J . Phys. Chem. 1994,98, 3300-3306
3300
Two-Dimensional Tunneling: Bifurcations and Competing Trajectories? V. A. Benderskii,' S. Yu. Crebensbchikov, and E. V. Vetoshkin Institute for Chemical Physics at Chernogolovka of the Russian Academy of Sciences, Chernogolovka, Moscow Region, 142432 Russia
C. V. Mil'nikov Institute of Structural Macrokinetics of the Russian Academy of Sciences, Chernogolovka. Moscow Region, 142432 Russia
D. E. Makarov*v* School of Chemical Sciences, University of Illinois, 505 S. Mathews Avenue, Urbana, Illinois 61801 Received: October 29, 1993; In Final Form: December 27, 1993'
Different types of tunneling trajectories contribute to 2D vibrationally assisted tunneling in semiclassical approximation. Two families of periodic trajectories compete on the potential energy surface (PES)with two symmetrically situated saddle points. Depending on the temperature, either 1D or 2D periodic orbits give the major contribution to the rate constant K( 2') for the incoherent transition. As a result, an additional crossover temperature Tczappears that corresponds to the bifurcation of the extremal tunneling trajectory. In the coherent case bath periodic and aperiodic paths contribute to the spectroscopic splitting A of degenerate vibrational levels, and competition between these trajectories occurs even on simple PES with the only saddle point. Contribution from aperiodic paths depends upon the quantum number of the promoting vibration. Bifurcation diagram for the two-proton transfer in free-base porphyrin is constructed. Its analysis show that at PES parameters, chosen to fit the experimental rate constant, the transfer follows the asynchronous mechanism closely up to low temperatures. Tunneling splitting in cyclopentanone is evaluated. Dominating aperiodic paths make the transition configuration of vibrationally excited cyclopentanone strongly bent, rather than planar. These paths become "switched off" at a small coupling strength between two coordinates.
Introduction
gated
Quantum chemical dynamics (QCD) allows the establishment of a unique view on such different phenomena as spectroscopic tunneling splitting (with corresponding frequencies of lo81012 s-1) and low-temperature chemical conversions (with rate constants up to less-l). Numerous examples of heavy-particle tunneling were discovered with modern high-resolutionvibrationinternal rotation spectroscopy of supercooled nonrigid molecules and molecular complexes, and with time-resolved spectroscopy of hydrogen transfer and other interconversion processes in the excited electronic states (see, for example, the review in ref 1). Although most of the tunneling systems have many degrees of freedom and QCD should be studied on multidimensional potential energy surfaces (PES), a set of QCD specific features can be elucidated by testing the reduced two-dimensional model PES. Three prototypes of actual PES were studied that differ in symmetry of coupling between the reaction coordinate (along which symmetric potential V(Y) has two wells) and the anharmonic vibrational mode X2
-
linear
t Dedicated with great appreciation to Professor J. Jortner on the occasion of his 60th birthday in recognition of his distinguished contribution to the tunneling acience. To whom the correspondence should be addressed. *Permanentaddress: Institute for Chemical Physics at Chernogolovka of the Russian Academy of Sciences, Chernogolovka, Moscow Region, 142432 Russia. Abstract published in Aduance ACS Abstracts, March 1, 1994.
