Noncovalent Interactions Descriptor Based on the Source Function of

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A: Molecular Structure, Quantum Chemistry, and General Theory

A Noncovalent Interactions Descriptor Based on the Source Function of Individual Localized Molecular Orbitals in Whole Space Alberto Baggioli J. Phys. Chem. A, Just Accepted Manuscript • DOI: 10.1021/acs.jpca.7b08774 • Publication Date (Web): 29 Mar 2018 Downloaded from http://pubs.acs.org on March 29, 2018

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The Journal of Physical Chemistry

A Noncovalent Interactions Descriptor Based on the Source Function of Individual Localized Molecular Orbitals in Whole Space Alberto Baggioli* Dipartimento di Chimica, Materiali e Ingegneria Chimica “Giulio Natta”, Politecnico di Milano, via Mancinelli 7, 20131 Milano, Italia.

AUTHOR INFORMATION Corresponding Author: Alberto Baggioli *Email: [email protected]

1 Environment ACS Paragon Plus

The Journal of Physical Chemistry 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

ABSTRACT

Partitioning of total electron density into localized molecular orbitals (LMOs) within the source function framework is presented as an exploit useful to the characterization of noncovalent interactions. The resulting approach uses only a converged LMO base as input, and does not require grid data handling nor numerical integration, making it viable to the study of large systems. Tests conducted on a series of prototypical hydrogen and halogen bonds demonstrate this descriptor retrieves chemically-intuitive interaction motifs beyond the classical two-centers picture. Advantages and disadvantages of different localization schemes are discussed in terms of their peculiar description of σ–π mixing, and in terms of their inherent basis set dependence.


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1. Introduction Noncovalent interactions play a vital role in a large number of chemical and physical phenomena occurring in the solid, liquid, and gaseous phases.1 They rule the aggregation of molecules in condensed matter, and prompt most of the physical properties that make complex molecular systems appealing for advanced technological applications. Physisorption, solvation, selfassembly and molecular recognition are only a few examples of noncovalent interactions-driven phenomena with applications in e.g. liquid-crystals technology,2 drug delivery,3 crystal engineering.4 Furthermore, the biology of living organisms is largely governed by noncovalent interactions, with three-dimensional structure modulation of nucleic acids and proteins representing one of the most impressive examples of self-organization known.5 Accurate rationalization of noncovalent interactions provides useful insights into the physics governing such phenomena, both toward direct understanding of the problem at hand and toward design of new and improved variants. The characterization of noncovalent interactions can be carried out in several different ways based on the desired level of detail and on the amount of computational resources available.6 In particular, however, whether ab initio calculations (e.g. as in SAPT7 and other energy decomposition approaches) or numerical integration (e.g. as for AIM-related descriptors8) are involved, most of the most popular techniques are indeed rather compute-intensive. Such approaches are evidently at a disadvantage against large molecular systems of interest to biochemists and materials scientists, among others. Notable exceptions comprise inspection of geometrical parameters,9 total electron density stationary points properties survey, NCI10 and DORI11 graphical tools (which, however, do require storage of large grid data), and natural bond orbital (NBO) analysis.12 These require little computational effort besides that involved in the



production of molecular geometries, wave functions, or grid data, and are thus viable for the investigation of remarkably large systems. In this work, a versatile and fundamentally inexpensive approach towards the characterization of noncovalent interactions will be introduced and thoroughly tested. Its definition follows from that of the source function descriptor proposed by Bader and Gatti,13 whereas individual localized orbitals, rather than atomic domains, are used as sources.

2. Theory 2.1. Source function Solving Poisson’s equation for a potential given by the electron density scalar field 𝜌(𝐫) within a volume Ω enclosed by a surface fulfilling the criterion of zero-flux in the gradient vector field of electron density ∇𝜌(𝐫), Bader and Gatti proposed13 that 𝜌(𝐫) at any point r corresponds to the average potential exerted at r by its Laplacian scalar field ∇2 𝜌(𝐫′), according to 𝜌(𝐫) = −

