Interactions of easily polarizable hydrogen bonds with polar solvents

the well with the larger lingering time only. The asym- metry of the potential energy curve in the proton ground state increases due to the polar envi...
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The Journal of Physical Chemistry, Voi. 83,

No. 19, 1979

described in terms of hydrogen atoms. Oxygens terminated with H atoms are not considered as models of the terminal hydroxyl groups of the zeolite but as skeletal oxygens which are described less adequately by the model. It is further assumed that the charges on the skeletal oxygens are correct then the error on oxygens terminated by hydrogen atoms is about O.leo, Le., about 20%. The error in the charges and further calculated characteristics of the skeletal oxygens and of the Na, Si, and A1 atoms, resulting from the imperfection of the zeolite model used, is probably much lower.

References and Notes (1) G. V. Gibbs, E. P. Meacher, J. V. Smith, and J. Pluth, ACS Symp. Ser., No. 40, 19 (1977). (2) V. I. Lygin and V. A. Seregina, Vest. I'&SCOW Vniv. Khim., 515 (1976). (3) V. A. Seregina, V. I. Lygin, and Z.V. Grazonova, Doki. Acad. Nauk, SSSR, 226, 604 (1976). (4) V. I. Lvoin and V. V. Smolikov. Zh. Fiz. Khim.. 49. 1526 (19751. (5) I.D. Mitheikin, I. A. Abronin, G.'M. Zhidomirov, and B. Kaianski, Kinet. Katal., 16, 1580 (1977). (6) I. D. Mikheikin, I. A. Abronin, G. M. Zhidomirov, and V. B. Kazanskii, J . Mol. Catal., 3, 445 (1978). (7) N. D. Chuvylkin and G. M. Zhidomirov, Kinet, Katal., 18, 903 (1977).

c.

H. Pfeiffer, G. Zundel,

and E. G. Weidemann

(8) N. D.Chuvylkin and V . B. Kazanskii, Kinet. Katal., 19, 99 (1978). (9) R . M. Barrer and R. M. Gibbons, Trans. Faraday Sac., 59, 2569 (1963);81, 948 (1965). (10) A. G.Berns, E. S. Dobrova, V. M. Katz, A. V. Kiselev, A. A. Lopatkin, and P. Q. Du, Kolloid. Zh., 37, 1045 (1975). (11) V. BosbEek and J. Dubskg, Collect. Czech. Chem. Commun., 40, 3281 (1975). (12) E. Dempsey, "Proceeding Molecular Sieves Conference", Society of Chemical Industry, London, 1968,p 293. (13) S. Beran, J. Dubskg, and 2. Slanina, Surface Sci., 79, 39 (1979). (14) S.Beran and R. Zahraddk, Kinet. Katal., 18, 359 (1977). (15) G. R. Eulenberger, D. P. Schoemaker, and J. G. Keil, J. fhys. Chem.,

71, 1812 (1967). (16)J. A. Pople and D. L. Beveridge, "Approximate Molecular Orbital Theory", McGraw-Hill, New York, 1970. (17) K. E. Wiberg, J. Am. Chem. Soc., BO, 59 (1968). (18) J. Dubskg, S.Beran, and V. BosbEek, J . Mol. Catal., submited for publication.

(19) eo is the elemental charge: e, = -1.60210 X lo-'' C. (20) For comparison, the Wiberg bond order for the Si-0 bond has a value of 0.75and for H-0, over 0.9. (21) A. Gupta and C. N. R, Rao, J . Phys. Chem., 77, 2888 (1973). (22) W. F. Cooper, F. K. Larsen, and P. Coppens, Am. Mineral., 56,21 (1973). (23) J. P. Suchet, "Chemical Physics of Semiconductors", van Nostrand, London, 1965. (24) R . Brill, H. G. Grimm, C. Hermann, and C. Peters, Ann. Phys., 34, 419 (1955).

Interactions of Easily Polarizable Hydrogen Bonds with Polar Solvents and Anions. Theoretical Considerations. 1. Interactions of a Single Bond Herbert Pfeiffer, Georg Zundel," Institut fur Physikalische Chemie, Universitat Munchen, 8000 Miinchen 2, West Germany

and Erich 6. Weidemann Sektion Physik, Universitxt Munchen, 8000 Munchen 2, West Germany (Received November 2, 1978; Revised Manuscript Received April 9, 1979)

We treat here the interaction of easily polarizable hydrogen bonds with polar solvents and anions. The degree of asymmetry of the double minimum potential is changed by the reaction potential caused by the polar environment and by the anion field. A self-consistent treatment is necessary since the reaction potential depends on the charge distribution in the hydrogen bond which, on the other hand, depends on the proton state. There exist at most four equilibrium states between the charge distribution in the polarizable hydrogen bond and the poiar environment if the proton is present in the two lowest states. One equilibrium state is usually unstable. In two of the equilibrium states corresponding to the proton ground state, the proton is largely localized in one of the potential wells. The third stable state corresponds to the first excited state of the proton. Due to the negative polarizability of this state the energy curve maintains its symmetry because of the interaction with the polar medium. To each equilibrium belongs a potential energy curve and a corresponding energy level scheme contributing to the continuous absorption observed in IR spectra of easily polarizable hydrogen bonds in solutions.

