Approximate quantum calculation of the d
281 zyxwvutsrq
Chemical Physics 104 (1986) 281-292
North-Holland,
Amsterdam
APPROXIMATE QUANTUM CALCULATION
OF THE DYNAMICS
OF GAS-PHASE REACTIONS OF A LIGHT-ATOM TRANSFER
IN THE TUNNELLING ENERGY REGION
M.V. BASILEVSKY,
G.E. CHUDINOV
and V.M. RYABOY
L. Ya. Karpov Institute of Physical Chemistry, ul. Obukha IO, 107120 Moscow B-120, USSR
Received
9 September
1985; in final form 30 December
1985
An approximate
method for calculating
the probabilities
of linear reactions of a light-atom
transfer based on the Born
distorted-wave
approximation
is proposed. The matrix element over the vibrational
coordinate
is numerically calculated, the
translational
motion is treated quasi-classically.
The reaction probability
expressed as a triple integral is evaluated by the
steepest descent method. All the integrands
are expanded at the saddle point, its position being determined
by iterations.
Complete quantum calculations
are compared with this model treatment for the following reactions: CH; +CH,,
CH; +
PhCH,, CH; + PhCH,D.
The results of the model and complete calculations
agree for both symmetrical
and asymmetrical
reactions with considerable
misfit of the resonance.
1. Introduction
Accurate quantum dynamical calculations
of linear triatomic reactions with the transfer of a light atom
have become available in the last few years. They involve rather laborious calculational
procedures dealing
either with numerical basis functions [1,2], or with a complicated curvilinear coordinate system [3-51. As is
known, the hamiltonian
of these systems contains a small parameter namely, the ratio of the masses of the
particles. In the scaled skew coordinates
this small parameter is the angle 28 between the asymptotic
directions
of the reactant and product valleys (fig. 1). It would be tempting
to exploit this fact for
simplifying the calculational
procedure.
Ovchinnikova
[6] used the smallness of the mass ratio for simplifying the classical equations of motion
in the classical S-matrix method. Approximate
calculations
of quantum transition
probabilities
made by
Babamov and co-workers [7-91 can be applied when vibrational
frequencies
of reactant and product
molecules are close or equal. Both approaches are invalid for calculating
the probabilities
of the reaction
between vibrational
states with considerable
energy difference.
In this communication
we develop a simple method of calculating
quantum
tunnelling
reaction
probabilities
for the system with small angles 8 and arbitrary
resonance
misfit. It is based on the
distorted-wave
Born approximation
(DWBA). The transition
matrix element is calculated using a quasiclassical representation
of the translational
wavefunctions
in the valleys of reactants and products. In the
systems under consideration
the adiabatic potentials of translational
motion for different vibrational
states
of reactants and products represent almost parallel curves. They are merely shifted along the energy axis
by the resonance misfit. The potentials of reactant and product levels intersect neither on the real axis of
translational
variable x, nor in its nearest neighbourhood.
Therefore,
the standard
treatment
[lo] of
quasi-classical
integrals is inapplicable.
Such an integral has been calculated for two cases when the transition
operator is independent
of x
(Landau and Lifshitz [lo]) and for an almost resonance situation (Babamov et al. (81). In the first case the
transition occurs far from the classical turning points of both translational
functions, in the second case it
0301-0104/86/$03.50
0 Elsevier Science Publishers
(North-Holland
Physics Publishing Division)
B.V.
M. V. Basilevsky et al. / Quantum dynamical calculations of linear exchange reactions
282
Fig. 1. Potential energy surface (schematically)
and coordinate
systems applied in the calculation
of the matrix element (I).
occurs in the vicinity of the turning point which is practically the same for both functions owing to their
resonance. For zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA
our method of calculating this integral the relative position of the transition region and
turning points is of no importance, so that the Landau and Babamov formulas are obtained as asymptotic
limit cases. This method works efficiently provided the relief of the potential energy surface (PES) in the
transition region is such that the pair of potential minima corresponding to reactant and product valleys
can be clearly distinguished.
Using this procedures we have calculated the probability of tunnelling transitions in thermoneutral and
exothermic radical reactions of hydrogen and deuterium atoms transfer:
R,H + R, -+ R, + HR,,
R,D+R,+R,+DR,
The results are compared with complete quantum calculations.
2. zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA
Formulation of the distorted-wave approximation
The amplitude of the transition
has the form [9,11]:
S,,(E)
= -27ri(x’*‘(E)
probability
from the reactant
(V,, Jx(‘)(E)).
to product
valley as given in the DWBA
(1)
The angular brackets denote integration in the coordinate plane, the upper indices 2 and 1 label the
wavefunctions in the reactant and product valleys, respectively. In each of the valleys the solutions x” ,*’
are products of the translational wavefunction ‘p,,* and the vibrational function $I,* and are most easily
represented in local coordinate frames (xi, y1 ), (x2, y2) (fig. 1) having one of the axes directed along the
valley bottom and the other being orthogonal to it:
X “‘=Vl(~,N,(YI~
$1,
x(*)=v*b*)~*(Y*~
x2).
(2)
h4. V. Basilevsky et al. / Quantum dynamical calculalions
oflinear exchange
(For brevity we omit in formulas (2) the index indicating
the
products.) The angle between the axes xi and x2, equal to 28,
reacting system. For matrix element (1) to be calculated, the valley
coordinate system (x, Y). Transformation
of the coordinates
has
reactions
vibrational
state of the reactants
and
is determined
by the mass ratio of the
solutions should be referred to a unique
the form:
~ 0~ e
*I
-:::)(
;)p
(;:)
= (E,”
-ZZ)(
;)=
sin B
i Yl )
i zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA
After passing to the frame (x, y) the integrand
angle B is assumed to be small. Then, neglecting
S,,(E)
283
(3)
in (1) can be expanded in a series of y sin 8, since the
the terms of the order of 8, we get:
= -2GA,
of translational
wavefunctions
generate a
(It can be shown that the terms - sin 8 in the arguments
- 8 in the result of integration.)
We rewrite formula (4) separating explicitly the integration
correction
over the vibrational
coordinate y:
where
B(x)
=JW &(Y,,
--m
For translational
vi(x)
+L(Y,,
wavefunctions
= ~zj’~k;‘/~
with c = (2m)‘12/fi.
x) dx.
we apply the quasi-classical
Ai( -zi),
Here Ai( -z)
Zi=($~)2’3,
~)G(Y,
q2(x)
representation
[12]:
= CZ?/~/C;~/~ Ai( -z2),
(6)
is the Airy function
w=A-‘/*[2m(Ea,
k,=dy/dx,
~i)]l’~
dx,
i= 1,2.
(7)
(v is the diabatic channel potential and a, is the coordinate of the turning point in the reactant or product
valleys.) The values entering into eq. (7) should have a second index indicating
the vibrational
state of
reactants or products, but it is omitted for the sake of brevity.
While calculating the integral over the translational
coordinate it is convenient
to choose the argument
of one of the Airy functions as an integration
variable. We choose z1 for definiteness.
