1118
J. Phys. Chem. 1996, 100, 1118-1128
Curve Crossing Formulation for Proton Transfer Reactions in Solution Daniel Borgis* Laboratoire de Physique The´ oriques des Liquides, UniVersite´ Pierre et Marie Curie, 75252 Paris Cedex 05, France
James T. Hynes* Department of Chemistry and Biochemistry, UniVersity of Colorado, Boulder, Colorado 80309-0215 ReceiVed: August 3, 1995; In Final Form: October 18, 1995X
A Landau-Zener curve crossing formulation is developed for proton transfer rate constants over the range of proton coupling. This approach spans the weak coupling regime, where the transfer is a nonadiabatic proton tunneling, to the strong coupling regime, where at the reaction transition state the proton motion is a quantized vibration lying above the barrier in the proton coordinate. The theory requires that there is strong coupling of the proton transfer solute to the surrounding polar solvent and includes the influence of the vibrational motion of the heavy particles between which the proton transfers. Analytic formulas are given for the rate constants in various limits of the proton coupling strength and the vibrational frequency of the heavy particle motion.
I. Introduction Perhaps the most striking regime of proton transfer reactions is the tunneling regime, with its nonclassical barrier penetration by the quantum proton.1-3 We have previously presented an analytic treatment of proton (and hydrogen atom) transfer tunneling reaction rate constants in solution, based on a dynamic and coupled description of the proton, the vibrations in the proton-containing solute, and the surrounding solvent.4-6 This theory was shown both to provide a route for proton transfer rate constants via molecular dynamics computer simulation and to provide accurate analytic formulas for those rate constants.6 Recently, this tunneling regime has been the subject of various theoretical7-15 and simulation studies.16-19 The tunneling regime is the limit of weak proton coupling, in which reaction can be viewed as a nonadiabatic transition between localized reactant and product states. The reaction rate constants depend quadratically on the proton coupling, whose small value arises from the small overlap of the tails of the (diabatic) proton wave functions localized in the reactant and product wells in the proton coordinate.4-6 But, there are important reaction classes where the proton coupling is larger, the tunneling limit does not apply, and an alternate description is required. These conditions are met, e.g., for proton transfers along strong symmetrical H bonds in solution.20,21 Another important reaction class is that of acid ionization in solution2,3
AH‚‚‚B f A- ‚‚‚HB+
(1.1)
Here, the reactant is an intermolecular hydrogen bonded complex between the acid AH and the base B, characterized by a hydrogen bond of moderate to large strength. The hydrogen bonding leads to an equilibrium value of the intermolecular A-B separation Q which is moderate to small in magnitude. (Intramolecular complexes typically have larger equilibrium Q values, due to the geometrical constraints imposed by the molecular framework.) This in turn is responsible for proton coordinate barriers of reduced height, increased overlap X
Abstract published in AdVance ACS Abstracts, December 15, 1995.
0022-3654/96/20100-1118$12.00/0
of the localized reactant and product proton vibrational wave functions, and increased proton coupling, compared to the tunneling regime. Acid-base proton transfer systems such as phenol-amine complexes in dipolar aprotic solvents22-25 and the acids HCl and HF in water26,27 provide examples. Such proton transfer systems are also typically characterized by large charge transfer, e.g., the contact ion pair product complex in eq 1.1, and strong electrostatic coupling of the reactant solute pair with surrounding polar solvent molecules. Thus, extensive solvent reorganization is involved in the reaction and is the source of an activation free energy for the rate.28,29 In the present paper, we develop an analytic approach for the proton transfer rate constant over the entire range of proton coupling, ranging from the weak coupling, nonadiabatic tunneling limit up to the strong coupling, adiabatic regimeswhere the reaction transition state motion of the proton is a quantized vibration above the barrier in the proton coordinate, through the span of intermediate coupling values. Our perspective is couched in a Landau-Zener curve crossing approach, which is a protonic analogue of the well-known electronic curve crossing problem.30,31 Early important efforts along these lines were undertaken by German et al.28 Our theory differs from these efforts by the fact that we adopt from the beginning an electronically adiabatic perspectivesin terms of an electronic Born-Oppenheimer surface describing the nuclear force field, proton includedsrather than a diabatic one; this is an appropriate representation for proton transfer reactions, with a usually large diabatic electronic coupling, so that no electronic curve crossing is involved.32 Furthermore, we consider the coupling of the reaction not only to the solvent but also to the internal elongation mode of the AH-B complex, together with its quantization. Finally, our approach is built on microscopic time-dependent correlation function formulas so that fully microscopic calculations based on, e.g., molecular dynamics simulations, are a natural and indeed feasible application, over the whole proton coupling rate.6,24-25,33 Along similar lines, nonadiabatic surface hopping algorithms have been proposed recently for studying proton transfer dynamics in solution.34 A quite different theoretical route also able to span the whole proton coupling range, from weak to strong, is the quantum path-integral method, © 1996 American Chemical Society
Curve Crossing Formulation of Proton Transfers
Figure 1. Vacuum proton potentials for a symmetric reaction for a range of values of the AB internuclear separation Q: (a) large Q, (b) intermediate Q, and (c) small Q; (s) adiabatic proton vibrational eigenlevels and (- -) the corresponding diabatic levels. In c, two different situations are exhibited (see the text).
either formulated in terms of the proton centroid coordinate7-19,18 or of the quantized solvent energy gap coordinate.10,19 We consider only reactions from the ground proton vibrational state and assume that the reaction asymmetry is sufficiently small that no excited proton vibrational states are produced. We restrict the description to the limit of strong coupling to the solvent, such that there is a barrier in the solvent coordinate; as noted above, reactions with moderate to strong proton coupling will usually be so characterized. The theory also accounts for the important Q coordinate describing the motion of the heavy nuclei between which proton transfer occurs, e.g., A and B in eq 1.1. We treat the span of Q vibrational frequencies ranging from lowsmost relevant to our major focus of intermolecular complexessto highsmost, although not exclusively, relevant to intramolecular complexes, always provided that there is strong coupling to the solvent. The outline of the paper is as follows. In section II, we give an overview of the proton coupling regimes, including the role of the Q vibration and the solvent. We begin the curve crossing formulation in section III, which focuses on the nonadiabatic, weak coupling tunneling regime. The agreement with rate constant results obtained by a completely different method4,5,25,33 establishes the validity of the curve crossing approach. This provides a “launching point” for a curve crossing formulation over the entire proton coupling range, given in section IV, where the Q vibration is described classically. Quantized Q vibration is treated in section V in the proton coupling regimes most relevant in real reaction systems possessing high Q vibrational frequencies. Concluding remarks are offered in section VI. II. Proton Coupling Overview It is useful for perspective to begin with a brief overview of the proton coupling and the variation between the weak coupling, nonadiabatic regime and the strong coupling, adiabatic regime. We begin by considering for simplicity a proton transfer system which is symmetric in the vacuum. We illustrate in Figure 1 the qualitative behavior of the proton potential and the associated quantized proton vibrational levels at different values of the vibrational coordinate Q. At large separations Q (Figure 1a), the proton barrier will be high and passage between the wells occurs via quantum nonadiabatic tunneling. The diabatic states localized in each well are weakly coupled to produce two nearly degenerate proton eigenstates. For intermediate Q values (Figure 1b), the barrier will be lower and less thick, and tunneling will be enhanced, resulting in a larger splitting 2C of the proton eigenlevels. Finally, at small Q (Figure 1c), the proton barrier can be lowered enough that either one or both of the proton vibrational eigenstates are above the barrier in the proton coordinate. The latter situation in particular is in the adiabatic limit.
