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Two-dimensional tunneling in a potential with two transition states
Volume 186, number 6
22 November 1991
CHEMICAL PHYSICS LETTERS
Two-dimensional tunneling in a potential with two transition states V.A. Bender&ii, V.I. Goldanskii and D.E. Makarov Instituteof Chemical Physics,Academy of Sciences of the USSR, 117334MOSCOW, USSR Received 19 April 1991;in final form 19 August 1991
Two-dimensional tunneling is studied for a model potential V(Q, X) = V,(Q) t tl(Q'-Qa)X*+ @X4. Vo(Q) is a symmetric double-well potential which has two transition states placed symmetrically in the dividing line. Extremal trajectories are shown to be of three types replacing each other at bifurcation values of the parameters Q0 and p= 1/k&F The doubly degenerate path near the saddle points (i) occurs in the Arrhenius region (b< /?,(QO)) and is replaced by the two-dimensional instanton (ii) at the cross-over temperature a( Q,,). At & less than the critical value, the one-dimensional instanton (iii) arises as the temperature drops. As Q, decreases, the region of existence of the two-dimensional instanton vanishes.
Modern quantum transition-state theory is based of multidimensional nuclear tunneling [ 11. One of the applications of this general theory is the description of solid-phase cryochemical reactions in which the low-temperature plateau of the rate constant [2] is due to the tunneling of a light particle between heavy fragments [3,4]. The effect of low-frequency local vibrations is regarded as a result of their coupling to the tunneling coordinate. In refs. [ 5,6], the strict WKl3 solution has been obtained for a model two-dimensional potential formed by two strongly shifted paraboloids. The genera1 methods [ 7-91 reduce the tunneling problem to solution of an integro-differential equation, i.e. minimizing non-local action along a one-dimensional trajectory. Such an approach supposes that there is a single minimum-energy path (MEP) connecting initial and final states. As first noted by Miller [lo], the potential energy surface (PES) with two transition states is a rather common case. The saddle points are placed symmetrically in the dividing plane and correspond to the double-well potential for the transversal coordinate_ While the initial and final states are not degenerate, the PES is split into two channels situated symmetrically in configuration space so that the calculation of quasiclassical periodic orbits [ 1] is beyond the scope of application of the usual instanton approximation [ 7-91. A number upon the concept
of chemical reactions are known in which such a situation takes place. The tunnel splitting of the ground state 3 ‘A, of tropolon is due to the proton transfer modulated by the low-frequency vibrations u31 (C=O 1C-O) and v34(C=C-C) (with coordinates R and 8, respectively ). Since the CZysymmetry breaks, the V(R, S) potential has two saddle points [ 111. The PES of two-proton exchange in the porphyrin ring [ 12,131 and in naphthazarin [ 141 comprises two MEPs corresponding to the consequent asynchronous transfer. The maximum between the saddle points corresponds to synchronous transfer. According to refs. [ 15,161, the asynchronous transfer prevails in the Arrhenius region. In the present paper, two-dimensional tunneling is studied for the potential modeling of the above-described situation. It is shown that the bifurcation of the extremal trajectory arises at a critical temperature depending on the potential parameters. We start with a model Hamiltonian chosen in the form H= V&+++~*t
V(Q,X)]
,
V(Q,X)=F’0(Q)t~~(Q’-Q;)X2t~~X4,
(1)
where V0is the barrier height, Q, X are dimensionless coordinates (Q=q/a, q is the reaction coordinate, 2a is the distance between the minima of the
O@J9-2614/9I /$ 03.50 0 1991 Elsevier Science Publishers B.V. All rights reserved.
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Volume 186, number 6
symmetric double-well potential VO(Q), I&,= V(O)=O, V,,=V(~I)=-l). The time in this expression is measured in dimensionless units T= coot, co,= (2V,//&$J*)“*, where ,u~ is the particle mass, corresponding to the reaction coordinate, and Q,, is the PES splitting parameter. At IQI > Qo. the minimum-energy path is one-dimensional (X=0) while at IQ I < Qo, the PES is split into two channels (fig. 1) with the MEP of the form X= k (n/cr)“’ (Q;-Q2)“2.