Then z2 and x
should be expressed as functions of z,. The change of the integration
variable is performed as follows:
dz, = (dz,/dW,)
dW, = ($ IV1)1’3k, dx =fi’
dx,
dx =fi
dz,,
(8)
where
f;=($v/k,3)1’3,
To express
z2 through
~2~~2(z:)+dzJdz,l,*(z,
i=1,2.
z, we expand
z2 in z, at some point
-z:)=z,(x*>+
x = x* = x(z:):
[.f,b*)/f2(~*>1(~,
-4%
Thus
z2 = 4(z, - t>,
(9)
M. V. Basilevsky et al. / Quantum dynamical calculations
284
of linear exchange
reactions
where
I= z1(x*)
4 =f,(-x*M(x*),
[fib*vfib*)l z:-
-
over the
Finally we represent
integral
B(x) (5) as a function
of z,. The result of integration
translational
coordinate depends on the choice of the basis functions and the interaction
potential. This is
discussed in appendix A. Here we employ the exponential
representation
of B(x), and expand the
exponential
index into a series at point x *, taking into account the relations
x =x*
B(x)
I,,( z1 -z:)
+ dx/dz,
= exp[ln
B(x*)
= x* +f,(x*)(z,
1xefiz:
- d In B(x)/dx
- z?),
- 7z1],
00)
where
r=
-dln
B(x)/dxl,,f,(x*).
The faster the integral over y changes,
of r. Now the integral (5) is:
A z
c2fi)/2f;/2exp{ln
on moving
along the translational
coordinate,
the larger the value
B(x*)
- [d In B(x*)/dx] zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLK
fi(x*)z:}A,,
- q(zl
- t))
where
A, =
J
dz, Ai( -zl)Ai(
By applying
triple one
A,
the integral
=-&///dz
representation
of the Airy functions
du du exp{i[+(a3
and evaluated by the steepest
obtain (see appendix B)
descent
exp( -rz,).
[13], integral
+ u’) - uz - uq(z - t)] -
method
simultaneously
A,
can be transformed
TZ}
over all the three variables.
into a
(11)
As a result we
(12)
where
(13)
The values q, t and r are
indicates that the quantity
for deriving formula (12),
estimation
of the integral.
equation
z1(x,)
=
functions of x*; they are determined
by formulas (9) and (10). The index s
U, is calculated at the saddle point x,. If the expansions (9) and (lo), applied
errors in the
are performed
at xS, they do not bring about any additional
According
to (B.3) the position of the saddle point is determined
from the
b4x*)12,or
W1(x,)
=$[ Us]‘.
(14)
While operating with formula (14) the equality x* = x, can be achieved by the iterative self-consistent
procedure,
For calculational
convenience we use the exponential
representation
of the potential in the reactant and
product valleys
V,(x)=Eexp[-c(x-a,)],
V,(x)=V,+A.
(15)
285
M. V. Basilevsky et al. / Quantum dynamical calculations of linear exchange reactions
(Here E is the translational
energy of reactants and A is the resonance misfit, i.e. the difference between
the positions of the energy levels in the reactant and product valleys.) For the LEPS-type PESs relation
(15) holds with an accuracy of 3-5s
in a wide range of the translational
coordinate
values. Such a
treatment
enables one to express the quantities
q and r through
elementary
functions.
In this case the
action integral is analytical.
3. zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA
Limiting cases of formula (12)
Let us start with elucidating
combination
of the parameters
-(q3-l)t=
into formula
in form:
(12). The
(2m/fi2)f,(F2-1;;)6,
where 6 is the distance
product valleys.
&=
the physical meaning of the values 7, q, t, entering
-(q3
- 1)t of (12) can approximately
be presented
-a~:/a~]~,,
(16)
along the translational
coordinate
between
the turning
points
i=l,2
in the reactant
and
07)
is the “force” at te turning point a,. It should be noted that both the force difference (F, - FI) and the
value 6 depend on the resonance misfit A.
Thus, the value of the transition
matrix element is mainly determined
by the variation
rate of the
integral B(x), the values of forces F, and F2 and by the resonance misfit A. There exist two limiting cases
which correspond
to two ways of expanding
the square root in eq. (13) into a series, depending
on the
value of the parameter
6 = T2/(1 - q3)t.
(18)
It can be shown that at a small resonance
thus represents the Messiah parameter.
misfit
(A/E -SK1) 5 = 5h2X2E/2mA2,
and the value of t-i/*
3.1. The Landau limit [lo]
T2e
1).
-r(q3-
(19)
If the dependence
of B(x)
calculated the overlap integral
acquires the form:
U2 - zi = 0,
on the translational
coordinate
is neglected, that is if we put T = 0 and
of two Airy functions,
the system of equations defining the saddle point
u2 - z2 = 0,
u dz,/dx
+ u dz,/dx
= 0.
(20)
(No approximation
related to the expansion of z2 over z, is made here.)
While analysing system (20) with the explicit form of potential (15) one can see that, in accordance with
the result of Landau and Lifshitz [lo], it has the solution x, = - co, and I/(x,) + cc. Expanding (12) and
(13) in the small parameter (E/V)‘/*,
we can obtain an analytical solution of equation (14), which then
turns into V(x*) = (A/2X)*. We introduce the notation:
h=
]dln
The coordinate
x,=:x*
B/dx]
=~/fi.
of the saddle point is
= -_c -* ln(A2/2X2E).
(21)
M. K Basilevsky et al. / Quantum dynamical calculations
286
Condition
oflinear exchange
reactions
(19) takes the form
Aw[(E+A)“~-E~/~]
and the transition
amplitude
becomes
s,, = (~~cA[(E+A)‘/~-E~~~])-~‘~~B(~)
The exponent of expression (22) coincides
vanishes at X 4 0. This is in agreement with
different energies (A + 0) are orthogonal.
It should be noted that for non-symmetric
always present when the mass of the central
(the reactant and product valleys are almost
cRelation
e,
72 - 8413 )
_ E112]).
(22)
[lo]. The pre-exponential
factor
of the Schradinger equation with
reactions (non-zero resonance misfit) the Landau limit is
particle tends to zero. We show (appendix C) that at small B
parallel) the quantity q is independent
of ti and
(23)
spontaneously.
A/E&l.
When the second inequality
d In B
>>
dx
I
A)‘/2
limit (81
~~x=- -t(q3-l),
A
~
(2my2
with the Landau exponent
the fact that two solutions
t-e-2’3.
(19) is thus fulfilled
3.2. The Babamov
exp( -(~TI~)[(E+
I
(24)
holds the first one takes the form
(25)
This inequality establishes a relation between the rate of change of the vibrational
matrix element and the
resonance misfit.
In this limiting case in deriving an expression for the transition matrix element, the exponent in (12) is
expanded in powers of A/E with inequality
(25) taken into account. All the A-dependent
terms in the
pre-exponential
factor are neglected. As a result we obtain:
(26)
The quantity X is determined in (21). For deriving (26) expansion (9) has been performed at point a,. Eq.
(26) differs from that reported by Babamov et al. [S] by three extra terms in the exponent index (the third,
fifth and the sixth terms are absent in ref. [8]). Eq. (26) coincides with the result of ref. [S] under the
additional condition of h2/E .szz1.
