Reaction path zero-point energy from diffusion Monte Carlo calculations
Jonathon K. Gregory, David J. Wales, and David C. Clarya)
Department of Chemistry, University of Cambridge, Cambridge CB2 1EW, United Kingdom
~Received 8 July 1994; accepted 19 October 1994!
A general diffusion quantum Monte Carlo method is described for accurately calculating the
zero-point energy of the vibrations orthogonal to a reaction path in a polyatomic system. The
method fully takes into account anharmonic and mode–mode coupling effects. The algorithm is
applied to the OH1H2→H2O1H reaction and the results are compared with a more approximate
calculation. The technique will have many useful applications to kinetic and spectroscopic problems
involving polyatomic molecules. © 1995 American Institute of Physics.
I. INTRODUCTION
Reaction paths are a common concept in chemistry. In
the fields of reaction kinetics and spectroscopy they are
widely used to interpret experimental results.1–3 Reaction
paths can be defined in several different ways but they all
connect saddle points and/or minima on potential energy surfaces. Miller et al.4 showed how to write the classical and
quantum Hamiltonian in terms of the arc length along the
reaction path, the normal modes for directions orthogonal to
this reaction path, and their conjugate momenta. This approach was further developed by Page and McIver5 and has
been used in several calculations on bimolecular reactions
and molecular isomerizations.6 – 8 Many of these reaction
path approaches involve the separation of the reaction path
from the orthogonal vibrational degrees of freedom which
are often treated by a Taylor expansion of the energy with
normal mode analysis of the second derivatives. Higher derivatives can also be used if they are available. Furthermore,
sophisticated transition state theories have been developed
based on reaction paths9–12 and these are used in a general
computer program of Truhlar and co-workers13 for calculating rate constants of polyatomic reactions. Many comparisons of such treatments with more accurate calculations have
demonstrated the accuracy of the reaction path
approximation.9–12
In several theoretical studies on chemical reactions, it
has become clear that the appropriate effective potential
V(s) for the reaction is not the potential energy V r (s) along
the reaction coordinate s, but is
V ~ s ! 5V r ~ s ! 1V z ~ s ! ,
where V z (s) is the zero-point energy of the coordinates $q%
orthogonal to the reaction path. The calculation of V z (s) is
fairly straightforward for simple reactions involving three
atoms.9 However, for reactions involving polyatomic molecules the accurate calculation of V z (s) is much more difficult and it has usually been necessary to resort to approximations such as the neglect of couplings between the internal
vibrational modes and the inexact treatment of
anharmonicities.11,12 In some polyatomic molecules, mode–
mode coupling14 can easily be as large as 200 cm21 and an
error of this amount in V z (s) will change a calculated room
a!
Author to whom correspondence should be addressed.
1592
J. Chem. Phys. 102 (4), 22 January 1995
temperature rate constant by more than a factor of 2. Furthermore, in van der Waals clusters and molecules with inversion
modes such as NH3 , reaction paths have been used to calculate spectroscopic observables such as tunneling
splittings,15–18 the calculation of which are also very sensitive to V z (s).15,17 For these reasons, it is important to develop practical methods for calculating V z (s) as accurately
as possible.
In this paper, an algorithm is described for the accurate
calculation of the total zero-point energy of the coordinates
orthogonal to a reaction path. The algorithm uses the diffusion Monte Carlo ~DMC! method which is a quantum mechanical technique for calculating ground state energies
accurately.19 As a DMC calculation is done using Cartesian
coordinates, it is quite straightforward to perform DMC calculations on systems containing several atoms ~e.g., clusters
of water molecules have been treated using DMC20,21!. This
is a significant advantage which remains to be exploited in
the field of reactive scattering where very complicated and
specialized systems of coordinates are normally applied.22
The algorithm we propose is general and should have wide
applications in both reaction kinetics and spectroscopy. Because an expansion in normal modes is not necessary in a
DMC calculation, the DMC algorithm is easily used in conjunction with different definitions of the reaction path.