J. Phys. Chem., Vol. 100, No. 4, 1996 1119
Figure 2. (a) Proton diabatic coupling C versus the AB internuclear separation Q for a semiempirical symmetric OH-O potential:36 (solid line) exact diagonalization results; (dotted line) eq 2.2 with C1 ) 1.36 kcal/mol, R1 ) 9.4 kcal/(mol‚Å), and R1 ) 2.4 Å. (b) Same for the logarithm of C: (solid line) exact diagonalization results; (dotted line) eq 2.1 with C0 ) 8.2 × 10-4 kcal/mol, R ) 27.9 kcal/(mol‚Å), and R0 ) 2.8 Å.
In the above simple symmetric case, the splitting between the proton vibrational eigenlevels is twice the proton coupling C and ranges from small values for large Q and large values for small Q. In the first situation (Figure 1a), the coupling is well described by an exponential function in Q4,5,35
C(Q) ) C0 exp[-R(Q - Q0)]
(2.1)
with Q0 an equilibrium position, reflecting an exponential decline with increasing Q of the overlap of the (diabatic) proton wave functions largely localized in the separate proton wells. The decay parameter R is in the range 25-35 Å-1, much larger than its analogue in electron transfer problems (≈1 Å-1) due to the larger proton mass. Equation 2.1 is often an adequate description for intramolecular proton transfer complexes with large equilibrium Q values. In the opposite limit of sufficiently small Q where one or both proton eigenstates are above the barrier, the splitting varies much less rapidly with changing Q and is approximately linear:
C(Q) ) C1 - R1(Q - Q1) + ...
(2.2)
On the one hand, this reflects the absence of a tunneling mechanism with its strong sensitivity to wave function overlap. On the other hand, it reflects the slowly increasing level gap for vibrational levels more and more above a diminishing central barrier. In Figure 2 we display the accuracy of the simple limiting formulas, eqs 2.1 and 2.2, when compared to exact diagonalization results (C is half of the calculated proton eigenstate splitting) for a semiempirical Q-dependent proton potential model.36 This figure illustrates quantitatively the trends discussed qualitatively above. Even for a proton transfer system that is symmetric in the gas phase, the electrostatic or hydrogen bonding interaction with a polar solvent will make the solute proton transfer potential asymmetric in, e.g., equilibrium configurations of the solvent. Figure 3a illustrates the situation for the weak proton coupling limit. In Figure 3a(i), the solvent is equilibrated to the reactant state, and no tunneling is possible, due to the solvent-induced asymmetry of the proton potential. However, a solvent fluctuation can restore the proton asymmetry (Figure 3a(ii)), and tunneling can occur. This fluctuation costs free energy and in Figure 3a(iii) is the cost to reach the intersection of the two diabatic solvent curves for the reactant (R) and product (P), which are simply the (free) energy level values for the R and P proton states. For weak coupling, these solvent curves are negligibly split at their intersection. Figure 3b portrays the
1120 J. Phys. Chem., Vol. 100, No. 4, 1996
Borgis and Hynes
HR ) 1/2mSS˙ 2 + 1/2kSS2 + 1/2mQQ˙ 2 + 1/2mQωQ2δQ2
(3.1)
in which the solvent force constant kS ) mSωS2 is directly related to the solvent reorganization energy ES,
kS ) 2ES
(3.2)
The deviation of the heavy particle oscillator coordinate Q, with mass mQ and frequency ωQ, from its equilibrium value QR in the R state is δQ ) Q - QR. The product diabatic Hamiltonian is (Figure 4)
HP ) 1/2mSS˙ 2 + 1/2kS(S - 1)2 + 1/2mQQ˙ 2 + Figure 3. Free energy curves illustrating the solvent influence for weak proton coupling (a) and strong proton coupling (b). In each case, (i) represents the proton potential versus the proton coordinate qH for solvation appropriate to the reactant solute state and (ii) that appropriate to the transition state. Curves iii represent the corresponding free energy curves in the solvent coordinate S.
/2mQωQ2(δQ - ∆Q)2 + ∆ES + ∆EQ (3.3)
1
where we have assumed for simplicity that the R and P state Q-vibrational frequencies are the same. The reaction asymmetries associated with the solvent and oscillator coordinates are ∆ES and ∆EQ, respectively. The solvent coordinate S has been defined so that its equilibrium position is SP ) 1 in the product and SR ) 0 in the reactant state. ∆Q ) QP - QR is the shift in the oscillator equilibrium position. The Q-oscillator reorganization energy is
EQ ) 1/2mQωQ2∆Q2 Figure 4. Definition curves for the asymmetries ∆ES and ∆EQ and reorganization (free) energies ES and EQ for (a) the solvent coordinate S and (b) the vibrational coordinate Q.
corresponding situation in the strong proton coupling limit. The splitting of the two adiabatic proton levels in the symmetric solvent configuration (Figure 3b(ii)) is now appreciable and is reflected in the splitting of the solvent free energy curves in Figure 3b(iii). Now there is a smooth lower adiabatic curve in the solvent coordinate, and the activation free energy is lowered, compared to the diabatic case, by the proton coupling C. Parts a(iii) and b(iii) of Figure 3, although they are simplified to the vacuum symmetric case and the vibrational coordinate was not considered explicitly, indicate that the proton transfer reaction can be considered as a curve crossing problem. We now pursue this in the remaining sections. III. Curve Crossing Approach for Weak Proton Coupling In this section, we set up a Landau-Zener curve crossing formulation for the proton transfer rate constant in the weak proton coupling limit. In this regime, the proton transfer act is a nonadiabatic event between the localized reactant (R) and product (P) states. We will confirm, by inspection of the two important limits of low and high temperature, βpωQ . 1 and βpωQ , 1, respectively, that the analytic results of refs 4 and 5 obtained by a quite different dynamical approach are recovered. (Here, ωQ is the Q-vibration frequency, and β ) (kBT)-1.) In addition, this nonadiabatic regime curve crossing formulation serves as a launching point for the curve crossing rate constant formulations over the complete proton coupling range to be described in subsequent sections. 3.A. Hamiltonian Description. We begin with the definitions of the diabatic Hamiltonia for the R and P states.4,5 The reactant Hamiltonian is (Figure 4)
(3.4)
and for symmetry with respect to the solvent definition eq 3.2, we will define an apparent oscillator force constant
kQ ) 2EQ
(3.5)
This is not the same as the true Q force constant mQωQ2, because of the differing dimensions of the solvent and oscillator coordinates in eqs 3.1 and 3.3, but no confusion should arise. An important role in all that follows will be played by the energy asymmetry gap
∆H ) HP - HR ) ∆ES + ∆EQ + ES + EQ - kSS - [2mQωQ2EQ]1/2δQ (3.6) which is linear in both the solvent and oscillator coordinates. Fluctuations in S and Q modulate the symmetry of the proton transfer system and are critical in determining the rate. This has been extensively described elsewhere.4-6 The above Hamiltonia are either classical or quantum (in the operator sense). It is important for further developments to have their quantized Q-oscillator versions, which are readily obtained with the quantization rule for the oscillator components HδQR,P of HR and HP, eqs 3.1 and 3.3,
HδQR,P f EVR,P ) (VR,P + 1/2)pωQ
(3.7)
with associated vibrational wave functions ψVR,P(Q). Finally, in the weak coupling limit of this section, the appropriate form of the proton coupling C is the exponential form, eq 2.1. As noted in the Introduction, the weak coupling limit is most likely to apply for intramolecular proton transfer complexes where the heavy particle vibration frequency ωQ is likely to be sufficiently high that a quantum treatment is appropriate: βpωQ > 1. We proceed now from this basic perspective, and recover the classical Q-vibration limit as a special case.