Neglection
of the sixth term is likely to be justified, at least due to the smallness of the numerical
coefficient. The fifth term can be neglected if h < 1. Apparently,
it seems inconsistent
to neglect the third
term at small A, retaining the fourth one. The difference between formula (26) and the results obtained by
Babamov et al. arises from the fact that they assumed q3 - 1 = 0 (or, in other words Fl = F2) whereas this
value is - A.
287
M. V. Basileusky et al. / Quantum dynamical calculations of linear exchange reactions
4. zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA
Results and discussion
The results of quantum [5] and model dynamical calculations
Ovchinnikova by the method of classical S-matrix [6]:
of the reactions (studied earlier by
CH; + CH,,
(a)
CH; + PhCH,,
(b)
CH; + PhCH,D,
(c)
are presented in fig. 2. The agreement between them depends essentially on the convergence of the iterative
search procedure of the model calculations providing the expansion point x* on the basis of eq. (14). The
convergence of iterations depends on the value of the resonance defect A, as well as on the translational
reactant energy. Iterations fail to converge if point x* turns out to be in the region where the cross section
of the PES along the y axis (fig. 1) loses its double-well shape, or more exactly, when the energy level
vanishes in one of the valleys. In this case the procedure applied for calculating the integral B(x) proves to
be invalid (appendix A). The condition of the iteration convergence can be formulated as follows: the
classical turning points of the potentials in the reactant and product channels involved in a given transition
must fall within that range of x values where the vibrational basis functions of the integral B(x) are well
defined.
For symmetrical reactions this condition is obeyed at all energy values and as can be seen from fig. 2a,
the results of complete quantum and model calculations are in a good agreement. In the case of
non-symmetrical reactions the condition of iteration convergence is fulfilled at relatively small values of
the resonance defect and translational energy (fig. 2b and 2c), and then the agreement between the
complete quantum and the model calculations is good as well. However, even if the iterations do not
converge it is still possible to obtain reasonable estimates of the transition probabilities in the model
a zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA
p9Plo
o-
-6 -
2 zyxwvutsrqponm
-Pas
I
I
lb
1.8
2.0
2.2
E&-q,
Fig. 2. Dependences
Solid lines represent
2.4
22
1.4
1.6
I.8
I
1.4
I.6
2.0
2.2
E.hw,,
E.kw,
of the partial transmission
probabilities
PO’, on the translational
energy
the complete quantum calculation,
dashed lines the model calculation.
18
of the reactants
for reactions
(a)-(c).
M. V. Basilevsky et al. / Quantum dynamical calculations of linear exchange reaciions
288
I
I
’ ‘.’ ‘13i?
Wa
1.0
E’
Fig. 3. Cross sections of the potential energy surface along the
lines x = constant and the respective energy levels for reactions
(b) and (c).
Fig. 4.
PO’,on
for the
dashed
Dependences
of the partial transmission
probabilities
the translational
energy of the reactants for reaction (c)
mass ratios (I)-(III).
Solid lines correspond
to n = 2,
lines to n = 3.
theory. An analysis of the dependence
of the transition probabilities
Pa, on the position of the expansion
point x* showed that the values of Pa, vary rather smoothly. If the iterations converge, the approximation
x* = max(a,,
a,),
(27)
where a, and a, are the coordinates of the classical turning points in the reactant and product channels,
respectively,
changes the true probability
no more than by an order of magnitude.
Based on this
observation,
we applied approximation
(27) for the cases when iterations did not converge.
As is seen from fig. 3, the resonance defect for reaction (b), when passing from the level n = 0 of
reactants to the level n = 2 of products (Ao2), is approximate
a factor of three smaller than that for the
n = 1 product level (A,,). The model calculation
converges for the probability
Paz at small values of the
translational
energy only and does not converge for P,,,. Fig. 2b presents the results of model calculations
performed in the approximation
(27). They agree with the results of complete quantum calculation
for the
range of energy values where the quantum calculation yields appreciably different partial probabilities
P,,
and Po2. With increasing energy, when the values of P,,, and Paz obtained by quantum calculation become
almost equal, the agreement
deteriorates.
The total reaction probability
is simulated
in those model
calculations
to an accuracy of an order of magnitude.
The intersection
of energy curves of the P,, and Paz probabilities
observed both in complete quantum
and model calculations
is a peculiar feature of reaction (b). To understand
the origin of this phenomenon
it is sufficient to consider the approximate
formula for the transition
probability
(26) and take into
account the fact that the reactant level n = 0 lies between the n = 1 and n = 2 product levels (fig. 3). While
estimating the quantity Pci, the values of F involved in eq. (26) are calculated by the prescription
(27) at
the point x* = a,, and at the point x* = a, when the values of PO2 are estimated. According to formulas
(15) and (17), in the first case we get F = cE, but in the second case F = c(E - A,,). Hence, it follows
In PO, - -A,,/cE
and In PO2- - zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA
A,,/c( E - A,,). One can see that P,,, > PO2 at small energy values
E = A,,, however, PO2> PO1 at E s=. Ao2, since A ,,1 > A ,,* (fig. 3).
In the case of reaction (c) the resonance defect has the least value, and the n = 2 product level is lower
h4. V. Basilevsky et al. / Quantum dynamical calculations of linear exchange reactions
289 zyxwvut
Table 1
V alues of angles 0 and resonance misfits A for reaction (c) for the mass ratios (I)-(III)
M ass ratio
0 (de@
I
14
0.35
0.5
II
11
0.2
0.65
0.4
0.45
III
5.5
than the n = 0 reactant level (fig. 3). The model calculations
of PO2 converge up to nearly the reaction
energy barrier and agree well with the results of quantum
calculation.
The model calculations
of PO3
converge at small energies only. Since in the whole energy range under study P,,) < PO*, the total reaction
probability
as obtained from model calculations
almost coincides with the exact quantum one.
In the example of reaction (c) it is convenient
to analyze the dependence
of transition probabilities
PO,,
on the value of the resonance misfit A and on angle 8. [In the limiting cases P,,,, is defined by formulas
(22) and (26).] Fig. 4 displays the energy dependences
of PO2 and PO3 for three mass ratios in the reacting
system:
*A
=
mc
=
15,
mA = 15, mc = 100,
*A
=
m, = 100.
(I)
(II)
(III)
The corresponding
values of B and A are listed in table 1. For the mass ratios (I) and (III) the values of 8
differ approximately
by a factor of 2.5, and the values of the resonance defects turn out to be almost equal.
Accordingly,
in the case of mass ratio (III) the probabilities
PO2 and PO3 are, respectively, two and three
orders of magnitude less than in case (I), which is due to the difference in the angles.
The difference in the mass ratio (I) and (II) results in a slight difference in the angle 0 (table 1).
However, the resonance misfit A,, for case (II) is less than for case (I), and for A,, the opposite is true. As
is seen from fig. 4 an increase in the resonance misfit under almost constant 0 always leads to a decrease in
the transition probability‘: so at the mass ratio (II) the probability
PoZ is greater and the probability
PO3 is
less than the corresponding
values for the mass ration (I).
The model calculations
thus confirm the qualitative regularities following from the asymptotic formulas
(22) and (26), but the latter formulas are not accurate enough to provide a quantitative
estimation of the
effect of the parameters
8 and A on the values of the probabilities.
