The cumulative
reaction probability
as eigenvalue
problem
Uwe Manthe and William H. Miller
Department of Chemistry, University of California, and Chemical SciencesDivision,
Lawrence Berkeley Laboratory, Berkeley, California 94720
(Received 13 April
1993; accepted 11 May 1993)
It is shown that the cumulative reaction probability for a chemical reaction can be expressed
(absolutely rigorously) as N(E) =Xkp#>,
where {pk} are the eigenvalues of a certain Hermitian matrix (or operator). The eigenvalues {pk} all he between 0 and 1 and thus have the
interpretation as probabilities, eigenreaction probabilities which may be thought of as the rigorous generalization of the transmission coefficients for the various states of the activated complex
in transition state theory. The eigenreaction probabilities {pk} can be determined by diagonalizing a matrix that is directly available from the Hamiltonian matrix itself. It is also shown how
a very efficient iterative method can be used to determine the eigenreaction probabilities for
problems that are too large for a direct diagonalization to be possible. The number of iterations
required is much smaller than that of previous methods, approximately the number of eigenreaction probabilities that are significantly different from zero. All of these new ideas are illustrated by application to three model problems-transmission
through a one-dimensional (Eckart potential) barrier, the collinear H +H, -+ H, + H reaction, and the three-dimensional version
of this reaction for total angular momentum J=O.
I. INTRODUCTION
The thermal rate constant for a chemical reaction is
expressed conveniently in terms of the cumdative reaction
probability N(E),’
k(T)=[2d@r(T)]m1~~a
dEe-m’TN(E),
(1.1)
where Q,.(T) is the reactant partition function (per unit
volume). (T is the temperature and E the total energy of
the molecular system.) The microcanonical rate constant,
usually of interest for unimolecular reactions, is also given
in terms of N(E),
k(E) = k-++,(E)
1-‘NE),
(1.2)
where pr is the density of reactant states per unit energy.
For a bimolecular reaction, N(E) is defined in terms of the
S matrix for the reaction
this type, but our goal is a rigorous procedure, to which
approximations may be added later if necessary for specific
applications.
The formal solution for a direct route to N(E) was
given as a byproduct of the work on reactive flux correlation functions,8 and this was recently put into a practical
form by Seideman and Miller. lo The purpose of the present
paper is to recast this into an even more useful form, one
which has important practical advantages as well as an
interesting conceptual interpretation.
Specifically, we show herein that N(E) can be expressed as
N(E) = ;P#),
where {p,JE)) are the eigenvalues of a certain Hermitian
matrix (operator) whose values all lie between 0 and 1,
O<p,@) ~1.
ME) = c I&p,,r(E) 1’9
(1.3)
“PP
where the sums are over all open channels (i.e., asymptotic
quantum states) of reactants (n,) and products (n,) at
total energy E. All of the averaging over initial and final
asymptotic states of the reactants and products is thus contained in N(E).
An ongoing goal of this research group has been the
development of practical ways to calculate N(E) directZy,2
i.e., without having to solve the complete state-to-state reactive scattering problem to obtain the S matrix as required by Eq. (1.3). melated work by several groups,3-7
using a reactive flux autocorrelation function,’ has focused
on the direct calculation of the thermal rate k(T) itself;
whether one wishes the primary calculation to be N(E) or
k(T) will depend on the application of interest.] Transition state theory9 (TST) is an approximate approach of
(1.4a)
(1.4b)
These eigenvalues thus have an interpretation as probabilities, which one may think of as the “eigenreaction probabilities” for eigenstates of an “activated complex.” This
language is borrowed from transition state theory, which in
one of its simplest versions gives N(E) asl$ll
(1.5)
n
where Pt,, (E,) is a one-dimensional tunneling (or transmission) probability as a function of the energy El in the
one-dimensional reaction coordinate; e”t are the eigenvalues for motion on a dividing surface, i.e., the molecular
system with one degree of freedom (the reaction coordinate) removed, nS= (nf ,nf ,...,n$- 1) labeling these energy
levels. Superficially, therefore, Eq. (1.4) has the same form
as the TST expression ( 1.5), though the index k simply
labels the different eigenvalues of the particular matrix
&ST(E)=
+(E-&,
0021-9606/93/99(5)/3411
/g/$6.00
@I 1993 American Institute of Physics
3411
J. Chem. Phys. 99 (5), 1 September 1993
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U. Manthe and W. H. Miller: Cumulative
3412
(vide infra) with no implied quantum number assignments; if the dynamics in the reaction region is sufficiently
simple, however, it should be possible to make such assignments, at least qualitatively.‘2 This is the same situation, of
course, in determining energy levels of a molecular
system-the energy levels are the eigenvalues of the Hamiltonian matrix and their assignment, or labeling by a set of
quantum numbers, relies on a qualitative correspondence
with some zeroth order Mamiltonian. One may think of
Eq. (1.4) as a rigorous quantum mechanical version of
transition state theory, though this is rather semantic; the
approach described in Sec. II, which leads to Eq. ( 1.4)) is
not a “theory” (i.e., approximation),
but rather a fully
rigorous quantum mechanical result for N(E) [e.g., equivalent to Eq. (1.3)].
After Sec. II presents the theoretical development, Sec.
III shows its application
to several standard test
problems-the
one-dimensional Eckart potential barrier,
the collinear H + HZ--+H2+H reaction, and this same reaction in three-dimensional space (for total angular momentum J=O). Section IV then discusses some of the
methodological aspects of applying this approach to
general/complex systems, and Sec. V concludes.
