J. Phys. Chem. 1993, 97, 13608-13614
13608
Catalyzed Cis/Trans Isomerization of Diazene. A Computational Study in the Gas and Aqueous Phases Michael L. McKee Department of Chemistry, Auburn University, Auburn, Alabama 36849 Received: June 22. 1993"
Several mechanisms for trans to cis isomerization for diazene (NzHz) were considered at reasonably high levels of a b initio theory. Inversion and N-H bond scission have activation barriers much too high to explain the rapid trans cis isomerization which must precede the facile dihydrogen transfer from the higher-energy cis isomer in solution. While one catalytic water has little effect on the activation barrier, two catalytic waters reduce the activation barrier to 26.9 kcal/mol (in the gas phase). When solvation effects are considered by combining A M l S M 2 solvation energies with a b initio free energies, one mechanism appears to be consistent with experimental observations. The solution phase proton affinity of truns-NzH2 is 3.2 kcal/mol smaller than that for HzO. This result indicates that trace amounts of acid can rapidly protonate trurrs-NzHz to form N2H3+. In turn, N2H3+ can lose a proton to form either the cis or the trans isomer in an equilibrium ratio determined by the free energy difference, which is calculated to be 4.9 f 2.1 kcal/mol at 298 K.
-
Introduction Diazene has great synthetic utility due to its ability to stereospecifically reduce olefinic bonds.'P2 The reaction occurs via an allowed pericyclic reaction which concertedly transfers two hydrogens from the cis- 1,2-N2H2 isomer, producing NZas a byprod~ct.~ A side reaction often encounteredis the dismutation of diazene, which produces molecular nitrogen and hydrazine. Both reactions, the reduction of olefinic bonds by diazene and the dismutation reaction, have been studied theoretically by ab initio calculations."I0 In all studies, the cis isomer is considered to be the dihydrogen donor rather than the trans isomer although the latter is known to be more stable. Earlier studies'9 calculated significant barriers (18.8-54.7 kcal/mol) for the concerted dihydrogen addition to ethylene, which is inconsistent with the known facile nature of the reaction in However, if electron correlation is introduced, the barrier is greatly reduced.I0 At theMP4/6-31-bG(Zd,p)//MP2/6-31G(d) level, theenthalpy of the transition state for transferring two hydrogens from cis1,Zdiazene to ethylene is 0.7 kcal/mol less than the enthalpy of separated reactants. With respect to thecomplex formed between the two reactants, the barrier is only 5.3 kcal/mol.I0 Numerous calculations have been reported on cis- and transdiazene and their interconver~ion.'~'~ In the gas phase, three unimolecular mechanisms for interconversion have similar activation barriers: cleavage/recombination, rotation about the N=N double bond, and inversion. Jensen et a1.l I used an MCSCF method to calculate barriers of 54.2, 62.9, and 63.4 kcal/mol, respectively, for the three mechanisms. Later calculations by McKeeet al.lOattheMP4/6-3 l+G(2d,p)//MP2/6-3lG(d) level reversed the order of the cleavage/recombination and inversion mechanisms (63.8 and 52.6 kcal/mol). Pople and Curtissis obtained a very similar value of the N-H bond energy, which is assumed to be identical with the cleavage/recombination barrier height, using G2 theory16 (64.1 kcallmol). Very few studies of the reactivity of tram-NzH2 have been carried out in the gas phase. Rate limiting trans cis isomerization was assumed in a study" of the first-order decomposition of tram-NzH2 as well as in a study'*of the reaction of trans-NzHz with olefins. While reaction of diazene (N2H2) occurs readily in solution, the reaction of substituted diazenes in solution occurs much more slowly. Several lines of evidence suggest that the inversion mechanism is operative rather than
-
@
Abstract published in Advance ACS Abstracts, November 15, 1993.
0022-3654/93/2097-13608$04.00/0
one-bond s c i s ~ i o n . 'Schmittel ~ ~ ~ ~ and RQchardtI9found in a study of substituted diazenes that the inversion mechanism was supported by structur+activity relationships. They measured activation barriers for several cis-NzR2 trans-NzRz reactions as well as isomerization enthalpies. The barrier for the least bulky group studied, isopropyl, yielded a barrier of 32.2 kcal/ mol for the cis trans reaction which is about 14 kcal/mol lower than the calculatedlo barrier of the unsubstituted diazene (46.6 kcal/mol). An important contribution to the understanding of the kinetics of the solution-phase reaction has been made by Stanbury and co-workers.21qz2They directly observed the second-order dismutation of N2H2 in aqueous solution between 10-40 "C by monitoring the ultraviolet absorption spectrum of N2H2 in a stopped-flow system.22 An activation energy of 3.3 f 0.1 kcal/ mol was calculated for the reaction. In addition, they were able to confirm that the diazene species initially generated in solution was the trans isomer by the good agreement of their spectrum in the near-ultravioletwith that of the gas-phasespectrumreported by Back et al.23 Computational and experimental studies of gas-phase NzHz as well as experimental studies of substituted diazenesin solution all suggest that a substantial barrier exists for the trans cis isomerization. The purpose of this contribution is to consider mechanisms for trans cis isomerization of N2H2 in aqueous solution.
