Download Transition state theory and its extension to include quantum

Transition state theory and its extension to
include quantum mechanical tunneling
Introduction - Review rate theory, i.e. transition state theory,
full (TST) and a harmonic approximation (HTST)
- How to extend to quantum mechanical systems?
Methodology
a.
b.
c.
d.
e.
Simple extension of HTST to quantum mechanical systems
Quantum statistical mechanics via Feynman path integrals
Extension of transition state theory to quantum systems
Harmonic, quantum TST (instantons, ImF)
Open questions and challenges
Applications
H2 desorption / H2 dissociative sticking at surfaces
Motivation
The tunneling mechanism will dominate the rate of
atomic rearrangements (chemical reactions, diffusion) at
low enough temperature
Typical temperature dependence of a rate constant
T
k
high T,
classical
low T, tunneling
1/T
1/Tc
cross over temp.
How can the cross over temperature, Tc, be estimated and
the contribution of tunneling to the transition rate evaluated?
Want to find the rate constant for a given temperature
Time dependent (wave packet) or time independent Schrödinger equation
can, in principle, be used to calculate the reaction probability
(transmission coefficient) as a function of energy, P(E),
and rate constant then found by taking a Boltzmann average
k(T) =
But, this is difficult to carry out.
- hard to solve the Shcrödinger equation for
many degrees of freedom (max. around 6, typically). Select?
- the range in energy that is relevant at around room temperature
is low, which makes wave packet calculations even
harder (more on that later) and extrapolations are needed.
Instead, apply a statistical approach to find k(T) directly
Even for a classical system (where Newton is OK),
a direct dynamical calculation of a rate constant is hard
Time scale problem:
• Most interesting transitions are rare events (ie, much slower than
vibrations). Typically there is clear separation of time scales!
• A transition with an energy barrier of 0.5
eV and a typical prefactor occurs 1000
times per second at room temperature.
A direct classical dynamics simulation
would require 1012 force evaluations
and thousands of years of CPU time to
cover the average time period between
such events.
1000/s
0.5 eV
Cannot simply heat the system, the mechanism can change!
Again, a statistical approach is needed
Transition State Theory (Wigner, Eyring 1930s)
Identify a 3N-1 dimensional dividing surface, a Transition State
(TS), that represents a bottle neck for going from an initial to a
final state:
Final state
Initial state
3N-1 dimensional
dividing surface, TS
The bottle neck
can be due to an
energy barrier
and/or entropy
barrier
kTST = (probability of being in TS) • (flux out of TS)
WARNING! Beware of P-Chem textbooks
Bruce Mahan, J. Chem. Education 51, 709 (1974):
“In view of [its] success, it is unfortunate that the theory [TST]
does not enjoy a better understanding and confidence among
non-specialists. Some of this difficulty can be traced to the rather
unconvincing derivations of the [TST] expression for the rate
constant which are found in many physical chemistry texts and
monographs on chemical kinetics.”
Not much seems to have changed since then. Textbooks on
Physical Chemistry that are commonly used today, illustrate the
assumptions of TST by showing a dip in potential energy
corresponding to ‘activated complex’ - Wrong message
AC
R
P
Transition State Theory
1.
2.
3.
4.
Born-Oppenheimer
Classical dynamics of nuclei (need to extend to quantum systems…)
Boltzmann distribution in R (OK if slow enough, kBT < Ea/5 )
No recrossings of TS, (often most serious, but can be fixed from trajectories)
Note:
-
TST gives the lifetime, τ=1/k,
of a given initial state - no
knowledge of final state(s)
Can run short time scale
dynamics starting from TS to
find the final state(s)
Such trajectories can be used to
take recrossings into account dynamical corrections
kexact = κ kTST
Need to create a TS dividing surface that
encloses the initial state, R
The probability of being in some subspace, S, of R is:
e"V (x ) / kB T dx
PS =
!
#S
Z
$ S
"V (x ) / k T
e
dx Z R
#R
configuration integrals
‡
B
A hyperplane ax+b=0 is a particularly simple
(but not necessarily good) choice for the
dividing surface
P
v"
R
thickness
!σ
kTST = (probability of being in TS) • (flux out of TS)
"
=
v" =
%
!
So, k
!
e#V (x) / kB T dx '
v%
‡
e#V (x ) / kB T dx "
$
$R
1
2
$ #&' 2 µ i v i / kB T '
v"
e i
dv
"
0
#$
1
2
1
2
$ # µ " v " / kB T
$ #&' 2 µ i v i / kB T '
e 2
dv"
e i
dv
#$
#$
$
1
# µ " v "2 / kB T
e 2
dv
%
TST
=
%
%
k BT Z‡
2 "µ# Z R
$
=
% 0 v"
$
%#$
1
# µ " v "2 / kB T
e 2
dv
1
# µ " v "2 / kB T
e 2
dv
"
3N-1 dimensional!
