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Tunnelling of two interacting particles moving parallel or antiparallel A comparative analysis
Chemical Physics ELSEVIER
ChemicalPhysics
183 (1994)
I-IO
Tunnelling of two interacting particles moving parallel or antiparallel. A comparative analysis Yuri I. Dakhnovskii Institute of Chemical Physics, Academy of Sciences
ofRussia,Kosygin Street 4,
II 7977Moscow GYP-l, Russian Federation
M&hail B. Semenov * De~r~Ineai
ofTheoreticaf Physics. h4ascow State ~niversi~, Lenin Hilts, 119899 Moscow, Russian Federadon Received 13 September 1993
Abstract The tunnellingtransition of two quantum particles int~mcting with each other and moving in parallel or antip~allel diction has been considered. The interaction between tunnelling particles has been taken into account for calculation of the twodimensional quasiclassical (instanton) action, the consideration being restricted to the exponential part only. At some critical temperature T, and value of the interaction constant for parallel moving particles there is a splitting of the two underbarrier degenerated trajectories from the single basic one. For the antiparallel case only the basic path takes place.
1. Introduction Quantum particle transfer by tunnelling is of importance in various physical and chemical processes [ l-321. Up to now usually quantum tunnelling of one particle interacting with phonons (a heat bath) has been considered. First
the temperature dependence of the probability for a metastabfe decay of two coupled Josephson junctions has been considered in Ref. [ lo]. For some two-Dimensions low-temperature adiabatic chemical reactions of ~ansition of protons in the condensed phase [ 17-19,22-24,261 it is of importance to take into account the tunnelling contribution to the reaction rate as well as a profound study of the tunnelling mechanism. Therefore it is of interest to investigate tunnelling of two particles interacting with each other and moving both parallel and antiparallel along the reaction coordinates. These types of tunnelling transitions are considered in this paper in the frame of the one-instanton approximation. When the interaction parameter is equal to zero the tunnelling motion of the particles may be considered as being independent and the semiclassical action is equal to twice the value of the action for a one-particle transition [ 30,3 11. For the strongly interacting particles, their dynamics can be approximately described as the motion of a single particle with double mass. In fact the interaction has intermediate values, and therefore it is necessary to study the system dynamics in detail as a function of the coupling parameter. * ~o~s~nding
author. Addressfor letters: Krasnayastreet 65, Apt. 8, Penza 440026,Russian Federation.
0301-0104/94/$07.00 0 1994Elsevier Science B.V. All rights reserved SSDIO301-0104 (94)00035-9
Yd.
Dakhnovskii,
M.B. Semenov /Chemical
on” the interaction)
l-10
b)
URJ
f
Fig. 1. The potential energy (before “switching (a), and antiparallel (b).
Physics 183 (1994)
as a function of the reaction coordinates
for the particles moving parallel
The aim of this work is to study the dependence of the tunnelling rate constant on the temperature and twoparticle interaction coefficient for two physically different cases: (a) the particles move in parallel directions; (b) the particles move antiparallel [ 321. for this purpose the general formalism of the instanton approach for quantum tunnelling with dissipation [ l-161 for chemical reactions has been used [30-321, thus giving the possibility to study the interaction with an oscillator bath too. The problem of two-dimensional tunnelling is more complicated than the problem of one-dimensional tunnelling, and it cannot be factorized, i.e. it is impossible to reduce the two-dimensional motion to the sum of two independent one-dimensional ones. The characteristic feature of the two-dimensional case is the appearance of two split equivalent tunnelling paths instead of a single one. Therefore the main question is to clarify the type of tunnelling path which gives the main contribution to the tunnel action. It is of interest to find the conditions of the transition between these two kinds of tunnelling mechanisms.
2. Formalism For parallel moving tunnelling [ 30-321: U,(R,,
R2) = $u2(R,
(R, -b)*] and for antiparallel
+u)~O( -R,)
B(R,) + $02(R2 +a)%( tunnelling
+ [ - AZ+ fco2 -r2)
+ [ - AZ+ +*(R,
-b)*]
B(R,) - ;a(R,
-R,)‘,
(1)
-R2)2,
(2)
(b)
U*(R,, R2) = fWZ(R, +a)2e( (R, -b)2]
particles let us choose the potential energy U, (R,, R2) in the following form (a)
-R,)
+ [ - AZ+ $0’
6(R,) + 4m2(R2 --u)~O(R,)
+ [ - AZ+ fu2(R,
+b)2] O( -R2)
- ;a(R,
where AZ= fw’( b* - u2) is the reaction heat; LYis the coefficient of interaction between the particles ( LY> 0). For the particles moving parallel (a), and antiparallel, (b) , the potential energy (before “switching on” the interaction) is depicted in Fig. 1 as a function of the reaction coordinates. We are interested in the transition probability per unit time, or, strictly speaking, its exponential part [ 31. We seek the transition probability in the Langer form [ 31 r=2TImZ/ReZ, where
(3)
I’d. Dakhnovskii, MB. Semenov/Chemical Physics I83 (1994) l-10
DRI
DR2
DQj exp[ -SIR,,
R2,
Qi II
3
(4)
is the partition function of the complete system. The appearance of the imaginary part comes from the decay of energy levels in the initial state well. S is the action of the total system. Following the reasoning of ref. [ 151 it is possible to evaluate approximatively the exponential part (i.e. the action) in closed form. Integrating out the phonon coordinates [ 71 (if each of the particles is assumed to interact linearly with an oscihator both) we obtain
Ii,”
SIR,, R2}=
1 +2
+ T
+V(RI,K~
PI:!
