J. Phys. Chem. 1987, 91, 6365-6373
6365
Nonbonded H*-H Repulsion Energy from ab Initio SCF Calculations of Methane, Ammonia, Water, and Methanol Dimers Donald E. Williams* and David J. Craycroft Department of Chemistry, University of Louisville, Louisville, Kentucky 40292 (Received: December 30, 1986)
Ab initio Hartree-Fock self-consistent-field wave functions at the 6-31G** and 6-31 1G** level were determined for Hm-H repulsive approaches of two methane, ammonia, water, or methanol molecules. A Morokuma decomposition scheme was used to obtain energy components. The exchange energy components were fitted with exponential functions and the results indicate that there are significant differences in the repulsion energy curves of a bonded hydrogen atom, with methane hydrogen being the most repulsive, ammonia hydrogen intermediate, and water hydrogen the least repulsive. The reduction in repulsion energy of the bonded hydrogen correlates with a decrease of electron density at the hydrogen and a corresponding increase in net atomic charge. The decrease in electron density reduces the amount of electron cloud overlap which leads to decreased repulsion going from methane to water. For a foreshortened model at the 6-311G** level, van der Waals diameters based on a force of N for hydrogen bonded to carbon, nitrogen, and oxygen were 2.562, 2.435, and 2.174 A, respectively. The electric potential around the molecules and their dimers was found and net atomic charges were obtained by least squares with and without foreshortening of the X-H bonds. The expected slight overestimation of the molecular dipole moment obtained with these basis sets was corrected by scaling the charges to the observed dipole moments of ammonia and water. The intermolecular Coulombic energy which was obtained by using the scaled charges was much less than the intermolecular electrostatic energy component as defined by Morokuma. When the dipole-scaled Coulombic energy was subtracted from the total calculated intermolecular energy, the residual energy was long range. Fitting this energy with an exponential repulsion function led to unexpectedly low values for the repulsion exponent. For water dimer a uetter fit to the intermolecular energy was obtained by upward scaling of the net atomic charges. The resulting Coulombic energy was closer to the value of the Morokuma electrostatic component, but these charges gave an overestimated value for the electric potential and dipole moment.
Introduction Since the periphery of organic molecules mostly consists of bonded hydrogen atoms, it is important to have an accurate description of the energy of Ha-H interaction as these molecules approach. Thus, for most of the surface of the molecule the H.-H interaction dominates the intermolecular repulsion energy. Whenever organic molecules interact, or even when distant parts of the same molecule interact, H-H repulsions are extremely important. As examples consider the folding of a protein molecule into its active conformation, or the complexation of a protein with a substrate molecule, or the intercalation of molecules into DNA structures. We seek accurate potential energy functions for H--H interactions. Since wave functions for large molecules are difficult or impossible to calculate, our approach is to model the intermolecular energy in large molecules with empirical functions obtained from small molecule dimers. A simple empirical representation of the intermolecular interaction energy is a sum of pairwise additive atom-atom terms’ with each term being the sum of several energy components. In this view V(tota1) = V(repu1sion) + V(dispersion) + V(Cou1ombic) summed over atom pairs from different molecules. These three types of energy can be represented by an (exp-6-1) nonbonded potential energy function where r is a nonbonded distance between atoms j and k in different molecules, q is a net atomic charge, and A, B, and Care adjustable parameters. The first quantity in the function represents shortrange repulsion and C i s called the repulsion exponent. The second quantity represents the leading term of the dispersion attraction energy in the functional form obtained by perturbation theory treatment at long distance. The last quantity represents Coulombic energy between net atomic charges of different monomers. It may be necessary or appropriate in some cases to include additional energy functions, such as for hydrogen bonding. The (exp-6- 1) function has enjoyed considerable success in modelling the interaction energy of molecules which are not hydrogen bonded in the crystal’” and has been used to model mo-
lecular clusters.’ If hydrogen bonding is present, the empirical form for V(hydrogen bond) which is compatible with (exp-6-1) functions for non-hydrogen-bonded nitrogen, oxygen, and hydrogen atoms is not well established. A clear separation of the repulsive, dispersion, and Coulombic energy components would assist future efforts to define the empirical form of V(hydrogen bond). Advances in computer technology have made possible ab initio calculation of wave functions for small dimers, as well as for monomers. The intermolecular energy may be obtained by subtraction of the monomer energy from the supermolecule energy: E(inter) = E(supermo1ecule) - E(monomers) At the Hartree-Fock (HF) level no correlation effects between electrons of opposite spin are included, and correlation effects between electrons of the same spin are only partially included.8 It is conventional to define the correlation energy as the difference between the HF energy and the exact nonrelativistic energy. Since the dispersion energy has its origin in electron correlation, the empirical dispersion energy term was neglected when modelling H F results. The H F calculation does include both the repulsion and electrostatic components, and the separation of these components in a physically meaningful way is the principal topic of this paper. Morokumag has developed a method of separating the HF energy into several components, including electrostatic, exchange repulsion, charge-transfer, and polarization components. We will compare the Morokuma components to the empirical energy (1) Kitaigorodsky, A. I. Molecular Crystals and Molecules; Academic: New York, 1973. (2) Williams, D. E.: Starr, T. L. Comput. Chem. 1977, I, 173. (3) Hsu, L. Y.; Williams, D. E. Acta Crystallogr., Sect. A 1980, A36, 277. (4) Cox, S. R.; Hsu, L. Y.; Williams, D. E. Acta Crystallogr., Sect. A 1981, A37, 293. (5) Williams, D. E. Top. Curr. Phys. 1981, 26, 3. (6) Williams, D. E.; Houpt, D. J. Acta Crystallogr., Sect. 8 1986, 842, 286. (7) Williams, D. E. Acta Crystallogr., Sect. A 1980, A36, 715. (8) Hehre, W. J.; Radom, L.; Schleyer, P. v. R.; Pople, J. A. Ab Initio Molecular Orbital Theory; Wiley: New York, 1986; p 29. (9) Morokuma, K. J . Chem. Phys. 1971. 55, 1236.