1 ∇2 𝜌(𝐫′) ∫ 𝑑𝐫′ = ∫ 𝐿𝑆(𝐫, 𝐫 ′ ) 𝑑𝐫′ = 𝑆(𝐫, Ω) 4𝜋 Ω |𝐫 − 𝐫 ′ | Ω

The local source LS operating at point 𝐫′ contributes to the source function S of the domain Ω at the reference point r and takes the form of an electron density Laplacian augmented by an influence function (Green’s function). This leads to the interpretation of the source function as the reconstruction of electron density at the reference point r by the influence of its Laplacian at any other point 𝐫′. However, integration of 𝐿𝑆(𝐫, 𝐫 ′ ) 𝑑𝐫′ over a domain Ω with boundaries at infinity evidently results in an identity. The impasse is broken once one realizes the topological definition of an atom in a molecule satisfies the criterion used to define domain Ω. Within


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Bader’s atoms in molecules framework, the electron density at reference point r is thus given by the summation of 𝑆(𝐫, Ω𝑖 ) terms over all atoms in a molecule or cluster thereof, such that 𝜌(𝐫) is reconstructed from atomic contributions instead. Alternatively, each atomic domain represents a source towards the value of 𝜌(𝐫), such that the summation of individual sources over all basins ultimately determines the effect, 𝜌(𝐫) itself. The interested reader is referred to Ref. 14–18 for details on significance and interpretation of source function analysis results. The location of the reference point r is evidently of paramount importance, and should be appropriately selected based on the case at hand. Since the AIM definition of atomic basin is customarily used in this context, critical points from the same quantum theory are a popular choice. Bond critical points (bcps), minimum electron density points along a path of maximum equatorial electron density connecting two atoms (bond path), have been extensively used for this purpose.

2.2. Choice of a set of sources So far, only a few variants of the original source function formulation have been proposed, and despite the warnings of the very authors about the arbitrariness of such operation,18-20 they all involve splitting of atomic contributions into orbital-related components. In a recent report, Gatti et al.19 introduced the spin density source function approach for the study of molecular magnetism, in which a deconstruction of atomic source functions into magnetic natural orbitals and reaction (or relaxation) orbitals is performed. Furthermore, Farrugia and Macchi20 reported on the partitioning of atomic contributions into individual canonical molecular orbitals (CMOs), and elaborated on the distinction between core and valence source function components. Partitioning of electron density into individual CMOs, without further distinction into atomic domains, was also considered by the authors.20 The use of CMOs as individual sources in source



function calculations, however, proved to be inconvenient due to their rather high degree of delocalization, such that the results presented for these specific calculations were quickly dismissed as irrelevant. Indeed, since no atomic domain partition was operated, using some other form of spatial segregation might have helped retrieve intelligible information. Due to the invariance of energy with respect to unitary transformations among occupied orbitals,21 a different set of MOs, more ‘localized’ in R3 space, could be selected to be used as source for the reconstruction of 𝜌(𝐫). LMOs, as the name eloquently suggests, are molecular orbitals localized within a restricted portion of space,22-26 and typically bear a qualitative resemblance to core, bond, and lone pair orbitals of chemical intuition.27 Due to the local nature of electron correlation,28 one of the main applications of LMOs is currently that of replacing CMOs in post-Hartree-Fock calculations in order to reduce computation time.29-32 England et al. also noted27 that LMOs are more appropriate than CMOs for comparison of corresponding states of related molecules, making them an ideal choice in the study of group transferability. Orbital localization methods selected comprise the Foster-Boys criterion (FB),22,23 which minimizes the sum of orbitals quadratic momenta about their centroid, the Pipek-Mezey criterion (PM),24 which maximizes partial charges at the nuclei, and the natural (N) LMO base by Reed and Weinhold,25 obtained from highly occupied natural bond orbitals (NBOs) by incorporation of delocalization tails from weakly occupied NBOs. These three approaches all feature a different treatment of σ and π symmetry. The FB procedure mixes multiple bonds and lone pairs to give equivalent bent bonds and ‘rabbit ear’ lone pairs, while the PM procedure retains σ‒π separation (which however leads to non-equivalent lone pairs). NLMOs, on the other hand,


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preserve σ and π symmetry in isolated molecules, but tend to mix lone pair orbitals whenever their symmetry suffers an external perturbation, such as a noncovalent interaction.