0022-3654/94/2098-3300%04.50/0
squeezed
Examples of chemical objects featuring (reduced) PES of the types 1-111 are extensively reviewed in refs 1, 3, and 4. Coupling of the tunneling coordinate with vibrational modes leads to either promotion (case 11) or hindering (I) of transition. The way nontunneling modes influence the transition probability depends upon the symmetry of coupling. In case I1 symmetrically coupled modes enhance tunneling (so-called vibrationally assisted tunneling (VAT)), shortening the tunneling distance. In case I11 VAT is purely dynamical: for C > 0 coupling widens the tunneling channel without decrease of the tunneling length. If the promoting vibration is classical (with frequency s2 < kBT/h),VAT for incoherent transition reveals itself in Arrhenius dependence of the rate constant with low apparent activation energy and prefactor. The low-temperature plateau is achieved at the crossover temperature Tc = h n / h k ~ . For the coherent case VAT leads to vibrational selectivity of the tunneling splitting A. For symmetrically coupled modes A increases with the quantum number n2 of the promoting mode, while antisymmetrically coupled modes suppress tunneling. In trajectory language, relevant for semiclassical approximation, VAT in cases I and I1 mainly results in deviationsof tunneling paths (extremal trajectoriesin imaginary time) from the minimum energy path (MEP). In the incoherent case the strength of these deviations depends on temperature. In case I11 for C > 0 the width of the tunneling channel centered around MEP increases 0 1994 American Chemical Society
The Journal of Physical Chemistry, Vol. 98, No. 13, 1994 3301
Two-Dimensional Tunneling with temperature (incoherent case) or quantum number n2 (coherent case). The temperature evolution of tunneling trajectory on PES with the only saddle point can be visualized as follows: At the crossover temperature Tcthe probability of underbarrier transition becomes equal to that of the overbarrier one, so that tunneling trajectory located near the saddle point appears with zero-point amplitude in the inverted 2D barrier. With decreasing temperature the tunneling path gradually elongates and deviates from MEP (in cases I and 11). The limiting trajectory corresponds to the lowtemperature limit of the rate constant. Coherent transition on simple PES 1-111 is the first problem where competing trajectories are encountered. We use the PES I11 to illustrate this phenomenon. In this potential there are two distinct families of tunneling trajectories: 1D (periodic) paths X = 0, and 2D aperiodic ones. In principle, both families could contribute to tunneling splitting. If relative contributions of these families depend on quantum number n2, the competition of trajectories leads to different transition-state configurations for different vibrational levels. However, there are chemical objects whose PES cannot be reduced to cases 1-111. Such systems have a t least two coupled double-well potentials, so that the total PES has two saddle points symmetrically located in the dividing plane. The list of such objects includes, in particular, nonrigid (floppy) molecules, such as methylamines (tunneling CH3 rotor coupled to the inverting NH3 group), methyl-substituted malonaldehydeb and acetylacetone' (hindered tunneling CH3 rotor coupled to tunneling proton), hydrogen-bonded dimers (H20)2 and (NH3)2* (two coupled generalized tunneling coordinates), formic acid dimer? and free-base porphyrin'OJ1 (two-proton transfer). Since in these systems there are several equivalent MEPs, the approximation of a single dominating family of tunneling paths is irrelevant, so that a set of competing underbarrier trajectories should be considered. This may cause a path bifurcation with varying temperature (or quantum number) mentioned earlier by Miller.12 The simplest PES with two saddle points, on which this effect can be seen, was proposed by de la Vega et al. in connection with the two-proton transfer in naphthazarin:"
V(X,Y) = V ( x )
+ V(Y) + C(X2 - X,z) Y2
(IV)