0022-365418712091-6365$01.50/00 1987 American Chemical Society
6366 The Journal of Physical Chemistry, Vol. 91, No. 25, 1987 components. Since our focus is on the H-H repulsion the major terms in the Morokuma decomposition are the electrostatic and exchange components. The charge-transfer and polarization components are of lesser importance in the repulsive configurations considered here but can become more important in hydrogenbonding configurations.*’ Our empirical Coulombic energy component is based on the assignment of net atomic charges. Net atomic charges can be set so as to reproduce the observed monomer dipole moments. The intermolecular electrostatic energy can then be found by applying Coulomb’s law between net atomic charges of different monomers. An alternative way of assigning net atomic charges is to require them to fit the electric potential surrounding the molecule. This electric potential may easily be found from the wave function, and net atomic charges obtained by least squares (potential-derived, or PD charge method). Cox and WilliamsIo investigated this method for a variety of small molecules. The PD net atomic charge method may be extended to explore net atomic charges in supermolecules,” such as methane, ammonia, water, and methanol dimers. In this approach the electric potential surrounding the dimer is found from the dimer wave function in a way that is completely analogous to that used for the monomer. Further, it is of interest to note whether the charges in the molecular components of the dimer remain essentially unchanged, or if there are signficant changes from monomer values as the molecules approach to form the dimer. The repulsion energy component of.the hydrogen atom is our primary interest. We wish to find suitable values for the repulsion parameters B and C that are physically reasonable and transferable to hydrogen atoms in other molecules. We anticipated that repulsion curves for hydrogen in methane H(C), ammonia H(N), and water H ( 0 ) might be different. Also, the scaling of the empirical Coulombic energy could affect the results for the repulsion energy. Repulsion is a short-range interaction dependent upon the molecular shape of the chemical system. The repulsive force between two or more molecules with filled orbitals is a quantum mechanical effect due primarily to the Pauli exclusion principle. As molecular electron clouds approach each other, the Pauli principle prohibits further occupation of the filled orbitals. The electron clouds must then distort to prevent any violation of the Pauli principle. The energy of distortion is approximately proportional to the overlap of the filled orbitals. Since the electron density declines exponentially from the nuclei, this atom-atom repulsion can reasonably be modelled by exponential function^.'^^^^ Since our goal was to obtain H.-H repulsion parameters, we selected dimer configurations which maximized He-H interaction (head-on approach). In this manner, the interacting hydrogens made a maximum contribution to the intermolecular energy, and the other atoms contributed only a small amount to the intermolecular repulsive energy. Figure 1 shows these head-on configurations for the dimers of methane, ammonia, water and methanol. These configurations eliminate any hydrogen bonding that could be present in other general orientations. The repulsion exponents of the H(H), H(C), H(N), and H ( 0 ) type may be derived from dimers of hydrogen, methane, ammonia, and water, respectively. Since water dimer gave results that may be unique among this set, methanol dimers were also utilized for comparison of H ( 0 ) interactions. Of course, the simplest case of nonbonded H-H repulsion is in the dimer of the hydrogen molecule. WilliamsI4 analyzed an early calculation of the intermolecular energy of molecular hydrogen dimers and found an atomatom repulsion curve for H.-H interaction. The hydrogen molecule was found more isotropic in its repulsion than the proton positions suggest. In order to get a good fit, it was necessary to shift the repulsion centers into the (10) Cox, S. R.; Williams, D. E. J . Comput. Chem. 1981,2, 304. (11) Williams, D. E.; Craycroft, D. J. J. Phys. Chem. 1985,89, 1461. (12) Kita, S.; Noda, K.; Inouye, H. J . Chem. Phys. 1976.64,3446. (13) Starr, T. L. Ph.D. Dissertation, Chemistry Department, University of Louisville, Louisville, KY, 1976. (14) Williams, D. E. J . Chem. Phys. 1965,43,4424.
Williams and Craycroft
&
u7
YnS Figure 1. Head-on configurations for dimers of methane, ammonia, water, and methanol.
bond by 0.07 8, (foreshortening). The idea of foreshortening is strongly supported by comparison of hydrogen atom positions obtained by X-ray and neutron diffraction. Since nonbonded repulsion is predominately an electronic effect, the foreshortened positions for hydrogen observed by X-rays are the appropriate positions to use for a spherical repulsion center. More recently, Starr and Williamsls repeated the hydrogen molecule dimer fitting calculation using energies obtained from an improved wave function and found a foreshortening of 0.16 8,. The best value to use for foreshortening in methane, ammonia, and water is difficult to establish from dimer energy calculations. We adopted a foreshortening value of 10% of the X-H bond length in the present work. We show below that the amount of foreshortening has a negligible effect on the value of the repulsion exponent. Williams and Starr2used an H-H exponent of 3.74 ,klalong with the above described foreshortening of 0.07 8, to describe repulsions in the experimentally determined crystal structures of 18 hydrocarbon molcules. Their nonbonded H-H repulsion function is E = 11677 exp(-3.74r) Because this function was experimentally based, we will use it in making comparisons of various nonbonded repulsion functions obtained purely from quantum mechanical calculations. In the head-on geometry the B coefficient may be adjusted for different assumptions about the amount of foreshortening. This adjustment of B shifts the repulsion curve by the amount of the foreshortening. With today’s computer power it is possible to calculate the ab initio energy of methane dimers, for instance, to determine the repulsion curve for H(C)-H(C) interactions. Further, ammonia dimers and water dimers can yield repulsion curves for H(N).H(N) and H(0)-H(0). To the first approximation, H(H), H(C), H(N), and H ( 0 ) repulsion curves should be transferable. However, the curves may be dependent upon the adjacent bonded atom, especially highly electronegative atoms such as oxygen or nitrogen. Thus, we might envision different curves for hydrogen bonded to carbon, nitrogen, and oxygen. Since hydrogen has no kernel electrons its electron cloud is highly deformable and its repulsion characteristics may depend on the attached atom. In particular, hydrogen acquires an electron deficit (positive charge) along the series H(H), H(C), H(N), and H ( 0 ) . This deficit reduces the repulsion in the same sequence. This work shows that (15) Starr, T. L.; Williams, D. E. J . Chem. Phys. 1977,66,2054.
He-H Interactions in Organic Molecules
The Journal of Physical Chemistry, Vol. 91, No. 25, 1987 6361
TABLE I: Values of Repulsion Exponents" researchers Kitaigorodsky et al. Williams and Houpt Starr Bohm and Ahlrichs
H...H 4.29 3.74 3.07 2.63
C--C 3.68 3.60 3.25 2.76
N.-.N
O...O
ref
3.78 3.78 3.91 3.24
4.18 3.96 4.35 3.71
16 6 13 17
"All values in ,k'. the relative sizes of bonded hydrogen atoms are H ( H ) > H(C) > H(N) > H ( 0 ) . We are particularly interested in establishing quantitative repulsion energy curves of different types of bonded hydrogen atoms since this information is difficult to derive from experimental crystal structure data. Table I lists values of repulsion exponents proposed by several researchers for H , C, N , and 0 nonbonded interactions. Note that the range for the hydrogen exponent is large. The center of interest here is the possible variation of the hydrogen exponent for H(H), H(C), H(N), and H ( 0 ) bond types. Kitaigorodskyi6 and Williams and Houpt6 proposed values based on analyses of crystal structure data. Starr13 utilized the method of Kita, Noda, and Inouyei2 to estimate an exponential repulsion potential between two nonbonded atoms. This method assumes that the repulsion is proportional to the electron density overlap. Rcently, Bohm and Ahlrichs" have proposed a method of treating nonbonded atom-atom interactions between closed-shell molecules. They also found that the repulsions can be represented by exponential functions. The value of the exponent was approximately proportional to the square root of the magnitude of the highest occupied atomic orbital energy. In the previously mentioned work of Starr and Williams15 on hydrogen dimer a He-H repulsion exponent 3.14 A-' was obtained. Price and Stonei8studied H2dimer repulsion based on a 130-point SCF-CI energy surface corrected for basis set superposition. They found a repulsion exponent of 3.43 based on repulsion sites foreshortened by 0.17 A. The use of SCF-CI includes some amount of dispersion energy. Price and Stone applied a dispersion function to offset this effect; however, the magnitude of the dispersion energy obtained from SCF-CI is uncertain. The outline of this paper is as follows. The Methods section describes the quantum mechanical and geometrical details of the model calculations, definition of the Morokuma energy decomposition terms, the potential-derived net atomic charge model, and the definition of the repulsion potentials. The Results section gives details for the Morokuma decomposition terms, the charge models, and the repulsion models.
