Comment on âNatural Bond Orbitals and the Nature of the Hydrogen

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Comment on "Natural Bond Orbitals and the Nature of the Hydrogen Bond" Frank Weinhold, and Eric D Glendening J. Phys. Chem. A, Just Accepted Manuscript • DOI: 10.1021/acs.jpca.7b08165 • Publication Date (Web): 06 Dec 2017 Downloaded from http://pubs.acs.org on December 6, 2017

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The Journal of Physical Chemistry

Comment on “Natural Bond Orbitals and the Nature of the Hydrogen Bond” F. Weinholda* and E. D. Glendeningb a Theoretical Chemistry Institute and Department of Chemistry, University of Wisconsin-Madison, Madison, Wisconsin 53706 b Department of Chemistry and Physics, Indiana State University, Terre Haute, Indiana 47809 Abstract We reply to the recent comments of Prof. A. J. Stone in this Journal [J. Phys. Chem. A 2017, 121, 1531-1534]. Introduction In a recent contribution to this Journal,1 Professor A. J. Stone (hereafter, AJS) argues forcefully that theoretical descriptors based on Natural Bond Orbital (NBO) analysis2-4 are “heavily contaminated with basis-set superposition error,” “meaningless,” and “seriously in error” for proper understanding of H-bonding and other intermolecular interaction phenomena. Here, we wish to clarify the conceptual differences between AJS’s preferred “symmetry adapted perturbation theory” (SAPT)5,6 vs. NBO-based description of supramolecular phenomena,7-9 present evidence from specific physical systems that underscores these differences, and offer direct replies to erroneous assertions concerning basis set superposition error (BSSE) artifacts in the basis of Natural Atomic Orbitals (NAOs)10 that underlie NBO analysis of charge transfer in H-bonding. Critical Comparison of SAPT vs. NBO Description SAPT-type methods can be traced back to the original Gordon-Kim local density method11 for approximating rare-gas interactions as a Coulombic interaction integral between the overlapping densities of isolated atoms, brought (without change) into the short-range attractive region. Early electrostatic-type models12-15 were formalized in perturbation-theoretic fashion by Buckingham and coworkers16 and numerically implemented in SAPT approximation methods6 that could be applied at Hartree-Fock or density functional levels. For present purposes we employ the same “DFT-SAPT (PBE0/AC)” program and theory level used in Ref. 1 to express ESAPT(R) (the

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SAPT model approximation to interaction energy) as the sum of four leading components, (1)

ESAPT(R) = EEL(R) + EIN(R) + EEX(R) + EDI(R)

identified as “electrostatics” (EL), “induction” (IN), “exchange” (EX), and “dispersion” (DI), respectively. An additional Eδ(method) term is often included that insures consistency with the quantum chemical HF/DFT method chosen for numerical evaluations, but we present equivalent “δ” (error compensation) information for the physical SAPT terms of Eq. (1) in figures to follow. A characteristic feature of separable “decompositions” such as Eq. (1) is the implicit assumption that each labeled component is a distinct physical factor that contributes independently to the property of interest. Coupling effects among the components are therefore presumed to be negligible or, if not, must be grouped (and thereby hidden) behind component labels by arbitrary assignment to one or another labeled term to maintain the additivity assumption. Uniquely among the many available energy decomposition analysis (EDA) schemes of this type,13,17-25 the SAPT approximation model (1) conspicuously lacks any labeled contribution for “charge transfer” effects (also identified as “orbital,” “covalency,” “donor-acceptor,” or “delocalization” effects in other EDA methods). Indeed, widely recognized quantummechanical charge-transfer interactions26-31 are dismissed by AJS as “an illdefined part of the induction (polarization) energy [that] vanishes in the limit of a complete basis”32 and “...not a well-defined quantity...merely a basisdependent part of the induction energy” (Ref. 5, p. 154). In contrast, NBO analysis3 is a localized interpretation of a chosen quantum chemical calculation, not a perturbative approximation model. NBO analysis is neither restricted to single-determinant HF/DFT description nor subject to frozen-monomer or other weak-interaction limitations, makes no distinction between intra- vs. intermolecular interaction phenomena, and makes no assumption that couplings between physical contributions (e.g., of steric, electrostatic, or CT type) are negligible. Instead, NBO analysis provides a localized interpretation of the input wavefunction or electron density for any chosen theoretical level, including, in principle, variational or perturbative approximation methods up to and including exact solution of Schrödinger’s equation. More specifically, the analyzed wavefunction or density is reexpressed in the complete orthonormal set of 1-center (NAO) and 2-center


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(NBO) localized orbitals that provide most rapidly convergent (maximaloccupancy “natural”) contributions to total electron density. In view of these fundamental differences, there can be no direct numerical comparisons or conflicts between SAPT (an approximation method that must be judged on its own merits) and NBO analysis (which yields localized NAO/NBO representation of the full wavefunction and energy for any chosen species and theory level). Assuming reasonable numerical agreement between ESAPT and EQC for the chosen species and theory level, the only possible conceptual conflicts between SAPT and NBO must concern the electrostaticsoriented labels attached to the four SAPT components in Eq. (1). These labels contrast strongly with the omitted CT-type donor-acceptor component that is consistently identified as the dominant attractive contribution to H-bonding in NBO analysis.7-9 The stage is thus set for fierce disagreements over which of these two competing labeling-schemes provides the more accurate characterization of H-bond attraction. In the following three sections, we raise title questions and present lines of evidence to address both specific aspects of AJS’s criticisms and the reliability of the SAPT approximation model itself. Primary emphasis is placed on four simple physical examples (He···He, Ar···Li+, F−···HF, F−···HSO4−) that represent distinct physical limits for which SAPT and NBO conceptions can be critically compared. Possible bridges between these conceptions are suggested in conclusion. In particular, the Supporting Information (SI) includes extensive comparisons of SAPT with NBO-based “natural energy decomposition analysis” (NEDA)17-20 for each of the physical examples considered herein, providing additional numerical details of unexplained anomalies in SAPT components with respect to alternative NBO-based formulation. Do SAPT Evaluations Contradict the Importance of Charge Transfer in H-Bonding? A fundamental assumption of SAPT-theoretic decomposition is that the transfer of electron density between monomers represents only a minor contribution to the “inductive” effect of classical electrostatic fields, rather than the quantum mechanical effect of stabilizing resonance-type interactions between available donor and acceptor orbitals (only weakly modulated by orbital polarization or other classical field effects). SAPT-based perceptions of the atomic origin or “transfer” of electronic charge density between monomers are intrinsically clouded by the ambiguities of overlapping monomer charge densities (common to all Gordon-Kim-type methods) as well as SAPT-specific