-
-
-
-
Metbod All ab initio calculations have been made using the GAUSSIAN 92 program systems.24 Geometries have been optimized at the MP2/6-3 IG(d) levels using standard algorithms.25 Vibrational frequencies are calculated at optimized geometries to determine the nature of the potential energy surface at that level and to calculate zero-point corrections. The zero-point energy was weighted by a 0.95 factor26 to compensate for the known overestimation of vibrational frequencies at the MP2/6-3 1G(d) level. All calculations which included electron correlation used the frozen-core approximation, i.e. excitation was not allowed from the core orbitals. The MP2/6-31G(d) geometries are fixed for several singlepoint calculations which are made to estimate relative energies at an even higher level of t h e ~ r y . ~ Specifically, ~.~* the basis set of the MP4/6-31G(d,p) method was supplemented by diffuse functions to give MP4/6-31+G(d,p) (6-31G(d,p) 6-31+G+
0 1993 American Chemical Society
Catalyzed Cis/Trans Isomerization of Diazene
The Journal of Physical Chemistry, Vol. 97, No. 51, 1993 13609
TABLE I: Absolute Energies (hartrees) of Various Species Calculated at MP2/6-3lG(d)-Optimized Geometries ~~~
sym
~
~
~
~
~
~
MP2/6-3 1G(d)
MP4/6-31G(d,p)
MP4/6-31+G(d,p)
QCISD(T)/6-3lG(d,p)
ZPE(N1F)"
-0.49823 -76.19685 -76.47510 -109.71103 -110.31518 -1 10.30473 -1 10.22328 -1 10.61894 -75.51314 -152.40536 -186.52496 -1 86.5 1386 -1 86.41 73 1 -186.43367 -186.43210 -262.73885 -262.73 148 -262.675 16
-0.49823 -76.231 15 -76.51884 -109.74869 -1 10.36247 -1 10.35293 -1 10.27160 -110.67615 -75.53476 -152.473 18 -1 86.60586 -186.59576 -186.49788 -1 86.5 1676 -186.51 547 -262.85268 -262.84584 -262.78770
-0.49823 -76.24452 -76.52121 -109.75659 -1 10.37234 -110.36112 -1 10.28358 -1 10.67926 -75.6 1066 -1 52.49824 -186.62657 -186.61734 -186.53360 -186.54495 -186.54329 -262.88430 -262.87667 -262.82075
-0.49823 -76.23157 -76.51937 -109.75302 -1 10.36294 -1 10.35363 -110,27291 -1 10.67754 -75.53514 -152.47390 -186.60652 -186.59668 -186.49831 -186.51158 -1 86.5 1053 -262.85362 -262.84694 -262.78768
13.48(0) 21.75(0) 9.82(0) 17.90(0) 17.46(0) 15.24( 1) 26.84(0) 5.12(0) 29.25(0) 34.1 l(0) 32.77(0) 30.84(2) 33.39(1) 33.24(1) 50.08(0) 49.82(0) 45.45( 1)
~
0.00
Zero-point energy in kcal/mol and number of imaginary frequencies calculated at the MP2/6-3 1G(d) level.