II. EIGENVALUES
OPERATOR
OF THE REACTION PROBABILITY
Seideman and Miller” showed that the cumulative reaction probability can be expressed as
N(E) =4 Tr[&E>t@(E)Q],
(2.1)
where
&E)=(E+iC-I?)-‘.
(2.2)
Here fi is the total Hamiltonian operator of the molecular
system, cr and $, are absorbing potentialsI in the reactant
and product regions, respectively, and 1~ gr+ iP is the total
absorbing potential. In Ref. 10, and also for our applications below in Sec. III, a basis set of grid points-i.e.,
a
discrete variable representation (DVR) 14-is used to evaluate the trace in Eq. (2.1), so all the operators become
matrices in grid point space. For example, all potential
energy operators, including 1, and iP, are diagonal matrices. In Eqs. (2.1), (2.2), and others expressions below, one
may thus think of the operators fi, G, Zr, etc., as DVR
matrices, though we will refrain from using matrix notation explicitly unless it $ needed. We also note that since
the operators/matrices Hand 1 are symmetric, the adjoint
o,f the Grp;en’s function is simply its complex conjugate
G(E)i=G(E)*.
It is useful to symmetrize the operand in the trace in
Eq. (2.1). Defining the reaction probability operator P by
&E)
=4i;‘2&(E)+@(E)C;‘2,
(2.3)
it is clear that Eq. (2.1) is equivalent (via cyclic permutation of operators inside the trace) to the following:
N(E) =Tr[?(E)]
(2.4a)
reaction probability
= ;p/LE),
(2.4b)
where (pk) are the eigenvalues of j. Since the absorbing
potentials are positive functions, there is no difficulty associated with the square roots in Eq. (2.3). It is easy to see
that I; is Hermitian, so its eigenvalues {pk) are real. Also,
the eigenvalues {pk} are invariant to an interchange of the
roles of reactants and products, i.e., rep in Eq. (2.3)
(though the eigenvectors are not). l5
The primary computational task in applying Eq. (2.1)
or Eq. (2.4) is the matrix inverse calculation required to
obtain the Green’s function (2.2); it is thus desirable to
minimize the effort related to this. We thereffre focus attention on determining the eigenvalues of the P(E) matrix,
hoping that there are not too many of them required. (We
will see below, for example, that for a one-dimensional
system, there is only one nonzero eigenvalue of P!) Furthermore, the matrix inverse calculation necessary to obtain the Green’s function can be avoided completely by
considering the inverse matrix P-r,
~-1=~,-1_/2(~+ii-E)i~1(~-ii-E)i,-1’2,
(2.5)
the eigenvalues of which are {l/pk3. Unlike ?, the matrix
Pm1 is directly available without having to determine the
Green’s function; i.e., *given the DVR matrix for the
Hamiltonian, that for P-’ is readily obtained. The strategy, therefore, is to determine the eigenvalues of P-i, the
reciprocals of which are the desired eigenreaction probabilities {pk3.
The re?der may have noticed a problem with the definition of P-’ in Eq. (2.5). As usually defined, the absorbing potential E, becomes vanishingly small at coordinates
outside the reactant absorbing region (and similarly for eP
outside the product absorbing region), and this will cause
numerical problems because of the factors g; 1’2 and i;’ in
Eq. (2.5). To avoid this problem, a constant “floor” is
added to the absorbing potentials
n ,.
G-+%+Eo,
(2.6a)
n A
++Ep+Eo,
(2.6b)
and thus
where e. is a small constant. In actual calculations, one
must check to see that e. is sufficiently small (vide infra).
At this point, it is easy to prove that the eigenvalues
{pk) are in fact probabilities, i.e., have values between 0
and 1. (The only task is to show that they are ~1 since it
is obvious that they are >O because ? has the form P= it i
with i=<j’2&F’2.)
C onsider the Hermitian operator 2,
n
A=ze,1..
l/2
.A a-1
A .A
(H+ze-E-221~&~
X (~---V--E+2i~~)~,T’“,
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(2.7)
1993
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U. Manthe and W. H. Miller: Cumulative reaction probability
which is a positive definite operator, and thus-has positive
ejgenvalues, since it is also of the form ,!?L [here with
L=i~‘/2(H-i~-E+2ii,)~,-“2].
Multiplying
out the
two binomals gives
+Z.!~-112[(a+jg-E) - (&-j~-E)]~~“2
4 r
+ 5-
lRip#;
l/2,
which is easily shown to reduce to the following:
j=i;-l-
1.
(2.13)
e
The matrix {Pi,,,} is thus of rank one, and therefore has
only ogle nonzero eigenvalue. [The eigenvalue is ] II ) 2, and
the corresponding eigenvector (unnormalized ) is u = {Ui].]
The components of the eigenvector are nonzero only at
points {Xi} in the reactant absorbing region, i.e., where
E,(Xi)#O.
If one furthermore
utilizes the WentzelKramers-Brillouin
(WKB) approximation”’ for the wave
function f(x), then one finds from Eq. (2.13) that the
square of the eigenvector components is given approximately by
(Ui12Ke,(xi)exp
1,
(2.9a)
and therefore
(2.9b)
Pd,
from which it follows that the individual eigenvalues pk are
<I.