Introduction Polar media interact strongly with easily polarizable hydrogen b ~ n d s , l -i.e., ~ with hydrogen bonds with a It was shown by infrared fluctuating proton. spectroscop~-'as well as by dielectric methodsgl0 that the symmetry of the energy surfaces in such hydrogen bonds with a double minimum potential well is strongly changed due to interactions with the environment. The influence of polar media with regard to proton transfer processes in hydrogen bonds was already treated theoretically by Kurz and Kurzll and by Vorotyntsev et The proton transfer processes in the charge relay system in chymotrypsin,13a 0022-3654/79/2083-2544$0 1 .OO/O

system formed by two easily polarizable hydrogen bonds, are controlled by changes of the polarity of the environment.14 Easily polarizable hydrogen bonds cause continuous absorptions in the infrared spectra. These continua can be calculated,15 starting from the vibrational transitions obtained by SCF-MO calculations of the electronic structure of the hydrogen bonds. Until now the interaction of the hydrogen bonds with the environment was described by a statistical distribution of bond lengths and by a distribution of the strength of the local electrical fields which affect strongly the proton transitions due to the large

0 1979 American

Chemical Society

Interactions of Hydrogen Bonds with Polar Solvents

polarizability.15J6 In this paper a more realistic model is treated in which the interdependence of the proton transitions and the polarization of the environment are considered.

2. The Model We treat a structurally symmetrical hydrogen bond BH+-B F= B-.H+B with a double minimum potential well for the proton motion. This bond is embedded in a polar solvent. Furthermore, the corresponding anion is present, The environment is described macroscopically by dielectric constants. Depending on the relative size of the proton fluctuation frequency up and the reorientation frequency of the molecules in the environment of the hydrogen bond uR,we have to distinguish two cases: (1) up < uR; ( 2 ) u > UR. (1)In the first case the polar environment can folpow the proton motion in the bond. Thus, the dipoles of the medium orientate themselves with respect to the well in which the proton is present lowering the potential energy at this point. The wells of the double minimum become deeper compared with the case of the isolated hydrogen bond. The symmetry of the potential energy curve is not disturbed by the polar solvent. (2) In the second case the polarization of the medium is divided into two parts: one part is assumed to follow the proton motion adiabatically whereas the other part cannot follow (inertial part). In all cases the first part contains the electron polarization. The second part is the orientation polarization of the polar medium. The atomic polarization may contribute to both parts. The second part of the polarization is not unordered but orientated with respect to the mean charge distribution in the hydrogen bond. The barrier increases due to the polar environment. A similar partition of the polarization was considered by Marcus1&in his work on electron transfer reactions, a work which has very much in common with the ideas of the present paper. It is also used in treatments on the electron-photon interaction in polar solids, Le., in polaron theories.16b When an anion is present in the neighborhood of the hydrogen bond the potential energy curve is, without polar medium, more or less asymmetrical. Then the proton has two different lingering times in the two wells: in the ground state the lingering time in the upper well is smaller, and in the lower well greater, than the lingering time in the symmetrical bond."-19 Two frequencies correspond to these lingering times, one for the transition from the upper to the lower well (upl) and one for the reverse transition (vp2). We have now to consider three cases: (1) v p l , up2 < VR; (2) v v 2 > VR; (3) up1 > VR > up2. (1)In this case d&ee of asymmetry is not influenced by the polar environment since the polar environment orientates with respect to the momentary position of the proton in each well, hence lowering both potential wells to the same degree. ( 2 ) Contrary to case 1,in case 2 the degree of asymmetry is changed by the polar environment. The inertial part of the polarization is orientated on the mean charge distribution in the hydrogen bond. This charge distribution is not symmetrical since the residence time is different in the two wells. Thus, also the contribution of the polar environment to the potential energy is different for both wells, changing the degree of asymmetry. (3) If vpl. > U R > vp2 the polar medium is orientated on the well with the larger lingering time only. The asymmetry of the potential energy curve in the proton ground state increases due to the polar environment, whereas, in the first excited proton state the degree of asymmetry

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The Journal of Physical Chemjstty, Vol. 83, No. 79, 1979 2545

decreases due to the polar environment, as discussed in the following. 2.1. The Hamiltonian. In order to separate the contributions of the inertial and the adiabatically following parts of the polarization, the potential of a proton at point r' in a medium with dielectric constant t may be written in the following form:

The first term is the potential of the proton screened by the adiabatically following polarization characterized by the dielectric constant cu. Hence the second term is the potential of the electrical field caused by the inertial polarization. If the proton is fluctuating the mean charge distribution pp of the proton along the hydrogen bond instead of the instantaneous position of the proton determines the inertial polarization. Thus, e must be substituted by pp in the second term of eq 1.