TABLE 11: Relative Energies (kcal/mol) of Relevant Species Optimized at the MP2/6-31G(d) Level MP4/ 6-3 lG(d,p)
2Hz0
0.0
MP4/ 6-31+G(d,p)
0.0
QCISD(T)/ 6-3 lG(d,p)
[QCISD(T)/ 6-3 1+G(d,p)]"
0.0
-6.8
-5.8
0.0 -6.8
HzO+H+ H@+
0.0 -180.5
0.0 -173.6
0.0 -180.6
0.0 -173.7
0.0 -165.9
t-NZHz+H+ NZH3'
0.0 -196.8
0.0 -192.6
0.0 -197.4
0.0 -193.2
0.0 -184.7
t-NzHz c-N~Hz N2H2 inv TS NzH+H
0.0 6.0 57.0 72.5 0.0 -7.7 -1.3 60.1 48.2 49.0 240.2 0.0 -6.8 -7.7 -1.3 -18.8 -13.2 23.3
0.0 7.0 55.7 73.7 0.0 -6.1 -0.3 52.2 45.1 46.2 205.2 0.0 -5.8 -6.1 -0.3 -17.5 -9.6 25.5
0.0
0.0 6.8 55.2 71.3
0.0
5.8 56.5 70.1
6.4 52.7 63.6
0.0 -7.5 -1.4 60.4 52.0 52.7 239.6
0.0 -5.9 -0.4 52.5 48.9 49.9 204.6
-3.3 0.9 52.0 50.8 51.7 205.2
0.0 -6.8 -7.5 -1.4 -14.4 -13.1 24.1
-5.8 -5.9 -0.4 -1 3.1 -9.5 26.3
(Hz0)z
t-NzHZ+HzO t-NzHyH20 c-N 2H H z0 NzHzmH20 SS NzH2.HzO TS 1 NzHyHzO TS2 NzH3+ OH-
+
t-NzH2+2H20 t-NzH2+(HzO)z t-N2Hz*HzO+H20 C-NZH~.H~O+HZO t-NzHr(HzO)z c-NzH~(Hz~)z NzHr(Hz0)z TS
0.0
+ZPCb
-5.8
0.0
-3.6
0.0
0.0 -3.6 -3.3 0.9 -8.1 -4.8 26.9
Relative energies at this level are approximated from additivity approximati~n:~' AE(QCISD(T)/6-3 l+G(d,p)) = AE(QCISD(T)/6-31G(d,p)) Zero-point corrections are weighted by 0.95 to account fcr the knownz6 overestimation of vibrational frequencies at the MP2 level of electron correlation. a
+ AE(MP4/6-31+G(d,p)) - M(MP4/6-3 lG(d,p)).
(d,p)) while the level of electron correlation was extended to QCISD(T)/6-31G(d.p) (MP4 QCISD(T)). The two effects can be combined to approximate the calculation at the QCISD(T)/6-31+G(d.p) level (eq l), which is much more expensive than the sum of the two smaller calculations. The entropy and +
AE(QCISD(T)/6-3 1+G(d,p)) = AE(QCISD(T)/6-3lG(d,p)) AE(MP4/6-3 l+G(d,p)) hE(MP4/6-3 lG(d,p)) (1)
+
heat capacity are calculated using the MP2/6-3 1G(d) geometries and vibrational frequencies (unscaled) by standard techniques within the GAUSSIAN pr0gram.2~Absolute energies (hartrees) and relative energies of relevant species are given in Tables I and 11. Selected geometric parameters are given in Figure 1. Although the importance of solvent effects in chemistry has long been the incorporation of solvent effects into theoretical calculations has only recently become commonplace.
An exceptionallydiverse set of approaches for includingsolvation reflects a number of different approximations as well as philosophies. While no means complete, refs 30-40 will give the flavor of severalstrategies for includingsolvation. The method selected in the present study is the implementation of the extended Born equations into a parameterized version of AM1 by Cramer and Truhlar called AM1-SM2.30,35,41Two terms (P-term and CDSterm) are included into the AM142 Hamiltonian and solved iteratively to self-consistency. The P-term accounts for the 'unrelaxed" electric polarizationof the solvent by the solute while the CDS-term accounts for (1) the free energy of creating a cavity in the solvent, (2) the dispersion energy of the solute caused by the solvent, and (3) changes in the first hydration shell of the solvent induced by the solute. In addition, the latter term also includes the 'relaxed" effects of mutual interaction of electric fields, i.e. the electric polarization of the solvent on (and by) the solute determined in a self-consistent manner. The A M l S M 2 method, which uses standard states of 1 M in both the gas and
_
_
_
McKee
13610 The Journal of Physical Chemistry, Vol. 97, No. 51, 1993
H
H 2 0 0.969, 104.0
1.261
H,O* 0.991, 111.4
OH
105.4
0.980
H NzH l,lSO(N-N) 1.O57(N-H) 121.2
178.0
H
H
1.031 105.3
H
0-0 2.915
1.036 105.4
1.575
N2HpH2O VSl)
125.2
H 1.263
H 2,263".
133.7 52.0
,'
'p,,'
H H
Figure 1. Geometric parameters of species calculated at the MP2/6-31G(d) level of theory.
solution phases at 298 K, is able to reproducethe solvation energies of a number of molecules to within 1-2 kcal/mol. A more recent i m p l e m e n t a t i ~ nof~ the ~ ~ method into the PM3 Hamiltonian is not appropriate for this study because solvation energies of nitrogen-containing compounds are poorly r e p r o d ~ c e d . ~ ~ The AM I S M 2 method has been parameterized to reproduce solvation energies of species in equilibrium with the solvent. Transition states may represent a special situation since (1) the theoretical method (AM1) may not beable toadequatelydescribe the electron density and (2) the solvent may not be able to fully adjust to the changing electronic environment as the molecule passes over a barrier (nonequilibrium solvation). While the AM 1SM2 method is used to estimate solvation energies of transition states in the present study, all conclusionsare based on equilibrium solvation effects.
