220
J . Phys. Chem. 1989, 93, 220-225
tightly and vigorously shaken for over 10 min. The remaining free acid was titrated potentiometrically with a 0.025 N sodium hydroxide solution, carefully protected from carbon dioxide. The titrations were carried out with a precision buret and an Orion Research 601 A digital p H meter, with use of a glass electrode. The instrument was calibrated every 2 days by using buffer solutions with pH values of 4.01 and 9.20. Notice that the samples were prepared both volumetrically and by weight, so that the only purely volumetric operation is the back-titration of the acid. The samples for the chromatographic titrations were prepared as follows: To a given volume of cyclohexane (ca. 8.5 mL) contained in a 10-mL volumetric flask at 20.0 OC, 1 mL of the solution was added. The flask was vigorously shaken and cyclohexane was added up to the IO-mL mark. These operations were also checked by weight. In some instances, the dilution was carried out by weight, with ca. 10 g of cyclohexane at 20.0 OC and ca. 1 g of the solution. In all cases, the final mixtures were filtered through a 0.5-pm Millipore filter. Samples of ca. 2 pL were injected into a Varian 3300 gas-chromatograph fitted with FID and equipped with a 15-m semicapillary Megabore column (either a DB-1 or a DB-WAX). Cyclohexane was used as the internal standard. With the exception of the systems shown in Table IX (Supplementary Material), the kinetic runs were carried out with solutions ca. 0.09 M in CH31and ca.0.12 M in (C2HJ3N. Activity Coefficients. We have closely followed Abraham’s method.17Gb In general, 2-pL liquid samples and 2-mL gas samples were injected. The chromatograph and the columns were the same used for the kinetic runs. Solubility of C in Organic Solvents. A sample of C was shaken for 48 h with 50 mL of the solvent and filtered through two 0.5-pm Millipore filters. The filtered solution was vacuum-evaporated to dryness in a round-bottomed flask attached to a Buchi Rotavapor. The flask was carefully rinsed with doubly distilled water, the ionic strength adjusted, and the iodide ion concentration determined with a Crison ionometer fitted with an Orion Research iodide-specific electrode, previously calibrated with NaI solutions. Blank tests using pure cyclohexane yielded a iodide concentration of 8.7 X lo-’ M, which is basically the detection threshold of the electrode. The lH N M R spectrum of C was obtained at 80.0 MHz in a Varian A80 FT N M R spectrometer. The refractive indexes were determined with an Atago refractometer. Its temperature was
kept constant by water circulation provided by a Lauda 2R thermostat.
Acknowledgment. This work is dedicated in memoriam to Dr. Mortimer J. Kamlet. The work of J.-L.M.A., M.J.A., and M.T.D was supported by a grant (0977/84) from CAICYT. Valuable discussions with Prof. M. H. Abraham (University of Surrey), E. M. Arnett (Duke University), J. Elguero (CSIC, Madrid), R. Gallo (IPSOI, Marseille), and R. W. Taft (University of California at Irvine) are acknowledged. The statistical analyses of the data have been carried out with the NEMROD package, kindly provided by Profs. D. Mathieu and R. Phan Tan Luu. Appendix: Treatment of the Experimental Data Both the potentiometric and the gas-liquid chromatographic methods allow the determination of the concentration of (C2H5),N and CH31 at any given time, t. Even in the absence of added cosolvents, the reagents act as “catalysts”, so that the complete kinetic equation can be written as U
= ko[(C2H&Nl [CH3II + kI[(C,H,),Nl W 3 I l 2 +
k2[(C2H5)3N12[CH311 (19 ) where
u = -d[(C,H,),N]/dt
= -d[CH,I]/dt
In the presence of the cosolvent, eq 20 applies: U
= kapp[(C&5)3Nl[CH311 -k ki[(C2H&N] [CH3II2
k2[(C2H5)3N12[CH311 (20) The integration of these equations as well as the results obtained in the study of 14 systems containing widely different initial concentrations of the reagents are given in the Supplementary Material. Registry No. CH31, 74-88-4; (C2H5),N, 121-44-8; Et,NMe-I, 99429-6; CH,C(O)N(CH,)Z, 127- 19-5; C-C~HI ICN, 766-05-2;C-C~HI IC(O)CH3, 823-76-7;CHjC(O)C2HS, 78-93-3; C,HSC(O)CH,, 98-86-2; (C~HS)~CO, 119-61-9;C6H5N02, 98-95-3; C6HsC(O)N(CH3)2. 61 174-5; tetradodecylammonium bromide, 14866-34-3. Supplementary Material Available: Tables I-IV and VI-IX ~ 16AG, . SK, Sexpt, and k,) and (giving values of pg and p1, ~ ~ kapp, a more detailed discussion (21 pages). Ordering information is given on any current masthead page.