Curve Crossing Formulation of Proton Transfers
J. Phys. Chem., Vol. 100, No. 4, 1996 1121
The definition diagrams in Figure 4 can be used to set up the problem. When the R and P Q-vibrations are quantized according to eq 3.7, the resulting picture in the classical solvent coordinate is given in Figure 5, in which the effective potentials in the solvent coordinate for the R and P states are, with VR ) n and VP ) m,
VR,n(S) ) VR(S) + (n + 1/2)pωQ, VR(S) ) 1/2kSS2 VP,m(S) ) VP(S) + (m + 1/2)pωQ, VP(S) ) 1/2kS(S - 1)2 (3.8) so that the R and P proton state curves are now offset by the total asymmetry
∆E ) ∆ES + ∆EQ
(3.9)
3.B. State-to-State Nonadiabatic Rate Constant. We first consider the weak proton coupling, nonadiabatic rate constant knm for a transition between Q-vibrational state n in the reactant to Q-vibrational state m in the product (cf. Figure 4). Inspection of Figure 5 shows that we can write an expression for knm as a transition state theory form. In particular, knm can be expressed as the average one-way flux in the solvent coordinate through the crossing point Snm of the two free energy curves for the n and m vibrational states, with the inclusion of wc giving the probability of a the transmission coefficient κnm successful curve crossing: wc (S˙ ,Snm)〉R knm ) 〈S˙ θ(S˙ )δ(S - Snm)κnm
(3.10)
Here the average is over the classical solvent distribution, normalized by the partition function of the solvent in the reactant region:
∫dS dS˙ e-βH (S,S˙ )(...) 〈(...)〉 ) ∫dS dS˙ e-βH (S,S˙ )
Figure 5. Diabatic free energy curves in the solvent coordinate S for several reactant (R) and product (P) vibrational states, with the overall reaction asymmetry ∆E indicated.
γnm )
2πCnm2 p(∂∆Vnm/∂S)SnmS˙
2πCnm2 pkSS˙
(3.14)
where ∆Vnm is the gap Vm - Vn and includes multiple pass effects on the transition probability.30,31 (Note that κnm f 1 in the strong coupling, adiabatic limit.) For weak proton coupling, γnm , 1, and we obtain the desired nonadiabatic limit result wc ) 2γnm κnm
(3.15)
The combination of eqs 3.10, 3.11, and 3.16 then gives the state-to-state rate constant as
knm )
2πCnm2 〈δ(S - Snm)〉R pkS
(3.16)
where we have used the fact that the thermal average of the velocity step function is 〈θ(S˙ )〉 ) 1/2. The remaining configurational average is straightforward with eq 3.11, and with eq 3.13 we find
knm )
R
)
]
(3.17)
1 (E + ∆E + ∆Enm)2 4ES S
(3.18)
[( )
β 2π 2 C p nm 4ESπ
1/2
e-β∆G nm q
R
HR(S,S˙ ) ) 1/2mSS˙ 2 + VR(S)
q is the activation free energy in which ∆Gnm
(3.11)
(Note that the constant Q-oscillator energy in eq 3.8 cancels in the average.) The location of the crossing point Snm in the solvent coordinate is given by the condition (cf. Figure 5 and eq 3.8)
VR(Snm) ) VP(Snm) + ∆Enm + ∆E ∆Enm ) Em - En
(3.12)
where ∆Enm is a vibrational asymmetry, so that
Snm )
(ES + ∆E + ∆Enm) 2ES
(3.13)
In order to find the appropriate nonadiabatic transmission wc for use in eq 3.10, we appeal to the coefficient factor κnm general Landau-Zener (LZ) transmission coefficient κnm,30,31 adapted for the present problem, and then take the weak proton coupling limit. The LZ factor, appropriate for a positive velocity approach to the crossing point, is
κnm ) [1 - 1/2 exp(-γnm)]-1[1 - exp(-γnm)]
q ) ∆Gnm
This result has the simple structure of the Golden Rule with the thermally solvent-averaged Franck-Condon factors given by the bracketed term. 3.C. Proton Transfer Tunneling Rate Constant. The thermal tunneling rate constant is given by the summation of the knm over the final Q-vibrational states m and the thermal average over the thermal populations Pn ) exp(-βEn)/∑n exp(-βEn) of the initial vibrational states:
k ) ∑∑Pnknm
( ) ∑∑
n
)
m
β π
p βES
1/2
n
PnCnm2e-β∆G nm q
(3.19)
m
To proceed to the explicit evaluation of k, we observe that, in the present nonadiabatic regime, the proton coupling is weak, and the exponential form C(Q) ) C0 exp[-R(Q - Q0)] is appropriate (cf. eq 2.1). The required coupling matrix elements can be computed,37 with the result
1122 J. Phys. Chem., Vol. 100, No. 4, 1996
Borgis and Hynes
([( ) ( ) ] ) [ ( )] ([( ) ( ) ] ) [ ( )] ER 1/2 pωQ n-m m! n-m EQ - ER L n! m pωQ
Cnm2 ) C02e-R∆Qe(ER-EQ)/pωQ EQ pωQ
1/2 2
ER pωQ
Cnm2 ) C02e-R∆Qe(ER-EQ)/pωQ EQ pωQ
1/2
2
, m e n (3.20)
+
n! m-n EQ - ER L m! n pωQ
1/2 2 m-n
2
, mgn
In this formula,
ER ) p2R2/2mQ
(3.21)
is a quantum energy associated with the coupling Q vibration,4,5 and L is the Laguerre polynomial.37 Equations 3.19 and 3.20 formally give the tunneling rate constant in the weak proton coupling, nonadiabatic regime and could be numerically evaluated. Since the nonadiabatic rate constant is already available from a different perspective,4-6 here we content ourselves with examining only two limits to establish that the curve crossing formulation correctly reproduces the available results: the “lowtemperature” quantum Q-vibration regime, βpωQ . 1, and the “high-temperature” classical Q-vibration regime, βpωQ , 1. Since the solvent is always assumed to be both liquid and classical, these two regimes are largely (although not completely) distinguished by high and low Q-vibrational frequencies. In the quantum regime βpωQ . 1, eq 3.19 reduces simply, when the reaction asymmetry magnitude |∆E| (cf. eq 3.9) is less than the solvent reorganization energy ES, to
( ) [
π β k ) C002 p βES
1/2
exp -
]
β (∆E - ES)2 4ES
(3.22)
where the (0,0) square coupling matrix element is
C002 ) C02e-R∆Qe(ER-EQ)/pωQ
(3.23)
In this result,4-6 the rate constant contains a familiar Dogonadze-Levich-Marcus38,39 term governing the probability that the solvent achieves the activation free energy (∆E + ES)2/4ES ) ∆Gq00. The proton characteristics enter solely via the prefactor C002, associated with the transition between the ground-R to ground-P Q-vibrational states. Other transitions are precluded by the conditions βpωQ . 1 and |∆E| < ES. The contribution to k from, e.g., the first Q-vibrationally excited state n ) 1 can be easily found from the formulas above. The high-temperature regime βpωQ , 1 rate constant follows from eq 3.19 via a lengthy analysis summarized in Appendix A. The result is
k)
( ) [
β〈C 〉 π p βEtot. 2
1/2
exp -
(
β ∆E + Etot. + 4Etot. 4 (E E )1/2 βpωQ Q R
)] 2
(3.24)
in which Etot. ) ES + EQ + ER and the thermally averaged square coupling is
[ (
〈C2〉 ) C02 exp
βpωQ ER 4 + + O((βpωQ)2) pωQ βpωQ 3
)]
(3.25)
