J . Phys. Chem. 1985,89, 4695-4700 This distribution is neither Poisson nor Wigner, but one that is characteristic of harmonic uncoupled systems,26 and is thus strongly indicative of nonstochastic behavior.
5. Comparison of the Dynamic Behavior of the Accurate and SCF Solutions From the fact that both the energy levels and the nearestneighbor-spacing histograms obtained by the S C F approach are very close to the exact energy spectra and histograms, respectively, it does not follow that the time-dependent properties derived from the S C F Hamiltonian will approximate the true properties. As noted in section 1, mode specificity means a nonstatistical distribution of vibrational energy among the molecular modes, but does not exclude oscillatory energy exchange among them; it has been shown that regular, Le., nonstochastic systems, can exhibit rapid oscillatory energy exchange between two mode^.^',^',^* However, in such cases the average energy of the individual modes, taken over a sufficiently long period of time, remains constant and does not decay rapidly to its equilibrium value, as it would if the system were irregular (stochastics). In Figures 6 arid 7 we present the results obtained by carrying out accurate variational calculations (not SCF) with 200 basis functions. These results show very rapid oscillations (-0.1 ps-’) between the two modes. Figure 6 depicts the time dependence of the energy in mode s: (4) where d(t)
=
C(nQ= 20, n, =
exp(-iEjt/h)+j
(5)
J
(26) M. V. Berry and M. Tabor, Proc. R. Sot. London, Ser. A, 356,388 (1977). (27) J. S.Hutchinson, E. L. Sibert, and J. T. Hynes, J . Chem. Phys., 81, 1314 (1984). (28) N. Moiseyev and P. R Certain, J . Phys. Chem., 86, 1149 (1982).
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in which ICJ and EJ respectively are the variational wave functions and eigenvalues. Figure 7 shows the fluctuations in the amplitudes of s expressed as the square root of the expectation value of s2: (s2)1/2 = [(+(t)ls21+(t))l”2
(6)
It can be seen from Figure 7b that these rapid oscillations are remarkably persistent. Even though the s and Q modes are very weakly coupled, fast oscillations might arise out of resonance between them, as suggested by Davis and Heller,29because their harmonic frequencies are nearly the same. However, there is no vibrational relaxation, the energy distribution between the two modes remaining nonstatistical after hundreds of oscillations. When the variational solutions QJand EJ are replaced by the corresponding orthogonalized S C F wave functions and the corresponding energy levels, both E, and ( s2) 1/2 turn out to be almost “constants of the motion” and do not show the correct dynamical behavior on a time scale of a few vibrational periods (0.1 ps). However, the S C F values of E, and (s2)lI2are almost time invariant and agree quite well with the time averages of the corresponding accurate values. These results show very clearly that the short-term time-dependent properties of the system cannot be quantitively studied within the framework of the static S C F approximation, though the dynamic S C F approximation” may hold promise. However, they also suggest that the strong short-period oscillations observed in the accurate calculations are not relevant to mode specificity.
Acknowledgment. This work was supported by the United States-Israel Binational Science Foundation, by the Technion V.P.R. Fund, and by the Lawrence Deutsch Research Fund. We thank Dr. E. Haller and Professor L. S. Cederbaum for allowing us to use their program for calculating the nearest-neighborspacing histograms. Registry No. HC02H, 50-00-0. (29) M. J. Davis and E. J. Heller, J . Chem. Phys., 75, 246 (1981).
Protonation of Nitrogen-Containing Bases in Solution: Continuum vs. Discrete-Continuum Models for Aqueous Solutions Enrique Sanchez Marcos,+ Bernard Terryn, and Jean-Louis Rivail* Unite de Recherche associee au CNRS No. 510, Laboratoire de Chimie theorique, UniuersitC de Nancy I , 54506 Vandoeuvre Les Nancy Cedex. France (Received: April 3, 1985)
The difference in the free energies of transfer from the gas phase to an aqueous solution of a nitrogen-containing base and its protonated conjugated acid is evaluated for a series of six systems (trimethylamine, triethylamine, N-methylpyrrolidine, pyridine, 2-fluoropyridine, and acetonitrile) with the help of a continuum cavity model described earlier. The cavity has a shape fitting the shape of the molecule, and the continuum has the static dielectric properties of liquid water. Full quantum chemical computations are performed on the isolated species and on the species surrounded by the solvent. The model reproduces pretty well the variations of free energy of the protonation reaction between the gas phase and the liquid phase. The induction effects on the solutes are not negligible, but their magnitude is the same for the base and for the corresponding protonated molecule so that these effects can be ignored in the evaluation of the solvent effect on the reaction. Finally, a study on the same series of solutes in which the directly hydrogen-bonded water molecule is explicitly taken into account (discrete-continuum model) does not yield a better description from the thermodynamic point of view. This surprising result which is probably tied in with the feeble accuracy of the semiempirical quantum chemical calculation (MNDO/H) emphasizes the major role of electrostatic interactions in aqueous solutions.