Theoretical and Computational Methods Geometry and Quantum Mechanics. The monomer structures were fixed a t their experimental geometries: methane, r(C-H) = 1.094 A, L(H-C-H) = 109So;l9ammonia, r(N-H) = 1.012 A, L(H-N-H) = 1O6.7O;l9water, r(O-H) = 0.957 A, L(H-0-H) = 104.52°;20 and methanol, r(0-H) = 0.963 A, L(C-0-H) = 108.03°.19 A repulsive dimer was taken as two monomers with a hydrogen directed at a hydrogen of the other monomer linearly and separated by an intermolecular hydrogen-hydrogen distance, r, measured between the nuclei. This head-on approach of the hydrogens is shown in Figure 1 for the dimers of methane, ammonia, water, and methanol. A collection of repulsive dimers was created in the same manner with only the intermolecular separation of the hydrogens, r in angstroms, varied (usually by 0.25-A increments).
H F wave functions for the monomers and each collection of dimers were computed by the ab initio self-consistent-field (SCF) molecular orbital method using the program of Singh and Kollmane21 Two standard Gaussian basis sets8 were used. All calculations utilized the 6-31G** basis set which is of split-valence type and includes polarization functions of d type on heavy atoms and p type on hydrogens. In addition, the triply split 6-311G** basis set was used for monomer charge analysis and dimer energy decomposition. This basis set was shown by Krishnan, Binkley, Seeger, and PopleZ2to give more flexibility in representing the atomic outer valence regions. Two approaches were used in deriving exponential repulsion curves from the S C F results. In the first approach, Morokuma decompositions of the total interaction energy were used to obtain the exchange (EX) component. Then, optimized exponential repulsion models were found which yielded the best fit to this component at each basis set level (6-31G** or 6-31G**). Since no charge interactions are included in the EX components, this optimization involved only the B and C parameters of the empirical potential function. The second approach was based on the total intermolecular energy. Values for the electrostatic energy component were obtained in two ways. The first estimate was obtained from the Morokuma electrostatic component, EL. The second estimate was obtained by applying Coulomb's law between atomic charges, and was designated COU. The net atomic charges on which COU is based were obtained by the potential-derived (PD) method. The repulsion curve for this approach involved B, C, and a charge scale factor K as adjustable parameters. With either approach there were questions about the effect of basis set choice, and basis set superposition correction23(BSSC). BSSC takes into consideration the fact that more basis functions are normally used in the dimer calculation. Corrected energies for the monomers as they exist in each dimer are obtained by using a basis set which includes "ghost" basis functions with zero nuclear charges at positions in space where the second molecule of the dimer would be located. The result is always a decrease in the energy of the monomer. The overall effect of BSSC is to increase the magnitude of the interaction energy. BSSC is time consuming since a separate monomer calculation with a large basis set must be made for each dimer configuration. To establish the relative importance of this effect, we carried out BSSC calculations for water dimer using the 6-311G** basis set. Morokuma Enero Decomposition. Each intermolecular energy for the collection of repulsive dimers was analyzed according to the Morokuma d e c o m p o s i t i ~ n . ~The ~ ~ ~Morokuma exchange repulsion component (EX) is the energy component caused by an exchange of electrons between monomers A and B. More physically, this is the short-range repulsion due to overlap of the electron distribution of A with that of B. The electrostatic component (EL) represents the interaction between the undistorted electron distribution of monomer A and that of monomer B. The polarization component (PL) refers to the effect of the electron distribution distortion of A by B, B by A, and the higher order coupling due to such distortions. Charge transfer (CT) is the energy component caused by a charge transfer from occupied M O s of A to vacant M O s of B and occupied MO's of B to vacant M O s of A including higher order coupled interactions. The mixed term (MX) is the difference between the total interaction energy and the sum of the above components. The total interaction energy (EI) is the energy of the supermolecule minus the energy of the monomers. E1 may be positive or negative. At infinite separations of the monomers, E1 must approach zero. The difference between E1 and the sum of the Morokuma components (EL, PL, CT, and EX) is defined as the
(16) Timofeeva, T. V.; Chernikova, N. Y . ;Zorkii, P . M. Rum. Chem. Rev.
1980, 49, 509.
(17) Bohm, H. J.; Ahlrichs, R. J . Chem. Phys. 1982, 7 7 , 2028. (18) Price, S. L.; Stone, A. J. Mol. Phys. 1980, 40, 805. (19) Harmony, M. D.; Laurie, V. W.; Kuczkowski, R. L.; Schwendeman, R. H.; Ramsay, D. A.; Lovas, F. J.; Lafferty, W. J.; Maki, A. G. J . Phys. Chem. Re$ Data 1979, 8, 619. (20) Benedict, W. S.; Gailar, N.; Plyler, E. K. J . Chem. Phys. 1956, 24, 1159.
(21) Singh, U. C.; Kollman, P. QCPE 1982, 15, 446. (22) Krishnan, R.; Binkley, J. S.;Seeger, R.; Pople, J. A. J. Chem. Phys. 1980, 7 2 , 650. (23) Szczesniak, M. M.; Scheiner, S. J . Chem. Phys. 1986, 84, 6326. (24) Stewart, R. F.; Davidson, E. R.; Simpson, W. T. J . Chem. Phys. 1965, 42, 3175. (25) Kitaura, K.; Morokuma, K. Int. J . Quanrum Chem. 1976, 10, 325.