When an anion is present, the potential of the electrical field caused by the inertial polarization is no longer determined by pp only, but also by the anion which is assumed to have unit charge. Separating the potential of the anion charge (-e) at the point rAanalogously to eq 1, one obtains

Hence the potential a0 of the inertial polarization orientated with respect to the mean charge distribution of the proton and with respect to the anion is given by

The potential of the field of the two charges screened by the adiabatically following part of the polarization is given by

In order to obtain the Hamiltonian for the motion of the proton in the hydrogen bond in the field of the anion and of the polarization of the medium, we calculate first the energy necessary to produce the inertial polarization, induced by proton and anion. We do this in two steps: In the first step, we build up the charge distribution in the dielectric medium; in the second step we remove the charge distribution, but with the inertial polarization considered fixed. The energy to form a charge density p(r) out of infinitesimal charge elements, from the infinite, is knownz0as follows: WI = l/zJp(r) @(r)d3r

(6)

Here, @(r)is the potential at point r produced by the charge density itself. @(r)consists of the two parts @o(r) and @"(r), eq 4 and 5. Hence

WI = l/zJp(r) @dr)d3r + X J p ( r )

@ A I )d3r

(7)

In order to remove the charge distribution with the inertial

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The Journal of Physical Chemistty, Vol. 83,

No. 79, 1979

polarization fixed, Le., with constant a0(r),the energy

W Z= -‘/zlp(r) @,(r)d3r - l d r ) M r ) d3r (8) must be supplied. The energy of the polarization field is thus given by

Wo = W1 + W2 = - ‘ / z l p ( r )ao(r)d3r

H. Pfeiffer, G. Zundel, and E. G. Weidemann

which is stored in the inertial polarization field, oriented with respect to the mean charge distribution of the proton. Thus, the Hamiltonian of the proton anion system, influenced by a polar solvent, becomes h2 7f = --A2M

+ W(rp)

(9)

In the case of one proton and one anion, p(r) consists of the charge distribution pp(r)and of the anion charge (-e) a t the point rA,hence

Substituting the potential of the inertial polarization, eq 4, we obtain for the energy of the polarization field

In the above equations we have always written the factor (1/4 - ( l / t , ) inside the integral since, for small distances, the saturation of the polarization must be taken into account. This can be done by using not a constant but distant-dependent e. Experimental results of Landis and Schwarzenbach21 were used to describe the functional dependence of t on the distance r. The experimental values can be fitted by the following function t(r) = 0.00008r4 + 0.0074r3 - 0.706r2

+ 16.8r - 45.6 (164

J (f

for 4 A -

Ir

- rAl

d3r (11)

We have omitted an infinite term which occurs because of the idealization of representing the anion charge by a point charge. This term arises because of the well-known infinite self-energy necessary to build up a point charge from infinitesimal charge elements. The energy of the polarization field Wois a part of the Hamiltonian for the motion of the proton in the hydrogen bond. To obtain the total Hamiltonian we have to move the proton and the anion (screened by the adiabatically following part of the polarization) to their final positions rpand rA,respectively. The total potential energy is thus given by

e2

W = Wo+ V(rp)+ eaO(rp)- eG0(rA)%lrp

- rAl

(12)

Here V(rp) is the double minimum in which the proton would move without the polar environment, eao(rp)is the modification of the potential energy curve due to the inertial part of the polarization, and -eG0(rA)is the potential energy of the anion in the field of the inertial polarization. The last term is the attractive force between proton and anion screened by the adiabatically following part of the polarization. Substituting in eq 1 2 Woand a0from eq 10 and 4 and combining terms of the same kind, we obtain the final expression for the total potential energy:

Here, we have used the abbreviation

Comparing this expression with eq 4 shows that chR(rp) is the potential of the inertial polarization field, oriented with respect to the charge distribution pp, a t the position rpof the proton producing the polarization. This potential is usually called the reaction potential. By means of compensation of several terms, the full static dielectric constant shows up in the third term of eq 13 instead of E, as in eq 12. The last term in eq 13 is, according to eq 11, the energy

< r < 20 A, t(r) = 80

(16b)

for r > 20 A, 2.2. Estimation of the Reaction Potential. The reaction potential is an important term in eq 13 since it describes the modification of the potential energy curve caused by the polarization of the environment. It was defined by eq 14. This equation can, however, not be used to calculate this potential because there is not even approximately a homogeneous isotropic medium in the immediate neighborhood of the proton, as assumed in the macroscopic description of the dielectric properties. The molecular structure of the environment and the saturation of the polarization has t o be taken into account. To estimate the reaction potential we apply the cavity method.22 We have to describe the medium outside the cavity by a dielectric constant which we denote by 7. If the cavity contains a charge distribution p(r) the potential of the polarization of the medium outside the cavity is given by