Results and Discussion Seven mechanismscan be written for trans +cis isomerization in aqueous solution (eqs 2-6). The first three reactions ( 2 4 ) are t-N,H,
t-N,H,
-
-
c-N,H, rotation about N-N
t-N,H, HN,
+H
-
-
c-N2H2inversion
bond
(2) (3)
c-N,H, cleavage/recombination (4)
t-N,H, C H2O t-N,H,
+H20
+
-
N2H3 + OH
N2H3'
-
+ OH -
c - N ~ H+~HZO (Sb) c - N ~ H ,+ H2O ( 5 ~ )
unimolecular reactions and have high activation barriers in the
gas phase. It is unlikely that the barriers could be significantly reduced in solution. Reactions 5 and 6 are catalyzed by one and two water molecules, respectively, while reaction 7 represents proton exchange with the solvent. The gas-phase mechanisms will be discussed first, followed by solvation effects estimated from A M I S M 2 calculations using fixed ab initio geometries. Reactions in the Cas Phase. Mechanism 2 was not considered because (1) the rotational barrier is calculatedk'by an MCSCF treatment to be very high (62.9 kcal/mol) and (2) experimental result^^^,^^ for isomerization of substituted diazenes support an inversion mechanism. The inversion barrier calculated at the highest level (52.7 kcal/mol at 0 K) is in good agreement with an earlier valuelo of 52.6 kcal/mol (at 298 K). Very recent calculation^^^ using density functional theory and the nonlocal spin density approach indicate a slightly lower barrier of 49 kcal/ mol for inversion. The cleavage/recombination barrier at the highest level in this work (63.6 kcal/mol at 0 K) is almost unchanged from the earlier worklo on NzH2 reactions at a somewhat lower level (63.8 kcal/mol at 298 K) and close to a G2-theory value of 64.1 kcal/mol (at 0 K) reported by Pople and Curtissl5 and a value of 62.3 kcal/mol (at 0 K) reported by Walch.13 The results indicate that the inversion barrier is about 10 kcal/mol below radical cleavage/recombinationand that both barriers are substantial. Mechanisms 5 and 6 involve catalytic H20 where a hydrogen is transferred from nitrogen via one or two water molecules to the same nitrogen but on the opposite side. Previous theoretical studies have also considered catalytic water?"' Two recent examples include the hydration of ketene46and formaldehyde4' by a water dimer. In both studies, it was concluded that the second water had a catalytic effect. Mechanisms 5a-5c are all catalyzed by HzO and differ in the nature of the potential energy surface between reactant and product. Mechanism 53 is a concerted hydrogen transfer, and 5b is a stepwise hydrogen-atom transfer, while 5c is a stepwise proton transfer. The hydrogen-bonded complex between HzO and rram-NzHz, bound by 3.3 kcal/mol, is characterized by two hydrogen-bonding interactions, one between oxygen and N-H and the other between nitrogen and 0-H (Figure 1). In the cis-NzHTHzO complex, both hydrogens on diazene interact equivalently with HzO to form a CZ,-symmetry complex which is bound by 5.5 kcal/mol,
Catalyzed Cis/Trans Isomerization of Diazene
The Journal of Physical Chemistry, Vol. 97, No. 51, 1993
13611
substantially more than the trans complex. The trans/& difference is reduced from 6.4 kcal/mol in NzHz to 4.2 kcal/mol in NzH2.H20. If a trans * cis equilibrium is established in solution, a reduction in the cis/trans difference will increase the concentration of the cis isomer in solution. A stationary point (SS)was located for mechanism 5a, the concerted hydrogen-atom transfer (see Figure l), where the H20+ N2H,' making/breaking 0-H bonds are both 1.550 A. The stationary point was 52.0 kcal/mol above trans-NzHz plus water and 55.3 kcal/mol above the complex. A frequency calculation revealed b) aqueous solution two imaginary frequencies,which signified the presence of lower4.W.1 energy transition states. The two transition states compose the 3.2 / H,O* + c-N2H2 two-step hydrogen-transfer mechanism (5b). The first transition 0.0 H20 + N,H,* state (TS1) corresponds to the transfer of a hydrogen to transH30* + t-N2H2 NzHz to form NzH3 while the second transition state (TS2) corresponds to the abstraction of a hydrogen from N2H3 by OH Figure 2. Relative energies in kcal/mol of the protonated/neutral pairs to form cis-NZH2 (Figure 1). It is