12220
J. Phys. Chem. 1995,99, 12220- 12224
Theoretical Treatment of the Spectroscopical Data of a Strong Hydrogen Bond with a Broad Single-Minimum Proton Potential S. Geppert, A. Rabold, and G. Zundel* Physikalisch-Chemisches Institut der Universitat Miinchen, Theresienstrasse 41, 0-80333 Miinchen, Germany
M. Eckert Theoretische Chemie, Technische Universitat Miinchen, 0-85747 Garching, Germany Received: March 30, 1995; In Final Form: May 25, 1995@
On the basis of ab initio MP2 calculations with the 3-21G* basis set, we simulated the interaction of the hydrogen bond complex methanesulfonic acid-dimethyl sulfoxide base with electrical fields simulating the environment. We compared the theoretical results with experimental ones. The potential energy surface is a broad flat single minimum also if strong electrical fields are present. The OH and 00 modes are almost uncoupled. The lowest energy levels and thus the force constant of the hydrogen bond vibrations are almost independent of the external electrical field. We also performed vibrational analysis for 13 investigated strong largely symmetrical hydrogen bonds with the semiempirical MNDO/H procedure. The calculation of the hydrogen bond vibration shows for all complexes very good agreement with the experimental results. It is shown that the hydrogen bond vibration is very complicated and does not include the whole mass of the complex.
1. Introduction Methanesulfonic acid-oxygen base systems have been already studied experimentally with regard to the nature of the hydrogen bond in these complexes in the middle-infrared (MIR) region as a function of the ApKa (PKa of the protonated oxygen base minus PKa of the acid).' The result of these investigations was that single-minimum proton potentials are present in these hydrogen bonds. With the weak base, the minimum of the well is located at the acid and it is relatively narrow. With increasing basicity of the base, the well shifts in direction of the base and becomes very broad. The infrared continuum which is observed, indicates that the systems show now large proton p~larizability.*-~ If the base becomes still stronger, the well shifts to the base and becomes again narrow. Recently, a large number of systems were studied experimentally5 which include the systems of ref 1. The same results as in ref 1 were obtained in the MIR. Additionally the farinfrared region was studied, with the result that with the symmetrical systems the continuum extends toward smaller wavenumbers in the FIR region and the hydrogen bond vibration in this region becomes strongly broadened and may merge in the background. In this paper we try to calculate the spectroscopical data of such a strong heteroconjugated almost symmetrical hydrogen bond. Finally, we compare these calculated data with the experimental ones.
Figure 1. Geometry of the system studied in this paper.
moment surface (Figure 2). 350 points were calculated from 0.7 to 2.0 8, in the OH direction and from 2.0 to 5.6 8, in the 00 direction, respectively. The dipole moment is given by
p ( m = px("
+ py(r,R%, + pz(r,R).Zz
(1)
,uzcan be neglected. The dipole moment surfaces pX and ,uy are shown in Figure 2. The gradient of the dipole moment surface ,uxis about 4 times larger than the gradient of ,uy. >-e interaction_ with the solvent is simulated by the term p F , whereby F is the total electrical field strength of the solvent which interacts with the hydrogen bond. The OH and 00 coordinates were transformed into Jacobi coordinates,6in order to reduce the three-body system by one degree of freedom. This transformation is performed as described in ref 6. It is
2. Theoretical Procedure The complex methanesulfonic acid (MSA) with dimethyl sulfoxide (DMSO) is treated by a SCF ab initio calculation using the MP2/3-21G* basis set. The geometry of the complex is illustrated in Figure 1. We introduced a Cartesian coordinate system in which the O H 0 hydrogen bond lies in the x coordinate and the SOHOS plane lies in the x,y plane. In a next step we calculated the 2-dimensional potential surface (Figure 3) and the dipole @
Abstract published in Advance ACS Abstracts, July 15, 1995.