I
~~‘~(~-~‘)[R,(~)+R~(~)I[R,(~‘)+R~(T’)I
(5)
- grz
where
(6) Y= 21&p is the Matsubara frequency, /3= h/k& T is the temperature, and D( T) is the phonon Green function. The two-dimensional semiclassical trajectory (instanton) minimizing the action functional S is determined from the equations of motion SS -=
SR,
0,
g=o. 2
We seek the solution of these equations in the following form: R,(r)=
~n~mR~l’exp(iv,,~lt
R2(7) = f
i n
RL2’ exp(iv,,r)
.
(8)
m
It should be noted that the solutions of Eqs. (7) are determined under the assumption of simultaneous motion of the particles. The times TV and 72 correspond to the instants when the particles pass the tops of the barriers [ 30-321, and can be derived from the equations R,(T~)=~,
R2(~2)=0.
(9)
Eqs. (9) permit to change the arguments of the &functions, instead of the O( &RI) and O( + R2) should be 6?(rfrpi- T,.~) + 6( T- T,,~). Such a substitution reduces Eqs. (7) to the linear ones which can be easily solved. substituting the trajectory found into Eq. (5) we obtain the instanton action. For antip~allel motion the tunnel action has been obtained in ref. 1321.
3. Results and discussion 3.1. Parallel motion of the tunnelling particles In the case of parallel motion of the tunnelling particles (potential energy U, (R,, R2), Eq. ( 1)) the action S (when the interaction with an oscillator bath is absent) as the function of parameters T,, 72 is equal to
Yu.I. Dakhnovskii, M.B. Semenov / Chemicul Physics 183 (1994) I-IO
S=2!a(b+a)(T,
W2(a+b)2(T,
+T2)W2-
- 2w4(a+b)2 P
-
+Tz)*
p
(sin u,ri +sin
EC
v,‘(
tl=l
vi
+
W4(a+b)2(ri -
v,r,)’
+(
(w2-2Cr)P
sin ~“‘,7,-sin 4
d)
-72)2
v;
~,r~)’
+W2-2Cz)
(10)
1.
Let us give the definitions: E= E*W= ( T, - r2)w; 7=27*0= (r, +T*)w; /!I*= @i/2; cu*=2a/w2; b*=bla, b&a. In the absence of interaction with phonons (an oscillator bath) the action S ( 10) as a function of the parameters E and T is given in the Appendix (IQ. (A. 1) ) Then Eqs. (9) look as sh E [ch rcth /l* - sh r- cth /3*] + --&sh(e~i?)[ch(&-?) 3---_
4
1
l+b*
1-a*
cth(p*J1_cu”)
-sh(7JI_CU*)
+cth(/3*41_dr)]
=O,
+che[shrcthP*-chr-l]+shrcthP*-chr
+ -&ch(eji?)[sh(&i?)cth(p*J1-orC)-ch(n/l-)+l]
- &
[sh(d=)
cth(p*J1_ol*)
-ch(rJ1_ol*)]
The solutions of Eqs. ( 11) are obtained in the following e=(r,
-T2)6J=0,
=O.
(11)
form:
trp, cY
At sufficiently low temperature co2<2(b-a)/(3b-a)=a,*, exp(-n/l-or*)=
x
3-
(o/3 >> 1) with exponential
&
-
(12) accuracy for 1
< 3; (b-a)
/2( b +a) < 2cz/
(l_a*)ll’l-Gx
l+(l_(y*)l/(l-m)
e -cz
+
1
exp( - 77/1_cy*)
,
(13)
wherecu*=2dW2;b*=bla. The solution ( 13) is valid for p>-
bln[exp(-rJ1_af)]=p,
(14)
The approximate solution can be written for large values of the parameter b * = b/a (and small a*), but we restrict ourselves to the more important physical solution ( 13). For the low temperature limit there is no perturbation solution of Eqs. ( 11) . For E= 0 (see ( 12) ) the action S has the following form (A. 1) :
Yu.1. Dakhnovskii,
M.B. Semenov /Chemical
Physics 183 (1994) l-10
Fig. 2. Two-dimensional trajectories (the basic path (E= 0) and splitting paths ( E# 0)) at o/3 B are the projections of the potential energy U, (R,, R2) minima.
+w(b+a)2
1 for two parallel moving particles.
( 1 )-( 4)
02(b2-a2)f? 2
ch(w/3/2)-{1+[(b*-1)/(b*+1)]2sh2(o~/2))”2
(15)
sh( op/2)
This expression coincides with that calculated in refs. [ 30,3 1 ] but multiplied Inthecaseofb=a(b*=l) ande=Oweobtain
by the factor two,
S=40a2th(wp/4).