2.3. Technical details Canonical molecular orbitals (CMOs), Foster-Boys localized molecular orbitals (FBLMOs), and Pipek-Mezey localized molecular orbitals (PMLMOs) wave functions were obtained at B3LYP/6-311G(d,p) level33-39 from MP2/aug-cc-pVTZ optimized geometries40-46 (where applicable) on the Gaussian 09 package.47 Natural localized molecular orbitals (NLMOs) were computed at the same level on the NBO 3.1 code.48 Default parameters were used throughout. AIM, 𝑆(𝐫, Ω𝑖 ), and 𝑆(𝐫, LMO𝑖 ) analyses were carried out on the Multiwfn 3.3.8 code.49 Since, by definition, the source function at a reference point r from an electron density distribution integrated over an infinite domain is equal to 𝜌(𝐫), exact 𝑆(𝐫, LMO𝑖 ) contributions were obtained as the orbital density of each LMO at r. As such, 𝑆(𝐫, LMO𝑖 ) ≥ 0 Ɐr,i. Orbital localization is executed numerically via an iterative algorithm subject to specific convergence criteria. As such, slight differences may result from AIM analysis of different orbital bases pertaining to the same geometry at the same level of theory. In order to ensure consistency, the value of 𝜌(𝐫) and the position of r in each specific LMO base were used as internal references. Consequently, 𝑆(𝐫, LMO𝑖 ) results are only discussed in terms of their relative magnitude %S, i.e. normalized against the appropriate value of 𝜌(𝐫).

3. Results and discussion 3.1. Test set 7 Environment ACS Paragon Plus


(H) 1.32

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(Oa) 52.6

(Ha) 30.8

1

(Ob) 9.05

(H) 2.43

(Hb) 7.73 (Cl) 55.1 (Ha) ‒18.3 (Hd) 1.43

2

(Oa) 33.8

(Ob) 35.1

(O) 55.6

3 (Cc) 3.99

(Ca) 8.72

(Hc) 4.05 (Hb) 9.55 (Ha) ‒34.6

(Ob) 71.9

(Ha) 4.81

(Cb) 4.41

(Oa) 27.0 (Hb) 3.76

4

(H) 13.1 (Hb) 28.6 (O) ‒15.8

(Ha) ‒87.9 (C) 50.9

5

(H) 33.5

(Hb)

Figure 1. Molecular graphs (orange paths and bcps), atom labelling, and atomic source contributions to reference points 1‒5 (green noncovalent bcps). Source functions are omitted for symmetry-related atoms.

Table 1. Relative source contributions in LMO space for bcps 1‒5.a FBLMO

PMLMO

σ(Oa–H) LP(Oa) core(Oa) 1s

0.57 85.5b 0.08b 0.11

1.99 82.4c 0.06d 0.04

σ(O–Ha) σ(O–Hb) LP(O) core(O) 1s LP(Cl) 3s LP(Cl) 3px LP(Cl) 3py

12.7 1.29 1.68b 0.22b 0.07 3.22 1.39 54.5

21.5 0.22 1.70c 0.00d 0.01 1.09 0.03 75.4

σ(Ca–Oa) π(Ca–Oa) σ(Ca–Ha)

0.12e 0.12e ‒ 0.29

0.14 0.00 0.12

NLMO 1 O···H H5O2+ (C2h) 0.51 σ(Ob–H) 81.3c 0.35d LP(Ob) 0.19 core(Ob) 1s 2 Cl···H Cl–/water 24.6 LP(Cl) 3pz 0.50 core(Cl) 1s 3.71c 0.00d core(Cl) 2s 0.01 core(Cl) 2px 0.07 core(Cl) 2py 0.00 core(Cl) 2pz 71.1 3 O···H malonaldehyde 0.44 σ(Ob–Hd) 0.00 LP(Oa) 0.07 LP(Ob)


FBLMO

PMLMO

NLMO

0.26 12.7b 0.02b 0.01

0.81 11.8c 0.07d 0.01

0.42 16.0c 0.28d 0.01

22.7 0.00 0.42 0.12 0.51 1.25

0.00 0.00 0.03 0.00 0.01 0.00

0.00 0.00 0.05 0.00 0.01 0.00

28.0 5.05b 62.2b 3.81b 0.01b

30.7 16.2c 51.3d 1.36c 0.00d

30.0 2.55c 63.2d 3.55c 0.00d

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σ(Ca–Cb) σ(Cb–Cc) π(Cb–Cc) σ(Cb–Hb) σ(Cc–Hc) σ(Cc–Ob)