This inapplicability
of the limiting
formulas for quantitative
estimates is accounted for by the fact that the value of the parameter E (18) for
all the reactions studied varies from 0.5 to 3.0. So, none of the limiting cases considered in section 3 can be
realized. A single exception is the transition probability
PO2 in reaction (c) for the mass ratios (I) and (II).
In the calculation of PO2 the value of 5 varies from 7 to 13, and the first inequality of (24) certainly holds.
Comparing
the results of the calculations
made in the framework of the present model and that of
Babamov and co-workers [7-91 it should be kept in mind that they differ in the following:
(1) In our model the complete action but not their linear expansions in x are used as arguments of the
Airy functions describing the translational
motion.
(2) The difference in the x-derivatives
of the translational
potentials
of the reactant and product
channels at their turning points is taken into account. This necessarily results in the iterative search of the
expansion point x*.
Refining
the treatment
of translational
motion,
these modifications
become insignificant
in the
symmetric case for which our procedure in its main part reduces to that of Babamov et al. [8]. We have
repeated the calculations
using Babamov formula and the matrix element B(x) obtained as outlined in
appendix A. The disagreement
with the preceding results did not exceed 10%. It should be born in mind
M. V. Basilevsky et al. / Quantum dynamical calculations of linear exchange reactions
290
that the turmelling
probabilities
are very sensitive to the quality of the vibrational
matrix element
calculation
and an error could appear at this stage due to the Morse approximation
of vibrational
PES
cross sections. That is why one should not reckon on obtaining quantitative
coincidence with the complete
quantum calculation until an accurate treatment of the vibrational
motion [8] is performed. In the case of
non-symmetrical
reactions the Babamov approximation
yields probability
values exceeding those obtained
in our model by one or two orders of magnitude
and its agreement
with the results of quantum
calculations
is always worse. The discrepancy
takes place even in the abovementioned
case when
4 = 10 X= 1. The reason is that in the calculation of the transition probability
PO2 the second inequality in
(24) is violated: A/E 2 0.3. The presence of the iteration search for the expansion point x* additionally
increases the divergence in the results of the calculations
obtained in the framework of the two models. zyxwvutsrq
Appendix A: Calculation of the integral B(x)
Proceeding from coordinate transformation
(3) and using the smallness
appearing in eq. (5) can be rewritten in the form
of the angle 8, the integral
B(x)
Here #i and q2 are the eigenfunctions
of the single-well potentials V,(y, x) and V,(y, x), corresponding
to the reactant and product valleys, and V,, ( y, x) is the difference between the real double-well potential
and one of single-well ones.
The integral B(x) can analytically
be calculated in the harmonic approximation,
but the results turn
out to be unsatisfactory
for quantitative
estimates. So we applied the eigenfunctions
of a Morse oscillator
as a vibrational
basis and approximated
the potentials V, and V2 by Morse curves. As in paper [9], the
quantity B(x) was obtained by numerical integration.
Appendix B: Calculation of the integral A,
Let us differentiate
the exponent
$=i[u’-q(r-l)]
$=i(u2-r)=O,
Cancelling
D=
index in (11):
=O,
z from the first two equations,
$=i(-u-uq)-*=O.
(B-1)
we obtain
(B.4
-u/q+iT/q
and then, using the third equation,
us=(-q3t+T2)/(7+
z = 24: = (
-?(1+
we obtain
the coordinates
of the saddle point
{q3[-(q3-l)t+TZ]}1’2},
43)
The value of u is determined
-
rq3(1
-
from (B.2)
43)
+ 27q3’2[
t(1 - 43) + 72]“2)/(1
- q3)2.
(B.3)
A4. V. Barilevsky
et al. / Quantum dynamical calculations of linear exchange reactions
The matrix of the second derivatives
(B.l):
required
for calculating
the pre-exponential
291
factor is derived
from
/
a2
-$$=ziu
-0
a*
a*
P=
a2=
, azau
a*
2=2iu
j&=0
-’
_.
a2
GE
l4
’
zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFED
.
-
avaz
2L
azav
-i
.
auat=
ihdu
-14
7
0
/
det p=2i(uq*+u).
As a result we get formula
05.4) zyxwvutsrq
(12).
Appendix C: On the e-dependence
of the main parameters of the problem
The LEPS-type
potential
energy surface of a triatomic system A-B-C
is constructed
in terms of
interatomic
distances R,, and R,,. The transition to the mass-weighed
Cartesian coordinates
is accomplished through well-known formulas [14]
p1 = M”*[R,a
+ (I-Q/~,)R~~]
a,
PZ = IU:/*Reca,
where
and a is an arbitrary
constant.
Mi =mAmc/(mA+mc),
and the coordinates
Once the mass of the central
atom is small (m, -K mA, m,),
we have
cl*=mg
pi and p2 coincide
with x and y, respectively:
If it is required that at any mass ratio the vibrational
frequency
w,, remains unchanged
under this
transformation,
then the scale along the y axis should be mass-independent.
This is achieved by the choice
ofa=m,
‘I* . Taking into consideration
that the equation of the x axis has the form R,, = R,, = R, we
obtain
x = 2[m,mc/mB(m,
+ m,)]“*=
R/d.
Therefore, upon varying the mass ratio in the reacting system and at a fixed value of Aw, the scale of the x
axis changes. It can easily be seen that the derivative av/ax
- 8 and wi - l/e. According to formulas (7)
and (8) zi - 6-2/3, f, - 6-‘/3. From relations (9) and (10) it follows that q is independent
of 8, and also
formulas (23) are derived.
References
[I] A. Kupperman,
J.A. Kaye and J.P. Dwyer, Chem. Phys. Letters
121 J. Riimelt, Chem. Phys. Letters 74 (1980) 263.
74 (1980) 257.
292
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[lo]
[ll]
[12]
[13]
[14]
M. V. Bosilevsky
et al. / Quantum dynamical calculations of linear exchange reactions
M.V. Basilevsky and V.M. Ryaboy, Chem. Phys. 50 (1980) 231.
M.V. Basilevsky, G.E. Chudinov and V.M. Ryaboy, Teor. Eksp. Khim. 21 (1985) 257.
M.V. Basilevsky, G.E. Chudinov and V.M. Ryaboy, Chem. Phys. 104 (1986) 265.
M.Ya. Ovchinnikova,
Chem. Phys. 36 (1979) 85.
V.K. Babamov and R.A. Marcus, J. Chem. Phys. 74 (1981) 1790.
V.K. Babamov, V. Lopes and R.A. Marcus, J. Chem. Phys. 78 (1983) 5621.
V. Lopes, V.K. Babamov and R.A. Marcus, J. Chem. Phys. 81 (1984) 3962.
L.D. Landau and E.M. Lifshitz, Quantum mechanics (Nauka, Moscow, 1974).
A. Messiah, Quantum mechanics (North-Holland,
Amsterdam,
1961).
S. Phigge, Practical quantum mechanics, Vol. 1 (Springer, Berlin, 1971).
M. Abramowitz
and I. A. Stegun, Handbook
of mathematical
functions (Dover, New York, 1972).
S. Glasstone, K. Laidler and H. Eyring, Theory of rate processes (McGraw-Hill,
New York, 1941).