6368 The Journal of Physical Chemistry, Vol. 91, No. 25, 1987 mixing or coupling term (MX) which accounts for higher order interactions. This term may be positive or negative. In the study by Umeyama and MorokumaZ6the value of EL was scaled by the ratio of the observed dipole moments divided by dipole moments calculated with the given basis set. They did not scale PL, CT, EX, or MX. We used a similar technique in this study. Potential-Derived Net Atomic Charges. The electric potential for a unit positive charge at a point r in the vicinity of the monomer or dimer containing nuclei A at positions RA and with charge ZA is given by
where P , is an element of the density matrix of the SCF mo~ the atomic wave functions used lecular wave function and $ Jare as a basis set. The first term is summed over the nuclei and the second term is summed over the electrons of the system. The set of points selected for the evaluation of the electric potential was a cubic grid of 1 A spacing in a 1.2 A thick shell around the molecule. The coordinate system was based on the directions of the principal axes of inertia. The inner surface of this shell was selected to represent the distance of closest normal intermolecular approach and was taken a t the van der Waals radius of the nearest atom plus the van der Waals radius of the smallest approaching atom, hydrogen. The van der Waals radii of oxygen, carbon, nitrogen, and hydrogen atoms were taken as 1.4, 1.7, 1.5, and 1.2 A, respectively. This procedure led to the definition of about 200 grid points around the monomers and approximately 300-400 grid points around each dimer configuration. The electric potential a t each grid point was calculated directly from the ab initio wave function. Potential-derived charges were obtained from these electric potential values by using the method of Cox and Williams.Io These atomic charges were determined by fitting them to the molecular electric potentials at the grid points by least squares. The goodness of fit was represented by the root-mean-square deviation of the potential or by the percentage relative root-mean-square deviation. For the repulsive dimers, the net atomic charge model was utilized with 10% electron density shift (foreshortening) on hydrogen. The electron density shift for a bonded hydrogen is well d o c ~ m e n t e d . ' ~ The ~ ' ~ charge ~ ~ ~ . ~model was further scaled as noted to fit the observed dipole moments or to optimize the repulsion curves, for each basis set level. Exponential Repulsion-Coulombic Models for the S C F Results. S C F supermolecule calculations are very time consuming even for the small monomer molecules considered here. It is desirable to have a simple empirical model for the intermolecular energy which can be readily applied to large molecules such as proteins. As noted above the dispersion energy is not included in the HF calculated energy. The largest components of E1 are EX and EL. A simple model for E1 is the (exp-1) function E1 = Bijexp(-Cijrij)
+ Pqiqjrij-'
where the sum is over intermolecular atomic distances. We will refer to the first energy term as REP and the second as COU; K is a charge scale factor applied to the net atomic charges. Optimization of B, C, and K was accomplished by a leastsquares fitting procedure. Since other intermolecular interactions besides H-H are present it was necessary to model these also. The parameters of Williams and co-workers6 were used for the C.-C, 0-0, and N-N potentials. These were fixed and held constant in all derivations. Heteroatomic nonbonded interactions were found by application of the geometric mean combining law. The sum of squares of weighted differences between calculated and expected energy is called RSUM; this goodness-of-fit indicator was monitored in the fitting calculations.
ymeyama,
H.;Morokuma, K, J , A ~(-hem, , sot. 1977, 99, 1316, (27) Singh, U. C.; Kollman, P. A. J . Chem. Phys. 1985, 83, 4033. (26)
Williams and Craycroft TABLE II: 6-31G** Morokuma Decomposition Analysis of Dimer Energies" H**.H EL PL CT EX MX E1
Methane Dimer 1.75 2.00 2.25 2.50 2.75 3.00 3.25 3.50 3.75 4.00
-3.371 -1.241 -0.361 -0.043 0.049 0.062 0.053 0.041 0.030 0.022
-0.614 -0.165 -0.048 -0.016 -0.006 -0.003 -0.002 -0.001 -0.001 -0.000
1.75 2.00 2.25 2.50 2.75 3.00 3.25 3.50 3.75
4.070 3.042 2.212 1.581 1.121 0.791 0.558 0.391 0.272
-1.283 -0.574 -0.9 -0.166 -0.101 -0.066 -0.044 -0.031 -0.022
1.75 2.00 2.25 2.50 2.75 3.00 3.25 3.50 3.75 4.00
15.984 11.435 8.407 6.324 4.852 3.790 3.008 2.422 1.974 1.628
-1.947 -1.001 -0.562 -0.338 -0.214 -0.141 -0.096 -0.067 -0.048 -0.035
1.75 2.00 2.25 2.50 2.75 3.00 3.25 3.50 3.75 4.75
16.144 11.522 8.441 6.322 4.827 3.751 2.960 2.370 1.921 0.919
-2.259 -1.180 -0.672 -0.409 -0.262 -0.175 -0.120 -0.085 -0.061 -0.019
-3.580 -1.368 -0.588 -0.308 -0.210 -0.170 -0.140 -0.106 -0.070 -0.040
17.852 7.365 2.974 1.179 0.460 0.177 0.066 0.024 0.009 0.003
1.465 0.498 0.191 0.083 0.038 0.018 0.008 0.003 0.001 0.000
1 1.753 5.087 2.169 0.894 0.332 0.085 -0.013 -0.038 -0.030 -0.015
9.645 3.745 1.422 0.529 0.193 0.069 0.024 0.008 0.003
1.215 0.444 0.190 0.096 0.048 0.022 0.008 0.003 0.001
10.394 4.996 2.529 1.386 0.839 0.556 0.393 0.286 0.209
5.389 1.993 0.721 0.256 0.089 0.031 0.010 0.003 0.001 0.000
1.401 0.612 0.276 0.122 0.050 0.018 0.006 0.002 0.000 0.000
17.450 1 1.262 7.848 5.810 4.481 3.545 2.851 2.321 1.909 1.585
1.494 0.654 0.296 0.131 0.053 0.019 0.006 0.001 0.000 0.000
17.009 11.005 7.680 5.688 4.382 3.457 2.770 2.246 1.839 0.899
Ammonia Dimer -3.252 -1.660 -1.002 -0.654 -0.422 -0.260 -0.154 -0.086 -0.044
Water Dimer -3.376 -1.778 -0.994 -0.554 -0.298 -0.154 -0.078 -0.038 -0.020 -0.008
Methanol Dimer -3.786 -2.002 -1.116 -0.616 -0.328 -0.170 -0.086 -0.044 -0.022 0.000
5.415 2.011 0.731 0.261 0.091 0.031 0.011 0.003 0,001 0.000
"The H...H distance is measured between nuclear sites, Le.. X-H distances are not foreshortened. Energies are in kJ/mol. In the first model we derived values of B and C from the Morokuma EX component. The weighting factors for the terms in RSUM were taken inversely proportional to the squares of the estimated errors in EX; these estimates were 10% plus 0.2 kJ/mol. In the second model values for B and C were derived from the net intermolecular energy obtained after subtraction of a COU component. The COU component was either fixed by scaling to the observed dipole moment or optimized with a charge scale factor K. The weighting factors were based on estimated errors of 5% plus 0.2 kJ/mol in EI. Summary of Labels for Energy Terms. EI, total Hartree-Fock intermolecular energy; EX, exchange component defined by Morokuma; EL, electrostatic component; PL, polarization component; CT, charge-transfer component; MX, mixed component; COU, empirical intermolecular Coulombic energy between net atomic charges; REP, empirical intermolecular repulsion energy between nonbonded atoms. Results and Discussion Energy Decomposition of Repulsive Dimers. The results of the ~~~~h~~ decomposition study at the 6-31G** and 6-31 1 ~ * * basis levels are shown in Tables I1 and 111. Methane was the only molecular dimer in this series having a negative E1 at large distances in the 6-31G** basis level. At the 6-31 1G** level all E1 values are positive. EL was small and attractive at short
H-H Interactions in Organic Molecules
The Journal of Physical Chemistry, Vol. 91, No. 25, 1987 6369
TABLE 111: Same as Table 11, but with 6-311G** Wave Functions
MX
E1
Methane Dimer -3.964 18.332 -1.612 7.765 -0.692 3.246 -0.306 1.344 -0.148 0.551 -0.088 0.224 -0.064 0.089 -0.048 0.035 -0.034 0.014
1.997 0.695 0.251 0.086 0.024 0.003 -0.001 0.000 0.000