likely that the intermediate H3O+/N2H2 and N2H,+/H20 (a) in the gas phase and (b) in aqueous solution at 298 K. If the amount of water is much larger than diazene is a tightly bound radical pair. The two-step mechanism (5b) is in the gas phase, the transfer of a proton from N2H3+to another diazene predicted to be only 0.3 kcal/mol lower in energy (see Table 11) will be much less likely. In solution,the rate of proton transfer is expected than the one-step mechanism (5a). In a solvent with a high to be rapid in both directionsand will lead to equilibrium concentrations dielectric constant (such as water), it is possible that mechanism of rrons-N2Ht and c ~ s - N ~ H ~ . 5c would be preferable to mechanism 5a or 5b since the solvent would be able to stabilize the charged species. However, in the spectrometric of N2Hz and NZH3 plus the heat of formation and IP of the hydrogen atom. gas phase, NzH3+ plus OH- is calculated to be uery endothermic A search was made for the gas-phasetransition state for proton (205.2 kcal/mol) with respect to trans-NzHz plus H20 (Table 11). transfer in the reaction H3O+ + c,t-NzHz HzO N2H3+. However, at both the HF/6-31G(d) and MP2/6-31G(d) levels, Mechanism 6 includes two catalytic water molecules which the proton was transferred without activation. While proton allows the concerted relay of a hydrogen to the other side of transfer from H30+ to N2H2 will be rapid in the gas phase to diazenevia a six-membered transition state (Figure 1). Acomplex form N2H3+, proton transfer from NzH3+ to H20 is expected to was calculated for trans-NzH2 with the two H20 molecules be very slow because the reaction is endothermic by 18.8 kcal/ coordinated in thesameorientation requiredtoreach the transition mol to form the trans isomer and endothermic by 25.2 kcal/mol state which was bound by 8.1 kcal/mol. While it is possible that to form the cis isomer (Figure 2a). the global minimum for two water molecules coordinatedto transReactions in the Aqueous Phase. Thus, it appears that all NzHz has not been found, the complex is bound by more than mechanisms considered are incapable of explaining rapid isomertwice the binding energy of one water and is stable with respect ization in solution based on ab initio gas-phase calculations. Since to eliminating a water dimer. Two water molecules are bound trans-NzHz decomposes in the gas phase while it undergoesrapid even more tightly in the complex of c ~ s - N ~(1H1.2 ~ kcal/mol), dismutation in solution, solvent effects must play an important which is probably due to the stronger interaction with the role in the reaction. In order to estimate solvation effects, it was permanent dipole moment of cis-NzHz. The trans-NzHz.2H~O decided to use the A M l S M 2 method in conjunction with the complex is calculated to be 3.3 kcal/mol more stable than the ab initio results. Solvation free energies are calculated with the cis-NzHz.2H20 complex, a 3.1 kcal/mol reduction from the A M l S M 2 method and combined with ab initio gas-phase free difference of the free NzH2 species. energies to give aqueous free energies at 298 K and 1 atm (eq In the transition state, the extent of bond formation and bond 8). The A M l S M 2 solvation free energies are taken as the breaking is nearly identical, Le. the transition state is neither early nor late (Figure 1). In both complexes and the transition AGo(aq,hybrid) = AGa(g,ab initio) state,all atomsarenearly coplanar except the twononparticipating AGO (solvation,AM 1 S M 2 ) (8) 0-H bonds which are above and below the approximate plane. The activation barrier with two waters is 26.9 kcal/mol, 25.1 difference between AM 1 S M 2 aqueous-phase energies and the kcal/mol lower than with one water (52.0 kcal/mol). AM1 gas-phase heats of formation calculated at fixed MP2/ The last mechanism considered is proton transfer between the 6-31G(d) geometries. Gas-phase free energies at 298 K are calculated using heat capacity corrections to the enthalpy plus solvent and trans-NzHz (eq 7). It is known that proton transfer the entropy. occurs with little or no activation barrier in the exothermic The A M l S M 2 solvation energy of HzO (-6.3 kcal/mol) is direction if