Tunneling paths on this PES evolve with temperature in the following manner. At Tcl two identical trajectories appear near each of the saddle point. These 2D paths elongate and deviate from MEP with decreasing temperature only up to the critical value Tc2 a t which another family of tunneling paths centered around the 1D trajectory, passing between the saddle points, becomes dominating. It is this phenomenon that is called "bifurcation of path"14J5 that can be regarded as another manifestation of competition between trajectories of different types. It is noteworthy that the appearance of the competing periodic paths is already "hidden" in the PES structure. Two crossover temperatures distinguishes the K( 7")dependence from that in the absence of bifurcation. This paper aims to give a description of trajectory competition in puckering of cyclopentanone (potential 111) and in two-proton transfer in free-base porphyrins (potential IV). Tunneling Splittings in Cyclopentanone Molecule
2.0
1.0
0.0
X
-1.0
-2.0
-1.8
-0.8
1.2
0.2
Y
Figure 1. Contour plot of the potential (2), the contours are spa& at
5.6 X 1012s-'. Dimensionlessparameters, 35Ocm-l; YO=752.2~m-~,wo= YO= 1.0, 01 = 0.6909, a = 0.0369, C = 0.2796, correspond to ref 17. Basic path I and aperiodicpaths I1 for the level (0,6) are shown. Dotted line is a mirror image of the path I1 in the "dividing plane". distinguishes cyclopentanone from other VAT systems and makes it especially attractive for the theoretical investigation. PES for cyclopentanone puckering, proposed in ref 17, is a function of two generalized coordinates that depend on twisting (5) and bending (7)angles, y = sin t,x = sin y. The Hamiltonian in mass-weighted coordinates reads'*
H = V,@2
+ Y2)+ V(X,Y)l
with the potential
V(X,Y) = (Y2 - Y t ) 2
+ '/2w12X2 +
+ CX2Y2
(2)
where C > 0 is the coupling constant; V0Yo4is the barrier height along the twisting coordinate Y. The time in this expression is measured in dimensionless units wot. The contour plot of the potential 2 using the parameters of ref 17 is shown in Figure 1. Transverse frequency at the saddle point is smaller than that in the well, in contrast with the "squeezed potential" C < 0 in ref 2. Coupling in the potential 2 is symmetric in both coordinates, so that the minimum energy path (MEP) is a straight line along the twisting coordinate at any C value. The 2D coherent tunneling problem (that is, the calculation of the tunneling splitting A) can be solved in two entirely different ways. The first method is the diagonalization of the Hamiltonian matrix. It can be used to check the accuracy of the second group of methods referred to as methods of classical under-barrier periodic orbits (instantons), introduced by Miller.19 The 2D version of the instanton method' enables one to calculate A for the ground level of the system. A possible way of calculation of A for the excited states was recently proposed in ref 20. It is based on 2D generalization of the well-known Lifshitz formula:21
a
Jdx w-%Y) ay Q ( ~ * Y ) l y - o '"In2
= 2hzJJq(X,Y)
Q*(X,Y) d X d Y
(3)
well
Experimental studies of the microwave and far-infrared spectra of cyclopentanone molecule, undertaken more than 20 years ag0,16.17 revealed that rise in the tunneling splitting A with the quantum number n2 of promoting (bending) vibration is anomalously sharp: when n2 varies from 0 to 6,A changes by lo2 times. This circumstance, caused by a strong VAT effect,
-
Here *(X,Y)is a semiclassical wave function of (nl, n2) level in an isolated well, with eigenvalue E,,,n2.nl and n2 are twisting and bending quantum numbers, respectively. The integral in the numerator is evaluated at the "dividng plane" Y = 0, while the integral in the denominator is taken over the isolated well. The
3302 The Journal of Physical Chemistry, Vol. 98, No. 13, 1994
Benderskii
(v)
TABLE 1: Semiclassical Quantization and Values of the Tunnelin Splitting A, Calculated w t Eqs 4 (Ackcl),5 (As), and 6 (!A*) for Two 3hts of PES (2) Parameters: (1) C = 0.2796, a = 0.0369, Corresponding to Cyclopentanone Molecule;17 (2) C = 0.0473, Q = 0.1107’ varying
Ix
PES
parameters
nl,m
1
(0,O)
(0,l) (0,2) (0,3) (0,4) (0,5) (0,6) 2
(0,l) (0,2) (0,3) (0,4) (0,5) (0,6)
Figure 2. Classically accessible region, bounded by the caustics, for the level (0,2)in the potential (2)with the parameters as in Figure 1. The point (O,Y,) is the starting point of the basic path I.
