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

€

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 :

€

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 :

€

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 :

€

qk+1( ) = A −Aqk−1( ) + 2 Hs q(k)[ ]

€

χ(k=0) = Gε,s(R) ψv1(r1) ψv2(r2) Gj1,j2,k1J,p q1,q2,ϕ( )

Initiation :

€

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

€

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.