stretched. Hence, the energy predicted by DFAs is much too low, and so are the barriers. The underestimation of band gaps of bulks is also a direct consequence of the delocalization error. 46 It can also be understood within the conceptual DFT framework from the wrong E vs N derivatives of DFAs. 51 Since Hartree-Fock (HF) leads to the opposite ill-behavior (over-localization - see Fig.

7), hybrid functionals may benefit from error cancellation, leading to good band gap predictions

in mid-size gaps. Novel DFAs designed to minimize delocalization error (such as MCY3 52 and

CAM-B3LYP 53 ) lead to significant improvement in the accuracy of predicted thermodynamical

properties.

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FIG. 7: Variation of the energy with respect to the number of electrons for the exact functional, most DFAs ( δ N-convex) and HF( δ N-concave). Reprinted with permission from Phys. Rev. Lett. 100, 146401.

Copyright 2008 American Physical Society.

2. Chemical insight - Visualizing delocalization error with ELF

To systematically examine the effect of delocalization error in more complex systems a series of model chemical systems based on hydrogen chains were designed. Firstly, several 16-hydrogen polymers where put together and lined up head-to-head in an equally displaced manner (Fig. 8a). 46 1D electron density difference map where constructed for 1, 2 and 3 of these units upon addition of an extra electron (Fig. 9). Differences between HF and LDA clearly appear: whereas HF localizes the electron on one unit, LDA deslocalizes it over several units. 46

ELF can be used in order to obtain a closer look at the relationship between the delocalization error and its effect on the electronic structure. For this purpose, (H 2 ) 10 chains with changing intermolecular distance d (see Fig. 8b) were constructed. 54 This distance acts as the parameter tunning delocalization error, while other main errors remain negligible or constant.

The effect of delocalization error on the energetics is assessed by analyzing the averaged de- viation of calculated adhesive energies (ADE), ∆E ad , with respect to highly accurate CCSD(T) data:

∆E ad = E ad DFTE ad CCSD(T )

n (18)

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FIG. 8: (a) Schematic diagram of a model hydrogen polymers. a) Polymer made of H

16

units. 2 units (M=2) are shown. b) Polymer made of 10 H

2

units. The intramolecular H − H bond length is fixed at r = 1.5 bohr, while the intermolecular distance d is varied.

where E ad = n E(H 2 ) − E(H 2n ).

Results for n = 10 are shown in Fig. 10a. At all values of d, the errors follow the ordering:

∆E ad LDA > ∆E ad GGA > ∆E ad B3LYP > ∆E ad

rCAM-B3LYP

(19) There is only one exception: at around d=3.0 bohr BLYP and B3LYP’errors become noticeably smaller, wheras HF remain noticeably big. As will be seen below, the system can be described as composed by H 2 units at this distance, so the lack of correlation in HF becomes more patent (see section III C). Hence, the big admixture with exact exchange at long distances in CAM-B3LYP, worsens the results.

In order to asses the increase in delocalization upon increasing d we have plotted the ELF=0.5

isosurface in Fig. 11. 54 At d = r the chain is encompassed by a continuous isosurface, indicating

that it behaves like a macromolecule, with bonding between H 2 units. As d increases, the ten H 2

units become more apparent and at d >> r, each H 2 unit can be considered as an independent

fragment. Each of these fragments are subject to delocalization error. Hence, one would expect

the error to increase with the distance. However, Fig. 10a shows the opposite trend. It would seem

that the error diminishes upon increasing the distance, opposite to what would have been expected

from the delocalization point of view. This is because the interaction between two neighboring

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FIG. 9: Density difference maps for (H

16

)

M

with HF and LDA. Reprinted with permission from Phys. Rev.

Lett. 100, 146401. Copyright 2008 American Physical Society.

H 2 units is weaker at a larger d. If relative errors are analyzed, ∆E ad DFT / | E ad CCSD(T) | (Fig. 10b), the expected increase in the error appears, indicating that the effect of delocalization error is indeed more prominent for a more diffusive electron distribution.

But is this error reflected in the electronic structure? can we have a glance at the delocalization provided by each functional in this model system? To compare results from each functional, one- dimensional (1D) ELF profiles along the internuclear axis are plotted in Fig. 11.