12.317 5.437 2.385 1.049 0.459 0.194 0.076 0.029 0.012
Ammonia Dimer -3.012 9.988 -1.498 3.996 -0.848 1.583 -0.524 0.622 -0.340 0.242 -0.220 0.093 -0.138 0.035 -0.082 0.013 -0.044 0.005
0.797 0.144 -0.007 -0.021 -0.009 0.000 0.003 0.003 0.002
10.427 5.105 2.680 1.536 0.961 0.649 0.463 0.341 0.257
Water Dimer -2.996 5.340 -1.700 2.000 -1.048 0.741 -0.664 0.273 -0.420 0.100 -0.206 0.025 -0.152 0.013 -0.084 0.004 -0.042 0.002
0.714 0.192 0.055 0.024 0.017 0.01 1 0.008 0.004 0.002
16.696 10.709 7.450 5.513 4.256 3.056 2.734 2.241 1.858
EL
PL
1.75 2.00 2.25 2.50 2.75 3.00 3.25 3.50 3.75
-3.185 -1.158 -0.344 -0.050 0.041 0.059 0.054 0.043 0.033
-0.863 -0.253 -0.076 -0.025 -0.006 -0.004 -0.002 -0.001 -0.001
1.75 2.00 2.25 2.50 2.75 3.00 3.25 3.50 3.75
4.020 3.064 2.252 1.626 1.168 0.840 0.606 0.437 0.3 15
-1.366 -0.601 -0.300 -0.167 -0.100 -0.064 -0.043 -0.030 -0.021
1.75 2.00 2.25 2.50 2.75 3.00 3.25 3.50 3.75
15.667 11.255 8.284 6.230 4.780 3.351 2.964 2.389 1.945
-2.029 -1.038 -0.582 -0.350 -0.221 -0.125 -0.099 -0.069 -0.049
H*-*H
CT
EX
distance, but became slightly repulsive beyond 3.00 A. From 1.75 to 3.25 A the major contributor to E1 was EX. Between the basis sets, EX terms were slightly larger a t the 6-31 1G** level. EL was consistently more positive at each distance. Hence, E1 for methane was slightly larger in the larger basis set. The negative sign of EL for methane below 2.5 8, may indicate that the electron cloud has been penetrated to expose the nuclear charge. This effect would be most important for methane because of the small charge on hydrogen and the relatively larger size of H(C). The Morokuma decomposition of ammonia head-on dimers showed no EL sign change. The C T terms were about the same magnitude as in methane, and E L and EX were the most significant terms in both basis sets. E L showed a longer range in ammonia than in methane and was the principal component at 2.25 A and beyond. In water E L became much more important and was always larger than EX. At the shortest distance, 1.75 A, EL was about 3 times EX. Again, CT was not significantly larger than in methane, but the magnitude of PL was slightly larger at all distances. Generally, E L terms using 6-3 11G** wave functions were smaller and EX terms larger (except at 1.75 A). Methanol dimer was very similar to water dimer at the 6-31G** basis level as Table I1 shows. Because of computer limitations we were not able to calculate methanol dimer at the 6-311G** level. A comparison in the value of E1 at each H(O)--H(O) distance between methanol and water dimers show a close similarity. This similarity suggests that H(O)-H(O) parameters may be transferable between water and methanol. At the distance sampled, methanol dimers had slightly less E1 and slightly more EX than water dimers. The larger P L in methanol is no doubt caused by replacement of a hydrogen by the larger and more polarizable methyl group. There was some cancellation among the PL, CT, and MX terms in all the dimers. Derivation of Repulsion Parameters from EX Components. The first approach in deriving B and C repulsion parameters for the various dimers used the Morokuma exchange energy. Tables I1 and I11 show that, at a given H--H distance, EX rapidly decreases in the order methane > ammonia > water. Note that in this approach no charge model was necessary because the physical interpretation of the exchange energy does not include any charge
TABLE I V Repulsion Function B and C Values for H a .HInteractions in Methane, Ammonia, Water, and Methanol Dimers Based on 6-316** and 6-3116.. Morokuma EX Components and either 10 or 0% Foreshortenin@ basis set 6-31G**
dimers Bij” methane 17325.6 H(C) (7937.1) ammonia 13619.8 H(N) (6303.7) water 9968.3 H(0) (4634.1) methanol 10048.9 (4595.0) H(0)
d(vdw)d 3.6313 2.548 (3.6302) (2.334) 3.8629 2.349 (3.8618) (2.150) 4.0392 2.181 (4.0386) (1.991) 4.0584 2.173 (4.0524) (1.983) Cif
6-311G** Bijb Cif d(vdw)d 12809.1 3.4775 2.562 (6071.5) (3.4761) (2.348) 9759.6 3.6870 2.358 (4680.5) (3.6859) (2.160) 8193.2 3.9511 2.174 (3871.5) (3.9505) (1.985)
In parentheses, parameters based on 0% X-H bond foreshortened kJ/mol. ‘In .&-I. dDistance at which the repulsive force is model. N.
I I
a 8
I 3 5 2 0
INTERNUCLEAR DISTANCE (ANGSTROMS)
Figure 2. Plots of He-H repulsion potential derived from EX versus the closest H.-H distance for the foreshortened case: (a) methane; (b) ammonia, (e) water or methanol; (d) Williams and Starr2 experimentally derived potential.
interaction component. Table IV shows the derived values of B and C for H(C), H(N), and H ( 0 ) interactions with either 0 or 10% foreshortening. The derivations were straightforward and there was a very good least-squares fit in each case, with RSUM always less than 0.1. As expected, foreshortening did not significantly affect the values obtained for C. The larger effect on B values is understandable in terms of the amount of foreshortening, Ar, by the relation B(foreshortened) = B(nonforesh0rtened) exp(CAr) The C values obtained using the 6-3 11G** energies were always smaller than those obtained with 6-31G** energies. This would correspond to a picture of the larger basis set allowing easier deformation of the electron clouds. There is a clear trend toward larger C values going from H(C) to H ( N ) to H ( 0 ) . Figure 2 shows plots of the He-H repulsion potential derived from EX versus the closest He-H distance for the foreshortened case. A plot for the nonforeshortened potentials would be shifted along the H-H distance by the amount of the foreshortening. Curves a, b, and c show curves for methane, ammonia, and water dimers, respectively. The curve for methanol is similar to the water curve. Curve d is the Williams and Starr2 H-H repulsion potential derived from 18 hydrocarbon crystal structures, corrected for 10% foreshortening. Of considerable interest is the decrease in size of the hydrogen atom going from H(C) to H ( N ) to H ( 0 ) . The difference of approximately 0.4 A in the apparent diameters of H(C) and H ( 0 ) is certainly significant. However, these curves are based only on the Morokuma EX components and it is highly desirable to verify this effect using the total energy, EI. It is of interest to compare these repulsion curves to familiar tabulated values of van der Waals diameters. Since van der Waals
6370 The Journal of Physical Chemistry, Vol. 91, No. 25, 1987
Williams and Craycroft
TABLE V Potential-Derived Net Atomic Charges at the 6-31G** Level for Methane, Ammonia, and Water Monomers and Their Dimers in the H e . .HRepulsive Configuration with 10%X-H Bond Foreshortening
net atomic charge4 dimers
methane
He *H,A 1.75 2.00 2.25 2.50 2.75 3.00 3.25 3.50 3.75 4.00
C1$2 -737 -738 -745 -762 -757 -750 -733 -73 1 -730 -739 -744
monomer
Hl,H5 185 183 184 189 187 186 181 182 181 184 186
ammonia N 1,N2
H2,H3,H4,H6,H7,H8 184 185 187 191 190 188 184 183 183 185 186
-1248 -1241 - 1240 -1232 -1235 -1230 -1230 -1218 -1210
H1.H4 386 387 392 386 391 392 396 394 392
H2.H3.H5.H6 43 1 427 424 423 422 419 417 412 409
-1191
397
397
net atomic charge" water
dimers
methanol
H*-*H,A 1.75 2.00 2.25 2.50 2.75 3.00 3.25 3.50 3.75 4.00 4.75
01,02 -853 -857 -865 -867 -87 1 -871 -872 -874 -874 -875
Hl,H3 383 390 399 405 415 415 419 424 427 428
H2,H4 470 467 466 462 456 456 453 450 447 447
monomer
-874
437
437
4Electronicunits
X
01,02 -628 -636 -658 -670 -692 -707 -73 1 -729 -727
c1,c2 105 90 128 159 213 252 332 323 303
H1,HS 399 412 426 433 443 453 457 458 46 1
H2,H6 86 92 86 76 66 54 34 36 41
H3,H4,H7,H8 19 21 9 1 -1 5 -26 -46 -44 -39
-740 -766
322 437
472 472
38 7
-46 -7 5
lo3.
