J . Phys. Chem. 1987, 91, 3179-3186
3179
A Simple Quantitative Model of Hydrogen Bonding. Application to More Complex Systems Mark A. Spackman? Department of Crystallography, University of Pittsburgh, Pittsburgh, Pennsylvania I5260 (Received: September 24, 1986;In Final Form: December 12, 1986)
The simple model of hydrogen bonding described previously is applied to a large number of dimers comprised of HF, HCl, H 2 0 , H2S, H2C0,NH,, and PH3. Equilibrium geometries and energies are generally in good agreement with ab initio or experimental results. A notable failure of the model occurs for NH,-HF, and reasons for this are explored. The neglect of the proton-acceptor repulsive term in the model appears to amount to implicit inclusion of a composite charge-transfer/polarization term of opposite sign.
1. Introduction We have recently described a simple model capable of quantitative estimates of equilibrium geometries, energies, and force constants for hydrogen-bonded complexes comprised of small linear molecules.’-* As our eventual aim is to model the interactions between systems of biological interest, using experimentally derived molecular electron densities, it is important to test the application of the model to more complex systems, particularly those containing water, and to calibrate the use of the repulsive potentials’ for many more systems containing second row atoms. In this work we extend the use of the model to include systems containing HF, HCl, H 2 0 , HIS, H2C0,NH3, and PH3. The scope of the study includes gas-phase dimers of small molecules for which the equilibrium structures and potential energy surfaces are well characterized experimentally and theoretically (e.g. (H20)2, H,O-HF), and the more strongly bound species NH3-HF and NH3-HC1, which are less well-known experimentally. The following section provides a brief description of the model and its major assumptions, and this is followed by a discussion of results obtained for the various systems studied. The breakdown of the model for NH3-HF provokes some detailed discussion reagarding the omission of the repulsive term for the proton and its acceptor in the hydrogen bond. 2. The Model The model has been described in detail elsewhere1p2and only a brief summary is given here. Emphasis is on the major assumptions and their expected limitations. The model is designed to describe the nonbonded interaction between two molecules. It’s essence is simplicity, and the major working assumptions are the following: (1) The interacting systems are not perturbed by the interaction. ( 2 ) The electron density and electrostatic potential of each monomer can be described by a sum of terms arising from the promolecule (an assembly of spherical ground-state atoms) and the classical electrostatic moments of the component atomlike fragments which describe local deviations from sphericity. (3) The electrostatic interaction energy, a quantity defined by first-order perturbation theory, is given simply by the classical interaction between the charge distribution of one monomer (including nuclei) and the electrostatic potential of the other. (4) Higher order terms in the energy arising from short-range exchange interactions and long-range dispersion can be approximated by an atom-atom potential of exp -6 form. ( 5 ) A hydrogen bond can be simply described by omitting the exp -6 potential terms between the proton and its acceptor. In this manner the total interaction energy is expressed as a sum of four terms:
Present address: Department of Chemistry, The University of New England, Armidale, N.S.W., 2351, Australia.
0022-3654/87/2091-3179!$01.50/0
E,, the classical electrostatic interaction between moments of the atomlike fragments, is generally attractive and is evaluated via expressions given by B ~ c k i n g h a m . ~E,, arises from the penetration of the fragment moments of one monomer inside the spherical charge distributions of the promolecule on the other monomer and is again entirely classical in origin, and generally repulsive. Ercpis the short-range repulsive term arising from overlap of the monomer promolecule charge distributions and is given by a sum of exponential atom-atom potentials.’r2 Edispis an approximation to the attractive long-range dispersion energy and is also given by atom-atom terms,’,* each proportional to R6. Each of the assumptions 1-5 has important implications, some of which have only been partially examined to date. The first is a key to the simplicity of the model, but it necessarily means that important terms such as charge transfer and polarization (or induction) are ignored. This did not appear to be a severe limitation in studies applied to small linear molecules,2 for reasons which will be discussed below. The second assumption stems from the exact partitioning of each monomer charge distribution into a sum of atomlike fragments, usually, but not necessarily, associated with the atomic nuclei, and a further partitioning of each fragment into a spherical atomic term, and a deformation term. The deformation terms could be expressed as multipole expansions, with complicated radial functions for each angular symmetry. However, the evaluation of the integrals E, and Em would be difficult and time-consuming. Instead, the deformation terms are summarized by their electrostatic moments. In this manner the integrals E , and E,, are simple, and exact for large separations, but increasingly less precise for smaller separations, since the radial “shape” of the deformations is being neglected here (Le. their mutual interpenetration). This could be a serious limitation of the present model, but the results presented earlier2 suggest that it is an excellent approximation in general. The separation at which E,, calculated via electrostatic moments becomes appreciably different from the true value, due to neglect of interpenetration, can be related to the separation at which the electrostatic potential computed from the sum of fragment moments differs significantly from the true electrostatic potential. According to Stone and A l d e r t ~ n the ,~ accurate calculation of the energy via moments on sites separated by R generally requires the same level of accuracy in the description of the electrostatic potential of either charge distribution at a distance of R / 2 . Thus we can judge the adequacy of any fragment multipole description of E , in terms of the electrostatic potentials of the components. Such a study is not part of the present work, but will be pursued elsewhere. It should therefore be borne in mind that assumptions 2 and 3 imply a certain tol(1) Spackman, M. A. J. Chem. Phys. 1986, 85, 6579. (2) Spackman, M. A. J . Chem. Phys. 1986, 85, 6587. (3) Buckingham, A. D. In Intermolecular Interactions: From Diatomics fo Biopolymers, Pullman, B., Ed.; Wiley: New York, 1978; p 1. (4) Stone, A. J.; Alderton, M. Mol. Phys. 1985, 56, 1047.
0 1987 American Chemical Society
3180 The Journal of Physical Chemistry, Vol. 91, No. 12, 1987
TABLE I: Results Obtained for (H20)3a R
6,
0,
-E,,,
linear, C, present work theoryb theory‘ theoryd experiment‘
2.67 2.96 2.96 2.91 2.96 (1)
52.1 55 49.7 56.8 57 (10)
1.2 7 3.4 4.5