0.12 0.01e 0.01e ‒ 0.01 0.12 0.01

0.05 0.02 0.00 0.04 0.08 0.02

σ(Oa–H) LP(Oa) core(Oa) 1s σ(Ob–Ha)

0.43 63.3b 0.29b 0.12 32.5

1.19 16.7c 45.0d 0.38 33.3

σ(C–Ha) σ(C–Hb) core(C) 1s

45.8 0.65 0.02

46.1 0.70 0.02

a

0.00 core(Ca) 1s 0.00 core(Cb) 1s 0.00 core(Cc) 1s 0.01 core(Oa) 1s 0.03 core(Ob) 1s 0.01 4 O···H water/water (Cs) 0.46 σ(Ob–Hb) 59.8b 0.23b LP(Ob) 0.11 core(Ob) 1s 34.6 5 O···H methane/water 47.5 σ(O–Hb) 0.73 LP(O) 0.07 core(O) 1s

0.00 0.00 0.00 0.09 0.00

0.00 0.00 0.00 0.02 0.01

0.00 0.00 0.00 0.13 0.02

1.21 0.88b 0.88b 0.00

1.10 1.18c 0.00d 0.0

1.43 2.83c 0.00d 0.01

1.67 47.8b 0.89b 0.12

2.91 26.7c 18.5d 0.76

1.68 43.5b 3.24b 0.09

See Figure 1 and relative caption for geometry information and atom labelling;

‘rabbit ear’ lone pairs;

c

Non-mixed lone pair of sp hybridization;

d

b

Equivalent

Non-mixed lone pair of p

hybridization; e Equivalent σ‒π mixed bent orbitals.

Table 1 collects 𝑆(𝐫, LMO𝑖 ) results obtained from five hydrogen-bonding (HB) interactions of rather different strength (displayed in Figure 1). These are the C2h H5O2+ (bcp 1) and the chlorate‒water (bcp 2) charge-assisted HBs, the Cs malonaldehyde (bcp 3) resonance-assisted HB, and isolated HBs in the Cs water dimer (bcp 4) and a weakly-bound methane‒water complex in which water acts as electron donor (bcp 5). Reference data regarding nature and inherent strength of these interactions are reported in Figure 1 in the form of 𝑆(𝐫, Ω𝑖 ) results. The O···H charge-assisted bond in the H5O2+ complex shows a strong covalent character, since the combined 𝑆(𝐫, Ω𝑖 ) contributions from the two atoms accounts to more than 80% of 𝜌(𝟏).14 Within the 𝑆(𝐫, LMO𝑖 ) framework, however, the bridging proton does not bear any contribution to the interaction, as only electron pairs are considered. As such, with reference to Table 1, a lone electron pair on Oa accounts for 81–86% of 𝜌(𝟏), while another lone pair on the accompanying water molecule provides an additional 12–16% with negligible help from σ(O–H) and core orbitals. Although a H2Ob···H+···OaH2 Lewis-like picture was indeed the most