(2)
An equipotential contour plot is shown in fig. 1. The barrier along the MEP is lower than in the one-dimensional case ( - AVxnZQd/4a) but the particle path length increases, which may be described in terms of effective mass [ 5,6,17] M*= I +1/a. If AV/ Lb-A/a, the Gamov factor (barrier transparency) along the MEP exceeds the one-dimensional one and splitting of the trajectory is expected. Comparing the time of transversal transition between the channels, tfx (a2V/&Y2) ;& =Q;‘A-‘/2, with the time of passing the split part of potential, rO=Qo/~,, and supposing, similarly to refs. [ 6,17 1, that splitting occurs at ro> TV,one obtains the same splitting condition, which can be rewritten in the form Q:z=Q:=v:lA.
I312
I
S=
Hdr,
(4)
p= 1/k, T .
The instanton equations (SS/SQ= SS/SX=O), ..
av
e= @+lQx2 ’ X=~X(Q~-Q;)+L~X~,
518
(6)
are subject to the periodic boundary conditions Q(/I)+Q(O),x(p)tX(O). Thereisasolutionofeq. (6) corresponding to the one-dimensional instanton: X=@
i’ Q=av/aQ.
(7)
The question is to find out in which region of parameters A,(Y,Qo, the solution (7) is stable and when the two-dimensional trajectory occurs. To study the stability of (7), it is suffkient to restrict oneself to a quadratic (Gaussian) approximation, whereas the quadratic corrections to the action along trajectory (6) lead to the prefactor [ 81. According to ref. [ 181, the solution (7) becomes unstable when the lowest eigenvalue of the differential operator -=-dr2+ 6X2(T)
d2
PV(Q,
X)
ax2
is negative. The equation for its eigenvalues is formally equivalent to the Schriidinger equation in a “periodic potential”: --
d2X(r) dT2
+n[Q2(~)-Q~IXn(~)=t,X,(r),
xl(~~=x(~+P)
Fig. I. Equipotential contour plot for the potential ( I), v(Q)=Q4-2QZ,Qr,=0.5,1=20,~=40.TheMEPsareshown.
(5)
-P/2
S2S
(3)
Here, v,,,= (2V)‘12 is the maximum imaginary-time under-barrier velocity. The transition probability between two minima is determined by the action along the periodic trajectory in the inverted barrier, named the instanton: rot exp(- voS/oo) ,
22 November 199I
CHEMICAL PHYSICS LETTERS
(9)
When LO, the “potential” is “shallow” so that the only “energy level” is situated near the maximum A(Q$ -Qi) =co, where Qm(/?) is the 1D instanton amplitude (maximum Q(z) for solution (7)). Hence, the one-dimensional motion is stable at Qi > Q8 and unstable in the opposite case. This situation is also realized for QmCK1. In a more interesting case of large transversal frequencies 1 B 1, the solution of (9) may be found in the harmonic ap proximation. Expanding the f( r) = LQ’( 7) function into a Taylor series around the “potential bottom”, T= $a one obtains the harmonic oscillator equation, .
Volume186,number 6
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CHEMICALPHYSICSLETTERS
(~n+WWn(~)>
22 November 1991
(10)
with the eigenvalues equal to (if”)-I’*
(c,+nQ;) =1+2n,
n=O, l,... , (11)
so that the splitting condition tn< 0 amounts to (3) since
The phase diagram of the system is shown in fig. 2, from which it follows that the 1D instanton stability region widens with decrease of T. Curve I, determined by (3), separates the region of stability of the ID instanton (i) and the 2D instanton region (ii). At Qi > (u,,,/~)‘/*= (2/A)l/*, the instanton, if it exists, is always two-dimensional. In the region (iii), the thermal activated transitions through the saddlepoint (Q=O, X=X,= rtQ&/a,)t/2) +I’take place. Curve 3 corresponds to the second-order phase transition and is described by the usual equation for
PI Wesupposethata2V/aQzI,=,o=o=4[1(A2/4~)Q$]~0. p2 The concrete potential chosen in fig. 2 is V(Q) = Q’-2Qz so that 8, =n[ I- (A/~cx)Q~]-~‘~.