diameters are usually determined from closest approach distances, it is reasonable to associate them with a certain repulsive force. N as a good choice for the Williams and Houpt suggested force defining the van der Waals diameter. Table IV shows these values, designated d(vdw). Pauling gives a value of 2.4 A for the hydrogen van der Waals diameter. Table IV shows values ranging from 1.983 to 2.348 A without foreshortening, and from 2.173 to 2.562 A with 10% foreshortening. It is apparent that these values are in the correct range. Most interesting is the change in going from H(C)-.H(C) to H(N)-H(N) to H(0)-H(O), where the size of the hydrogen decreases significantly. The larger size of the hydrogen atom in methane correlated with a relatively small hydrogen net atomic charge of 0.186e. For ammonia the hydrogen net atomic charge increased to 0.309, and for water the hydrogen net atomic charge was 0.366. As the net atomic charge increases, the electron density and its consequent overlap in the dimer decreases. Therefore it is quite reasonable that this size difference should exist. Potential-DerivedCharge Model for Repulsive Dimers. Charge scale factors were used because it is known that there is a systematic over- or underestimation of chargesl0 caused by the particular quantum mechanical basis set for which the electrostatic potential surface was generated. The ratio of the observed to the calculated dipole moment of the monomers was used as the initial scale factor for charges derived from that basis set. We used the following observed dipole moments: ammonia,28 1.4718; water,30 1.827; and methanol:9 1.662 D. Methane has zero dipole moment and the scale factor for methane dimers was set at 1. The 6-3 1G** scale factors (dipole ratios) were 0.7788 for ammonia, 0.8364 for water, and 0.8685 for methanol. The 6-31 1G** scale factors were 0.8326,0.8431, and 0.8724, respectively. In all cases, the 6-31G** and 6-31 1G** basis sets overestimated the observed dipole moment. These values for the charge scale factors bring the magnitude of the net atomic charges into agreement with the observed dipole moment. (28) Marshal, M. D.; Muenter, J. S. J . Mol. Speczrosc. 1981, 85, 322. (29) Amano, T. J. Mol. Spectrosc. 1981, 88, 194. (30) Kuze, H.; Amano, T.; Shimizu, T. J . Chem. Phys. 1981, 75, 4869.
Table V shows the PD charges of methane dimers with 10% foreshortening. At all distances, the fits were excellent in the range of 0.2-0.3 kJ/mol or 12.3-17.1%. The symmetry of the systems required equal charges on several atoms as shown in the table. The variation of the charge on the carbon atom is between -0.730 to -0.762e. The observation that methane shows little change in atomic charges with intermonomer distance suggests only a slight charge polarization with H(C)-H(C) distance. The ammonia dimer foreshortened PD charges are also shown in Table V. The fits were in the range 2.6-3.1 kJ/mol or 9.8-1 2.0%. The nitrogens were always highly negative ranging from -1..191e at the monomer to -1.248e at the shortest distance. The trend was to increase the negative charge on nitrogen as the molecules approached. The nearest hydrogen, H1, showed only a small decrease in charge from 0.397 to 0.386e. The distant hydrogen, H2, showed a larger change than the nearest-neighbor hydrogen. The increase from 0.397e in the monomer to 0.431 in the dimer at close approach correlated with the increasing negativity of the nitrogen. For the repulsive water dimer the fit was in the range of 2.3-3.1 kJ/mol or 7.5-10.9%. In contrast to the ammonia nitrogen, the water oxygen becomes less negative as the monomers approach. The approaching hydrogens, H1 and H3, have a charge decrease from 0.437 to 0.383 e. The distant hydrogens, H 2 and H4, increase their charge by an amount larger than what H1 and H 3 lose. For methanol dimer the fits to the S C F potential with the charges shown in Table V were in the range 1.7-2.1 kJ/mol or 7.8-11.2%. As with water, the oxygen charge (-0.766 in the monomer) becomes more positive as the H(O)-H(O) distance decreases. The positive charge on carbon (0.437e in the monomer) decreases at shorter distances. Also, q(H1), which is 0.472e in the monomer, decreases at shorter distances and H 2 and H3 have only small charges. The shifts in net atomic charges were larger for methanol than in water. The C-0 and H-C bonds show stronger charge polarization. PD net atomic charges were also obtained with the 6-31G** basis set for the monomers only. Using the same 10% foreshortening, the methane monomer charges increased on carbon to
The Journal of Physical Chemistry, Vol. 91, No. 25, 1987 6371
H--H Interactions in Organic Molecules TABLE VI: Empirical Coulombic Energy (COU) for Methane, Ammonia, Water, and Methanol Dimers Compared to Scaled 6-31G** Morokuma Electrostatic Components (EL)'
methane dimer H**H,A 1.75 2.00 2.25 2.50 2.75 3.00 3.25 3.50 3.75 4.00
COUb 0.480 0.279 0.177 0.125 0.081 0.056 0.036 0.028 0.019 0.015
ELC -3.371 -1.241 -0.361 -0.043 0.049 0.062 0.053 0.041 0.030 0.022
water dimer H.-.H,A 1.75 2.00 2.25 2.50 2.75 3.00 3.25 3.50 3.75 4.00 4.75
COU 3.769 2.855 2.271 1.831 1.600 1.243 1.045 0.903 0.774 0.646
EL 13.369 9.564 7.032 5.289 4.058 3.170 2.516 2.026 1.651 1.362
ammonia dimer COUb 0.873 0.444 0.253 0.004 -0.045 -0.083 -0.087 -0.099 -0.111
EL' 3.170 2.369 1.723 1.231 0.873 0.616 0.435 0.305 0.212
TABLE VI1 Repulsion Function B and C Values at Various Charge Scale Factors for Methane, Ammonia, Water, and Methanol Based on the Total 6-31G** Interaction Energy with 10%X-H Bond Foreshortening
charge scale factor
0.000 0.500 1.000 1.111 1.500
cou
EL 14.021 10.007 7.331 5.491 4.192 3.258 2.571 2.058 1.668
0.621
0.798
RSUM'
6831.3 7272.4 8859.1 945 1.8 12748.3
3.4239 3.4576 3.5637 3.5986 3.7594
0.58 0.69 1.10 1.26 2.10
486.4 441.0 366.0 226.8 182.2
2.3356 2.3030 2.2410 2.0823 2.0101
3.12 3.56 4.42 6.55 7.47
Water Dimer 0.000 0.500 0.836 0.929 1.549d
29.4 28.9 30.3 32.3 343.4
0.8762 0.9098 1.0079 1.0654 2.2301
3.24 3.01 2.55 2.34 0.49
Methanol Dimere 0.000 0.434 0.869 0.912 0.ooof 0.8699 1.085d,g