The eigenvalues {pk(E))
may thus be thought of as
eigenreaction probabilities, the cumulative reaction probability being their sum. By analogy with transition state
theory, they correspond to transmission probabilities for
different states of an activated complex, except that they
incorporate the reaction dynamics fully exactly. And finally, they can be calculated as the reciprocals of the eigenvalues of the matrix 3-l [Eq. (2.5)], which is readily
obtained from only the Hamiltonian matrix itself.
To conclude this presentation of the general theoretical
development, it is useful to consider the one-dimensional
case explicitly to sh?w theAnature of the eigenvalues (and
eigenfunctions) of P and P-‘. Specifically, we consider a
one-dimensional potential barrier (cf. the Eckart potential
barrier of the next section), for which x--t - 03 is the “reactant” and x+ + COthe “product” (i.e., reactants to the
left, products to the right). In a coordinate representation,
the Green’s function has the form
<xl&E)
““f(Xj)
(2.8)
Since a>O, one has
P-l>
UiO: Er(Xj)
3413
Ix’> =fb<
(2.10)
>g(x> >,
where f(x) is the solution of the time-independent S&r&
dinger equation (for total energy E) with outgoing waves
toward reactants, g(x) is the solution with outgoing waves
toward products, and x< (x, ) is the smaller (larger) of x
and x’. The coordinate representation of the P operator is
then easily shown [by substituting Eq. (2.10) into Eq.
(2.3)] to be
(x(~Ix’)=E,(x)“2f(x))k
[ j- dx”Idx”)
12+n)]
(2.11)
xf(x’)e~,(X’)*‘2,
where it is assumed that E,(X) and r+(x) do not overlap. If
the coordinate x is now discretized to the grid {xi}, then
the DVR matrix of 3 has the form
p. v .* = u”u!
1 I)
where the vector {Ui) = u proportional
(2.12)
to
[ -- iu j-r
d=r(x)li
(2.14)
where x0 is the boundary of the reactant absorbing region;
i.e., e,(x)#O only for x<x,.
1Zlil 2 thus increases from
zero for Xi< ~0, approximately proportional to er(Xi), and
then decreases to zero for smaller Xi as the exponential
damping factor takes over [cf. Ref. 13 (b) for a discussion
of the time-dependent analog of such exponential damping]. The eigenvector is thus a simple function of the coordinate, with no nodes. This qualitative behavior will be
seen in examples in the next section (cf. Fig. 3).
The above exsmple also illustrates the situation for the
inverse operator P-‘. If, e.g., 20 grid points are used, then
the matrix CPi,i,} of Eq. (2.12) is 20x 20, with one_nonzero
eigenvalue and 19 zero eigenvalues. The matrix P-’ thus
has one nonzero eigenvalue and 19 infinite ones. This is
why it is necessary to modify the absorbing potentials as in
Eq. (2.6). The eigenvalues of P- ’ are then the one relevant
finite one, which is essentially independent of the constant
~6, and 19 others that are large, proportional at least to
l/e,, for small eO. One is thus interested in only the lowest
eigenvalue of the matrix P-‘, or in the multidimensional
case, the few lowest eigenvalues.
III. EXAMPLES
To illustrate the theoretical development of the previous section and support the physical interpretations that
are suggested from the properties of P(E) and P(E)-‘,
several standard test problems are treated in this sectionthe one-dimensional Eckart potential barrier, the collinear
model of the H + H2 -+ H2 + H reaction, and the full threedimensional H + H, + H2 + H reaction for total angular
momentum J=O. For all examples, the eigenvalues (and
eigenvectors, if presented) of P-’ have been calculated by
direct library programs. For calculations that involve the
Green’s function, i,t has been obtained by direct numerical
inversion of (E---H-j-z?) using standard library programs.
A. The Eckart barrier
The Eckart potential function
V(x) = V. sech2(x/a)
(3.1)
is investigated first, with parameters that have been used
before”(“) to correspond approximately to the H+H,
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U. Manthe and W. H. Miller: Cumulative
3414
reaction probability
1015
10-l
1012
lo-;!
2
3
s
IO9
.Q
lo6
a
IO3
lOA
IO3
IO0
FIG. 1. The lowest, next to lowest, and the highest eigenvaluesof I;-’ as
function of Ec (in electron volts) for energiesEz0.3 (fulI line), 0.4 (long
dashed line), 0.5 (dashed line), and 0.6 eV (dotted line) for the onedimensional Eckart potential barrier described in Sec. III.
FIG. 2. The fractional error A as function of q, (in electron volts) for
energiesE=0.3 (full line), 0.4 (long dashed line), 0.5 (dashed line), and
0.6 eV (dotted line) for the one-dimensional Eckart potential.
reaction--Vo=0.425
eV and a=0.734ao, with a mass
,u= 1061m,. The absorbing potential is also taken to be
that used before’0(a)
dependence on e. for the higher eigenvalues is seen to be
- l/e0 for the second lowest eigenvalue and - l/G for the
highest. (Deviations from this behavior for the smallest
values of e. are due to numerical round-off error, but this
causes no practical difficulties.) Since the eigenreaction
probabilities {pk( E)) are the reciprocals of these eigenvalues, it is clear that there is only one such nonzero value and
thus only one term in Eq. ( 1.4a) for the cumulative reaction probability.
Figure 2 shows the fractional error in the cumulative
reaction probability
ep(x) =2ACl+expt
(x,,-x)/q13-1,
E,(X) =eJ-x)
(3.2)
with /2=1.5 eV, 17=1.5ao, and xmax=27.5ao. Other analytic forms for the absorbing potentials have also been
used, and similar results obtained with them.