The polymerized system (d = r) is reflected in very low difference between the ELF value at the maxima and the minima, as expected for a H n system. As the distance in between hydrogen molecules increases (d > r), fragments are reflected in a lowering of ELF at the minima. At around d=3.0 bohr, the system can be understood as a chain of weakly interacting H 2 molecules and this is reflected in ELF ≃ 0 at the minima. 33

Moreover, ELF can also be used to understand the delocalization error along the series. HF

leads to the lowest values of ELF in the minima, highlighting the description with this index of the

system as more localized H 2 units at each step. As d increases and the system becomes fragment

like, ELF from HF calculations approaches CISD results. As pointed out above, the HF error at

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1.50 1.75 2.00 2.25 2.50 2.75 3.00 -8

-4 0 4 8 12

1.50 1.75 2.00 2.25 2.50 2.75 3.00 -0.08 -0.04 0.00 0.04 0.08 0.12

(b) E ad

(kcal/mol)

d (bohr)

LDA

BLYP

PBE

B3LYP

HF

rCAM-B3LYP

(a)

E ad /|E ad

|

d (bohr)

FIG. 10: ∆E

adDFA

and (b) ∆E

adDFA

/ | E

adCCSD(T)

| for (H

2

)

10

at various intermolecular distances d. Reprinted with permission from J. Chem. Phys. 137, 214106. Copyright 2012. AIP Publishing LLC.

FIG. 11: Evolution of ELF profiles upon stretching of the (H

2

)

10

chain for the indicated d values. Top:

isosurfaces for ELF=0.5 (homogeneous electron gas value). Bottom: 1D profiles around the saddle point.

Reprinted with permission from J. Chem. Phys. 137, 214106. Copyright 2012 AIP Publishing LLC.

this stage is not related to delocalization but rather to the lack of correlation. Conversely, LDA

(and GGA to a lesser extent) leads to values of ELF which are too high at the minima. In this case,

the delocalization between units is overestimated. This simple model reflects that delocalization

error is not only reflected in the energy, but can be easily casted from QCT.

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B. Fractional spins

1. The error

One of the most well known failures of DFAs is their inability to describe degenerate or near- degenerate states (static correlation), such as arise in bond breaking, strongly correlated materials or transition metal. These are manifestations of another error that can be easily identified with a model system, H 2 , in terms of fractional spins. 48,55 Opposite to the previous case, when even- electron bonds, as in H 2 , are stretched, the dissociation limit is predicted to have too high energy.

Following the same ensemble interpretation used in H + 2 , the dissociation of H 2 can be described with a single-reference state using fractional orbital occupancies. This leads to stretched H 2 as beign described as two H atoms holding an electron of each spin each. 45,50,55–58 From this line of thought, it is deduced that the exact energy behaves like a piecewise-flat-plane as a function of fractional charges and fractional spins. 58 In the case of hydrogen, the exact energy is then given by:

E (n) =

nE(1, 0) for 0 ≤ n ≤ 1

(2 − n)E(1, 0) + (n − 1)E(1, 1) for 1 < n ≤ 2, (20) where n α and n β are the α and β -spin orbital occupations, and n = n α + n β it the total number of electrons. Thus, a plot of the energy versus n α and n β should consist of two planes which intersect at n α + n β = 1), leading to a derivative discontinuity. In very rare cases, a second type of flat plane condition appears, in which the planes intersect at n αn β = 0. 59,60 In both cases, violation of the exact discontinuity condition underlies the failure of common functionals for strongly-correlated systems. 58

Thus, looking at the ability of a functional to provide the correct flat planes behavior is a mea- sure of their lack of fractional charge and spin errors. Identification of zones of error in functionals and use of chemical information can lead to improve functionals. Let’s look for example at Becke’s successful non-dynamical correlation (NDC) functional, B03: 61–63

E NDC B03 = 1 2 f

Z ρ α U X β + ρ β U X α

dr (21)

where U X σ is the HF exchange potential and f is a fraction of the effective opposite-spin exchange hole. This approach recovers a behavior very close to the flat plane (Fig. 12a).

B03 gives accurate energies in the range 0 < (n α + n β ) < 1 ( f = 1). The derivative discon-

tinuity is well reproduced. However, a dip appears in the region (n α , n β ) = (0.75, 0.75), where

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0.0 0.2

0.4 0.6

0.8 1.0 0.0

0.2 0.4

0.6 0.8

1.0 -0.5

-0.4 -0.3 -0.2 -0.1 0

Electronic Energy (a.u.)

n α n β

Electronic Energy (a.u.)

-0.5 -0.4 -0.3 -0.2 -0.1 0

0.0 0.2

0.4 0.6

0.8 1.0 0.0

0.2 0.4

0.6 0.8

1.0 -0.5

-0.4 -0.3 -0.2 -0.1 0

Electronic Energy (a.u.)

n α n β

Electronic Energy (a.u.)

