J. Phys. Chem. 1990, 94, 4449-4452
represent the crystal in this respect. The NQCC value for 2H in the free molecule exhibits excellent agreement with its experimental counterpart, as shown in Table 11. In the case of the cluster, the atoms which are symmetry related in the model (as shown in Figure 1 ) are designated identically. Also here, the agreement between hydrogen atoms which are symmetry unrelated in the model is convincing. The solid-state shift can be attributed to the major changes induced in the individual NH3 molecules by condensation to the solid state, viz. (a) an increase in the ionic character of the N-H bonds and (b) a redistribution of electron density around the N atom, causing a density decrease in the direction of the lone pair
4449
and a density increase in directions bisecting the N-H bonds.' This density rearrangement leads to a more spherical electron density distribution around the nitrogen nucleus, and thus to a reduction in the I4N quadrupole coupling constant. The present results hence confirm the interpretation of the electron density rearrangements in similar systems put forward in our previous work.'-4 Acknowledgment. This research was supported by the Swedish Natural Science Research Council (NFR). S.P.G. expresses his gratitude to the Swedish Institute for the award of a guest research scholarship and to Dr. Jean-Louis Calais for valuable discussions.
Theoretical Study of the Short Asymmetric [OmH-0] Hydrogen Bond in Solid Potasslum Hydrogen Diformate, Including Electron Correlation Shridhar P. Gejji, Oscar E. Taurian,+and Sten Lunell* Department of Quantum Chemistry, Uppsala University, Box 51 8, S- 751 20 Uppsala, Sweden (Received: June 27, 1989; In Final Form: January 2, 1990)
The short intermolecular hydrogen bond ([O-.O], 2.437 A at 120 K) between the two formate groups in solid KH(HC00)2 was studied by employing ab initio second-order Maller-Plesset perturbation theory with two different basis sets viz., (a) a double zeta basis with polarization functions added on all atoms (DZP) and (b) with bond functions instead of the 3d polarization functions in the DZP basis (DZB(*)). The potential function for the hydrogen bond proton was calculated for a H(HCO0); dimer in the crystalline environment. Given the heavy atom positions, an asymmetric potential energy curve with a single minimum was obtained, with the proton displaced 0.053 A from the midpoint, in excellent agreement with the experimental result 0.0515 f 0.001 A. It is suggested that quantum chemical calculations of the present type may be used for good quantitative predictions of the positions of hydrogen atoms in crystals where only the heavy atom positions are known, e.g., from X-ray experiments, as a substitute for more expensive and cumbersome neutron diffraction measurements. A comparison of the deformation densities obtained from the two different basis sets is also presented.
The asymmetry of strong [X--H-.X] hydrogen bonds and their sensitivity to molecular environment has been studied extensively in the recent literature, with the aid of neutron diffraction experiment~'-~ as well as a b initio theoriese* in the case of hydrogen-bonded dimers. The properties of short hydrogen bonds, where the [O-01 distance lies in the interval 2.40-2.50 A, have been reviewed by Olovsson and Jonsson.' Most of the bonds of this type that have been studied by diffraction methods were found either to have intramolecular character or to connect crystallographically equivalent groups. Hermansson et aL9 have reported the results of neutron diffraction experiments at low temperature for the potassium hydrogen diformate crystal. The detailed geometry of this compound was determined with high accuracy and particular attention was paid to the short intermolecular [O.-O] hydrogen bond (2.437 A at 120 K) between the two formate groups. An unusual property of this bond in K H ( H C 0 0 ) 2 is that it connects two chemically equivalent but crystallographically nonequivalent groups, admitting either a centered or a noncentered bond. Somewhat unexpectedly, for such a short hydrogen bond, it was found to be noncentered, with the proton displaced about 0.05 A from the midpoint of the bond. A theoretical analysis of the hydrogen bonding in this compound therefore is of particular interest. From a computational point of view, the potassium hydrogen diformate crystal, whose building units are K+ and H ( H C 0 O ) c ionsgJOas shown in Figure 1, is well suited for a theoretical study owing to the fact that the diformate anion contains relatively few *The author to whom all correspondence should be addressed. 'Present address: Universidad Nacional de Rio Cuarto, Centro de Computacion, 5800 Rio Cuarto, Argentina.