classically accessible region in a well is confined by a boundary, referred to as caustic, contained inside the energy shell V(X,Y) = E,,,,22 (see Figure 2). q(X,Y) should be built separately in the forbidden and accessible regions and then smoothly seamed at the caustics. Seaming is facilitated by freedom in decomposition of the total energy E,,,, between the Hamilton-Jacobi and transport equations.20 A similar procedure was utilized by Schmid23in calculation of the ground-state wave function for the multidimensional decay rate problem. One might expect that the main contribution to the wave function in the forbidden region comes from the trajectory that starts at the caustic and passes the “dividing plane” at the right angle, in much the same fashion as in the decay rate problem. It follows from the symmetry of the potential 2 that the only periodic trajectory minimizing the action is the path X = 0 (path I in Figure 1). This path coincides with the periodic instanton path with the energy E = V(O,Y,) (see Figure 2). Assuming that the wave function falls off rapidly in the Xdirection, we may consider only a narrow pack of paths in the vicinity of this basic trajectory, arriving a t the following result: (4)
Here A&l) is the splitting in the 1D potential v(0,Y) calculated as in ref 1; X(t) is the transverse deviation from the basic path I obeying the linearized equation of motion; TOis the quarter of the full period of the basic path I. In approximate methods the 2D potential is reduced to the effective 1D. In the limit of slow transverse mode, sudden approximation (SA), well-known in low-temperature rate constant calculation^,^^ can be used (see also ref 2). In this case the wave function can be decomposed as \k(X,Y) = \kl(Y,X992(X), like in the usual Born-Oppenheimer approximation for fast (\kl) and slow (q2) subsystems ( q l ( Y , x * )is found from the SchrMinger equation with fixed slow coordinate Xvalue). Inserting this into the Schriidinger equation gives tunneling splitting as
Here q,,(X) is the wave function of slow motion in the potential given by the solution of the SchrBdinger equation for the fast subsystem. Trajectories in SA are straight lines ( X = constant) connecting two wells weighted by the probability of finding a particle a t a given X coordinate, so that one may expect that SA would work well for the potential 2 in which extrema1 trajectory has zeroth curvature.
(0,O)
Ecua
178.32 273.24 367.42 460.83 553.42 645.15 734.85 166.87 239.86 314.41 390.43 467.85 546.59 626.60
lO3&, 103&.0~ lo3& 103A* 178.33 1.05 0.81 1.03 1.70 275.29 1.36 1.56 1.61 2.54 369.19 3.36 2.25 2.75 3.70 462.61 8.02 3.60 4.6 6.74 554.69 20.1 5.61 7.63 12.90 646.44 52.2 8.63 12.7 29.80 740.03 140 14.10 22.1 70.50 167.58 0.91 0.77 0.88 1.2 242.42 0.98 0.88 0.90 0.68 315.55 1.07 0.97 0.93 0.31 391.20 1.18 1.12 1.01 0.11 468.74 1.31 1.33 1.15 0.03 550.79 1.47 1.74 1.44 8 X lt3 627.67 1.67 2.04 1.61 1 X l t 3
a Other PES parameters are as in Figure 1. Energies are given in cm-1.
TABLE 2: Actions alon the Basic Path I (W,/h),and along the Aperiodic Path11 (W*/h) Calculated for Two Sets of PES (2) Parameters: (1) C = 0.2796, a = 0.0369, Corresponding to Cyclopentanone Molecule;17(2) C = 0.0473, a = 0.1107’ case 1.
2. a
WI/h WZ/h WI/h W*/h
0 5.85 5.66 5.77 5.86
1 5.99 5.50 5.77 6.16
2 6.15 5.31 5.79 6.55
3
4
5
6
6.31 5.01 5.81 7.05
6.47 4.68 5.83 7.67
6.63 4.26 5.84 8.36
6.78 3.80 5.86 9.25
Other PES parameters are as in Figure 1.