the transfer is from protonated nitrogen or oxygen and if there is no significant solvent reorganization r e q ~ i r e d . ~ * . ~ ~in good agreement with an experimentalvalue53of -6.4 kcal/mol (at 298 K). However, the A M l S M 2 value refers to the process For proton transfer from H 3 0 + to H20, a reaction which is HzO(g) HzO(aq) (i.e. where water is a solute molecule) while thermoneutral (H30++ HzO+ HzO H3O+),a small activation the desired process is HZO(g) HzO(1) (Le. where water is the barrier (2.4 kcal/mol) has been measured in solution.50Therefore, solvent). Rather thancorrect theAMlSM2value for thechange if the proton affinities of HzO and trans/& NzH2 are similar, in standard state, the experimental value54will be used for this then a catalytic amount of H+ (perhaps present from autoionprocess (-2.0 kcal/mol). Of note in this regard is a recent ization of water) could allow rapid exchange to take place. The molecular-dynamicsstudyby Corongiu and Clementis5on solvated calculated gas-phase proton affinities of HzO and trans-NzHz water molecules which provides insight into the view of a water are 165.9 and 184.7 kcal/mol (at 0 K), respectively (Table 11). molecule solvated in an aqueous medium. Experimentalsolvation For comparison, the G2-theory values are 163.l5' and 183.515 free energies were also used for a hydrogen atom (2.1 kcal/ kcal/mol (at 0 K)for H2O and frans-NzHz, respectively. There is also good agreement with experimental values for HzO (165.1 mol)56 and a proton (-260.5 kcal/m01).~' All other values are kcal/mol at 0 K)Is and trans-NzHz(186.2 kcal/mol at 0 K). The AM 1SM2-calculated values. latter value was derived using the heats of formation of transAs a test of the hybrid method, the water dimer was calculated. NzH2 and NZH3 and the IP of N2H3 determined in a recent mass Numerous calculations of the water dimer have appeared, and
+
+
+
-
-
McKee
13612 The Journal of Physical Chemistry, Vol. 97, No. 51, 1993
TABLE III: Thermodynamic Corrections to Electronic Energies Calculated at the Estimated QCISD(T)/6-3l+G(d,p)//MP2/ 6-31G(d) Level ~
ab initio
AH (g,O K)
entropy
AH (g,298 K) 0.0 -3.4 0.0 -166.8 0.0 -185.6
AG (g,298 K) 0.0 2.5 0.0 -159.4 0.0 -178.5
AMlSM2 AGsoiv" -4.0' -9.5 -262.5'~~ -104.2
~~~
hybrid AG (aq,298 K) 0.0 -3.0 0.0 -1.1
90.2 0.0 2H20 (H20)2 -3.6 70.5 71.lC 0.0 H20+H+ -165.9 46.2 H,O+ 78.2c -270.1d 0.0 0.0 t-N2H2+H+ 54.3 -89.5 2.1 -184.7 N2H3+ 52.2 0.0 -9.6 0.0 0.0 0.0 t-NzH2 6.4 6.4 52.2 6.4 -9.0 7.0 c-N~H~ 52.7 52.8 53.4 52.5 -12.2 49.9 N2H2 inv TS 63.6 64.5 80.9c 56.0 -5.8' 59.8 N2H+H 0.0 0.0 97.3 0.0 -1 1.6' 0.0 t-N2H2+H20 70.2 4.6 -12.6 3.6 t-N2H2*H20 -3.3 -3.5 76.5 6.6 -11.8 6.4 c-N~Hz'H~O 0.9 0.4 63.7 60.8 -30.4 42.0 N2Hz.Hz0 SS 52.0 50.8 50.8 49.8 63.9 59.8 -28.4 43.0 NzHzeH20 TSl 63.7 60.6 -26.5 45.7 N2H2*H20TS2 51.7 50.6 95.5 204.4 -197.5 18.5 N2H3+ + OH205.2 204.9 142.4 0.0 -13.6' 0.0 0.0 0.0 t-N2H2+2HzO 86.2 8.5 -14.9 7.2 t-NzH24H20)2 -8.1 -8.3 86.0 11.8 -14.1 11.3 4.8 -5.0 c-N~H~W~O)~ 26.9 25.1 73.8 45.6 -18.0 41.2 N2H2*(H20)2TS The solvation free energy, AG,I,, is calculated with the A M l S M 2 methcd30g35*41 at fixed MP2/6-31G(d) geometries. The experimental value ) ~ ~used for water rather than the A M l S M 2 value o f d . 3 of the free energy change at 298 K for the reaction HzO(g) HzO(l) (-2.0 k ~ a l / m o lwas kcal/mol. The entropies of H and H+ are 27.4 and 26.0 kcal/m01.~~The free energy of solvation of a proton (H+(g) H+(aq) is -260.5 kcal/mol at 298 K.57 The solvation free energy of the hydrogen atom (H(g) H(aq) was taken to be 2.1 kcal/m01.~~
-
-
SCHEME I Thermal Cycle to Determine Free Energy (kcallmol) of Solvatlon
-159.4 ab lnltlo
H*(g)
+
H2W)
-2.0 exptl
-260.5 exptl
I
I
H")
H30W
+
H20V)
-104.2 AMI-SMZ
-
I
-1.1
I
H3O'(aq)