"All energies are in kJ/mol. bBased on 10% X-H bond foreshortened PD charges scaled by dipole ratio. 'Morokuma electrostatic energy scaled by dipole ratio. -0.768e and on hydrogens to 0.192e with a fit of 0.3 kJ/mol or 18.8%. The change in the monomer charges between 6-31G** and 6-311G** was small (less than 0.03e). At the 6-31 1G** basis level, ammonia monomer gave atomic charges on nitrogen and hydrogens as -1.122e and 0.374e with a fit of 3.5 kJ/mol or 13.2%. The changes in the net atomic charges between 6-31G** and 6-311G** basis sets again were small (for nitrogen 0.007e, and for hydrogens 0.023e. For water monomer, the 6-31 1G** basis set gave optimum net atomic charges of -0.866e on oxygen and 0.433e on the hydrogens with a fit of 3.1 kJ/mol or 9.6%. The use of this larger basis set on water monomer decreased the magnitude of the charges on oxygen and hydrogen by only 0.008e and 0.004e. Methanol charges were not evaluated with the 6-311G** basis set. Comparison of Coulombic Models. Comparisons of the intermolecular Coulombic energy, COU, for methane, ammonia, water, and methanol dimers with the Morokuma EL components are made in Table VI. The COU component was calculated by using PD charges based on a foreshortened model and scaled to reproduce the observed dipole moment. A nonforeshortened model for COU was also evaluated, in which case the COU values were slightly larger. The EL Morokuma components were scaled by the dipole ratio, as described above. At a given Ha-H distance, the EL components rapidly increase in going from methane to ammonia to water. Methanol shows values similar to water, but slightly larger. For methane COU tracks E L fairly well a t distances of 3 8, or greater. Below 3 A COU continues positive while EL goes negative. The latter effect may be caused by the electron cloud penetration effect mentioned previously. For ammonia COU remains small and is even negative at 2.75 A and beyond. E L is significantly larger than COU, especially at close distance. For water and methanol COU shows larger values but is still generally much less than EL. Differences between values of COU and EL caused problems in deriving exponential repulsion parameters from EI, especially for water and methanol as further discussed in the following section. Note that the persistent repulsive energy at long distance shown by E1 for water and methanol dimers largely originates in the EL component.
C,,b
Ammonia Dimer 0.000 0.300 0.500 0.779 0.865
methanol dimer 5.752 4.539 3.678 2.938 2.413 2.080 1.670 1.367 1.149
B,;"
Methane Dimer
17.5 16.0 19.5 24.3 16.9 19.5 497.7
0.5932 0.6186 0.9156 1.0363 0.5871 0.9175 2.4476
0.50 0.58 1.34 1.37 0.53 1.37 0.24
'In kJ/mol. A-'. CDimensionlessunits. dOptimum charge scale factor with Blj and C,. 'Unless noted, B and C of H(C)-..H(C) are 16073.67 and 3.74. fL3 and C for H(C)...H(C) from methane dimer with K = 0.0 scale factor. gAs i n J but with K = 1.0. Derivation of Repulsion Parameters from EI. In this approach the total HF intermolecular energy, EI, was modelled by R E P and COU. It was proposed that the sum of REP and COU add to E1 and individually REP and COU need not necessarily correspond exactly to Morokuma components. In particular we showed in the previous section that although EX may be associated with REP, there are problems with associating EL with COU. As noted earlier, the dimer configurations are such that hydrogen bonding is not present. In order to retain the simple (exp-1) or REP COU model the PL, CT, and MX Morokuma components are not explicitly considered, but these components partly cancel and also may be partly absorbed into the derived (exp-1) functions through adjustment of the coefficients of REP and COU. Table VI1 shows optimized values for B and C obtained by fitting the 6-31G** E1 with an (exp-1) model. As previously noted, PD net atomic charges were obtained directly from the dimer wave functions and used to obtain COU with Coulomb's law. In some of the derivations a charge scale factor, K , was varied. The COU component is affected according to @. The table shows that optimized values of C for methane dimer ranged from 3.42 to 3.76 and were therefore in the same range as exponents derived from the EX component. For ammonia dimer the optimized values of C ranged from 2.59 to 2.38, which were smaller than exponents derived from EX components. For water and methanol dimer unreasonably small values of C (around 1 A-') were obtained, much smaller than exponents derived from EX components. Variation of the charge scale factor for methane and ammonia dimers showed that the lowest values of RSUM (best fits) were with zero charge. This meant that all of E1 was being represented by REP, with no COU. For water and methanol, however, there was a nonzero optimum value for K. For water the optimum K was 1.549, which corresponded to COU being scaled up by 2.399 before being subtracted from EI. There was a drastic increase in C (to 2.23 A-1) accompanying this scaling of COU. For methanol, there was a smaller optimum K value of 1.085, but the
+
6372 The Journal of Physical Chemistry, Vol. 91, No. 25, 1987
Williams and Craycroft TABLE VIII: Repulsion Function B and C Values at Various Chatge Scale Factors for Methane, Ammonia, and Water Dimers Based on the Total 6-311G** Energy with 10%X-H Bond Foreshortening
8 0 fi
r
”\
\\
i
3
0.500 1.000 1.500
CL
4 0
C,b
Bun
RSUM‘
Methane Dimer 0.000
6 8
>
5
charge scale factor
6246.2 5056.3 6246.2 9272.1
3.3823 3.2707 3.3823 3.5908
0.15 0.03 0.15 0.61
Ammonia Dimer 296.5 287.0 268.7 284.8
0.000 0.416 0.833 1.082
2 0
8 8
effect on C (to 2.45 A-’) was similar to water. Thus it appeared necessary to deviate from the dipole ratio charge scale factor in order to supply sufficient COU to allow a short-range REP. The use of the larger charge scale factors brings COU closer to that obtained from Morokuma’s scaled EL terms. Singh and K01lman~~ studied hydrogen-bonded water dimers using the Morokuma decomposition. When they fitted the EL EX MX components to an (exp-1) model they also obtained a long-range exponential repulsion. In methanol dimers more distant H(C)-H(C) interactions are present as well as the short H(O)-.H(O) interaction. We determined, however, that allowing a reasonable range of variation of the H(C)-.H(C) potential did not significantly affect the H(O)--H(O) potential. For instance, Table VI1 shows that if the potential derived from the methane EX components is used instead of the Williams and Starr potential there is almost no change in the derived C(H(0)-.H(O)) values obtained with the same charge model (see the K = 0.0 and K = 0.869 entries). Figure 3 shows potential energy curves for H-H interactions in the dimers