The results of calculations of A with eqs 4and 5 are summarized in Table 1 and are compared with the exact values obtained by diagonalization. The level spacing is nearly constant in progression nl = 0, n2 = 0-6, to show that the potential in the wells is almost separable. In this respect, transverse wave functions \k,(X) in eq 5 could be chosen as eigenfunctions of the harmonicXoscillator. For the low-lying levels, n2 = 0, 1,2, the both approximations agree reasonably with LXaa. For higher nz levels (nl = 0, n2 1 4)the situation changes drastically: all approximate expressions underestimate A by 3-10 times. Taking that eqs 4 and 5 are similar in structure and are roughly the 1D splitting multiplied by the transverse prefactor, one may consider the discrepancy as caused by underestimation of one of these constituents. However, none of these reasons seems to be adequate in this case. AIDis nearly constant for all the levels in the same progression, while disagreement increases rapidly with n2. On the other hand, SA should give the upper limit for A, if the Gaussian channel is centered around the basic path I. We should admit that the basic path I in Figure 1 does not provide the only maximum for the wave function at the “dividing plane” Y = 0. This means that the trajectories with a smaller action exist, which do not belong to the family of paths in the vicinity of the basic path I. Up to date we know no regular way of a search for the “lost” families. However, the guess is relatively simple, since the answer, is known. The most likely candidates representing distinctly different families of trajectories are paths I1 emerging from the classical turning points (where momenta p x = p y = 0). These paths are essentially aperiodic (see Figure 1) and have nothing to do with the usual instanton trajectories, but it turns out (see Table 2) that they do produce smaller action w* up to the “dividing plane” than does the basic path I. Thus, the amplitude of the wave function at the “dividing
The Journal of Physical Chemistry, Vol. 98, No. 13, 1994 3303
Two-Dimensional Tunneling plane” calculated using these paths is not small. The contribution from such trajectories to the tunneling splitting can be estimated to exponential accuracy in a well-known manner:25
2h
A*nln22 -exp(-2W*/h) r,l”2
It is noteworthy that the eq 6 can be interpreted in terms of periodic Miller’s orbits. Factor 2 in the exponent allows one to combine the segment of aperiodic path I1 with its mirror image in the “dividing plane” into a single “trajectory” (see Figure l), the action on which determines This “trajectory” is periodic but nonclassical, since it does not obey the classical equations of motion. In this respect the terms “aperiodic paths” and “nonclassical paths” are interchangeable in the present context. A*,,,, values are given in Table 1. Paths I1 contribute to the splitting even more than does a basic classical periodic family. This competition between different families of trajectories in cyclopentanone puckering means that for strongly excited bending levels n2 = 3,4, ...,where aperiodic paths (with large amplitude a t the dividing plane Y = 0) “win”, the interconversion between two twisted conformations goes via bent rather than planar form. For example, bending angle at Y = 0 amounts to -35O for nz = 6. Thus, the actual trajectory becomes closer to pseudorotation than to inverconversion through a planar form. Different types of trajectories dominate at different PES parameters. When the coupling between two coordinates goes weaker, the aperiodic trajectories become “switched off“.26 For example, for C = 0.0473, a = 0.1 107, the action W , is greater than that along the basic path I, and the difference increases with n2 (see Table 2). In this case path I prevails, and both S A and the semiclassical expression 4 agree excellently with 4exact (Table 1). This testifies indirectly that it is these aperiodic paths I1 that are responsible for the anomalously fast growth of A values with the bending quantum number n2 in cyclopentanone.
Synchronous versus Asynchronous Two-Proton Transfer Two-proton transfer (TPT) is normally classified as synchronous or asynchronous. The former was observed in carbonic acid dimers27 and is characterized by a distinct low-temperature plateau for the rate constant. Asynchronous TPT was mostly extensively studied in free-base porphyrins, in which the rate constant decreases from 105 to 10-5 s-l in the temperature range 320-95 K, and the activation energy drops from 10.4 (200-320 K)lo to 6.4 kcal/mol (95-110 K). For D atoms low-temperature activation energy is 3.3 kcal/mol greater than for H atoms.” The persistence of the Arrhenius behavior up to low temperatures in combination with large isotope H / D effect distinguishes this system from others and appeals for a theoretical explanation. For the system that includes two linear O-He-0 hydrogen bonds, TPT reaction is described using symmetric (Y) and antisymmetric ( X ) coordinates given by9
+ r2- r3- r4 X = rl - r2- r3+ r4 Y = rl
(7)
where rl are the O-H bond distances. Two N-He-N fragments are nonlinear in porphyrin (see Figure 3), and four coordinates are required for the description of the plane motion of two protons. If the coordinates 7 are used, the transition-state geometry a t different temperatures is reproduced correctly; however, this reduction of 4D PES means that only 2D projection of actual proton motion can be obtained. The reduced mass along each of these coordinates equals m ~ / 8 . The 2D Hamiltonian reads
H = Vo(’/3r’+ ‘/p + V(X,Y))
(8)
where time is measured in the dimensionless units T = oot. The
b
a
Figure 3. Two-proton transfer in free-base porphyrin molecule: (a) intramolecular coordinates rl-r4 in the initial state; r3 - rl = r4 - r2 =
1.25 A; (b) temperature evolution of the transition state geometry, calculated usin the trajectories 1-3 depicted in Figure 4: T = 180 K, rl - r3 = 1.24 (solid circles, trajectory 1); T = 80 K,rl - r3 = 1.15 A (open circles, trajectory 2 ) ; hypothetical case T = 0 K,rl = r3 (solid squares, trajectory 3). Conjugated bonds of frame skeleton are shown for trans (a) and cis (b) forms.