Quack and Suhm have proposed a ‘‘clamped coordinate
quasiadiabatic channel’’ approximation23 which has some
similarities to the method proposed here. This method was
applied to the unimolecular dissociation and isomerization of
the van der Waals molecule ~HF!2 . The algorithm proposed
in the present paper is more general and requires only Cartesian coordinates. Furthermore, we apply our algorithm to a
real bimolecular reaction.
This paper is organized as follows. In Sec. II we describe
the new DMC algorithm which involves two simple constraints on a normal DMC calculation to ensure that motion
along the reaction path is not included in the calculation of
V z (s). In Sec. III we give numerical details on the application of the algorithm to calculate V z (s) for the
OH1H2→H2O1H reaction. We choose this ‘‘benchmark’’
reaction for several reasons. There have been some rate constant calculations on this reaction using variational transition
state theory11,24 that incorporate an approximate computation
of V z (s). Also, as this is a reaction involving four atoms it
illustrates well the advantages of the DMC approach in ac-
0021-9606/95/102(4)/1592/5/$6.00
© 1995 American Institute of Physics
Gregory, Wales, and Clary: Reaction path zero-point energy
curately calculating vibrational mode–mode coupling effects. This reaction has also recently been the subject of
many quantum reactive scattering calculations of reaction
probabilities, rate constants, and state-selected cross
sections.25–29 We wish to emphasize that the DMC approach
will be applicable to reactions with more atoms than four, but
the aim of this paper is to describe the algorithm and apply it
to a well-tested system. Section IV presents the results for
V z (s). Also reported are tunneling probabilities and transition state theory rate constants for the OH1H2→H2O1H
reaction obtained with our calculated V z (s), and compared
with those obtained using a more approximate form of V z (s)
calculated by Isaacson and Truhlar.11 Conclusions are in Sec.
V in which the future prospects of the technique are considered.
II. METHOD
1593
where s50 is the transition state. The vector e(s) was simply
taken as the component of the gradient of the potential in the
direction of the eigenvector following step.
B. Diffusion Monte Carlo calculation
The algorithm we use to calculate V z (s) is based on the
conventional DMC method19 that has been widely applied to
the calculation of rovibrational states of strongly and weakly
bound molecules and clusters.20,21,23,34 –38 However, modifications of the conventional algorithm are needed and so we
briefly describe here the DMC approach we have used and
then discuss our modifications to it.
If the substitution t5it is made in the time-dependent
Schrödinger equation, an equation analogous to the diffusion
equation is obtained:
N
]c ~ q, t !
1
5
“ 2k c ~ q, t ! 2 @ U ~ q! 2U ref# c ~ q, t ! ,
]t
2m
k
k51
(
A. Calculation of reaction path
Our DMC algorithm is applicable to any definition of a
reaction path for an N atom system such that a vector e(s)
can be defined which is tangential to the reaction path at
point s. Here, e(s) has components for each Cartesian coordinate (x i ,y i ,z i ) of atom i, with i51,2,...,N. The potential
energy at point s, V r (s), is also needed together with the
vector f(s), which specifies the Cartesian coordinates as a
function of s. Here we use one approach for calculating f(s),
e(s), and V r (s), but we wish to emphasize that other definitions of the reaction path can also be employed.
In the present work we performed geometry optimizations for the transition state and minima and calculated approximate steepest descent paths using eigenvector
following.30 The approach we used has been described before in numerous studies on clusters31 and follows Baker and
Hehre32 in employing Cartesian coordinates and projection
operators to remove overall translation and rotation. Numerical first and second derivatives of the energy were employed
at each step combined with a maximum step size condition;
precise details of the step may be found elsewhere.31
Once the transition state had been located, approximate
steepest descent paths were found by perturbing the transition state geometry slightly according to the Hessian eigenvector corresponding to the unique negative Hessian eigenvalue. Eigenvector following energy minimization was then
employed for both positive and negative displacements along
the reaction coordinate. All of these calculations were performed in Cartesian coordinates without any mass weighting.