This result, previously obtained by a quite different method, is discussed at some length in refs 4 and 5. Here, we simply note that the apparent activation free energy in eq 3.24 is now determined both by the solvent and the Q vibration but is not related to the proton barrier height. In addition, the apparent non-Arrhenius T behavior arising from the average square coupling 〈C2〉, eq 3.25, is in practice quite weak for typically realistic parameter values5 so that, for all practical purposes, k displays Arrhenius behavior, despite the fact that the intrinsic reactive event is quantum proton tunneling. It is also worth pointing out that eq 3.24 is not strictly a classical limit (p f 0), due to, e.g., the appearance of the quantum coupling factor ER in Etot.; this feature will be relevant for the classical Q description in section IV. Finally, the state-to-state rate constants eq 3.17 could be used to discuss Q-vibrationally state selected experiments in solution and in clusters.40 IV. Curve Crossing Rate Constant Formulation over the Full Coupling Range: Classical Q Vibration As noted in the Introduction, the weak proton coupling, nonadiabatic limit will be inappropriate for many proton transfer reactions of interest. In particular, for proton transfers within intermolecular, hydrogen-bonded complexes, the heavy particle internuclear separation Q need not have a large value such as that enforced by the molecular skeleton of intramolecular complexes, and the attractive hydrogen-bonding interactions can lead to smaller equilibrium Q values, where the coupling C(Q) is no longer weak. It is therefore necessary to have a formulation capable of treating a range of C(Q) values, up to and including C(Q) magnitudes for which the proton adiabatic regime applies. Among such proton transfer systems, acid ionizations AH + B f A- + HB+, eq 1.1, are of special importance. The transfer will occur in a hydrogen-bonded complex to produce a contact ion pair, and the accompanying large charge displacement will lead to a large electrostatic coupling to the surrounding polar solvent, i.e., a large solvent reorganization energy ES. Examples include phenol-amine complexes in dipolar aprotic solvents, for which ES ≈ 15 kcal/mol,24 and HCl ionization in water, for which ES ≈ 13 kcal/mol.26 At the same time, the absence of an intramolecular framework will generally result in heavy particle intermolecular vibration frequencies ωQ on the low side, typically a few hundred cm-1 or less. (Exceptions will be discussed in section V.) In summary then, it is necessary to develop a rate constant formulation for this reaction class over a wide proton coupling range, with the conditions of strong solvent coupling and lowfrequency, classical Q vibrations:
βES . 1; βpωQ , 1
(4.1)
4.A. General Formulations. Before writing the rate constant appropriate over the entire coupling range and examining its limits, it will prove to be important to shift our perspective from Q and S to new energy variables. The motivation for this is provided by Figure 6, where it is seen that, from a diabatic perspective, the energy gap variable ∆H ) HR - HP, eq 3.6, renders the curve crossing problem one-dimensional. This important feature will then allow us to use a one-dimensional LZ formula for the curve crossing transition probability, a route not available in the two-dimensional (δQ, δS) formulation. The rate constant over the entire coupling range will then be
k ) 〈∆H˙ θ(∆H˙ ) δ(∆H - ∆Hq)κ(Q)〉R
(4.2)
the net one-way flux in ∆H, at the transition state location ∆H
Curve Crossing Formulation of Proton Transfers
J. Phys. Chem., Vol. 100, No. 4, 1996 1123 To begin then, we recast the diabatic Hamiltonian eqs 3.1 and 3.3 in the two energy variables
∆H ) ∆H0 - kQδQ - kSδS ∆HQ ) -kQδQ +
kQ ∆H kQ + kS 0
(4.5)
with ∆H0 ) ∆ES + ∆EQ + ES + EQ; see eq 3.6. The result is
HR )
KS
2ωS
∆H˙ +
[
KQ
2
2ωQ
]
2 2π C (Q) p ∆H˙
KS
F ) [∫dΓR e-βHAD]-1 exp(-βHAD)
∆H 1 - [(∆H)2 + 4C2(Q)]1/2 2 2
where we have defined two new force constants
KS ) kS-1 ) (2ES)-1; KQ ) kQ-1 ) (2EQ)-1
(4.7)
more suitable for the energy coordinates. Again, from eqs 4.24.7, the rate can be computed numerically by performing the double integral in the ∆H, ∆HQ variables in eq 4.2. To gain some insight, we now consider different limiting regimes where analytical results can be derived. 4.B. Nonadiabatic Limit. It is useful to check the above formulation in the nonadiabatic, weak coupling limit, where
in which the adiabatic Hamiltonian is
HAD ) HR -
)
q ) [4(ES + EQ)]-1∆H02 ∆GNA
(4.3)
which is the classical analogue of eq 3.14. The average is to be taken with respect to the adiabatic equilibrium distribution
∆H˙ Q2 +
HP ) HR - ∆H
) ∆Hq, incorporating the LZ transmission coefficient (κ(Q)) for successful passage to the product state,
γ(Q) )
(
2
KSKQ ∆H ∆H (4.6) KS + KQ 0
Figure 6. (a) Intersecting diabatic free energy surfaces in the S and Q coordinates. (b) Top view of a, with the direction of the energy gap coordinate ∆H indicated. (c) Free energy along ∆H, with the proton coupling C0 (at the transition state) producing a ground adiabatic free energy curve.
κ(Q) ) {1 - 1/2 exp[-γ(Q)]}-1{1 - exp[-γ(Q)]}
+
KS
∆H˙ ∆H˙ Q + 2ωS ωS2 KS KS + KQ q + ∆H2 + ∆HQ2 + KS∆H∆HQ ∆GNA 2 2 2
(4.4)
and ΓR denotes the phase space variables in the reactant region. HAD reduces respectively to HR for ∆H , 0 and to HP for ∆H . 0. Note that, at this stage, eq 4.2 is a general statistical formula, which makes it possible to compute the rate numerically and over the whole proton coupling range, from weak to strong. In the present model, the general form of the coupling C(Q) is known (see section II and Figure 2) and the calculation of k amounts simply to a double integral in S and Q. The applicability of the microscopic rate formula (4.2), in a molecular dynamics context, has also been demonstrated, for adiabatic or nonadiabatic (or intermediate) proton transfer reactions; see refs 24 and 25. In the context of the rate constant expression eq 4.2, the transition state location ∆Hq will vary with the proton coupling C(Q): in the nonadiabatic limit, ∆Hq ) 0, while, in the adiabatic limit, ∆Hq must be determined, in a fashion described below, from the surface eq 4.4. At this point, one might be concerned, in connection with eq 4.2, that ∆H is not the reaction coordinate in the adiabatic limit (which should in principle be determined by a normal mode analysis of HAD). For the moment, we will simply note that it is shown below that in fact ∆H remains the reaction coordinate in the adiabatic regime for the proton coupling values relevant for proton transfers with strong solvent coupling.