1
choice of the potential V(X,Y) relevant for the description of TPT in free-base porphyrins is still an open question. Twodimensional PES with four minima and four saddle points was proposed in ref 28 on the strength of the results of quantumchemical calculations. However, the obtained barriers (36.643.2 and 60.8-66.5 kcal/mol for synchronous and asynchronous transfer, respectively) are too high, so that a VAT mechanism including an additional promoting vibration with adjustable parameters was postulated in order to fit the experimental findings.” Butenhoff and Moore” suggested that the cut of PES along the reaction coordinate for asynchronous transfer has an additional minimum a t the barrier top. In this paper we choose a simplified version of PES, that is, PES of the type IV, which a t the same time reflects the main features of tunneling TPT and, in particular, enables one to reproduce the experimentally observed rate constant” with reasonable accuracy (see below). Thus, we have
V(X,Y) = I‘4 - 2Y2
+ (C/2)(Y2-
Yt)X2
+
(9)
Here coordinates and parameters C and a are dimensionless; Us is referred to as the PES splitting parameter. The potential 9 is a symmetric double-well potential with the minimum located at points ( 0 , f l ) in the (X,Y) plane. MEP satisfied the condition dV(X,Y)/dX = 0. Two saddle points are located at X* = fYs(C/a)I/z and V = 0. The MEP coincides with the straight line X = 0 a t Ill > YSand is split into two channels at Ill < Ys:
x = f(C/a)1’2(Y:
- Y2y2
(10)
so there are two equivalent ways for the stepwise transfer, corresponding to the order in which the protons are transferred. A contour plot of the potential 8 is shown in Figure 4. The exponent in the expression for the transition probability between the two minima is determined by the action along the extreme &periodic trajectory in the upside-down potential (instanton) satisfying the Euler-Lagrange equations SS/SX = SS/SY = 0 with t h e periodic boundary conditions X(@)= X ( 0 ) ; Y(p) = Y(0)(0 l/keT). The periodic trajectory is unstable in the sense that trajectories having slightly different initial conditions perform infinite aperiodic motions. The inclusion of fluctuations of the path in the direction normal to the coordinate along the instanton trajectory leads to an additional preexponential factor in the rate constant:29-32
Benderskii
3304 The Journal of Physical Chemistry, Vol. 98, No. 13, 1994 2.00
Z
w
2
1.50
0.00 0.00
, , , , , , , , , 3!* 7 0.60 0.80 1 .oo
I I I I , I I , I I , , , , , , , , , , , , , '
0.20
0.40 V
'S
Y
Figure 4. 2D PES (eq 9) at C = 2.50, a = 2.94, YS= 0.96. Curves 1-3
correspond to numerically calculated trajectories at different temperatures: 180 K (l), 80 K (2); hypothetical case T - 0 K (3).