As Banerjee and Adams have carefully explained,33 steepest
descent paths are invariant to the choice of the coordinate
system. However, pathways are often calculated using massweighted coordinates.33 In the present calculations on the
OH1H2 reaction, in order to have a definition of s equivalent to that of Truhlar and co-workers,11 the Cartesian coordinates f(s) of the atoms for each point on the reaction coordinate were mass-scaled in the same way as described in
Eq. ~2! of Ref. 11. Then, if u(s) are the mass-scaled coordinates along the reaction coordinate, s is defined explicitly by
s5 $ @ u~ s ! 2u~ s50 !# –@ u~ s ! 2u~ s50 !# %
1/2
,
~1!
~2!
where m k is the mass of atom k, U~q! is the potential energy
surface, and U ref is a reference energy which is introduced to
stabilize the state of lowest energy. The wave function c~q,t!
is interpreted as a probability density in a diffusion process
with diffusion constant D5\ 2 /2m. A random walk algorithm can be used to solve this diffusion equation. The linearity of the equation allows many noninteracting random
walkers ~replicas! to be used; a replica defines a particular
geometry for a system and is used to partially describe the
wave function.
The formal solution to Eq. ~2! can be expanded in terms
of a complete set of eigenfunctions fn with eigenvalues E n :
c ~ q, t ! 5 ( c n f n exp@ 2 ~ E n 2U ref! t # .
~3!
n
At sufficiently large values of t, only the contribution from
the ground state energy survives. Therefore, the ground state
wave function and energy are obtained by averaging during
propagation along t using finite values of t. The calculation
involves a random walk to simulate diffusion ~the kinetic
energy term! and source/sink terms ~by weight adjustment of
each replica! to represent the effect of U~q!.
At each time step, all Cartesian coordinates of each atom
are subject to random displacements of the form
Dq c 5G(D t ,m), which are chosen from a Gaussian distribution centered at zero with a standard deviation of
~Dt/m!1/2. The weight of each replica W i is then adjusted
continuously according to
W i 5W i exp~ 2 @ U i 2U ref# D t ! ,
~4!
i
where U is the potential energy of replica i. Replicas with
W i ,0.001 are removed and the highest weighted replica is
divided into two replicas which evolve independently. The
reference potential is adjusted at each time step by
U ref5U av2 a
S
D
W2N 0
,
N0
J. Chem. Phys., Vol. 102, No. 4, 22 January 1995
~5!
1594
Gregory, Wales, and Clary: Reaction path zero-point energy
where W is the sum of the weights of all replicas, N 0 is the
number of replicas, U av is the average potential energy of all
replicas, and a is an adjustable parameter which is chosen to
minimize statistical fluctuations and sequential correlation.21
Essentially all the numerical work in the algorithm involves repeated calls to the potential energy surface as the
replicas are propagated, and the whole calculation is done in
Cartesian coordinates. It is this feature that makes DMC particularly attractive in calculations on the rovibrational ground
states of polyatomic molecules.
In our application of DMC to the calculation of V z (s)
for a given point on the reaction path s5s 0 , all replicas are
given the initial Cartesian coordinates f~s 0!. Random Cartesian displacements Dqc are then calculated for all the atoms
as described above with a time step Dt. To ensure that the
Cartesian displacement is orthogonal to the reaction path at
s 0 , we employ a simple algorithm that is similar to an approach used by Elber39 in molecular dynamics calculations
on biomolecules. At each time step, the Cartesian displacements Dqc are modified by the formula
Dq5Dqc 2e~ s 0 !~ Dqc –e~ s 0 ! / @ e~ s 0 ! –e~ s 0 !# !
~6!
so that Dq is orthogonal to e~s 0!.
Because of the finite time step used in the calculation
there is a very small chance that a replica might have an
energy smaller than V r (s 0 ) even after application of Eq. ~6!.
Such a replica would begin to dominate the population @see
Eq. ~4!#. This is dealt with simply by removing the replica
from the population as all replicas must have a potential
energy greater than V r (s 0 ).
The final energy calculated from this procedure for long
times is V(s 0 ), where V(s 0 )5V r (s 0 )1V z (s 0 ), and repetition of the calculation for a grid of s 0 values gives the V(s)
that can be used in tunneling probability calculations.