commonly obtained, a H2Ob···[H–OaH2]+ one could also be crafted depending on the initial SCF guess wave function. In the latter case, the 𝑆(𝟏, σ(H– O𝑎 )) contribution is only a few percentiles larger than that of LP(Oa) in the former at the expense of LP(Ob)’s. The interaction between a chlorate anion and a water molecule is an example of a chargeassisted HB with a lower covalent character compared to 1. Both heavy atoms contribute in fact to about 55% of 𝜌(𝟐) in terms of atomic source functions. In LMO bases however, the electrondonating chlorine atom contributes to 70–77% with its lone electron pairs, while the bridging σ(O–Ha) orbital fills the gap with a source function of about 13–24%. A small contribution from chlorine core orbitals is noticeable in the FBLMO base, slightly in excess of 2%. This is most likely the result of a sub-optimal localization, which left a few delocalization tails. The O···H interaction in ground state malonaldehyde was chosen as an example of a resonance-assisted HB. Similarly to what obtained for the chlorate–water complex, both protondonating and proton-accepting atoms account for 33–35% of 𝜌(𝟑), with the bridging atom showing a small positive source function.14 Results from 𝑆(𝐫, LMO𝑖 ) analyses show a 66–68% contribution to 𝜌(𝟑) from lone pairs on the electron-donating Oa atom and a 28–31% contribution from the bridging σ(Ob–Hd) orbital. The only other significant source (1.3–3.8%) is represented by a non-pure-p lone electron pair on the proton-donating Ob atom. Although, the information on resonance ‘assistance’ provided by 𝑆(𝐫, Ω𝑖 ) analysis14 is lost within the 𝑆(𝐫, LMO𝑖 ) framework, the results obtained for 3 are still within the expected trend in relation to interaction strength, as it will be discussed later. The water dimer is probably the most representative case of hydrogen bond. The mainly electrostatic character of this isolated HB makes it rather weak compared to interactions 1–3, as shown by the large 𝑆(𝐫, Ω𝑖 ) contribution to 𝜌(𝟒) from the proton-donating Ob atom and the


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strength of the sink from the bridging hydrogen. Data in Table 1 highlight for the same complex a 𝑆(𝟒, LMO𝑖 ) contribution of 60–62% from lone pairs on Ob, and of 32–35% from the bridging σ(Ob–Ha) orbital. The remaining three σ orbitals act as weak sources, with a combined effort of about 3% of 𝜌(𝟒). Lastly, the methane–water complex represents a remarkably weak, dispersion-based HB-like interaction.50,51 Accordingly, the bridging atom is revealed in this case as a strong sink within the 𝑆(𝐫, Ω𝑖 ) framework, with a contribution of about –88%. The rest of the proton-donating molecule acts as a source toward 𝜌(𝟓). Results of 𝑆(𝐫, LMO𝑖 ) calculations return contributions to 𝜌(𝟓) in the neighborhood of 46–49% from both electron donor lone pairs on oxygen and the bridging σ(C–H) orbital, the striking balance of which is interpreted as a symptom of the notably low interaction energy holding water and methane together. The picture presented for these five prototypical noncovalent interactions by source function analysis over LMO space lends itself to a straightforward interpretation based on chemical intuition. Although virtual orbitals are not considered, the corresponding bonding orbitals (e.g. the σj in a lone pair → σj* interaction such as that at 4) yield the largest contribution out of all LMOs of the electron-accepting species, whereas the largest contribution from the electron donor species comes from predictable sources (i.e. lone electron pairs). This aspect can be exploited in order to define quantitative indexes of interaction strength within the 𝑆(𝐫, LMO𝑖 ) framework. For this purpose, it is useful to define DM and AM as the summation of all 𝑆(𝐫, LMO𝑖 ) contributions from the electron-donating and electron-accepting molecules respectively, where applicable. Similarly, Dm and Am will refer to the summation of the single largest 𝑆(𝐫, LMO𝑖 ) contribution to 𝜌(𝐫) and all others of comparable order of magnitude, from the electron donor and acceptor respectively.



Table 2. Values of DM, AM, Dm, and Am for bcps 1–5.