This is the difference between the potential of the charge screened by the medium with dielectric constant Z, minus the potential of the unscreened charge. Comparing this with the reaction potential eq 14, we obtain for 7

Taking water for the polar solvent, we have to use c = 80. For E,, we choose 3.5, a mean value between n2 = 1.78 and the dielectric constant in the microwave region t, = 5.25. Hence a part of the atomic polarization is included in the E , value used. With these values we obtain 7 = 1.38. This value becomes so small since, according to eq 14 and 17, the medium with the dielectric constant 7 has to be polarized by the unscreened charge distribution to the same degree as the real medium by the screened charge distribution. The reaction of the polar medium is quenched, because the fluctuating proton is always screened by t u . The determination of the reaction potential is known for arbitrary distributions inside the cavity.22 For simplicity we replace the continuous charge distribution p ( x ) along the hydrogen bond by two point charges e‘ anc! e’’ localized in the potential wells at x = - d / 2 and x = +d/2, respectively. The two point charges result from the charge

The Journal of Physical Chemistry, Vol. 83, No. 79, 1979 2547

Interactions of Hydrogen Bonds with Polar Solvents

‘I

X

i

2

Flgure 1. Maximum energy difference of the two wells of the potential curve as function of the cavity radius a for various values of the dielectric constant 7 defined in eq 18.

density p ( x ) = el$(x)12 by integrating over the individual wells

e’+ e “ = e Specializing the general formula, given in ref 22, to the case of the two point charges, we obtain the following formula in polar coordinates

(20) whereby a is the cavity radius and PLare Legendre polynomials of order l. The polar axis is chosen to coincide with the hydrogen bond axis, i.e., with the line connecting the two charges eland e ” a t r = d / 2 and 29 = or 29 = 0, respectively. The energy difference AV of the two wells produced by the reaction potential

follows from eq 20 as

The potential remains symmetric if e ’ = e”or 7 = 1. The maximum asymmetry AV, is obtained if a proton is localized completely in one well (e’ or e” = e). The occupied well is in fact lowered as shown by eq 19, 21, and 22. This is the case since the dipoles are orientated with respect to the well in which the proton is mainly present. In Figure 1 AV, is plotted as function of cavity radius a for various 7 and a distance between the wells of d = 1 A. With 7 = 1.38 and a = 3 A, a maximum asymmetry of AV,.= 400 cm-l is obtained. This value depends on the special system and the effects discussed in the following become more or less pronounced, dependent on the above parameters. We use in the following calculation as maximal asymmetry caused by the reaction potential AV, = 400 cm-’.. 2.3. Treatment of the Model. We consider the proton motion along the hydrogen bond axis (2) and neglect all

Figure 2. Schematic representation of the potential energy curve, eigenvalues and eigenfunctions.

coupling effects with other vibrations, especially also the coupling with the bond stretching vibration which we considered in ref 23, since we are here only interested in the influence of the polar solvent. We divide the Hamiltonian, eq 15, into two parts replacing the coordinate rp by x and Jrp- r A l by rpA. 7f = H , H1 (23) whereby

+

H, =

ti’ d2 2Mdx2

+ V(x)

C is the last term in eq 13 which does not depend on the proton coordinate. H o is the Hamiltonian of the hydrogen bond without polar environment and H1describes the influence of polar environment and anion. The potential energy curve and the four lowest energy levels of Ho are shown in Figure 2 in a schematic representation. To determine the eigenstates $ of H we expand these states with respect to the eigenstates of Ho. The approximation is that we restrict the expansion to the four lowest states $= + c2$C- + c3$1+ “b c4$l-(26) In this basis set the Hamiltonian is represented by a four by four matrix. The wave functions in a double minimum with high barrier are essentially concentrated in the two wells; the continuous charge densities can therefore be approximated by point charges in the wells as discussed in detail in ref 24. The point charges have to be chosen equal to the integrated charge density in each well. Considering Figure 2, we can thus use the following approximations:

The transition densities between states 0 and 1 vanish approximately since the wave functions have a node in each well, so that the products have negative and

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The Journal of Physical Chemistry, Vol. 83, No. 19, 1979

H. Pfeiffer, G. Zundel, and E.

G. Weidemann

positive values averaging almost to zero in one well. Thus, the matrix consists of two submatrices with two rows related to the lower and upper pair of levels (eq 29). Here, C

-

.