Chemical Physics 104 (1986) 281-292
North-Holland,
Amsterdam
APPROXIMATE QUANTUM CALCULATION
OF THE DYNAMICS
OF GAS-PHASE REACTIONS OF A LIGHT-ATOM TRANSFER
IN THE TUNNELLING ENERGY REGION
M.V. BASILEVSKY,
G.E. CHUDINOV
and V.M. RYABOY
L. Ya. Karpov Institute of Physical Chemistry, ul. Obukha IO, 107120 Moscow B-120, USSR
Received
9 September
1985; in final form 30 December
1985
An approximate
method for calculating
the probabilities
of linear reactions of a light-atom
transfer based on the Born
distorted-wave
approximation
is proposed. The matrix element over the vibrational
coordinate
is numerically calculated, the
translational
motion is treated quasi-classically.
The reaction probability
expressed as a triple integral is evaluated by the
steepest descent method. All the integrands
are expanded at the saddle point, its position being determined
by iterations.
Complete quantum calculations
are compared with this model treatment for the following reactions: CH; +CH,,
CH; +
PhCH,, CH; + PhCH,D.
The results of the model and complete calculations
agree for both symmetrical
and asymmetrical
reactions with considerable
misfit of the resonance.
1. Introduction
Accurate quantum dynamical calculations
of linear triatomic reactions with the transfer of a light atom
have become available in the last few years. They involve rather laborious calculational
procedures dealing
either with numerical basis functions [1,2], or with a complicated curvilinear coordinate system [3-51. As is
known, the hamiltonian
of these systems contains a small parameter namely, the ratio of the masses of the
particles. In the scaled skew coordinates
this small parameter is the angle 28 between the asymptotic
directions
of the reactant and product valleys (fig. 1). It would be tempting
to exploit this fact for
simplifying the calculational
procedure.
Ovchinnikova
[6] used the smallness of the mass ratio for simplifying the classical equations of motion
in the classical S-matrix method. Approximate
calculations
of quantum transition
probabilities
made by
Babamov and co-workers [7-91 can be applied when vibrational
frequencies
of reactant and product
molecules are close or equal. Both approaches are invalid for calculating
the probabilities
of the reaction
between vibrational
states with considerable
energy difference.
In this communication
we develop a simple method of calculating
quantum
tunnelling
reaction
probabilities
for the system with small angles 8 and arbitrary
resonance
misfit. It is based on the
distorted-wave
Born approximation
(DWBA). The transition
matrix element is calculated using a quasiclassical representation
of the translational
wavefunctions
in the valleys of reactants and products. In the
systems under consideration
the adiabatic potentials of translational
motion for different vibrational
states
of reactants and products represent almost parallel curves. They are merely shifted along the energy axis
by the resonance misfit. The potentials of reactant and product levels intersect neither on the real axis of
translational
variable x, nor in its nearest neighbourhood.
Therefore,
the standard
treatment
[lo] of
quasi-classical
integrals is inapplicable.
Such an integral has been calculated for two cases when the transition
operator is independent
of x
(Landau and Lifshitz [lo]) and for an almost resonance situation (Babamov et al. (81). In the first case the
transition occurs far from the classical turning points of both translational
functions, in the second case it
0301-0104/86/$03.50
0 Elsevier Science Publishers
(North-Holland
Physics Publishing Division)
B.V.
M. V. Basilevsky et al. / Quantum dynamical calculations of linear exchange reactions
282
Fig. 1. Potential energy surface (schematically)
and coordinate
systems applied in the calculation
of the matrix element (I).
occurs in the vicinity of the turning point which is practically the same for both functions owing to their
resonance. For zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA
our method of calculating this integral the relative position of the transition region and
turning points is of no importance, so that the Landau and Babamov formulas are obtained as asymptotic
limit cases. This method works efficiently provided the relief of the potential energy surface (PES) in the
transition region is such that the pair of potential minima corresponding to reactant and product valleys
can be clearly distinguished.
Using this procedures we have calculated the probability of tunnelling transitions in thermoneutral and
exothermic radical reactions of hydrogen and deuterium atoms transfer:
R,H + R, -+ R, + HR,,
R,D+R,+R,+DR,
The results are compared with complete quantum calculations.
2. zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA
Formulation of the distorted-wave approximation
The amplitude of the transition
has the form [9,11]:
S,,(E)
= -27ri(x’*‘(E)
probability
from the reactant
(V,, Jx(‘)(E)).
to product
valley as given in the DWBA
(1)
The angular brackets denote integration in the coordinate plane, the upper indices 2 and 1 label the
wavefunctions in the reactant and product valleys, respectively. In each of the valleys the solutions x” ,*’
are products of the translational wavefunction ‘p,,* and the vibrational function $I,* and are most easily
represented in local coordinate frames (xi, y1 ), (x2, y2) (fig. 1) having one of the axes directed along the
valley bottom and the other being orthogonal to it:
X “‘=Vl(~,N,(YI~
$1,
x(*)=v*b*)~*(Y*~
x2).
(2)
h4. V. Basilevsky et al. / Quantum dynamical calculalions
oflinear exchange
(For brevity we omit in formulas (2) the index indicating
the
products.) The angle between the axes xi and x2, equal to 28,
reacting system. For matrix element (1) to be calculated, the valley
coordinate system (x, Y). Transformation
of the coordinates
has
reactions
vibrational
state of the reactants
and
is determined
by the mass ratio of the
solutions should be referred to a unique
the form:
~ 0~ e
*I
-:::)(
;)p
(;:)
= (E,”
-ZZ)(
;)=
sin B
i Yl )
i zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA
After passing to the frame (x, y) the integrand
angle B is assumed to be small. Then, neglecting
S,,(E)
283
(3)
in (1) can be expanded in a series of y sin 8, since the
the terms of the order of 8, we get:
= -2GA,
of translational
wavefunctions
generate a
(It can be shown that the terms - sin 8 in the arguments
- 8 in the result of integration.)
We rewrite formula (4) separating explicitly the integration
correction
over the vibrational
coordinate y:
where
B(x)
=JW &(Y,,
--m
For translational
vi(x)
+L(Y,,
wavefunctions
= ~zj’~k;‘/~
with c = (2m)‘12/fi.
x) dx.
we apply the quasi-classical
Ai( -zi),
Here Ai( -z)
Zi=($~)2’3,
~)G(Y,
q2(x)
representation
[12]:
= CZ?/~/C;~/~ Ai( -z2),
(6)
is the Airy function
w=A-‘/*[2m(Ea,
k,=dy/dx,
~i)]l’~
dx,
i= 1,2.
(7)
(v is the diabatic channel potential and a, is the coordinate of the turning point in the reactant or product
valleys.) The values entering into eq. (7) should have a second index indicating
the vibrational
state of
reactants or products, but it is omitted for the sake of brevity.
While calculating the integral over the translational
coordinate it is convenient
to choose the argument
of one of the Airy functions as an integration
variable. We choose z1 for definiteness.
