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

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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

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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)

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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( )

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Hs = as H + bs

Then the “cosine” equation loses its cosine :

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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?