“midbond density,” associated with ghost basis functions that are arbitrarily positioned midway between monomer mass-centers to assist SAPT convergence. Such overlap- and midbond-based ambiguities necessarily affect each SAPT contribution in Eq. (1), so that SAPT-theoretic misjudgment of the nature or extent of CT-dependence extends far beyond its supposed ill-defined effect on the inductive (EIN) term, as widely assumed by SAPT adherents. In NBO theory, atomic and monomer charge assignments are based on natural population analysis (NPA),10 where each natural atomic charge is simply determined from the summed NAO populations on the atom. The total charge transfer (QCT) of H-bond formation has been shown33 to exhibit robust numerical correlations (Pearson χ2 ≈ 0.9) with H-bond strength and other known signatures of H-bonding, such as hydride IR red-shift and bond elongation, downfield NMR proton shielding, and the like. Such correlations, although strongest for NPA-based QCT(NPA), are found to be similarly strong for Bader’s QCT(AIM) 34,35 and other popular atomic-charge measures, which in turn are known to exhibit strong correlations with one another.36,37 Hence, if ESAPT model values are reasonably consistent with other measures of H-bond strength, they must exhibit similar strong correlations with NPA or other reasonable measures of CT, whether or not such dependence is reflected in the assigned SAPT labels. Indeed, Scheiner38 has demonstrated such numerical correlations of total ESAPT (and its individual components) with NBO-based measures of charge transfer for a wide variety of H-bonded species. These facts alone serve to discount AJS’s assertions that CT is an unimportant factor in Hbonding. Are SAPT Approximations and Labels Reliable? It is evident that errors of any non-variational approximation to a given computational level could in principle have the same or opposite sign as those with respect to deeper levels of theory. In the latter case, fortuitous error cancellations allow the approximation to exhibit unexpectedly high numerical accuracy with respect to experiment or higher-level theory. Here, we employ simple numerical examples, representative of idealized component types, to critically examine both the numerical error (δ ≡ ESAPT − EQC) of SAPT approximation with respect to conventional quantum-chemical calculation (i.e., at the same method and basis level employed for SAPT calculation of monomer properties or comparison NBO descriptors of the composite system) and the conceptual accuracy of SAPT labels with respect to common chemical understanding. The examples include both weakly bound rare-gas species (He⋅⋅⋅He, Ar⋅⋅⋅Li+), where SAPT perturbative assumptions should be as


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defensible as possible, and strong ionic (F−⋅⋅⋅HF) and doubly ionic (F−⋅⋅⋅HSO4−) H-bonded species, where SAPT assumptions appear inherently unrealistic. If the SAPT approximation model itself is failing, the labeling of its components becomes immaterial. Exchange Repulsion Limit: He⋅⋅⋅He The helium dimer can be taken as an exceptionally favorable case for SAPT assumptions and the idealized limit of pure steric repulsion, particularly at HF level where attractive dispersion-type contributions are absent. Tang and Toennies39 have shown the high accuracy of HF-level description of the repulsive portion of experimentally determined He⋅⋅⋅He potentials, indicating the aptness of this level for discussion of pure exchange repulsion effects. In this case, the SAPT model appears to give excellent performance over a wide range of distances, with negligible deviation from conventional quantumchemical treatment down to R ≈ 1.5 Å.40 For present purposes, we can therefore assume that unambiguous comparisons of the SAPT exchange component EEX with other theoretical and empirical measures of He⋅⋅⋅He steric repulsion can be safely drawn throughout the region ∞ > R ≥ 1.5 Å, where breakdown of the SAPT model is not considered to be an issue. In NBO analysis, assessment of steric repulsion is provided (via the STERIC keyword) in terms of steric exchange energy ESXE, a characteristic “orthogonalization energy” associated with occupied orbitals of the HF/DFT determinant. It is well known that steric forces are intrinsically of non-classical origin, associated with the energy-raising effects of exchange-antisymmetry that distinguishes simple orbital product (Hartree) and determinantal (HartreeFock) forms of the wavefunction. However, it was recognized41-43 that the energy differences between Hartree- and Hartree-Fock wavefunctions are essentially removed if the orbitals are required to be mutually orthogonal (consistent with an elementary mathematical requirement for eigenorbitals of Hermitian operators, and for proper Fermi-Dirac anticommutation properties of the associated field operators in 2nd-quantized formulation). This observation suggests the close numerical connection between steric exchange forces and the calculable energy differences between non-orthogonal and orthogonal orbitals that underlies NBO evaluation of ESXE. The simple physical picture of “kinetic energy pressure” that connects Paulitype steric repulsions with interatomic orbital orthogonality was admirably sketched by Weisskopf as one of the fundamental physical principles that govern all natural phenomena.44 This physical picture is the direct starting



point for numerical evaluation of energy differences between overlapping (“pre-orthogonal”) PNAOs and orthonormal NAOs and resulting ESXE steric exchange energy in natural steric analysis.45 Excellent harmony between NBObased ESXE and empirical steric concepts is demonstrated by close agreement of ESXE with total interaction energy of rare-gas dimers in the long-range limit where Pauli repulsion is expected to dominate, as well as consistency of ESXEderived “natural van der Waals radii” with empirical Pauling values and other common steric measures.46,47 Figure 1 compares both the NBO-based ESXE (dashed line, x’s) and the SAPT-based EEX (dotted line, squares) with the full HF-level interaction potential ∆E (solid line, circles) in the weak-repulsion region (2.5 > R > 1.5 Å). Significant discrepancies between the two “exchange” estimates are immediately apparent. As found for other rare-gas potentials ,45 NBO-based ESXE accurately tracks with exact ∆E at long-range, consistent with the conventional steric picture of rare-gas repulsions. In contrast, SAPT-based EEX is conspicuously “over-repulsive” (EEX > ∆E) at all distances, requiring physically implausible off-setting long-range attractions from other components (other than dispersion, which is strictly absent at HF level) to maintain overall ESAPT ≈ ∆E consistency.