%𝑆(𝐫, LMO𝑖 ) FBLMO base DM AM Dm Am PMLMO base DM AM Dm Am NLMO base DM AM Dm Am

1

2

bcp 3

4

5

86.8 13.2 85.5 12.7

84.1 15.9 81.8 12.7

/ / 67.2 28.0

64.6 35.4 63.6 32.5

52.2 47.8 47.8 45.8

86.5 13.5 82.4 11.8

76.5 23.5 76.5 21.5

/ / 67.5 30.7

64.5 35.5 61.7 33.3

51.8 48.2 45.3 46.1

82.9 17.1 81.3 16.0

71.2 28.8 72.7 24.6

/ / 65.7 30.0

61.1 38.9 60.1 34.6

50.2 49.8 46.8 47.5

With reference to data collected in Table 2, DM and Dm both decrease with decreasing interaction strength from about 87% for a quasi-covalent bond to about 50% for a dispersionbound complex, while AM and Am increase symmetrically from about 12% to about 48% along the same direction. Overall, DM and AM share similar values with Dm and Am, respectively, although as far as whole ligands are concerned, electron donors always retain a DM larger than 50%. Charge-assisted HBs, with their strong covalent character, feature 71-87% of 𝜌(𝐫) from donors and 12-30% from acceptors. The source function of the resonance-assisted HB at reference point 3 is split 67% to 30% between donor and acceptor. 𝑆(𝟒, LMO𝑖 ) for the Coulombic force-driven HB in the water dimer are partitioned 60% to 33% between donor and acceptor, while the London force-driven HB between methane and water shows at bcp 5 a nearly evenly matched source from donor and acceptor. Similar indexes have been shown to be relevant to the study of HB interactions within 𝑆(𝐫, Ω𝑖 ) analysis, whereas the source function of the


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bridging hydrogen, of the accompanying atom, of the electron-donating atom, as well as of their combinations, are considered.2 The results produced by the three different localization schemes are overall quantitatively comparable to one another within this test set. The preservation of σ and π symmetries in the Pipek-Mezey and NLMO schemes is ideal for chemical systems involving multiple bonds. On the contrary, mixed ‘rabbit ear’ lone pair orbitals from Foster-Boys and NLMO schemes are better suited than unmixed ones in the description of electron donors. However, as it is shown in Table 1, contributions from the σ orbitals of multiple bonds are not negligible compared to those of the associated π systems. Moreover, geminal unmixed lone pair orbitals acting as electron donors tend to yield contributions of comparable order of magnitude. As such, these groups of orbitals have been considered as single entities in the evaluation of Dm, which helped to level off data obtained from different LMO bases, as shown in Table 2. Nevertheless, each localization scheme relies on unique criteria in order to yield a set of LMOs. Consequently, comparison between 𝑆(𝐫, LMO𝑖 ) results for different molecular systems are only relevant if the corresponding wave functions were consistently obtained using the same localization scheme.



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6

10 7

8

9

12 11

X

16

13, X = Cl 14, X = Br 15, X = I

17

Figure 2. Geometries and reference points 6–17 (in green) location for the additional noncovalent interactions comprising the ‘practical’ test set.

Table 3. Values of DM, AM, Dm, and Am for bcps 6–17.a

%𝑆(𝐫, LMO𝑖 ) FBLMO base DM AM Dm Am PMLMO base DM AM Dm Am NLMO base DM AM Dm Am a

bcp 6

7

8

9

10

11

12

13

14

15

16

17

56.1 43.9 56.0 38.5

66.2 33.8 64.0 30.0

63.3 36.7 62.3 32.3

63.2 36.8 62.5 31.4

63.5 36.5 62.7 29.3

44.7 55.3 42.9 51.4

62.5 37.5 58.3 34.8

41.6 58.4 40.4 57.3

/ / / /

/ / / /

/ / / /

/ / / /

56.7 43.3 56.7 39.1

67.1 32.9 64.5 29.9

37.8 62.2 37.2 61.7

32.7 67.3 32.0 65.5

47.5 52.5 46.8 51.5

63.8 36.2 56.4 34.4

62.2 37.8 56.0 33.1

55.6 44.4 57.0 41.0

41.7 58.3 40.8 53.8

40.3 59.7 39.4 46.6

42.8 57.2 42.6 52.3

63.1 36.9 60.7 31.8

57.9 42.1 57.7 35.8

67.1 32.9 65.6 28.2

64.4 35.6 62.5 30.2

64.7 35.3 62.6 30.4

66.4 33.6 65.7 26.8

59.4 40.6 58.4 39.8

60.1 39.9 57.6 37.0

57.9 42.1 52.4 43.2

58.8 41.2 57.9 40.2

58.5 41.5 57.6 39.6

53.3 46.7 53.0 44.1

51.5 48.5 50.2 38.6

Data in italics indicate a pruned basis sets had to be used (n=1); blank entries indicate no

converged LMO base was obtained even with severely pruned basis sets.