Q,Z
Fig. 3. Maximum instanton velocity v,,,= 2”‘*k/( I + k’) and instanton amplitude Qm=2”zk( I t/c*)-“’ versus p=2( 1t k2)‘/*K(k) for the Q4-2Q2 potential [6]; K(k) is the first-order elliptic integral.
the cross-over temperature (see for example refs. 1341) ‘*> fi, =~~T(~~I’/~Q~) -I’*.
(12)
Near the mutual point of all three curves, Qm==zz 1 and the quasiclassical approximation for (9) breaks down so that one should use the “geometric” splitting condition, Q,,,= Q,, (curve 2), as in the case of small 1. At Qi<(2/n) ‘I*, the second cross-over temperature corresponding to the instanton splitting appears which satisfies eq. (3 ). We show further that transition across curve 1 is also similar to the second-order phase transition and find the split trajectory near the ID-2D cross-over point. The eigenfunction of (lo), corresponding to the unstable mode, is a Gaussian function at z< 1: X0(s)= exp( --fr2 v,L’/*) ,
(13)
and falls off strongly during the time, t% (v,A”*)-“*, which is shorter than the characteristic instanton time (?d< 1). Introducing the characteristic dynamical splitting length, Id= v,z~< Q,, allows one to rewrite (3) in the form Qoal,,, which covers both the LZP 1 and ,J-ZK1 cases. We search for the split trajectory in the form X(r)=&,(t), Fig. 2. Bifurcation diagram for the potential (I ), V(Q)= Q’-2Q*, a=90,1= 16. The curves (l)-(3) satisfy the equations Q$=u,L-“~, Qa=Q,&9) and @c( 1-dQi/4@)-‘/‘, respectively, and separate the 1D instanton (i), 2D instanton (ii) and the Arrhenius (iii) regions.
QW=dW+~QW.
(14)
Steepest-descent integration over the unstable mode (which takes into account the fourth-order terms in the expansion of the action by the coefficient c) gives rise to 519
Volume 186,number 6
In order to check on the approximate approach, we nowderive the criticalcondition for V(Q) = Q4- 2Q’ at T=O.Substitution of Q(T)=tanh(z$) for (9) leads to the “Schriidinger equation” for the Eckart potential,
!3/2
X
22 November I991
CHEMICALPHYSICSLETTERS
I
X;(T) dr+$tc4 >I -P/Z
d*&(r) JX”(~) -- dz2 - cosh2(& = ita +4Qi
A&$ (Q:-Qf)?’
1 X;(r)dT. -I@/2
(15)
,
(211
with the “zero-level” eigenfunction, X,,(r)= [cosh(zJi)]-[“+2~““-‘~‘2,
The coefficient r includes all the fourth-order terms and F corresponds to the minimum action. Substituting (14) for (6), we obtain
-
- 1) l&(7)
$$+AX(@-Qf)tAX(Q,z-Q;,
tuQx3Q+ax3=0.
(16)
From multiplying the second equation by X(Z)= 2X0(2) and integratingit over the period, we obtain the followingequation for E*:
(22)
at the critical splitting parameter, (23) At /~-PO,Q,,=Qc.At 1~ 1, (23) coincides with (3) since u,= 2’/‘. The dependence S(8) is shown in fig. 4. In accordance with (3), as Q, grows and approaches the critical value a, the two-dimensionalinstanton &gion widens and the I D-2D cross-over temperature tends to zero (&co). At Q; %QZ, the action can be evaluated as (24)
812 812 1 X&)drt2AF2 oX:8Qd7 -P/Z -B/Z
J
P(Qf-Q;) P/2 t&Y
5
X;(z) dr=O .