1 dimensional!
"
=
k BT
2'µ"
Harmonic approximation to TST
Typically a good approximation for
solids (low enough temperature,
but not too low).
When energy of second order
saddle points is much higher
than kBT over the energy of
first order saddle points and
when the potential is smooth
enough that a second order
Taylor approximation to the
PES is good enough in the
region with large statistical
weight.
Analogous to the standard
approximation for diatomic
molecules:
1
2
V (r) " V (r0 ) + k ( r # r0 )
2
2nd order SP
> kBT
1st order SP
R
V(r)
r
Derivation of HTST: Taylor expand PES around minimum, find normal modes
3N
VR (q) " Vmin +
#
i=1
1
2
k R, i qR,
i
2
R
Similarly, Taylor expand PES around SP:
3N #1
$
V‡ (q) " VSP +
!
i=1
1
k ‡ , i q‡2, i
2
Then, the TST rate constant becomes:
k
HTST
=
!
=
k BT
2"µ#
k BT Z ‡
2 "µ# Z R
,
3N $1 1
&
)
$(VSP + % k‡, i q‡2, i + / kB T
'
*
i =1 2
,
3N 1
&
)
$(Vmin + % k R, i q 2R, i + / kB T
'
*
i =1 2
-$, e
-$, e
Define: " = # / 2$ =
!
which finally gives:
!
1
2$
k
µ
‡
'
dq‡
dq R
3N
" R, i
#
$(V
HTST
i=1
k
=
e
3N $1
#i=1"‡ , i
SP $Vmin
) / kB T
Agrees with the
empirical Arrhenius
law
• Need to find all relevant saddle points
on the potential energy rim
surrounding the energy basin
corresponding to the initial state.
• The transition state is approximated as
a set of hyperplanes going through the
saddle points with the unstable mode
normal to the hyperplane, and a second
order Taylor approximation to the PES
at minimum and saddle points is used.
• For each saddle point region:
TS
× Saddle
point
D
k
!
HTST
" R, i
#
$(V
i=1
=
e
D$1
#i=1"‡ , i
SP $Vmin
) / kB T
Temperature and entropy are
taken into account within the
normal mode approximation
HTST is more than 103
faster than full TST!
How can HTST be extended to include
quantum mechanical effects?
Replace νi with qcl h/kBT to introduce the classical vibrational partition
function, qcl, and then replace it with the quantum mechanical
vibrational partition function for each normal
D mode, to get
" R, i
#
$(V $V ) / k T
i=1
k HTST =
e
D$1
" ,i
#
i=1not tunneling. This will
This includes zero-point energy effects, but
SP
min
B
‡
be referred to as ʻquasi-quantumʼ approximation.
Note: A simple zero point energy ʻcorrectionʼ, i.e. simply replacing
ΔE = VSP - Vmin
!
by
ΔEZPC
in the classical HTST expression is not appropriate at high T and can
be a worse approximation than no quantum correction at all!
Example: CH4 dissociative adsorption on Ni, activation energy (eV)
Classical
cl. with ZPC
0.82
0.66
quasi-quantum (at 500 K)
0.72
Extend TST to quantum mechanical systems using
Feynman Path Integrals
Assume thermalization and decoherence after each transition
* Statistical mechanics of a quantum particle is
mathematically equivalent (isomorphic) to classical
statistical mechanics of a distribution of images of
the particle connected with springs with stiffness
proportional to T2
ksp = mP (2πkBT/h)2
* Instead of the potential surface, work with an effective
potential
V(r)
Derivation of FPI representation of quantum partition function
The thermally averaged expectation value for an observable is
Where Q is the partition function is
Use Trotter formula for operators
and insert the identity operator to rewrite the partition function as
With the solutions for a free particle going from xi to xi+1, it becomes
Veff Equivalent to a classical configurational partition function
for a ring polymer with P particles connected by springs.
Isosurfaces of the effective potential, Veff: Example
1D Eckart potential
Consider Feynman paths of the
form ,
, only the first two
Fourier components included
q(!)
R
P
!
At high T, T>Tc , quantum delocalization
(non-zero x1) increases the effective potential
The contour plot shows
the value of the effective
potential as a function of
(x0,x1)
Isosurfaces of the effective potential, Veff
Consider Feynman paths of the form q(τ) = x0 + x1sin(2π τ/Τ)
T=βh/2π, only first two Fourier components
q(τ)
x1
R
P
x0
At high T, T>Tc , quantum
delocalization (non-zero x1)
increases the effective potential
τ
Contour plot shows
the value of the
effective potential as a
function of (x0,x1)
At lower temperature new features appear - tunneling
Summarize the temperature dependence of the effective QM
potential surface, Veff for an Eckart potential barrier
Example:
Eckart
potential
High T,
classical overthe-barrier
transitions
Below cross
over T, new
saddle points
form instantons
instanton
RAW-QTST: an extension of TST for quantum
mechanical systems, includes tunneling mechanism
Quantum transition state theory
based on Feynman path integrals
Estimate the rate at which closed
Feynman paths move from the reactant
region, R, to the product region, P.