All operators are represented in the DVR scheme of
Colbert and Miller, l7 i.e., on a uniform grid xi=iAx, i=O,
f 1, =!=2,..., for which the kinetic energy matrix is
A= jN(E,eo) --N(E)
T,, = &
[ 2py==i,],
and all potential energy matrices are diagonal
Vf,,it=S,j, V(Xjt),
(3.3)
and similarly for (E,) i,i,, etc. The grid is truncated to the
region 1x I < 27.5ao, and the grid parameters AX are chosen
to have four grid points per de Broglie wavelength, i.e.,
A.x=;(2dk),
(3.4)
where k = ,/m,
which has been seen in earlier
work’“” to be sufficient to produce accuracy of three to
four significant figures.
We*fust show how the eigenvalues of the inverse operator P-‘(E)
[Eq. (2.5)] depend on the parameter e. [cf.
Eq. (2.6)] that is introduced to eliminate its singular character. Figure 1 shows t$e lowest, the next lowest, and the
highest eigenvalue of P- ‘, as a function of eo, for four
values of the energy E from 0.3 to 0.6 eV. As anticipated
from the discussion in Sec. II, all eigenvalues except the
lowest one become unboundedly large as e. decreases,
while the lowest one quickly settles to a finite value. The
(/N(E)
(3.5)
as a function of the parameter Ed. That is, N(E) is the
converged value for eo=O, and N(E,e,) is the value given
cy Eq. (1.4a) with the reciprocals of the eigenvalues of
P-‘(E,eo)
used for the eigenreaction probabilities {pk).
The results are shown for the same four energies E as in
Fig. 1. A is seen to decrease approximately linearly with
decreasing eo, a dependence that is easily understood from
Eqs. (2.3) and (2.6); deviations for small e. are again due
to numerical roundoff. [Note that this simple dependence
of N(E,c,) on e. could also be used to extrapolate values
calculated for finite e. to eo=O.] Convergence with respect
to decreasing e. is thus readily achieved.
Finally, Fig. 3 shows the eigenvector of ?-’ corresponding to its lowest eigenvalue, the physically relevant
one. It has precisely the form discussed at the end of Sec.
II, i.e., Eq. (2.14). (The eigenuectors of @ and 3-l are th’e
same. )
For this one-dimensional example, therefore, the cumulative reaction probability is determined by the lowest
eigenvalue of the operator P-‘(E)
[Eq. (2.5)J (for a suitably chosen value of ~~0).The DVR matrix of P- ’ is readily
obtained from the matrix of the Hamiltonian, requiring no
matrix inversions.
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1993
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U. Manthe and W. H. Miller: Cumulative
10-l
1o-*
IO4
lOA
IO”
-25
-10
-15
-20
x [a.u.]
FIG. 3. The square modulus of the lowest eigenfunction of 8-l as a
function of coordinate for energies E=0.3 (long dashed line), 0.4
(dashed line), and 0.5 eV (dotted line) for the one-dimensional Eckart
potential. The solid line indicates the reactant absorbing potential (in
electron volts).
B. The collinear
H+H,
reaction
The collinear model of the H +H,+ H,+H reaction,
involving two degrees of freedom, is the next step toward a
realistic model of a chemical reaction and thus a logical
test for the present methodology. The DVR grid and absorbing potentials for our study are the same as those in
Ref. 10(a); the parameters of the Woods-Saxon absorbing
potential are /2= 1.1 eV, ~=0.4a,, and qmax=6ao, with the
coordinates for the absorbing potential chosen as in Eq.
(3.3a) of Ref. 10(a). An energy cutoff I’,=3 eV and four
grid points per de Broglie wavelength were used.
Figure 4 shows the eigenreaction
probabilities
{pk(E))
as a function of energy E, and also their sum,
which is the cumulative reaction probability N(E). These
1
I
I ...7 _
0.0 ’
0.4
0.6
0.8
1.0
1.2
E WI
FIG. 4. The cumulative reaction probability N(E) (solid line) and the
individual eigenvalues of P (dotted lines) as a function of energy (in
electron volts) for the collinear H+H, reaction.
3415
reaction probability
values {p,JE)}
were obtained as the reciprocals of the
eigenvalues of the matrix k’(E)
with ~c= 10B6 eV, a
value which assured convergence with respect to it. Up to
an energy of E=0.85 eV, there is essentially only one nonzero reaction probability, at which energy a second one
becomes significant. In contrast to the first eigenreaction
probability, the second one does not approach unity (at
least in the energy range displayed), already decreasing
beyond E= 1 eV after reaching a maximum value of
-0.55.
A third eigenvalue becomes significant
at
E= 1.2 eV.
The behavior seen in Fig. 4 is indeed qualitatively reminiscent of transition state theory [cf. Eq. (1.5)] and one
can assign the various eigenreaction probabilities to individual states of the activated complex, i.e., v,=O, 1, and 2
quanta in the symmetric stretch mode. The primary qualitative difference from transition state theory is that in TST
the transmission probabilities of various states of the activated complex increase monotonically from 0 to 1 as a
function of E (essentially a rounded-off step function at the
various energies of the activated complex). The nonmonotonic behavior of the eigenreaction probabilities is a manifestation of the transition state theory-violating (i.e., nondirect) dynamics in this reaction. In this case, this is
largely associated with a scattering resonance at E~0.88
eV that arises because of a short-lived collision complex,
behavior that is indeed in contradiction to the “direct dynamics” assumption of transition state theory. At higher
energies, there is also rebound dynamics that violates the
dynamical approximation inherent in TST.