C. Calculation of tunneling probabilities and rate
constants
Once V(s) has been calculated using the above procedures, tunneling probabilities P(E), where E is the collision
energy, can be calculated. From P(E), and other parameters,
rate constants k(T) can be computed from transition state
theory.13 The aim of this paper is to describe a new and more
accurate way of computing V(s). We are not proposing here
new methods for calculating P(E) and k(T) from V(s); this
can be done using established techniques.
In calculating P(E), there are several different ways for
defining an appropriate effective mass m(s) in the simplified
reaction path Hamiltonian9–12
H5
2\ 2 d 2
1V ~ s ! .
2 m ~ s ! ds 2
~7!
The most appropriate choice of m(s) depends on the curvature of the reaction path.4,9,40 For example, the small curvature method of Truhlar and co-workers has often given good
results.9–13 Here, we simply choose a constant value for m
and solve numerically, with standard scattering boundary
conditions, the one-dimensional Schrödinger equation with
the Hamiltonian of Eq. ~7!. This is done for V(s) calculated
using both the DMC and the more approximate method of
FIG. 1. The reaction path potential energy V r (s) for the OH1H2→H2O1H
reaction obtained in the present calculations described in Sec. II A and by
Isaacson and Truhlar ~Ref. 11!. The zero on the energy scale corresponds to
the energy of OH1H2 with infinite separation.
Isaacson and Truhlar which uses a Morse quadratic quartic
~MQQ! approximation with no mode–mode coupling.11 This
P(E) is then used in standard transition state theory calculations of k(T) with this tunneling correction.11 The aim is not
to perform a highly accurate calculation of k(T) with the
most sophisticated transition state theory but rather to illustrate the effect of the different forms for V(s) on P(E) and
k(T).
III. NUMERICAL DETAILS
The calculations on the OH1H2→H2O1H reaction
were done with the potential energy surface of Schatz and
Elgersma.41 In the calculation on the reaction path for
OH1H2→H2O1H, a step length of 0.01 a.u. was used. This
small value was not adopted for reasons of accuracy, but to
enable a large number of points on the reaction path to be
obtained for use in the transition state calculations. Figure 1
shows a comparison of the V r (s), calculated for the OH1H2
reaction using the procedures described in Sec. II A, with
that calculated by Isaacson and Truhlar ~called V MEP in their
paper!.11 It can be seen that the two forms obtained for V r (s)
are almost identical even though somewhat different definitions of the reaction path have been employed.
The DMC calculations used a grid of 13 s 0 values for
negative s ~the OH1H2 channel! and 13 for positive s ~the
H2O1H channel!. For each chosen value of s 0 , 1000 replicas
were used. The population was initially propagated for
10 000 steps with a time step of 1 a.u. The time step was then
lowered to 0.2 a.u. and the propagation was continued for an
additional 5000 time steps. V(s 0 ) was then calculated by
averaging the reference energy U ref over the second stage of
the simulation. The whole simulation for each value of s 0
was repeated for a total of ten runs and the error in the
average value of V(s 0 ) was estimated as a standard deviation
of the ten values. These errors were all in the range 10–20
cm21, which is easily sufficient for the present purposes.
The cpu time needed to calculate one value of V(s) was
about 30 min on a Dec Alpha 3000– 600 workstation. This is
much more time than is used in more approximate calcula-
J. Chem. Phys., Vol. 102, No. 4, 22 January 1995
Gregory, Wales, and Clary: Reaction path zero-point energy
FIG. 2. The reaction potential energy V(s) for the OH1H2→H2O1H reaction obtained in the present DMC calculations and by Isaacson and Truhlar
~MQQ! ~Ref. 11!. The zero on the energy scale corresponds to the zero-point
energy of OH1H2 with infinite separation.
1595
FIG. 3. Tunneling reaction probabilities P(E) for the OH1H2→H2O1H
reaction plotted against collision energy E. Results calculated with the DMC
and MQQ ~Ref. 11! forms for V(s) are compared.
tions of V(s), but is still quite reasonable. The calculated
DMC energy of H2O was 2712 cm21. This should be compared with the value of 2560 cm21 obtained with the more
approximate MQQ procedure11 which neglects the coupling
between different vibrational modes in the H2O molecule.