Then z2 and x
should be expressed as functions of z,. The change of the integration
variable is performed as follows:
dz, = (dz,/dW,)
dW, = ($ IV1)1’3k, dx =fi’
dx,
dx =fi
dz,,
(8)
where
f;=($v/k,3)1’3,
To express
z2 through
~2~~2(z:)+dzJdz,l,*(z,
i=1,2.
z, we expand
z2 in z, at some point
-z:)=z,(x*>+
x = x* = x(z:):
[.f,b*)/f2(~*>1(~,
-4%
Thus
z2 = 4(z, - t>,
(9)
M. V. Basilevsky et al. / Quantum dynamical calculations
284
of linear exchange
reactions
where
I= z1(x*)
4 =f,(-x*M(x*),
[fib*vfib*)l z:-
-
over the
Finally we represent
integral
B(x) (5) as a function
of z,. The result of integration
translational
coordinate depends on the choice of the basis functions and the interaction
potential. This is
discussed in appendix A. Here we employ the exponential
representation
of B(x), and expand the
exponential
index into a series at point x *, taking into account the relations
x =x*
B(x)
I,,( z1 -z:)
+ dx/dz,
= exp[ln
B(x*)
= x* +f,(x*)(z,
1xefiz:
- d In B(x)/dx
- z?),
- 7z1],
00)
where
r=
-dln
B(x)/dxl,,f,(x*).
The faster the integral over y changes,
of r. Now the integral (5) is:
A z
c2fi)/2f;/2exp{ln
on moving
along the translational
coordinate,
the larger the value
B(x*)
- [d In B(x*)/dx] zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLK
fi(x*)z:}A,,
- q(zl
- t))
where
A, =
J
dz, Ai( -zl)Ai(
By applying
triple one
A,
the integral
=-&///dz
representation
of the Airy functions
du du exp{i[+(a3
and evaluated by the steepest
obtain (see appendix B)
descent
exp( -rz,).
[13], integral
+ u’) - uz - uq(z - t)] -
method
simultaneously
A,
can be transformed
TZ}
over all the three variables.
into a
(11)
As a result we
(12)
where
(13)
The values q, t and r are
indicates that the quantity
for deriving formula (12),
estimation
of the integral.
equation
z1(x,)
=
functions of x*; they are determined
by formulas (9) and (10). The index s
U, is calculated at the saddle point x,. If the expansions (9) and (lo), applied
errors in the
are performed
at xS, they do not bring about any additional
According
to (B.3) the position of the saddle point is determined
from the
b4x*)12,or
W1(x,)
=$[ Us]‘.
(14)
While operating with formula (14) the equality x* = x, can be achieved by the iterative self-consistent
procedure,
For calculational
convenience we use the exponential
representation
of the potential in the reactant and
product valleys
V,(x)=Eexp[-c(x-a,)],
V,(x)=V,+A.
(15)
285
M. V. Basilevsky et al. / Quantum dynamical calculations of linear exchange reactions
(Here E is the translational
energy of reactants and A is the resonance misfit, i.e. the difference between
the positions of the energy levels in the reactant and product valleys.) For the LEPS-type PESs relation
(15) holds with an accuracy of 3-5s
in a wide range of the translational
coordinate
values. Such a
treatment
enables one to express the quantities
q and r through
elementary
functions.
In this case the
action integral is analytical.
3. zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA
Limiting cases of formula (12)
Let us start with elucidating
combination
of the parameters
-(q3-l)t=
into formula
in form:
(12). The
(2m/fi2)f,(F2-1;;)6,
where 6 is the distance
product valleys.
&=
the physical meaning of the values 7, q, t, entering
-(q3
- 1)t of (12) can approximately
be presented
-a~:/a~]~,,
(16)
along the translational
coordinate
between
the turning
points
i=l,2
in the reactant
and
07)
is the “force” at te turning point a,. It should be noted that both the force difference (F, - FI) and the
value 6 depend on the resonance misfit A.
Thus, the value of the transition
matrix element is mainly determined
by the variation
rate of the
integral B(x), the values of forces F, and F2 and by the resonance misfit A. There exist two limiting cases
which correspond
to two ways of expanding
the square root in eq. (13) into a series, depending
on the
value of the parameter
6 = T2/(1 - q3)t.
(18)
It can be shown that at a small resonance
thus represents the Messiah parameter.
misfit
(A/E -SK1) 5 = 5h2X2E/2mA2,
and the value of t-i/*
3.1. The Landau limit [lo]
T2e
1).
-r(q3-
(19)
If the dependence
of B(x)
calculated the overlap integral
acquires the form:
U2 - zi = 0,
on the translational
coordinate
is neglected, that is if we put T = 0 and
of two Airy functions,
the system of equations defining the saddle point
u2 - z2 = 0,
u dz,/dx
+ u dz,/dx
= 0.
(20)
(No approximation
related to the expansion of z2 over z, is made here.)
While analysing system (20) with the explicit form of potential (15) one can see that, in accordance with
the result of Landau and Lifshitz [lo], it has the solution x, = - co, and I/(x,) + cc. Expanding (12) and
(13) in the small parameter (E/V)‘/*,
we can obtain an analytical solution of equation (14), which then
turns into V(x*) = (A/2X)*. We introduce the notation:
h=
]dln
The coordinate
x,=:x*
B/dx]
=~/fi.
of the saddle point is
= -_c -* ln(A2/2X2E).
(21)
M. K Basilevsky et al. / Quantum dynamical calculations
286
Condition
oflinear exchange
reactions
(19) takes the form
Aw[(E+A)“~-E~/~]
and the transition
amplitude
becomes
s,, = (~~cA[(E+A)‘/~-E~~~])-~‘~~B(~)
The exponent of expression (22) coincides
vanishes at X 4 0. This is in agreement with
different energies (A + 0) are orthogonal.
It should be noted that for non-symmetric
always present when the mass of the central
(the reactant and product valleys are almost
cRelation
e,
72 - 8413 )
_ E112]).
(22)
[lo]. The pre-exponential
factor
of the Schradinger equation with
reactions (non-zero resonance misfit) the Landau limit is
particle tends to zero. We show (appendix C) that at small B
parallel) the quantity q is independent
of ti and
(23)
spontaneously.
A/E&l.
When the second inequality
d In B
>>
dx
I
A)‘/2
limit (81
~~x=- -t(q3-l),
A
~
(2my2
with the Landau exponent
the fact that two solutions
t-e-2’3.
(19) is thus fulfilled
3.2. The Babamov
exp( -(~TI~)[(E+
I
(24)
holds the first one takes the form
(25)
This inequality establishes a relation between the rate of change of the vibrational
matrix element and the
resonance misfit.
In this limiting case in deriving an expression for the transition matrix element, the exponent in (12) is
expanded in powers of A/E with inequality
(25) taken into account. All the A-dependent
terms in the
pre-exponential
factor are neglected. As a result we obtain:
(26)
The quantity X is determined in (21). For deriving (26) expansion (9) has been performed at point a,. Eq.
(26) differs from that reported by Babamov et al. [S] by three extra terms in the exponent index (the third,
fifth and the sixth terms are absent in ref. [8]). Eq. (26) coincides with the result of ref. [S] under the
additional condition of h2/E .szz1.