The definition of the quantum
transition state needs to involve both
position and shape of the closed
Feynman paths in order to
confine the system to the bottle neck
region around the instanton.
RAW-QTST:
a full free energy calculation of such a transition state.
Harmonic QTST: make a second order expansion of the effective potential
around the instanton
Final expression for the rate constant (next slide) is equivalent to Im F (Langer,
1969) and instanton theory (Miller, 1975, Callan and Coleman, 1977)
Analogy between HTST and HQTST
HTST
(N dimensions)
Vsp
HQTST
(NP dimensions)
Vins
Spreading of replicas lowers effective
activation energy - Accounts for tunneling
Calculated sticking coefficient for H2 on Cu(110)
Use EAM potential function with little input on the H-Cu interaction, dynamical surface atoms
min.
energy
path
NEB method
100 K
instanton path
While wave packet calculations for
energy that is large compared to
T=500 K show clear quantum
effects in state-to-state cross
sections, the thermal average that
determines S is classical at T>300K
Mills et al. Surf. Sci. 324, 305 (1995)
Compare with the Harmonic approximation to QTST
The full RAW-QTST calculation requires a large number of force evaluations.
So, it can currently only be applied to systems described by analytical
potential functions.
Implemented using minimum mode following method (Henkelman & J, ‘99).
Requires 500-1000 force evaluations per system replica and can be
used with ab initio calculations of the atomic forces.
A very brief intro to H2-Cu studies
The sticking of H2 on Cu surfaces has become a benchmark problem in
surface science due to extensive experimental and theoretical studies.
Molecular beam studies:
Balooch, Cardillo, Miller, Stickney (Surf. Sci. 1974)
…..
Auerbach, Rettner, Michelsen (Surf. Sci. 1993)
375 citations
Sitz and coworkers
Wave packet calculations:
Jackson and Metiu (Surf. Sci. 1974)
Holloway and coworkers
…..
Kroes, Olsen and coworkers (JCP 2005) 6D calculations with PES
fitted to DFT calculations,
frozen surface
Compare wave packet and HQTST calculations
H2 dissociative sticking on Cu(100)
QD: 6-d wave packet calculations by Roar Olsen and Geert-Jan Kroes
quasi-quantum
classical
Extrapolation to the relevant energy range
QD: 6-d wave packet calculations by Roar Olsen and Geert-Jan Kroes.
They had to work hard and use much computer time to push the
calculations to this low energy.
different extrapolations
Some open questions
A more general prefactor needs to be derived for RAW-QTST, so far only simple
transitions with a simple energy barrier can be treated.
How can dynamical corrections be evaluated given a RAW quantum mechanical
transition state?
Is there a variational principle for quantum transition state theory?
Is there perhaps an even better way to approach the problem of estimating tunneling
rates in complex systems, does a transition state theory approach optimal?
Perhaps go back to
?
k(T) =
Summary
RAW-QTST is an extension of classical transition state theory which includes
quantum mechanical effects, such as tunneling by using closed Feynman paths.
It can be applied to large systems with hundreds of degrees of freedom and there is
no need to choose beforehand special treatment for some but not others.
The definition of the transition state needs to be generalized, no transition state
defined in the space of 3N coordinates will give a good rate estimate, a quantum
mechanical transition state needs to be defined in the 3NP coordinate space of
closed Feynman paths.
A harmonic approximation to RAW-QTST can be used when the transition involves
a significant energy barrier. A quadratic approximation around the first order saddle
point, the instanton, is then used for the energy surface of the Feynman paths.
Applications to H2/Cu dissociative adsorption and associative desorption show that
quantum mechanical transition state theory can be used to estimate rates, even
sticking probability of molecules at surface (good agreement with wave packet
propagation results). The harmonic approximation even gives good results in these
test problems.
Several open questions remain in the formulation of quantum transition state theory
and several improvements in the methodology could be, and should be, pursued.
Acknowledgements
Andri Arnaldsson and Greg Mills, former Ph.D. students
Contributions were also made by Thomas Bligaard and
Stefan Andersson (postdocs)
Some of the calculations are in collaboration with Greg Schenter (PNNL),
and with Roar Olsen and Geert-Jan Kroes (University of Leiden)
Funded by European network on ʻPredicting catalysisʼ and
European integrated project ʻNESSHyʼ on hydrogen storage.