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3.3 Practical examples A set of twelve additional noncovalent interactions have been built in order to further test the proposed 𝑆(𝐫, LMO𝑖 ) approach. Structures are shown in Figure 2 along with the location of the bcps used as reference points 6–17. The ESI holds molecular graphs and source function data in both atomic and LMO terms. Obtained values of DM, AM, Dm, and Am are collected in Table 3. The usage of wave functions generated at a different level of theory than that at which geometries were optimized, along with the presence of rather heavy atoms and strongly delocalized π orbitals throughout the set, led to several convergence issues for FB and PM localization schemes. A popular remedy, routinely suggested in quantum chemistry codes’ manuals, requires a selective pruning of the basis set in use. In particular, the n most diffuse functions of each angular momentum of each atom are to be removed prior to wave function generation. Although this exploit did solve the convergence problem in some cases, the resulting LMO bases were completely inconsistent. The use of explicit ‘diffuse’ functions was avoided in the original basis set to prevent convergence issues beforehand. Therefore, high angular momentum ‘polarization’ functions, which were among the most diffuse basis functions present, were systematically removed. Data obtained thanks to these pruned basis sets have been put on display (in italics) both in Table 3 and in the ESI, for the sole purpose of showing how detrimental this exploit is to 𝑆(𝐫, LMO𝑖 ) calculations. These data need no commentary and will be ignored in the remainder of the discussion. Consequently, usage of NLMO bases is recommended in virtue of the notoriously low basis set dependence of NBO-related calculations. Bcp 6 describes a weak HB-like interaction in which the carbon atom of methane acts as a nucleophile on water. Atomic source functions computed compare well with a recent report on a



methane‒iodine complex of similar coordination,52 with carbon acting as a strong sink, and all four hydrogens of methane yielding positive source contributions. Within the 𝑆(𝐫, LMO𝑖 ) framework, each σ(C–H) orbital contributes positively to 𝜌(𝟔), and the bridging σ(O–H) orbital yields a contribution of 36–39%. According to Table 3, the strength of this HB is correctly located between interactions 4 and 5 in Table 2.50,51 The π system of ethylene and a proton from water engage at bcp 7 in a HB-like interaction of medium-low strength. 𝑆(𝐫, LMO𝑖 ) analysis shows that about 94% of the electron density at 7 is reconstructed from the bridging σ(O–H) and both σ and π C‒C orbitals in a near 1:2 ratio. The interaction is thus readily assigned to a π···H–O bond. Due to the presence of several weak contributions from neighboring orbitals, however, Dm and Am indexes would overestimate the binding energy of this complex, which is instead better described by DM and AM as being comparable in strength to interaction 4. A set of three additional π···H–O interactions from the S66 benchmark dataset53 were also considered. These feature benzene as an electron donor, and water, methanol, and acetic acid as electron acceptors. The corresponding interaction energies increase in order from water to acetic acid. Bonding patterns for the three complexes are reported in Figure 2 along with the location of bcps 8–10. Similarly to the ethylene–water case, main contributors within the 𝑆(𝐫, LMO𝑖 ) framework comprise σ and π C‒C orbitals of benzene for the electron donor, and one σ(O–H) for the electron acceptor. According to Table 3, only the Am index in FBLMO base and all indexes in NLMO base reproduce the expected trend for interaction energies. It should be pointed out, however, that binding energies for the three complexes are rather close to each other, all falling within less than a 1.5 kcal mol–1 range.53 In addition, a single π orbital (the closest to r)


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contributes to about 75% of the DM index. Perhaps a different approach should be used for conjugated π orbitals, such as a n-centers n-electrons collective π system. In order to experiment with donor–acceptor interactions involving common chemical species other than water, a pyridine–acetylene complex was summoned from the S66 dataset. 𝑆(𝐫, LMO𝑖 ) calculations reveal a lone pair on N and the electron-accepting σ(C–H) as the only two sensible contributors to 𝜌(𝟏𝟏). Results in Table 3 correctly describe the interaction as slightly weaker than that in 4. A N-methylacetamide dimer was selected from the S66 dataset as an example of an HB involving a N–H electron acceptor. 𝑆(𝐫, LMO𝑖 ) analysis yields σ(N–H) and oxygen lone pairs as the sole quantitative contributors to 𝜌(𝟏𝟐). Data in Table 3 do not compare well with an expected binding energy about twice as large as that of the Cs water dimer (interaction 4 in Table 2).53 However, these two peptides also interact via a weak N···H–C bond and are likely to be interested by relevant London forces, as shown in the ESI by the relative molecular graph. In addition, it would also be possible for Di and Ai (i=M,m) indexes to be sensitive to the actual nature of electron donor and acceptor moieties, rather than just the interaction strength. Halogen bonding (XB) was also investigated, at first via a set of halobenzene–trimethylamine complexes from the X40 benchmark dataset.54 The corresponding interaction energies increase in order from chloro- to bromo- and then iodobenzene. Bonding patterns for the three complexes are reported in Figure 2 along with the location of bcps 13–15. 𝑆(𝐫, LMO𝑖 ) analysis highlights one orbital featuring a source function in excess of 50%, nitrogen’s lone electron pair, which can be safely sorted out as electron donor. Halobenzenes contribute to 𝜌(𝐫) via both σ(C–X) and the sp hybridized lone pair on X, in a ratio of about 1:3. Core orbitals contributions on halogen atoms are limited to (n – 1)s and (n – 1)d, and their combined magnitude never exceeds 1.3%.



