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Intrinsic atomic orbitals: An unbiased bridge between quantum theory and chemical concepts

Modern quantum chemistry can make quantitative predictions on an immense array of chemical systems. However, the interpretation of those predictions is often complicated by the complex wave function expansions used. Here we show that an exceptionally simple algebraic construction allows for defining atomic core and valence orbitals, polarized by the molecular environment, which can exactly represent self-consistent field wave functions. This construction provides an unbiased and direct connection between quantum chemistry and empirical chemical concepts, and can be used, for example, to calculate the nature of bonding in molecules, in chemical terms, from first principles.


I. INTRODUCTION
Chemical concepts as fundamental as atomic orbitals (AOs) in molecules, covalent bonds, or even partial charges, do not correspond to physical observables and thus cannot be unambigously defined in pure quantum theory. This leads to the unpleasing situation that quantum chemistry can tell us benzene's heat of formation with ¡2 kJ/mol accuracy, 1 but, strictly speaking, neither that it has twelve localized σ -bonds and a delocalized π-system, nor what the partial charges on the carbons are. Chemical bonds have even been compared to unicorns-mythical creatures of which everyone knows how they look, despite nobody ever having seen one. 2 However, qualitative concepts are of essential importance for practical chemistry, and thus a large number of competing techniques were developed for extracting them from quantum chemical calculations. In particular, Bader's atoms in molecules 3 and Weinholds natural atomic/bond orbital analysis (NAO/NBO) 4,5 are widely used for interpreting molecular electronic structure. Nonetheless, the former is known to produce counter-intuitive results in many cases, 6 and the latter, while undoubtedly having brought countless successes in chemical interpretation, is complicated and pre-imposes various non-trivial assumptions. In particular, NBO analysis is based on the two notions that atomic orbitals (AOs) in molecules have spherical symmetry and can be obtained by a particular complex series of transformations, 4 and that a Lewis-like bonding pattern for any given molecule exists and only needs to be found-by comparing the wave function to all possible Lewis patterns. 7 While normally applicable, violations of both assumptions are conceivable in unusual bonding situations, and might then lead to erratic interpretations.
We here present a new technique to connect quantitative self-consistent field (SCF) wave functions to a qualitative chemical picture. This technique is essentially free of empirical input, allows for computing the nature and shape of chemical bonds, and is not biased towards any preconceived notion of bonding. This is achieved by first defining a new intrinsic minimal basis (IMB), a set of perturbed core-and valence AOs which can exactly describe the occupied molecular orbitals (MOs) of a previously computed SCF wave function. We will show that the intrinsic AOs (IAOs) thus defined can be directly interpreted as the chemical AOs, and that partial charges and bond orbitals (IBOs) derived from them perfectly agree with both experimental data and intuitive chemical con-cepts. In particular, we find a natural emergence of the Lewis structure of molecules.
While other IMB constructions have been developed before, [8][9][10][11][12][13][14][15] most are rather complex and so far none has found widespread use comparable to Bader or NAO analysis. Our contribution is a IMB which is simple and efficient, its use in constructing bond orbitals, and the demonstration that this combination shows excellent promise for interpreting chemical bonding and reactivity. The technique thereby provides a firm quantum mechanical basis for ubiquitous fundamental concepts.

II. CONSTRUCTION OF INTRINSIC ORBITALS
Assume that we have computed a molecular SCF wave function |Φ . |Φ is defined by its occupied MOs |i = ∑ µ |µ C µ i , where µ ∈ B 1 are basis functions from a large basis set B 1 . The key problem in interpreting wave functions is that the basis functions |µ cannot be clearly associated with any atom; each function will contribute most where it is most needed, and due to B 1 's high variational freedom, this often is not on the atom it is placed on. On the other hand, if one were to expand the MOs over a minimal basis B 2 of free-atom AOs (i.e., a basis of accurate AOs, but e.g., with only AOs 1s,2s,2px-2pz for each C atom), the wave function would be easy to interpret. But it would be inaccurate, and might be even qualitatively incorrect, because free-atom AOs contain no polarization due to the molecular environment. We thus propose to first calculate an accurate wave function |Φ , and then to form a set of polarized AOs |ρ / ∈ B 2 which can exactly express |Φ s occupied MOs |i . For this, we first split the free-atom AOs |ρ ∈ B 2 into contributions corresponding to a depolarized occupied spaceÕ = ∑ĩ |ĩ ĩ | and its complement 1 −Õ. The depolarized MOs are obtained as by projecting the accurate MOs |i from the main basis B 1 onto the minimal basis B 2 (which cannot express polarization) and back. 16 We can then get the polarized AOs |ρ from the free-atom AOs |ρ by simply projecting their contributions iñ O and 1−Õ onto their polarized counterparts O = ∑ i |i i| and 1 − O: Thus, to construct the polarized AOs it is sufficient to load a free-atom basis, calculate its overlap with the main basis and within itself, and perform the numerically trivial projection (2). Contrary to the related approach of Ref. 12, no functional optimization or reference to virtual orbitals is required. In this article we will also symmetrically orthogonalize the vectors obtained by (2), to arrive at an orthonormal minimal basis which divides the one-particle space into atomic contributions; the latter will be referred to as intrinsic atomic orbitals (IAOs).