This example illustrates the qualitative relation of the
present rigorous theoretical description to transition state
theory-cf.
Eqs. (1.4) and (1.5)~and the reason we refer
to the eigenvalues {pk(E)}
as the eigenreaction probabilities of the system.
C. The three-dimensional
H+H2 reaction
for J=O
The final example is the H+H,
reaction in its full
dimensionality, for total angular momentum J=O. Our
treatment follows that of Ref. 10 (b) which used Eq. (2.1)
to-calculate N(E) . Thus the same coordinate system, DVR
Hamiltonian, and absorbing potentials were used. The parameters of the Woods-Saxon absorbing potential are
q=2.5a,,,
qmax= 5Oa,, and A=O.6E, and the energy cutoff
is V,=E+1.2
eV; 3.6 and 2.7 grid points per de Broglier
wavelengths were used in the Q2 and Q1 coordinates, respectively, and five Gauss-Laguerre DVR grid points were
used for the bending degree of freedom. These parameters
are sufficient to yield accuracy comparable to that of Fig. 4
of Ref. 10(b).
Figure 5 shows the eigenreaction
probabilities
{pk(E)}
[obtained by diagonalizing k’(E)
with a sufficiently small value of ee] as a function of energy; Fig. 5 (a)
shows the results on the usual linear scale and Fig. 5 (b) on
a logarithmic scale. (The values shown here are for the
distinguishable atom case, so the overall result should be
multiplied by 2 if comparisons are made to other calculations for the total reaction probability for this system.) The
sum of these reaction probabilities gives the cumulative
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1993
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U. Manthe and W. H. Miller: Cumulative
3416
0.8
0.8
2
3
reaction probability
2
3 0.6
z
8 0.4
$.. *,
0.2
0.6
iz
.zb 0.4
a
0.0
0.6
0.8
1.0
1.2
E WI
1.4
E WI
FIG. 6. The same as Fig. 5(a) except the connecting lines are drawn to
indicate “diabatic” curves. Numbers, ( V,,uJ beside the lines indicate the
approximate assignment, u, and u,, being the number of quanta in the
symmetric stretch and bending vibrations, respectively, of the activated
complex.
.;.
.-
:
0.6
0.8
1.0
1.2
1.4
E WI
FIG. 5. The eigenvalues of fi as a function of energy (in electron volts)
for the three-dimensional H+H, (J=O) reaction. Solid points are the
calculated values and the solid lines are drawn to connect the eigenvalues
for the different energies. (a) depicts values on a linear scale; (b) depicts
values on a logarithmic scale.
reaction probability [cf. Eq. (1.4)], which agrees with previous calculations, so N(E) itself is not shown here.
Just as for the collinear example discussed above, the
results in Fig. 5 look qualitatively like transition state theory transmission probabilities, except that they also do not
all increase monotonically from 0 to 1 as a function of
energy-because the dynamics is not completely “direct.”
Particularly interesting in Fig. 5 is the appearance of
avoided crossing structures in the eigenreaction probabilities; i.e., just as one-dimensional potential energy curves
for electronic states of the same symmetry cannot cross,
neither do the various curves pk(E) vs E. These avoided
crossings are a manifestation of Fermi resonance interactions between different states of the activated complex that
have been noticed earlier in a semiclassical transition state
theory. *’ They obvious 1y frustrate a simple prescription for
assigning individual eigenreaction probabilities to specific
states of the activated complex.
To help in the qualitative interpretation of the results
in Fig. 5-i.e., to make an “assignment” of the various
eigenreaction probabilities-Fig.
6 shows a crude “diabatiia$G” of th$ resul,ts, obtained simply by connecting the
points in what-appears intuitively to be the most reasonable
way. By knowing the symmetric stretch and bending frequencies at the transition state (ti,=O.256
eV and %.~g
= O.l”l2 eV), one can pick out progressions and assign the
curves to vatious states (u,,ud) of the activated complex, as
-indicated in Fig. 6. (Only even values of the bending vibrational quantum number ub are allowed for J=O.) These
are the same assignments deduced by Chatfield et al. I2 in
analyzing plots of (d/dE) N (E) vs E using the results from
scattering calculations and Eq. (1.3) to obtain N(E). The
energy spacings attributed to the bending progression,
AEr0.20-0.22
eV, agree quite well with the harmonic
bend frequency at the transition state 2hbTO.224 eV, but
the energy spacings identified as symmetric stretch progression, AE~O.31-0.33 eV, are significantly larger than
the harmonic symmetric stretch at the transition state ?Lx,
=0.256 eV. This latter effect was seen earlier in a semiclassical calculation’g based on the “instanton model,“20 a
periodic orbit in imaginary time on the upside-down potential energy surface, where the effective symmetric
stretch frequency-the
stability frequency of the periodic
orbit-was
seen to be intermediate between the harmonic
frequency at the transition state and that of the reactant H2
molecule ( htin2 = 0.55 eV). Finally, we also note in Fig.
5(b) an avoided crossing/Fermi resonance interaction in
the tunneling region, at E-O.92 eV, between the state with
one quanta of symmetric stretch and that with four quanta
of bend.