This illustrates well the magnitude of the coupling between
the vibrational modes in this system.
The transition state theory calculations were based on
the approach described in Ref. 11 @apart from the calculation
of V(s) and the choice of m(s)#. The harmonic vibrational
frequencies given in Ref. 11 for s520.0982 a.u. @i.e., at the
maximum in V(s)# were used to calculate the transition state
partition functions. It was assumed that the transition state is
a symmetric top to calculate its rotational partition
function.27 The procedures for dealing with spin–orbit coupling of Ref. 11 were also applied.
Figure 4 shows a plot of the rate constants k(T) calculated as described in Secs. II and III. Rate constants obtained
from P(E) calculated with the DMC and MQQ forms for
V(s) are compared @and it should be emphasized that all
other parameters in the two calculations of k(T) are the
same#. It can be seen that the agreement between the two sets
of k(T) is good with the DMC results falling slightly below
the MQQ values at lower temperatures. This is because the
DMC P(E) are slightly smaller than the MQQ results. For
example, at 298 K the DMC rate constant is 0.40 3 10214
cm3 s21 molecule21 while the MQQ value is 0.58 3 10214
cm3 s21 molecule21. The difference between these results is
not very large and suggests that the MQQ procedure is quite
reasonable for calculating V(s) for this reaction. However, it
is quite possible that the DMC approach will be needed for
applications to some reactions of more complicated polyatomic molecules.
IV. RESULTS
V. CONCLUSIONS
A. V ( s )
Figure 2 shows a plot of the reaction potential V(s)
against s for the OH1H2→H2O1H reaction. Our DMC calculations are compared with the V(s) calculated by Isaacson
and Truhlar in their MQQ approximation.11 It can be seen
that the agreement between the two sets of calculations is
quite good, especially at the maximum in V(s) close to s5
20.1 a.u. The exit H1H2O channel shows the largest differences between the results of the two methods and this is
probably due to the neglect of coupling between the different
vibrational modes of the four-atom reaction in the MQQ calculation.
A diffusion quantum Monte Carlo algorithm has been
described for accurately calculating the zero-point energy of
B. P ( E ) and k ( T )
Figure 3 shows the tunneling probabilities P(E) calculated with the DMC and MQQ forms for V(s) and with m51
amu. It can be seen that the agreement between the two sets
of P(E) is quite good. The DMC values for P(E) fall just
below the MQQ results as the V(s) is slightly wider ~see
Fig. 2!.
FIG. 4. Rate constants k(T) for the OH1H2→H2O1H reaction. Results
calculated with the DMC and MQQ ~Ref. 11! forms for V(s) are compared.
J. Chem. Phys., Vol. 102, No. 4, 22 January 1995
1596
Gregory, Wales, and Clary: Reaction path zero-point energy
the coordinates orthogonal to a reaction path. The method is
general and reasonably inexpensive and should be applicable
to many reactions involving polyatomic molecules. Given
that the DMC method has been used successfully in calculations on the vibrations of polyatomic van der Waals molecules, our technique should also be directly applicable to
reactions in clusters, where anharmonic and mode–mode
coupling effects can be very important. Furthermore, reaction path methods have also been used in spectroscopic applications such as the calculation of tunneling splittings and
our DMC method will also have useful applications to problems such as these. The present example of the OH1H2 reaction serves as a good first demonstration of the algorithm.
We are currently working on applying the method to reaction
path problems involving more complicated polyatomic molecules.
ACKNOWLEDGMENTS
We would like to acknowledge very helpful suggestions
from V. Buch and R. Elber ~Hebrew University, Jerusalem!.
This work was supported by the Engineering and Physical
Sciences Research Council ~EPSRC! and the National Environmental Research Council ~NERC!. J. K. G. acknowledges
a Research Studentship from EPSRC and D. J. W. acknowledges a Research Fellowship from the Royal Society.
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