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Reply to “Comment on: ‘Natural Bond Orbitals and the Nature of the Hydrogen Bond’ ” Anthony J. Stone, and Krzysztof Szalewicz J. Phys. Chem. A, Just Accepted Manuscript • DOI: 10.1021/acs.jpca.7b09307 • 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

Reply to “Comment on: ‘Natural Bond Orbitals and the Nature of the Hydrogen Bond’ ” Anthony J. Stone∗,† and Krzysztof Szalewicz‡ †University Chemical Laboratory, Lensfield Road, Cambridge, CB2 1EW, U.K. ‡Department of Physics and Astronomy, University of Delaware, Newark, Delaware 19716, USA E-mail: [email protected]

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Introduction Most of the Comment by Weinhold & Glendening (WG) does not address the criticism of the natural bond orbital (NBO) approach to charge transfer set out in Stone’s paper, 1 but seeks to find fault with symmetry-adapted perturbation theory (SAPT), although Stone’s criticism does not depend on comparisons with SAPT. The only part of their Comment that does address the issue is the discussion of their Figure 6, and that misses the point entirely, as we shall explain. The charge-transfer interaction is generally viewed as the consequence of electron density being transferred from one molecule (the electron donor) to the other (the electron acceptor), and the charge-transfer energy is the lowering in energy associated with this transfer. The reference electron densities are those of the separate molecules comprising the dimer, normally in their ground states. However the assignment of electron density to one molecule or the other in the dimer is not well defined. Accordingly, SAPT itself does not attempt to define a charge-transfer energy, though there are several methods for estimating it, 2–5 all of which give much smaller values than the NBO approach. The core of the problem with the NBO approach concerns basis set superposition error (BSSE). This is the term that has been used for many years, but “basis set deficiency error” might be a better term. It normally arises in a calculation of the interaction energy between two molecules when the basis set used for each monomer is inadequate or deficient. The energy calculated for each monomer is then higher than would be obtained with a more complete basis set, but in the calculation on the dimer the deficiency can be reduced, and the energy improved to some degree, by using the basis functions of the partner. The energy calculated for the dimer then includes a correction for the initial poor energy of the monomers as well as the true interaction energy. This use of the partner molecule’s basis functions to improve what is really the energy of the monomer has been recognized in the study of intermolecular interactions since the 1970s. 6 Modern basis sets are much better than they were then, and BSSE is much less of a problem in modern supermolecule calculations, and is often small enough to be ignored. However, basis set deficiency arises in the NBO procedure in a different way. The NBO ap2


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proach operates by constructing atomic and bond orbitals for each molecule separately, forcing them to be orthogonal, and forcing orbitals on the two molecules to be orthogonal to each other. This procedure assigns the one-electron functions unambiguously to one or other molecule, but because of the orthogonalization they can actually contain contributions from the orbitals of the partner molecule. Moreover the orthogonalization distorts the molecular orbitals, resulting in monomer wavefunctions with substantially higher energy, as explained in Stone’s paper. 1 Thus, even if the original basis set would have been adequate for a normal supermolecule calculation, the rise in energy shows that the orthogonalized monomer basis set is seriously deficient. This increased energy is taken as the reference point for calculating the NBO charge-transfer energy, which therefore is substantially exaggerated in magnitude. Even moving towards a complete monomer basis set does not help, because the NBO procedure always creates a severely deficient set of monomer orbitals. (For example, the energy shown by the green curve in Figure 1 of Stone’s paper, which gives a measure of the basis set deficiency with the aug-cc-pVTZ basis, is 41.0 kJ mol−1 at the HF dimer equilibrium geometry. The corresponding figure for the aug-cc-pV5Z basis is 52.5 kJ mol−1 .) That is why the discussion around Figure 6 in WG does not provide any justification for the NBO method’s supposed charge-transfer energy. Note that this assessment of the failure of the NBO method does not depend in any way on comparisons with other methods such as SAPT. Consequently WG’s attack on SAPT is irrelevant. However their criticism is flawed in several important respects, and although it is irrelevant to the issue under discussion we should comment on these.