(17)
-P/2
To solve the first of eqs. ( 16) , we note that at 1~ 1, X,(r) varies much faster than o(r) so that one can replace a2V/aQ2and Q by a2V/aQzIaE,= LJi and v,,,r,respectively.Since d2/dt2 w1- ‘I2a L#, the correction in Q(T) is equal to RIP0f 6Q(r)=- 2 exp( -s2u,l’/*) 512
0
(18) a
Substitution of (18) for (17) gives, finally,
(19) A&-f 520
n’l=v~2R-‘f=(Q~_Qc2)E2.
(20)
4
6
s
B
Fig. 4. Action S versus fiat a splitting parameter Q0 equal to 0, 0.330, 0.346, 0.350 for the curves l-4, respeotively. Potential Q”-2Q2, A= 16, a=90. In the Arrhenius region S=/3( I d*Q$4a), the 1D instanton action equals: S- 16( l+k2)-3’2 x~(ltk2)E(k)-(I-kz)K(k)]+~[1-4kz(l+kZ)-Z]. The dashed line correspondsto the two-dimensionalinslanton region.
Volume 186, number 6
CHEMICAL PHYSICS LETTERS
22 November 1991
paths and the temperature dependence of the rate constant, I&CCexp[ -S(B) 1, contains a wide region of low apparent activation energy (compared to V,) (fig. 4). Seemingly, this situation is realized in refs. [ 12,13] since the low-temperature plateau of KCcorresponding to the ID instanton is not reached. The synchronous transfer can be observed when AV is small as compared to V, and the temperature region of existence of the 2D instanton (/&dp
Fig. 5. Action Sversus AV=d*Qg/4a at 8=4.0,5.0,6.0 for curves 1-3, respectively.
so that the action decrease is determined by the static barrier lowering along the two-dimensional trajectory and the dynamical instanton fall-off time, rd, which does not correspond to the time of passage of the split part ofthe potential, so N3.According to (24) the relative decrease of action compared to the onedimensional case is smaller than the relative decrease of potential AV/ Vby the factor of z r&O. Fig. 5, where the dependence of S versus AV is represented for different temperatures demonstrates this fact. The dashed line for the 2D instanton connects the 1D instanton and Arrhenius regions where the action value is readily obtainable. At A V/ V60.5, the relative action decrease compared to the one-dimensional instanton action makes up w 0.15. As seen from our study, an alternative consideration of synchronous (along trajectory (7) ) and asynchronous (along the MEP) transfer is rather conventional. When the 2D instanton arises at /3> fl,, the split trajectory coincides with neither of these two x3 It follows from ( 13) that at 1~ 1,the r., time is the geometric average of T,,and the transversal passage time so that the condition of splitting can be expressed as 7, c T, = (T,s,) “’ c q,. At a-0, rd=r@
References (11 W.H. Miller, I. Chem. Phys. 61 (1974) 1823; 62 (1975) 1899; 63 (1975) 996,1166. [2] V.I. Goldanskii, Dokl. Akad. Nauk SSSR 124 (1959) 1261; 127 (1959) 1039. [3] V.A. Bender&ii, V.I. Goldanskii and A.A. Gvchinnikov, Chem. Phys. Letters 73 ( 1980) 492. [4] V.I. Goldanskii, V.A. Benderskii and L.I. Trakhtenberg, Advan. Chem. Phys. 75 (1989) 349. [5 ] V.A. Benderskii, V.I. Goldanskii and D.E. Makarov, Chem. Phys. Letters 171 (1990) 91. [6] V.A. Benderskii, V.I. Goldanskii and D.E. Makarov, Chem. Phys. 154 ( 1991) 407. [ 71 A.O. Caldeira and A.J. Leggett, Ann. Phys. 149 ( 1983) 374. [8] A.J. Leggett, S. Chakravarty, A.T. Dorsey, M.P.G. Fisher, A. Garg and M. Zverger, Rev. Mod. Phys. 59 ( 1987) 1. [9] H. Grabert, P. Olschovskii and U. Weiss, Phys. Rev. B 36 (1987) 1931. [ 101 W.H. Miller, J. Phys. Chem. 87 (1983) 3811. [ 1 I ] R.L. Redington, J. Chem. Phys. 92 ( 1990) 6447. [ 121T.J. Buttenhoff and C.B. Moore, J. Am. Chem. Sot. 110 (1988) 8336. [ 131 B. Wehrleand H.H. Limbach, Chem. Phys. 36 (1989) 223. [ 141J.R. de IaVega, J.H. Bush, J.H. Schaube, K.L. Kunze and B.E. Haggert, J. Am. Chem. Sot. 104 (1982) 3295. [ 1S] Z. Smerdachina, W. Siebrand and F. Zebretto, Chem. Phys. 136 (1989) 298. [ 161Yu.1. Dakhnovskii and M.B. Semenov, J. Chem. Phys. 92 (1989) 7606. [ 171J. Sethna, Phys. Rev. B 24 ( 1981) 692. [ 181B.I. Ivlev and Yu.N. Ovchinnikov, JETP 93 ( 1987) 668.
