5170
J. Phys. Chem. 1987, 91, 5170-5177
*
Proton Polarizabiiity, Dipole Moment, and Proton Transitions of an AH***B A-***H+B Proton-Transfer Hydrogen Bond as a Function of an External Electrical Field: An ab Initio SCF Treatment Michael Eckert and Georg Zundel* Physikalisch-Chemisches Institut der Universitat Munchen, 0-8000 Miinchen, FRG (Received: January 5, 1987)
-~
An ab initio calculation of the energy surface E(BrH,BrN) and the respective dipole moment surfaces is performed for the hydrogen bond in the BrH-CH3H2N system. These surfaces were used to calculate the eigenvalues, eigenfunctions, dipole moments, proton polarizabilities, and the line spectra as functions of an electrical field in the direction of the hydrogen bond. The energy surface without field has two minima. They have the same energy if an electrical field of -0.6 X 10' V/cm is present at the H bond directed from Br to N. Under this condition the hydrogen bond shows a proton polarizability which is about 2 orders of magnitude larger than the polarizabilities due to distortion of electron systems and thus comparable to that of homoconjugated hydrogen bonds. Due to the double minimum potential many intense transitions occur which are strongly influenced by electrical fields with regard to their wavenumber and intensity. Thus, if one considers a distribution of electrical fields a continuous absorption occurs in the region 600-1600 cm-l. The sum of the intensities is, however, not largest if the two minima are energetically equalized but if the right one is deeper. Finally, on the basis of the obtained results it is discussed why double minimum energy surfaces occur with hydrogen bonds between much weaker acids and bases if these bonds are present in environments.
Introduction It was shown by analyticall as well as by SCF-MO-LCG02s3 calculations that the proton polarizability of the structurally symmetrical B+H-.B e B.-H+B bond within the H,O2+ group is about 2 orders of magnitude larger than the usual polarizabilities caused by the distortion of electron systems. The same is true with structurally symmetrical hydrogen-bonded systems4 These proton polarizabilities arise due to the motion of the proton within such hydrogen bonds. Here, the asymmetrical charge distribution within the hydrogen bond can easily be induced by an electrical field since the distance between the two lowest energy levels with symmetrical double minimum potentials is small, and the ground state has a symmetrical and the excited state an antisymmetrical wave function.' This symmetry behavior of the wave functions was already proved by Hund.j Thus, if by an electrical field the first excited state is admixed to the ground state, an asymmetrical charge distribution arises' and the shape of the proton potential becomes more or less asymmetricaL2 Such hydrogen bonds cause continua in the infrared spectra,6 whereas they are indicated by intense Rayleigh wings a t the excitation line when Raman spectra are taken from systems in which B'H-B B-H'B bonds are present.' The IR continua occur since due to their large proton polarizability such hydrogen bonds strongly interact with their environments.2.6 The following interactions are significant: (1) Such hydrogen bonds are strongly influenced by the local fields of their environments. ( 2 ) Dynamical interactions, for instance, the dipole4ipole coupling of the proton transitions of the hydrogen bond with other transitions of its environment. Herewith, for the continua observed in the case of solide-state systems9-" the ( I ) Weidemann, E. G., Zundel, G . Z. Naturforsch. A 1976, 25a, 627. (2) Janoschek, R.: Weidemann, E. G.; Pfeiffer, H.; Zundel, G. J . A m . Chem. Soc. 1972. 94. 2387. (3) Janoschek, R.; Weidemann, E. G.; Zundel, G. J . Chem. Soc.. Faraday Trans. 2, 1973, 69, 505. (4) Fritsch, J.; Zundel, G.; Hayd, A.; Maurer, M . Chem. Phys. Lett. 1984, 107. 65. (5)Hund. F. 2. Phys. 1927, 43, 805. (6) Zundel, G . In The Hydrogen Bond - Recent Deuelopments in Theory and Experiments; Schuster, P., Zundel, G., Sandorfy, C., Eds.; North Holland: Amsterdam, 1976: Vol. 11, Chapter 15, pp 683-766. (7) Danninger, W.; Zundel, G. J . Chem. Phys. 1981, 74, 2769. (8) Zundel, G.; Fritsch, J. J . Phys. Chem. 1984, 85, 6295. (9) Grech, E.; Malarski, Z.; Ilczyszyn, M.; Czupiuski, 0.;Sobczyk, L.; RoziBre. J.: Bonnet, B.: Potiers. J. J . Mol. Srruct. 1985, 128. 249. ~~
0022-3654/87/2091-5170S01.50/0