Figure 1. Short-range region of He⋅⋅⋅He repulsive potential (HF/aug-cc-pVTZ level; solid line, circles), with comparison SAPT-based EEX (dotted line, squares) vs. NBO-based ESXE (dashed line, x’s) assessments of Pauli-type exchange repulsion.


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For example, at RHe⋅⋅⋅He = 1.6 Å, where slight ∆E − ESXE deviations are first perceptible in Fig. 1, the corresponding ∆E − EEX deviations require large canceling attractions (ca. 4 kcal/mol!) to maintain ESAPT ≈ ∆E. However, the actual weak induction dipoles at each center (0.06D magnitude) are easily shown to be oppositely oriented, so the expected leading dipole-dipole contribution must also be repulsive at this separation (consistent with the sign of ∆E − ESXE). The cancellation of errors must therefore arise from some compensating He⋅⋅⋅He attractive terms (of unknown physical origin) in other SAPT components, calling the accuracy of the associated labels into similar question. The unphysically large SAPT estimate EEX of steric repulsions can evidently be attributed to coupling terms mentioned above. It is evident that NAO orthogonalization (associated with Weisskopf’s “kinetic energy pressure” picture of steric repulsions) leads to the onset of non-spherical distortions (nonvanishing multipoles of many orders) that are strongly coupled to “induction” (EIN) and other electrostatic effects. However, such “sterics-induced induction” leads to oppositely oriented dipoles, apparently included as an additional repulsive coupling correction in the EEX component, which in turn mandates that cancelling “attractions” must arise in remaining components to maintain overall ESAPT ≈ ∆E consistency. Surprisingly, the SAPT EIN component remains relatively negligible throughout the range of Fig. 1, so the compensating attractions appear primarily in EEL (and EdHF correction component 40). All such physical and numerical inconsistencies point to the implausibility of SAPT-based definitions of “exchange” and other labels, even in this most favorably simplified limit. Note that significant over-repulsiveness of the SAPT EEX exchange component (leading to typical δ = ESAPT − EQC errors of positive sign, opposite to the energy-lowering effect of variational improvement) seems to be a rather general tendency of ESAPT modeling of intermolecular forces, as found in other examples to be considered below. Induction Limit: Ar⋅⋅⋅Li+ Let us next consider interaction of a small ion (He-like Li+) with a polarizable rare-gas (Ar) as a prototype of ion-induced dipole interaction, the claimed “induction” (EIN) component of SAPT theory. We first consider the accuracy of the SAPT model itself for estimating the quantum-chemical EQC (here taken as PBE0/aug-cc-pVTZ level) Ar⋅⋅⋅Li+ potential curve. Figure 2 displays the comparison of ESAPT with Efull throughout the region (2Å < R < ACS Paragon Plus Environment


6Å) of the binding well. In this case, the energy error δ ≡ ESAPT − EQC is again of consistent positive sign (with EEX > ESXE at all R), but with steadily increasing magnitude as the binding minimum is approached (as shown by the vertical dotted lines marking error thresholds of 0.1 or 1.0 kcal/mol in Fig. 2). Despite the rapidly increasing errors of the SAPT model in the neighborhood of the equilibrium minimum, we can take 2Å < R < 5Å as the approximate region where SAPT induction (EIN) is the dominant attractive component of the potential energy curve, in accordance with the classical ion-induced dipole picture of the binding mechanism.

Figure 2. Comparison of full potential energy curve (PBE0/aug-cc-pVTZ; solid line, circles) vs. corresponding SAPT model (DFT-SAPT (PBE0/AC; dashed line, +’s) for ion-induced dipole binding of Ar⋅⋅⋅Li+. The vertical dotted lines mark approach distances where the errors of the SAPT approximation exceed 0.1 kcal/mol (near 3.9Å) or 1.0 kcal/mol (near 2.2Å).

Within this region, the induced dipole moment on the Ar atom (µAr) is the expected observable signature of induction strength. This property is readily calculable at any R and can be analyzed (via the DIPOLE keyword48) in terms of localized “NBO dipole” or delocalized “NLMO dipole” contributions, whose difference is the resonance-type charge transfer (CT) contribution. Such “CTinduced induction” is another important coupling term, which in this case must be absorbed into the EIN component (because SAPT makes no provision for ECT).


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Figure 3 shows the calculated µAr induction dipole, comparing the classicallike NBO dipole (+’s) with full NLMO dipole (circles) values. The onset of strong CT contribution to the induction dipole is evident near 4Å, steadily increasing toward ca. 50% CT-enhancement near the potential minimum. NBO dipole analysis is thereby able to explicitly quantify the strong CTdependencies of the induction dipole that are systematically dismissed or ignored by SAPT adherents, both in this simple limiting case and more complex H-bonding interactions.

Figure 3. Localized analysis of induced electronic dipole moment on Ar (µAr, Debye) in Ar⋅⋅⋅Li+ interaction (PBE0/aug-cc-pVTZ level), comparing the full value (“NLMO”, circles) with an idealized localized description (“NBO”, squares) that neglects chargetransfer (“CT”) contributions.

