quantum dynamics of four-atom reactions within the real wave packet framework stephen k. gray...
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Quantum Dynamics of Four-Atom Reactions within the Real Wave Packet
Framework
Quantum Dynamics of Four-Atom Reactions within the Real Wave Packet
Framework
Stephen K. GrayChemistry Division
Argonne National LaboratoryArgonne, Illinois 60439
gray@tcg.anl.gov
AcknowledgementsAcknowledgements
Gabriel Balint-Kurti: co-developer of the RWP method
Evelyn Goldfield: co-developer of the four-atom implementation
OutlineOutline
• Introductory Remarks
• Real Wave Packet Framework:• Cosine Iterative Equation
• Modified Schrödinger Equation
• Inferring Physical Observables
• Four-Atom Systems:• Representation
• Dispersion Fitted Finite Differences
• Initial Conditions and Final State Analysis
• Cross Sections and Rate Constants
• Concluding Remarks
Introductory RemarksIntroductory Remarks
• Real wave packet (RWP) method: An approach for obtaining accurate quantum dynamics involving the real part of a wave packet and Chebyshev iterations [Gray and Balint-Kurti]
• Can view it as a highly streamlined version of Tal-Ezer and Kosloff’s propagator
• Shares features with: Mandelshtam and Taylor’s Chebyshev expansion of the Green’s operator, Kouri and co-workers’ “time-independent” wave packets, Chen and Guo’s Chebyshev propagator
Cosine Iterative EquationCosine Iterative Equation
€
ihddtΨ(t) = HΨ(t)
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Ψ(t+τ) = exp(−iHτ/h) Ψ(t)
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Ψ(t−τ) = exp(+iHτ/h) Ψ(t) implies
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Ψ(t+τ) = − Ψ(t−τ) + 2 cos(Hτ/h) Ψ(t)
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Ψ(t) = Q(t) + i P(t)
If H is time-independent,
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Q(t+τ) = − Q(t−τ) + 2 cos(Hτ/h) Q(t)
Including absorption, A:
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Q(t+τ) = A −AQ(t−τ) + 2 cos(Hτ/h) Q(t)[ ]
• Cosine equation was successful,
S. K. Gray, J. Chem. Phys. 96, 6543 (1992)
• However, cos(H) must still be evaluated
in some way
• Can we do better?
Modified Schrödinger EquationModified Schrödinger Equation
€
ihd
duχ (u) = f (H )χ (u)
• Underlying time-independent Schrödinger equationhas the same bound states (and scattering states)
• Solutions of the modified equation contain the sameinformation as the more standard one
χ(u) = q(u) + i p(u)
€
q(u+δ) = A −Aq(u−δ) + 2 cos(f(H)δ/h) q(u)[ ]Let
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f(H) = −hδ cos−1Hs( )
€
Hs = as H + bs
Then the “cosine” equation loses its cosine :
€
q(u+δ) = A −Aq(u−δ) + 2 Hs q(u)[ ]
Inferring Physical ObservablesInferring Physical Observables
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Φ(E) = c(E) 12πh dt exp(iEt/h) Ψ(t)
−∞
+∞∫ = c(E) δ(E−H) Ψ(0)
Use
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δ(E−H) =df(E)dE δf(E)−f(H)[ ]
To obtain a connection to f(H) dynamics :
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Φ(E) = c(E) df(E)dE δf(E)−f(H)[ ] Ψ(0)
= c(E) df(E)dE 1
2πh du expif(E)u/h[ ] χ(u)−∞
+∞∫
But χ(u) is still complex -- how to relate to
q(u) = Re[χ(u)] ?
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du exp if (E)u / h[ ]−∞
+∞
∫ χ (u) = 2 du exp if (E)u / h[ ]−∞
+∞
∫ q(u)
If χ has no f(E) components for f(E) < 0 (or f(E) > 0)
Allows energy-resolved scattering and related quantities,e.g., S matrix elements and reaction probabilities, to beobtained from Fourier analysis of q.
