pressure dependent kinetics: single well reactions · single well reactions simple models •...
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Pressure Dependent Kinetics:Single Well Reactions
Simple Models• Lindemann-Hinshelwood• RRKM Theory• Modified Strong ColliderThe Master Equation• 1-dimensional (E)• 2d Master Equation (E,J)• Energy Transfer• Troe Fitting• Product Channels CH3 + OHTheory of Unimolecular and Recombination Reactions, R. G. Gilbert and S. C.
Smith, Blackwell, 1990Unimolecular Reactions, K. A. Holbrook, M. J. Pilling, S. H. Robertson,
Wiley, 1996
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Recombination KineticsRecombination is a Multistep Process - not single elementary step
A + B → AB(E) kf(E) [k(T) = ∫ k(E) P(E)]
But, E is above dissociation threshold so AB just redissociates
AB(E) → A + B kd(E)
Need some process to take away energy and stabilize ABCollisions with bath gas M (or photon emission)
AB(E) + M → AB(E') + M' kc x P(E → E')
Effective rate constant is some mix of kf(E), kd(E), kc, and P(E → E')Dissociation is related to recombination through equilibrium constant
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Simple Models Lindemann-HinshelwoodAssume every collision leads to stabilizationTreat association and dissociation on canonical levelA + B → AB* kf(T)AB* → A + B kd(T)AB* + M → AB + M' kc
Steady state for [AB*] =>d[AB]/dt = keff [A] [B]keff = kf kc [M] / ( kd + kc [M] ) = kf Pstabilization
High Pressure limit ([M] → ∞)keff = kf
Low Pressure limit ([M] → 0)keff = kf kc / kd
Not accurate but good for qualitative thought
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Simple Models RRKM TheoryTreat energy dependence of association and dissociation rate constantskf(E) and kd(E)keff (T,P) = dE keff (E) P(E) = dE kf(E) P(E) Pstabilization (E,P)Use transition state theory with quantum state counting to evaluate kf, kd
keff = dEN ± E( )
hρreac tan t (E)∫
ρreac tan t E( )exp(−βE)QAQB
kc M[ ]kd (E) + kc M[ ]
keff =1
hQAQB
dEN ± E( )exp(−βE) kc M[ ]kd (E) + kc M[ ]∫
keff∞ =
1hQAQB
dEN ± E( )exp(−βE) = kBT
hQAQB
dEρ± E( )exp(−βE)∫∫
keff∞ =
kBT
h
Q±
QAQB
Consider High Pressure Limit; [M] → ∞
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Simple Models Modified Strong ColliderAssume only a fraction βc of collisions lead to stabilization
keff =1
hQAQB
dEN ± E( )exp(−βE) βckc M[ ]kd (E) + βckc M[ ]∫
Consider low pressure limit; [M] → 0
keff0 =
1
hQAQB
dEN ± E( )exp(−βE) βckc M[ ]kd (E)
∫
keff0 =
βckc[M]
QAQB
dEρAB E( )exp(−βE)0
∞
∫keff
0 does not depend on transition state! Only the threshold Ematters
βc is a fitting parameter -typical value ~ 0.1
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Master EquationConsider n(E,t) = time-dependent population of AB molecule at
energy EMaster equation Irreversible Formulation
dn(E)
dt= kc[M] dE ' P E,E '( )n E ', t( ) − P(E ',E)n E,t( )[ ]∫ − kd E( )n E,t( )
Replace n(E,t) with normalized population x(E,t) = n(E,t)/ dEn(E,t)
Steady state for x =>
−k(T, p)x(E) = kc[M] dE 'P E,E '( )x E '( ) − kc[M]x E( )∫ − kd E( )x E( )
Master equation Reversible Formulation
dn(E)
dt= kc[M] dE ' P E,E '( )n E ', t( ) − P(E ',E)n E,t( )[ ]∫ − kd E( )n E,t( ) +
k f (E)ρreac tan t E( )exp(−βE)
QAQB
nAnB
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Master Equation Symmetrized Formf2(E) = ρ(E) exp(-βE)=F(E)Q(T)y(E) = x(E)/f(E)Discretize master equation
d y
dt= G' y
Gij' = kc[M]P Ei,E j( ) f E j( )
f Ei( ) δE − 1+kd E( )kc[M]
⎡
⎣ ⎢
⎤
⎦ ⎥ δ ij
Diagonalize
Eigenvalues are all negativeOne with smallest magnitude defines the rate coefficientk(T,p) = -ξ1Others are related to rate of energy transfer - form continuum
y t( ) = expj=1
N
∑ ξ j t( ) g j' g j
' y 0( )
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Master Equation Problems at Low T
numerical difficulties with diagonalization due tolarge dynamic rangeVarious Solutions
1. Integrate in time2. Quadruple Precision3. Reformulate with sink for complex => Matrix