Another feature of NBO analysis can be used to exhibit the qualitative importance of CT interaction for a broader range of energetic, structural, and spectroscopic properties in Ar⋅⋅⋅Li+ binding. Specifically, NBO energetic analysis ($DEL keylist input, Ref. 48, Sec. B.5) allows one to selectively delete some or all Fock/Kohn-Sham matrix elements associated with intermolecular donor-acceptor (CT) interactions and variationally re-calculate the potential energy surface as though these interactions are absent. Table 1 shows the powerful effect of such “no-CT” deletions on the calculated properties of the [ArLi]+ complex: binding energy (∆E) reduced by ca. 70%, equilibrium bond length (Req) elongated by ca. 0.6Å, and vibrational stretching frequency (ν) redshifted by ca. 260 cm−1. These signatures of qualitatively reduced binding in the no-CT complex confirm and extend the conclusions drawn from dipole analysis.



property ∆E (kcal/mol) Req (Å) ν (cm−1)

full No-CT 7.22 2.29 2.35 2.98 395 134

Table 1. Comparison properties of [Ar···Li]+ in “full” PBE0/aug-cc-pVTZ vs. “no-CT” ($DEL deletion of intermolecular CT) optimizations, showing significant reduction of binding energy ∆E, elongation of bond length Req, and red-shift of vibrational stretching frequency ν... At the optimum no-CT separation (RAr···Li = 2.98Å), the calculated energy-change of CT-deletion is 2.23 kcal/mol (ca. 0.0007% of total E).

Ion-Dipole Limit: F−⋅⋅⋅HF The bifluoride ion [FHF]− was among the earliest recognized and still most influential examples of the H-bonding phenomenon.49,50 Due to its unusual D∞h-symmetric [F···H···F]− geometry, bifluoride is also the prototype of strong “charge-assisted” H-bonding.51 Formally, bifluoride is the H-bonded complex of closed-shell F− ion with neutral HF molecule (NPA charges: QF = −QH = −0.555). Accordingly, H-bond formation must involve significant CT in this case, altering the initial fluoride ion (QF = −1) and neutral HF (QF = −0.555) to achieve the final D∞h-symmetric charge distribution. Given the manifestly strong CT of bifluoride formation and the contrary SAPT-based supposition that CT is merely “an ill-defined part of the induction [that] vanishes in the limit of a complete basis,”32 one might expect complete failure of the SAPT numerical model in this case. It is therefore remarkable that the SAPT numerical model actually provides a qualitatively reasonable representation of the frozen-monomer potential for F−⋅⋅⋅HF throughout the region of the binding minimum, as shown in Figure 4. As usual, the SAPT model is most accurate in the long-range limit, with errors increasing at smaller RF···F as shown by the vertical dotted lines for various δ values. The behavior of individual SAPT components (not shown; see SI) is also rather typical of that in other species, with exaggerated “steric” EEX and off-setting “electrostatic” EEL, EIN components combining to give surprisingly reasonable net binding. Apparently, the composite SAPT model somehow captures important physical features of H-bonding despite the misleading assumptions expressed in the labels of its individual components.


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Figure 4. Similar to Fig. 2, for F−···HF interaction (frozen monomer geometry, RHF = 0.9197Å). The asterisk and short solid bar mark the geometry and binding energy of fully relaxed optimization where the error of frozen-monomer approximation is avoided.

Compared to a fully optimized PBE0/aug-cc-pVTZ description (as shown by the asterisk and small horizontal energy level marked near 2.3Å), the SAPT model underestimates the actual binding by more than 10 kcal/mol. However, most of this error is attributable to the frozen-monomer approximation that is inherent in SAPT-level modeling and severely limits its quantitative accuracy in all but the weak-interaction limit. The smaller δ ≈ +3 kcal/mol deviation with respect to quantum-chemical EQC near 2.3Å is also somewhat fortuitous, because the sign of characteristic δ errors switches from +1 kcal/mol at the “δ=1” marker to −10 kcal/mol at the “δ=10” marker, with favorably reduced error magnitude as the cross-over is approached near the binding minimum. Despite such numerical peculiarities, the qualitative reasonableness of the SAPT binding well for this large-CT species suggests that the ambiguities of SAPT overlap and midbond densities somehow permit partial capture of physically important CT effects, even if the labels attached to SAPT components suggest otherwise. As further confirmation of the importance of CT to bifluoride binding, we can again re-optimize the geometry with variational E($DEL) deletion of CT contributions, as summarized in Table 2. As usual, “no-CT” reoptimization leads to significant binding loss (ca. 55%), bond elongation (ca. 0.8Å), and IR red-shift (ca. 400 cm−1). At the elongated no-CT geometry, the E($DEL)



binding energy comes into much closer agreement with the idealized ion-dipole energy for monomer dipole µHF = 1.8081D, and thus into better correspondence with SAPT labels. property full No-CT 38.92 17.50 ∆E (kcal/mol) Req (Å) 2.31 3.10 a −1 633 230a ν (cm ) a

for fully relaxed geometry

Table 2. Similar to Table 1, comparing “full” vs. “no-CT” properties for [F···HF]−.... At the optimum no-CT separation (RF···F = 3.10Å), the calculated energy-change of CTdeletion is 6.82 kcal/mol (ca. 0.005% of total E).