κ(Q) f 2γ(Q) )
2 4π C (Q) p ∆H˙
(4.8)
and we should recover the classical Q limit result of eq 3.24. In this regime, the transition state location is ∆Hq ) 0, the adiabatic Hamiltonian eq 4.4 reduces to its diabatic version, and the proton coupling has its limiting form C(Q) ) C0 exp[-RδQ] (cf. eq 2.1). A straightforward but lengthy calculation of eq 4.2, with eqs 4.6 and 4.8, then gives
k)
[
exp -
4π 〈θ(∆H˙ )〉R〈δ(∆H)C2(∆HQ)〉R p
[ ( )]( )(
ER β ) C02 exp p pωQ
(
πKSKQ 4 βpωQ KS + KQ
)
1/2
( )
×
4 β ∆E + ES + EQ (E E )1/2 βpωQ Q R 4(ES + EQ)
)] 2
(4.9) which is exactly eq 3.24 (with the neglect of the quantum term ER). The structure of the averages in eq 4.9 shows that this result has the simple basic structure
1124 J. Phys. Chem., Vol. 100, No. 4, 1996
k)
1/2 β 2π 2 q ) 〈C 〉∆H)0 (ES + EQ)-1 exp(-β∆GNA p 4π
[
]
Borgis and Hynes
(4.10)
involving the average square proton coupling at the nonadiabatic transition state. 4.C. Adiabatic Limit. In this limit, the proton coupling is sufficiently large that γ . 1 and κ(Q) ) 1 in eq 4.2. But, according to our basic focus, the solvent reorganization energy ES is large (.kBT). Anticipating the result below (eq 4.13) for the thermal one-way velocity average of ∆H˙ ) [(kBT/π)(ωS2ES + ωQ2EQ)]1/2 and ignoring the Q contribution for simplicity, an estimate of γ is
γ)
( ) ( )
2C π 1 (2βC) (βES)1/2 2 βpωS ES
(4.11)
With typical solvent frequencies ≈ 10 ps-1,24,25,41 solvent reorganization energies of a few kilocalories per mole and adiabatic proton transfer couplings C ∼ 1-2 kcal/mol (refs 24 and 26; see also Figure 2), the required conditions are readily met. With κ ) 1 then, the adiabatic limit of eq 4.2 for the rate constant is
k ) 〈∆H˙ θ(∆H˙ ) δ(∆H - ∆H )〉R q
(4.12)
and we can make some progress formally before explicit evaluation is required. The kinetic part of the average can be performed with eq 4.6, with the result
from which we conclude that, as anticipated at the beginning of section 4.A, the “unstable” reaction coordinate is indeed still ∆H in the adiabatic limit, given that the solvent reorganization energy is large, 4C0KS , 1. More precisely, at zeroth order in 4C0KS, the eigenvectors at the transition state (∆Hq, ∆HQq ) are the unstable mode δ∆H and the nonreactive transverse mode δ∆HQ. The corresponding force constants, valid throughout first order in 4C0KS, are
(
[
]
(ωS2ES + ωQ2EQ) πβ
〈δ(∆H - ∆Hq)〉R
(4.13)
)
where K denotes the kinetic energy contribution. We have defined the coupling derivatives Cn ) dnC/d∆HQ2|0 [C0,1 have here a different meaning than in sections II and III]. The first point to note is that, under the condition that (twice the) proton coupling is small compared to the solvent reorganization energy 2C0 , ES (or 4C0KS , 1), the matrix of second derivatives with respect to δ∆H ) ∆H - ∆Hq and δHQ ) ∆HQ - ∆HQq (cf. Appendix B) is
)
(4.16)
[(
∆HQq ) D-1 KS -
]
)
1 C - KSx 4C0 1
(4.17)
D ) (KS + KQ - C2)(KS - 1/4C0) - KS2; x ) [KSKQ/(KS + KQ)]∆E, which in the regime KS , (4C0)-1 reduces to
∆Hq = -4C0
[
KSC1 KSKQ∆E KS + KQ KS + KQ - C2
]
C1 KS + KQ - C2
(4.18)
1/2
q - C0 + HAD ) K + GAD ) K + ∆GNA 1 1 /2 KS ∆H2 + 1/2(KS + KQ - C2)∆HQ2 + 4C0 KSKQ∆E KS∆H∆HQ ∆H - C1∆HQ (4.14) KS + KQ
(
)
∆Hq ) D-1[(KS + KQ - C2)x - KSC1]
∆HQq =
It remains to find the configurational average, which will produce an activation free energy Boltzmann factor, together with entropic factors related, e.g., to configurational contributions transverse to the reaction coordinate at the transition state. We proceed by noting that since the coupling is not absolutely large, the transition state saddle point location (∆Hq, ∆HQq ) should not be shifted very much from its diabatic location (0,0), and a quadratic expansion of HAD in ∆H and ∆HQ can be performed. The algebra for this is relegated to Appendix B, with the result that
(
)
1 - KS 0 4C0 KS + KQ - C2 0
The maximum of the adiabatic free energy GAD in eq 4.14 is located at the shifted transition state values
k ) 〈∆H˙ θ(∆H˙ )〉R〈δ(∆H - ∆Hq)〉R
)
(
-
(
)
1 KS K 1 -1 0 4C0 S ≈ 4C0(KQ - C2) 4C 0 0 KS KS + KQ - C2 (4.15)
confirming that the coupling induces only fairly small shifts. For the adiabatic rate constant eq 4.13, we only need the free energy evaluated at ∆Hq, and consistent with the expansion eq 4.14 through second order, the free energy in the Hamiltonian HAD ) K + GAD at ∆Hq is q q (∆Hq,∆HQ) ) ∆GAD + 1/2(KS + KQ - C2) × GAD
(∆HQ - ∆HQq )2 (4.19) q ) GAD(∆Hq,∆HQq ) is the reaction activation free where ∆GAD energy, to be given below. The configurational average in eq 4.13 can now be carried out, to give
k)
[
ωS2ES + ωQ2EQ 1 2π ES - EQ - 2ESEQC2
]
1/2 q exp(-β∆GAD )
(4.20)
The prefactor (without the coupling gradient term) is directly analogous to that which appears in electron transfer theory.30 Its structure in the present problem may be comprehended in terms of the general formula for a TST rate prefactor42
ω|Rω⊥ 2πω⊥q
)
ωSωQ 2πω⊥q
(4.21)
where | and ⊥ indicate directions parallel and transverse to the reaction coordinate in the reactant and transition state regions. (The second member of (4.21) follows from the invariance of the frequency product to rotations.) The transverse TS frequency ω⊥q is that associated with the δ∆HQ motion. From eqs
Curve Crossing Formulation of Proton Transfers
J. Phys. Chem., Vol. 100, No. 4, 1996 1125 pωQ ) 1 in Figure 7. Well below this line, the Q vibration is fast compared to the interconversion of the proton diabatic vibrational states: pωQ > 2C. In this regime, a BornOppenheimer-like description43 is appropriate in which the Q vibration is fast enough to “see” the individual R and P diabatic proton vibrational states, even in the region of their intersection where they interconvert with a frequency proportional to the coupling between them. More precisely, the relevant proton coupling in this regime is C00 ) 〈0R|C(Q)|0P〉, i.e., the coupling matrix element between the R and P ground Q-vibrational diabatic states (see also eqs 3.22 and 3.23). Within this region then, we have a curve crossing problem in the solvent coordinate involving the two Hamiltonian HR,0 and HP,0 with
HR,0 ) 1/2mSS˙ 2 + VR,0(S) Figure 7. Proton coupling-Q-vibrational frequency diagram indicating the location of several regimes of proton transfer for quantized Q vibration. [See text.]