Theory of “Dynamic” Hydrogen Bonding: Vibronic Effect in Proton Dynamics Akitomo Tachibana,* Takayuki Inoue, Masataka Nagaoka, and Tokio Yamabet Department of Hydrocarbon Chemistry and Division of Molecular Engineering, Faculty of Engineering, Kyoto University, Kyoto 606, Japan (Received: December 21, 1987)
Based on a quantum mechanical treatment of proton, a theory of “dynamic” hydrogen bonding is presented. The vibronic effect which is characteristicof proton dynamics plays an important role. If the vibronic force satisfies the condition of attraction, then there is a contraction of the site distance in a hydrogen-bonded system. This is well-known experimentallyas the Ubbelohde effect. A correlation between the degree of the isotope effect and the hydrogen-bond length is found. Moreover, a novel “inverse” isotope effect is predicted, which is also observed experimentally.
I. Introduction Recently, the ferroelectric phase transition in hydrogen-bonded crystals has received much attention.’-I0 In these crystals, dielectric properties are related closely to the order-disorder arrangements of protons constituting the hydrogen bonds. This fact was verified by the large isotope effect at the transition ternEspecially in potassium dihydrogen phosphate, perature Tc.*-5910 KH2P04 (KDP), it was found that Tc for the crystal ‘Also belongs to Institute for Fundamental Chemistry, 15 Morimoto-cho, Shimogamo, Sakyo-ku, Kyoto 606, Japan
0022-3654/89/2093-0220$01.50/0
deuterons was about 1.7 times as large as that for the crystal including Protons. (1) (a) Blinc, R.; Zeks, 8 . Soft Modes in Ferroelectrics and Antijerroelectrics; North-Holland: Amsterdam, 1974. (b) Lines, M. E.; Glass, A. M. Principles and Applications of Ferroelectrics and Related Materials; ClarOxfordi U.K., 1977. (2) (a) Hamilton, W. C.; Ibers, J. A. Hydrogen Bonding in Solids: Methods of Molecular Structure Determination;Benjamin: New York, 1968. (b) Olovsson, I.; Jonsson, P.-G. In The Hydrogen B o d , Schuster, P., Zundel, G., Sandorfy, C., Eds.; North Holland: Amsterdam, 1976; Vol. 2, Chapter 8.
0 1989 American Chemical Society
Theory of Dynamic Hydrogen Bonding The active participation of the proton in “chemical” bonding is also demonstrated by another fact that the crystal structure of KDP itself changes by extensive deuteriation which exceeds 98%: This structural isotope effect was first found by Ubbelohde,’ and thus is called the Ubbelohde effect. It is true that the so-called Ubbelohde effect, which will be called the “normal” Ubbelohde effect hereafter, is an observation in which the hydrogen bond is lengthened by deuteriation. However, there remain open questions with respect to the Ubbelohde effect. There seems to be a correlation between the degree of the isotope effect and the hydrogen-bond length, and hence the mechanism of the isotope effect is not known d e f i n i t e l ~ .The ~ difficulty originates largely from the smallness of magnitude of the isotope effect (-0.001 A), which implies that it must be completely a quantum effect which is related to the mass dependence of the de Broglie wavelength on mass. Moreover, it remains unresolved that the “inverse” isotope effect is also found? the contraction of hydrogen-bond lengths has been observed when deuterons are substituted for protons in a crystaL3 So far, Anderson and Lippincott have revealed a large vibronic effect in hydrogen bonding for the system X-H-X.* They extended the Born-Oppenheimer adiabatic approximation so as to include the hydrogen kinetic energy term beyond the original simple treatment which considers solely electronic energy. As a result, they found that there is a large isotope effect in the X-H-X system. Recently, Matsushita and Matsubara have presented a detailed theory for the isotope effect in hydrogen-bonded crystals.I0 It was based on the electrostatic potential energy determined by taking the quantum mechanical motion of the proton into consideration. In this treatment, the equilibrium distance of the X-X bond, R,, was determined by d -(V+ dR
MODEL SYSTEM Figure 1. Model system with C,, symmetry for the hydrogen-bonded crystal, and the key of the structural parameters.