coupling between proton transitions is probably of particular significance (proton dispersion forces).I 2 , l 3 In the meantime with hundreds of systems containing interas well as intramolecular heteroconjugated, i.e., AH-B + A--.H+B bonds, IR continua have been observedi4-I6 (further references are given there). These types of hydrogen bonds are of decisive significance for the proton-transfer processes in electrochemical14 and biological s y s t e m ~ . ' ~ It was concluded from the IR continua observed with these hydrogen bonds that not only homoconjugated B+H.-B F= B-H'B bonds but also heteroconjugated AH-B A---H'B bonds show large proton polarizabilities if the proton potentials of these bonds are not too asymmetrical. However, no theoretical treatment of AH-B F= A--.H+B bonds was performed which justifies this conclusion. The proton-transfer equilibria in AH-B + A--H'B bonds are determined by the properties of A H and B, but to the same extent in solutions and solid state by the interaction of these hydrogen bonds with their environments. As discussed in ref 8 and 14, these equilibria are usually completely shifted to the left if they are studied without environment. This fact is shown by gas-phase I R and N M R measurement^'^-'^ as well as by S C F calculations.20-22 In ref 21 it is shown that in the HCI-NH, (IO) Videnova-Adrabinska, V.; Baran, J.; Ratajczak, H. Spectrochim. Acta, Part A 1986, 42A, 641. ( 1 I ) Videnova-Adrabinska, V.; Baran, J.; Ratajczak, H.; Orville-Thomas, W. J. Can. J . Chem., in press. (12) Weidemann, E. g.; Zundel, G. Z . Phys. 1967, 198, 283. (1 3) Weidemann, E. G. In The Hydrogen Bond - Receni Developments in Theory and Experiments; Schuster, P., Zundel, G., Sandorfy, C., Eds.: North Holland: Amsterdam, 1976; Vol. I, Chapter 5 , pp 245-293. (14) Zundel, G.; Fritsch, J. In Chemical Physics of Sobation; Dogonadze, R. R., Kglmln, E., Kornyshev, A. A,, Ulstrup, J., Eds.; Elsevier: Amsterdam, 1986; Vol. 11, Chapters 2 and 3, pp 21-117. (15) Zundel, G. In Methods in Enzymology; Packer, L., Ed.; Academic: New York, 1986; Vol. 126, pp 431-455. (16) Brzezinski, B.; Zundel, G. J . Phys. Chem. 1982, 86, 5133. (17) Kulbida, A. I.; Schreiber, V. M . J . Mol. Srruct. 1978, 47, 32. (18) Golubev, N. S.; Denisov, G . S. Chem. Phys. ( U S S R ) 1983, 5 , 563. (19) Denisov, G. S.; Golubev, N. S . J . Mol. Srrucr. 1981, 75, 31 1, (20) Schuster, P.; Wollschan, P.; Fortschanoff, K. In Molecular Biology, Biochemistry and Biophysics; Springer: Heidelberg, FRG, 1977; Val. 24, p 114. (21) Brzic, A.; Karpfen, A.; Lischka, H.; Schuster, P. Chem Phys. 1984, 89. 337. (22) Kollman, P. Presented at the 6th Workshop on Horizons in Hydrogen Bonding. Leuven. 1982.
0 1987 American Chemical Society
r--Tl:~ ---=
Proton Polarizability in the BrH-CH3H2N System
;1
The Journal of Physical Chemistry, Voi. 91, No. 20, 1987 5171
where x and y are Jakobi coordinates instead of the difference coordinates BrH and BrN of the atomic distances. This transformation is performed as described in ref 27. The dipole moment Y ( 3 ) W ,-.~-surfaces are represented analogously. '.. . By use of . these surfaces the. Schrodinger equation is solved for . Y(2) the coupled vibrational modes of the proton motion and hydroa l l m n a21mn gen-bond vibration as a function of an external electrical field F Y(1) L ~ ( i ) ~ ( 2 ) x(3j-Y in the hydrogen-bond direction using atomic units ....
7 ~
~
i
Figure 1. The energy and dipole moment areas are divided in rectangular integration areas ( k J ) . So it is possible to approximate the energy and dipole moment surfaces by bicubic spline functions.
HWX,Y) = . W x , y )
(2)
is given in our special case as
-system a single minimum energy suuface, E(HCI,CIN), is obtained, in which this minimum is at CI. Only with pairs of very strong acids as HBr and strong bases as methylamine these proton transfer equilibria are more or less shifted to the right if these hydrogen bonds, formed between acid and base, are considered without environment. The S C F calculations in ref 21 have shown that a double minimum energy surface is present in this system. In contrast to this fact in solutions hydrogen bonds between much less stronger acids and bases cause continua. From these continua it has been concluded that double minimum potentials are present in these hydrogen bonds with environment.1e16 The fact that in such hydrogen bonds really double minima occur which are caused by the environments will be discussed on the basis of our results. An exact a b initio treatment of hydrogen bonds with environment is impossible. Therefore, we studied the proton polarizability, the dipole moment caused by proton transfer, and the transition moments of the proton as a function of an external electrical field in the case of the BrH-.NH2CH3 Br--.H+NH2CH3 hydrogen bond. From S C F calculations of Karpfen et aLZ1it is known that a double minimum proton potential is present if these groups are considered without environment.
Method of Calculation Analogously to ref 2 the S C F a b initio calculations are per~ - ~the~ basis formed using a modified HONDO p r ~ g r a m ~with functions given in ref 21. These functions are products of polynomials and Gauss functions. The geometry of the hydrobromide-methylamine system is chosen in such way that Br, N , and C are in one plane.