-0.5 -0.4 -0.3 -0.2 -0.1 0

’b05-error.dat’

0.0 0.2 0.4 0.6 0.8 1.0

n

α

0.0

0.2 0.4 0.6 0.8 1.0

n

β

-0.05 -0.04 -0.03 -0.02 -0.01 0.00 0.01

’new4-error.dat’

0.0 0.2 0.4 0.6 0.8 1.0

n

α

0.0

0.2 0.4 0.6 0.8 1.0

n

β

-0.05 -0.04 -0.03 -0.02 -0.01 0.00 0.01

FIG. 12: Hydrogen-atom electronic energies, as a function of orbital occupation (n

α

,n

β

), for (a) the orig- inal B03 density functional and (b) a modified version to fix the flat plane condition. Lower panels show deviations from planarity. Reprinted with permission from J. Chem. Phys. (Communication) 135, 081103.

Copyright 2011 AIP Publishing LLC.

the model predicts too much correlation energy. This is due to the fact that Eqn. 21 models non-dynamical correlation potential energy, but disregards the kinetic contribution. 61,62 This leads to an overestimation of the NDC energy when kinetic contributions are important. We will use chemical information to correct this behavior in the coming subsection.

2. Chemical insight - correcting E

c

with LOL

In order to correct the ill behavior of B03, it is important to first understand its origin. The

model is correct at both extremes [(n α , n β ) = (1, 1), and (n α , n β ) = (0.5, 0.5)]. These two points

correspond to the extremes of the binding well: equilibrium and dissociation. But the model

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overestimates the correlation energy for intermediate coupling strengths [(n α , n β ) = (0.75, 0.75)].

It is also important to recognize that B03 only contains NDC potential energy information. Hence the kinetic energy part is being lost. Thus it can be corrected by using chemical information: the fractional occupation interpretation together with the virial theorem.

By virtue of the virial theorem, the total NDC energy is half the NDC potential energy. Hence, the behavior for medium-coupling range can be fixed interpolating between strongly- and weakly- interacting regimes. Assuming a linear interpolation, a new version for NDC can be constructed, E NDC JJ : 64

E NDC JJ = 1 2 f g

Z ρ α U + ρ β U X α

dr (22)

where g is the coupling factor, defined as follows:

g = min

2 − N X e f f αN X e f f β , 1

(23) Eq. 23 leads to the following values for g:

g = 0 at (n α , n β ) = (1, 1)

g = 0.5 at (n α , n β ) = (0.75, 0.75) and,

g = 1 at (n α , n β ) = (0.5, 0.5) and for 0 < (n α + n β ) < 1, preserving the exact energy results for this region

Fig. 12b shows the results for the correction. The features that were well described are conserved, and the dip at high spin is corrected. This is specially relevant for stretched bonds (medium range correlation).

But can we go even further using the chemical understanding of the system? It is important to realize that perfectly-localized electron pairs are well described by the monodeterminantal- integer charges scheme. Thus, they are not subject to non-dynamical correlation. This goes back to a well-known concept in chemistry: only valence electrons are affected by stretching. Core electrons behavior is constant upon stretching. In other words, the effective hole normalization in the core is one everywhere whereas in the valence region, the effective normalization is equal to the fractional occupancy, which can be described by the above corrected model.

The core-valence separation can be easily done in terms of the Lewis-style descriptors intro-

duced above (e.g. laplacian, kinetic energy density measures). The laplacian is known to have

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0.0 0.2

0.4 0.6

0.8 1.0 0.0

0.2 0.4

0.6 0.8

1.0 -0.2

-0.1 0.0

Relative Energy (a.u.)

n α n β

Relative Energy (a.u.)

-0.20 -0.15 -0.10 -0.05 0.00

0.0 0.2

0.4 0.6

0.8 1.0 0.0

0.2 0.4

0.6 0.8

1.0 -0.2

-0.1 0.0

Relative Energy (a.u.)

n α n β

Relative Energy (a.u.)

-0.20 -0.15 -0.10 -0.05 0.00

FIG. 13: Relative electronic energies for multielectronic atoms, as a function of orbital occupation (n

α

,n

β

), using our modified non-dynamical correlation functional. (a) Lithium atom (b) Sodium atom. It should be noted that in the case of Na, point-wise effective exchange-hole normalizations with the inverse-Becke- Roussel procedure

61,62

, were necessary to eliminate the curvature. Atomic units. Reprinted with permission from J. Chem. Phys. (Communication) 135, 081103. Copyright 2011 AIP Publishing LLC.

failures in identifying the shell structure of heavy elements. 65 Thus, an indicator based on the ki- netic energy densities is more suited. We have used the outer-most LOL maximum to define a distance R core which divides an atom into core and valence regions. Then the effective exchange- hole normalizations is defined differently for these two regions: 64

N X e f f σ = 1 for the core (r < R)

N X e f f σ = n σ for the valence (r > R).