- 112.6095
-112.6100
> -112.6105 W
z W
-112.6120 \ \
-112.6125
-1 12.6130
(v)) = 0
where Vdenotes the binding potential contributed from all atoms other than the proton under consideration or, in other words, from the lattice, and (v) is the average binding potential u of the proton with respect to the proton wave function which is a function of the bond distance between two X atoms R.lo Under these circumstances, it is emphasized in the present paper than eq 1 should be modified further as d -(V+ dR
The Journal of Physical Chemistry, Vol. 93, No. 1, 1989 221
e) = 0
2.60
2.65
2.70
R(A)
Figure 2. Energy of the hydrogen-bonded model system. Bottom curve is the binding potential V. Total energy E given by eq 13 is also shown: Ht (top curve) and D+ (middle curve).
T
reaction p a t h
with
where e denotes the energy eigenvalue of the proton as a function of the X-X bond distance R;: is obtained by the average of both the kinetic energy operator k and the binding potential u of the proton with respect to the wave function of the proton: the theory is based on the “dynamic” potential energy determined by the quantum mechanical motion of the proton in a fixed lattice orientation. One can understand that this is a self-consistent treatment for the proton which is regarded rightly as a quantum mechanical entity. It is worth noting that a similar vibronic treatment has also been executed by Anderson and Lippincott.* The mechanism to produce the vibronic attraction of the X-X bond will be described in section 11. Proton dynamics in the present system is discussed precisely by utilizing the notion of the
>
I
r Figure 3. Schematic feature of potential surface.
backflow of the medium. After the condition for the appearance of the vibronic attraction is settled, it will be demonstrated that if the vibronic force satisfies the condition of attraction, then contraction of the X-X bond distance takes place. Further, a large isotope effect is predicted essentially.
(3) (a) Ichikawa, M.; Acta Crystallogr. 1972, 828, 755. (b) Ibid. 1974, 830, 651. (c) Ibid. 1978, 834, 2074. (d) Chem. Phys. Lett. 1981, 79, 583. ( 4 ) Novak, A. Strucr. Bonding 1974, 18, 177. (5) Saito, T.; Mori, K.; Itoh, R. Chem. Phys. 1981, 60, 161. ( 6 ) Nelmes, R. J. Phys. Status Solidi 1972, B52,K89. ( 7 ) (a) Robertson, J. M.;Ubbelohde, A. R. Proc. R . SOC.(London) 1939, 170,222. (b) Ubbelohde, A. R. Ibid. 1939, A173, 417. (c) Ubbelohde, A. R.; Woodward, I. Ibid. 1942, A179, 399. (8) Anderson, G.R.; Lippincott, E. R. J . Chem. Phys. 1971, 55, 4077. (9) Cowley, R. A. Phys. Rev. Lett. 1976, 36, 744. (10) Matsushita, E.; Matsubara, T. Prog. Theor. Phys. 1982, 67, 1 .
The above-mentioned
problem with respect to the correlation between the degree of the isotope effect and the hydrogen-bond length is also resolved, and the normal isotope effect is explained together with the prediction of the “inverse” isotope effect. Finally, in section 111, we present brief concluding remarks. 11. Dynamic Hydrogen Bonding A . Proton Dynamics and Backfow of the Medium. In the present paper, we analyze a proton-transfer mechanism based on
222 The Journal of Physical Chemistry, Vol. 93, No. 1 . 1989
Tachibana et al.
TABLE I: Structural Parameters Optimized for the System Shown in Finure 1’
R, = 2.729
R n = 2.595
1.086 1.008 1.004 109.70 109.68
1.291 1.006 1.006 109.96 109.96
r 11 11 01
a2
R, is the equilibrium bond length, while Rn is the bond length in the transition state (see text). The units of length and angle are angstrom and degree, respectively. (I
I
-
R: 2 729 R - 2 660 R= 2 595
-112.600
3
1
i? LLi
-112.605 .
1
0.0
BACKFLOW Figure 5. Backflow of sphere moving in liquid.