Quantum chemical techniques are accurate enough nowadays to be applied to the study of a large variety of molecular properties provided that the molecule under study can be considered as ‘On leave from the Departamento de Quimica-Fisica, Universidad de Sevilla, Tramontana s/u Sevilla 41012, Spain.
0022-3654/85/2089-4695$01.50/0
perfectly isolated. This limitation may be found very severe if one thinks that most current chemical experiments are performed in a condensed phase, very often the liquid state. Then the molecule cannot be considered as independent of its environment which may modify to a greater or lesser extent the molecular properties, as shown by the comparison of some measurements 0 1985 American Chemical Society
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The Journal of Physical Chemistry, Vol. 89, No. 22, 1985
performed both in the gas phase and in solution on the same molecular system. Chemical thermodynamics is in principle the only way to study a macroscopic system; for instance, a solution and thermodynamic functions can be computed with the help of statistical mechanics. Modern computers allow such theoretical calculations by means of simulation techniques such as Monte Carlo computations.] Nevertheless, in a sample which consists of 100 or more molecules, it becomes necessary to simplify the description of each individual molecule via an appropriate modelization. This procedure is not always compatible with the necessity of studying in great detail the electronic structure of one molecule submitted to the influence of an averaged environment, for the calculation of molecular properties such as spectroscopic constants for instance. This is the reason why some models which allow a quantum chemical computation on a system which consists of a single molecule perturbed by modelized surroundings have been d e ~ e l o p e d . ~ - ~ The whole solvation process consists of creating the cavity and then placing the solute within it. The first step implies work being done, which may be considered as mainly dependent on the size and to a lesser extent on the shape of the cavity. The energetics of the second step can be related to the usual terms of intermolecular interactions: electrostatic, polarization, and dispersion energies.6 The latter energy, which is by no means negligible, can be related to the electronic polarizabilities of the interacting molecules6 and evaluated if necessary.' It can be shown that dispersion interactions do not play a major role in SCF calculations,8 contrarily to the first two terms which are strongly dependent on the electronic structure of the solute. The protonation of molecules is a very simple but very important process, since it may characterize the basicity or the nucleophilicity of the molecule. These processes have been extensively studied in various solvents and in the gas phase as ~ e l l . ~ 9In ' ~addition, since the number of electrons does not vary during the process, it may be considered that the molecular volume remains almost constant and therefore that there is no need to vary the volume of the cavity in a continuum model. These reactions appear then as good candidates to test our model as a tool for predicting the magnitude of the solvent effect on a reaction. In order to test this model in the less favorable case, we shall apply it to aqueous solutions in which hydrogen bonds are often invoked to account for solvent effects." This is the reason why we shall represent the environment of the solute by two different models: the pure continuum one, in which water is only described by its static dielectric permittivity at room temperature, and the mixed discrete-continuum model12in which the hydrogen-bonded water molecules of the first solvation shell are explicitly considered and calculations are performed on a polymolecular supersystem. In the evaluation of the solvation energies, the latter model (1) W. L. Jorgensen, J . Phys. Chem., 87, 5304 (1983). (2) J. L. Rivail and D. Rinaldi, Chem. Phys., 18, 233 (1976). (3) (a) S. Miertus, E. Scrocco, and J. Tomasi, Chem. Phys., 55, 117 (1981); (b) S.Miertus and J. Tomasi, Chem. Phys., 65, 239 (1982); (c) R. Bonaccorsi, R. Cimiraglia, and J. Tomasi, J . Comput. Chem., 4, 567 (1983);
(d) R. Bonaccorsi, R. Cimiraglia, and J. Tomasi, J . Mol. Struct.: THEOCHEM, 107, 197 (1984). (4) D. Rinaldi, M. F. Ruiz-Lopez, and J. L. Rivail, J . Chem. Phys., 78,
Marcos et al. implies the calculation of the difference between two S C F energies: that of the solvated supermolecule and that of the isolated molecules composing the supersystem. The solvation energies calculated in this way are always afflicted by nonnegligible uncertainties which increase with the number of water molecules taken into account. In addition, one should pay attention to the entropy of such a cluster and the evaluation of this term would be a tedious task, especially when the cluster is more complex. This is the reason why we selected a series of nitrogen-containing bases which are able to be hydrogen bonded directly with only one water molecule (tertiary amines, pyridines, and acetonitrile).
Methodology The continuum model used in this study has been described in great detail in a previous paper.s The solvent is portrayed by its static dielectric permittivity