inversion
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Master Equation Problems at high TDissociation occurs on same time scale as energy
relaxation
Nonequilibrium factor fne
fne =dEc E( )∫( )2
dEc 2 E( )F(E)∫
⎛ ⎝ ⎜
⎞ ⎠ ⎟ 2 c(E) = steady state distribution
Deviation of fne from unity indicates how muchdissociation happens before relaxation
Detailed balance is still satisfied for fraction thathappens after relaxation
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Boltzmann Distributions CH4
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Boltzmann Distributions C2H5O2
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Nonequilibrium Factor
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Master Equation 2-DimensionalTotal Angular Momentum J - conserved between
collisionsMaster equation in E and J
n(E,J,t) or x(E,J,t)P(E,J,E',J')k(E,J)
Numerical solution timeconsumingNeed more information on energy transfer than we
have
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Approximate Reduction from 2D to 1DE model
P(E,J,E',J') = P(E,E') ϕ(E,J)Rotational energy transfer like vibrationalJ distribution given by phase space volumeϕ(E,J)=(2J+1) ρ(E,J) / ρ(E)ρ(E) = ∑J (2J+1) ρ(E,J)k(E) = ∑J (2J+1) N‡(E,J) / hρ(E)Use k(E) and P(E,E’) in 1D Master Eqn
Does not resolve J dependent thresholdsAll rotational degrees of freedom are activeIncorrect low pressure limit
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2D Master Equation E,J ModelE,J model
like E model, but treat k(E,J) properlyk(E) = ∑J k(E,J) y(E,J) / ∑J y(E,J)y(E,J) = ϕ(E,J) / { kc[M] + k(E,J) }x(E) = ∑J x(E,J)
x(E,J) =kc[M]ϕ(E,J)Z + k(E,J)
dE 'P E,E '( )x E '( )∫Proper treatment of J dependent thresholdsProper zero-pressure limitProper high-pressure limit Consistent with detailed balance
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2D Master Equation ε,J Modelε,J model
Active energy - does not include overall rotation ε = E - EJEJ = BJ(J+1)P(ε,J,ε',J') = P(ε,ε')Φ(ε,J)
Φ(ε,J) = (2J+1)ρ(ε,J)exp(-βEJ)/∑J(2J+1)ρ(ε,J)exp(-βEJ) ρ(ε,J) = density of states for active degrees of
freedomThermally equilibrated J distributionSatisfies Detailed balance
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Steady State Distribution CH4
E model E,J model
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Low Pressure Limit CH4
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Low Pressure Limit
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Reduced Falloff Curves
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Collision RatesHard Sphere
kcHS =
8kBT
πμπd2
Lennard-Jones
kcLJ = kc
HSΩ2,2*
Underestimates collision rate Correct with larger average energy transferredDipole Corrections
Ω2,2* =
1.16145
T*( )0.14874+
0.52487
exp 0.7732T*( ) +2.16178
exp 2.437887T*( ) T* = kBT/ε
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Energy Transfer FormsExponential Down
P E,E '( ) = 1
CN E '( ) exp −ΔE /α( )
P E,E '( ) = 1
CN E '( ) exp − ΔE /α( )2[ ]
P E,E '( ) = 1
CN E '( ) 1− f( )exp −ΔE /α1( ) + f exp −ΔE /α2( )[ ]Double Exponential Down
Gaussian Down
α = α0(T /298)n
α0 ~ 50-400 cm-1
n ~ 0.85Fit to experiment
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Energy Transfer MomentsAverage Energy Transferred
ΔE = dE E '−E( )∫ P E,E '( )
for exponential down
Average Downwards Energy Transferred
ΔE d = dE ' E '−E( )0
E '
∫ P E,E '( ) / dE '0
E '
∫ P E,E '( )
ΔE 2 = dE E − E '( )∫2P E,E '( )
ΔE d ≈α
Average squared energy transfer
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Fits to Experiment H + C2H2 Addition
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Fits to Experiment C2H3 Dissociation
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Fits to Experiment T dependent ΔEd
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Energy Transfer from Trajectories
Collisional energy transfer in unimolecularreactions: Direct classical trajectoriesfor CH4=CH3+H in Helium
