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Overview We will learn the “nuts and bolts” of six- dimensional AB + CD ABC + D reactive scattering Within the RWP framework but many issues apply to other propagation schemes System: H 2 (v 1, j 1 ) + OH(v 2, j 2 ) H 2 O + H using the old, but much studied “WDSE” potential
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Tutorial: Quantum Dynamics of Four-Atom Reactions within the Real Wave
Packet Framework
Stephen GrayChemistry Division
Argonne National LaboratoryArgonne, Illinois 60439
Outline
• Overview• Representation and Evaluation of H q• Initial Conditions, Propagation, Analysis• Computational Experiments
Overview
• We will learn the “nuts and bolts” of six-dimensional AB + CD ABC + D reactive scattering
• Within the RWP framework but many issues apply to other propagation schemes
• System: H2(v1, j1) + OH(v2, j2) H2O + H using the old, but much studied “WDSE” potential
Representation
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qJ ,K ,p(R,r1 ,r2 ,θ1 ,θ1 ,ϕ ) =
C j1 ,k1 , j2J ,K ,p
j1 ,k1 , j2
∑ R,r1 ,r2( ) G j1 ,k1 , j2J ,K ,p θ1 ,θ 2 ,ϕ( )
R, r1 : H2 -- OH distance and H2 internuclear distance : Large, evenly spaced grids
r2 : OH distance -- remains bound so a PODVR is convenient since it is a (small) set of grid points consistent with a set of vibrational states
Rotational basis: j1 = 0, 2, 4, .. ; j2 = 0, 1,.. ; k1 = determined o be consistent. E.g., for K = 0, even parity, k1 = 0, 1, .., min(j1, j2)
H q = (Td+Trot) q + V q• Use Dispersion Fitted Finite Differences
(DFFD’s) to evaluate R and r1 terms in Td
If C(iR, i1, i2, j1, j2, k1) denotes the wave packet,the R kinetic energy would involve
do over k1, j2, j1, i2, i1
do iR = 1, NR
do s = -n, n C’’(iR, ..) = C’’(iR, ..) + d(s) C(iR+s, ..)
PODVR for r2 : A small kinetic energy matrix in the PODVR points acts on the r2 part of C
Trot : Not strictly diagonal with our rotational basis -- tridiagonal in k1 -- however this is irrelevant in terms of actual numerical effort which is dominated by Td and V
Numerical effort for Td q ?
Nrot {N1 N2 NR (2n+1) + N2 NR N1 (2n+1) + NR N1 N22 }
= Ntot { 4n+ 2 + N2 }, near linear scaling
V q
• Diagonal in the three radial distances, so loop over them and (i) transform from rotational basis to angular grid :
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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( )
• Multiply by a diagonal V and convert back to basis :
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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( )
Effort : about 2 NR N1 N2 Nj1 Nj2 Nk1 (Nj1+Nj2+Nk1) -- again near linear
Propagation and AnalysisCore propagation :
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qk+1( ) = A −Aqk−1( ) + 2 Hs q(k)[ ]
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χ(k=0) = Gε,s(R) ψv1(r1) ψv2(r2) Gj1,j2,k1J,p q1,q2,ϕ( )
Initiation :
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q(k=0) = Rεχ(0)[ ]q(k=1) = Hs q(0) − 1−Hs2 Im χ(0)[ ]
Analysis
Use H2 distance to separate reactants from
products : S(r1) = r1 – r1* = 0
Write out q(R, r1*, r2, j1,j2,k1) and
q(R, r1, r2, j1,j2,k1)/ r1 | r1=r1*
every L time (or iteration) steps
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P(E) ∝ exp[ if (E)k] qk r1*
( ) k∑ exp[if (E)k] ∂qk /∂r1
k∑ r1
*( )
Computational Experiments
• Incoming Gaussian wave packet centered at R = 10 ao, with ε = 0.25 eV and a width, s = 0.3 ao
• Reactants in ground vibration-rotation states• r1* = 3 ao for analysis line
• Absorption in last 3 ao of R and r1 grids
Run 1 :Radial representation
R/ ao : 1 – 14, NR = 130 pointsr1/ao : 0.5 – 6 , N1 = 35 points
.2 PODVR points in r2, based, on diagonalization of a primitive grid Hamiltonian with r2/ao : 1 – 4, 32 points
(r2e = 1.85 ao; the PODVR pts are 1.79 and 2.07 ao)Nj1 = 5 H2 rotational states, j1 = 0, 2, .., 4
Nj2 = 10 OH rotational states, j2 = 0, 1, .., 9
A total of 180 rotational states. (The number of angular grid points should be about 10 for each angle.)
Rotational Representation
Reaction Probability
Run 1 requires 2.6 hrs on a 1 GHz Linux workstation(and 65 MB)
The reaction probabilitiy P(E) is essentially converged over the E = 0.5 to 1.1 eV total energy range (energy relativeto separated H2 + OH)
General aim of the experiments: to experiment with therepresentation details to see how they affect P(E) and ifsmaller grids or bases can be used for some purposes
Run 2 : Try just one PODVR point, N2 = 1 all else the same. Result: good, but still rather long for our purposes. (PODVR pt 1.90 ao, CPU, memory both halved.)
Run 3: NR = 80 and N1 = 25, N2 = 1
CPU time has been further reduced by 0.5 to 0.6 hrs or 36 minutes. Result -- still reasonably good
Run 4 : Like Run 3 but Nj1 = 4 (j2 = 0, 2, 4 and 6) and Nj1 = 9 (j1 = 0,1,..,8) the total number of allowed rotational states decreases from 180 to 94.
Result: CPU time of just 19 minutes. Noticeable high error in P(E) :
Run 5 : Like Run 4 but now we have reduced the number of R points from NR = 80 to NR = 60. This leads to about a 33% speedup and the calculation requires about 14 min.
• Consider a setup like either Run 4 or Run 5. Experiment further with reductions in the number of grid points in R and r1.
• Investigate, with Run 4 (or 5), how the quality of the calculated reaction probability varies with DFFD approximation.
• Experiment, with Run 4 (or 5), the role of rotational excitation in the reactants
Questions and Further Runs
Appendix I: Making and Running the Codes --
See also “README” files• To make the propagation program, abcd.x : make -f makefile.abcd• To run it :
abcd.x <abcd.run5.in > abcd.run5.out&• Making, running the (flux-based) probability
program :g77 -O3 prodflux.f -o prodflux.xprodflux.x <prodflux.run5.in >prodflux.run5.out
Appendix II : DFFD files
• Subdirectory dffd has various (2n+1) DFFD’s of various overall accuracies ε
E. g., fd11.e-3 is a (2n+1) = 11 DFFDwith accuracy 10-3. See Gray-Goldfieldpaper for more details [JCP 115, 8331
(2001).]
Appendix III: PODVR’s (Echave and Clary; Wei and Carrington)
• Define a finite grid representation of some 1D potential problem in x: Ho = To+ Vo
• Diagonalize, obtain numerical eigenstates {<xi|v>, Ev} , xi = grid, v = 0, 1,..• Now represent “x” in a finite vibrational basis, xv,v’ = <v| x | v’>, v,v’ = 0,..,Npo-1
• The eigenvalues of the xv,v’ matrix are the PODVR grid pts
Can use, e.g., T0 = H0 - V0 where H0 = Npo x Npo rep. of H0 in PODVR eigenstates and V0 = diagonal potential in PODVR to approximate KE operator for the x degree of freedom within the PODVR.