0022-3654/95/2099- 12220$09.00/0
and the reduced masses are
and X H are the coordinates of the acid, the base, and the proton, respectively (Figure 1). mA, mB, and mH are the masses of acid, base, and proton. We used the most frequent isotopic mass of each atom in order to calculate the reduced masses M, and MR. In a next step, the Schrodinger wave equation was
XA, XB,
0 1995 American Chemical Society
Spectroscopical Data of a Strong Hydrogen Bond
J. Phys. Chem., Vol. 99, No. 32, 1995 12221
4.0
2.0 Figure 3. Two-dimensional potential energy maps V(r,R) dependent on extemal fields F, (upper number) and Fy (lower number). Equipotential contours are separated by an energy difference of loo0 cm-I. The field strengths are given in lo7 Vlcm.
4.0
F y = 5.9 * 1 0 7 V / c m
Energy [7/cm/
i
5000
3000
I
moo -5
-2 5
0
F,
5
[IO? ~ / c m ]
Figure 4. Energy levels as a function of the extemal field F , with constant F) = 5 x 10' Vkm.
2.0
branch can be calculated precisely. A test has shown that 1000 experimental eigenfrequencies of microwave cavities agree with calculated eigenvalues for at least four digits.* The total dipole transition moment is given by
Figure 2. Dipole moment map for px (a) and p, (b)
solved for the coupled vibrational modes of the proton motion and hydrogen bond vibration.The Hamiltonian of the twodimensional system is W , R ) = T, + TR
2 5
+ VS&,R) + Vi"t@,R)
whereby
(5)
Tdesignates the kinetic energies, Vsc~(rJ2) is the calculated gasphase potential and finally Knt(r,R)represents the electrostatic interaction between complex and solvent, which is given by
We used a new computational method for solving partial differential equations, especially eigenvalue problem^:^ The Schrodinger equation is solved by a grid method with asymmetric sparse matrix. Also higher eigenstates in the Morse
pxmn
= f f ~ m ( r , R p p ~ ( r , R p ~ ~ ( rd, R r)
pymn
= f f ~m(r,Rppy(r,Rp~n(r,R) dr d~
Amn = IPmnI
2
(8)
(9)
The relative absorption intensities I,, as a function of temper-
Geppert et al.
12222 J. Phys. Chem., Vol. 99, No. 32, 1995
3.5
2.0
0.7
1.7
(A)
7Figure 5. Eigenfunctions of the potential for external field strengths of (a, top left) F, = F) = 0 V/cm and (b, top right) F, = F, = 5 x lo7 Vlcm for MR = 44. In (c, bottom) the eigenfunctions of F, = F) = 5 x lo7 V/cm are given for MR = 7. The difference between two energy equipotential contours amounts to 4000 cm-'.
ature are given by I,, = A,,exp(-E,/kQ/xexp(-E,/kT)
(10)
i
3. Results and Discussion
3.1. Methanesulfonic Acid-Dimethyl Sulfoxide Complex. Figure 3 shows the ab initio SCF calculated potential energy surface of the system methanesulfonic acid (MSA)-dimethyl sulfoxide (DMSO). Herewith the abscissa r is the OH distance and the ordinate R is almost the 00 distance. Equipotential contours are separated by an energy difference of 1000 cm-'. In a next step the Schrodinger equation for the two-dimensional system is solved whereby_t&e complex-solvent interaction is simulated by the term -pF. In all cases a single-minimum potential is observed, also if the electrical field strength amounts
to f 5 x lo7 Vlcm. The dissociation energy of the hydrogen bond without field amounts to 7000 cm-I. In the gas-phase potential the morse branch is directed parallel to the ordinate R. Figure 4 shows the energy levels for various electrical field strengths. The six lowest levels in Figure 4 are almost equidistant. Thus, the potential is almost harmonic for these lowest levels. Figure 5a shows the first 20 eigenfunctions without electrical field and Figure 5b shows the eigenfunctions for the electrical field strengths F, = 5 x lo7 Vlcm and F, = 5 x lo7 Vlcm. The fact that such a lot of wave functions of the hydrogen bond vibration exist is caused by the large reduced mass of the hydrogen bond vibration; therefore, the levels are deep in the potential well. The number of nodes of the 00 vibration is much larger than for the OH vibration (Figure 5). The nodes of the wave functions are either in the 00 or OH
Spectroscopical Data of a Strong Hydrogen Bond
J. Phys. Chem., Vol. 99, No. 32, 1995 12223
ram