Comments on Symmetry-Adapted Perturbation Theory Weinhold & Glendening treat SAPT as a low accuracy method and “benchmark” it against plain density-functional theory (DFT). This is inappropriate. There are many published results where SAPT was benchmarked against the coupled-cluster method with single, double, and non-iterative triple excitations [CCSD(T)] and the discrepancies were only a few percent. In particular, in a

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recent blind test of DFT-based methods for computing interaction energies, 7 the difference between a version of SAPT based on a DFT description of monomers, SAPT(DFT) 8–15 and CCSD(T) in the same basis sets was only 1.4%. This high performance is independent of the size of system and of basis set. For example, at the complete basis set (CBS) limit, the difference between SAPT and CCSD(T) interaction energies of He2 at the van der Waals minimum is 1.8% (Ref. 16). Studies on small systems show that SAPT in an infinite-order approach recovers the exact solution of the Schrödinger equation. For example, for the interaction of Li and H atoms, SAPT can recover the full configuration (FCI) interaction energies with errors smaller than 0.0001% (Ref. 17). Thus, if WG want to compare SAPT and NBO, the reference energies should be provided by CCSD(T) or FCI. Ironically, when WG discuss short-range repulsive effects for He2 , the best benchmarks to use are SAPT results from Ref. 16. There are undoubtedly major differences in approach between symmetry-adapted perturbation theory (SAPT) and energy decomposition analyses such as that based on the NBO approach. However many of the statements about SAPT made by WG are seriously incorrect or misleading. SAPT does not originate, as WG claim, from the Gordon-Kim method, which is a supermolecular approach, but has its roots in London and Eisenschitz’s remarkable 1930s papers, 18–20 still valid today in the long-range limit. SAPT is a perturbation approach that obtains the interaction energy directly. In the long-range region, the unperturbed states of the system are simple products ΨiA ΨBj , where the ΨiA are eigenfunctions of the Hamiltonian for molecule A, fully antisymmetrized with respect to the electrons of A, and ΨBj likewise for molecule B. ΨiA and ΨBj describe observable states of the isolated molecules A and B. In principle the products should be antisymmetrized with respect to all electrons, but in the long range that is unnecessary because it does not change the energy or other properties. If no antisymmetrization is used, SAPT reduces to the Rayleigh-Schrödinger (RS) perturbation theory with the zeroth-order Hamiltonian being the sum of Hamiltonians of monomers A and B and the perturbation operator collecting all Coulomb interactions of particles of system A with those of B. Therefore, the modern terminology is to call this the RS expansion.

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At shorter distances, full antisymmetrization becomes necessary, but attempts by many workers over several decades to find a perturbation expansion in terms of the antisymmetrized products, A(ΨiA ΨBj ), all ran into serious difficulties. 21 The solution was the simple but ground-breaking observation that for moderate interaction energies the simple products could be used without antisymmetrization in the perturbation expansion of the wavefunction, and the antisymmetrization was only needed for the evaluation of the energy. 22 Since the simple products form an orthogonal set, the perturbation expansion is straightforward. Each term in the RS expansion then has an exchange correction at short range resulting from the antisymmetrization. The energy at first order comprises the classical electrostatic energy between the unperturbed charge densities, and its exchange counterpart, resulting from the presence of the antisymmetrizer, which describes the repulsive energy due to Pauli exclusion. These are both well-understood physical effects. The second-order terms are the induction energy, readily understood and described in classical terms, and the dispersion energy, a quantum effect understood in terms of correlated fluctuations of the electron densities of the two molecules. These too are well-understood physical effects. They are both modified at short range by exchange corrections, arising from Pauli exclusion effects, again well understood. SAPT does not identify an energetic contribution of charge transfer, but the physical effect of charge transfer is included in the induction energy and in the induction wave function which can be used to determine the change of the charge density upon interaction at any point in space. A good example showing that charge-transfer effects are fully included in SAPT interaction energies is the fact that SAPT predicts the interaction energy of the water dimer at the van der Waals minimum as −21.13 kJ mol−1 whereas the CCSD(T) result is −21.38 kJ mol−1 , both results at the CBS limit. 23 However, so far, nobody has shown that a specific energetic contribution from this charge transfer can be rigorously defined. Nevertheless, SAPT does not question the importance of charge transfer for hydrogen bonding. This importance can be evaluated by computing the difference between perturbed and non-perturbed charge densities. Some aspects of the SAPT approach need to be emphasized here. First, the individual ΨiA and ΨBj can be calculated accurately for the separated monomers, using any suitable basis set. For each