Results for Li and Na atoms are shown in Fig.13. The flat-plane behavior is recovered for s atoms by this simple chemical input. 64 This is similar to other chemical informations introduced for treating medium-range separation, 66 where the local density and gradient information was used to examine chemical bonding.

C. Non-covalent interactions

1. The error

With boosted computational capacities, it has become strikingly clear that the description of

dispersion interactions is necessary to capture the chemistry of big structures which are now at

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reach. Morevoer, it has been shown that dispersion is also crucial in the understanding of smaller systems. Highly branched structures, 67 and estimations of enantiomeric excess 68 are some exam- ples. And still, most commonly used functionals are not able to retrieve long-range dispersion interactions due to their inherent non-local nature.

From the 3D point of view, two very different cases appear: intermediate and long distances.

Whereas semilocal functionals are not able to correctly describe the latter, some functionals (e.g.

PW91) 14 are able to provide at least qualitatively correct interaction potentials if the fragment electron densities overlap (e.g. at equilibrium).

One interesting equilibrium case which has given rise to much controversy in the literature are 1,3-interactions. It is known that DFAs give systematic errors for very simple reactions. Moreover, just like the H chains analyzed in Section III A, these error increase with system size.

An interesting example of this size dependency is the isodesmic reaction where linear alkanes of growing size are fragmented into ethane molecules: 69–73

CH 3 (CH 2 ) n CH 3 + nCH 4 −→ (n + 1)C 2 H 6 (24) The atoms involved are main elements from the first row and the number and type of bonds is conserved. Hence, these reactions would be expected to be well described by basically any DFA.

This is far from being the case. Results for LDA 14 and GGAs 13,74,75 are presented in Fig. 14.

Whereas LDA performs extremely well, GGAs dramatically fail (errors of up to ca. 12 kcal/mol are found for n-octane!). Similar errors have also been reported for isomerization reactions. 76–80 .

Let’s try to understand the reasons for this apparent failure of the gradient expansion . First of all, if we dissect LDA’s behavior into exchange and correlation, we see that the good perfor- mance of LDA is entirely due to the exchange term. LDA (only exchange) and LDA-xc (including correlation) curves nearly overlap (Fig. 14). Thus, this is not a correlation effect and the failure must come from the exchange term. Since GGAs are built from a gradient expansion over LDA, the gradient contribution to the exchange term must be responsible for the incorrect description of isodesmic reactions. 80,81 Moreover, the error is systematic (reaction energies underestimated) and grows with system size.

Once the part of the functional that is failing is located, we need to understand what it is doing

wrong about the chemistry of the system. The main difference when looking at the interactions

in alkanes and ethane, is the absence of 1,3-interactions in the latter. Hence, another isodesmic

reaction can be conceived where these interactions are still present in order to check whether the

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FIG. 14: Isodesmic reaction energies (Eqn.24) for the n-alkane series from propane through octane. Values are expressed as a function of carbon chain length for the LDA exchange-correlation (XC) functional and exchange-only GGA functionals. Calculations were performed with the NUMOL program.

82

Experimen- tal data were taken from Ref. 83. Reprinted with permission from J. Chem. Theory Comput. 8, 2676.

Copyright 2012 American Chemical Society.

error is coming from a wrong description of these interactions. If we consider the fragmentation of alkanes into propane units:

CH 3 (CH 2 ) n CH 3 + (n − 1)C 2 H 6 −→ nC 3 H 8 . (25)

the 1,3-interaction is preserved in reactants and products. Table II presents the errors per propane

observed with different functionals for Eqns. 24 and 25. Now, both GGAs and LDA provide a

correct description of Eqn. 25. This highlights the fact that the error is indeed localized in the

1,3-interaction, known as protobranching. 70

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TABLE II: Errors in calculated isodesmic reaction energies (Eqn.24 and Eqn.25) relative to experiment.

83

for linear alkanes up to octane. Results are shown for selected exchange-correlation (XC) or exchange-only functionals. Errors are expressed per propane unit in kcal/mol.