Neglection
of the sixth term is likely to be justified, at least due to the smallness of the numerical
coefficient. The fifth term can be neglected if h < 1. Apparently,
it seems inconsistent
to neglect the third
term at small A, retaining the fourth one. The difference between formula (26) and the results obtained by
Babamov et al. arises from the fact that they assumed q3 - 1 = 0 (or, in other words Fl = F2) whereas this
value is - A.
287
M. V. Basileusky et al. / Quantum dynamical calculations of linear exchange reactions
4. zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA
Results and discussion
The results of quantum [5] and model dynamical calculations
Ovchinnikova by the method of classical S-matrix [6]:
of the reactions (studied earlier by
CH; + CH,,
(a)
CH; + PhCH,,
(b)
CH; + PhCH,D,
(c)
are presented in fig. 2. The agreement between them depends essentially on the convergence of the iterative
search procedure of the model calculations providing the expansion point x* on the basis of eq. (14). The
convergence of iterations depends on the value of the resonance defect A, as well as on the translational
reactant energy. Iterations fail to converge if point x* turns out to be in the region where the cross section
of the PES along the y axis (fig. 1) loses its double-well shape, or more exactly, when the energy level
vanishes in one of the valleys. In this case the procedure applied for calculating the integral B(x) proves to
be invalid (appendix A). The condition of the iteration convergence can be formulated as follows: the
classical turning points of the potentials in the reactant and product channels involved in a given transition
must fall within that range of x values where the vibrational basis functions of the integral B(x) are well
defined.
For symmetrical reactions this condition is obeyed at all energy values and as can be seen from fig. 2a,
the results of complete quantum and model calculations are in a good agreement. In the case of
non-symmetrical reactions the condition of iteration convergence is fulfilled at relatively small values of
the resonance defect and translational energy (fig. 2b and 2c), and then the agreement between the
complete quantum and the model calculations is good as well. However, even if the iterations do not
converge it is still possible to obtain reasonable estimates of the transition probabilities in the model
a zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA
p9Plo
o-
-6 -
2 zyxwvutsrqponm
-Pas
I
I
lb
1.8
2.0
2.2
E&-q,
Fig. 2. Dependences
Solid lines represent
2.4
22
1.4
1.6
I.8
I
1.4
I.6
2.0
2.2
E.hw,,
E.kw,
of the partial transmission
probabilities
PO’, on the translational
energy
the complete quantum calculation,
dashed lines the model calculation.
18
of the reactants
for reactions
(a)-(c).
M. V. Basilevsky et al. / Quantum dynamical calculations of linear exchange reaciions
288
I
I
’ ‘.’ ‘13i?
Wa
1.0
E’
Fig. 3. Cross sections of the potential energy surface along the
lines x = constant and the respective energy levels for reactions
(b) and (c).
Fig. 4.
PO’,on
for the
dashed
Dependences
of the partial transmission
probabilities
the translational
energy of the reactants for reaction (c)
mass ratios (I)-(III).
Solid lines correspond
to n = 2,
lines to n = 3.
theory. An analysis of the dependence
of the transition probabilities
Pa, on the position of the expansion
point x* showed that the values of Pa, vary rather smoothly. If the iterations converge, the approximation
x* = max(a,,
a,),
(27)
where a, and a, are the coordinates of the classical turning points in the reactant and product channels,
respectively,
changes the true probability
no more than by an order of magnitude.
Based on this
observation,
we applied approximation
(27) for the cases when iterations did not converge.
As is seen from fig. 3, the resonance defect for reaction (b), when passing from the level n = 0 of
reactants to the level n = 2 of products (Ao2), is approximate
a factor of three smaller than that for the
n = 1 product level (A,,). The model calculation
converges for the probability
Paz at small values of the
translational
energy only and does not converge for P,,,. Fig. 2b presents the results of model calculations
performed in the approximation
(27). They agree with the results of complete quantum calculation
for the
range of energy values where the quantum calculation yields appreciably different partial probabilities
P,,
and Po2. With increasing energy, when the values of P,,, and Paz obtained by quantum calculation become
almost equal, the agreement
deteriorates.
The total reaction probability
is simulated
in those model
calculations
to an accuracy of an order of magnitude.
The intersection
of energy curves of the P,, and Paz probabilities
observed both in complete quantum
and model calculations
is a peculiar feature of reaction (b). To understand
the origin of this phenomenon
it is sufficient to consider the approximate
formula for the transition
probability
(26) and take into
account the fact that the reactant level n = 0 lies between the n = 1 and n = 2 product levels (fig. 3). While
estimating the quantity Pci, the values of F involved in eq. (26) are calculated by the prescription
(27) at
the point x* = a,, and at the point x* = a, when the values of PO2 are estimated. According to formulas
(15) and (17), in the first case we get F = cE, but in the second case F = c(E - A,,). Hence, it follows
In PO, - -A,,/cE
and In PO2- - zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA
A,,/c( E - A,,). One can see that P,,, > PO2 at small energy values
E = A,,, however, PO2> PO1 at E s=. Ao2, since A ,,1 > A ,,* (fig. 3).
In the case of reaction (c) the resonance defect has the least value, and the n = 2 product level is lower
h4. V. Basilevsky et al. / Quantum dynamical calculations of linear exchange reactions
289 zyxwvut
Table 1
V alues of angles 0 and resonance misfits A for reaction (c) for the mass ratios (I)-(III)
M ass ratio
0 (de@
I
14
0.35
0.5
II
11
0.2
0.65
0.4
0.45
III
5.5
than the n = 0 reactant level (fig. 3). The model calculations
of PO2 converge up to nearly the reaction
energy barrier and agree well with the results of quantum
calculation.
The model calculations
of PO3
converge at small energies only. Since in the whole energy range under study P,,) < PO*, the total reaction
probability
as obtained from model calculations
almost coincides with the exact quantum one.
In the example of reaction (c) it is convenient
to analyze the dependence
of transition probabilities
PO,,
on the value of the resonance misfit A and on angle 8. [In the limiting cases P,,,, is defined by formulas
(22) and (26).] Fig. 4 displays the energy dependences
of PO2 and PO3 for three mass ratios in the reacting
system:
*A
=
mc
=
15,
mA = 15, mc = 100,
*A
=
m, = 100.
(I)
(II)
(III)
The corresponding
values of B and A are listed in table 1. For the mass ratios (I) and (III) the values of 8
differ approximately
by a factor of 2.5, and the values of the resonance defects turn out to be almost equal.
Accordingly,
in the case of mass ratio (III) the probabilities
PO2 and PO3 are, respectively, two and three
orders of magnitude less than in case (I), which is due to the difference in the angles.
The difference in the mass ratio (I) and (II) results in a slight difference in the angle 0 (table 1).
However, the resonance misfit A,, for case (II) is less than for case (I), and for A,, the opposite is true. As
is seen from fig. 4 an increase in the resonance misfit under almost constant 0 always leads to a decrease in
the transition probability‘: so at the mass ratio (II) the probability
PoZ is greater and the probability
PO3 is
less than the corresponding
values for the mass ration (I).
The model calculations
thus confirm the qualitative regularities following from the asymptotic formulas
(22) and (26), but the latter formulas are not accurate enough to provide a quantitative
estimation of the
effect of the parameters
8 and A on the values of the probabilities.