521
22 November 1991
CHEMICAL PHYSICS LETTERS
Two-dimensional tunneling in a potential with two transition states V.A. Bender&ii, V.I. Goldanskii and D.E. Makarov Instituteof Chemical Physics,Academy of Sciences of the USSR, 117334MOSCOW, USSR Received 19 April 1991;in final form 19 August 1991
Two-dimensional tunneling is studied for a model potential V(Q, X) = V,(Q) t tl(Q'-Qa)X*+ @X4. Vo(Q) is a symmetric double-well potential which has two transition states placed symmetrically in the dividing line. Extremal trajectories are shown to be of three types replacing each other at bifurcation values of the parameters Q0 and p= 1/k&F The doubly degenerate path near the saddle points (i) occurs in the Arrhenius region (b< /?,(QO)) and is replaced by the two-dimensional instanton (ii) at the cross-over temperature a( Q,,). At & less than the critical value, the one-dimensional instanton (iii) arises as the temperature drops. As Q, decreases, the region of existence of the two-dimensional instanton vanishes.
Modern quantum transition-state theory is based of multidimensional nuclear tunneling [ 11. One of the applications of this general theory is the description of solid-phase cryochemical reactions in which the low-temperature plateau of the rate constant [2] is due to the tunneling of a light particle between heavy fragments [3,4]. The effect of low-frequency local vibrations is regarded as a result of their coupling to the tunneling coordinate. In refs. [ 5,6], the strict WKl3 solution has been obtained for a model two-dimensional potential formed by two strongly shifted paraboloids. The genera1 methods [ 7-91 reduce the tunneling problem to solution of an integro-differential equation, i.e. minimizing non-local action along a one-dimensional trajectory. Such an approach supposes that there is a single minimum-energy path (MEP) connecting initial and final states. As first noted by Miller [lo], the potential energy surface (PES) with two transition states is a rather common case. The saddle points are placed symmetrically in the dividing plane and correspond to the double-well potential for the transversal coordinate_ While the initial and final states are not degenerate, the PES is split into two channels situated symmetrically in configuration space so that the calculation of quasiclassical periodic orbits [ 1] is beyond the scope of application of the usual instanton approximation [ 7-91. A number upon the concept
of chemical reactions are known in which such a situation takes place. The tunnel splitting of the ground state 3 ‘A, of tropolon is due to the proton transfer modulated by the low-frequency vibrations u31 (C=O 1C-O) and v34(C=C-C) (with coordinates R and 8, respectively ). Since the CZysymmetry breaks, the V(R, S) potential has two saddle points [ 111. The PES of two-proton exchange in the porphyrin ring [ 12,131 and in naphthazarin [ 141 comprises two MEPs corresponding to the consequent asynchronous transfer. The maximum between the saddle points corresponds to synchronous transfer. According to refs. [ 15,161, the asynchronous transfer prevails in the Arrhenius region. In the present paper, two-dimensional tunneling is studied for the potential modeling of the above-described situation. It is shown that the bifurcation of the extremal trajectory arises at a critical temperature depending on the potential parameters. We start with a model Hamiltonian chosen in the form H= V&+++~*t
V(Q,X)]
,
V(Q,X)=F’0(Q)t~~(Q’-Q;)X2t~~X4,
(1)
where V0is the barrier height, Q, X are dimensionless coordinates (Q=q/a, q is the reaction coordinate, 2a is the distance between the minima of the
O@J9-2614/9I /$ 03.50 0 1991 Elsevier Science Publishers B.V. All rights reserved.