A. W. Jasper, J. A. Miller, J. Phys. Chem. A 113,5612 (2009).
α0=110 cm-1 n=0.81
Barker is studying P(E,J,E’,J’)
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Troe FittingNeed to represent k(T,P) for Global ModelsStandard is Troe Fitting
k(T, p) =k0[M]k
∞
k∞ + k0[M]F log10 F =
log10 Fcent
1+log10(p*) + c
N − d log10 p*( ) + c( )⎡
⎣ ⎢ ⎢
⎤
⎦ ⎥ ⎥
2
c = −0.4 − 0.67log10 Fcent N = 0.75 −1.27log10 Fcent
d = 0.14p* = k0[M]/k∞
Fcent = 1− a( )exp −T /T***( ) + aexp −T /T*( ) + exp −T** /T( )
Fit k0 & k∞ to modified Arrhenius k0 = A0Tn 0 exp(−E0 /T)
Fit Fcent to:
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Troe Fitting ProblemsLimited AccuracyTypical Errors ~ 10 to 20%Improved Fitting Formulas
A Fitting Formula for the Falloff Curves of UnimolecularReactions, P. Zhang, C. K. Law, Int. J. Chem. Kinet. 41,727 (2009)
Still problems for tunnelingMultiple channels - actual P dependence is dramatically
different from Troe Form
Use Log Interpolation
logk = logki + logki+1 − logki( ) log p − log pi( )log pi+1 − log pi( )
Part of Current ChemKin
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CH3 + OH
Potential EnergySurface
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G2M MRCI//CASQCISD(T)/CBS
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Reactions with Products: CH3 + OHExperiment:Triangles - De AvillezPereira, Baulch, Pilling,Robertson, and Zeng, 1997Circles - Deters, Otting,Wagner, Temps, László,Dóbé, Bérces, 1998Theory: Master EquationsDotted - De Avillez Pereiraet al.Solid & Dashed - PresentWork
<ΔEd> = 133 (T/298)0.8 cm-1
± 25%
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CH3 + OH: Higher T and P ~1 atm
Shock tube studies• 1991, Bott and
Cohen (1 atm)• 2004, Krasnoperov
and Michael (100−1100 torr)
• 2006, Srinivasan,Su, and Michael(200−750 torr)
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Methanol decomposition: Low pressure limit
CH3OH → CH3 + OH Experimental• 2004, Krasnoperov and Michael• 2006, Srinivasan, Su, and Michael• 1981−2000, Many others• k independent of P (100−1000 torr)• 60−90% CH3 + OHPrevious theory• 2001, Xia, Zhu, Lin, and Mebel
(shown at 1 atm)• Falloff below 1 atm• ~33% CH3 + OH
~52% CH2 + H2O~15% H2 + HCOH
Present theory• Low-P limit at 1 atm• ~75% CH3 + OH
~20% CH2 + H2O< 5% H2 + HCOH
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Methanol decomposition: Product branching
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Secondary kinetics of methanol decompositionShock tube OH absorption profilesensitivities for CH3OH → CH3 + OH
Srinivasan, Su, and Michael, JPCA, 2007
Well characterizedOH + OH → O + H2OH + OH → O + H2
Not well characterized3CH2 + OH → CH2O + H
Ambiguous experiments3CH2 + 3CH2 → C2H2 + 2HCH3 + 3CH2 → C2H4 + H
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Secondary kinetics: OH Time Traces
Michael et al.