(16)
This expression clearly shows the temperature dependence of S( p) . The character of this dependence is practically unchanged for b* > 1. We do not present the cumbersome solution for Scczo) obtained from the substitution of (13) into (A.l). Comparing Sce=O) and ScEZO), it is easy to show that S~azO)
k----------_____--______-_-___(I) \
i.’\ \
\
\
I 0
Fig. 3. The dependence of the action on the interaction path,c=O,(l);andonthesplittingpaths,~+O,(b-a)/(b+a)=0.1
‘\
0.05
parameter
(2)
0.1
((I * = 2a/o*) (2)).
0.15a&c/&+ at op z+ 1 for the parallel moving particles,
(on the basic
Yu.1. Dakhnovskii, M.B. Semenov/Chemicul
6
Physics 183 (1994) I-10
( EZ 0) , as shown in Fig. 2. For /3 > PC (T < T,) the latter takes place (E # O), because from the comparison of the action values SCrzO) and SCs+O) the latter is seen to be smaller (see Fig. 3). For p < PC (T> T,) and (Y> CY,(see Eq. ( 13) ) there is only a single trajectory. In the case of a symmetric potential (b = a) the single trajectory solution always takes place for the whole temperature range. 3.2. Antiparallel
motion of the tunnelling particles
The case of antiparallel motion of the tunnelling particles is more applicable to the investigation of the two-proton low-temperature adiabatic transfer [ 17-19,22-261. In this case (model (b) , Eq. (2) ) the action as the function of E and r has been calculated in ref. [ 321 (see (A.2) ) . The parameters E and rare governed by the following system of equations (see Eqs. (9) ) : -she(cthp*+chrcth/3*-shr) + -&
sh(e\/1-*)[cth(/l*J1_ol*) 4
-l-
1 + 1-a*
(l+b*)(l-a*>
-ch(dm)
cth(p*{=)
+sh(7JI_CU*)l
=O,
+(che-l)(shrcthP*-chr)+che
+ -&-([ch(eJI-ol*)+l][sh(iJ1--(yl)cth(~*\ll-a*)-ch(7J1-a*)l -ch(edl--*)}=O.
(17)
The solutions of Eqs. ( 17) are obtained in the following e=(7,
-7*)w=O,VP,
form:
cr
1 20 = 20\/1--cy* 7
r=r2=
arcsh
Similar to (a) at low temperature exp( - rJiYP)
e’z
=
A-[1/(1-a*)]
+ $.
( wp x=- 1)) and with exponential
(18)
accuracy,
A(l-r~*)“~ ’
l-(l-a*)“YIA/y-ll(l-a*)] exp(-dg)
(19)
exp( -rJCGP) where AZ-~-
4 (l+b*)(l-a*)
-. 3 + 1--a*
y=l-&cy* and cr *, b *, E, 7 are determined as for case (a). This solution is correct for cr; < LY*< cr,*z,where (Y:, and LY,*~ are derived from a cumbersome trancendental equation (which is not given here). But for a= b, I < 2alw2 < 1. The approximate solution can be written for the larger values of the parameter b * = b/a (and small but we restrict ourselves to the more important physical solution ( 19). The solution (19) is correct for
L*),
A( 1 -a*) l-(l-a*)“YIA/y-l/(l-~*)]
“7
(20)
Yu.I. Dakhnovskii, MB. Semenov / Chemical Physics 183 (1994) l-10
Fig. 4. The two-dimensional (l)-(4) are the projections
trajectories (the basic path ( E = 0) and the splitting ones ( E # 0) ) at o/3 3p 1 for two antiparallel of the potential energy U,(R,, R2) minima.
moving particles.
where A and y were defined above. For wp B 1 the solution of Eqs. ( 17) can also be found from the perturbation theory (for small E) at some values of the parameters (b -a) / (b + a) and cy*. For E=O (solution (18)) the action S (Eq. (A.2)) has the following form
o(b+a)* + (l__o*)s/* For the symmetric 4wa2 ‘=
(1_a*)3/2
ch(;o&?p)-{l+[(b-a)l(b+a)]*sh*(+~=p)}*’* sh( $od=p)
(21)
potential when b = a and E= 0 (see Fig. 5)) th($wd=)
.
(22)
SCEZO) > &=O) for fi> PC (see (20)) and for appropriate (Y*. We do not present the cumbersome expression for s Cc+“) obtained from the substitution of solutions for rand E into (A.2). As for the case of parallel motion the twodimensional tunnelling paths are found from Eqs. ( 18), ( 19). These trajectories on the (R,, R,)-plane are depicted in Fig. 4. As for the parallel case there is a splitting of the single trajectory into the two generated ones for p > /3,. In contrast to the previous case such splitting takes place for any potential parameters. For /3> PC, SCczo) > SC6=O)r and therefore the latter determines the tunnelling rate. If p < PC the degenerated paths are transformed to a single one.
4. Conclusions As has been mentioned in the Introduction there is a qualitative difference between tunnelling of one and two interacting particles. The problem of two-dimensional tunnelling is more complicated than that of the one-dimensional case because it can not be factorized into two one-dimensional motions. For one-particle tunnelling there is only one tunnel path (instanton) at which the action has a minimum value. In the two-dimensional case there are two types of trajectories, i.e. the main contribution to the instanton action is determined by the single path or by the doubly degenerated ones. In the latter case for parallel tunnelling the particles do not simultaneously (7, # T*) pass the tops of the barriers. It means, that the tunnelling transfer is asynchronous.
8
Yd.