517
Volume 186, number 6
symmetric double-well potential VO(Q), I&,= V(O)=O, V,,=V(~I)=-l). The time in this expression is measured in dimensionless units T= coot, co,= (2V,//&$J*)“*, where ,u~ is the particle mass, corresponding to the reaction coordinate, and Q,, is the PES splitting parameter. At IQI > Qo. the minimum-energy path is one-dimensional (X=0) while at IQ I < Qo, the PES is split into two channels (fig. 1) with the MEP of the form X= k (n/cr)“’ (Q;-Q2)“2.
(2)
An equipotential contour plot is shown in fig. 1. The barrier along the MEP is lower than in the one-dimensional case ( - AVxnZQd/4a) but the particle path length increases, which may be described in terms of effective mass [ 5,6,17] M*= I +1/a. If AV/ Lb-A/a, the Gamov factor (barrier transparency) along the MEP exceeds the one-dimensional one and splitting of the trajectory is expected. Comparing the time of transversal transition between the channels, tfx (a2V/&Y2) ;& =Q;‘A-‘/2, with the time of passing the split part of potential, rO=Qo/~,, and supposing, similarly to refs. [ 6,17 1, that splitting occurs at ro> TV,one obtains the same splitting condition, which can be rewritten in the form Q:z=Q:=v:lA.
I312
I
S=
Hdr,
(4)
p= 1/k, T .
The instanton equations (SS/SQ= SS/SX=O), ..
av
e= @+lQx2 ’ X=~X(Q~-Q;)+L~X~,
518
(6)
are subject to the periodic boundary conditions Q(/I)+Q(O),x(p)tX(O). Thereisasolutionofeq. (6) corresponding to the one-dimensional instanton: X=@
i’ Q=av/aQ.
(7)
The question is to find out in which region of parameters A,(Y,Qo, the solution (7) is stable and when the two-dimensional trajectory occurs. To study the stability of (7), it is suffkient to restrict oneself to a quadratic (Gaussian) approximation, whereas the quadratic corrections to the action along trajectory (6) lead to the prefactor [ 81. According to ref. [ 181, the solution (7) becomes unstable when the lowest eigenvalue of the differential operator -=-dr2+ 6X2(T)
d2
PV(Q,
X)
ax2
is negative. The equation for its eigenvalues is formally equivalent to the Schriidinger equation in a “periodic potential”: --
d2X(r) dT2
+n[Q2(~)-Q~IXn(~)=t,X,(r),
xl(~~=x(~+P)
Fig. I. Equipotential contour plot for the potential ( I), v(Q)=Q4-2QZ,Qr,=0.5,1=20,~=40.TheMEPsareshown.
(5)
-P/2
S2S
(3)
Here, v,,,= (2V)‘12 is the maximum imaginary-time under-barrier velocity. The transition probability between two minima is determined by the action along the periodic trajectory in the inverted barrier, named the instanton: rot exp(- voS/oo) ,
22 November 199I
CHEMICAL PHYSICS LETTERS
(9)
When LO, the “potential” is “shallow” so that the only “energy level” is situated near the maximum A(Q$ -Qi) =co, where Qm(/?) is the 1D instanton amplitude (maximum Q(z) for solution (7)). Hence, the one-dimensional motion is stable at Qi > Q8 and unstable in the opposite case. This situation is also realized for QmCK1. In a more interesting case of large transversal frequencies 1 B 1, the solution of (9) may be found in the harmonic ap proximation. Expanding the f( r) = LQ’( 7) function into a Taylor series around the “potential bottom”, T= $a one obtains the harmonic oscillator equation, .