IV. ITERATWE
CALCULATION
OF N(E)
All calculations of the eigenreaction probabilities described in the previous section were performed by direct
J. Chem. Phys., Vol. 99, No. 5, 1 September
1993
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U. Manthe and W. H. Miller: Cumulative reaction probability
diagonalization of the matrix $-‘(E)
[Eq. (2.5)] using
standard library routines. This is certainly the most convenient procedure when it is possible to store the matrix
p- ’ in the core memory of the computer (as required by
direct library routines), because finding all the eigenvalues
of an MXM
Hermitian matrix requires only M3/3 operations, the same number that direct library routines require
to solve one set of simultaneous linear equatio?s. Furthermore, we do not need all the eigenvalues of P-l, only a
small subset of them (the smallest ones), so that further
efficiencies are also possible.
If the matrix is too large to be stored in core, thoughe.g., MZ2000
for a workstation with 64 Mbytes of
memory-then
one needs to consider iterative methods for
determining the eigenvalues. Such methods do not require
storage of the relevant matrix, only that it be possible to
multiply the matrix into a vector, something that is extremely efficient for DVR or fast fourier transform
(FFT) 21 approaches (matrix-vector multiplication is typically an M2 process, but because of the special structure of
DVR or Fl?T schemes, it requires only approximately
M log M operations in this casei7>. Iterative methods thus
have the advantage that they scale with the size of the
matrix much more benignly than M3. The primary disadvantage of iterative methods is that they are not always
‘black box” procedures and may require attention to the
specific calculation being undertaken.
We first consider an iterative calculation for the eigenvalues of the matrix fi-‘, which are the reciprocals of the
eigenreaction probabilities, i.e., {l/pk}. There are a few
small eigenvalues (the ones of physical significance) and
many large eigenvalues (of order l/es or so) that are physically irrelevant. Though this at first seems simple, this
structure
of the matrix
?’
makes an iterative
calculation-at
least one based on
a Krylov
representatio$2-unfeasible.
This is because the large eigenvalues of P-’ are so much larger than the small ones of
interest that the vector space generated by the Krylov
procedure+i.e., the vectors generated by successive multiplication of some initial vector by the matrix $-‘-picks
out the vector space spanned by the largest eigenvalues of
@-‘, precisely the opposite of the vector space one is interested in. Therefore only negligible components of the lower
eigenvectors are incorporated in the Krylov space until the
dimension of the Krylov space approaches the full dimension of the matrix, so that these procedures give the lowest
eigenvalues accurately only when the number of iterations
approaches the order of the matrix, behavior that we have
verified in numerical calculations.
This poor behavior of iterative methods for the matrix
j-’ means, on the other hand, that they will be extremely
efficient for the matrix 3(E) itself, because one is interested
in its largest eigenvalues. Therefore the Krylov vectors
generated by successive multiplications by P will s?an the
space of eigenvectors with nonzero values of pk, precisely
the vector space of interest. In fact, if i is a matrix rank n,
i.e., has only n nonzero eigenvalues, then only n Lanczostype iterations are required to produce these n eigenvalues
exactly. And as we have seen, the number n of nonzero
3417
eigenreaction probabilities {pk} iS typically much smaller
than the size of the matrix.
Iterative meihods will thus be optimum for finding the
eigenvalues of P, the only down side to this procedure
being that it is necessary to have the Green> function
G(E) [Eg. (2.211 multiply a vector [and also G(E)*] everytime P multiplies a vector. That is, for each Lanczos
iteration, when P multiplies a vector to generate the next
Krylov vector, it is necessary for 6 and &* to act on a
vector. This is considerably more difficult than when working with the inverse matrix 3-i which only requires that
the HamiZfPnian matrix multiply a vector twice for each
action of P-‘, but it is the price which must be paid in
using a Lanczos-type iterative procedure&for I;.
Though a Lanczos calculation for P does not avoid
having to deal with the Green’s function, it does greatly
reduce the number of Green’s function operations from
that required in previous calculations based on Eq. (2.1).
Thus to evaluate the trace to obtain N(E) via Eq. (2.1),
one must have G(E) act on each basis function (i.e., grid
point) in the reactant (or product) absorbing region; this
is typically” -20% of the total number of grid points. The
discussion in the preceding paragraph, however, shows
that only -2n operators of the Green’s function are re,
quired if there are it nonzero eigenvalues of the matrix P,
and this will typically be many fewer. The remainder of
this section describes the particular iterative procedure
(Lanczos with reorthogonalization)
we have used to test
these ideas and discusses the results obtained.
Starting with an arbitrary initial vector q. which is at
least partially localized in the reactant absorbing region, a
Krylov basis is constructed by successivz multiplication
with the reaction probability operator P. Employing a
Lanczos scheme with full reorthogonalization, the following recursion relation:
I-l
Yl(Vr=&-1- l,zoY;?$P9Y:;“=wIPI3r-*>
(4.1)
was used to generate a Lth order set of basis functions
{&,,Q~,...,$J~). (For infinite numerical accuracy, $” vanishes for I’ < 1-2 and the conventional Lanczos recursion
relation is obtained.) The 3 operator is then represented in
the basis of {q),
pr,l~=bw%P)7
(4.2)
and its eigenvalues obtained by diagonalizing the small (L
+ 1) by ( L-t- 1) matrix. Alternatively, N(E) is calculated
by directly evaluating the trace of ? in this matrix representation
L
L
(4.3)
N(E)
= /z.(1crrl~I
$1)
= /z.rj”“.