1
,
i :
0.2 R/Z- r
0.4
Figure 4. Binding potential v of proton as a function of r for fixed R. At each fixed R, the underlying binding potential V of the N-N bond is added to the curve. Vis a constant for fixed R and is shown in Figure 2.
the model system H3N-H+-NH3, as shown in Figure 1. The six hydrogen atoms are in a staggered conformation within the system. The proton (H3+) is assumed to move along the C3, symmetric axis from one site (N,) to another (NJ. This type of hydrogen-bond pair is also of biological importance and has been examined recently in various fields of We have optimized the structure by an ab initio MO calculation with a 4-31G basis set using the GAUSSIAN 80 program.I6 The curve of the binding potential Vis elucidated in the bottom of Figure 2. Structural parameters can be obtained by optimization for an assumed bond distance R between a couple of N atoms. The schematic representation of the potential energy surface is shown in Figure 3, where there appear two kinds of characteristic sites R , and RTS with respect to R . R, is an equilibrium bond length such that the binding potential V takes its minimum value and has been found to be 2.729 A by an optimization procedure. On the other hand, R n is equilibrium bond length such that the proton is situated in the middle of the N-N bond, where the system is unstable, because Vincreases as R decreases and then the hydrogen-bond length R-r increases. At the center of the N-N bond, the proton is said to be located at the transition state (TS) from one site of the N-N bond to another. The optimized value of Rn is 2.595 A. The structural parameters for the two characteristic structures are tabulated in
R . Actually, we obtained the adiabatic potential energy V + u as a function of r for five different bond lengths: R = 2.739, 2.700, 2.660,2.630, and 2.595 A. In Figure 4, three of them ( R = 2.729, 2.660, and 2.595 A) are demonstrated and are recognized to be typical of the symmetric doubleminimum potentials. For smaller R , the maximum of the adiabatic potential decreases and the minimum increases. The dependency would make the proton transfer from one site to another much easier. In addition, we were able to recognize that in the region R > 2.660 %I the decreasing rate of the maximum is greater than the increasing rate of the minimum. To analyze the vibronic effect, the energy-transfer mechanism between the hydrogen-bond pair N-H+-N, Le., one part of the whole system, and the other remaining six hydrogen atoms has ~~ been studied by introducing the notion of the b a ~ k f l o wwhich was presented by Feynman and C0her1.l~~Originally, the backflow was defined as a current of the medium that appears when a sphere moves through a liquid medium as shown in Figure 5. It is natural that the effect depends on the velocity of the sphere and its results in a heavier effective mass of the sphere. Incidentally, a similar treatment has been applied to study the solvent effect of a chemical reaction in terms of the dynamical potential field.ls In the proton-transfer mechanism in the system shown in Figure 1, it is a natural matter of course that the six hydrogen atoms move as the proton transfer proceeds. As a result, their motion as a whole makes the mass of the proton heavier effectively than the bare mass itself. In other words, the six hydrogen atoms behave as if they were a “medium” in the dynamics of the proton-transfer process. However, it must be noted that the situation is different slightly from that in Figure 5 because the backflow effect in our system is not uniform and thus it depends on the position of proton. In order to elucidate the methodology with the idea of backflow, we shall formulate our own theory. At first, we define (Xi,&Zi) (i = 1-6) to be the Cartesian coordinates of the ith hydrogen atom and (XG,Y&G) the center of mass of the whole system; then the coordinates of the ith hydrogen atom in the center-of-mass system are obtained from
If we suppose that the N-N bond is along the x axis and N, is taken as its origin, then the position s and the kinetic energy of the proton k(O) along x axis are given respectively by
Table I.
The binding potential u of the proton is represented by the quantity V + u in Figures 4, because Vis a constant for a fixed (1 1) Bell, R. P. The Tunnel Effect in Chemistry; Chapman and Hall: London, 1980, and references cited therein. (12) (a) Scheiner, S . J. Am. Chem. SOC.1981, 103, 315; (b) Int. J. Quantum Chem.: Quantum Biol. Symp. 1981,8, 221. (13) Scheiner, S.; Harding, B. J. Am. Chem. SOC.1981, 103, 2169. (14) Larsson, S . Int. J. Quantum Chem.: Quantum Biol. Symp. 1982, 9,
s=r-XG
(5)
and
k(O) = LmH(ds/dt)2 2
where mH denotes the mass of bare proton. s is considered to be the reaction coordinate in our treatment. The “backflow” of the
385.
(15) Staab, H. A.; Saupe, T.; Krieger, C. Angew. Chem. Int. Ed. Engl.
1983. 22. 731.
(16) Binkley, J. S.; Whiteside, R. A.; Krishnan, R.; Seeger, R.; DeFrees, D. J.; Schlegel, H. B.; Topiol, S.; Kahn, L. R.; Pople, J. A. QCPE 1981, 13, 406.