Data collected in Table 3 correctly describes the increase in binding energy from 13 to 15. On an absolute scale, however, the mismatch with Table 2 data again suggests Di and Ai indexes obtained from fundamentally different interactions are not directly comparable to each other. Finally, an XB (bcp 16) and a HB (bcp 17) within the same ethane–iodine complex were considered.52 Data from 𝑆(𝐫, LMO𝑖 ) analysis of bcp 16, similarly to what commented on bcp 6, do not provide any additional clue compared to 𝑆(𝐫, Ω𝑖 ): the latter has hydrogen atoms on methane as sources of 𝜌(𝟏𝟔), while the former specifies that their contribution comes from the corresponding σ(C–H) orbitals, which is evidently the only reasonable outcome. Regarding the HB interaction, iodine lone electron pairs and σ(I–I) contribute to 𝜌(𝟏𝟕) in excess of 50%, while the bridging σ(C–H) represents the main contributor in the electron acceptor. NLMO-based data reported in Table 3 for 16 and 17 agree with the low binding energy of the complex42 in comparison to stronger halogen bonds and hydrogen bonds previously discussed. On the contrary, the use of a PMLMO base led to a qualitatively inaccurate description of 16 and an overestimation of the strength of 17.

4. Concluding remarks Source function integration over localized molecular orbitals in whole space was described as an inherently linearly-scaling noncovalent interactions analysis tool. The partition of total electron density into easily recognizable entities such as covalent bonds and lone electron pairs allowed for a straightforward interpretation based on chemical intuition. Quantitative indexes based on the total source function of specifically defined sets of orbitals were introduced, singling out the contribution of whole molecules in donor–acceptor complexes (where applicable), or just a short list of major contributors within either electron-accepting or electron-donating moiety. Although 18 Environment ACS Paragon Plus

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further validation may be in order, the reliability of these indexes was successfully tested against 17 noncovalent interactions comprising hydrogen bonds of rather different character and a few simple examples of halogen bond. Foster-Boys, Pipek-Mezey, and Natural LMO bases were considered throughout. The different handling of σ–π symmetry separation among these localization schemes inevitably leads to peculiar discrepancies in 𝑆(𝐫, LMO𝑖 ) results. However, it was found that the very elements that distinguish one scheme from another, i.e. the way multiple bonds and lone electron pairs are described, could not be reliably considered in terms of their isolated constituents. In particular, σ orbitals in multiple bonds acting as electron-rich ligands, and lone electron pairs facing away from the electron-accepting ligand, all yield non-negligible contributions. The necessity to consider more than a single orbital as main electron-donor, thus, leveled out the differences between the three localization schemes discussed. The reported data also highlighted stability and robustness of the NLMO criterion compared to FB and PM schemes, whereas the convergence issues encountered while using the latter limited their usefulness and reliability. Attempts at using pruned basis sets to remedy convergence failure were unsuccessful.

ASSOCIATED CONTENT Supporting Information. Complete 𝑆(𝐫, Ω𝑖 ) and 𝑆(𝐫, LMO𝑖 ) data relative to bcps 6–17.

ACKNOWLEDGEMENTS The author declares no competing financial interests.



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Noncovalent Interactions Descriptor Based on the Source Function of

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