While the construction makes reference to free-atom AOs through basis B 2 , it must be stressed that these are not empirical quantities. Free-atom orbitals can be calculated with any high-level quantum chemistry program. However, in practice this is not even required because they are already tabulated as part of several standard basis sets; here we take the AO functions of cc-pVTZ. 17 Since IAOs are directly associated with atoms, they can be used to define atomic properties like partial charges. Let us denote the closed-shell SCF density matrix as γ = 2 ∑ i |i i|, where i are the occupied MOs. We can then define as the partial charge on atom A, where Z A is the atom's nuclear charge and ρ its IAOs. Tab. I shows that the partial charges obtained are insensitive to the basis set, follow trends in electronegativities, and some defects seen in other methods (e.g., Bader's description of the CN bond in HCN as ionic) are absent. Partial charges will be further analyzed below.
IAOs provide access to atomic properties, but it is often desirable to get a clearer picture of molecular bonding. We now show that by combining the IAOs with orbital localization in the spirit of Pipek-Mezey (PM), 18 with respect to U ii . Here n A (i ) = 2 ∑ ρ∈A ρ|i i |ρ is the number of |i 's electrons located on the IAOs ρ of atom A. This construction effectively minimizes the number of atoms an orbital is centered on. The exponent 4 is preferred over the exponent 2 of PM because it leads to discrete localizations in aromatic systems; for other systems both exponents lead to effectively identical results. Fig. 1 shows IBOs computed for the acrylic acid molecule. Here 16 of 19 occupied MOs can be expressed to ¿99% by charge with IAOs on one or two centers, respectively. The three other MOs are part of a π-system: The oxygen p lone pairs (which have about 7% bonding character), and the C=C π-bond (which has about 3% contributions on the third C atom, and about 1% on the doubly-bonded O). In total, we see a direct correspondence of the obtained IBOs with the classical bonding picture: σ -bonds, π-bonds, and lone pairs are exactly where expected, and the π-system is slightly delocalized. We stress again that these 19 IBOs are exactly equivalent to the occupied MOs they are generated from: Their anti-symmetrized product is the SCF wave function, and this is a valid representation of its electronic structure. Note that the IBO construction makes no reference to the molecule's Lewis structure whatsoever; the classical bonding picture thus arises as an emergent phenomenon rooted in the molecular electronic structure itself, even if not imposed.
The major improvement of IBOs over PM orbitals is that they are based on IAO charges instead of the erratic Mulliken charges (cf. CH 4 in Tab. I). As a result, IBOs are always well-defined, while PM orbitals are unsuitable for interpretation because they are unphysically tied to the basis set (they do not even have a basis set limit). IBOs lift this weakness while retaining and even improving on PM's computational attractiveness. 17

III. CONSISTENCY WITH EMPIRICAL FACTS
Our hypothesis is that IAOs offer a chemically sound definition of atoms in a molecule. But since these are not physically observable, this claim can only be backed by consistency with empirical laws and facts. 19 We thus now investigate whether partial charges derived from IAOs follow expected trends based on electronegativies, C 1s core level shifts, and linear free-energy relationships for resonance substituent effects (Taft's σ R ). We then go on to see how IBOs reflect bonding in some non-trivial molecules.
