Theory of hydrogen-transfer reactions based on a harmonic two-mode tunneling model: an analytical treatment

Chemical Physics North-Holland

161 (1992)

313-326

Theory of hydrogen-transfer reactions based on a harmonic two-mode tunneling model: an analytical treatment U. Schnabel and H. Gabriel Fachhererch Physrk, Frele Unwersdt Received

8 August

Berim, Arnvnaiiee 14, W-1000 Berhn 33, Gernranl

199 1: in final form 7 January

1992

Based on a hydrogen-transfer model proposed by Siebrand et al. an exact analytical expression for the temperature-dependent rate constant IS derived. In this model the potential barner separating the posittons of the transferred particle in the imttal and final configuratton is modulated by the reactant-product oscillation, the frequency of which is assumed to be small compared to the frequencies of the collinear mtramolecular vlbrattons ofthe hydrogen atom before and after the transfer. The possible changes in the latter is neglected. Moreover, adiabattc separabihty of the slow and fast oscillatton IS used. Thus, in the harmomc approximation used here, the dynamics of the system is that of two independent harmonic oscillators with frequenctes w and Q. There IS, however, explicit mode interaction in the “two-mode tunneling model” vta the vibrattonal overlap integral of the fast H osctllattons centered at distances which are specific for particular transfer reacttons. The analytical results show explicitly in which way the physical parameters enter. In the symmetric case, which IS met when the exothermtcity AE of the reactron vamshes and the condition fiw> ksTis fulfilled, the result of Slebrand et al. is regamed. The variation of the transfer rate constant at T=O as a function of the tunneling distance, the line width of the final denstty of states and AE 1s discussed. Moreover, numerical calculattons which can be easily done on the basis of the analyttcal result, are performed for different combinations of values for the two frequencies involved.

1. Introduction The theory of tunneling-transfer reactions has been the subject of many publications the references of which can be found in refs. [ l-61. The most successful attempts to explain the main features of the nonclassical behaviour of hydrogen or deuterium transfer is based on a formulation analogous to the theory of radiationless transitions [ 3,5]. It is usually developed under the additional assumption that the nuclear subsystem may be divided into two parts with fast (intramolecular) and slow (intermolecular) vibrations, respectively. An example of this approach is the theory of the rate constants for chemical reactions in the solid phase formulated by Trakhtenberg et al. [ 2,3 1. Another one is the theory of the transfer rate developed by Siebrand et al. [ 4,5 J. The latter is a golden rule approach where the transition between the initial and final states - taken as vibrational eigenstates of the non-interacting reactants and products - is induced by an interaction operator JAB. A barrier is obtained from the potentials governing the hydrogen motion in the interacting system of reactant and product. This barrier is modulated by the low-frequency modes of the reactant and product vibrations. The barrier oscillations turned out to be of major importance for the temperature dependence of the rate constant. In view of the key position of the tunneling-transfer model proposed in ref. [ 4 ] and despite its inherent limitations which arise from the oversimplification of the reaction dynamics, it seemed worthwhile to complete the numerical investigations presented in ref. [ 51 by an analytical treatment. Two recent investigations should be mentioned since both deal with tunneling-barrier fluctuations. Benderskii et al. [ 81 use the quasi-classical path-integral method to calculate the transition probability in two different double-well potentials modulated by both symmetric and antisymmetric coupling to intermolecular vibrations. 0301-0104/92/$05.00

0 1992 Elsevier Sctence Publishers

B.V. All rights reserved.

314

lJ. Schnabel, H. Gabrtel/ Theory ofhydrogen-transfer reactions

The results thus received for collinear particle exchange are valid for arbitrary flip conditions, temperatures, frequencies and coupling parameters. Suarez and Silbey [ 91 contributed to the theory of hydrogen tunneling in condensed media on the basis of the spin-boson Hamiltonian (two levels of an asymmetric double-well potential ). Fluctuations in both the tunneling matrix element and the asymmetry of the system are caused by coupling to local molecular vibrations. In the present paper the dynamical behaviour of the “two-dimensional” tunneling model is represented by two adiabatically decoupled harmonic oscillators which enables us to apply operator-algebraic methods. The paper is organized as follows: section 2 briefly describes the model proposed by Siebrand et al. [ 4,s 1. In section 3 the golden-rule-based expression for the transfer rate is adjusted to this model and rewritten in the language of second quantization. An outline of the theoretical treatment is given in section 4, followed by a discussion of the results. Several appendices concerning mathematical techniques and intermediate steps of the calculation are added.