Ion-Ion Limit: F−⋅⋅⋅HSO4− As a final example, we consider the paradoxical case of “anti-electrostatic” H-bonding (AEHB) between ions of like charge,52 contrary to the fundamental “like charges repel” tenet of classical electrostatics. Theoretical and experimental evidence is now available53-56 for a wide variety of AEHB species (including polyanionic clusters up to net charge −4), signaling the importance of this counterintuitive class of H-bonding interactions in a broad range of gaseous and condensed-phase phenomena. Here, we consider the AEHB interaction of elementary fluoride (F−) and bisulfate (HSO4−) anions, which is predicted57 to exhibit a robust binding well of ca. 7 kcal/mol that is kinetically stabilized by a broad potential barrier opposing “Coulomb explosion” to separated ions. Figure 5 shows the calculated PBE0/aug-cc-pVTZ(frozen monomer) binding curve for F−⋅⋅⋅HSO4− (circles), together with its SAPT model approximation (+’s) and leading “electrostatic” component (EEL, squares). Once again, the SAPT frozen-monomer approximation severely degrades quantitative accuracy, with the metastable well-depth reduced to only about half its fully optimized value. However, within this limitation the SAPT model is again seen to provide a crude qualitative picture of the location (RF⋅⋅⋅O ≈ 2.5Å) and non-zero depth (ca. 2 kcal/mol) of the AEHB binding well. This unexpected success of the SAPT model may again be attributed to CT-type contributions that enter indirectly through the ambiguities of overlap, midbond densities, and “hidden” coupling terms, consistent with the general CTdependence of SAPT results as found in other cases.38


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Figure 5. Comparison of quantum-chemical (frozen-monomer geometry) potential energy curve (PBE0/aug-cc-pVTZ; solid line, circles) vs. corresponding SAPT model (DFT-SAPT (PBE0/AC; dashed line, +’s) for AEHB binding of F−⋅⋅⋅HOSO3−. Vertical dotted lines mark approach distances where errors of the SAPT approximation exceed 0.1 kcal/mol (near 4.2Å) or 1.0 kcal/mol (near 2.5Å). Corresponding SAPT “electrostatic” EEX component values (squares) are shown for comparison in the region of the binding well.

Still more remarkable (for opposite reasons) is what SAPT identifies as the “electrostatic” contribution (squares). The EEL component is strongly overrepulsive at longer range (i.e., by ca. 2 kcal/mol near RF⋅⋅⋅O ≈ 5Å), necessarily cancelled by equally mysterious “long-range attractions” from other components. However, EEL abruptly reverses direction near 3.5Å, plunging sharply (by >60 kcal/mol!) to become net attractive near 2Å. Based on common conceptions of Coulombic point-charge electrostatics, one might conclude that the erratic behavior of the SAPT EEL component is at least as mysterious as the H-bonding phenomenon the SAPT model purports to rationalize. Consistent with other examples discussed above, it is found that E($DEL) deletion of CT removes any trace of the H-bonding well in F−⋅⋅⋅HSO4−, with attempted re-optimization leading promptly to the Coulomb explosion predicted by classical electrostatics. Thus, each of these examples illustrates numerical limitations of the SAPT model and misleading aspects of its “CTdenial” component labels. Overall, we can conclude that SAPT results and component labels provide no reliable basis for conclusions about the negligibility of CT in H-bonding.



Can SAPT-NBO Conceptual Differences be Attributed to Superposition Error of NAOs vs. AOs? AJS’s primary criticism of NBO analysis1 focuses on supposed BSSE associated with transformation from AOs to NAOs. This criticism is certainly invalid, because the transformation from AOs to NAOs is numerically exact, and no possible “incompleteness” or loss of numerical accuracy can occur in this basis-set transformation. But what of possible BSSE artifacts in the AO basis itself (which of course must be preserved in the transformation to NAOs)? Serious concern over BSSE artifacts was certainly reasonable for the near-minimal AO basis sets of pioneering quantum chemical studies, when the Boys-Bernardi counterpoise method for BSSE estimation was first introduced.58 However, it is wellrecognized that BSSE artifacts must vanish as the AO basis set is extended to completeness. Convergence studies, including complete basis set extrapolations, have demonstrated that such artifacts are relatively negligible in the augmented multi-zeta basis sets now in common usage.59 Furthermore, Mentel and Baerends60 have recently shown at the theorem level that the counterpoise estimate of BSSE is unreliable, and the associated “corrections” should be avoided in studies based on reasonably current AO basis standards. It is noteworthy in this respect that NBO analysis of H-bonded species has presented a consistent picture of the dominant CT character of H-bonding from the initial pioneer applications61-63 to the high-level computational studies (and experimental comparisons) 64,65 of the present day, despite remarkable advances in all aspects of computational methods and basis set technology. Hence, neither current understanding of BSSE nor its demonstrably marginal effect on the NBO-inferred CT picture of H-bonding (as confirmed over more than a quarter-century of computational studies) provides support for AJS’s repeated assertions that NBO analysis is “swamped” or “heavily contaminated” with BSSE. Of course, we can explicitly demonstrate such basis-insensitivity of NBO CT-descriptors for any particular case, such as the HF···HF example selected by Stone.1 For this species, Figure 6 shows the variability of key NBO CTdescriptors (QCT; Enσ*(2); upper panels), as well as total calculated binding energy (−∆Eint; lower panel) for two distinct DFT choices (PBE0, circles, solid line; B3LYP, x’s, dashed line) and all available aug-cc-pVnZ basis sets, n = 26. Although both NBO descriptors and total calculated ∆Eint (for both DFT choices) exhibit slight even-odd oscillations with increasing n, it is clear that NBO-based QCT , Enσ*(2) descriptors of charge transfer remain tightly clustered


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around values that are widely separated from corresponding SAPT assessments, no matter which n-cardinality is selected over the range of ca. 10-fold variations in basis size. We challenge AJS or other SAPT adherents to identify any species for which similar n-dependent studies would lend support to AJS’s baseless claim that NBO vs SAPT differences are attributable to “heavy contamination” of NAO/NBO results by basis-set superposition errors.

Figure 6. Computed NBO charge transfer QCT (upper panel), NBO 2nd-order estimate Enσ*(2) of the nF →σHF* interaction energy (middle panel), and total DFT interaction energy −∆Eint (lower panel) in HF···HF for PBE0 (circles, solid line) or B3LYP (x’s, dashed line) DFT levels as a function of basis set extension from aug-cc-pVDZ (n = 2. nbf = 64) to aug-cc-pV6Z (n = 6, nbf = 632), showing the close clustering of NBO results around values that remain far separated from corresponding SAPT estimates throughout ca. 10-fold expansion of basis size toward the complete basis set limit. The double-circles mark the computational level used throughout the present work.