(17) (a) Feynman, R. P.; Cohen, M . Phys. Rev. 1956, 102, 1189. (b) Ghassib, H. B.; Awwad, K. Y.; Lewis, L. J. Presented at the XVIII International Conference on Low Temperature Physics, Kyoto, Japan, 1987; paper BH10. (18) Tachibana, A.; Fukui, K. Theor. Chim. Acta (Berlin) 1979,51,275.
-
The Journal of Physical Chemistry, Vol. 93, No. 1, 1989 223
Theory of Dynamic Hydrogen Bonding TABLE II: Backflow Effect in Terms of 76 per Mass of H+’
H+ RIA 2.729 (=R,) 2.700 2.660 2.630 2.595 (‘Rn)
D+
Fb
TC
Fb
TC
0.437 0.449 0.341 0.378 0.381
0.874 0.894 0.701 0.772 0.762
1.566 1.395 1.341 1.495 1.542
2.586 2.454 2.168 2.238 2.351
’The F region and the T region are shown in Figure 6. *The F region. cThe T region.
2.6
2.8 -3.0
3.2
r(A)
2.6
2.8
3.0
3.2
r(i)
Figure 7. Energy level E of (a) H+,and (b) D+ at R = 2.660 A.
Figure 6. The F region and the T region. six hydrogen atoms can be described by a set of coordinates representing their positions: xi = xi(s), yi = yi(s), zi = zi(s) (i = 1-6) (7) Then, the backflow effect can be taken into account by introducing the kinetic energy of six hydrogen atoms k’
neling. The backflow in the T region is about 1.6 times as great as that in the F region for the case of H+, and about twice as great as that for the case of D+. In other words, H+ or D+ moves the six hydrogen atoms more vigorously in the T region than in the F region. Moreover, comparing H+ with D+,the backflow effect of D+ is 3-4 times larger in the F region and is about 3 times larger in the T region than each of H+. By interpolating the binding potential u of proton in the above cases for each R by spline functions, we solved the following type of one-dimensional Schriidinger equation for the proton by using Runge-Kutta-Gill integration method
(-&5
(12)
where m, \k, and e denotes, respectively, the effective mass of proton defined by eq 11 taking the backflow effect into consideration, the wave function of the proton, and the energy eigenvalue given in eq 3. Thus, we have calculated the total vibronic energy E E=V+e
where
((
m’= E6 m H i= 1
dxi)’
+
ds
(
dYi)2 ds
+
(dzi)’) ds
(9)
Therefore, the total kinetic energy of an “effective” proton, reflecting the backflow effect, can be represented as follows k = kC0) + k’ =
“)’
iF( 2
dt
where the effective mass I.I of proton is defined by p = mH m’
+
(11)
It is understood that the effective mass of the proton becomes heavier than its bare mass by the amount of additional mass m’. Actually, it originates from the dynamical motion of the six hydrogen atoms as a surrounding medium and is further a function of the reaction coordinate s of the proton. In the following, the same treatment will be used for the case of D+ in place of H+. The increase in mass by the backflow effect is calculated by eq 9 for five N-N bond lengths, and ratios between m’and mH in % are shown in Table 11, where the F region and the T region are defined pictorially in Figure 6. In the F region, the proton can move freely, while in the T region it can move only by tun-
(13)
and we can compare it with the purely “electronic” binding potential V. Since Vis constant for a fixed R, we can rewrite eq 12 as \k(s) = E\k(s)
(14)
Hence it can be understood that the vibronic energy E serves as the vibronic binding potential for the N-N bond by the analogy of the original BO adiabatic approximation. We can assume that, at low temperature, only the zero-point vibrational level of the proton is populated. Then, if the vibronic effect is excluded, as depicted by the curve in the bottom of Figure 2, the energy of the system takes its minimum value when R is 2.729 A. However, when the vibronic effect is included, the minimum value appears at a rather small R which is about 2.660 8, for both H+and D+ cases, namely, in E(H+) and E(D+)as shown in Figure 2 (upper two curves). Comparing this with the quantity R, - RTS,the contraction of R for H+ is slightly stronger than that for D+and has been found to be 59.4% for H+ and 58.0% for D+,respectively. Moreover, the energy levels for R = 2.660 A are shown in Figure 7. There is one energy level for H+ and two for D+ below the barrier. We shall call the contraction effect the dynamic hydrogen bonding hereafter. The dynamic hydrogen bonding leads to
224
The Journal of Physical Chemistry, Vol. 93, No. 1, 1989
Tachibana et al.
v +
0
0.010
0.005
V,(au)
0
0005
0.010
tr, (au)
Figure 9. Zero-point energy level E of (a) H+and (b) Dt as a function of V, and V2 ( V2 2 VI) for the analytical model potential of a = 0.4 au and b = 0.6 au. Equienergy curves are drawn with spacing 0.001 au.