2. The model We consider a reaction

scheme

A-H+B+A+H-B, i.e.. two molecules A and B at a distance R of which A carries initially a light particle, say a hydrogen atom, which is transferred to its final position at molecule B. We assume that the inverse reaction is suppressed by rapid energy loss into appropriate medium modes. The arguments given in refs. [ 4,5 ] to justify the ingredients of the model use the specific situation that a C-H bond is initially present in molecule A which is transformed into a C-H bond in molecule B. It is assumed that the heavy carrier molecules (or atoms) oscillate harmonically around their equilibrium positions with frequency 52 which is low in comparison to the frequency w of the collinear A-H or H-B stretching vibrations. The coordinates of the system are depicted in fig. 1. R (I?) denotes the instantaneous (equilibrium) separation of A and B, respectively, and r refers to the coordinate of the H atom. The (moving) equilibrium positions in the initial and final configuration are related to the instantaneous A-B separation through r,-r,=R-25,

(1)

where 2< is the sum of the A-H and H-B equilibrium separations. As a result of ( 1) the two oscillations are not independent of each other. It can be easily shown that the fast and the slow oscillations are adiabatically separable if the conditions Equlllbrlum R

A-H

rB Instantaneous

Fig. 1. Coordinates

of the reactant-product

system.

Ii. Schnabel.H. Gabriel/ Theory of h.vdrogen-transfer reactlons

m<<4M

and

Q2<
315

(2)

hold. In this case the displacement coordinates (R-l?) and (r-r,,,) are proportional to the normal coordinates of the system which will allow straightforward reformulation of the rate expression in terms of boson destruction and creation operators, later. With (2) being fulfilled the potential energies are sums of two independent harmonic oscillator potentials L’A.B=jmw2(r-rA,B)Z+fMS22(R-~)2+E~~B,

(3)

where constant terms E,A, Et are added, whose difference AE= E,B-E$ denotes the exothermicity of the reaction (m and Mare reduced masses). In the initial as well as in the final configuration the corresponding eigenfunctions 1iA) and 1f B, are written as products of eigenfunctions for the intramolecular A-H or H-B vibrations ( 1p) ) and for the intermolecular A-B oscillation ( 1/i ) ) : Ii”>=

lfB>

iO?r’(r-rA))

=

lpdr-rB)

In,

>

(4a)

(R-R)),

IA&R-RI

>,

tab)

where v, Vand w, Wdenote vibrational quantum numbers for the initial and final states, respectively. oscillator eigenfunCtiOnS 1f&( r - ru ) ) can be expressed as

The H-B

(5)

Ia?*,(r-rB))=UI~l,(r-rA)>,

where the unitary transformation Ci=exp[-(rr,-r+)a/ar]

(6)

depends on R according to eq. ( 1). The interaction operator J 48 also carries an explicit R dependence and is therefore modulated by the low-frequency intermolecular vibration. It accounts for the electronic overlap integrals of the relevant molecular orbitals or, approximately, to the orbitals of those atoms to which the H atom is bound (e.g. the 2s- and 2p-orbitals in the case of C-H). In refs. [ 4,5] it is chosen as an exponential function JAB =Iexp[

- (ilao)

(R-R)

1,

(7)

where Jand [ are constants and a0 is the Bohr radius. This completes the description of the model used by Siebrand et al. As in refs. [ 45 ] we restrict our treatment to the case that the A-H and the H-B stretching modes have the same frequency o. For harmonic potentials extension of the model to different frequencies is, in principle, possible. Since the calculations become much more involved we keep the assumptions of ref. [ 41 also in this respect.

3. The transfer rate The temperature-dependent

transfer rate is in the present investigation

based on the expression

(8) i.e. on the golden rule rate constant for a particular transition from state 1i “) of the initial configuration to state I f B, of the product configuration, summed over all accessible final states and thermally averaged over the initial states. Energy conservation is ensured by the line shape function pr( E) if the energies of the initial and final states differ by E. In the numerical calculations a Gaussian will be used. (As usual we have set /3= 1/k,T, where k, is the Boltzmann constant and T the temperature. ) The transition matrix element occurring in (8 ) takes

U. Schnabel, H.

316

Gabrrel/ Theory of hydrogen-transfer reactions

different forms depending on whether adiabatic or nonadiabatic transitions are considered (see chapter 3 of ref. [ 21 for a discussion of this matter). As demanded by the aim of the present investigation we follow the approach proposed in refs. [ 4,5 1. Inserting eqs. (4)-( 7 ) into (8 ) yields (in units fi k(T)=2n

F ;, T C I I I w

(9)

with Z=

(9a)

$ Fexp[-_B(EV+e,)]=[l-exp(-&?)I-

and

(9b)

E W,R-E,,,,=o(w-v)+Q(W--V)+AE. This was evaluated analytically in ref. [ 41 only for a special case (hw> P= w=O). We will now treat the general case. We introduce two mutually commuting sets of creation and annihilation the fast and slow vibrations by a +:=$(x_-!L),

b+:= j=& Both sets obey the commutation J&n=Jexp[

-K(b+b+)]

U=exp[-3

,

withx=fi

A), relations

k,T and a symmetric operators

transfer

(a, a + ) and (b. b+ ) for

(r-r*),

with X=@ for boson operators.