0

hv,

-+ h v s t 2

because of eq 28, 21, and 25

I

1

Hence 2 W12is the energy difference of the minima of the two wells determined by the reaction potential (AV given by eq 22) and the influence of the anion. Regarding eq 27 all matrix elements of H1 in the main diagonal are equal and have been omitted by choosing an appropriate zero point of the energy scale. The zero point of the energy scale has been shifted to the midpoint of the lowest level pair. The frequencies vo, vl, and ugt are explained in Figure 2. According to the form of energy matrix, eq 29, the eigenvalue problem for the determination of the energy levels decomposes into the two equations:

whereby the lower pair of the energy levels is given by the first equation and the upper pair of levels by the second equation. According to eq 30 the’nondiagonal matrix element W12 contains the energy difference of the potential wells produced by the reaction potential and is therefore, according to eq 22, a linear function of e’- e’! W12 = p(e’- e’? + g (33) Here, p is a constant depending on the parameters used in the calculation of the reaction potential and q depends on the position of the anion. Hence W12 = Ae’+ B A = 2p B = q-ep (34) e’is given by the charge distribution in the proton states considered according to eq 19. Substituting for the lower pair of levels, we obtain the expansion of the wave function with respect to the eigenfunctions in the symmetrical potential, $ = cl$,,+ + cz$@ in eq 19 (35)

if we use

The first two equations follow from the symmetry and the normalization of the wave functions whereas the third is, according to eq 28, approximately valid in the case of a symmetrical double minimum potential with a high barrier (Figure 2). According to eq 31-35 the matrix which has to be diagonalized is itself dependent on the eigenstates. Thus, we have to solve a nonlinear problem. The physical reason for the nonlinearity is the mutual dependence of the proton state and the reaction potential caused by the polar environment. A usual procedure for solving such nonlinear equations is the iterative method. We start with an arbitrary value for e’, diagonalize the matrix, determine cy and c2 for the corresponding eigenstates, and calculat,e e ’ from these coefficients by eq 35. The value obtained for e’ is called e’ and we repeat the procedure starting with this value. This procedure is continued until finally E’ = e’, Le., until self-consistency is reached. By applying this method we have two possibilities: in calculating the parameter e’ we can either choose the amplitudes el, c2 of the ground state or of the first excited state. In the first case we assume that the polar medium is orientated with respect to the charge distribution of the hydrogen bond in the proton ground state. In the second case we assume an equilibrium between the charge distribution in the first excited state and the polar solvent. A necessary condition for the existence of the latter equilibrium is that the lifetime of the excited proton state is longer than the relaxation time for the reorientation of the polar environment. The calculations show that up to four self-consistent solutions of eq 31 exist. In the first case up to three self-consistent solutions exist, in the second case only one. To these solutions correspond parameters e’ by eq 35 which are usually different. By these parameters four reaction potentials are determined according to eq 20. Thus, there exist various equilibria between the charge distributions in the hydrogen bond and the orientation of the polar medium, To each equilibrium belongs a potential energy curve for the proton motion and a corresponding energy level scheme. The determination of the self-consistent solutions is illustrated in Figure 3. In this figure E”, the e’value after one iteration step, is plotted as function of e’ for two distances between the proton and anion. The anion is situated a t x = R, Le., at a distance R from the center of the hydrogen bond on the hydrogen bond axis. Hence e’ is the charge in the well opposite to the anion, according to eq 19, and e ”the charge in the well directed to the anion. The solid lines correspond to the charge distribution in the proton ground state, the dashed curve to the first excited state. The self-consistency condition is e’ = e’. Thus, solutions are obtained by the intersection of these curves with the diagonal. For a distance 8 A there exist three intersection points with the solid line (Pl,P3,and I‘d) and one with the dashed line (P2).With the solutions PI and P4the proton is almost completely localized in one of the potential wells, P1 corresponds to a localization of the proton in the well opposite to the anion, P4to a localization in the well in the direction of the anion. Wilh the solutions P2 and P3 the proton is distributed with almost equal probabilities over both wells. Hence, with the latter solutions, the potential energy curve remains largely symmetrical. With the distance of 6 A only two solutions exist,

The Journal

Interactions of Hydrogen Bonds with Polar Solvents

of Physical Chemistry, Vol. 83, No. 19, 1979 2549

to eq 34 and 42. From eq 41 we then obtain 4e2A2Z4+ [(hvo)2 - e2A2]Z2= 0

(43)

with the solutions 22 = 0

- Z). Thus, the following e’ According to eq 39 e ’ = values correspond to the solutions (eq 44) e’ = e 1 2

e f --[e2A2 l e’ = 2 2A Figure 3. e’is the representative point charge in one of the two wells of the potential (eq 19). G’is the same quantity as obtained after one step of the iteration procedure described in the text. The intersection points G’ = e’correspond to the self-consistent solutions. The arrows denote the directian in which the iteration procedure follows the curves.