references can be found in several recent s t ~ d i e s . ~ *The -~~ calculated association energy (no zero-point correction) is 5.8 kcal/mol, which is slightly higher than a recent estimate based on higher-level calculations (5.1 k c a l / m ~ l ) .A~ ~study has been made by Bertran et al.64of the water dimer in liquid water using the self-consistent reaction field model. As their solvation free energies do not include the cavitation or dispersion terms, direct comparison is difficult. In addition, their discussion is focused on the interaction energy in solution, which is the electronic energy plus the solvation free energy, while the present discussion is in terms of free energies in solutions. However, both studies concur that solvation should stabilize the water dimer. The free energy of dimerization (2H20 (H20)2) in the gas phase at 298 K is unfavorable due to a large negative entropy of association. When the solvation free energies are included, the dimer is bound by a free energy of 3.0 kcal/mol. It must be remembered that the dimer represents oneof four strong hydrogen-bonding interactions that each water molecule experiences in liquid water. One interaction is calculated explicitly, while the other three are modeled implicitly by AM 1-SM2 solvation energies. A second test of the hybrid model was made by calculating the free energy of eq 9. In aqueous solution, H+(aq) and H3O+(aq)
-
H+(aq) + H,0(1)
-
H,O+(aq)
(9)
have the same meaning (as do HsOz+(aq),H703+(aq), H904+-
-
(aq), etc.) since there is no intent to specify the degree of aggregation in solution. Therefore, the free energy change for eq 9 should be zero. A thermochemical cycle can be constructed consisting of ab initio, A M l S M 2 , and experimental values (Scheme I) which gives a free energy change of -1.1 kcal/mol (Table 111),close to the expected value. What are the solvation effects of mechanism 3-6, and in particular, what is the solvation effect on the cis/trans energy difference? Table I11 tabulates the calculated free energies of the various species. Since the trans isomer has a dipole moment of zero by symmetry while the cis isomer has a significant dipole moment (3.40 D, HF/6-31+G(d,p)), it might be expected that solvation effects would reduce the difference. However, solvation effects actually favor the trans isomer, probably due to the contribution of the quadrupolar moment. An analogous conclusion was drawn toexplain thesimilar solvation effects observed for trans- and cis-1 ,Zdichloroethene, where the trans isomer also has a dipole moment of zer0.65366 Solvation free energies also favor the trans isomer with one or two explicitly coordinated water molecules. But as noted above, the explicit water molecules stabilize the cis isomer relative to the trans. Withoneexplicit water molecule, the freeenergy difference is 2.8 kcal/mol, and with two explicit water molecules, the difference is 4.1 kcal/mol. If the spread in values reflects the accuracy of the method, then the free energy difference between cis and trans is 4.9 f 2.1 at 298 K. It is worthwhile noting that the geometry of diazene optimized with one or two explicit waters is an approximation of the relaxed solution-phase geometry. The calculated solution-phase free energy of isomerization may vary with the number of waters complexed because there is no compensating change in the free energies of solvation which are determined by AM 1-SM2 without geometry relaxation. Mechanisms 3 (inversion) and 4 (cleavage/recombination) still have significant free energies of activation of 49.9 and 59.8 kcal/ mol, respectively, in solution. The observed competition between inversion and decomposition in solution for substituted diazenesIg is likely due to the fact that the N-R bond strength is weaker. Trans to cis isomerization catalyzed by one water (mechanisms 5a,b) is predicted to occur by a two-step mechanism in the gas
The Journal of Physical Chemistry, Vol. 97, No.51. 1993 13613
Catalyzed Cis/Trans Isomerization of Diazene
TABLE I V Summary of Activation Barriers (kcal/mol) of Trans ~~
reaction t-NzH2 C-NZHZ t-NzHz HNz+H C-NZHZ t-NzHz+HzO C-NZHZ+HZO t-N2H2+HzO NzH~+OH t-NzHZ+HzO N2H3+ + OH-+ t-N2H2+2Hz0 c-NzHz+~H~O t-N;H;+H,O+ c - N ~ H ~ + H ~ O +
- -
+
+
-
+ +
0
-
Cis Isomerization of N2H2 in the Cas and Solution P
k
~
gas phase AH' (298 K) 52.8 64.5"
mechanism inversion cleavage/recombination cat. lHzO concerted cat. lHzO stepwise cat. 1H20 ionic cat. 2HzO concerted proton transfer
solution phase AG*(298 K) 49.9 59.8"
42.0 45.7 18.5" 41.2
50.8
49.8 204.9" 25.1 25.2"
4.9
* 2.1"
Assuming no reverse activation barrier.