If Ψ(t) satisfies
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ihddtΨ(t) = HΨ(t)
and χ(u) satisfies
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ihdduχ(u) = f(H)χ(u)
Then if each have the same initial condition, Taylor seriesexpansion of f(H) shows that
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Ψ(t) ≈ expiα t/β[ ] χu=t/β[ ]
with
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β(E ) = ashδ 1−E s2
Let u = k δ then physical time and Chebyshev iteration k arerelated by
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t ≈ ash k1−E s2
Four-Atom SystemsFour-Atom Systems
Diatom-diatom
Jacobi coordinates,
body-fixed z-axis
is the R vector
θ1
R
θ2r2
r1 ϕ
AB + CD ABC + D
RepresentationRepresentation
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qJ ,K ,p(R,r1 ,r2 ,θ1 ,θ1 ,ϕ ,k) =
C j1 ,k1 , j2J ,K ,p
j1 ,k1 , j2
∑ R,r1 ,r2 ,k( ) G j1 ,k1 , j2J ,K ,p θ1 ,θ 2 ,ϕ( )
J = total angular momentum quantum numberp = parityK = projection of total angular momentum on a body-fixed axis (often an approximately good quantum number)
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G j1 ,k1 , j2J ,K ,p
Gatti and co-workers; Goldfield; Chen and Guo
Note: Most applications so far have assumed Kto be good (centrifugal sudden approximation)
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Gj1,k1,j2J,K,p : Parity-adapted angular functions –
linear superpositions of allowed “primitive” functions based on (J, K), (j1 ,k1) and (j2, k2),
K = k1 + k2.
H and H qH and H q
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ˆ H = −h22μ
∂2∂R2 −h2
2m1∂2∂r12
−h22m2
∂2∂r22
+ (̂ J − ˆ j 1 − ˆ j 2)22μR2 + ˆ j 12
2m1r12 + ˆ j 22
2m2r22 + V(R,r1,r2,θ1,θ2,ϕ)
H q = T q + V q = (Td + Trot) q + V q
Comments on H q :Comments on H q :
• Three distance (or radial) kinetic energy
contributions evaluated with either
dispersion fitted finite differences (DFFD’s) or potential-optimized discrete-variable representations (PODVR’s)
DFFD: Gray and Goldfied
PODVR: Echave and Clary; Wei and
Carrington
DFFD’sDFFD’s
Can obtain signifcantly betterAccuracy than standard FDapproximation
Error in reaction probability for 3D D + H2 reaction
V q V q
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v j1 , j2 ,k1( ) = q j1 , j2 ,k1( )
v j1 , j2 ,k1( ) T1 ⏐ → ⏐ v θ1 , j2 ,k1( )
v θ1 , j2 ,k1( ) T2 ⏐ → ⏐ v θ1 ,θ 2 ,ϕ( )
v θ1 ,θ 2 ,k1( ) Tϕ ⏐ → ⏐ v θ1 ,θ 2 ,ϕ( )
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v θ1 ,θ 2 ,ϕ( ) = V θ1 ,θ 2 ,ϕ( ) v θ1 ,θ 2 ,ϕ( )
v θ1 ,θ 2 ,ϕ( ) Tϕ
−1
⏐ → ⏐ ⏐ v θ1 ,θ 2 ,k1( )
v θ1 ,θ 2 ,ϕ( ) T2−1
⏐ → ⏐ ⏐ v θ1 , j2 ,k1( )
v θ1 , j2 ,k1( ) T1−1
⏐ → ⏐ ⏐ v j1 , j2 ,k1( )
Basis to grid, multiplyBy diagonal V, thenConvert back to basis
A key “trick” thatallows large rotationalbases to be treated
Favorable, near linearscaling with problemsize
Propagation and AnalysisPropagation and Analysis
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qk+1( ) = A −Aqk−1( ) + 2 Hs q(k)[ ] , k = 0, 1, …
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χ(k=0) = Gε,σ(R) ψv1(r1) ψv2(r2) Gj1,j2,k1J,p θ1,θ2,ϕ( )