What then could be the misunderstanding underlying such assertions? A hint is provided by AJS’s surprising remarks on orbital orthogonality (Ref. 1, p. 1533): At long range, the orthogonalization has no effect because the orbitals of the two molecules have negligible overlap, but at shorter distances its effect is to increase the energy, by at least 40 kJ mol−1 at the equilibrium geometry.



That is, the NBO orthogonalization causes a deterioration in the quality of the basis set on each molecule. As discussed elsewhere,10 “NBO orthogonalization” begins at the NAO level, thereby insuring compliance with general mathematical requirements for Hermitian eigenorbitals throughout the NAO → NHO → NBO → NLMO → MO sequence. As noted by AJS, AO → NAO orthogonalization indeed leads to significant energy differences, but such differences are readily attributed to the important physical effects of steric repulsions,41-45 rather than to any “deterioration in the quality of the basis set.” Such steric opposition must always be offset against the purely attractive effects of donor-acceptor stabilization to gain an estimate of net H-bond strength. The close association of Pauli exchange repulsions with orbital orthogonalization could also be inferred from the contrasting commutation properties for orthogonal vs. nonorthogonal orbital operators in 2nd-quantization formulation, where the former lead automatically to proper Fermi-Dirac anticommutation relations, but the latter give rise to a complex assortment of intra- and inter-orbital contributions66-68 that defy simple physical interpretation. As shown in the He···He example, the NBO-based steric exchange energy accurately approximates the repulsive wall of closed-shell interaction potentials, whereas SAPT portrays a puzzling mix of components beyond the exchange-repulsion contribution that is expected to dominate this region. In this light, the energy differences between AOs and NAOs should not be considered as “BSSE contamination” of the NBO description, but rather as the proper remedy for unphysical neglect of steric exchange effects in the nonorthogonal basis AOs of SAPT description. For example, such neglect may give rise to the variational absurdity of final occupied MOs that are all higher in energy than an equal or greater number of basis AOs that contribute to their LCAO-MO description, or to the similar absurdity of basis AOs on one atom that apparently lower their energy when overlapping basis AOs on another atom.69 Proper recognition of the steric exchange contribution to overall NBO energetics also serves to correct the misimpression1 that the NBO procedure has “nothing to say” about contributions to the total interaction energy other than CT. Summary and Conclusions In the preceding three sections, we have attempted to confront the misleading aspects of Prof. Stone’s SAPT-based criticisms [1] in terms of three distinct lines of evidence, each offered to address the section title question:


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• NBO-based CT descriptors demonstrate strong correlative relationships with all known experimental signatures of H-bonding, contradicting SAPT-based conclusions that CT plays a negligible role in H-bonding; • SAPT labels are inconsistent with broader chemical understanding of the binding in specific species (He···He, Ar···Li+, F−···HF, F−···HSO4−) that represent well-defined limiting cases; • BSSE is demonstrably irrelevant to assessing the role of CT in Hbonding, but steric exchange effects (particularly, in AO → NAO transformation) and ambiguous SAPT handling of overlap and coupling terms adequately account for the profoundly different NBO vs. SAPT perceptions of this role. In each case, the presented evidence indicates that the title question of the section must be answered negatively. As a possible way forward, we suggest that SAPT programmers make provision for introducing the 4-index AO → NAO transformation (easily obtained from NBO program output; NBO Manual, p. B-9) as a constructive step toward resolving SAPT vs. NBO disagreements. Simple AO → NAO relabeling of SAPT components could clarify what are effectively wasteful rehashes of disputes between AO-based (i.e., Mulliken-type) vs. NAO-based (NPA) assignments of atomic charge and charge transfer, inherently ambiguous in the current SAPT framework. Further progress can result from closer comparisons of SAPT components with corresponding features of NBO-based NEDA decomposition (see SI), where the importance of CT and CT-induced couplings is evident throughout. In addition, it seems possible that the many suggested higher-order elaborations of SAPT theory (see, e.g., Lao et al.50) could contribute to partial resolution of numerical anomalies associated with the low-order evaluation of terms of Eq. (1) as employed in Ref. 1 and the present work. Nevertheless, the SAPT numerical evaluations that underlie Ref. 1 seem manifestly inadequate to support the extreme conclusions drawn by AJS concerning the nature of hydrogen bonding and the “meaningless” quality of NBO analysis. Acknowledgments We thank Mary Van Vleet and J. R. Schmidt for assistance with the DFT-SAPT (PBE0/AC) calculations. Discussions with C. R. Landis are also gratefully acknowledged.



Supporting Information The supporting information is available free of charge on the ACS Publications website at .... “Direct SAPT Comparisons with Natural Energy Decomposition Analysis (NEDA)” provides additional numerical comparisons and commentary on SAPT vs. NEDA components for each of the four examples considered in this work. References (1) Stone, A. J. Natural bond orbitals and the nature of the hydrogen bond, J. Phys. Chem. A 2017, 121, 1531-1534. (2) Glendening, E. D.; Landis, C.R.; Weinhold, F. Natural bond orbital methods, WIREs Comput. Mol. Sci. 2012, 2, 1-42. (3) Weinhold, F.; Landis, C.R.; Glendening, E. D. What is NBO analysis and how is it useful? Int. Rev. Phys. Chem. 2016, 35, 399-440. (4) Weinhold, F.; Landis, C. R. Discovering Chemistry with Natural Bond Orbitals; Wiley-VCH: Hoboken NJ, 2012. (5) Stone, A. J. The Theory of Intermolecular Forces, 2nd ed.; Oxford U. Press: London, 2013. (6) Jeziorski, B.; Moszynski, R.; Szalewicz, K. Perturbation theory approach to intermolecular potential energy surfaces of van der Waals complexes. Chem. Rev.1994, 94, 1887-1930. (7) Reed, A. E.; Curtiss, L. A.; Weinhold, F. Intermolecular interactions from a natural bond orbital, donor-acceptor viewpoint. Chem. Rev. 1988, 88, 899-926. (8) Weinhold, F. Resonance character of hydrogen-bonding interactions in water and other H-bonded species. In, R. L. Baldwin, R. L.; Baker, D. (Eds.), Peptide Solvation and H-Bonds: Advances in Protein Chemistry, Vol. 72; Elsevier: Amsterdam, 2006, pp. 121-155. (9) Weinhold, F.; Klein, R. A. What is a hydrogen bond? Resonance covalency in the supramolecular domain. Chem. Educ. Res. Pract. 2014, 15, 276-285 2014.