We saw in Tab. I that, unlike Mulliken charges, IAO charges are insensitive to the employed basis set, and unlike Bader charges, IAO do not erronously describe the CN bond in HCN as ionic. We now follow Ref. 6 and investigate IAO charges in relation to electronegativity χ diffences. We start with the series CH 3 X (X=F, Cl, Br, H). Due to the (Allen 20 ) electronegativities (F: 4.193, Cl: 2.869, Br: 2.685, C: 2.544, H: 2.300), we expect halogens to have a negative charge, getting smaller in the series, and hydrogen to have a positive charge. As shown in Fig. 2a, this is what we find. In Fig. 2b, we show the series YH 4 (Y=C, Si, Ge). We find charges in close correspondence with χ (C: 2.544, Si: 1.916, Ge: 1.994), and the inversion that χ(Si) < χ(Ge) is properly reflected. If we extrapolate the curves in a and b to ∆χ = 0, we find in both cases that q(X) ≈ 0 and q(Y ) ≈ 0, respectively. That is, if there is no difference in electronegativity, IAO partial charges predict no bond polarization. This consistency with empirical electronegativies is further reflected in the almost linear shapes of the curves. In the series CH 4−n F n (n=0. . .4), we find C partial charges of -0.52, -0.01, 0.44, 0.85, 1.23. The charge increase by ≈0.5e − per fluorine atom agrees with the understanding of CF bonds in organic chemistry 21 and earlier calculations, 22 contrary to the much smaller charges found in Hirshfeld and Voronoi deformation density (VDD) analysis. 6 A different test of IAO charges can be performed by comparing to experimental data which are known to be highly correlated with charge states of specific atoms A. A prime example for this is the C 1s core-level ionization energy shift due to the molecular environment. This shift can be estimated 23 as where the second term is an estimate for the electrostatic po-  tential of the other atoms B, the last term is a contribution due to core orbital relaxation, and k is a (hybridization dependent) proportionality constant. This model has been employed to calibrate widely used electronegativity equilibration models, 24 and has been found to be perfectly correlated with both experimental 25 and theoretical 26 mean dipole derivatives (which for the molecules studied here can be interpreted as charges, 26 but not generally 27 ). In Fig. 3 we show the results obtained with IAO partial charges based on Hartree-Fock wave functions for all the sp 3 hybridized molecules as studied in Ref. 25. IAO charges where used both for the abscissa and V , the second term in Eq. (5). ∆E relax was calculated at Hartree-Fock level by a ∆SCF procedure. We here obtain a linear regression coefficient of r = 0.997, or r = 0.9995 if the two outliers CF 3 CF 3 and CH 3 C≡CH 3 are excluded. This is the same level of correlation as obtained with dipole moment derivatives, 25 and much higher than for CHELPS, Bader, or Mulliken charges. 26 One advantage of Hilbert-space based partial charges over real-space partial charges is that they can be split not only into atomic contributions, but also orbital contributions. Recently Ozimiński and Dobrowolski 28 used this freedom to introduce a set of descriptors for the electronic σ -and π-substituent effects, called sEDA and pEDA, and showed that they are both internally consistent and highly correlated with empirical substituent effect parameters like Taft's σ R . Concretely, for a substituted benzene R-C 6 H 5 , the pEDA parameter is defined as the number of p z electrons on the six carbon atoms of the benzene ring, relative to the unsubstituted benzene: where Ozimiński defined this quantity based on NAO population analysis 4 with a specified type of wave function and basis set. In order to demonstrate the potential of IAO charges in the interpretation of chemical reactivity, in Fig. 4 we show that the same kind of correlation with empirical substituent constants is also obtained when calculating pEDA from IAO charges (r = 0.966) instead of NAO charges (r = 0.943 28 ). Calculating such EDA values is computationally trivial, which makes them an attractive quantity in the study of unusual substituents not be contained in common tables, or for testing if the substituents behave differently for different hosts than benzene. Indeed, a similar idea to Ozimiński's has been considered previously, 30 but was much less practical due to being based on carefully crafted real-space integration because Hilbert space approaches were considered unreliable. 30 A deeper insight into the nature of a molecule's bonding can be obtained by calculating its bond orbitals. As previously noted, IBOs are an exact representation of SCF wave functions, and we have seen in Fig. 1 that they normally reflect the classical bonding concepts one to one. However, in many molecules the Lewis structure does not tell the entire story. Therefore we now probe how IBOs reflect bonding in some well known, but in different senses non-trivial molecules.