The eigenfunctions \k were represented by basis functions 9] (4)
with the coefficients cj and n basis functions. To obtain the matrix representation we have to multiply by pi and integrate over the whole area
with the abbreviations
one obtains with formula 5
U
or in short
H*c= E S-c H ' h
By this procedure 250 points of the energy surface (Figure 2c) and the dipole moment surfaces are obtained. The dipole moment surface in the hydrogen-bond direction is shown in Figure 3a. The dipole moments perpendicular to the hydrogen bond (in the molecular plane) are smaller by a factor of about 3 X (Figure 3b) and can be neglected in the following calculations. Those perpendicular to the plane are zero. From the S C F points the energy and the dipole moment surfaces are represented analytically since afterwards the matrix elements of the treatments are calculated analytically. These representations are performed as follows: The energy surface cannot be represented by a single analytical expression; therefore, the surface is subdivided in rectangular sectors (see Figure 1). The cross sections of this representation are SCF points. A bicubic spline function was determined for each sector ( k , l ) of the raster. They are represented in the following way 3
E ~ / ( X > Y=) C
(9)
S = 1 in the case of orthonormal functions. In our case nonorthogonal wave functions must be used, however, to obtain larger flexibility, necessary with regard to the difficult field-dependent energy surfaces. The two-dimensional basis functions are built up by the product of two one-dimensional oscillator functions. P,(X>Y)= 9,,,,,)(X) 9y,JJ4 Ix(i) and ly(i) are degrees of the one-dimensional oscillator functions. These one-dimensional oscillator functions in x direction are given by 9,,(x) = H*~,((mx.%,)1'2(X- 4)) exP(-1/2mx%,(x - c x j ) 2 ) (10)
where h = 1; the indexj = Ix(i); Hao is the Hermite polynomial of b ( j ) degree; u',, is the shift of the one-dimensional oscillator function; and
3
Cak/mnXmYn m=On=O
(1)
(23) Pitzer, R. M. J . Chem. Phys. 1973, 58, 3111. (24) Hsu, H. L.; Davidson, E. R.; Pitzer, R. M. J . Chem. Phys. 1976, 65, 609. (25) Dupuis, M.; Rys, J.; King, H. F. J . Chem. Phys. 1976, 65, 111. (26) King, H. F.; Dupuis, M. J . Compur. Phys. 1976, 21, 144.
where
is mass of CH3NH2, and
w,,
is the width parameter
(27) Blochinzew, D. 1. Grundhgen der Quantenmechunik; Harri Deutsch: Frankfurt, 1963; pp 41 1 ff.
5172
The Journal of Physical Chemistry, Vol. 91, No. 20, 1987
of the one-dimensional basis functions. u,, and uy are adjusted by minimizing the energy eigenvalues using the k i t z variation principle. The parameters w and u allow to adjust the basis set to our problem most exactly. For the y direction analogous expressions are obtained. The Hamilton matrix elements HI, are determined as follows:
Eckert and Zundel TABLE I: Positiom BrH and BrN for the Five Enerev Surfaces"
distance, A
field F, V/cm -1.4 x 107
right
left right left right left right
0 1.0 x 107
1.4 x 107
__
BrN
enercrv E . cm-'
1.42 1.88 1.43 1.88 1.44 1.88 1.49 1.88
3.40 2.99 3.31 2.99 3.31 2.99 3.24 2.99
1790 2450 1490 1750 720 0 -100 -1750
1.88
2.99
-2450
BrH
left
-1.0 x 107
-
left
right
OParameter is the electrical field F. The position of the right minimum is field-independent.
With regard to the representation of the surface by squares, these matrix elements are calculated between the boundaries x ( k ) , x ( k + l ) , y ( l ) ,y ( l + 1) of rectangular integration areas (Figure 1). h, and I, are the numbers of the one-dimensional integration boundaries and (k,,, - l)(lma,- 1) the number of integration areas of the whole surface. The single parts of the Hamilton matrix elements are given by the following expressions: kinetic energy, x direction:
Hence, the energy surface is deformed by the electrical field. The mean dipole moments in hydrogen-bond direction as a function of the temperature T are obtained by
whereby wxmm
hmn
potential energy: (ixiylXmynliJy)kl= (ixlx"bx)~(iylu"li,)~ = Xr,kmK,/n
(14)
First, the one-dimensional matrix elements were calculated analytically. By use of these results the whole Hamiltonian matrix H is calculated according to formula 11, whereas the overlap matrix S is obtained with a similar procedure (see formula 7 ) . In this way the eigenvalue problem eq 9 was solved. It is assumed that the I R transitions are fast compared to the changes of the local electrical fields in the environment of the hydrogen bonds, an assumption which is justified by the fact that bands of the groups of both proton limiting structures AH-B A--.HfB are observed in the middle infrared Thus, the treatment of the problem using the time-independent Schrodinger equation is justified. Then, the interaction term in the Hamiltonian of the dipole of the hydrogen bond with the electrical field is given by AV(x,y) = -hx(X,Y)F, - Py(X,Y)F,
(15)
where the fix and fi, values are given by the dipole moment surfaces obtained from the S C F calculations. Here we obtain
In this equation the term containing the b, coefficients can be neglected since by