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(21) Bickelhaupt, F. M.; Baerends, E. J. Kohn-Sham density functional theory: Predicting and understanding chemistry. Rev. Comput. Phys.2000, 15, 1-86. (22) Nakashima, K.; Zhang, X.; Xiang, M.; Lin, Y.; Lin, M.; Mo, Y. Blocklocalized wavefunction energy decomposition (BLW-ED) analysis of σ/π interations in metal-carbonyl bonding. J. Chem. Theor. Comput 2008,.7, 639654. (23) R. Z. Khaliulin, R. Z.; Lochan, R.; Cobar, E.; Bell, A. T.; Head-Gordon, M. Unravelling the origin of intermolecular interactions using absolutely localized molecular orbitals. J. Phys. Chem. A 2007, 111, 8753-8765. (24) Michalak, A.; Mitoraj, M.; Ziegler, T. J. Bond orbitals from chemical valence theory. J. Phys. Chem. A 2008, 112, 1933-1939. (25) Frenking, G.; Bickelhaupt, F. M. The EDA perspective of chemical bonding. In, Frenking, G.; Shaik, S. S. (Eds.), The Chemical Bond: Fundamental Aspects of Chemical Bonding; Wiley: New York, 2014, pp. 121157. (26) Mulliken, R. S. Molecular compounds and their spectra: III. The interaction of electron donors and acceptors. J. Phys. Chem. 1952, 56, 801-822. (27) Coulson, C. A. The hydrogen bond - review of the present position. Research 1957, 10, 149-159. (28) Herzberg, G. Molecular Spectra and Molecular Structure. III. Electronic Spectra and Electronic Structure of Polyatomic Molecules; Van Nostrand Reinhold: New York, 1966, p. 427. (29) Pimentel, G. C.; McClellan, A. L. The Hydrogen Bond; W. H. Freeman: San Francisco, 1960. (30) Ratajczak, H. Charge-transfer properties of the hydrogen bond. I. Theory of the enhancement of dipole moment of hydrogen-bonded systems. J. Phys. Chem. 1972, 76, 3000-3004. (31) Weinhold, F.; Landis, C. R. Valency and Bonding: A Natural Bond Orbital Donor-Acceptor Perspective; Cambridge U. Press: New York, 2005, Chap. 5. (32) A. J. Stone, A. J. Computation of charge-transfer energies by perturbation theory. Chem. Phys. Lett. 211, 101-109 (1993). ACS Paragon Plus Environment

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(33) Weinhold, F.; Klein, R. A. What is a hydrogen bond? Mutually consistent theoretical and experimental criteria for characterizing H-bonding interactions. Mol. Phys. 2012, 110, 565-579. (34) Bader, R. F. W. Atoms in Molecules: A Quantum Theory; Oxford U. Press: London, 1994. (35) Weinhold, F. Natural bond critical point analysis: Quantitative relationships between NBO-based and QTAIM-based topological descriptors of chemical bonding. J. Comp. Chem. 2012, 33, 2440-2449. (36) Wiberg, K. B.; Rablen, P.R. Comparison of atomic charges derived via different procedures. J. Comp. Chem. 1993, 14, 1504-1518. (37) Gross, K. C.; Seybold, P. G.; Hadad, C. M. Comparison of different atomic charge schemes for predicting pKa variations in substituted anilines and phenols. Int. J. Quantum Chem. 2002, 90, 445-458. (38) Scheiner, S. (oral presentation to International Conference on Chemical Bonding, Kauai, HI, 2014). (39) Tang, K. T.; Toennies, J. P. A simple theoretical model for the van der Waals potential at intermediate distances. I. Spherically symmetric potentials. J. Chem. Phys.1977, 66, 1496-1506. (40) This assessment is misleading, because the standard DFT-SAPT (PBE0/AC) program values plotted in Fig. 1 include an additional ad hoc “EdHF” component [not shown in Eq. (1), but defined as EdHF ≡ EHF − ESAPT] that makes “δ = ESAPT − EQC = 0” tautologically true in the HF case. According to this EdHF correction factor, Fig. 1 should be marked with “δ = 0.1” kcal/mol near 2.60Å, “δ = 1.0” kcal/mol near 1.72Å, and “δ = 10.0” kcal/mol near 1.12Å. These additional numerical issues, although concerning, will be ignored for present purposes. (41) Christiansen, P. A.; Palke, W. E. A study of the ethane internal rotation barrier. Chem. Phys. Lett.1975, 31, 462-466. (42) Christiansen, P. A.; Palke, W. E. Effects of exchange energy and orbital orthogonality on barriers to internal rotation. J. Chem. Phys. 1977, 67, 57-63.