(b)
Figure 8. Analytical model potential: (a) asymmetric and (b) symmetric ( c = a b).
+
contraction of the lattice spacing. The detail of the mechanism of this dynamic hydrogen bonding will be discussed in the next subsection. B. Mechanism of Dynamic Hydrogen Bonding. To discuss essentially the dynamic hydrogen bonding, we use a simple analytic model potential. Let us start with the general asymmetric potential shown in Figure 8a. This double-minimum square potential well, which has been used also in a theory of chemical reactions,lg is characterized by three distinct regions A, B, and C, of which the boundary values are given by s = -a, 0, b, and c. Then, the one-dimensional Schriidinger equation (1 2) may be reduced to a set of three equations for each region 1 d2
--2~ ds2 -\kA(s)
= e\kA(s)
0
0.005
0.010
VI ( a u )
Figure 10. Underlying mechanism of vibronic attraction (see text for the
detail).
(15a)
where vl and u2 denote two parameters for the potential barrier height as characterized in Figure 8a, and superscripts A, B, and C denote the regions A, B, and C , respectively. Since we want to deal with the case where the efficiency of proton transfer is at its maximum, it is sufficient for eq 15 to be solved in the special case of a symmetric double-minimum square potential well shown in Figure 8b. Furthermore, it is convenient to treat eq 14 instead of eq 12 and then to introduce new parameters VI and V, instead of uI and v2 for the potential as shown in Figure 8b. In addition, the boundary values are now found to satisfy c = a b. We have calculated the energy of the system as a function of VI and V2: The dependency of the zero-point energy E on Vl and
+
(19) Fukui, K.; Tachibana, A.; Yamashita, K. Int. J . Quantum Chem.: Quantum Chem. Symp. 1981, IS, 621. (20) (a) Kunze, K. L.;de la Vega, J. R. J . Am. Chem. SOC.1984, 106, 6528. (b) Hameka, H. F.; de la Vega, J. R. Ibid. 1984, 206, 7703.
I 2.60
2.65
R(h
2.70
Figure 11. Isotope effect of e given by eq 12 as a function of' R . V2is of particular interest at low temperature and is depicted in Figure 9a for H+ and in Figure 9b for D+,respectively. These figures were both prepared for a particular condition of a = 0.4 and b = 0.6 au. However, in both of them, V2decreases if the system conserves the total energy as VI increases. This process is shown schematically in Figure 10. Imagine that the system is initially in the condition represented by the point P in Figure 10. If VI increases by AV, and the system moves to the point Q, it means that the total system energy increases. Accordingly, in
J. Phys. Chem. 1989, 93, 225-229 order to stabilize the system energy, Vz has to decrease by AV2 to move to the hatching region. Therefore, the energy of the system will be stabilized by the decrease of Vz even if V, increases. This is the underlying mechanism in the dynamic hydrogen bonding which was elucidated in section IIA. Indeed, in Figure 4, we have shown the vibronic attraction using the change of the binding potential energy as R decreases from 2.729 to 2.660 A. Consequently, the contraction of lattice spacing suggested by Anderson and Lippincott will be explained by this vibronic effect.* C. Isotope Effects. In this subsection, the isotope effects are explained. Apparently, the isotope effect of the dynamic potential energy E is solely due to e. In Figure 11, we shall show the e which has been obtained as the eigenvalue of eq 12 for each case H+ and D+. It should be noted that the derivative de/dR corresponds to the vibronic attractive force for the N-N bond. There are two regions: (I) R > 2.65 A, where the attractive force of H+ is greater than that of D+, and (11) R C 2.65 A, where the attractive force of D+ is greater than that of H+. The origin of the clear separation into I and I1 may be explained as follows. Using the analytical parameters of the potential, the force may be decomposed into two terms: -de= - - de dV1 dR dVl dR
de dv2 +--dV2 dR
Now, in the region I, the H+ is bound little with respect to the top of the barrier (see Figure 7a). In this region, the vibronic force de/dR given by eq 16 may be approximated as de dR
de dvz dV2 dR
N - -
because we find from Figure 4 that the relation IdVl/dRI