4.16 and 4.6, this frequency is
ω⊥q )
[
) ωSωQ
[
]
KS + KQ - C2 KQ/2ωQ2
1/2
]
ES + EQ - ESEQC2 ωQ2EQ + ωS2ES
1/2
(4.22)
which, with eq 4.21, reproduces the prefactor in eq 4.20. q ) GAD(∆Hq, Finally, the activation free energy ∆GAD q ∆HQ) is
[ ( )]
q q ) ∆GNA - C0 1 - 1/2 ∆GAD
∆Hq 2C0
2
-
(and the same for P; cf. eq 3.8) and the coupling C00. The rate constant is then
k00 ) 〈S˙ θ(S˙ ) δ(S - S00)κ00(S˙ ,S00)〉R
k00 )
( )
ωS q exp(-β∆GAD,00 ) 2π
The low-temperature, high-Q-oscillation frequency regime βpωQ . 1 has already been discussed in the weak proton coupling limit in section 3. Here, we consider the quantized Q-vibration situation more generally, for different regimes of the proton coupling. The discussion is best initiated by consideration of Figure 7, which characterizes the proton coupling and Q-vibrational frequency space through the variables 2C/kBT and pωQ/kBT, respectively. For perspective, the treatment of section 3 dealt with the proton transfer rate for small values of 2C/kBT, along the pωQ/kBT axis, while section 4 dealt with the rate for small values of pωQ/kBT, along the 2C/kBT axis. Our concern now is with the region of Figure 7, where pωQ/kBT is large, along the proton coupling axis 2C/kBT. For simplicity, we restrict our remaining discussion to the situation where the reaction asymmetry is sufficiently small (|∆E| < 2ES) that only transitions involving the ground reactant (VR ) n ) 0) and the ground product (VP ) m ) 0) Q-vibrational states are involved (as in section 3C). For reasons to be described in more detail presently, the coupling Q-frequency space is naturally divided by the line 2C/
(5.3)
q is the activation free energy derived from the where ∆GAD,00 adiabatic solvent Hamiltonian involving the C00 coupling
HAD,00 ) 1/2mSS˙ 2 + 1/2[HR,0(S) + HP,0(S)] - 1/2[[HP,0(S) HR,0(S)]2 + 4C002]1/2 (5.4)
/2(KS + KQ - C2)(∆HQq )2 (4.23)
V. Quantized Q Vibration
(5.2)
where the LZ transmission coefficient κ00 is given by eq 3.14 with n ) m ) 0; eq 5.2 is clearly the generalization of eq 3.10 for finite coupling. The limits of the rate constant eq 5.2 are, for weak proton coupling, eq 3.17 for n ) m ) 0, and for stronger proton coupling C00 (but still such that 2C00/pωQ < 1),
1
where it is to be understood that the leading order values eq 4.18 for ∆Hq and ∆HQq are to be taken. Even if the proton coupling is constant, the activation free energy is modified from q - C0 due to any finite a simple symmetric result ∆GNA reaction asymmetry ∆E.
(5.1)
where eq 5.1 gives the diabatic Hamiltonian. We now turn to the region well above the 2C/pωQ ) 1 dividing line in Figure 7, again for pωQ/kBT in the quantum regime. Here, the interconversion, at their intersection, of the two diabatic proton states is fast on the time scale of the Q-vibration frequency, and a self-consistent description will apply;43 the slower Q vibration will see, at and in the neighborhood of the crossing, a resonance mixture of the two coupled diabatic proton states. The reaction rate constant in this regime is
k00 )
( )
ωS q exp(-β∆GAD,sc,00 ) 2π
(5.5)
q has the where the effective activation free energy ∆GAD,sc,00 following meaning. The underlying Hamiltonian is proton adiabatic, but now at the level of both the solvent S and vibrational Q coordinates,
HAD,sc ) 1/2[HR(S,Q) + HP(S,Q)] - 1/2[[HR(S,Q) HP(S,Q)]2 + 4C2(Q)]1/2 (5.6) as in eq 4.4 The two diabatic Hamiltonia HR(S,Q) and HP(S,Q) are coupled by the proton coupling C(Q) at each Q value. Equation 5.6 defines, via
HAD,sc ) KS + KQ + GAD,sc(S,Q)
(5.7)
a two-dimensional free energy surface GAD,sc(S,Q) in S and Q.
1126 J. Phys. Chem., Vol. 100, No. 4, 1996
Borgis and Hynes
At any given S value, there is a minimum Qeq(S) value defined by ∂GAD,sc(S,Q)/∂Q|S ) 0. The quantization of Q can then be effected by a harmonic expansion in Q around Qeq(S) at each S up to quadratic order, and that harmonic Q potential can be quantized at each S value, to give for the ground vibrational state (labeled 00)
GAD,sc,00 ) GAD,sc[S,Qeq(S)] + 1/2pωQ,00(S)
(5.8)
The harmonic frequency will reduce to the simple constant Q-oscillator value ωQ in the R and P solvent wells but otherwise varies with the solvent coordinate. In particular, ωQ,00 will exceed ωQ in the neighborhood of the barrier top in GAD,sc[S,Qeq(S)]; for there, where HP - HR ≈ 0, the strong Qdependent coupling becomes very important in eq 5.6 and steepens the potential in the Q direction. Finally, the required activation free energy in eq 5.5 is the difference between the maximum value of eq 5.8 and the value in the reactant solvent well. All of this is illustrated in ref 24 via a molecular dynamics computer simulation of a model of the phenol-trimethylamine proton transfer in methyl chloride solvent. It seems appropriate to limit the presentation for quantized Q vibration to those cases described above. Thus, for example, cases in the borderline region 2C/pωQ ) 1 for a quantum Q vibration, while conceivable, do not seem very likely. For example, if the reaction is to be proton adiabatic with 2C/kBT . 1,
( )( )
2C kBT 2C ) pωQ kBT pωQ
(5.9)
very difficult for any relevant6 solvent dynamics to be slower than this. For strong proton coupling, the solvent frictional recrossing of the adiabatic reaction barrier top will, for barriers of modest to large height, be described by Grote-Hynes theory.45 Indeed, this has been shown for a molecular dynamics study of the model phenol-amine reaction in ref 24. This should be true even for adiabatic reactions when the solvent barriers are nearly cusped, in analogy to the electron transfer case.46 If instead the barrier is very low, of order kBT, as it evidently is for acid ionization in water,26 a diffusional description seems called for, but this awaits future analysis. Acknowledgment. This work was supported by NSF Grants CHE-88-07858 and CHE-93-12267 and an NIH Shannon Award, and a CNRS-NSF U.S.-France International collaboration Grant (UA0765-D01). This manuscript was completed while J.T.H. was an Iberdrola Invited Professor at Departamant de Quimica, Unita de Quimica Fisica, Universitat Autonoma de Barcelona. Appendix A In this appendix, we sketch the derivation of the hightemperature regime βpωQ , 1 result of eq 3.24 starting from q given by eq 3.18, Cmn2 given by eq 3.20 eq 3.19 with ∆Gmn and with the equilibrium Q-oscillator reactant state distribution given by Pn ) (1 - e-βpω)e-βpω, where we drop the “Q” subscript in ωQ for ease of notation. We present only the case where the asymmetry ∆E ) 0 and the Q-reorganization energy EQ ) 0 to simplify the algebra. It is convenient to define
( ) ( ) [ ( )]
can only be ≈1 if pωQ/kBT is quite large. But this is only plausible (see the Introduction) for intramolecular complexes, and there the proton coupling cannot be large.