Method Eqn.24 Eqn.25 LDA XC 0.15 0.02

LDA 0.15 0.11 PBEsol 1.39 0.20 PBE 1.76 0.23 B88 2.14 0.20

2. Chemical insight - Choosing the correct functional with the reduced density gradient

Since the problem is arising from the inability of the GGA exchange term to describe non- covalent 1,3-interactions, we can try to understand the origin of the failure of GGAs in terms of the reduced density gradient, which is the 3D information input in this term (see Eqn. 6). The different performance of LDA and GGA should mainly originate from regions with large changes in gradient between ethane and a larger alkanes, leading to very different contributions in the GGA term. Fig. 15 shows the s( ρ ) plot for ethane and propane. The points that are giving rise to this differential behavior are shown in the inset of Fig. 15. As expected, they correspond in real space to the 1,3-interaction. 84

In order to further verify that the 1,3-interaction is at the origin of the GGA missperformance, we have compared the DFT integration grid points that were leading to different energetics in between LDA and GGA. These points are shown in blue in Fig. 15: 84

• In 3D, they correspond indeed to the protobranching interaction

In 2D, they correspond to the dip in the s( ρ ) diagram

Since the reduced gradient values are larger for ethane than propane, the exchange energy will be lower. In other words, propane is destabilized with respect to ethane. This leads to the systematic energy error in protobranching. Moreover, since the error is related to a given local interaction, it will systematically grow with the number of interactions involved.

But, once again, can we use our understanding of the problem for obtaining better results?

Functionals with a reduced magnitude of the gradient term (such as PBEsol) will lead to smaller

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a) b)

FIG. 15: a) Overlayed s(ρ) plots for ethane and propane. Key grid points centered on the left-most CH

3

group, where the reduced gradient changes by 10% or more between ethane and propane, are highlighted.

These points are plotted in real space for the propane molecule in the right inset. The left inset shows the non-covalent interaction region, which is an s = 1.5 au isosurface containing those points that fall below the lower edge of the propane s(ρ) curve. b) s

4

(ρ) plots for ethane and propane. Electron densities were calculated with B3LYP/6-31G*

15,49,86

and isosurfaces were obtained using the NCIplot program.

87

Reprinted with permission from J. Chem. Theory Comput. 8, 2676. Copyright 2012 American Chemical Society.

errors. However, errors are still too large (see Fig. 14). Another option within semilocal func- tionals, is to reduce the dependency on the reduced density gradient. Higher order terms are less sensitive to gradient changes (Fig. 15b). Thus, functionals such as B97D, 85 which expands PBE to 4 th order:

E x PBE = E x LDA + c 1Z ρ 4/3 1 + s 2 γ s 2 (26)

E x B97 = E x LDA + c 1Z ρ 4/3 1 + s 2 γ s 2 + c 2Z ρ 4/3 (1 + s γ 4 s 2 ) 2 (27)

should be expected to provide a better answer. This is indeed the case, yielding errors of 0.23

kcal/mol per propane unit. It should be noted that inclusion going beyong the generalized gradi-

ent approximation (inclusion of dispersion, long-range exact exchange, or MP2 correlation) also

restores the good agreement with experimental data. 69–73

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IV. CONCLUSIONS

In this Perspective we have reviewed two parallel scientific frameworks which exploit similar information: the development of semilocal functionals and the topological analysis of the elec- tronic structure. These two frameworks are built on 3D scalar functions (the electron density, kinetic energy densities, etc.), which they translate respectively into energies and bonding frame- works. We aimed at merging the information from both approaches and try to understand the information coded in functionals in real space. This information provides insight into what type of bonds we can expect to describe along each set of variables in semilocal functionals. Moreover, it can be used to understand the typical failures of DFT. Here, we have worked through examples the case of fractional electrons, showing that topological indexes are able to reveal delocalization errors. We have also seen how this can be taken one step further, proposing variations of ex- isting functionals to minimize the fractional spin error in alkalis. Finally, the wrong description of non-covalent interactions has been exemplified in 1,3 interactions in alkanes. The unexpected regression along Jacob’s ladder for this case has been explained in terms of the NCI contribu- tions, and has served to guide the choice of approximate functionals . All in all, understanding the chemical bonding , and the behavior of functionals for this specific situation should be infor- mative for users in choosing functionals and developers inidentifying the error and improving the performance.

Acknowledgements

We would like to acknowledge Dr Roberto Boto for Figures 1 and 2 and R. Laplaza for valuable help with the layout. This work was supported partially by the framework of CALSIMLAB un- der the public grant ANR-11-LABX-0037-01 overseen by the French National Research Agency (ANR) as part of the “Investissements d’Avenir” program (reference: ANR-11-IDEX-0004-02), by the National Science Foundation (CHE-1362927) and by the National Institute of Health (R01- GM061870).

Corresponding author: contrera@lct.jussieu.fr

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