(R-l?). The operators

(6) and (7) now have the form

~=(/a~~,

(b+b++P)(a-a+)]

with

(10)

,

B= gQ.

P=m

(11)

(R-20.

B,

K and Pare the basic physical parameters of the model. In the two-mode harmonic oscillator model the energies occurring commuting Hamiltonians

H,=w(a+a+t)

and

in (9) are connected

to eigenvalues

of the

H,=R(b+b+i).

Functions of the latter will in the language of second quantization occur in place of the energy dependent quantities among them pr(E). To have the option of adaptingp,( E) to either a Lorentzian or a Gaussian line shape function we define an operator .U’(I.E,~)U(I)]=&

s dtexp[ifE-((r/2)]t]]f(t), --oo

(12a)

(12b) which, forf( t) = 1, is just the Fourier representation of a Lorentzian ( 12a) and a less common representation of a Gaussian ( 12b ). The latter is based on the identity (B. 3 ) of Appendix B. (The parameter t must be set to zero after having performed the differential operation on exp (i?E) f(t). ) In this way can the energy dependence of the two line shape functions be treated on the same footing. This enables us to write the transfer rate as

317

U. Schnabel, H Gabrrel / Theory of hydrogen-transfer reactlons

U+J+

JUexp[it(oa+a+Qb+b)]

>I

exp[ -(it+/?)(oa+a+Qb+b)] Z(o)

Z(Q)

=:27rY(t,AE,T)[Y(t)] in the formalism

(13)

of second quantization.

The main task is to evaluate the transition

function

Y(t) just defined.

4. Outline of the operator-algebraic treatment In the following we will make use of various identities and formulas valid both for the algebra of boson operators and for so-called hyperalgebras. The latter were used by Heinzel [ lo] in an algebraic treatment of FranckCondon transitions. They turn out to be a powerful auxiliary method in many cases where direct operatoralgebraic treatment is difficult. A collection of the most important formulas is given in Appendices A and B. With the help of (A.4) and by introducing g,(b, b+):=$ g2(b,

(b+b++P)=:g,

b+):=fi

[bexp(-itQ)+b+

the transfer function Y(t)=Tr,

)

Y(t)

(Ida)

exp(itQ)+P]=:g,,

(14b)

is written as exp( -PsZb’b)

S(o)exp{-~[bexp(--it~)+b+exP(itQ)l}

J*exp[-K(b+b+)]

Z(Q)

>.

(15)

The inner trace exp[-g,(b,

b+)(a-a+)]

exp{-g2(b,

b+)[uexp(-ito)-a+

exp(i*o)]}exp(~~~~tu) >

(16) will be evaluated first. Complications arise from the appearance of the noncommuting operator functions g, (b, b) and g,(b, b). They are circumvented by using the hyperalgebraic representation of the expression under the trace. As described in Appendix B in greater detail we introduce mutually commuting parameters j, and j2 and the corresponding differential operators (?l/aj, ) and (a/aj,) which obey the commutation relations (B. 1) and (B.2 ). Then we make use of several identities like #i exp[-_gl(~-~+)l=exp(-_g,a/aj,~expLil(~-~+)lI,=o.

In this way we arrive at S(w)=exp(-gia/aj,) xTr(expti,(u-a+)]

exp(-Mlai,) expu,[uexp(

The trace can now be calculated S(o)=exp(-gialaj,)

-ito)-u+exp(ito)]}

exp( -pwu+u)/Z(w))

exactly by using Bloch’s theorem

exp(-g2aiaj2)

exp[-C(o)

o’f+ji)l

],1,,2=0.

(17)

(A.8 ), yielding ew[-v(o)

@(60)j~j21,

(18)

with C(o)=4 ”

coth(t/?w),

n(w)=Z(o)

Even if not explicitly noted, it 1s understood corresponding differential operators.

exp(-$/3o), that all hyperalgebraic

(18a) parameters,,

cr,

must be set to zero after having applied

the

U. Schnabel. H.

318

Gabnel/ Theory of hydrogen-transfer reactions

(18b)

@(t,f3)=exp[(it+t/3)o]+exp[-(it+{P)w]. For the following it will turn out to be much easier, to use eq. ( 18 ) in the form

(a/aMi

S(w)=ew[-C(o)

x ew[---C(w)

exp[rl(o)

@(t,~) wahwaj,)i

waa2)2i exp[(al+jlk,l

(19)

ew[(~2+j2k211

where we have introduced a new set of parameters (a/&!,, a,) and (a/as,, (Y?) which again commute with (a/+,, j, ) and ( a/ajz, j, ) . (The equality of eq. ( 18 ) and ( 19 ) will be shown in Appendix C. ) Comprising all differential operators in ~(a,j,

[(am,

t, 0) =exp{ -K(o)

and inserting

the definitions

Y(t)=.PH(a,j,

V+ (a/a~,m

ewIBv(w)

fw,

~4 (a/ah 1wh)

1

(20)

of g, and g,, we obtain

t, w) exp[ (a, +j,)P]

exp[ (cwZ+j2)P] exp( -&%+b)

exp[((Y,+~,-~)(b+b+)]exp{(a,+j,-K)[bexp(-itB)+b~exp(it~)]}

Z(Q)

>.