Volume186,number 6
=
CHEMICALPHYSICSLETTERS
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22 November 1991
(10)
with the eigenvalues equal to (if”)-I’*
(c,+nQ;) =1+2n,
n=O, l,... , (11)
so that the splitting condition tn< 0 amounts to (3) since
The phase diagram of the system is shown in fig. 2, from which it follows that the 1D instanton stability region widens with decrease of T. Curve I, determined by (3), separates the region of stability of the ID instanton (i) and the 2D instanton region (ii). At Qi > (u,,,/~)‘/*= (2/A)l/*, the instanton, if it exists, is always two-dimensional. In the region (iii), the thermal activated transitions through the saddlepoint (Q=O, X=X,= rtQ&/a,)t/2) +I’take place. Curve 3 corresponds to the second-order phase transition and is described by the usual equation for
PI Wesupposethata2V/aQzI,=,o=o=4[1(A2/4~)Q$]~0. p2 The concrete potential chosen in fig. 2 is V(Q) = Q’-2Qz so that 8, =n[ I- (A/~cx)Q~]-~‘~.
Q,Z
Fig. 3. Maximum instanton velocity v,,,= 2”‘*k/( I + k’) and instanton amplitude Qm=2”zk( I t/c*)-“’ versus p=2( 1t k2)‘/*K(k) for the Q4-2Q2 potential [6]; K(k) is the first-order elliptic integral.
the cross-over temperature (see for example refs. 1341) ‘*> fi, =~~T(~~I’/~Q~) -I’*.
(12)
Near the mutual point of all three curves, Qm==zz 1 and the quasiclassical approximation for (9) breaks down so that one should use the “geometric” splitting condition, Q,,,= Q,, (curve 2), as in the case of small 1. At Qi<(2/n) ‘I*, the second cross-over temperature corresponding to the instanton splitting appears which satisfies eq. (3 ). We show further that transition across curve 1 is also similar to the second-order phase transition and find the split trajectory near the ID-2D cross-over point. The eigenfunction of (lo), corresponding to the unstable mode, is a Gaussian function at z< 1: X0(s)= exp( --fr2 v,L’/*) ,
(13)
and falls off strongly during the time, t% (v,A”*)-“*, which is shorter than the characteristic instanton time (?d< 1). Introducing the characteristic dynamical splitting length, Id= v,z~< Q,, allows one to rewrite (3) in the form Qoal,,, which covers both the LZP 1 and ,J-ZK1 cases. We search for the split trajectory in the form X(r)=&,(t), Fig. 2. Bifurcation diagram for the potential (I ), V(Q)= Q’-2Q*, a=90,1= 16. The curves (l)-(3) satisfy the equations Q$=u,L-“~, Qa=Q,&9) and @c( 1-dQi/4@)-‘/‘, respectively, and separate the 1D instanton (i), 2D instanton (ii) and the Arrhenius (iii) regions.
QW=dW+~QW.
(14)
Steepest-descent integration over the unstable mode (which takes into account the fourth-order terms in the expansion of the action by the coefficient c) gives rise to 519
Volume 186,number 6
In order to check on the approximate approach, we nowderive the criticalcondition for V(Q) = Q4- 2Q’ at T=O.Substitution of Q(T)=tanh(z$) for (9) leads to the “Schriidinger equation” for the Eckart potential,
!3/2
X
22 November I991
CHEMICALPHYSICSLETTERS
I
X;(T) dr+$tc4 >I -P/Z
d*&(r) JX”(~) -- dz2 - cosh2(& = ita +4Qi
A&$ (Q:-Qf)?’