Under these conditions the number of nodes of the 00 vibration wave functions, and thus the number of energy levels are strongly reduced (Figure 5c). However, the wavenumber of the OH vibration remains the same because the system is nearly uncoupled. Only the wavenumbers of the 00 vibrations increase. Thus, we get the interesting result that the reduced mass MR has nearly no influence on the wavenumbers of the OH vibration, and thus it has no influence on the theoretically calculated continua. The line spectra with mass MR = 44 are shown for different electrical field strengths in Figure 6. We took into consideration only the first 25 wave functions. Figure 6a shows the relative intensities as a function of the wavenumber for the temperature T = 276 K. The 0-1 proton transition always dominates. With increasing electrical field strength, this proton transition shifts toward smaller wavenumbers and the intensity increases. With increasing positive electrical field strength, the number of transitions with noticeable intensity increases, which are overtones of the OH vibration. The 0-1 transition of the 00 vibration has much less intensity than the OH vibration and is seen weakly at 240 cm-'. The ratio of the intensity between the 0-1 transition of the 00 vibration and the OH vibration is approximately 1:30. In Figure 6b, the relative intensities are shown in a logarithmic scale to demonstrate that a large number of transitions are also present in the region of 1000-300 cm-', which are overtones of the 00 vibration. The intensities and the number of transitions increase with increasing positive electrical field strength. The comparison with the experiments5shows that the largest intensity of the continua is found in the region 1500-800 cm-*. From the calculated line spectra the main absorbance is expected at higher wavenumbers. This may have the following reason: The 00 d_iis_tancecould be stronger increased by the environment than the ,wF term takes into account. The solvent weakens the hydrogen bond, and thus the OH modes shift toward smaller wavenumbers. The experimental results5 show that with the most symmetrical system, Le., the systems with the largest proton polarizabilities, the continua extend to the far-infrared region and the hydrogen bond vibration is extremely broadened. These effects cannot be explained by the calculated model. This model has, however, a lot of low-wavenumber transitions, with less intensitiy, this intensity may be increased if the interaction with the thermal bath would be taken into account.1° 3.2. Nature and Assignement of the Hydrogen Bond Vibration of MSA-Oxide Complexes. The nature of the hydrogen bond vibration was investigated with the semiempirical MNDO/H method for various methanesulfonic acid (MSA)oxygen base complexes (Figure 7). We obtained the important result that with these systems the hydrogen bond vibration is no pure hydrogen bond stretching vibration but has a very complex nature. Therefore, the reduced mass of this vibration is much smaller than expected with regard to the mass of the atoms, as already mentioned. Furthermore, we performed a vibrational analysis for a great number of different complexes and compared the calculated wavenumbers of the FIR hydrogen bond vibrations with the experimental values of ref 5 (Table 1). We see that with these systems the MNDO/H procedure gives astonishingly good results for the hydrogen bond vibration. This good agreement of experimental and calculated data becomes understandable as follows: The MNDO/H vibrational analysis procedure neglects anharmonic forces and calculates the hydrogen bond vibration without influence of the solvent. Since the 00
r-''[ll--rJ
00
0.135
0
4000
2000
c
00
111
I
2000
4000
V
(Ucm)
Figure 6. Relative absorption intensities (a, top) and the logarithms of the relative absorption intensities (b, bottom) as a function of the external electrical fields. The wavenumbers (abscissa) reach from 0 to 4000 cm-'. Only the first 25 transitions were taken into consideration.
Figure 7. Hydrogen bond vibration.
direction. Therefore, in zero approximation one can separate both motions, i.e.
We get the important result that the modes of the complex are almost uncoupled. Therefore, we introduce two quantum numbers which describe the OH and the 00 mode. If the nodes are directed parallel to the abscissa, the wave functions describe an 00 mode. If the nodes are directed perpendicular to the abscissa, we have an OH mode. The reduced mass of the hydrogen bond vibration MR amounts to 44 proton masses. But, not the whole mass takes part in the ~ibration.~ Therefore, we performed the calculations also with a reduced mass MR = 7. The result is shown in Figure 5c.
Geppert et al.
