Ab Initio Direct Chemical Dynamics

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Ab Initio Direct (Chemical) Dynamics
Nonadiabatic Reactions X Statistical Methods
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Adiabatic Reactions
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What do we want to know?
Products Rate coefficients, reaction cross se
ctions, initial state dependence, final state di
stribution, state to state cross sections, Me
chanism
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Remember what?
The Potential Energy Surface, of course!
A brief reminder of the interpolated PES method
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More learning as we go
As the size of the data set grows,
we can use the data itself to do a
statistical (Bayesian) analysis of
the errors that result from each
Taylor series. So we learn how much trust we c
an have in each Taylor series.
So we know how to more accurately
calculate the interpolation weights.
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H3O
R. P. A. Bettens, T. A. Hansen, and M. A.
Collins, JCP 111 (1999) 6322
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M. A. Collins and D. H. Zhang, JCP 111 (1999)
9924.
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H H2O
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Rate coefficientOH H2 a H2O H
M. Yang, D. H. Zhang, M. A. Collins and S.-Y.
Lee, JCP 114 (2001) 4759. D. H. Zhang, M. A. Coll
ins and S.-Y. Lee, Science 290 (2000) 961.
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How many data points does Grow need?
(1) You can get an answer with 20-100 mep points
(2) You can get the right answer with
4 atoms BeH3, BH3, CH3, NH3, OH3, NeCOH,
ArCOH, etc 400-1500 OH3 , OH3- (overkill) 20
00 5 atoms FHCOH 1500 6 atoms H2OCOH
, CH5 2000-3000
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Adiabatic Reactions Solve dynamical Equations

Born-Oppenheimer Non Born-Oppenheimer
(semi-)Classical Quantum
Car-Parinello Extended Lagrangian
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Car-Parinello Extended Lagrangian
Benefits Automated. Cheap electronic calc
ulation.
Potential Limitations DFT New calculation
at every configuration (use it and los
e it).
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Born-Oppenheimer Methods
Traditional quantum Uses a preset grid, so
needs to know the mechanism. Probably needs a P
ES. Can be exact. Diffusion Monte Carlo - bou
nd states only?
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How many geometries for a simple quantum case?
Quantum grid 3 atoms 104 4 atoms 107 - 10
8 5 atoms ? ( 109 in 7 dimensions) N atoms
103N-6 x the number of attempts
The number is proportional to the relevant volum
e.
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Born-Oppenheimer Methods
Classical, semiclassical, spawning
Can use any ab initio method.
Can be direct or use a PES. Do not need to know
the mechanism. Dynamical limitations.
How many geometries in a classical study?
(103 - 105 trajectories) x (103 - 105 - 106 time
steps) x (the number of different initial states
) Is this nearly independent of dimensionality
?
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Direct Methods vs PES
(1) Both methods can be automated, but the
direct method is simpler. (2) If the dynamics
is complicated in the sense that
many molecular configurations are involved,
then a PES may be much less expensive.
(3) Point (2) is true a fortiori if high level a
b initio methods are necessary. (4) PES metho
d benefits from CNP symmetry.
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The products and mechanism may be unclear
DHO OCH
DHO HOC
H2O DOC
DH2O OC
H2O OCD
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How many geometries?
Classical, spawning, extended Lagrangian
Direct Number µ Number of trajectories
x collision time x number of initial st
ates
PES based Number Number of data points
needed to span the
relevant volume
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How big is the relevant volume?
E(R) If there are nS spectator degrees of freedom
and nL large amplitude degrees of freedom
with nS nL 3N-6, then Volume µ dnS DnL
nL may be 6 the number of torsions. The
identity of the large amplitude degrees of
freedom changes from region to region.
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DHO OCH
6
DHO HOC
6
H2O DOC
5
DH2O OC
6
H2O OCD
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H2O DOC
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Interpolated PES If, on average, just one secon
d-order Taylor expansion is sufficiently accur
ate over lengths of order d, then the number of
data points required for an interpolated PES i
s Ndata Ndata µ (D/d)nL (where nL
6 the number of torsions)
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Direct dynamics If a direct method Monte Carlo
samples 3N - 6 - 1 degrees of freedom in the i
nitial configurations, then the number of grad
ient calculations is Ndirect Ndirect µ Ntraj
x the average number of steps per trajectory.
Ntraj may thus be virtually independent of nS.

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However, the more complex the dynamics and the
less probable the observables, the larger Ntraj
must be. Moreover, it is not clear how the time
scale for the collisions scales with the numbe
r of atoms. While the direct method can give re
sults for large systems, the question is whethe
r feasible calculations can yield results which
are both converged and accurate.
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Clusters and Fluids
Both direct and PES approaches can be applied to
clusters and fluids via a many-body expansion.
However, a PES for (say) water is transferable t
o many different problems, while the direct meth
od must be repeated.
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Conclusions?
The direct method has the advantage of
simplicity, and may scale well with dimensionalit
y. Storing and re-using the ab initio data in a
n interpolated PES is more efficient for small
systems, and is currently practical for high
levels of electronic structure. It is unclear
how these two methods will compare
For large systems.