Dakhnovskii,
M.B. Semenov /Chemical
Physics 183 (1994) I-IO
But, at a definite value of the interaction parameter (small values of CX*(see ( 13) ) and for temperatures /3 < pL. (see ( 14)), the symmetry of the two-dimensional path can be changed, i.e. there is no splitting of paths from the single one (R, =&); the particles pass the tops of the barriers along both the reaction coordinates at the same instants (7, = r2). Thus the tunnelling transfer of particles is synchronous. The dependence of the action on the interaction parameter for parallel transfer is depicted in Fig. 3. The type of interaction chosen in ( 1) and (2) does not influence the motion along the “center of mass” coordinate (R, =R,). For this reason the semiclassical action does not depend on the interaction parameter in the case of parallel transfer along the basic path. Since the state of the interacting system happens to be energetically preferable when the value of the relative coordinate (R, = -I&) has a maximum, it is clear why the semiclassical action decreases with the increase of the interaction parameter in the case of parallel transfer along the degenerated tunnelling trajectories and why it increases with the increase of the interaction parameter in the case of the antiparallel transfer. For antiparallel tunnelling the synchronous transfer takes place, but asynchronous transfer is energetically forbidden, because it gives a greater contribution to the semiclassical action (Fig. 5). For parallel and antiparallel cases the influence of the oscillator bath does not qualitatively change the character of tunnelling. But quantitatively the interaction with phonons differently affects on antiparallel and parallel transfer. From Eq. (5) it follows, that a medium affects the motion of the center of mass (R, = R2). Therefore, in the case of antiparallel motion there is no bath influence on the rate constant. For the parallel transfer it always takes place. Thus, expressions for the action (5) in the one-instanton approximation have been found in the models with adiabatic potential energies ( 1) and (2)) and a comparative analysis of the tunnelling of two interacting particles, moving parallel ( 1) and antiparallel (2)) has been performed. Appendix 1. In the case of parallel motion of the tunnelling form
-
o(a+6)2
-cth
2 -
p*+
-&
(
w(a+b)2 2(1-,*>3’*
[ch(P*-
( -cth(p”dg)
particles the action S as a function of parameters E and r has the
$che+ch(p*-T)-ch(/3*1
+
16‘1)]
ch[(p*-r)d=]
ch(ed=)
sh(p*da)
-ch[(p*-r)JI--;;;;‘J+ch[(p*-
,e,)d=,l).
(A.])
where the parameters E, r, (Y*, /?* were determined above. 2. In the case of antiparallel motion of the tunnelling particles the action S as a function of parameters E and T has the form s= -
W7(!P-a2) l-a”
-
+ ch(e\lI-CU*) (l-a”)3’2 +
$$
+ 1
[ch(/?*-7)
[ E( ’ ’ l-
“(a:b)2 1 sh(p*dm)
+ch fl*l
&)+
{ch[(p*-r){=
1.
““:i”~’
-shIel
] -ch(p*d=
(A.21
Yu.1. Dakhnovskii, MB. Semenov /Chemical Physics 183 (1994) I-10
Fig. 5. The temperature dependence of the action of antiparallel (2) a*=O.l. (3)isthecaseofparallelmovingparticles(e=O,b=a).
moving particles ( l = 0, b = a) for different interaction parameters:
(1) a* = 0.5;
The single particle action for piece-wise harmonic bistable potentials has been obtained first in ref. [ 151
References [ 11 I. Affleck, Phys. Rev. Letters 46 (1981) 388. [2] P.G. Wolynes, Phys. Rev. Letters 47 (1981) 968. [3] J.S. Langer, Ann. Phys. (NY) 41 (1967) 108. [4] S. Coleman, Phys. Rev. D 15 (1977) 2929. [5] A. Schmid, J. Low Temp. Phys. 49 (1982) 609. [6] J.P. Sethna, Phys. Rev. B 25 (1982) 5050. [7] A.O. Caldeira and A.J. Leggett, Ann. Phys. (NY) 149 (1983) 374. [8] A.I. Larkin and Yu.N. Ovchinnikov, Phys. Rev. B 28 (1983) 6281. [9] B.l. Ivlev and V.I. Melnikov, Zh. Eksp. Teor. Fiz. 91 (1986) 1944. [IO] B.I. Ivlev and Yu.N. Ovchinnikov, Zh. Eksp. Teor. Fiz. 93 (1987) 668. [ 111 H. Grabert, U. Weiss and P. Hanggi, Phys. Rev. Letters 51 (1984) 2242. [ 121 A. Schmid, Ann. Phys. (NY) 170 (1986) 333. [ 131 A.J. Leggett, S. Chakravarty, A.T. Dorsey, M.P.A. Fisher, A. Garg and W. Zwerger, Rev. Mod. Phys. 59 (1987) 1. [ 141 A.D. Zaikin and S.V. Panyukov, Zh. Eksp. Teor. Fiz. 89 (1985) 1890. [ 151 II. Weiss, H. Grabert, P. Hiinggi and P. Riseborough, Phys. Rev. B 35 (1987) 9535. [ 161 B.I. Ivlev, Zh. Eksp. Teor. Fiz. 94 (1988) 333. [ 171 K. Tokumura, Y. Watanabe and M. Itoh, J. Phys. Chem. 90 (1986) 2362. [ 181 Z. Smedarchina, W. Siebrand and T.A. Wildman, Chem. Phys. Letters 143 (1988) 395. [ 191 Z. Smedarchina, W. Siebrand and F. Zerbetto, Chem. Phys. 136 (1989) 285. [20] R.L. Hudson, M. Shiotani and F. Williams, Chem. Phys. Letters 48 (1977) 193. [21] S.L. Buchwalter and G.L. Closs, J. Am. Chem. Sot. 101 (1979) 4688. [22] B.H. Meier, F. Graf and R.R. Ernst, J. Chem. Phys. 76 (1982) 767. [23] A. Stockli, B.H. Meier. R. Kreis, R. Meyer and R.R. Ernst, J. Chem. Phys. 93 (1990) 1502. [24] R. Meyer and R.R. Ernst, J. Chem. Phys. 93 (1990) 5518. [25] R.P. Bell, The tunnel effect in chemistry (Chapman and Hall, London, 1980). [26] S.Ya. Ishenko, M.V. Vener and V.M. Mamaev, Theomt. Chim. Acta 68 (1985) 351. [27] V.A. Benderskii, V.I. Goldanskii and A.A. Ovichinnikov, Chem. Phys. Letters 73 ( 1980) 492. [28] K.I. Zamaraev and R.F. Khairutdinov, Usp. Khim. 47 (1978) 992. [29] V.A. Benderskii, P.G. Philippov, Yu.1. Dakhnovskii and A.A. Ovchinnikov, Chem. Phys. 67 (1982) 301. [30] Yu.1. Dakhnovskii, A.A. Ovchinnikov and M.B. Semenov, Zh. Eksp. Teor. Fiz. 92 (1987) 955. [Sov. JETP 65 (1987) [31] Yu.1. Dakhnovskii, A.A. Ovchinnikov and M.B. Semenov, Mol. Phys. 63 (1988) 497. [32] Yu.1. Dakhnovskii and M.B. Semenov, J. Chem. Phys. 91 (1989) 7606.
5411
ChemicalPhysics
183 (1994)
I-IO
Tunnelling of two interacting particles moving parallel or antiparallel. A comparative analysis Yuri I. Dakhnovskii Institute of Chemical Physics, Academy of Sciences
ofRussia,Kosygin Street 4,
II 7977Moscow GYP-l, Russian Federation
M&hail B. Semenov * De~r~Ineai
ofTheoreticaf Physics. h4ascow State ~niversi~, Lenin Hilts, 119899 Moscow, Russian Federadon Received 13 September 1993
Abstract The tunnellingtransition of two quantum particles int~mcting with each other and moving in parallel or antip~allel diction has been considered. The interaction between tunnelling particles has been taken into account for calculation of the twodimensional quasiclassical (instanton) action, the consideration being restricted to the exponential part only. At some critical temperature T, and value of the interaction constant for parallel moving particles there is a splitting of the two underbarrier degenerated trajectories from the single basic one. For the antiparallel case only the basic path takes place.
1. Introduction Quantum particle transfer by tunnelling is of importance in various physical and chemical processes [ l-321. Up to now usually quantum tunnelling of one particle interacting with phonons (a heat bath) has been considered. First
the temperature dependence of the probability for a metastabfe decay of two coupled Josephson junctions has been considered in Ref. [ lo]. For some two-Dimensions low-temperature adiabatic chemical reactions of ~ansition of protons in the condensed phase [ 17-19,22-24,261 it is of importance to take into account the tunnelling contribution to the reaction rate as well as a profound study of the tunnelling mechanism. Therefore it is of interest to investigate tunnelling of two particles interacting with each other and moving both parallel and antiparallel along the reaction coordinates. These types of tunnelling transitions are considered in this paper in the frame of the one-instanton approximation. When the interaction parameter is equal to zero the tunnelling motion of the particles may be considered as being independent and the semiclassical action is equal to twice the value of the action for a one-particle transition [ 30,3 11. For the strongly interacting particles, their dynamics can be approximately described as the motion of a single particle with double mass. In fact the interaction has intermediate values, and therefore it is necessary to study the system dynamics in detail as a function of the coupling parameter. * ~o~s~nding
author. Addressfor letters: Krasnayastreet 65, Apt. 8, Penza 440026,Russian Federation.
0301-0104/94/$07.00 0 1994Elsevier Science B.V. All rights reserved SSDIO301-0104 (94)00035-9
Yd.
Dakhnovskii,
M.B. Semenov /Chemical
on” the interaction)
l-10
b)
URJ
f
Fig. 1. The potential energy (before “switching (a), and antiparallel (b).
Physics 183 (1994)
as a function of the reaction coordinates
for the particles moving parallel
The aim of this work is to study the dependence of the tunnelling rate constant on the temperature and twoparticle interaction coefficient for two physically different cases: (a) the particles move in parallel directions; (b) the particles move antiparallel [ 321. for this purpose the general formalism of the instanton approach for quantum tunnelling with dissipation [ l-161 for chemical reactions has been used [30-321, thus giving the possibility to study the interaction with an oscillator bath too. The problem of two-dimensional tunnelling is more complicated than the problem of one-dimensional tunnelling, and it cannot be factorized, i.e. it is impossible to reduce the two-dimensional motion to the sum of two independent one-dimensional ones. The characteristic feature of the two-dimensional case is the appearance of two split equivalent tunnelling paths instead of a single one. Therefore the main question is to clarify the type of tunnelling path which gives the main contribution to the tunnel action. It is of interest to find the conditions of the transition between these two kinds of tunnelling mechanisms.