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Gε,σ(R) = complex-valued incoming Gaussian wave packet
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q(k=0) = Reχ(0)[ ]q(k=1) = Hs q(0) − 1−Hs2 Im χ(0)[ ]
Reaction ProbabilitiesReaction Probabilities
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PI E( ) = ΦI(E) ˆ F ΦI(E)
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ˆ F = h2msiδ(s−s0)∂∂s −
∂∂sδ(s−s0)
⎡ ⎣ ⎢ ⎤
⎦ ⎥
Write ΦI as FT of q (Meijer et al.) -- problem reduces to saving certain dq/ds and q at s0 as a function of effectivetime and then constructing PI afterwards
Cross Sections, Rate ConstantsCross Sections, Rate Constants
Since we can compute PI(E), I = some initial state, there is nothing special about
constructing cross sections and rate constants
The problem is the large number of I states
that must be considered: I = J, p, K, j1, j2,
k1, v1, v2
A State-Resolved Cross Section:A State-Resolved Cross Section:
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σv1 , j1 ,v2 , j2 (ε ) = π
2με (2J +1) Pv1 , j1 ,v2 , j2
J
J
∑ ε( )
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Pv1 , j1 ,v2 , j2J ε( ) =
1
(2 j1 +1)(2 j2 +1) Pv1 , j1 ,k1 ,v2 , j2
J ,K ,p ε( )K ,p,k1
∑
Rate ConstantsRate Constants
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kT( ) = gel(T)Qr(T) gj1,j2exp[−εv1,j1,v2,j2/(kBT)]v1,j1,v2,j2
∑ kv1,j1,v2,j2(T)
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kv1,j1,v2,j2T( ) = 8kBTπμ ⎛ ⎝ ⎜ ⎞
⎠ ⎟1/2 1
(kT)2 dε ε exp−ε/(kBT)[ ]0
∞∫ σv1,j1,v2,j2(ε)
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Qr(T) = gj1,j2exp[−εv1,j1,v2,j2/(kBT)]v1,j1,v2,j2∑
Approximation: J-ShiftingApproximation: J-Shifting
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pv1,j1,v2,j2ε() ≡ Pv1,j1,v2,j2Jref ε()
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Pv1,j1,v2,j2J ε() ≅ pv1,j1,v2,j2ε' = ε + (EJref−EJ) ⎡
⎣ ⎢ ⎤ ⎦ ⎥
with
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EJ = 1nJ B J(J+1)+(A−B )K2( ) nJKK∑
Use result for a “reference” J to extrapolate to other J
Bowman has extensively discussed J-shiftingThe idea of using non-zero J values to base the J-shifting is not new -- previous work along related lines includes
• S. L. Mielke, G. C. Lynch, D. G. Truhlar, and D. W. Schwenke, Chem. Phys. Lett. 216, 441 (1993).
• H. Wang, W. H. Thompson, and W. H. Miller, J. Phys. Chem. A 102, 9372 (1998).
• J. M. Bowman and H. M. Shnider, J. Chem. Phys. 110, 4428 (1999).
• D. H. Zhang and J. Z. H. Zhang, J. Chem. Phys. 110, 7622 (1999).
Concluding RemarksConcluding Remarks
For accurate quantum dynamics of three and four-atom systems, the RWP method is a good choice of methods -- less memory and more efficient than comparable complex wave packet calculations
However, to go beyond four-atoms requires (most likely) abandoning the detailed scattering theory
approach involving complicated angular momentum bases and detailed state-resolved considerations
Cumulative reaction probability and related approaches to direct evaluation of averaged quantities (Miller, Manthe)
The use of parallel computers and Cartesian coordinates?
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