βC02 2π A) p βkS
VI. Concluding Remarks
cnm ) C0 exp
In this paper, we have presented a curve crossing formulation for the rate constants of proton transfer reactions. The coupling of the proton transfer solute to the surrounding solvent is supposed to be strong enough that there is a barrier in the solvent coordinate which provides at least in part the activation free energy of the reaction. The rate constant is found over the entire proton coupling regime, from weak to strong, when the Q-vibrational motion of the heavy particles between which the proton transfers can be treated classically. When instead it must be treated quantum mechanically, the rate constant is given in the most important regions of the proton coupling. We expect that the analytical formulas given here will prove useful in the interpretation of experimental reaction results. Further, numerical implementation of the formal rate formulas should prove to be similarly useful in those intermediate proton coupling strength regions when closed form analytical rate constants are not available. We have focused within on what could be termed transition state theory results in the sense that no solvent-induced recrossing of the reaction barriers (totally or in part) in the solvent coordinate was considered. Here, we briefly indicate the nature of the “frictional” barrier recrossing transmission coefficient corrections that can arise from this, sorted according to the proton coupling strength.44 As described elsewhere6 for weak proton coupling, dynamical solvent effects arising from dynamics in the reactant and product wells are expected to be negligible; the slow intrinsic event in this nonadiabatic tunneling regime is the proton tunneling occurring when the diabatic solvent curves intersect, and it is
1/2
ER pω
exp
-1
ER 2pω
Cnm
(A.1)
to write eq 3.21 as q ) k ) A∑Pncnm2 exp(-β∆Gnm
(A.2)
n,m
and distinguish, according to eq 3.2, between n e m and n > m. On switching to the summation variables (n, ∆n ) m - n), this gives the two contributions k ) k1 + k2, with ∞
0
k1 ) A ∑
∑ e-β∆G ∆n)-n
Pncnm2
nm(∆n)
q
n)0 ∞
k2 ) A ∑ e
-β∆Gqnm(∆n)
∆n)0
∞
Pncnm2 ∑ n)0
(A.3)
The n summation in the k2 contribution can be effected immediately47 to give
{[
k2 ) A exp
ER
( )[ pω
∞
∑ exp ∆n)0
βpω∆n 2
( )]}
-1 + coth I∆n
βpω 2 ER
×
]
pω sinh(βpω/2)
q exp(-β∆Gnm (∆n′))
(A.4)
where I is the modified Bessel function of the first kind.47 The contribution k1 can be put into a similar form by extending the
Curve Crossing Formulation of Proton Transfers
J. Phys. Chem., Vol. 100, No. 4, 1996 1127
lower limit of the ∆n summation from -n to -∞; this will be valid in the present high-temperature limit where the major contributions are from large n, but with only modest ∆n (the coupling factor Cmn2 falls off as |∆n| increases). Then setting ∆n′ ) -∆n in k1, performing the n summation as for k2, then returning to ∆n ) -∆n′, and using the relation I+∆n(z) ) I-∆n(z) for ∆n an integer, we find an expression similar to that in eq A.4, with a summation over ∆n running from -∞ to 0 (instead of 0 to +∞). Then, with the definitions
ER ER βpω pj ) ζ ; qj ) ; ζ ) coth pω pω 2
( )
(A.5)
the expressions for k1 and k2 can be combined to give
∑
k ) A[exp(-qj + pj)]
( ) pj - qj
∞
∆n)-∞
-∆n/2
I∆n[(pj2 - qj2)1/2] ×
pj + qj
q (∆n)) exp(-β∆Gnm
)
∞
A
q exp(-β∆Gnm (∆n)) × ∑ ∆n)-∞
[exp(-qj + pj)]
2π
∫-π+πdτ cos[qj sin τ - jr] exp[pj cos τ]
(A.6)
The integral (denoted J) in (A.6) can be evaluated by the stationary phase method. The result is
J)
(
2π |ψ′′(τS)|
)
1/2
exp[ψ(|τS|)]
(A.7)
where ψ(τ) ) iqj sin τ - jrτ + pj cos τ. Here, τS is the saddle point of ψ in the complex plane, defined by the solution of ψ(τS) ) 0, which is
[
iτS ) ln
jr + [rj2 + (pj2 - qj2)]1/2 qj + pj
]
≈ (rj - pj)/pj
(A.8)
for the high-temperature regime (cf. eq A.5), so that
J ) (2π/pj)1/2 exp(pj) exp[-(qj - jr)2/2pj]
(A.9)
When this integral is inserted into eq A.6 and the sum over ∆n converted to an integral over jr ) (∆n/pω), the result after considerable straightforward algebra is, with eq A.1,
k)
[
] [
β〈C2〉 π h β(ER + ES)
1/2
exp -
]
β(ER + ES) 4
(A.10)
which, with eq 3.25 for βpω , 1 is eq 3.24 in the case of ∆E, EQ ) 0 stated at the beginning of the Appendix. Appendix B Here we give various results associated with the reduction of the adiabatic Hamiltonian eq 4.4 to eq 4.14 and related information on the analysis of eq 4.14. With eq 4.4 and the diabatic Hamiltonian eq 4.6, the adiabatic Hamiltonian is
KS 2 1 ∆H + (KS + KQ)∆HQ2 2 2 KS KS + KQ 1 ∆EδH + ∆H∆HQ - [∆H2 + 4C2]1/2 (B.1) KSKQ 2 2
q + HAD ) K + ∆GNA
(
)
The various derivatives with respect to ∆H and ∆HQ evaluated
at ∆H, ∆HQ ) 0, 0 are
∂HAD/∂∆H|0 ) (KS + KQ)∆E/(KSKQ) ∂HAD/∂∆HQ|0 ) -∂C/∂∆HQ|0 ≡ -C1 ∂2HAD/∂∆H2|0 ) KS - (4C0)-1 ∂2HAD/∂∆HQ2|0 ) KS + KQ - ∂2C0/∂∆HQ2|0 ≡ KS + KQ - C2 ∂2HAD/∂∆H ∂∆HQ|0 ) KS
(B.2)