Since only eigenvectors of j with nonvanishing eigenvalues
contribute to the basis functions @l added to the Krylov
space by the recursion (4. l), the basis {@o,+l,...,~L}
should contain all relevant eigenvectors if L equals or exceeds the number of nonzero eigenvalues. Therefore one
can expect to obtain the correct N(B) from Eq. (4.3) if L
J. Chem. Phys., Vol. 99, No. 5, 1 September 1993
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3418
U. Manthe and W. H. Miller: Cumulative reaction probability
r”
a ..-m
” .7 *., 1*
5.
t
I
12 /
9 s
z6-
-c-+.--A-~--*-~-,1.2eV
fiil;l;i
/
r
-_
1
/I
l.QeV
r, / *--c--.--*--+.-C--*--C.-+--,
/fJ
3
ib
;Y
._._,.II....
~ ..........-._..
.__...._
I ._..,....
..... .........................*.0.8eV
.,...
* *
.
--
I
,~--+--
--
3
, --
+
--
r
6
--
r
--.
+-+:,.-J&y,
--
9
12
-.
1
--
order of Lanczos scheme
FIG. 7. The result obtained for N(E) (the three-dimensional H+H,
reaction) as a function of the order of the Lanczos iteration scheme for
different values of the energy as indicated. The lines simply connect the
points (which are the calculated values).
is at least as large as the number of eigenstates of the
activated complex that contribute significantly.
Figure 7 shows the results given by this iterative procedure for the three-dimensional H+H,
(J=O) reaction
treated in Sec. III C. The cumulative reaction probability
N(E) is calculated for different orders L of the Lanczos
scheme for energies E=0.6, 0.8, 1.0, 1.2, and 1.4 eV. One
fmds that converged values of N(E) are obtained for the
different energies if the order L is at least 1, 2, 4, 7, or 10,
respectively. These numbers correspond quite nicely to the
number of effectively nonzero eigenreaction probabilities
depicted in Fig. 5 (Sec. III C), where one, two, four, six,
and nine nonzero reaction probabilities are found at the
different energies, respectively. The slight deviation of one
for the two highest energies may result from the fact that a
few extra iterations are required to obtain the smallest
eigenreaction probabilities. Also important is the fact seen
in Fig. 7 that the result for N(E) is stable with respect to
order even after convergence has been achieved.
The primary a:complishment of this iterative treatment of the matrix P(E) is therefore that it greatly reduces
the number of Green’s function operations (i.e., solutions
of a set of linear equations) that are required. For the
three-dimensional H + H2 reaction described above, this
amounts to almost an order of magnitude fewer Green’s
function operations that the previous calculations’0(b)
based directly on Eq. ( 2.1) . For systems with more degrees
of freedom, this savings will be even more dramatic, especially in the low energy regime where not too many eigenreaction probabilities will contribute.
We end this section with a technical note on the use of
iterative methods to treat >; i.e., why did we use the reorthogonalized Lanczos method (4.1) rather than the normal Lanczos scheme? The problem in the use of the normal
Lanczos scheme can be the appearance of spurious eigenvalues due to numerical round-off error.22 It can be expected that a Lanczos scheme based on the @ matrix is
especially sensitive to this problem. If the order of the
Lanczos scheme exceeds the number of nonzero eigenvalues, vectors generated by the recursion relation become
very small and therefore very sensitive to numerical roundoff error. This expectation was confirmed by calculations
which were performed employing the normal (nonorthogonalized) Lanczos scheme. Spurious eigenvalues have been
found for some examples if the order of the Lanczos
scheme exceeds the number of nonzero eigenvalues by only
two. Therefore considerable attention is necessary in the
normal Lanczos scheme to avoid miscalculating N(E) due
to added contributions of spurious eigenvalues, but since
the order of the Lanczos scheme is quite low, the difference
in effort
r .= between the normal and the reorthogonalized
LanczZ% scheme-is small com@recj to the numerical effort
required for the multiplication by P that is required. Therefore the reorthogonalized Lanczos scheme is our method of
choice.
V. CONCLUDING REMARKS
Following previous theoretical work” on the cumulative reaction probabil$y, we have introduced the reaction
probability operator P(E) [Eq. (2.3)], which is Hermitian
and positive definite, and whose trace is the cumulative
reaction probability N(E) . Its eigenvalues all lie between 0
and 1, and by analogy with transition state theory, have an
interpretation as the eigenreaction probabilities of an activated complex (or reactive intermediate). In favorable
cases it is possible to make qualitative assignments of the
individual eigenreaction probabilities to specific zeroth order states of the activated complex, but other times the
states are so strongly mixed that such assignments are
quite ambiguous.
Introduction of the reaction probability operator also
leads to significant computational efficiencies for calculation of the cumulative reaction probability. If the Hamiltonian matrix for the system of interest can be stored efficiently enough to permit direct diagonalization, then the
procedure of choice is diagonalization of the inverse matrix
Pm1 [Eq. (2.5)], the reciprocals of whose eigenvalues are
the desired eigenreaction probabilities. The matrix 8-r is
readily available from the Hamiltonian matrix itself-i.e.,
no operations with the Green’s function are necessaryand its diagonalization requires no more effort than one
matrix multiplication
by the Green’s function matrix
(many of which were required in earlier approaches).