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molecule, the same basis set is used throughout, so there is no basis set superposition error. The basis set can and usually does include the basis functions of the partner molecule. This allows the description of each monomer to be variationally improved to take account of the presence of these functions in the dimer, without them contributing spuriously to the binding energy, whatever basis set is used. When this is done, both molecules use the same basis set, but this doesn’t lead to any difficulties because the electronic wavefunctions of the two molecules are described using different sets of electron coordinates. ‘Midbond’ functions, located in the region between the two molecules, can be included in the basis to improve convergence; they do not affect the physical interpretation of SAPT, and there are no ambiguities associated with their use, as alleged by WG. Moreover, SAPT components can be extrapolated to the CBS limit, allowing basis-set independent interpretations. 16,23,24 The effects of electron correlation can be included in the unperturbed molecular wavefunctions; in the original SAPT this was done using the Møller–Plesset perturbation expansion or CC methods for each molecule, but it is now more usual to use DFT, as in the SAPT(DFT) 8–10 and DFT-SAPT 12–15 methods (developed independently but equivalent). WG assert that the physical interpretation of SAPT components is arbitrary, but this is wholly false. The SAPT or SAPT(DFT) energy terms are closely related to observable properties of the individual molecules — charge densities and polarizabilities. The induction is described at long range by the permanent multipole moments and static polarizabilities, and the dispersion by polarizabilities at imaginary frequency, obtainable in principle from spectroscopic frequencies and oscillator strengths 25 but in practice easy to calculate. At short range, the multipole moments are replaced by unperturbed charge densities and the polarizabilities by density–density response functions. The exchange–repulsion turns out to be nearly proportional to the overlap integral of the molecular electron densities, 26 suggesting that it is a more physically justifiable representation than the NBO exchange term. All of these features make SAPT a very suitable starting-point for the construction of model intermolecular force fields. A recent example is the application to pyridine: a force field has been constructed in this way for pyridine dimer 26 and used sucessfully to predict the crystal structure of a previously unreported polymorph of crystalline pyridine. 27

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The NBO treatment is an energy decomposition analysis, and there is no reason to expect any one-to-one relationship between terms that arise in the two methods. In fact there is every reason to expect otherwise, because the reference wavefunction for each molecule in the NBO picture is built from the distorted basis of orthogonalized natural bond orbitals, and has a substantially higher energy than the isolated molecule. 1 Any energy decomposition analysis gives energy components that, by construction, sum to the correct total energy. WG point out that the omission of their charge-transfer term leads to an incorrect account of the bifluoride (F− ···HF) interaction. Omitting any large component obviously leads to incorrect results, but the fact that the omission of the NBO charge-transfer energy does so says nothing about the usefulness of this particular decomposition or the validity of this definition of charge transfer energy. WG’s remarks about the electrostatic energy reveal only a lack of understanding of the behaviour of the electrostatic interaction between overlapping atomic charge densities. The electrostatic term in SAPT is the classical interaction between the unperturbed molecular charge densities. It does become negative at very short range, even for negative ions, though the nucleus-nucleus repulsion eventually wins, and the first-order exchange term always outweighs the electrostatic term at short distances. For all these reasons, we do not believe that it would be helpful to comment in detail on the particular examples discussed by WG, since they are all seriously flawed as discussed above.

Conclusion We conclude that nothing in Weinhold & Glendening’s Comment refutes our view that the chargetransfer energy as defined in NBO theory is dominated by basis-set superposition error, arising not from any inadequacy in the original monomer basis sets, but from a deficiency in the basis sets caused by the NBO orthogonalization, and is physically meaningless. We repeat that this view is based on an examination of the NBO procedure itself and does not depend on comparisons

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with symmetry-adapted perturbation theory or any other method. However SAPT does provide an accurate, physically understandable and practically useful account of intermolecular interactions.

Acknowledgements K.S. acknowledges support from NSF Grant No. CHE-1566036.