This inapplicability
of the limiting
formulas for quantitative
estimates is accounted for by the fact that the value of the parameter E (18) for
all the reactions studied varies from 0.5 to 3.0. So, none of the limiting cases considered in section 3 can be
realized. A single exception is the transition probability
PO2 in reaction (c) for the mass ratios (I) and (II).
In the calculation of PO2 the value of 5 varies from 7 to 13, and the first inequality of (24) certainly holds.
Comparing
the results of the calculations
made in the framework of the present model and that of
Babamov and co-workers [7-91 it should be kept in mind that they differ in the following:
(1) In our model the complete action but not their linear expansions in x are used as arguments of the
Airy functions describing the translational
motion.
(2) The difference in the x-derivatives
of the translational
potentials
of the reactant and product
channels at their turning points is taken into account. This necessarily results in the iterative search of the
expansion point x*.
Refining
the treatment
of translational
motion,
these modifications
become insignificant
in the
symmetric case for which our procedure in its main part reduces to that of Babamov et al. [8]. We have
repeated the calculations
using Babamov formula and the matrix element B(x) obtained as outlined in
appendix A. The disagreement
with the preceding results did not exceed 10%. It should be born in mind
M. V. Basilevsky et al. / Quantum dynamical calculations of linear exchange reactions
290
that the turmelling
probabilities
are very sensitive to the quality of the vibrational
matrix element
calculation
and an error could appear at this stage due to the Morse approximation
of vibrational
PES
cross sections. That is why one should not reckon on obtaining quantitative
coincidence with the complete
quantum calculation until an accurate treatment of the vibrational
motion [8] is performed. In the case of
non-symmetrical
reactions the Babamov approximation
yields probability
values exceeding those obtained
in our model by one or two orders of magnitude
and its agreement
with the results of quantum
calculations
is always worse. The discrepancy
takes place even in the abovementioned
case when
4 = 10 X= 1. The reason is that in the calculation of the transition probability
PO2 the second inequality in
(24) is violated: A/E 2 0.3. The presence of the iteration search for the expansion point x* additionally
increases the divergence in the results of the calculations
obtained in the framework of the two models. zyxwvutsrq
Appendix A: Calculation of the integral B(x)
Proceeding from coordinate transformation
(3) and using the smallness
appearing in eq. (5) can be rewritten in the form
of the angle 8, the integral
B(x)
Here #i and q2 are the eigenfunctions
of the single-well potentials V,(y, x) and V,(y, x), corresponding
to the reactant and product valleys, and V,, ( y, x) is the difference between the real double-well potential
and one of single-well ones.
The integral B(x) can analytically
be calculated in the harmonic approximation,
but the results turn
out to be unsatisfactory
for quantitative
estimates. So we applied the eigenfunctions
of a Morse oscillator
as a vibrational
basis and approximated
the potentials V, and V2 by Morse curves. As in paper [9], the
quantity B(x) was obtained by numerical integration.
Appendix B: Calculation of the integral A,
Let us differentiate
the exponent
$=i[u’-q(r-l)]
$=i(u2-r)=O,
Cancelling
D=
index in (11):
=O,
z from the first two equations,
$=i(-u-uq)-*=O.
(B-1)
we obtain
(B.4
-u/q+iT/q
and then, using the third equation,
us=(-q3t+T2)/(7+
z = 24: = (
-?(1+
we obtain
the coordinates
of the saddle point
{q3[-(q3-l)t+TZ]}1’2},
43)
The value of u is determined
-
rq3(1
-
from (B.2)
43)
+ 27q3’2[
t(1 - 43) + 72]“2)/(1
- q3)2.
(B.3)
A4. V. Barilevsky
et al. / Quantum dynamical calculations of linear exchange reactions
The matrix of the second derivatives
(B.l):
required
for calculating
the pre-exponential
291
factor is derived
from
/
a2
-$$=ziu
-0
a*
a*
P=
a2=
, azau
a*
2=2iu
j&=0
-’
_.
a2
GE
l4
’
zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFED
.
-
avaz
2L
azav
-i
.
auat=
ihdu
-14
7
0
/
det p=2i(uq*+u).
As a result we get formula
05.4) zyxwvutsrq
(12).
Appendix C: On the e-dependence
of the main parameters of the problem
The LEPS-type
potential
energy surface of a triatomic system A-B-C
is constructed
in terms of
interatomic
distances R,, and R,,. The transition to the mass-weighed
Cartesian coordinates
is accomplished through well-known formulas [14]
p1 = M”*[R,a
+ (I-Q/~,)R~~]
a,
PZ = IU:/*Reca,
where
and a is an arbitrary
constant.
Mi =mAmc/(mA+mc),
and the coordinates
Once the mass of the central
atom is small (m, -K mA, m,),
we have
cl*=mg
pi and p2 coincide
with x and y, respectively:
If it is required that at any mass ratio the vibrational
frequency
w,, remains unchanged
under this
transformation,
then the scale along the y axis should be mass-independent.
This is achieved by the choice
ofa=m,
‘I* . Taking into consideration
that the equation of the x axis has the form R,, = R,, = R, we
obtain
x = 2[m,mc/mB(m,
+ m,)]“*=
R/d.
Therefore, upon varying the mass ratio in the reacting system and at a fixed value of Aw, the scale of the x
axis changes. It can easily be seen that the derivative av/ax
- 8 and wi - l/e. According to formulas (7)
and (8) zi - 6-2/3, f, - 6-‘/3. From relations (9) and (10) it follows that q is independent
of 8, and also
formulas (23) are derived.
References
[I] A. Kupperman,
J.A. Kaye and J.P. Dwyer, Chem. Phys. Letters
121 J. Riimelt, Chem. Phys. Letters 74 (1980) 263.
74 (1980) 257.
292
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[lo]
[ll]
[12]
[13]
[14]
M. V. Bosilevsky
et al. / Quantum dynamical calculations of linear exchange reactions
M.V. Basilevsky and V.M. Ryaboy, Chem. Phys. 50 (1980) 231.
M.V. Basilevsky, G.E. Chudinov and V.M. Ryaboy, Teor. Eksp. Khim. 21 (1985) 257.
M.V. Basilevsky, G.E. Chudinov and V.M. Ryaboy, Chem. Phys. 104 (1986) 265.
M.Ya. Ovchinnikova,
Chem. Phys. 36 (1979) 85.
V.K. Babamov and R.A. Marcus, J. Chem. Phys. 74 (1981) 1790.
V.K. Babamov, V. Lopes and R.A. Marcus, J. Chem. Phys. 78 (1983) 5621.
V. Lopes, V.K. Babamov and R.A. Marcus, J. Chem. Phys. 81 (1984) 3962.
L.D. Landau and E.M. Lifshitz, Quantum mechanics (Nauka, Moscow, 1974).
A. Messiah, Quantum mechanics (North-Holland,
Amsterdam,
1961).
S. Phigge, Practical quantum mechanics, Vol. 1 (Springer, Berlin, 1971).
M. Abramowitz
and I. A. Stegun, Handbook
of mathematical
functions (Dover, New York, 1972).
S. Glasstone, K. Laidler and H. Eyring, Theory of rate processes (McGraw-Hill,
New York, 1941).