0022-3654/90/2094-4449$02.50/0
light atoms. In our earlier paper'' we compared the deformation density maps obtained theoretically by a b initio calculations at the self-consistent-field (SCF) level with experimental low-temperature X-N maps, showing excellent agreement between the theoretical and experimental deformation maps. In this paper we study in detail the nature of the hydrogen bond by calculating the potential energy curve obtained by moving the H(3) atom along the [0(3)-.0(bond 1)](cf. Figure 1). Computational Method The crystal was modeled, as in ref 11, by (i) evaluating the complete electronic structure for the H(HC00)2- unit, (ii) assuming a +1 charge on the K+ sites, and (iii) invoking a self(1) Olovsson, I.; Jonsson, P. G . In The Hydrogen Bond Recenf Deuelopmenrs in Theory und Experiments; Schuster, P., Zundel, G., Sandorfy, C., Eds.; North Holland: Amsterdam, 1976; Vol. 11. (2) Williams, J. M.; Schneemeyer, L. F. J . Am. Chem. SOC.1973, 95, 5780. (3) Takusgawa, F.; Koetzle, T. F. Acra Crysralbgr., Secr B 1979,35,2126. (4) Frisch, M . J.; Del Bene, J. E.; Binkley, J. S.; Schaefer 111, H. F. J . Chem. Phys. 1986,84, 2279. (5) Scheiner. S.; Redfern, P.; Szczeniak, M. M. J . Phys. Chem. 1985, 89,
262.
(6) Szcz@niak, M. M.; Scheiner, S . J . Phys. Chem. 1985, 89, 1835. (7) Seel, M.; Del Re, G . Inr. J . Quantum Chem. 1986, 34, 8 5 . (8) Tostes, J. G . R.; Taft, C. A.; Ramos, M. N.; Lester, Jr., W. A. Inr. J . Quunrum Chem. 1986, 34, 85. (9) Hermansson, K.; Tellgren, R.; Lehmann, M. S. Acra Crystollogr.,Secr. C 1983, 39, 1507. (10) Larsson, G.; Nahringbauer, I. Acta Crysfallogr.,Secr. B 1968, 24, 666. ( I 1 ) Taurian, 0. E.; Lunell, S . ; Tellgren, R. J . Chem. Phys. 1987, 86, 5053.
0 1990 American Chemical Society
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\
Gejji et al.
!
2.803
',
I
11.226
1.
03 1.1671
H ( 3 ) displacement from t h e midpoint of 0 1 - 0 3 .
(A)
Figure 2. Hartree-Fock and MP2 energies for a free H(HCO0)Y ion as a function of H(3) displacement from the midpoint of 0(1).-0(3). Dotted and dashed lines denote the H F results in the DZP and DZB(*) basis sets, respectively. The corresponding MP2 profiles are shown by continuous and dash-dotted lines, respectively. All the curves are plotted on a relative scale.
Figure 1. Atomic arrangement in the diformate ion in H(HC00);, its nearest neighbors (from ref 9 ) .
with
consistent determination of the point charges representing the remaining crystal. The self-consistent calculation of the point charges was performed by employing the A L C H E M Y ~package ~ according to the following scheme: (a) A free H ( H C 0 0 ) 2 - dimer calculation was carried out. (b) A Mulliken population analysis was performed and the resulting charges were distributed at the appropriate atomic positions as determined by neutron diffraction and out to a distance of about I 3 8, from the midpoint of the ion. The cluster, i.e., the H ( H C 0 0 ) y ion plus the point charges, was electrically neutral. (c) The one electron integrals were recalculated. (d) A new calculation was performed for the H ( H C 0 0 ) y ion in the crystal field created by the surrounding point charges. Steps b-d were repeated until a self consistent set of point charges was obtained. The self consistency criterion used for the point charges was 0.001. In order to study the nature of the hydrogen bond in the crystal in more detail, considering separately the influence of the crystal field, two different potential functions were calculated. One set of computations was thus performed for the H(HCOO)*- dimer in the crystal field of the point charges, including Hartree-Fock (HF) as well as second-order M~ller-Plesset'~(MP2) perturbation theory calculations (with the inner shells frozen), with the atom H(3) at different positions along the [0(1)-0(3)] bond. A corresponding set of calculations was also carried out for the free dimer for the same geometric configurations but without the inclusion of the crystal field. The GAUSSIAN-86 program14 was used for these calculations. Two different types of basis sets were employed for the present work: (a) Dunning's (9s5p)/ [4s2p] functions of double zeta15 type, augmented with 3d polarization functions16 with exponents 0.85 and 0.75 for oxygen and carbon atoms, respectively, and a 2p (12) Almof, J.; Bagus, P. S.;Liu, S.; McLean, D.; Wahlgren, U. I.; Yoshimine, M. IBM, San JOSE Research Laboratory (unpublished work). (13) Maller, C.; Plesset, M. S. Phys. Reu. 1934, 46, 618. (!4) Frisch, M. J.; Binkley, J. S.; Schlegel, H. B.; Raghavachari, K.; Mel~us,C. F.: Martin, R. L.; Stewart, J. J. P.; Bobrowicz, F. W.; Rohlfing, c. M.; Kahn, L. R.; Defrees, D. J.; Seeger, R.; Whiteside, R. A.; Fox,D. J.; Fleuder, E. M.; Pople, J. A. GAUSSIAN-86; Carnegie-Mellon Quantum Chemistry Publishing Unit: Pittsburgh PA, 1984. ( I S ) Dunning, T. H. J . Chem. Phys. 1970.53, 2823. (16) Rws, B. and Siegbahn, P. Theor. Chim Acto 1970, 17. 199.