With eq B.2, the expansion of HAD through second order in ∆H and ∆HQ is eq 4.14 and eq B.2 also gives the second derivative matrix eq 4.15. [In the text, that matrix is quoted as the second derivative matrix with respect to the variables ∂∆H and ∂∆HQ defined with the shifted transition state ∆Hq and ∆HQq as the new origin. The second derivative matrix is the same as that determined by eq B.2 since the variables (δ∆H, δ∆HQ) differ from (∆H, ∆HQ) by constants only.] The maximum of HAD is located at the solutions to the vanishing first derivative conditions for eq 4.14, which are eq 4.17. References and Notes (1) Bell, R. P. The Tunnel Effect in Chemistry; Chapman and Hall: London, 1980. (2) Caldin, E. F.; Gold, V. Proton Transfer Reactions; Chapman Hall: London, 1975. (3) Hibbert, F. AdV. Phys. Org. Chem. 1986, 22, 113. (4) Borgis, D.; Lee, S.; Hynes, J. T. Chem. Phys. Lett. 1989, 162, 19. (5) Borgis, D.; Hynes, J. T. Chem. Phys. 1993, 170, 315. (6) Borgis, D.; Hynes, J. T. J. Chim. Phys. (Paris) 1990, 87, 819; J. Chem. Phys. 1991, 94, 3619. (7) Gillan, M. J. Phys. C 1987, 20, 3621; Phys. ReV. Lett. 1987, 58, 563. (8) Voth, G. A.; Chandler, D.; Miller, W. H. J. Phys. Chem. 1989, 93, 7009; J. Chem. Phys. 1989, 91, 7749. Voth, G. A. J. Phys. Chem. 1993, 97, 8365. (9) Li, D.; Voth, G. A. J. Phys. Chem. 1991, 95, 10425. BhattacharyaKodali, I.; Voth, G. A. J. Phys. Chem. 1993, 97, 11253. (10) Warshel, A.; Chu, Z. T. J. Chem. Phys. 1990, 93, 4003. (11) Makri, N.; Miller, W. H. J. Chem. Phys. 1987, 87, 5781; 1989, 91, 4026. (12) Cukier, R.; Morillo, M. J. Chem. Phys. 1989, 91, 857; 1993, 98, 4548. (13) Bosch, E.; Moreno, M.; Lluch, J. M. Chem. Phys. 1992, 159, 99. (14) Schenter, G. K.; Messina, M.; Garrett, B. C. J. Chem. Phys. 1993, 99, 1674. Liu, Y. P.; Lynch, G. C.; Truong, T. N.; Lu, D. H.; Truhlar, D. G.; Garrett, B. C. J. Am. Chem. Soc. 1993, 115, 2408. (15) Suarez, A.; Silbey, R. J. Chem. Phys. 1991, 94, 4809. (16) Truong, T. N.; McCammon, J. A.; Kouri, D. J.; Hoffman, D. K. J. Chem. Phys. 1992, 96, 8136. Bala, P.; Leysyng, B.; Truong, T. N.; McCammon, J. A. In Molecular Aspects of Biotechnology: Computational Models and Theories; Bertran, J., Ed.; Kluwer: Amsterdam, 1992. (17) Mavri, J.; Berendsen, H. J. C. J. Phys. Chem. 1993; 97, 13464; J. Mol. Struct. 1994, 82, 322. (18) Lobaugh, J.; Voth, G. A. Chem. Phys. Lett. 1992, 198, 311; J. Chem. Phys. 1994, 100, 3039. (19) Warshel, A. J. Phys. Chem. 1982, 86, 2218. Hwang, J. K.; Chu, Z. T.; Yadav, A.; Warshel, A. J. Phys. Chem. 1991, 95, 8445. (20) Borgis, D.; Tarjus, G.; Azzouz, H. J. Phys. Chem. 1992, 96, 3188; J. Chem. Phys. 1992, 97, 1390. (21) Laria, D.; Ciccotti, G.; Ferrario, M.; Kapral, R. J. Chem. Phys. 1992, 97, 378. (22) Ratajczak, H. J. Phys. Chem. 1972, 76, 3000, 3991. Ratajczak, H.; Orville-Thomas, W. H. J. Chem. Phys. 1973, 58, 991. (23) Ilczyszyn, M.; Ratajczak, H.; Ladd, J. A. Chem. Phys. Lett. 1988, 15385; J. Mol. Struct. 1989, 198, 499. (24) Staib, A.; Borgis, D.; Hynes, J. T. J. Chem. Phys. 1995, 102, 2487. (25) Azzouz, H.; Borgis, D. J. Chem. Phys. 1993; 98, 7361; J. Mol. Liq. 1995, 61, 17; J. Mol. Liq. 1995, 63, 89. (26) Ando, K.; Hynes, J. T. In Structure, Energetics and ReactiVities in Aqueous Solution; Cramer, C. J., Truhlar, D. G., Eds.; American Chemical Society: Washington, DC, 1994. Ando, K.; Hynes, J. T. J. Mol. Liq., in press; J. Phys. Chem., submitted for publication. (27) Laasonen, K.; Klein, M. J. Am. Chem. Soc. 1994, 116, 11620.
1128 J. Phys. Chem., Vol. 100, No. 4, 1996 (28) German, E. D.; Kuznetsov, A. M.; Dogonadze, R. R. J. Chem. Soc., Faraday Trans. 2 1980, 76, 1128. (29) Ulstrup, J. Charge Transfer Processes in Condensed Media; Springer: Berlin, 1979. (30) Nikitin, E. E. Theory of Elementary Atomic Molecular Processes in Gases; Clarendon: Oxford, U.K., 1974. (31) Newton, M. D.; Sutin, N. Annu. ReV. Phys. Chem. 1984, 35, 437. (32) Juanos i Timoneda, J.; Hynes, J. T. J. Phys. Chem. 1991, 95, 10431. (33) Borgis, D. In Ultrafast Reaction Dynamics and SolVent Effects; Gauduel, Y., Rossky, P., Eds.; AIP Press: New York, 1994. (34) Hammes-Schiffer, S.; Tully, J. C. J. Chem. Phys. 1994, 101, 4657. (35) Trakhtenberg, L. I.; Klochikin, V. L.; Pshezhetsky, S. Ya. Chem. Phys. 1982, 69, 121. (36) Matsushita, E.; Matsubara, T. Prog. Theor. Phys. 1981, 67, 1. (37) Gradstein, I. S.; Ryzhik, I. M. Table of Integrals, Series and Products; Academic Press: New York, 1980; p 838. (38) Dogonadze, R. R.; Kuznetsov, A. M.; Levich, V. G. Electrochim. Acta 1968, 13, 1025. (39) Marcus, R. A. Faraday Symp. Chem. Soc. 1975, 10, 60.
Borgis and Hynes (40) Syage, J. J. Phys. Chem. 1993, 97, 12523; 1995, 99, 5772. (41) Hynes, J. T. J. Phys. Chem. 1986, 90, 370. Maroncelli, M. J. Chem. Phys. 1991, 94, 2084. Carter, E. A.; Hynes, J. T. J. Chem. Phys. 1981, 94, 5961. (42) Van der Zwan, G.; Hynes, J. T. J. Chem. Phys. 1983, 76, 4174. (43) Gehlen, J. N.; Chandler, D.; Kim, H. J.; Hynes, J. T. J. Phys. Chem. 1992, 96, 1748. Kim, H. J.; Hynes, J. T. J. Chem. Phys. 1990, 93, 5194; 1992, 96, 5088. (44) For recent reviews on barrier crossing dynamics, see: Hynes, J. T. In Ultrafast Dynamics of Chemical Systems; Simon, J. D., Ed.; Kluwer: Amsterdam, 1994. Hanggi, P.; Talkner, P.; Borkovec, M. ReV. Mod. Phys. 1990, 62, 251. Whitnell, R. M.; Wilson, K. R. AdV. Comput. Chem. 1993, 4, 67. (45) Grote, R. F.; Hynes, J. T. J. Chem. Phys. 1980, 76, 2715. (46) Smith, B. B.; Staib, A.; Hynes, J. T. Chem. Phys. 1993, 176, 521. (47) Abramowitz, M.; Stegun, I. A. Handbook of Mathematical Functions; Dover: New York, 1970.
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