one for the ground and one for the excited state. The molecular reasons for the various equilibria between charge distribution in the bond and the environment are discussed in the following sections. 2.4. Analytical Solution. Even an analytical solution is possible for the nonlinear problem, eq 31 with 35. It can be shown, for instance, by means of the variational principle, that the solution of eq 31 for a given reaction potential, i.e., for a fixed value of W12,satisfies the equation (37) This result is obtained by varying the expectation value of the Hamikonian

with respect to cI and c2, taking the normalization condition c I 2 cZ2 = 1 into account. The second condition which must be satisfied by the solution is the self-consistency condition (see eq 35):

+

e’ = e(Y2 - c1c2) = e ’

(39)

From this condition and eq 36 follows

ITl2= Ae(X - c1c2) + B

(40)

Substituting this expression in eq 37 we obtain an equation by which the self-consistent solution can be determined. A convenient way to solve this equation is to introduce clc2 =: Z as an independent variable. The condition can be expressed in these variables by taking the squares of eq 37 and of the normalization condition. The result is the following fourth degree equation in Z: 4e2A2Z4- 8eAKZ3 -+ [ ( h q J 2 - e2A2+ 4K2]Z2+ 2eAKZ - K2 = 0 (41) whereby K is the abbreviation K = (e/2)A

+B

(42)

A fourth degree equation has at most four real solutions. These solutions correspond to the points P1-P4 in Figure 3. The problem simplifies considerably if we remove the anion, i.e., if we put q = 0 in eq 33. Then K = 0 according

(45)

Two of the solutions yield a symmetrical potential. The other two solutions yield asymmetrical potentials, whereby either one or the other well is lowered. These solutions are real only if eA > hvo (46) i.e,, only if this condition is fulfilled can the potential become asymmetrical due to the polar environment. According to eq 30,33, and 34 we obtain for the energy difference of the potential wells caused by the reaction field only ( q = 0) AV = 2W12 = 2p(e’- e”) = A(e’- e’? (47) the maximal AV is obtained with e’ = e, e ” = 0. Hence AV, = eA. According to eq 46

AV,

> hvo

(48)

Thus asymmetrical potentials due to the reaction field only exist if the maximum energy difference caused by the polar medium is larger than the tunneling splitting of the lowest levels in the symmetrical potential.

3. Discussion of the Results We can summarize the results of the numerical and the analytical calculation as follows: the discussion of Figure 3, as well as the analytical calculations (eq 44 and 45) demonstrate that there exist at most four equilibrium states between the charge distribution in a polarizable hydrogen bond and a polar environment if the proton is present in one of the two lowest states. To each equilibrium corresponds a potential energy curve for the motion of the proton along the hydrogen bond which are generally different. In three of these equilibrium states the proton is present in the ground state, in the fourth it is in the first excited state. In two of the equilibrium states corresponding to the proton ground state the proton is largely localized in one of the two wells of the energy curve. The other two equilibrium states have largely symmetrical potential energy curves. The proton is distributed with almost equal probability on both sides. The reason for the various energy curves is the reaction potential caused by the polar medium. Figure 4 shows the energy difference of the potential minima for the four equilibria dependent on the distance between the hydrogen bond and the anion. The anion is located as in Figure 3. According to eq 22, AV > 0 means e ’ > e”, Le., that the proton is mainly present in the well opposite to the anion. With AV < 0 the proton is mainly present in the well next to the anion. With iarge distances between the hydrogen bond and the anion all curves in Figure 4 become independent of R approaching AV = 0 or f400 cm-l. The asymmetry is caused in the two latter

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The Journal of Physical Chemistry, Vol. 83, No. 79, 1979

H. Pfeiffer, G.

Zundel, and E. G. Weidemann

6

2000

4

I

b

a

16

12 R [AI

Figure 4. The energy difference of the two minima of the potential A V as function of the distance R between hydrogen bond and anion arranged as in Figure 3: (-) ground state, (--) metastable state, (---) proton in first excited state, (- - -) A V without reaction potential.