(21) The remaining

trace in (21) can again easily be calculated

Y(t)=pH(a,j,

t, 0) exp[(cy,

+j, +a2+jZ)Pl

exp[C(Q)

with the help of (A.8) yielding

Cd+1411 exp[rl(Qn)@(L Q2)717211

where we have used ( 18a) and ( 18b) with o replaced by D and y,= a,+j,rewrite eq. (22) as Y(t)=J2exp{--K(w)

K.

(22)

For the following it is useful to

[(a/aa,)2+(a/acuz)2]}s(t)exp[(cr,+j,+CY2+j?)P]exp[C(n)

(y?+y:)], (23a)

where s(t)=exp[Brl(w)

@(t, 0) (aiaj,)(a/aj,)]

exp[rl(Q)

@(t, Q) YM

(23b)

comprises all time-dependent quantities. Proceeding with the evaluation of the hyperalgebraic expressions first, which is feasible, would lead to functions for which the action of the operator U( t, AE, r) (see eq. ( 12 ) ) could not be performed exactly. This is, however, easily done by noticing that the form of @( t, !2) given in eq. ( 18b) allows application of Watson’s generating function for the modified Bessel function of nth order

exp[fx(y+ll.v)l=

f

n= --m

which leads immediately U(t,AE,r)

9(t)=

xexp[f/?(wn+Qm)]