polarization function with exponent 0.8 for the hydrogen atoms (DZP). (b) Dunning's (9s5p)/[4s2p] bases of double zetaI5 type for the heavier atoms, augmented with bond functions according to Rothenberg and Schaeferl' and Vladimiroff.18 For the C-O bonds a Is (exponent 1.17) and a 2p (Le., one 2px, one 2py, and one 2pz, with exponent 0.64) were used, placed at the midpoint of the bonds. For the C-H and 0 - H bonds two 1s (exponent 1.27) bond functions, placed at distances 1/3 and 2/3 down the C-H, 0-H, and 0 - H bonds, respectively, were used. The location of the latter bond functions therefore varied with the position of H(3). As in type a, Dunning's (4s)/[2s] basis set augmented with a 2p function with exponent 0.8 was used for the hydrogen atoms. This is the basis used in our earlier deformation density study" and will be denoted as the DZB(*) basis. An entire geometry optimization of the H(HCOO)*- ion was not performed, so that the positions obtained by neutron diffraction9 were used for all atoms except H(3).
Results and Discussion The H F and MP2 potential energy (PE) functions were obtained by calculating the total energy of the system for different positions of the hydrogen bond proton H(3), using the two different basis sets described in the Computational Method section. The following discussion will in principle be based on the DZP results, since this basis set is the bigger one of the two, and we will supplement this with a brief comparison of the DZP and DZB(*) results. A . H F and MP2 Potential Energy Functions for the Hydrogen-Bonded Proton. The Hartree-Fock energies, using the DZP basis, for an isolated H(HC00)2- dimer without the inclusion of the crystal field are shown in Figure 2 as a function of H(3) displacement from the midpoint of O(1)--0(3), along with their MP2 counterparts, by a dotted and a continuous line, respectively. A slight asymmetry can be seen to be present already for the free dimer, the reason being the crystallographic inequivalence of the two formate groups in the real crystal. The PE function obtained from H F theory shows two minima at -0.155 and 0.166 A, corresponding to the energies -377.092 13 and -377.092 75 au (1 au = 627.51 kcal/mol), respectively, and a maximum at -0.009 A having energy 0.0018 au higher than the lower minimum. The PE profile from MP2 theory, in contrast to H F theory, shows only one minimum at 0.032 A having the total electronic energy -378.079 32 au. The PE profiles obtained from the H F and MP2 theories in presence of the crystal field are depicted in Figure 3 by dotted (17) Rothenberg, S . ; Schaefer 111, H . F. J . Chem. Phys. 1971, 54, 2765. (18) Vladimiroff, T. J . Phys. Chem. 1973, 77, 1983.
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Study of the [O-.H-.O] Hydrogen Bond
H ( 3 ) displacement from the midpoint of 0 1 - 0 3 .