1 X;(r)dT. -I@/2
(15)
,
(211
with the “zero-level” eigenfunction, X,,(r)= [cosh(zJi)]-[“+2~““-‘~‘2,
The coefficient r includes all the fourth-order terms and F corresponds to the minimum action. Substituting (14) for (6), we obtain
-
- 1) l&(7)
$$+AX(@-Qf)tAX(Q,z-Q;,
tuQx3Q+ax3=0.
(16)
From multiplying the second equation by X(Z)= 2X0(2) and integratingit over the period, we obtain the followingequation for E*:
(22)
at the critical splitting parameter, (23) At /~-PO,Q,,=Qc.At 1~ 1, (23) coincides with (3) since u,= 2’/‘. The dependence S(8) is shown in fig. 4. In accordance with (3), as Q, grows and approaches the critical value a, the two-dimensionalinstanton &gion widens and the I D-2D cross-over temperature tends to zero (&co). At Q; %QZ, the action can be evaluated as (24)
812 812 1 X&)drt2AF2 oX:8Qd7 -P/Z -B/Z
J
P(Qf-Q;) P/2 t&Y
5
X;(z) dr=O .
(17)
-P/2
To solve the first of eqs. ( 16) , we note that at 1~ 1, X,(r) varies much faster than o(r) so that one can replace a2V/aQ2and Q by a2V/aQzIaE,= LJi and v,,,r,respectively.Since d2/dt2 w1- ‘I2a L#, the correction in Q(T) is equal to RIP0f 6Q(r)=- 2 exp( -s2u,l’/*) 512
0
(18) a
Substitution of (18) for (17) gives, finally,
(19) A&-f 520
n’l=v~2R-‘f=(Q~_Qc2)E2.
(20)
4
6
s
B
Fig. 4. Action S versus fiat a splitting parameter Q0 equal to 0, 0.330, 0.346, 0.350 for the curves l-4, respeotively. Potential Q”-2Q2, A= 16, a=90. In the Arrhenius region S=/3( I d*Q$4a), the 1D instanton action equals: S- 16( l+k2)-3’2 x~(ltk2)E(k)-(I-kz)K(k)]+~[1-4kz(l+kZ)-Z]. The dashed line correspondsto the two-dimensionalinslanton region.
Volume 186, number 6
CHEMICAL PHYSICS LETTERS
22 November 1991
paths and the temperature dependence of the rate constant, I&CCexp[ -S(B) 1, contains a wide region of low apparent activation energy (compared to V,) (fig. 4). Seemingly, this situation is realized in refs. [ 12,13] since the low-temperature plateau of KCcorresponding to the ID instanton is not reached. The synchronous transfer can be observed when AV is small as compared to V, and the temperature region of existence of the 2D instanton (/&dp
Fig. 5. Action Sversus AV=d*Qg/4a at 8=4.0,5.0,6.0 for curves 1-3, respectively.
so that the action decrease is determined by the static barrier lowering along the two-dimensional trajectory and the dynamical instanton fall-off time, rd, which does not correspond to the time of passage of the split part ofthe potential, so N3.According to (24) the relative decrease of action compared to the onedimensional case is smaller than the relative decrease of potential AV/ Vby the factor of z r&O. Fig. 5, where the dependence of S versus AV is represented for different temperatures demonstrates this fact. The dashed line for the 2D instanton connects the 1D instanton and Arrhenius regions where the action value is readily obtainable. At A V/ V60.5, the relative action decrease compared to the one-dimensional instanton action makes up w 0.15. As seen from our study, an alternative consideration of synchronous (along trajectory (7) ) and asynchronous (along the MEP) transfer is rather conventional. When the 2D instanton arises at /3> fl,, the split trajectory coincides with neither of these two x3 It follows from ( 13) that at 1~ 1,the r., time is the geometric average of T,,and the transversal passage time so that the condition of splitting can be expressed as 7, c T, = (T,s,) “’ c q,. At a-0, rd=r@
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