Y”Mx)=

ff

n= -cc

~~~~~j!~~)f*k;l,

(24)

to

f fj wmw n=-cc rn-cc p,(wn+SZm+AE)

(WI)

(w2))

~m(247(-Q)~3)

,

(25)

where we have used Y(t,AE.T)

exp[it(wn+SZm)]=p,(wn+&n+m).

In this way the Lorentzian or Gaussian line shape function is restored. Energy conservation terms of the summation indices n (m) which enumerate the quantum transitions involved.

(26) is now expressed in

319

U. Schnabel, H. Gabriel / Theory of hydrogen-transfer reactions

The expression

k(T)=

for the transfer rate which we have found so far is given by

[(a/ac#+(a/aa,)*]j

2nJ’-exp{-BC(w)

~wm4

(a/ah) ww)

xew[tB(on+Qm)l which is obviously thus can write

f II--m

~mwh~~~

exp[(~,+_h +az+j2)Pl

symmetrical

in the indices

x f [~(~)12k+‘“’ {exp[ -KY(w) k=,, k!(k+ Irnl )!

ewLC(Q2) k4+24)1 y

1 and 2. Representing

(a/acu)2](a/aj)2~+ifl’y2k+‘m’

being left with the final task to calculate the expression C. The final result is given by k( T)=2flD-’

x

exp( --E/D)

I=(o)

f

d

f pr(~~+f2m+AE) In=--or,

f

12’+‘n’exp( _por)

the Bessel functions

exp[P(a+J)l

Y*]}*,

(28)

in curly brackets { } =: 2. This is also done in Appendix

exp{tp[o(n-Inl)+~n(m-lml)l}

f

[Z(Q) 12k+‘m’exp(-jlQk)[~(2r+~n~,2k+~m~)]*

k=o k!(k+

by its power series, we

exp[C(Q)

f

n=-cc lx-cc

r!(r+ In\ )!

(27)

/ml)!

(29) where

with 1=2r+

InI ,

a=2k+

Irnl ,

and X, = (P-2C,tc)/2( C,=C(w)=i D= 1+4BC,C,,

-DC’n)“*,

coth($j?w),

X, = (~BC,P+K)/~(DBC,)“~, C,=C(Q),

E=2BC,P2-2CQ~*-8BC,CnP~,

and B, K and P defined in eqs. ( 12) and ( 13 ). H,(x) is a Hermite polynomial of order n. Eq. (29) is an exact solution for k(T) within the frame of the underlying model. It shows explicitly in which way the physical parameters enter. Eq. (29) is well suited for numerical calculations. We will give a few examples in section 5.

U. Schnabel. H Gabrrel / Theory of hydrogen-transfer reacttons

320

5. Results and discussion

5.1. Resonant transfer in the symmetric case As already mentioned the analytical result given in ref. [ 41 applies to the totally symmetric case (AE= 0) and to frequencies for which Ao XDk,T is fulfilled. Resonant transfer then occurs between the ground states of the fast oscillations. In order to derive the transfer rate kS( T) for this special case we must therefore set AE=O, r= 0 and n = r= 0 in our general expression (29). Energy conservation, now guaranteed by the delta function also leads to m = 0. We thus receive (31) which is the result obtained

in ref. [ 41 as was proven in ref. [ 111.

5.2. The low temperature limit. Resonant and nonresonant transitions The T+O limit of the low-temperature plateau is obtained indices which are connected with nonvanishing Boltzmann C,=C,-tf,Z(o)=Z(Q)~l,sothat k(0)=2fi&Bexp

-Bp2T:z2BpK

n, m)]2pr(no+mQ+AE).

For the set of parameters used in ref. [ 41 and in our numerical exponential function is BP2, so that ks(O)=2rrl’$----exp

from eq. (29) by retaining only those summation factors, i.e. r= k=O and n, m>O. We also have

-BP2::2sf2BPK

calculations

%2X.77 +Bexp[-fmc0(R-2S)‘l, >

(32)

(see below), the leading term in the

(33)

which is also the cofactor of the double-sum expression (32). Thus, the T= 0 value of the rate constant is dominated by the overlap of the vibrational ground states in the initial and final positions of the hydrogen atom. The R dependence of the coupling JAB can to a good approximation be neglected ([= K=O) at least at very low temperatures (see also ref. [ 5 ] ). So far, we have merely discussed the symmetric case (A,!?= 0, r= 0) in which only resonant transfer can occur. We now represent the final density of states by a normalized Gaussian line shape function in the form C&T)-’ exp[ - (E/r)2] #* and calculate the behaviour of k( 0) versus AE for two values of r (see fig. 2). As expected, there is strong variation of k( 0) for small values of ras a function of A,5 which is smoothed out as r increases. This behaviour reflects the particular importance of resonant versus nonresonant transfer in the former case (energy conservation). We observe from fig. 2 maxima of the transfer rate located at values AE= ml2 (notice that Q=200 cm-’ was chosen). This results from the fact that the leading terms in eq. (32) are for AE values well below o given by n = 0 and the remaining sum over m. The maximum of the envelope occurring at hEr 1200 cm-’ results from two opposite effects: With increasing AE, we have a larger energy transfer from the o oscillation to the Q oscillations in the final state, i.e. a stronger modulation of the barrier and therefore a larger transfer probability. On the other hand, if higher Q states are populated after the transfer than before, the overlap of these states is smaller, i.e. the Franck-Condon factors are reduced, resulting in a decrease of the transfer probability.

” Note that the half-width

of the Gaussian

is equal 2,,/1n 2 K

U. Sehnabel. H. Gabriel / Theory of hydrogen-transfer reactrons

321

-25