Good agreementat long timesusing ourpredicted ratesfor
3CH2 + OH3CH2 + 3CH2
CH3 + 3CH2
CH3 + OH
OH
con
cent
ratio
n
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Multiple-Well Multiple-ChannelTime Dependent Master Equation
1. The Kinetic Model2. Collisionless Limit3. CH + N24. Time Dependent Populations5. Kinetic Phenomenology6. C2H5 + O27. Reduction in Species at High Pressure8. C3H3 + H9. Radical Oxidation10.C3H3+C3H3
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Multiple-Well Multiple-Channel Master Equation
M Wells Np ProductsM+1 Chemical SpeciesnB>>nm>>nR B=Bath, m=Molecule, R=RadicalLinear Master Equation
dni(E)
dt= kcnB dE 'Pi E,E '( )ni E '( ) − kcnBni E( )∫ − kdi E( )ni E( ) − kpi E( )
p=1
N p
∑ ni E( ) −
k jiisom E( )
j≠ i
M
∑ ni E( ) + kijisom
j≠ i
M
∑ E( )n j E( ) + Keqikdi E( )Fi E( )nRnm
dnRdt
= dEkdi E( )ni E( )∫i=1
M
∑ − nRnm KeqidEkdi E( )Fi E( )∫
i=1
M
∑
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Collisionless LimitConsider Z→ 0
d n(E,J)
dt= −K E,J( ) n E,J( ) + nRnm b(E,J) ρRm (E,J)exp −βE( ) /QRm
Steady State for n(E,J)
d P(E,J)
dt= D E,J( ) n E,J( )
d P(E,J)
dt= D E,J( )K −1 E,J( ) b E,J( ) nRnmρRm E,J( )exp −βE( ) /QRm (T)
k0(T) =1
QRm (T)2J +1( )
J∑ dED E,J( )∫ K −1 E,J( ) b E,J( ) ρRm E,J( )exp −βE( )
Flux coefficients
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CH + N2 Prompt NO• 1971 Fenimore 2CH + N2 → HCN + 4N• 1991 Dean, Hanson & Bowman -- shock tube measurements of
rate for 2CH + N2 → Products• 1991 Manaa & Yarkony -- located minimum crossing point for
doublet to quartet transition• 1996 Miller & Walch -- found maximum on spin forbidden path
corresponds to dissociation of the quartet complex; not thedoublet-quartet crossing; presume rapid ISC and fit experimentaldata
• 1999 Qui, Morokuma, Bowman & Klippenstein -- predicted spin-forbidden reaction to be less than observed rate by at least 102
• 2000 Moskaleva, Xia & Lin -- predicted new spin allowedmechanism,
2CH + N2 → HNCN → 2H + 3NCN• 2007 Szpunar, Faulhaber, Kautzman, Crider & Neumark --
observed the photodissociation of DNCN to CD+N2 and D+NCNwith 1:1 branching ratio
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Recent Modeling• Williams, Fleming
Proc. Comb. Inst. 31, 1109-1117, 2007NO severely underpredicted in CH4 and C3H8 flames
• El Bakali, Pillier, Desgroux, Lefort, Gasnot, Pauwels, da Costa,Fuel 85, 896, 2006Increasing CH + N2 rate by 1-2 orders of magnitude over the1000 to 1500 K range yields good predictions for NO in naturalgas flames
• Sutton, Williams, Fleming,Comb. Flame, 2008, in press.Improved modeling for CH4/O2/N2 flames with rates of ElBakali et al.
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Moskaleva, Xia and Lin (2000)
Present
HN C N
N N N N
H H
N N
H
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CAS+1+2+QC/aug-cc-pvtz
Contour Increments: Thick- 5 kcal/mol, Thin- 1 kcal/mol
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CASPT2 CAS+1+2+QC
CCSD(T)B3LYP
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656, 1029, 1062,1424, 2985, 361i
CASPT2:
B3LYP:507, 540, 1059,
1494, 2505, 1225i
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Discretize Energy Levels
Transition Matrix; Renormalize → real, symmetric; G
Diagonalize
Time-Dependent Populations
( ) ( )d w t G w tdt
=
( ).