2. Formalism For parallel moving tunnelling [ 30-321: U,(R,,
R2) = $u2(R,
(R, -b)*] and for antiparallel
+u)~O( -R,)
B(R,) + $02(R2 +a)%( tunnelling
+ [ - AZ+ fco2 -r2)
+ [ - AZ+ +*(R,
-b)*]
B(R,) - ;a(R,
-R,)‘,
(1)
-R2)2,
(2)
(b)
U*(R,, R2) = fWZ(R, +a)2e( (R, -b)2]
particles let us choose the potential energy U, (R,, R2) in the following form (a)
-R,)
+ [ - AZ+ $0’
6(R,) + 4m2(R2 --u)~O(R,)
+ [ - AZ+ fu2(R,
+b)2] O( -R2)
- ;a(R,
where AZ= fw’( b* - u2) is the reaction heat; LYis the coefficient of interaction between the particles ( LY> 0). For the particles moving parallel (a), and antiparallel, (b) , the potential energy (before “switching on” the interaction) is depicted in Fig. 1 as a function of the reaction coordinates. We are interested in the transition probability per unit time, or, strictly speaking, its exponential part [ 31. We seek the transition probability in the Langer form [ 31 r=2TImZ/ReZ, where
(3)
I’d. Dakhnovskii, MB. Semenov/Chemical Physics I83 (1994) l-10
DRI
DR2
DQj exp[ -SIR,,
R2,
Qi II
3
(4)
is the partition function of the complete system. The appearance of the imaginary part comes from the decay of energy levels in the initial state well. S is the action of the total system. Following the reasoning of ref. [ 151 it is possible to evaluate approximatively the exponential part (i.e. the action) in closed form. Integrating out the phonon coordinates [ 71 (if each of the particles is assumed to interact linearly with an oscihator both) we obtain
Ii,”
SIR,, R2}=
1 +2
+ T
+V(RI,K~
PI:!
I
~~‘~(~-~‘)[R,(~)+R~(~)I[R,(~‘)+R~(T’)I
(5)
- grz
where
(6) Y= 21&p is the Matsubara frequency, /3= h/k& T is the temperature, and D( T) is the phonon Green function. The two-dimensional semiclassical trajectory (instanton) minimizing the action functional S is determined from the equations of motion SS -=
SR,
0,
g=o. 2
We seek the solution of these equations in the following form: R,(r)=
~n~mR~l’exp(iv,,~lt
R2(7) = f
i n
RL2’ exp(iv,,r)
.
(8)
m
It should be noted that the solutions of Eqs. (7) are determined under the assumption of simultaneous motion of the particles. The times TV and 72 correspond to the instants when the particles pass the tops of the barriers [ 30-321, and can be derived from the equations R,(T~)=~,
R2(~2)=0.
(9)
Eqs. (9) permit to change the arguments of the &functions, instead of the O( &RI) and O( + R2) should be 6?(rfrpi- T,.~) + 6( T- T,,~). Such a substitution reduces Eqs. (7) to the linear ones which can be easily solved. substituting the trajectory found into Eq. (5) we obtain the instanton action. For antip~allel motion the tunnel action has been obtained in ref. 1321.
3. Results and discussion 3.1. Parallel motion of the tunnelling particles In the case of parallel motion of the tunnelling particles (potential energy U, (R,, R2), Eq. ( 1)) the action S (when the interaction with an oscillator bath is absent) as the function of parameters T,, 72 is equal to
Yu.I. Dakhnovskii, M.B. Semenov / Chemicul Physics 183 (1994) I-IO
S=2!a(b+a)(T,
W2(a+b)2(T,
+T2)W2-
- 2w4(a+b)2 P
-
+Tz)*
p
(sin u,ri +sin
EC
v,‘(
tl=l
vi
+
W4(a+b)2(ri -
v,r,)’
+(
(w2-2Cr)P
sin ~“‘,7,-sin 4
d)
-72)2
v;
~,r~)’
+W2-2Cz)
(10)
1.
Let us give the definitions: E= E*W= ( T, - r2)w; 7=27*0= (r, +T*)w; /!I*= @i/2; cu*=2a/w2; b*=bla, b&a. In the absence of interaction with phonons (an oscillator bath) the action S ( 10) as a function of the parameters E and T is given in the Appendix (IQ. (A. 1) ) Then Eqs. (9) look as sh E [ch rcth /l* - sh r- cth /3*] + --&sh(e~i?)[ch(&-?) 3---_
4
1
l+b*
1-a*
cth(p*J1_cu”)
-sh(7JI_CU*)
+cth(/3*41_dr)]
=O,
+che[shrcthP*-chr-l]+shrcthP*-chr
+ -&ch(eji?)[sh(&i?)cth(p*J1-orC)-ch(n/l-)+l]
- &
[sh(d=)
cth(p*J1_ol*)
-ch(rJ1_ol*)]
The solutions of Eqs. ( 11) are obtained in the following e=(r,
-T2)6J=0,
=O.