-

cases by the reaction field due to the polar environment as can be seen by comparing with the dashed-dotted line where the reaction potential has been omitted. The particular value of 400 cm-' corresponds to our choice of the model parameters at the end of section 2.2. The three curves show the following behavior, dependent on the distance between hydrogen bond and anion: Case a (solid line in Figure 4). The proton is largely localized in the well directed to the anion. This well is lowered further by the anion field, Case b (dotted curve in Figure 4). The proton is largely localized in the well opposite to the anion. The field of the anion acts against the influence of the polarization field of the environment. The deeper well becomes somewhat raised by the anion. AV becomes smaller. The asymmetry, caused by the polar medium, decreases due to the interaction with the anion. This curve comes to an end a t about R = 6 A. Figure 3 shows that at R = 6 8, no selfconsistent solution, P4,exists. The reason is that with this small anion-proton distance the anion field is so strong that the anion field overcompensates the influence of the reaction field of the polar medium. This equilibrium may be called metastable. Case c. The dashed line in Figure 4 corresponds to the equilibrium in which the proton is present in the first excited state. The potential energy curve remains largely symmetrical up to relatively short proton-anion distances. Thus, the influence of the anion is compensated by the influence of the polar medium. The reason for this behavior is the negative polarizability of the hydrogen bond in the first excited proton stateP2whereby the proton is shifted to the higher well by the anion field. Then the polarization of the medium orientates on this well lowering its energy. This is illustrated by comparing the dashed curve with the dashed dotted curve. With the latter the influence of the polar medium is omitted. Thus, the symmetry of the energy curve is maintained by the polar medium. The fourth equilibrium is omitted in Figure 4 since this equilibrium is unstable under the conditions chosen. This is demonstrated by the results in Figure 3. The arrows on the curves denote the directions in which we follow the curves with the iteration procedure for the determination of the self-consistent solutions. Hence Figure 3 shows that P3 cannot be reached by iteration. The nonconvergence of the iteration procedure reflects the instability of this equilibrium. A deviation of the Polarization from the equilibrium causes a change of the charge distribution in the hydrogen bond which increases the deviation of the polarization. Therefore, this equilibrium is not stable.

1000

-I t

\

200 0

- 200

/ 8

'2 R[A]

Figure 5. Energy level scheme in the potentials: case a (-), b (.e.), c (---), d, ( * * * ) w i t h o u t reaction potential. (v, = 50 cm-', v 1 = 600 cm-' and vSt = 2200 cm-'; see Figure 2.)

There may exist, however, systems, in which this equilibrium state is stable, whereas states a and b are unstable as shown by the analytical calculations in section 2.4. The existence of equilibrium states a and b depends on the condition of eq 48, since these equilibrium states no longer exist if the maximum energy difference of the potential wells caused by the polar medium becomes smaller than the tunneling splitting of the two lower levels. This may be the case with short hydrogen bonds in less polar media. To each stable equilibrium between charge distribution in the hydrogen bond and the polar environment one potential energy curve and hence one energy level scheme exists. The energy levels dependent on the proton-anion distance for all three stable cases are plotted in Figure 5. This figure also contains the levels of the proton without polar environment. In this scheme radiative transitions are only possible between levels of the same potential energy curve. The relative positions of the energy levels in each of the three potential energy curves are independent of the energy of the polarization field which is expressed by the last term in eq 13. This term does not depend on the proton coordinate because the polar medium is orientated with respect to the charge distribution in the hydrogen bond averaged over the proton fluctuations. The relative position of the three energy level schemes is, however, influenced by this term. This is not taken into account in Figure 5, where we have chosen as an origin of each level scheme the midpoint of the lower level group. As long as we are only interested in wavenumber values of the transitions the absolute energy values are not of interest since no radiative transitions occur

Hydrogen Atom Abstraction by Triplet Pyrazine

between the various schemes. The energy values would be relevant if we intended to calculate the intensities of radiative transitions between the levels. To this aim, however, the classical description of the polar medium used in the present paper seems no longer appropriate but rather an approach along the lines followed by Jortner and his g r o ~ p . *He~ performed ~~~ a general quantum mechanical calculation of transition probabilities for electron transfer processes treating the electron donor, the electron acceptor, and the polar solvent as one “supermolecule”. Thereby he established the dominant role of low-frequency polar solvent modes for electron transfer reactions between ions in solutions. In order to apply an approach of this kind to our problem, far more detailed knowledge of the spectral properties of the “supermolecule” would, however, be needed than is available at the present time.

4. Conclusions In earlier publication~,l-~J~J~ we have shown that easily polarizable hydrogen bonds cause IR continua since the distribution of local electrical fields in solution polarize the hydrogen bonds more or less strongly, since transitions in the hydrogen bonds couple with low-frequency vibrations and since a distribution of the hydrogen bond lengths is present. The results of the calculations presented here give insight how these local fields interact with the polarizable hydrogen bonds. The local electrical fields in polar media are particularly the reaction fields. The distribution of the local electrical fields is changed and indeed broadened by these interaction effects. If all protons were present in the ground state the hydrogen bonds would be largely polarized by the anion fields. This polarization is partially enhanced by the interaction with the polar environment but also partially weakened. In the first excited proton state the proton is delocalized by the compensation effect of the reaction and anion fields. Thus, the distribution of local fields is broadened in the direction of larger, as well as smaller field strengths. All these effects occur, however, only if the reorientation of the polar environment is slow compared with the proton

The Journal of Physical Chemistry, Vol. 83, No. 79, 7979

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fluctuation between the two wells. If the reorientation is faster than the fluctuation of the proton, then the environment follows the proton motion and lowers both wells without changing the degree of asymmetry. Acknowledgment. Our thanks are due to the Deutsche Forschungsgemeinschaft and to the Fonds der Chemischen Industrie for providing the facilities for this work.