If the basis set (e.g., the set of grid points in a DVR)
is too large for a direct diagonalization, then an iterative
procedure is necessary. (The iterative approach may be
preferable even if the matrix fits in the computer because it
scales with matrix size approximately as M log M, rather
than M3, though with a much larger proportionality constant.) Iterative procedures of the Lanczos type, i.e., those
based on the Krylov vector space,*must be applied to the
matrix P(E) itself, rather than to P-'(E). The number of
iterations required is approximately the number of eigenvalues { pk} ( eigenreaction probabilities) that are significantly different from zero, i.e., the number of states of the
activated complex that contribute significantly to the reac-
J. Chem. Phys., Vol. 99, No. 5, 1 September
1993
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U. Manthe and W. H. Miller: Cumulative reaction probability
tion. At low energies, this will typically be a very modest
number. Each iteration for the P matrix requires two operations of the Green’s function on a vector, but this will
typically be many fewer such operations than previous approaches require.
ACKNOWLEDGMENTS
U.M. gratefully acknowledges support by a fellowship
from the Fond der Chemischen Industrie. This work has
been supported by the Director, Office of Energy Research,
Office of Basic Energy Sciences, Chemical Sciences Division of the U.S. Department of Energy under Contract No.
DE-AC03-76SF00098 and also in part by the National Science Foundation, Grant No. CHE-8920690. We are also
pleased to acknowledge helpful discussions with Professor
Claude Leforestier and Dr. Peter Saalfrank.
‘W. H. Miller, J. Chem. Phys. 62, 1899 (1975); here, however, the
cumulative reactive probability was called P(E) [cf. Eqs. (2.28)(2.30)]. N(E) seems preferable since it has the interpretation as the
number of reactive states, and also it is the numerator for the microcanonical rate [Eq. (1.2)]
*For a recent review, see Act. Chem. Res. 26, 174 (1993).
‘(a) J. W. Tromp and W. H. Miller, J. Phys. Chem. 90, 3482 (1986);
Faraday Discuss. Chem. Sot. 84, 441 (1987); (b) K. Yamashita and
W. H. Miller, J. Chem. Phys. 82, 5475 (1985).
4R. E. Wyatt, Chem. Phys. Lett. 121, 301 (1985).
‘(a) T. J. Park and J. C. Light, J. Chem. Phys. 85, 5870 (1986); 88,
4897 (1988); 91, 974 (1989); 94, 2946 (1991); (b) M. Baer, D. Neuhauser, and Y. Oreg, J. Chem. Sot. Faraday Trans. 86, 1721 (1990);
(c) D. Brown and J. C. Light, J. Chem. Phys. 97, 5465 (1992).
6(a) G. Wahnstriim, B. Carmeli, and H. Metiu, J. Chem. Phys. 88,247s
(1988); (b) K. Haug, G. Wahnstriim, and H. Metiu, ibid. 92, 2083
(1990).
3419
‘P. N. Day and D. G. Truhlar, J. Chem. Phys. 94, 2045 (1991); 95,
5097 (1991).
‘W. H. Miller, S. D. Schwartz, and J. W. Tromp, J. Chem. Phys. 79,
4889 (1983).
‘For reviews, see (a) D. G. Truhlar, W. L. Hase, and J. T. Hynes, J.
Phys. Chem. 87, 2664 (1983); (b) P. Pechukas, in Modern Theoretical
Chemistry, edited by W. H. Miller (Plenum, New York, 1976), Vol. 2,
Chap. 6.
lo (a) T. Seideman and W. H. Miller, J. Chem. Phys. 96, 4412 (1992);
(b) 97, 2499 (1992).
‘*R. A. Marcus, J. Chem. Phys. 45, 2138 (1966).
12D. C. Chatfield, R. S. Friedman, D. G. Truhlar, and D. W. Schwenke,
J. Phys. Chem. 96, 2414 (1992).
I3 (a) A. Goldberg and B. W. Shore, J. Phys. B 11, 3339 (1978); (b) C.
Leforestier and R. E. Wyatt, J. Chem. Phys. 78, 2334 (1983); (c) R.
Kosloff and D. Kosloff, J. Comput. Phys. 63, 363 (1986); (d) D.
Neuhauser and M. Baer, J. Chem. Phys. 90, 4351 ( 1989).
14(a) D. 0. Harris, G. G. Engerholm, and W. D. Gwinn, J. Chem. Phys.
43, 1515 (1965); (b) J. V. Lill, G. A. Parker, and J. C. Light, ibid. 85,
900 (1986).
l5 If one has already carried out a complete reactive scattering calculation
and has the S matrix, then a representation of the reaction probability
operator in the basis of asymptotic reactant states is P,,,,;(E)
=z nps .,,npW)$,,nr l(E), or similarly in terms of the asymptotic product states.-The main point of this paper, however, is to be able to
construct P(E), evaluate its trace, obtain its eigenvalues, etc., without
having to carry out the complete state-to-state reactive scattering calculation.
16The same as in Ref. 10(b), Sec. II B.
“D T. Colbert and W. H. Miller, J. Chem. Phys. 96, 1982 (1992).
“Ml J. Cohen, N. C. Handy, R. Hernandez, and W. H. Miller, Chem.
Phys. Lett. 192,407 (1992).
“S Chapman, B. C. Garrett, and W. H. Miller, J. Chem. Phys. 63,271O
(i975).
*‘W. H. Miller, J. Chem. Phys. 62, 1899 (1975).
2*D. Kosloff and R. Kosloff, J. Comput. Phys. 52, 35 (1983).
22C. Lanczos, J . Res. Natl. Bur. Stand. 45, 255 (1950); J. K. Cullum and
R. A. Willoughby, Lanczos Algorithms for Large Symmetric Eigenvalue
Computations (Burkhauser, Boston, 1985).
J. Chem. Phys.,
Vol. to
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5, I September
1993see http://jcp.aip.org/jcp/copyright.jsp
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