(11)
form:
trp, cY
At sufficiently low temperature co2<2(b-a)/(3b-a)=a,*, exp(-n/l-or*)=
x
3-
(o/3 >> 1) with exponential
&
-
(12) accuracy for 1
< 3; (b-a)
/2( b +a) < 2cz/
(l_a*)ll’l-Gx
l+(l_(y*)l/(l-m)
e -cz
+
1
exp( - 77/1_cy*)
,
(13)
wherecu*=2dW2;b*=bla. The solution ( 13) is valid for p>-
bln[exp(-rJ1_af)]=p,
(14)
The approximate solution can be written for large values of the parameter b * = b/a (and small a*), but we restrict ourselves to the more important physical solution ( 13). For the low temperature limit there is no perturbation solution of Eqs. ( 11) . For E= 0 (see ( 12) ) the action S has the following form (A. 1) :
Yu.1. Dakhnovskii,
M.B. Semenov /Chemical
Physics 183 (1994) l-10
Fig. 2. Two-dimensional trajectories (the basic path (E= 0) and splitting paths ( E# 0)) at o/3 B are the projections of the potential energy U, (R,, R2) minima.
+w(b+a)2
1 for two parallel moving particles.
( 1 )-( 4)
02(b2-a2)f? 2
ch(w/3/2)-{1+[(b*-1)/(b*+1)]2sh2(o~/2))”2
(15)
sh( op/2)
This expression coincides with that calculated in refs. [ 30,3 1 ] but multiplied Inthecaseofb=a(b*=l) ande=Oweobtain
by the factor two,
S=40a2th(wp/4).
(16)
This expression clearly shows the temperature dependence of S( p) . The character of this dependence is practically unchanged for b* > 1. We do not present the cumbersome solution for Scczo) obtained from the substitution of (13) into (A.l). Comparing Sce=O) and ScEZO), it is easy to show that S~azO)
k----------_____--______-_-___(I) \
i.’\ \
\
\
I 0
Fig. 3. The dependence of the action on the interaction path,c=O,(l);andonthesplittingpaths,~+O,(b-a)/(b+a)=0.1
‘\
0.05
parameter
(2)
0.1
((I * = 2a/o*) (2)).
0.15a&c/&+ at op z+ 1 for the parallel moving particles,
(on the basic
Yu.1. Dakhnovskii, M.B. Semenov/Chemicul
6
Physics 183 (1994) I-10
( EZ 0) , as shown in Fig. 2. For /3 > PC (T < T,) the latter takes place (E # O), because from the comparison of the action values SCrzO) and SCs+O) the latter is seen to be smaller (see Fig. 3). For p < PC (T> T,) and (Y> CY,(see Eq. ( 13) ) there is only a single trajectory. In the case of a symmetric potential (b = a) the single trajectory solution always takes place for the whole temperature range. 3.2. Antiparallel
motion of the tunnelling particles
The case of antiparallel motion of the tunnelling particles is more applicable to the investigation of the two-proton low-temperature adiabatic transfer [ 17-19,22-261. In this case (model (b) , Eq. (2) ) the action as the function of E and r has been calculated in ref. [ 321 (see (A.2) ) . The parameters E and rare governed by the following system of equations (see Eqs. (9) ) : -she(cthp*+chrcth/3*-shr) + -&
sh(e\/1-*)[cth(/l*J1_ol*) 4
-l-
1 + 1-a*
(l+b*)(l-a*>
-ch(dm)
cth(p*{=)
+sh(7JI_CU*)l
=O,
+(che-l)(shrcthP*-chr)+che
+ -&-([ch(eJI-ol*)+l][sh(iJ1--(yl)cth(~*\ll-a*)-ch(7J1-a*)l -ch(edl--*)}=O.
(17)
The solutions of Eqs. ( 17) are obtained in the following e=(7,
-7*)w=O,VP,
form:
cr
1 20 = 20\/1--cy* 7
r=r2=
arcsh
Similar to (a) at low temperature exp( - rJiYP)
e’z
=
A-[1/(1-a*)]
+ $.
( wp x=- 1)) and with exponential
(18)
accuracy,
A(l-r~*)“~ ’
l-(l-a*)“YIA/y-ll(l-a*)] exp(-dg)
(19)
exp( -rJCGP) where AZ-~-
4 (l+b*)(l-a*)
-. 3 + 1--a*
y=l-&cy* and cr *, b *, E, 7 are determined as for case (a). This solution is correct for cr; < LY*< cr,*z,where (Y:, and LY,*~ are derived from a cumbersome trancendental equation (which is not given here). But for a= b, I < 2alw2 < 1. The approximate solution can be written for the larger values of the parameter b * = b/a (and small but we restrict ourselves to the more important physical solution ( 19). The solution (19) is correct for
L*),
A( 1 -a*) l-(l-a*)“YIA/y-l/(l-~*)]
“7
(20)
Yu.I. Dakhnovskii, MB. Semenov / Chemical Physics 183 (1994) l-10
Fig. 4. The two-dimensional (l)-(4) are the projections
trajectories (the basic path ( E = 0) and the splitting ones ( E # 0) ) at o/3 3p 1 for two antiparallel of the potential energy U,(R,, R2) minima.
moving particles.
where A and y were defined above. For wp B 1 the solution of Eqs. ( 17) can also be found from the perturbation theory (for small E) at some values of the parameters (b -a) / (b + a) and cy*. For E=O (solution (18)) the action S (Eq. (A.2)) has the following form
o(b+a)* + (l__o*)s/* For the symmetric 4wa2 ‘=
(1_a*)3/2
ch(;o&?p)-{l+[(b-a)l(b+a)]*sh*(+~=p)}*’* sh( $od=p)
(21)
potential when b = a and E= 0 (see Fig. 5)) th($wd=)
.