(A)
Figure 3. Hartree-Fock and MP2 energies for a H(HC00); ion as a function of H(3) displacement from the midpoint of 0(1)-0(3) in,the presence of the crystal field. Dotted and dashed lines denote the HF profiles with the DZP and DZB(*) basis, respectively. The corresponding MP2 profiles are shown by continuous and dash-dotted lines. All the curves are plotted on a relative scale.
and continuous lines, respectively. The PE function from H F theory shows two minima at -0,149 and 0.172 A having energies -388.31542 and -388.31721 au, respectively. The maximum in the H F curve occurs at -0.027 A which corresponds to an energy of -388.3 14 76 au. The difference between these two minima in the H F curve, although being rather small, increases by a factor of 3 in the presence of the crystal field. The crystal field has practically no influence on the positions of the minima in the potential energy function. The MP2 calculations, on the other hand, show a clearly asymmetric PE curve with only one minimum at 0.053 8, (having the energy -389.303 1 1 au) which is in excellent agreement with its experimental9 counterpart, 0.0515 f 0.001 A. Although this almost perfect agreement may be considered to be to some extent fortuitous, it gives strong credibility to the model used in the present calculations. A comparison of the H F and MP2 results show that the correlation effects are most important near the midpoint of the curve, where both the O(I)-H(3) and the 0(3)-H(3) distances are longer than the normal 0-H bonds. This reflects the well-known inability of H F wave functions to describe bond dissociation correctly. An inclusion of electron correlation through, e.g., MP2 theory will therefore produce a lowering of the central barrier and
a shift of the minima toward the midpoint, thus improving the agreement with experiment remarkably. Since the two formate groups are chemically equivalent, it is clear that the noncentered nature of the hydrogen bond is due to the influence of more distant neighbors, primarily the K+ ions. This influence is both indirect, by changing the geometry of the formate groups, and direct, through the creation of an unsymmetrical electrostatic field at H(3). A comparison of Figures 2 and 3 shows that the latter effect is clearly dominating. B. Comparison of the DZP and DZB(*) Basis Sets. The PE profiles derived for an isolated H(HC00)2- dimer, using the DZB(*) basis, without the inclusion of the crystal field, can be seen to be very similar to the corresponding ones obtained from the use of the DZP basis (cf. Figure 2). H F theory here again yields a double-well PE function (as depicted by the dashed line). Two minima are found at -0.155 and 0.162 A, corresponding to the energies -377.069 02 and -377.069 58 au, respectively, and a maximum occurs at -0,010 A, having an energy 0.0016 au higher than that for the lower minimum. The corresponding MP2 curve (dash-dotted line), however, has only one minimum at 0.040 A with energy -377.86401 au. The PE profiles from H F and MP2 calculations in presence of the crystal field, and using the DZB(*) basis, are depicted by a dashed line and a dash-dotted line, respectively, in Figure 3. The H F curve shows two minima at -0.145 and 0.168 A, having the energies -388.292 36 and -388.293 98 au, respectively, and a maximum at -0.027 A with energy 0.0022 au higher than that for the lower minimum. The positions of the minima are thus, as before, not greatly affected by the presence of the crystal field. The MP2 curve, in contrast to its H F counterpart, shows only one minimum at 0.067 8, (with energy -389.087 76 au). Thus, also in the DZB(*) basis, the hydrogen bond becomes clearly noncentered with the minimum on the correct side of the midpoint, as found by neutron diffraction experiments. The quantitative agreement is, however, not quite as good as when the DZP basis was employed. It should be remarked in this connection that the positions of the minima in the PE curves, strictly speaking, are not directly comparable with the experimentally observed position of H(3), because of the vibrational effects. The anharmonicity of the PE curve will cause the vibrationally averaged position to be shifted closer to the center, which will improve the agreement of the DZB(*) result with experiment. A simple order-of-magnitude estimate from Figure 3 ( k T i= 0.24 kcal/mol at 120 K), however, suggests that the shift will be rather small and affect the theoretical position only in the third decimal. A full treatment of the vi-
(a! -+---Figure 4. (a, left) Theoretical deformation density map in the plane of one of the formate groups in KH(HCOO)*, obtained in the DZP basis. Contour interval 0.1 e/A3. Dashed lines denote negative levels. Zero contour omitted. Crystal field included. (b, center) Theoretical deformation density map in the plane of one of the formate group in KH(HC00)2, obtained in the DZB(*) basis. Same contours as in (a). (From ref 1 1 .) (c, right) Experimental X-N deformation density map in the same plane as in (a). Contour interval 0.05 e/A. (From ref 20.)
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TABLE I: Comparison of Theoretical and Experimental Peak Heiehts in the Deformation Densitv Maw (e/A3) D Z P basis DZB(*) basis” expt X-Nb C(2)-H(2) 0.5 0.8 0.45 c(2)-0(4) 0.6 0.9 0.50 C(2)-0(3) 0.5 0.7 0.40 0(3)-H(3) 0.3 0.4 0.30 Reference 11.