where WI is the longitudinal frequency at the minimum: W? = (d2V/dYZ)y,tt. The transverse prefactor Bt is equal to3' (see Appendix, where a simple derivation of eq 12 is given)
B, =
sinh(u,p/2) sinh(X/2)
where ut2 = (d2V/dX)X=0,y,*l = C(l - Ys2) is the transverse frequency at the potential minima; X is the stability parameter showing how fast and in which fashion the trajectory deviates from the periodic instanton solution after a small initial perturb a t i ~ n .Purely ~ ~ imaginary A means a stable instanton solution, while real X corresponds to infinite motion along the transverse coordinate. The exact value of longitudinal prefactor for various potentials is of the order of unity. It is calculated in the standard literature on 1D instanton.1J4 There are two types of trajectories that may be realized in the potential 9: the first of them is the 1D solution to the equations of motion, which always exists:
X=O,
Y = aviau
(13)
Whether or not a 2D trajectory exists, which is characterized by a greater barrier transparency, is a result of the tradeoff between the length of the path and the barrier height along it. At high temperatures (j3 0) the barrier height factor always wins and the transition goes through the saddle point. The barrier along the MEP is AV = -C2Y4Vo/4a lower than the 1D barrier. The relative increase in the path length along the MEP compared to the 1D path is -C/2a. Naively assuming that a 2D trajectory appears when Gamov factor along the MEP exceeds that of 1D trajectory, one obtains an approximate criterion for the appearance of two 2D instantons:
-
AV/2V = C/a where Vis the barrier height. A more rigorous approach is based on the observation that at the bifurcation point the stability parameter X should vanish.15 When X passes through zero, the tunneling channel centered around an extreme 1D trajectory becomes infinitely wide, and transverse fluctuations are no longer Gaussian. This manifests
Figure 5. Dependence of the stability frequency X/@WOon the splitting parameter Ys for the potential (9) with parameters C = 2.50, a = 2.94 at different temperatures; &oh equals 4.0 (curve 1); 7.0 (curve 2); 10.75 (curve 3).
the appearance of a new type of trajectory (2D) with smaller action. The mathematical formulation of this transition is given in ref 14. Trajectories with a given period are found computationally with the algorithm described e l s e ~ h e r e . 'The ~ stability parameter X was calculated from eigenvalues of the monodromy matrix M.I.15 Numerical calculations were performed for the potential 9 with C = 2.50, a = 2.94, V, = 33 kcal/mol, 00 = 1.34 X 1014 s-l, YS = 0.96. These parameters arechosen to fit the experimental K(T)dependence. In this case WI and utare equal to 3.6 X loi4 and 0.6 X 1014 s-I, respectively; the potential barrier VC and real frequency at saddle points are 14 kcal/mol and 1.65 X 1014 s-I, respectively. The first step in numerical analysis is the construction of the bifurcation diagram. To this end the values of splitting parameters YS a t which stability parameter X is zero were determined at various inverse temperatures j3* E hwo/kBT (see Figure 5). Collecting the points X(Y0) = 0 for different temperatures, we can plot the YO@*) dependence determining the separatrix between the 1D and 2D solution regions. The bifurcation diagram is presented in Figure 6. The maximum value achieved in j3* m limit is 0.76, and YS = 0 a t j3* = r, in agreement with predictions made in ref 14. The curve separating the activated and tunneling behavior regions
-
p , = (2r)-l(a2vp~2)1/2
(14)
that is, the dependence of the usual crossover temperature on the splitting parameter, is also indicated in Figure 6 . Thus, three distinct domains with different Ys values can be recognized in Figure 6. This diagram covers all possible mechanisms of TPT reaction in 2D case. 1. The distance between the saddle points in this case is greater than in the two above cases, and tunneling trajectories are always two-dimensional. Therefore, there is no 1D-2D transition in this case, and the low-temperature plateau of the rate constant is achieved slowly. Above the second crossover point but below the first one, the 1D and 2D instantons coexist, the action on the latter being the smallest. At a given YSthe difference between their actions decreases with decreasing temperature. The point at which the 1D action is equal to the 2D action is a trajectory bifurcation point which is always followed by 1D trajectories. With increasing Ys values this point moves to lower temperatures. Note however, that for YS> 1D and 2D actions will never coincide, except for 0 = =. Thus, although a 1D trajectory exists at arbitrary 0, the contribution to the transfer probability from 2D trajectories should be taken into account above the second crossover point, where the 2D action is smaller than the 1D one, or even slightly below this point when the two instanton contributions have similar actions. The fact that YSvalue, adequate for free-base porphyrin PES, equals =0.96 > means that TPT in this system can never be synchronous. The rate constant calculated using the eq 11, along with the experimental findings,” is presented in Figure 7. Both curves agree reasonably in the temperature range 270-95 K. The analysis performed enables not only the calculation of the TPT rate constant itself but also the extraction of information on the transition-state geometry. The initial state configuration is drawn in Figure 3a. The only simplifying assumption that we make is that the tunneling motion of protons in real molecular space is only slightly curved and proceeds closely to the sides of a square depicted in Figure 3b with the dashed lines. It is found that the bond distances in the initial state ( X = 0, Y = -1) are such that r3 - rl = 1.25 A, while the value corresponding to the molecule geometry is 1.34 A.28 Purely concerted transfer means that rl = r2 = r3 = r4 in the transition state. Transition-state geometry at different temperatures is shown in Figure 3b. In particular, a t T = 180 and 80 K, corresponding to the trajectories 1 and 2 in Figure 4, (rl - r3) equals 1.24 and 1.15 A, respectively. With decreasing temperature, the position of the saddle points in rl - r4 space gradually approaches the center of the square. In this respect, TPTin free-base porphyrins is neither concerted nor stepwise: the trajectories in the investigated temperature
range correspond to an intermediate situation. Nevertheless, the mechanism of the reaction is closer to the asynchronous one, since even at 80 K the saddle point is strongly shifted away from the point rl = r2 = r3 = 1-4.
y
y
Conclusions The examples considered abovedemonstrate that in polyatomic chemical systems with multidimensional PES the tunneling transition from the initial to the final state can proceed in several alternative ways. In the trajectory language this signifies the possibility of competition (and possible bifurcation) between extrema1 paths of different types. As a result, the transition-state geometry of a system depends on the temperature (for incoherent processes), or on quantum numbers of vibrational level (for coherent processes). It is significant that for the incoherent case different extreme trajectories also imply different temperature dependences of the reaction rate constant. The field is now open for more close investigations. PESs of actual systems (even reduced to 2D) are more complex than those, described above. This brings up the question: How the path bifurcation is connected with the topology of PES? Evidently, the number of competing families of paths increases with increasing number of the saddle points. For example, for the incoherent transition on 2D PES with two global, two local minima, and four saddle points between them, the tunneling paths will be borne around each saddle point. Coalescing of these trajectories by pairs will give rise to a new crossover temperature. At the same time, the role of PES topology is less obvious in the coherent problem, since the dominating aperiodic (or periodic nonclassical, see above) paths are detected even for simple PES. The natural problem here is to invent a regular procedure of search for the “lost” families in the case, when simple symmetry reasoning is unhelpful.
Acknowledgment. V.A.B.,S.Yu.G.,G.V.M.,andE.V.V.wish to thank the Russian Fund of Fundamental Investigations for the financial support (Grant No. 93-03-4539). Appendix. Proof of Eq 12 for the Transverse Prefactor B,
Let u be the coordinate running along the instanton path; is normal to u. The transverse prefactor Bt (eq 12) reads35
det(4:
+ w:)
x
(All
where W ? ( U ( T ) ) = a2V/dx2is the time-dependent frequency of
Benderskii
3306 The Journal of Physical Chemistry, Vol. 98, No. 13, 1994 the transverse vibrations along the instanton path ~ ( 7 )cot; is the transverse frequency at the potential minima; at2 d2/dr2. Determinant in the numerator of (Al) can beexpressed through the path integral (det(4:
+ u:(T)))-"~
= N K d x l(x,x,P)
(A21
where N is the normalizing factor, which will eventually cancel out in ratio A l , and36 cp
+ u?(7)x2/2)
~ D [ x ( T ) ]exp(-J'(X2/2
d7 =
(1 / ~ T ) ' / ~ I X ~ ( O ) exp(-So) I-~/~ ( ~ 3 ) x(0) = x(B) = x with SO= Jg(x