(22)
SCEZO) > &=O) for fi> PC (see (20)) and for appropriate (Y*. We do not present the cumbersome expression for s Cc+“) obtained from the substitution of solutions for rand E into (A.2). As for the case of parallel motion the twodimensional tunnelling paths are found from Eqs. ( 18), ( 19). These trajectories on the (R,, R,)-plane are depicted in Fig. 4. As for the parallel case there is a splitting of the single trajectory into the two generated ones for p > /3,. In contrast to the previous case such splitting takes place for any potential parameters. For /3> PC, SCczo) > SC6=O)r and therefore the latter determines the tunnelling rate. If p < PC the degenerated paths are transformed to a single one.
4. Conclusions As has been mentioned in the Introduction there is a qualitative difference between tunnelling of one and two interacting particles. The problem of two-dimensional tunnelling is more complicated than that of the one-dimensional case because it can not be factorized into two one-dimensional motions. For one-particle tunnelling there is only one tunnel path (instanton) at which the action has a minimum value. In the two-dimensional case there are two types of trajectories, i.e. the main contribution to the instanton action is determined by the single path or by the doubly degenerated ones. In the latter case for parallel tunnelling the particles do not simultaneously (7, # T*) pass the tops of the barriers. It means, that the tunnelling transfer is asynchronous.
8
Yd.
Dakhnovskii,
M.B. Semenov /Chemical
Physics 183 (1994) I-IO
But, at a definite value of the interaction parameter (small values of CX*(see ( 13) ) and for temperatures /3 < pL. (see ( 14)), the symmetry of the two-dimensional path can be changed, i.e. there is no splitting of paths from the single one (R, =&); the particles pass the tops of the barriers along both the reaction coordinates at the same instants (7, = r2). Thus the tunnelling transfer of particles is synchronous. The dependence of the action on the interaction parameter for parallel transfer is depicted in Fig. 3. The type of interaction chosen in ( 1) and (2) does not influence the motion along the “center of mass” coordinate (R, =R,). For this reason the semiclassical action does not depend on the interaction parameter in the case of parallel transfer along the basic path. Since the state of the interacting system happens to be energetically preferable when the value of the relative coordinate (R, = -I&) has a maximum, it is clear why the semiclassical action decreases with the increase of the interaction parameter in the case of parallel transfer along the degenerated tunnelling trajectories and why it increases with the increase of the interaction parameter in the case of the antiparallel transfer. For antiparallel tunnelling the synchronous transfer takes place, but asynchronous transfer is energetically forbidden, because it gives a greater contribution to the semiclassical action (Fig. 5). For parallel and antiparallel cases the influence of the oscillator bath does not qualitatively change the character of tunnelling. But quantitatively the interaction with phonons differently affects on antiparallel and parallel transfer. From Eq. (5) it follows, that a medium affects the motion of the center of mass (R, = R2). Therefore, in the case of antiparallel motion there is no bath influence on the rate constant. For the parallel transfer it always takes place. Thus, expressions for the action (5) in the one-instanton approximation have been found in the models with adiabatic potential energies ( 1) and (2)) and a comparative analysis of the tunnelling of two interacting particles, moving parallel ( 1) and antiparallel (2)) has been performed. Appendix 1. In the case of parallel motion of the tunnelling form
-
o(a+6)2
-cth
2 -
p*+
-&
(
w(a+b)2 2(1-,*>3’*
[ch(P*-
( -cth(p”dg)
particles the action S as a function of parameters E and r has the
$che+ch(p*-T)-ch(/3*1
+
16‘1)]
ch[(p*-r)d=]
ch(ed=)
sh(p*da)
-ch[(p*-r)JI--;;;;‘J+ch[(p*-
,e,)d=,l).
(A.])
where the parameters E, r, (Y*, /?* were determined above. 2. In the case of antiparallel motion of the tunnelling particles the action S as a function of parameters E and T has the form s= -
W7(!P-a2) l-a”
-
+ ch(e\lI-CU*) (l-a”)3’2 +
$$
+ 1
[ch(/?*-7)
[ E( ’ ’ l-
“(a:b)2 1 sh(p*dm)
+ch fl*l
&)+
{ch[(p*-r){=
1.
““:i”~’
-shIel
] -ch(p*d=
(A.21
Yu.1. Dakhnovskii, MB. Semenov /Chemical Physics 183 (1994) I-10
Fig. 5. The temperature dependence of the action of antiparallel (2) a*=O.l. (3)isthecaseofparallelmovingparticles(e=O,b=a).
moving particles ( l = 0, b = a) for different interaction parameters:
(1) a* = 0.5;
The single particle action for piece-wise harmonic bistable potentials has been obtained first in ref. [ 151
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