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(8) Misquitta, A. J.; Szalewicz, K. Intermolecular forces from asymptotically corrected density functional description of monomers. Chem. Phys. Lett. 2002, 357, 301–306. (9) Misquitta, A. J.; Jeziorski, B.; Szalewicz, K. Dispersion energy from density-functional theory description of monomers. Phys. Rev. Lett. 2003, 91, 033201. (10) Misquitta, A. J.; Szalewicz, K. Symmetry-adapted perturbation-theory calculations of intermolecular forces employing density-functional description of monomers. J. Chem. Phys. 2005, 122, 214109. (11) Misquitta, A. J.; Podeszwa, R.; Jeziorski, B.; Szalewicz, K. Intermolecular potentials based on symmetry-adapted perturbation theory with dispersion energies from time-dependent density-functional calculations. J. Chem. Phys. 2005, 123, 214103. (12) Hesselmann, A.; Jansen, G. First-order intermolecular interaction energies from Kohn–Sham orbitals. Chem. Phys. Lett. 2002, 357, 464–470. (13) Hesselmann, A.; Jansen, G. Intermolecular induction and exchange-induction energies from coupled-perturbed Kohn–Sham density functional theory. Chem. Phys. Lett. 2002, 362, 319– 325. (14) Hesselmann, A.; Jansen, G. Intermolecular dispersion energies from time-dependent density functional theory. Chem. Phys. Lett. 2003, 367, 778–784. (15) Hesselmann, A.; Jansen, G.; Schütz, M. Density-functional-theory–symmetry-adapted intermolecular perturbation theory with density fitting: a new efficient method to study intermolecular interaction energies. J. Chem. Phys. 2005, 122, 014103. (16) Jeziorska, M.; Cencek, W.; Patkowski, K.; Jeziorski, B.; Szalewicz, K. Pair potential for helium from symmetry-adapted perturbation theory calculations and from supermolecular data. J. Chem. Phys. 2007, 127, 124303.

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(17) Patkowski, K.; Jeziorski, B.; Szalewicz, K. Unified treatment of chemical and Van der Waals forces via symmetry-adapted perturbation expansion. J. Chem. Phys. 2004, 120, 6849–6862. (18) Eisenschitz, L.; London, F. About the relationship of the van der Waals forces to the covalent bonding forces. Z. Phys. 1930, 60, 491–527. (19) London, F. Zur theorie und systematik der molekularkräfte. Z. Phys. 1930, 63, 245–279. (20) London, F. The general theory of molecular forces. Trans. Faraday Soc. 1937, 33, 8–26. (21) Stone, A. J. The theory of intermolecular forces, 2nd ed.; Oxford University Press: Oxford, 2013. (22) Jeziorski, B.; Szalewicz, K.; Chałasinski, G. Symmetry forcing and convergence properties of perturbation expansions for molecular interaction energies. Int. J. Quantum Chem. 1978, 14, 271–287. (23) Cencek, W.; Szalewicz, K. Erratum “on asymptotic behavior of density functional theory", [J. Chem. Phys. 139, 024104 (2013)]. J. Chem. Phys. 2014, 140, 149902. (24) Jeziorska, M.; Cencek, W.; Patkowski, K.; Jeziorski, B.; Szalewicz, K. Complete basis set extrapolations of dispersion, exchange, and coupled-clusters contributions to the interaction energy: a helium dimer study. Int. J. Quantum Chem. 2008, 108, 2053–2075. (25) Meath, W. J.; Margoliash, D. J.; Jhanwar, B. L.; Koide, A.; Zeiss, G. D. Accurate molecular properties, their additivity, and their use in constructing intermolecular potentials. Intermolecular forces. Dordrecht, 1981; pp 101–115. (26) Misquitta, A. J.; Stone, A. J. Ab initio atom–atom potentials using CamCASP: theory and application to many-body models for the pyridine dimer. J. Chem. Theory Comput. 2016, 12, 4184–4208. (27) Aina, A. A.; Misquitta, A. J.; Price, S. L. From dimers to the solid-state: distributed intermolecular force-fields for pyridine. J. Chem. Phys. 2017, 147, 161722. 10


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