Benzene: A straight application of the IBO construction produces the orbitals a-e shown in Fig. 5. As expected, both the CC and the CH σ -bonds of the system are completely localized, and can be be expressed to ¿99% with IAOs on only the two bonded centers. As a prototypical delocalized system, this does, however, not hold for the π system. Two aspects are important: (i) The three π orbitals cannot be expressed with IAOs on less than four centers each (in c-e having the weights 1.000, 0.444 (ortho), and 0.111 (para)), and (ii) there are two different maximal localizations of the functional Eq. (4), orbitals c-e and orbitals c-e rotated by 60 • in real space. If in Eq. (4) we had chosen to maximize ∑ n A (i) 2 instead of ∑ n A (i) 4 , there would even be a continuum of maximal localizations, also including orbitals f-h (which look closer to classical π-bonds), and everything in between. In this case the classical resonance structure reflects the nature of the bonding well.
Cyclopropane: According to its Lewis structure, cyclopropane is a simple alkane. However, due to the massive ring strain one could expect that some bonding effects are inevitable. Nevertheless, if we calculate the IBOs of this molecule (Fig. 6, a and b), we find six CH single bonds and three CC single bonds, all perfectly localized (to ¿99%) on the two bonded centers, with no delocalization whatsoever. However, a closer look reveals that while the carbon part of the CH bond orbitals has about 28% s character and 72% p character, (close to the ideal sp 3 hybrid values of 1/4s + 3/4p), the CC bonds only have 18% s character and 82% p character. So although they are localized single bonds, they must be considered an intermediate between a regular sp 3 -hybrid σbond and a π-bond. This explains the well-known similarity in reactivity to alkenes.
Diborane: B 2 H 6 has been a serious challenge to the classical bonding picture, with even scientists like Pauling championing an ethane-like structure until proven wrong irrefutably. 31 Its bridged structure was popularized in 1943, 32 and spawned investigations culminating in Lipscomb's 1976 Nobel price for his "studies on the structure of boranes illuminating problems of chemical bonding". One could think that this molecule presents a challenge to a IBO bonding analysis. However, IBOs are just the most local exact description of a first-principles wave function, and their construction does not make any reference to any perceived nature of the bonding. Consequently, for IBO analysis diborane is not different than other molecules, and it uncovers diborane's two two-electron three-center bonds (Fig. 6d) just as its six standard σ -bonds (c), without any problems.
Sulfur trioxide: SO 3 is one of the simplest "hypervalent" molecules, apparently violating the octet rule. An IBO analysis finds two oxygen lone pairs, one σ -and one π-bond (Fig. 6e) per oxygen. Formally this calls for describing the SO bonds as double bonds. However, the π-bonds have only a small bonding component (83% on oxygen, 15% on sulfur), so it is a matter of taste whether they should be called true πbonds or not. But in any case, they are highly localized (98% on two centers) and clearly not resonating, so the resonance structure -commonly found in textbooks-is at best misleading. 33 Bifluoride anion: We see a similar discrepancy to textbook knowledge in the description of FHF − . This molecule is alternatively cited as the strongest known hydrogen bond, 34 or as an example for a 3-center 4-electron bond (since an influential paper by Pimentel 35 ). However, IBO anaylsis reveals that it can be perfectly described by six F lone pairs and two HF single bonds, all completely localized. Since the bonds are highly polarized, there is again no violation of the octet rule: In fact, the H orbital has a population of only 0.6 electrons total (out of the up to two electrons it theoretically could harbor), and the nature of bonding in this molecule is not very different than in HF.

IV. CONCLUSIONS AND OUTLOOK
The proposed IAOs offers a simple and transparent way to relate chemical intuition to quantum chemistry. In particular, the fact that most simple bonds can be expressed to ¿99% with IAOs on only two atoms strongly indicates that IAOs can be interpreted as chemical valence orbitals in molecules. That properties of individual such orbitals can then be directly calculated, as shown in Fig. 4, may turn out to be a decisive factor in future research on chemical reactivity. Similarly, IAOs may greatly simplify the construction of realistic tight-binding model Hamiltonians and their use in eludicating complex correlated electronic structure phenomena. 36,37 The proposed IBOs can help to uncover the nature of bonding in molecules-due to their unbiased nature also in unusual cases. However, the IBO construction's simplicity, ease of implementation, and high runtime efficiency make it an excellent choice also where localized orbitals are used for purely technical reasons (e.g., in local electron correlation methods).
We acknowledge funding through ERC Advanced Grant 320723 (ASES).