Fig. 2. Plot of In k( T) against AE for two values of the hne-width IY r=20 cm-‘: strongly varying curve. T= 100 cm-‘: smooth curve starting at a lower rate constant for AE=O due to normalization factor I/$ Tof the Gaussian line profile. Other parameters: same as for reference curve in fig. 3.

5.3. The influence of the model parameters on the transfer rate As already pointed out in ref. [ 5 ] and confirmed by the upper curve displayed in fig. 3 the absolute magnitude of the rate constant is very sensitively affected by relatively small changes of the equilibrium tunneling distance R- 2r (see also eqs. (32) and (33 ) ). Uncertainties in I?- 2l are automatically connected with corresponding uncertainties in Jif k(0) is fixed. Since it is the purpose of this paper to elucidate the behaviour of k( T) under more general conditions as met in ref. [ 5 ] rather than fitting the theoretical curves to appropriate experimental data, we have chosen a particular set of parameters already used in ref. [ 41. It is given in the figure caption of fig. 3. The corresponding reference curve is marked by open circles in figs. 3 and 4. The effect of lowering the frequency of the stretching mode is shown in the two upper curves of fig. 4. We observe an increase in the absolute value of the rate constant in comparison to the reference curve and a decrease in the slope of d [In k(T) ] /dT. The importance of the slow intermolecular vibration for the temperature variation of k(T), especially for the onset of the strong nonexponential rise at temperatures around 50 K, is well known and discussed in refs. [4,5]. By lowering the frequency Q (with o fixed) we find a drastic increase of k(0) and the slope d [ In k(T) ] /dT. Moreover, there is a distinct downward shift of the temperatures where k( T) merges into its low-temperature plateau if Q is lowered. The frequency dependence of k( 0) can qualitatively be understood as follows: At the classical turning points r,,,the potential energy of the vibrating hydrogen atom equals its zero point energy jkw=

imo’(r!#)

-rP)*

(,u=A or B) ,

(34)

rWkj =rP 5 (h/mw)“2. The minimal

tunneling

(34a) separation

is then estimated

as

ArT.~r~~)_)-r~~,q?)=rg-rTA-2(Z2/mw)”*. Analogous

reasoning

(35)

leads to an estimate of the A-B separation

R,,,_, =I?- (A/M@“‘.

(36)

U Schnabel, H. Gabrrel/ Theory of hydrogen-transfer reactlons

322

-5 -10

-15

- 20 -25

50

100

150

200

250

300

T/K I

I

I

I

I

50

100

150

200

250

300

T/K

Fig 3. Dependence of In k(T) on the system parameters I? and m. Reference curve (O-O-O) for the parameter set: J’p(O)= 103 cm-‘, i= 1.625,l?=3.35 A, r= 1.09 A, m=0.923m0 (m, mass of a free hydrogen atom), M= 22m,, w= 3000 cm-‘, Q=200cm-‘.Uppercurve (+-+-+): same.excepta=3.2.A. Lowest curve (a-a-* ): isotope effect in the deuterated referencesystemwithmD=1.714m,,andwD=2200cm-’.

Fig. 4. Dependence of In k(T) on the frequencres o and R. Parameters: same as for reference curve in fig. 3 (O-O-O ). Curves (*-*-~):w=3000cm-‘,G!=300cm~‘(lowercurve),~=150 cm-’ (upper curve). Curves (+-+-+ ): Q=200 cm-‘, w=2400 cm-’ (lower curve). o= 1800 cm-’ (upper curve).

Thus lower values of both, o and 52 diminish the separation of the reactants resulting in an increase of the transfer probability. The different behaviour of the slope d [ In k(T) ] /dT as a function of the frequencies Sz and w results from different mechanisms. The dependence on &simply results from the influence of thermal activation. Decreasing Sz leads to higher level populations and larger changes with increasing temperature thus increasing k(T) as well as its slope. The effect of the hydrogen stretching frequency w occurs for a completely different reason: The delocalization of the oscillating H atom around the equilibrium positions r, or r, diminishes with increasing frequency. The result of this enhanced localization is a significant drop of k( T) at a given temperature for larger values of w. However, at the same time the overlap becomes more sensitive to changes of the modulation induced by the oscillatory changes of R. This results in an increase of the slope with rising temperature. Finally we recall that the choice of the linewidth r has an effect on the absolute rate constants k(T). This is clear from section 5.2. It leads to a shift of the k(T) curves without changing their shapes. 5.4. Comments andfinal remarks The results of the present analytical study of Siebrand’s two-mode fluctuating barrier model cover the entire temperature range and are valid for arbitrary mass and frequency ratios as long as the conditions (2) are not violated. These limitations result from the adiabatic decoupling of the two collinear vibrations involved. They can, as done in the semiclassical treatment by Benderskii et al. [ 8 1, be removed by transforming to proper

U. Schnabel, H Gabrrel I Theory of hydrogen-transfer reactions

323