12224 J. Phys. Chem., Vol. 99,No. 32, I995 TABLE 1: Comparison of Wavenumbers of the Experimental Data and the Calculated (MNDO/H) Ones of Methanesulfonic Acid-Oxide Base Systems base
exueriment
calculation
diphenyl sulfoxide di-4-tolyl sulfoxide methylphenyl sulfoxide dimethyl sulfoxide triphenylphosphine oxide tributylphosphine oxide 2,6-chloropyridine N-oxide 3-bromopyridine N-oxide 3-chloropyridine N-oxide 4-chloropyridine N-oxide 4-phenylpyridine N-oxide 4-methylpyridine N-oxide 4-methoxypyridine N-oxide
245 228 255 220 215 299 238 282 290 238 252 243 251
247 220 212 210 21 1 298 231 216 211 238 253 246 255
potential is almost harmonic with all complexes and is almost independent of the electrical solvent field strength, the MNDO/H procedure is a good approximation.
4. Conclusions To understand the MIR continua of strong heteroconjugated hydrogen bonds with broad flat largely symmetrical singleminimum potential we studied the MSA-DMSO system. We calculated the energy surface for different electrical fields simulating the environment. In a next step we solved the twodimensional Schrodinger wave equation. The first important result is that the OH and 00 modes are always only very weakly coupled. A second result is that the lowest energy levels (00 vibrations) are almost equidistant and independent from the extemal field. Hence, the hydrogen bond vibration is nearly harmonic. In the line spectra the proton transition shifts with increasing positive field strength toward smaller wavenumbers. The main absorbance is in the region 2500-1500 cm-'. The experimental main continuous absorbance is at lower wavenumbers. This discrepancy may have two reasons: (1) The 00 distance-increased more strongly by the solvent than what the term p F takes into account. The solvent weakens the hydrogen bond and thus the OH modes shift toward smaller wavenumbers. (2) The hydrogen bond vibration is different
from a stretching vibration and the coordinate R is only an approximation. The strong broadening of the hydrogen bond vibration, which was found in the experiment for the almost symmetrical hydrogen bonds, could not be explained by this model. Therefore, only a model which takes the thermal bath of the environment into account can explain this effect. Due to the pK, dependence of this broadening effect5 this interaction may also depend on the proton potential, Le., on the proton polarizability of the hydrogen bond. To get a better impression of the nature of the hydrogen bond vibration modes found in the experiment, we performed a normal-coordinate analysis of 13 MSA-oxide systems with the semiempirical MNDO/H method. This method gives good results for strong hydrogen-bonded complexes. The agreement of the semiempirically calculated wavenumber values of the hydrogen bond vibration with the experimental ones is excellent. The hydrogen bond vibration is, however, very complicated and does not include the whole mass of the complex.
Acknowledgment. Our thanks are due to the Deutsche Forschungsgemeinschaft and the Fonds der Chemischen Ind. for their support of this work. We express our sincerest thanks to the Leibniz Rechenzentmm for providing the necessary computational time. References and Notes (1) Bohner, U.: Zundel, G. J. Phys. Chem. 1985, 89, 1408. (2) Zundel, G.The Hydrogen Bond-Recent Developments in Theory and Experiments; Schuster, P., Zundel, G.,Sandorfy, C., Eds.; North Holland: Amsterdam, 1976; Vol. 11, Chapter 15, pp 683-766. (3) Eckert, M.; Zundel, G . J. Phys. Chem. 1987, 91, 5170. (4) Borgis, D.; Tarjus, G.; Azzouz, H. J. Chem. Phys. 1992, 97, 1390. (5) Langner R.; Zundel, G. J. Phys. Chem. 1995, 99, 12214. (6) Blochinzew, D. I. Grundlagen der Quantenmechanik, Hani Deutsch: Frankfurt, 1963; pp 411ff. (7) Eckert, M. J. Comput. Phys., in press. ( 8 ) Graf, H. D.; et al. Phys. Rev. Lett. 1992, 69, 1296. (9) Rabold, A.; Zundel, G. J. Phys. Chem., in press. (10) Bala, P.; Leysing, B.; McCammon J. A. Chem. Phys. 1994, 180, 27 1 Jp950923U