Dynamic deformation density, ref 20.
brational motion must necessarily include the coupling between 0-H and 0-0 stretching modes. which is known to be strong,19 which places it beyond the scope of the present investigation. A further comparison of the DZP and DZB(*) basis sets is given in Figure 4, which shows calculated deformation densities in the two different basis sets together with the experimental X-N map.20 The theoretical deformation densities were in both cases obtained by subtracting spherically averaged S C F atomic ground-state densities, calculated in the same basis sets, but without polarization or bond functions, from the calculated total electron density. Both theoretical deformation density maps clearly give essentially the same general picture of the bonding in the H(HCOO)*- ion. Quantitatively, the DZB(*) basis gives consistently higher deformation densities in the bonding regions than the DZP basis, by 25-60%, which is exactly the effect to be anticipated from the addition of extra basis functions in those regions. (The difference in the total electron density obtained in the two basis sets is much smaller, of the order 2-6%.) A comparison with the experimental X-N map in Figure 4c would seem to indicate that the bond deformation densities given by the DZB(*) basis are somewhat exaggerated, while the DZP results look quite satisfactory (cf. Table I). However, considering the fact that the experimental maps are thermally averaged, leading to a significant lowering in peak heights,2’ it is questionable whether the more expensive DZP basis indeed gives better deformation densities than the DZB(*) one.
Conclusions The present calculations have been found to reproduce the noncentered nature of the [O.-H.-O] hydrogen bond in K(HC00)2in the crystalline environment, predicting an asymmetric (19) See, for example: Berglund, B.; Lindgren, J.; Tegenfeldt, J. J . Mol. Struct. 1978, 43, 179. (20) Hermansson, K.; Tellgren, R. Acta Crystallogr.,Sect. B 1989, 45, 252. _. -
(21) Cf.. e.g.: Hermansson, K.; Lunell, S . Acta Crystallogr., Sect. B 1982, 38, 2563.
Gejji et al. position for the H(3) proton. The H F theory yields a double-well potential energy curve. The difference between the two minima, however, is rather small. With inclusion of electron correlation by means of MP2 theory, an asymmetric curve with a single minimum at 0.053 8, from the midpoint of the 01-03 bond is obtained. This shows a very good quantitative agreement with the experimentally observed displacement (0.051 5 f 0.010 A). It can be seen from Figures 2 and 3 that the introduction of electron correlation is most important when H(3) is in the midpoint of the [0(1)-.0(3)] bond. Inclusion of electron correlation thus removes the barrier completely in Figure 2 and shifts the energy minimum closer to the midpoint in both Figures 2 and 3. As mentioned above, this is to be expected, since for this H(3) position neither the [0(1)-.H(3)] or the [0(3).-H(3)] bonds are normal 0-H bonds but are stretched beyond the normal OH bond length. For such stretched bonds, the H F method is expected to give too high energies because of its well-known erroneous behavior in the dissociation limit. It has also been demonstrated that the use of bond functions instead of atom-centered polarization functions leads to deformation densities as well as potential energy curves which provide essentially the same information as the larger DZP basis. The advantage of DZB basis sets is a significant reduction of computing requirements, as has been emphasized, e.g., by Schweig and coworker~.~~-~~ X-ray diffraction experiments are now routinely performed in order to determine atomic positions in crystalline solids. Although yielding accurate information about heavier atoms, the information that those experiments give about hydrogen atom positions is very limited. As illustrated in this paper, quantum chemical calculations can be used to give such information, in a quantitative way, thereby providing an alternative to other techniques, such as neutron diffraction experiments, for this purpose. Acknowledgment. This research was supported by the Swedish Natural Science Research Council (NFR). S.P.G. expresses his gratitude to the Swedish Institute for the award of a guest research scholarship. The computations were done on the Alliant Fx-80 minisupercomputer of the Faculty of Science and the BASF 7/75 computer at the Uppsala University Computer Center (UDAC). We thank Mario Natellio for his assistance during the course of the calculations and the referees for valuable comments. (22) Breitenstein, M.; Dannohl, H.; Meyer, H.; Schweig, A.; Seeger, R.; Seeger, U.; Zittlau, W. Int. Reu. Phys. Chem. 1983, 3, 335. (23) Hase, H. L.; Schweig, A. Angew. Chem. 1977,89, 264. (24) Lauer, G.; Meyer, H.; Schulte, W.; Schweig, A,; Hase, H. L. Chem. Phys. Lelr. 1979, 67, 503.