1
. . 1
( ) 0I M
jN N
tj j
jw t e g g wλ
+
=
+
= ∑
w(t) = yI (E0 I),L,yI (Emax ),L,yi(E0 i
),L,yi(Emax ),L,nm
QRmδE⎛ ⎝ ⎜
⎞ ⎠ ⎟ 1/ 2
XR ,L⎡
⎣ ⎢ ⎢
⎤
⎦ ⎥ ⎥
T
yi E,t( ) = xi E,t( ) / fi E( )
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Kinetic PhenomenologyExperimental Viewpoint
• Find regimes of single exponential decay (λ)– λ implies total rate coefficient– Eigenvector corresponding to λ implies branching– Branching implies individual rate coefficient (ktot)
• When is decay close enough to single exponential?– Suppose 2nd eigenvector contributes to only 1% of
the initial decay but that λ2/λ1 = 100– Rate coefficient will differ by a factor of two from
apparent exponential decay– Branching similarly incorrect
• Difficult to find single exponential decay regimes inmultiple well situations
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Eigenvalues C3H3 + C3H3
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A Simple Solution: Separation of Timescales• M+1 modes corresponding to chemical change have
least negative eigenvalues.• λ’s for chemical modes well separated from remaining
λ’s for energy transfer• After energy relaxation can treat populations as
•
•
• Eigenpairs (λi, ΔXi) correspond to Normal modes ofchemical relaxation
( ) jtj jj
1
=1
M+ ët e g g (0)Aw w= Σ ll
( ) ( ) ( )1
1; (0)j A
ij
M t Ai jj ij A i jj i
d XX e X g w E f E gdt
λ δλ+
= ∈= − Σ Δ = −Δ Σ l l
l
λ
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Method 1 t=0 Limit and Start in Well A
• Phenomenology
• Master Equation
•
• Similarly, consider dXi/dt implies
•
•
( ) ( )0 0ATA A
dX k Xdt
= −
( ) ( )1
10
MAA
j Ajj
dX Xdt
λ+
== − Σ Δ
( )1
1
MA
TA j Ajjk Xλ
+
== Σ Δ
( )1
1
MA
Ai j ijjk Xλ
+
== − Σ Δ
( )1/ 2
1
1; (0)
MA Rm
AR j Rj Rj j j Ajm
Q Ek X X g g wnδλ
+
=
⎛ ⎞= − Σ Δ Δ = −⎜ ⎟
⎝ ⎠l
( )1
1; 0
M MA
Ap j pj R p ij i Ij
k X X X Xλ+
= =
⎛ ⎞= − Σ Δ Δ + Δ + Σ Δ =⎜ ⎟⎝ ⎠
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Method 2 Long time limit
•
•
•
•
( )1 1
0 0
jM Mt
i ij ij jj jX t a e a vλ+ +
= == Σ ≡ Σ X A v= v B X=
1 2
0 1
M Mi
j ij jj
dX a b Xdt
λ+ +
= == Σ Σ l l
l
ii i ii i
dX k X k Xdt ≠ ≠
= Σ − Σl l ll l
1
0
M
i j ij jjk a b iλ
+
== Σ ≠l l l
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C2H5 + O2 Potential Energy Surface
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C2H5 + O2
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C2H5 + O2 T Dependent Rate Coefficients
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C2H5 + O2 P Dependent Rate Coefficients
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C2H5O2 → C2H4 + HO2 P Dependent Rate Coefficients
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C3H7 + O2 Formally Direct Pathways; QOOH
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C3H4 Potential Energy Surface
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C3H4 Rate Coefficients
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C3H3 + C3H3 Eigenvalues
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C3H3 + C3H3 Rate Coefficients
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C3H3 + C3H3 Isomerization Rate Coefficients
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C3H3 + C3H3 Product Branching in 1,5-Hexadiyne Pyrolysis
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C3H3 + C3H3 Product Branching
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C3H3 + C3H3 Product Branching
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C3H3 + C3H3 Rate Coefficients
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C6H6 Dissociation Rates
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Master Equation CodesEigenvalue Eigenvector Methods
VariFlex KlippensteinResearch Code - Not usable without personal training
MESMER Pilling (Leeds)http://sourceforge.net/projects/mesmer/
Stochastic Master Equation SolversExperimental Perspective onlyMultiwell Barker(Michigan)
http://esse.engin.umich.edu/multiwell/MultiWell/MultiWell%20Home/MultiWell%20Home.html
Vereecken and Peeters (Leuven)Steady State Solvers
ChemRate Tsang (NIST)http://www.mokrushin.com/ChemRate/chemrate.html