phase (TS1 and TS2) rather than a one-step mechanism (SS). However, the enthalpy difference between TS2 and SS is only 0.2 kcal/mol at 298 K. Solvation effects favor SS over TSl and TS2 (possibly due to greater local charges in the concerted transition state) sufficientlyto givea lower free energy of activation for the one-step process (42.0 kcal/mol at 298 K). Mechanism 5c is stabilized substantially because two ionic species, N2H3+ and OH-, are formed. However, even with a calculated solvation free energy of -1 97.5 kcal/mol, the reaction is still predicted to be endothermic by 18.5 kcal/mol (Table 111). Under the observed reaction conditioqz2isomerizationvia 5c is predicted to be slower than the measured rate of reaction. When two waters participate in the isomerization,the enthalpy of activation is 25.1 kcal/mol (at 298 K), a reduction of 25.7 kcal/mol compared to the case for one catalytic water. However, the much greater entropy of the transition state and the smaller solvation free energy greatly destabilize the two-water pathway relative to the one-water pathway. At 298 K the two-water catalyzed pathway (AG*= 41.2 kcal/mol) is favored, but by only 0.8 kcal/mol. Using the same procedure, the free energy change for eq 10 is calculated to be. 2.1 kcal/mol (Table 111). In acid/base H+(aq)
+ rrans-N,H,(aq)
-
N,H,+(aq)
(10)
terminology, in aqueous solution, H20 is a stronger base than trans-NzHz by 3.2 kcal/mol (or equivalently 2.4 pKb units). Another simple calculation is the pK, of NzH3+ in solution which is -1.5 (AG = 2.1 kcal/mol for the reverse of eq 10). Of course, it should be recognized that the calculated solution-phase free energy changes may be in error by several kcal/mol, which will have a significant effect on the pKa or pKb. Deprotonation of NzH3+ can give either ds-NzH2, which is slightly uphill, or rrans-NzH2,which is slightly downhill (Figure 2b). Thus, rapid proton exchange according to eq 11 will yield H,O+(aq)
+ trans-N,H,(aq)
present results will encourage further experimental work to determine conditions under which isomerization is rate-determining.
Conclusions Accurate ab initio calculations have been applied to the determination of the trans to cis isomerization for diazene. All mechanisms considered in the gas phase appear to be unable to rationalize the rapid isomerization which takes place in solution. When ab initiogas-phase free energies arecombinedwith solvation free energies calculated by the A M l S M 2 method at fixed ab initio geometries,one mechanism is feasible, namely rapid proton exchange with the solvent. The solution-phase free energy of protonation of trans-NzHz is only 3.2 kcal/mol smaller than that for water, making it a slightly weaker base. Rapid proton transfer from trace acid can form NzH3+ which deprotonates to give the cis and trans isomers in an equilibrium amount based on the free energy difference which is 4.9 i 2.1 kcal/mol at 298 K. The water-catalyzedisomerizations are predictedto have much higher free energy barriers.
Acknowledgment. Computer time for this study was made availableby the Alabama Supercomputer Networkand theNSFsupported Pittsburgh Supercomputer Center. I would like to thank Dr. Christopher Cramer, Dr. George Ford, and Mikhail Glukhovtsev for helpful discussions and Dr. David Stanbury for providing a preprint of his experimentalwork on diazenereactions and for his insight into the kinetics of isomerization.
-
Appendix: Comparison of Obsened RatG2 to Predicted Rate of Trans Cis Isomerization at 0.001 M t-NzH2 and pH = 5
+
k-i
(1)
-
K = kL - 3 X lo-' k-1
+
HzO(1) + N2H3+(aq)+ H,O+(aq)
ki
+
t-N2H2 H30+ N2H3+ H 2 0 AGO = 2.1 kcal/mol
+ cis-N,H,(aq)
(1 1)
an equilibrium mixture of cis- and truns-NzH2 in proportion to their free energy difference (Figure 2). A summary of the gasphase and solution-phaseactivation barriers is given in Table IV. In the experimental study of the dismutation reaction,22the same rate was observed in solutions of pH = 1.5 and pH = 5.1. While this could be taken as evidence against an acid-catalyzed mechanism, since changing the concentration of H3O+ by over 3 orders of magnitudedoes not change the rate, it may also indicate that proton transfer is not the ratelimiting stepevenat thesmallest H+ concentration (pH = 5.1). A rough estimate of the relative rates of isomerization is made in the Appendix under experimental conditionsz2(0.001 M diazene at pH = 5 ) and compared with the observedz2rate constant of 4 X 104. The isomerization rate is predicted to be about 100 times faster than the observed rate and, thus, should not be rate-determining. It is hoped that the
k-,
N
10''
(from free energy difference) (2)
(pseudo-first-order diffusion-controlled
k,
-
rate constant4*) (3)
3
x
lo8
(4)
isomerization rate at pH = 5 = k,[H,O+] [t-N2H2] ( 5 )
= (3 x 10~)(10-~)(10-~) = 3 M s-, observed rate22 = k[N2HJ2
= (4 X 104)(10-3)2= 4
X
(6)
lo-' M s-l
Conclusion: is0 rate >> obs rate22
References and Notes (1) Saljoughian, M.;Williams, P. G.; Morimoto, H.;Goodlett, D. R.; Breemen, R. B. v. J . Chem. Soc., Chem. Commun. 1993,414.
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