References and Notes (1) E. G. Weidemann and G. Zundel, Z. Naturforsch. A, 25, 627 (1970). (2) R. Janoschek. E. G. Weidemann. H. Pfeiffer, and G. Zundel, J . Am. Chem. Soc., 94, 2387 (1972). G. Zundel in “The Hydrogen Bond-Recent Developments in Theory and Experiments”, P. Schuster, G. Zundel, and C. Sandorfy Ed., North Holland Publishing Co., Amsterdam, 1976. R. Lindemann and G. Zundel, J. Chem. Soc., Faraday Trans 2, 73, 788 (1977). R. Lindemann and 0. Zundel, Biopolymers, 16, 2407 (1977). G. Zundel and A. Nagyrevi, J. Phys. Chem., 82, 685 (1978). R. Lindemann and 0. Zundel, Biopolymers, 17, 1285 (1978). J. Jadiyn and J. MaYecki, Acta Phys. Polon. A , 41, 599 (1972). 2. PaweYka and L. Sobczyk, unpublished results. L. Sobczyk in “The Hydrogen Bond-Recent Developments in Theory and Experiments”, P. Schuster, G. Zundel, and C. Sandfory, Ed., North Holland Publishing Co., Amsterdam, 1976. J. L. Kurz and L. C. Kurz, J . Am. Chem. Soc., 94, 4451 (1972). M. A. Vorotyntsev, R. R. Dogonadze, and A. M. Kuznetsov, Dokl. Akad. Nauk USSR, 209, 1135 (1973). D. M. Blow, Acc. Chem. Res., 9, 145 (1976). G. Zundel, J. Mol. Sfrucf., 45, 55 (1978). A. Hayd, E. G. Weidemann, and G. Zundel, J. Chem. Phys., 70, 86 (1979). R. Janoschek, A. Hayd, E. G. Weidemann, M. Leuchs, and G. Zundel, J. Chem. Soc., Faracky Trans. 2, 74, 1238 (1978). (a) R. A. Marcus, J . Chem. Phys., 24, 966, 979 (1956). (b) 0. Madelung, “Introduction to Solid-state Theory”, Springer-Veriag, Berlin, 1978. E. G. Weidemann and G. Zundel, Z. Naturforsch. A , 28, 236 (1973). E. G. Weidemann in “The Hydrogen Bond-Recent Developments in Theory and Experiments”, P. Schuster, G. Zundel, and C. Sandorfy, Ed., North Holland Publishing Co., Amsterdam, 1976. J. Brickmann and H. Zirnmermann, J. Chem. phys., 50, 1608 (1969). E. M. Prucell, “Electricity and Magnetism”, Berkeley Physics Course, Vol. 2, McGraw-Hill, New York, 1965, p 53. Th. Landis and G. Schwarzenbach, Chimia, 23, 146 (1969). C. J. F. Bottcher, “Theory of Electric Polarization”, Elsevier, Amsterdam, 1952, Section 16. R. Janoschek, E. G. Weidemann, and G. Zundel, J . Chem. Soc., Faraday Trans. 2 , 69, 505 (1973). E. G. Weidemann and G. Zundel, Z . Phys., 98, 288 (1967). R. Kestner, J. Logan, and J. Jortner, J. Phys. Chem., 78, 2148 (1974). J. Jortner, J . Chem. Phys., 64, 4860 (1976).

Hydrogen Atom Abstraction Reaction by Triplet Pyrazine at Low Temperature Mamoru Jlnguji, Yoshihlko Hosako, and Klnichi Obi” Department of Chemistty, Tokyo Institute of Technology, Ohokayama, Meguro-ku, Tokyo, Japan (Received March 7, 1979)

The hydrogen atom abstraction reaction of the triplet pyrazine at low temperature was studied. Activation energies for pyrazyl stabilization and phosphorescence quenching agree with each other and are 2.1-2.3 kcal mol-I. This value shows the activation energy of the hydrogen abstraction reaction by the triplet pyrazine. The effect of the solvent cage is negligible in this case.

The lowest triplet state, 3 n ~ *of, pyrazine is known to abstract a hydrogen atom from hydrogen-containing solvent molecules and to form the pyrazyl radical at room temperature. It is reported that hydrogen atom abstraction reactions by the triplet pyrazine from alcohols are E 100 times faster than corresponding reactions by the 3 n ~ state * of phenones and related aromatic compounds.lI2 Very little work has been carried out so far on the pho0022-3654/79/2083-255 1$01.OO/O

tochemical reaction of pyrazine at low temperature. In this paper, the hydrogen atom abstraction reaction of the lowest triplet state of pyrazine at low temperature was studied. Activation energies for pyrazyl radical formation and for quenching of the triplet pyrazine were measured. The results are discussed by comparison with the results of the triplet benzophenone reported in a previous paper.3

0 1979 American

Chemical Society