normal coordinates. The evaluation of the transition rate is feasible but will become more complicated because of the particular form of the shift operator U (eq. ( 11) ). The latter reflects the fact that the distance between the A-H and H-B oscillators is itself modulated by the low-frequency intermolecular vibration. Extending the quantum mechanical treatment to the case which avoids the adiabatic decoupling approximation, would provide the proper basis for comparing the results with that obtained by the path-integral approach. In contrast to refs. [ 2,3,6,9] coupling of the reactive subsystem to the modes of a heat bath is not included in Siebrand’s model. The influence of a dissipative environment is modelled by a phenomenological final density of states. It is obvious that coupling to a harmonic-oscillator heat bath could also be handled by the operator technique used in this paper. A general formal treatment of this problem was already given in ref. [ 3 1, extended in ref. [ 6 ] and taken up by ref. [ 9 ] in the context of a modified spin-boson model for hydrogen tunneling in condensed media. It is well known that simplifying assumptions concerning the properties of the phonon spectrum are necessary even for an ideal solid heat bath in order to exploit the formal theory for practical purposes. A common feature of all quantum mechanical models under consideration is that the “tunneling matrix element” depends exponentially on, say Q, the coordinate of the barrier-modulating (intermolecular) vibration #3. Together with the common assumption that all oscillations of the system may be treated in the harmonic approximation and despite the different physical pictures used in refs. [2,3,8], respectively, one would expect equivalent results. In fact, equating the parameters one notices that eq. ( 13) of ref. [ 91 for the rate is essentially equal to eq. ( 10.1) of ref. [ 21 (the latter being a special case of eq. (5.24) ). The expressions differ only by a subtracted term proportional to the squared thermal average of the transition operator. This results from the definition of the rate used in ref. [ 91 which by its restriction to a second-order description is equivalent to a golden-rule-like thermally averaged second-order transition probability. It corresponds to eq. ( 13) in this work. As expected, and in fact, demonstrated by analysis of the few real systems in which hydrogen transfer was observed in a sufficiently wide temperature range [ 71, the two-mode tunneling model also used in this paper turns out to be of limited validity. The basic assumption, that the coordinates of the adiabatically decoupled inter- and intra-molecular vibrations as well as the reaction coordinate are altogether collinear, is an oversimplification of the dynamics of the reacting molecules. As already mentioned the interaction with their surroundings was given particular attention in ref. [ 3 ] and also discussed in ref. [ 2 1. These references also contain a model for the type of reaction considered here. It is based on the use of two counter-directed Morse potentials which are modulated in the same way as the harmonic potentials in our model. An exact analytical solution cannot be found in this case. Siebrand et al. [7] discuss the effect of an anharmonicity of the low-frequency mode on the rate constants which is expected to be significant for the temperature variation of k( T). Whenever closed molecular model potentials (like the Morse potential) are chosen, the operator-algebraic method cannot be used and direct numerical evaluation of the rate expression is demanded. If anharmonicities are introduced approximately via the anharmonic lowest-order terms of a power series of the potential, the analytical approach can, in principle, be applied, but will go along with rather involved mathematical manipulations. We have not attempted to take anharmonicities into account, since priority was given to find an exact analytical expression for the generalized transfer rate of the original harmonic two-mode model.

Appendix A. Some operator-algebraic formulas With use of the well-known exp(xA)

B exp( -xA)

Lie series

=B+x[A,

B] + $

[A, [A, B] ] +... ,

(A.1)

113The extended version of ref. [2] that the matrix element is proportional to exp(a I+‘+ bQ2) (a, b constants) will not be discussed here. We restrict to the case b= 0. As pointed out in ref. [ 6 ] inclusion of the quadratic term improves the high-temperature behaviour of the transfer rate m comparison wtth the numerical results gained on the basis of a Morse potenttal.

U. Schnabel. H. Gabriel / Theory of hydrogen-transfer reactions

324

and the Baker-Hausdorffformula

(if [A, [A, B] ] = [B, [A, B] ] =O)

exp(A+B)=exp(A)exp(B)

exp(--f[A,B])

or

exp(A)

exp(B)=exp(B)exp(A)exp(+[A,B]),

(A.2)

the following formulas hold: exp(xa)f(a,

u+)=f(a,

a++~)

exp( -xu+)f(u,

u+)=f(u+x,

exp(xu+u)f(u,

u+)=f(uexp(

whenever the functionfcan

exp(xu)

,

(A.3)

a+) exp( -xu+)

,

-x),

exp(xu+u)

u+ exp(x))

be expanded

(A.4) (A-5)

,

in a power series. By complete induction

one can also easily show that

(A-6) The trace expressions Tr[exp(c,u)

occurring

exp(clu+)

exp[c,(u?u+)] =exp[