(43) Wilson, C. W., Goddard, W. A. III Ab initio calculations on the H2 + D2 → 2HD four-center exchange reaction. II. Orbitals, contragradience, and the reaction surface. J. Chem. Phys.1972, 56, 5913-2920. (44) Weisskopf, V. F. Of atoms, mountains, and stars: A study in qualitative physics. Science 1975, 187, 605-612. (45) Badenhoop, J. K.; Weinhold, F. Natural bond orrbital analysis of steric interactions. J. Chem. Phys. 1997, 107, 5406-5421. (46) Badenhoop, J. K.; Weinhold, F. Natural steric analysis: Ab initio van der Waals radii of atoms and ions. J. Chem. Phys. 1997, 107, 5422-5432. (47) Badenhoop, J. K.; Weinhold, F. Natural steric analysis of internal rotation barriers. Int. J. Quantum Chem. 1999, 72, 269-280. (48) Weinhold, F.; Glendening, E. D. NBO 6.0 Program Manual (http://nbo6.chem.wisc.edu/nbo6ab_man.pdf), Sec. B.6.3. (49) G. A. Jeffrey, G. A. An Introduction to Hydrogen Bonding. Oxford U. Press: London, 1997, p. 3. (50) Lao, K. U.; Schäffer, R.; Jansen, G.; Herbert, J. M. Accurate description of intermolecular interactions involving ions using symmetry-adapted perturbation theory. J. Chem. Theory Comput. 2015, 11, 473-2486. (51) Gilli, P.; Bertolasi, V.; Gilli, G. Covalent nature of the strong homonuclear hydrogen-bond -- Study of the O-H···O system by crystal-structure correlation methods. J. Am. Chem. Soc. 1994, 116, 909-915. (52) Weinhold, F.; Klein, R. A. Anti-electrostatic hydrogen bonds. Angew. Chem. Int. Ed. 2014, 53, 11214–11217. (53) Knorr, A.; Ludwig, R. Cation-cation clusters in ionic liquids: Cooperative hydrogen bonding overcomes like-charge repulsion. Sci. Rep. 2015, 5, 17505. (54) Knorr, A.; Stange, P.; Fumino, K.; Weinhold, F; Ludwig, R. Spectroscopic evidence for clusters of like-charged ions in ionic liquids stabilized by cooperative hydrogen bonding, ChemPhysChem 2015, 17, 458–462.


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(55) Fatila, E. M.; Twurm, E. B.; Sengupta, A.; Pink, M.; Karty, J. A.; Raghavachari, K.; Flood, A. H. Anions stabilize each other inside macrocyclic hosts. Angew. Chem. Int. Ed.2016, 55, 14057–14062. (56) Strate, A.; Niemann, T.; Michalik, T.; Ludwig, R. When like charged ions attract in ionic liquids: Controlling the formation of cationic clusters by the interaction strength of the counterions. Angew. Chem. Int. Ed.2017, 56, 496– 500. (57) Weinhold, F. Theoretical prediction of robust 2nd-row oxyanion clusters in the metastable domain of anti-electrostatic hydrogen bonding (Inorg. Chem., submitted for publication). (58) Boys, S. F.; Bernardi, F. The calculation of small molecular interactions by the differences of separate total energies. Some procedures with reduced errors. Mol. Phys. 1970, 19, 553 -566. (59) Xantheas, S. S.; Burnham, C. J.; Harrison, R. J. Development of transferable interaction models for water. II. Accurate energetics of the first few water clusters from first principles. J. Chem. Phys. 2002, 116, 1493-1499. Specifically in Table V, the CP-“corrected” estimate of binding energy is seen to have larger absolute error (with respect to CBS-extrapolated basis limit) than the direct CP-free estimate in the vast majority of tabulated cases (25/30, 83%) , often by large margins (viz, 11.8% CP-“corrected” vs. 3.9% CP-free error for cyclic water hexamer at MP2/aug-cc-pVDZ level). The few exceptions to this pattern refer to highly extended basis sets (aug-cc-pV5Z) where both CP-free and CP-corrected errors are negligibly small and of comparable magnitude. (60) Mentel, L. M.; Baerends, E. J. Can the counterpoise correction for basis set superposition effect be justified? J. Chem. Theory Comput.2014, 10, 252267. (61) Reed, A. E.; Weinhold, F. Natural bond orbital analysis of near-HartreeFock water dimer. J. Chem. Phys. 1983, 78, 4066-4073; (62) Reed, A. E.; Weinhold, F.; Curtiss, L. A.; Pochatko, D. J. Natural bond orbital analysis of molecular interactions: Theoretical studies of binary complexes of HF, H2O, NH3, N2, O2, F2, CO, and CO2 with HF, H2O, and NH3. J. Chem. Phys.1986, 84, 5687-5705.



(63) Curtiss, L. A.; Melendres, C. A.; Reed, A. E.; Weinhold, F. Theoretical studies of O2−:(H2O)n clusters. J. Comp. Chem. 1986, 7, 294-305. (64) Belpassi, L.; Tarantelli, F.; Pirani, F.; Candori, P.; Cappelletti, D. Experimental and theoretical evidence of charge transfer in weakly bound complexes of water. Phys. Chem. Chem. Phys. 2009, 11, 9970-9975. (65) Cappelletti, D.; Ronca, E.; Belpassi, L.; Tarantelli, F.; Pirani, F. Revealing charge-transfer effects in gas-phase water chemistry. Acc. Chem. Res. 2012, 45, 1571-1580. (66) Jørgensen, P.; Simons, J. Ab initio analytic molecular gradients and Hessians. J. Chem. Phys. 1983, 79, 334-357 (1983), App. A. (67) Surjan, P. Second Quantization Approach to Quantum Chemistry: An Elementary Introduction; Springer: Berlin, 1989, Chap. 13. (68) Weinhold, F.; Carpenter, J. E. Some remarks on nonorthogonal orbitals in quantum chemistry. J. Mol. Struct. (Theochem) 1988, 165, 189-202. (69) Specific numerical examples of such unrealistic AO energetics are given in Part II of the website tutorial, “MO vs. NBO Analysis: What’s the Difference?” (http://nbo6.chem.wisc.edu/tut_cmo.htm), where additional details of the contrasts between NAO-based vs. conventional AO-based description can be found.


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