based upon fitting E1 using a foreshortened model and dipole moment scaling. The repulsion potentials for water and methanol (curves c and d) are long range and seem physically unrealistic. The form of these curves can be attributed to the large values of E1 at long distances; these large values are not well represented by COU and therefore must appear in a long-range REP. Methane dimer (curve a) does not exhibit this behavior. In fact, curve a is close to curve e, the Williams and Star$ potential derived from observed hydrocarbon crystal structures. Methane dimer potentials were the most stable in all the derivations. Curve b for ammonia dimer shows some increased repulsion at large distance, but the effect is much less than for water or methanol dimers. It was apparent that upward scaling of COU by increasing the net atomic charges was necessary because only these larger charges resulted in a physically reasonable REP. Unfortunately, these larger charges do not reproduce the observed dipole moments of the monomers nor do they well represent the calculated electric potentials around the dimers. To further investigate this difference between COU and EL we considered the effects of using a larger basis set (6-31 lG**) for the S C F calculation and the basis set superposition correction. Table VI11 shows the derived repulsion parameters obtained by using various charge scale factors for methane, ammonia, and water repulsive dimers based on the 6-31 1G** E1 values. Again, the lowest value of RSUM occurs with a zero charge model for ammonia dimer. Methane dimer showed a minimum RSUM with K = 0.5. As before, the methane R E P was well behaved while ammonia and water showed an anomalously long-range REP. As with the smaller basis set, the fitting of water dimer E1 produced a long-range REP unless K was optimized in addition to B and C. With optimized K , the resulting values for C were the largest obtained in this study (4.60 A-’). The larger 6-3 11G**
+
+
3.83 4.20 5.45 6.73
Water Dimer
INTERNUCLEAR DISTANCE (ANGSTROMS)
Figure 3. Potential energy curves for H-H interactions in the dimers: (a) methane; (b) ammonia; (c) water; (d) methanol; (e) Williams and Starr2 potential derived from hydrocarbon crystal structures.
2.1326 2.1446 2.2025 2.3182
0.000 0.422 0.843 1.179d
29.8 26.0 16.1 19330.1
0.9004 0.9021 0.9376 4.5989
4.00 3.72 3.11 0.79
Water Dimer‘ 0.000 0.422 0.843 1.223d
34.3 30.6 20.6 4228.4
0.9120 0.9 159 0.9532 3.8030
2.82 2.52 1.85 0.02
“In kJ/mol. k’.CDimensionlessunits. dOptimum charge scale factors with B , and C,. ‘BSSE corrected 6-311G** interaction energies for repulsive water dimers used for this derivation (see text).
r
r
INTERNUCLEAR DISTANCE (ANGSTROMS)
Figure 4. Foreshortened water dimer H(O).-H(O) REP curves: (a) obtained by fitting the Morokuma EX values; (b) obtained by fitting E1 with dipole-scaled COU; (c) with K optimized; (d) obtained by fitting E1 with BSSC and dipole-scaled COU; (e) with K optimized.
basis set yielded a smaller optimized value of K , 1.179, as compared to a value of 1.549 in the 6-31G** basis. This suggested the possibility that an even larger basis set might further decrease K toward the dipole ratio value. Note that only monomer-derived net atomic charges were used at the 6-31 1G** level. The use of this charge model for the dimer neglected any change in the net atomic charges with distance. To further examine the long-range repulsion problem in water dimer, calculations were made to correct E1 for basis set superposition (BSSC). This correction affected mostly the close distances. The derived repulsion parameters based on 6-31 1G** E1 with BSSC are given in Table VIII. With BSSC included the long-range REP persisted with dipole-scaled charges. As before, allowing K to increase resulted in a more normal repulsion range. Figure 4 shows the foreshortened water dimer H(O)-H(O) REP curves obtained by fitting the 6-3 11G** basis set calculations. Curve a shows REP obtained by fitting the Morokuma EX values. Curve b shows REP obtained by fitting E1 with dipole-scaled COU. In curve c K was optimized. Curve d shows REP obtained by fitting E1 with BSSC and dipole-scaled COU. In curve e K was optimized. It is the treatment of COU which differs between
J. Phys. Chem. 1987,91, 6373-6380 these curves. Models based on 6-31 1G**E1 with and without BSSC showed long-range REP when dipolescaled COU was used. This apparent long-range R E P disappeared when COU was optimally scaled. Overall, the results obtained by fitting E1 were consistent with those from fitting EX, showing hydrogen diameters H(C) > H(N) > H ( 0 ) . The results obtained by fitting EX were more consistent and physically reasonable. However, we recommend a cautious interpretation of the repulsion parameters listed in Table IV. The
6373
establishment of differing diameters for bonded hydrogen by theoretical methods lays groundwork for improving methods of quantitative derivation of empirical nonbonded potential parameters from experimental data. Acknowledgment. This research was supported by National Institutes of Health Research Grant No. GM37453. Registry No. Methane, 74-82-8; ammonia, 7664-41-7; water, 773218-5; methanol, 67-56-1.
Photochemistry of 3,4,9,1O-Perylenetetracarboxylic Dlanhydrlde Dyes. 3. Singlet and Triplet Excited-State Properties of the Bls(2,5-di-ferf -butylphenyl)imide Derivative' William E. Ford*+and Prashant V. Kamat Radiation Laboratory, University of Notre Dame, Notre Dame, Indiana 46556 (Received: February 27, 1987; In Final Form: July 10, 1987)
Fluorescence lifetime and triplet-state spectral, kinetic, and energetic properties are reported for the first time for a perylenebis(dicarboximide) dye, N,N'-bis(2,5-di-tert-butylphenyl)-3,4,9,lO-~~lenebis(dicarboximide) (DBPI). The fluorescence lifetime (3.8 i 0.2 ns), quantum yield (10.95), and singlet-state energy (54 f 1 kcal mol-') were determined for DBPI in a variety of organic solvents (24 "C). Triplet sensitization was used to characterize the triplet state of DBPI by laser flash photolysis and pulse radiolysis. The triplet-triplet absorption spectrum of DBPI had a maximum in the range 495-505 nm with an extinction coefficient of (6.3 i 0.7) X lo4 M-' cm-' , and the triplet-state energy and lifetime were 27.5 f 2 kcal mol-' and -100 ~ s respectively. , Direct photoexcitation produced the DBPI triplet with very low quantum yields (