&C(o)

are of a form such that Bloch’s theorem can be applied. It reads

exp(-_pou+u)/Z(w)]=exp[Z(w)c,cz]

where cl, c2 are constants, different form: Tr

in our treatment

p= l/k,T

and Z(w) = [ 1 -exp(

exp[ &q(w)

where we have used the abbreviations

-PO)]

(A.7)

-‘. This theorem

is also used in a slightly

exp( -/kf~~+u)

exp{c,[uexp(-ito)*u+exP(ifo)]} (c:+c:)]

,

Z(o)

>

(‘4.8)

@(t, w) clcz] ,

defined by eqs. ( 18a), and ( 18b).

Appendix B. An outline of the hyperalgebraic method In many cases the operator expressions are too complicated to be handled with techniques like those given in Appendix A. We will then use a representation used by Heinzel [ lo]. He introduces a set of parameters ((~i, (Y*, ...) and a related set of differentiation-operators ( (i!l/acu, ), (a/&,), ...). which obey the commutation relations

[am,, c-u,1= 1 ,

(B.1)

[a/aa,,a,]=[a/aa,,a/aa;]=[cr,,a;]=o,

fori#j,

(B.2)

and are therefore called hyperulgebru. Its use is demonstrated exp(Cu2)

exp[D(u+)21=exp[c(a/aa,)21

((.u, and CT?must be set to zero after same commutation relations as (a, a In many cases it will not be necessary (A.6) to bring a/as and (Yin normal We note some important formulas

exp[B(a/aa)2]

exp(Cor2)

exp(a,u)

by

exp[o(a/aa,)21

exp(a2u+)l,,,,,=o.

(B.3)

having applied the differential operators.) Because (a/as,, (Ye) obey the + ) we can use all formulas given above by replacing alaa,-+u, and (~,-a+. to really execute the differentiation - one can also use the formulas (A. l)form, and then set all parameters (Yequal to zero. for hyperalgebraic expressions. We have

exp(Da)I,=,=(~~)-‘exp[BD2/(1-4BC)l,

if I4BCI < 1. The proof can be found in ref. [ 10 1,

(B.4)

U. Schnabel, H. Gabrrel/ Theory of hydrogen-transfer reactlons

325

n

exp( -Ca’)

exp(Da)

= (,/@)“K(DI2&

(B.5)

.

Cl=0

where Hm (x) is the Hermite polynomial of nth order. With relation (B.2) and by using (A.4) and (A.6) we have

We can also evaluate “mixed” algebraic methods:

expressions,

consisting

of different operators

(a, a + ) and (b, b + ) with hyper-

expV;(a,a+)gl(b,b+)lexpU;(a,a+)g~(6,b+)l =ewCfiWh) expCfiW2) wLhg,(b,b+)l

explj2gZ(~,b+)ll,,,,230.

This representation even holds for non-commuting Finally we have used the expression

=ew(--Cfi)

(B-7)

operator functions.

nzof V;)Y-vhYexp(-CfS),

(B.8)

which is derived with the help of (A.3 ).

Appendix C. Intermediate The expression S(w)

steps in the calculation of the transfer rate

given in eq. ( 18 ) is with (B.8 ) directly rewritten

=exp[ -C(w)

gf]

E

‘II(O) :I”’ O) I”

PI=0

as

(g,)“(gz)“exp[-CC(o)g:l.

(Note that g, and g, are non-commuting operator functions ! ) Now introducing eters (alaa,, a,) and (a/do!,, az), (a/aj,, j,) and (a/aj,, jz), wecan write

(C.1) new sets of hyperalgebra

param-

cc.21 which by using the mutual commutivity Evuluutlon

of the expression

of the different parameter

sets leads immediately

to eq. ( 19 ).

59.

(C.3) We introduce

a new parameter

,Uto write

2k+m Y

w(w)

3

and can therefore express 3 as

(C.4)

326

U. Schnabel, H. Gabnel / Theory of hydrogen-transfer reacttons

dl=exp[-BCw($)2]($)2,+,io2*imexp(py) exp[P(a+~)] exp(CQr2). Now we insert y= cu+jJ=

exp(-$D)

K

and use relation

(.$(g!)expM/D)

(B.4) which leads to

ew{[Cd2+ (f’-2cQ~)A/Dl

xexpj-[~C,~2+(2~C,~+~)~]/D}. where we have introduced 1=2r+n,

a=2k+m,

The remaining

(C.6)

the abbreviations D=1+4BC,C,,

differential

t-1

operations

(C.6a)

E=2BC,P2-2cCn~2-%BC,CnP~.

are performed

with the identity

(B.6) yielding

,~ = exp ( - E/2D) m’n(L,a) A CT Jo z. (F)(S)S!~(~~-‘exp([Cnj’+(P-2C~K)/l/nl

c-5

a xiiji(

WK'~'+K)PI/D).

em{-[BG,p’+

)

which is transformed

to the final expression,

1 = exp ( - E/2D) m’n(i,o) 1 Jo

x~)‘“-““Ho_A

z.

(s)(;)s!&(-

(-

(y;;;;K)).

using (B.5 ) that introduces

(C.7)

the Hermitian

polynomials

~)‘A-s”2Hi-x(2’;~a)

(C.9)

Note that the appearance of negative arguments in the square-roots of the sum 8(A, a) is just formally. The prefactor ( - CQ/D)“-F and the powers of the Hermitian polynomial HA-,(X, ) together yield a real value.

References [ I ] R.P. Bell, The tunnel effect in chemistry (Chapman and Hall, London, 1980). [2] V. Goldanskit, V.N. Fleurov and L.I. Trakhtenberg, Tunneling phenomena in chemical physics (Gordon and Breach, New York, 1989). [ 31 L.I. Trakhtenberg, V.L. Klochtkm and S.Ya. Pshezhetsky, Chem. Phys. 69 ( 1982) 12 I, [4] W. Siebrand. T.A. Wildman and M.Z. Zgierski, Chem. Phys. Letters 98 (1983) 108. [ 51 W. Siebrand, T.A. Wildman and M.Z. Zgierski. J. Am. Chem. Sot. 106 (1984) 4083. [ 61 Z.K. Smechardina, Chem. Phys. 150 ( 1991) 47. [ 71 W. Srebrand. T.A. Wildman and M.Z. Zgierski, J. Am. Chem. Sot. 106 (1984) 4089. [ 81 V.A. Benderskii, V.I. Goldanskii and D.E. Makarov, Chem. Phys. Letters 17 1 ( 1990) 9 1; Chem. Phys. 154 ( 199 1) 407. [9] A. Suarez and R. Silbey, J. Chem. Phys. 94 (1991) 4809. [ lo] W. Hemzel, Phys. Cond. Matter I7 ( 1974) 99. [ 111 U. Schnabel, Dtplomarbeit, Fachbererch Physik, FU Berlin ( 1990).